Annales Henri Poincaré - Volume 1

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Author: Vincent Rivasseau (Chief Editor)

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Ann. Henri Poincar´ e 1 (2000) 1 – 57 c Birkh¨

auser Verlag, Basel, 2000 1424-0637/00/010001-57 $ 1.50+0.20/0

Annales Henri Poincar´ e

Continuous Constructive Fermionic Renormalization M. Disertori and V. Rivasseau Abstract.We build the two dimensional Gross-Neveu model by a new method which requires neither cluster expansion nor discretization of phase-space. It simply reorganizes the perturbative series in terms of trees. With this method we can define non perturbatively the renormalization group differential equations of the model and at the same time construct explicitly their solution.

I Introduction The popular versions of renormalization and the renormalization group in field theory are based on differential equations (among which the most famous one is the Callan-Symanzik equation). However no non-perturbative version of these differential equations has been given until now. On the other hand the renormalization group in statistical mechanics, for instance for spin systems after the works of Kadanoff and Wilson, relies on closely related but discretized equations. When block spinning or other discretization of momentum space is used, the result is a discretized evolution of the effective action step by step. This point of view, in contrast with the first one, has led to rigorous non perturbative constructions for various models which have renormalizable power counting. In particular the two dimensional Gross Neveu model has been built by two groups [FMRS][GK]; also the infrared limit of the φ44 , a bosonic theory, has been controlled (see [R] and references therein). In all these cases the methods always involved some discretization of phase space and the outcome is a discrete (not differential) flow equation. Furthermore, the rigorous discretization of phase space came with a price, namely the use of some technical tools such as cluster or Mayer expansions which are neither popular among theoretical physicists nor among mathematicians. The proposal of Manfred Salmhofer to build a continuous version of the renormalization group for Fermionic theories [S] is therefore very interesting and welcome. Indeed Fermionic series with cutoffs are convergent (in contrast with Bosonic ones, which are Borel summable at best), and the continuous version of renormalization group which works so well at the perturbative level should therefore apply to them1 . In this paper we realize the Salmhofer proposal on the particular example of the two-dimensional Gross-Neveu model. We rearrange Fermionic perturbation 1 The continuous limit of the discretized non perturbative RG equations has been also studied for a certain many fermion system in 1+1 dimension in reference [C].

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theory according to trees, an idea first developed in [AR2], perform subtractions only when necessary according to the relative scales of the subgraphs, and obtain (to our own surprise, quite easily) an explicit convergent representation of the model without any discretization or cluster or Mayer expansion. To prove the convergence requires only some well-known perturbative techniques of parametric representations (“Hepp’s sectors”), Gram’s bound on determinants and a crucial but rather natural concatenation of some intervals of integration for loop lines. Therefore we can now consider that constructive theory for Fermions has been “reduced” to perturbation theory. Remark also that since the representation we use is an “effective” representation in the sense of [R], hence with subtractions performed only when necessary according to the relative scales of the subgraphs, we never meet the so-called problem of “overlapping divergences” or classification of Zimmermann’s forests. In this sense constructive renormalization is easier than ordinary perturbative renormalization (which, from the constructive point of view, is flawed anyway because it generates renormalons; these renormalons are the reason for which the ordinary renormalized perturbation theory of this model is only Borel summable [FMRS], as stated in Theorem 1). Having an explicit convergent representation of the theory with a continuously moving cutoff, it is trivial both to define the continuous renormalization group equations which correspond to the variation of this cutoff and, at the same time, to check that our explicit representation is a solution of these equations. Remark however we have not yet found the way to short-circuit our representation and to prove that the equations and their solutions exist by a purely inductive argument `a la Polchinski [P] which would avoid an explicit formula for the solution. This is presumably possible but this question as well as the extension to other models, in particular to interacting Fermions models of condensed matter physics, is left for future investigation. It is also important to recall that we do not see at the moment how to extend this method to Bosons, since there are no determinant and Gram’s bound for them.

II Model and main result We consider the massive Gross-Neveu model GN2 , which describes N types of Fermions. These Fermions interact through a quartic term. Actually, the GN2 action also requires a quadratic mass counterterm and a wave function counterterm in order for the ultraviolet limit to be finite. Therefore the bare action in a finite volume V is (using the notations of [FMRS]): SV

=

λ N

Z

X

d2 x [ V

a

Z 2

+δm

ψ¯a (x)ψa (x)]2

X

d x[ V

a

(II.1) Z

ψ¯a (x)ψa (x)] + δζ

X

d2 x [ V

a

ψ¯a (x)i 6 ∂ψa (x)]

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Continuous Constructive Fermionic Renormalization

3

where λ is the bare coupling constant, δm and δζ are the bare mass and wave function counterterms, and a is the color index: a = 1, . . . , N . The action (II.1) and the power counting of the GN2 model are like the ones for the Bosonic φ44 theory, except that, unlike the latter, the GN2 theory is asymptotically free for N ≥ 2, a condition which we assume from now on. The free covariance in momentum space is 1 6p + m γδ Cab (p) = δa,b = δa,b (II.2) − 6 p + m γ,δ p2 + m2 γ,δ where γ, δ are the spin indices, and a, b are the color indices. Most of the time we skip the inessential spin indices to simplify notation. The mass m is the renormalized mass. To avoid divergences, according to the notations of [KKS] we introduce an ultraviolet cut-off Λ0 and (for later study of the renormalization group flow) a scale parameter Λ which plays the role of an infrared cutoff: 2 2 2 2 (p +m ) (p +m ) CΛΛ0 (p) = C(p) η − η . (II.3) 2 2 Λ0

Λ

The cutoff function η might be any function which satisfies η(0) = 1, which is smooth, monotone and rapidly decreasing at infinity (this means faster than any fixed power). For simplicity in this paper we restrict ourselves to the most standard case η(x) = e−x . In this case both CΛΛ0 and its Fourier transform have explicit socalled parametric representations: Z Λ−2 2 2 Λ0 CΛ (p) = (6 p + m) e−α(p +m ) dα Λ−2 0

Z

CΛΛ0 (x

− y)

= π

Λ−2

Λ−2 0

(6 x− 6 y) m −αm2 −|x−y|2 /4α i + dα e 2α2 α

(II.4)

We define now the connected truncated Green functions, also called vertex functions, which are the coefficients of the effective action. The partition function with external fields ξ, ξ¯ is Z ¯ ΛΛ0 ¯ ¯ −SV (ψ,ψ)+<ψ,ξ>+<ξ,ψ> ZV (ξ, ξ) = dµC Λ0 (ψ, ψ)e Λ Z ¯ < ψ, ξ > := d2 x ψ(x)ξ(x). (II.5) V

The vertex function with 2p external points is: ΛΛ0 0 ΓΛΛ 2p ({y}, {z}) : = Γ2p (y1 , . . . , yp , z1 , . . . , zp )

= lim

δ 2p

¯ 1 )...δ ξ(y ¯ p) V →∞ δξ(z1 )...δξ(zp )δ ξ(y

ξ, CΛΛ0 ξ

(II.6) (ln ZVΛΛ0 − F )(CΛΛ0 )−1 (ξ) ξ=0

where F (ξ) =< > is the propagator, and color indices are implicit. Please do not confuse these vertex functions, which are connected, with the one-particle

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irreducible functions, since the latter are usually also called Γ in the literature. These functions (in fact distributions) form the coefficients of the effective action (expanded in powers of the external fields) at energy Λ with UV cutoff Λ0 . Developing the exponential in Z and attributing prime and double prime indices respectively to the mass and wave function counterterms we have: ZVΛΛ0 (ξ) = Z 2

∞ X 1 p!2 p=0 2

∞ X

0

n,n0 ,n00 =0

2

2

00

(−1)n+n +n n!n0 !n00 ! 2

2

d y1 ...d yp d z1 ...d zp d x1 ...d

X

λ N

n

0

00

(δm)n (δζ)n

xn d2 x01 ...d2 x0n0 d2 x001 ...d2 x00n00

V

(

(II.7)

ai bi ci di p Y

ξdi (zi )ξ¯ci (yi )

i=1

y1,c1 z1,d1

... yp,cp ... zp,dp

x1,a1 x1,a1

x1,b1 x1,b1

... xn,an ... xn,an

x01,a0 1 x01,b0

xn,bn xn,bn

1

... x00n00 ,a0000 n ... x00n00 ,b0000

)

n

where we used Cayley’s notation for the determinants: ui,a = det(Dab (ui − vj )) vj,b

(II.8)

and ai , bi , a0i , b0i , a00i , b00i , ci , di are the color indices. By convention Dab (ui − vj ) := Cab (ui − vj )

(II.9)

except when the second index is the one of a ψ field hooked to a δζ vertex. In this particular case the vertex has a so-called derivative coupling, and therefore the propagator D bears a derivation, namely Dab00 (ui − vj ) := i 6 ∂vj Cab00 (ui − vj ):= 0 Cab 00 (ui − vj ). This derived propagator is explicitly 0

CΛΛ0 (x − y)

Z =π

Λ−2 Λ−2 0

|x − y|2 im(6 x− 6 y) 1 + − 2 3 2 4α 2α α

e−αm

2

−|x−y|2 /4α

dα (II.10)

Expanding the determinant in (II) one obtains the usual perturbation theory in terms of Feynman graphs with the three types of vertices corresponding to the three terms of the action (II.1), and the logarithm is simply the sum over connected graphs. To see if a graph is connected, it is not necessary to know its whole structure but only a tree in it. Based on this remark the logarithm of (II) was computed in [AR2] using an expansion which is intermediate between the determinant form (II) and the fully expanded Feynman graphs. This expansion is based on a forest formula. Such formulas, discussed in [AR1], are Taylor expansions with integral remainders. They test the coupling or links (here the propagators) between n ≥ 1 points (here the vertices) and stop as soon as the final connected components are built. The result is therefore a sum over forests, which are simply defined as union of disjoint trees. A forest is therefore a (pedantic, but poetic) word for a Feynman graph without loops, and our point of view is that these are the natural objects to express Fermionic perturbation theory.

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Here we use the most symmetric forest formula, the ordered Brydges-Kennedy Taylor formula, which states [AR1] that for any smooth function H of the n(n − 1)/2 variables ul , l ∈ Pn = {(i, j)|i, j ∈ {1, . . . , n}, i 6= j}, ! ! k k X Z Y Y ∂ H|ul =1 = dwq H (wlF (wq ), l ∈ Pn ) (II.11) ∂u l 0≤w1 ≤...≤wk ≤1 q=1 q q=1 o−F

where o − F is any ordered forest, made of 0 ≤ k ≤ n − 1 links l1 , . . . , lk over the n points. To each link lq q = 1, . . . , k of F is associated the parameter wq , and to each pair l = (i, j) is associated the weakening factor wlF (wq ). These factors Q replace the variables ul as arguments of the derived function kq=1 ∂u∂l H in (II.11). q

These weakening factors wlF (w) are themselves functions of the parameters wq , q = 1, . . . , k through the formulas F wi,i (w) = 1 F (w) = wi,j

if i and j are connected by F

inf wq ,

F lq ∈Pi,j

F is the unique path in the forest F connecting i to j where Pi,j F wi,j (w) = 0

if i and j are not connected by F.

(II.12)

We apply this formula to the determinant in (II), inserting the interpolation parameter ul in the cut-off (but only between distinct vertices, so not for the “tadpole” lines): CΛΛ0 (x, y, u)

=

δ(x − y)CΛΛ0 (x, x) + [1 − δ(x − y)]CΛΛ0 (x, y, u) Λ (u)

:= CΛ 0 (x, y) Z Λ−2 (6 x− 6 y) m −αm2 −|x−y|2 /4α i := π + e dα (II.13) 2α2 α Λ−2 0 (u) where

−2 −2 Λ−2 + u(Λ−2 ). 0 (u) = Λ 0 −Λ

(II.14)

We use similar interpolation for the C 0 propagators. When u grows from 0 to 1, the ultraviolet cut-off of the interpolated propagator (between distinct vertices) grows therefore from Λ to Λ0 . We define ∂ Λ0 (6 x− 6 y)Λ40 (u) Λ0 ,u 2 CΛ (x, y) := C (x, y, u) = π i + mΛ0 (u) ∂u Λ 2 −2

−m Λ0 (u)−|x−y| Λ0 (u)/4 · (Λ−2 − Λ−2 (II.15) 0 )e 2 6 0 ∂ 0 Λ0 |x − y| Λ0 (u) CΛΛ0 ,u (x, y) := C (x, y, u) = π ∂u Λ 4 4 2 2 im(6 x− 6 y)Λ0 (u) −m2 Λ−2 0 (u)−|x−y| Λ0 (u)/4 − Λ40 (u) (Λ−2 − Λ−2 + 0 )e 2 2

2

2

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impulsions

Λ0 Λ0 (w)

Λ

0

positions

Figure 1 The derivative of η fixes C u at an energy near Λ0 (u). We observe that for any fixed 0 we have the scaled decay: 0

−|x−y|(1− )Λ0 |C u (x, y)| ≤ KΛ30 (u)(Λ−2 − Λ−2 0 )e 0

0

m

−|x−y|(1− )Λ0 |C u (x, y)| ≤ KΛ40 (u)(Λ−2 − Λ−2 0 )e

(u)−0 m2 Λ−2 0 (u)/2

(II.16)

(u)−0 m2 Λ−2 0 (u)/2

(II.17)

m

where K is a constant depending only on 0 and Λm 0 (u) := sup[m, Λ0 (u)].

(II.18)

Applying this interpolation and the ordered forest formula (II.11) to the propagators in the determinant of (II) we obtain ZVΛΛ0 (ξ) = Z

∞ X 1 p!2 p=0 2

∞ X n,n0 ,n00 =0 2

X X X n 0 00 1 λ (δm)n (δζ)n (F, Ω) N 0 00 n!n !n !

2

o−F Col

2

2

Ω 2

d y1 . . . d yp d z1 . . . d zp d x1 . . . d xn¯ V

Z

Y k

0≤w1 ≤...≤wk ≤1 q=1

p Y

ξdr (zr )ξ¯cr (yr )

r=1

Λ ,w DΛ0 q (¯ xlq , xlq )dwq [det]left (wF (w))

(II.19)

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Continuous Constructive Fermionic Renormalization

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where for simplicity the position of any vertex is simply denoted by the letter x and n ¯ := n + n0 + n00 . x ¯lq and xlq are the ends of line lq . [det]left is the remaining determinant. Its entries correspond to the remaining fields necessary to complete each vertex of the forest into a quartic or quadratic vertex, according P to its type (interaction or counterterm). For this model, remark that the sum o−F is performed only over the ordered forests that have, for each point xi coordination number n(i) ≤ 4 or n(i) ≤ 2 depending of the type of the vertex (all other terms being zero). The additional sums over Col and Ω correspond to coloring choices at each vertex and “fields versus antifields” choices at each line and vertex [AR2]. The sign (F, Ω) comes in from the antisymmetry of Fermions and is computed in [AR2]: here we only need to know that it factorizes over the connected components of F. To find the expression for ln Z we write Z as an exponential. In equation (II), the determinant factorizes over the ordered trees T1 . . . Tj forming the forest. Indeed one can resum all orderings of the ordered forest F compatible with fixed orderings of its connected components, the trees T1 . . . Tj . Furthermore the “weakening factor” wF vanishes between vertices belonging to different connected components. Hence: ZVΛΛ0 (ξ) ∞ X 1 = 2 p! p=0

∞ X

n

n,n0 ,n00 =0

X

X 1 1 n!n0 !n00 ! j=0 j!

X

X

n1 ,...nj n1 +···+nj =n

X

00 n01 ,...n0j n00 1 ,...nj 00 00 n01 +···+n0j =n0 n00 1 +···+nj =n

n!n0 !n00 ! p!2 p0 ! 0 0 00 00 n1 ! . . . nj !n1 ! . . . nj !n1 ! . . . nj ! p1 !2 . . . pj !2 p0 !2

p1 ,...pj ,p0

p1 +···+pj +p0 =p 0

(ξ, CΛΛ0 ξ)p

j h Y

λ N

n

i

i 0 00 (δm)ni (δζ)ni A(ni , n0i , n00i , pi )

(II.20)

i=1

where A(ni , n0i , n00i , pi ) Z X X (Ti , Ωi ) d2 y1 . . . d2 ypi d2 z1 . . . d2 zpi d2 x1 . . . d2 xn¯ i = Ti Coli ,Ωi p Y

ξdr (zr )ξ¯cr (yr )

r=1 Ti

[det]left,i (w (w))

n¯Y i −1

Z 0≤w1 ≤...≤wn ¯ i −1 ≤1

Λ ,w DΛ0 q (¯ xlq , xlq )dwq

q=1

(II.21)

¯ i −1 where n ¯ i is the number of vertices in the ordered tree Ti , which has therefore n lines, pi is the number of external fields of type y (and z) attached to the Ti , and p0 is the number of free external propagators (not connected to any vertex) in the

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forest. This can be written as an exponential, hence ln ZVΛΛ0 (ξ) ∞ ∞ X X 0 00 n 1 1 λ (δm)n (δζ)n N 2 0 00 p! n!n !n ! p=0 n,n0 ,n00 =0 Z p X X Y 2 2 2 2 2 2 (T , Ω) d y1 . . . d yp d z1 . . . d zp d x1 . . . d xn¯ ξdr (zr )ξ¯cr (yr )

= (ξ, CΛΛ0 ξ) +

o−T Col,Ω

Z

V

n¯Y −1

0≤w1 ≤...≤wn−1 ≤1 q=1 ¯

r=1

Λ0 ,wq DΛ (¯ xlq , xlq )dwq [det]left (wT (w))

(II.22) where T is an ordered tree over n ¯ points, and the external points are all connected to the tree. Now, applying the definition (II.6), we obtain the vertex functions, for which the limit V → ∞ can be performed (because the external points hooked to the tree ensure convergence). The set E

= {(i1 , . . . ip , j1 , . . . , jp )|i1 , . . . ip , j1 , . . . jp ∈ {1, . . . , n ¯ }}

(II.23)

fixes the internal points to which the 2p external lines hook. We recall the well-known fact that the vertex functions in x-space are in fact distributions. For instance it is easy to see that when some of the external points ik , jk in the previous sum coincide, one has to factor out the product of the corresponding delta functions of the external arguments to obtain smooth functions. This little difficulty can be treated either by considering the vertex functions in momentum space (they are then ordinary functions of external momenta, after factorization of global momentum conservation), or by smearing the vertex functions with test functions. Here we adopt this last point of view. The quantity 0 under study is then ΓΛΛ 2p smeared with smooth test functions φ1 (y1 ), . . . , φp (yp ), φp+1 (z1 ), . . . , φ2p (zp ): 0 ΓΛΛ 2p (φ1 , . . . φ2p ) Z = d2 y1 . . . d2 yp d2 z1 . . . d2 zp 0 ΓΛΛ 2p (y1 , . . . , yp , z1 , . . . , zp )φ1 (y1 ) . . . φp (yp )φp+1 (z1 ) . . . φ2p (zp ).

(II.24) where we asked the test functions to have compact support: φ ∈ D(R2 ). Remark that when some external antifield hooks to a δζ vertex, the amputation by C instead of C 0 leaves a δ 0 distribution, which means a derivative acting on the corresponding test function.

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Continuous Constructive Fermionic Renormalization

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We obtain the formula: ∞ X

0 ΓΛΛ 2p (φ1 , . . . φ2p ) =

XX X o−T

Z

E Col,Ω

λ N

n

0

n,n0 ,n00 =0

Z

(T , Ω) n¯Y −1

0≤w1 ≤···≤wn−1 ≤1 q=1 ¯

00

(δm)n (δζ)n

1 n!n0 !n00 !

(II.25)

d2 x1 . . . d2 xn¯ φ1 (xi1 ) . . . φ2p (xjp ) Λ ,w DΛ0 q (¯ xlq , xlq )dwq

[det]left (wT (w), E)

where the propagator D is now C or C 0 according to the discussion above. When renormalization is introduced, it will be convenient to use the BPHZ subtraction prescription at 0 external momenta, which corresponds to integrate the vertex functions over all arguments except one. In this prescription one defines the renormalized coupling constant as the 4-vertex function of the full theory at zero external momenta: Z λren b ΛΛ0 (0, 0, 0, 0) = d2 x2 d2 x3 d2 x4 ΓΛΛ0 (0, x2 , x3 , x4 ) (II.26) := Γ 4 4 N Moreover we want the renormalized mass and wave function constant to be respectively m and 1. This means that we impose the additional renormalization conditions: Z b ΛΛ0 (0, 0) = d2 x2 ΓΛΛ0 (0, x2 ) = 0 δmren := Γ (II.27) 2 2 Z b ΛΛ0 (0, 0) = d2 x2 i 6 x2 ΓΛΛ0 (0, x2 ) = 0 (II.28) δζren :=6 ∂ Γ 2 2 With these conditions the whole theory (at fixed renormalized mass m) becomes parametrized only by λren , hence not only λ but also δm and δζ in (II.1) become functions of λren . This of course has a precise meaning only if we can construct the theory and solve the renormalization group flows, which is precisely what we are going to do. We can express the main result of this paper as a theorem on the existence of the ultraviolet limit of the vertex functions and of the renormalization group flows. Recall that the theory is not directly the sum but the Borel sum of the renormalized perturbation theory. In summary 0 Theorem 1 The limit Λ0 → ∞ of ΓΛΛ 2p (φ1 , . . . φ2p ) exists and is Borel summable

in the renormalized coupling constant λren , uniformly in N (where N is the number of colors). Since the parameter Λ varies continuously, the continuous renormalization group equations and in particular the β function are also well defined in the limit Λ0 → ∞. The first part of the theorem is similar to [FMRS], but the second part (the existence of the continuous renormalization group equations) is new. The rest of the paper is devoted to the proof of this theorem.

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The precise bounds on the smeared vertex functions are given in Theorem 3 below. They are uniform in N (and in fact proportional to N 1−p ). Let us discuss also briefly the dependence in m, the renormalized mass. For m 6= 0 fixed, we can define the physical scale of the system by putting m = 1. The theorem is then uniform in the infrared cutoff Λ, including the point Λ = 0. In the case m = 0 our method requires a nonzero infrared cutoff Λ 6= 0. Since this cutoff is the only scale of the problem, we can then put it to 1: Λ = 1. In this last case, improperly called the “massless theory”, we know that there should be a non-perturbative mass generation [GN]. This mass generation has been proved rigorously for the model with fixed ultraviolet cutoff and large number N of components in [KMR], using the Matthews-Salam formalism of an intermediate Bosonic field and a cluster expansion with a small/large field expansion. Our result in the massless case m = 0 with a finite infrared cutoff Λ should therefore glue with the method and results of [KMR] to obtain at large N the mass generation of the full model without ultra-violet cutoff.

III The expansion III.1

The continuous band structure

Remark that in (II.25) w1 ≤ w2 ≤ · · · ≤ wn¯ −1 =⇒ Λ0 (w1 ) ≤ Λ0 (w2 ) · · · ≤ Λ0 (wn¯ −1 ).

(III.1)

This naturally cuts the space of momenta into n ¯ bands B = {1, . . . , n ¯ } (see Figure 2). Looking at equation (II.4), we see that the covariance can be written as a sum of propagators restricted to single bands: CΛΛ0 (p) =

n ¯ X

Λ (w )

k CΛ00(wk−1 ) (p) =

k=1

n ¯ Z X k=1

Λ−2 0 (wk−1 )

Λ−2 0 (wk )

(6 p + m)e−α(p

2

+m2 )

dα = C(p)

n ¯ X

ηk

k=1

(III.2) where we defined ηk

−2

:= e−Λ0

(wk )(p2 +m2 )

−2

− e−Λ0

(wk−1 )(p2 +m2 )

(III.3)

and we adopted the convention −2 w0 = 0 ⇒ Λ−2 0 (w0 ) = Λ

−2 wn¯ = 1 ⇒ Λ−2 ¯ ) = Λ0 . 0 (wn

(III.4)

Similar formulas hold for C 0 but with an additional 6 p. To cure the ultraviolet divergences we have to combine the divergent local parts of some subgraphs with counterterms and reexpress the series for G as an effective series in the sense of [R]. For that purpose we use the band structure to distinguish the divergent subgraphs from the convergent ones and hence decide where renormalization is necessary. Please recall that in contrast with usual

Vol. 1, 2000

Continuous Constructive Fermionic Renormalization

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momenta Λ0 n ¯ Λ0 (wn¯ −1 ) n ¯−1 Λ0 (wn¯ −2 )

links of the tree Λ0 (w1 ) 1

Λ

0

positions

Figure 2 perturbation theory we never develop explicitly the loop lines of these subgraphs. Contrary to naive expectation, one does not need to know the particular loop structure to perform renormalization!

III.2

Notations

Now we fix some notations. It is convenient to give indices to the fields variables or the half-lines which correspond to these fields after Grassmann integration. We observe that there are several types of such variables, the half-lines which form the lines of the tree, the external variables (which correspond to amputated lines) and the entries (rows or columns) in the determinant detleft . These entries will be called “loop fields” or “loop half-lines” since they form the usual loop lines of the Feynman graphs if one expands the determinant. We define E and L as the set of all external and loop half-lines. For each level i there is a tree-line li with two ends corresponding to two half-lines called fi and gi (to fix ideas let’s say that fi is the end corresponding to the field and gi the end corresponding to the anti-field, as ¯ f ) are called hf and decided by the index Ω in (II)). The loop fields ψ(xg ) and ψ(x ¯ hg , and (when expanded) the loop line ψ(xf )ψ(xg ) is called lf g (it corresponds to a particular coefficient in the determinant detleft ). Each tree half-line fi or gi , each loop field hf or hg is hooked to a vertex called vfi or vgi or vf or vg . We need also

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to care about the set S of special fields (or antifields) which are hooked to the δζ vertices and correspond to propagators C 0 which have different “power counting” than C. Finally the index of the highest tree-line hooked to a vertex v is called iv . Now [det]left is the determinant of a matrix (n + 1 − p) × (n + 1 − p). The corresponding loop fields can be labeled by an index a = 1, . . . , 2n + 2 − 2p. The matrix elements are D(xf , xg , wvTf ,vg (w)). Therefore in terms of bands the line lf g is restricted by the weakening factor wvTf ,vg (w) to belong to the bands from 1 to T T the lowest index in the path Pf,g (this path Pf,g is defined in equation (II.12)). T We call if,g this index: T } iTf,g = inf {q | lq ∈ Pf,g

(III.5) iT f,g

D(xf , xg , wvTf ,vg (w)) = D(p)

X

η k (p)

(III.6)

k=1

By multilinearity one can expand the determinant in (II.25) according to the different bands in the sum (III.6) for each row and column. X [det]left (wT (w), E) = det M(µ) (III.7) µ

where we define the attribution µ as a collection of band indices for each loop field a: µ = {µ(f1 ), . . . µ(fn+1−p ), µ(g1 ), . . . µ(gn+1−p )} , µ(a) ∈ B for a = 1 . . . 2n+2−2p. (III.8) Now, for each attribution µ we need to exploit power counting. This requires notations for the various types of fields or half-lines which form the analogs of the quasi local subgraphs of [R] in our formalism. We define: Tk ITk ILk EEk ETk

= {l ∈ T | ivl ≥ k} = {fi , gi ∈ T | i ≥ k} = {a ∈ L| µ(a) ≥ k} = {f, g ∈ E|ivf , ivg ≥ k} = {fi |ivfi ≥ k, i < k} ∪ {gi |ivgi ≥ k, i < k}

ELk Nk Nk0 Nk00 ¯k N Gk Ek Ek00

= {a ∈ L|iva ≥ k, µ(a) < k} = {v of type λ |iv ≥ k} = {v of type δm |iv ≥ k} = {v of type δζ |iv ≥ k} = Nk ∪ Nk0 ∪ Nk00 = ITk ∪ ILk = EEk ∪ ETk ∪ ELk = Ek ∩ S , Tk = {li | i ≥ k}

(III.9)

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where we recall that S in the last definition is the set of those fields hooked to a vertex of type δζ which bear a derivation. We note |A| the number of elements in the set A. For instance the reader can check that |IT1 | = 2¯ n − 2 and that ¯ − 1. Each Gk has c(k) connected components Gjk , j = 1, . . . , c(k). |T1 | = n To help the reader understand better these technical definitions, let’s say that: • T stands for “tree” • IT stands for the set of “internal tree half lines” of a subgraph; • IL stands for the set of “internal loop half lines” of a subgraph. Together ITk and ILk form the full subgraph Gk ; • EE stands for the set of “half lines which are external both for the subgraph and for the whole graph”; • ET stands for the set of “half lines which are external for the subgraph but not for the whole graph, and which belong to the tree”; • EL stands for the set of “half lines which are external for the subgraph but not for the whole graph, and which are loop lines in the full graph”; • N N 0 and N 00 are used for the different types of vertices in a subgraph. All the definitions in (III.9) can be restricted to each connected component. Applying power counting, the convergence degree for the subgraph Gki is ω(Gki ) =

0 1 (|Eik | + 2|Ni k | − 4) 2

(III.10)

where we assumed that no external half-line hooked to a vertex of type δζ bears a i 6 ∂ . To assure this for any Gki , we apply, for each vertex v 00 , the operator i 6 ∂ (or −i ∂6 ) to the highest tree half-line hooked to v 00 (there is always at least one). In this way for all k |Ek00 | = 0, and no loop line bears a gradient. Then M(µ) is a matrix whose coefficients are Z d2 p −ip(xf −xg ) ) Mf g (µ)(xf , xg ) = δµ(f ),µ(g) e C(p) η µ(f ) (p)Wvµ(f (III.11) f ,vg (2π)2 where k Wv,v 0

= 1 if v and v 0 are connected by Tk = 0 otherwise

(III.12)

since we always have D = C in the matrix M(µ). From (III.10) we see that there are three types of divergent subgraphs: 0

• for |Eik | = 4, |Ni k | = 0 we have logarithmic divergence (ω(Gki ) = 0); 0 • for |Eik | = 2, |Ni k | = 0 we have linear divergence (ω(Gki ) = −1); 0 • for |Eik | = 2, |Ni k | = 1 we have logarithmic divergence (ω(Gki ) = 0). In fact the divergent graphs are only those for which the algebraic structure of the external legs is of one of the three types in (II.1). For instance not all four-point

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subgraphs are divergent, but only those for which the flow of spin indices follows the flow of color indices [GK][FMRS]. Using the invariance of L under parity and charge conjugation one finds that all counterterms which are not of the three types in (II.1) are zero (this means that the corresponding subgraphs have 0 local part). Then renormalizing these subgraphs we improve power counting without generating new counterterms. In what follows, for simplicity, “divergent subgraph” always means subgraph with two or four external legs (this means we will renormalize some subgraph which does not need it but this does not affect the convergence of the series). Also for simplicity we change the definition of convergence degree (III.10) in 1 ω0 (Gki ) = (|Eik | − 4) . (III.13) 2 To cure divergences, we apply to the amplitude of each divergent subgraph g the operator (1−τg )+τg . In the momentum space τg is the Taylor expansion at order −ω(g) of the amplitude gˆ(p) at p = 0. The operator 1−τg makes the amplitude convergent when the UV cut-off is sent to infinity. The remaining term τg gˆ gives a local counterterm for the coupling constant that depends on the energy of the external lines of g. At each vertex v, we can resum the series of all counterterms obtained applying τg to all divergent subgraphs (for different attributions µ) that have the same set of external lines as v itself. In this way we obtain an effective coupling constant which depends on the energy Λ0 (wiv ) of the highest tree line hooked to the vertex v. This is true because after applying the 1 − τg operators, for each graph with nonzero amplitude the highest index at each vertex coincides with the highest tree index iv at each vertex! Indeed at vertices v for which this is not true, there are loop fields with attribution µ higher than iv . By (III.12) they must contract together forming tadpoles, which are set to zero by the 1 − τg operators. The corresponding graphs therefore disappear from the expansion. For each attribution µ we define the set of divergent subgraphs as Dµ := { Gki |ω0 (Gki ) ≤ 0}.

(III.14)

The action of τg is −ω 0 (g)

τg gˆ(p1 , . . . , pk ) =

X 1 dj gˆ(tp1 , . . . tpk )|p=0 j! dtj j=0

k = 2, 4.

(III.15)

With this definition the effective constants λw , δmw , δζw turn out to be the vertex functions Γ4 , Γ2 and 6 ∂Γ2 for an effective theory with infrared parameter Λ = Λ0 (w):

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Continuous Constructive Fermionic Renormalization

15

Effective constants

In the space of positions, the operator τg is computed by partial integration on the product of external propagators a(x1 , . . . xve ), as in [R]: −ω 0 (g)

τg∗ (ve )a(x1 , . . . xve )

X 1 dj a(x1 (t), . . . , xve (t))|t=0 j! dtj j=0

=

(III.16)

where xi (t) = xve +t(xi −xve ), and ve is an external vertex of g chosen as ‘reference vertex’. This formula means that τg∗ for each divergent tree subgraph g moves all external half-lines to a single reference vertex in the subgraph, hence computes a local couterterm. The choice of this reference vertex is given in Section IV.3.1. As announced we find the three possible counterterms of (II.1). For |Ei | = 4 we have τ ∗ (x1 )

4 Y

C(xi , yi ) =

i=1

4 Y

C(x1 , yi ),

(III.17)

i=1

so the counterterm is Z Z 4 Y ai ai 2 Cαi α0 (x1 , yi ) d2 x2 d2 x3 d2 x4 g(0, x2 , x3 , x4 )aα11aα22aα33aα44 d x1 i

i=1

Z =

2

d x1

4 Y

Cαaii aαi0 (x1 , yi ) gˆ(0, . . . , 0) i

(III.18)

i=1

This gives a coupling constant counterterm. For |Ei | = 2 we have ∗ µ ∂ C(x1 , y2 ) . τ (x1 )[C(x1 , y1 )C(x2 , y2 )] = C(x1 , y1 ) C(x1 , y2 ) + (x2 − x1 ) ∂xµ1 (III.19) Integrating over internal points, we obtain a mass counterterm from the first term in the sum: Z Z 2 Y d2 x1 Cαaii aαi0 (x1 , yi ) d2 x2 gαa11aα22 (0, x2 ) i

Z

i=1

d2 x1

= Z =

d2 x1

2 Y

Cαaii aαi0 (x1 , yi ) gˆα1 α2 (0)δa1 ,a2 i

i=1 2 Y

Cαaii aαi0 (x1 , yi ) δa1 ,a2 f1 (0) i

(III.20)

i=1

where we applied the development gˆ(p) = f1 (p2 ) + γ5 f2 (p2 )+ 6 pf3 (p2 ) + γ5 6 pf4 (p2 )

(III.21)

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and we adopted for the gamma matrices the conventions in [FMRS]. By invariance under charge conjugation and parity f2 (0) = f4 (0) = 0. For the second term in the sum we obtain a wave function counterterm: Z Z ∂ d2 x1 Cαa11 aα10 (x1 , y1 ) µ Cαa22 aα20 (x1 , y2 ) d2 x2 (x2 − x1 )µ gαa11aα22 (x1 − x2 ) = 1 2 ∂x1 Z ∂ ∂ = d2 x1 Cαa11 aα10 (x1 , y1 ) µ Cαa22 aα20 (x1 , y2 )i µ gˆ(p)|p=0 δa1 a2 1 2 ∂x1 ∂p Z a a a a = d2 x1 Cα11 α10 (x1 , y1 )i 6 ∂Cα12 α10 (x1 , y2 )f3 (0). (III.22) 1

2

Theorem 2 If we apply to each divergent subgraph g ∈ Dµ , for any attribution µ, the operator (1 − τg ) + τg = Rg + τg , the function (II.25) can be written as

Z XX X 1 = (T , Ω) d2 x1 . . . d2 xn¯ n!n0 !n00 ! 0 00 n,n ,n =0 o−T E,µ Col,Ω " #" # #" Z n ¯Y −1 Y λw(v) Y Y dwq δmw(v0 ) δζw(v00 ) N 0≤w1 ≤···≤wn−1 ≤1 q=1 0 00 ¯ v ∞ X

0 ΓΛΛ 2p (φ1 , . . . φ2p )

Y Gk i ∈Dµ

RGki

n¯Y −1

v

Λ ,w DΛ0 q (¯ xlq , xlq )

v

det M(µ)φ1 (xi1 ) . . . φ2p (xjp )

(III.23)

q=1

where the constants λw , δmw , δζw are the ‘effective constants’, defined as: Z λw Λ0 (w),Λ0 Λ (w),Λ0 ˆ = Γ4 (0, 0, 0, 0) = d2 x2 d2 x3 d2 x4 Γ4 0 (0, x2 , x3 , x4 ) N Z ˆ Λ0 (w),Λ0 (0, 0) = d2 x2 ΓΛ0 (w),Λ0 (0, x2 ) δmw = Γ 2 2 Z ˆ Λ0 (w),Λ0 (p)|p=0 = d2 x2 i 6 x2 ΓΛ0 (w),Λ0 (0, x2 ) (III.24) δζw = 6 ∂ Γ 2 2 The effective constants are the vertex functions Γ4 , Γ2 and 6 ∂Γ2 for an effective theory with infrared parameter Λ0 (w), and the renormalized constants correspond to the effective ones at the energy Λ. (For the massive theory recall that we can use Λ = 0.) λw=0 δmw=0 δζw=0

= λr = δmr = 0 = δζr = 0

(III.25)

The reshuffling of perturbation theory performed by Theorem II can be proved by standard combinatorial arguments as in [R] (the only difficulty was discussed above, when we remarked that the parameter w of the effective constants always

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Continuous Constructive Fermionic Renormalization

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corresponds to the highest tree line of the vertex. Otherwise the effective vertex generates a tadpole graph whose later renormalization gives 0). This reshuffling is similar to the reorganization of renormalized perturbation theory according to the formalism of Gallavotti and coworkers [GN].

IV Convergence of the series Theorem 3 Let > 0 be fixed. Suppose Λm (defined below) belongs to some fixed compact X of ]0, +∞). The series (III.23) is absolutely convergent for |λw |, |δmw |, |δζw | ≤ c, c small enough. This convergence is uniform in Λ0 and N (actually Γ2p ΛΛ0 is proportional to N 1−p ). The ultraviolet limit ΓΛ exists and 2p = limΛ0 →∞ Γ2p satisfies the bound: 5/2 |ΓΛ [K(c, , X)]p (Λm )2−p N 1−p 2p (φ1 , . . . φ2p )| ≤ (p!)

||φ1 ||1

2p Y

||φi ||∞,2 e−(1−)Λ

m

(IV.1)

dT (Ω1 ,...Ω2p )

i=2

where ||φi ||∞,2

Λm := sup [Λ, m] , 0 00 := ||φi ||∞ + ||φi ||∞ + ||φi ||∞ ,

(IV.2) (IV.3)

Ωi is the compact support of φi , K(c, , X) is some function of c and R X, which tends to zero when c tends to 0, ||φi ||∞ = supx∈Ωi |φi (x)|, ||φ1 ||1 = d2 x|φ1 (x)|, and dT (Ω1 , . . . Ω2p ) := dT (x1 , . . . x2p ) :=

inf dT (x1 , . . . x2p ) X inf |¯ xl − xl |.

xi ∈Ωi u−T

(IV.4)

l∈T

where in the definition of dT (x1 , . . . x2p ), called the “tree distance of x1 , . . . x2p ”, the infimum over u − T is taken over all unordered trees (with any number of internal vertices) connecting x1 , . . . x2p . This bound means that one can construct in a non perturbative sense the ultraviolet limit of either the massive theory with any infrared cutoff Λ including Λ = 0, or the weakly coupled massless theory with nonzero infrared cutoff Λ. To complete Theorem 1 from Theorem 3, one needs only to check Borel summability by expanding explicitly at finite order n in λren and controlling the Taylor remainder. This additional expansion generates a finite number of Taylor operators τg for a finite number of non quasi-local subgraphs, which are responsible for the n! of Borel summability [R]. Since this is rather standard we will not include this additional argument here. Finally the renormalization group equations are discussed in Section V. The rest of the section is devoted to the proof of this theorem.

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IV.1

Ann. Henri Poincar´ e

Plan of the proof

To prove the theorem we show that the absolute value of the term (n, n0 , n00 ) in the sum (excluding the effective constants) is bounded by K n¯ . The strategy for the proof consists in moving the absolute value inside all sums and integrals, bounding the product of effective constants, #" # #" " Y Y Y λw(v) |δmw(v0 ) | |δζw(v00 ) | ≤ cn¯ , (IV.1) v

v0

v00

then taking c < K −1 . The loop determinant will be bounded by a Gram inequality, and we shall use the tree lines decay to bound the spatial integrals. Actually, we cannot move the absolute value directly inside the sum over attributions because #{µ} ' n ¯ !. In other words fixing the band index for each single half-line develops too much the determinant. The way to overcome this difficulty is to remark that the attributions contain much more information than necessary. We can in fact group the attributions into packets to reduce the number of determinants to bound. We observe that, if for the level i a connected component Gki has |EEik | + |ETik | ≥ 5, the subgraph is convergent and we do not need to know the band indices for the loop lines in that connected component. So for each convergent Gki : • if |EEik | + |ETik | ≥ 5, we do not want to know anything on loop lines; • if |EEik | + |ETik | < 5, we just want to fix 5 − |EEik | − |ETik | half-lines with energy lower than i, but we are not interested in knowing the band index (' energy) of the other half-lines because we need to know whether a subgraph is convergent or not (i.e. has more than 4 external half lines or not), but we do not care about its exact number of external half lines; whether it has 10 or 25 does not matter, and it is in fact precisely the extraction of this information that could be dangerous for convergence of the expansion. Instead of expanding the loop determinant over lines and columns as a sum over all attributions X det M = det M(µ, E) (IV.2) µ

we write it as a sum over a smaller set P (called the set of packets). These packets are defined by means of the function φ : {µ} −→ P µ 7→ C = φ(µ)

(IV.3)

but this function must respect some constraints related to the future use of Gram’s inequality. This motivates the following definition:

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Continuous Constructive Fermionic Renormalization

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Definition 1 The pair (P, φ) is called a “Gram-compatible pair” if ∀ C ∈ P, ∀a, ∃ Ja (C) ⊂ B

(IV.4)

with the property φ−1 (C) = {µ|µ(a) ∈ Ja (C) ∀ a}. This definition means that for any packet C the attributions in the packet exactly correspond to a particular set of band indices allowed for each loop line. It ensures that there exists a matrix M0 such that X det M(µ) = det M0 (C) (IV.5) µ∈φ−1 (C)

because each loop line a is a matrix entry. This in turn ensures that Gram’s inequality can be applied to det M0 (C), as shown in Lemma 4.

IV.2

Construction of P

We build first the partition P of the set of attributions into packets. These packets should contain the informations we need over |Ekj |. In contrast with attributions there should be few of them; more precisely they should satisfy #P ≤ K n¯ . Finally, together with the function φ, they should form a Gram-compatible pair. To define P we introduce some preliminary definitions and notations. To each ordered tree o − T we can associate a rooted tree RT , which pictures the inclusion relation of the Gjk [R]. We can picture this tree with two types of vertices: crosses and dots. We recall that the leaves of a rooted tree are the vertices of the rooted tree with coordination number one. The leaves in our case are the dot-vertices and correspond exactly to the vertices v, v 0 or v 00 of the initial ordered tree T . The other vertices of RT are crosses. Each cross i corresponds to a line li of the initial ordered tree T , and has coordination number three, except the root which has coordination number two. To build RT we take the lowest line in T , l1 , as root 1.

T0

T 00

l1 l1 b1 Λ

Figure 3

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Ann. Henri Poincar´ e

This line l1 , or root, separates T into two connected components T 0 and T 00 possibly reduced to a single vertex. When T 0 or T 00 is a single vertex, it gives a dot connected to 1. Otherwise it gives a cross, which is the lowest line of T in it. This procedure is repeated at each cross-vertex obtained, and generates RT . T0

T 00 v l2 2 1

l1 b1 Λ

Figure 4 Finally to complete the picture to each dot of RT we hook all loop half-lines hooked to the corresponding vertex (there could be none). We define the ancestor of i A(i) as the cross-vertex just under i in RT and we call va , the dot-vertex to which the half-line a is hooked and ia the cross-vertex connected to va (which represents a line of the initial tree!). For each cross-vertex i we define ti := {lj ∈ T |j ≥ i, lj connected to li by Ti }

(IV.6)

This is the spanning tree in the connected component of Gi containing the line li . va

i

a

ia

A(i)

Figure 5 An example of a tree with its associated RT is given in Figure 6: For each tree line (cross-vertex) i and each connected component Gki , no new 0 00 line connects to ti in the interval between i and A(i). Hence ω 0 (Gki0 ) ≥ ω0 (GkA(i)+1 ) 0

0

∀i ≥ i0 > A(i) and Tik0 ⊂ Gki0 ⊂ Gki . Therefore we can neglect what happens in this interval and generalize the definitions of (III.9) for the internal lines of a subgraph Gki .

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Continuous Constructive Fermionic Renormalization

v0

Λ0 4

3 2 1 5 6

4

5

= 8

7

v 00

v 00 8

21

4

v0

2 2

v0

v 00

3

3

5

7

8

7

6 1

T

Λ

RT

6 1

Figure 6 We define: gi egi eti eei eli

:= ti ∪ {a ∈ L|ia ≥ i, va ∈ ti , µ(a) ≥ A(i) + 1} := eti ∪ eei ∪ eli := {fi0 |ivf ≥ i, vf ∈ ti , i0 < i} ∪ {gi0 |ivg ≥ i, vg ∈ ti , i0 < i} := {f, g ∈ E|ivf , ivg ≥ i, vf , vg ∈ ti } := {a ∈ L|ia ≥ i, va ∈ ti , µ(a) ≤ A(i)}

(IV.7)

This set of definitions (IV.7) concerns the connected component gi above line i. Remark that we defined as loop internal lines of gi , all loop lines higher than A(i). We also need some additional definitions concerning the other connected components: i(k) := inf j {j≥i,vj ∈Tik }

gik := gi(k)

(IV.8)

egik := etki ∪ eeki ∪ elik etki := eti(k)

eeki := eei(k)

elik := eli(k)

This second set of definitions is used only much later in the bounds when all connected components are considered at once. Definition 2 A chain Ca,i is the unique path in RT from the half-line a to the cross-vertex i with ia ≥T i: Cai := {i0 |i ≤T i0 ≤T ia } ∪ {a}

(IV.9)

In the following, we write ia ≥T i to specify that va and vi are connected by ti . Definition 3 A class C is a set of chains over RT with the properties: ∀Cai ∈ C, ∀Ca0 i0 ∈ C one has a 6= a0 ∀ i ci ≤ max[0; 5 − |eei | − |eti | − c0i ]

(IV.10)

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M. Disertori and V. Rivasseau

a

Ann. Henri Poincar´ e

va

Cai

i

Figure 7 where we defined: ci c0i

= #{Cai ∈ C| i fixed} = #{Ca0 i0 ∈ C| ia0 ≥T i, i0 < i}

(IV.11)

So ci is the total number of chains arriving at i and c0i is the total number of chains passing through i and continuing further below. This definition ensures therefore that there are at most five chains passing through each cross i. Definition 4 The partition P is the set of all possible classes C over RT . To verify that this is a good definition, we have to prove three lemmas. Lemma 1 The cardinal of P is bounded by K n¯ . Proof. We prove that P ⊆ P 0 and #P 0 ≤ K n¯ . We define P 0 as the set of all sets of chains D, that are unions of five subsets (possibly empty) Yj , where Yj is a set of completely disjoint chains (this means they have no cross and no dot in common). P 0 := {D}

D := ∪5j=1 Yj .

(IV.12)

To build a set of disjoint chains Yj , we have at most three possible choices for each vertex: at each cross-vertex we can have no chain passing, a chain going right or left; at each dot-vertex touched by a chain, we have to choose among three (at most) loop half-lines. Putting all this together we have: #P 0 ≤ (35 )n¯ −1 (35 )2n+2−2p ≤ K n¯

(IV.13)

where the number 5 comes because each element of C is made of five sets Yj . Figure 8 shows an example of disjoint sets built in this way.

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Continuous Constructive Fermionic Renormalization

L L

L

R

L

23

R R L

L Yj

Figure 8 Now we prove that P ⊆ P 0 by induction on i. For each C ∈ P we define C(i) as the subset of C that contains only chains ending in some point (cross-vertex) of the unique path connecting i to the root. C(i) := {Cai0 ∈ C| i0 ≤T i}

(IV.14)

This set satisfies the following induction law: if, for A(i) there are five sets (eventually empty) of disjoint chains Y1 (A(i)). . . Y5 (A(i)) with C(A(i)) = ∪5j=1 Yj (A(i)),

(IV.15)

then there are five sets Y1 (i),. . . ,Y5 (i) with C(i) = ∪j Yj (i). This can be seen observing that C(i) can be written as C(i) = C(A(i)) ∪ {Cai0 ∈ C|i0 = i}.

(IV.16)

Among the five sets Yj forming C(A(i)) there are c0i ones containing chains passing through i: Y1 (A(i)),. . . , Yc0i (A(i)). If c0i + |eei | + |eti | ≥ 5, there are no chains ending at i so C(i) = C(A(i)) ⊂ P 0 . If c0i + |eei | + |eti | < 5 there are ci chains ending at i Ca1 ,i , . . . Caci ,i , with ci ≤ 5 − c0i , so we can define Yj (i) = Yj (A(i)) for j ≤ c0i , Yc0i +j (i) = Yc0i +j (A(i)) ∪ {Caj i } j = 1, . . . , ci , Yj (i) = Yj (A(i)) for j > c0i + ci .

(IV.17)

With these definitions we have C(i) = ∪5j=1 Yj ⊂ P 0

(IV.18)

Now, the hypothesis (IV.15) is true for the root r. In fact, by construction, we have at most five chains ending at r: Ca1 ,r , . . . Ca5 ,r . If we define: Y1 (r) = {Ca1 r }, . . . , Y5 (r) = {Ca5 r }

(IV.19)

we have C(r) = Y1 ∪ · · · ∪ Y5 ⊂ P 0 . Working the induction up to the leaves of RT completes the proof of the lemma.

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Lemma 2 There exists a function φ : {µ} −→ P which associates to each attribution µ = (µ(1), µ(2), . . .) a class C in P. To define φ we fix an order over the half-lines and the lines of RT . We do it turning around RT clockwise and we call n(a) the index of a in the ordering and si the index of the line in RT connecting i to A(i). 4

3 4 2 1

5

5

9

8 3 2

7

6 7 8

11 12 10

1

6

Figure 9 We build the class φ(µ) as a union of chains by induction, defining first the chains in φ(µ) ending at the root, then the ones ending at the cross connected to the root by the line 1, and so on, following the ordering si . Therefore for each i we consider the set Ai = {a ∈ eli | 6 ∃Cai0 ∈ φ(µ) with i0 < i} which is the set of loop half-lines that are external lines for gi and are not connected to a chain in φ(µ) ending lower than i. • If [5−|eei |−|eti |−c0i ] > 0 and #Ai < [5−|eei |−|eti |−c0i ] we have a divergent subgraph, and we add to the part already built of φ(µ) all the chains starting at an element of Ai and ending at i, so ci = #Ai .

(IV.20)

c0i ],

we have a convergent subgraph, so we • If #Ai ≥ max[0, 5 − |eei | − |eti | − put ci = max[0, 5 − |eei | − |eti | − c0i ] (IV.21) and we add to the part already built of φ(µ) the ci chains Ca0 ,i , with a0 = aji , j = 1 . . . , ci , which start at the ci elements in Ai that have the lowest values of n(a), and end at i. In this way we obtain a set of chains with the two properties (IV.10). For each µ, φ(µ) is an element of P and {φ−1 (C)}C∈P is a partition of the set of attributions. We call Bi the set of half-lines in Ai which are the starting points of chains in φ(µ) ending at i (see Figure 10). Therefore in the divergent case Bi = Ai and in the convergent case Bi = {aji , j = 1 . . . , ci }. We also define egi (C) := eti ∪ eei ∪ {a|ia ≥T i and a ∈ Bi0 for some i0 ≤T i}

(IV.22)

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Continuous Constructive Fermionic Renormalization

25

a1i a2i aci i

i

Figure 10 With this definition we have |egi (C)| = ci + c0i + |eti | + |eei |. Remark that in the divergent case |egi | ≤ 4, one has |egi | = |egi (C)|, and in the convergent case one has |egi | ≥ |egi (C)| ≥ 5. The next lemma describes the structure of the classes C. Lemma 3 For each class C ∈ P and each half-line a = 1, . . . , 2n + 2 − 2p there exists a subset of band indices Ja (C) ⊆ B such that φ−1 (C) = {µ|µ(a) ∈ Ja (C) ∀ a}.

(IV.23)

Proof. The existence of the ci chains Cai for a ∈ Bi ending at i implies a certain set of constraints on attributions. We distinguish two situations. 1) If |egi (C)| ≤ 4 (divergent case) • ∀a ∈ Bi , µ(a) ≤ A(i); • ∀ a 6∈ Bi with ia ≥T i, µ(a) > A(i). 2) If |egi (C)| ≥ 5 (convergent case) • ∀a ∈ Bi , µ(a) ≤ A(i); • ∀a 6∈ Bi with ia ≥T i, and n(a) < maxa0 ∈Bi0 n(a0 ), µ(a) > A(i). In any other case, there is no particular constraint. We observe that the underlined constraints for µ(a) are therefore determined by the chain structure and the ordering, but the crucial point is that they are independent from each other. Hence Ja (C) is an interval in terms of band indices. Remark that if some chain in C starts from a, it must end at some unique i, called i0a . In this case we define M (a, C) = A(i0a ). Otherwise we define M (a, C) = ia . Moreover for each i0 such that a 6∈ Bi0 we have two different lower bounds on µ(a), depending whether gi0 is divergent or convergent. So the constraints in cases 1 and 2 simply mean m(a, C) ≤ µ(a) ≤ M (a, C), where M (a, C)

= A(i0a ) if a ∈ Bi0a

,

M (a, C) = ia otherwise

26

M. Disertori and V. Rivasseau

m(a, C)

=

Ann. Henri Poincar´ e

sup [A(i0 ) + 1]

i0 ∈I(a,C)

m(a, C) = 1 if I(a, C) = ∅

(IV.24)

and n(a0 )} I(a, C) := {i0 |ia ≥T i0 , a 6∈ Bi0 , and, if |egi0 (C)| ≥ 5, n(a) < max 0 a ∈Bi0

(IV.25) In summary the constraints are expressed by φ−1 (C) = {µ| µ(a) ∈ Ja (C) ∀ a} Ja (C) = [m(a, C), M (a, C)]

(IV.26)

Example. As the definition of the interval Ja (C) = [m(a, C), M (a, C)] is certainly hard to grasp, let us give an example. In Figure 11 we pictured a class C made of two chains Ca1 ,i1 and Ca2 ,i2 with a1 ≤ a2 in the clockwise ordering. The allowed interval for a1 has maximum M (a1 , C) = M1 , the cross just below i1 , since the presence of the chain forces the half line a1 to be an external leg strictly below i1 . The minimum is m(a1 , C) = m1 , the cross where the second chain ends. Indeed, since a2 ≥ a1 , the attribution for a1 cannot go below m1 , otherwise a longer chain Ca1 ,j1 with j1 ≤ m1 would have been chosen earlier, lower in the tree. Finally suppose a3 is a loop line with index bigger than a2 in the clockwise ordering, and suppose that the cross m1 corresponds to a divergent subgraph G1 , for which the number of external legs is fixed. Then m(a3 , C) = m1 , since the leg a3 cannot go below m1 ; this would add a forbidden external leg to the divergent subgraph G1 . We invite the reader to check his understanding on further examples. a1 i1 a2 a3

M1

i2 = m1 M2

Figure 11 We observe that, after packing the attributions into classes, the sets Ti , ti , eei , eti are still well defined, but we can no longer define gi and eli . We already

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Continuous Constructive Fermionic Renormalization

27

defined egi (C) in (IV.22). We add further definitions gi (C) ili (C) eli (C)

:= ti ∪ ili (C) := {a ∈ L|ia ≥T i, M (a, C) ≥ A(i) + 1} := {a ∈ L|ia ≥T i, M (a, C) ≤ A(i)}

(IV.27)

which generalize the notions of internal and external loop lines. Remark that egi (C) = eti ∪ eei ∪ eli (C), and |eli (C)| = ci + c0i . In the same way we extend these definitions to the other connected components gik (C) := gi(k) (C) , ilik (C) := ili(k) (C) , elik (C) := eli(k) (C)

(IV.28)

Furthermore the generalized definitions for the convergence degree and the set of divergent subgraphs after packing the attributions into classes become: ω(gi (C)) := (|egi (C)| − 4)/2. DC := {gi (C) | ω(gi (C)) ≤ 0}.

(IV.29)

We return now to the loop determinant in (III.23). Lemma 3 ensures that X detM(µ) = detM0 (C) (IV.30) µ∈φ−1 (C)

and that for each loop half-line a there exists a characteristic function 0 if k 6∈ Ja (C) k χa (C) : k ∈ B → {0, 1} χa (C) = 1 if k ∈ Ja (C) .

(IV.31)

Therefore the matrix elements for M0 (C) can be written M0 f g (xf , xg ) =

Z Z

=

X d2 p −ip(xf −xg ) e C(p) χka(f ) χka(g) η k (p)Wvkf ,vg 2 (2π) k∈B

2

d p F ∗ (p)Gg (p) (2π)2 f

X

χka(f ) χka(g) η k (p)Wvkf ,vg

k

where we omit for simplicity to write the dependence in C, and we defined: Ff (p) = eixf p

1 (p2

+

1 m2 ) 4

Gg (p) = eixg p

(− 6 p + m) 3

(p2 + m2 ) 4

.

(IV.32)

va is the vertex to which the half-line a is hooked and η k is the cutoff restricted to the band k (see equation (III.3)). Finally W k is the n ¯×n ¯ matrix defined in equation (III.12). Our next lemma is crucial since it bounds the loop determinant without generating any factorial.

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Lemma 4 The matrix M0 (C) satisfies the following Gram inequality: 0

| det M (C)| ≤

Y Z f

d2 p f η (p)|Ff (p)|2 (2π)2 C

12 Y Z g

d2 p g η (p)|Gg (p)|2 (2π)2 C

12

(IV.33) where the cutoff functions ηCf (p) and ηCg (p) corresponding to fields f and g are defined in equation (IV.44) below. Proof. The Gram inequality states: If M is a n × n matrix with elements Q Mij =< Q fi , gj > and fi , gj are vectors in a Hilbert space, we have | det M | ≤ ni=1 ||fi || nj=1 ||gj ||. To apply Gram’s inequality, the matrix elements must be written as scalar products. We introduce the q × q matrix 1q which is not the identity, but the matrix with all coefficients equal to 1. It is obviously a non-negative symmetric k matrix. We observe P that the matrix Wv,v0 is block diagonal with diagonal blocks of type 1qj , and qj = n ¯ . Each block corresponds to all the vertices in a given connected component of Tk . Therefore W itself is non-negative symmetric. We can define the symmetric matrix (2n + 2 − 2p) × (2n + 2 − 2p): k Rab := χka χkb

(IV.34)

where a and b are the indices for the loop half-lines. By a permutation of field indices, we can list first the q half-lines for which χka (C) = 1. In this way the matrix R becomes 2 × 2 block diagonal non-negative of the type

1q 0

0 0

.

(IV.35)

Now we can group W and R in a unique matrix (tensor product) k k k k Wv,a;v 0 ,b := χa χb Wv,v 0

(IV.36)

that is still non-negative as we can diagonalize separately W and R. Hence the matrix X k η k Wv,a;v (IV.37) 0 ,b = Tv,a;v 0 ,b k

is non-negative symmetric, as it is a linear combination (with positive coefficients η k ) of non-negative symmetric matrices; therefore we can take its square root (which is also non-negative symmetric): Tv,a;v0 ,b =

X w,c

Uv,a;w,c Uw,c;v0 ,b .

(IV.38)

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Continuous Constructive Fermionic Renormalization

29

Now, we can write M0f g as Z

d2 p F ∗ (p)Gg (p)Tvf ,a(f );vg ,a(g) (2π)2 f Z X d2 p = Ff∗ (p)Gg (p) Uvf ,a(f );v0 ,s Uv0 ,s;vg ,a(g) 2 (2π) 0

M0f g =

(IV.39)

v s

If we introduce the vectors Fvf0 s (p) = Ff (p)Uv0 ,s;v(f ),a(f )

Gvg0 s (p) = Gg (p)Uv0 ,s;v(g),a(g)

(IV.40)

we can write M0f g as M0f g

Z =

d2 p X f ∗ g ~ f , G~ g > . Fv0 s Gv0 s =< F (2π)2 0

(IV.41)

v ,s

Now we can apply Gram’s inequality: | det M0f g | ≤

n+1−p Y

~ f || ||F

n+1−p Y

||G~ g ||

(IV.42)

g=1

f =1

where Z ~ f ||2 = ||F Z

d2 p X f t f (Fv0 s ) (Fv0 s ) (2π)2 0 v ,s

d2 p X 0 U 0 U |Ff |2 = (2π)2 0 v(f ),a(f );v ,s v ,s;v(f ),a(f ) v s Z Z d2 p d2 p X k 2 k k 2 = T |F | = χa(f ) χka(f ) Wv(f f ),v(f ) η |Ff | (2π)2 v(f ),a(f );v(f ),a(f ) (2π)2 k Z Z 2 d2 p X k d p = ( η (p)χka(f ) )|Ff (p)|2 = η a(f ) (p) |Ff (p)|2 (IV.43) (2π)2 (2π)2 k

as (χka(f ) )2 = χka(f ) , and, as the bands in χa are adjacents, the cut-offs sum up (using equations (III.2–III.3) to give ηCa (p)

:= η

p2 + m2 Λ0 (wM (a,C) )

−η

p2 + m2 Λ0 (wm(a,C)−1 )

We can treat in the same way G and this achieves the proof of (IV.33).

(IV.44)

30

IV.3

M. Disertori and V. Rivasseau

Ann. Henri Poincar´ e

Bound on the series

We are now in the position to bound the series (III.23). After packing the attributions into packets we can put the absolute value inside the integrals and the sums and boundPthe product of effective constants by cn¯ . Moreover, we observe that the two sums Col,Ω in (III.23) are bounded by taking the supremum over Col and Ω and multiplying by the number of elements. We have 0

00

#{Ω} ≤ 22n+n +n −p < 4n¯ 2−p #{Col} ≤ N n+1−p

(IV.45)

Indeed to estimate #{Col} remark that, once T and Ω are known, the circulation of color indices is determined. If there are no external color indices fixed (vacuum graph), the attribution of color indices costs N 2 at the first four-point vertex (taken as root) and climbing inductively into the tree layer by layer a factor N for each of the remaining four-point vertices of the tree (see [AR2]). The two-point vertices do not contribute as color is conserved at them. When we have fixed the p independent external colors for the 2p external fields only N n+1−p choices remain. We introduce some notations. Recalling the definitions (IV.27) and (IV.29) we say that a divergent subgraph gi (C) ∈ DC is ‘D1PR’ (‘dangerous one particle reducible’) if, by cutting a single tree line, we can cut it into two subgraphs gj (C) and gj 0 (C), one of them, say gj (C), being a two legged subgraph. The line to cut is then the tree line lA(j) . In Figure 12 we show some examples of D1PR subgraphs, where tree lines are solid lines and loop half-lines are wavy.

gj 0

gj

lA(j)=A(j 0 )

gj

gj 0

lA(j)=A(j 0 )

Figure 12 All subgraphs that cannot be cut in this way are called D1PI (‘dangerous one particle irreducible’). We say that a four-point D1PI subgraph gi (C) is ‘open’ (as in [R]) if there exists a two-point subgraph gj (C) ∈ DC (called its closure) with j ≤T i (then gi (C) ⊂ gj (C)) and they have two external lines in common (see Figure 13). A four-point subgraph is called ‘closed’ if it is D1PI but not open. A two-point D1PI subgraph is always closed by definition. This classification of subgraphs is Q useful, as only closed subgraphs contribute in the product g∈DC (1−τg∗ ). Applying the definition of τg in the momentum space one can see that:

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Continuous Constructive Fermionic Renormalization

31

gi gj

Figure 13 • if gi (C) is D1PR and gj (C) is the corresponding divergent subgraph, then τgi (C) (1 − τgj (C) ) = 0

(IV.46)

so the renormalization of gi (C) is ensured by that of gj (C); • if gi (C) is four-point and open, and gj (C) is the associated two-point subgraph containing it, then (1 − τgj (C) )(τgi (C) ) = 0. (IV.47) For any gi (C) ∈ DC we know exactly which loop half-lines are external lines, therefore we can still apply the operator 1 − τg∗ = Rg∗ to the external propagators, and distinguish closed subgraphs. Hence we define DCc := {gi (C) ∈ DC |gi (C) closed}

(IV.48)

and we apply Rg∗ only to g ∈ DCc . By the relation of partial order in RT we see that for each pair gi (C),gi0 (C) ∈ DCc we can only have that gi (C) ∩ gi0 (C) = ∅, or gi (C) ⊆ gi0 (C) (if i0 ≤ i). Hence DCc has a forest structure. Following [R] we define the ‘ancestor’ of gi (C) ∈ DCc , called B(gi (C)), as the smallest subgraph in DCc containing gi (C): B(gi (C)) := gi0 (C),

i0 =

max

c , g (C)⊆g gi00 (C)∈DC i i00 (C)

i00 .

(IV.49)

With all these bounds and definitions, the sum (III.23) becomes: 0 |ΓΛΛ 2p (φ1 , . . . , φ2p )| ≤

Z

Z d2 x1 . . . d2 xn¯

∞ X

N 1−p (cK)n¯

n,n0 ,n00 =0

XX 1 0 00 n!n !n !

(IV.50)

o−T E,C

n ¯Y −1

0≤w1 ≤...≤wn−1 ≤1 q=1 ¯

dwq

Y n¯Y −1 Λ ,w Rg∗ DΛ0 q (¯ xlq , xlq ) det M0 (C) φ1 (xi1 ) . . . φ2p (xjp ) g∈DCc q=1

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Ann. Henri Poincar´ e

Q Before performing any bound we must study the action of g∈Dc Rg∗ on the C tree propagators, the loop determinant and the external test functions. As the external half-lines for any subgraph cannot be of type C 0 we will write C instead of D in the formulas. We distinguish two situations. 1) If |eg(C)| = 4 then ω(gi (C)) = 0 and the action of Rg∗ is: Rg∗ (x1 )

4 Y i=1

C(xi , yi ) := 

= C(x1 , y1 )   = C(x1 , y1 ) 

4 X

0 Rgi (x1 )[

i=2 4 X Y

4 Y

C(xi , yi )]

(IV.51)

i=1

C(xj , yj )[C(xi , yi ) − C(x1 , yi )]

i=2 2≤j
Y i<j≤4

C(xj , yj )[δ0 C(xi , x1 , yi )]

i=2 2≤j
Y

 C(x1 , yj ) 

C(x1 , yj )

i<j≤4

0 as the operator where we took as reference vertex x1 and we defined Rgi that moves the external line with i on the reference vertex x1 , and applies a difference δ0 C(xi , x1 , yi ) between two covariances on the line i.

2) If |egi (C)| = 2 then ω(gi (C)) = −1 and the action of Rg∗ is: Rg∗ (x1 )C(x1 , y1 )C(x2 , y2 ) := Rg1 (x1 )C(x1 , y1 )C(x2 , y2 ) ∂ = C(x1 , y1 ) C(x2 , y2 ) − C(x1 , y2 ) − (x2 − x1 )µ µ C(x1 , y2 ) ∂x1 (IV.52) = C(x1 , y1 ) [δ1 C(x2 , x1 , y2 )] where we took as reference vertex x1 . IV.3.1 Choice of the reference vertex Now, for each gi ∈ DCc we call the reference vertex ve (gi (C)). In this paper the choice of this vertex is adapted to the tree T , and is different from previous rules such as [R], chap.II. We adopt the following rule. We call the first external vertex of the graph, the one with position xi1 the root of the tree. We define DC0c and DC1c as the subsets of four-point and two-point subgraphs in DCc . For every subgraph g ∈ DC0c and any vertex of g there is a single path in T joining this vertex to the root. This path must contain a single well defined external line of g. The vertex to which this external line hooks is by definition our reference vertex for g (see Figure 14). This rule leaves us free of choosing in a different way the reference vertex for any two-point D1PI subgraph DC1c . The rule must ensure that open subgraphs and D1PR subgraphs are automatically renormalized by renormalization of their closure or proper parts. We decide to take as border vertex of any subgraph g ∈ DC1c

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Continuous Constructive Fermionic Renormalization

33

root

root

root ve (g 0 )

ve (g 0 ) g0 ve (g) = ve (g 0 )

g

0

g0 g

ve (g)

g

g

a)

ve (g)

b)

c)

Figure 14 the one to which the highest of the two external half-lines of g hooks. Remark that this external half-line is always a tree half-line in T , so we know its scale. This rule, shown in Figure 15, fulfills the desired requirement, as will be shown below.

g

ve

Figure 15 Finally we add a rule which is not strictly speaking necessary but simplifies the discussion: to compute the action of the renormalization operator we perform first all operations corresponding to two point subgraphs, then all operations corresponding to four point subgraphs, starting from the smallest graphs towards the largest. This rule ensures that any external half-line of a subgraph g bearing one or two gradients because of the action of the Taylor operator for g, cannot bear additional gradients from the later action of another Taylor operator for a different subgraph g 0 . IV.3.2 Processes Returning to equations (IV.51) and (IV.52), we start the renormalization for the two point subgraphs from the leaves of RT (hence from the smallest subgraphs at highest energy) and go down. Then we perform the renormalization of four point subgraphs. Some of them may be already convergent due to renormalization of two point subgraphs. Also, even after fixing the reference vertex for each closed

34

M. Disertori and V. Rivasseau

Ann. Henri Poincar´ e

subgraph, there remains an arbitrary ordering of the other external lines for each four-point graph and a sum over three possible terms as shown in IV.51. Again some of these terms may themselves renormalize some lower four point subgraphs, so the outcome of the renormalization is difficult to capture in a single formula. We index the terms finally obtained by an index P , called the process, which summarize all these choices made for the four point subgraphs. Hence X Y Y Y (Rg ) = [Rg (P )] Rg1 (IV.53) c g∈DC

0c (P ) P g∈DC

1c g∈DC

∗ 6= 1, where DC0c (P ) is the subset of DC0c made of the subgraphs for which Rg(C) hence for which there is a non-trivial renormalization.

Rg (P )

= Rg0 i(P ) , i(P ) ∈ {2, 3, 4} if g ∈ DC0c (P )

(IV.54)

Hence in equation (IV.50), the absolute value inside the integrals can be bounded by: Y n¯Y −1 Λ ,w Rg DΛ0 q (¯ xlq , xlq ) det M0 (C) φ1 (xi1 ) . . . φ2p (xjp ) (IV.55) g∈DCc q=1 ≤

−1 X n¯Y P

r,Λ0 ,wq

|DΛ

(¯ xlq , xlq )|| det M0 (C)| |φr1 (xi1 ) . . . φr2p (xjp )| r

q=1

r 0r r where Q we defined D , M , φ as the functions obtained after the application of g∈Dc (P ) Rg (P ). Again we bound the sum over processes P by the supremum C times the number of possible processes. This number is bounded by 3n−1 . Indeed we recall that for any forest F of closed four-point subgraphs, we have f (F) ≤ n − 1, where f (F) is the number of four-point subgraphs in F [CR, Lemma C1]; this maximal number is not changed by adding n0 + n00 two point vertices because of one particle irreducibility of the closed subgraphs. From now on we work therefore with a fixed process P . We introduce some notations. We define L0 (P ) and L1 as the set of loop half-lines which bear some 0 single or double gradient respectively by some Rgi (P ) or Rg1 operator, Lr0 (P ) as 0 (P ) and Lu (P ) the set of loop half-lines moved to the reference vertex by some Rgi the loop half-lines left unchanged. In the same way we define the sets T 0 (P ), T 1 , T r0 (P ) and T u (P ) for the tree half-lines, and E 0 (P ), E 1 , E r0 (P ) and E u (P ) for the external half-lines. To avoid confusion, from now on we call fi and f¯i the two half-lines forming the tree-line li .

IV.3.3 Interpolations of the lines For a four-point subgraph the difference δ0 C is expressed by Z 1 d δ0 C(x, xv , y) = dt C(x(t), y) dt 0

(IV.56)

Vol. 1, 2000

Continuous Constructive Fermionic Renormalization

35

For a two-point subgraph δ1 C is expressed by Z

1

dt(1 − t)

δ1 C(x, xv , y) = 0

d2 C(x(t), y) dt2

(IV.57)

This means that the external line hooked to x has been hooked to the point x(t) on any piecewise differentiable path joining x to xv and has now a propagator (see Figure 16) y4

y4

x4 x3

x3 x2

x1 y1

y4

x4

y3

x1

x2

y1

y2

x4

x4 x4 (t) y3

y1

y2

x3

x3 x1

x2 x3 (t) y3

x2 x2 (t)

x1 y1

y2

Rg04

g

y4

y3

y2

Rg02

Rg03

Figure 16

C 0 (x(t), y) :=

d C(x(t), y) dt

(IV.58)

or (see Figure 17) C 1 (x(t), y) := (1 − t)

d2 C(x(t), y). dt2

(IV.59)

g R1

= x2

x2

x1 y1

x1 y2

x2 (t)

y1

y2

Figure 17 In previous perturbative or constructive works, this path x(t) is always defined to be the linear segment connecting x to xv hence is parametrized by x(t) := xv + t(x − xv ) x(0) = xv x(1) = x

(IV.60)

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M. Disertori and V. Rivasseau

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But with the continuous band structure this obvious choice when applied to tree half-lines leads to difficulties. It is therefore more convenient to treat differently the loop, tree and external half-lines. Loop lines and external half-lines (except the root) do not affect spatial integration (recall that this spatial integration is performed using the decay of the tree lines). So for them we can choose the obvious linear interpolation that makes easier to factorize the matrix elements of M0 as scalar products and to apply Gram’s inequality. For the tree lines it will be convenient to exploit the existence of T to choose a different path. IV.3.2.A: Loop lines Now for each ha ∈ L0 (P ) ∪ L1 we define xa (t) := xve + t(xa − xve )

(IV.61)

which is the obvious linear path. The propagator for the line bearing the difference becomes respectively for a four-point and a two-point subgraph C 0 (xa (t), y) :=

d ∂ C(xa (t), y) = (xa − xv )µ µ C(xa (t), y) dt ∂x

(IV.62)

d2 ∂ ∂ C(xa (t), y) = (xa − xv )µ (xa − xv )ν µ ν C(xa (t), y). dt2 ∂x ∂x (IV.63) If the second end of the line y is also moved we just apply the same formulas to C 0 (xa (t), y) and C 1 (xa (t), y), interpolating y. Introducing these formulas into the matrix element M0 rf g we can factorize it as the scalar product of Ffr and Grg where C 1 (xa (t), y) := (1 − t)

Ffr = Ff for hf 6∈ L0 (P ) ∪ L1 (IV.64) Z ∂ Ffr = dt(xf − xv )µ µ Ff (xf (t)) for hf ∈ L0 (P ) ∂x Z ∂ ∂ r Ff = dt(1 − t)(xf − xv )µ (xf − xv )ν ν µ Ff (xf (t)) for hf ∈ L1 ∂x ∂x where Ff , Gg are defined in (IV.32). The same definitions hold for Grg . With these definitions the determinant is bounded by Y Y r | det M0 (C)| ≤ ||Ffr || ||Grg || (IV.65) f

g

Now we can bound the norms using the following lemma. Lemma 5 The norms of Ff and Gg satisfy the bounds ||Ff ||2C

≤ K [Λ0 (wM (f,C) ) − Λ0 (wm(f,C)−1 )]

||Gg ||2C

≤ K [Λ0 (wM (g,C) ) − Λ0 (wm(g,C)−1 )]

(IV.66)

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Continuous Constructive Fermionic Renormalization

37

Proof. Applying the definition (IV.44), ||Ff ||2C is written d2 p 1 p2 + m2 p2 + m2 η − η (2π)2 (p2 + m2 ) 12 Λ20 (wM (f,C) ) Λ20 (wm(f,C)−1 ) Z Z Λ−2 0 (wm(f,C)−1 ) d2 p 2 2 12 (p + m ) dα −η0 [(p2 + m2 ) α] = 2 (2π) Λ−2 0 (wM (f,C) ) Z ∞ Z Λ−2 0 (wm(f,C)−1 ) √ − 32 dα π α dv v[−η0 (v)] ≤ Z

||Ff ||2C

=

Λ−2 0 (wM (f,C) )

0

≤ K [Λ0 (wM (f,C) ) − Λ0 (wm(f,C)−1 )]

(IV.67)

where, in the third line, we performed the change of variable v = α(p2 + m2 ). The same result holds for ||Gg ||2C . A similar argument can be performed for loop lines with some gradient ap1 plied. Each derivative adds a factor α− 2 in the integral. With these definitions the determinant is bounded by Y 1 [Λ0 (wM (a,C) ) − Λ0 (wm(a,C)−1 )] 2 a∈Lu (P )∪Lr0 (P )

Y

1

|xa − xv(a) |[Λ30 (wM (a,C) ) − Λ30 (wm(a,C)−1 )] 2

a∈L0 (P )

Y

1

|xa − xv(a) |2 [Λ50 (wM (a,C) ) − Λ50 (wm(a,C)−1 )] 2

(IV.68)

a∈L1

IV.3.2.B: External lines. The only external line essential in spatial integration is the root y1 , then we can choose this point as reference vertex for the whole graph so that it is never interpolated. For the other external lines we take again the easiest formula, the linear one. All gradients generated by moving the external lines in fact apply to the test functions. Therefore the product is bounded by Y Y ||φie ||∞ |xie − xv |||φ0ie ||∞ hie ∈E u (P )∪E r0 (P ),ie 6=1

hie ∈E 0 (P )

Y

|xie − xv |2 ||φ00ie ||∞

(IV.69)

hie ∈E 1

IV.3.2.C: Tree lines. Now we consider tree lines. We observe that the set of fi , f¯i ∈ T r0 modifies the tree T but does not disconnect it in the sense that it simply changes the hooking vertices of some line. On the other hand, interpolating each fi , f¯i ∈ T 0 (P ) ∪ T 1 with the linear rule (equation (IV.60)) in an intuitive sense “disconnects” the tree, since the point x(t) in general no longer hooks to some point on a segment corresponding to a tree

38

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Ann. Henri Poincar´ e

line. This defect would lead to difficulties when integrating over spatial positions. To avoid it we express the differences δ0 C and δ1 C using the connection between external vertices of any subgraph which is provided by the tree T itself. But, as the tree T is itself modified by renormalization, this process has to be inductive, starting from the smallest two point subgraphs of DC1c and proceeding towards the biggest, then again from the smallest four point subgraphs of DC0c (P ) and proceeding towards the biggest. Our induction creates progressively a new tree T (P, J). To describe it, we number the subgraphs to renormalize in the order the operations are performed as g1 , . . . gr . At the stage 1 ≤ p ≤ r, before renormalization of gp , the tree is called T (P, Jp−1 ) (we put T (P, J0 ) = T ). If the renormalization of gp as specified by the process P does not act on a tree half-line external to gp , we neither modify Jp−1 nor T (P, Jp−1 ). If the renormalization of gp results in some half-tree line fi or f¯i belonging to T r0 (P ), as shown in Figure 18, case 1), we do not modify Jp−1 , so we put Jp = Jp−1 , but we modify T (P, Jp−1 ), that is we define T (P, Jp ) as T (P, Jp−1 ) but with the half line now hooked to the reference vertex of gp . Finally when the renormalization of gp interpolates a tree half-line fi or f¯i , we T (P,J ) modify both Jp−1 and T (P, Jp−1 ). There exists a unique path Pxf ,xv p−1 joining the vertex xf where the half-line hooked to the fixed vertex of gp . This path is made of q lines and goes through q + 1 vertices with positions x0 = xv , x1 , . . . , xq = xf . We interpolate the half line using this path instead of the linear path. This means that we write, if gp is a four point subgraph, q X

δ0 C(xf , xv , y) =

δ0 C(xj , xj−1 , y)

j=1 q X

Z

(xj − xj−1 )µ

=

j=1

1

dt 0

∂ C(xj (t), y) ∂xj (t)µ

(IV.70)

and if gp is a two point subgraph we write ∂ δ1 C(xf , xv , y) = δ0 C(xf , xv , y) − (xf − xv )µ µ C(xv , y) ∂xv " q q X X ∂ = δ0 C(xj , xj−1 , y) − (xj − xj−1 )µ C(xj−1 , y)− µ ∂x j−1 j=1 j=1 "j−1 ## X ∂ ∂ − C(xk , y) − µ C(xk−1 , y) ∂xµk ∂xk−1 k=1 Z 1 X q ∂2 = dt (1 − t)(xj − xj−1 )µ (xj − xj−1 )ν µ ν C(xj (t), y) ∂ ∂ 0 j=1 Z +

1

dt 0

q X j−1 X

(xj − xj−1 )µ (xk − xk−1 )ν

j=1 k=1

∂2 C(xk (t), y) ∂µ∂ν

(IV.71)

Vol. 1, 2000

Continuous Constructive Fermionic Renormalization

39

where for each j (and k ≤ j) we defined xj (t) := xj−1 + t(xj − xj−1 ) , xk (t) := xk−1 + t(xk − xk−1 ) .

(IV.72)

Then we update J and T . In the first case, where g is a four point subgraph, we define Jp = Jp−1 ∪ {j}, where j is the index of the line of T (P, Jp−1 ) chosen in IV.70. In the second case, where g is a two point subgraph, we define Jp = Jp−1 ∪ {j} ∪ {k}, where j and k are the indices of the lines of T (P, Jp−1 ) chosen in IV.71. Finally we update the tree according to Figure 18, case 2 and 3. xk−1

xj−1 fi xv Case 1

xv

xj

fi

xv

xf

Case 2

xk

fi

xj−1 xj

xf

Case 3

Figure 18 This means that, for gp a four point subgraph, in T (P, Jp ) the external line hooked to xf is now hooked to the point xj (t) on the tree segment [xj−1 , xj ] and has propagator C 0 (xj (t), y) := (xj − xj−1 )µ

∂ C(xj (t), y). ∂µ

(IV.73)

For gp a two point subgraph, the external line previously hooked to xf is now hooked to the point xk (t) on the tree segment [xk−1 , xk ] (with k ≤ j) and has propagator C 1 (xk (t), y) := (xj − xj−1 )µ (xk − xk−1 )ν

∂2 C(xk (t), y) ∂µ∂ν

C 1 (xj (t), y) := (1 − t)(xj − xj−1 )µ (xj − xj−1 )ν

k 6= j

∂2 C(xj (t), y). ∂µ∂ν

(IV.74)

Remark that the new tree T (P, Jp ) has therefore one additional vertex and one line more than T (P, Jp−1 ). The final tree built inductively in this way, T (P, J) is still a tree connecting all initial vertices, with at most n − 1 new vertices and new lines (as n − 1 is the maximal number of closed divergent subgraphs). The treatment of two or four point subgraphs and the rules for their fixed vertex being different, we write J = J 0 ∪ J 1 , where J 0 is the set of indices j for the interpolations associated to the renormalization of four point graphs, and J 1 is the set of indices jg and kg for the interpolations associated to the renormalization of two point graphs.

40

M. Disertori and V. Rivasseau

Ann. Henri Poincar´ e

IV.3.4 Bound on the sum over J 1 and on the associated distance factors Now, before going on, in order to reproduce the spatial decay between supports of Theorem 3, we take out a fraction (1 − ) of the exponential decay of each tree line in (II.16) and (II.17): this factor is bounded by n ¯Y −1

e−(1−)|¯xi −xi |Λ0

m

(wi )

≤ e−Λ

m

(1−)dT (Ω1 ,...Ω2p )

(IV.75)

i=1

and we keep the remaining decay e−(/2)|¯xi −xi |Λ0 (wi ) (adjusting 0 = /2) for two purposes: a fraction of this decay is used to perform spatial integration and the other to bound the sum over J 0 and the distance factors generated by equations (IV.62)–(IV.63) and (IV.73)–(IV.74). Therefore we have to bound, for a fixed process P XX Z dx A(x, J, P, T ) B(x, J, P, T ) (IV.76) m

J1

J0

where

Y

A(x, J, P, T ) :=

fi ,f¯i ∈T 1

Y

|xie − xv |

hie ∈E 1

ha

Y

2

Y

|xj − xj−1 ||xk − xk−1 |

|xa − xv |2

∈L1

−(/4)|¯ xl −xl |Λm 0 (wl )

e

(IV.77)

l∈T (P,J)

Y

B(x, J, P, T ) := Y

|xj − xj−1 |

hie ∈E 0 (P )

Y

|xa − xv |

ha ∈L0 (P )

fi ,f¯i ∈T 0 (P )

|xie − xv |

Y

e−(/4)|¯xl −xl |Λ0

m

(wl )

(IV.78)

l∈T (P,J)

where J specifies in particular the distance factors |xj − xj−1 | and |xk − xk−1 |, as explained above. The strategy of the bound is to write Z XXZ XX A(J, P, T ) dx B(x, J, P, T ) dx A(x, J, P, T ) B(x, J, P, T ) ≤ J1

J0

J1

J0

(IV.79) where A(J, P, T ) := supx A(x, J, P, T ). For each divergent subgraph gi ∈ DCc we define t(i) as the index of the lowest tree line in the path of T (P, J) joining xv(gi ) to the interpolated half-line which renormalize it. The next lemma proves that A(J, P, T ) is bounded by something independent of J: Lemma 6 A(J, P, T ) ≤ K n

Y c1 gi ∈DC

where K is some dependent constant.

[Λ0 (wt(i) ) − Λ0 (wA(i) )]−2

(IV.80)

Vol. 1, 2000

Continuous Constructive Fermionic Renormalization

41

Proof. For each loop or external line the difference |x−xv | can be bounded, applying several triangular inequalities, by the sum over the tree lines on the unique path in T (P, J) connecting x to xv . We observe that the same tree line lj can appear in several paths connecting different pairs of points xv , x. Using the same fraction of its exponential decay many times might generate some unwanted factorials since supx xn exp(−x) = (n/e)n . To avoid this problem we define Dj as the set of subgraphs gi ∈ DC1c ∪ DC0c (P ) which use the tree distance |¯ xlj − xlj | to bound the norm of |x − xv(gi ) | or its square and we apply the relation e− 4 |¯xlj −xlj |Λ0

m

− 4 |¯ xlj −xlj |

(wj )

≤ e

P

gi ∈Dj [Λ0 (wt(i) )−Λ0 (wA(i) )]

(IV.81)

With this expression a different decay factor is used for each subgraph. Applying this result and the inequalities xe−x ≤ 1, x2 e−x ≤ 1 completes the proof of the lemma. It is proved in the next section that the sum and spatial integral XZ

dx B(x, J, P, T )

J0

is in fact independent of P and J 1 (and of the interpolation parameters t that we omitted). Therefore the sum over J 1 will simply lead to the bound of the next section multiplied by the cardinal of the set J 1 , that is by |J 1 |. This is done thanks to the following lemma: Lemma 7 We have |J 1 | ≤ e2¯n . Proof. We consider the graphs g1 , . . . gr1 of DC1c in the order used for their renormalization in Subsection IV.3.2.C. We define, for each such two point subgraph g ∈ DC1c the set A(g) of maximal subgraphs g 0 , g 0 ∈ DC1c , g 0 ⊂ g, and the reduced graph g/DC1c where each g 0 ∈ A(g) has been reduced to a single point. We also count the number Lg of lines on the unique path in the tree T ∩ g/DC1c which joins the two P external vertices of g. Remark that this number Lg is independent of J and that g∈D1c Lg ≤ n ¯ − 1. Finally we define the subset A0 (g) C 0 of A(g) made of those g in A(g) which appear as reduced points on this unique path (see Figure 19). By induction one can bound the number of choices in J 1 by

|J | ≤ 1

Y 1c g∈DC

P Lg + g0 ∈A0 (g) kg0

X

jg X

jg =1

kg =1

1

(IV.82)

42

M. Disertori and V. Rivasseau

g1 ∈ A g2 ∈ A0

Ann. Henri Poincar´ e

g1 g2

g3 ∈ A0

g3

Figure 19 where this product is again ordered from the smallest to the largest graphs. Now R 1+L P for any positive increasing function f we have L f (x)dx so that j=1 f (j) ≤ 1 |J | ≤ 1

Y Z 1c g∈DC

≤

1+xg

dxg

1

Y Z

1c g∈DC

Z

P 1+Lg + g0 ∈A0 (g) yg0

dyg 1

Z

P 1+Lg + g0 ∈A0 (g) yg0

1+xg

dxg

0

dyg ≤ e

2r1

0

Y

(IV.83) Lg

e

≤e

2¯ n

1c g∈DC

R 1+xgr1 dygr1 by e1+xgr1 where in the last inequality, we bounded the last integral 0 and then effectuate each integral exactly and bound each difference ex − 1 by ex . Finally we used the fact that every subgraph g 0 is in A0 (g) for at most one g, and ¯ − 1 (again [CR, Lemma C1]). the fact that 2r1 ≤ n Remark. This lemma does not apply to J 0 . For a counter-example, the reader can look at the following graph and tree, for which J 0 can be of order K n (n/5)!. interpolated lines

graph

tree

Figure 20

IV.3.5 Bound on the sum over J 0 , on the associated distance factors, and spatial integration of the vertices It remains now to perform the sum over J 0 and the of Q integral over the position m internal vertices, using the remaining tree decay l∈T (P,J) e−(/4)|¯xl −xl |Λ0 (wl ) in

Vol. 1, 2000

Continuous Constructive Fermionic Renormalization

43

B, and to check that the result, as announced, is independent of P , J and of the interpolation parameters t. We recall that Λm 0 (w) was defined in (II.18). Lemma 8 XZ dx B(x, J, P, T ) ≤ K n¯

Y

[Λ0 (wt(i) ) − Λ0 (wA(i) )]−1

c0 (P ) gi ∈DC

J0

n ¯Y −1

−2 (Λm 0 (wq ))

q=1

(IV.84) where K is some dependent constant. Proof. First we divide one half of our remaining tree decay as inQIV.81. This half will Q be used to bound each distance factor in ha ∈L0 (P ) |xa − xv | hi ∈E 0 (P ) |xie − xv | e Q as in Lemma 6, the sum over J 0 and the distance factors fi ,f¯i ∈T 0 (P ) |xj − xj−1 |. Q Q As in Lemma 6, each distance factor in ha ∈L0 (P ) |xa − xv | hi ∈E 0 (P ) |xie − e xv | leads to a bound K[Λ0 (wt(i) ) − Λ0 (wA(i) )]−1 . Then we perform the spatial integrals from the leaves of the tree T (P, J) towards the root x1 , using the other half of the tree decay. In this inductive process when we meet an interpolated line hooked at some interpolated point xj (t) or xk (t) two different situations can occur, as pictured in Figure 21. The second situation (interpolated point not towards the root) can occur only for interpolations of two point subgraphs, hence associated to the J 1 indices. The first situation (interpolated point towards the root) must occur for all interpolations associated to four point subgraphs plus possibly some interpolation associated to two point subgraphs. This is the consequence of our rule for the preferred vertices of four point subgraphs (the interpolations associated to four point subgraphs always bring nearer to the root). The sum over J 0 is performed in pieces; the sum over each index jg in J 0 is performed right after the spatial integration which has used the corresponding interpolated line. By the remark above, each sum over jg in J 0 occurs in the first situation of Figure 21. xj−1 y xj (t)

root xj

a)

xj−1

y

xj (t)

0

xj

y y0

root

b)

Figure 21 The decay of any tree line li = [x, y] with both ends fi , f¯i ∈ T u (P ) ∪ T r0 (P ) gives, by translation invariance, a factor: Z m −2 (IV.85) e−(/8)|x−y|Λ0 (wi ) d2 x = 128π−2 [Λm 0 (wi )]

44

M. Disertori and V. Rivasseau

Ann. Henri Poincar´ e

Surprisingly, the same result holds when one or both ends of the line have been interpolated, as we explain now2 . • In the first situation of Figure 21, y, the other end of the interpolated line, is connected to the root through the interpolated point (see Figure 21a). We can integrate over y before integrating over xj and xj−1 . We use translation invariance to cancel the dependence from xj (t) in the interpolated covariance. The integral over the variable t is bounded by 1. The spatial integral over y then gives the same factor as in (IV.85). Then we perform the corresponding sum over j in J 0 using q X

|xj − xj−1 |e− 8 [

Pq j=1

|xj −xj−1 |][Λ0 (wt(i) )−Λ0 (wA(i) )]

≤ K[Λ0 (wt(i) ) − Λ0 (wA(i) )]−1

j=1

(IV.86) • When the point x(t) is connected to the root through y (see Figure 21b) we have to compute the integral Z

Z

1

I=

dt

d2 xj d2 xj−1 e−(/8)|xj −xj−1 |Λ0

m

(wj ) −(/8)|xj (t)−y|Λm 0 (wk )

e

(IV.87)

0

where xj (t) = xj−1 + t(xj − xj−1 ). Performing, for t 6= 0 the change of variables v1 = xj (t) = txj + (1 − t)xj−1

v2 =

1 (xj−1 − xj (t)) = (xj−1 − xj ) t

(IV.88)

the integral becomes Z I

Z

1

dt

=

2

−(/8)|v1 |Λm 0 (wk )

d v1 e

Z

d2 v2 e−(/8)|v2 |Λ0

m

(wj )

0 −2 m = [128π−2 ]2 [Λm [Λ0 (wk )]−2 0 (wj )]

(IV.89)

This is again the same contribution as (IV.85), hence the same contribution as if the line had not been interpolated! Following the tree from the leaves to the root we can perform the integrals over all positions in this way, except for xi1 . This last point is integrated using the test function φ1 , which gives a factor ||φ1 ||1 . Hence, the result of all spatial Qn¯ −1 −2 integrations is K n q=1 (Λm and the sum over J 0 has been performed at a 0 (wq )) n 0 cost of K (although |J | can be very large). This completes the proof of Lemma 8. Since the result of Lemma 8 is independent of J 1 , as announced, we can apply Lemma 7. Let us collect the result of Lemmas 6-8, together with the other factors remaining after spatial integration. The product over tree lines propagators, is 2 We thank V. Mastropietro for explaining to us that this surprising fact was known to him and can be found in his work [BM]

Vol. 1, 2000

Continuous Constructive Fermionic Renormalization

bounded by

n ¯ −1 K n¯ (Λ−2 − Λ−2 e−(/4)m 0 )

2

·

n ¯Y −1

Y

Λ30 (wq )

Λ−2 0 (w1 )

Λ0 (wq )

fq ,f¯q ∈T 0 ∪T 0 (P )

q=1

45

Y

(IV.90)

Λ20 (wq )

fq ,f¯q ∈T 1

where K is some dependent constant,and we included the scaling prefactors in (II.16)–(II.17). These factors are • • • •

a factor Λ0 (w)3 for each tree line, a factor Λ0 (w) for each half-line in T 0 (P ), a factor Λ0 (w)2 for each half-line in T 1 , a factor Λ0 (w) for each half-line in T 0 , the set of half-lines hooked to a δζ vertex which bears a derivative, hence has covariance C 0 .

Remark that we kept for only one line, the lowest one, the massive decay 2 −2 e−(/4)m Λ0 (w1 ) from bounds (II.16)–(II.17). It will be useful only to conclude the bound in the massive case when Λ < m. In this case the n0 vertices of type δm may create some infrared difficulties if we were to replace directly for them the −2 by (Λ0 (wq ))−2 . We introduce the set T 0 of the tree lines used factor (Λm 0 (wq )) for integration of the δm vertices. There are n0 of them (or n0 − 1 if the root is of type δm, a case we will exclude for simplicity). Recall that by (II.18) we have −2 (Λm ≤ (Λ0 (wq ))−2 0 (wq )) −2 (Λm 0 (wq ))

−1

≤m

(IV.91)

−1

(Λ0 (wq ))

(IV.92)

0

We use the bound (IV.92) only for the lines of T when Λ < m. For all other cases we use the bound (IV.91). IV.3.6 Integration over the parameters wi Now, putting everything together, we describe first the bound when Λ > m, hence Λm = Λ. Equation (IV.50) is bounded by |ΓΛ 2p (φ1 , . . . φ2p )| ≤ ||φ1 ||1 N

2p Y

||φi ||∞,2

i=2 1−p −(1−)Λm dT (Ω1 ,...Ω2p )

e

∞ X

(cK)n¯

n,n0 ,n00 =0 n ¯Y −1

1 n!n0 !n00 !

n ¯ −1 (Λ−2 − Λ−2 0 ) X XZ

o−T E,C

Y

Λ0 (wq )

fq ,f¯q ∈T 0 ∪T 0 (P )

q=1

Y a∈Lu (P )∪Lr0 (P )

n ¯Y −1

0≤w1 ≤...≤wn−1 ≤1 q=1 ¯

Λ0 (wq )

Y fq ,f¯q ∈T 1

Λ20 (wq )

1 Λ0 (wm(a,C)−1 ) 2 Λ0 (wM (a,C) ) 1 − Λ0 (wM (a,C) ) 1 2

dwq

46

M. Disertori and V. Rivasseau

Y

Λ3 (wm(a,C)−1 ) Λ0 (wM (a,C) ) 1 − 0 3 Λ0 (wM (a,C) ) 3 2

a∈L0 (P )

Ann. Henri Poincar´ e

12

1 Λ50 (wm(a,C)−1 ) 2 Λ0 (wM (a,C) ) 1 − 5 Λ0 (wM (a,C) ) a∈L1 Y Λ0 (wA(i) ) −1 Λ−1 (w ) 1 − t(i) 0 Λ0 (wt(i) ) 0c Y

5 2

gi ∈DC

Y 1c gi ∈DC

Λ−2 0 (wt(i) )

Λ0 (wA(i) ) −2 1− Λ0 (wt(i) )

(IV.93)

The differences 1 − Λ0 (wA(i) )/Λ0 (wt(i) ) are dangerous as they appear with a negative exponent. They are the price to pay for implementing continuous renormalization group. Indeed, in this continuous formalism one has to perform renormalization even when the differences between internal and external energies of subgraphs are arbitrarily small. However, there is a natural solution to this problem: each subgraph to renormalize has necessarily loop lines and these loop lines, when evaluated in the continuous formalism by Gram’s inequality, give small factors precisely when the differences between internal and external energies of subgraphs become arbitrarily small. In other words, we can cancel the dangerous differences with a negative exponent against the analogous differences with a positive exponent given by the loop lines. This is the purpose of the next lemma. Lemma 9 If gi ∈ DCc0 (P ) there is at least one loop line internal to gi which satisfies Λ0 (wM (a,C) ) ≤ Λ0 (wt(i) ) and Λ0 (wm(a,C)−1 ) ≥ Λ0 (wA(i) ). If gi ∈ DCc1 there are at least two loop lines internal to gi and which satisfy Λ0 (wM (a,C) ) ≤ Λ0 (wt(i) ) and Λ0 (wm(a,C)−1 ) ≥ Λ0 (wA(i) ). q q √ √ 3 1−x5 Assuming the lemma true, and using the relations 1−x ≤ 3 and 5, 1−x 1−x ≤ one obtains 1 1 Y Λ0 (wm(a,C)−1 ) 2 Y Λ30 (wm(a,C)−1 ) 2 1− 1− 3 Λ0 (wM (a,C) ) Λ0 (wM (a,C) ) u r0 0 a∈L (P )∪L

a∈L (P )

(P )

1 −1 Y Λ50 (wm(a,C)−1 ) 2 Y Λ0 (wA(i) ) 1− 5 1− Λ0 (wM (a,C) ) Λ0 (wt(i) ) 1 0c

a∈L

(IV.94)

gi ∈DC

−2 Y Λ0 (wA(i) ) ≤ 5n−1 1− Λ0 (wt(i) ) 1c

gi ∈DC

where we bounded by 1 the loop lines differences that were not used to compensate −1 some 1 − Λ0 (wA(i) )/Λ0 (wt(i) ) factor.

Vol. 1, 2000

Continuous Constructive Fermionic Renormalization

47

Proof of Lemma 9. We observe that the lowest tree line lt(i) in Ti (J, P ) joining the interpolated line and the reference vertex is external line for the two subgraphs of gi , gt(i)1 and gt(i)2 . One of these two subgraphs has for external line the reference external line of gi and the other has for external line the interpolated line moved by the renormalization Rg∗i . But gt(i)1 and gt(i)2 must both have at least some additional external lines, otherwise gi would be D1PR. By parity gt(i)1 and gt(i)2 must both have at least two such additional external lines. We distinguish two cases: • If gi ∈ DCc0 (P ), since there are at most two additional external lines of gi , we find that there must be at least two external half-lines of gt(i)1 ∪ gt(i)2 different from lt(i) which are internal in gi . If they are both loop half-lines we are done. If some of them is a tree half-line, the other half is external line for another subgraph of gi , g 0 . Repeating the argument for g 0 (as |eg| = 1 is forbidden) we finally must find an associated loop half-line (see Figure 12). • If gi ∈ DCc1 , since there was no additional external line of gi , then both gt(i)1 and gt(i)2 must have at least two external half-lines different from lt(i) which are internal in gi . Either these four half-lines are loop half-lines and we are done, or some of them are tree lines, which we follow as above until we find the corresponding loop half-lines. After applying the bound (IV.94) we can take the limit Λ0 → ∞. Performing the change of variable ui = 1 − wi equation (IV.93) becomes: |ΓΛ 2p (φ1 , . . . φ2p )| ≤ ||φ1 ||1

2p Y

||φi ||∞,2 e−(1−)Λ

m

dT (Ω1 ,...Ω2p )

i=2

n ¯Y −1 X XZ 1 N Λ (cK) duq n!n0 !n00 ! ≤···≤u1 ≤1 q=1 ¯ n,n0 ,n00 =0 o−T E,C 0≤un−1   "n¯ −1 # Y 1 Y Y 1 1  √ √  u u u q q q q=1 fq ,f¯q ∈T 0 ∪T 0 (P ) fq ,f¯q ∈T 1  "  # Y Y Y 1 3 5 − − −  (uM (a,C) ) 4   (uM (a,C) ) 4  (uM (a,C) ) 4 1−p

∞ X

2−p−n0

a∈Lu (P )∪Lr0 (P )

 

Y

0c gi ∈DC



(ut(i) ) 2   1

n ¯

a∈L0 (P )

Y



a∈L1

(ut(i) )

(IV.95)

1c gi ∈DC

To factorize the integrals we perform the change of variable: ui = βi ui−1

βi ∈ [0, 1]

(IV.96)

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where by convention u0 = 1. The Jacobian of this transformation is the determinant of a triangular matrix hence it is given by: J = β1 (β1 β2 ) . . . (β1 β2 . . . βn¯ −2 ) =

n ¯Y −1

βin¯ −1−i .

(IV.97)

i=1 0

We absorb Λ−n into the term K n¯ since we recall that Λ > m hence that 0 Λ−n = (Λm )−n , and that in Theorem 3 Λm remains in the compact X, hence is bounded away from 0. Then the integral (IV.95) becomes 0

|ΓΛ 2p (φ1 , . . . φ2p )| ≤ ||φ1 ||1

2p Y

||φi ||∞,2 e−(1−)Λ

m

dT (Ω1 ,...Ω2p )

(IV.98)

i=2

Z 1 n¯Y −1 X XZ 1 1 N (cK) . . . dβi (Λ ) n!n0 !n00 ! 0 i=1 n,n0 ,n00 =0 o−T E,C 0    n ¯Y −1 Y Y 1 1 −1+ 12 (¯ n−i)− 12 |Ni00 |    p βi βq . . . β1 βq . . . β1 0 1 ¯ ¯ i=1 fq ,fq ∈T (P ) fq ,fq ∈T    Y Y 1 3  (βM (a,C) . . . β1 )− 4   (βM (a,C) . . . β1 )− 4 ) m 2−p

"

1−p

∞ X

n ¯

a∈Lu (P )∪Lr0 (P )

a∈L0 (P )

  # Y Y Y 5 1 (βM (a,C) . . . β1 )− 4  (βt(i) . . . β1 ) 2   (βt(i) . . . β1 ) 0c gi ∈DC

a∈L1

1c gi ∈DC

Each βi appears with the exponent −1 + xi . xi

=

1 1 1 [¯ n − i] − |Ni00 | − |ILi (C)| 2 2 4 1 0 0 + [|Si (C)| − |ITi | − |IL0i (C)|] + [|Si1 (C)| − |ITi1 | − |IL1i (C)|] 2 (IV.99)

where we defined ITi0 ITi1 ILi (C) IL0i (C) IL1i (C) Si0 (C) Si1 (C)

:= := := := := := :=

{fj , f¯j ∈ T 0 (P )|j ≥ i} {fj , f¯j ∈ T 1 |j ≥ i} {a ∈ L|M (a, C) ≥ i} {a ∈ L0 (P )|M (a, C) ≥ i} {a ∈ L1 |M (a, C) ≥ i} {gj ∈ DC0c (P )|t(j) ≥ i} {gj ∈ DC1c |t(j) ≥ i}.

(IV.100)

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c(i) is the number of connected components in Ti (P ). All these definitions can 1k be restricted to the connected components: ITi0k ,ITi1k , ILki (C), IL0k i (C) ILi (C), 0k 1k Si (C) and Si (C). We observe that ILi (C) corresponds to the set of half-lines that could have, in the class C, µ(a) ≥ i and it is the equivalent of ILi defined in (III.9). ITi0 (respectively ITi1 ) and IL0i (C) (respectively IL1i (C)) are the set of tree half-lines and loop half-lines at a level higher or equal to i, which are the interpolated external lines for some divergent subgraph in DC0c (P ) (respectively in DC1c ), Si0 (C) (respectively Si1 (C)) is the set of subgraphs in DC0c (P ) (respectively in DC1c ) that have the internal tree line lt(j) of a level higher or equal to i. In the same way, we can define the equivalent of ELi and Ei as c(i)

ELi (C) := ∪k=1 elik (C)

(IV.101)

which is the set of loop half-lines that are forced to have µ(a) ≤ i, and c(i)

Ei (C) := ∪k=1 egik (C).

(IV.102)

The integral in the variable dβi can be performed only if the exponent of βi is bigger than −1. Using the relations n ¯−i =

c(i) X

0

[|Nik | + |Ni k | + |Ni00k | − 1]

k=1

|Ei (C)| = |ELi (C)| + |ETi | + |EEi | = 2|Ni | + 2 − |ILi (C)| , the exponent of βi can be written as −1 + 0 1 1 k xi := (|Eik (C)| + 2|Ni k | − 4) 2 2

Pc(i) k=1

(IV.103)

xki , where

(IV.104) 0k 0k 0k 1k 1k 1k +[|Si (C)| − |ITi | − |ILi (C)|] + 2[|Si (C)| − |ITi | − |ILi (C)|]

Remark that for any level i we have [|Si0 (C)| − |ITi0 | − |IL0i (C)|] ≥ 0 [|Si1 (C)| − |ITi1 | − |IL1i (C)|] ≥ 0

(IV.105)

as each half-line in ITi0 (ITi1 ) or IL0i (C) (IL1i (C)) is the external interpolated line for a subgraph gj . This subgraph gj must have j > i hence have t(j) > i. Therefore for each half-line in one of these sets there is always at least one corresponding half-line in Si0 (C) (Si1 (C)). Lemma 10 For any connected component in Tik we have xki ≥ 1/2. Proof. We distinguish three situations.

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• If |Eik (C)| ≥ 5, in fact, by parity of the number of external half-lines of any subgraph, |Eik (C)| ≥ 6 and then xki ≥ (1/4)(|Eik (C)| − 4) ≥ 1/2.

(IV.106)

• If |Eik (C)| = 4, then there must be a subgraph gj ∈ DC0c (P ) with j ≥ i (j = i only if li belongs to the connected component Tik (J, P )) and A(j) < i. Hence the interpolated line for gj does not belong to ITi0k or IL0k i (C), but the corresponding internal line lt(j) belongs to Si0k . Then |Si0k | − |ITi0k | − |IL0k i (C)| ≥ 1

(IV.107)

and 1 1 1k 1k 1k [|Ni0k | + [|Si0k | − |ITi0k | − |IL0k i (C)|] + 2[|Si | − |ITi | − |ILi (C)|] ≥ . 2 2 (IV.108) • Finally if |Eik (C)| = 2 one can see, by the same arguments, that xki =

|Si1k | − |ITi1k | − |IL1k i (C)| ≥ 1

(IV.109)

and xki =

0 1 1k 1k 1k [−1 + |Ni k | + [|Si0k | − |ITi0k | − |IL0k i (C)|] + 2[|Si | − |ITi | − |ILi (C|) 2

≥ [−1 + 2[|Si1k | − |ITi1k | − |IL1k i (C)|] ≥ 1/2.

(IV.110)

Now we can perform the integrals in equation (IV.98) and we obtain |ΓΛ 2p (φ1 , . . . φ2p )| ≤ ||φ1 ||1

2p Y

||φi ||∞,2 e−(1−)Λ

m

dT (Ω1 ,...Ω2p )

(IV.111)

i=2 m 2−p

(Λ )

N

1−p

∞ X

−1 X X X n¯Y 1 1 (cK) Pc(i) k 0 !n00 ! n!n n,n0 ,n00 =0 u−T σ E,C i=1 k=1 xi n ¯

where we wrote the sum over ordered trees as sum P over unordered trees and sum over all possible orderings σ of the tree. The sum C is over a set whose cardinal is bounded by K n¯ so it’s sufficient to bound them with the supremum over this set, as we are interested in a theorem at weak coupling λ. However the sum over E to attribute the 2p external lines to particular vertices runs over a set of at most n ¯ 2p (this is an overestimate!). This will lead to the factorial in Theorem 3. We remark however that a better bound on the behaviour of the vertex functions at large p can presumably be obtained when the external points are sufficiently spread (not too closely packed), but we leave this improved estimate to a future study.

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P n−2 ¯ Moreover, we bound (¯n1)! u−T f (T ) by n¯ n¯ ! supu−T |f (T )| using Cayley’s theorem. Therefore, by Stirling’s formula, it’s enough to consider the unordered tree PT which gives the maxu−T |f (T )|. The sum that could still give some factorial is σ . To bound it we use the product of fractions obtained after integration on the βi . • if |ETik | ≥ 5 we have (|ETik | + |EEik | + |ELki (C)| − 4)/4 ≥ (|ETik | − 4)/4 ≥ • if |ETik | < 5 we have xki ≥ 1/2 ≥

|ETik | + 1 (IV.112) 24

|ETik | + 1 24

(IV.113)

Now |ETi | depends on the (now unordered) tree T and on its ordering σ. Therefore we call it from now on |ETiσ |. Recall that it is the total number of external tree half-linesP of the subset Tiσ of T made of the n ¯ − i highest lines in the permutation σ. Since k (|ETik | + 1) ≥ |ETiσ | + 1, equation (IV.111) becomes |ΓΛ 2p (φ1 , . . . φ2p )|

≤ ||φ1 ||1

2p Y

||φi ||∞,2 e−(1−)Λ

m

dT (Ω1 ,...Ω2p )

i=2 ∞ X

(Λm )2−p N 1−p

0

00

n ¯ 2p (cK)n+n +n

n,n0 ,n00 =0

−1 X n¯Y σ

i=1

1 (IV.114) |ETiσ | + 1

At this point we can apply a result of [CR] (Lemma A,1, B.3, B.4) which states that for any tree we have −1 X n¯Y σ

1

|ETiσ | i=1

+1

≤ 4n¯ .

(IV.115)

For completeness let us recall the proof of this result. For each tree T we can define a mapping ξ of T on a chain-tree with the same number of vertices: ξ : T → ξT

(IV.116)

To define ξ, we turn around T starting from an arbitrary end line, and we number the lines in the order we meet them for the first time. The lines of ξT are numbered in the same way and ξT associates the lines with the same number. Now we observe that the sum over the orders on T corresponds to the sum over all permutations of the indices in ξ(T ). Moreover Lemma B.3 in [CR] proves that for any connected or disconnected subgraph R of T , we have ET (R) + 1 ≥ c(ξT (R))

(IV.117)

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1 2 3 4

9 10 7

ξ(T )

6

5 1

5

8

4

2 T

6 7

3

8 9 10

Figure 22 where c(ξT (R)) is the number of connected components of the image of R ξT (R) and ET (R) is the number of external half-lines of R in T . Finally we note that ξ(Ti ) is the set of lines with number j ≥ n ¯ − i so we can write −1 X n¯Y σ

i=1

n ¯ −1

XY 1 1 = := ∆n¯ , σ |ETi | + 1 c(Diσ ) σ i=1

(IV.118)

¯ − i (after where Diσ is the set of lines in the chain-tree ξ(T ), that have σ(j) ≥ n the permutation σ). Now, applying Lemma B.4. in [CR], we obtain ∆n¯ ≤ 4n¯

(IV.119)

We recall that this can be proved by remarking that ∆n¯ satisfies the inductive equation n ¯ −1 X ∆n¯ = ∆p ∆n¯ −k , (IV.120) k=1

so that equation (IV.114) becomes |ΓΛ 2p (φ1 , . . . φ2p )| ≤ ||φ1 ||1

2p Y

||φi ||∞,2 e−(1−)Λ

m

dT (Ω1 ,...Ω2p )

i=2

(Λm )2−p N 1−p

∞ X n,n0 ,n00 =0

n ¯ 2p (4cK)n¯

(IV.121)

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where K depends only on . Taking small enough completes the proof of the P c 2p theorem in the case Λ > m, since n¯ n ¯ e−¯n ≤ K p (p!)2 . In the case Λ < m, we have a few changes to perform. Replacing the lines of T 0 2 −2 in (IV.90) by the bound (IV.92), keeping the massive decay factor e−(/4)m Λ0 (w1 ) in (IV.90) and passing to the limit Λ0 → ∞ we have the following changes: in (IV.95) we add the factors 0

(Λ/m)n

Y

2 −2 (uq )−1/2 e−(/4)u1 m Λ

(IV.122)

lq ∈T 0 0

0

0

0

The factor (Λ/m)n exactly changes Λ2−p−n into Λ2−p m−n = Λ2−p (Λm )−n . 0 The factor (Λm )−n is absorbed in K n¯ since Λm in the hypothesis 3 Q of Theorem −1/2 remains in the compact X. Passing to the variables βi , the factor lq ∈T 0 (uq ) Q |N 0k | is bounded by the factor i βi i in (IV.104), which was previously bounded by 1, hence not used at all. Finally the last integral over β1 becomes bounded, for p > 2 by: Z 1 (p−2)/2 dβ1 −(/4)m2 β1 Λ−2 Λ2−p β1 e (IV.123) β1 0 Changing to the variable v = (/4)m2 β1 Λ−2 we obtain for the final bound a factor Z (4/m2 )(p−2)/2

m2 Λ−2 /4

v (p−2)/2 0

p dv −v e ≤ (Λm )2−p K p p! v

(IV.124)

The case p = 2 is easy and √left to the reader. Hence Theorem 3 holds in every case, by combining the factor p! with the factor (p!)2 coming from the sum over E. Remark that in the case m = 0 we have never Λ < m, hence the factor (p!)5/2 can be replaced by (p!)2 .

V The renormalization group equations In this section we establish the renormalization group equations obtained when varying Λ and we check that for a fixed and small renormalized coupling constant, the effective constants remain bounded and small as predicted by the well known perturbative analysis of the model, which is asymptotically free in the ultraviolet regime [MW]. The derivative as:

ΛΛ0 ∂ ∂Λ Γ2p (φ1 , . . . φ2p )

can be written, using the expression (III.23),

∂ ΛΛ0 ΛΛ0 0 Γ (φ1 , . . . , φ2p ) = T ΓΛΛ 2p (φ1 , . . . , φ2p ) + LΓ2p (φ1 , . . . , φ2p ). ∂Λ 2p

(V.125)

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0 The first term T ΓΛΛ 2p (φ1 , . . . , φ2p ) is the series obtained when the derivative falls on a tree line propagator (see Figure 23a):

Z XX X 1 (T , Ω) d2 x1 . . . d2 xn¯ 0 !n00 ! n!n n,n0 ,n00 =0 o−T E,µ Col,Ω " #" # #" Z n ¯Y −1 Y λw(v) Y Y dwq δmw(v0 ) δζw(v00 ) N 0≤w1 ≤···≤wn−1 ≤1 q=1 0 00 ¯ v

0 T ΓΛΛ 2p (φ1 , . . . φ2p ) =

Y

RGki

Gk i ∈Dµ

Y

nX ¯ −1

q0 =1

∞ X

v

v

∂ Λ0 ,wq (¯ xlq0 , xlq0 ) D ∂Λ Λ

Λ ,w DΛ0 q (¯ xlq , xlq )

det M(µ) φ1 (xi1 ) . . . φ2p (xjp )

(V.126)

q6=q0

a)

b)

Figure 23 0 The second term LΓΛΛ 2p (φ1 , . . . , φ2p ) is the series obtained when the derivative falls on a loop line in the determinant (see Figure 23b):

Z XX X 1 = (T , Ω) d2 x1 . . . d2 xn¯ n!n0 !n00 ! 0 00 n,n ,n =0 o−T E,µ Col,Ω " #" # #" Z n ¯Y −1 Y λw(v) Y Y dwq δmw(v0 ) δζw(v00 ) N 0≤w1 ≤···≤wn−1 ≤1 q=1 0 00 ¯ v ∞ X

0 LΓΛΛ 2p (φ1 , . . . φ2p )

Y

RGki

Gk i ∈Dµ

X

n¯Y −1

v

Λ ,wq

DΛ0

q=1

(−1)(f,g)

hf ,hg |µ(f )=µ(g)

v

(¯ xlq , xlq )φ1 (xi1 ) . . . φ2p (xjp )

∂ Λ0 (wµ(f )−1 ) CΛ0 (wµ(f ) ) (xf , xg )detlef t M(µ) ∂Λ

(V.127)

where (f, g) is a sign coming from the development of the determinant. The convergence proofs of course extend to both terms of equation (V.125). Indeed, in the first one, the sum over the tree lines is bounded by a factor n ¯ , and in

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Continuous Constructive Fermionic Renormalization

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the second one the sum is over the set of loop half-lines which is bounded by a factor n ¯ 2 . Therefore these sums cannot generate any factorial. Then we obtain the same bound as in (IV.114), with an additional factor 1/Λ. This factor disappears when, as usual, the renormalization group equations are written as derivatives with respect to log Λ rather than Λ. From these equations one can derive also equations for the flow of the effective constants defined in (III.24). For instance to obtain the flow of the effective coupling constant λ which is the four-point vertex function at zero external momenta, we can use equations (V.125)–(V.127) in which we let φ1 → δ(0), φ2 , φ3 , φ4 → 1. This is compatible with our L1 -L∞ bounds, so that everything remains bounded. We obtain in this way the famous continuous flow equation which gives the derivative of the coupling constant with respect to log Λ: ∂ ∂ bΛ NΓ λΛ = β2 λ2Λ + O(c3 ) + λ2Λ O(Λ−α ) 4 (0, 0, 0, 0) = ∂log Λ ∂log Λ

(V.128)

where β2 = −2(N − 1)/π

(V.129)

is the first non trivial term corresponding to the four-point graph with one tree line and one loop line, and the last term λ2Λ O(Λ−α ) is an infrared correction to the asymptotic flow (see [FMRS]). The negative sign of β2 is responsible for the asymptotic freedom of the model. Similar equations hold for the flow of δm and δζ (which remain bounded). For these equations up to one loop, see [MW] [GN] [GK] [FMRS]. For the computation up to two loops, we refer to [W]. From these renormalization group equations one can control the behavior of the effective constants and check that they remained bounded (until now this was assumed). The reader might be afraid that there is something circular in this argument. In fact this is not the case. Let us discuss for simplicity the massless case b ΛΛ0 (0, 0, 0, 0) is only a function of Λ0 /Λ and of where the renormalized coupling Γ 4 the bare coupling λ. We know that it is analytic at the origin as function of the bare coupling λ [AR2]. Therefore from (V.125)–(V.128) it is for small bare λ and Λ0 /Λ a monotone increasing function of the ratio Λ0 /Λ (although this function might blow up in finite time). Inverting the map from bare to renormalized couplings, one can prove that b ΛΛ0 (0, 0, 0, 0) all the effective conconversely for small renormalized coupling Γ 4 stants λw remain bounded by the renormalized one. Therefore one can pass to the ultraviolet limit Λ0 → ∞, in analogy with the completeness of flows of vector fields on compact manifolds. Furthermore one can compute the asymptotic behavior of the bare coupling which tends to 0 as 1/(|β2 | log(Λ0 /Λ))). Similar arguments hold for the mass and wave function effective constants and achieve the proof of Theorem 1. We recall for completeness that it is easy to build the Schwinger functions from the vertex functions and that the Osterwalder-Schrader axioms of continuous Euclidean Fermionic theories hold for the massive Gross-Neveu model at Λ = 0.

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The simplest proof is to remark that being the Borel sum of the renormalized expansion, the Schwinger functions we build are unique. Building them as limits of theories with different kinds of cutoffs prove the axioms since different sets of cutoffs violate different axioms [FMRS].

Acknowledgements It is a pleasure to thank D. Brydges, C. de Calan, H. Kn¨ orrer, C. Kopper, G. Poirot and M. Salmhofer for discussions related to this paper. In particular we thank C. Kopper for his careful reading of this preprint, which lead us to correct the initial version of Subsections IV.3.4 and IV.3.5.

References [AR1]

A. Abdesselam and V. Rivasseau, Trees, forests and jungles: a botanical garden for cluster expansions, in Constructive Physics, ed by V. Rivasseau, Lecture Notes in Physics 446, Springer Verlag, 1995.

[AR2]

A. Abdesselam and V. Rivasseau, Explicit Fermionic Cluster Expansion, preprint: cond-mat/9712055.

[BM]

F. Bonetto, V. Mastropietro, Beta Function and Anomaly of the Fermi Surface for a d=1 system of interacting fermions in a periodic potential, Commun. Math. Phys. 172, 57–93 (1995).

[C]

P. Contucci, Ph. D. Thesis.

[CR]

C. de Calan and V. Rivasseau, Local existence of the Borel transform in Euclidean φ44 , Commun. Math. Phys. 82, 69 (1981).

[FMRS] J. Feldman, J. Magnen, V. Rivasseau and R. S´en´eor, A renormalizable field theory: the massive Gross-Neveu model in two dimensions, Commun. Math. Phys. 103, 67 (1986). [G]

G. Gallavotti, Renormalization Theory and ultraviolet stability for scalar fields via renormalization group methods, Rev Mod. Phys. 57, 471 (1985).

[GK]

K. Gawedzki and A. Kupiainen, Commun. Math. Phys. 102, 1 (1985).

[GN]

D. Gross and A. Neveu, Dynamical Symmetry breaking in asymptotically free field theories, Phys. Rev. D10, 3235 (1974).

[KKS]

G. Keller, Ch. Kopper and M. Salmhofer, Helv. Phys. Acta 65 32 (1991).

[KMR]

C. Kopper, J. Magnen and V. Rivasseau, Mass Generation in the Large N Gross-Neveu Model, Commun. Math. Phys. 169, 121 (1995).

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[MW]

P.K. Mitter and P.H. Weisz, Asymptotic scale invariance in a massive Thirring model with U(n) symmetry, Phys. Rev. D8, 4410 (1973).

[P]

J. Polchinski, Nucl. Phys. B 231, 269 (1984).

[R]

V. Rivasseau, From perturbative to constructive renormalization, Princeton University Press (1991).

[S]

M. Salmhofer, Continuous renormalization for Fermions and Fermi liquid theory, ETH preprint, cond-mat/9706188.

[W]

W. Wetzel, Two-loop β-function for the Gross-Neveu model, Phys. Lett. 153B, 297 (1985).

M. Disertori and V. Rivasseau Centre de Physique Th´eorique CNRS UPR 14 Ecole Polytechnique F-91128 Palaiseau Cedex, France Email: [email protected], [email protected] Communicated by J. Bellissard submitted 02/03/98, revised 08/09/98; accepted 18/11/98

Ann. Henri Poincar´ e 1 (2000) 59 – 100 c Birkh¨

auser Verlag, Basel, 2000 1424-0637/00/010059-42 $ 1.50+0.20/0

Annales Henri Poincar´ e

Correlation Asymptotics of Classical Lattice Spin Systems with Nonconvex Hamilton Function at Low Temperature V. Bach, T. Jecko and J. Sj¨ ostrand Abstract. The present paper continues Sj¨ ostrand’s study [14] of correlation functions of lattice field theories by means of Witten’s deformed Laplacian. Under the assumptions specified in the paper and for sufficiently low temperature, we derive an estimate for the spectral gap of a certain Witten Laplacian which enables us to prove the exponential decay of the two-point correlation function and, further, to derive its asymptotics, as the distance between the spin sites becomes large. Typically, our assumptions do not require uniform strict convexity and apply to Hamiltonian functions which have a single, nondegenerate minimum and no other extremal point. Keywords. Correlation Function, Lattice Spin Systems, Exponential Decay, Witten Laplacian.

I Introduction and results The present paper can be viewed as a continuation of works by Helffer-Sj¨ ostrand [10] and Sj¨ ostrand [14] on Laplace integrals Z e−2βH(x) u(x) dx (I.1) Rm

in the limit m → ∞ and for large β > 0. In particular, we are interested in the two-point correlation functions ETβ (xj ; xk ) := Eβ (xj · xk ) − Eβ (xj ) Eβ (xk )

(I.2)

when |j − k| → ∞ (and more precise assumptions will be given below), where Z −1 Z −2βH(x) Eβ (xj ) := e dx xj e−2βH(x) dx (I.3) Rm

Rm

is the expectation of xj . In [10], the authors studied exponential decay of the correlations under assumptions on the function H containing that of uniform strict convexity. They exhibited a certain matrix Schr¨ odinger operator for gradients and studied it by means of a maximum principle. The global convexity was quite crucial for the maximum principle to apply.

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In [14], one of us identified the matrix operator (up to a conjugation) with the Witten Laplacian in degree 1, i.e., with the Hodge Laplacian associated to the conjugated de Rahm complex β −1 e−βH d eβH , d = exterior derivative ,

(I.4)

and an explicit identity ((I.27) below) was given for the correlations. A more systematic use of L2 -methods, together with the use of Grushin-Feshbach reductions led (essentially) to an asymptotic formula for the correlations when |j − k| → ∞. Unfortunately, some use of the maximum principle remained and consequently it was still necessary to impose uniform strict convexity, as well as some other unnatural assumptions. The purpose of the present paper is to completely eliminate the maximum principle and to work entirely with L2 -methods. This allows us to weaken the assumptions on H considerably. The present assumptions (see below) imply that H has a non-degenerate minimum and that this is the only critical point. Away from the minimum, however, H is allowed to be non-convex. The novelty (see Section III) is the use of certain weighted estimates on quantities related to H (like, for instance, H 00 (x) − H 00 (0)) in terms of the derivatives ∂H/∂xj . Similar ideas have recently been developed by Helffer [6, 7, 8, 9], who, for a wide range of parameters, derives exponentially decaying upper bounds for the two-point correlation function in case that the interaction is strictly convex or quadratic (Gaussian), while H may be non-convex. In other parts of the paper, we roughly follow [14]. Physically, H is the Hamiltonian (energy) function for a continuous spin system on the lattice ΛL ⊆ Zd which one may either derive directly from first principles or from a discrete spin system by a Sine-Gordon transformation. Even though we have weakened the assumptions on H compared to [14], our results imply that the system represented by this Hamiltonian function does not exhibit phase transitions, and the extension of our method to include the description of multiple phases, our ultimate goal, is not obvious. We remark that continuous spin systems with multiple phases have been successfully studied by other methods, e.g., the Pirogov-Sinai theory and contour methods [5, 18]. We consider a system of real-valued spins on the sequence {ΛL }L∈N of finite, n-dimensional lattices ΛL := (Z/LZ)n . Given L ∈ N, the corresponding spin configuration space is R|ΛL | , and the energy of a spin configuration is determined by a Hamilton function HL ∈ C 2 (R|ΛL | , R). To ensure the existence of the thermodynamic limit, we shall generally assume the following hypothesis. 0 Hypothesis 1. There exist constants C(H1) > 0, δ 0 = δ(H1) ≥ δ = δ(H1) > 0, |ΛL | , independent of L ∈ N, such that, for all x = (xj )j∈ΛL ∈ R

X X 0 |xj |δ − C(H1) ≤ HL (x) ≤ |xj |δ + C(H1) . j∈ΛL

j∈ΛL

(I.5)

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0 Under this assumption there exists a constant m = m(β, δ(H1) , δ(H1) ) such that, for any inverse temperature 2β > 0, Z e−2βHL (x) dx ≤ exp 2βm|ΛL | . (I.6) exp −2βm|ΛL | ≤ Ξ(2β) :=

Thus, replacing HL and C(H1) by HL,β := HL (x) + (2β)−1 log Ξ(2β) and 0 0 ) := C(H1) + m(β, δ(H1) , δ(H1) ), respectively, we obtain that C(H1) (β, δ(H1) , δ(H1) HL,β fulfills Hypothesis 1, as well, and Z

e−2βHL,β (x) dx = 1 .

(I.7)

We note that this replacement does not affect the derivatives of HL,β . Henceforth, we often neither display the dependence of HL,β on L nor β and simply write H = HL,β . Thus we have that e−βH ∈ H(0) := L2 (R|ΛL | ) and ke−βH k = 1. Equivalently, dµ(x) := e−2βH(x) dx defines a probability measure, the Gibbs measure, on R|ΛL | . Given a polynomially bounded observable u, i.e., a polynomially bounded, measurable function R|ΛL | → R, we define its expectation by Z u(x) e−2βHL,β (x) dx . (I.8) EL,β (u) := By (I.7), EL,β (1) = 1. The truncated correlation of two polynomially bounded observables u, v is defined by ETL,β (u ; v) := EL,β (u · v) − EL,β (u) · EL,β (v) .

(I.9)

To formulate our first main result, we assume the following specific hypothesis on HL,β , remarking that below and henceforth we use the notation Fi0 (x) := ∂i F (x) =

∂F (x) , ∂xi

00 Fi,j (x) := ∂i ∂j F (x) =

∂2F (x) , ∂xi ∂xj

(I.10)

for any F ∈ C 2 (R|ΛL | ; C). Hypothesis 2. HL,β ∈ C 2 (R|ΛL | ; R) has a unique minimum at x = 0, and for any other critical point xc ∈ R|ΛL | \ {0} of HL,β , we have HL,β (xc ) ≥ HL,β (0) + C(H2) , 00 for some constant C(H2) > 0. Furthermore, the Hessian HL,β (0) of HL,β at x = 0 is bounded by 00 (0) ≤ λmax · 1 , 0 < λmin · 1 ≤ HL,β

(I.11)

for two constants λmax ≥ λmin > 0. These constants C(H2) , λmin , and λmax neither depend on L nor β.

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Hypotheses 1–2 guarantee that, for fixed L, the Gibbs measure EL,β ( · ) is concentrated about x = 0, as β → ∞. We do not expect that the system described by HL undergoes a phase transition. Rather, we expect to have exponentially decaying correlations, T EL,β (xj ; xk ) ≤ Cβ exp −µβ d(j − k) , 1 d(j − k) L , (I.12) for some Cβ ≥ 0, µβ > 0. Indeed, we give such an upper bound in Theorem I.6, and we derive the precise asymptotics of ETL,β (xj ; xk ) in Theorem I.9 below. In Eqn. (I.12), we use the natural euclidean distance function d : (R/LZ)n → R on the torus, given by q n o ˜ Rn = k˜2 + . . . + k˜2 k˜ ∈ π −1 (k) , d(k) := min |k| (I.13) 1 d where π : Rn → ΛL = (R/LZ)n is the canonical projection. In other words, if we (box) := [−L/2 , L/2)n ∩ Zn then identify ΛL with the fundamental domain ΛL d(k) := J(k) Rn (I.14) (box)

is the euclidean length of J(k), where J : ΛL → ΛL given by J −1 = π|Λ(box) .

is the natural bijection

L

For the derivation of the asymptotics of ETL,β (xj ; xk ), for large d(j − k), the following summability hypothesis, which depends on a weight function G : ΛL → [0, +∞), is an important requirement. Hypothesis 3. [G] For an even function G : ΛL → [0, +∞), there exist weights aij (k) = aji (k) ≥ 0, bij (k) = bji (k) ≥ 0, where i, j, k ∈ ΛL , such that, for all i, j ∈ ΛL and x ∈ R|ΛL | , o Xn 00 00 (0) ≤ aij (k)|Hk0 (x)| + bij (k)|Hk0 (x)|2 . (I.15) Hi,j (x) − Hi,j k∈ΛL

These weights fulfill the summability condition that n X n X o o max aij (k) + bij (k) eG(i−j) aij (k) + bij (k) , max j∈ΛL

i,k∈ΛL

k∈ΛL

(I.16)

i,j∈ΛL

is bounded above by some constant C(H3) (G), which neither depends on L nor β. In Theorem I.4 we only require Hypothesis 3 with the trivial weight G ≡ 0 to prove a spectral estimate which implies the existence of a spectral gap for the relevant operator. To turn this gap estimate into an exponential decay estimate similar to (I.12), we need to make a slightly stronger assumption, namely, that Hypothesis 3 holds for G = µd, for some µ > 0, where d is the euclidean distance function on ΛL specified in (I.13). Additionally, we require Hypothesis 4[νd] below, for some ν > 0, which is an estimate on the Hessian of H at x = 0.

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Hypothesis 4. [G] Assume Hypothesis 2. For an even function G : ΛL → [0, +∞), there exists a constant 1 > C(H4) (G) > 0, neither depending on L nor β, such that the Hessian of H at 0 satisfies X 00 00 λmin ∀L, ∀i ∈ ΛL : eG(i−j) Hi,j (0) ≤ 1 − C(H4) (G) Hi,i (0) . λmax j∈ΛL \{i}

(I.17) For the derivation of the precise asymptotics of ETL,β (xj ; xk ), our requirement for G in Hypothesis 3 is even stronger. Indeed, starting from the norm Srv : Rn → [0, +∞) given by the support function Srv defined in (I.40), for some r > 1, we assume that Hypothesis 3[G] holds with G ≡ θerS , where n o ˜ k˜ ∈ π −1 (k) . θerS (k) := inf Srv (k) (I.18) We note here that in general, if S : Rn → [0, +∞) is a semi-norm then dS : (R/LZ)n → [0, +∞) defined similarly to (I.18) by dS (x) := inf x˜∈π−1 (x) {S(˜ x)} defines a semi-metric which obeys the triangle inequality, dS (x+y) ≤ dS (x)+dS (y). The choice of G in the three cases decribed above can be expressed in terms of the underlying semi-norm on Rn , namely, G ≡ dS ≡ 0, for S ≡ 0, G ≡ dS = νd, for S = ν| · |Rn , and G ≡ dS = θerS , for S ≡ Srv . We remark that Hypotheses 1 and 3 partially strengthens Hypothesis 2, as they imply that there is only one critical point, namely at the minimum. To see this, we observe that Hypothesis 3 imposes that any critical point is a strictly relative minimum. If there were two different critical points, there would exist a saddle point by the Mountain Pass Lemma (see [3, 15]) in contradiction to Hypothesis 3. Nevertheless, by using Hypothesis 3[G], we avoid Sj¨ ostrand’s requirement [14] of uniformly strict convexity of H, i.e., H 00 (x) ≥ c · 1 > 0, for all x ∈ R|ΛL | . The main example we have in mind is a pair interaction Hamilton function of the following form, for some ν > 0. Example I.1. [ν] There exist 0 < g ≤ 1, f ∈ C 2 (R; R), obeying |t|δ − c ≤ f (t) ≤ 0 |t|δ + c, for some δ, δ 0 , c > 0, and wij ∈ C 2 (R2 ; R), for all i, j ∈ ΛL , wii ≡ 0, such that X X HL (x) = f (xj ) + g e−ν d(i−j) wij (xi , xj ) . (I.19) j∈ΛL

i,j∈ΛL

00

Furthermore f (0) > 0, and f and {wij }i,j∈ΛL obey | ∂s wij (s, t) | , | ∂t wij (s, t) | ≤ |f 0 (s)| + |f 0 (t)| , (I.20) 00 00 0 0 2 | f (t) − f (0) | ≤ |f (t)| + |f (t)| ,  2 2 |∂s wij (s, t) − ∂s wij (0, 0)| ,  |∂t2 wij (s, t) − ∂t2 wij (0, 0)| , ≤ |f 0 (s)| + |f 0 (s)|2 + |f 0 (t)| + |f 0 (t)|2 ,  |∂s ∂t wij (s, t) − ∂s ∂t wij (0, 0)| , |∂s2 wij (0, 0)| , |∂t2 wij (0, 0)|

,

|∂s ∂t wij (0, 0)| ≤ 1 .

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The function given by (I.19) satisfies Hypotheses 1, 2, and 4[µd], for some 0 < µ < ν small enough. Note that in our example, we require that f 00 (xc ) = f 00 (0) > 0, for any critical point xc . Thus, f has its minimum at t = 0 and no other critical point. Furthermore, we remark that the restriction to small values Q of g ≥ 0 should ensure that e−2βHL (x) j dxj is close to a product measure of Q the form j e−2βf (xj ) dxj . For such a Hamilton function, we prove the following lemma in Section A. Lemma I.2. Assume √ that HL is a Hamilton function as in Example I.1[ν], and let Mα := 2n (1 − e−α/ n )−n , for α > 0. Then, for 0 ≤ g < Mν−3 /24 and any 0 ≤ µ < ν, the Hamilton function HL fulfills Hypothesis 3[µd], with C(H3) = 2 + 12gMν−µ and X aij (k) := bij (k) := e cij (l)Rlk , (I.21) l∈ΛL

where e cij (l) and Rlk are defined in (A.8) and (A.12) in Section A below. To state our first main result, we introduce some more notation. We define two operators, (1)

∆H

A(1)

2 00 H (x) , β 2 00 (0) H (0) , := ∆H ⊗ 1 + β (0)

:= ∆H ⊗ 1 +

(I.22) (I.23)

on H(1) := L2 (R|ΛL | ) ⊗ C|ΛL | , the space of square-integrable one-forms on R|ΛL | , where H 00 (x) is the multiplication by the Hessian matrix of H at x, and (0)

∆H

:= =

o X n 1 ∂2 2 1 00 (x) − 2 2 + Hj0 (x) − Hj,j β ∂xj β j∈ΛL X Zj (H)∗ Zj (H) ,

(I.24)

j∈ΛL

with Zj (H) := e−βH β −1 ∂j eβH = β −1 ∂j + Hj0 (x) , Zj (H)∗ := eβH −β −1 ∂j e−βH = −β −1 ∂j + Hj0 (x) . (1)

(I.25) (I.26)

Under the assumption of Hypotheses 1 and 2, both ∆H and A(1) are strictly positive, invertible operators on H(1) . While for A(1) , this follows simply from (0) A(1) ≥ 2β −1 λmin 1, which is implied by the positivity of ∆H , the strict positivity (1) (0) (1) of ∆H is less obvious. It origins from the fact that ∆H and ∆H can be viewed as restrictions to the space of square-integrable zero- and one-forms on R|ΛL | ,

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respectively, of the Witten Laplacian, ∆H , the Hodge Laplacian conjugated with e−βH , which acts on forms of all degrees [17, 4]. We outline the argument in Section II. It is convenient to introduce the set O(1) of observables u ∈ C 1 (R|ΛL | ; R), for which both u and ∇u are polynomially bounded. We remark that e−βH ∇u (1) ∈ H(1) , for any u ∈ O(1) . The importance of the Laplacian ∆H lies in the following identity, used implicitly by Helffer and Sj¨ ostrand [10] and stated explicitly in [14], and for which we give a new derivation in Section II. (1)

Lemma I.3. Assume Hypotheses 1 and 2. Then ∆H is strictly positive on H(1) , and, for any two observables u, v ∈ O(1) , the following identity holds: E 1D (1) −1 −βH ETL,β (u ; v) = 2 e−βH ∇u ∆H e ∇v (1) . (I.27) β H Lemma I.3 allows us to express the truncated correlations by matrix elements (1) of the resolvent of ∆H . Thus, the analysis of the truncated correlation traces back (1) to the spectral analysis of ∆H . The latter is not entirely trivial, a priori, as the 00 Hessian H (x) may become small or even negative, for some x ∈ R|ΛL | . Our first main result, Theorem I.4 below, shows that, under the additional assumption of Hypothesis 3 without exponential weights, i.e., G ≡ 0, the values of the Hessian H 00 (x), for x away from the origin, are irrelevant. Theorem I.4. Assume Hypotheses 1, 2, and 3[0]. Then there exist constants C ≥ 0 and β0 ≥ 0, both independent of L, such that, for all β ≥ β0 , C C (1) (I.28) 1 − 1/2 A(1) ≤ ∆H ≤ 1 + 1/2 A(1) β β holds in the sense of quadratic forms on Q(1) ⊆ H(1) , the form domain of A(1) (1) and ∆H . We begin the discussion of Theorem I.4 by deriving a corollary which immediately follows from A(1) ≥ 2β −1 H 00 (0). Corollary I.5. Assume Hypotheses 1, 2, and 3[0]. Then there exist constants C ≥ 0 and β0 ≥ 0, both independent of L, such that, for any observable u ∈ C 1 (R|ΛL | ; R), for which u and ∇u are polynomially bounded, and any β ≥ β0 , we have −1 −βH E 2 C D −βH ETL,β (u ; u) ≤ ∇u H 00 (0) e ∇u (1) e 1 + 1/2 β H β 2 1 C ≤ EL,β |∇u|2 . (I.29) 1 + 1/2 β λmin β We compare this result to the Brascamp-Lieb inequality [2, 14, 7, 12], which states that −1 −βH E 2 D −βH ∇u H 00 (x) e ∇u (1) , (I.30) ETL,β (u ; u) ≤ e β H

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for strictly convex H, i.e., H 00 (x) ≥ λmin (x) > 0, for all x ∈ R|ΛL | , where λmin (x) may become very small, for certain values of x. Our result in Corollary I.5 is stronger in the sense that it only requires H 00 (0) ≥ λmin and a certain control of H 00 (x) − H 00 (0) by |H 0 (x)|, specified in Hypothesis 3[0]. Our second main result concerns the low-temperature asymptotics of the twopoint correlation function ETL,β (xj ; xk ). An application of Lemma I.3 with u := xj and v := xk yields E 1 D (1) −1 −βH ETL,β (xj ; xk ) = 2 e−βH ⊗ ej ∆H e ⊗ ek (1) , (I.31) β H where {ei }i∈ΛL denotes the standard basis in C|ΛL | . On the other hand, we trivially have D E 1 00 −1 H (0) = e−βH ⊗ ej (A(1) )−1 e−βH ⊗ ek (1) , (I.32) 2β j,k H (1)

and Theorem I.4 asserts that ∆H agrees with A(1) up to a relative error which becomes small, as β → ∞. It is thus reasonable to believe that 1 00 −1 ETL,β (xj ; xk ) ≈ H (0) , (I.33) 2β j,k as β → ∞, in a suitable sense made precise in Lemma I.8 and Theorem I.9 below. In fact, under the additional requirement of Hypotheses 3 and 4, it is fairly straightforward to turn the spectral estimates of Theorem I.4 into an upper bound for |ETL,β (xj ; xk )| with an exponential decay in d(j − k), as the following theorem makes explicit. Theorem I.6. Assume Hypotheses 1, 2, 3[µd], and 4[νd], for some µ, ν > 0, i.e., G = µd in Hypothesis 3 and G = νd in Hypothesis 4. Then there exist constants C ≥ 0 and β0 ≥ 0, both independent of L, such that T EL,β (xj ; xk ) ≤ C exp − min(µ ; ν) d(j − k) , (I.34) β for all β ≥ β0 . To prove and quantify the relation (I.33), we assume the translation invariance of the Hamilton function. Hypothesis 5. The Hamilton function is translation invariant. That is, HL (τm x) = HL (x), for any m ∈ ΛL , where (τm x)j := xj−m denotes the shift on the lattice by m. Note that translation invariance of HL implies the translation invariance of the Hessian of HL at x = 0. Indeed, since 00 00 (0) = Hj−k,0 (0) , Hj,k

(I.35)

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the Hessian H 00 (0) operates on C|ΛL | as convolution with H 00·,0 (0). We remark that 00 H0,0 (0) ≥ λmin > 0, assuming Hypothesis 2. Furthermore, we assume the Hessian H 00 (0) to be ferromagnetic, of finite range, and independent of L, for L sufficiently large. More precisely, we require the following additional hypothesis. Hypothesis 6. Assume Hypotheses 2 and 5, and define a function vL by setting 00 00 vL (k) := −Hk,0 (0)/H0,0 (0), for k 6= 0, and vL (0) := 0. There exists an even, nonnegative function v : Zn → [0, +∞), v(k) = v(−k) ≥ 0, of bounded support such that vL = v ◦ J (where J is defined in (I.14)), for all L larger than 2 times the diameter of the support of v. Moreover, the subgroup of Zn generated by the support of v is Zn , Gr supp{v} = Zn , (I.36) i.e., the smallest nontrivial subgroup of Zn , which contains supp{v}, is Zn itself. We list two important consequences of Hypotheses 5 and 6. First, they imply that there exists a set of linearly independent vectors {k1 , . . . , kn } ⊆ Zn and a constant δ > 0, such that v(kν ) ≥ δ, for all 1 ≤ ν ≤ n. Secondly, Hypotheses 5 and 6 and a Perron-Frobenius argument imply that the lowest eigenvalue of H 00 (0) is given by X 00 0 < λmin = H0,0 (0) 1 − v(k) . (I.37) k∈Zn

Moreover, this eigenvalue is nondegenerate, and the corresponding eigenvector has constant entries. Note further that, under Hypothesis 6 and the additional assumpP tion that k∈Zn v(k) < 1/2, we can find some ν > 0 such that Hypothesis 4[νd] is satisfied. Using v, we define a function Fv : Rn → [0, +∞) by the following finite sum, X eη·k v(k) , (I.38) Fv (η) := k∈Zn

for all η ∈ Rn . We point out that Fv (η) = vˆ(iη), where vˆ is the Fourier transform of v. Moreover, under Hypothesis 2, Eqns. (I.37) and (I.38) imply that 1 − Fv (0) > 0. Next, for r > 1 − Fv (0), we introduce the open level sets and their boundaries Dv (r) := η ∈ Rn Fv (η) < r , Σv (r) := ∂Dv (r) , (I.39) and by means of these we define the support function Srv : Rn → [0, +∞) as Srv (x) := sup η · x η ∈ Dv (r) . (I.40) Finally, we need to make use of the following closed subset of Rn , Ar := x ∈ Rn Srv (x) = minn {Srv (x + qL)} , q∈Z

(I.41)

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implicitly using that the minimum is attained. The definitions of Fv , Dv (1), Σv (1), and S1v , in this context, go back to [14] (although no finite range condition (see Hypothesis 6) is imposed there), and most of the properties collected in Lemma I.7 below can already be found there. We give a proof of Lemma I.7 at the end of Section VI. Lemma I.7. Assume Hypotheses 5 and 6. Let 0 < δ0 < min{Fv (0) , 1 − Fv (0)}. Then (i) the function Fv is strictly convex, and there exist constants C, C 0 ≥ 0 such that, for any ε > 0 and any η ∈ Rn , 0 (I.42) Fv (η) + C|η|2 ε ≤ Fv (1 + ε)η ≤ eC |η|ε Fv (η) ; (ii) for every 1 − δ0 ≤ r ≤ 1 + δ0 , Dv (r) is a strictly convex, bounded, open set with smooth boundary Σv (r) := ∂Dv (r). More specifically, r 7→ Dv (r) is monotonically increasing, and there exist two constants R1 , R2 > 0 such that B(R1 , 0) ⊆ Dv (r) ⊆ B(R2 , 0) ;

(I.43)

(iii) the support function Srv : Rn → R+0 defines a norm on Rn , for each 1 − δ0 ≤ r ≤ 1 + δ0 . Furthermore, Srv (x) = ηv (x) · x, where ηv (x) ∈ Σv (r) is uniquely determined by ∇η Fv (ηv (x)) = µx, for some µ > 0, and we have ∇x Srv (x) = ηv (x). Moreover, there exist constants C, C 0 ≥ 0 such that, for any 0 < ε < 1, v (1 + ε)S(1−Cε)r ≤ Srv ≤ (1 + ε)Srv exp(−C 0 ε) ;

(I.44)

(iv) the set Ar is star-shaped. There exist two constants R10 , R20 > 0 such that B(R10 L, 0) ⊆ A1 ⊆ B(R20 L, 0) .

(I.45)

Moreover, there is a fundamental domain (Ar )◦ ⊂ Aer ⊆ Ar for the canonical S projection π : Rn → (R/LZ)n . That is, Rn = q∈Zn Aer + qL, and Aer + qL ∩ Aer + q0 L = ∅, for q 6= q 0 , q, q 0 ∈ Zn . In [14], these definitions are used to prove the following asymptotics for the inverse of the Hessian H 00 (0). Lemma I.8. Assume Hypotheses 2, 5, and 6. Then there exists a constant 0 < δ ≤ 1/2 such that, for j ∈ ΛL with 1 d(j) = |J(j)| ≤ δL, −1 = (I.46) H 00 (0) j,0

1 + O 1/|J(j)| d−1 00 (0) 2π |J(j)| 2 H0,0

d−3 2 h i ∂k Fv ηv (J(j)) v exp −S (J(j)) , i 1 1/2 2 F η (J(j)) det ∂⊥ v v

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where O 1/d(k) ≤ C/d(k), for some constant C ≥ 0 which is uniform in L → ∞ and β → ∞. Here ∂k (resp. ∂⊥ ) represents the derivative along the direction of J(j) (resp. along the directions orthogonal to J(j)). The next theorem quantifies the relation (I.33), as it asserts a formula for the low-temperature asymptotics of the two-point correlation function ETL,β (xj ; xk ) very similar to (I.46). Theorem I.9. Assume Hypotheses 1, 2, 5, 6, and Hypothesis 3[Srv ], for some 1 < −1 = π|Ae1 r < 2 − Fv (0). Denote by JA : ΛL → Ae1 the natural bijection given by JA (see Lemma I.7(iv)) and fix some 0 < λ < 1. Then, for β sufficiently large, for j ∈ ΛL such that d(j) is sufficiently large and that JA (j) ∈ λAe1 , we have ETL,β (xj ; x0 ) = 1 + O 1/|JA (j)| d−1 00 (0) 2π |J (j)| 2 H0,0 A

(I.47) d−3 2 h i ∂k Fw ηw (JA (j)) exp −S1w (JA (j)) , i 1/2 2F det ∂⊥ w ηw (JA (j))

where the functions Fw , ηw , and S1w are defined below in Sections VI–VII and fulfill ∂ηα Fw (ηw (k)) = ∂ηα Fv (ηv (k)) + O(β −1/2 ) and S1w = S1v + O(β −1/2 ), so that we furthermore have 1 + O |JA (j)|−1 + O β −1/2 T (I.48) EL,β (xj ; x0 ) = d−1 00 (0) 2π |J (j)| 2 H0,0 A d−3 2 h i ∇η Fv ηv (JA (j)) · exp − 1 + O(β −1/2 ) S1v (JA (j)) , i 1/2 2 Fv ηv (JA (j)) det ∂⊥ where the O-symbols in (I.48) are uniform in L → ∞ and β → ∞. Comparing (I.48) to (I.46), we finally notice that the decay rate of the twopoint correlation function ETL,β (xk ; x0 ) agrees with the decay rate of the resolvent matrix elements of the Hessian H 00 (0), given by the support function S1v of Dv (1), modulo a factor of 1 + O(β −1/2 ). Furthermore, we note that in Theorem I.9 it is natural to use the bijection provided by JA : ΛL → Ae1 rather than J. Using JA instead of J amounts to (box) projecting onto the fundamental domain Ae1 rather than ΛL . The difference between these two maps becomes important when studying the two-point function (box) ETL,β (xj , x0 ) for J(j) which are close to the boundary of ΛL . We conclude this introduction with a brief survey on the organization of the following sections. In the next section, we introduce a deformed Dirac operator

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DH , by means of which we rederive Lemma I.3. In Section III, we prove our first main result, the operator inequalities (I.28) in Theorem I.4. They derive from a form bound proven in slightly greater generality, as to accommodate for the case of distorted operators, which we have to deal with in Sections VI–VII when deriving the asymptotics of the two-point function. In Section IV we give a short proof of exponential decay of the two-point function under a least hypothesis. Section VI is devoted to the analysis of the support function Srw which determines the precise rate of exponential decay of the two-point function. The estimates derived in Section VI are then used in Section VII to prove the asymptotics as claimed in Lemma I.8 and Theorem I.9. Finally, this paper has three appendices. Appendix A contains the verification of the admissibility of Example I.1[ν], asserted in Lemma I.2. In Appendix B, the self-adjointness and other basic spectral properties of the Dirac operator DH are proven, and Appendix C contains an example of a non-ferromagnetic (i.e., nonpositivity preserving) Hessian of H at x = 0, for which yet the decay of the matrix elements of the resolvent can be made precise.

II Dirac operator and Witten Laplacian In the present section, we prove Lemma I.3, i.e., we show that, for any two observables u, v ∈ O(1) , the following relation (I.27) holds ETL,β (u ; v) =

(1) 1 −βH ∇u (∆H )−1 e−βH ∇v H(1) , e 2 β

(II.1)

(1)

where ∆H was defined in (I.22). Before turning to the proof, we introduce some more notation. The fermion Fock space Ff [C|ΛL | ] over C|ΛL | is defined to be the orthogonal sum of the N (N) fermion sectors Ff , for N ∈ {0, 1, . . . , |ΛL |}, |ΛL |

Ff [C|ΛL | ] =

M

(N)

Ff

.

(II.2)

N=0 (N)

For each N ∈ {0, 1, . . . , |ΛL |}, an orthonormal basis in Ff is given by ∗ ∗ cj1 cj2 · · · c∗jN Ω j1 , j2 , . . . , jN ∈ ΛL , j1 < j2 < . . . < jN ,

(II.3)

where “<” is the order on ΛL induced by some bijection ΛL → {1, . . . , |ΛL |}. Here c∗j and cj are the standard fermion creation and annihilation operators on Ff [C|ΛL | ] obeying the canonical anticommutation relations (CAR), ci , c∗j = δi,j , (II.4) ci , cj = c∗i , c∗j = 0 , where {A, B} := AB + BA, and Ω is the unique (up to a phase) normalized vector in Ff [C|ΛL | ] such that cj Ω = 0, for all j ∈ ΛL . Note that Ff [C|ΛL | ] is nothing but

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V(N) |ΛL | the space of antisymmetric N -forms over R|ΛL | , frequently denoted (R ) (0) (1) (see, e.g., [4]). In particular, Ff = C · Ω, and Ff may be naturally identified with C|ΛL | . We may therefore identify L2 (R|ΛL | ) with H(0) and L2 (R|ΛL | ; C|ΛL | ) with H(1) , where H(N) := L2 (R|ΛL | ) ⊗ Ff

(N)

(II.5)

is the space of square-integrable N -forms, by means of the natural identification maps L2 (R|ΛL | ) → H(0) ,

I (0) : I

(1)

2

|ΛL |

: L (R

|ΛL |

;C

) → H

ψ 7→ ψ ⊗ Ω ,

(0)

, (ψj )j∈ΛL 7→

X

(II.6) ψj ⊗

c∗j Ω

.

(II.7)

j∈ΛL

Equipped with this notation, we return to the proof of Lemma I.3. We introduce the rank-one projection P0 = |e−βH ihe−βH | = P02 = P0∗ and P 0 = 1 − P0 , and we observe that

ETL,β (u ; v) = e−βH u e−βH v L2 − e−βH u e−βH L2 e−βH e−βH v L2

= e−βH u P 0 e−βH v 2 , (II.8) L

where L2 is a shorthand notation for L2 (R|ΛL | ). Using the isomorphism I (0) defined in (II.6) and P0 ⊗ PΩ := 1 − P0 ⊗ PΩ , we find that

(II.9) ETL,β (u ; v) = e−βH u ⊗ Ω P0 ⊗ PΩ e−βH v ⊗ Ω H(0) . Next, we use Zj (H) := β −1 ∂j + Hj0 (x) and Zj (H)∗ = −β −1 ∂j + Hj0 (x), as defined in (I.25)–(I.26), to introduce a deformed exterior differential dH and its adjoint d∗H on H by X X Zj (H) ⊗ c∗j , d∗H := Zj (H)∗ ⊗ cj . (II.10) dH := j∈ΛL

j∈ΛL

Adding these two, we obtain the Dirac operator, X DH := dH + d∗H = Zj (H) ⊗ c∗j + Zj (H)∗ ⊗ cj .

(II.11)

j∈ΛL

Squaring DH yields the Witten Laplacian ∆H , and using d2H = (d∗H )2 = 0 and the CAR (II.4), one easily sees that X 2 X 2 00 ∆H := DH = Zj (H)∗ Zj (H) ⊗ 1 + Hi,j (x) ⊗ c∗i cj . (II.12) β j∈ΛL

i,j∈ΛL

In Lemma B.1 in the Appendix, we prove that DH is essentially self-adjoint on C0∞ (R|ΛL | ) ⊗ Ff [C|ΛL | ]. Moreover, we show in Lemma B.3 that (II.13) Ker{DH } = Ker{∆H } = C · e−βH ⊗ Ω = Ran{P0 ⊗ PΩ } .

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We observe that ∆H leaves the N -fermion sectors invariant, ∆H : H(N) → H(N) , e (N) . Recalling the definitions and we denote the restriction of ∆H to H(N) by ∆ H (1) (0) (I.22) and (I.24) of ∆H and ∆H , we find that e (0) = I (0) ◦ ∆(0) ◦ (I (0) )−1 , ∆ H H

e (1) = I (1) ◦ ∆(1) ◦ (I (1) )−1 , ∆ H H

(II.14)

e (1) and ∆(0) , ∆(1) anymore. e (0) , ∆ and henceforth we do not distinguish between ∆ H H H H Equation (II.13) implies that DH is invertible on Ran{P0 ⊗ PΩ }. Thus we have 1 = DH (∆H )−1 DH on Ran{P0 ⊗ PΩ }, and using this identity in Eqn. (II.9), we obtain (II.15) ETL,β (u ; v) =

−βH −1 −βH DH P0 ⊗ PΩ e u ⊗ Ω ∆H DH P0 ⊗ PΩ e v⊗Ω H. Furthermore, an easy computation shows that 1 X −βH e ∂j u ⊗ c∗j Ω ∈ H(1) . DH P0 ⊗ PΩ e−βH u ⊗ Ω = β

(II.16)

j∈ΛL

We may hence restrict ∆H to H(1) in (II.15), and inserting (II.16), we arrive at X 1 X −βH (1) −1 ETL,β (u ; v) = 2 e ∂j u ⊗ c∗j Ω ∆H e−βH ∂j v ⊗ c∗j Ω H(1) , β j∈ΛL

j∈ΛL

(II.17)

which is equivalent to (I.27).

III Summable weights In this section, we show that the operators 2 X (0) 00 Hi,j (x) ⊗ c∗i cj ∆H := ∆H ⊗ 1 + β

and

(III.1)

i,j∈ΛL

(0)

A := ∆H ⊗ 1 +

2 X 00 Hi,j (0) ⊗ c∗i cj β

(III.2)

i,j∈ΛL

are close to each other, even after deformation by suitable weights. This will be important in Sections IV, VI, and Appendix B. In particular, the main result of this section, Theorem III.1 below, implies Theorem I.4 by restricting the asserted bound to the invariant subspace H(1) ⊆ H. Theorem III.1. Assume Hypotheses 1, 2, and 3[0]. Then there exist constants C ≥ 0 and β0 ≥ 0, both independent of L, such that, for all β ≥ β0 , C C (III.3) 1 − 1/2 A ≤ ∆H ≤ 1 + 1/2 A β β holds in the sense of quadratic forms on Q ⊆ H, the form domain of A and ∆H .

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Theorem III.1 is a simple consequence of Theorem III.3 below, in which we derive a bound on the difference between ∆H and A relative to A. We observe that this difference, ∆H − A, is given by 2β −1 W , where W is the operator of multiplication by the x-dependent matrix X β (1) 00 00 W = ∆H − A0 = Hi,j (x) − Hi,j (0) ⊗ c∗i cj . (III.4) 2 i,j∈ΛL

Theorem III.3 below gives an estimate on W conjugated by a diagonal xindependent matrix of weights, and Theorem III.1 follows in the special case where the weights are all equal to 1. As a preparation, we develop some estimates of independent interest. Recall by (I.24) that, for all j ∈ ΛL , Zj (H)∗ Zj (H) = −

0 2 1 00 1 ∂2 00 Hj (x) − Hj,j (x) ≥ Hj0 (x) 2 − 1 Hj,j + (x). 2 β 2 ∂xj β β (III.5)

We improve this inequality in the following lemma. Lemma III.2. Under Hypotheses 1, 2, and 3[0], for any β0 > 3C(H3) (0) and β ≥ β0 , we can find a β-dependent matrix (Cj,k (β))j,k∈ΛL with nonnegative entries, satisfying o o nX nX 3C(H3) (0) −1 max Cj,k (β) + max Cj,k (β) ≤ 3C(H3) (0) 1 − , j∈ΛL k∈ΛL β j∈ΛL

k∈ΛL

(III.6) uniformly in L, such that, for β ≥ β0 , j ∈ ΛL , and x ∈ R|ΛL | , X 0 2 Hj (x) ≤ δjk + β −1 Cj,k (β) Zk (H)∗ Zk (H) + Cβ −1 ,

(III.7)

k∈ΛL

where C := λmax + C(H3) (0)/2. Proof. By means of (III.5) and (I.15), we obtain the estimate 0 2 Hj (x)

2 1 00 1 00 1 00 00 ≤ Hj0 (x) − Hj,j (x) + Hj,j (0) + Hj,j (x) − Hj,j (0) β β β 1 00 ∗ ≤ Zj (H) Zj (H) + Hj,j (0) (III.8) β 2 1 X 1 X ajj (k) Hk0 (x) + bjj (k) Hk0 (x) . + β β k∈ΛL

k∈ΛL

Next, using 0 Hk (x) ≤ 1 Hk0 (x) 2 + , 0 < < ∞ , 2 2

(III.9)

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with = 1, and thanks to Hypothesis 2, we get 0 2 Hj (x)

≤ Zj (H)∗ Zj (H) +

1 1 X ajj (k) λmax + β 2β k∈ΛL

2 1 X 1 + ajj (k) + bjj (k) Hj0 (x) , β 2

(III.10)

k∈ΛL

which is equivalent to X k∈ΛL

2 1 1 X δj,k − Mj,k Hj0 (x) ≤ Zj (H)∗ Zj (H) + λmax + ajj (k) , β 2β k∈ΛL

(III.11) where Mj,k := ajj (k)/2 + bjj (k). In view of Hypothesis 3[0], bounded by C(H3) (0) and o o X X Mj,k + max Mj,k ≤ 3 C(H3) (0) , max j∈ΛL

k∈ΛL

k∈ΛL

P k∈ΛL

ajj (k) is

(III.12)

j∈ΛL

so that the L(lp ; lp )-norm of the matrix M ≡ (Mj,k )j,k∈ΛL is bounded by 3C(H3) (0), uniformly in L and p ∈ [1; ∞]. Then, for β0 > 3C(H3) (0) and β ≥ β0 , setting 1 := (δj,k )j,k∈ΛL , −1 = δj,k + β −1 Cj,k (β) , (III.13) 1 − β −1 M j,k

where the matrix C(β) := (Cj,k (β))j,k∈ΛL satisfies (III.6). Multiplying (III.11) by the nonnegative numbers δjk + β −1 Cj,k (β) and summing over j ∈ ΛL , we arrive at (III.7), with C = λmax + C(H3) (0)/2. Theorem III.3. Let S be a semi-norm on Rd and assume Hypothesis 3[dS ] in addition to Hypotheses 1 and 2, where dS is the corresponding semi-metric on ΛL . For weights θ : ΛL −→ R satisfying ∀j, k ∈ ΛL : we denote W (θ) := W (x, θ) :=

X

|θ(j) − θ(k)| ≤ dS (j − k) ,

(III.14)

00 00 eθ(i)−θ(j) Hi,j (x) − Hi,j (0) ⊗ c∗i cj .

(III.15)

i,j∈ΛL

Then, there exist C, β0 > 0, such that, uniformly in L, θ, and β ≥ β0 , for all u, v ∈ C0∞ (R|ΛL | ; Ff [C|ΛL | ]),

hu| W (θ) viH ≤ C β 1/2 (A(1) )1/2 u (A(1) )1/2 v , (III.16) H H where H = L2 (R|ΛL | ; Ff [C|ΛL | ]).

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We remark that on H(1) , W (θ) := eθ W e−θ is the conjugation of W by the diagonal matrix eθ := (δjk eθ(j) )j,k∈ΛL (i.e., e−θ := (eθ )−1 ). Proof of Theorem III.3. First, we control the perturbation W (θ) by a diagonal 00 perturbation. To this end, we fix x ∈ R|ΛL | , denote Xi,j = Xj,i := |Hi,j (x) − 00 (0)|, F := Ff [C|ΛL | ], and we observe that, pointwise in x ∈ R|ΛL | , Hi,j X hu(x)| W (x, θ) v(x)iF ≤ eθ(i)−θ(j) Xi,j kci u(x)kF kcj v(x)kF i,j∈ΛL

f (x, θ) u(x)i1/2 hv(x)| W f (x, θ) v(x)i1/2 , ≤ hu(x)| W F F where we use the Cauchy-Schwartz inequality, Eqn. (III.14), and XX f (x, θ) := W exp[dS (i − j)] Xi,j ⊗ c∗i ci . i∈ΛL j∈ΛL

Due to Hypothesis 3[dS ], we have Xi,j ≤ thus, for any 0 < ε ≤ 1/2, f (x, θ) ≤ W XX i∈ΛL j∈ΛL

(III.17)

(III.18)

P 0 0 2 k aij (k)|Hk (x)| + bij (k)|Hk (x)| and

(III.19) o n 1 ε edS (i−j) aij (k) + bij (k) |Hk0 (x)|2 + aij (k) ⊗ c∗i ci . 2ε 2

Next, we pass from F to H = L2 (R|ΛL | )⊗F. We abbreviate Qij := δij +β −1 C(β)i,j , and we insert (III.7) and 0 ≤ c∗i ci ≤ 1 into (III.19). This yields nQ X X kl f (θ) ≤ W edS (i−j) aij (k) + bij (k) Zl (H)∗ Zl (H) + Cβ −1 2ε i∈ΛL j,k,l∈ΛL o ε + δkl aij (k) ⊗ c∗i ci (III.20) 2 X X 1 ≤ edS (i−j) Qkl aij (k) + bij (k) Zl (H)∗ Zl (H) ⊗ 1 2ε l∈ΛL i,j,k∈ΛL ε C X X dS (i−j) + e Qkl aij (k) + bij (k) ⊗ c∗i ci . + 2 βε i∈ΛL j,k,l∈ΛL

Then (III.6), (I.16), and the choice ε := β −1/2 give

f (θ) ≤ C 0 β 1/2 ∆(0) ⊗ 1 + β −1 1 ⊗ NL , W H

(III.21) 0 −1 where C := (1 + C (0) + λ ) C (d ) 1 + C (0)(β − C (0)) , and max S (H3) (H3) (H3) (H3) P NL := i∈ΛL c∗i ci is the number operator. On the other hand, 2 X 2 λmin (0) (0) 00 A = ∆H ⊗ 1 + Hi,j (0) ⊗ c∗i cj ≥ ∆H ⊗ 1 + 1 ⊗ NL , β β i,j∈ΛL

(III.22) and we arrive at the claim.

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IV Exponential decay of the two-point function In this section, we give a direct proof of exponential decay of the two-point function ETL,β (xj ; xk ) under least requirements. For the derivation of its asymptotics in Section VII, however, our assumptions will be somewhat stronger. As in [8, 10, 14], we use a Combes-Thomas type argument to prove Theorem I.6, i.e., the estimate (I.34) for some C > 0, T EL,β (xj ; xk ) ≤ C exp − min{µ, ν} d(j − k) , (IV.1) β for β large enough. Proof of Theorem I.6. Starting from (I.31), we introduce diagonal weights of the form used in Section III. Using the notation introduced in Theorem III.3, we write E −1 −βH 1 D (1) ETL,β (xj ; xk ) = 2 e−βH ⊗ e−θ ej ∆H (θ) e ⊗ eθ ek (1) , (IV.2) β H where θ satisfies (III.14), for dS = min(µ; ν)d, and ∆H (θ) := (1 ⊗ eθ )∆H (1 ⊗ e−θ ) = ∆H ⊗ 1 + (1)

(1)

(0)

2 (1 ⊗ eθ )H 00 (x)(1 ⊗ e−θ ) , β (IV.3)

temporarily assuming its invertibility. Using (III.1)–(III.2) and the definition of W (θ) given in Theorem III.3, we can write (1)

∆H (θ) = A(1) +

2 (1) 2 W (θ) + 1 ⊗ B(θ) . β β

(IV.4)

where W (1) (θ) denotes the restriction of W (θ) to H(1) and B(θ) := eθ H 00 (0) e−θ − H 00 (0) .

(IV.5)

By Theorem III.3, we can bound W (1) (θ) by (III.16). Furthermore, X X 00 B(θ)i,j = sup sup (0) eθ(j)−θ(i) − 1 Hi,j i∈ΛL

i∈ΛL

j∈ΛL

≤

sup i∈ΛL

j∈ΛL \{i}

X

00 e|θ(j)−θ(i)| Hi,j (0) ,

(IV.6)

j∈ΛL \{i}

and interchanging i and j, we convince ourselves that a similar estimate holds P for supj i B(θ)i,j . Since θ satisfies (III.14) for dS = min(µ; ν)d and H satisfies Hypothesis 4[min(µ; ν)d], we see that, setting C := C(H4) (min(µ; ν)d), kB(θ)k ≤

00 λmin (1 − C) sup Hi,i (0) ≤ λmin (1 − C) . λmax i∈ΛL

(IV.7)

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77

we obtain

(1) −1/2 2

(A ) 1 ⊗ B(θ) (A(1) )−1/2 k ≤ (1 − C) , β

(IV.8)

uniformly in L. Thus, for β0 large enough and β ≥ β0 , we can explicitly construct (1) the resolvent of ∆H (θ) as X ∞ −1 −1/2 (1) (1) −1/2 n A(1) = A Q(θ) , (IV.9) ∆H (θ) n=0

where Q(θ) =

−1/2 −1/2 2 A(1) W (θ) + 1 ⊗ B(θ) A(1) β

(IV.10)

is a bounded operator of norm less than 1 − C + O(β −1/2 ). Hence, for β0 > 0 suffi−1 (1) 2 is bounded by 1+C k(A(1) )−1/2 k2 , ciently large, the operator norm of ∆H (θ) which, in turn, is bounded by β/λmin (1 + C). Inserting this bound in (IV.2), we obtain T 1 EL,β (xj ; xk ) ≤ eθ(k)−θ(j) , (IV.11) βλmin (1 + C) uniformly w.r.t. L and β ≥ β0 , and for θ satisfying (III.14) with dS = min(µ; ν)d. Now we choose θ = θk defined by ∀l ∈ ΛL , θk (l) = min(µ; ν) d(l − k) .

(IV.12)

Thanks to the triangle inequality, θj satisfies (III.14) for dS = min(µ; ν)d, and (IV.11) for θ = θk yields (IV.1), i.e., (I.13).

V The Feshbach operator In this section, we introduce a suitable Feshbach operator. The Feshbach map is a key tool in Sections VI and VII and some properties derived in this section and concerning this Feshbach operator are used in Section VI. Let P := | e−βH ihe−βH | ⊗ 1 and P = 1 − P .

(V.1) (0)

Note that P is the orthogonal projection onto the ground state of ∆H ⊗ 1. So we expect that (1)

∆H

(1)

:= P ∆H P ,

(V.2)

restricted to RanP , has a much larger spectral gap above zero than 2λmin /β, which (1) is the spectral gap of ∆H . Indeed, in the following proposition, we show that the (1)

(1)

spectral gap of ∆H is almost twice as big as the one of ∆H .

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Proposition V.1. Under Hypotheses 1,2, and 3[0] and for β large enough, (1)

∆H

≥

4 λmin 1 − O(β −1/2 ) P . β

(V.3)

Proof. According to Lemma B.5 (see also [14, 12]), we have that (0) (1) P ∆H ⊗ 1 P ≥ inf σ(∆H ) P , (1)

(V.4)

(1)

where σ(∆H ) denote the spectrum of ∆H . Using this and (I.28), we therefore obtain (1)

P ∆H P

≥ ≥

C P A(1) P β 1/2 2 C (1) 1 − 1/2 inf σ(∆H ) + λmin P , β β

1−

by (I.23) and Hypothesis 2. Using (I.28) again, this leads to C C 2 (1) 1 − 1/2 + 1 λmin P , P ∆H P ≥ 1 − 1/2 β β β

(V.5)

(V.6)

which proves (V.3). (1)

The Feshbach operator of ∆H associated to P is defined by FP

−1 (1) (1) (1) (1) := P ∆H P − P ∆H P ∆H P ∆H P = (0)

(V.7) −1 2 00 2 4 (1) H (0) P + P W (1) P − 2 P W (1) P ∆H P W (1) P , β β β (1)

(1)

since ∆H P = 0. Thanks to the invertibility of ∆H on RanP , the inverse of ∆H is given by −1 −1 (1) −1 (1) (1) (1) (1) −1 P − P ∆H P ∆H ∆H = P − P ∆H P ∆H P FP P −1 (1) P + P ∆H

(V.8)

(see [1]). If we insert (V.8) in (I.31), we obtain D E ETL,β (xj ; xk ) = e−βH ⊗ ej FP−1 e−βH ⊗ ek

H(1)

,

(V.9)

since P e−βH ⊗ el = 0, for l = j, k. So we do not loose any information if we (1) replace ∆H by FP in (I.31). Furthermore, FP is related to the Hessian at 0 in the following way.

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Proposition V.2. Assume Hypotheses 1,2, 3[0]. For β large enough, we have

FP − 2 H 00 (0) P = O(β −3/2 ) , β

(V.10)

where the O-symbol is uniform in L → ∞ and β → ∞. Proof. Using the fact that P = (A(1) )−1/2 P ( β2 H 00 (0))1/2 , we obtain the estimate 2 2 kP W (1) P k ≤ λmax · C β −1/2 , β β

(V.11)

by Hypothesis 2 and Theorem III.3. Thanks to (V.3), we also have

−1 4 2 C2 β

(1) (1) (1) P W P ∆ P W P ≤ λmax 1 − O(β −1/2 ) .

H β2 β2 4λmin (V.12)

Using (V.7), the estimates (V.11) and (V.12) imply (V.10).

VI Analysis of the support function In this section, we analyze the support function Srw which we identify in Section VII to be the rate of exponential decay of the two-point function ETL,β (xj , xk ). Throughout this section we require Hypotheses 1, 2, 5, 6 and Hypothesis 3[Srv ], for some 1 < r < 2 − Fv (0). We note that 1 − Fv (0) > 0, thanks to (I.37)–(I.38). We recall that π : Rn → (R/LZ)n is the canonical projection and that J : (R/LZ)n → [−L/2 , L/2)n is the natural identification map from the torus to the fundamental domain. In particular, (box) J is a bijection from ΛL to ΛL := Zn ∩ [−L/2 , L/2)n . Another natural bijection is provided by JA : ΛL → Ae1 , where JA is de−1 termined by JA = π|Ae1 and Ae1 is described in Lemma I.7(iv). Using JA instead of J amounts to projecting onto the fundamental domain Ae1 rather than [−L/2 , L/2)n . The difference between these two maps becomes important when studying the two-point function ETL,β (xj , x0 ) for J(j) close to the boundary of (box)

ΛL

. Next, we recall from Eqns. (V.7)–(V.9) that D E ETL,β (xj , xk ) = β −2 e−βH ⊗ ej FP−1 e−βH ⊗ ek ,

(VI.1)

where (1) −1

FP = 2β −1 P ⊗ EL,β (H 00 ) − 4β −2 P W (1) P ∆H

P W (1) P ,

(VI.2)

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and P = |e−βH ihe−βH | ⊗ 1. Thus ETL,β (xj ; xk ) = (2β)−1 Q−1 j,k , where Q is the real, self-adjoint matrix D E (1) −1 00 ) − 2β −1 e−βH ⊗ ej W (1) P ∆H P W (1) e−βH ⊗ ek . Qj.k = EL,β (Hj,k (VI.3) Since H is assumed to be translation invariant, Qj,k = Qj−k,0 . We define a real, even function w : Zn → R by ( 00 (0) if k ∈ Ae1 \ {0}, −Qπ(k),0 /H0,0 (VI.4) w(k) := 0 otherwise. 00 In other words, w ≡ 0 outside Ae1 and solves Qπ(k),0 = H0,0 (0) δk,0 − w(k) inside Ae1 . We remark that w is similar to v, which also vanishes outside Ae1 and solves 00 00 Hπ(k),0 (0) = H0,0 (0) δk,0 − v(k) inside Ae1 . Comparing w to v, we observe that −1 00 00 00 (0) EL,β Hπ(k),0 − Hπ(k),0 (0) (VI.5) w(k) = v(k) − H0,0 D E −1 (1) −1 00 + 2 H0,0 (0)β P W (1) e−βH ⊗ e0 , e−βH ⊗ eπ(k) W (1) P ∆H for any k ∈ Ae1 . Given η ∈ Rn and an even function u : Zn → R of compact support, we define a smooth (in fact, analytic) function Fu ∈ C ∞ (Rn ) by X X Fu (η) := eη·k u(k) = cosh(η · k) u(k) , (VI.6) k∈Zn

k∈Zn

Du (r) :=

η ∈ Rn Fu (η) < r .

and, for any r ∈ R, we set (VI.7)

Then the support function Srv : Rn → [0, +∞) is defined by Sru (z) := sup η · z η ∈ Du (r) .

(VI.8)

Recall from Lemma I.7, which we shall prove at the end of this section, that Fv (η) is strictly convex and that B(C, 0) ⊆ Dv (r) ⊆ B(C 0 , 0), for some constants C, C 0 and all 1 − δ0 ≤ r ≤ 1 + δ0 , provided δ0 < Fv (0). Thus Dv (r) = −Dv (r) is a strictly convex, bounded set with smooth boundary Σv (r) = ∂Dv (r) ⊆ {Fv (η) = r}. Moreover, for any z ∈ Rn \ {0}, there is a unique vector η(z) ∈ Σv (r) such that η(z) = λ z, for some λ > 0, and this vector realizes the supremum in (VI.8), Srv (z) = η(z) · z. Moreover, an easy computation shows that ∇z Srv (z) = η(z). Our goal in this section is to prove the following theorem.

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Theorem VI.1. Assume Hypotheses 1, 2, 5, 6 and Hypothesis 3[Sru ], for some 1 < r < 2 − Fv (0). There exist constants C ≥ 0 and β0 which are independent of L, such that, for any β > β0 and η ∈ Dv (r), α ∂η Fw (η) − ∂ηα Fv (η) ≤ C β −1/2 , (VI.9) where α ∈ {0, 1, 2, 3}n is a multiindex. This theorem has several important consequences, obtained from Taylor expansions and (VI.9). We collect these in the following corollary. Corollary VI.2. Assume Hypotheses 1, 2, 5, 6 and Hypothesis 3[Srv ], for some 1 < r < 2 − Fv (0). There exist constants C ≥ 0 and β0 which are independent of β and L, such that, for any β > β0 , 1 − δ0 ≤ r ≤ 1 + δ0 and 0 < δ0 < min{Fv (0), 1 − Fv (0)}, (VI.10) Dv r − Cβ −1/2 ⊆ Dw r ⊆ Dv r + Cβ −1/2 , (VI.11) 1 − Cβ −1/2 Srv ≤ Srw ≤ 1 + Cβ −1/2 Srv . Moreover, Dw (r) is a strictly convex, bounded, open set, with a smooth boundary Σw (r) = ∂Dw (r). We break up the proof of Theorem VI.1 into several lemmata. Our general strategy is to apply a Combes-Thomas argument. To formulate this, we introduce some suitable exponential weights on (R/LZ)n . We construct these weights from a family of uniformly Lipschitz-continuous functions θer : (R/LZ)n → [0, +∞), where 1 − δ0 ≤ r ≤ 1 + δ0 , for some δ0 > 0. That is, we require that, for some constant C which neither depends on r nor L, we have θer (x) − θer (x) ≤ C d(x − y) , (VI.12) for all x, y ∈ (R/LZ)n . Furthermore, we assume that θer (0) = 0 and that, for almost every x ∈ (R/LZ)n , ∇x θer (x) ∈ Dv (r) .

(VI.13)

The main example we bear in the back of our mind is the following. Lemma VI.3. Let θerS : (R/LZ)n → [0, +∞) be defined by the seminorm ˜ ∈ π −1 (x) , x) x θerS (x) := inf Srv (˜

(VI.14)

for 1 − δ0 ≤ r ≤ 1 + δ0 , with δ0 ≤ Fv (0). Then {θerS }1−δ0 ≤r≤1+δ0 is a family of uniformly Lipschitz-continuous functions obeying (VI.12)–(VI.13).

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Proof. We first recall that Srv : Rn → [0, +∞) is a norm, so µr |˜ x| ≤ Srv (˜ x) ≤ (2) (1) (2) n v µr |˜ x|, for some 0 < µr ≤ µr < ∞ and all x ˜ ∈ R . Therefore, Sr (J(x)) ≤ (2) (1) Srv (˜ x), whenever |˜ x| > 2d/2 µr L/µr , and we conclude that the infimum in (VI.14) is actually a minimum, attained for some x˜ in the finite set π −1 (x) ∩ (2) (1) B(2d/2 µr L/µr , 0), (1) ˜ ∈ π −1 (x) ∩ B(2d/2 µ(2) x) x (VI.15) θerS (x) := min Srv (˜ r L/µr , 0) . Next, we observe that Srv ∈ C 1 (Rn \{0}) and ∇x˜ Srv (˜ x) = η(˜ x) ∈ Dv (r) ⊆ B(C 0 , 0), 0 for some C ≥ 0 which is independent of r and L. We conclude that θerS is the minimum over finitely many C 1 functions and, as such, it is Lipschitz-continuous. Indeed, for almost every x ∈ (R/LZ)n , its gradient is given by ∇x θerS (x) = ∇x˜ Srv (˜ x), (2) (1) for some x ˜ ∈ B(2d/2 µr L/µr , 0). Thus, ∇x θerS (x) ∈ Dv (r) ⊆ B(C 0 , 0), for almost every x ∈ (R/LZ)n . R Next, we pick χ ∈ C0∞ (B(1, 0) ; [0, +∞)) such that χ(x)dn x = 1, and we define χR : (R/LZ)n → [0, +∞) by χR (x) := R−d χ(x/R), for R < L/2n . Convolving θer and χR , we obtain a family of smooth functions θr : (R/LZ)n → [0, +∞), θr := θer ∗ χR ,

(VI.16)

having the following important properties: Lemma VI.4. Assume {θer }1−δ0 ≤r≤1+δ0 to be a family of uniformly Lipschitz continuous functions described in (VI.12)–(VI.13), and define θr := θer ∗ χR , as in (VI.16). Then, ∇x θr (x) ∈ Dv (r) ,

(VI.17)

for all x ∈ (R/LZ) . Moreover, there exists a constant C ≥ 0, neither depending on r nor L, such that θr (x) − θer (x) ≤ C R (VI.18) n

and

θr (x) − θr (y) − ∇x θr (y) · J(x − y) ≤ C d(x − y)2 R−1 ,

(VI.19)

for all x, y ∈ (R/LZ)n with d(x − y) ≤ R. Proof. Since the convolution with χR defines a probability measure and Dv (r) is convex, the assertion (VI.17) follows from (VI.13). From the Lipschitz-continuity of θer , we obtain that Z θr (x) − θer (x) ≤ e θr (y) − θer (x) χR (x − y)dn y ≤ C R , (VI.20) and thus (VI.18).

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Finally, (VI.19) follows from a Taylor expansion and the observation that

∂x ∂x θr ≤ ∇x θer ∇x χR ≤ C R−1 . k l ∞ ∞ ∞

(VI.21)

We now come to defining the exponential weights mentioned above. Given a family of functions θr with the properties (VI.17)–(VI.19), we denote by e±θr the |ΛL | ±θr multiplication operators on C , acting as e ϕ (k) := e±θr (k) ϕ(k). Next, we |ΛL | introduce various operators on C and H(1) obtained from conjugating by e±θr ±θr and 1 ⊗ e , respectively: 1 ⊗ eθr W (1) (x) 1 ⊗ e−θr ,

W (1) (θr )

≡

B(θr ) (1) ∆H (θr )

:= e H (0) e − H (0) , (1) θr := 1 ⊗ e ∆H 1 ⊗ e−θr

W (1) (θr , x) := θr

=

00

A0 + 2β

−θr

−1

00

1 ⊗ B(θr ) + W

(VI.22) (VI.23)

(1)

(θr ) .

(VI.24)

We derive various norm estimates on these operators that allow us to analyze Fw , Dw (r), and Srw . We first estimate the norm of 2Re{B(θr )} = B(θr ) + B(−θr ). Lemma VI.5. Let {θr : (R/LZ)n → [0, +∞)}1−δ0 ≤r≤1+δ0 , for some δ0 > 0, be a family of functions with the properties (VI.17)–(VI.19), and assume Hypotheses 2, 5, and 6. Then, for some constant C ≥ 0,

00

Re{B(θr )} ≤ H0,0 (0) eC/R r − Fv (0) .

(VI.25)

Proof. Since Re{B(θr )} is a self-adjoint matrix, we may estimate its norm by

Re{B(θr )} ≤ ≤

00 (0) H0,0

sup

X

i∈ΛL

( Re{B(θr )} )i,j

(VI.26)

j∈ΛL

Xn i o h sup cosh θr (i) − θr (j) − 1 vL (i − j) i∈ΛL

j∈ΛL

Xn i o h 00 ≤ H0,0 (0) sup cosh ∇x θr (i) · J(i − j) + δ(i, j) − 1 vL (i − j) , i∈ΛL

j∈ΛL

where |δ(i, j)| ≤ Cd(i − j)2 R−1 , and the last inequality derives from (VI.19), provided R is chosen larger than C 0 , where C 0 is a constant such that supp{v} ⊆ e −1 where C e := C(C 0 )2 . B(C 0 , 0). Therefore, |δ(i, j)| ≤ CR

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Next, we observe that cosh(a + b) ≤ e|b| cosh(a), for arbitrary a, b ∈ R. Inserting this estimate in (VI.26), we obtain

Re{B(θr )} X e 00 ≤ H0,0 (0) eC/R sup cosh ∇x θr (i) · J(i − j) vL (i − j) − Fv (0) i∈ΛL

j∈ΛL

n o e 00 (0) eC/R sup Fv ∇x θr (i) − Fv (0) ≤ H0,0 i∈ΛL e C/R 00 ≤ H0,0 (0) e r − Fv (0) ,

(VI.27)

since ∇x θr (i) ∈ Dv (r), for all i ∈ ΛL , by (VI.17).

(1)

Next we need the following estimate on the norm of the resolvent of ∆H (θr ) (1) := P ∆H (θr ) P on RanP , relative to A(1) . Lemma VI.6. Let {θr : (R/LZ)n → [0, +∞)}1−δ0 ≤r≤1+δ0 , for some δ0 > 0, be a family of functions with the properties (VI.17)–(VI.19), and assume Hypotheses 2, 5, and 6. Then, for any r < 2 − Fv (0), there exist constants C, β0 ≥ 0, independent of L, such that, for any β ≥ β0 and R ≥ C,

−1 1/2

(1) 1/2

(1) ∆H (θr ) P ≤ C. (VI.28) A(1)

A Proof. We recall form (IV.4) and (VI.22)–(VI.24) that (1)

∆H (θr ) = A(1) +

2 (1) 2 W (θr ) + 1 ⊗ B(θr ) , β β

(VI.29)

and hence, using Theorem III.3 and the fact that A(1) P ≥ 4λmin β −1 P (see Proposition V.1), we obtain o 1/2 n (1) 1/2 Re ∆H (θr ) A(1) P (VI.30) A(1) 1

Re B(θr ) − Cβ −1/2 · P . ≥ 1− 2λmin 00 Since, according to (I.37), λmin = H0,0 (0)(1 − Fv (0)), Lemma VI.5 yields that

−1 1 e

Re B(θr ) ≤ 1 eC/R r − Fv (0) 1 − Fv (0) . 2λmin 2

(VI.31)

e

By assumption, r < 2 − Fv (0), and thus also eC/R r < 2 − Fv (0), provided R is sufficiently large. Hence, for some small positive constants c, c0 , we have that n o (1) (VI.32) Re (A(1) )1/2 ∆H (θr ) (A(1) )1/2 P ≥ c − Cβ −1/2 ≥ c0 · P , proving the claim.

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Proof of Theorem V I.1. First, we recall that the family {θrS : (R/LZ)n → [0, +∞)}1−δ0 ≤r≤1+δ0 ,

for some δ0 > 0,

is a family of functions with the properties (VI.17)–(VI.19), according to Lemmata VI.3 and VI.4. Since we have furthermore assumed Hypotheses 2, 5, 6, and δ0 < min{Fv (0), 1 − Fv (0)}, we may apply Lemma VI.6. We first recall from Eqn. (VI.5) that E −1 D −βH 2 00 e (VI.33) (0) ⊗ ek W (1) e−βH ⊗ e0 w(k) − v(k) = H0,0 β E −1 D −βH 4 (1) −1 00 − 2 H0,0 (0)β ⊗ ek W (1) P ∆H P W (1) e−βH ⊗ e0 , e β S

for any k ∈ ΛL . Using the conjugation by eθr , we may rewrite the terms in Eqn. (VI.33) as D E D E S e−βH ⊗ ek W (1) e−βH ⊗ e0 = e−θr (k) e−βH ⊗ ek W (1) (θr ) e−βH ⊗ e0 , (VI.34) and similarly E D (1) −1 P W (1) e−βH ⊗ e0 = (VI.35) e−βH ⊗ ek W (1) P ∆H D E −1 S (1) P W (1) (θr ) e−βH ⊗ e0 . e−θr (k) e−βH ⊗ ek W (1) (θr ) P ∆H (θr ) Using these two identities and the fact that

2

2

00 1/2 −βH (0) 2 H0,0

(1) 1/2 −βH e ⊗ ek = A(1) e ⊗ ek = ,

A β

(VI.36)

we obtain from (VI.33) that

−1/2 (1) −1/2

eθr (k) w(k) − v(k) ≤ 2β −1 A(1) W (θr ) A(1)

2 −1 (1) 1/2 −1/2 (1) −1/2 (1) 1/2 (1)

+4β −2 A(1) W (θr ) A(1) ∆H (θr ) P A

A ≤ Cβ −1/2 + Cβ −1 ≤ 2Cβ −1/2 ,

(VI.37)

for any 1 − δ0 ≤ r ≤ 1 + δ0 , provided that δ0 < min{Fv (0), 1 − Fv (0)}. Here we use Lemma VI.6 and Theorem III.3 to derive the second inequality. To make use of (VI.37), we recall that according to Lemma I.7(iii) there exist constants C, C 0 ≥ 0 such that, for any 0 < ε < 1, v (1 + ε)S(1−Cε)r ≤ Srv ≤ (1 + ε)Srv exp(−C 0 ε) ,

(VI.38)

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provided that 1 − δ0 ≤ r ≤ 1 + δ0 . Thus, for any η ∈ Dv (r), 0 < ε < 1, and any multiindex α ∈ {0, 1, 2, 3}n , we have the estimate X α ∂η Fw (η) − ∂ηα Fv (η) ≤ exp[η · k] w(k) − v(k) |k||α| X

≤

e1 ∩Zn k∈A

X

≤

e1 ∩Zn k∈A

exp[Srv (k)] w(k) − v(k) |k||α| exp (1 + C(r − 1))S1v (k) w(k) − v(k) |k||α|

e1 ∩Zn k∈A

≤

X

exp −εS1v (k) |k||α|

(VI.39)

e1 ∩Zn k∈A

sup

e1 ∩Zn k∈A

n o exp (1 + C(r − 1) + ε)S1v (k) w(k) − v(k) .

The important point about using Ae1 rather than [−L/2, L/2)n for the definition of w is that on Ae1 we have S1v = θe1S . Moreover, by (VI.38) and (VI.18), there exists a constant C 0 ≥ 0 such that (1 + C(r − 1) + ε)S1v (k) = minn (1 + C(r − 1) + ε)S1v (k + qL) q∈Z v S ≤ minn S(1+C = θe(1+C 0 (r−1+ε)) (k + qL) 0 (r−1+ε)) (k) q∈Z

S 0 ≤ θ(1+C 0 (r−1+ε)) (k) + C .

(VI.40)

Thus, if r > 1 is sufficiently small such that 1 + C 0 (r − 1 + ε) < 1 + δ0 , we obtain n o α S w(k) − v(k) , (VI.41) ∂η Fw (η) − ∂ηα Fv (η) ≤ Cε,α sup exp θ1+δ (k) 0 e1 ∩Zn k∈A

which is bounded by C β −1/2 , due to Eqn. (VI.37). This proves (VI.9).

We close this section with a Proof of Lemma I.7. (i) First, H¨older’s inequality implies that, for η, η0 ∈ Rn and α ∈ (0, 1), X α 1−α 0 Fv αη + (1 − α)η0 = eη·m v(m) · eη ·m v(m) k∈Zn \{0}

≤ Fv (η)α · Fv (η 0 )1−α ,

(VI.42)

and hence convexity of ln Fv . To obtain strict convexity, we observe that equality holds in (VI.42) if and only if there is a constant µ > 0 such that eη·k v(k) = 0 µ eη ·k v(k), for all k ∈ Zn . But since Gr(supp{v}) = Zn , by Hypothesis 6, this is equivalent to η = η 0 . Since ln Fv is strictly convex, so is Fv .

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Again by Hypothesis 6, there exists a set of linearly independent vectors {k1 , . . . , kn } ⊆ Zn and a constant δ > 0, such that v(kν ) ≥ δ, for all 1 ≤ ν ≤ n. Thus, there exist a constant C such that max1≤ν≤n |η · kν | ≥ C|η|, and we observe that, for ε > 0, h i 1 Fv (1 + ε)η − Fv η ≥ δ cosh (1 + ε)Cη − cosh Cη ≥ δC|η|ε , 2 (VI.43) implying the first inequality in (I.42). The second inequality in (I.42) follows from cosh(a + b) ≤ cosh(a) e|b| and the boundedness of supp{v}. 2 (ii) Similarly to (VI.43), we obtain p that Fv (η) − Fv (0) ≥ C|η| , for some constant C > 0 and thus Dv (r) ⊆ B( (r − Fv (0))/C , 0), for any r ≥ 1 − δ0 > Fv (0). Conversely, since Fv (η) ≤ eC|η| Fv (0) ≤ r, for any r ≥ 1 − δ0 > Fv (0), provided that |η| is sufficiently small, we have B(C, 0) ⊆ Dv (r), for some C > 0. Smoothness and strict convexity of Dv (r) follows from (i). (iii) The support function Srv obviously defines a semi-norm on Rn , for each 1 − δ0 ≤ r ≤ 1 + δ0 , and, by additionally using Hypothesis 6, one checks that Srv (x) = 0 implies x = 0. The vector η(x) is the solution of the Euler-Lagrange equation ∇η Fv (η(x)) = µx for Srv , and ∇x Srv (x) = η(x) follows from differentiating Fv (η(x)) = r. The estimate (I.44) follows from Eqn. (I.42) with the additional observation that we assume |η| ≥ C for some C > 0 in this estimate. For example, Fv (1 + 0 00 ε)η ≤ eC |η|ε Fv (η) implies that Dv (r) ⊆ (1+ε)Dv e−C ε r which, in turn, implies v that Srv ≤ (1 + ε)Sexp(−C 00 ε)r . (iv) From the definition of Ar it is immediate that there is a fundamental S domain (Ar )◦ ⊂ Aer ⊆ Ar for π. That is, Rn = q∈Zn Aer + qL, and Aer + qL ∩ Aer + q 0 L = ∅, for q 6= q 0 , q, q 0 ∈ Zn . Next, we note that since Srv (k) is a norm on Rn , there exist two constants 0 < µr1 ≤ µr2 such that µr1 |k| ≤ Srv (k) ≤ µr2 |k|. Thus, for given x ∈ [−L/2, L/2)n and q ∈ Zn \ {0}, we have the estimate Srv (x + qL) − Srv (x) ≥ µr1 |q| L − 2µr2 /µr1 |x| . (VI.44)

Therefore, B µr1 L/2µr2 , 0 ⊆ Ar ⊆ B 2µr2 L/µr1 , 0 . Finally, to prove that Ar is star-shaped, we notice that, for λ > 0, x ∈ Rn , and q ∈ Zn \ {0}, d = η(λx) − η(λx + qL) · λx (VI.45) λ Srv (λx) − Srv (λx + qL) dλ = Srv (λx) − η(λx + qL) · λx > 0 . Since Srv (0) − Srv (qL) < 0, Eqn. (VI.45) proves that Ae is star-shaped.

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VII Asymptotics of the two-point function This section is devoted to the proof of Theorem I.9. We recall from Sections IV, Eqns. (V.7)–(V.9) and VI, Eqn. (VI.1) that, for j, k ∈ ΛL , the two-point function is given by D E (VII.1) ETL,β (xj , xk ) = β −2 e−βH ⊗ ej FP−1 e−βH ⊗ ek . Thus, Fourier’s inversion formula gives ETL,β (xi , xj ) =

X

1 2β

00 (0) Ln H0,0 ξ∈Λ∗ L

exp[ihξ, i − ji] , 1 − w J(i − j)

(VII.2)

where w is defined in (VI.5), and Λ∗L = 2π(Z/LZ)n is the dual lattice to ΛL . A Poisson formula, derived in [14, Eqn. (4.32)], yields ETL,β (xi , xj ) =

1

2β

X

Ew (k) 00 (0) (2π)n H0,0 k∈π−1 (i−j)

where

,

(VII.3)

Z

eiξ·k dn ξ , (VII.4) Tn 1 − Fw (iξ) P and we recall from (VI.6) that Fu (η) = k∈Zn cosh(η · k) u(k), so Fu (i · ) is the Fourier transform of u (u having bounded support), X X X Fu (iξ) = cosh(iξ · k) u(k) = cos(ξ · k) u(k) = e−iξ·k u(k) . Ew (k) =

k∈Zn

k∈Zn

k∈Zn

(VII.5) In [14] the following asymptotics was shown to hold for Ev . Lemma VII.1. Assume Hypothesis 6. Then there exists C0 > 0 such that, for any |k| ≥ C0 , we have h i n−3 ∇η Fv (ηv (k)) 2 exp −S1v (k) . (VII.6) Ev (k) = 1 + O |k|−1 n−1 2 2 F (η (k))] 1/2 2π |k| det[∂⊥ v v Theorem VI.1 ensures that Fv and Fw and their derivatives only differ by terms of order O(β −1/2 ), and using this, we show Theorem VII.2 below. We note that while both Theorem VI.1 and Lemma VII.1 require Hypothesis 6, particularly v ≥ 0 and Gr suppv = Zn , it is possible to derive an asymptotic formula for Ev like (VII.6) also for some cases of v without assuming its positivity. We make the requirements that substitute for Hypothesis 6 more precise in Appendix C.

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Theorem VII.2. Assume Hypotheses 1, 2, 5, 6 and Hypothesis 3[Srv ], for some 1 < r ≤ 2 − Fv (0). Then there exist β0 > 0 and C0 > 0 such that, for any β > β0 and |k| ≥ C0 , we have Ew (k) = 1 + O |k|−1 + β −1/2 (VII.7) h i n−3 ∇η Fv (ηv (k)) 2 exp − 1 + O(β −1/2 ) S1v (k) . n−1 2 2 F (η (k))] 1/2 2π |k| det[∂⊥ v v In view of the Poisson summation formula (VII.3), Theorem VII.2 is the main ingredient for the proof of Theorem I.9, and we demonstrate below how Theorem I.9 derives from it. Proof of Theorem I.9. First, fixing 0 < λ < 1, the homogeneity of Srv implies the existence of a constant c > 0 such that S1v (x) ≤ S1v (x + qL) − c|q|L, for all x ∈ λAe and q ∈ Zn \ {0}. Thus, by Theorem VII.2, for sufficiently large β, we have (VII.8) Ew (x + qL) ≤ exp −ε L |q| Ew (x) , for some ε > 0. Hence, we obtain ETL,β (xj , x0 ) = = and thus Theorem I.9.

1 00 (0) (2π)n 2β H0,0

X

E(k)

(VII.9)

k∈π−1 (j)

1 −εL E J (j) , 1 + O e A 00 (0) (2π)n 2β H0,0

Proof of Theorem V II.2. We first recall that Fw is a finite sum of exponentials and thus entire. Moreover, Theorem VI.1 implies that (VII.10) Re Fw (η + iξ) ≤ Fv (η) + Cβ −1/2 < 1 , for all η + iξ ∈ Dv (r) + iTn , r < 1, and β sufficiently large. Thus the integrand in (VII.4) is regular, and we may deform the contour C0 := 0 + iTn into n o C1 := (1 − ε + τ ξ 2 )ηw (k) + iξ ∈ Rn + iTn ξ ∈ Tn , (VII.11) where ηw (k) ∈ Σw (1) is the unique vector such that ηw (k) · k = S1w (k). Here, τ, ε > 0 are two sufficiently small numbers chosen below (independently of β, k, and L). Moreover, we identify the Torus Tn with the rectangular domain [−π, π)n . Thus we obtain a new representation for Ew (k) by the following integral, Z exp (1 − ε + τ ξ 2 )ηw (k) · k + iξ · k dn ξ . (VII.12) Ew (k) = 1 − Fw (1 − ε + τ ξ 2 )ηw (k) + iξ [−π,π)n We introduce χ ∈ C0∞ (Rn ; [0, 1]), χ ≡ 1 on B(1/2, 0), and suppχ ⊆ B(1, 0). We denote χ ¯ := 1 − χ.

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Lemma VII.3. For any N > 0 there exists constants CN and αN such that Z ξ exp(1 − ε + τ ξ 2 )η (k) · k + iξ · k dn ξ w χ ¯ lim ε→0 [−π,π)n R 1 − Fw (1 − ε + τ ξ 2 )ηw (k) + iξ ≤ CN exp[−S1w (k)] |k|−N(VII.13) ,

where R > 0 is defined by R2 := αN max β −1/2 , |k|−1 ln |k| }.

Proof. We first recall from Hypothesis 6 that v(kν ) ≥ δ > 0, for some linearly independent {k1 , . . . , kn }. Hence, we have n X

v(kν ) cosh(η · kν ) cos(ξ · kν )

(VII.14)

ν=1

≤ ≤

n X ν=1 n X

v(kν ) cosh(η · kν ) −

n o δ min |ξ · kν |2 2 1≤ν≤n

v(kν ) cosh(η · kν ) − δ 0 ξ 2 ,

ν=1

for some δ 0 > 0, and we thus obtain < Fv (η + iξ) ≤ Fv (η) − δ 0 ξ 2 . Next, we observe that η˜ := (1 − ε − τ ξ 2 )ηw (k) ∈ Dw 1 + Cτ ξ 2

(VII.15)

⊆ Dv 1 + Cτ ξ 2 + C 0 β 1/2 , (VII.16)

for some constants C and C 0 , by Theorem VI.1 and Corollary VI.2. Furthermore we remark that ξ 2 ≥ R2 /16 ≥ (αN /16)β −1/2 . Using this observation, (VII.15), and (VII.16), we get the following estimate, η + iξ) ≤ Fv (˜ η + iξ) + C β −1/2 ≤ Fv (η) − δ 0 ξ 2 + C β −1/2 Re Fw (˜ δ0 2 ξ , ≤ 1 − (δ 0 − C 0 τ − 16C 0 /αN )ξ 2 ≤ 1 − (VII.17) 2 provided τ and 1/αN are sufficiently small. We insert this estimate and S1w (k) ≥ C|k| into (VII.13), which yields Z ξ exp(1 − ε + τ ξ 2 )η (k) · k + iξ · k dn ξ w χ ¯ R n 1 − F (1 − ε + τ ξ 2 )ηw (k) + iξ w [−π,π) Z C exp[− C |k| ξ 2 ] dn ξ ≤ exp[−(1 − ε)S1w (k)] √ ξ2 R/2≤|ξ|≤ nπ ≤ CN exp[−(1 − ε)S1w (k)] |k|−N ,

(VII.18)

as one easily verifies using the properties of R and choosing αN sufficiently large.

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Proof of Theorem V II.2 (continued). By Lemma VII.3 and (VII.12), we have Z ξ exp(1 − ε + τ ξ 2 )η (k) · k + iξ · k dn ξ w Ew (k) = χ R n 1 − F (1 − ε + τ ξ 2 )ηw (k) + iξ w [−π,π) + exp[−(1 − ε)S1w (k)] O |k|−N , (VII.19) where R2 := αN max β −1/2 , |k|−1 ln |k| }. Since |ξ| ≤ 2R 1 on the support of the integrand in (VII.19), we can now find a smooth change of coordinates ξ 7→ φ(ξ) = ξ + O(|ξ|2 ), ∂j φk (ξ) = δjk + O(|ξ|), and ∂ α φ(ξ) = O(|ξ|), for |α| ≥ 2, such that 1 − Fw (1 − ε + τ |ξ|2 )ηw (k) + iξ = 1 − Fw (1 − ε)ηw (k) + i∇η Fw (1 − ε)ηw (k) · φ(ξ) 1

(VII.20) − φ(ξ) , Fw00 (1 − ε)ηw (k) φ(ξ) . 2 Integrating first along the direction of ∇η Fw (1 − ε)ηw (k) and using the cancellation due to the sign change of the variable, we convert the integral into an absolutely convergent one. Then, using the stationary phase method as in [14] to compute the asymptotics of the oscillatory integral in (VII.19) and taking ε → 0, we arrive at i h Ew (k) = 1 + O |k|−1 (VII.21) exp −S1w (k) ! n−3 ∇η Fw (ηw (k)) 2 −N . 1/2 n−1 + O |k| det[∂η2⊥ Fw (ηw (k))] 2π |k| 2 Now, the claim follows by absorbing the errors made by approximating ∂ α Fw ηw (k) = ∂Fv ηv (k) + O(β −1/2 ) and by choosing N ≥ (n + 1)/2.

Appendix A Admissibility of example I.1[ν] In this appendix, we prove Lemma I.2, i.e., we justify that Example I.1[ν] satisfies Hypothesis 3[µd], for all ν > µ. We recall that in Example I.1[ν] the Hamilton function HL is assumed to be of the form X X f (xj ) + g e−ν d(i−j) wij (xi , xj ) , (A.1) HL (x) = j∈ΛL

i,j∈ΛL

where ν > 0, and g > 0 is sufficiently small, wij = wji , wii = 0, and furthermore, for m = 1 or m = 2, f and {wij }i,j∈ΛL obey Eqns. (I.20). Lemma I.2 is equivalent to the following one.

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√

Lemma A.1. Let Mα := 2n (1 − e−α/ n )−n , for α > 0. Then, for any µ > ν and 0 ≤ g < Mν−3 /24, the Hamilton function HL fulfills Hypothesis 3[µd], with C(H3) = 2 + 12gMν−µ and X aij (k) := bij (k) := e cij (l)Rlk , (A.2) l∈ΛL

where e cij (l) and Rlk are defined in (A.8) and (A.12) below. Proof. Denoting by ∂1 w (resp. ∂2 w) the derivative with respect to the first (resp. second) variable of w, the derivatives of H are given by o n X Hk0 (x) = f 0 (xk ) + g e−ν d(k−l) ∂1 wkl (xk , xl ) + ∂2 wlk (xl , xk ) , (A.3) l∈ΛL

00 Hi,j (x) 00 Hi,i (x)

o n = ge ∂1 ∂2 wij (xi , xj ) + ∂1 ∂2 wji (xj , xi ) , o n X = f 00 (xi ) + g e−ν d(i−l) ∂12 wil (xi , xl ) + ∂22 wli (xl , xi ) , −ν d(i−j)

(A.4) (A.5)

l∈ΛL

for k 6= k0 . Furthermore, we note that, for α > 0, n Pn X X √ 2 ˆ √ e−α d(k) ≤ e(−α ν=1 |kν |/ n) ≤ =: Mα , 1 − e−α/ n k∈Λ ˆ n L

(A.6)

k∈Z

uniformly in L. Thus, using Eqns. (I.20), we obtain that X 00 00 (x) − Hi,j (0) ≤ e cij (k) |f 0 (xk )| + |f 0 (xk )|2 , eµ d(i−j) Hi,j

(A.7)

k∈ΛL

where e cij (k) := 2g e−(ν−µ) d(i−j) (1 − δij ) (δik + δjk ) n o + δij δik 1 + gMν + g e−ν d(i−k) .

(A.8)

Next, we derive a bound that enables us to control the derivatives of the form |f 0 (xk )|m k∈Λ in (A.7) by |Hk0 (x)|m k∈Λ . By Eqn. (A.3), we may estimate L L the differences by o n X 0 Hk (x) − f 0 (xk ) ≤ 2g e−ν d(k−l) |f 0 (xk )| + |f 0 (xl )| , (A.9)

l∈ΛL \{k}

for each k ∈ ΛL . Hence, for m = 1, 2 and k ∈ ΛL , we obtain from Mν ≥ 1 that m (A.10) |f 0 (xk )|m ≤ 3 Hk0 (x) + 12gMν2 |f 0 (xk )|m X 2 −ν d(k−l) 0 m + 12gMν e |f (xl )| , l∈ΛL \{k}

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where we have used Schwarz’ inequality, in case that m = 2. Since Mν ≥ 1 in (A.6), this implies that 0 X 0 2 δnl − Qnl |f 0 (xl )| + |f 0 (xl )|2 ≤ Hn (x) + Hn (x) , (A.11) l∈ΛL

for all n ∈ ΛL , where Q is the self-adjoint matrix on C|ΛL | with matrix elements given by Qkl := 12gMν2 e−ν d(k−l) . Since these matrix elements only depend on the P difference k − l, the norm of Q is bounded by 12gMν2 k∈ΛL e−ν d(k) = 12gMν3 . Thus, for g < Mν−3 /12, the matrix 1 − Q is invertible, and its inverse R is given by the norm-convergent Neumann series R :=

∞ −1 X = Qp . 1−Q

(A.12)

p=0

Since Qkn ≥ 0, also Rkn ≥ 0, and Inequality (A.11) is preserved under left multiplication by R. That is, multiplying (A.11) by Rkn and summing over n, we obtain X m 0 |f 0 (xk )|m ≤ Rkn Hn (x) . (A.13) l∈ΛL

Inserting (A.13) into (A.7), we arrive at X 00 2 00 eµ d(i−j) Hi,j (x) − Hi,j (0) ≤ cij (k) Hk0 (x) + Hk0 (x) ,

(A.14)

k∈ΛL

with cij (k) :=

X

e cij (l) Rlk .

(A.15)

l∈ΛL

It remains to check the summability condition (I.16). We first observe that, for fixed i, j ∈ ΛL , X X X X cij (k) = e cij (l) Rlk = e cij (l) Rlk k∈ΛL

k,l∈ΛL

= |ΛL |−1

X l∈ΛL

l∈ΛL

X e cij (l) Rkl ,

k∈ΛL

(A.16)

k,l∈ΛL

since Rlk depends only on the difference k − l. Using the normalized vector η ∈ C|ΛL | , ηk := |ΛL |−1/2 , we estimate the sum of the Rkl as follows: X

1 −1 Rkl = η R η ≤ R ≤ . (A.17) |ΛL | 1 − 12gMν3 k,l∈ΛL

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Inserting (A.17) into (A.16) and summing over i and l, we have X X 4g e−(ν−µ) d(i−j) 1 + 2gMν δij cij (k) ≤ + (1 − δ ) ij 1 − 12gMν3 1 − 12gMν3 i∈ΛL

i,k∈ΛL

≤

1 + 6gMν−µ , 1 − 12gMν3

(A.18)

for any j ∈ ΛL . Next, we note that e cii (l) = δil (1 + gM ) + ge−ν d(i−l) only depends on the difference i − l. Similar to (A.16)–(A.18), we obtain X

cij (k) ≤

i,j∈ΛL

1 + 6gMν , 1 − 12gMν3

(A.19)

for any k ∈ ΛL .

B Self-adjointness of DH In this appendix, we show that DH is essentially self-adjoint on C0∞ (R|ΛL | ) ⊗ Ff [C|ΛL | ], under Hypotheses 1 and 2. To this end, we follow the strategy in [16], p. 113. Since we assume that the function H is only C 2 , we need to adapt the arguments, especially the elliptic regularity. Still assuming Hypotheses 1 and 2, we derive some useful properties of DH , in particular a kind of integration by parts. This allows us to prove (II.13). Using further some results from [12], we also show (V.4). Here, we work in the representation |ΛL |

H :=

M

H(N) ∼ = L2 R|ΛL | , Ff [C|ΛL | ] .

(B.1)

N=0

Lemma B.1. The Dirac operator, given by (II.11), is essentially self-adjoint on C0∞ R|ΛL | , Ff [C|ΛL | ] .

Proof. It suffices to show that each equation (DH ±i)u = 0, for u ∈ L2 R|ΛL | , Ff ] , in the distributional sense, implies u = 0. (Here and henceforth we abbreviate Ff := Ff [C|ΛL | ].) First, we note that, for v ∈ C0∞ R|ΛL | , Ff , DH v = D0 v + MH v ,

(B.2)

where D0 = DH for H = 0 and where MH is a multiplication operator by a matrix with C 1 entries. Let u ∈ L2 with (DH ± i)u = 0. Since MH u ∈ L2loc R|ΛL | , Ff , we see that (D0 ± i)u ∈ L2loc R|ΛL | , Ff . Since, for any integer k, −1 D02 + 1 : H k R|ΛL | , Ff −→ H k+2 R|ΛL | , Ff ,

(B.3)

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is bounded (H k being the kth Sobolev space), we have that −1 D02 + 1 χ(D0 ± i)u ∈ H 2 R|ΛL | , Ff ,

95

(B.4)

for any χ ∈ C0∞ (R|ΛL | , C). Since the commutator [χ, D0 ] is a multiplication by a bounded matrix, we find that −1 D02 + 1 (B.5) (D0 ± i)χu ∈ H 2 R|ΛL | , Ff . Thus, the following equality in H 1 R|ΛL | , Ff , −1 −1 (D0 ∓ i) D02 + 1 χ(D0 ± i)u = (D0 ∓ i) D02 + 1 [χ, D0 ]u (B.6) −1 2 + (D0 ∓ i) D0 + 1 (D0 ± i)χu holds, and the last term equals in fact χu. This yields χu ∈ H 1 R|ΛL | , Ff and u ∈ 1 1 1 Hloc R|ΛL | , Ff . Since C 1 R|ΛL | , Ff ⊆ Hloc R|ΛL | , Ff , MH u ∈ Hloc R|ΛL | , Ff . 2 Following the previous lines again, we prove that u ∈ Hloc R|ΛL | , Ff . ∞ |ΛL | Now, let f ∈ C0 (R , R) with f (x) = 1 for |x| ≤ 1 and fn (x) = f (x/n). A direct computation shows that (DH ± i)fn u = [DH , fn ⊗ 1]u + 0 1 X −1 β (∂xj f )(x/n) ⊗ c∗j − cj u . = n

(B.7)

j∈ΛL

Thus

2 kfn uk2L2 + DH fn u L2 ≤

2 1 sup ∇f l∞ kuk2L2 . 2 n x

(B.8)

Since the norm of fn u converges to the norm of u, as n goes to infinity, we obtain u = 0. Lemma B.2. (“Integration by parts.”) The domain of DH is D(dH )∩D(d∗H ), where n o D(dH ) = u ∈ L2 R|ΛL | , Ff [C|ΛL | ] dH u ∈ L2 R|ΛL | , Ff [C|ΛL | ] , n o u ∈ L2 R|ΛL | , Ff [C|ΛL | ] d∗H u ∈ L2 R|ΛL | , Ff [C|ΛL | ] . D(d∗H ) = 2 2 ) ⊆ D(DH ) and, for u ∈ D(DH ) and v ∈ D(DH ), Moreover D(DH 2 u|vi = hDH u|DH vi = hdH u|dH vi + hd∗H u|d∗H vi . hDH

(B.9)

Proof. We follow a standard argument, e.g., given in [11]. Let u ∈ D(DH ), fm (x) := f (x/m), and Fn (x) := n|ΛL | f (nx), where f ∈ C0∞ (R|ΛL | ; [0, 1]), as in the proof of Lemma B.1. Define Z (B.10) um,n (x) := fm (Fn ∗ u) (x) = fm (x) Fn (y − x) u(y) d|ΛL | y .

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Then um,n ∈ C0∞ (R|ΛL | ; Ff ), and thus we have that

dH um,n 2 + d∗H um,n 2 = DH um,n 2 .

(B.11)

Since limm,n→∞ um,n = u (the order of the limits is irrelevant) and DH is closed, it follows that limm,n→∞ DH um,n = DH u. Next, as in the proof of Lemma B.1, we have that

#

d um,n − fm d# (Fn ∗ u) ≤ C1 , H H m

(B.12)

# −1 ∗ ), where d# for some constant C1 , since [d# H , fm ] = O(m H = dH or dH = dH . Similarly, we observe that # d# (B.13) H (Fn ∗ u)(x) − (Fn ∗ dH u)(x) Z X |ΛL | Hj0 (x) − Hj0 (y) (c# y. = Fn (x − y) j u)(y) d j∈ΛL

Therefore, for any ϕ ∈ H, an application of Schwarz’ inequality yields ϕ fm d# (Fn ∗ u) − (Fn ∗ d# u) ≤

≤

H |ΛL |2

n

H

sup

00 (x)| max |Hi,j

(B.14)

|x|≤m+1 i,j∈ΛL

Z

Fn (x − y) kϕ(x)kFf ku(y)kFf d|ΛL | x d|ΛL | y .

C2 (β, L, m) kϕk kuk , n

for some constant C2 (β, L, m). Choosing the number m ∈ N sufficiently large such that C1 /m ≤ ε/2, and afterwards choosing n ∈ N large enough so that C2 (β, L, m)/n ≤ ε/2, we observe that for any given ε > 0 we have

#

d um,n − fm (Fn ∗ d# u) ≤ ε , (B.15) H H provided that m and n are sufficiently large. Therefore, dH u, d∗H u ∈ H and

dH u 2 + d∗H u 2 = DH u 2 .

(B.16)

Thus D(DH ) ⊆ D(dH ) ∩ D(d∗H ), and since the opposite inclusion is trivial, we have that D(DH ) = D(dH ) ∩ D(d∗H ) .

(B.17)

The remaining parts of Lemma B.2 are now an immediate consequence of this fact.

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Lemma B.3. For sufficiently large β > 0, the kernels of the Dirac operator DH 2 and the Witten Laplacian ∆H := DH are given by (II.13), i.e., (B.18) Ker{DH } = Ker{∆H } = C · e−βH ⊗ Ω . 2 }, since DH is self-adjoint (see Proof. First, we note that Ker{DH } ⊆ Ker{DH −βH [13]). Second, we observe that e ⊗ Ω ∈ Ker{DH }, and hence we have 2 } = Ker{∆H } . span{e−βH ⊗ Ω} ⊆ Ker{DH } = Ker{DH

(B.19)

Theorem III.1 implies that Ker{∆H } = Ker{A}, for β > 0 sufficiently large. Furthermore we recall from Eqn. (III.22) that (0)

A ≥ ∆H ⊗ 1 +

2 λmin 1 ⊗ NL . β

(B.20)

Since the number operator NL act as multiplication by the form degree, (1 ⊗ NL )|H(`) = `(1 ⊗ 1)|H(`) , we conclude that \ (0) Ker{Zj (H)} ⊗ Ω . (B.21) Ker{A} = Ker{∆H } ⊗ Ω ⊆ j∈ΛL

T

then fj0 (x) 2 |ΛL |

So, if f ∈ j∈ΛL Ker{Zj (H)} 2 tells us that f ∈ Hloc (R|ΛL | ) ∩ L (R 2 |ΛL | ) and Hloc (R

= −Hj0 (x)f (x), and elliptic regularity ), since Hj0 ∈ C 1 (R|ΛL | ). Moreover, eβH f ∈

∇x eβH f

= 0,

which implies that eβH f = const, and this in turn gives f ∈ span{e−βH }.

(B.22)

From the proofs of Lemmata B.2 and B.3 we also derive the following corollary Corollary B.4. For sufficiently large β > 0, X n o D(DH ) = Q(∆H ) = ψ ∈ H kZj (H)ψk2 < ∞ .

(B.23)

j∈ΛL

Finally we derive the following lemma. Lemma B.5. Under Hypothesis 1–2, the inequality (V.4) holds, that is (0) (1) P ∆H ⊗ 1 P ≥ inf σ(∆H ) P ,

(B.24)

where P = P0 ⊗ 1, P0 is the orthogonal projection of L2 onto e−βH , and where (1) (1) σ(∆H ) denotes the spectrum of ∆H . (0)

(1)

Proof. Since ∆H ≥ 0, (B.24) is true if inf σ(∆H ) = 0. Let us then assume that it is strictly positive. Note further that P projects onto the kernel of ∆H , which is (0) also the kernel of the restriction dH of dH to H(0) . Thanks to [12], the restriction (0) of ∆H to the orthogonal of this kernel is unitarily equivalent to the restriction of (1) (0) ∆H to the range of dH . This proves (B.24).

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C About the ferromagnetic condition In this section of the Appendix, we want to give another situation where we can obtain a similar asymptotics as in Lemma I.8. In Hypothesis 6, we require, as in [14], that the Hessian at 0 is of ferromagnetic type (i.e v ≥ 0). This is not necessary to obtain the asymptotics of Ev as we shall show it in Lemma C.1 below, using the arguments in [14]. We consider a function v satisfying all the conditions in Hypothesis 6 but the non-negativity requirement. Starting from (VI.4) for v, we follow the reasoning in [14]. Recall that Fv (i · ) is the Fourier transform of v. The key point is to move the contour of integration into Td + iRd until we meet the singularities set {z ∈ Td + iRd ; vˆ(z) = 1}.

(C.1)

To this end, we introduce the function Fev given by (I.38) for u = |v|, which satisfies X |v(n)|cosh(n · η) = Fev (η). (C.2) Re vˆ(k + iη) ≤ n∈Zd

Since we assume that v has compact support, Fev is well defined and analytic on Rd . By (I.36), Fev is strictly convex everywhere and Σ = {η ∈ Rd ; Fev (η) = 1}

(C.3)

is the boundary of a strictly convex, relatively compact domain in Rd (see [14] or Lemma I.7 (ii)). Thus we have the same property as in [14] but with the new function Fev . A good situation to derive the rate of the exponential decay appears when equality in (C.2) holds at a unique point k0 ∈ Td . Let us describe this condition. Equality in (C.2) holds if and only if ∀n ∈ Zd , v(n) cos(n · k) = |v(n)|,

(C.4)

If we choose v such that the group generated by {y ∈ Z , v(y) > 0} is Z and such that there exists some y0 ∈ Zd with v(y0 ) < 0, then equality in (C.2) holds for no point k. On the other hand, let us construct a function v, with non-constant sign, for which equality in (C.2) holds for exactly one point k0 ∈ Td . We split Zd into the direct sum Zd−1 ⊕ Z and write d

d

Zd 3 n = (¯ n, nd ) ∈ Zd−1 ⊕ Z. We choose v with (−1)nd v(n) ≥ 0,

(C.5)

such that its support is bounded and satisfies (I.36). Let k0 = (0, · · · , 0, π) ∈ Td , we have, according to (C.2), X Re vˆ(k + k0 + iη) = |v(n)| cos(n · k)cosh(n · η). n∈Zd

Thus, equality in (C.4) holds if and only if k = 0 in Td .

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The method in [14] allows us to derive Lemma C.1. Let v as above. Then the asymptotics given in Lemma I.8 holds with a new function Fev defined by Re vˆ(k0 +iη) = Fev (η), where k0 = (0, · · · , 0, π) ∈ Td .

Acknowledgements We are especially indebted to B. Helffer for many useful discussions we have had with him during the progress of our work and for making his results accessible to us prior to publication. Furthermore we are grateful to A. Bovier, J.-D. Deuschel, L. Erd˝os, J. Fr¨ohlich, J. Johnsen, M. Klein, T. Spencer, H. T. Yau for various discussions and for their helpful comments. Finally, we thank V. Ivrii and M. Zworski at the Fields Institute in Toronto, T. Paul at the Universit´e Paris-Dauphine (IX), and A. Laptev at the KTH Stockholm for their hospitality.

References [1] V. Bach, J. Fr¨ ohlich, and I. M. Sigal. Renormalization group analysis of spectral problems in quantum field theory. Adv. in Math., 137:205–298, 1998. [2] H.J. Brascamp and E.H. Lieb. On extensions of the Brunn-Minkowski and Pr´ekopa-Leindler theorems including inequalities for log concave functions, and with applications to the diffusion equation. J. Funct. Anal., 22:366–389, 1976. [3] R. Courant. Dirichlet’s Principle, Conformal Mapping and Minimal Surface. Springer-Verlag (reprinted from Interscience, NY, 1950), Berlin, 1977. [4] H. Cycon, R. Froese, W. Kirsch, and B. Simon. Schr¨ odinger Operators. Springer, Berlin, Heidelberg, New York, 1 edition, 1987. [5] R. Dobroushin and M. Zahradnik. Phase diagrams for continuous spin models: An extension of the Pirogov-Sinai theory. In R. Dobroushin, editor, Mathematical Problems of Statistical Mechanics. Reidel, Dordrecht, 1986. [6] B. Helffer. On Laplace integrals and transfer operators in large dimension: examples in the non-convex case. Lett. Math. Phys., 38(3):297–312, 1996. [7] B. Helffer. Remarks on the decay of correlations and Witten’s Laplacians – I. Brascamp-Lieb inequalities and the semiclassical limit. J. Funct. Anal., 155:571–586, 1998. [8] B. Helffer. Remarks on the decay of correlations and Witten’s Laplacians – II. analysis of the dependence of the interaction. Rev. Math. Phys. (in press), 1999.

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[9] B. Helffer. Remarks on the decay of correlations and Witten’s Laplacians – III. applications to logarithmic Sobolev inequalities. Ann. de l’Inst. H. Poincar´e (Sect. Proba-Stat) (to appear), 1998. [10] B. Helffer and J. Sj¨ ostrand. On the correlations for Kac like models in the convex case. J. Stat. Phys., 74:349–369, 1994. [11] L. H¨ ormander. Introduction to Complex Analysis in Several Variables. Elsevier (North-Holland Mathematical Library, Vol. 7), Amsterdam, 3 edition, 1990. [12] J. Johnsen. On the spectral properties of Witten Laplacians, their range projections and Brascamp-Lieb’s inequality. Preprint, 1998. [13] M. Reed and B. Simon. Methods of Modern Mathematical Physics: Functional Analysis, volume 1. Academic Press, San Diego, 2 edition, 1980. [14] J. Sj¨ostrand. Correlation asymptotics and Witten Laplacians. St. Petersburg Math. J. (AMS), 8:123–148, 1997. Also in St. Petersburg Math. J. 8 (1996), no. 1, pp. 160–161. [15] M. Struwe. Variational Methods. Springer-Verlag, Berlin, 1990. [16] B. Thaller. The Dirac Equation. Springer-Verlag, Heidelberg, New York, 1992. [17] E. Witten. Supersymmetry and Morse theory. J. Diff. Geom., 17:661–692, 1982. [18] M. Zahradnik. Contour methods and Pirogov-Sinai theory for continuous spin models. Preprint, 1998. V. Bach, T. Jecko FB Mathematik MA 7-2 TU Berlin; Strasse des 17 Juni 136 D-10623 Berlin, Germany Email :[email protected], [email protected] J. Sj¨ostrand Centre de Math´ematiques Ecole Polytechnique CNRS, URA 169 F-91128 Palaiseau Cedex, France Email :[email protected] Communicated by J. Bellissard submitted 04/09/98; revised 22/02/99; accepted 02/03/99

Ann. Henri Poincar´ e 1 (2000) 101 – 172 c Birkh¨

auser Verlag, Basel, 2000 1424-0637/00/010101-72 $ 1.50+0.20/0

Annales Henri Poincar´ e

Complex Angular Momentum in General Quantum Field Theory J. Bros and G.A. Viano Abstract. It is proven that for each given two-field channel – called the “t-channel” – with (off-shell) “scattering angle” Θt , the four-point Green’s function of any scalar Quantum Fields satisfying the basic principles of locality, spectral condition together with temperateness admits a Laplace-type transform in the corresponding complex angular momentum variable λt , dual to Θt . This transform enjoys the following properties: a) it is holomorphic in a half-plane of the form Reλt > m, where m is a certain “degree of temperateness” of the fields considered, b) it is in one-to-one (invertible) correspondence with the (off-shell) “absorptive parts” in the crossed two-field channels, c) it extrapolates in a canonical way to complex values of the angular momentum the coefficients of the (off-shell) t-channel partial-wave expansion of the Euclidean four-point function of the fields. These properties are established for all space-time dimensions d + 1 with d ≥ 2.

I Introduction The complex angular momentum analysis was widely used in the sixties, in particle physics, for describing the high-energy asymptotic behaviour of the scattering amplitude. With the arrival of QCD much attention was diverted away from the “old-fashioned” approach to the strong interactions. Interest was reignited (see [1] and references therein) within the particle physics community with the arrival of colliders capable of delivering very large centre-of-mass energies (e.g. the HERA collider at DESY and the Tevatron collider at FNAL); from the theoretical viewpoint, this revival was made possible by the much earlier important results of BFKL[2] who discovered and characterized the Regge-like asymptotic properties of appropriate resummations of perturbative amplitudes 1 in QCD. In the deep inelastic lepton-proton scattering a central role is played by the so-called “structure-functions” which parametrize the structure of the target as “seen” by the virtual photon. They are usually denoted by Fi x, Q2 , where µ Q2 = −q2 and qµ = kµ − k0 is the momentum transfer (k µ and k0µ being re2 spectively the incoming and outgoing lepton four-momenta), while x = Q 2ν , ν = p · q p2 = M2 , M being the proton mass . It is now possible to explore the structure functions in a region where the momentum transfer is much smaller than the centre-of-mass energy, i.e. for small values of x. In the parton model one can show that the x-dependence, in the limit x → 0, is related to the behaviour of hadronic scattering cross-sections at high energy [4]. This behaviour, which ap1 see

also [3] for similar results in the case of scalar fields

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pears to exhibit Regge-like asymptotic properties, is reminiscent of the concept of “exchange of families of particles with different spins”. A detailed analysis of small x structure function measurements, at fixed target energies [5], show that they are indeed approximately consistent with the predictions of such a model. We can thus say that, on one side, the phenomenology calls for an extension of the conventional exchange process and suggests an exchange mechanism involving families of particles; on the other side, from a theoretical viewpoint, these families could be described by “moving poles”, namely, poles in a certain “complex angular momentum plane”. The complex angular momentum theory originated long ago in connection with some problems of classical mathematical physics, mainly the diffraction [6]. Then Regge [7] extended these methods to quantum mechanics and specifically to the scattering by Yukawian potentials (see also [8]). In these works, the complex angular momentum analysis was produced by a direct analytic interpolation to complex values of the angular momentum variable of the relevant differential equations for partial waves. Then several authors (see [9] and references therein) conjectured that the results proved by Regge, at non-relativistic level, might as well be applied to the high energy relativistic dynamics where the method could really display all its power. This relativistic extension, which of course could no more be justified in a simple framework of differential equations, was given a tentative formulation [9] in the approach of the so-called “S-matrix theory” based on the general, but rather loose concept of “maximal analyticity”. However, it must be emphasized that since that time no genuine relativistic complex angular momentum theory relying on the general principles of Quantum Field Theory (Q.F.T.) has been given at all. In view of the considerations developed above, one is then led to set the following question whose conceptual interest is of primary importance: Is it possible to find in the framework of general Q.F.T. a mathematical structure which leads to poles moving in the complex angular momentum plane and that are responsible for an exchange mechanism involving families of particles and giving rise to Regge-like asymptotic properties, as suggested by the analyses of BFKL[2] and Berg`ere et al. [3] in the philosophy of resummations of perturbative Q.F.T? We have already announced and briefly sketched a positive answer to this question in a previous work [10]. Here we shall provide a detailed proof of the first basic result of [10], namely the existence of a field-theoretical off-shell version of the Froissart-Gribov representation of the partial-waves [11,12]; the latter had been discovered by these authors in 1961 in the analytic S-matrix approach of particle physics, requiring that the scattering amplitude should satisfy the Mandelstam representation. In order to prove our field-theoretical result we make use of a basic analyticity property of the four-point function F implied by the standard axioms of locality, spectrum and Lorentz invariance; moreover we use a majorization of F which is a consequence of the “temperateness axiom” of quantum field theory. The result which can be derived from these properties is the following: for each

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given two-field channel called the t-channel, with total squared energy-momentum t and (off-shell) scattering angle Θt , there exists an appropriate Fourier-Laplace type transform of F with certain analyticity properties in the complex angular momentum λt which is the natural conjugate variable of Θt . One thus obtains a generalization of the relationship which exists in the standard Laplace transform theory between analyticity properties (including possible poles) of the transform and the asymptotic behaviour of the original function. From our viewpoint this Laplace-type transform can be regarded as the mathematical structure which relates the complex angular momentum poles (moving poles) to the high-energy asymptotic behaviour. Moreover this approach presents further advantages: i) The analysis is completely worked out in the complex momentum space scenario appropriate to Q.F.T. [13] (see, on this point, our comment below). ii) It is the joint exploitation of harmonic analysis on orbital manifolds of the Lorentz group together with basic analyticity properties of Q.F.T. which entails the complex angular momentum structure; this method holds in any space-time dimension d + 1 with d ≥ 2. iii) By the use of our Fourier-Laplace-type transformation one can perform a partial diagonalization (namely a diagonalization with respect to the angular variables) of the convolution product involved in the Bethe-Salpeter integral equations. This rigorous mathematical structure, which pertains to the general framework of Q.F.T., is thereby directly responsible for the existence of poles in the complex angular momentum variable. This is the content of our second basic result presented in [10], whose detailed proof will be given elsewhere[14]. One can specify the advantage mentioned in i) under two respects: a) with respect to the S-matrix approach. The absorptive parts of F in the crossed two-field channels have their supports inside regions of appropriate one-sheeted hyperboloids determined by the future cone ordering relation (in view of the spectral conditions). This geometrical property can be properly specified in terms of energy-momentum configurations, which are of more controllable interpretation than the sets of Lorentz invariants, as they were used in the Mandelstam representation. As a matter of fact, the Mandelstam double spectral region (used in [11, 12]) corresponds to complex energymomentum configurations which have no simple physical interpretation. b) with respect to the approach of Euclidean Q.F.T. The fact that the “Euclidean partial-waves” admit an analytic interpolation in the complex angular momentum variables is explicitly shown to be equivalent to the property of analytic continuation of the four-point function from Euclidean momentum-space to Minkowskian momentum-space (through a domain which is permitted by the requirements of locality and spectral conditions). We now wish to stress that the conceptual interest of the present study can be envisaged from two viewpoints, according to whether the fundamental fields

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considered are those of the QCD-theory or the “elementary meson and baryon fields” used at the age of dispersion theory. In the latter case, which is the traditional case of application of the axioms of Q.F.T., our results appear as “off-shell results”; but in order to get rid of this restriction, one can use the analytic continuation technique adopted in the proof of dispersion relations (see [15] and references therein) and/or positivity constraints (analogous to those used by Martin[16]) to reach the mass-shell values and possibly a positive interval in the energy variable t, so that a range of possible bound-states might be included in our analysis. We now conjecture that our results might be applicable with even more interest to the former case, in which the phenomenon of confinement is present, so that the off-shell character of our study not only remains relevant but is even the only one to be relevant! In fact, it seems admitted that the general principles used here (locality, spectrum, Lorentz covariance, temperateness) still apply to theories of QCD-type in suitable gauges: our results on complex angular momentum analysis then follow without requiring the existence of asymptotic elementary particles of the fields and are fully consistent with confinement. Moreover, the possible production of a discrete spectrum of composite particles (namely hadrons and possibly “glueballs”) appearing as “Regge-type particles” via appropriate Bethe-Salpeter-type equations is built-in [10,14] in this general field-theoretical framework. In the present study, we only considered (for simplicity) the case of scalar fields; but one can expect that the joint exploitation of harmonic analysis on Lorentz orbital manifolds together with axiomatic analyticity still yields similar results for more general fields in Lorentz covariant gauges. The paper is organized as follows. Section 2 is devoted to an appropriate analysis of the complex geometry associated with a given two field t-channel. In Section 3 we derive axiomatic analyticity properties and bounds of the four-point functions with respect to the (off-shell) scattering angle Θt in manifolds bordered by the s-cut and u-cut of the crossed channels. It is then shown in Section 4 that these properties of the four-point function are equivalent to the existence of a Laplace-type transform of the latter with respect to the corresponding complex angular momentum variable λt . This transform, which is explicitly defined in terms of the (off-shell) absorptive parts of the crossed s- and u-channels, is studied in arbitrary space-time dimension d+1 (d ≥ 2): analyticity and bounds in a half-plane Reλt > m and the property of Carlsonian interpolation of the Euclidean partialwaves satisfied by this Laplace-type transform (Froissart-Gribov-type equalities) are established. The inverse of the transformation is also described and, as a byproduct, the connection (mentioned above) between the analytic continuation from Euclidean to Minkowskian space and the analytic interpolation in the complex angular momentum plane is displayed. In Appendix A, we give mathematical tools used for the analytic completion of Section 3. Appendix B is devoted to primitives and derivatives of non-integral order in a complex domain and to their Laplace transforms: it provides a complete treatment of the distribution-like character of the Green functions and absorptive parts in Section 4.

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II Complex geometry associated with a two-field channel Space-time and energy-momentum space are (d + 1)-dimensional, with d ≥ 2. Vec− → tors in (d + 1)-dimensional Minkowskian space are represented by k = (k (0) , k ) = (k(0) , k(1) , · · · , k(d) ); the corresponding scalar product is denoted k.k0 = k(0) k0 (0) − − → (1) (d) k(1) k0 · · · − k(d) k0 and k2 = k.k = k(0)2 − k 2 . In all the following, a special role is played by a given two-field channel (called the t-channel) in which the pairs of incoming and outgoing complex energymomenta are denoted respectively (k1 , k2 ) and (k10 , k20 ); we choose the correk1 − k2 sponding set of independent vector-variables K = k1 + k2 = k10 + k20 , Z = , 2 0 0 k − k2 Z0 = 1 . K is the total energy-momentum vector of this t-channel, whose 2 squared energy is t = K 2 . In this paper we shall always assume that K is fixed ˆ K (resp. M ˆ (c) ) the space of all real and space-like , i.e. t 6 0. We shall call M K real (resp. complex) momentum configurations [k] = (k1 , k2 , k10 , k20 ) such that ˆ K (resp. M ˆ (c) ) is isomorphic to the real (resp. complex) k1 + k2 = k10 + k20 = K. M K space R2(d+1) (resp. C2(d+1) ) of the couple of vectors (Z, Z 0 ), Z and Z 0 being respectively the relative incoming and outgoing (off-shell) (d + 1)-momenta of the t-channel. Choosing once for all a time-axis with unit vector e0 e20 = 1 determines ˆ (c) in which all the energy-momenta are of the the “Euclidean subspace” EˆK of M K (0) → 0(0) →0 (0) 0(0) → → form ki = iq , − pi , k0 = iq , − pi (with − pi 0 , q , q real). pi , − i

i

i

i

i

We shall mainly √ consider the case K 6= 0, and choose K along the d-axis of coordinates: K = −t ed , where ed denotes the corresponding unit vector e2d = −1 . We also introduce the (off shell) “scattering angle” Θt of the t-channel as being the angle between the two-planes π and π 0 spanned respectively by the pairs of vectors (Z, K) (or (k1 , k2 )) and (Z 0 , K) (or (k10 , k20 )). It is convenient to introduce (real or complex) unit vectors z, z 0 , (uniquely determined up to a sign) orthogonal to K and belonging respectively to π and π 0 , such that the following orthogonal decompositions hold: Z = ρz + w K, Z 0 = ρ0 z 0 + w0 K,

(2.1.a)

with z.K = z 0 .K = 0, z 2 = z 0 = −1, 2

(2.1.b)

or equivalently: 1 1 k1 = ρz + w + K, k2 = −ρz − w − K, 2 2 1 1 K, k20 = −ρ0 z 0 − w0 − K. k10 = ρ0 z 0 + w0 + 2 2

(2.2.a) (2.2.b)

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Then, the “scattering angle” Θt of the t-channel is defined by the equation: cos Θt = −z.z 0

(2.3)

(note that Θt = 0 for z = z 0 ). The parameters ρ, w (resp. ρ0 , w0 ) introduced in Equations (2.1), (2.2) can 0 be computed in terms of the scalar products Z 2 , Z.K, K 2 (resp. Z 2 , Z 0 .K, K 2 ) or, 2 0 equivalently, in terms of the Lorentz invariants ζi = ki (resp. ζi = ki02 ), i = 1, 2, and t. One readily obtains: Z.K Z 0 .K ζ1 − ζ2 . ζ 0 − ζ20 . = , w0 = = 1 t 2t t 2t Λ (ζ1 , ζ2 , t) 2 2 2 ρ = −Z + w t = 4t 0 0 Λ (ζ 1 , ζ2 , t) ρ02 = −Z 02 + w02 t = , 4t w=

(2.4) (2.5) (2.6)

where: 2

Λ (a, b, c, ) = a2 + b2 + c2 − 2 (ab + bc + ca) = (a − b) − 2 (a + b) c + c2

(2.7)

Finally, the variable cos Θt is also a Lorentz invariant which can be expressed as follows in terms of ζi , ζi0 (i = 1, 2) , t and the squared momentum transfer 2 2 s = (k1 − k10 ) = (Z − Z 0 ) : s + ρ2 + ρ02 − (w − w0 ) t 2ρρ0 2

cos Θt = or cos Θt =

2st + t2 − (ζ1 + ζ2 + ζ10 + ζ20 ) t + (ζ1 − ζ2 ) (ζ10 − ζ20 ) [Λ (ζ1 , ζ2 , t) Λ (ζ10 , ζ20 , t)]

1/2

The following alternative expression also holds: − u + ρ2 + ρ02 + (w + w0 )2 t cos Θt = , 2ρρ0

(2.8.a)

(2.8.b)

(2.9)

where u denotes the squared momentum transfer in the crossed channel, namely 2 2 u = (k1 − k20 ) = (Z + Z 0 ) , which is such that: u = −s − t + ζ1 + ζ2 + ζ10 + ζ20 . For K = 0, Equations (2.1)–(2.9) reduce to the following ones: k1 = −k2 = Z = ρz, ρ = −ζ1 = −ζ2 , 2

k10 = −k20 = Z 0 = ρ0 z 0 , 02

ρ =

−ζ10

=

−ζ20

(2.10) (2.11)

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and cos Θt = −

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Z.Z 0 s − ζ1 − ζ10 u − ζ1 − ζ10 = = − ρρ0 2(ζ1 ζ10 )1/2 2(ζ1 ζ10 )1/2

(2.12)

The space of Lorentz invariants: ˆ (c) , we call I ([k]) the corresponding set of For any point [k] = (k1 , k2 , k10 , k20 ) in M K Lorentz invariants, namely {ζ = (ζ1 , ζ2 ), ζ 0 = (ζ10 , ζ20 ), (s, t, u) with s + t + u = ζ1 + ζ2 + ζ10 + ζ20 }, which vary in a complex space C6(I) . In this space, the choice of variables adapted to the t-channel is specified as follows: I([k]) = (It ([k]) , cos Θt ) with It ([k]) = (ζ1 , ζ2 , ζ10 , ζ20 , t). ˆ K be the subset of all points [k] in For each K with t = K 2 ≤ 0, let Ω (c) 0 0 ˆ M K whose parameters ρ, w, ρ , w in the representation (2.2) are real-valued. This reality condition is equivalent (in view of Equations (2.4), (2.5), (2.6)) to the fact that ζi , ζi0 , (i = 1, 2) are real and satisfy the following inequalities: Λ (ζ1 , ζ2 , t) 6 0,

Λ (ζ10 , ζ20 , t) 6 0,

which imply, for K 6= 0, that the points ζ = (ζ1 , ζ2 ) and ζ 0 = (ζ10 , ζ20 ) belong to the following parabolic region (see Fig. 1): n o 2 ∆t = ζ = (ζ1 , ζ2 ) ∈ R2 ; (ζ1 − ζ2 ) − 2 (ζ1 + ζ2 ) t + t2 6 0 (2.13) For K = 0, the corresponding set ∆0 is (in view of Equations (2.11)) the half-line ζ1 = ζ2 ≤ 0.

II.1

Lorentz foliation and the associated complex quadrics; the case K 6= 0

Using the (d − 1)-dimensional unit complex quadric: For K 6= 0, the range of each vector z, z 0 in Equations (2.2) is (in view of Equations (c) (2.1.b)) a (d − 1)-dimensional complex quadric Xd−1 in the subspace orthogonal to K, namely: n o (c) Xd−1 = z = z (0) , z (1) , . . . z (d−1) ∈ Cd ; z (0)2 − z (1)2 − · · · − z (d−1)2 = −1 (2.14) (c) Two real submanifolds of Xd−1 play an important role: (c)

a) the one-sheeted hyperboloid Xd−1 = Xd−1 ∩ Rd , obtained by restricting z (0) , . . . , z (d−1) to real values in Equation (2.14). (c)

b) the “euclidean sphere” Sd−1 = Xd−1 ∩ iR × Rd−1 , obtained by putting z (0) = iy (0) and z (1) . . . z (d−1) real in Equation (2.14). ˆ In view of Equations (2.2)–(2.6), each point [k] = (k1 , k2 , k10 , k20 ) in M K can thus be represented by (It ([k]) , (z, z 0 )) , with It ([k]) = (ζ, ζ 0 , t) , (ζ, ζ 0 ) ∈ C4 and the (c) (c) pair (z, z 0 ) in Xd−1 × Xd−1 . (c)

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ζ2 t

t 4

0 ζ1 t 4

∆t

t

Figure 1 ˆ K of M ˆ (c) : We introduce the following Cauchy-Riemann submanifold Ω K n o ˆ K = [k] ≡ (It ([k]) , (z, z 0 )); (ζ, ζ 0 ) ∈ ∆t × ∆t , (z, z 0 ) ∈ X (c) × X (c) Ω d−1 d−1 ˆK: We then distinguish the following two maximal real submanifolds of Ω ˆ (sp) of M ˆ K characa) (z, z 0 ) in Xd−1 × Xd−1 : this submanifold is the subset M K 0 terized by the condition that the two-planes π and π determined respectively by the real vectors (k1 , k2 ) (or (z, K)) and (k10 , k20 ) (or (z 0 , K)) are space-like. ˆ (c) . b) (z, z 0 ) in Sd−1 × Sd−1 : this is the Euclidean subspace EˆK of M K We note that in the representation (It ([k]) , (z, z 0 )) of [k] the pair (z, z 0 ) still contains one Lorentz invariant, namely cos Θt = −z.z 0 , replaced equivalently by s or u according to Equations (2.8), (2.9), and that three situations are of special interest: i) [k] ∈ EˆK : the corresponding condition (z, z 0 ) ∈ Sd−1 × Sd−1 then implies that −1 6 cos Θt 6 1

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ˆ (sp) and s = (k1 − k10 )2 > 0: Equation (2.8) implies that cos Θt − 1 > ii) [k] ∈ M K 0; the corresponding pair (z, z 0 ) lies in Xd−1 × Xd−1 in such a way that the two-plane spanned by z and z 0 is time-like (i.e. Θt = iv with v real, and z.z 0 = − cosh v). ˆ (sp) and u = (k1 − k20 ) > 0: Equation (2.9) implies that cos Θt + 1 < 0 iii) [k] ∈ M K and one has: (z, z 0 ) ∈ Xd−1 × Xd−1 with z.z 0 = cosh v (i.e. Θt = π + iv). Let G be the connected Lorentz group acting in the Minkowskian space Rd+1 , namely G ≈ SO0 (1, d) and let G(c) ≈ SO0 (1, d)(c) be the complexified of G, acting (c) on Cd+1 . Let then GK (resp. GK ) be the stabilizer of K in G (resp. G(c) ). Since K (c) is real and space-like, one has GK ≈ SO0 (1, d − 1) and GK ≈ SO0 (1, d − 1)(c) ; (c) (c) GK and GK act transitively respectively on Xd−1 and Xd−1 . We also introduce (c)

the maximal orthogonal subgroup OK ≈ SO (d) of GK which acts transitively on (c) the euclidean sphere Sd−1 of Xd−1 . With each (ζ, ζ 0 , K) , ζ ∈ ∆t , ζ 0 ∈ ∆t , we associate the manifold o n ˆ (ζ,ζ 0 ,K) = [k] ∈ M ˆ (c) ; [k] = [k](ζ,ζ 0 ,K) (z, z 0 ) ; (z, z 0 ) ∈ X (c) × X (c) , (2.15) Ω K d−1 d−1 where the mapping [k] = [k](ζ,ζ 0 ,K) (z, z 0 ) is defined by Equations (2.2), with the parameters ρ, w, ρ0 , w0 reexpressed in terms of (ζ, ζ 0 , t) via Equations (2.4)–(2.6), namely (c) (c) [k] = [k](ζ,ζ 0 ,K) (z, z 0 ), (z, z 0 ) ∈ Xd−1 × Xd−1 : (2.16.a) 1 Λ(ζ1 , ζ2 , t) 2 ζ1 − ζ2 + t z + K, k1 = 4t 2t (2.16.b) 1 Λ(ζ1 , ζ2 , t) 2 ζ1 − ζ2 − t k2 = − z − K, 4t 2t 1 Λ(ζ10 , ζ20 , t) 2 0 ζ 0 − ζ20 + t 0 K, z + 1 k1 = 4t 2t (2.16.c) 1 Λ(ζ10 , ζ20 , t) 2 0 ζ10 − ζ20 − t 0 k2 = − z − K. 4t 2t ˆ K whose sheets Ω ˆ (ζ,ζ 0 ,K) ˆ (ζ,ζ 0 ,K) ; (ζ, ζ 0 ) ∈ ∆t ×∆t } defines a foliation of Ω The set {Ω ˆ (ζ,ζ 0 ,K) have the following interpretation: for ζ and ζ 0 ∈ / ∂∆t , each submanifold Ω is the product of two (d − 1)-dimensional complex quadrics and can be seen as an (c) (c) orbit of the group GK × GK via the action (ki , ki0 ) → (gki , g 0 ki0 ) , i = 1, 2, (g, g 0 ) ∈ (c)

(c)

GK × GK . (Note that a similar foliation could be defined for the whole set

ˆ (c) ; it is not used in the present paper). M K ˆ (ζ,ζ 0 ,K) (obWe also note that the “Euclidean spheres” of the manifolds Ω 0 tained by restricting Equation (2.16.a) to the set {(z, z ) ∈ Sd−1 × Sd−1 }) define ˆK. correspondingly a foliation of the Euclidean subset EˆK of Ω

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Choice of a base point (c)

(c)

Since the group GK acts transitively on Xd−1 it is convenient to introduce a “base-point” z0 on the latter which we choose on the (d − 1)-axis of coordinates, namely z0 = (0, . . . 0, 1, 0). By now assuming that the point z 0 is fixed at z 0 = z0 in Equations (2.2), one obtains a set of definitions which parallel those of the previous paragraph. (c) ˆ (c) (resp. Ω ˆ K ) in which One thus defines MK (resp. ΩK ) as the subset of M K 0 0 the vectors (k1 , k2 ) are real and belong to the (zd−1 , zd )-plane of coordinates. One also associates with each (ζ, ζ 0 , K) , ζ ∈ ∆t , ζ 0 ∈ ∆t , the manifold n o (c) (c) (2.17) Ω(ζ,ζ 0 ,K) = [k] ∈ MK ; [k] = [k](ζ,ζ 0 ,K) (z, z0 ) ; z ∈ Xd−1 . If ζ ∈ / ∂∆t , Ω(ζ,ζ 0 ,K) is a (d − 1) dimensional complex quadric which is an orbit (c) (c) 0 0 of the (ki , ki ) → (gki , ki ) , i = 1, 2, g ∈ GK . The group GK 0via the action set Ω(ζ,ζ 0 ,K) ; (ζ, ζ ) ∈ ∆t × ∆t thus defines a foliation of ΩK ; ΩK is a Cauchy(c) Riemann submanifold of MK whose complex structure is parametrized by the (c) variable z in Xd−1 and which contains as maximal real submanifolds: (sp) (c) ˆ (sp) obtained for z varying in Xd−1 a) the real submanifold MK = MK ∩ M K and characterized by the property that the plane π defined by the (real) points k1 , k2 is space-like. (c) b) the Euclidean subspace EK = MK ∩ EˆK obtained for z varying in Sd−1 .

Finally, the passage from the vectors to the invariants is summarized in Proposition 1 Let I be the projection which associates with each configuration ˆ (c) the set of invariants I ([k]) = (It ([k]) , cos Θt ). [k] ≡ (It ([k]) , (z, z 0 )) in M K ˆi

This projection is implemented by the mapping (z, z 0 ) → cos Θt = −z.z 0 which (c) (c) projects Xd−1 × Xd−1 onto C. (c) ˆ (c) is impleCorrespondingly, the restriction of I to the subspace M of M K

i

K

(c)

mented by the mapping z → cos Θt = −z.z0 = z (d−1) which projects Xd−1 onto the complex z (d−1) -plane.

II.2

Lorentz foliation and the associated complex quadrics; the case K = 0:

For K = 0, the range of the vectors z and z 0 is the complex quadric: n o 2 2 2 (c) Xd = z = z (0) , z (1) , . . . , z (d) ∈ Cd+1 ; z (0) − z (1) − · · · − z (d) = −1 (2.18) (c) and one similarly introduces the real one-sheeted hyperboloid X = X ∩ Rd+1 d d (c)

and the Euclidean sphere Sd = Xd ∩ iR × Rd .

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0 ˆ Each point [k] = (k1 , −k1 , k10 , −k10 ) in M 0 is represented by (I0 ([k]), (z, z )) 0 0 0 0 0 where I0 ([k]) = (ζ, ζ , 0) with ζ = (ζ1 , ζ1 ) , ζ = (ζ1 , ζ1 ) , ζ1 ∈ C, ζ1 ∈ C, and (z, z 0 ) ∈ Xd(c) ×Xd(c) . We note the degeneracy of the representation (I0 ([k]) , (z, z 0 )) ˆ (c) , namely the fact that the number of Lorentz invariants in I0 ([k]) for the space M 0 reduces from four to two, while the number of “orbital variables” (z, z 0 ) increases correspondingly from 2 (d − 1) to 2d. ˆ 0 is the following Cauchy-Riemann manifold The set Ω n o ˆ 0 = [k] ≡ (I0 ([k]) , (z, z 0 )); (ζ, ζ 0 ) ∈ ∆0 × ∆0 , (z, z 0 ) ∈ X (c) × X (c) , Ω d d (c)

which contains as maximal real submanifolds the Minkowskian and Euclidean subˆ (c) , obtained respectively for the ranges ˆ (sp) and Eˆ0 of M manifolds M 0 0 {z, z 0 ∈ Xd × Xd } and {(z, z 0 ) ∈ Sd × Sd }. All the previous considerations concerning the variable cos Θt = −z.z 0 remain valid in the case K = 0. With each point (ζ, ζ 0 , 0) , with ζ = (ζ1 , ζ1 ) , ζ1 ≤ 0, ζ 0 = (ζ10 , ζ10 ) , ζ10 ≤ 0, one now associates the manifold n o ˆ (c) ; [k] = [k](ζ,ζ 0 ,0) (z, z 0 ) ; (z, z 0 ) ∈ X (c) × X (c) ˆ (ζ,ζ 0 ,0) = [k] ∈ M Ω 0 d d n o ˆ (ζ,ζ 0 ,0) ; ζ = (ζ1 , ζ1 ) , ζ1 ≤ 0, ζ 0 = (ζ10 , ζ10 ) , ζ10 ≤ 0 defines a foliation of The set Ω ˆ 0 ; in this foliation, each sheet Ω ˆ (ζ,ζ 0 ,0) is the product of two d-dimensional comΩ plex quadrics and can be seen as an orbit of the group G(c) × G(c) via the action: (ki , ki0 ) → (gki , g 0 ki0 ) , i = 1, 2, (g, g 0 ) ∈ G(c) × G(c) . We can make use of the same base point z0 as before (z0 = (0, . . . , 0, 1, 0)) (c) ˆ (c) in which k10 = −k20 is real and along and introduce the subspace M0 of M 0 0 the zd−1 -axis. Then for all (ζ, ζ ) ∈ ∆0 × ∆0 the following d-dimensional complex manifolds n o (c) (c) Ω(ζ,ζ 0 ,0) = [k] ∈ M0 ; [k] = [k](ζ,ζ 0 ,0) (z, z0 ) ; z ∈ Xd , (c) 0 0 (c) are orbits of the i , ki ) , i = group G via the action (ki ,0ki ) →0 (gk 1, 2, g ∈ G . 0 0 The set Ω(ζ,ζ 0 ,0) ; ζ = (ζ1 , ζ1 ) , ζ1 ≤ 0; ζ = (ζ1 , ζ1 ) , ζ1 ≤ 0 defines a foliaˆ 0 in which the vector k10 = −k20 is real and along the zd−1 tion of the subset Ω0 of Ω axis. Ω0 is a Cauchy-Riemann manifold whose complex structure is parametrized (c) (sp) by z (z ∈ Xd ), and which contains the real Minkowskian submanifold M0 and (c) the Euclidean subspace E0 of M0 (obtained respectively for z ∈ Xd and z ∈ Sd ). (c)

(c)

Proposition 1 remains true up to obvious changes (Xd−1 being replaced by Xd ).

II.3

The spectral sets Σs and Σu

We define the s-channel and u-channel spectral sets Σs and Σu associated with a given field theory as the following analytic hypersurfaces in complex momentum

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3(d+1)

space C(k)

: n o 2 Σs = [k] ≡ (K, Z, Z 0 ) ∈ C3(d+1) ; s = (Z − Z 0 ) = s0 + τ ; τ ≥ 0 , (2.19) n o 2 Σu = [k] ≡ (K, Z, Z 0 ) ∈ C3(d+1) ; u = (Z + Z 0 ) = u0 + τ ; τ ≥ 0 , (2.20)

where s0 and u0 are positive numbers interpreted as the mass thresholds of the corresponding channels. Since Σs and Σu are Lorentz-invariant sets, their projections onto the space of Lorentz invariants (It ([k]) ; cos Θt ) are analytic hypersurfaces I (Σs ) and I (Σu ) whose equations result from Equations (2.8), (2.9) namely: i h   s0 + (ρ − ρ0 )2 − (w − w0 )2 t + τ  ; cos Θt − 1 = I (Σs ) (2.21) 2ρρ0   with τ > 0. i h  2 2  −u0 − (ρ − ρ0 ) + (w + w0 ) t − τ  cos Θt + 1 = I (Σu ) (2.22) 2ρρ0   with τ > 0. ˆ (ζ,ζ 0 ,K) Let us now consider the intersections of Σs and Σu with any orbit Ω ˆ in ΩK ; it readily follows from Equations (2.21) and (2.22) that these intersections can be parametrized by the variables z, z 0 in the following way: ˆ (ζ,ζ 0 ,K) = Σs ∩ Ω n (c) (c) [k]; [k] = [k](ζ,ζ 0 ,K) (z, z 0 ) ; (z, z 0 ) ∈ Xd−1 × Xd−1 , ˆ (ζ,ζ 0 ,K) = Σu ∩ Ω n (c) (c) [k]; [k] = [k](ζ,ζ 0 ,K) (z, z 0 ) ; (z, z 0 ) ∈ Xd−1 × Xd−1 ,

−z.z 0 = cosh v,

o v > vs (2.23)

o v > vu (2.24) where vs = vs (ζ, ζ 0 , t) and vu = vu (ζ, ζ 0 , t) are defined by the equations: z.z 0 = cosh v,

cosh vs − 1 =

s0 + (ρ − ρ0 ) − (w − w0 ) t , 2ρρ0

(2.25)

cosh vu − 1 =

u0 + (ρ − ρ0 ) − (w + w0 ) t , 2ρρ0

(2.26)

2

2

2

2

with ρ, w, ρ0 , w0 expressed by Equations (2.4), (2.5), (2.6). We then see that the images of these sets in the cos Θt -plane (by the projection ˆi introduced in Proposition 1) are the two real half-lines σ+ (vs ) = cosh vs , +∞ and σ − (vu ) = − ∞, − cosh vu (2.27)

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In the next section, the previous sets will appear as “cuts” bordering analyticity domains, namely the following “cut orbits” (for each (ζ, ζ 0 , K)): ˆ 0 ˆ (cut)0 Ω (ζ,ζ ,K) = Ω(ζ,ζ ,K) \ (Σs ∪ Σu ) ;

(2.28)

We also introduce correspondingly in ΩK the cut orbits: (cut)

Ω(ζ,ζ 0 ,K) = Ω(ζ,ζ 0 ,K) \ (Σs ∪ Σu ) ;

(2.29)

each of them is represented in the parametric variables z by the complex quadric (c) Xd−1 minus the cuts n o (c) (c) (2.30) Σ+ (vs ) = z ∈ Xd−1 ; z (d−1) ∈ [cosh vs , +∞[ and

n o (c) (c) Σ− (vu ) = z ∈ Xd−1 ; z (d−1) ∈ ] − ∞, − cosh vu ] .

(2.31)

As an immediate consequence of Proposition 1 one then has: (cut) ˆ (cut)0 ˆ Lemma 1 The projection of each set Ω (ζ,ζ ,K) by i and of each set Ω(ζ,ζ 0 ,K) by i onto the cos Θt -plane is the corresponding cut-plane Π(ρ,w,ρ0 ,w0 ,t) = C\ σ + (vs ) ∪ σ − (vu ) . (2.32)

entirely specified by formulas (2.25), (2.26) and (2.4)–(2.6).

III Perikernel structure of four-point functions in complex momentum space III.1

Four-point functions of local fields: Primitive analyticity domain and bounds

We here recall some basic results of the theory of four-point functions in the axiomatic framework of quantum field theory (see e.g. [13] and references therein). In this theory, one deals with the set of “generalized retarded functions” which are built from vacuum expectation values of the form: W (i) (x1 , x2 , x3 , x4 ) = hΩ, φi1 (xi1 ) φi2 (xi2 ) φi3 (xi3 ) φi4 (xi4 ) Ωi , (i) = (i1 , i2 , i3 , i4 ) denoting any permutation of (1, 2, 3, 4); here, the φj 0 s (j = 1, 2, 3, 4) denote local fields which satisfy mutually the postulate of local com2 mutativity: [φj (x) , φ` (y)] = 0 if (x − y) < 0 (j, ` = 1, 2, 3, 4). In view of the translation invariance of the theory, the so-called “Wightman functions” W (i) (x1 , x2 , x3 , x4 ) are defined in the space R4(d+1) /Rd+1 of the vector variables ξ = (ξi = xi − xi+1 , i = 1, 2, 3); in the standard formulation of Wightman field theory, they are defined as tempered distributions.

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The construction of the generalized retarded functions (g.r.f.) in terms of the Wightman functions requires the use of the algebra generated by multiple commutators of the fields together with step functions of the time-coordinates (0) (0) θ xi − xj [17,18,19]. The g.r.f. are special elements rα (x) of this algebra which have minimal support properties in the configuration space R4(d+1) in the following sense. In ξ-space (i.e. R4(d+1) /Rd+1 ) the support Γα of each g.r.f. rα (x) = ˆ α is a salient cone with rα (ξ) is a Lorentz-invariant cone whose convex hull Γ apex at the origin: each cone Γα is determined explicitly as a consequence of the postulate of local commutativity. It is assumed that the set of g.r.f. rα can be defined 2 as tempered distributions on R4(d+1) /Rd+1 satisfying the previously mentioned support properties. Analyticity and bounds in the tubes Tα : Analyticity in complex momentum space is readily obtained by introducing the Fourier-Laplace transforms of the g.r.f. rα . Due to translation invariance, the Fourier transforms of the g.r.f. rα (x) are of the form: δ (p1 + p2 + p3 + p4 ) r˜α (p) , where each r˜α (p) is a tempered distribution on the linear space M = {p = (p1 , . . . , p4 ) ; p1 + p2 + p3 + p4 = 0}. Let M (c) be the complexified of M, whose points are denoted by k = p + iq = (k1 , k2 , k3 , k4 ) , with k1 + k2 + k3 + k4 = 0. The support properties of the distributions rα (ξ) = rα (x) imply that one can define the corresponding Fourier-Laplace transforms (still denoted by) r˜α (k) , formally given by Z 1 r˜α (k) = ei(k.ξ) rα (ξ) dξ1 dξ2 dξ3 , (2π)2(d+1) with (k.ξ) = k1 .ξ1 + (k1 + k2 ) .ξ2 + (k1 + k2 + k3 ) .ξ3 ≡

4 X

ki xi ,

i=1

as holomorphic functions in the respective domains Tα = M + i Cα of M (c) , called “the tubes Tα with bases Cα ”. For each α, Cα is the (open) 3 dual cone ˆ α of the latter), namely: of the support Γα of rα (or of the convex hull Γ n o ˆα . Cα = q ∈ M ; (q.ξ) > 0 for all ξ ∈ Γ Moreover, as a consequence of the tempered character of rα (ξ) , r˜α (k) satisfies a global majorization of the following form in its domain Tα : h i |˜ rα (k)| 6 C max (1+ k k k)m , [d (q, ∂Cα )]−n (3.1) 2 The use of “sharp” time-ordered or retarded products (involving formally the product of distributions with the “sharp” step-function θ x(0) ) necessitates an extra-postulate with respect to the Wightman axioms (see e.g. the axiomatic presentations of [20] and [21]). 3 C is open and non-empty, since Γ ˆ α is a salient cone. α

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where k k denotes a euclidean norm in M (c) , d (q, ∂Cα ) denotes the corresponding distance from q to the boundary ∂Cα of Cα , and m ≥ 0, n ≥ 0. These numbers characterize the “degrees of temperateness” of the theory by taking into account respectively the dominant ultraviolet behaviour and the highest degree of local singularities of the four-point function in momentum space. In view of the role which they will be shown to play in complex angular momentum analysis, it is better to assume that they are general real numbers, i.e. not necessarily integers (as often assumed in standard Q.F.T.). Under these assumptions, each Fourier transform r˜α (p) is then rigorously characterized as the “distribution-boundary value” of the corresponding holomorphic function r˜α (k) from the tube Tα namely: Z lim r˜α (p + iq) ϕ (p) dp = h˜ rα , ϕi q→0, q∈Cα

for all test-functions ϕ (p) in the Schwartz space S (M ). We now recall the definition of the cones Cα (see [13] and references therein). Let α = ((αi ; i = 1, 2, 3, 4) , (αj` ; j, ` = 1, 2, 3, 4, j 6= `)) , where the αi and αj` are equal to +1 or −1. The corresponding cone Cα is defined by the following conditions: (3.2) αi qi ∈ V + , αj` (qj + q` ) ∈ V + ; i, j, ` = 1, 2, 3, 4, j 6= `; the condition of non-emptiness of Cα puts obvious constraints on the set α, such as αj` = −αmn if (j, `, m, n) = (1, 2, 3, 4) , αj` = αj if αj = α` , αn = −αj if αj = α` = αm etc. . . . Each cone Cα is represented conveniently by a simplicial triedron in R3(s1 ,s2 ,s3 ) whose faces are contained in three of the planes with equations si = 0, i = 1, 2, 3, s4 = − (s1 + s2 + s3 ) = 0, si + sj = 0, i, j = 1, 2, 3, or equivalently by a triangular cell determined by these planes on the unit sphere s21 + s22 + s23 = 1: we thus obtain the so-called “Steinmann sphere” representation of the tubes Tα of the four-point function in complex momentum space. The coincidence region R: It follows from the spectral conditions of the field theory considered that all the distributions r˜α (p) coincide in the following region R of M : R = p = (p1 , p2 , p3 , p4 ) ∈ M ; p2j < M2j , 1 ≤ j ≤ 4, o t ≡ (p1 + p2 )2 < t0, s ≡ (p1 + p3 )2 < s0 , u ≡ (p1 + p4 )2 < u0 , where the numbers Mj , 1 ≤ j ≤ 4 are mass thresholds associated with the corresponding fields, and t0 , s0 , u0 are the mass thresholds of the corresponding two-field channels. The region R is a star-shaped region with respect to the origin in M. The four-point function H (k): Since all the holomorphic functions r˜α (k) have boundary values on the reals which

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coincide on the region R, they admit a common analytic continuation denoted by H (k) whose existence results from the “edge-of-the-wedge theorem” (see [22] and references therein). This function H (k) , called the analytic four-point function in complex momentum space of the set of fields considered, is holomorphic in the following complex domain D of M (c) : ! [ [ D= Tα N (R) , α

where N (R) is a certain complex neighborhood of the region R (chosen for example as N (R) = {k = p + iq; p ∈ R, k q k< ε0 }). The bounds (3.1) on H (k) in the tubes Tα imply4 similar majorizations in N (R) , namely: h i |H (p + iq)| 6 C 0 max (1+ k p k)m , [d (p, ∂R)]−n , (3.3) where d (p, ∂R) is the distance from the real point p to the boundary of R. D is called the primitive axiomatic domain of the four-point function. D is not a “natural” holomorphy domain; this means that it admits a holomorphy envelope H (D) in which all functions which are holomorphic in D can be analytically continued [22]. Some general properties of the holomorphy envelope H (D): The problem of the determination of (parts of) the holomorphy envelope H (D) of D by means of various methods (such as the tube theorem, etc . . . [22]) is called “the analytic completion problem”. Although the complete knowledge of H (D) has not been obtained, the following general properties of this domain have been established [13] (the proof of a) and c) requires all the mass thresholds Mj , 1 ≤ j ≤ 4, s0 , t0 , u0 to be strictly positive). Theorem a) H (D) is a domain of M (c) which is star-shaped with respect to the origin, b) H (D) is invariant under the diagonal action of the complex Lorentz group (c) G(c) = SO0 (1, d) , namely if k = (k1 , k2 , k3 , k4 ) ∈ H (D) , then gk = (gk1 , gk2 , gk3 , gk4 ) ∈ H (D) , for every g in G(c) , c) For any fixed choice of the time-axis, the corresponding Euclidean subspace n o (0) E = k = (ki ; 1 6 i 6 4) ∈ M (c) ; ki = pi + iqi ; pi = (0, p~i ) , qi = (qi , 0) , is contained in H (D). d) The seven “spectral sets” Σs = {k ∈ M (c) ; s = (k1 + k3 )2 ≥ s0 }, Σu = {k ∈ M (c) ; u = (k1 + k4 )2 ≥ u0 }, Σt = {k ∈ M (c) ; t = (k1 + k2 )2 ≥ t0 }, Σ(j) = {k ∈ M (c) ; kj2 ≥ M2j }, 1 ≤ j ≤ 4, do not intersect H(D). 4 This

result can be obtained as a direct application of Proposition A.3.

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Absorptive parts: The axiomatic framework also provides a complete description of the structure and primitive analyticity domains of the off-shell absorptive parts, which are the discontinuity functions ∆s H, ∆u H, ∆t H of H in the respective s, u and t-channels. To be specific ∆s H (k) is a holomorphic function of k1 and k2 which is defined in the “face” {k; Im (k1 + k3 ) = 0} of the complex momentum space “triangulation” described above; its support is the intersection of the face Im (k1 + k3 ) = 0 with the corresponding “spectral set” Σs = k ∈ M (c) ; s ≥ s0 (also introduced with the notations of Section 2) in (2.19)). The primitive analyticity domain Ds of ∆s H in (k1 , k2 ) −space is the union of the four tubes {(k1 , k2 ) ; Imk1 ∈ ε1 V + , Imk2 ∈ ε2 V + ; ε1 , ε2 = + or −} connected together by a complex neighborhood of the region: (3.4) Rs = k real; k ∈ Σs ; kj2 < M2j , 1 ≤ j ≤ 4 . “Sections of maximal analyticity” or “cut-submanifolds” We shall say that a complex submanifold L of M (c) provides a section of maximal analyticity or a cut-submanifold of the domain D (resp. of its holomorphy envelope H (D)) for the s and u-channels if L∩D (resp. L∩H (D)) is equal to L\ (Σs ∪ Σu ). Such sections of H (D) will be produced below (see §3.3); in these sections, the jumps of H (k) across Σs and Σu are always equal to the analytic continuations of the corresponding absorptive parts ∆s H, ∆u H. In fact, the existence of the analytic continuation of H in L implies that the jumps ∆s H, ∆u H are obtained there as distributions in the real submanifold of L; they are defined as differences of boundary values of holomorphic functions from the complex regions of L, namely from new directions of Imk-space which belong to H (D) , although not to D. If L is one-dimensional, L\ (Σs ∪ Σu ) is called a “cut-plane section” of H (D). Complex Lorentz invariance of H (k): In the following, we shall restrict ourselves to the case of scalar local fields. In this case, the g.r.f. rα (x) are invariant under the (diagonal) action of the real connected Lorentz group G = SO0 (1, d); this invariance property is then satisfied by the corresponding Fourier-Laplace transforms r˜ (k) and therefore by H (k) in its analyticity domain D (D being itself invariant under this group). By a standard argument (based on the uniqueness of analytic continuation), it follows that H (k) is also invariant under the complex connected Lorentz group G(c) , i.e. H (ki , . . . , k4 ) = H (gki , . . . , gk4 ) for all g in G(c) , this property being satisfied in the whole holomorphy envelope H (D).

III.2

A simple step in the analytic completion problem

From now on, we adopt the notations of the t-channel kinematics given in Section 2, 0 0 0 0 namely we put k1 = −k3 , k2 = −k4 so that k1 + k2 = k1 + k2 = K, and we replace the notation k = (k1 , k2 , k3 , k4 ) of §3.1 by [k] = (k1 , k2 , k10 , k20 ); accordingly, the four-point function is now denoted H([k]).

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The step of the analytic completion problem which we shall perform will yield (c) domains in any subspace MK such that K 2 = t 6 0, with the coordinatization √ of Section 2: K = −t ed and (as specified in §2.1), k10 and k20 are real vectors varying in the (ed−1 , ed )-plane of the completed coordinate system. This step can be said to be “simple” because all the new points obtained are boundary points of the primitive domain D described in §3.1: in fact, the simple (c) geometrical property which we use is that each subspace MK is a linear manifold containing a common part of the boundaries of the following two tubes5 + T(1) = [k]; Imk10 ∈ V − , Imk20 ∈ V − , Imk1 ∈ V + and

− = [k]; Imk10 ∈ V + , Imk20 ∈ V + , Imk2 ∈ V − . T(2)

This “common face” (carried by the linear manifold Imk10 = Imk20 = 0) is the tube (c)

TK+ = {[k] ∈ MK ; Imk1 = −Imk2 ∈ VK+ }, where VK+ denotes the intersection of the hyperplane orthogonal to K (namely the space Rd spanned by e0 , e1 , . . . , ed−1 ) with the forward light cone V + of Rd+1 . Similarly we introduce the opposite tube TK− = {[k] ∈ MK ; Imk1 = −Imk2 ∈ VK− }, (c)

− + = −T(1) where VK− = −VK+ . TK− is the common face (in MK ) of the tubes T(1) + − and T(2) = −T(2) of the primitive domain D. The following statement is then contained in Theorem 4 of [23], but for simplicity and self-consistency of the present paper, we prefer to give here a direct 6 proof of this result (with the help of Appendix A). (c)

Proposition 2: a) H (D) contains the set of all points [k] in TK+ ∪TK− ; moreover, at all the points in TK+ (resp. TK− ), H([k]) admits a common analytic continuation from both + − − + and T(2) (resp. T(1) and T(2) ) of the primitive domain D. tubes T(1) b) The two sets TK+ and TK− are connected in H (D) by N (R) ∩ MK . (c)

ˆ = (kˆ1 , kˆ2 , kˆ0 , kˆ0 ) be any real momentum configuration in M (c) conProof. Let [k] 1 2 K ˆ belongs to the region tained in the (e1 , e2 , . . . , ed )-hyperplane of coordinates ; [k] 0 R, since all quantities kˆi2 , kˆi2 , (kˆi − kˆj0 )2 , i, j = 1, 2 and (kˆ1 + kˆ2 )2 = t are ≤ 0. 5 these tubes are the analyticity domains of the Laplace transforms of the ordinary advanced and retarded four-point functions a(2) and r(1) 6 the proof given in [23] makes use of a theorem by Bremermann and relies on a condition of coincidence for adjacent tubes which, for simplicity, we have omitted in §3.1.

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+ We shall now exhibit a two-dimensional (complex) section of the tubes T(1) and − T(1) of the domain D by putting 0

0

0

0

0

0

k1 = kˆ1 + ηe, k1 = kˆ1 − η e, k2 = kˆ2 − η e,

(3.5)

with e fixed in VK+ . This two-dimensional section is represented by the union of the tubes T+ , T− 0 of η, η space described in Proposition A-1. Now, it is clear that there exists 0 a square of the form |η| < a, η < a in R2 whose image by the mapping (3.5) 0

belongs to the region R (since the point η = η = 0 represents the configuration ˆ which belongs to R). [k] 0 0 Corollary A-2 then implies that all the points [k] = (k1 = ηe + kˆ1 , kˆ1 , kˆ2 ) such that either Imη > 0 or Imη < 0 or η ∈ ]−a, +a[ lie in H (D) ; since this holds for every choice of e in VK+ and kˆ1 in the (e1 , e2 , . . . , ed )-hyperplane, it is thus proved that all points in TK+ (resp. TK− ) appear as points of analyticity for H([k]) obtained + − from the tube T(1) (resp. T(1) ). − + A similar argument based on a two-dimensional section of T(2) ∪ T(2) would + − exhibit all the points in TK (resp. TK ) as points of analyticity obtained from the − + tube T(2) (resp. T(2) ). The fact that the analytic continuations of H([k]) obtained by these two procedures coincide results from the principle of uniqueness of analytic continuation, since both of them coincide in the intersection of TK± with the edge-of-the-wedge neighborhood N (R) (contained in D). Point a) of Proposition 2 is thus proved and point b) is then trivial. We shall now restate the result of Proposition 2 in terms of the variables (c) introduced in Section 2. A parametrization of MK is given by Equations (2.2) in which ρz = k = (k(0) , . . . , k(d−1) , 0) varies in Cd , z 0 = z0 = (0, . . . , 0, 1, 0) and w, ρ0 , w0 are real (ρ0 ≥ 0), namely [k] = [k](k; w, ρ0 , w0 , K) ≡ 1 1 k1 = k + (w + )K, k2 = −k − (w − )K, 2 2 0 1 1 0 0 0 0 k1 = ρ z0 + (w + )K, k2 = −ρ z0 − (w0 − )K, 2 2

(3.6)

Proposition 3 Let D(w,w0 ,ρ0 ) be the following domain in the space Cd of the complex vector k: D(w,w0 ,ρ0 ) = T + ∪ T − ∪ N R(w,w0 ,ρ0 ) , where: a) T ± = Rd + iV ± , with h i1 2 2 2 V + = −V − = {q ∈ Rd ; q (0) > q (1) + · · · + q (d−1) }

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b) N R(w,w0 ,ρ0 ) is a suitable complex neighborhood of the following region R(w,w0 ,ρ0 ) = k ∈ Rd ; k2 < µ2 , (k − ρ0 zo )2 < µ2s , (k + ρ0 zo )2 < µ2u ; in the latter, the constants, µ2 , µ2s , µ2u are defined in terms of the mass thresholds M21 , M22 , s0 , u0 by the following expressions 2 2 ! 1 1 2 2 2 , , M2 − t w − µ = min M1 − t w + 2 2 0 2 0 2 µ2s = s0 − t w − w , µ2u = u0 − t w + w . Then H (D) contains the union of the following sets: n o ˆ (w,w0 ,ρ0 ) = [k] ∈ M (c) ; [k] = [k] (k; w, ρ0 , w0 , K) ; k ∈ D(w,w0 ,ρ0 ) , D K 0

0

for all real values of w, w and ρ

0 ρ >0 .

Proof. This statement is a direct consequence of Proposition 2 since (in view of the parametrization (3.6)) R(w,w0 ,ρ0 ) is the trace of R in complex k-space, for fixed 0 0 values of w, w , ρ . We shall now prove that bounds of the type (3-1) are satisfied by the analytic continuation of the four-point function H ([k]) in the regions described in Proposition 3; this is a simple example of the extension to points of H (D) of bounds which are prescribed in the primitive domain D. Proposition 4. Bounds of the following form are satisfied by H ([k](k; w, ρ0 , w0 , K)) for k varying in the domains D(w,w0 ,ρ0 ) of Proposition 3. −n m 0 0 |H ([k] (k; w, ρ , w , K))| 6 Cw,ρ0 ,w0 max (1 + kkk) , d k, ∂D(w,w0 ,ρ0 ) (3.7) where k k k2 =

X

(i) 2 k

06i6d−1

and m, n are the same numbers as in formula (3.1) (m ≥ 0, n ≥ 0). Proof. For w, w0 , ρ0 fixed, we consider the section of the primitive domain D by the following complex submanifold parametrized by k and η, k ∈ Cd , η ∈ C: k1 = k + w + 1 K + ηe0 k10 = ρ0 z0 + w0 + 1 K + ηe0 2 2 [k] = [k] (k, η) k2 = −k − w − 12 K + ηe0 k20 = −ρ0 z0 − w0 − 12 K + ηe0 (3.8) Let us first consider the case when k varies in the tube T + . One then checks that if η varies in a strip 0 < Imη < h (k) such that Imk − h (k) e0 ∈ ∂V+ ,

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− the corresponding point [k] = [k] (k, η) defined by (3.8) varies in the tube T(2) = {[k]; Imk10 ∈ V + , Imk20 ∈ V + , Imk2 ∈ V − } of the domain D. Similarly, for η varying in the strip −h (k) < Imη < 0, the corresponding point [k] = [k](k, η) varies in the + tube T(1) = {[k]; Imk10 ∈ V − , Imk20 ∈ V − , Imk1 ∈ V + } .

Moreover, when η varies in a real interval [−a, +a] such that k102 , k202 and t = 2 (k10 + k20 ) remain negative, the boundary values of H ([k]) from the two previous tubes (i.e. from the sides Imη > 0, and Imη < 0) define a common analytic continuation of the corresponding two branches of H ([k](k, η)) across the interval [−a, +a]: this follows from the application of Proposition 2 to all situations such that k10 + k20 = K + 2ηe0 (which is legitimate for η ∈ [−a, +a]). − We shall now consider the majorizations (3.1) of H ([k]) in the tubes T(2) and + T(1) and give their expressions in terms of the complex variables k and η when [k] belongs to the submanifold (3.8). For k in T + , the following majorizations follow from (3.1), if η varies in the set {η ∈ C; |Reη| < a, 0 < |Imη| < h (k)}: h i |H ([k] (k, η))| 6 C max (1+ k k k)m , |Imη|−n , (h (k) − |Imη|)−n

(3.9)

where C is a suitable constant. In order to obtain a bound for H ([k](k, η))at η = 0, i.e. at the corresponding ˆ (w,w0 ,ρ0 ) , it is appropriate to point [k] = [k] (k; w, ρ0 , w0 , K) (see (3.6)) of the set D apply Proposition A-3 to the function f (η) = H ([k] (k, η)) in a square ∆b such −m √ , (1+ k k k) n , a . In fact, one checks that in this domain ∆b that b = min h(k) 2 √ the majorization (A-1) is implied by (3.9) (with M = C( 2 − 1)−n ). We can therefore apply the majorization (A-2) which yields (for the chosen value of b): "

m

|H ([k] (k, 0))| 6 cn C max (1+ k k k) ,

h (k) √ 2

−n

# , a−n .

(3.10)

h (k) Since d k, ∂T + = √ and the constant a is independent of k, the inequality 2 (3.10) can be replaced by h −n i m |H ([k] (k, 0))| 6 C 0 max (1+ k k k) , d k, ∂T + ,

(3.11)

(C 0 being a new constant). The previous argument holds similarly, when k varies in the tube T − (the + − tubes T(1) and T(2) being now replaced by their opposites) and yields: h −n i m |H ([k] (k, 0))| 6 C 0 max (1+ k k k) , d k, ∂T −

(3.110 )

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Finally, when k belongs to N R(w,w0 ,ρ0 ) , the majorization (3.3) yields im7

(by restriction to the submanifold (3.8)): −n H [k] p + iq, 0 6 C 00 max 1+ k p k m , d p, ∂R(w,w0 ,ρ0 )

mediately

(3.12)

The set of majorizations (3.11), (3.110 ), (3.12) is equivalent to the global majorization (3.7) in the domain D(w,w0 ,ρ0 ) .

III.3

ˆ (c) The perikernel structure in the space M K

In this subsection, we shall establish the analyticity properties and bounds of the four-point function H ([k]) which are necessary for introducing (in the next section) an appropriate Laplace-type transform of H in a complex angular-momentum variable associated with the t-channel. These analyticity properties and bounds are direct applications of the results of Propositions 3 and 4, which will be completed in a second step by making use of the property of complex Lorentz invariance of H ([k]). We first consider the following family of one-dimensional complex submani(c) folds ω(ζ,ζ 0 ,K) in MK . With each (ζ, ζ 0 , K) such that t < 0, ζ ∈ ∆t \∂∆t , ζ 0 ∈ ∆t (see Equation (2.13)), we associate the complex hyperbola ω(ζ,ζ 0 ,K) = [k]; [k] = [k](ζ,ζ 0 ,K) (z, z0 ) ; z = (−i sin θ, 0, . . . , 0, cos θ) , θ ∈ C (3.13) where [k] = [k](ζ,ζ 0 ,K) (z, z 0 ) is the mapping defined by Equations (2.16). Each hyperbola ω(ζ,ζ 0 ,K) appears to be the meridian hyperbola in the (eo , ed−1 ) plane of the corresponding hyperboloid Ω(ζ,ζ 0 ,K) (see Equation (2.17)). We shall then prove: Proposition 5 a) For each (ζ, ζ 0 , K) with ζ ∈ ∆t \∂∆t , ζ 0 ∈ ∆t , the submanifold ω(ζ,ζ 0 ,K) provides a section of maximal analyticity of H (D) which is the cut-domain (cut) ω(ζ,ζ 0 ,K) = ω(ζ,ζ 0 ,K) \ (Σs ∪ Σu ) ; Σs , Σu are the spectral sets defined by Equations (2.19) (2.20). (cut) b) The domain ω(ζ,ζ 0 ,K) is represented in the 2π-periodic θ-plane as the following cut-plane: Π(ρ,w,ρ0 ,w0 ,t) = C\ {σ+ (vs ) ∪ σ− (vu )} , (3.15) where: σ+ (vs ) = {θ ∈ C; θ = iv + 2`π, |v| > vs , ` ∈ Z} , σ− (vu ) = {θ ∈ C; θ = iv + (2` + 1) π, |v| > vu , ` ∈ Z} .

(3.16) (3.17)

(with vs , vu defined by Equations (2.25), (2.26)). 7 The majorization (3.12) can also be obtained directly from (3.11), (3.110 ) and the analyticity of H in R(w,w0 ,ρ0 ) by applying again Proposition A.3.

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c) The restriction of the function H ([k]) to each submanifold ω(ζ,ζ 0 ,K) is well defined as a 2π-periodic function: , (3.18) Hω(ζ,ζ0 ,K ) (θ) = H [k](ζ,ζ 0 ,K) (z, z0 ) | z=(−i sin θ,0,...,0,cos θ) which is holomorphic in the domain Π(ρ,w,ρ0 ,w0 ,t) and satisfies bounds of the following form: −n (3.19) Hω(ζ,ζ0 ,K ) (θ) 6 C(ζ,ζ 0 ,K) em∗ |Imθ| [d (θ, σ+ (vs ) ∪ σ− (vu ))] . in the latter m∗ = max (m, n) and C(ζ,ζ 0 ,K) is a suitable constant. Proof. a) We shall prove that for every (ζ, ζ 0 , K) with ζ ∈ ∆t \∂∆t , ζ 0 ∈ ∆t , the (cut) ˆ (w,w0 ,ρ0 ) of H (D) cut-domain ω(ζ,ζ 0 ,K) is contained in the corresponding subset D obtained in Proposition 3. In fact, each point [k] in ω(ζ,ζ 0 ,K) is such that [k] = [k] (k; w, ρ0 , w0 , K) (see Equations (3.6)), with k = ρz, z = (−i sin θ, 0, . . . , 0, cos θ). By putting θ = u + iv, we check that: (Imk)2 = ρ2 (Imz)2 = ρ2 sin2 u > 0

(3.20)

Since we have assumed that ζ ∈ / ∂∆t , i.e. ρ 6= 0, we see from (3.20) that all the complex points [k] of ω(ζ,ζ 0 ,K) are represented by vectors k such that (Imk)2 > 0, which means that k belongs either to T + or to T − and therefore to the domain D(w,w0 ,ρ0 ) of Proposition 3. Moreover, the real points of ω(ζ,ζ 0 ,K) are represented by real vectors k, such that k2 = ρ2 z 2 = −ρ2 < 0; therefore, they belong to D(w,w0 ,ρ0 ) , i.e. to the region R(w,w0 ,ρ0 ) , if and only if they do not belong to the union of the spectral sets Σs (cut)

and Σu . This proves that the domain ω(ζ,ζ 0 ,K) = ω(ζ,ζ 0 ,K) \ (Σs ∪ Σu ) is contained in H (D). Since all points in Σs ∪Σu are outside H (D) (see d) of Theorem in §3.1), (cut) ω(ζ,ζ 0 ,K) is actually the intersection of H (D) with ω(ζ,ζ 0 ,K) . (cut)

b) In view of (3.20), all the complex points of ω(ζ,ζ 0 ,K) are represented in the θ-plane by the set {θ = u + iv; u 6= `π, ` ∈ Z}; the real points form two disjoint sets, represented respectively by {θ = iv + 2`π;|v| < vs ,` ∈ Z} and {θ = iv + (2` + 1)π; (cut) |v| < vu , ` ∈ Z}. This shows that ω(ζ,ζ 0 ,K) is represented by the periodic cut-plane Π(ρ,w,ρ0 ,w0 ,t) . c) We shall apply the majorizations of Proposition 4 to the present situation, in which k = ρz, z = (−i sin θ, 0, . . . , 0, cos θ) , θ = u + iv. Since Rek = (ρ cos u sinh v, 0, . . . , 0, ρ cos u cosh v) Imk = (−ρ sin u cosh v, 0, . . . , 0, −ρ sin u sinh v) we get:

k k k2 =k Rek k2 + k Imk k2 = ρ2 2 cosh2 v − 1

(3.21)

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On the other hand, for k ∈ T ± , we have: d k, ∂T ± = inf Im k (0) + k(d−1) , Im k(0) − k(d−1) = ρ |sin u| e−|v| . (3.22) In this situation, which corresponds to sin u 6= 0, the majorizations (3.11), (3.110 ) therefore yield (in view of Equations (3.21), (3.22)): m √ , Hω(ζ,ζ0 ,K ) (u + iv) 6 C max 1 + ρ 2 cosh v

en|v| n , ρn |sin u|

(3.23)

which implies a bound of the form (3.19) when θ varies in Π(ρ,w,ρ0 ,w0 ,t) by staying outside neighborhoods of the intervals {θ = iv + 2`π; |v| < vs , ` ∈ Z} and one {θ = iv + (2` + 1) π, |v| < vu ; ` ∈ Z}. When θ varies in theseneighborhoods, makes use of the majorization (3.12) (since in this case k ∈ N R(w,w0 ,ρ0 ) ), which completes the proof of the bound (3.19). Remark. In the limiting case where ρ = 0, i.e. ζ ∈ ∂∆t , the set ω(ζ,ζ 0 ,K) reduces to a single point [k], which belongs to R, and therefore to H (D) , but the statement of Proposition 5 has a trivial content; we notice that (according to the expressions (2.25), (2.26) of vs , vu ) the cuts σ+ (vs ) , σ− (vu ) of Π(ρ,w,ρ0 ,w0 ) are shifted up to infinity when ρ tends to zero. We shall now extend the previous analyticity properties of H ([k]) to the ˆ (ζ,ζ 0 ,K) (see Equations (2.17), (2.15)) by exploiting the manifolds Ω(ζ,ζ 0 ,K) and Ω Lorentz invariance of H. We shall first use the invariance of H ([k]) under the subgroup of complex Lorentz transformations which leave the vectors k10 , k20 un(c) changed. When K 6= 0, this is the subgroup G(c) = SO0 (1, d − 2) which leaves the (ed−1 , ed )-plane of coordinates unchanged. In this case, H ([k]) is then holomorphic and constant at all points [k] = [k] (gk; w, ρ0 , w0 ) deduced from the points (cut) [k] (k; w, ρ0 , w0 ) in ω(ζ,ζ 0 ,K) by the action of any element g in G(c) . For K = 0, the (c)

analysis is similar, except that the group G(c) is now the subgroup SO0 (1, d − 1) which leaves the point z0 (i.e. ed−1 ) unchanged. ˆ in ω(ζ,ζ 0 ,K) , represented by a vector kˆ = ρˆ In particular, for each point [k] z ˇ = [k] k; ˇ w, ρ0 , w0 , with zˆ = (−i sin θ, 0, . . . , 0, cos θ) , the corresponding point [k] obtained by the symmetry θ → −θ (namely such that kˇ = ρˇ z , with zˇ = (i sin θ, 0, ˆ w; ρ0 , w0 )} of G(c) (in fact, one can . . . , 0, cos θ)) belongs to the orbit {[k] = [k](g k; (c) ˆ ˆ such that: kˇ = gk→ find an element gk→ ˆ k ˇ of G ˆ k ˇ (k)). It follows that H([k]) = ˇ H([k]); correspondingly Hω(ζ,ζ,K) (θ) is an even function of θ and therefore a holomorphic function of cos θ which we denote by H (ζ,ζ 0 ,K) (cos θ). The domain of the latter, which is the image of Π(ρ,w,ρ0 ,w0 ,t) onto the cos θ-plane, is the cut-plane Π(ρ,w,ρ0 ,w0 ,t) introduced in (2.32).

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In view of Lemma 1 one can now define H([k]) in each cut-domain Ω(ζ,ζ 0 ,K) of Ω(ζ,ζ 0 ,K) (see Equations (2.17), (2.29)) as the G(c) -invariant function H([k]) = H (ζ,ζ 0 ,K) (cos Θt ) ,

(3.24)

where [k] ≡ ((ζ, ζ 0 , K), (z, z0 )) and cos Θt = −z.z0 = z (d−1) ; Θt is the off-shell scattering angle introduced in (2.3) (with here z 0 = z0 ) and cos Θt therefore coincides with the variable cos θ of the parametrization (3.13) when z belongs to the meridian hyperbola ω(ζ,ζ 0 ,K) of Ω(ζ,ζ 0 ,K) . We can thus state Proposition 6 For every submanifold Ω(ζ,ζ 0 ,K) in ΩK (with ζ ∈ ∆t \∂∆t , ζ 0 ∈ (cut) ∆t ), the corresponding “cut-submanifold” Ω(ζ,ζ 0 ,K) belongs to H (D) . In each of these submanifolds, the restriction of the function H ([k]) is invariant under the group G(c) and can be identified with a holomorphic function H (ζ,ζ 0 ,K) (cos θ) whose domain is the cut-plane Π(ρ,w,ρ0 ,w0 ,t) . Moreover, the jumps of H ([k]) across the two cuts Σs and Σu in Ω(ζ,ζ 0 ,K) (or equivalently the jumps of H (ζ,ζ 0 ,K) (cos θ) across the cuts σ + (vs ) and σ − (vu )) are the corresponding restrictions of the absorptive parts ∆s H and ∆u H of H. (For a complete justification of the last statement in Proposition 6, we refer the reader to the paragraph “Absorptive parts” in §3.1.) ˆ (cut)0 One similarly extends H([k]) to the cut-domains Ω (ζ,ζ ,K) of the submanifolds ˆ (ζ,ζ 0 ,K) (see Equations (2.15), (2.16), (2.28)) by now using formula (3.24) with Ω [k] ≡ ((ζ, ζ 0 , K), (z, z 0 )) and cos Θt = −z.z 0 . By also taking into account the bounds (3.19) on Hω(ζ,ζ0 ,K) (Θt ) = H (ζ,ζ 0 ,K) (cos Θt ) , one can then state: ˆ K (with ζ, ζ 0 ∈ ˆ (ζ,ζ 0 ,K) in Ω Theorem 1 For every submanifold Ω / ∂∆t ), the corre(cut) ˆ 0 sponding “cut-submanifold” Ω (ζ,ζ ,K) belongs to H (D). The restriction of H ([k]) to each of these submanifolds defines an “invariant perikernel of moderate growth with distribution boundary values” on the corresponding complexified hyperboloid (c) (c) Xd−1 (if K 6= 0) or Xd (if K = 0). This invariant perikernel H [k](ζ,ζ 0 ,K) (z, z 0 ) is holomorphic on the domain (c) (c) (c) (c) of Xd−1 × Xd−1 (resp. Xd × Xd ) which is defined as the complement of the 0 0 union of the cuts {(z, z ) ; z.z 6 − cosh vs } and {(z, z 0 ) ; z.z 0 > cosh vu }. It can be identified with the holomorphic function H (ζ,ζ 0 ,K) (−z.z 0 ) of the single variable cos Θt = −z.z 0 , Θt being the off-shell scattering angle of the t-channel. The domain of this function is the cut-plane Π(ρ,w,ρ0 ,w0 ,t) and its growth is controlled by the following bounds in terms of u = ReΘt and v = ImΘt : −n (3.25) H (ζ,ζ 0 ,K) (cos (u + iv)) 6 C (ζ,ζ 0 ,K) em∗ |v| |sin u| if cos Θt ∈ / R, and: −n −n H (ζ,ζ 0 ,K) (cos Θt ) 6 C (ζ,ζ 0 ,K) |cos Θt − cosh vs | |cos Θt + cosh vu | , (3.26)

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if cos Θt belongs to a neighborhood of the real interval ]− cosh vu , cosh vs [. In these bounds, m∗ = max(m, n), m and n being the “degrees of temperateness” of the theory (introduced in (3.1)). The notion of “invariant perikernel of moderate growth on a complexified hyperboloid” has been introduced in [24,25] as an appropriate notion for studying the Laplace transformation associated with the complexified Lorentz group. While the term “perikernel” refers to the analyticity property of H [k](ζ,ζ 0 K) (z, z 0 ) in the cut-domain described above, its “invariant” character means that it satis (c) fies the condition H [k](ζ,ζ 0 K) (gz, gz 0 ) = H [k](ζ,ζ 0 K) (z, z 0 ) for all g in GK . Finally the property of “moderate growth”, characterized by the bounds (3.25), (3.26), fits with the definition given in [25] as far as the behaviour at infinity is concerned. However, the present perikernels have distribution-like (instead of continuous) boundary values on the reals. / ∂∆t in Proposition 6 and Theorem 1 simRemark. The condition ζ (and ζ 0 )∈ ply expresses the non-degeneracy of the corresponding submanifolds Ω(ζ,ζ 0 ,K) and ˆ (ζ,ζ 0 ,K) . In the degenerate cases these sets are trivially contained in H(D) (see Ω our previous remark after Proposition 5).

IV Harmonic analysis of the four-point functions of scalar fields Having established in Theorem 1 the perikernel structure of a four-point function H ([k]) relatively to a given t-channel we are now in a position to apply the results of [25] which concern the harmonic analysis of invariant perikernels of moderate (c) growth on the complexified (unit) hyperboloid Xd−1 . We shall give a self-contained account of these results in §4-1 for the case d = 2 and in §4-2 for the general case d > 2. As a matter of fact, we need to present an extended version of the results of [25] which includes: a) the presence of two cuts (instead of one, as in [25]) in the definition of the analyticity domain of the perikernels (the corresponding results have already been announced and their derivation outlined in [10]) b) the occurrence of perikernels with distribution-like boundary values on the reals (as previously noticed). Concerning the rigourous treatment of b), our arguments will make use of results proved in Appendix B. In §4-3, we come back to our analysis of the four-point function H([k]) of scalar local fields, also written in terms of the t-channel variables as follows: H([k]) = F (ζ, ζ 0 ; t, cos Θt ) .

ˆK Ω

(4.1)

ˆ (ζ,ζ 0 ,K) of the “Lorentz-foliation” of The restrictions of H to the manifolds Ω defined in Equations (2.15), (2.16) can then be identified with the perikernels

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of Theorem 1 (in their reduced form H), namely: for all (ζ, ζ 0 , t) with (ζ, ζ 0 ) in ∆t × ∆t , F (ζ, ζ 0 ; t, cos Θt ) = H (ζ,ζ 0 ,K) (cos Θt )

(4.10 )

Applying the results of §4.1 and §4.2 will then directly lead us to introduce and describe the properties of a Fourier-Laplace-type integral transform F˜ (ζ, ζ 0 ; t, λt ) of F (ζ, ζ 0 ; t, cos Θt ) which interpolates analytically in an appropriate way the set of (off-shell) t-channel partial-waves: 0

Z

f` (ζ, ζ , t) = ωd−1

+1

−1

(d)

P`

(cos Θt ) F (ζ, ζ 0 ; t, cos Θt ) [sin Θt ]d−3 d cos Θt

(4.2) (d)

in a half-plane of the complex variable λt . In Equation (4.2), the functions P` are the “ultraspherical Legendre polynomials” considered in [26] 8 (chapter 1 §2), which reduce to cos `Θt for d = 2 and to the Legendre polynomials P` for d = 3; they are given (for d ≥ 3) by the following integral representation (see Equation (III.18) of [25c)]): Z ωd−2 π (d) P` (cos t) = (cos t + i sin t cos φ)` (sin φ)d−3 dφ. (4.3) ωd−1 0 In (4.2) and (4.3), ωd−1 denotes the area of the sphere Sd−2 . In §4.4, it is shown that the previous property of analytic interpolation of the f` in the variable λt is indeed equivalent to the property of analytic continuation of ˆ K : more precisely, the Euclidean four-point function into the Lorentz-foliation of Ω this structure is characterized by kernels of the Euclidean sphere-foliation of EˆK which are analytically continued into perikernels.

IV.1

Fourier-Laplace transformation on cut-domains (c) of the complexified hyperbola X1

(0) (1) = z = −isin θ, z = cos θ, θ = u + iv , one (c) (c) (c) considers the domain D = X1 \ Σ+ ∪ Σ− (Fig. 2), whose representation in the θ-plane is the periodic cut-plane Π = C\ (σ+ ∪ σ− ) . The cuts σ+ , σ− are of the form (3.16), (3.17) (with vs = v+ , vu = v− ) and the corresponding subsets (c) (c) (c) Σ+ , Σ− of X1 are given (as in (2.30) and (2.31)) by: (c)

On the complex hyperbola X1

n o (c) (c) Σ± = z = z (0) , z (1) ∈ X1 ; ±z (1) ∈ [cosh v± , +∞[

(4.4)

polynomials are proportional to the Gegenbauer polynomials C`p considered in chapter IX of [27] (see in the latter Equation (6) of §4.7, which coincides with (4.3) for p = d−2 up to 2 the normalization constant). 8 These

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z (0) Σ+

iy S1 − cosh v−

cosh v+ x

z (1) X1

Σ−

Figure 2 (c) Note that the circle S1 = z = (iy0 , x1 ) ; y02 + x21 = 1; y0 , x1 real of X1 , represented by the periodic u-axis in Π, is contained in the domain D. (c) An invariant perikernel K (z, z 0 ) on X1 is identified with a function F (z) = (c) K (z, z0 ) (where z0 = (0, 1)) holomorphic in the domain D of X1 and depending (1) only on the variable z = cos θ. Representing F by the even periodic function f (θ) = F (z (θ)) , holomorphic in the cut-plane Π, one then defines [25a)] the following Fourier-Laplace-type transform F˜ = L (f ): Z F˜ (λ) = eiλθ f (θ) dθ (4.5) γ

with the prescription of Fig. 3 for the contour γ. (γ “encloses” the components of σ+ , σ− inside the half-strip: v > 0, −α m} provided F is a perikernel of moderate growth satisfying the following bound in D: m |F (z)| 6 cst 1 + z (1) , (4.6) or equivalently provided f satisfies the following one in Π: |f (u + iv)| 6 cst em|v| ,

(4.7)

Let us first assume that f admits continuous boundary values (from both sides) on the cuts σ+ and σ− and call ∆f+ (v) , ∆f− (v) the corresponding jumps of if , which it is sufficient to consider in the upper half-plane (v > 0): ∆f+ (v) = i lim (f (ε + iv) − f (−ε + iv)) ,

(4.8)

∆f− (v) = i lim (f (π + ε + iv) − f (π − ε + iv))

(4.9)

ε→0 ε→0

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v σ−

σ+ γ+ v+

γ−

v−

−α

0

π

2π−α 2π u

Figure 3

Assuming that the bound (4.7) is uniform in Π and therefore applies to the discontinuity functions ∆f+ , ∆f− of if, the Laplace transforms of ∆f+ , ∆f− Z∞ F˜± (λ) =

e−λv ∆f± (v) dv

(4.10)

0 (m)

are holomorphic in C+ . Then applying a simple contour-distortion argument to (m) the integral (4.5) yields the following relations, valid in C+ : i)

F˜ (λ) = F˜+ (λ) + eiπλ F˜− (λ) .

This follows from replacing the contour γ by a pair of contours (γ+ , γ− ) enclosing respectively the cuts σ+ , σ− and then from flattening them (in a folded way) onto the cuts (see Fig. 3). ii)

F˜ (`) = f` , for all integers ` such that ` > m, 2π−α Z

ei`u f (u) du.

where f` = −α

This follows from choosing γ = γα with support ]−α + i∞, −α] ∪ [−α, 2π − α] ∪ [2π − α, 2π − α + i∞[ , and taking into account the 2π-periodicity of the integrand of (4.5) for λ = ` integer. We note that the Fourier coefficients f` of f (u) are associated with the (ro(c) tational invariant) kernel K (z, z 0 ) on the “imaginary circle” S1 of X1 which is

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obtained by taking the restriction of the perikernel K (z, z 0 ) , namely K = K|S1 ×S1 and f (u) = K (z, z 0 ) with cosu = −z.z 0 = y0 y00 + x1 x01 . We now state in a more detailed form an extension of the previous properties which applies to the case when f (resp. F or K) admits distribution-like boundary values (and discontinuities) on the cuts which border its domain. Theorem 2 Let f (θ) be a (2π-periodic) even holomorphic function in the cut(c) plane Π (representing an invariant perikernel of moderate growth K (z, z 0 ) on X1 ) satisfying uniform bounds of the following form (with m and β fixed, m ∈ R, β ≥ 0): |f (u + iv)| 6 Cη−β emv ,

(4.11)

+ in all the corresponding subsets Π+ η (η > 0) of Π = Π ∩ {θ ∈ C; Imθ > 0} : + Π+ η = Π \ {θ ∈ C; θ = u + iv, |u − 2nπ| < η, n ∈ Z, v > v+ − η}

\ {θ ∈ C; θ = u + iv, |u − (2n − 1) π| < η, n ∈ Z, v > v− − η}

(4.12)

Then, i) The “discontinuity functions” ∆f+ , ∆f− of if across the cuts σ+ , σ− are well defined in the sense of distributions, and admit Laplace-transforms F˜+ (λ) , (m) F˜− (λ) which are holomorphic in C+ and satisfy uniform bounds of the 0 following form (for all ε, ε > 0): 0 (ε,ε0 ) ˜ |λ − m|β+ε e−[Reλ−(m+ε)]v± (4.13) F± (λ) 6 C± (m+ε)

in the corresponding half-planes C+

.

ii) The transform F˜ = L (f ) of f, namely F˜ (λ) =

R

eiλθ f (θ) dθ, is holomorphic

γ (m)

in C+ a)

and satisfies the following properties: F˜ (λ) = F˜+ (λ) + eiπλ F˜− (λ)

(4.14)

b) for all integers ` such that ` > m, the Fourier coefficients of f|R , namely 2π−α Z

ei`u f (u) du,

f` =

(4.15)

−α

are given by the following relations: f` = F˜ (`)

(4.16)

Proof. i) The validity of the bounds (4.11) on the function f (which characterize it as a “function of moderate growth” near its boundary set σ+ ∪σ− ) is equivalent (see e.g. [28]) to the fact that f admits boundary values in the sense of distributions

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on iR and π + iR (from both sides of each of these lines) and therefore that the discontinuities ∆f+ , ∆f− are defined as distributions with respective supports σ+ , σ− . We refer the reader to Proposition B.4, for a comprehensive study of holomorphic functions of this type, considered as derivatives (of integral or nonintegral order) of holomorphic functions with continuous boundary values. In view of the exponential factor in (4.11), the Laplace transforms F˜+ (λ) , (m) ˜ F− (λ) of ∆f+ , ∆f− can always be defined as holomorphic functions in C+ by the following contour integrals: Z F˜+ (λ) =

eiλθ f (θ) dθ, γ+

F˜− (λ) = e−iπλ

(4.17) Z

Z iλθ

e

γ−

eiλθ f (θ + π) dθ

f (θ) dθ =

(4.18)

γ+

(γ+ , γ− being chosen as in Fig. 3 with suppγ− = {θ = π + θ0 , θ0 ∈ suppγ+ }). When γ± is flattened onto σ± , the limit of the r.h.s. of (4.17) (resp. (4.18)) can now be seen as the action of the distribution ∆f+ (v) (resp ∆f− (v)) on the test-function e−λv (the latter being admissible for Reλ > m). The derivation of the bounds (4.13) on F˜± (λ) , which relies on a technique of Abel transforms (or primitives of non-integral order) is given in Proposition B.4. The latter must be applied to the functions fm+ (θ) = eimθ f (θ) and fm− (θ) = (cut) eim(θ+π) f (θ+π), which (in view of (4.11) and (4.12)) belong to the class Oβ (Ba ) (cut) (e.g. a = π2 ) as described in Appendix B (see of an appropriate domain Ba Fig. B1). The majorization (B.19) then applies to the Laplace transforms F˜m± of fm± , which are such that F˜± (λ) = F˜m± (λ − m), thus yielding the desired result (4.13). ii) The proof of the relations (4.14) and (4.16) relies on the contour-distortion argument presented above in the case where f (θ) has continuous boundary values. Remark. In view of Equation (4.14), the relations (4.16) yield: for ` even, for ` odd,

f` = F˜+ (`) + F˜− (`) f` = F˜+ (`) − F˜− (`)

(4.19) (4.20)

Since the holomorphic functions F˜± (λ) satisfy the bounds (4.13), which are in (m) particular dominated by any exponential function eε|λ| (ε > 0) in C+ , these functions appear respectively as the (unique) Carlsonian interpolations [29] of the corresponding sequences {F˜± (`) ; ` ∈ N, ` > m}. However the function F˜ (λ) itself (m) (which behaves like e−πImλ in C+ ) does not satisfy the Carlsonian property with respect to the sequence {f` } which it interpolates.

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This remark suggests the introduction of the following “symmetrized and antisymmetrized quantities”: ∆f (s) (v) = (∆f+ ) (v) + (∆f− ) (v) , ∆f (a) (v) = (∆f+ ) (v) − (∆f− ) (v) , (4.21) whose respective Laplace transforms are: F˜ (s) (λ) = F˜+ (λ) + F˜− (λ) , F˜ (a) (λ) = F˜+ (λ) − F˜− (λ) ;

(4.22)

We can then give the following alternative version of Theorem 2 ii): Proposition 7 The transform F˜ of f has the following structure: πλ ˜ (s) πλ ˜ (a) iπ λ ˜ 2 cos , F =e F − i sin F 2 2 (m) F˜ (s) and F˜ (a) being Carlsonian interpolations in the half-plane C+ of the respective sets of even and odd Fourier coefficients of f|R ; namely, one has:

for 2` > m, for 2` + 1 > m,

f2` = F˜ (s) (2`) f2`+1 = F˜ (a) (2` + 1)

(4.23) (4.24)

(s) (a) (m) F˜λ and F˜λ satisfying bounds of the form (4.13) in C+ .

Inversion formulae: a) The discontinuities (∆f )± (v) (considered as distributions with support in {v > 0}) can be recovered from the corresponding functions F˜± (λ) by the following inverse Fourier formulae (equivalent in view of the Cauchy formula applied (m) to F˜± (λ)e−λv in C+ ):

(for v > 0)

or

1 (∆f )± (v) = π 1 (∆f )± (v) = π

+∞ Z F˜± (m + iν) cos [(ν − im) v] dν;

(4.25)

−∞ +∞ Z F˜± (m + iν) sin [(ν − im) v] dν;

(4.250 )

−∞

(note that under the assumptions of Theorem 2, Equation (4.25) has to be understood in the sense of tempered distributions) On the other hand, there exists a well-defined integral representation of the holomorphic function f (θ) in its domain Π in terms of the Laplace transforms

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F˜+ (λ) , F˜− (λ) namely (assuming that m is positive) 1 f (θ) = − 2π

+∞ Z F˜+ (m + iν) cos [(m + iν) (θ ± π)] dν sin π (m + iν)

−∞

+∞ Z 1 F˜− (m + iν) cos [(m + iν) θ] 1 X − f` cos `θ dν + 2π sin π (m + iν) 2π

(4.26)

|`|<m

−∞

In fact, the first term at the r.h.s. of (4.26) can be seen to define a pair of holomorphic functions in the respective strips 0 < u < 2π and −2π < u < 0 (corresponding to the choice of the sign n− or + in the cosine factor), while the second term defines a holomorphic function in the strip −π < u < π: this follows from the bounds (4.13) on F˜± (λ). The proof of (4.26) consists in showing that for θ = u real, it reduces to the Fourier series of f|R , namely: f (u) =

1 X f` cos `θ 2π

(4.27)

`∈Z

As a matter of fact, by using a standard contour distortion argument and resumma(m) tion of residues at integral points inside C+ (known as the Sommerfeld-Watson resummation method [30,31]), one shows that the first two terms at the r.h.s. of Equation (4.26) are respectively equal to the sums of the series 1 X ˜ F+ (`) cos `θ π `∈N l>m

and

1 X (−1)` F˜− (`) cos `θ, π `∈N l>m

which therefore (in view of Equations (4.19), (4.20)) reconstitute the r.h.s. of (4.27). Remarks. i) Equation (4.250 ) for the discontinuities can also be recovered from (4.26) (taken in the limits Reθ → 0 or π), at first in a formal way, and more rigorously by using the techniques of primitives, presented in Appendix B. ii) If Equation (4.26) is used for m integer, its r.h.s. must be understood as the 1 action of the distribution limε→0,ε>0 sin π(m−ε+iν) on the numerator of the integrand.

IV.2

Fourier-Laplace transformation on cut-domains (c) of the complexified hyperboloid Xd−1 , d > 2

We present a geometrical treatment of the d-dimensional case (d > 2) which is very close in its spirit to the one given above for the case d = 2. This treatment (see [25b),c)]) provides the connection, via analytic continuation, between

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Fourier analysis on the sphere Sd−1 ≈ SO (d) /SO (d − 1) and an appropriate realization of Fourier-Laplace analysis on the unit one-sheeted hyperboloid Xd−1 ≈ SO◦ (1, d − 1) /SO◦ (1, d − 2). Analytic continuation takes place on the complexified unit hyperboloid n o 2 2 2 (c) Xd−1 = z = z (0) , . . . , z (d−1) ∈ Cd ; z 2 ≡ z (0) − z (1) − · · · − z (d−1) = −1 , (c)

which contains Sd−1 and Xd−1 as submanifolds of real type, namely Sd−1 = Xd−1 ∩ (c) iR × Rd−1 and Xd−1 = Xd−1 ∩ Rd . One then considers classes of functions which enjoy analyticity, and invariance properties in the “cut power boundedness (c) (c) (c) (c) (c) domain” D = Xd−1 \ Σ+ ∪ Σ− , where Σ+ , Σ− are given (as in Equations (2.30), (2.31)) by: n o (c) (c) Σ± = z = z (0) , z (1) , . . . , z (d−1) ∈ Xd−1 ; ±z (d−1) ∈ [coshv± , +∞[ (4.28) More specifically, the functions F (z) considered are supposed to be invariant (c) under the stabilizer Gz0 (isomorphic to SO0 (1, d − 2)) of the base point z0 = (0, . . . , 0, 1) and therefore only depend on z (d−1) = cos θ, so that one can again put F (z) = f (θ) , with f even, 2π-periodic and holomorphic in the cut-plane Π = C\ (σ+ ∪ σ− ) (Fig. 3). The analyticity domain D of these functions F is (c) the preimage of Π in Xd−1 (through the mapping z → z (d−1) = cos θ → ±θ). In 2 particular, the sphere Sd−1 (z (0) = iy (0) , z (j) = x(j) real for all j 6= 0, y (0) + 2 2 x(1) + · · · + x(d−1) = 1) is embedded in D, and projects onto the interval [−1, +1] in the z (d−1) -plane. The cuts σ+ and σ− are the images of subsets Σ+ and Σ− of Xd−1 , defined respectively by the conditions z (0) > 0, z (d−1) > coshv+ and z (0) < 0, z (d−1) 6 − coshv− ; (see Fig. 4). The jumps ∆f+ , ∆f− of if across σ+ , σ− can now be considered as functions (or distributions) on Xd−1 (depending only on the coordinate z (d−1) = coshv or − coshv) with supports contained respectively in Σ+ , Σ− . Each function F (z) = f (θ) also represents an invariant perikernel K (z, z 0 ) (such that K (gz, gz 0 ) = K (z, z 0 ) for all g in SO0(c) (1, d − 1) , and K (z, z0 ) = (c) (c) (c) ˆ+ F (z)) which is holomorphic in Xd−1 × Xd−1 minus the union of the cuts Σ = (c) 0 0 0 0 ˆ {(z, z ) ; z · z 6 − coshv+ } and Σ− = {(z, z ) ; z · z > cosh v− } (these notations being similar to those used in Theorem 1). The restriction of K to the sphere Sd−1 , namely K = K|Sd−1 ×Sd−1 is an analytic invariant kernel on Sd−1 , represented by F|Sd−1 = f|R = f . While the Fourier analysis of K (z, z 0 ) on the sphere Sd−1 is given [26,27] by the following set of coefficients of the generalized Legendre expansion of K (d) involving the polynomials P` (see Equation (4.3)): Zπ (d)

f` = ωd−1

P` 0

(cos θ) f (θ) (sin θ)d−2 dθ,

` ≥ 0,

(4.29)

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z z (0)

− cosh v−

−1

0

Σ+

cosh v+ cosh v 1 cosh w z (d−1)

Σ−

Figure 4

the introduction of Laplace transforms associated with K along the same line as in the case d = 2 (see §4-1) necessitates a special geometrical study. Before presenting the latter, we note that the discontinuities (∆f )+ (v) , (∆f )− (v) of if represent correspondingly the discontinuities (∆F)+ (z) , (∆F)− (z) of iF (z) on the cuts (c)

(c)

Σ+ , Σ− , which we can consider (after restriction to the real hyperboloid Xd−1 ) as functions (or distributions) with support contained respectively in the regions Σ+ , Σ− : these functions (depending only on zd−1 = coshv) also represent Volterra kernels K+ (z, z 0 ) , K− (z, z 0 ) (such that K± (z, z0 ) = ∆F± (z)) on the hyperboloid Xd−1 , namely kernels with causal support properties on Xd−1 × Xd−1 which are stable by the composition product [24,32] (this structure will be exploited in [14]). Laplace transformation on Xd−1 for functions of moderate growth P with support ± Two systems of local coordinates on Xd−1 , are equally valid in a neighborhood of the set Σ+ = z ∈ Xd−1 ; z (d−1) > coshv+ , z (0) > 0 , namely: a) The polar coordinates: z (0) = sinh w cosh ϕ, z (d−1) = cosh w, [~z] = z (1) , . . . , z (d−2) = sinh w sinh ϕ [~ α] , [~ α] ∈ Sd−3

(4.30)

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b) The horocyclic coordinates: z (0) = sinhv +

1 2

k ~x k2 ev ,

z (d−1) = coshv− 12 k ~x k2 ev ,

~x ∈ Rd−2

[~z] = ~x ev ,

(4.31)

The sections v = cst are paraboloids in the hyperplanes z (0) + z (d−1) = ev , called horocycles. For classes of functions F+ (z) with support in Σ+ which are invariant under the stabilizer of z0 , namely F+ (z) ≡ F+ z (d−1) = f+ (w) (supp. f+ ⊂ [v+ , +∞[), and which moreover satisfy a bound of the form: m |F+ (z)| 6 cst z (d−1) or |f+ (w)| 6 cst em|w| , (4.32) the Laplace transform F˜+ (λ) of F+ is defined as follows: Z F˜+ (λ) = e−λv F+ [cosh w] d~x dv

(4.33)

Σ+

In the latter we have used “mixed coordinates” v, w, [~x] (see Fig. 4); from (4.30), (4.31) one gets: ~x = e−v/2 [2 (cosh v − cosh w)]

1/2

[~ α] , with [~ α] ∈ Sd−3 ,

which allows one to rewrite Equation (4.33) as follows (ωd−2 being the area of Sd−3 ): Z ∞ d−2 ˜ F+ (λ) = ωd−2 e−λv e−( 2 )v Ad f+ (v) dv, (4.34) v+

Z

with

v

f+ (w) [2 (cosh v − cosh w)]

Ad f+ (v) =

d−4 2

sinh w dw

(4.35)

v+

It has been proved in [25] that under the moderate growth condition (4.32) (m) the Laplace transform F˜+ (λ) of F+ (z) is holomorphic in C+ . This follows from the fact that provided m > −1 the exponential bound (4.32) on F+ is preserved d−2 by the transformation f+ (w) → e− 2 v [Ad f+ ] (v) (see Proposition II-2 of [25b)] for a precise formulation of this statement). (d) On the other hand, by introducing the second-kind function Qλ via the integral representation (valid for w 6= 0 and Reλ > −1) 9 Z ∞ d−4 d−2 ωd−2 (d) −1 Qλ (cosh w) = ωd−1 e−(λ+ 2 )v [2 (coshv − coshw)] 2 dv, d−3 (sinh w) w (4.36) 9 We

use here a normalization for these functions which is appropriate to our joint consideration (d) (d) of Pλ and Qλ ; for d = 3, the discrepancy with the standard normalization of the second-kind Legendre function [36a)] is a factor π1 .

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we obtain (by inverting the integrations in (4.34)) the following alternative expres(m) sion of F˜+ (λ) in its domain C+ Z ∞ (d) ˜ F+ (λ) = ωd−1 f+ (w) Qλ (cosh w) (sinhw)d−2 dw. (4.37) v+

(Note that the previous restriction m > −1 can be seen to be produced by the (d) pole of the function λ → Qλ (cosh w) at λ = −1.) By now replacing w by w + iπ and v by v + iπ in (4.30), (4.31), we obtain similar systems of local coordinates which are valid in a neighborhood of the set Σ− = z ∈ Xd−1 ; z (d−1) 6 − cosh v− , z (0) < 0 . Then one can consider similarly the invariant function F− (z) ≡ F− z (d−1) = f− (w) with support contained in Σ− (supp. f− ⊂ [v− , +∞[) and satisfying the growth condition (4.32). This function (m) F− admits the following Laplace transform which is also holomorphic in C+ : Z Z ∞ d−2 e−λv F− [− cosh w] d~xdv = ωd−2 e−λv e−( 2 )v Ad f− (v)dv, F˜− (λ) = v−

Σ−

(4.38) Z

with

v

f− (w) [2 (cosh v − cosh w)]

Ad f− (v) =

d−4 2

sinh w dw,

(4.39)

v−

or equivalently in view of Equation (4.36): Z ∞ (d) F˜− (λ) = ωd−1 f− (w) Qλ (cosh w) (sinh w)d−2 dw

(4.40)

v−

(c)

Laplace transformation on Xd−1 for holomorphic functions in D with continuous boundary values With every Gz0 -invariant holomorphic function F(z) = f (θ) defined in the cut(c) domain D of Xd−1 and satisfying moderate growth condition of the form |F(z)| 6 cst |zd−1 |m (or |f (u + iv)| 6 cst em|v| ), we shall now associate a Fourier-Laplace(m) type transform F˜ (λ), holomorphic in the half-plane C+ , by a formula similar to (4.34) and (4.38), except that a complex integration contour is used, namely Z d−2 (c) F˜ (λ) = ωd−2 ei(λ+ 2 )θ Ad f (θ) dθ; (4.41) γ

here γ is the same contour as for the case d = 2 (see Fig. 3 and Equation (4.5)), (c) and the definition of Ad requires the following procedure. We introduce a decomposition of f (θ) of the form f (θ) = f+ (θ) + f− (θ) where f+ and f− have the same analyticity and symmetry properties as f, but enjoy the following additional property: f+ (resp. f− ) admits a single cut, namely σ+ (resp. σ− ) across

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which its discontinuity coincides with the corresponding one of f, denoted unambiguously by ∆f+ (v) (resp. ∆f− (v)). Such a decomposition can be done by considering the representation f (cos θ) = f (θ) of f as a holomorphic function f in C\ {[coshv+ , +∞ [∪] − ∞, − coshv− ]} bounded by cst |cos θ|m and defining f ± (cos θ) = f± (θ) through appropriate Cauchy integrals involving the respec tive weights ∆f + (coshv)/(coshv)E(m)+1 and ∆f − (coshv)/(coshv)E(m)+1 on the corresponding cuts of f (i.e. in physical terms by the method of “subtracted dispersion relations”). The decomposition is non-unique, but defined up to a polynomial in cos θ with degree E(m). We then define: (c) (c) (c) Ad f (θ) = Ad+ f+ (θ) + Ad− f− (θ) , (4.42) (c)

(c)

where Ad+ f+ and Ad− f− are respectively defined as holomorphic functions in the periodic cut-planes C\σ+ and C\σ− by the following integrals: Z d−4 (c) Ad+ f+ (θ) = − f+ (τ ) [2 (cos θ − cos τ )] 2 sin τ dτ (4.43) γ(π,θ) Z d−4 (c) f− (τ ) [2 (cos θ − cos τ )] 2 sin τ dτ. (4.44) Ad− f− (θ) = − γ(0,θ)

In the latter, the path γ (π, θ) (resp. γ (0, θ)) with end-points π and θ (resp. 0 and θ) has to belong to the domain C\σ+ (resp. C\σ− ) and the function d−4 [2 (cos θ − cos τ )] 2 is determined by the condition that it is positive for θ = iv, τ = iw, 0 < w < v. Let us first assume that the boundary values of F(z) = f (θ) on the cuts P(c) ± (resp. σ± ) are continuous (from both sides). One then checks that the jumps (c)

of iAd f (θ) across the cuts σ+ and σ− are respectively equal to Ad ∆f+ (v) and d−2 e−iπ( 2 ) Ad ∆f− (v) (in view of Equation (4.35) and (4.39)). By using the same contour distortion argument as for the case d = 2, namely by replacing γ by γ+ + γ− , and then flattening γ+ , γ− on the respective cuts σ+ , σ− , one can then rewrite the integral at the r.h.s. of (4.41) as: Z d−2 (c) ei(λ+ 2 )θ Ad f (θ)dθ γ+ +γ− Z Z d−2 d−2 (c) (c) = ei(λ+ 2 )θ Ad+ f+ (θ) dθ + ei(λ+ 2 )θ Ad− f− (θ) dθ (4.45) γ+ γ− +∞ +∞ Z Z d−2 d−2 e−(λ+ 2 )v (Ad ∆f+ ) (v) dv + eiπλ e−(λ+ 2 )v (Ad ∆f− ) (v) dv = v+

v−

In view of Equations (4.34), (4.38), the latter can be rewritten (as for d = 2): F˜ (λ) = F˜+ (λ) + eiπλ F˜− (λ)

(4.450 )

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(m) where the functions F˜+ , F˜− , holomorphic in C+ now denote the Laplace transforms of the discontinuities ∆f+|Σ+ , ∆f−|Σ− taken on the corresponding sets Σ+ , Σ− according to formulae (4.33), (4.38) (with F± = ∆f±|Σ± ).

Remark. The Laplace transform F˜ (λ) that we have introduced only depends on the function F (z) through its discontinuities on Σ+ , Σ− ; it therefore does not depend (c) on the particular decomposition f = f+ +f− , in spite of the fact that Ad f actually depends on the latter. Link with the Fourier expansion on the sphere Sd−1 (Froissart-Gribov-type equalities) For λ = ` integer (with ` > m), we rewrite Equation (4.41) with the choice of contour (as in the case d = 2, see Fig. 3); by taking the periodicity of γ = γαd−2 (c) i( 2 )θ into account, this yields: Ad f (θ). e Z F˜ (`) = ωd−2

2π−α

ei(`+

d−2 2

−α

)u A(c) f (u)du d

(4.46)

By choosing α = π, the latter can be rewritten in view of Equation (4.42): Z π h i d−2 (c) (c) F˜ (`) = ωd−2 ei(`+ 2 )u Ad+ f+ (u) + Ad− f− (u) du (4.47) −π

Then by applying Equations (4.43), (4.44), inverting the order of integrations and using obvious symmetries in the double integrals, one obtains: Z π d−2 (c) ei(`+ 2 )u Ad+ f+ (u) du ωd−2 −π

Z

Z

π

f+ (t) sintdt

= ωd−2 0

and ωd−2

Z

π

ei(`+

−π

Z

d−2 2

0

(4.48) i(`+ d−2 2 )u

e −t

[2 (cos u − cos t)]

d−4 2

du

)u A(c) f (u) du d− − Z

π

2π−t

f− (t) sintdt

= ωd−2

t

(4.480 ) d−2 i(`+ d−2 2 )u

(−i)

e

[2 (cos t − cos u)]

d−4 2

du

t

We now use the following integral representations of the ultraspherical Legendre polynomials, which are consequences of the representation (4.3) (see in [25c)] the derivation of Equations (III-25) and (III-250 ) from (III-18)): Z t d−4 ωd−2 (d) −(d−3) (sin t) ei(`+(d−2)/2)u [2(cos u − cos t)] 2 du P` (cos t) = ωd−1 −t Z 2π−t d−4 ω d−2 −(d−3) = (−i)d−2 ei(`+(d−2)/2)u [2(cos t − cos u)] 2 du. (sin t) ωd−1 t (4.49)

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The latter imply that Equations (4.48) and (4.480 ) can be rewritten as follows: Zπ ωd−2

i(`+ d−2 2 )u

e

Zπ (c) (d) Ad± f± (u) du = ωd−1 f± (t) P` (cos t) (sin t)d−2 dt.

−π

0

(4.50) In view of Equation (4.50), Equation (4.47) can now be rewritten Zπ (d)

F˜ (`) = ωd−1

(f+ (t) + f− (t)) P`

(cost) (sint)d−2 dt,

(4.51)

0

and since f = f+ + f− , by comparing to Equation (4.29): (for ` > m)

F˜ (`) = f (`) .

(4.52)

These Froissart-Gribov-type equalities can also be given the following more precise form (in view of Equation (4.450 )): f` = F˜+ (`) + (−1)` F˜− (`)

(4.53)

Note that if one calls f`± the Legendre coefficients of the corresponding functions f± (θ) = F± (z) on the sphere Sd−1 , it can be easily checked that F˜+ (`) = f`+ and (−1)` F˜− (`) = f`− . The case of perikernels with distribution-like boundary values As in §4.1 (Theorem 2), we shall now give a detailed version of the previous properties under the assumption that F (z) = f (θ) admits distribution-like boundary (c) values (and discontinuities) on the cuts Σ± (resp. σ± ). Theorem 3 Let F (z) = f (θ) represent an invariant perikernel of moderate growth (c) on Xd−1 , satisfying uniform bounds of the following form |f (u + iv)| 6 cη−β emv in all the subsets

Π+ η (η

(4.54)

+

> 0) of Π

(see Equation (4.12)), or equivalently: m (4.540 ) |F (z)| 6 Cη −β z (d−1)

0 in the subsets Dη , defined as the preimages of Π+ η in D. In (4.54), (4.54 ), we assume that m > −1 and β ≥ 0. Then (c) i) The discontinuities (∆F)± (z) = (∆f )± (v) of iF (resp. if ) across the cuts Σ± (resp. σ± ) are well defined as distributions. They admit Laplace-transforms F˜± (λ) on the hyperboloid Xd−1 defined for Reλ > m by: Z +∞ (d) d−2 F˜± (λ) = ωd−1 ∆f± (v) Qλ (coshv) (sinhv) dv, (4.55) v±

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where these integrals are understood as the action of the distributions ∆f± on (d) the (admissible) test-functions Qλ (coshv) (sinhv)d−2 . F˜± (λ) are holomorphic in (m) C+ and satisfy uniform bounds of the following form (for all ε, ε0 > 0): d−2 0 ˜ (ε,ε0 ) |λ − m|β− 2 +ε e−[Reλ−(m+ε)]v± (4.56) F± (λ) 6 C± (m+ε)

in all the corresponding half-planes C+

.

ii) The Laplace-transform F˜ = Ld (F) of F is defined as F˜ = L fˆ , where d−2 (c) fˆ (θ) = ω ei( 2 )θ (A f ) (θ) and L is the Fourier-Laplace transformation (4.5); d−2

(c)

d

(Ad f ) (θ) is defined by means of formulae (4.42), (4.43), (4.44) involving a decomposition f = f+ + f− of f into “single-cut functions” f+ , f− . This transform (m) F˜ (λ) is holomorphic in C+ and satisfies the following properties: a)

F˜ (λ) = F˜+ (λ) + eiπλ F˜− (λ)

(4.57)

b) for all integers ` such that ` > m, the Legendre coefficients f` of F|Sd−1 , defined by Equation (4.29), are given by the following (Froissart-Gribov-type) relations: f` = F˜ (`) = F˜+ (`) + (−1)` F˜− (`) (4.58) Z (d) (4.59) c) F˜ (λ) = (−i)d−2 ωd−1 f (θ) Qλ (cos θ) (sin θ)d−2 dθ γ

Proof. As in Theorem 2, the validity of bounds of the form (4.540 ) (resp (4.54)) is equivalent to the existence of distribution boundary values and discontinuities (c) on Σ± (resp. σ± ). The corresponding discontinuities ∆f± , now defined as distributions with support σ± , can still be used for introducing a decomposition f = f+ + f− of f into “single-cut functions” f± by means of Cauchy integrals in the cos θ-plane; in the latter, the “weights” ∆f ± (cosh v)/(cosh v)E(m)+1 act as distributions on the Cauchy kernel considered as a test-function. The expressions (c) (4.43), (4.44) of Ad± f± (θ) then remain well defined in the corresponding cutplanes C\σ± and we can introduce the functions d−2 (c) fˆ± (θ) = ωd−2 ei( 2 )θ Ad± f± (θ) (4.60) which allow us to write F˜ = L fˆ , with fˆ = fˆ+ + fˆ− . The functions fˆ+ , fˆ− and fˆ are 2π-periodic and holomorphic in Π+ and we claim that the assumed bounds (4.54) on f (with m > −1 and β ≥ 0) imply that fˆ satisfies the assumptions of Theorem 2, namely bounds of the form (4.11) with the same value of m (although not the same value of β). This fact is fully justified in the detailed analysis given below for proving the bounds (4.56); it relies on the interpretation of the transfor(c) mation Ad as a primitive of non-integral order with respect to the variable cos θ

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(see Appendix B, Proposition B.6). The conclusions of Theorem 2 then imply the expression (4.57) of F˜ (λ) , with Z

+∞

F˜± (λ) =

∆fˆ± (v)e−λv dv,

(4.61)

v±

these integrals being (if necessary) understood as the action of the distributions ∆fˆ± on the exponential function e−λv (as specified in the proof of Theorem 2). The proof of Equation (4.58) has already been given above in full generality (see the computation after (4.46)) which is independent of the continuous or distributionlike character of the boundary values of f. Proof of the bounds (4.56). One applies the results of Proposition B.6 with the following specifications. The holomorphic function f (θ) of Proposition B.6 plays (α) the role respectively of f+ (θ) and f− (θ + π). One then considers the function fˆm d−2 studied in Proposition B.6 for the value α = 2 . Then, in view of Equation (4.60), (α) fˆm (θ) coincides respectively (up to a constant factor) with fˆm+ (θ) = eimθ fˆ+ (θ) and fˆm− (θ) = eim(θ+π) fˆ− (θ+π) and the corresponding Laplace transforms F˜m± (λ) of fˆm± (θ) are such that F˜± (λ) = F˜m± (λ − m). According to the results of Proposition B.6, one is then led to distinguish three cases: (cut) β− d−2 ˆ ˆ 2 (Bπ a) β > d−2 ). 2 : in this case, fm+ and fm− belong to the class O ˆ ˆ ˆ (Note that f = f+ + f− then satisfies uniform bounds of the form d−2 ˆ f (u + iv) 6 Cη−(β− 2 ) emv + in all the corresponding subsets Π+ η (η > 0) of Π ).

As in Theorem 2 i), the corresponding majorization (4.56) of F˜± (λ) (namely ˜ of Fm± (λ − m)) then follows from Proposition B.4 iii), formula (B.19), with (in the present case) β replaced by β − d−2 2 . (cut) 0∗ ˆ ˆ b) β = d−2 ), (which then 2 : the functions fm+ and fm− belong to O (Bπ mv + ˆ ˆ ˆ implies that f = f+ + f− is bounded by C |log η| e in each Πη > 0). Proposition B.4 iii) still applies and yields again the corresponding majorization ε0 (4.56) (involving the power |λ − m| ). ˆ c) β < d−2 2 : in this case, fm± (θ) admit continuous boundary values; more precisely, Proposition B.5 shows that fˆm± (θ) belongs to the class

O d−2 −β (Bπ(cut) ); 2

therefore, in view of Proposition B.3, F˜± (λ) again satisfy the bound (4.56). It remains to show that the expressions (4.61) of F˜± (λ) imply the corresponding alternative form (4.55). Considering F˜+ (λ), one rewrites (4.61) (as in

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σ+

γ+

γ+ v+ θ γ(π, θ)

0

π

Figure 5

Theorem 2) as Z

Z F˜+ (λ) =

iλθ

e γ+

fˆ (θ) dθ =

Z

F˜+ (λ) = ωd−2

eiλθ fˆ+ (θ) dθ, i.e. γ+

ei(λ+

γ+

d−2 2

)θ A(c) f (θ) dθ + d+

(4.62)

(c) with Ad+ f+ (θ) expressed by Equation (4.43). By choosing γ+ and γ (π, θ) such that, for all θ, supp. γ (π, θ) ⊂ supp γ+ (as e.g. in Fig. 5), one can treat the (m) resulting expression for F˜+ (λ) as a double integral (convergent for λ in C+ ) in which the order of the integrations can be inverted. This yields: Z (d) F˜+ (λ) = (−i)d−2 ωd−1 f+ (τ ) Qλ (cos τ ) (sin τ )d−2 dτ (4.63) γ+

Now, by the very definition of boundary values of holomorphic functions in the sense of distributions, the expression (4.63) of F˜+ (λ) can be rewritten in the distribution form (4.55) (by flattening the folded contour γ+ onto the cut σ+ ). A similar argument holds for F˜− (λ). Moreover, by plugging the expression (4.63) of F˜+ (λ) and the analogous one for F˜− (λ) into Equation (4.57) and then noticing that the corresponding integration paths γ+ and γ− can be replaced by γ, one obtains the expression (4.59) of F˜ (λ) in terms of f = f+ + f− .

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As in the case d = 2, one still defines the quantities F˜ (s) (λ), F˜ (a) (λ) by formula (4.22): they are respectively the Laplace transforms on Xd−1 of the distributions ∆f (s) , ∆f (a) defined (on Xd−1 ) by Equations (4.21). One can thus complete the second part of Theorem 3 by the Proposition 7bis The statement of Proposition 7 is valid without modification in the d-dimensional case (d ≥ 3) apart from the bounds on F˜ (s) (λ), F˜ (a) (λ) which are now given by the r.h.s. of (4.56). Inversion formulae We shall give formulae which express F (z) = f (θ) and its discontinuities (∆F)± (z) = (∆f )± (v) in terms of the Laplace transforms F˜± (λ) of the latter. The formulae exactly parallel the inversion formulae (4.25), (4.26), (4.27) of the two-dimensional case; they only differ from the latter by the fact that the trigonometric kernel cos λθ is (d) (d) replaced by hd (λ) Pλ (cos θ) , where Pλ is the d-dimensional first-kind Legendre function Z θ d−4 ωd−2 (d) −(d−3) Pλ (cos θ) = 2 (sin θ) cos[(λ + (d − 2)/2)τ ][2(cos τ − cos θ)] 2 dτ ωd−1 0 (4.64) and (2λ + d − 2) Γ (λ + d − 2) hd (λ) = . . (4.65) (d − 2)! Γ (λ + 1) (d)

Pλ (cos θ) is defined as a holomorphic function in the cut plane C\ ]−∞, −1]. The following formula is shown to hold in the open set 0 < |Reθ| < π: with the specification εθ = sgn (Reθ) F (z) ≡ f (θ)

Z

(d) F˜+ (m + iν) hd (m + iν) Pm+iν (cos θ − εθ π) dν sin π (m + iν) −∞ Z +∞ ˜ (d) F− (m + iν) hd (m + iν) Pm+iν (cos θ) 1 − dν 2ωd −∞ sin π (m + iν) 1 X (d) + f` hd (`) P` (cos θ) . ωd

=−

1 2ωd

+∞

(4.66)

06`<m

As in Equation (4.26), the first term at the r.h.s. of (4.66) defines a pair of holomorphic functions in the respective strips 0 < u < 2π and −2π < u < 0 (corresponding to the choice εθ = + or − in the argument of the cosine), while the second term defines a holomorphic function in the strip −π < u < π: this follows

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(d) from the bounds (4.57) on F˜± (λ) and the power behaviour of Pm+iν (cos θ) as

|ν|

d−1 2

(d)

(easily derived from the representation (4.64) of Pλ ). The restriction of F to the sphere Sd−1 , namely F|Sd−1 (z) = f|R (θ) , is also expressed by the generalized Legendre (or “partial-wave”) expansion: f|R (θ) =

1 X (d) f` hd (`) P` (cos θ) ωd

(4.67)

`∈N

Finally, the discontinuities ∆f+ (v) and ∆f− (v) of f across the cuts σ+ and σ− are given by the following (identical) formulae: (∆f )± (v) =

1 ωd

Z

+∞ −∞

(d) F˜± (m + iν) hd (m + iν) Pm+iν (coshv) dν

(4.68)

which (in view of the polynomial increase in ν of all factors of the integrand) must be understood in the sense of distributions according to Appendix B. All these formulae have been established in [25c)] (under assumptions of continuity for the boundary values of F) in the case where a single cut, namely σ+ , is present. The proof given in [25c)] applies equally well to the derivation of Equation (4.66) under the present assumptions; however, for tutorial reasons, we will sketch the derivation of this result which relies on the inversion of the two transformations (4.41) and (4.42), . . . (4.44). We must treat separately the cases of even and odd dimensions d. a) d even (d > 4): the Abel-type transformations (4.43), (4.44) can be inverted as follows: d−2 h i 2 1 1 d (c) f± (θ) = ωd−2 − Ad± f± (θ) (4.69) 2π sin θ dθ and the inversion of the Fourier integrals over γ± in (4.45) yields (by taking into account that f` = f`+ + f`− ): (c) (c) (c) ωd−2 Ad f (θ) = ωd−2 Ad+ f+ (θ) + ωd−2 Ad− f− (θ) d Z (θ − εθ π) (−1) 2 ∞ F˜+ (m + iν) cos m + iν + d−2 2 dν = 2π sin π (m + iν) −∞ Z ∞ ˜ F− (m + iν) cos m + iν + d−2 θ 2 − dν sin π (m + iν) −∞ X f` cos(` + (d − 2)/2)θ + −m−d+2<`<m

(4.70) By applying the differential operator at the r.h.s. of (4.69) to both sides of Equation (4.70), we then directly obtain Equation (4.66), thanks to the integral representa-

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tion (see [25b)] Equation (II.85)): (d) hd (λ) Pλ

(cos θ) =

d−2 2 1 d − [cos(λ + (d − 2)/2)θ] sin θ dθ

2ωd (2π)d/2

(4.71)

b) d odd (d > 3): the Abel inversion formulae (4.69) are replaced by d−1 Z θ 2 1 1 d (c) f+ (θ) = −2ωd−2 − Ad+ f+ (τ )[2(cosθ − cosτ )]−1/2 sinτ dτ 2π sinθ dθ εθ π (4.72) d−1 Z θ 2 1 1 d (c) f− (θ) = −2ωd−2 − Ad− f− (τ )[2(cosθ − cosτ )]−1/2 sinτ dτ 2π sinθ dθ 0 (4.73) and correspondingly the inversion of the Fourier integrals in (4.45) yields the fol(c) lowing expressions for Ad± f± in the open set {θ; 0 < |Reθ| < π}: (c) ωd−2 Ad+ f+ (θ) ( Z d−1 ∞ F+ (m + iν) sin m + iν + d−2 (θ − εθ π) iεθ (−1) 2 2 − dν = (4.74) 2π sin π (m + iν) −∞ ) X + f`+ sin[(` + (d − 2)/2)θ − εθ ((d − 2)/2)π] −m−d+2<`<m

i (c) ωd−2 Ad− f− (θ) = 2π

( Z −

∞

−∞

X

+

F˜− (m + iν) sin m + iν + d−2 θ 2 dν sin π (m + iν) )

(4.75)

f`− sin(` + (d − 2)/2)θ

−m−d+2<`<m

By applying the integro-differential operator at the r.h.s. of (4.72) (resp. (4.73)) to both sides of Equation (4.74) (resp. (4.75), we then directly obtain Equation (4.66), thanks to the integral representation (see [25b)] Equation (II-86)): (d) hd (λ) Pλ

(cos θ) =

4iωd (2π)

d+1 2

−1 d sin θ dθ

d−1 Z 2 0

θ

sin λ +

d−2 2

τ sin τ

[2 (cos θ − cos τ )]1/2

dτ

(4.76)

For θ = u real (i.e. z ∈ Sd−1 ), formula (4.66) can be seen to reduce to the expansion (4.67) by using the same contour distortion argument [30,31] in the λplane as in the case d = 2 (see Equations (4.26), (4.27)); this can be done in two equivalent ways: i) by proceeding directly with the r.h.s. of (4.66) thanks to the properties of (d) Pλ (cos θ) as a holomorphic function of λ in the right-hand plane.

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ii) by proceeding with the r.h.s. of (4.70) (resp. (4.74), (4.75)) which only involves trigonometric functions and then applying the inverse Abel operator of Equation (4.69) (resp. (4.72), (4.73)) which will restore the ultraspherical (d) polynomials hd (`) P` (cos θ) term by term in the expansion. Finally, the formulae (4.68) for the discontinuities ∆f± are obtainable from (d) (4.66) either directly (thanks to relevant discontinuity formulae for the Pλ (cos θ) (c)

on ]−∞, −1]) or by computing the corresponding discontinuities of Ad± f± (θ) from their representations (4.70) (resp. (4.74), (4.75)) and then applying the inverse Abel operators of Equation (4.69) (resp. (4.72) (4.73)). The case of distribution-like discontinuities requires a suitable regularization corresponding to the application of a “cut-off” to F˜± (m + iν). Remarks i) If Equation (4.66) is used for m integer, its r.h.s. must be understood as the 1 action of the distribution limε→0,ε>0 sin π(m−ε+iν) on the numerator of the integrand. ii) We refer to Theorem 4 of [25c)] for the possible use of formula (4.66) in a precise range of negative values of m. However, it is only for m > −1 that Equation (4.66) is exactly the inverse of the transformation (4.55) defined under the assumptions of Theorem 3 (for m ≤ −1, this transformation can generate poles of F˜ (λ) located as those of Qλ at all the negative integers). The following complement of Theorem 3 which emphasizes the reciprocal property of the transformation follows from the previous study of the inversion used conjointly with the principle of uniqueness of analytic continuation (it is the adaptation of Theorem 3 of [25c)] to the case with two cuts) Theorem 4 Let K(z, z 0 ) be an SO(d)-invariant kernel on the sphere Sd−1 with set of Legendre coefficients f` (defined by Equation (4.29) in terms of the function f (θ) = F(z) = K(z, z0 ), z.z0 = − cos θ). Let us then assume that the sets of even and odd coefficients f` admit respectively analytic interpolations F˜ (s) (λ) and (m) F˜ (a) (λ) in C+ satisfying uniform bounds of the form (4.56) with m > −1. (c) Then there exists an invariant perikernel K(z, z 0 ) of moderate growth on Xd−1 represented by a holomorphic function F(z) = K(z, z0 ) = f (θ) satisfying all the assumptions of Theorem 3 such that K|Sd−1 ×Sd−1 = K and correspondingly f|R = f ; moreover, the functions F˜ (s) (λ) and F˜ (a) (λ) appear respectively as the symmetric and antisymmetric combinations of the Laplace transforms F˜± of the discontinuities ∆f± of f defined by formula (4.55).

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IV.3

Ann. Henri Poincar´e

Complex angular momentum analysis of the four-point functions

Starting from the basic postulates of Q.F.T., we have established in Theorem 1 that the four-point function H ([k]) of any set of scalar fields enjoys a structure ˆ (ζ,ζ 0 ,K) of the set Ω ˆ associated of invariant perikernel in each submanifold Ω √ K with any space-like energy-momentum vector K = 0, . . . , 0, −t , with t 6 0. In particular, we have shown that the temperateness assumption expressed by the bounds (3.1) results in the properties of moderate growth (3.25), (3.26) of these perikernels: K(ζ,ζ 0 ,K) (z, z 0 ) ≡ H [k](ζ,ζ 0 ,K) (z, z 0 ) = H (ζ,ζ 0 ,K) (cos Θt ), with cos Θt = −z.z 0 . (4.77) One can therefore apply the results of Theorem 3 to the latter, for which the notations of §4.2 and the identification (4.2) can also be used: H (ζ,ζ 0 ,K) (cos Θt ) = f(ζ,ζ 0 ,K) (Θt ) = F (ζ, ζ 0 ; t, cos Θt )

(4.78)

As in §4.2 (for d > 2) or §4.1 (for d = 2), we introduce the discontinuities ∆f(ζ,ζ 0 ,K) ± (v) of the function if(ζ,ζ 0 ,K) across the respective cuts σ+ (vs ),σ− (vu ) with thresholds vs = vs (ζ, ζ 0 , t) , vu = vu (ζ, ζ 0 , t) given by Equations (2.23), (2.24). These discontinuities are interpreted as the s- and u-channel “absorptive parts” ˆ (ζ,ζ 0 ,K) : of F in the submanifold Ω (4.79) ∆f(ζ,ζ 0 ,K) + (v) = ∆s F (ζ, ζ 0 ; t, coshv) 0 (4.80) ∆f(ζ,ζ 0 ,K) − (v) = ∆u F (ζ, ζ ; t, coshv) ˆ (ζ,ζ 0 ,K) , namely In view of the two possible dimensions of the manifolds Ω 2(d − 1) for K 6= 0 and 2d for K = 0 (see §2.2 and Theorem 1), the application of Theorem 3 (resp. Theorem 2 for the case d = 2) allows one to define correspondingly two different Laplace transforms of H([k]). However, by considering the case K 6= 0 (i.e. t < 0), one obtains the generic complex angular momentum analysis of H([k]) whose results are specified in the following theorem; the peculiarities of the case K = 0 will be briefly commented at the end. Theorem 5 Let F (ζ, ζ 0 ; t, cos Θt ) ≡ H ([k]) be any four-point function of local scalar fields satisfying bounds of the form (3.25), (3.26) in each section of maximal ˆ (cut)0 , with K 2 = t < 0, (ζ, ζ 0 ) ∈ ∆t × ∆t . analyticity (or cut-submanifold) Ω (ζ,ζ ,K) Then there exists a function F˜ (ζ, ζ 0 ; t, λt ) which is holomorphic with respect to λt (m∗ )

in C+ a)

and satisfies the following properties: F˜ = F˜s + eiπλt F˜u ,

where, in the general case d > 2: F˜s (ζ, ζ 0 ; t, λt ) = ωd−1 (d) Qλt

Z

+∞ vs (ζ,ζ 0 ,t)

(4.81)

∆s F (ζ, ζ 0 ; t, coshv)

(coshv) (sinhv)

d−2

dv,

(4.82)

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Complex Angular Momentum in General Quantum Field Theory

F˜u (ζ, ζ 0 ; t, λt ) = ωd−1

Z

+∞

vu (d) Qλt

(ζ,ζ 0 ,t)

∆u F (ζ, ζ 0 ; t, coshv)

(coshv) (sinhv)

d−2

149

(4.83)

dv.

(m ) C+ ∗

which satisfy uniform b) F˜s and F˜u are holomorphic functions of λt in bounds of the following form: d−2 0 ˜ ε,ε0 |λt − m∗ |n− 2 +ε e−[Reλt −(m∗ +ε)]vs,u (4.84) Fs,u (ζ, ζ 0 ; t, λt ) 6 Cs,u (m∗ +ε)

in the corresponding half-planes C+

.

c) for ` > m∗ , the off-shell partial-wave functions f` (ζ, ζ 0 , t) of F , defined for ζ, ζ 0 ∈ ∆t × ∆t , t < 0 by Equation (4.2), are given by the following (FroissartGribov-type) relations: f` (ζ, ζ 0 , t) = F˜ (ζ, ζ 0 ; t, `) (4.85) Moreover, the “symmetric and antisymmetric Laplace transforms” F˜ (s) (ζ, ζ 0 ; t, λt ) and F˜ (a) (ζ, ζ 0 ; t, λt ) defined by

10

F˜ (s) = F˜s + F˜u , F˜ (a) = F˜s − F˜u (m∗ +ε)

are Carlsonian interpolations in C+ partial-waves of F , namely one has: for 2` > m, for 2` + 1 > m,

(4.86)

for the respective sets of even and odd

f2` (ζ, ζ 0 , t) = F˜ (s) (ζ, ζ 0 ; t, 2`) f2`+1 (ζ, ζ 0 , t) = F˜ (a) (ζ, ζ 0 ; t, 2` + 1) .

(4.87) (4.88)

d) The four-point function F and the absorptive parts ∆s F, ∆u F are reobtained in terms of F˜s and F˜u by the following formulae: F (ζ, ζ 0 ; t, cos Θt ) Z +∞ ˜ (d) Fs (ζ, ζ 0 ; t, m∗ + iν) hd (m∗ + iν) Pm∗ +iν (cos (Θt − εΘt π)) 1 =− dν 2ωd −∞ sin π (m∗ + iν) Z +∞ ˜ (d) Fu (ζ, ζ 0 ; t, m∗ + iν) hd (m∗ + iν) Pm∗ +iν (cos Θt ) 1 − dν 2ωd −∞ sin π (m∗ + iν) 1 X (d) f` (ζ, ζ 0 , t) hd (`) P` (cos Θt ) + ωd 06`<m∗

∆s,u F (ζ, ζ 0 ; t, coshv) =

1 ωd

Z

(4.89) +∞

−∞

(d) Pm∗ +iν

F˜s,u (ζ, ζ 0 ; t, m∗ + iν) hd (m∗ + iν)

(4.90)

(coshv) dν.

10 Note that when F is the four-point function of a single scalar field Φ or of two fields Φ, Φ0 in such a way that the t-channel is (Φ, Φ) → (Φ0 , Φ0 ), one has ∆s F = ∆u F ; in such cases F˜ (a) = 0, f2`+1 = 0, and only Equation (4.87) survives.

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e) When [k] belongs to the Euclidean region EˆK , (i.e. for cos Θt ∈ [−1, +1]), formula (4.89) can be replaced by the partial wave expansion of H|EˆK , namely: H([k]) = F (ζ, ζ 0 ; t, cos Θt ) =

1 ωd

X

f` (ζ, ζ 0 , t) hd (`) P`

(d)

(cos Θt ) ,

(4.91)

06`<∞

f) In the case d = 2, all the previous results are valid provided the following replacements are performed: In Equations (4.82), (4.83): (d)

ωd−1 Qλt (cosh v)

by

e−λt v ,

In Equation (4.89): 1 (d) hd (m∗ + iν) Pm∗ +iν (cos Θt ) ωd

by

1 cos[(m + iν)Θt ], π

and similar ones in Equations (4.90), (4.91). The proof of properties a) and b) is a direct application of Theorem 3, since the bounds (3.25), (3.26) established in Section 3 are equivalent in an obvious way to (4.54), (4.540 ) with the pair (m, β) replaced by (m∗ , n). Property c) follows from Eq(4.58) in Theorem 3 completed by Proposition 7bis. Properties d) and e) directly follow from the inversion formulae (4.66), (4.68) and (4.67) derived after Theorem 3. The case d = 2 (property f)) is obtained similarly as an application of Theorem 2 and of the inversion formulae (4.25), (4.26), (4.27). In the case K = 0, it is possible to define two Laplace transforms, namely: a) the function F˜ (ζ, ζ 0 ; 0, λt ) obtained as the limit of F˜ (ζ, ζ 0 ; t, λt ) (for (ζ, ζ 0 ) ∈ ∆0 × ∆0 ) when t tends to zero (and satisfying exactly all the formulae of Theorem 3, with t = 0), b) the function F˜0 (ζ, ζ 0 ; λt ) obtained by formulae similar to Equations (4.81)– (4.83), with d replaced by d + 1 (a complete substitute to Theorem 5 being equally valid). The connection between these two functions, both defined for (ζ, ζ 0 ) ∈ ∆0 × (m) ∆0 and λt ∈ C+ , and the possible exploitation of this connection for the structure of the four-point function H([k]) will be treated elsewhere. Remark. In view of the field-theoretical interpretation of m∗ = max(m, n) and n as degrees of temperateness of the four-point function, the properties obtained in Theorem 5 have only made use of the results of harmonic analysis of Theorem 3 for the case m ≥ 0. However it may occur that negative values of this parameter, corresponding to extended analyticity or meromophy properties of the four-point functions in the complex angular momentum plane, be of relevant use in a second step where the “Bethe-Salpeter structure” will be taken into account [14].

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Analytic continuation of Euclidean field theory and complex angular momentum analysis

In this last subsection, let us forget about the Minkowskian framework of Q.F.T. considered throughout this paper and adopt the viewpoint of Euclidean field theory. The starting point of such a theory is the set of n-point “Schwinger functions”, considered as tempered distributions on the corresponding n-point Euclidean ((d + 1)-dimensional) space-time, whose Fourier transforms are the n-point Green functions taken in the Euclidean energy-momentum space. In particular, the data which concern the four-point Green function are encoded in the properties of HE = H|E , where E is the Euclidean subspace of M (c) introduced in §3.1 (see the Theorem at the end of §3.1). The crucial problem of Euclidean field theory, which is the problem of reconstructing Minkowskian Q.F.T. by analytic continuation of the Schwinger functions, has been solved under a certain set of sufficient conditions called the Osterwalder-Schrader axioms [33,34]. Here we wish to stress the special and unexpected relationship which exists between this problem and the validity of complex angular momentum analysis, as a direct corollary of our Theorem 4. In fact, from the assumed SO(d+1)-invariance of the theory it follows that the Euclidean four-point function HE is represented by the function F (ζ, ζ 0 ; t, cos Θt ) considered as given on the set {(ζ, ζ 0 , t, cos Θt ); t < 0, (ζ, ζ 0 ) ∈ ∆t × ∆t , cos Θt ∈ [−1, +1]}. It is therefore equivalent to give oneself HE or the corresponding set of “partial-wave-functions” {f` (ζ, ζ 0 , t), ` ≥ 0} defined by Equation (4.2) (for t < 0, (ζ, ζ 0 ) ∈ ∆t × ∆t ). We can then state: Theorem 6 Let HE ([k]) = F (ζ, ζ 0 ; t, cos Θt ) be any SO(d + 1)-invariant Euclidean four-point function whose sets of even and odd partial-wave functions f` (ζ, ζ 0 , t) ad(m) mit Carlsonian analytic interpolations in a given half-plane {λt ∈ C+ }, denoted respectively by F˜ (s) (ζ, ζ 0 ; t, λt ) , F˜ (a) (ζ, ζ 0 ; t, λt ) and satisfying Equations (4.87), (4.88) and bounds of the form (4.84) with 11 vs = vu . Then for each vector K ∈ Rd+1 , and each unit vector e0 orthogonal to K fixing the time-direction, the restriction of HE to EˆK , admits an analytic continuation ˆ K \(Σs ∪ Σu ) (represented by the cut ˆ (cut) = Ω in the corresponding cut-manifold Ω K 0 cos Θt -plane, with ζ, ζ varying in ∆t × ∆t ). This analytic continuation is defined explicitly via formula (4.89), which reduces in EˆK , to the partial wave expansion (4.91) of HE . ˆ (cut) (for all K orThe analytic continuation of HE in the cut-manifolds Ω K thogonal to a given e0 ) is indeed interpretable as an analytic continuation which reaches the Minkowskian momentum space (spanned by the hyperplane orthogonal to e0 and the vector −ie0 ) and moreover generates the “absorptive part structure” with specified mass thresholds.

11 for

the sake of simplicity

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Appendix A Analytic completion and propagation of bounds A property of analytic completion We recall the following result, first obtained by V. Glaser by a method based on the Cauchy integral, then extended to the case of two polydisks in arbitrary situations by the tube method presented below [35]. Proposition A.1 Let T+ = (η, η0 ) ∈ C2 ; Imη > 0, Imη 0 > 0 and T− = {(η, η 0 ) ∈ C2 ; Imη < 0, Imη 0 < 0 and let R be the “coincidence region” R = (η, η 0 ) ∈ R2 ; a < η < b, a0 < η0 < b0 . Then the holomorphy envelope of the “edge-of-the-wedge domain” ∆ = T+ ∪T− ∪R can be defined as follows: H (∆) = ∪ Tα , where each domain Tα is the following 06α6π

polydisk:

Tα = (η, η 0 ) ∈ C2 ; η ∈ Γab (α) , η0 ∈ Γa0 b0 (α) ;

Γab (α) and Γa0 b0 (α) respectively denote the disks whose bordering circles make the angle α with the real axis and intersect the latter respectively at a, b and a0 , b0 (see Fig A.1); η−b η0 − b0 Proof. Let χ = log . One easily checks that the images of , χ0 = log 0 η−a η − a0 0 T+ , T− in the space of variables χ, χ are the respective tubes: T+ = (χ, χ0 ) ∈ C2 ; 0 < Imχ < π, 0 < Imχ0 < π , T− = (χ, χ0 ) ∈ C2 ; π < Imχ < 2π, π < Imχ0 < 2π , while the image of R is the set R = (χ, χ0 ) ∈ C2 , Imχ = π, Imχ0 = π , which is the common edge to T+ and T− . Then, in view of the tube theorem (applied in the limiting “edge of the wedge situation”, illustrated by Fig A.2), the holomorphy envelope H (T+ ∪ T− ∪ R) is the convex tube Tˆ = ∪ Tα whose basis in (Imχ, Imχ0 )-space is represented 06α6π

on Fig A.2. The result of Proposition A.1 is readily obtained by taking the inverse image Tα of each tube Tα in the variables (η, η 0 ). Corollary A.2 The holomorphy envelope H (∆) contains all the points of the form (η, 0) , with η varying in the cut-plane C\{η real; η ∈]a, / b[ }. Proof. These points (η, η0 = 0) have images (χ, χ0 ) such that: 0 < Imχ < 2π and Imχ0 = π and therefore belong to the convex tube Tˆ (see Fig. A.2). Propagation of bounds in the analytic completion procedure We need the following extension of the “maximum modulus principle”.

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α a

b0

a0

153

α

b η 0 -plane

η-plane

Figure A.1

Proposition A.3 Let f (η) be holomorphic in the domain ∆b = {η ∈ C; |Reη| < b, |Imη| < b} , ¯ b of ∆b . Let the following majorization hold in ∆ ¯b : and continuous in the closure ∆ |f (η)| 6 M |Imη|−n ,

(A.1)

where n is a given positive number and M is a constant. Then for all β with −b ≤ β ≤ b, one has: √ n |f (iβ)| 6 5 M b−n

(A.2)

Proof. One considers the function g(η) = (b2 − η2 )n f (η), which is also holomorphic ¯ b . One directly deduces from (A.1) the following uniform in ∆b and continuous in ∆ majorization for g on the boundary of ∆b : n

|g(η)| ≤ 5 2 M bn . In view of the maximum modulus principle, this majorization extends to all points in ∆b ; by writing it at η = iβ, one then obtains: n n n b 2 2 −n ≤ 5 2 M b−n . |f (iβ)| = |g(iβ)|(b + β ) ≤ 5 2 M 2 2 b +β

B Primitives and derivatives of non-integral order in a complex domain and Laplace transformation (cut)

a being a given positive number, we define in C the following subset Ba = Ba \σ, where Ba = {θ ∈ C; θ = u + iv, |u| < a, v > 0} and σ = {θ ∈ C; θ = iv, v > v0 }, v0 > 0 (see Fig. B1).

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Imχ0 2π

T− π+α

R

π Tα Tˆ α T+ 0

π

π+α

Imχ

Figure A.2

We then introduce the space of holomorphic functions denoted O∞ (Ba which is generated by

(cut)

(cut)

i) all functions f (θ) holomorphic in Ba

)

and satisfying bounds of the form

|f (u + iv)| 6 Cη (v)

(B.1)

in the corresponding subsets Ba(η) = Ba \ {θ ∈ C; θ =u+iv, |u| < η, v > v0 − η}

(B.2)

of C, for all η > 0. In (B.1), Cη (v) denotes an increasing and locally bounded function with at most power-like behaviour for v tending to infinity. ii) the products of functions of the previous type by θρ , with ρ real > 0. Laplace transforms Let γ0 and γ00 be two infinite paths with origin 0 in the respective half-strips u > 0 and u < 0 of Ba , and whose infinite branches are asymptotically parallel to the (cut) imaginary axis of the θ-plane. We associate with each function f (θ) ∈ O∞ (Ba )

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the Laplace-type transforms: Z L0 (f ) (λ) = L00 (f )(λ) =

Z

eiλθ f (θ) dθ,

(B.3)

eiλθ f (θ) dθ.

(B.30 )

γ0

γ00

In view of (B.1), the latter are holomorphic in the half-plane C+ = {λ ∈ C; Reλ > 0} and admit bounds of the form cε,η eη|Imλ| , for all ε > 0, η > 0, in the correspond(ε) ing half-planes C+ = {λ ∈ C; Reλ > ε} (as it results from a suitable distortion of the paths γ0 , γ00 in the integrals (B.3), (B.30 )). v

σ

Ba

v0 γ00

γ0

−a

0

a

u

Figure B.1 If f has continuous boundary values (from both sides) on σ, the corresponding discontinuity function ∆f (v) = i lim [(f (η + iv)) − f (−η + iv)] admits the η→0,η>0

Laplace transform: L (∆f ) (λ) = L0 (f ) − L00 (f ) =

Z

Z γ0 −γ00

∞

eiλθ f (θ)dθ =

e−λv ∆f (v)dv

(B.4)

0

In the general case, we still say that L(∆f )(λ) = L0 (f ) − L00 (f ) represents the Laplace-transform of the discontinuity ∆f of f, now considered as a hyperfunction

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with support σ; L(∆f )(λ) is holomorphic in C+ and such that: |L(∆f )(λ)| 6 2cε,η e−(Reλ−ε)(v0 −η) eη|Imλ|

(B.5)

(ε)

in each subset C+ of C+ , for all η > 0. Primitives of non integral order in the complex domain For every real positive α, we associate with any function f (θ) of the previous class (cut) O∞ (Ba ) the following function Z 1 α−1 [Pα f ] (θ) = (θ − θ0 ) f (θ0 ) dθ0 , (B.6) Γ (α) γ(0,θ) (cut)

where θ varies in Ba and γ(0,θ) is a path with end-points 0, θ, (homotopous to (cut) the linear segment [0, θ]) whose support is contained in Ba . By choosing for γ(0,θ) the linear segment [0, θ] and making the change of variables θ0 = θt, with t ∈ [0, 1] in (B.6), one checks that each function Pα f is the product of the ramified function θα (case i)) or more generally θα+ρ (case ii)) by a (cut) function holomorphic in Ba and that it also satisfies bounds of the form (B.1) (cut) (with functions Cηα (v) = v α Cη (v)) and therefore belongs to the class O∞ (Ba ). The same change of variables also shows that the integral (B.6) reduces to a Riemann-Liouville integral. Therefore, by using the standard properties of the latter (see [36b)], p 181–182), we obtain the following properties of the operators Pα : for all α, β > 0, Pα ◦ Pβ = Pβ ◦ Pα = Pα+β (B.7) and for all positive integers n:

d dθ

n [Pn f ] (θ) = f (θ) ,

from which it follows that, for all α > n: n d [Pα f ] (θ) = [Pα−n f ] (θ) . dθ

(B.8)

(B.8a)

These equations lead one to call Pα f (for general α) a “primitive of order α of f in the complex domain”. 12 Derivatives

n d (cut) act on the class O∞ (Ba ) for all integers n, dθ it is natural to extend Equation (B.8a) to the case α < n and to introduce the derivation Dν of non-integral order ν = n − α by the following formula: Since the operations Dn =

Dν f ≡ Dn Pα f = Dn+r Pα+r f, 12 Note

(B.8b)

that all the primitives Pα f are still well defined (via Equation (B.6)) for functions f (cut) ). such that, for some ε > 0, θ1−ε f (θ) ∈ O∞ (Ba

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in which the last equality holds, in view of (B.7),(B.8), for every integer r such that α + r > 0. We then have: Proposition B.1 (cut) a) For any function f (θ) in O∞ (Ba ) and any \ real positive number ν, the (cut) O∞ (Ba−δ ) by θ−ν . derivative Dν f is the product of a function in δ; δ>0

b) Equations (B.8a), (B.8b) admit the following generalizations, valid for all positive numbers β and ν: Dβ Pβ f = f, (B.8c) if

β > ν, Dν Pβ = Pβ−ν ,

(B.8d)

if

β < ν, Dν Pβ = Dν−β .

(B.8e)

c) If the function f (θ) is holomorphic in positive integers n, one has: [Dn Pα f ](θ) − [Pα Dn f ](θ) = θα−n

(cut) Ba ,

then for all α > 0 and for all

n−1 X

[Dp f ](0)

p=0

θp Γ(α − n + p + 1)

and for all ρ > 0:

(B.9)

(B.90 )

DPα (θρ f ) = Pα D(θρ f ).

Proof. a) Since Dν f = Dn Pα f , with ν = n − α, and since Pα f is the product of (cut) (cut) θα by a function in O∞ (Ba ) (for any α > 0 and f in O∞ (Ba )), the usual −ν derivation Dn yields the analytic structure with the factor θ and the Cauchy (cut) inequalities imply bounds of the form (B.1) in Ba−δ , for all δ > 0. b) In view of (B.8b) and (B.7), one can always write: Dν Pβ = Dn Pα+β with ν = n − α, n integer. Applying Equations (B.8) or (B.8a) or the definition (B.8b) according to whether n = α + β or n < α + β or n > α + β yields respectively Equations (B.8c),(B.8d) and (B.8e). c) If f is holomorphic at the origin, one can apply integration by parts to Equation (B.6) with f replaced by any derivative Dp f (with 0 ≤ p ≤ n − 1); one gets: [Pα Dp f ](θ) = [Pα+1 Dp+1 f ](θ) + [Dp f ](0)

θα , Γ(α + 1)

and therefore in view of (B.8a): [Dn−p Pα Dp f ](θ) = [Dn−p−1 Pα Dp+1 f ](θ) + [Dp f ](0)

θα−n+p . Γ(α − n + p + 1)

Using the latter recursively with 0 ≤ p ≤ n−1 then yields Equation (B.9). Equation (B.90 ) is obtained as (B.9) for n = 1, the r.h.s. of (B.90 ) being still meaningful (see (cut) footnote 11) since θ1−ρ D(θρ f ) belongs to O∞ (Ba−δ ) (for all δ > 0).

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Remark. Property c) extends the usual Taylor expansion (obtained for α = n in Equation (B.9)). In particular, the holomorphic (ramified) function at the r.h.s. of Equation (B.9) has no discontinuity across σ; therefore, Dn Pα f and Pα Dn f have “the same discontinuity” across σ (i.e. represent the same hyperfunction with support σ) which we denote Pα−n ∆f when n < α and Dn−α ∆f when n > α. (cut) Since Dn f is holomorphic at the origin, Pα Dn f belongs to O∞ (Ba−δ ) (for all δ > 0). In other words, if f (in O∞ (Ba )) is holomorphic at the origin, all the \ (cut) ∞ O (Ba−δ ). hyperfunctions Dν ∆f admit a representative in (cut)

δ; δ>0

Laplace transforms of the primitives Pα f and derivatives Dν f : We first notice that since all the primitives Pα f of a function f in O∞ (Ba ) remain in the same space, they all admit well-defined Laplace-type transforms L0 (Pα f ), L00 (Pα f ) (defined via Equations (B.3), (B.30 )). On the contrary, the operations L0 and L00 do not act in general on the corresponding derivatives Dν f , since the latter may contain non-integrable factors θ% (with % ≤ −1). However, the Laplace transforms L(∆Dν f ) ≡ L(Dν ∆f ) are always well defined via the following (cut) procedure. One uses the fact that for functions f (and Pα f ) in O∞R(Ba ), the iλθ defining formula (B.4) can be alternatively replaced by L(∆f )(λ) = γ e f (θ)dθ, (cut)

whose support avoids the where γ is a cycle homotopous to γ0 − γ00 in Ba (cut) origin (i.e. lies in the interior of Ba ). Since each derivative Dν f = Dn Pα f (cut) is holomorphic and of power-like growth at infinity in the interior of Ba , the previous formula applies and defines Z eiλθ [Dν f ](θ)dθ L(Dν ∆f )(λ) = (cut)

γ

as a holomorphic function in C+ . The following statement extends to the primitives Pα and derivatives Dν the usual property of Laplace transforms. Proposition B.2 For any holomorphic function f (θ) in the space O∞ (Ba ) and for the corresponding (hyperfunction) discontinuity ∆f, there holds the following property of the Laplace transforms in the half-plane C+ : a) for all α > 0, (cut)

L0 (Pα f ) (λ) = e 2 α λ−α L0 (f ) (λ), iπ

iπ α

L00 (Pα f ) (λ) = e 2 λ−α L00 (f ) (λ), iπ α

L (Pα ∆f ) (λ) = e 2 λ−α L (∆f ) (λ),

(B.11)

L (Dν ∆f ) (λ) = e− 2 ν λν L (∆f ) (λ).

(B.12)

b) for all ν > 0, iπ

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Proof. a) It is sufficient to prove the first equation in (B.11); the r.h.s. of this equation can be written for all α > 0: Z Z 1 α−1 iλθ e dθ (θ − θ0 ) f (θ0 ) dθ0 (B.13) L0 (Pα f ) (λ) = Γ (α) γ0

γ(0,θ)

For simplicity, we choose γ0 such that its support is a convex (infinite) curve (see Fig. B1) and we specify γ(0,θ) by the condition that its support is contained in the support of γ0 . For λ in C+ , the integral in (B.13) is absolutely convergent and can be rewritten (by inverting the integrations and putting θ00 = θ − θ0 ): Z Z 00 1 α−1 iλθ 0 0 0 L0 (Pα f ) (λ) = e f (θ ) dθ eiλθ (θ00 ) dθ00 , (B.14) Γ (α) γ0 (θ 0 )

γ0

where the support of γ0 (θ0 ) is the set θ00 ∈ C; θ00 + θ0 ∈ supp γ0 \ supp γ(0,θ0 ) . Since this (infinite) path γ0 (θ0 ) is homotopous to [0, i∞[ , the subintegral of (B.14) iπ α R∞ π 2 is independent of θ0 and equal to ei 2 α 0 e−λv v α−1 dv = e λαΓ(α) . b) Let ν = n − α, with α < n; in view of (B.10), we have: Z eiλθ [Dn (Pα f )](θ)dθ L (Dν ∆f ) (λ) = L (Dn Pα ∆f ) (λ) = γ

Z = (−iλ)n

eiλθ [Pα f ](θ)dθ = (−iλ)n L (Pα ∆f ) (λ). γ

Equation (B.12) then readily follows from the latter and from (B.11). The case of distribution-like boundary values on σ We shall now restrict our attention to functions of the class O∞ (Ba ) which are “of moderate growth” near the cut σ. More precisely, we introduce for each real (cut) (cut) number β, with β > 0, the class Oβ (Ba ) by the same definition as O∞ (Ba ), except for the uniform bounds (B.1) which are replaced by: (cut)

|f (u + iv)| 6

C (v) , ηβ

(B.15)

(η)

in the corresponding subsets Ba of Ba , (C (v) being again a locally bounded function with power-like behaviour for v tending to infinity). When the bounds (B.15) are replaced by logarithmic bounds of the form: |f (u + iv)| 6 C (v) |ln η| ;

(B.16) (cut)

the corresponding class of holomorphic functions f (θ) is called O0∗ (Ba

).

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We also need to consider functions f of the class O0 (Ba ) which have continuous boundary values on σ, as well as all their derivatives Dν 0 f for all ν 0 < ν, ν being a given positive number. each of these derivatives Dν 0 f is \ If moreover 0 (cut) O0 (Ba−δ ) by θ−ν , we say that f belongs to the product of a function in δ; δ>0

the class

(cut) Oν (Ba ).

Functions in these classes satisfy the (cut)

Proposition B.3 If f belongs to Oν (Ba ), then the Laplace transform L (∆f ) of the discontinuity ∆f of f satisfies uniform bounds of the following form 0

|L (∆f ) (λ)| 6 cεε0 |λ|−ν+ε e−(Reλ−ε)v0

(B.17)

in the corresponding half-planes C+ , for all ε > 0, ε0 > 0. (ε)

Proof. In view of Proposition B.2 b), we can write for any ε0 > 0: 0

0

L (Dν−ε0 ∆f ) (λ) = ei 2 (ε −ν ) λν−ε L (∆f ) (λ). π

(B.18)

Since |[Dν−ε0 f ] (u + iv)| 6 C (v) , for |u| < a − δ, v > δ (with 0 < δ < v0 ), the expression of L(Dν−ε0 ∆f )(λ) given by (B.10) (with γ flattened onto σ from both sides) can be uniformly bounded in modulus by cεε0 e−(Reλ−ε)v0 in any half-plane (ε) C+ (ε > 0). This implies the bound (B.17) in view of Equation (B.18). (cut)

We now study the properties of the functions in the classes Oβ (Ba ), β > 0, (cut) and O0∗ (Ba ) and characterize in a precise way their distribution-like boundary values on the cut σ and their Laplace transforms. (cut)

Proposition B.4 Let f (θ) belong to a class Oβ (Ba ), with β > 0, or (for β = 0) (cut) to O0∗ (Ba ). Then i) The various “primitives” Pα f (α > 0) satisfy the following properties: (cut) a) if α < β, Pα f belongs to the class Oβ−α (Ba ), (cut) b) if α = β, Pα f belongs to the class O0∗ (Ba ), (cut) c) if α > β, Pα f belongs to the class Oα−β (Ba ); ii) The boundary values f+ , f− of f on iR from the respective sides Reθ > 0, Reθ < 0, and the corresponding discontinuity ∆f = i (f+ − f− ) (with support contained in σ) are defined in the sense of distributions and such that f± = Dp F± ,

∆f = Dp ∆F,

with F± continuous on iR, supp ∆F ⊂ σ and p = E(β) + 1; iii) The Laplace transform L(∆f ) of the distribution ∆f satisfies uniform bounds of the following form (for all ε > 0, ε0 > 0) 0

|L (∆f ) (λ)| 6 cεε0 |λ|β+ε e−(Reλ−ε)v0 (ε)

in the corresponding half-planes C+ .

(B.19)

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Proof. i) For all α (α > 0), the expression (B.6) of [Pα f ](θ) can be rewritten with the following choice: supp γ(0,θ) = [0, b] ∪ [b, b + iv] ∪ [b + iv, u + iv] , where θ = u + iv and b is a fixed number such that 0 < |b| < a. As seen below, this choice is suitable for showing that [Pα f ](θ) satisfies bounds of the form (B.15) or (B.16) (η) on the part u = ±η, v ≥ v0 − η of the border of a given region Ba (estimates on the remaining “small” part |u| < η, v = v0 − η are similar 13 ). (cut) Let us first assume that f belongs to Oβ (Ba ), with β ≥ 0. In view of (B.15), one readily obtains that the first two contributions to [Pα f ](θ) (given by the integrations on [0, b] and [b, b + iv]) admit uniform bounds of the form c(v), where c(v) is locally bounded and power-like behaved for v tending to infinity. The third contribution (given by the interval [b + iv, u + iv]) can be majorized by the following expression (written for the case 0 < u = η < b): Z b 1 α−1 −β (u0 − η) (u0 ) du0 . (B.20) C (v) Γ (α) η a) If α < β, the integral in (B.20) is bounded by cst η −(β−α) and therefore Pα f (cut) belongs to Oβ−α (Ba ). b) If α = β, the integral in (B.20) is bounded by cst |ln η|. This shows that Pβ f (cut) belongs to the class O0∗ (Ba ). c) If α > β, the integral in (B.20) is bounded by a constant and therefore Pα f (cut) belongs to O0 (Ba ). In order to show that Pα f admits continuous boundary values on σ, one writes Pα f = Pε Pα−ε f for a given ε > 0 such that α−ε > β. Since (cut) g = Pα−ε f is then itself in O0 (Ba ), one is led to apply directly the following result to the expression (B.6) of [Pε g](θ) (with the choice of the linear segment (cut) [0, θ] for supp γ(0, θ), θ being or on the cut σ ): for every ε > 0, the a R x either in Bε−1 Abel transform gε (x) = 0 f (y) (x − y) dy of a locally bounded function f is continuous. Moreover the previous argument holds for every derivative Dν 0 (Pα f ) such that ν 0 < α − β, since in this case (in view of (B.8d)) Dν 0 (Pα f ) = Pα−ν 0 f . (cut) We have thus proved that Pα f belongs to the class Oα−β (Ba ). In order to complete the study of the case c), let us now assume that f (cut) belongs to O0∗ (Ba ); in view of (B.16), the majorization (B.20) on the third contribution to [Pα f ](θ) is now replaced by Z b 1 α−1 (u0 − η) |ln u0 | du0 (B.200 ) C (v) Γ (α) η (cut)

which is bounded by cst C(v). This proves that Pα f belongs to O0 (Ba ), and since the result holds for every α0 , with 0 < α0 < α, the same argument as above (cut) in c) shows that Pα f belongs to the class Oα (Ba ) for all α > 0. 13 For |u| < η, v = v − η, one chooses the path with support [0, u] ∪ [u, u + i(v − η)], which 0 0 yields two contributions to (B.6): while the first one is bounded by a constant, the second one is bounded (up to a constant factor) by the same integral as in (B.20) or (B.200 ) whose dependence on η yields the desired result.

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(cut)

(cut)

ii) If f belongs to a class Oβ (Ba ), or also (for β = 0,) to O0∗ (Ba ), let p = E(β) + 1; it follows from i)c) that the function F = Pp f admits continuous boundary values F+ , F− on σ. Then it results from the standard definition of distribution-like boundary values of holomorphic functions that the function p d f (θ) = F (θ) admits boundary values on σ which are the corresponding dθ p 1 d F± denoted Dp F± . Since derivatives in the sense of distributions f± = i dv F+|[0,v0 [ = F−|[0,v0 [ , the discontinuity ∆F = i (F+ − F− ) of F is a continuous function with support contained in σ, which yields the desired structure for the distribution ∆f = Dp ∆F. iii) Let us consider, for any ε0 > 0, the Laplace transform L (Pβ+ε0 ∆f ) ; in view of i)c), Pβ+ε0 ∆f is a continuous function with support contained in σ and satisfying a bound of the following form: |[Pβ+ε0 ∆f ](v)| 6 Cε0 (v) , where Cε0 (v) has power-like behaviour at infinity. Therefore the corresponding expression of L (Pβ+ε0 ∆f ) (λ) given by (B.4) (ε) can be uniformly bounded by cεε0 e−(Reλ−ε)v0 in any half-plane C+ (ε > 0). Since we have (in view of Proposition B.2, Equation (B.11)): 0

0

L (∆f ) (λ) = e− 2 (β+ε ) λβ+ε L (Pβ+ε0 ∆f ) (λ) , iπ

the majorization (B.19) follows from the previous bound on L (Pβ+ε0 ∆f ). We now complete the statements of Proposition B.4 i) by considering the action of derivatives Dν of arbitrary order ν. (cut)

Proposition B.5 Let f (θ) belong to a class Oβ (Ba ), with β ≥ 0. Then, for all (cut) ν > 0, the product θν Dν f (θ) belongs to the class Oβ+ν (Ba−δ ) for any δ > 0. Proof. Putting ν = n − α, with n integer and 0 < α < 1, we can write in view of Equations (B.9),(B.90 ): [Dν f ](θ) = [Dn Pα f ](θ) = [Dn−1 (Pα Df )](θ) + f (0)

θ−ν . Γ(1 − ν)

Since the second term at the r.h.s. of this equation has no discontinuity across σ, (cut) we are led to prove that if f belongs to Oβ (Ba ), then Dn−1 (Pα Df ) belongs (cut) β+ν to O (Ba−δ ) for any δ > 0. At first, simple estimates based on the Cauchy formula for the derivative of a holomorphic function show that Df belongs to (cut) Oβ+1 (Ba−δ ) for all δ > 0. Then, since α < 1 < β + 1, the case a) of Proposition B.4 i) applies to Pα Df and implies that this function belongs to the corresponding classes Oβ+1−α ; applying now again the Cauchy formula to the derivative Dn−1

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of the previous function implies that Dn−1 (Pα Df ) belongs to Oβ−α+n (Ba−δ ) for all δ > 0. The rest of this Appendix is devoted to proving the following result which is of direct use for our Theorem 3 (see Sec 4). Although very close to Proposition B.4 i) in its form, this result is technically more sophisticated since its statement involves conjointly primitives Pα with respect to the complex variable z = cos θ, together with the previous derivatives Dν with respect to θ (involved in the definition of (cut) the classes Oα−β (Bπ ) used again below). Proposition B.6 Let α, β and m be fixed real numbers such that α > 0, β ≥ 0 and (cut) and such that: m > −1. With each function f holomorphic in Bπ i) f satisfies uniform bounds of the following form |f (θ)| ≤ (η)

CemImθ ηβ (cut)

in all the corresponding subsets Bπ of Bπ , ii) f (u) = f (−u) for 0 ≤ u ≤ π, one associates the following function: (α) (θ) = ei(m+α) fˆm

θ

1 Γ(α)

(cut)

where θ varies in Bπ

Z

cos θ

−1

(cos θ − cos τ )α−1 f (τ ) d cos τ,

(B.21)

.

Then a) If α < β, b) If α = β, c) If α > β,

(α) (cut) fˆm belongs to the class Oβ−α (Bπ ), (α) (cut) fˆm belongs to the class O0∗ (Bπ ), (α) (cut) fˆm belongs to the class Oα−β (Bπ ).

The proof of the latter relies on two auxiliary lemmas, for which we need the following notations. Let CA = C\{z real; z ≥ A}\{z real; z ≤ −1} with A ≥ 1. For every function f (z) , holomorphic in CA and continuous on the cut z ≤ −1, and for every α > 0, we put Z z 1 α−1 [P α f ](z) = f (z 0 ) (z − z 0 ) dz 0 . (B.22) Γ(α) −1 (cut)

By choosing A = cosh v0 , the cut-plane CA appears as the image of the set Bπ by the mapping z = cos θ. Considering the function f (θ) = f (cos θ), holomorphic (cut)

in Bπ

(and such that f (u) = f (−u) for −π ≤ u ≤ π), we then also put:

[Pα f ](θ) = [P α f ](cos θ) = −

1 Γ(α)

Z

θ

π

(cos θ − cos τ )α−1 f (τ ) sin τ dτ

(B.23)

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Lemma B.7 closely parallels the results of Proposition B.4 i) but it involves primitives P α taken in the cut-plane CA and a corresponding new specification of the increase properties of the holomorphic functions considered. Lemma B.7 Let f (z), holomorphic in CA and continuous on the cut z ≤ −1, satisfy uniform bounds of the following form m f (z) 6 C (1 + |z|) , φβ

with m > −1 and β ≥ 0

(B.24)

in the corresponding regions (see Fig. B2) φ (φ) CA = CA \ z ∈ C; z = ρeiψ , ρ > A(1 − ), |ψ| < φ π for all φ (0 < φ < π). Then [P α f ](z) is holomorphic in CA and satisfies uniform bounds of the following (φ)

form for z ∈ CA : a) If α < β,

b) If α = β, c) If α > β,

|z|)m+α [P α f ](z) 6 Cα (1 +β−α φ

(B.25)

[P α f ](z) 6 Cα (1 + |z|)m+α |ln φ|

(B.26)

[P α f ](z) 6 Cα (1 + |z|)m+α

(B.27)

and [P α f ](z) is continuous in the closure of CA (from both sides of the cuts). Proof. In order to obtain the bounds (B.25)–(B.27), it is sufficient to consider two typical geometrical situations: i) z is of the form z = A(1− πφ )eiψ , with 0 ≤ |ψ| ≤ φ; the integration path in (B.22) is then chosen as the union of two linear paths with supports {z 0 real; −1 ≤ z 0 ≤ 0} 0 and {z 0 ∈ C; z 0 = A(1 − φπ )eiψ , φ ≤ φ0 ≤ π}. By using the assumption (B.24), one checks that the first contribution to [P α f ](z) is bounded by a constant, while Rπ the second one is majorized (up to a constant factor) by φ (φ0 − φ)α−1 (φ0 )−β dφ0 , which is of the same form as the integral in (B.20). In view of the analysis after (B.20) (cases a), b), c)), we then obtain the corresponding bounds (B.25)–(B.27) (with |z|m+α replaced by a constant). ii) z is of the form z = ρeiφ , with ρ > A(1 − πφ ); the integration path in (B.22) is then chosen as the union of three paths (see Fig. B2) with respective supports 0 {z 0 real; −1 ≤ z 0 ≤ 0}, {z 0 = ρ0 eiφo ; 0 ≤ ρ0 ≤ ρ} and {z 0 = ρeiφ ; φ ≤ φ0 ≤ φ0 } (φ0 being a fixed angle with 0 < φ0 ≤ π). In view of (B.24), the corresponding first two contributions to P α f (z) are majorized respectively by cst|z|α−1 and cst|z|m+α and

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z = ρeiφ

φ0 φ −1

³

0

A 1−

φ π

A

´

Figure B.2 therefore (since m > −1) both by cst|z|m+α . The contribution given by the third Rφ path is majorized by cst|z|m+α φ 0 (φ0 − φ)α−1 (φ0 )−β dφ0 . By applying again the results described after Equation (B.20), we then obtain the majorizations (B.25)– (B.27) in the corresponding cases a), b) and c). Finally the continuity of P α f on σ (from both sides) in case c) is again justified as in Proposition B.4. d Lemma B.8 shows that the identity h(θ) = dθ [P1−ν (Pν h)] (θ) is replaced by an equally regular operation when Pν is replaced by Pν and intertwining exponentials are added. (cut)

Lemma B.8 For every function h(θ) holomorphic and uniformly bounded in Bπ and admitting continuous boundary values on σ, the following transform i d h (θ) [Kν,µ,r h](θ) = P1−ν ei(r+µ)θ Pν e−i(µ−ν)θ h dθ \ (cut) O0 (Bπ−δ ) by θ−ν , and also admits continuous is the product of a function in δ; δ>0

boundary values on σ, provided one has 0 < ν < 1, µ > ν − 1, r ≥ 0. Proof. In view of Equations (B.6) and (B.23), we have: Z θ 0 −1 d [Kν,µ,r h](θ) = (θ − θ0 )−ν ei(r+µ)θ dθ0 . . . Γ(1 − ν)Γ(ν) dθ 0 # "Z 0 θ

0

ν−1 −i(µ−ν)τ

(cos θ − cos τ )

... π

e

h(τ ) sin τ dτ

(B.28)

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which is well defined for 0 < ν < 1 and can be rewritten as a sum of two terms [Kν,µ,r h](θ) = h1 (θ) + h2 (θ)

(B.29) R0 corresponding to the following splitting of the integral over τ : π = 0 + π . We shall then study h1 and h2 separately and prove that they both satisfy the property to be shown for Kν,µ,r h. We first treat the term h1 by inverting the order of the integrations over θ0 and τ , which yields: Z θ d K(θ, τ ) ei(r+ν)τ h(τ ) sin τ dτ, (B.30) h1 (θ) = − dθ 0 R θ0

R θ0

where the kernel K is defined as follows: Z θ 0 1 K(θ, τ ) = (θ − θ0 )−ν (cos θ0 − cos τ )ν−1 ei(r+µ)(θ −τ ) dθ0 . (B.31) Γ(1 − ν)Γ(ν) τ The validity of Equation (B.30) is submitted to the proof of the regularity of K given below; in particular, the following alternative to Equation (B.30) will be justified after checking the regularity of K on the diagonal: Z θ ∂K i(r+ν)θ h1 (θ) = −K(θ, θ) e h(θ) sin θ − (θ, τ ) ei(r+ν)τ h(τ ) sin τ dτ. (B.32) 0 ∂θ Study of K: by putting θ0 = τ + t(θ − τ ) and passing to the integration variable t (0 ≤ t ≤ 1) in Equation (B.31), we can rewrite the latter as follows: Z 1 t(θ − τ ) i(r+µ)t(θ−τ ) 1 Φ τ, (1 − t)−ν tν−1 dt, (B.33) K(θ, τ ) = e Γ(1 − ν)Γ(ν) 0 2 where:

ν−1 sin ζ Φ(τ, ζ) = − . sin(τ + ζ) ζ

(B.34)

One immediately obtains that K(θ, θ) = Φ(θ, 0) = (− sin θ)ν−1 , so that the first contribution to h1 (θ) in Equation (B.32) is equal to (− sin θ)ν ei(r+ν)θ h(θ). Since (cut) h is holomorphic and bounded, this function belongs to O0 (Bπ ) (under the assumptions ν > 0 and r ≥ 0); it also admits continuous boundary values on σ like h. One notices that this contribution is the exact analogue of the reproducing d expression dθ [P1−ν (Pν h)] (θ) = h(θ). The study of the second contribution to Equation (B.32) relies on the following expression of ∂K ∂θ (deduced from (B.33)): 1 ∂K (θ, τ ) = ··· ∂θ Γ(1 − ν)Γ(ν) Z 1 1 ∂Φ i(r + µ)Φ(τ, ζ) + ei(r+µ)t(θ−τ ) (1 − t)−ν tν dt. (τ, ζ) 2 ∂ζ 0 |ζ= t(θ−τ ) 2

(B.35)

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Since ν − 1 < 0, Equation (B.34) implies that the expression inside the bracket {. . . } in the latter integral is (for each t ∈ [0, 1]) a holomorphic function of τ and θ in the domain ∆ = {(θ, τ ) ∈ C2 ; θ ∈ Bπ , τ ∈ Bπ , 0 < Imτ < Imθ} which ν−2 is uniformly bounded by cst τ ν−2 e(tIm(θ−τ )+Imτ )(ν−1) . up to peaks in |(π ± θ)| near θ − ±π (their contributions can be factored out in the bounds). It directly follows that, under the conditions r ≥ 0, µ > ν − 1, the complete integrand of (B.35) and thereby the kernel ∂K ∂θ (θ, τ ) are themselves holomorphic and uniformly bounded by cst τ ν−2 e(ν−1)Imτ in ∆. One then sees (by using again the condition r ≥ 0) that in the second contribution to the r.h.s. of Equation (B.32), the integrand is uniformly bounded by cst τ ν−1 ; this contribution is therefore holomorphic (cut) in Bπ (except for a branch-point with behaviour θν at θ = 0), and uniformly ν−2 bounded there by cst θν up to the previous peaks in cst |π ± θ| near θ = ±π. (cut) 0 It therefore belongs to O (Bπ−δ ) for all δ > 0. Moreover, since K is analytic for θ = iv, τ = iw, 0 < w ≤ v, this contribution admits (like h) continuous boundary values on σ. We have thus proved that h1 (θ) satisfies the desired properties. The term h2 (θ) is treated directly by writing (in view of (B.28)): h2 (θ) =

d −1 Γ(1 − ν)Γ(ν) dθ

with

Z Ψ(θ) = −

π

Z

θ

0

(θ − θ0 )−ν ei(r+µ)θ Ψ(θ0 )dθ0 ,

(B.36)

0

(cos θ − cos τ )ν−1 e−i(µ−ν)τ h(τ ) sin τ dτ.

(B.37)

0

In fact, both functions Ψ(θ) and θ ∂Ψ ∂θ (θ) are holomorphic in Bπ (except for branchpoints at 0, π and −π) and uniformly bounded by cst e(ν−1)Imθ in this domain, 2ν−1 up to peaks in |(π ± θ)| near θ = ±π. By passing to the integration variable 0 θ t = θ , 0 ≤ t ≤ 1, in Equation (B.36), which allows one to derive with respect to θ under the integral and to factor out θ−ν , one can make use of the previous bounds. In view of the conditions r ≥ 0, µ > ν − 1, one checks that the integral is (cut) uniformly bounded in Bπ−δ and therefore that θν h2 belongs to O0 (Bπ−δ ) for all δ > 0; moreover, h2 is holomorphic on σ (like Ψ, it has no cut). Proof of Proposition B.6. One easily checks that the function f (z) defined by f (cos θ) = f (θ) is holomorphic in CA , with A = cosh v0 , and that it satisfies (η)

the assumptions of Lemma B.7. This follows from the fact that the sets Bπ (see (φ) Equation (B.2)) are equivalent to the sets CA of Lemma B.7 by the mapping (aη) (η) (bη) Z : θ → z = cos θ, (in the following sense: CA ⊂ Z(Bπ ) ⊂ CA with 0 < a < 1 < b) and that a|e−iθ | < (1 + |z|) < b|e−iθ | (a, b, a and b being fixed numbers). It then follows from these facts and from the conclusions of Lemma B.7 (for(α) mulae (B.25)–(B.27)) that the corresponding functions fˆm (θ) = ei(m+α)θ (Pα f )(θ) (cut) and enjoy the following properties: are holomorphic in Bπ

168

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|fˆm (θ)| ≤

a) If α < β,

C

Ann. Henri Poincar´e

(η) (α) for θ ∈ Bπ , which entails that fˆm ∈

η β−α

(cut)

Oβ−α (Bπ

), (α) (η) (α) |fˆm (θ)| ≤ C| ln η| for θ ∈ Bπ , which entails that fˆm ∈

b) If α = β, (cut) O0∗ (Bπ ), (α) (cut) c) If α > β, fˆm is bounded and continuous in the closure of Bπ . In order (α) (cut) to establish that fˆm belongs to the class Oα−β (Bπ ), we shall now prove α (θ) is the that for all real ν such that 0 < ν < α − β, the function Dν fˆm (cut) product of θ−ν by a holomorphic function belonging to O0 (Bπ−δ ) for all δ > 0. This will be done in three steps: we first give a proof for the case of ordinary derivatives, i.e. ν = r integer; then we reduce the proof for a general non-integral value of ν to that of a similar property for the corresponding value ν1 = ν − E(ν) and finally we show the latter property for all values of ν1 with 0 < ν1 < 1. 1) ν = r integer: we claim that a relation of the following form holds: i h (α) ](θ) ≡ Dr ei(m+α) θ (Pα f ) (θ) [Dr fˆm = =

r X r0 =0 r X

0

0

X (r ) (eiθ )ei(m+α−r ) θ [Pα−r0 f ](θ)

(B.38)

0 (α−r0 ) X (r ) (eiθ ) fˆm (θ),

r0 =0 0

where each X (r ) is a polynomial of degree 2r0 . This relation (which is a variant of Equation (II.43) of [25b)]) is obtained by taking the derivative of order r with respect to θ in the integral of (B.21): this is justified since α > β + r > r. (α) Equation (B.38) immediately shows that [Dr fˆm ](θ) is bounded and contin(cut) (α−r0 ) (where α − r0 > β) uous in the closure of Bπ , since each of the factors fˆm 0

and X (r ) (eiθ ) (with |eiθ | = e−v < 1 in Bπ

(cut)

) satisfies this property individually.

2) for non-integral ν , let ν = ν1 + r with r = E(ν) ≥ 0, 0 < ν1 < 1. We apply α (θ) is holomorphic at θ = 0): Equation (B.9) (which is legitimate since fˆm α α α [Dν fˆm ](θ) = [Dr+1 P1−ν1 fˆm ](θ) = [DP1−ν1 Dr fˆm ](θ) +

r−1 X [Dp f ](0) p=0

θp−ν Γ(p − ν + 1)

(B.39) The sum at the r.h.s. of Equation (B.39) is the product of θ−ν by a function in (cut) O0 (Bπ ) (with no cut on σ). In view of Equation (B.38) we are thus led to show the following property (in which we have put α0 = α−r0 , with α0 ≥ α−r > β +ν1 ):

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Let 0 < ν1 < 1; then for every α0 iwith α0 > β + ν1 and for every r0 (r0 ≥ 0), h 0 α0 the function DP1−ν1 X (r ) (eiθ ) fˆm (θ) is the product of θ−ν1 by a function in \ (cut) O0 (Bπ−δ ) and it admits continuous boundary values on σ. δ; δ>0 α (θ)]). (When ν = ν1 < 1 (i.e. r = 0), one just uses the latter for DP1−ν [fˆm

3) The proof of the previous statement relies in a crucial way on Lemma B.8. In 0 fact, it is sufficient to replace the polynomial X (r ) (eiθ ) by a typical term eirθ , r ≥ 0, and to study the expression: i i h 0 d h α0 Dν1 eirθ fˆm (θ) = P1−ν1 ei(r+m+α ) θ Pα0 (f ) (θ). dθ

(B.40)

We can now write Pα0 f = P α0 f = P ν1 g, with g = P α0 −ν1 f and notice that, since 0 α0 − ν1 > β, Lemma B.7 c) implies that one can put g(cos θ) = e−i(m+α −ν1 )θ h(θ), (cut)

with h holomorphic and uniformly bounded in Bπ , and admitting continuous boundary values on σ. Equation (B.40) then becomes: i i h 0 0 d h α0 Dν1 eirθ fˆm (θ) = P1−ν1 ei(r+m+α )θ Pν1 (e−i(m+α −ν1 )θ h) (θ). dθ = Kν1 ,m+α0 ,r h(θ)

(B.41)

Since α0 > ν1 and m > −1, the assumptions of Lemma B.8 are satisfied and the announced result follows, which ends the proof of Proposition B.6.

Acknowledgements We are very grateful to the Italian Istituto Nazionale di Fisica Nucleare (I.N.F.N.), Sezione di Genova, for its financial support to one of us (J.B.) during the elaboration of this research project.

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References [1] J.R. Forshaw and D.A. Ross, “QCD and the Pomeron”, Cambridge University Press, 1997. [2] V.S. Fadin, E.A. Kuraev and L.N. Lipatov, Phys. Lett. B 60 (1975) p. 50; I.I. Balitsky and L.N. Lipatov, Sov. J. Nucl. Phys. 28 (1978) p. 822. [3] M.C. Berg`ere and C. Gilain, “Regge-pole behaviour in φ3 -field theory”, J. Math. Phys. 19 (1978) p. 1495–1512; M.C. Berg`ere and C. de Calan, “Reggepole behaviour from perturbative scalar field theories”, Phys. Rev. D 20 (1979) p. 2047–2067. [4] R.K. Ellis, W.J. Stirling and B.R. Webber “QCD and Collider Physics”, Cambridge University Press, 1996. [5] A. Donnachie and P.V. Landshoff, Z. f¨ ur Phys. C 61, (1994) p. 139. [6] H.M. Nussenzweig, “Diffraction effects in semi-classical Physics”, Cambridge University Press, 1992. [7] V, de Alfaro and T. Regge, “Potential Scattering”, North Holland, 1965. [8] R.G. Newton, “The complex j-plane”, W.A. Benjamin, 1964. [9] P.D.B. Collins, “Introduction to Regge theory and High Energy Physics”, Cambridge University Press, 1977. [10] J. Bros and G.A. Viano, “Complex angular momentum in axiomatic quantum field theory”, in Rigorous methods in particle physics, S.Ciulli, F. Scheck, W. Thirring Eds. (Springer tracts in Mod. Phys. 119 (1990)) p. 53–76. [11] M. Froissart, “Asymptotic behaviour and subtractions in the Mandelstam representation”, Phys. Rev. 23 (1961) p. 1053–1057. [12] V.N. Gribov, “Partial waves with complex angular momenta and the asymptotic behaviour of the scattering amplitude”, J. Exp. Theor Phys. 14 (1962) p. 1395. [13] H. Epstein, “Some Analytic Properties of Scattering Amplitudes in Quantum Field Theory”, in Axiomatic Field Theory, M. Chr´etien and S. Deser Eds, Gordon and Breach, New York, 1966, p. 3–133. [14] J. Bros and G.A. Viano, “From Bethe-Salpeter Equation to Regge-poles in General Quantum Field Theory”, in preparation. [15] R. Omnes, “D´emonstration des relations de dispersion” in Relations de dispersion et particules ´el´ementaires, C. De Witt and R. Omnes Eds, Hermann, Paris, 1960, p. 317–385.

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[16] A. Martin, Nuovo Cimento 42 (1965) p. 930 and 44 (1966) p. 1219. [17] O. Steinmann, Helv. Phys. Acta 33 (1960) p. 257. [18] D. Ruelle, Nuovo Cimento 19 (1961) p. 356 and Thesis, Z¨ urich, 1959. [19] H. Araki, J. Math. Phys. 2 (1961) p. 163. [20] H. Epstein, V. Glaser, R. Stora, “General properies of the n-point functions in local quantum field theory” in Structural Analysis of Collision Amplitudes, R. Balian and D. Iagolnitzer Eds, North-Holland, Amsterdam, 1976, p. 7–93. [21] O. Steinmann, Commun. Math. Phys. 10 (1968) p. 245. [22] A.S. Wightman, “Analytic functions of several complex variables” in Relations de dispersion et particules ´el´ementaires, C. De Witt and R. Omnes Eds, Hermann, Paris, 1960, p. 227–315. [23] J. Bros, H. Epstein and V. Glaser, “Some rigorous analyticity properties of the four-point function in momentum space”, Nuovo Cimento 31 (1964) p. 1265–1302. [24] J. Bros and G.A. Viano, “Connection between the algebra of kernels on the sphere and the Volterra algebra on the one-sheeted hyperboloid: holomorphic perikernels”, Bull. Soc. Math. France 120 (1992) p. 169–225. [25] J. Bros and G.A. Viano, “Connection between the harmonic analysis on the sphere and the harmonic analysis on the one-sheeted hyperboloid: an analytic continuation viewpoint”, Forum Math. a) 8 (1996) p. 621–658, b) 8 (1996) p. 659–722, c) 9 (1997) p. 165–191. [26] J. Faraut, “Analyse Harmonique et Fonctions Sp´eciales”, Ecole d’´et´e d’Analyse Harmonique de Tunis, 1984. [27] N.Ja. Vilenkin, “Special Functions and the Theory of Group Representations”, Amer. Math. Soc. Transl. 22, Providence R.I., 1968. [28] R.F. Streater and A.S. Wightman, “PCT, Spin and Statistics and all that”, W.A. Benjamin, New York, 1964. [29] R.P. Boas, “Entire Functions”, Academic Press, New York, 1954. [30] A. Sommerfeld, “Partial Differential Equations in Physics”, Academic Press, New York, 1949. [31] G.N. Watson, “The diffraction of electric waves by the earth” Proc. Roy. Soc., London, 95 (1918) p. 83–99. [32] J. Faraut and G.A. Viano, “Volterra algebra and the Bethe-Salpeter equation” J. Math. Phys. 27 (1986) p. 840–848.

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[33] K. Osterwalder, R. Schrader, Commun. Math. Phys. 33 (1973) p. 83 and 42 (1975) p. 281. [34] V. Glaser, Commun. Math. Phys. 37 (1974) p. 257. [35] J. Bros and V. Glaser, “L’enveloppe d’holomorphie de l’union de deux polycercles”, mimeographed report, Saclay 1961. [36] A. Erd´elyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, a) “Higher Transcendantal Functions”, Vol.1, McGraw-Hill, New-York, 1953, b) “Tables of Integral Transforms”, Vol.2, McGraw-Hill, New-York, 1954. J. Bros Service de Physique Th´eorique CEA – Saclay F-91191 Gif-sur-Yvette Cedex France G.A. Viano Istituto Nazionale di Fisica Nucleare (I.N.F.N.) Sezione di Genova Dipartimento di Fisica dell’Universit` a I-Genova, Italy Communicated by J. Bellissard submitted 04/01/99, accepted 02/07/99

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auser Verlag, Basel, 2000 1424-0637/00/010173-19 $ 1.50+0.20/0

Annales Henri Poincar´ e

Solution of Certain Integrable Dynamical Systems of Ruijsenaars-Schneider Type with Completely Periodic Trajectories F. Calogero and J.P. Fran¸coise Abstract. The first main result Pn of this paper is the solution of the (complex) equations of motion z¨j +iΩ z˙j = z˙ z˙ f (zj −zk ) with f (z) = 2a cotgh (az)/[1+ k=1,k6=j j k r2 sinh2 (az)], and the consequent confirmation of the conjecture that all the trajectories of this dynamical system are completely periodic with period (at most) T 0 = T n!, T = 2π/Ω. We also discuss a symplectic reduction scheme which features new Lie-theoretic aspects for these systems. These developments are introduced here in the perspective of applying them in future studies to implement geometric quantization techniques.

I Introduction It was recently pointed out [1] that the dynamical systems of Ruijsenaars-Schneider (RS) type characterized by the equations of motion n X

z˙j z˙k f (zj − zk ), j = 1, . . . , n,

(1.1)

“case (i)”,

(1.2a)

“case (ii)”, “case (iii)”, “case (iv)”,

(1.2b) (1.2c) (1.2d)

f (z) = 2acotgh(az)/[1 + r2 sinh2 (az)] “case (v)”, f (z) = −aP 0 (az)/[P(az) − P(ab)] “case (vi)”,

(1.2e) (1.2f)

z¨j + iΩ z˙j =

k=1,k6=j

are “integrable” or “solvable” [2], if f (z) = 2/z 2 2

f (z) = 2/[z(1 + r z )] f (z) = 2acotgh(az) f (z) = 2a/sinh(az)

and it was conjectured that, if and only if the constant Ω is real and nonvanishing, all their trajectories zj (t), j = 1, . . . , n, are periodic with period (at most) T 0 = T n! , T = 2π/Ω.

(1.3)

This conjecture was proven in Ref. [1] (on the basis of previous findings [3]) for cases (i) and (iii), by solving the corresponding equations of motion: zj (t) are

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given by (in case (i); in case (iii), they are closely related to) the n (complex) zeros of a (monic) polynomial of degree n in z whose n coefficients are explicitly known functions of time, all periodic with period T . The availability of this method of solution suggests calling these two models “solvable” [2]. The main contribution of this paper is to also solve, rather explicitly, the equations of motion (1) for all the other cases listed above, except case (vi): the validity of the conjecture [1] is thereby validated for these cases. The technique of solution is based on the possibility to put these equations of motion in “Lax pair” form [4,1] (for this reason these systems are called “integrable” [2]): the solutions zj (t) are then given by (or are closely related to) the n eigenvalues of an (explicitly known) (n×n)-matrix which is periodic in time with period T . Of course the solutions zj (t) of (1.1) move in the complex plane; and indeed all the constants appearing in (1.2), namely r, a and b, as well as the constants ω and ω 0 implicit in the definition of the Weierstrass function P(z) ≡ P(z|ω, ω 0 ), might be complex. Note that in case (i) it is justified to identify the complex plane with the physical plane, obtaining thereby a (real) rotation-invariant model which describes the motion of particles in the plane, and which is in fact the special case of a more general solvable model of this type [5]. More generally, a reinterpretation of the complex model (1.1) as a real rotation-invariant model describing motions in the real (“physical”) plane is also possible for any choice of the function f (z) [6]. In the following Section II we describe a simple trick [2], namely a change of the dependent variable (“time”), which allows to relate equations of type (1.1) to analogous equations of motion, but with Ω = 0. In Section III we prove our main result for case (v), and then by specialization for cases (i)–(iv) as well: these are indeed all special subcases of case (v), see below (while case (v) is itself a subcase of case (vi)). We also discuss tersely the connection between “solvable” and “integrable” [2] models and the corresponding techniques of solution, for the solvable cases (i) and (iii). In Section IV we analize the possible usefulness of “fake Lax pairs” [1], which exist for systems of type(1.1) for arbitrary odd f (z), to solve the corresponding equations of motion (which are all Hamiltonian [1]). The results reported herein draw heavily on previous findings; indeed the techniques of solution we employ are not new, neither for solvable [3] nor for integrable [7] systems, although they had not been previously applied to solve the equations of motion (1.1) (but they have been certainly applied to analogous systems, indeed we like to record explicitly, for the integrable cases, the pioneering work by S.N.M. Ruijsenaars and his school [8]). The elliptic system (namely, case (vi)) requires a separate treatment. Indeed, even in the simpler (“nonrelativistic”) context (namely, for the so-called CalogeroMoser (CM) systems [7]), no finite-dimensional framework is available to solve the elliptic case: the only method known so far is due to I.M. Krichever and it is based on the study of solutions of the Kadomtsev-Petviashvili (KP) nonlinear partial differential equation in 2+1 dimensions [9]. Similar techniques were re-

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cently developed by I.M. Krichever and A.V. Zabrodin for RS systems [10]. Also, an infinite-dimensional Hamiltonian reduction scheme was recently introduced for the elliptic CM system [11, 12]. In the present paper we have restricted our investigation to the most general RS system which can be treated by finite-dimensional techniques, including its integrable deformation characterized by the remarkable property to feature only completely periodic trajectories (namely, case (v) as defined above, see (1.1) with (1.2e)). Under this limitation, we also provide below, in Section V, a Hamiltonian reduction scheme analogous to the Kazhdan-Kostant-Sternberg (KKS) [13] construction for CM systems. Such a scheme was used by one of us (JPF) to demonstrate the existence of a symplectic action of the torus associated with the rational CM system with an external quadratic potential [14] (see also [15], where the KKS framework was used to demonstrate the complete integrability of the rational CM system with an external quartic potential). More recently this approach has featured in several papers. While the quick pace of development in this area entails that we cannot provide an exhaustive list of contributions, we would like to mention that this Hamiltonian reduction has been discussed by H.W. Braden and R. Sasaki for case (iv) (with Ω = 0) [16]. To the best of our knowledge, it was not yet considered for case (v) (even for Ω = 0); and of course the treatment given here, of the case Ω 6= 0 featuring completely periodic orbits, is new. The Hamiltonian reduction framework provides the tools to demonstrate the existence of a symplectic action of the torus which leaves the system invariant. We provide this proof for the rational case in Section VI. Hamiltonians which are invariant under a symplectic action of the torus feature specially interesting properties; in particular, contributions by M. Atiyah, Y. Colin de Verdiere, V. Guillemin and S. Sternberg [17] are then available to investigate the quantized systems. Hence the findings reported in Sections V and VI provide the foundations for an analytic approach, which we plan to provide in a subsequent paper, to the quantized systems corresponding to (1.1) with (1.2). We do not consider here the R-matrix approach, which provides an alternative route to deal with quantization. For this approach we refer to several recent papers [18–23], none of which however considers the completely periodic case introduced in [1], on whose treatment the present paper is instead focussed.

II A simple trick In this section we report a simple trick [2] that relates the solutions of (1.1) to the solutions of the same equations of motion, but with Ω = 0. Consider the (more general) equations of motion z¨j (t) − α(t)z˙j (t) =

n X k,m=1

z˙k (t)z˙m (t)fjkm [z(t)], j = 1, . . . , n,

(2.1)

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where of course z is the n-vector of components zj . Now set zj (t) = ζj (τ ), τ = τ (t).

(2.2)

Then clearly τ (t) − α(t)τ˙ (t)]ζj0 (τ ) = ζj00 (τ ) + [τ˙ (t)]−2 [¨

n X

0 ζk0 (τ )ζm (τ )fjkm [ı(τ )], j = 1, . . . , n.

k,m=1

(2.3) Hence the choice of a function τ (t) which satisfies the (linear) equation τ¨(t) = α(t)τ˙ (t),

(2.4)

transforms the equations of motion (2.1) for zj (t) into the same equations of motion for ζj (τ ) (of course with z˙j (t) replaced by ζj0 (t) ≡ dζj (τ )/dτ , and so on), but without the second term in the left-hand side. Note that this conclusion holds for arbitrary fjkm (z). In particular the position τ (t) = (i/Ω)[exp(−iΩt) − 1]

(2.5)

transforms (1.1) into ζj00 (τ ) =

n X

ζj0 (τ )ζk0 (τ )f [ζj (τ ) − ζk (τ )], j = 1, . . . , n.

(2.6)

k=1,k6=j

Hereafter we will therefore firstly solve (1.1) with Ω = 0, since the solution of (1.1) with Ω 6= 0 can then be recovered by replacing t with τ , see (2.5). Note that the finding described in this section reinforces the plausibility of the conjecture proffered in [1], but does not quite prove its validity.

III Solution of the equations of motion In this section we obtain the (rather explicit) solution of the equations of motion z¨j =

n X

z˙j z˙k f (zj − zk ), j = 1, . . . , n ,

(3.1)

k=1,k6=j

with f (z) given by (1.2e) (“case (v)”). The solution of (1.1) with f (z) given by (1.2a,b,c,d,e) is then obtained by using the trick of the preceding Section II and by appropriate specialization of the constants, as we indicate below. The starting point of the analysis is the observation [4,1] that (3.1) with (1.2e) is equivalent to the following “Lax-type” (n×n)-matrix equation: L˙ = [L, M ]− ,

(3.2)

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with Ljk = δjk z˙j + (1 − δjk )(z˙j z˙k )1/2 α(zj − zk ), Mjk = δjk

n X

z˙m β(zj − zm ) + (1 − δjk )(z˙j z˙k )1/2 γ(zj − zk ),

(3.3) (3.4)

m=1,m6=j

and α(z) = sinh(aµ)/ sinh[a(z + µ)],

(3.5a)

β(z) = −acotgh(aµ)/[1 + r sinh (az)], γ(z) = −acotgh(az)α(z), 2

2

(3.5b) (3.5c)

where sinh(aµ) = i/r.

(3.5d)

We then introduce the diagonal matrix E(t) = diag{exp[2azj (t)]},

(3.6)

and we note that there holds the matrix formula E˙ = [E, M ]− + a[E, L]+ ,

(3.7)

whose validity can be very easily verified by explicit computation using (only!) (3.6), (3.3) and (3.4) with (3.5c). Above and throughout of course [A, B]− ≡ AB − BA and [A, B]+ ≡ AB + BA. One can now use a technique introduced, in the CM context, by M.A. Olshanetsky and A.M. Perelomov [7]. Set ˜ = U LU −1 , M ˜ = U M U −1 , E ˜ = U EU −1 , L

(3.8)

and note that the last of these equations, together with (3.6), entails that the ˜ quantities exp[2azj (t)] are the n eigenvalues of the matrix E(t). Define now the (n×n)-matrix U (t) via the equations U (0) = I, Ujk (0) = δjk , ˜ (t)U (t), M ˜ (t) = U˙ (t)[U (t)]−1 . U˙ (t) = M

(3.9a) (3.9b)

These two equations define U (t) uniquely; the fact that we do not know how to ˜ (t) nor how to solve compute this matrix, since we neither know the matrix M (3.9b), is immaterial. Indeed (3.9b), together with (3.8) and with (3.2) and (3.7), clearly entail the equations ˜˙ L(t) = 0, ˜˙ ˜ ˜ +. E(t) = a[E(t), L(t)]

(3.10a) (3.10b)

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Hence, from (3.10a) and (3.9a), ˜ = L(0) ˜ L(t) = L(0),

(3.11)

and, from (3.10b), (3.11) and (3.9a), ˜ = exp[aL(0)t]E(0) exp[aL(0)t]. E(t)

(3.12)

The matrices E(0) and L(0) are explicitly given, in terms of the initial data ˜ zj (0), z˙j (0), j = 1, . . . , n, by (3.6) and (3.3) with (3.5a). Hence the matrix E(t) is rather explicitly given by this formula, (3.12), and its eigenvalues provide the quantities exp[2azj (t)] (see the remark after equation (3.8)). Replacement of t by τ , see (2.5), in the right-hand side of (3.12) entails ˜ that the matrix E(t) becomes periodic in t with period T , see (1.3); hence its eigenvalues exp[2azj (t)] are also periodic in time, with period (at most) T 0 , see (1.3). And the periodicity of the quantities exp[2azj (t)] entails periodicity, with the same period, of the quantities zj (t). [From the explicit time-evolution (3.12) ˜ of E(t) it is clear that its determinant does not vanish over time: hence none of its eigenvalues vanishes, and their logarithms are uniquely defined by continuity.] We have thereby proved the conjecture of Ref.[1] for case (v), namely for the equations of motion (1.1) with (1.2e). It is easily seen that this proof holds equally for the 4 cases (i)–(iv), which are in fact all subcases of case (v). Indeed cases (iv) respectively (iii) obtain from case (v) by setting r2 = 1 (and changing a into a/2) respectively r = 0 (no change in a); case (ii) obtains by replacing r with r/a and then letting a → 0, which entails the replacement of (3.12) by the formula ˜ = Z(0) + L(0)t Z(t)

(3.13)

Z = diag(zj ),

(3.14)

with Ljk = δjk z˙j + (1 − δjk )(z˙j z˙k )

1/2

/[1 + ir(zj − zk )],

(3.15)

and of course the property that the n quantities zj (t) are the n eigenvalues of the ˜ matrix Z(t). Finally, case (i) is merely the special case of the results we just detailed for case (ii), corresponding to r = 0 (see (3.15)). Two remarks are appropriate in this connection (somewhat analogous observations apply to case (iii); we leave their elaboration as a task for the diligent reader; but see also the treatment given in the following Section IV). Firstly we observe that, in the r = 0 case, the matrix L, see (3.15), is highly degenerate (separable of rank 1); this corresponds to the fact that this case is not only “integrable” but also “solvable” [2]. Indeed the quantities zj (t), being the ˜ eigenvalues of the matrix Z(t), are the roots of the following polynomial of degree n in z: n n Y X ˜ det[zI − Z(t)] = [z − zj (t)] = z n + cm (t)z n−m . (3.16) j=1

m=1

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This polynomial is linear in t (see below), hence c¨m (t) = 0,

m = 1, . . . , n.

(3.17)

But this last equation, together with the nonlinear one-to-one mapping between the zj ’s and the cm ’s entailed by the second relation (3.16), is precisely the basis of the technique of solution of case (i) [3]. To complete this argument there remains to show that the polynomial (3.16) is linear in t. This is a consequence of the special structure of the matrix L in this case, Ljk = (z˙j z˙k )1/2 (3.18) (see (3.15) with r = 0). Indeed (see (3.13) and (3.14)) ˜ det[zI − Z(t)] =

n Y

(v(m) , {diag[z − zj (0)] − L(0)t}v(m) ),

(3.19)

m=1

where the n-vectors v(m) constitute an (arbitrary) orthonormal vector basis. Now choose a t-independent basis such that (1)

vj

= [z˙j (0)]1/2 /[V (0)]1/2 , V (0) =

n X

(0)

z˙j ,

(3.20a)

j=1

(assuming for simplicity the quantity V (0) does not vanish), so that (v(1) , v(m) ) = δ1m , m = 1, . . . , n ,

(3.20b)

hence (see (3.18) and (3.20), (3.21)) L(0)v(m) = δ1m V (0)v(m) , m = 1, . . . , n ,

(3.21)

hence (see (3.19)) ˜ det[zI − Z(t)] = [(v(1) , diag[z − zj (0)]v(1) ) − V (0)t]

n Y

(v(m) , diag[z − zj (0)]v(m) ).

m=2

(3.22) This completes the argument, except for the special case in which V (0) vanishes (“center of mass at rest”). We leave the analysis of this exceptional case as an exercise for the diligent reader, who will also note the special behavior of the system in this case, as discussed in Section 4.F of Ref.[5]. The second remark notes that, in the r = 0 case, the matrix L, see (3.15), corresponds to a “fake Lax pair” [1], yet the technique of solution we have just exhibited works in this case as well (it does yield the solution to the equations of motion). This fact is sufficiently intriguing to justify a detailed discussion in the following Section IV.

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But before ending this section, let us return to the dynamical system (3.1) with (1.2e) (case (v)), to report the following Remark. The explicit expressions (3.3) with (3.5a) of L, and (3.6) of E, entail that the matrix S ≡ cosh(aµ)[E, L]− + sinh(aµ)[E, L]+ (3.23) is symmetrical and has rank 1: Sjk = sj sk

(3.24)

sj = [2 sinh(aµ) z˙j ]1/2 exp(azj ).

(3.25)

This suggests an alternative route to treat this dynamical system. One starts ˜ ˜ from the evolution equations (3.10a,b) for the two (n×n)-matrices L(t) and E(t), which of course entail (3.11) and (3.12). Then one introduces the matrix U (t) as ˜ the one that diagonalizes E(t) (see the third equation (3.8), and (3.6)), and then the matrix L(t) via the first of the 3 equations (3.8). Then one imposes the validity of (3.23) and (3.24) (subject to a final consistency check). It is then easily seen that the explicit expressions of sj , see (3.25), and of L, see (3.3) and (3.5a), can be derived, as well as the equations of motion (3.1) with (1.2e). [This can be achieved ˜ via (3.9b) and then M via the second of the via the following steps: introduce M (3.8); then (3.10 a,b) entail (3.7) and (3.2); then the diagonal part of (3.7), together with (3.6), yield the diagonal part of L, see (3.3); then equate the right-hand sides of (3.23) and (3.24), and thereby get firstly (3.25) (from the diagonal part, using (3.6) and the already evaluated diagonal part of L), and then the off-diagonal part of L (from the off-diagonal part); then, from the off-diagonal part of (3.7), evaluate the off-diagonal part of M (using (3.6) and the now known off-diagonal part of L); and finally, from the diagonal part of (3.2), get the equations of motion (3.1) with (1.2e) (using the now known off diagonal parts of L and M )]. This method of obtaining the equations of motion (3.1) with (1.2e), as well as their Lax-pair structure and their solution, might appear tortuous. But it shows that one can obtain this dynamical system starting from the extremely simple (n×n)-matrix evolution equations (3.10a,b), and supplementing them with the (a posteriori compatible) constraint (3.23) with (3.24) (with E and L defined via (3.8) and (3.6)). This provides the framework for a reduction procedure which is discussed in some detail in Sections V and VI.

IV Natural ansatz for finding the “angle” equation Complete integration of a Hamiltonian system needs finding a Lax matrix (which provides the conserved quantities) and an extra-equation (cf. 3.7) which may be called the “Angle” equation (in reference to the classical terminology of the ActionAngles coordinates here adopted in its broad sense). It seems quite interesting to note that in the examples we consider here, the Lax equation may be “fake” (in the

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sense that it does not provide the expected number of conserved quantities) and nevertheless it may be implemented with an extra-equation (the “Angle” equation) which still allows the full integration of the system. In this sense, even a “fake” Lax pair may be useful. We provide here a complete discussion of the ansatz for finding the “Angle” equation. We derive the system (1.2e) as only solution. This shows clearly that the elliptic system (1.2f) cannot be reached with our techniques and that a separated study for finding the “Angle” equation should be developed (perhaps based on the ideas of [9] and [1 ]). It is easy to verify that the equations of motion z¨j =

n X

z˙j z˙k fjk (zj − zk ) , j = 1, . . . , n ,

(4.1)

k=1

correspond to the (n×n)-matrix “Lax equation” L˙ = [L, M ]− ,

(4.2a)

with [1] (4.2b) Ljk = δjk z˙j + (1 − δjk )(z˙j z˙k )1/2 = (z˙j z˙k )1/2 , 1 Mjk = (δjk − 1)(z˙j z˙k )1/2 fjk (zj − zk ). (4.2c) 2 Note that the only condition required for the equivalence of (4.1) and (4.2) is that the matrix-valued function fjk (z) be “odd,” in the following sense: fkj (−z) = −fjk (z).

(4.3)

Incidentally, this condition is also sufficient to guarantee that the equations of motion (4.1) are Hamiltonian [1]; indeed a Hamiltonian whose equations of motion, q˙j = ∂H/∂pj , p˙j = −∂H/∂qj ,

(4.4a)

yield (4.1) (with zj = qj ) reads as follows: H=

n X

hj (spj , q) ,

s = arbitrary nonvanishing constant,

(4.4b)

j=1

hj (pj , q) = exp{pj +

n X

Fjk (qj − qk )} ,

(4.4c)

k=1,k6=j 0 0 fjk (q) = Fkj (−q) − Fjk (q) .

(4.4d)

Note that the last equation entails (4.3). Moreover, the condition (4.3) is clearly also sufficient to guarantee that the velocity of the center of mass, V /n, V =

n X j=1

z˙j ,

(4.5a)

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is a constant of motion, V˙ = 0 .

(4.5b)

The “Lax pair” (4.2b,c) is a “fake Lax pair:” indeed the time-independence of the eigenvalues of L implied by (4.2a) only entails (4.5b), due to the highly degenerate character of the matrix L, see (4.2b), which is clearly separable of rank 1. But in the preceding Section III we have seen that a Lax pair of this kind may be instrumental to solve the corresponding equations of motion. It is therefore natural to investigate whether the technique of solution described in the preceding section is more generally applicable to a system of type (4.1). Two ingredients play a crucial role in the technique of solution described in the preceding section. One is the Lax equation, see (3.2); we now have an analogous formula, see (4.2). The other is the matrix equation (3.7). Let us therefore see whether we can now manufacture an equation analogous to (3.7). To this end we introduce the matrix G = diag[g(zj )], (4.6) with g(z) a function to be determined, and we require that it satisfy the equation G˙ = [G, M ]− + a[G, L]+ + b[L, G]− + cG + dL + h

(4.7)

with a, b, c, d and h five arbitrary (scalar) constants (the justification for this ansatz for the right-hand side of (4.7) is that, as it can be easily seen, it allows to perform all subsequent steps in the technique of solution described in the preceding Section III). The compatibility of (4.7) with (4.6) and (4.2b,c) is easily seen to imply the following results: firstly, from the diagonal part of (4.7), c = h = 0,

(4.8a)

g(z) = C exp(2az) − d/(2a), C arbitrary,

(4.8b)

and then, from the off-diagonal part of (4.7) (note the disappearance of C and d, as well as of the indices j and k, in the right-hand side) fjk (z) = 2a cotgh(az) + 2b .

(4.8c)

But the condition (4.3) then entails b = 0.

(4.8d)

Hence we have merely reobtained the “solvable” [1,3] case (iii), see (1.2c); and of course, via the limit a → 0, case (i) can be reobtained as well, see (3.13). [The diligent reader may repeat this calculation starting from the more general ansatz for G that results from the replacement of g(zj ) with gj (zj ) in (4.6). The more general result obtained in this manner corresponds merely to the freedom to perform, in the equations of motion (see (4.1) or (3.1)), the translations zj → zj + wj , the n quantities wj being arbitrary constants].

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V The Hamiltonian reduction procedure The general idea of the “Hamiltonian reduction procedure” is to start with a “large” initial phase space and a “simple” Hamiltonian possessing a symmetry group. Factorizing the corresponding motion by this symmetry yields a nontrivial dynamical system defined on a reduced phase space. Let us tersely outline how it works here. For the models treated in this paper, namelycases (i) to (v) (see (1.1) with (1.2a,b,c,d,e)), one can use the cotangent bundle T ∗ G over the Lie group G = Gl(n, C). The space T ∗ G is naturally isomorphic to G × G ∗ where G ∗ is dual to the Lie algebra G = M at(n, C). Let (E, L) be an element of T ∗ G, where E belongs to the image of the exponential mapping: exp : G→G. The group G acts on itself and this action gets lifted into a Hamiltonian action on T ∗ G. Write E = exp(2aZ),

(5.1)

where Z belongs to an orbit of maximal dimension of the adjoint action of G on G, and it is diagonalizable, Z = W ZW −1 , Z = diag(zj ).

(5.2a) (5.2b)

Note that one is now generalizing the Kazhdan-Kostant-Sternberg reduction techniques [13], by replacing the lifted action of G to T ∗ G by an action “weighted on the left and on the right”, of the image of the exponential mapping: Write U = exp(T ) then U.E = exp[(α + β)/2]T E exp[(−α + β)/2]T,

(5.3a)

so that the corresponding momentum map is (E, L) 7−→ α[E, L]− + β[E, L]+ .

(5.3b)

Note that one is assuming here that G is not only a Lie algebra but also an associative algebra; namely that not only the commutator [E, L]− = EL − LE is defined but the anticommutator [E, L]+ = EL + LE as well. The two complex parameters α and β are fixed and determine the weights of the action. Let us remind the reader that for the case of rational and trigonometric CM systems, the fiber of the momentum map corresponds to the specific rank-one matrix c ⊗ c† , with c† = (1, . . . , 1) [13]. Here the analogous role is played by the more general rank-one matrices Σ = σ⊗σ 0 ,

(5.4)

where σ and σ 0 are two (appropriately chosen, see below) complex n-vectors. The manifold M on which the dynamics unfolds is the inverse image by the momentum

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map of the set of matrices Σ. Of course, we factorize this inverse image by the action of the subgroup that leaves this set invariant. Indeed, let us consider the simple dynamical system defined on T ∗ G by the differential equations E˙ = a[E, L]+ , L˙ = iΩL,

(5.5a) (5.5b)

where a is a complex parameter and Ω is real. This differential matrix system can be easily integrated: L = L(0) exp(iΩt), (5.6a) E = exp{(−ia/Ω)[exp(iΩt)−1]L(0)}E(0) exp{(−ia/Ω)[exp(iΩt)−1]L(0)}. (5.6b) Note that if

α[E(0), L(0)]− + β[E(0), L(0)]+ = σ(0) ⊗ σ0 (0),

(5.7)

then (see (5.6a,b)) α[E(t), L(t)]− + β[E(t), L(t)]+ = σ(t)⊗σ 0 (t),

(5.8)

σ(t) = exp(iΩt/2) exp{(−ia/Ω)[exp(iΩt) − 1]L(0)}σ(0), σ0 (t) = σ 0 (0) exp(iΩt/2) exp{(−ia/Ω)[exp(iΩt) − 1]L(0)}.

(5.9a) (5.9b)

with

Hence we see that the dynamics leaves globally invariant the manifold M defined as the inverse image of the set of rank-one matrices Σ, see (5.4). We need finally to factorize the manifold M by the action of the group. We describe a parametrization of the reduced space which yields precisely the Lax matrix of Ref. [4]. To this end, we first diagonalize the matrix Z (see (5.2)) and introduce (see (5.1)) E = W −1 EW = diag[exp(2azj )].

(5.10)

σj σk0 .

For generic values of The entries of the matrix Σ, see (5.4), are Σjk = σ and σ 0 , there are a diagonal matrix D and a vector s such that D−1 ΣD = s⊗s.

(5.11)

Indeed, if σj 6= 0 for all j = 1, . . . , n, the elements of the diagonal matrix D are dj = (σj0 /σj )1/2 , which yields sj = (σj σj0 )1/2 . Then we can set E = U −1 EU = diag[exp(2azj )],

(5.12a)

with U = DW , and we also define L = U −1 LU.

(5.12b)

From here on, one can proceed as described in Section III, see (3.23) and the discussion following it; note that (3.23) and (5.6) entail the identification α = cosh(aµ), β = sinh(aµ).

(5.13a) (5.13b)

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VI The associated Poisson action of the torus for the rational case In this section we demonstrate, in the simpler case characterized by a rational f (z), see (1.2a) and (1.2b), how one can introduce a Poisson action of the torus which leaves the motion invariant, and thereby identify explicitly n constants of the motion. We moreover show that, for Ω real and nonvanishing, all the flows of the “particle coordinates” zj (t) induced by these constants (considered as Hamiltonians) are completely periodic with period (at most) T 0 , see (1.3). Let us first of all obtain, via the a→0 limit, the equations relevant to the rational case. They read, in place of (5.3), Z˙ = L, L˙ = iΩL,

(6.1b)

˙ Z¨ = iΩ Z.

(6.2)

(6.1a)

entailing The equations (6.1) are susceptible of Hamiltonian interpretation, with the matrix Z as canonical variable and the matrix P, defined by L = exp(P),

(6.3)

as conjugated canonical momentum. Here we are of course assuming that L belongs to the image of the exponential mapping, which is consistent with (5.6a) or (6.1b). The corresponding Hamiltonian reads H = tr[exp(P) − iΩZ],

(6.4)

ω = tr[dZ∧dP] = tr[dZ∧L−1 dL],

(6.5a)

˙ = dH = tr[dL − iΩdZ], ˙ −1 dL − dZL−1 L] tr[ZL

(6.5b)

and the symplectic form is

entailing

so that Hamilton’s equations read ˙ −1 = I, ZL L−1 L˙ = iΩI,

(6.6a) (6.6b)

namely they reproduce (6.1). Note that the Hamiltonian (6.3) coincides with that considered in Ref.[1]. Before proceeding, let us review what was done in Ref.[14] for the rational CM system with an external quadratic potential, characterized by the equations of motion x˙ j = yj , y˙j = −Ω 2 xj +

(6.7a) n X k=1,k6=j

(xj − xk )−3 .

(6.7b)

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This system was obtained as Hamiltonian reduction of the matrix “pure harmonic oscillator” system X˙ = Y, Y˙ = −Ω 2 Y.

(6.8a) (6.8b)

We may now compare this system with (6.1). Both systems obviously display periodic solutions. One of the specific features of (6.1) is that, in contrast with (6.8), it only makes sense for complex matrices Z and L, so that in fact the real system associated with (6.8) is obtained by considering the time evolution of Re(Z) and Im(Z). Hamiltonian reductions of (6.8) yield systems of particles on the line while Hamiltonian reductions of (6.1) correspond to systems of particles on the plane [2, 6]. The quantization of the harmonic oscillator is completely classical in the framework of geometric quantization theory. The quantization of (6.1) is less standard and we plan to discuss it in a future paper. Let us now pursue the analogy between the two systems. In Ref.[14] a new matrix variable was introduced,

with the Hamiltonian

Ξ = X + iΩY, Ξ ∗ = X − iΩY,

(6.9a) (6.9b)

H = tr[ΞΞ ∗ ],

(6.10)

and the KKS [13] symplectic form ω = tr{−[i/(2Ω)]dΞ∧dΞ ∗ }.

(6.11)

It was then noted that not only is H a constant of the motion, but in fact the whole matrix A = ΞΞ ∗ is conserved by the flow (6.8). An analogous phenomenon occurs for (6.1). Indeed, setting B = L − iΩZ,

(6.12)

one immediately sees that (6.1) entails B˙ = 0.

(6.13)

One of the main results obtained in Ref.[14] was that all the eigenvalues of A, seen as functions of the matrix variables, generate via the symplectic form (6.11) commuting Hamiltonian systems, all of which possess only completely periodic orbits, with period T = 2π/Ω. This set of commuting Hamiltonians defines a symplectic action of the torus, which leaves invariant the CM system with external quadratic potential (and eventually explains why the corresponding quantum spectrum coincides with its semi-classical approximation).

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We prove here the following analogous result: the set of eigenvalues βm of the matrix B, see (6.12), generate, via the symplectic form (6.5), commuting Hamiltonian flows for the matrix variables, all of which are completely periodic, with period T = 2π/Ω. The proof of this statement goes as follows. Let βm be an eigenvalue of the matrix B. Let Ψ (m) be the corresponding eigenvector and let N (m) be the projector on Ψ (m) . The dynamical system associated with βm is then characterized by the matrix differential equations Z˙ = LN (m) , L˙ = iLN (m) .

(6.14a) (6.14b)

Hence again, under all these flows, the matrix B, see (6.12), is constant: B˙ = iΩLN (m) − iΩLN (m) = 0.

(6.15)

This clearly entails that all the eigenvalues of B are constants of the motion under all these flows; hence different eigenvalues Poisson commute pairwise, and define commuting flows. Finally, let us prove that all these matrix flows are in fact periodic with period T , see (1.3). Let U be a (time-independent) matrix which diagonalizes the (time-independent) matrix B, and let us define L0 = U LU −1 , Z 0 = U ZU −1 , B0 = U BU −1 , N 0(m) = U N (m) U −1 .

(6.16)

Here the matrix B0 is by definition diagonal, and clearly 0(m)

Njk

= δjm δkm .

(6.17)

L˙ 0 = iΩL0 N 0(m) ,

(6.18a)

L0 (t) = L0 (0) exp(iΩtN 0(m) ).

(6.18b)

Z˙ 0 (t) = L0 (0) exp(iΩtN 0m )N 0(m) ,

(6.19a)

Then (4.16b) yields which entails Hence (6.14a) yields

which admits the solution: Z 0 (t) = (iΩ)−1 Z 0 (0)L0 (0) exp(iΩtN 0(m) )N 0(m) ,

(6.19b)

entailing (see (6.17)) [Z 0 (t)]jk = [Z 0 (0)]jk + (iΩ)−1 exp(iΩt)[L0 (0)]jm δkm ,

(6.20)

which displays the explicit time-evolution of the matrix Z 0 (t) under the flow generated by the m-th eigenvalue of B. Obviously this evolution is periodic, with period T = 2π/Ω. The coordinates zj (t) are just the eigenvalues of this matrix Z 0 (t) (see (6.16) and (5.2)), hence they are all periodic with period (at most) T 0 , see (1.3).

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VII Outlook This is meant to be the first paper of a series. In subsequent papers we plan to treat case (vi), to report analogous results for systems with “nearest neighbor” interactions, to focus on formulations of these models that can be properly interpreted as (real) many-body problems in the plane featuring only completely periodic trajectories [6], and to discuss quantum versions of such many-body problems. The quantization of the Hamiltonian system (6.4) is nontrivial even in the simplest case n = 1. In this case, the system is defined via the complex coordinates (z, p) or equivalently via the real ones: (x, y; px , py ) : z = x + iy, p = px − ipy

(7.1).

The Hamiltonian H reads then: H(z, p) = exp(p) − iΩz = exp(y)[cos(py ) − isin(py )] − iΩ(x + iy)

(7.2)

and this yields (see [6]) the 2-dimensional Hamiltonian system characterized by the real Hamiltonian: H1 (x, px ; y, py ) = exp(y)cos(py ) + Ωy.

(7.3)

All the trajectories of this dynamical system are completely periodic with period T = Ω/2π as it is easily seen from the second order differential equation associated with the Hamiltonian (7.2): z¨ = iΩ z˙

(7.4)

It is expected that the corresponding quantum model feature an equispaced energy spectrum (to be properly defined). But the system (7.4) is not trivially equivalent to a harmonic oscillator. Thus the usual schemes of geometric quantization shall have to be appropriately revisited.

Acknowledgements It is a pleasure to acknowledge with thanks the contributions of the CNRS in France and of the CNR in Italy, as well as the hospitality provided on various occasions by the Universities Paris VI, Roma La Sapienza and Roma Tre.

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References [1] Calogero, F.: A class of integrable Hamiltonian systems whose solutions are (perhaps) all completely periodic. J.Math. Phys. 38, 5711–5719 (1997). [2] Calogero, F.: Tricks of the trade: relating and deriving solvable and integrable dynamical systems. To appear in the Proceedings of the International Workshop on Calogero-Moser-Sutherland type models, held at the Centre de Recherches Math´ematiques de l’Universit´e de Montr´eal, Canada, in March 1997 (Springer, in press). [3] Calogero, F.: Motion of poles and zeros of nonlinear and linear partial differential equations and related many-body problems. Nuovo Cimento 43B, 177–241 (1978). [4] Bruschi, M., Calogero, F.: The Lax representation for an integrable class of relativistic dynamical systems. Commun. Math. Phys. 109, 481–492 (1987). [5] Calogero, F.: A solvable N-body problem in the plane. I. J. Math. Phys. 37, 1735–1759 (1996). [6] Calogero, F.: Integrable and solvable many-body problems in the plane via complexification. J. Math. Phys., 39, 5268–5291 (1998). [7] Olshanetsky, M.A., Perelomov, A.M.: Classical integrable finite-dimensional systems related to Lie algebras. Phys. Rep. 71, 313–400 (1981). [8] See, for instance: Ruijsenaars, S.N.M., Schneider, H.: A new class of integrable systems and its relation to solitons. Ann. Phys. (NY) 170, 370–405 (1986); Ruijsenaars, S.N.M.: Systems of Calogero-Moser type. Proceedings of the 1994 Banff Summer School on Particles and Fields, CRM Proceedings and Lecture Notes (in press), and the papers referred to there; van Diejen, J.F.: Families of commuting difference operators. PhD Thesis, University of Amsterdam, 28 June 1994. [9] Krichever, I.M.: Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles. Funct. Anal. Appl. 14, 282–289 (1981). [10] Krichever, I.M., Zabrodin, A.V.: Spin generalizations of the RS model, nonabelian 2D Toda chain and representations of Sklyanin algebra. Russ. Math. Surv. 50, 1101–1150 (1995). [11] Gorsky, A., Nekrasov, N.: Hamiltonian systems of Calogero-type, and twodimensional Yang-Mills theory. Nuclear Phys. B414, 213–238 (1994); Relativistic Calogero-Moser model as gauged WZW theory. Nuclear Phys. B436, 582–608 (1995).

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[12] Arutyunov, G.E., Frolov, S.A., Medvedev, P.B.: Elliptic Ruijsenaars-Schneider model via the Poisson reduction of the affine Heisenberg double. hepth/9607170 (1996); Elliptic Ruijsenaars-Schneider model from the cotangent bundle over the two-dimensional current group. hep-th/9608013 (1996). [13] Kazhdan, D., Kostant, B., Sternberg, S.: Hamiltonian group actions and dynamical systems of Calogero type. Commun. Pure Appl. Math. 31, 481–507 (1978). [14] Fran¸coise, J.-P.: Canonical partition functions of Hamiltonian systems and the stationary phase formula. Commun. Math. Phys. 117, 37–47 (1988). [15] Fran¸coise, J.-P., Ragnisco, O.: Matrix differential equations and Hamiltonian systems of quartic type. Ann. Inst. H. Poincar´e 49, 369–375 (1989). [16] Braden, H.W., Sasaki, R.: The Ruijsenaars-Schneider model. Prog. Theor. Phys. 97, 1001–1015 (1997). [17] Atiyah, M.: Convexity and commuting Hamiltonians. Bull. London Math. Soc. 14, 1–15 (1982); Colin de Verdi`ere, Y.: Spectre joint d’op´erateurs pseudodiff´erentiels qui commutent. II – Le cas int´egrable. Math. Zeit. 171, 169– 182 (1980); Guillemin, V., Sternberg, S.: Symplectic Techniques in Physics. Cambridge University Press, Cambridge (1984). [18] Arutyunov, G.E., Frolov, S.A.: Quantum dynamical R-matrices and quantum Frobenius group. Commun. Math. Phys. (in press); On Hamiltonian structure of the spin RS model. J. Phys. A: Math. Gen. (submitted to). [19] Arutyunov, G.E., Chekhov, L.O., Frolov, S.A.: R-matrix quantization of the elliptic RS model. Theor. Math. Phys. 111, 182–217 (1997). [20] Avan, J., Babelon, O., Billey, E.: The Gervais-Neveu-Felder equation and the quantum Calogero-Moser systems. Commun. Math. Phys. 178, 281–299 (1996). [21] Felder, G.: Conformal field theory and integrable systems associated with elliptic curves. Proceedings of the International Congress of Mathematicians, 1247–1255, Z¨ urich (1994). [22] Nijhoff, F.W., Kuznetsov, V.B., Sklyanin, E.K., Ragnisco, O.: Dynamical Rmatrix for the elliptic Ruijsenaars-Schneider model. J. Phys. A: Math. Gen. 29, L333–L340 (1996). [23] Suris, Yu.B.: Why is the Ruijsenaars-Schneider hierarchy governed by the same R-matrix as the Calogero-Moser one? Phys. Rev. A225, 253–261 (1997).

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F.Calogero Visiting Professor (Permanent position: Professor of Theoretical Physics University of Rome I “La Sapienza”) on leave while serving as Chairman of the Pugwash Council Pugwash Conferences on Science and World Affairs J.P. Fran¸coise Equipe “G´eom´etrie Diff´erentielle Syst`emes Dynamiques et Applications” Universit´e de Paris 6 UFR 920, tour 45–46 4, place Jussieu B.P. 172 F-75252 Paris, France Communicated by J. Bellissard submitted 09/06/98, revised 08/01/99; accepted 22/01/99

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auser Verlag, Basel, 2000 1424-0637/00/010193-8 $ 1.50+0.20/0

Annales Henri Poincar´ e

On the Perturbation Series in Large Order of Anharmonic Oscillators T. Koike Abstract. We present a mathematically rigorous proof of the Bender-Wu (BanksBender) formula based on exact WKB analysis.

I Introduction We will consider the large n asymptotics of a perturbation series of an eigenvalue ∞

EK = K +

1 X K n + A λ , 2 n=1 n

of a generalized anharmonic oscillator d2 1 2 − 2+ x + λx2N+2 ψ = Eψ, dx 4

(1)

(2)

where λ > 0, K = 0, 1, 2, · · · and N = 1, 2, · · ·. In [3](for N = 1) and [2](for N ≥ 2) the following interesting formula is presented: 3 1 −K− 12 −nN (K+ 12 )/N B( 2 , N ) K n+1 N 4 √ An = (−1) 3 2N K! 2π 1 1 ×Γ K + + nN 1+O , (n → ∞), (3) 2 n where B(x, y) and Γ(x) respectively denote the beta and the gamma function. For N = 1 there are some rigorous proofs of the above formula; [8] in connection with the resonance problem, and [5] from the resurgent theory. Generalization for a higher dimensional case was considered in [9]. The purpose of this article is to give another proof of the above formula for general N using the exact WKB analysis ([1], [4], [5]) along the original idea in [3]. By the help of Stokes geometry, our argument becomes simpler and more transparent; it clearly explains why complex turning points do not give any effects on the above formula (cf. [2]). I would like to thank Prof.Kawai and Prof.Takei for giving me many valuable advise and comments. I also thank Prof.Aoki for providing me with a computer program for drawing Stokes curves, some of which I used in this article.

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II Calculation of eigenvalues Following Bender-Wu, we start our argument with the relation ((2.7) in [3]) Z

0

∆E K (λ) dλ, (4) λn+1 −∞ where ∆E K (λ) denotes the difference of E K arg λ=π and E K arg λ=−π . This relation was derived from the analyticity of an eigenvalue. (See [6],[8].) By this relation, our problem is reduced to the determination of an eigenvalue for small λ and for arg λ = ±π. Keeping this fact in mind, we introduce a large parameter η in (2) by √ setting λ 7→ η−N eiθ (θ = arg λ) and x 7→ ηx so that (2) becomes d2 2 −1 − 2 + η Q(x) − η E ψ = 0, (5) dx AK n

1 = 2πi

where

1 2 (6) x + eiθ x2N+2 . 4 One important point is, however, Stokes geometry is degenerate for θ (= arg λ) = ±π: that is, there exists a Stokes curve connecting turning points. (See the left of Fig. 1, for example.) To resolve this degeneracy, we rotate the contour in (4) in the following way: Q(x) =

AK n =

1 2πi

Z

0

−∞ ei

∆E K (λ) dλ, λn+1

(7)

where is a sufficiently small positive number. (See the right of Fig. 1. We can show this modified relation in a similar way as that of the original one. See also §V in [2].) Hence we determine an eigenvalue of (5) for θ = ±π + . There are one double turning point at x = 0 and 2N simple turning points at x = exp(i((2k + 1)π − θ)/2N ) (k = 0, 1, 2, · · · , 2N − 1). For θ = −π + First we compute E K (λ) when θ (= arg λ) = −π + . The examples of Stokes curves are given in the right of Fig. 1 (for N = 1) and Fig. 2 (for N = 2). (The left of Fig. 1, 2 are that for = 0.) We can determine analytically continued eigenvalues by the rotating sector condition (cf.[3], see also [8]); we require that the solutions of (5) tend to 0 in the sectors π θ Σ± (θ) = x ∈ C ; arg(±x) + < . (8) 2(N + 2) 2(N + 2) As a path of the analytic continuation of solutions connecting these sectors Σ± (θ) near ∞, we can choose and fix the following path Γ throughout this article: We define Γ with the help of Stokes curves for = 0. If is equal to 0, x = ±1 are

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Perturbation Series in Large Order of Anharmonic Oscillators

γ0

γ−

195

γ+

γ 0+

Figure 1: Stokes curves (N = 1) for = 0 (left) and for > 0 (right).

γ 0−

γ−

γ0

γ+

γ 0+

Figure 2: Stokes curves (N = 2) for = 0 (left) and for > 0 (right). turning points, and x = ±i also become turning points if N is even. Let γ0 be a Stokes curve emanating from x = 0 in the direction of positive imaginary axis, γ+ (resp. γ− ) a Stokes curve emanating from x = 1 (resp. x = −1) and penetrating into the upper (resp. lower) half plane (cf. Fig. 1, 2). Then we take Γ as a path which is sufficiently close to γ+ ∪ {x; −1 < x < 1} ∪ γ− , crosses two Stokes curves {x; −1 < x < 0} and γ0 just once, and never crosses any other Stokes curves. The examples of Γ are given in Fig. 3. For > 0, we find that this Γ crosses two Stokes curves when N is odd, and four Stokes curves when N is even. We denote them by γ1 and γ2 (or γ1 , · · · , γ4 ) as is shown in Fig. 3. In the following we use WKB solutions which are normalized as Z x Z x p p 1 ψ± = √ exp ±η Q(x)dx exp ± Sodd − η Q(x) dx , (9) Sodd 0 ∞

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γ2 γ 1

Γ

γ3

γ2 γ4

Figure 3: Path of analytic continuation for θ = −π + . (left: N = 1, right: N = 2. Wiggly lines indicate cuts.) where

∞ X

Sodd =

η −j Sodd,j = η

j=−1

p

E + ··· Q(x) − p 2 Q(x)

(10)

is the odd degree part of a solution S of the Riccati equation associated with (5). (See [1].) p Here we choose the branch of Q(x) so that • we place cuts from each simple turning point to ∞ (See Fig. 3.); p 1 • Q(x) = x + O(x2 ). 2 Then ψ− satisfies the boundary condition in Σ± (θ). We find that ψ− is dominant on γ1 (resp. γ2 ) and subdominant on γ2 (resp. γ1 , γ3 , γ4 ), which implies that only γ1 (resp. γ2 ) is relevant for the calculation of the Jost function. As a matter of fact, using the connection formula near a double turning point, we find that ψ− in Σ+ (θ) becomes J+ (η, E)ψ+ + J− (η, E)ψ− in Σ− (θ) after the analytic continuation along Γ, where √ 2π J+ (η, E) = iC (11) e−iπF η F . Γ(F + 12 ) Here C

=

∞ X

η −j Cj = eE(log 4−iπ) + · · · ,

(12)

j=0

F

= Res Sodd = −E + · · · , x=0

(13)

are infinite series which are determined by the connection formula near a double turning point(cf.[5], [7], [12]. In the above “· · ·” means the negative degree part

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with respect to η.) Hence the eigenvalue E must satisfy J+ (η, E) = 0. We solve this equation by the help of the implicit resurgent function theorem[10]. Since this equation is equivalent to 1 1 1 = 0 ⇔ F + 2 = −K (K = 0, 1, 2, · · · .), Γ(F + 2 ) we obtain

(14)

∞

EK = K +

1 X K −j + E η . 2 j=1 j

(15)

Remark. Many terms of {EjK } actually vanish; we can show by induction that ∞

EK = K +

1 X e K iθ −N j E e η , + 2 j=1 j

(16)

e K is a constant independent of θ (cf. λ = eiθ η −N ). where each E j For θ = π + Stokes curves are the same as in the case of θ = −π + (cf. Fig. 1, 2). We define a path Γ0 of the analytic continuation in a similar way as that for θ = −π + , that 0 0 is, let γ+ (resp. γ− ) be the Stokes curve emanating from x = 1 (resp. x = −1) and penetrating into the lower (resp. upper) half plane. (See the left of Fig. 2.) Then 0 0 we take Γ0 as a path which is sufficiently close to γ+ ∪ {x; −1 < x < 1} ∪ γ− and never crosses any Stokes curves except for {x; 0 < x < 1} and γ0 . In this case Γ0 cross 4 Stokes curves for > 0. We denote them by γ1 , · · · , γ4 as is shown in Fig. 4. We can verify that ψ− is dominant on γ1 , γ3 and γ4 and subdominant on γ2 .

γ4

γ4

γ3

γ3

γ2

γ1

γ2

Γ0

γ1

Γ0

Figure 4: Path of analytic continuation for θ = π + . (left: N = 1, right: N = 2. Wiggly lines indicate cuts.)

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We now consider the analytic continuation of ψ− along Γ0 . By using the connection formula near a simple turning point (cf. [1], [4], [5]) together with the connection formula near a double turning point, we obtain the following expression of the Jost function: 0 (η, E) = i (D2 + A− − A+ + A+ D1 D2 + A+ A− D1 ) , J+

(17)

where A± D1 D2

Z ±a Z ±a p p = exp −2η Q(x)dx exp −2 Sodd − η Q(x) dx , 0 ∞ √ 1 2π = η−F , C Γ(−F + 1/2) √ 2π = C e−iπF η F . Γ(F + 1/2)

−i/N Here a = Z ae p , and C, F are the same infinite series as in (12) and (13). Let ω denote Q(x)dx. (A simple calculation shows 0

ω=

e−i/N 3 1 B( , ), 2N 2 N

(18)

0 where B(x, y) denotes the beta function.) Then we find that J+ (η, E) has the following form: 0 0 0 0 J+ (η, E) = J+,0 (η, E) + J+,1 (η, E)e−ηω + J+,2 (η, E)e−2ηω ,

(19)

0 where J+,l (η, E) contains no exponential term. Let us suppose E has the form

E = E0 (η) + E1 (η)e−ηω + E2 (η)e−2ηω + · · · .

(20)

Then by substituting (20) into (19) and comparing the same exponential terms, we find the following: • Since E0 (η) must satisfy D2 (η, E0 ) = 0, or equivalently F + 1/2 = −K (K = 0, 1, 2, · · ·), its solution E0K = K +

1 + ···, 2

(21)

is equal to (15). • Next E1 (η) must satisfy ∂D2 e e e = 0, E1 + A− − A+ + D1 D2 A+ ∂E E=E K 0

(22)

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Perturbation Series in Large Order of Anharmonic Oscillators

Z e± = exp −2 A

where

±a

∞

199

p Sodd − η Q(x) dx .

(23)

Let us first note that the identity ∂ 1 = (−1)n n! (n = 0, 1, 2, · · ·) ∂z Γ(z) z=−n implies

(24)

√ 1 ∂D2 = −i 2πCη −K− 2 K! + · · · . ∂E

Secondly, since

Z

a

Sodd,0 dx = −

∞

and e−1 e− A A +

(25)

iEπ 2N

(26)

= exp −2πi Res Sodd x=0

θ=π+−0

−2πiF

= e

(27)

(which can be verified by deforming the contour appropriately), we obtain e− − A e− A e−1 e+ = A e+ A A − 1 + = eiEπ/N + · · · e−2πiF − 1 . (28) That is, e− − A e+ A

E=E0K

Thirdly we have D1 D2 |E=E K 0

=

= −2 exp

iπ N

1 K+ + ···. 2

(29)

2π −iπF 1 1 e Γ(F + 2 )Γ(−F + 2 ) E=E K 0

= 0.

(30)

Through (22), (25), (29) and (30), we obtain 1 2i 1 E1K = √ η K+ 2 4(K+ 2 )/N (1 + · · ·). 2π 3 K!

(31)

Combining the above results for θ = ±π + , we conclude that the leading term of ∆E K is 1 1 2i √ (32) 4(K+ 2 )/N η K+ 2 e−ηω . 3 2π K! Hence by using (7) we have the desired result.

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References [1] T. Aoki, T. Kawai and Y. Takei, The Bender-Wu analysis and the Voros theory. ICM-90 Satellite Conf. Proc. “Special Functions”, Springer-Verlag, 1991, pp. 1–29. [2] T.I. Banks and C.M. Bender, Anharmonic oscillator with polynomial selfinteraction. J. Math. Phys., 1972, 13, pp. 1320–1324. [3] C.M. Bender and T.T. Wu, Anharmonic oscillator. II. A study of perturbation theory in large order. Phys. Rev. D, 1973, 7, pp. 1620–1636. [4] E. Delabaere, H. Dillinger and F. Pham, R´esurgence de Voros et p´eriodes des courves hyperelliptique. Annales de l’Institut Fourier, 1993, 43, pp. 163–199. [5] E. Delabaere, H. Dillinger and F. Pham, Exact semiclassical expansions for one dimensional quantum oscillators. J. Math. Phys., 1997, 38, pp. 6126–6184. [6] E. Delabaere and F. Pham, Unfolding the quartic oscillator. Ann. Phys., 1997, 261, pp. 180–218. [7] A.O. Jidoumon, Mod`eles de r´esurgence param´etrique: fonctions d’Airy et cylindroparaboliques. J. Math. Pures et Appl., 1994, 73, pp. 111–190. [8] E. Harrell and B. Simon, The mathematical theory of resonances whose width are exponentially small. Duke math. J., 1980, 47, pp. 845–902. [9] B. Hellfer and J. Sj¨ostrand, R´esonances en limite semi-classique. M´emoire de la Soci´et´e Math´ematique de France. N◦ 24/25. Suppl´ement au Bulletin de la S.M.F., 1986, 114, fascicule 3. [10] F.Pham, Fonctions r´esurgentes implicites. C. R. Acad. Sci. Paris, 1989, 309, s´erie I, pp. 999–1001. [11] B. Simon, Coupling constant analyticity for the anharmonic oscillator. Ann. Phys., 1976, 58, pp. 76–136. [12] Y. Takei, On the connection formula for the first Painliv´e equation. RIMSKˆokyˆ uroku, 1995, 931, pp. 70–99. T. Koike Member of JSPS fellows Research Institute for Mathematical Sciences Kyoto University Kyoto, 606-8502 JAPAN Communicated by J. Bellissard submitted 17/02/98, revised 29/08/98; accepted 23/11/98

Ann. Henri Poincar´ e 1 (2000) 201 – 202 c Birkh¨

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Annales Henri Poincar´ e

Editorial

Two well-known and old journals, Annales de l’Institut Henri Poincar´e (section de physique th´eorique) and Helvetica Physica Acta have now merged into a single journal under the name Annales Henri Poincar´ e (AHP). This new journal is owned jointly by the Institut Henri Poincar´e and by the Swiss Physical Society. It is published by Birkh¨ auser. The primary reason for this union is to enlarge the scope of the two merging journals by creating a truly international periodical. Our goal is to serve the international scientific community by publishing original research papers meeting the highest professional standards in the field. AHP will be a journal of theoretical and mathematical physics. It will publish original research papers in mathematical physics, including theoretical physics papers of interest for mathematically inclined readers; it will also publish papers on purely mathematical methods but with direct applications to physics. The journal will accept long papers of exceptional quality with no limitation in size; it may also include special issues on particular subjects, and invited papers. It will not in general publish letters, review papers or conference proceedings. In mathematical physics, which itself has become a very large field, the journal will put emphasis on analytical theoretical and mathematical physics, in a broad sense, and on rigorous results, rather than on algebraic or geometrical aspects, or on physical results of a more speculative nature. This is of course not a judgement of value; it seems simply that the mathematical physics community now needs a good journal with that particular orientation, which has also been strongly rooted in the two previous merging journals. The full editorial board is responsible for the scientific policy of the journal. Within that board, the section editors accept responsibility for a particular domain; within their section they organize the refereeing process, and decide on acceptance or refusal of papers, within limits negotiated with the editor in chief. The typical refereeing process goes as follows : when submitted to a section editor, a paper is transmitted to a single referee. The section editor also reads the paper. On the basis of the referee report he is responsible for acceptance or rejection. The paper appears with the section editor name mentioned: “communicated by...” Double refereeing is used only when needed. In exceptional cases the final decision is taken by the editor in chief, in concertation with the section editor concerned. A paper may also be submitted directly to the editor in chief, who can trans-

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mit it to a section editor or directly to a referee. The help of an associate editor may be also asked for. We will put emphasis on the quality of the refereeing process, which justifies the very existence of journals such as Annales Henri Poincar´e in a world of increasingly immediate availability of preprints in data banks. We will also maintain the journal at a very reasonable price per page, in order not to strain the budget of scientific libraries or institutions, at a time of increasing number of journals. Finally let us mention that under a particular agreement with Birkh¨ auser, all authors of contributions to AHP will have their intellectual work protected under the usual rules of copyright, but they will also retain the right to reproduce in part or total, their contributions elsewhere, provided they inform the journal. An electronic version of the journal will be available online to members of the subscribing institutions. There will be an annual prize funded by Birkh¨auser for the best article published in the journal. The jury which awards that prize is composed of the associate editors of the journal. The structure of the journal is flexible and open to further eventual unions, or to the participation of additional institutions. We hope that this journal, born in the symbolic year 2000, will serve the expanding mathematical and theoretical physics community in the next century, and we wish a happy growth to it!

Vincent Rivasseau

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Two Dimensional Magnetic Schr¨odinger Operators : Width of Mini Bands in the Tight Binding Approximation H.D. Cornean and G. Nenciu Abstract. The spectral properties of two dimensional magnetic Schr¨ odinger operators are studied. It is shown in the tight-binding limit that when a nonzero constant magnetic field is perturbed by an infinite number of magnetic and scalar ”wells”, the essential spectrum continues to have gaps and moreover, it can be nonempty in between the Landau levels and is localized near the one well Hamiltonian eigenvalues which develop into mini-bands whose width is believed to be optimally controlled. R´esum´e. On va ´ etudier les propri´ et´ es spectrales de l’operateur de Schr¨ odinger pour une particule bidimensionelle qui se trouve dans un champ magn´etique, dans l’approximation tight-binding. On va montrer que, pour un champ magn´etique constant, diff´erent de z´ero, perturb´ e par un nombre infini de puits magn´etiques et ´ el´ ectriques, le spectre essentiel continue de pr´ esenter des lacunes spectrales et qu’il peut ˆ etre non vide entre les niveaux de Landau. Plus encore, chaque valeur propre de l’hamiltonien avec un seul puits se transforme dans une bande spectrale dont la largeur est control´ee de mani` ere pr´ ecise.

I Introduction In this paper we continue the study (begun in [C-N]) of the spectral properties of two dimensional magnetic Schr¨odinger operators. In [C-N] we considered the ”one well problem” i.e. (1.1) H = (p − a0 − a)2 + V, where a0 corresponds to a nonzero constant magnetic field, B0 , the magnetic perturbation B0 (x) = curl a(x) is bounded in the sense that: b ≡ max{||Dα B 0 ||∞ , |α| ≤ 1} < ∞

(1.2)

and the scalar perturbation V = V1 + V2 obeys: V1 ∈ L2 (R2 ),

V2 ∈ L∞ (R2 )

(1.3)

It is known that if both the magnetic and the scalar perturbations are vanishing at infinity, then (see [I; H]): σess (H) = σL (B0 ) = {(2n + 1)B0 | n = 0, 1, ...}

(1.4)

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It was proved in [C-N] that if dist(z, σL (B0 )) = d > 0, then for sufficiently small b, ||V1 ||2 and ||V2 ||∞ we have that z ∈ / σ(H) and (H − z)−1 is an integral operator with a kernel which obeys: |K(x, x0 )| ≤ const(d) exp (−µ|x − x0 |),

|x − x0 | > 1

(1.5)

where µ goes to infinity when b, ||V1 ||2 and ||V2 ||∞ go to zero. (Actually, in [CN] the above estimate was given in the absence of the scalar potential, but the extension is straightforward). When the perturbations are vanishing at infinity, an important consequence of (1.5) (proved in [C-N]) is that if E ∈ σdisc (H), its corresponding eigenfunctions decay quicker than any exponential as |x| → ∞. Under more restrictive conditions imposed on V and B0 , a quicker (eventually Gaussian) decay can be proved (see [E; Na 2; S; C-N]). In particular, it is easy to see that if V and B0 vanish outside a compact set, then the decay is Gaussian. In this paper we shall deal with the multiple well case. The reason for considering this case is that when adding to a nonzero constant magnetic field a magnetic field perturbation and a scalar potential both having no decay at infinity a rich structure of the spectrum arise: the Landau spectrum suffers a radical change and one is expecting to find essential spectrum and gaps in between the Landau levels; moreover in the tight binding limit , there is a remarkable enhancement in the localization of the spectrum in comparison with a higher dimensional case ( see Section 3 for precise formulation of our main result). The multiple well problem has been considered both in the zero and nonzero magnetic field case but (see [H-S 1,2; C; B-C-D; N-B; H-H; Na ]) mainly below the essential spectrum of the ”unperturbed” Hamiltonian; what we add to the existent results is that in the two dimensional nonzero magnetic field case the width of the ”mini-bands” located below or in between the Landau levels shrinks Gaussian like in the limit when the inter well distance goes to infinity. Notice that the limit considered in [H-H; Na 1] is the strong field case i.e. the magnetic field outside the wells goes to infinity. The contents of the paper is as follows: Section 2 fixes some notations and gives a few results needed in the next section. Lemma 2.1 outlines the Gaussian decay of the kernel of the ”free resolvent” (the magnetic field is constant here and the scalar potential is absent); in Lemma 2.2 the localization of eigenfunctions of magnetic Schr¨odinger operators is briefly discussed. Propositions 2.1 and 2.2 give explicit examples of one well Hamiltonians with discrete eigenvalues in between the Landau levels. Section 3 contains the main result of this paper (namely Theorem 3.1) and it is devoted to the multiple well case, when the wells are far apart one from each other. For simplicity, we shall consider only the case of identical wells (but not necessarily arranged in a periodic lattice). The heuristics behind the proofs is the same as in the zero magnetic field case: due to the ”interactions” between wells, each eigenvalue of the one well Hamiltonian develops into a mini-band whose width

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205

shrinks to zero as the separation between wells tends to infinity. From the technical side our proof is in the spirit of the ”geometric perturbation theory ” in [B-C-D]. As in the zero magnetic field case, the size of the width of the ”mini-bands” is dictated by the decay of the one well eigenfunctions and that’s where the difference from higher dimensions appears: while in higher dimensions the width is shrinking exponentially with the inter-well distance, in our setting the width has a Gaussian decay (see Theorem 3.1 for the precise statement and the remark before its proof). Finally, Corollary 3.1 gives the existence of essential spectrum in between the Landau levels provided the one well Hamiltonian has discrete eigenvalues there.

II Preliminaries As already said, we shall consider only the two dimensional case (i.e. the particle is confined in the plane x3 = 0 and the magnetic field is orthogonal to that plane). Let B(x) ∈ C 1 (R2 ). We shall use the following family of vector potentials corresponding to B(x): 0

Z

1

a(x, x ) =

ds s B(x0 + s(x − x0 )) ∧ (x − x0 )

(2.1)

0

For x0 = 0, this is nothing but the usual transversal gauge (see e.g. [T]): Z a(x, 0) ≡ a(x) =

1

ds s B(s x) ∧ x

(2.2)

0

If we define f (x, x0 ) = a(x) − a(x, x0 )

(2.3)

then there exists ϕ(x, x0 ) such that ∇x ϕ(x, x0 ) = f (x, x0 )

(2.4)

The additional requirement ϕ(x0 , x0 ) = 0

(2.5)

gives ϕ(x, x0 ) =

Z

x1

x01

dt f1 (t, x2 ; x0 ) +

Z

x2

x02

dt f2 (x01 , t; x0 )

(2.6)

where xi , x0i , fi are the Cartesian components of x, x0 and f respectively. Performing the path integral of f (y, x0 ) on the segment γ(x, x0 ) = {y(t) = x0 + t(x − x0 )|t ∈ [0, 1]}

(2.7)

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and because a(y(t), x0 ) · (x − x0 ) = 0 for all t, one obtains: Z ϕ(x, x0 ) = a(y) · dy

(2.8)

γ(x,x0 )

The last equation shows that ϕ(x, x0 ) = −ϕ(x0 , x) and ϕ(x0 , 0) = ϕ(0, x) = 0, therefore Z 0 0 0 ϕ(x, x ) = ϕ(x , 0) + ϕ(x, x ) + ϕ(0, x) = a(y) · dy (2.9) ∆

where ∆ is the triangle γ(x0 , 0) ∪ γ(x, x0 ) ∪ γ(0, x). The last equality says (via the Stokes theorem) that −ϕ(x, x0 ) equals the flux of the magnetic field through ∆. Using (2.8), after a little calculation one obtains (and this is true in three dimensions, too): ϕ(x, x0 ) = −

Z

Z

1

1

dt 0

ds s B(s t (x − x0 ) + s x0 ) · (x ∧ x0 )

(2.10)

0

If B(x) = B0 is constant, then 1 ϕ0 (x, x0 ) = − B0 (x1 x02 − x01 x2 ) 2 1 0 a0 (x, x ) = B0 ∧ (x − x0 ) 2

(2.11)

The Hamiltonian of a particle in the presence of the magnetic field and a scalar potential V is (in the transversal gauge): = (p − a(x))2 + V (x) ∂ ∂ , −i p = −i ∂x1 ∂x2 Z 1 Z ds s B(s x), x1 a(x) = −x2 H

0

(2.12) 1 ds s B(s x)

0

In the case of a constant magnetic field, one has the Hamiltonian H0

= (p − a0 (x))2 where

p = −i∇x

and a0 (x) =

1 1 − B0 x2 , B0 x1 2 2

(2.13) (2.14)

which is essentially self-adjoint on C0∞ R2 and its spectrum is the well known Landau spectrum σ(H0 ) = σess (H0 ) ≡ σL (B0 ) = {(2n + 1)B0 | n = 0, 1, 2, . . .}

(2.15)

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207

For z 6∈ σ(H0 ) and g ∈ L2 R2 , we write (H0 − z)−1 g (x) =

Z

dx0 K0 (x, x0 ; z)g(x0 )

(H0 − z)K0 (x, x0 ; z) = δ(x − x0 )

(2.16)

Then takes place (see e.g. [J-P]): Lemma II.1 Let ϕ0 (x, x0 ) = −

B0 (x1 x02 − x2 x01 ) 2

B0 |x − x0 |2 4 1 z α = − − 1 6= −1, −2, . . . 2 B0

ψ(x, x0 ) =

Then 0

K0 (x, x0 ; z) = ei ϕ0 (x,x ) G0 (x, x0 ; z) ≡ Γ(α) i ϕ0 (x,x0 ) −ψ(x,x0 ) ≡ e U (α, 1; 2 ψ(x, x0 )) e 4π

(2.17)

where Γ is the Euler function and U (α, γ; ξ) is the confluent hyper-geometric function [A-S]. From Lemma II.1 one sees that K0 (x, x0 ; z) has a Gaussian decay as |x − x0 | → ∞. We shall use this in the following form: Corollary II.1 Let χ1 , χ2 ∈ L∞ (R2 ) such that |χ1 |, |χ2 | ≤ M Then for all 0 < δ <

B0 4

and

dist{supp χ2 , supp χ1 } = d > 0.

and z ∈ ρ(H0 ), one has that −1

||χ1 (H0 − z)

χ2 || ≤ M 2 const(z) exp(−δ d2 )

(2.18)

Proof. Use the explicit form of K0 and Young inequalities (see [C-N] for further discussions). Remark. Since under a gauge transformation (Uχ f ) (x) = ei χ(x) f (x) and (2.19) Z Uχ∗ (H0 − z)−1 Uχ f (x) = dx0 Kχ (x, x0 ; z) = R2 Z 0 dx0 e−i χ(x) K0 (x, x0 ; z)ei χ(x ) f (x0 ) = R2

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one has

Ann. Henri Poincar´e

|Kχ (x, x0 ; z)| = |K0 (x, x0 ; z)|

(2.20)

i.e. the Gaussian decay is valid for an arbitrary gauge. Suppose now that the scalar potential V and the magnetic field which corresponds to a describe the one well case studied in [C-N] i.e. satisfy the following conditions : B = B0 + B 0 , B0 > 0 B 0 ∈ C 1 R2 ; lim ||B 0 ||C 1 (R2 \{|x|≤n}) = 0 n→∞ V = V1 + V2 ; V1 ∈ L2 R2 , V2 ∈ L∞ R2 lim sup |V2 (x)| = 0 n→∞ |x|≥n

(2.21)

In particular, under these conditions H is essentially self-adjoint on C0∞ R2 (see e.g. [C-F-K-S] ). Moreover, V is relatively compact with respect to (p − a)2 [C-F-K-S] which together with the results in [I, H] it implies that σess (H) = σ(H0 ) = {(2n + 1)B0 | n = 0, 1, 2, . . .}

(2.22)

In the rest of this section, g ∈ C ∞ R2 ; R ; ||g||C 2 (R2 ) = M < ∞

(2.23)

Let E ∈ σdisc (H) (the discrete spectrum of H) and let ψ be a normalized eigenfunction corresponding to E. We are interested now in controlling as good as possible the term ||[H, g]ψ||. Under the conditions (2.21), one has D(H) = D((p − a)2 ) and (pj − aj )(H + −1 i) is bounded, j ∈ {1, 2}. Moreover, because

it follows that

[H, g] = −i{(p − a) · ∇g + ∇g · (p − a)}

(2.24)

||[H, g](H + i)−1 || ≤ const(M )

(2.25)

which gives the following rough result: ||[H, g]ψ|| ≤ const(M ) (E 2 + 1)1/2

(2.26)

In order to obtain a sharper estimate on this term, we use the following form of the I.S.M. localization lemma [C-F-K-S]: Lemma II.2 Let ϕ ∈ D(H). Then: < ϕ, gHgϕ >= <(< ϕ, g 2 Hϕ >)+ < ϕ, |∇g|2 ϕ >

(2.27)

< gψ, (H − E)gψ >=< ψ, |∇g|2 ψ >

(2.28)

and

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209

Under the conditions (2.21), there exists a constant c > 0 independent of ψ such that: 1 (2.29) | < gψ, V gψ > | ≤ < gψ, (p − a)2 gψ > +c||gψ||2 2 From (2.28) and (2.29) it follows that: ||(p − a)gψ||2 ≡< gψ, (p − a)2 gψ >≤ Z ≤ 2M 2 [|E| + c + 1] dx|ψ(x)|2

(2.30)

supp g

After a little calculation, from (2.24) and (2.30) one obtains: ! Z ||[H, g]ψ||2 ≤ const(E, M )

dx|ψ(x)|2

(2.31)

supp |∇g|

which is the needed estimate. We’ll show now that there are many examples of one well Hamiltonians with discrete spectrum in between the Landau levels. We put this in the form of two propositions: the first one constructs a purely electric well which gives an eigenvalue located anywhere we want outside σL (B0 ) and the second one states that any sufficiently “small” purely magnetic well with definite sign creates eigenvalues near any Landau level we choose. Proposition II.1 Take λ ∈ R, λ ∈ / σL (B0 ). Then there exists a bounded, compactly supported potential V ∈ L∞ R2 such that λ is a discrete eigenvalue for the operator sum H = H0 + V . Proof. Fix λ as mentioned above. From Lemma II.1, one can easily see that K0 (x, 0; λ) = K0 (0, x; λ) = K0 (x, 0; λ)

(2.32)

where the over-line means complex conjugation. Because the confluent hypergeometric function U (α, 1, ξ) is analytic in {ξ ∈ C, <ξ > 0} and together with the realty of K0 (x, 0; λ) one obtains the existence of A > 0, 0 < < A and 0 < δ < 1 such that if A− ≤ |x| ≤ A+, then K0 (x, 0; λ) is not changing sign and moreover, one can suppose without loss that K0 (x, 0; λ) ≥ δ

(2.33)

Define now: η1 ∈ C0∞ (R2 ), 0 ≤ η1 ≤ 1 and 1 if |x| ≤ A η1 (x) = 0 if |x| ≥ A +

(2.34)

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η2 ∈ C ∞ (R2 ), 0 ≤ η2 ≤ 1 and 0 if |x| ≤ A − η2 (x) = 1 if |x| ≥ A ψ(x) = η1 (x) + η2 (x)K0 (x, 0; λ)

Ann. Henri Poincar´e

(2.35)

(2.36)

We also require η1 and η2 to be radially symmetric. Using (2.33) and the definitions of the cut-off functions, one obtains that ψ ∈ L2 R2 and ψ(x) ≥ δ if |x| ≤ A + . Take now Φ ∈ L∞ R2 such that: Φ(x) =

if |x| ≤ A + 1 if |x| > A +

1 ψ(x)

(2.37)

Finally, the potential we are looking for will be: V = −Φ · {(H0 − λ)η1 + [H0 , η2 ]K(·, 0; λ)}

(2.38)

Due to the fact that a0 is written in the transversal gauge (which implies a0 (x) · x = 0), it follows that H0 maps radially symmetric functions into real functions, and that V is real, bounded and compactly supported. Moreover, H ψ = λ ψ. Proposition II.2 Let B 0 ∈ C01 R2 ; R be a nonnegative, compactly supported function and let a0 (x) be the transversal gauge which gives B 0 . For b > 0, define Hb = (p − a0 − ba0 )2 . Let En = (2n + 1)B0 be the n-th Landau level. Then for b sufficiently small, Hb will have at least one eigenvalue near En . Proof. Because B 0 has compact support, one has |a0 (x)| ≤ const· < x >−1 where 1 < x >≡ 1 + x2 2 . Denote with W (b) = −b(p − a0 ) · a0 − ba0 · (p − a0 ) + b2 a02 and with V = −(p − a0 ) · a0 − a0 · (p − a0 ). It is easy to see that W (b) is relatively bounded to H0 ; one can then apply the analytic perturbation theory around En if b is kept small enough. The reduced operator defined in RanPn (Pn being the projector associated with En ) will have the form: Hef f (b) = En Pn + Pn T (b)Pn T (b) ≡ bPn V Pn + O(b2 ).

(2.39)

The only thing we should check is that Pn T (b)Pn is not zero; this would imply that Hef f (b) − En has nonzero spectrum, therefore Hb will have (discrete) spectrum

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near En . To achieve that, one can compute < fn , V fn > where fn stands for the spherically symmetric, real eigenfunction of H0 , corresponding to En . This computation gives: Z Z |x| < fn , V fn >= 2 < fn , a0 · a0 fn >= B0 dx fn2 (x) dρ ρB 0 (ρ, θ) (2.40) 0

This quantity is not zero because B 0 is not changing sign; therefore, if b is small enough, Pn T (b)Pn 6= 0. Remark. This type of argument also works in the case of a purely electric well; one only has to check that the term in (2.40) (where V stands now for the scalar potential) is different from zero. In conclusion, it is not difficult to give examples of one well Hamiltonians with discrete spectrum outside the Landau levels; the really hard problem is to study their behaviour near the essential spectrum.

III Gaps in the essential spectrum Consider 2 ΓN = {x(i)}N i=1 ⊂ R ,

N ≤∞

(3.1)

Without loss of generality, one can always take x(1) = 0. The main assumption about ΓN is that: inf |x(j) − x(k)| = r > 0 (3.2) j6=k

and since the limit to be considered is r → ∞, we assume for technical reasons that r is sufficiently large, say r ≥ 1000. Concerning the magnetic field and the potential, we assume: B0 > 0 , B 0 ∈ C 1 (R2 ) and supp B 0 ⊂ {|x| ≤ 1} V ∈ L2 (R2 ), supp V ⊂ {|x| ≤ 1}

(3.3)

Let BN (x) = B0 + Z

N X

B 0 (x − x(j)), VN (x) =

j=1

N X

V (x − x(j)),

j=1

1

ds s BN (s x) ∧ x

aN (x) =

(3.4)

0

Consider now for N = 1, 2, . . . , ∞ the following family of Hamiltonians: HN = (p − aN )2 + VN .

(3.5)

These operators are essentially self-adjoint on C0∞ (R2 ) and for N < ∞ σess (HN ) = σL (B0 )

(3.6)

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In particular, H1 = (p − a1 )2 + V1

(3.7)

is the ”one well” Hamiltonian. The main result of this section is contained in Theorem III.1 Let 1 ≤ N < ∞, c < d, K ≡ [c, d], K ∩ σL (B0 ) = ∅ and suppose that σ(H1 ) ∩ K = {E1 < . . . < Es } ⊂ σdisc (H1 ), s ≥ 1 mult(Ej ) = mj , j ∈ {1, . . . s} If c, d are not eigenvalues for H1 , then there exist r0 (K, m1 , . . . , ms ), C < ∞ and u > 0 independent of N such that i) σ(HN ) ∩ K ⊂

s [

[Ej − δ, Ej + δ], 0 ≤ δ ≤ Ce−u r , for all r ≥ r0 2

(3.8)

j=1

ii) dim{Ran PN [σ(HN ) ∩ K]} = N

s X

mj

(3.9)

j=1

where PN is the spectral measure associated with HN . Remark. If one drops the compactness condition in (3.3) but imposes additional conditions to (2.21) in order to ensure the finiteness of the ”total perturbations” in H∞ , such as: max{|B 0 (x)|, |V (x)|} ≤ const (1 + |x|)−β , β > 2

(3.10)

then the proof of Theorem 3.1 can be adapted such that ii) remains true and i) is changing in the sense that instead of a Gaussian decay in r, we can only say that δ goes to zero when r goes to infinity and this comes from the fact that in this case, the wells are no longer well individualized. Proof of i). Define : ΣN ≡ σ(HN ) ∩ K (3.11) Because of (3.6), ΣN is discrete if not empty. For simplicity, we suppose s = 1 and m1 = 1; the proof in the general case is similar. During the proof of this theorem, E ∈ ΣN will denote an eigenvalue of HN and ψ a corresponding normalized eigenfunction. We give first a few helpful technical lemmas and we start with some definitions. For (p1 , p2 ) ∈ Z2 and δ > 0, define: δ r 1 2 K(p1 , p2 ; δ) = x ∈ R | |xj − pj + | ≤ , j ∈ {1, 2} (3.12) 100 2 2

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r It is easily seen that for any δ ≥ 100 , [ K(p1 , p2 ; δ) = R2

213

(3.13)

(p1 ,p2 )∈Z2

If m ∈ R2 , then the translation tm : L2 → L2 , (tm f )(x) = f (x − m) is an unitary operator. Given any j ∈ {1, . . . , N }, there exists (pj1 , pj2 ) ∈ Z2 such that: x(j) ∈ K(pj1 , pj2 ; r/100) and x(k) 6∈ K(pj1 , pj2 ; r/100) if j 6= k If β, γ ∈ {−1, 0, 1} then define [ K(pj1 + β, pj2 + γ; δ), Kj (δ) =

δ>0

(3.14)

(3.15)

β,γ

By construction, dist{x(j), ∂Kj (r/100)} ≥ r/100. Denote with FN =

N [

Kj (r/100)

(3.16)

(3.17)

j=1

Lemma III.1 Take

Denote with m =

r 100

K(p, q; r/100) 6⊂ FN and η ∈ C0∞ (R2 ), supp η ⊂ K(p, q; r/98). r p + 12 , 100 q + 12 . Then:

HN η = ei ϕN (.,m) tm H0 t−m e−i ϕN (.,m) η

(3.18)

Proof. If x ∈ supp η, then : aN (x) = aN (x, m) + ∇ϕN (x, m) VN (x) = 0 Z 1 aN (x, m) = ds s BN (m + s(x − m)) ∧ (x − m)

(3.19)

0

Because for all y ∈ {m + s(x − m), 0 ≤ s ≤ 1} one has BN (y) = B0 then aN (x, m) HN η

= a0 (x − m) and = ei ϕN (x,m) [p − a0 (x − m)]2 e−i ϕN (x,m) η = ei ϕN (.,m) tm H0 t−m e−i ϕN (.,m) η

(3.20)

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Lemma III.2 Fix j ∈ {1, . . . , N }. Take ηj ∈ C0∞ (R2 ) and supp ηj ⊂ Kj (r/98). Then : HN ηj = ei ϕN (.,x(j)) tx(j) H1 t−x(j) e−i ϕN (.,x(j)) ηj (3.21) Proof. As before, Since r −

√ 3 2 98 r

aN (x) = aN (x, x(j)) + ∇ϕN (x, x(j))

(3.22)

> 1 and x ∈ supp ηj , then

|x(j) − x(k) + s(x − x(j))| ≥ |x(j) − x(k)| − |x − x(j)| > 1,

j 6= k

(3.23)

therefore: aN (x, x(j)) = a0 (x − x(j)) + Z 1 N X ds s B0 (x(j) − x(k) + s(x − x(j))) ∧ (x − x(k)) + 0

k=1

= a1 (x − x(j)) + Z 1 X + ds s B0 (x(j) − x(k) + s(x − x(j))) ∧ (x − x(k)) 0

k6=j

= a1 (x − x(j))

(3.24)

If x ∈ supp ηj , then VN (x) = V (x − x(j)); putting all these together, (3.21) follows. Lemma III.3 Under the same assumptions made in Lemma 3.1, suppose now that 0 ≤ η ≤ 1 and 1 if x ∈ K(p, q; r/99) η(x) = 0 if x ∈ 6 K(p, q; r/98)

(3.25)

Then there exist C1 > 0 and u > 0 (which are independent of N , (p, q) and E ∈ ΣN ) such that: Z Z 2 dx |ψ(x)|2 ≤ e−ur C1 dx |ψ(x)|2 (3.26) K(p,q;r/100)

K(p,q;r/98)\K(p,q;r/99)

Proof. (3.18) implies that

If

[HN , η]ψ = ei ϕN (.,m) tm (H0 − E) t−m e−i ϕN (.,m) ηψ or ηψ = ei ϕN (.,m) tm (H0 − E)−1 t−m e−i ϕN (.,m) [HN , η]ψ

(3.27)

x ∈ K(p, q; r/100) and x0 ∈ K(p, q; r/98) \ K(p, q; r/99)

(3.29)

(3.28)

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then r2 |x − x | ≥ 4 0 2

1 1 − 99 100

215

2 (3.30)

1 1 2 0 and using Corollary 2.1, one has (0 0, C < ∞ with the properties given in Lemma III.3 such that: Z 2 dx |ψ(x)|2 ≤ C e−ur (3.32) (FN )c

Proof. Adding the contributions given by all K(p, q; r/100) 6⊂ FN in (3.26) and because XZ dx |ψ(x)|2 ≤ 4 (3.33) (p,q)

K(p,q;r/98)\K(p,q;r/99)

the result follows. We are now able to prove the first affirmation of Theorem 3.1. For j ∈ {1, . . . , N } take ηj ∈ C0∞ (R2 ), 0 ≤ ηj ≤ 1 and 1 if x ∈ Kj (r/99) ηj (x) = 0 if x ∈ 6 Kj (r/98) Let ψ˜ =

N X

ηj ψ

(3.34)

(3.35)

j=1

then from Lemma 3.2 one has: (HN − E)ψ˜ =

N X

ei ϕN (.,x(j)) tx(j) (H1 − E) t−x(j) e−i ϕN (.,x(j)) ηj ψ

(3.36)

j=1

or

N X j=1

||(H1 − E)t−x(j) e−i ϕN (.,x(j)) ηj ψ||2 =

N X

||[HN , ηj ]ψ||2

(3.37)

j=1

But ||(H1 − E)t−x(j) e−i ϕN (.,x(j)) ηj ψ||2 ≥ dist2 {E, σ(H1 )}||ηj ψ||2

(3.38)

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therefore, together with (3.37) and (2.31) one obtains: dist2 {E, σ(H1 )}

N X

Z ||ηj ψ||2 ≤ C

j=1

or

!

Z dist {E, σ(H1 )} 1 − 2

dx|ψ(x)| c

2

dx |ψ(x)|2

(FN )c

(3.39)

Z ≤C

(FN )

dx|ψ(x)|2

(3.40)

c

(FN )

and together with (3.32), the affirmation stated in (3.8) follows. Proof of ii). Let’s show first that dim{Ran[PN (ΣN )]} ≥ N.

(3.41)

Denote with ψ1 the normalized eigenvector of H1 corresponding to E1 : H1 ψ1 = E1 ψ1 ,

||ψ1 || = 1

(3.42)

With the notations introduced in (3.34), let n o VN = ψ˜j = ei ϕN (.,x(j)) ηj tx(j) ψ1

(3.43)

j=1,N

be an orthogonal system. Because

Z

||ψ˜j ||2 = ||(t−x(j) ηj )ψ1 ||2 ≥ 1 −

(F1 )c

dx |ψ1 (x)|2 ∼ 1 − Ce−ur

2

(3.44)

then for r large enough, VN is an ”almost orthonormal system” and dim VN = N,

r ≥ r0

(3.45)

(notice that r0 does not depend upon N ). Suppose now that (3.41) were false; this would imply the existence of an r > r0 such that dim{Ran[PN (ΣN )]} ≤ N − 1. (3.46) (3.46) and (3.8) would imply then that: dim{Ran[PN (K)]} ≤ N − 1.

(3.47)

˜ = 1 such that: Then there exists ψ˜ ∈ VN , ||ψ|| ψ˜ ∈ {Ran[PN (K)]}⊥

(3.48)

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Because < ψ˜j , ψ˜k >= 0 if j 6= k and using (3.44), one has that ψ˜ =

N X

Cj ψ˜j ,

j=1

N X

|Cj |2 ∼ 1 − Ce−ur0

2

(3.49)

j=1

Without loss, suppose that there exists δ > 0 such that [E1 − δ, E1 + δ] ⊂ K. Then (3.48) implies ˜ ≥ δ. ||(HN − E1 )ψ|| (3.50) But using Lemma 3.2, one obtains: (HN − E1 )ψ˜ =

N X

Cj (HN − E1 )ψ˜j

j=1

=

N X

Cj ei ϕN (.,x(j)) tx(j) (H1 − E1 )t−x(j) ηj tx(j) ψ1

j=1

=

N X

Cj ei ϕN (.,x(j)) tx(j) (H1 − E1 )(t−x(j) ηj )ψ1

(3.51)

j=1

Using that < tx(j) (H1 − E1 )(t−x(j) ηj )ψ1 , tx(k) (H1 − E1 )(t−x(k) ηk )ψ1 >= 0, j 6= k one has ˜ 2= ||(HN − E1 )ψ||

N X

|Cj |2 ||[H1 , (t−x(j) ηj )]ψ1 ||2 .

(3.52)

(3.53)

j=1

From (2.31), (3.32) and (3.49) it follows that ˜ 2 ∼ e−ur02 ||(HN − E1 )ψ||

(3.54)

which can be made arbitrarily small and then contradicting (3.50). Let’s prove now that dim{Ran[PN (K)]} ≤ N

(3.55)

In order to prove (3.55), we shall construct a finite rank operator PN0 (not necessary an orthogonal projector) such that: dim Ran PN0 ≤ N and ||PN (K) − PN0 || < 1, for all r ≥ r0 Proposition III.1 Suppose (3.56) fulfilled. Then (3.55) takes place.

(3.56)

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Proof. Assume that dim{Ran[PN (K)]} ≥ N + 1 for some r ≥ r0 . Then there exists ψ ∈ Ran[PN (K)], ||ψ|| = 1 such that:

But and

ψ ∈ [Ran(PN0 )]⊥

(3.57)

| < ψ, (PN (K) − PN0 )ψ > | ≤ ||PN (K) − PN0 || < 1

(3.58)

| < ψ, (PN (K) − PN0 )ψ > | = | < ψ, PN (K)ψ > | = 1

(3.59)

which contradicts (3.58). Let’s construct now PN0 . Using (3.8), one obtains the existence of r0 () such that {|z − E1 | = } ∩ σ(HN ) = ∅ (3.60) as soon as r ≥ r0 () ( being chosen sufficiently small then kept fixed). The idea consists (see for similar reasoning [B-C-D] and [Na 1]) in approximating the resolvent (HN − z)−1 for |z − E1 | = and then integrating over the contour. Let m(p, q) = (r/100(p + 1/2), r/100(q + 1/2)) if K(p, q; r/100) 6⊂ FN and Γ∞ = ΓN

[ {m(p, q)}(p,q)

(3.61) (3.62)

It is possible to construct a quadratic partition of unity which has the following properties (see [C-F-K-S]): •

X

ηm ∈ C0∞ (R2 ),

0 ≤ ηm ≤ 1

(3.63)

ηm (x) = 0 if x 6∈ Km (r/99) and m ∈ ΓN

(3.64)

2 ηm = 1,

m∈Γ∞

• •

ηm (x) = 0 if x 6∈ K(p, q; r/99)

• ||(∂ηm /∂xi )||C 1 ≤

const , r

and m ∈ {m(p, q)}(p,q) m ∈ Γ∞ ,

i = 1, 2

(3.65)

(3.66)

Lemma III.5 The operator X ei ϕN (.,m) ηm tm (H1 − z)−1 t−m ηm e−i ϕN (.,m) AN (z) = m∈ΓN

+

X

m∈Γ∞ \ΓN

ei ϕN (.,m) ηm tm (H0 − z)−1 t−m ηm e−i ϕN (.,m) (3.67)

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is bounded and moreover

const where the constant in (3.68) is independent upon N . ||AN (z)|| ≤

(3.68)

Proof. Take f ∈ L2 (R2 ). Then, from (3.60) and (3.63), AN (z)f consists in a sum in which each term is bounded and from (3.64) and (3.65) there results that each term is orthogonal to all others except at most 16 ”neighbours”. AN (z) is our approximation of the resolvent. From Lemma 3.1 and Lemma 3.2, one obtains: (HN − z)AN (z) X = ei ϕN (.,m) tm (H1 − z)t−m ηm tm (H1 − z)−1 t−m ηm e−i ϕN (.,m) m∈ΓN

+

X

ei ϕN (.,m) tm (H0 − z)t−m ηm tm (H0 − z)−1 t−m ηm e−i ϕN (.,m)

m∈Γ∞ \ΓN

= 1+

X

ei ϕN (.,m) tm [Hm , (t−m ηm )](Hm − z)−1 t−m ηm e−i ϕN (.,m)

m∈Γ∞

≡ 1 + TN (z)

(3.69)

where

Hm =

H1 H0

if m ∈ ΓN if m ∈ Γ∞ \ ΓN

(3.70)

By essentially the same argument used in deriving (2.25), one obtains: ||[Hm , (t−m ηm )](Hm − z)−1 || ≤

const() . r

(3.71)

As in Lemma 3.5, one finally obtains ||TN (z)|| ≤

const() . r

Take r1 () ≥ r0 () such that ||TN (z)|| ≤

1 2

(3.72)

if r ≥ r1 (); then one can write:

(HN − z)−1 = AN (z) − AN (z)TN (z)[1 + TN (z)]−1

(3.73)

Integrating over {|z − E1 | = }, it follows that: PN ([E1 − , E1 + ]) X = ei ϕN (.,m) ηm tm P1 ({E1 })t−m ηm e−i ϕN (.,m) + RN m∈ΓN

≡ PN0 + RN

(3.74)

where ||RN || < 1,

r ≥ r2 () ≥ r1 ()

and the proof is completed, due to the fact that

PN0

(3.75)

has its rank equal to N .

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Corollary III.1 Let N = ∞. Then Theorem III.1 remains true (with N formally replaced with ∞). Proof. Define ΓN(R) = {x ∈ Γ∞ | |x| ≤ R}

(3.76)

Then : HN(R) → H∞

in the strong sense

(3.77)

From the essential self-adjoint-Ness on the same core and from (3.77), one obtains HN(R) → H∞

in the strong resolvent convergence sense

(3.78)

(see e.g. [K] ). Then Theorem VIII.1.4 in [K] and (3.8) imply that (3.8) remains true for H∞ . Finally, to show that dim{Ran P∞ [σ(H∞ ) ∩ K]} = ∞

(3.79)

one can use an ad-absurdum argument as that used in proving (3.41). Remark. It is easy to show now that when increasing r, one still can find essential spectrum of H∞ near a part of σL (B0 ). Take for example the old eigenvalue B0 ; an eigenfunction of H0 which corresponds to it reads as: r ψ0 (x) =

B0 |x|2 B0 exp − . 2π 4

Take > 0 sufficiently small and let’s prove that (K = [B0 − , B0 + ]): dim{Ran P∞ [σ(H∞ ) ∩ K]} = ∞ The ad-absurdum argument used in proving (3.41) can be applied again, replacing H1 , ψ1 , E1 and Kj with H0 , ψ0 , B0 and K(p, q).

Acknowledgements The first version of this paper was written during a visit to Erwin Schr¨ odinger Institute in Vienna. The financial support of ESI is gratefully acknowledged.

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[B-C-D]

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Nakamura, S.: Band spectrum for Schr¨ odinger operators with strong periodic magnetic fields. Operator Theory: Advances and Applications 78, 261–270 (1995)

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H. D. Cornean Institute of Mathematics of the Romanian Academy P.O. Box 1-764 70700 Bucharest, Romania e-mail [email protected] G. Nenciu Department of Theoretical Physics University of Bucharest P.O. Box MG 11, 76900 Bucharest, Romania e-mail [email protected] Communicated by J. Bellissard submitted 19/02/98, revised 21/10/98 ; accepted 18/11/98

Ann. Henri Poincar´ e 1 (2000) 223 – 248 c Birkh¨

auser Verlag, Basel, 2000 1424-0637/00/020223-26 $ 1.50+0.20/0

Annales Henri Poincar´ e

Invariant Tori, Effective Stability, and Quasimodes with Exponentially Small Error Terms I – Birkhoff Normal Forms G. Popov Abstract. The aim of this paper (part I and II) is to explore the relationship between the effective (Nekhoroshev) stability for near-integrable Hamiltonian systems and the semi-classical asymptotics for Schr¨ odinger operators with exponentially small error terms. Given a real analytic Hamiltonian H close to a completely integrable one and a suitable Cantor set Θ defined by a Diophantine condition, we are going to find a family Λω , ω ∈ Θ, of KAM invariant tori of H with frequencies ω ∈ Θ which is Gevrey smooth in a Whitney sense. Moreover, we obtain a symplectic Gevrey normal form of the Hamiltonian in a neighborhood of the union Λ of the invariant tori which can be viewed as a simultaneous Birkhoff normal form of H around all invariant tori Λω . This leads to effective stability of the quasiperiodic motion near Λ. As an application we obtain in part II (semiclassical) quasimodes with exponentially small error terms which are associated with a Gevrey family of KAM tori for its principal symbol H. To do this we construct a quantum Birkhoff normal form of the Schr¨ odinger operator around Λ in suitable Gevrey classes starting from the Birkhoff normal form of H.

Consider the Schr¨ odinger operator Ph = −h2 ∆ + V (x), x ∈ Rn ,

(.1)

with a real analytic potential V (x), where 0 < h ≤ h0 is a small parameter. Let us set E∞ = lim inf V (x) |x|→∞

and suppose that E∞ > −∞. Denote by Ph also the corresponding Friedrichs extension in L2 (Rn ). Let H(x, ξ) = |ξ|2 + V (x) be the corresponding classical Hamiltonian. Suppose that E0 = min V (x) < E∞ . Then Ph has a discrete spectrum in any interval [E0 , E] with E0 < E < E∞ . The motivation for this paper comes from the semi-classical behavior of the low lying eigenvalues of Ph when the potential V has a non-degenerate minimum E0 = V (0) < E∞ , i.e. near the bottom of a well. In that case, after a linear change of variables in x we may assume that H(x, ξ) = E0 +

n X αj0 j=1

2

(ξj2 + x2j ) + O |(x, ξ)|3 , for small (x, ξ) ,

(.2)

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where 0 < α10 ≤ · · · ≤ αn0 . Under these assumptions, Helffer and Sj¨ ostrand [11] determined modulo O(exp(−c/h)), c > 0, the semi-classical asymptotics (as h & 0) of all the eigenvalues of Ph lying in an interval [E0 , E0 + Ch], where C > 0 is arbitrary but fixed (see also [27]). Sj¨ ostrand [28] investigated the semi-classical behavior of all the eigenvalues lying in any interval [E0 , E0 + hδ ], where 0 < δ < 1 is fixed. It turned out that it depends essentially on the Birkhoff normal form (BNF) of H at the origin. Let us exclude resonances of some finite order N ≥ 4, i.e. assume that hα0 , ki 6= 0, 1 ≤ |k1 | + · · · + |kn | ≤ N, k ∈ Zn ,

(.3)

where α0 = (α10 , . . . , αn0 ). Then the Hamiltonian can be written in a BNF near 0. In other words, there exist analytic symplectic coordinates in a neighborhood of 0 such that the Hamiltonian becomes H(x, ξ) = H 0 (I) + H1 (x, ξ), where I = (I1 , . . . , In ), Ij = 12 (x2j + ξj2 ), and 1 H 0 (I) = E0 + hα0 , Ii + hQI, Ii + · · · 2

(.4)

is a polynomial of degree ≤ (N − 1)/2, while H1 (x, ξ) = O(|I|(N+1)/2 ), as |I| → 0 . Moreover, the BNF is non-degenerate, which means that det Q 6= 0. If there are no resonances of any order, then H 0 stands for a smooth function in I with a prescribed Taylor expansion at 0 and (x, ξ) are smooth symplectic coordinates, while H1 satisfies the above estimate for each N ≥ 4. Conjugating Ph by a suitable unitary h-Fourier integral operator (h-FIO), Sj¨ ostrand [28] obtained a microlocal normal form Ph0 of Ph . The complete Weyl symbol of Ph0 has the form p(I, h) + O(|I|(N+1)/2 ), where ∞ X p(I, h) ∼ p0j (I) hj , j=0

and p0 = H 0 . Moreover, one can take N = +∞ if there are no resonances of any order. The operator Ph0 is referred to as a quantum Birkhoff normal form (QBNF) of Ph . It leads immediately to quasimodes for Ph concentrated in the region |I| < hδ with polynomially small error terms ON (hN ), N > 0, and gives semi-classical asymptotics for all the eigenvalues of Ph in [E0 , E0 + hδ ]. Stronger result has been proved recently for Gevrey smooth potentials V (x) by Bambusi, Graffi and Paul [1]. They obtained a quantization formula modulo O(h∞ ) for all the eigenvalues of the operator Ph in an interval [E0 , E0 + ϕ(h)] where ϕ(h)b ln h → 0 as h & 0 and b is an explicitly determined constant. Sj¨ ostrand conjectured in [28] that given a real analytic Hamiltonian H one can expect semi-classical asymptotics with exponentially small errors.

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Our aim is to study the semi-classical behavior of the eigenvalues in a larger interval [E0 , E], where E > E0 is close to E0 but independent of h. We are interested in quasimodes for Ph with exponentially small error terms. Such quasimodes have the form Q = {(um (·, h), λm (h)) : m ∈ Mh }, where um (·, h) ∈ C0∞ (Rn ) has support in a fixed bounded domain independent of h, λm (h) are real valued functions of h in an interval (0, h0 ], Mh is a finite index set for each fixed 0 < h ≤ h0 , and 1/% , ||Ph um − λm (h)um ||L2 = O e−c/h 1/% hum , ul iL2 − δm,l = O e−c/h , for m, l ∈ Mh . Here % > 1 and c are positive constants, and δm,l is the Kronecker index. To construct quasimodes for Ph with quasi-eigenvalues λm (h) ∈ [E0 , E] we need more information on the dynamics of the classical Hamiltonian H in the compact H −1 ([E0 , E]). It was observed by Lazutkin [14], [15], and by Colin de Verdi`ere [5] that certain families of invariant tori of the classical Hamiltonian system can be quantized asymptotically. In other words, one can associate quasimodes with them with polynomially small errors. Suppose as above that (.3) holds for some fixed N ≥ 4. After a polar symplectic change of the variables p p xj = 2Ij sin ϕj , ξj = 2Ij cos ϕj , j = 1, . . . , n , the Birkhoff normal form of H can be written as follows H(ϕ, I) = H 0 (I) + H1 (ϕ, I) , H1 (ϕ, I) = O(|I|(N+1)/2 ) , as |I| → 0 ,

(.5)

where ϕ = (ϕ1 , . . . , ϕn ) ∈ Tn = (R/2πZ)n and I belongs to a proper open cone Γ ⊂ Rn+ with a vertex at 0. Hence, we can consider H as a small analytic perturbation of the completely integrable analytic Hamiltonian H 0 in any domain Tn × D, where D ⊂ Γ is close to the origin but still away of it. Moreover, we suppose that H 0 is non-degenerate in D which amounts to det ∂ 2 H 0 /∂I 2 6= 0 in D .

(.6)

More generally we consider a real analytic non-degenerate completely integrable Hamiltonian H 0 (I) in Tn × D, where D is a bounded domain in Rn and (ϕ, I) are the ”angle-action” coordinates in T ∗ (Tn ). The classical Kolmogorov-ArnoldMoser (KAM) Theorem asserts that the invariant tori Tn ×{I} of the Hamiltonian flow of H 0 having frequencies ω = ∇H 0 (I), ∇ = (∂/∂I1 , . . . , ∂/∂In ), in a suitable

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Cantor set sustain small real analytic perturbations H of H 0 . Fix κ > 0 and τ > n − 1. The frequencies ω satisfy the Diophantine condition |hω, ki| ≥

κ , for all 0 6= k ∈ Zn , |k|τ

(.7)

where |k| = |k1 | + · · · + |kn |. It is also known [22] that the corresponding invariant tori Λω of H are analytic Lagrangian submanifolds of Tn × D depending smoothly on ω in the sense of Whitney. Our aim in this paper is to prove that for each µ > τ + 2 there exists a family of Lagrangian KAM tori Λω , ω ∈ Ωκ , which is Gµ -Gevrey regular with respect to ω in a Whitney sense. Here, Ωκ is a suitable subset of frequencies of a positive Lebesgue measure in Rn satisfying (.7). Moreover, we shall obtain a symplectic Gevrey normal form of H in a neighborhood of that family of invariant tori. More precisely, we shall prove that there exist symplectic coordinates (ϕ, I) in Tn × D and a Gµ diffeomorphism ω : D → Ω, Ω being a neighborhood of Ωκ , such that the Hamiltonian becomes H(ϕ, I) = K(I) + R(ϕ, I), where K is of Gevrey class Gµ (for a definition of Gevrey classes see Sect. 1), R is analytic with respect to ϕ and Gµ with respect to I, and R and ∇K(I) − ω(I) are flat at the Cantor set Tn × Eκ , Eκ = {I ∈ D : ω(I) ∈ Ωκ }. This normal form can be viewed as a BNF of H around the union Λ of the invariant tori Λω = Tn × {I}, ω = ω(I) ∈ Ωκ . A C ∞ analogue of it has been proved by P¨ oschel [22] (see also [5], Theorem 11.1, and [8], [9],[10], and [29]). Denote by XH the Hamiltonian vector field of H and by exp(tXH ) its flow. The above Gµ normal form of H leads immediately to effective (Nekhoroshev) stability of the action on any Hamiltonian trajectory starting near Λ. Even more, it gives effective stability of the quasiperiodic motion near Λ (see Corollary 1.2). In particular, the KAM tori Λω , ω ∈ Ωκ , trap nearby trajectories over exponentially long time intervals. This leads to a semi-classical concentration near Λ of ”quasi-eigenfunctions” of Ph with exponentially small error terms with respect to 1/h. Indeed, in part II we are going to construct a quasimode of Ph with an exponentially small error term, the Gevrey micro-support of which coincides with Λ. To do this we shall obtain a QBNF of Ph around Λ in suitable Gevrey classes starting from the BNF of its principal symbol H. In the C ∞ case such a QBNF of Ph was obtained by Colin de Verdi`ere [5]. Effective stability for analytic perturbations of completely integrable Hamiltonians was first studied by Nekhoroshev. The Nekhoroshev theorem [19], [20], states that the variation of the action on each orbit of an analytic Hamiltonian Hε remains ε-small in a finite but exponentially long time interval 0 ≤ t ≤ T exp(εa ), T > 0, a > 0, if H0 satisfies certain generic steepness conditions (see also [16], [23]). Gevrey type estimates for the BNF near an elliptic equilibrium point were obtained in [6]. Gevrey BNFs around invariant tori were predicted by Lochak [16]. Effective stability of the billiard flow near the boundary of a strictly convex bounded domain in Rn , n ≥ 2, with analytic boundary was obtained in [7]. A link between Nekhoroshev’s stability for the classical system and the semi-classical

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asymptotics with exponentially small error term of the low lying eigenvalues of the corresponding Schr¨odinger operator is suggested by Sj¨ ostrand [28]. An extension of Nekhoroshev’s theorem in quantum mechanics was proposed by Bellissard and Vittot [2], see also [1]. The paper is organized as follows: The main results are stated in Sect. 1. In Sect. 2 we derive a Gevrey symplectic normal form of H around Λ from a suitable KAM Theorem. In Sect. 3 we prove the existence of a family of the invariant tori Λω , ω ∈ Ωκ , which is Gµ -regular with respect to ω. We follow the classical construction of Moser as it was presented recently by P¨ oschel [21]. Sect. 4 is devoted to a suitable Whitney extension theorem in Gevrey classes which is essential also for the second part of this paper.

I KAM tori and BNF Let us recall some basic properties of Gevrey classes. For each µ ≥ 1, we denote by Gµ (D) the space of all Gevrey functions in a domain D ⊂ Rn of index µ, namely f ∈ Gµ (D) if f ∈ C ∞ (D) and for every compact subset Y of D there exists C = C(Y ) > 0 such that sup |∂Iα f (I)| ≤ C |α|+1 α! µ , ∀ α ∈ Zn+ ,

I∈Y

where Z+ stands for the set of all nonnegative integers and α!µ = (α1 ! . . . αn !)µ , α = (α1 , . . . , αn ). Evidently G1 (D) coincides with the space of all analytic functions in D, while for µ > 1 there are nontrivial compactly supported Gµ functions. Given σ, µ ≥ 1, we say that f ∈ Gσ,µ (Tn × D) if for every compact subset Y of D there exists C = C(Y ) > 0 such that sup (ϕ,I)∈Tn ×Y

|∂ϕβ ∂Iα f (ϕ, I)| ≤ C |α|+|β|+1 β! σ α! µ , ∀ α, β ∈ Zn+ .

(I.1)

The Taylor series of any Gevrey functions f ∈ Gµ (D), µ > 1, has the following property: For each compact set Y ⊂ D there exist η > 0, c > 0 and C1 > 0 depending only on the constant C = C(Y ) such that X f (I0 + r) = fα (I0 ) rα + R(I0 , r) , |α|≤η|r|1/(1−µ)

where fα (I0 ) = ∂ α f (I0 )/α! and − 1 µ−1 β 1+|β| β! µ e−c |r| , 0 < |r| ≤ r0 , ∂I R(I0 , r) ≤ C1

(I.2)

uniformly with respect to I0 ∈ Y (see for example [16], Appendix 2). In particular, if the function f ∈ Gµ (D) is flat at I0 (fα (I0 ) = 0 for each α), then it is exponentially small in a neighborhood of I0 . More generally, let σ ≥ 1, µ > 1 and

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R ∈ Gσ,µ (Tn × D) satisfy (I.1) in a compact Tn × Y . Suppose that R is flat on Tn × E, where E is a compact subset of Y . Then using (I.2), we can find two positive constants C1 and c depending only on the constant C in (I.1) such that for every α, β ∈ Zn+ the following estimate holds |α|+|β|+1

|∂ϕβ ∂Iα R(ϕ, I)| ≤ C1

1 β! σ α! µ exp −c |E − I| − µ−1 ,

(I.3)

/ E, ∀(ϕ, I) ∈ Tn × Y, I ∈ where |E − I| = inf I 0 ∈E |I 0 − I| is the distance to the compact set E. Recall that a smooth function Φ in Tn × D is a generating function of a symplectic map χ of Tn × D into itself, if det (Id − ΦθI ) 6= 0, and χ (ΦI (θ, I), I) = (θ, Φθ (θ, I)) , (θ, I) ∈ Tn × D . Let Θ be a compact subset of Rn and D a bounded domain in Rn . Let H(ϕ, I) be a real analytic Hamiltonian in Tn × D and µ > 1. It will be said that H admits a Gµ -BNF around a family of invariant tori with frequencies in Θ if the following holds: (BF ) There exists a neighborhood Ω of Θ, a Gµ -diffeomorphism ω : D → Ω, and an exact symplectic transformation χ ∈ G1,µ (Tn ×D, Tn ×D) defined by a generating function Φ ∈ G1,µ (Tn × D), such that H(χ(ϕ, I)) = K(I) + R(ϕ, I) in Tn × D, where K ∈ Gµ (D) and R ∈ G1,µ (Tn × D) satisfy DIα R(ϕ, I) = 0 and DIα (∇K(I) − ω(I)) = 0 for any (ϕ, I) ∈ Tn × ω −1 (Θ) and α ∈ Zn+ . Moreover, ||Id − ΦϕI (ϕ, I)|| ≤ ε in Tn × D for some 0 < ε < 1. Here || · || is the usual sup-norm in the space of n × n matrices. Let Ω 3 ω → I(ω) ∈ D be the inverse map to the diffeomorphism I → ω(I) and E = ω −1 (Θ). Then each Λω = χ(Tn × {I(ω)}) , ω ∈ Θ , is an invariant torus of H with a frequency ω, and in view of (I.2) K(I) can be regarded as a simultaneous Birkhoff normal form of H around all the invariant tori Λω . Moreover, R satisfies (I.3) with E = ω −1 (Θ). To explain the ”smallness” condition in the KAM theorem we need some additional notations. Let D0 be a bounded domain in Rn and r0 > 0. Let H 0 be a real analytic Hamiltonian in D0 + r0 = {z ∈ Cn : dist (z, D0 ) ≤ r0 } . Suppose that H 0 is non-degenerate in D0 + r0 . In other words we assume that the 0 Hessian matrix Hzz (z) of H 0 is non-degenerate, i.e. 0 0 −1 Hzz 0 0 , Hzz (I.4) 0 0 ≤ R D +r D +r

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for some R > 0, where | · |D0 +r0 stands for the sup-norm in D0 + r0 . We suppose also that the map D0 3 I −→ ∇H 0 (I) ∈ ∇H 0 (D0 ) = Ω0

(I.5)

is a diffeomorphism. Given r0 > r > 0, s > 0, and a subdomain D ⊂ D0 , we set Tn + s = {z ∈ Cn /2πZn : |= z| ≤ s} , Us,r = Us,r,D = (Tn + s) × (D + r) , the latter being equipped with sup-norm | · |s,r and denote Ω = ∇H 0 (D). Let Ξκ be the set of all frequencies ω ∈ Ω satisfying the Diophantine condition (.7) and also having distance ≥ κ to the boundary of Ω. We define Ωκ to be the set of points of a positive Lebesgue density in Ξκ . In other words, ω ∈ Ωκ if for any neighborhood U of ω in Ω the Lebesgue measure of U ∩ Ξκ is positive. Obviously, Ωκ and Ξκ have the same Lebesgue measure, which is positive if 0 < κ ≤ κ e(D) is sufficiently small. Fix the constant τ > n − 1 in the small divisor condition (.7) and chose τ 0 > max (5/2, τ ) and s > 0. Notice that the condition τ 0 > 5/2 is required only for dimensions n ≤ 3. Fix an integer N ≥ 1. Theorem I.1 Assume that H 0 is real analytic and non-degenerate in D0 + r0 , and the map (I.5) is a diffeomorphism. Let D be a subdomain of D0 . Choose κ e=κ e(D) > 0 so that the Lebesgue measure of Ωκ˜ is positive. Fix κ and r such that 0 < κ ≤ κ e(D) and κ ≤ r ≤ r0 . Then there exists δ > 0 independent of the domain D ⊂ D0 and of the parameters κ and r, such that any real analytic Hamiltonian H on Us,r,D with δH = κ−2 |H − H 0 |s,r ≤ δ def

admits a Gµ -Birkhoff normal form (BF ), where µ = τ 0 + 2, Θ = Ωκ and Ω = ∇H 0 (D). Moreover, for any fixed 0 < q < 1 and N ≥ 1, there exists L > 0 independent of κ, r, δH , and of D such that the function Φ and the diffeomorphism ω in (BF ) satisfy β α Dϕ DI (Φ(ϕ, I) − hϕ, Ii) + DIα (ω(I) − ∇H 0 (I)) q , ∀β ∈ Zn+ , ≤ L|β|+1 κ1−|α| β! δH

for (ϕ, I) ∈ Tn × D and |α| ≤ N . e Denote Eκ = {I ∈ D : ω(I) ∈ Ωκ } and set H(ϕ, I) = H(χ(ϕ, I)), where χ is the symplectic transformation in (BF ). The symplectic normal form (BF ) obtained in Theorem 1.1 leads immediately to effective stability of the quasiperiodic motion around the invariant tori. Indeed, taking σ = 1 and E = Eκ in (I.3) we easily obtain:

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Corollary I.2 For any 0 ≤ p ≤ c and 0 < ε ≤ 1, and for any integral curve exp(tXH e )(ϕ0 , I0 ) = (ϕ(t), I(t)) e with initial data of the Hamiltonian vector field H (ϕ(0), I(0)) = (ϕ0 , I0 ) , |Eκ − I0 | ≤ ε/2 , we have

0 ε exp −p ε−1/(τ +1) , 2 0 |ϕ(t) − ϕ0 − t∇K(I0 )| ≤ Cε exp −(2p − c) ε−1/(τ +1) , |I(t) − I0 | ≤

for some C > 0 provided that

e ε exp (c − p) ε−1/(τ 0 +1) , |t| ≤ C

e = (2√nC 2 )−1 and c and C1 are the constants in (I.3). where C 1 Consider for example an elliptic equilibrium at some %0 = (x0 , ξ0 ), and denote by ±iα10 , . . . , ±iαn0 the corresponding characteristic exponents which are purely imaginary. To apply Theorem 1.1 we fix some N ≥ 4 and exclude the resonances of order ≤ N , i.e. we assume that (.3) holds. Consider the BNF H 0 of H given by (.5) near %0 and fix D0 = {I ∈ Rn : |Ij | < C1 a0 , j = 1, . . . , n}, where 0 < a0 1 and C1 > 1. We suppose that H 0 is nondegenerate in D0 + a0 . For each 0 < a ≤ a0 set D = Da = {I ∈ Γ : C1−1 a ≤ Ij ≤ C1 a, j = 1, . . . , n}. Next we choose κ = κa = εa, where 0 < ε 1 is fixed, then fix s > 0 and take r = a. The perturbation H1 satisfies |H1 |s,r ≤ κ2 C2 a(N−3)/2 ≤ κ2 δ,

(I.6)

for each 0 < a ≤ a0 choosing a0 small enough. Hence, applying Theorem 1.1 we obtain: Corollary I.3 Let %0 be an elliptic equilibrium of a real analytic Hamiltonian H. Suppose that (.3) is satisfied, and denote by H(ϕ, I) = H 0 (I) + H1 (ϕ, I) the corresponding BNF. Let det Q 6= 0. Then for each 0 < a ≤ a0 , a0 sufficiently small, there exists a symplectic diffeomorphism χ0 of Gevrey class G1,µ , µ = τ 0 + 2, mapping Tn × D into itself, a Gµ -diffeomorphism ω : D → Ω, and K ∈ Gµ (D) e e = H ◦ χ0 ∈ G1,µ (Tn × D) has the form H(ϕ, I) = such that the Hamiltonian H n K(I) + R(ϕ, I), where R and ω(I)−∇K(I) are flat on T ×Eκ and Eκ = ω−1 (Ωκ ) respectively. The symplectic map χ0 has a generating function Φ ∈ G1,µ (Tn × D) and for each |α| ≤ (N − 1)/2 and 0 < q < 1, we have β α Dϕ DI (Φ(ϕ,I) − hϕ,Ii) + DIα (ω(I) − ∇H 0 (I)) ≤ C |α|+|β|+1 aq(N−1)/2−|α| β!, ∀β ∈ Zn+ , for any (ϕ, I) ∈ Tn × D, where C does not depend on a.

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II Proof of the BNF The inverse map ψ0 : Ω0 → D0 of the frequency map ∇H 0 : D0 → Ω0 is given by ψ0 (ω) = ∇g(ω), where g is the Legendre transform of H 0 defined by g(ω) = supI∈D0 (hI, ωi − H 0 (I)). Given a subdomain D ⊂ D0 we set Ω = ∇H 0 (D). Fix s > 0 and r0 ≥ r > 0, and for given τ 0 > max (5/2, τ ) denote µ = τ 0 + 2. We shall deduce Theorem 1.1 from the following: Theorem II.1 Suppose that H 0 and H satisfy the assumptions of Theorem 1.1. Then there exists a constant δ > 0 independent of κ, r and of the domain D ⊂ D0 , and for each real analytic Hamiltonian H in Us,r,D with δH ≤ δ, there is a map f : Tn × Ω → D of Gevrey class G1,µ , such that any torus Λω = {(θ, f (θ, ω)) : θ ∈ Tn } , ω ∈ Ωκ , is invariant with respect to XH and the restriction of exp(tXH ) to Λω is conjugated to the linear flow on Tn with frequency ω. Moreover, for any integer N ≥ 1 and 0 < q < 1, there exists C > 0 independent of κ, r, δH , and of D, such that F (θ, ω) = f (θ, ω) − ψ0 (ω) satisfies β α q (II.1) Dθ Dω F (θ, ω) ≤ C |β|+1 κ1−|α| β! δH , ∀ (θ, ω) ∈ Tn × Ω , ∀ β ∈ Zn+ , |α| ≤ N . Recall that Ωκ is of a positive Lebesgue measure if κ is sufficiently small. Theorem 2.1 will be proved in Sect. 3. Moreover, for any 0 < q < 1 we shall obtain the following Gevrey type estimates : β α e |α+β|+1 κ1−|α| β! α! µ δ q , (II.2) Dθ Dω F (θ, ω) ≤ C H ∀ (θ, ω) ∈ Tn × Ωκ , ∀ α, β ∈ Zn+ , where C˜ is independent of κ, r, δH , and D. Proof of Theorem 1.1. Fix an integer N ≥ 1. Recall that each torus Λω , ω ∈ Ωκ , is a Lagrangian submanifold of T ∗ (Tn ) (see also [8], Sect. I.3.2). Denote by p : Rn → Tn the natural projection. Lemma II.2 There exist a function ψ ∈ G1,µ (Rn × Ω) and a map R ∈ Gµ (Ω; Ω) such that (i) ∇x ψ(x, ω) = f (p(x), ω)), for each (x, ω) ∈ Rn × Ωκ , (ii) The function Q(x, ω) = ψ(x, ω) − hx, R(ω)i is 2π-periodic in x, and for each 0 < q < 1, there exists a constant C independent of κ, r, δH , and of Ω = ∇H 0 (D), such that for each |α| ≤ N we have β α Dx Dω Q(x, ω) + |Dωα (R(ω) − ψ0 (ω))| ≤ C |β|+1 κ1−|α| β! δ q , ∀β ∈ Zn+ , (II.3) H for any (x, ω) ∈ Rn × Ω.

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First we shall find a function ψe of Gevrey class G1,µ (Rn × Ωκ ) such that e ω) = f (p(x), ω), ∀ (x, ω) ∈ Rn × Ωκ ∇x ψ(x,

(II.4)

(for a definition of Gevrey classes on Rn × Ωκ see Sect. 4). To this end we set γx = {(tx, f (p(tx), ω)) : 0 ≤ t ≤ 1}, and consider the action on it e ω) = ψ(x,

Z

Z

1

hf (p(tx), ω), xi dt,

σ = γx

0

where σ = ξdx is the canonical one-form on T ∗ (Rn ). Obviously, ψe ∈ G1,µ (Rn ×Ωκ ) in the sense of Whitney. Moreover, as Λω , ω ∈ Ωκ , is a Lagrangian torus, we get Z e e σ , ∀ y ∈ Rn , ψ(x + y, ω) − ψ(x, ω) = l(x,y)

where l(x,y) = {(x + ty,f (p(x + ty),ω)) : 0 ≤ t ≤ 1}. The integral above is equal to Z 1 hf (p(x + ty),ω),yidt = hf (p(x),ω),yi + O(y 2 ), 0

e which implies (II.4). In particular, ∇x ψ(x,ω) is 2π-periodic in x for each ω ∈ Ωκ , which implies e + 2πm,ω) − ψ(x,ω) e e ψ(x = h2πm, R(ω)i , m ∈ Zn , ω ∈ Ωκ , e can be written as follows where the components of R e ej (ω) = ψ(2πe 2π R j ,ω), {ej } being an unit basis in Rn . Then the function e e e Q(x,ω) = ψ(x,ω) − hx, R(ω)i, (x,ω) ∈ Rn × Ωκ , e ∈ G1,µ (Rn × Ωκ ) in the Whitney sense. is 2π-periodic in x ∈ Rn , and we have Q n e e Moreover, using (II.1) it is easy to see that R−ψ 0 and Q satisfy (II.3) on T ×Ωκ . Now, we apply Proposition 4.2 to the functions q −1 e q −1 e (κδH ) Q(θ,κω) , (κδH ) (R(κω) − ψ0 (κω)), (θ,ω) ∈ Tn × (κ−1 Ωκ ).

In this way we obtain Q ∈ G1,µ (Tn × Ω) and R ∈ Gµ (Ω) such that e e Q(x,ω) = Q(p(x),ω), R(ω) = R(ω), ∀(x,ω) ∈ Rn × Ωκ , and (II.3) holds. Then ψ(x,ω) = Q(p(x),ω) + hx,R(ω)i is the desired function.

2

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In particular, choosing δ sufficiently small, we obtain a diffeomorphism Ω 3 ω → ∇x ψ(x,ω) ∈ D for any x fixed. In this way we extend the family Λω , ω ∈ Ωκ , to a G1,µ -foliation of Lagrangian tori Λω = {(p(x),∇x ψ(x,ω)) : x ∈ Rn } , ω ∈ Ω, of Tn × D. We define the action I = (I1 ,...,In ) of Λω by Z −1 σ , j = 1,...,n, Ij (ω) = (2π) γj (ω)

where γj (ω) = {(p(tej ), ∇x ψ(tej ,ω)) : 0 ≤ t ≤ 1}. Now, we have Ij (ω) = (2π)−1 (ψ(2πej ,ω) − ψ(0,ω)) = Rj (ω),

(II.5)

where I(ω) − ψ0 (ω) is estimated by (II.3). Hence, we can suppose that the ”frequency” map Ω 3 ω → I(ω) ∈ D is a Gµ -diffeomorphism and we define the map D 3 I → ω(I) ∈ Ω as its inverse. Then ω is a Gµ -map as well. Denote by Eκ = I(Ωκ ) the image of Ωκ via I. Notice that for any |α| ≤ N we have α DI (ω(I) − ∇H 0 (I)) ≤ Lκ1−|α| δ q , I ∈ D, H where L does not depend on κ, r, δH and D. We set Φ(x,I) = ψ(x,ω(I)) , (x,I) ∈ Rn × D. Then Φ ∈ G1,µ (Rn × D) (the class Gµ is invariant under composition [25], Appendix), and using (II.5) we obtain Φ(x,I) = hx,Ii + φ(p(x),I), where φ(x,I) = Q(x,ω(I)) is 2π-periodic in x in view of Lemma 2.2, (ii). Then φ ∈ G1,µ (Tn × D), and for each |α| ≤ N we have β α |β|+1 1−|α| q κ β!δH , (θ,I) ∈ Tn × D, (II.6) Dθ DI φ(θ,I) ≤ L0 where L0 is independent of κ, r, δH and D. We have obtained the desired estimates for Φ and its derivatives. Next we observe that for δH small enough, Φ is a generating function of an exact symplectic transformation χ : Tn × D → Tn × D of Gevrey class G1,µ , defined by χ (∇I Φ(θ,I),I) = (θ,∇θ Φ(θ,I)). Indeed, we have χ(ϕ,I) = (y(ϕ,I),η(ϕ,I)), where θ = y(ϕ,I) solves the equation ϕ = (∇θ Φ)(θ,I).

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Using (II.6) we get an unique solution θ which belongs to G1,µ and such that |θ−ϕ| < π (see [13] or [25], Appendix). Moreover, η(ϕ,I) = (∇θ Φ)(y(ϕ,I),I) belongs to G1,µ as well. On the other hand, using Lemma 2.2, (i), we obtain that χ−1 (Λω ), ω ∈ Ωκ , coincides with the torus Tn × {I(ω)}. Set e e I Φ(0,I),I) = H(0,∇θ Φ(0,I)), H(ϕ,I) = H(χ(ϕ,I)), K(I) = H(∇ Eκ = {I(ω) : ω ∈ Ωκ }. e Since H is constant on each invariant torus Λω , ω ∈ Ωκ , we get H(ϕ,I) = K(I) on e − K(I) is flat at Tn × Eκ since any I ∈ Eκ is a point of a Tn × Eκ . Then H(ϕ,I) positive Lebesgue density in Eκ (see [5] and [24], Lemma 2.1). In particular, the frequency of Λω(I) is ∇K(I). By Theorem 2.1 the frequency on Λω(I) equals ω(I) for I ∈ Eκ . Hence, ∇K(I) = ω(I), I ∈ Eκ , and we obtain that ∇K(I) − ω(I) is flat at Eκ . The proof of Theorem 1.1 is complete. Moreover, using (II.2) one can prove 2 Gevrey type estimates for φ(ϕ,I) and for ω(I) − ∇H 0 (I) on Tn × Eκ .

III Gevrey regular families of KAM tori To prove Theorem 2.1 we shall follow the classical approach of Moser [18] as it was presented by P¨ oschel [21]. We shall deduce Theorem 2.1 from another KAM theorem concerned with perturbations of a family of linear Hamiltonians where the frequencies are taken as independent parameters. One of the advantages of this approach is that it makes comparatively easy to check the regularity of the corresponding family of invariant tori with respect to the frequencies. Consider the Hamiltonian H(θ,z) = H 0 (z) + F (θ,z). Expanding H 0 (z) near given z0 ∈ D ⊂ D0 , we can write Z H (z) = H (z0 ) + h∇z H (z0 ),Ii + 0

0

1

(1 − t)h∇2z H 0 (zt )I,Iidt,

0

0

where zt = z0 +tI, I varies in a small ball B in Rn centered at the origin, and ∇2z H 0 stands for the Hessian matrix of H 0 . Let us recall that Ω0 3 ω → ψ0 (ω) = ∇g(ω) is the inverse to the frequency map D0 3 z → ∇H 0 (z), where g(ω) is the Legendre transform of H 0 (z). Setting ω = ∇H 0 (z0 ), we get z0 = ∇g(ω) and we can write H 0 (z) = e(ω) + hω,Ii + PH 0 (I;ω), F (θ,r) = F (θ,∇g(ω) + I) = PF (θ,I;ω),

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where e(ω) = H 0 (∇g(ω)) while PH 0 stands for the quadratic term in I in the expression of H 0 . In this way we obtain a family of Hamiltonians H(θ,I;ω) = N (I;ω) + P (θ,I;ω) in Tn ×B depending on the frequency ω ∈ Ω. Here, N (I;ω) = e(ω)+hω,Ii is a linear normal form with respect to I which admits the invariant torus T0 = Tn × {0} for any ω ∈ Ω, while P (θ,I;ω) = PH 0 (θ,I;ω)+PF (θ,I;ω) is the perturbation. The idea is to find for each τ 0 > max(5/2,τ ) a family of symplectic transformations F(θ,I;ω) 0 which are analytic with respect to (θ,I), and of Gevrey regularity Gτ +2 with respect to the frequency parameter ω ∈ Ω, and such that the Hamiltonian H(θ,I;ω) is transformed by F to a linear normal form with a quadratic in I remainder term for any fixed ω ∈ Ωκ . As in [21], we are looking for a transformation of the form F(θ,I;ω) = (Φ(θ,I;ω),φ(ω)), Φ(θ,I;ω) = (U (θ;ω),V (θ,I;ω)), 0

having Gevrey regularity G1,τ +2 (analytic in (θ,I) and Gτ Φ(θ,I;ω) is a linear form with respect to I, and the map

0

+2

(III.1)

in ω), such that

F(.,.;ω) : T ∗ (Tn ) → T ∗ (Tn ) is symplectic for any fixed ω ∈ Ω. The transformations F form a subgroup of the group of diffeomorphisms in Tn × B × Ω which will be denoted by G. To state the main result we introduce the complex domains Ds,r = {θ ∈ Cn /2πZn : |=θ| < s} × {I ∈ Cn : |I| < r}, Oh = {ω ∈ Cn : |ω − Ωκ | < h}. The sup-norm of functions in Ds,r × Oh will be denoted by | · |s,r,h . Set ν = τ + n + 1 and recall that τ 0 > max(5/2,τ ). We have the following Gevrey analogue of Theorem A, [21]: Theorem III.1 Let the Hamiltonian H(θ,I;ω) = N (ω) + P (θ,I;ω) be real analytic on Ds,r × Oh , where κsν ≤ h and 0 < r,s,h < 1. Then there exists c > 0 depending only on n, τ , τ 0 , s, such that if the perturbation P satisfies the inequality |P |s,r,h ≤ cκr, then the following holds: There exists F : Ds/2,r/2 × Ω −→ Ds,r × Ω of the form (III.1) such that F ∈ G, F belongs to the Gevrey class G1,τ Ds/2,r/2 × Ω, and H ◦ F(θ,I;ω) = ee(ω) + hω,Ii + Pe(θ,I;ω),

0

+2

on

236

where

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DIα Pe(θ,0;ω) = 0, ∀(θ,ω) ∈ Tn × Ωκ ,

for each |α| ≤ 1. Moreover, for any 0 < q < 1 there exists Cq > 0 such that s−1 |Dωα (U (θ;ω) − θ)| + r−1 |Dωα (V (θ,I;ω) − I)| + κ−1 |Dωα (φ(ω) − ω)| ≤ Cq|α|+1 κ−|α| (α!)τ

0

−1

uniformly on Ds/2,r/2 × Ωκ , where δP = (κr) n, τ, τ 0 , and q.

+2 q δP

,

|P |s,r,h , and Cq depends only on

Proof of Theorem 2.1. We write as above H = N + P , where P = PH 0 + PF is real analytic on Dr,s × Oh with κsν ≤ h, r ≥ κ. Given r0 ≥ r ≥ % > 0 we obtain |PH 0 |s,%,h ≤ cR %2 , where cR ≥ 1 depends on the constant R in (I.4), but not on κ, r, and D. We suppose also that 2cR ≥ c, where c is the small constant in Theorem 3.1, which does not depend on κ, r, and D either. Fix δ > 0 by δ 1/2 = c(2cR )−1 and set % = κδ 1/2 ≤ r. Then we have |P |s,%,h ≤ |PH 0 |s,%,h + |PF |s,r,h ≤ cR (%2 + κ2 δ) = cκ%. Hence, we can apply Theorem 3.1. Set Ψ(θ,ω) = (U (θ;ω),W (θ,ω)), where W (θ,ω) = (∇g)(φ(ω)) + V (θ,0;ω). Then R 3 t −→ Ψ(θ + tω,ω),

(θ,ω) ∈ Tn × Ωκ ,

is an integral curve of the Hamiltonian vector field XH . Moreover, Ψ : Tn × Ω → Tn × D , Ψ(θ,ω) = (U (θ,ω),W (θ,ω)), e > 0 independent is of Gevrey class G1,µ . Moreover, for any 0 < q < 1 there exists C of κ, δH , r, and D, such that for any multi-indices α and β we have β α β Dθ Dω (U (θ,ω) − θ) + κ−1 Dθ Dωα (W (θ,ω) − ψ0 (ω)) e |α|+|β|+1 κ−|α| β!(α!)µ δ q , ≤C H for each (θ,ω) ∈ Tn × Ωκ , since % ≤ κ. Fix N ≥ 1. Using Proposition 4.2 we extend U and W on Tn × Ω so that

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β α β Dθ Dω (U (θ,ω) − θ) + κ−1 Dθ Dωα (W (θ,ω) − ψ0 (ω)) q ≤ C |β|+1 κ−|α| β!δH ,∀β ∈ Zn+ ,

for each (θ,ω) ∈ Tn × Ω and |α| ≤ N , where C > 0 is independent of κ, δH , r, and D. In particular, for any δH ≤ δ sufficiently small, the map Tn 3 θ → U (θ,ω) ∈ Tn is a diffeomorphism close to the identity for each ω ∈ Ω. Using the implicit function theorem ([13], [25], Appendix A.2), we obtain f on Tn × Ω and prove (II.1). 2 To prove Theorem 3.1 we first rescale κ to 2 multiplying ω, h, N and P by 2/κ, and then we denote by Ω∗ the set of all ω ∈ Ω such that |ω − Rn \ Ω| ≥ 2, and (.7) holds with κ = 2. We set Oh = {ω ∈ Cn : |ω − Ω∗ | < h}, with h already rescaled. As in [21] we are going to use the following notations: if u < cv with some positive constant depending only on n and τ , then we write u < ·v, and if c < 1 we write u· < v. Given σ,r > 0 we set ! σ −1 Id 0 , W= 0 r−1 Id where Id is the identity matrix. We keep the same notation W for the corresponding linear map in Cn × Cn , and denote by id the identity map. To prove Theorem 3.1 we use the KAM step established by P¨ oschel [21]: Proposition III.2 [21]. Suppose that |P |s,r,h ≤ ε with ε· < ηrσ ν , ε· < hr, h ≤ K −τ −1 ,

(III.2)

0 < s,r < 1, 0 < η < 1/8, 0 < σ < s/5, K ≥ 1.

(III.3)

for some Then there exists a real analytic transformation F = (Φ,φ) : Ds−5σ,ηr × Oh/4 −→ Ds,r × Oh in the group G such that H ◦ F = N+ + P+ with N+ (I;ω) = e+ (ω) + hω,Ii, and |P+ |s−5σ,ηr,h/4 < ·

ε2 + (η2 + K n e−Kσ )ε, rσ ν

|N+ − N ◦ φ|s,ηr,h/4 < ·ε. Moreover, |W (Φ − id)|, |W (DΦ − Id)W −1 | < · ε |φ − id|, h|Dφ − Id| < · , r uniformly on Ds−5σ,ηr × Oh/4 and Oh/4 , respectively.

ε , rσ ν

(III.4)

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As in [18] and [21] we are going to iterate the KAM step infinitely many times choosing appropriately the parameters 0< s,r,σ,h,η < 1 and so on. Our goal 0 is to get a convergent scheme in the Gevrey spaces G1,τ +2 . Assume that s,r and σ are chosen as in (III.3). Next we choose a “weighted error” 0 < E < 1 and set η = E 1/2 , ε = Erσ ν . Now (III.3) requires 0 < E < 1/64. We define K and h by K n e−Kσ = LE, h = K −τ −1 , where L = ([ν] + 2)! and [ν] is the integer part of ν = τ + n + 1. Setting x = Kσ we get the equation (III.5) xn e−x = LEσ n , which has an unique solution with respect to x ∈ [n + 1,+∞) provided that 0 < E < min{

1 (n + 1)n e−n−1 , }. 64 L

(III.6)

Then K = xσ−1 > 1, and we have ([ν] + 1)! ε h ν −x = x e ≤h · < h. r L L Hence (III.2) is satisfied and we can apply the KAM step. We set r+ = ηr, s+ = s − 5σ, σ+ = δσ,

2 < δ < 1. 3

Now the KAM step gives the estimate |P+ |s+ ,r+ ,h/4 < ·rσ ν (E 2 + (η2 + K n e−Kσ )E) ν 3/2 = (2 + L)rσ ν E 2 < ·r+ σ+ E .

Hence there is a constant c1 > L > 1 depending only on n and τ such that 1/2

ν 3/2 E . |P+ |s+ ,r+ ,h/4 ≤ c1 r+ σ+ 1/2

ν We fix the weighted error for the iteration as E+ = c1 E 3/2 , set ε+ = E+ r+ σ+ , 3/2 and then define η+ , x+ , K+ , and h+ as above. Notice that, c1 E+ = (c1 E) . We require also c1 E < 1 which leads to an exponentially converging scheme. Then E+ < E and x+ > x ≥ n + 1. Suppose that h+ ≤ h/4. Then we obtain

|P+ |s+ ,r+ ,h+ ≤ ε+ .

(III.7)

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Consider now the normal form N+ (I;ω) = e+ (ω) + hω,Ii given by the KAM step. We have ε |e+ − e|h+ < · (1 + |e|h ) , |e+ |h+ < · ε + |e|h . (III.8) rh Indeed, using the KAM step and the Cauchy theorem we obtain |e+ − e|h+ ≤ |N+ − N ◦ φ|r+ ,h/4 + r|φ − id|h/4 + |e ◦ φ − e|h/4 < · (1 + |e|h )

ε . rh

On the other hand, using the relation φ(Oh/4 ) ⊂ Oh we obtain the second inequality in (III.8). It remains to check the inequality h+ ≤ h/4 choosing appropriately δ = σ+ /σ, and taking into account the assumption τ 0 ≥ 5/2. Lemma III.3 Let max(5/2,τ ) < τ 0 < 2τ +1. Then there exist constants C(n) ≥ n+1 and 0 < δ < 1 such that for any 0 < σ < 1, and any x satisfying x > C(n), xτ we have

0

−τ

στ +1 > 1,

0

τ +1 > 1. x+ > C(n), xτ+ −τ σ+

Moreover, 1 h, 4 where d1 > d0 > 1 and c0 > 0 depend only on n,τ,τ 0 and c1 . d0 x ≤ x+ ≤ d1 x and c0 h ≤ h+ ≤

Proof. We fix 0 < q < 1/2 such that (3/2 − q)7 > 16 and then define

3 δ= −q 2

0 ττ−τ +1

.

Obviously, 2/3 < δ < 1. Consider the function g(t) = t−nlogt. Set d = − 21 log(c1 /L), and fix C(n) ≥ n + 1 such that nlogt < qt − |d|

(III.9)

for each t > C(n). We have n xn+ e−x+ = Lσ+ E+ = δ n

c 1/2 1

L

σ−n/2 (xn e−x )3/2 .

Hence

1 1 g(x+ ) − g(x) = (g(x) + nlogσ) + nlog + d. (III.10) 2 δ Since τ 0 −τ ≤ τ +1 and x ≥ C(n) > 1, we get xσ ≥ 1 from the inductive assumption. Now, (III.9) implies g(x) + nlogσ ≥ x − 2nlogx ≥ (1 − 2q)x + 2|d|.

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Using x+ > x ≥ C(n) we obtain x+ ≥ x + g(x+ ) − g(x) ≥ (3/2 − q)x, and we set d0 = 3/2 − q > 1. Since τ 0 > 5/2 we get τ +1 τ +1 −τ 0 −1 −7/2 h+ 1 x K 3 3 τ +1 = −q −q =δ < < < . h K+ x+ 2 2 4 It is easy to see that 1 (x+ − x) n+1 since x+ > x ≥ n + 1. On the other hand, g(x) + nlogσ < x. Using (III.10) we find d1 > d0 such that x+ ≤ d1 x for x ≥ C(n), and as above we obtain h+ ≥ c0 h, where c0 depends only on n,τ,τ 0 ,c1 . Moreover, we have g(x+ ) − g(x) ≥

0

τ +1 xτ+ −τ σ+ ≥

3 −q 2

τ 0 −τ δ τ +1 xτ

0

−τ

σ τ +1 = xτ

0

−τ

στ +1 > 1, 2

and the proof of the Lemma is complete. We are ready to set up our parameters. Let σj+1 = δσj , sj+1 = sj − 5σj , with σ0 = s0 /20 < c0 (n) < 1. Let 1/2

c1 Ej+1 = (c1 Ej )3/2 , ηj = Ej , rj+1 = ηj rj . Suppose that E0 ≤ γ0 , where

γ0 < min (n + 1)n e−n−1 L−1 , 1/64,c−1 , L = ([ν] + 2)!. 1

Then c1 E0 < 1, and for any j ∈ Z+ there exists an unique xj ≥ n + 1 such that xnj e−xj = LEj σjn < (n + 1)n e−n−1 . We set

Kj = xj σj−1 , hj = Kj−τ −1 , εj = Ej rj σjν , j ∈ Z+ .

Choosing γ0 sufficiently small as a function of s0 , n, τ and τ 0 we can assume that (τ +1)/(τ −τ 0 ) x0 > min C(n), σ0 , where C(n) ≥ n + 1 is given in Lemma 3.3. Now Lemma 3.3 yields for any j ∈ Z+ the inequalities 1 d0 xj ≤ xj+1 ≤ d1 xj , c0 hj ≤ hj+1 ≤ hj , 4

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xτj

0

−τ

σjτ +1 > 1.

(III.11)

The last inequality will be used to gain Gevrey regularity with respect to ω. It shows as well that εj /rj hj < Cσjν and hj < σjν , where C depends only on n,τ and τ 0 . Indeed, we have n(2τ −τ 0 +1)/(τ 0 −τ )

hj σj−ν < σj

1 ν −xj < Cσjν x e L j

< 1 , εj /rj hj = Ej σjν /hj =

with some C large enough. We can formulate now the iterative lemma. Set Dj = {θ ∈ Cn /2πZn : |=θ| < sj } × {I ∈ Cn : |I| < rj }, ej = {ω ∈ Cn : |ω − Ω∗ | < hj /2}, Oj = {ω ∈ Cn : |ω − Ω∗ | < hj } , O and put



σ0−1 Id 0

 W0 =   0 0

 0

r0−1 Id 0 0

h−1 0 Id

 . 

We are going to formulate a Gevrey analogue of the Iterative Lemma in [21]. Proposition III.4 Assume max(5/2,τ ) < τ 0 < 2τ + 1 and suppose that P0 is real analytic on D0 × O0 with |P0 |s0 ,r0 ,h0 = ε0 ≤ γ0 r0 σ0ν , where γ0 is sufficiently small, depending only on s0 , n, τ, and τ 0 . Then for each j ≥ 0 there exists a normal form Nj (I;ω) = ej (ω)+ hω,Ii and a real analytic transformation F j defined by F 0 = Id, and for j ≥ 1 by F j = F0 ◦ ···◦ Fj−1 : Dj × Oj −→ D0 × O0 , F j (θ,I;ω) = (uj (θ;ω),v j (θ,I;ω),φj (ω)), which belongs to the group G and such that H ◦ F j = Nj + Pj with |Pj |sj ,rj ,hj ≤ εj = Ej rj σjν . Moreover, |ej+1 − ej | < · (1 + |e|h0 )εj /rj hj , and |W0 (F j+1 − F j )|, |T F j+1 − T F j ◦ Fj | < ·

εj rj hj

(III.12)

in Dj+1 × Oj+1 , and for each 0 < q < 1 there exists a constant Cq > 0 depending only on n,τ,τ 0 , and q such that 1−|β|

|W0

Dωα DIβ (F j+1 − F j )| ≤ Cq|α|+1 (α!)τ

0

+2

E0q x−1 j ,

(III.13)

ej+1 , for any j ≥ 0, α ∈ Zn+ , and each β with 0 ≤ |β| ≤ 1. uniformly on Dj+1 × O

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Proof. We proceed as in [21]. Set F 0 = Id and define Fj , j ≥ 0, successively by Proposition 3.2. We have already checked the assumptions of the KAM step for each j ≥ 0. Thus we obtain for any j ≥ 0 a transformation Fj : Dj+1 × Oj+1 −→ Dj × Oj taking Hj = Nj + Pj into Hj+1 = Hj ◦ Fj = Nj+1 + Pj+1 , while (III.7) and (III.8) imply |Pj+1 | ≤ εj+1 and |ej+1 − ej | < ·(1 + |e|h0 )εj /rj hj . Moreover, the KAM step yields |φj − id| < ·εj /rj . Notice that F j+1 = F0 ◦ ··· ◦ Fj takes H to Hj+1 = Nj+1 + Pj+1 . The estimate (III.12) follows from the arguments in [21]. Let us evaluate Jjα = W0 Dωα (F j+1 − F j ) ej+1 for any j ≥ 0 and |α| ≥ 1. Recall that (III.12) is valid in Dj+1 ×Oj+1 . in Dj+1 × O Then Cauchy’s estimate leads to Jjα < ·

2|α| α!εj |α|

=

rj hj hj+1

2|α| α!Ej σjν |α|

.

hj hj+1

According to (III.11) we have hj+1 ≥ c0 hj , hence Jjα < ·

2 c0

|α|

α!Ej σjν |α|+1

.

hj

Recall that hj = Kj−τ −1 = (σj−1 xj )−τ −1 and Ej = L−1 (σj−1 xj )n e−xj . Then for any fixed 0 < q < 1 we obtain Jjα < ·

2 c0

|α|

α!xνj (σj−1 xj )(τ +1)|α|−nq e−(1−q)xj Ejq . (τ 0 +1)/(τ +1)

Taking into account (III.11) we get 1 < xj σj−1 < xj E0 for j ≥ 0, we obtain Jj,α < ·

2 c0

|α|

ν+(τ 0 +1)|α|

e−(1−q)xj xj

< Cq|α|+1 (α!)τ

0

and using that Ej ≤

α! E0q

+2 −1 q xj E0 ,

where Cq depends only on n, τ , τ 0 , and q. In the same way we prove (III.13) when |β| = 1. 2 To prove Theorem 3.1 we apply the Iterative lemma to H = N + P as in [21], letting P0 = P , r0 = r, s0 = s, h0 = h. Set |P0 |s0 ,r0 ,h0 = ε0 ≤ γ0 r0 σ0ν . The linear (in I) maps F j and their derivatives T F j with respect to I converge uniformly to certain mappings F∗ and T F∗ in D∗ × Ω∗ , D∗ = 0 × {|=θ| < s/2}. In this way we obtain a

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linear in I map F = F∗ +(T F∗ )I in Ds/2,r/2 ×Ω∗ . The corresponding normal forms e (I;ω) = ee(ω) + hI,ωi, and we have H e =H ◦F =N e + Pe, where Nj converge to N |Pe|s0 /2,0,0 = |DI Pe|s0 /2,0,0 = 0. Moreover, 0 |W0 Dωα (F∗ − Id)| , |Dωα (T F∗ − Id)| ≤ Cq|α|+1 (α!)τ +2

ε0 r0 h0

q ,

P −1 in Ds0 /2,0 × Ω∗ since xj < ∞ in view of (III.11). The Theorem is proved for κ = 2. Finally, we scale back ω to κω/2 and multiply h0 , P0 and ε0 by κ/2 in order to complete the proof of Theorem 3.1. 2

IV On Whitney’s extension theorem in Gevrey classes Let E be a compact subset in Rn . To simplify the notations about Whitney jets we suppose that each I ∈ E is a point of a positive Lebesgue density of E (see Sect. 1). Then each f ∈ C ∞ (Rn ) with f = 0 on E is a flat function on E (see [24], Lemma 2.1). Moreover, the usual definition of differentiation can be extended on E. For each multi-index α we denote by ∂Iα f (I) the corresponding partial derivative (if it exists) of a given function f on E. We say that f ∈ Gµ (E) in the sense of Whitney if f ∈ C ∞ (E) and |∂Iα f (I)| ≤ C |α|+1 (α!)µ , I ∈ E , for each multi-index α. Let U be a compact neighborhood of E in Rn and σ ≥ 1 and µ > 1. It is known that there may not exist a continuous linear extension operator from Gµ (E) to Gµ (U ). Nevertheless, the restriction map % : Gµ (U ) → Gµ (E) is surjective [3], and using an argument of Bruna [4] we obtain for each bounded set V1 in Gµ (E) a bounded set V2 in Gµ (U ) such that %(V2 ) = V1 . This leads to the following result which will be essential for the construction of QBNFs in the second part of this paper. Theorem IV.1 Let F be a set of smooth functions on Tn × E in the sense of Whitney with the following property: There exists C > 0 such that β α ∂ϕ ∂I f (ϕ,I) ≤ C |α+β|+1 β! σ α! µ , (ϕ,I) ∈ Tn × E , (IV.1) for each f ∈ F and any multi-indices α and β. Then there exists C0 > 0 and a map j : F → Gσ,µ (Tn × U ) such that for each f ∈ F we have ∂Iα j(f )(ϕ,I) = ∂Iα f (ϕ,I) , ∀(ϕ,I) ∈ Tn × E , ∀α ∈ Zn+ , β α ∂ϕ ∂I j(f )(ϕ,I) ≤ C |α+β|+1 β! σ α! µ , ∀(ϕ,I) ∈ Tn × U , ∀α,β ∈ Zn+ . 0

(IV.2) (IV.3)

244

Proof.

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For each f ∈ F consider the Fourier coefficients Z −n fk (I) = (2π) e−ihk,ϕi f (ϕ,I)dϕ , k ∈ Zn . Tn

Using (IV.1) we obtain for some r = r(n,σ,C) > 0 er|k|

1/σ

|∂Iα fk (I)| ≤ 2C |α|+1 α! µ , I ∈ E

for any α,k and f . Then gf k (I) = er|k|

1/σ

fk (I), I ∈ E , k ∈ Zn , f ∈ F ,

is a bounded family in Gµ (E). On the other hand, the restriction map of Gµ (U ) to Gµ (E) is surjective (see [3], Theorem 3.9, and Remark 3.13). Then an argument of Bruna ([4], Lemma 3.4) holds, and we obtain a family of functions Gf k ∈ Gµ (U ) such that ∂Iα Gf k (I) = ∂Iα gf k (I) , ∀I ∈ E , f ∈ F , k ∈ Zn , α ∈ Zn+ , and

|α|+1

|∂Iα Gf k (I)| ≤ C2

α! µ , ∀I ∈ U , f ∈ F , k ∈ Zn , α ∈ Zn+ ,

with some C2 > 0. Set j(f )(ϕ,I) =

X

eihk,ϕi−r|k|

1/σ

Gf k (I).

(IV.4)

k∈Zn

Obviously (IV.2) is satisfied. Moreover, it is easy to see that (IV.3) holds with a suitable C0 > 0. 2 Unfortunately we have no control on the constant C0 in (IV.3). In particular, C0 may depend on the compact set E. Proposition IV.2 Let f be a G1,µ function on Tn × E in the sense of Whitney such that (IV.1) holds with σ = 1. Then for each integer N ≥ 1 there exists F ∈ G1,µ (T ∗ (Tn )) and a positive constant L depending only on µ, N , the dimension n, and on the constant C in (IV.1) but not on the set E such that ∂Iα F (ϕ,I) = ∂Iα f (ϕ,I) , (ϕ,I) ∈ Tn × E , for each multi-index α, and β α ∂ϕ ∂I F (ϕ,I) ≤ L|β|+1 β!, (ϕ,I) ∈ T ∗ (Tn ), for each multi-index β and any α with |α| ≤ N .

(IV.5)

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Proof. Denote by U a ball in Rn containing E in its interior and such that the distance between E and the boundary of U is ≥ 2. Consider the Whitney extension g = j(f ) of f in G1,µ ((Tn +r0 )×U ) given by (IV.4), where r0 = r/2 and r = r(n,C). There exists an open neighborhood W of E in U and some L1 > 0 depending only on n,µ,N, and C (choosing C > 1 one can take L1 = C N+2 (N !)µ ) such that the distance from W to the boundary of U is ≥ 1 and |∂Iα g(ϕ,I)| ≤ L1 , ∀(ϕ,I) ∈ (Tn + r0 ) × W , |α| ≤ N . By Cauchy’s estimate, β α ∂ϕ ∂I g(ϕ,I) ≤ L1 r−|β| β!, ∀(ϕ,I) ∈ Tn × W , ∀β ∈ Zn , |α| ≤ N . 0 Then there exists r1 > 0, depending only on n and C, such that the Fourier coefficients gk of g can be estimated by er1 |k| |∂Iα gk (I)| ≤ 2L1 , ∀I ∈ W , |α| ≤ N , ∀k ∈ Zn . Using the Whitney extension theorem in [21] we extend each hk (I) = er1 |k| gk (I), I ∈ W, to a C N function Hk in Rn such that |∂Iα Hk (I)| ≤ c0 L1 , I ∈ U , |α| ≤ N ,

(IV.6)

where c0 > 0 depends only on n and N . Fix 0 < ε < 1 such that 2ε is smaller than ek the distance between E and the boundary of W . For each k there is a function H µ in G (U ) such that α e (IV.7) ∂I (Hk (I) − Hk (I)) ≤ εN+1 , I ∈ U , |α| ≤ N . Choose ψ ∈ Gµ (Rn ) such that ψ = 1 on E and ψ = 0 outside U . We can suppose that the constant C in the Gevrey estimates for the derivatives of ψ on U does not depend on E since the distance between E and the boundary of U is ≥ 2. Choose ψj ∈ Gµ (Rn ), j = 1,2, with supports in U such that ψ1 + ψ2 = 1 in a neighborhood of suppψ, suppψ1 ⊂ W , ψ1 = 1 in a neighborhood of E, and |∂Iα ψj (I)| ≤ ε−|α| , I ∈ U , |α| ≤ N , j = 1,2.

(IV.8)

e k )ψ, and define Set Fk = (ψ1 hk + ψ2 H X F (ϕ,I) = eihk,ϕi−r1 |k| Fk (I). k∈Zn

Taking into account (IV.6), (IV.7) and (IV.8) we prove that F has the desired properties. 2 Remark 4.3 In the same way one can see that Theorem 4.1 and Proposition 4.2 hold for any compact subset E 6= 0 of Rn using the usual definition of Whitney jets in Gevrey classes (see [3], Sect. 3).

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Acknowledgements I would like to thank J¨ urgen P¨ oschel and Michael Herman for the helpful discussions on the KAM theorem and Nekhoroshev’s theorem and Reinhold Meise for discussions on the Whitney extension theorem in nonquasianalytic spaces.

References [1] D. Bambusi, S. Graffi and T. Paul, Normal forms and quantization formulae, preprint, 1998. [2] J. Bellissard and M. Vittot, Heisenberg’s picture and non commutative geometry of the semi classical limit in quantum mechanics, Ann. Inst. Henri Poincar´e, Phys. Theor., Vol. 52, 1990, 3, pp. 175–235. [3] J. Bonet, R. Braun, R. Meise and B. Taylor, Whitney’s extension theorem for nonquasianalytic classes of ultradifferentiable functions, Studia Math., Vol. 99, 1991, 2, pp. 155–194. [4] J. Bruna, An extension theorem of Whitney type for non quasi-analytic classes of functions, J. London Math. Soc., Vol. 22, 1980, 2, pp. 495–505. [5] Y. Colin de Verdi`ere, Quasimodes sur les vari´et´es Riemanniennes, Inventiones Math., Vol. 43, 1977, pp. 15–52. [6] A. Giorgilli, A. Delshams, E. Fontich, L. Galgani and C. Sim´ o, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three-body problem, J. Differential Equations, Vol. 77, 1989, 1, pp. 167–198. [7] T. Gramchev and G. Popov, Nekhoroshev type estimates for billiard ball maps, Ann. Inst. Fourier (Grenoble), Vol. 45, 1995, 3, pp. 859–895. [8] M. Herman, In´egalit´es a priori pour des tores lagrangiens invariants par des diff´eomorphismes symplectiques, Publ. Math. I.H.E.S., Vol. 70, 1989, pp. 47– 101. [9] M. Herman, Existence et non existence de tores invariants par des ´ diff´eomorphismes symplectiques, Ecole Polytechnique, 1988, preprint [10] M. Herman, On the dynamics of Lagrangian tori invariant by symplectic ´ diffeomorphisms, Ecole Polytechnique, 1990, preprint [11] B. Helffer and J. Sj¨ ostrand, Multiple wells in the semi-classical limit I, Comm.P.D.E., Vol. 9, 1984, pp. 337–408. [12] L. H¨ ormander, The analysis of linear partial differential operators, I-IV, Springer-Verlag, Berlin, 1985.

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[13] H. Komatsu, The implicit function theorem for ultradifferentiable mappings, Proc. Jap. Acad., Ser. A, Vol. 55, 1979, pp. 69–72 [14] V. Lazutkin, Asymptotics of the eigenvalues of the Laplacian and quasimodes, Math. USSR Izvestija, Vol. 7, 1973, pp. 185–214. [15] V. Lazutkin, KAM theory and semiclassical approximations to eigenfunctions, Springer-Verlag, Berlin, 1993. [16] P. Lochak, Canonical perturbation theory:an approach based on joint approximations, Uspekhi Mat. Nauk, Vol. 47, 6, 1992, pp. 59–140 (in Russian); translation in: Russian Math. Surveys, Vol. 47, 6, 1992, pp. 57-133. [17] P. Lochak and A. Neishtadt, Estimates of stability time in nearly integrable systems with a quasiconvex Hamiltonian, Chaos, Vol. 2, 4, 1992, pp. 495–499. [18] J. Moser, On invariant curves of area preserving mappings of an annulus, Nachr. Acad. Wiss. G¨ ot., Math. Phys. KI., 1962, pp. 1–20 [19] N. Nekhoroshev, Exponential estimate of the stability time of near-integrable Hamiltonian systems I, Russ. Math. Surveys, Vol. 32, 6, 1977, pp. 1–65. [20] N. Nekhoroshev, Exponential estimate of the stability time of near-integrable Hamiltonian systems II, Trudy Sem. Petrovs., Vol. 5, 1979, pp. 5–50 (in Russian) [21] J. P¨ oschel, Lecture on the classical KAM Theorem, School on dynamical systems, May 1992, International center for science and high technology, Trieste, Italy [22] J. P¨ oschel, Integrability of Hamiltonian systems on Cantor tori, Comm. Pure Appl. Math., Vol. 35, 1982, pp. 653–695. [23] J. P¨ oschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems, Math. Z., Vol. 213, 1993, pp. 187–217. [24] V. Petkov and G, Popov, On the Lebesgue measure of the periodic points of a contact manifold, Math. Z., Vol. 218, 1995, pp. 91–102. [25] G. Popov, Invariant tori effective stability and quasimodes with exponentially small error terms II - Quantum Birkhoff normal forms, AHP Vol.1(2), 2000, pp. 249–279. [26] L. Rodino, Linear partial differential operators in Gevrey spaces, World Scientific, Singapore, 1993 [27] B. Simon, Semiclassical analysis of low-lying eigenvalues, Ann. Inst. Henri Poincar´e, Phys. Theor., Vol. 38B, 1983, pp. 295–307.

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[28] J. Sj¨ ostrand, Semi-exited states in nondegenerate potential wells, Asymptotic Anal., Vol. 6, 1992, pp. 29–43. [29] J.-C.Yoccoz, Travaux de Herman sur les tores invariants, S´eminaire Bourbaki, Vol. 714, 1991–92

Georgi Popov* D´epartement de Math´ematiques UMR 6629 Universit´e de Nantes - CNRS B.P. 92208 F-44322 Nantes-Cedex 03, France e-mail: [email protected] *Author partially supported by grant MM-706/97 with MES, Bulgaria Communicated by J. Bellissard submitted 15/09/98 ; accepted 06/01/99

Ann. Henri Poincar´ e 1 (2000) 249 – 279 c Birkh¨

auser Verlag, Basel, 2000 1424-0637/00/020249-31 $ 1.50+0.20/0

Annales Henri Poincar´ e

Invariant Tori, Effective Stability, and Quasimodes with Exponentially Small Error Terms II Quantum Birkhoff Normal Forms G. Popov Abstract. The aim of this paper is to obtain quasimodes for a Schr¨ odinger type operator Ph in a semi-classical limit (h & 0) with exponentially small error terms which are associated with Gevrey families of KAM tori of its principal symbol H. To do this we construct a Gevrey quantum Birkhoff normal form of Ph around the union Λ of the KAM tori starting from a suitable Birkhoff normal form of H around Λ. As an application we prove sharp lower bounds for the number of resonances of Ph defined by complex scaling which are exponentially close to the real axis. Applications to the discrete spectrum are also obtained.

Let M be either Rn or a compact real analytic manifold of dimension n ≥ 2 and let J X Ph = Pj (x, hD)hj , 0 < h ≤ h0 , (.1) j=0

be a formally selfadjoint h-differential operator acting on half densities in 1 C ∞ (M, Ω 2 ), where Pj (x, ξ) are polynomials of ξ with analytic coefficients, and D = (D1 , . . . , Dn ), Dj = −i∂/∂xj . We denote the principal symbol of Ph by H(x, ξ) = P0 (x, ξ), (x, ξ) ∈ T ∗ (M ), and suppose that its subprincipal symbol is zero. Our main example will be the Schr¨ odinger operator Ph = − h2 ∆ + V (x) , where ∆ is the Laplace-Beltrami operator on M , associated with a real analytic Riemannian metric and V (x) is a real analytic potential on M bounded from below. Given % > 1, we define a G% (Gevrey) quasimode Q of Ph as follows: Q = {(um (·, h), λm (h)) : m ∈ Mh }, where um (·, h) ∈ C0∞ (M ) has a support in a fixed bounded domain independent of h, λm (h) are real valued functions of h ∈ (0, h0 ], Mh is a finite index set for each fixed h, and (i) ||Ph um − λm (h)um ||L2 ≤ C e−c/h

1/%

(ii) |hum , ul iL2 − δm,l | ≤ C e−c/h

1/%

, m ∈ Mh ,

, m, l ∈ Mh ,

for 0 < h ≤ h0 . Here C and c are positive constants, and δm,l is the Kronecker

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index. Recall that for any C ∞ quasimode Q the right hand side in (i) and (ii) is ON (hN ) for each N ≥ 0. We define the G% micro-support M S % (Q) ⊂ T ∗ (M ) of Q as follows: (x0 , ξ0 ) ∈ / M S % (Q) if there exist compact neighborhoods U of x0 and V of ξ0 in a given local chart such that for any G% function v with support in U Z 1/% e−ihx,ξi/h v(x)um (x, h)dx = O e −c/h , as h & 0 , uniformly with respect to m ∈ Mh and ξ ∈ V . We are going to find a Gevrey quasimode Q of Ph , the Gevrey micro-support of which coincides with the union Λ of a suitable Gevrey family of KolmogorovArnold-Moser (KAM) invariant tori Λω , ω ∈ Ωκ , of H, obtained in [19]. For this aim we construct a Quantum Birkhoff Normal Form (QBNF) of Ph around Λ in suitable Gevrey classes starting from the Birkhoff Normal Form (BNF) of its principal symbol H obtained in Theorem 1.1 [19]. In other words, conjugating Ph with an unitary h-Fourier Integral Operator (h-FIO) we transform it to a suitable h-pseudodifferential operator (h-PDO) Ph0 acting on sections in C ∞ (Tn ; L), where L is a flat Hermitian linear bundle of Tn = (R/2πZ)n associated to the Maslov class of the invariant tori. The operator Ph has a Gevrey symbol p0 (ϕ, I, h) ∼

∞ X

(Kj (I) + Rj (ϕ, I))hj , (ϕ, I) ∈ Tn × D ,

j=0

such that each Rj is flat at the Cantor set Tn × Eκ , Eκ = {I ∈ D : ω(I) ∈ Ωκ } , where ω : D → Ω is a Gevrey diffeomorphism, Ω is a neighborhood of Ωκ , ω(I) − ∇K0 (I) is flat at Eκ , and K0 (I)+R0 (ϕ, I) is just the BNF of H around Λ obtained in [19]. Then Rj turn out to be exponentially small around Tn × Eκ and we obtain a Gevrey quasimode of Ph (see Corollary 1.2). In the C ∞ case a similar QBNF was first obtained by Colin de Verdi`ere [7] for the Laplace-Beltrami operator ∆ on a compact Riemannian manifold M . As a consequence, C ∞ quasimodes for ∆ were obtained in [7]. Quasimodes provide information about the spectrum of Ph . If Ph has discrete spectrum, we can find eigenvalues of Ph exponentially close to the quasi-eigenvalues λj (h). Moreover, the total multiplicity of the part of the spectrum of Ph approximated by Q modulo an exponentially small error term is given asymptotically by (2πh)−n Vol (Λ) as h & 0. The notion “total multiplicity” will be explained in Sect. 1.2. In the case of “scattering”, using a result of Stefanov [25], we shall find a large set of resonances of Ph (defined by complex scaling) which are exponentially close to the real axis (see Sect. 1.3). Quasimodes associated to a Cantor family of invariant tori were first obtained by Lazutkin [14] for the Laplace operator in strictly convex bounded domains in R2 (see also [15] and references there) and for n ≥ 2 by Colin de Verdi`ere [7] who also

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constructed a QBNF around a family of invariant tori in the C ∞ case. Quasimodes associated with invariant tori of the classical Hamiltonian have been obtained also in [5], [6], [8], [18]. An extension of Nekhoroshev’s theorem in quantum mechanics is proposed by Bellissard and Vittot [3]. They investigate the rate of divergence in the Rayleigh-Schr¨odinger perturbation series when the unperturbed Hamiltonian is given by a family of harmonic oscillators whose frequencies satisfy a small divisor condition. If 0 is a nondegenerate minimum of V , Sj¨ ostrand [22] obtained a quantization formula for all eigenvalues of Ph in an interval [0, hδ ], where δ > 0 is fixed. Stronger result has been proved recently for Gevrey smooth potentials V (x) by Bambusi, Graffi and Paul [1]. They obtained a quantization formula modulo O(h∞ ) for all eigenvalues of Ph in an interval [0, ϕ(h)] where ϕ(h)b ln h → 0 as h & 0 and b is an explicitly determined constant. A link between Nekhoroshev’s stability for the classical system and the semi-classical asymptotics with exponentially small error term of the low lying eigenvalues of the corresponding Schr¨ odinger operator is suggested by Sj¨ ostrand [22]. The techniques developed in the present paper could be used to obtain quasimodes with exponentially small error terms for the Laplace operator −∆ with Dirichlet (Neumann) boundary conditions in a domain Ω ⊂ Rn with a compact analytic boundary which are associated to Gevrey families of invariant tori of the broken bicharacteristic flow. The paper is organized as follows: The main results are formulated in Sect. 1. In Sect. 2 we define suitable classes of Gevrey symbols, h-PDOs and h-FIOs. We conjugate Ph with an elliptic h-FIO Th to a h-PDO P˜h of Gevrey class acting on sections in C ∞ (Tn ; L), the principal symbol of which is just the BNF of H and the subprincipal symbol is 0. In Sect. 3 we obtain a QBNF of P˜h conjugating it with an elliptic h-PDO Ah . We first find the full symbol of Ah on the Cantor set Tn × Eκ and then use a suitable Whitney extension theorem in Gevrey classes. To obtain the full Gevrey symbol of Ah on Tn × Eκ we have to solve the homological equation h∇K0 (I), ∂ϕ if (ϕ, I) = g(ϕ, I) , ϕ ∈ Tn , uniformly with respect to I ∈ Eκ and to provide the corresponding Gevrey estimates for the solution. Here g(ϕ, I) is a Gevrey function in Tn × Eκ in the sense of Whitney, and Z g(ϕ, I) dϕ = 0 . Tn

The analysis of the solution of the homological equation is done in Sect. 4. In Sect. 5 we complete the construction of the normal form of Ph near Λ.

I QBNF around KAM tori and quasimodes 1.1 Main results. We are going to formulate the main assumptions on the principal symbol H of Ph . Fix κ > 0 and τ such that τ > n − 1 when n ≥ 3 and τ > 3/2 when n = 2. Given a bounded domain Ω ⊂ Rn we consider the set Ξκ of all

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ω ∈ Ω having distance ≥ κ to the boundary of Ω and satisfying the Diophantine condition κ |hω, ki| ≥ , for all 0 6= k ∈ Zn , (I.1) |k|τ where |k| = |k1 | + · · · + |kn |. We denote by Ωκ the set of points of a positive Lebesgue density in Ξκ , namely, ω ∈ Ωκ iff for any neighborhood U of ω the Lebesgue measure of U ∩ Ωκ is positive. Fix s = τ 0 + 2 with τ 0 > max{τ, 5/2}. We suppose that there exists a real analytic exact symplectic diffeomorphism χ1 : Tn × D −→ U ⊂ T ∗ (M ) , where D is a domain in Rn such that the Hamiltonian def e H(φ, I) = (H ◦ χ1 )(ϕ, I)

(I.2)

admits a Gs -BNF around a family of invariant tori with frequencies in a suitable e Ωκ . In other words, we assume that the Hamiltonian H(φ, I) satisfies : (BF ) There exists a domain Ω, a Gs -diffeomorphism ω : D → Ω, and an exact syme 0 (ϕ, I)) = plectic transformation χ0 ∈ G1,s (Tn × D, Tn × D) such that H(χ n s 1,s K0 (I)+R0 (ϕ, I) in T ×D, where K0 ∈ G (D) and R0 ∈ G (Tn ×D) satisfy DIα R0 (ϕ, I) = 0 and DIα (∇K0 (I) − ω(I)) = 0 for any (ϕ, I) ∈ Tn ×ω −1 (Ωκ ) and α ∈ Zn+ . Moreover, there exists a generating function Φ ∈ G1,s (Tn × D) of χ0 such that ||Id − ΦϕI (ϕ, I)|| ≤ ε in Tn × D for some 0 < ε < 1. Here || · || is the usual sup-norm in the space of n × n matrices. Recall that Φ is a generating function of χ0 if χ0 (∇I Φ(ϕ, I), I) = (ϕ, ∇ϕ Φ(ϕ, I)) for any (ϕ, I) ∈ Tn × D. Theorem 1.1 in [19] shows that any real analytic Hamiltoe nian H(ϕ, I), (ϕ, I) ∈ Tn × D, which is a sufficiently small perturbation of a non-degenerate real analytic completely integrable Hamiltonian H 0 (I), satisfies (BF ) with Ω = ∇H 0 (D). The map χ1 provides “action-angle” coordinates for the completely integrable Hamiltonian H 0 and it can be constructed by the LiouvilleArnold theorem. For example we can take M = Rn and suppose that V has a non-degenerate minimum E0 = V (0) and that there are no resonances of order 4 (see (0.3), [19]). Then Corollary 1.3, [19], holds. In this case χ1 transforms H to its Birkhoff normal form. Set χ = χ1 ◦ χ0 : Tn × D −→ U ⊂ T ∗ (M ). Let Λ be the union of the invariant tori Λω = χ(Tn × {I(ω)}) of H with frequencies ω ∈ Ωκ , where Ω 3 ω → I(ω) ∈ D is the inverse to the frequency map D 3 I → ω(I) ∈ Ω. The Maslov class of Λω , ω ∈ Ωκ , can be identified to an element ϑ of H 1 (Tn ; Z) = Zn via the symplectic map χ. Notice that ϑ = (2, . . . , 2) in the case when V has a nondegenerate minimum E0 = V (0). As in [7] we consider the flat Hermitian line bundle L over Tn which is associated to the class ϑ. The sections f in L can be identified canonically with functions fe : Rn → C so that π fe(x + 2πp) = ei 2 hϑ,pi fe(x)

(I.3)

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for each x ∈ Rn and p ∈ Zn . It is easy to see that an orthonormal basis of L2 (Tn ; L) is given by em , m ∈ Zn , where eem (x) = exp (ihm + ϑ/4, xi) . Set ν = τ + n + 1 and fix τ 0 such that τ + n − 1 > τ 0 > max(τ, 5/2). Then fix ν > µ > τ 0 + 2, choose σ > 1 sufficiently close to 1 such that ν > µ > σ(τ 0 + 1) + 1 ,

(I.4)

and set % = σν. Thus % could be any number bigger than ν and sufficiently close to ν. Set ` = (σ, µ, %) and consider the corresponding class of Gevrey symbols 0 S` (Tn × D) (see Sect. 2). Starting from the Gτ +1 -BNF of H around Λ given by (BH), we are going to find a QBNF of Ph around Λ in the class of h-PDOs in L2 (Tn ; L) with a symbol in S` (Tn × D), conjugating Ph with a suitable h-FIO with canonical relation C = graph (χ). Recall that Ph is a selfadjoint h-differential 1 operator acting on half densities in C ∞ (M, Ω 2 ) of the form (.1) with analytic coefficients in M and with a subprincipal symbol equal to zero. Theorem I.1 Suppose that there exists a real analytic exact symplectic map χ1 : e I) = H(χ1 (ϕ, I)), Tn × D → U ⊂ T ∗ (M ) such that the Hamiltonian H(ϕ, (ϕ, I) ∈ Tn × D, satisfies (BF ) for s = τ 0 + 2. Then there exist a family of uniformly bounded h-FIOs Uh : L2 (Tn ; L) → L2 (M ), 0 < h ≤ h0 , associated with the canonical relation C such that the following holds: (i) Uh∗ Uh − Id is a pseudodifferential operator with a symbol in the Gevrey class S` (Tn × D) which is equivalent to 0 on Tn × D0 , where D0 is a subdomain of D containing the union Λ of the invariant tori (ii) Ph ◦ Uh = Uh ◦ Ph0 , and the full symbol p0 (ϕ, I, h) of Ph0 has the form 0 p (ϕ, I, h) = K 0 (I, h) + R0 (ϕ, I, h), where the symbols X X K 0 (I, h) = Kj (I)hj and R0 (ϕ, I, h) = Rj (ϕ, I)hj 0≤j≤ηh−1/%

0≤j≤ηh−1/%

belong to the Gevrey class S` (T ∗ (Tn )), η > 0 is a constant, K 0 is real valued, and R0 is equal to zero to infinite order on the Cantor set Tn × Eκ . As a consequence we obtain a G% - quasimode Q of Ph with an index set Mh = {m ∈ Zn : | Eκ − h(m + ϑ/4)| ≤ hε } where ε = ε(µ) ∈ (0, 1). It is easy to see that #{m ∈ Mh } =

1 Vol (Tn × Eκ )(1 + o(1)) (2πh)n

1 Vol (Λ)(1 + o(1)) , h & 0 , (I.5) (2πh)n where Vol (Λ) stands for the Lebesgue measure of the union Λ of the invariant tori in T ∗ (M ). =

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Corollary I.2 Let um (x, h) = Uh (em )(x), and λm (h) = K 0 (h(m + 14 ϑ), h), for m ∈ Mh . Then Q = {(um (x, h), λm (h)) : m ∈ Mh } is a G% -quasimode of Ph . Moreover, M S % (Q) = Λ.

(I.6)

To prove Corollary 1.2 we write Ph0 = Kh0 + Rh0 , where the symbols of Kh0 and Rh0 are K 0 (I, h) and R0 (ϕ, I, h) respectively. It is easy to see that Ph0 (em )(ϕ) = (λm (h) + R0 (ϕ, h(m + ϑ/4), h)) em (ϕ) for any m ∈ Mh . On the other hand, |Dϕβ DIα R0 (ϕ, I, h)| ≤ C |α|+|β|+1 β! σ α! µ , ∀ (ϕ, I, h) ∈ Tn × D × (0, h0 ] , because of (II.3). Then there exist two positive constants C1 and c depending only on the constant C such that for every α, β ∈ Zn+ the following estimate holds 1 |α|+|β|+1 |∂ϕβ ∂Iα R0 (ϕ, I, h)| ≤ C1 β! σ α! µ exp −c |Eκ − I| − µ−1 , / Eκ , where |Eκ −I| = inf I 0 ∈Eκ |I 0 −I| is the for each (ϕ, I, h) ∈ Tn ×D×(0, h0 ], I ∈ distance to the compact set Eκ (see [19], (1.3)). Using the inequality µ < ν < %, and choosing appropriately ε we prove that Q satisfies (i) in the introduction. On the other hand (ii) and (I.6) follow directly from the definition of the index set Mh , the orthogonality of em , and (i) in Theorem 1.1. 2

1.2 Applications to the discrete spectrum. Consider now the Schr¨ odinger operator Ph = −h2 ∆ + V (x) in M , where ∆ is the Laplace-Beltrami operator associated with a real analytic Riemannian metric on M which coincides with the Euclidean metric when M = Rn . Suppose that Ph satisfies the assumptions of Theorem 1.1 in a bounded subdomain of T ∗ (M ). Set E1 = max{H(x, ξ) : (x, ξ) ∈ Λ}. Suppose that H −1 ((−∞, E2 ]) is compact for some E2 > E1 and fix E ∈ (E1 , E2 ) and E0 < min{H(x, ξ) : (x, ξ) ∈ Λ}. We need that assumption only when M = Rn . Then Ph , 0 < h ≤ h0 , has only a discrete spectrum in [E0 , E]. Hereafter h0 > 0 is chosen sufficiently small. Fix c ≥ ε ≥ 0 and C 0 > C, where c and C are the constants in the definition of Q. Denote by Πh the spectral projector of Ph and for each 0 < h ≤ h0 and m ∈ Mh set h i 1/% 1/% ∆hε,m = λm (h) − C 0 e−(c−ε)/h , λm (h) + C 0 e−(c−ε)/h . Then there exists at least one eigenvalue of Ph in ∆h0,m , and we have ||Πh (∆hε,m )um − um || ≤ e−ε/h

1/%

, 0 < h ≤ h0 1 , m ∈ Mh

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(see [15], Proposition 32.1 and (32.2)). Set I h = ∪{∆hε,m : m ∈ Mh } and fix A > 2(2π)−n Vol(Λ). Taking into account (I.5) we obtain that I h ⊂ [E0 , E] is a finite union of disjoint intervals Ijh of length h Ij ≤ AC 0 h−n e−(c−ε)/h1/% . Denote by Lhj the span of all um (·, h) such that m ∈ Mh and λm (h) ∈ Ijh . Then ||Πh (Ijh )v − v|| ≤ A0 h−n e−ε/h

1/%

||v|| , 0 < h ≤ h0 1 ,

for each v ∈ Lhj and some constant A0 > 0. Then it is natural to call Nh∗ (I h ) =

X

dim Πh (Ijh )Lhj

j

total multiplicity of the part of spectrum of Ph in I h which is approximated by the quasimode Q modulo an exponentially small error term (for C ∞ quasimodes see [15] ). Moreover, we have dim Πh (Ijh )Lhj = dim Lhj , 0 < h ≤ h0 1 , hence, Nh∗ (I h ) = #{m ∈ Mh } =

1 Vol (Λ)(1 + o(1)) , h & 0 . (2πh)n

(I.7)

Recall that the function Nh ([E0 , E]) counting with multiplicities the eigenvalues of Ph in [E0 , E] has a semiclassical asymptotic Nh ([E0 , E]) = (2πh)−n C1 (1 + o(1)), where C1 = Vol (H −1 ([E0 , E]) is the Lebesgue measure of H −1 ([E0 , E]) in T ∗ (M ). 1.3 Applications to resonances. Consider a selfadjoint second order differential operator in Rn X Ph = aα (x)(hD)α hj . |α|+j≤2

As in [26] we impose the following hypothesis: (H1 ) The coefficients aα (x) are real analytic and they can be extended holomorphically to {rω : ω ∈ Cn , dist(ω, Sn ) < ε, r ∈ C, |r| > R, arg r ∈ [−ε, θ0 − ε]} for some ε > 0 and θ0 > 0 and the coefficients of − h2 ∆ − Ph tend to zero as |x| → ∞ in that set uniformly with respect to h.

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(H2 ) For some C > 0 we have X aα (x)ξ α ≥ C |ξ|2 , (x, ξ) ∈ T ∗ (Rn ) . |α|=2

Then the resonances Res Ph of Ph close to the real axis can be defined in a conic neighborhood Γ of the positive half axis in the lower half plain by the method of complex scaling (see [23] and [24]). They coincide in Γ with the poles of the meromorphic continuation of the resolvent 2 (Ph − z)−1 : L2comp (Rn ) → Hloc (Rn ) , Im z > 0 .

Thang and Zworski [26] obtained a result which implies lower bounds of the the resonances Res Ph of Ph close to the real axis for any h ∈ (0, h0 ], provided that there exists a quasimode Q for Ph . Stefanov [25] obtained sharp lower bounds, he showed that for each h ∈ (0, h0 ] the number of the resonances of Ph close to the real axis is not less than the cardinality of the index set Mh of the quasimode Q. We set Nh = #{λ ∈ Res Ph : Re λ ∈ [E0 , E], 0 < −Im λ ≤ h−n−2 e−c/h

1/%

},

where the resonances are counted with multiplicities, c > 0 is the constant in the definition of Q and E0 < E are as in 1.2. Burq [4] showed that there exists ε > 0 and C > 0 such that there are no resonances of Ph , 0 < h ≤ h0 , in {λ ∈ C : Re λ ∈ [E0 , E], 0 < −Im λ ≤ εe−C/h } . Combining Corollary 1.2 with Theorem 1.1 in [25] (which holds also for noncompactly supported perturbations of −h2 ∆ satisfying (H1 ) and (H2 )), and using (I.5), we obtain the following: Theorem I.3 Suppose that Ph satisfies (H1 ), (H2 ), and the assumptions of Theorem 1.1. Then 1 Vol (Λ)(1 + o(1)) , h & 0 . Nh ≥ (2πh)n

II Gevrey symbols h-PDOs and h-FIOs 2.1. Gevrey symbols. We are going to put the operator Ph in a QBNF around the union of the invariant tori Λ conjugating it by an elliptic h-FIO with a suitable Gevrey symbol. The resulting operator will be a h-PDO with a Gevrey symbol. First we define the class of Gevrey symbols that we need. Denote by D a bounded domain in Rn . Let X be either Tn or a bounded domain in Rm , m ≥ 1. Fix σ, µ > 1, % ≥ σ + µ − 1, and set ` = (σ, µ, %). We introduce a class of formal Gevrey symbols F S` (X × D) as follows. Consider a sequence of smooth functions

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pj ∈ C0∞ (X × D), j ∈ Z+ such that supp pj is contained in a fixed compact subset of X × D. We say that ∞ X pj (ϕ, I) hj (II.1) j=0

is a formal Gevrey symbol in F S` (X × D) if there exists a positive constant C such that pj satisfies the estimates sup |∂ϕβ ∂Iα pj (ϕ, I)| ≤ C j+|α|+|β|+1 β! σ α! µ j! %

(II.2)

X×D

for any α, β and j. The function p(ϕ, I; h), (ϕ, I) ∈ X × Rn , is called a realization of the formal symbol (II.1) in X × D if for each 0 < h ≤ h0 it is smooth with respect to (ϕ, I) and has compact support in X × D, and if there exists a positive constant C1 such that N X sup ∂ϕβ ∂Iα (p(ϕ, I, h) − pj (ϕ, I)hj ) Q

j=0

≤ h

N+1

N+|α|+|β|+2

C1

β! σ α! µ (N + 1)! %

(II.3)

for any multi-indices α, β and N ∈ Z+ , where Q = X × D × (0, h0 ]. For example, one can take X p(ϕ, I, h) = pj (ϕ, I) hj , j≤ εh−1/%

where ε > 0 depends only on the constant C1 and the dimension n (for σ = µ = 1 see [22], Sect. 1). We denote by S` (X × D) the corresponding class of symbols. Given g ∈ S` (X × D), we say that g ∈ S`−∞ (X × D) if sup |∂ϕβ ∂Iα g(ϕ, I; h)| ≤ hN C N+|α+β|+1 β! σ α! µ N ! % Q

for 0 < h ≤ h0 , ∀N ∈ Z+ , and for any multi-indices α, β ∈ ZN + , or equivalently |α+β|+1

sup |∂ϕβ ∂Iα g(ϕ, I; h)| ≤ C1

β! σ α! µ exp(−ch−1/% )

Q

for some C1 , c > 0, and any h ∈ (0, h0 ], α, β ∈ Zn+ . Moreover, given f, g ∈ S` (X × D), we say that f is equivalent to g (f ∼ P g) if f − g ∈ S`−∞ (X × D). ∞ It is not hard to prove that any two realizations of j=0 pj hj in S` (X × D) are σ equivalent. When σ = µ and % = 2σ − 1, we set S = S` and S σ,−∞ = S`−∞ . Having two symbols p, q ∈ S` (X × D) we denote their composition by p ◦ q ∈ S` (X × D) which is the realization of ∞ X j=0

cj hj ∈ F S` (X × D),

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where

X

cj (ϕ, I) =

r+s+|γ|=j

Ann. Henri Poincar´ e

1 γ D pr (ϕ, I) ∂ϕγ qs (ϕ, I). γ! I

(II.4)

In particular, S` becomes an algebra under this composition. Having a symbol p ∈ S` (X × D) associated to the formal symbol (II.1), we define its conjugate p∗ as the realization of the formal symbol ∞ X

cj hj ∈ F S` (X × D),

j=0

where cj (ϕ, I) =

X r+|γ|=j

1 γ γ D ∂ pr (ϕ, I). γ! I ϕ

To each symbol p ∈ S` (X × D) we associate an h-pseudodifferential operator (h-PDO) by Z −n Ph u(x) = (2π h) eihx−y,ξi/h p(x, ξ, h) u(y)dξdy, u ∈ C0∞ (X). R2n

It is well defined modulo exp(−ch−1/% ). Indeed, for any p ∈ S`−∞ we have ||Ph u||L2 ≤ C exp(−ch−1/% )||u||L2 , u ∈ C0∞ (X), with some positive constants c and C. Then the composition of two h-PDOs Ph and Qh with symbols p, q ∈ S` (X × D) is a h-PDO of the same class with a symbol p ◦ q, and the L2 -adjoint of Ph has a symbol p∗ . Moreover, h-PDOs with symbols of the class S σ = S` , ` = (σ, σ, 2σ − 1), σ > 1, remain in the same class after a Gσ change of the x variables, and they can be defined as well on any Gσ compact manifold (see Theorem 2.3 [9]). Let u(x, h) be a family of smooth functions in M for 0 < h ≤ h0 . The G% micro-support M S % (u) ⊂ T ∗ (M ) of u is defined as follows: (x0 , ξ0 ) ∈ / M S % (u) if there exists c > 0 and compact neighborhoods U of x0 and V of ξ0 in a given local chart such that for any G% function v with compact support in U Z −1/% e−ihx,ξi/h v(x)u(x, h)dx = O e−ch , as h & 0 , uniformly with respect to ξ ∈ V . Obviously, (x0 , ξ0 , x0 , −ξ0 ) does not belong to the G% microsupport of the distribution kernel of the h-PDO Ph above if its amplitude p ∈ S` belongs to S`−∞ in a neighborhood of (x0 , ξ0 ). 2.2. Quantization of χ1 . We are going to quantize the real analytic symplectic transformation (x, ξ) = χ1 (y, η) defined by (I.2). Set C1 = {(χ1 (y, η), y, η) : (y, η) ∈ Tn × D}

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and denote

259

C10 = {(x, y, ξ, η) : (x, ξ, y, −η) ∈ C1 }.

Recall that χ1 : Tn × D → T ∗ (M ), D ⊂ Rn , is exact symplectic , hence C10 is an exact Lagrangian submanifold of T ∗ (M × Tn ). In other words, the pull-back ı∗ α of the canonical one-form α of T ∗ (M × Tn ) via the inclusion map is an exact form, ı∗ α = df (II.5) for some analytic function f on C10 . This means that the Liouville class [ı∗ α] of C10 is trivial in the first cohomology group H 1 (C10 ; R) which allows us to quantize χ1 . Given σ > 1, we are going to define a class of h-FIOs the distribution kernels of which are oscillatory integrals in the sense of Duistermaat [5] associated with C10 and having Gevrey symbols in S σ . Locally C10 can be defined by a nondegenerate real analytic phase function as follows. Let us fix some ζ0 = (x0 , y0 , ξ0 , η0 ) in C10 . Choosing suitable analytic local coordinates x in a neighborhood U0 of x0 , we can parameterize (locally) the Lagrangian manifold C1 by (y, ξ) ∈ U1 × U2 , where U1 is a local chart of Tn and U2 is a neighborhood of ξ 0 in Rn . Then there exists a real analytic function φ(y, ξ) in U1 × U2 such that C1 = {(φ0ξ , ξ, y, φ0y )} and det ∂ 2 φ/∂y∂ξ 6= 0 in U1 × U2 (see [12], Proposition 25.3.3). It is uniquely defined up to a constant, and we fix it by φ(y0 , ξ0 ) = hx0 , ξ0 i − f (ζ0 ), where f is given by (II.5). The real analytic phase function Ψ(x, y, ξ) = hx, ξi − φ(y, ξ) defines locally the Lagrangian manifold C10 , namely, rank d(x,y,ξ) dξ Ψ = n on OΨ = {(x, y, ξ) : dξ Ψ = 0}, and the map 0 ıΨ : OΨ 3 (x, y, ξ) −→ (x, y, Ψ0x , Ψ0y ) ∈ CΨ

is a local diffeomorphism in C10 . Moreover, we have Ψ(x0 , y0 , ξ0 ) = f (ζ0 ).

(II.6)

We are ready to define h-FIOs associated to C1 and mapping C ∞ (Tn ; Ω ×L) 1 to where Ω 2 is the corresponding half density bundle and the sections in L are defined by (I.3). Fix σ > 1 and choose a symbol a ∈ S` (U × U2 ) = S σ (U × U2 ), ` = (σ, σ, 2σ − 1), where U = U0 × U1 . We extend a for y ∈ Rn by 1 2

1 C0∞ (M, Ω 2 ),

e a(x, y + 2πp, ξ, h) = e−i 2 hϑ,pi a(x, y, ξ, h) , (x, y, ξ) ∈ U × Rn , p ∈ Zn , π

and we extend φ as a 2π periodic function with respect to y in U0 ×(U1 +2πZn )×U2 . Then given a section u ∈ C ∞ (Tn ; L) of the linear bundle L we set Z Z −n Th u(x) = (2π h) eiΨ(x,y,ξ)/h e a(x, y, ξ, h) u e(y)dξdy, (II.7) Rn

U1

where u e satisfies (I.3). Notice that e a(x, y, ξ, h)e u(y) is 2π periodic with respect to y in Rn and we can replace U1 by U1 + 2πp for any p ∈ Zn . Denote by Kh (Ψ, a) the distribution kernel of Th . We define a class of h-FIOs Th : C ∞ (Tn ; Ω 2 ⊗ L) → C0∞ (M, Ω 2 ) 1

1

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with Gσ Gevrey symbols as a finite sum of operators given microlocally by (II.7), where the half density bundles have been trivialized by dividing with the corresponding canonical half densities. We denote the class of the distribution kernels Kh of Th by Iσ (M × Tn , C10 ; 1 Ω 2 ⊗ L0 , h), where L0 is the dual bundle to L. One can show that the definition does not depend on the choice of the phase functions. Indeed, fix ζ0 ∈ C10 as above and choose as above a real analytic nondegenerate phase function Φ(x, y, ξ) such that C10 = CΦ locally near ζ0 and such that (II.6) holds. It can be proved that there exists a symbol g ∈ S σ such that ζ0 ∈ / M S 2σ−1 (Kh − Kh (Φ, g)), where Kh (Ψ, a) denotes the distribution kernel of (II.7) (one can also take more general phase functions as in [5], Proposition 1.3.1). Here we use the following stationary phase lemma: Lemma II.1 Let Φ(x, y) be a real analytic function in a neighborhood of (0, 0) in Rm1 +m2 . Assume that Φ0x (0, 0) = 0 and that Φ00xx (0, 0) is non-singular. Denote by x(y) the solution of the equation Φ0x (x, y) = 0 with x(0) = 0 given by the implicit function theorem. Then for any g ∈ S σ (U ), where U is a suitable neighborhood of (0, 0) we have Z eiΦ(x,y)/h g(x, y, h)dx = eiΦ(x(y),y)/h G(y, h), where G ∈ S σ . To prove the lemma we first use the Morse lemma with parameters for real analytic functions which can be proved as in [12], Lemma C.6.1, and then we follow the proof of Lemma 7.7.3 in [12] (see also [9]). Actually it could be proved that Lemma 2.1 holds also when Φ ∈ Gσ . The principal symbol of Th (see [5], [16]) is of the form eif (ζ)/h Υ(ζ), where 1 1 ∗ Υ is a smooth section in Ω 2 (C10 ) ⊗ MC ⊗ πC (L0 ). Here Ω 2 (C10 ) is the half density ∗ bundle of C10 , MC is the Maslov bundle of C10 , and πC (L0 ) is the pull-back of L0 1 0 n via the canonical projection πC : C1 → T . The bundle Ω 2 (C10 ) is trivialized by the pull-back of the canonical half density of Tn × D via the canonical projection ∗ π2 : C10 → Tn × D. As in the proof of Theorem 2.5, [7], πC (L0 ) can be canonically 0 identified with the dual MC of the Maslov bundle. Hence, the principal symbol of Th can be canonically identified with a smooth function b on C10 . Moreover, for any Th of the form (II.7) we have b(φ0ξ (y, ξ), y, ξ, −φ0y (y, ξ)) = a0 (φ0ξ (y, ξ), y, ξ)| det ∂ 2 φ/∂y∂ξ(y, ξ)|−1/2 , where a0 is the leading term of the amplitude a. We choose an operator T1h as above with a principal symbol equal to one in a neighborhood of the pull-back via π2 of the union of the invariant tori Λ of H ◦ χ1 , given by (BF ). ∗ Using Lemma 2.1 it can be proved 1h T1h is a h-PDO in P∞ that Qhj = T ∞ n σ C (T , L), with a symbol q(x, ξ) = j=0 qj (x, ξ)h in S (Tn × D). Moreover, its

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principal symbol is equal to 1 in a neighborhood U of Λ and we can assume that q1 (x, ξ) = 0 in U . To do this we write T1h = Ah + hBh , where the principal symbol of Ah is equal to 1 in U , and then we solve a linear equation for the real part of the principal symbol of B. Let us conjugate Ph by an operator T1h defined as above. ∗ Using Lemma 2.1 it can be proved that Ph1 = T1h Ph T1h is a h-PDO in C ∞ (Tn , L), −1 σ n with a symbol in S (T × D). Moreover, we have Ph1 = T1h Ph T1h + h2 Rh , where Rh is a h-PDO. As in Lemma 2.9, [7], we obtain that the principal symbol of Ph1 is equal to H ◦ χ1 and that its subprincipal symbol is zero. 2.3. Quantization of χ0 . We are going to conjugate Ph1 with a h-FIO T2h : L2 (Tn ; L) → L2 (Tn ; L) associated to the canonical relation graph (χ0 ), where (x, ξ) = χ0 (y, I) is given by (BF ). The distribution kernel of T2h has the form Z −n (2π h) ei(hx−y,Ii+φ(x,I))/h b(x, I, h) dI, where φ(x, I) = Φ(x, I) − hx, Ii, and Φ ∈ G1,s (Tn × D) is given by (BF ), while b is a symbol of Gevrey class Se (Tn × D) with `e = (σ, µ, σ + µ − 1) and µ > s = ` 0 τ + 1 > σ > 1 is fixed in (I.4). We suppose that the principal symbol of T2h is equal to 1 in a neighborhood of Tn × D. Set Th = T1h ◦ T2h . Proposition II.2 The operator Peh = Th∗ ◦ Ph ◦ Th is a h-PDO with a symbol in the class Se , where `e = (σ, µ, σ + µ − 1). Moreover, the principal symbol of Peh equals ` e = H ◦ χ, and its sub-principal symbol is zero. H ∗ Proof. We are going to show that Peh = T2h ◦ Ph1 ◦ T2h is a h-PDO with a symbol in σ n Se . Denote by a ∈ S = S(σ,σ,2σ−1) (T × D) the amplitude of Ph1 and recall that ` b ∈ Se (Tn × D). Choosing a suitable partition of the unity in Tn , we suppose that ` the support of b(z, η, h) with respect to z is contained in a fixed local chart of Tn . Then the Schwartz kernel of the operator Ph1 ◦ T2h can be written in the form Z (2π h)−n ei(hx−y,ηi+φ(x,η))/h

Z −n × (2πh)

Rn

iψ(x,z,ξ,η)/h

e

q(x, z, ξ, η, h) dzdξ

dη ,

(II.8)

Rn ×D

where q(x, z, ξ, η, h) = a(x, ξ, h)b(z, η, h), and ψ(x, z, ξ, η) = hx − z, ξ − ηi + φ(z, η) − φ(x, η) = hx − z, ξ − η + φez (x, z, η)i . e h) belongs to the symbol Setting x e = (x, z) and ξe = (ξ, η) we obtain that q(e x, ξ, . Consider the inner integral u(x, η, h) in (II.8). Changing the variables in class Se ` it we obtain Z −n u(x, η, h) = (2πh) eihz,ξi/h Q(x, z, ξ, η, h) dzdξ (mod S σ,−∞ ) ,

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where Q is again in Se . Applying the Taylor formula at z = 0 and then integrating ` by parts with respect to ξ we obtain that u belongs to Se . Now we can write the ` e Schwartz kernel of the operator Ph in the form −n

Z eihx−y , ξi/h

(2π h)

Rn

Z × (2π h)−n

eiΨ(y,z,ξ,η)/h b(z, ξ, h)u(z, η, h)dzdη dξ.

(II.9)

Rn ×D

The phase function Ψ can be written as follows Ψ(x, y, z, ξ, η) = hy − z , ξ − ηi + φ(z, η) − φ(z, ξ) = hy − z − φeξ (z, ξ, η) , ξ − ηi where φeξ (z, ξ, η) =

Z

1

∂φ/∂ξ(z, ξ + t(η − ξ)) dt 0

is analytic with respect to z and Gµ with respect to (ξ, η), i.e. φeξ ∈ G1,µ with respect to (z, ζ), ζ = (ξ, η). The stationary points with respect to (z, η) are η = ξ and z = y − φeξ (z, ξ, ξ) in view of (BF ). Integrating by parts with respect to (z, η) in the inner integral we can suppose that |z − y + φeξ (z, ξ, η)| , |η − ξ| 1 on the support of b(z, ξ, h)u(z, η, h). On the other hand dz φeξ (0, ξ, ξ) is nondegenerate in view of (BF ) and there exists z = ze(y, ϕ, ξ, I) given by the implicit function theorem such that ϕ = z − y + φeξ (e z , ξ, η). Moreover, one can show that ze(y, ϕ, ξ, η) is real analytic with respect to (y, ϕ) and Gµ with respect to (ξ, η) and that for any function g(z, ξ, η) of class Gσ,µ with respect to (z, ζ), ζ = (ξ, η), the function g(e z (y, ϕ, ξ, η), ξ, η) is Gσ,µ with respect to ((y, ϕ), (ξ, η)) (see Appendix A.2). We make a change of the variables in the inner integral in (II.9) setting ϕ = z − y + φeξ (z, ξ, η) and I = η − ξ. Then the inner integral becomes Z v(y, ξ, h) = (2πh)−n eihϕ,Ii/h R(ϕ, y, ξ, I, h) dϕdI , where R ∈ Se . Using the Taylor formula at ϕ = 0 and integrating by parts with ` respect to I we obtain that v belongs to Sel . Moreover, choosing the subprincipal −1 ∗ = T2h + O(h2 ), we obtain that the subprincipal symbol symbol of T2h so that T2h of Peh is 0. 2

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III Normal form of Peh We can suppose now that Peh is a selfadjoint pseudodifferential operator with a symbol p ∈ Se (Tn × Γ), `e = (σ, µ, σ + µ − 1), where ` p(ϕ, I; h) ∼

∞ X

pj (ϕ, I)hj ,

j=0

and p0 (ϕ, I) = p0 (I) = K0 (I) ,

p1 (ϕ, I) = 0,

∀(ϕ, I) ∈ Tn × Eκ .

Recall that Eκ is a Cantor set in a bounded domain D such that each I ∈ Eκ is of a positive Lebesgue density, i.e. the Lebesgue measure of Eκ ∩ U is positive for any neighborhood U of I. Then given ` = (σ, µ, %), we can define S` (Tn × Eκ ) as above, where the derivatives with respect to I in Eκ are taken in the sense of Whitney. On the other hand, having a (formal) symbol p(ϕ, I) =

∞ X

pj (ϕ, I)hj ∈ F S` (Tn × Eκ ) ,

j=0

we can extend it to a formal symbol pe ∈ F S` (Tn × D) using a suitable Whitney extension theorem in Gevrey classes (see [19], Theorem 4.1). In other words, using that theorem we can extend simultaneously all pj to Gevrey functions of the same class in Tn × D with a Gevrey constant C independent on j. Recall that for any f ∈ C ∞ (Tn × D) with f (ϕ, I) = 0 for (ϕ, I) ∈ Tn × Eκ , we have ∂Iα ∂ϕβ f (ϕ, I) = 0,

for all (ϕ, I) ∈ Tn × Eκ ,

for any multi-indices α, β ∈ Zn+ . Hence, if X (k) p(k) = pj hj , k = 1, 2 , η > 0 , 0≤j≤ηh−1/%

are two extensions of the formal Gevrey symbol p in F S` (Tn × D), then p(1) − p(2) is a flat function on Tn × Eκ for each 0 < h ≤ h0 . We are going to transform Peh to a normal formal Ph0 conjugating it with an elliptic pseudodifferential operator Ah with a symbol a(ϕ, I, h) in S` (Tn × Γ) where ` = (σ, µ, %), % = σν and ν = τ + n + 1. To this end we consider p(ϕ, I; h) as (Tn × Eκ ), where `e = (σ, µ, σ + µ − 1). The main technical a symbol of the class Se ` part in the proof is the following: Theorem III.1 There exist symbols a and p0 in S` (Tn × Eκ ), ` = (σ, µ, %), given by ∞ ∞ X X a(ϕ, I, h) ∼ aj (ϕ, I)hj , p0 (I, h) ∼ p0j (I)hj , j=0

j=0

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with a0 = 1, p00 = K0 , p01 = 0, such that p ◦ a − a ◦ p0 ∼ 0 in S` (Tn × Eκ ). Theorem 1.1 follows from the result above. First, using [19], Theorem 4.1, we extend a to a symbol of a pseudodifferential operator Ah in S` (Tn ×Γ), ` = (σ, µ, %) so that a0 = 1, and set Vh = Th ◦Ah . Then we have Ph ◦Vh = Vh ◦(Ph0 +Rh ), where Ph0 and Rh have the desired properties. Unfortunately, Vh may not be an unitary operator. For this reason weP consider the pseudodifferential operator Wh = Vh∗ ◦Vh j n with a symbol w(ϕ, I, h) = ∞ j=0 wj (ϕ, I)h in S` (T × Γ). Then w0 = 1 and we have: Lemma III.2 For each j the function p0j (I) is real valued on Eκ and wj (ϕ, I) does not depend on ϕ for each I ∈ Eκ . Proof. We have w0 = 1, p00 (I) = K0 (I), p01 = 0. Moreover, it is easy to see that Wh ◦ (Ph0 + Rh ) = (Ph0∗ + Rh∗ ) ◦ Wh , since Ph is selfadjoint. Then we have p0 ◦ w = w ◦ p0 on Tn × Eκ . This equality implies 1 Lω w1 (ϕ, I) + p02 (I) − p02 (I) = 0, I ∈ Eκ , i where Lω stands for the derivative along the vector field ω(I) = ∇K0 (I), namely, def

Lw =

n X

ωj (I)∂ϕj .

(III.1)

j=1

Integrating in ϕ ∈ Tn we obtain that the imaginary part =p02 = 0 and w1 (ϕ, I) = w1 (0, I). In the same way we get by induction 1 Lω wj (ϕ, I) + p0j+1 (I) − p0j+1 (I) = 0, I ∈ Eκ , i and as above we prove that p0j is real valued and that wj+1 does not depend on I ∈ Eκ . 2 The symbol q(ϕ, I, h) of Qh = (Vh∗ ◦ Vh )−1/2 belongs to S` (Tn × Γ), ` = (σ, µ, %), and q(ϕ, I, h) − q(0, I, h) has a zero of infinite order at Tn × Eκ in view of Lemma 3.2. Now Uh = Vh ◦ Qh is the desired unitary operator. 2

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IV On the homological equation in Gevrey classes The aim of this Section is to solve the equation Lω u = f in Gevrey classes in Eκ . Lemma IV.1 Let ω ∈ C ∞ (Eκ ; Rn ) satisfy the following Gevrey type estimates: |α|

|Dα ω(I)| ≤ C1 α! τ

0

+2

, ∀ I ∈ Eκ , α ∈ Zn+ \ {0},

|hω(I), ki| ≥ κ |k|−τ , ∀ I ∈ Eκ , k ∈ Zn \ {0} .

(IV.1) (IV.2)

Then there exists a positive constant C0 depending only on n, κ, τ 0 , and C1 , such that α DI hω(I), ki−1 ≤ C |α|+1 α! max |k|τ j+τ +j (|α| − j)! τ 0 +1 , (IV.3) 0 0≤j≤|α|

for any I ∈ Eκ , 0 6= k ∈ Zn and α ∈ Zn+ . Proof. Set gk (I) = hω(I), ki for 0 6= k ∈ Zn . Applying the Leibnitz rule to the identity DIα (gk gk−1 ) = 0, |α| ≥ 1, we get X α β α −1 −1 DI gk (I) DI gk (I) DIα−β gk (I)−1 . = − gk (I) β 0<β≤α

Assuming that (IV.3) is valid for |α| < m, we shall prove it for |α| = m. In view of (IV.1) there exists C2 > 0 depending only on C1 and τ 0 such that ≤

|DIα ω(I)|

|α| C2

α! |α|

τ 0 +2 , ∀ I ∈ Eκ , α ∈ Zn+ \ {0}.

Set C0 = ε−1 C2 with some ε > 0 which will be determined later. Then using the above inequality, (IV.2), as well as the estimate x!y! ≤ (x + y)!, we obtain 0 X β! τ +1 |β| |α−β|+1 α DI gk (I)−1 ≤ κ−1 |k|τ +1 α! C2 C0 |β|

0<β≤α

×

0≤j≤|α−β|

|α|+1

≤ dε C0

max

α!

max

0≤j≤|α|−1

|α|+1

≤ dε C0

0 (|α − β| − j)! τ +1 |k|τ j+τ +j 0 (|α| − j − 1)! τ +1 |k|τ (j+1)+τ +j+1

α! max

where dε = κ−1

0≤j≤|α|

X 0<β

0 (|α| − j)! τ +1 |k|τ j+τ +j ,

ε|β| < ε κ−1

∞ X s=2

sn εs−2 < 1

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2

choosing ε sufficiently small.

For any m > 0 we set hkim = 1 + |k1 |m + · · · + |kn |m , k ∈ Zn . Next, for any j ∈ Z+ we denote m(j) = [(τ 0 + 1)j + τ ] + n + 1 , where [p] stands for the integer part of p ∈ R. Set W (k) = (1 + |k|)n+ε with 0 < ε ≤ min{τ 0 − τ, [τ ] − τ + 1}. Then

τ j + j + τ + n + ε < m(j) ≤ (τ 0 + 1)j + ν , ∀j ∈ Z+ ,

with ν = τ + n + 1. Taking into account the inequality |k|m ≤ nm hkim , m > 0, k ∈ Zn , and using Lemma 4.1, we get 0 |α|+1 α! max (|α| − j)! τ +1 hkim(j) , (IV.4) W (k) DIα hω(I), ki−1 ≤ C0 0≤j≤|α|

for any I ∈ Eκ , α ∈ Zn+ , and 0 6= k ∈ Zn , with a constant C0 > 0 depending only on n, κ, and C1 . Suppose that f ∈ C ∞ (Tn × Eκ ) satisfies α β DI Dϕ f (ϕ, I) ≤ d0 C µ|α|+|β| Γ(µ|α| + σ|β| + q) (IV.5) for any I ∈ Eκ , α, β ∈ Zn+ , and some q > 0, where Γ(x), x > 0, is the Gamma function and σ and µ are suitable positive constants. Let Z f (ϕ, I)dϕ = 0 . (IV.6) TN

We are going to solve the equation Lω u(ϕ, I) = f (ϕ, I) , u(0, I) = 0 ,

(IV.7)

and provide the corresponding estimates for the derivatives of u, where Lω is defined in (III.1) and ω(I) satisfies (IV.1), (IV.2) and (IV.4) on Eκ . Proposition IV.2 Let f ∈ C ∞ (Tn ×Eκ ) satisfy (IV.5) and (IV.6), where σ > 1 and µ−1 > σ(τ 0 +1). Then the equation (IV.7) has a unique solution u ∈ C ∞ (Tn ×Eκ ) and there is c0 = c0 (n, C0 ) > 1, C0 being the constant in (IV.4), such that if C > c0 , then the solution u of (IV.7) satisfies the estimate α γ DI Dϕ u(ϕ, I) ≤ R d0 C µ|α|+|γ|+ν Γ (µ|α| + σ|γ| + σν + q) , (IV.8) for any I ∈ Eκ , and α, γ ∈ Zn+ , where R > 0 depends only on n, τ , τ 0 and C0 .

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Proof. Consider the Fourier expansions of f and u X eihk,ϕi fk (I), f (ϕ, I) = k∈Zn

u(ϕ, I) =

X

eihk,ϕi uk (I),

k∈Zn

where fk (I) = (2π)−n

Z

e−ihk,ϕi f (ϕ, I)dϕ,

Tn

and uk (I) is defined in the same way. Now, u0 = 0 in view of (IV.6), and uk (I) = hω(I), ki−1 fk (I), I ∈ Eκ , 0 6= k ∈ Zn . Integrating by parts, and using (IV.5) we get for any γ ∈ Zn+ and m ∈ Z+ the following estimate for the Fourier coefficients of f : |kγ hkim DIα fk (I)| ≤ (n + 1)d0 C µ|α|+|γ|+m Γ(µ|α| + σ|γ| + σm + q) , for any I ∈ Eκ , k ∈ Zn , and any α, γ ∈ Zn+ . Now, taking into account (IV.4) we estimate the quantity Ak = W (k) |k γ DIα uk (I)| X α 0 |β|+1 ≤ β ! C0 max (|β| − j)! τ +1 kγ hkim(j) DIα−β fk (I) β 0≤j≤|β| 0≤β≤α

≤ (n + 1)d0

X 0≤β≤α

0 α! |β|+1 C0 max (|β| − j)! τ +1 Γ(s) C t . (α − β)! 0≤j≤|β|

Here, t = µ|α − β| + |γ| + m(j) and we write def

s = µ|α − β| + σ|γ| + σm(j) + q. Using the inequality µ − 1 > σ(τ 0 + 1) we get s ≤ µ|α − β| + σ(τ 0 + 1))j + σ|γ| + σν + q ≤ µ|α| − |β| − σ(τ 0 + 1))(|β| − j) + σ|γ| + σν + q . On the other hand, by Stirling’s formula we have (x !) τ

0

+1

≤ C2x Γ((τ 0 + 1)x) , x ≥ 1,

with some constant C2 > 0. Using the relations Γ(s + 1) = s Γ(s) , Γ(s)Γ(u) ≤ Γ(s + u) , ∀ s , u ≥ 1 ,

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and the inequalities σ > 1 and s ≥ 1, we obtain for each 0 ≤ j < |β| 0 α! (|β| − j)! τ +1 Γ(s) (α − β)!

|β|

≤ C2 (s + |β| − 1) · · · s Γ(s) Γ((τ 0 + 1)(|β| − j)) |β|

≤ C2 Γ(s + |β| + σ(τ 0 + 1)(|β| − j)) |β|

≤ C2 Γ(µ|α| + σ|γ| + σν + q) . Obviously, the same inequality holds for j = |β|. Moreover, t ≤ µ|α| + |γ| − |β| + ν. Hence, Ak ≤ (n + 1)d0 C0

X

(C0 C2 C −1 )|β| C µ|α|+|γ|+ν Γ(µ|α| + σ|γ| + σν + q) .

0≤β≤α

We choose c0 > C0 C2 > 1 and set ε = C0 C2 c−1 0 . Then for any C > c0 we obtain Ak ≤ d0 R1 C µ|α|+|γ|+ν Γ(µ|α| + σ|γ| + σν + q) , where R1 = (n + 1)C0

∞ X

sn εs−1 .

s=1

Finally, we obtain X

α γ DI Dϕ u(ϕ, I) ≤

W (k)−1 Ak

k∈Zn \0

≤ d0 R C µ|α|+|γ|+ν Γ(µ|α| + σ|γ| + σν + q) , where R = R1

X

W (k)−1 .

k∈Zn

The proof of the proposition is complete.

2

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269

V Proof of Theorem 3.1 Set ` = (σ, µ, %), where % = σν. We are looking for symbols a and p0 in S` (Tn ×Eκ ) of the form ∞ ∞ X X aj (ϕ, I)hj , p0 ∼ p0j (I)hj , a ∼ j=0

j=0

where aj ∈ C ∞ (Tn × Eκ ) and p0j ∈ C ∞ (Eκ ). Consider the symbol r = p ◦ a − a ◦ p0 ∼

∞ X

rj (ϕ, I)hj .

j=0

We have a0 = 1, p00 (I) = p0 (I) = K0 (I), and p01 = p1 = 0 in Tn × Eκ . Then r0 = r1 = 0 and for any j ≥ 2 we get rj (ϕ, I) =

1 (Lw aj−1 )(ϕ, I) + pj (ϕ, I) − p0j (I) + Fj (ϕ, I) . i

Here F2 (ϕ, I) = 0, and for j ≥ 3, we have Fj (ϕ, I) = Fj1 (ϕ, I) − Fj2 (ϕ, I) , Fj1 (ϕ, I) =

j−2 X

X

s=1

r+|γ|=j−s

Fj2 (ϕ, I) =

j−2 X

1 γ D pr (ϕ, I) ∂ϕγ as (ϕ, I) , γ! I

as (ϕ, I) p0j−s (I) .

s=1

We solve the equations rj = 0, j ≥ 2, as follows: First we put Z p0j (I) = (2π)−n (pj (ϕ, I) + Fj (ϕ, I)) dϕ ,

(V.1)

Tn

then, using Proposition 4.2, we find aj−1 from the equations 1 Lw aj−1 (ϕ, I) = fj (ϕ, I) , i Z aj−1 (ϕ, I)dϕ = 0 , Tn

where fj (ϕ, I) = p0j (I) − pj (ϕ, I) − Fj (ϕ, I). For j = 2 we obtain p02 (I)

−n

Z

= (2π)

p2 (ϕ, I) dϕ, Tn

(V.2) (V.3)

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G. Popov

and

1 Lw a1 (ϕ, I) = p02 (I) − p2 (ϕ, I) , i

Ann. Henri Poincar´ e

Z a1 (ϕ, I) dϕ = 0.

(V.4)

Tn

On the other hand, we can suppose that pj , j ∈ Z+ , satisfy the estimates j+|α|+|β|+1

|∂Iα ∂ϕβ pj (ϕ, I)| ≤ C1 j+|α+β|+1

≤ C0

α!µ β!σ (j!)σ+µ−1

α! β! Γ+ ((µ − 1)|α| + (σ − 1)|β| + (σ + µ − 1)(j − 1)) ,

(V.5)

for any multi-indices α, β ∈ Zn+ , j ∈ Z+ , where Γ+ (x) = Γ(x) for x ≥ 1 and Γ+ (x) = 1 for x ≤ 1. In particular, using Proposition 4.2 we find a solution a1 of (V.4) such that |∂Iα ∂ϕβ a1 (ϕ, I)| ≤ 2RC0 C µ|α|+|β| Γ(µ|α| + σ|β| + %) , choosing C > c0 . Fix j ≥ 3 and suppose that there exist p0k (I), 2 ≤ k ≤ j − 1, satisfying (V.1) and ak (ϕ, I), 1 ≤ k ≤ j − 2, satisfying (V.2) and (V.3), and such that |∂Iα p0k (I)| ≤ dk−3/2 C µ|α| Γ(µ|α| + (k − 1)%) , 2 ≤ k ≤ j − 1 ,

(V.6)

|∂Iα ∂ϕβ ak (ϕ, I)| ≤ dk C µ|α|+|β| Γ(µ|α| + σ|β| + k%) , 1 ≤ k ≤ j − 2 ,

(V.7)

for any (ϕ, I) ∈ Tn × Eκ and α, β ∈ Zn+ , where d ≥ 2RC0 . Choosing appropriately d as a function of n, τ, µ, σ, C0 and C only, we shall prove that p0j and aj−1 satisfy the same estimates. First we estimate the derivatives of Fj . Lemma V.1 Let C > 4C0 . Then for any α and β in Zn+ we have |DIα Dϕβ Fj1 (ϕ, I)| ≤ R1 dj−2 C µ|α|+|β| Γ(µ|α| + σ|β| + (j − 1)%) , (ϕ, I) ∈ Tn × Eκ , where R1 depends only on n, τ, µ, σ, C0 and C. Proof.

Set Br,s,γ (ϕ, I) =

1 γ ∂ pr (ϕ, I)∂ϕγ as (ϕ, I), γ! I

(V.8)

where 3 ≤ r + s + |γ| = j ,

1 ≤ s ≤ j − 2.

(V.9)

Then |γ| + r ≥ 2, and by (I.4) we have (µ−1)|γ|+(σ+µ−1)(r−1) ≥ (µ−1)(|γ|+r−1)−σ ≥ µ−σ−1 > στ 0 > 1 . (V.10) Taking into account the above inequality, (V.5) and (V.7) we obtain |∂Iα ∂ϕβ Br,s,γ (ϕ, I)|

Vol. 1, 2000

≤

Invariant Tori, II

271

X X 1 α β γ+α |∂I 1 ∂ϕβ1 pr (ϕ, I)| |∂Iα−α1 ∂ϕγ+β−β1 as (ϕ, I)| β1 γ! α1

α1 ≤α β1 ≤β

≤ d

s

X X α1 ≤α β1 ≤β

β1 ! (γ + α1 )! γ!

α α1

β β1

×Γ((µ − 1)|γ + α1 | + (σ − 1)|β1 | + (σ + µ − 1)(r − 1)) |γ+α1 |+|β1 |+r+1

≤ Γ(µ|α − α1 | + σ|γ + β − β1 | + s%) C0

C µ|α−α1 |+|γ+β−β1 | .

Now Lemma A.2 yields |∂Iα ∂ϕβ Br,s,γ (ϕ, I)| ≤ X X

ds C µ|α|+|β|

Γ(µ|α| + σ|β| + (σ + µ − 1)(|γ| + r − 1) + s%)

α1 ≤α β1 ≤β |γ|+r+1

×(2C0 /C)|α1 +β1 | C0

(2C)|γ| .

Set δ = % − σ − µ + 1. Since ν > µ we have δ = σν − µ − σ + 1 > (µ − 1)(σ − 1) > 0 . On the other hand, (j − 1)% − δ(|γ| + r − 1) = (σ + µ − 1)(|γ| + r − 1) + s% ≥ 1 . Hence, using Lemma A.1 we get Γ(µ|α| + σ|β| + (σ + µ − 1)(|γ| + r − 1) + s%) = Γ(µ|α| + σ|β| + (j − 1)% − δ(|γ| + r − 1)) ≤

Γ(µ|α| + σ|β| + (j − 1)%) . δ Γ(δ(|γ| + r − 1))

Suppose that C > 4C0 . Then, for any r, s, γ satisfying (V.9) we obtain |∂Iα ∂ϕβ Br,s,γ (ϕ, I)| ≤ R0 dj−2 C µ|α|+|β| Γ(µ|α| + σ|β| + (j − 1)ν) ×

C0 C 2|γ|+2r , δ Γ(δ(|γ| + r − 1))

where 1/2

R0

=

X α∈Zn +

2−|α| .

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Hence, we obtain j−2 X

|DIα Dϕβ Fj1 (ϕ, I)| ≤

X

|DIα Dϕβ Br,s,γ (ϕ, I)|

s=1 |γ|+r=j−s

≤ R1 dj−2 C µ|α|+|β| Γ(µ|α| + σ|β| + (j − 1)%) , where R1 =

R0 C0 δ

X p∈Zn+1 +

(V.11)

C 2|p|+2 < ∞. Γ(δ(|p| + 1)) 2

We have proved the lemma. Now we can estimate p0j (I), j ≥ 3, given by (V.1). Notice that Z Fj2 (ϕ, I)dϕ = Tn

j−2 X

Z p0j−s (I)

s=1

as (ϕ, I)dϕ = 0 Tn

in view of (V.3). Hence, p0j (I) = (2π)−n

Z (pj (ϕ, I) + Fj1 (ϕ, I))dϕ , Tn

and taking into account (V.5) and (V.11) we obtain for any j ≥ 2 the following inequality: |∂Iα p0j (I)| ≤ R1 dj−2 C µ|α| Γ(µ|α| + (j − 1)%) |α|+j+1

+ C0

Γ(µ|α| + (j − 1)(σ + µ − 1))

≤ dj−3/2 C µ|α| Γ(µ|α| + (j − 1)%) , since % = σν > µ + (σ − 1)ν > σ + µ − 1. Here we choose d sufficiently large as a function of n, τ, µ, σ, C0 and C. This proves (V.6). It remains to estimate Fj2 (ϕ, I) and aj−1 (ϕ, I). Lemma V.2 For any α and β in Zn+ we have 3

|DIα Dϕβ Fj,2 (ϕ, I)| ≤ M2 dj− 2 C µ|α|+|β| Γ(µ|α|+σ|β|+(j −1)%) , (ϕ, I) ∈ Tn ×Eκ , where M2 depends only on n, τ, µ, σ, C0 and C.

Vol. 1, 2000

Proof.

Invariant Tori, II

273

In view of (V.6) and (V.7) we have X α γ |DIα Dϕβ (as (ϕ, I)p0j−s (I))| ≤ |DI Dϕβ as (ϕ, I)||DIα−γ p0j−s (I)| γ γ≤α X α j− 32 µ|α|+|β| C Γ(µ|γ| + σ|β| + s%) ≤ d γ γ≤α

×Γ(µ|α − γ| + (j − s − 1)%). Recall that 1 ≤ s ≤ j − 2 and µ > τ 0 + 2 > 9/2. Using Lemma A.3 and the inequalities −1 j−2 B(σ|β| + s%, (j − s − 1)%) < B(s, j − s − 1) < , s−1 we obtain |DIα Dϕβ (as (ϕ, I)p0j−s (ϕ, I))| j− 32

≤ Md

C

µ|α|+|β|

X |α| −1/6 γ≤α

|γ|

B(σ|β| + s%, (j − s − 1)%)1/3 j− 32

×Γ(µ|α| + σ|β| + (j − 1)%) < M1 d

C

µ|α|+|β|

j −2 s−1

−1/3

×Γ(µ|α| + σ|β| + (j − 1)%), where M1 = 2M

P γ∈Zn +

2−|γ|/6 . On the other hand

−1/3 j−2 X j−2 s=1

s−1

≤ 2

+∞ X

2−p/3 < ∞ .

p=0

Then we get 3

|DIα Dϕβ Fj,2 (ϕ, I)| ≤ M2 dj− 2 C µ|α|+|β| Γ(µ|α| + σ|β| + (j − 1)%) , 2

which proves the lemma.

Finally, combining Lemma 5.1 and Lemma 5.2 we estimate the right hand side of (V.2) as follows: 3

|∂Iα ∂ϕβ fj (ϕ, I)| ≤ M3 dj− 2 C µ|α|+|β| Γ(µ|α| + σ|β| + (j − 1)%) , ∀α, β ∈ Zn+ , where M3 depends only on n, τ, µ, σ, C0 and C. Now applying Proposition 4.2 we find a solution aj−1 of (V.2) and (V.3) which satisfies (V.7) for k = j − 1, choosing d = d(n, τ, µ, σ, C0 , C) sufficiently large. 2

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Appendix A.1. We are going to recall certain properties of the Gamma function Z ∞ Γ(x) = e−t tx−1 dt , x > 0. 0

We have the following relation Γ(x)Γ(y) = Γ(x + y)B(x, y) , x, y > 0 , (see [2]), where

Z

1

(1 − t)x−1 ty−1 dt .

B(x, y) = 0

In particular, B(x, y) ≤ y −1 for any x ≥ 1 and y > 0, and we obtain Lemma A. 1 For any x ≥ 1 and y > 0 we have 1 Γ(x + y) . y x we set = y

Γ(x) Γ(y) ≤ For any 0 ≤ y ≤ x, x, y ∈ Zn+

x! y!(x−y)!

where 0! = 1 by

convention. Lemma A. 2 For any α1 ≤ α, β1 ≤ β, and γ ∈ Zn+ and for any s ≥ 1, r ≥ 0 with |γ| + r ≥ 2, we have β1 ! α β (γ + α1 )! Γ((µ − 1)|γ + α1 | + (σ − 1)|β1 | + (σ + µ − 1)(r − 1)) α β γ! 1 1 × Γ(µ|α − α1 | + σ|γ + β − β1 | + s%) |γ+α1 |

≤ 2 Proof.

Γ(µ|α| + σ|β| + (σ + µ − 1)(|γ| + r − 1) + s%) .

Using the equality x Γ(x) = Γ(x + 1), x > 0, we obtain β1 ! α β (γ + α1 )! Γ(µ|α − α1 | + σ|γ + β − β1 | + s%) β α γ! 1 1 ≤ 2|γ+α1 |

|β|! |α|! Γ(µ|α − α1 | + σ|β − β1 | + σ|γ| + s%) |α − α1 |! |β − β1 |!

≤ 2|γ+α1 | Γ(|α| + |β| + (µ − 1)|α − α1 | + (σ − 1)|β − β1 | + σ|γ| + s%) = 2|γ+α1 | Γ(µ|α| + σ|β| − (µ − 1)|α1 | − (σ − 1)|β1 | + σ|γ| + s%) . On the other hand, s% > 1 and by (V.10) (µ − 1)|γ + α1 | + (σ + µ − 1)(r − 1) > 1 , and applying Lemma A.1 we complete the proof of the assertion.

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275

Lemma A. 3 Let µ ≥ 9/2. Then there exists a positive constant M such that for any x, y ∈ Z+ and p ≥ 1, q ≥ 1, we have

Proof.

x+y x

7/6 Γ(µx + p) Γ(µy + q) ≤ M Γ(µ(x + y) + p + q) B(p, q)1/3 .

Suppose that x ≥ 1 and y ≥ 1. We have Γ(µx + p)Γ(µy + q) = Γ(µ(x + y) + p + q)B(µx + p, µy + q).

On the other hand Z

1

tµx+p−1 (1 − t)µy+q−1 dt ≤ B(µx, µy),

B(µx + p, µy + q) = 0

and in the same way we get B(µx + p, µy + q) ≤ B(p, q). Hence Γ(µx + p) Γ(µy + q) ≤ Γ(µ(x + y) + p + q) B(µx, µy)2/3 B(p, q)1/3 . By Stirling’s formula there exists L > 0 such that for any x ≥ 1 we have L−1 ≤ Γ(x)(2π)−1/2 x 2 −x ex ≤ L . 1

Then

Γ(µx) ≤ L(2π)1/2 xµx− 2 e−µx µµx− 2 1

≤ Lµ+1 Γ(x)µ

1

x µ−1 1 2 µµx− 2 . 2π

In the same way we get Γ(µy) ≤ Lµ+1 Γ(y)µ

y µ−1 1 2 µµy− 2 , 2π

−1

−µ

Γ(µ(x + y))

≤ L

µ+1

Γ(x + y)

Hence B(µx, µy) ≤ L

3µ+3

(2π)

≤ M

1−µ 2

xy x+y

−1/2

µ

2π x+y xy x+y

µ−1 2 B(x, y)

µ−1 2

µ−1 2

µ−1 2

µ−µ(x+y)+ 2 . 1

B(x, y)µ =

(A.1)

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G. Popov

= M

x+y x

(1−µ)/2

≤ M

x+y x

Ann. Henri Poincar´ e

−7/4 ,

since µ ≥ 9/2. This proves the assertion for x , y ≥ 1. On the other hand, if x = 0 and y ≥ 0, we have Γ(p)Γ(µy + q) = Γ(µy + p + q)B(p, µy + q) ≤ ≤ Γ(µy + p + q)B(p, q) ≤ Γ(µy + p + q)B(p, q)1/3 , 2

which completes the proof of Lemma A.3.

A.2. At the end of this section, we collect some more or less known facts about the composition of Gevrey functions. Fix µ ≥ σ ≥ 1. Let f ∈ Gσ in a neighborhood of 0 ∈ Rn1 and g = (g1 , . . . , gn1 ) ∈ Gσ,µ with respect to (x, y) ∈ Rn2 × Rn3 in a neighborhood of (0, 0), g(0, 0) = 0. Following an argument in [11] (see also [9]), we shall show that h(x, y) = f (g(x, y)) belongs to Gσ,µ in a neighborhood of (0, 0) ∈ Rn2 × Rn3 . Set F (z, x, y) = f (z) and denote L = (L1 , . . . , Ln2 ) and K = (K1 , . . . , Kn3 ), where Lj = ∂/∂xj + h∂g/∂xj , ∂/∂zi , Then given (α, β) ∈ Zn+2 × Zn+3 , we obtain (∂/∂x)α (∂/∂y)β h(x, y) =

Kj = ∂/∂yj + h∂g/∂yj , ∂/∂zi . Lα K β F (g(x, y), x, y) .

(A.2)

Set n = n1 + n2 , m = n3 , and t = (z, x), and denote by U a compact neighborhood of (0, 0) in Rn × Rm . Consider gk as functions in U and denote by A the finite set of functions a = 1, ∂gk /∂xj , and ∂gk /∂yj defined in U . Fix C > 0 such that (∂/∂t)α (∂/∂y)β a(t, y) ≤ C |α+β|+1 α! σ β! µ in U for any a ∈ A and any (α, β) ∈ Zn+ × Zm + . We suppose that F (t, y) satisfies the same inequalities in U . Notice that the right hand side of (A.2) is a sum of at most (n + m)N , N = |α + β|, terms of the form Dγ,δ (t, y) = P1γ1 Qδ11 · · · PNγN QδNN F (t, y) , where γj , δj ∈ {0, 1} , γj + δj = 1 , |γ| =

N X j=1

γj ≥ |α| , |δ| =

N X

δj ≤ |β| , |γ| + |δ| = N ,

j=1

and Pj = aj (t, y)∂/∂tkj , Qj = bj (t, y)∂/∂ymj , with aj and bj in A. We use the convention Pj0 = Q0k = 1. Then |γ|! σ |δ|! µ ≤ (|α|+|β|)! σ |β|! µ−σ ≤ C0N+1 |α|! σ |β|! µ , and the statement follows from the following lemma, which is a variant of [11], Lemma 5.3 and [9], Lemma 3.1.

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Lemma A. 4 There exists a constant C1 > 0 independent of γ and δ such that γ,δ D (t, y) ≤ (C1 C)N+1 |γ|! σ |δ|! µ . (A.3) To prove (A.3), we notice that γ,δ D (t, y) ≤ C N+1 |γ|! σ |δ|! µ #BN , where BN = {u ∈ ZN−1 : u1 + · · · + uj ≤ j, 1 ≤ j ≤ N − 1} , + and #BN stands for its cardinality. Setting w1 = u1 and wj = u1 + · · · + uj , 2 ≤ j ≤ N −1, we obtain 0 ≤ w1 ≤ 1 and 0 ≤ wj ≤ wj+1 ≤ j +1 for 1 ≤ j ≤ N −2. Assigning to any such w = (w1 , . . . , wN−1 ) the unit cube [w1 , w1 + 1] × · · · × [wN−1 , wN−1 + 1] in RN−1 , we estimate #BN from above by the volume of WN = {s = (s1 , . . . , sN−1 ) ∈ RN−1 : 0 ≤ sj ≤ sj+1 +1 ≤ N +1, 1 ≤ j ≤ N −2} . On the other hand, vol WN ≤ 2N−1 (N − 1)(N−1) /(N − 1)! ≤ C1N+1 , and we obtain the desired inequality. In the same way one can prove that h(x, y) = f (x, g(y)) is a Gσ,µ function if f ∈ Gσ,µ and g ∈ Gµ . Using a similar argument one can prove also the implicit function theorem in Gevrey classes (see also [13]). More precisely, let f = (f1 , . . . , fn1 ) ∈ Gσ,σ,µ , µ ≥ σ ≥ 1, with respect to (z, x, y) ∈ Rn1 × Rn2 × Rn3 in a neighborhood of (0, 0, 0). Suppose that f (0, 0, 0) = 0 and that dz f (0, 0, 0) is nondegenerate. Let z = ze(x, y), ze(0, 0) = 0, be the function given by the implicit function theorem. Then we obtain ze ∈ Gσ,µ in a neighborhood of (0, 0).

Acknowledgements I would like to thank Fernando Cardoso and Todor Gramchev for helpful discussions on Gevrey classes of pseudodifferential operators and quasimodes, and Johannes Sj¨ ostrand and Yves Colin de Verdi`ere for discussions on semi-classical asymptotics and quasimodes.

References [1] D. Bambusi, S. Graffi, T. Paul, Normal forms and quantization formulae, preprint, 1998. [2] H. Bateman and A. Erd´elyi, Higher transcendental functions, Vol. 1, New York, Mc Grow-Hill, 1953.

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[3] J. Bellissard and M. Vittot, Heisenberg’s picture and non commutative geometry of the semi classical limit in quantum mechanics, Ann. Inst. Henri Poincar´e, Phys. Th´eor., Vol. 52, 1990, 3, pp. 175–235. [4] N. Burq, Absence de r´esonance pr`es du r´eel pour l’op´erateur de Schr¨ odinger, Seminair de l’Equations aux D´eriv´ees Partielles, no 17, Ecole Polytechnique, 1997/1998 [5] J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfolding of singularities, Comm. in Pure and Appl. Math., Vol. 27, 1974, pp. 207–281 [6] F. Cardoso and G. Popov, Rayleigh quasimodes in linear elasticity, Comm. in Part. Diff. Equations, Vol. 17, 1992, pp. 1327–1367. [7] Y. Colin de Verdi`ere, Quasimodes sur les vari´et´es Riemanniennes, Inventiones Math., Vol. 43, 1977, pp. 15–52 [8] S. Graffi and T. Paul, The Schr¨ odinger equation and canonical perturbation theory, Comm. Math. Phys., Vol. 108, 1987, pp. 25–40 [9] T. Gramchev, The stationary phase method in Gevrey classes and Fourier Integral Operators on ultradistributions, PDE, Banach Center Publications, Warsaw, 1987 [10] T. Gramchev, M. Yoshino, Rapidly convergent iteration method for simultaneous normal forms of commuting maps, Math. Z., (to appear) [11] L. H¨ ormander, Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients, Comm. Pure Appl. Math., Vol. 24, 1971, pp. 671–704 [12] L. H¨ ormander, The analysis of linear partial differential operators, I-IV, Springer-Verlag, Berlin, 1985 [13] H. Komatsu, The implicit function theorem for ultradifferentiable mappings, Proc. Jap. Acad., Ser. A, Vol. 55, 1979, pp. 69–72 [14] V. Lazutkin, Asymptotics of the eigenvalues of the Laplacian and quasimodes. A series of quasimodes corresponding to a system of caustics close to the boundary of the domain. Math. USSR Izvestija, Vol. 7, 1973, pp. 185– 214 1974, pp. 439–466 [15] V. Lazutkin, KAM theory and semiclassical approximations to eigenfunctions, Springer-Verlag, Berlin, 1993 [16] V. Petkov and G. Popov, Semi-classical trace formula and clustering of eigenvalues for Schr¨odinger operators, Ann. Inst. Henri Poincar´e, Phys. Theor., Vol. 68, 1998, 1, pp. 17–83.

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[17] V. Petkov and G. Popov , On the Lebesgue measure of the periodic points of a contact manifold, Math. Z., Vol. 218, 1995, pp. 91–102. [18] G. Popov, Quasimodes for the Laplace operator and glancing hypersurfaces, Proceedings of the Conference on Microlocal Analysis and Nonlinear Waves, Ed. M. Beals, R. Melrose, J. Rauch, Springer Verlag, 1991. [19] G. Popov, Invariant tori effective stability and quasimodes with exponentially small error terms I - Birkhoff normal forms, AHP, Vol.1(2), 2000, pp. 223–248. [20] L. Rodino, Linear partial differential operators in Gevrey spaces, World Scientific, Singapore, 1993. [21] J. Sj¨ ostrand, Singularit´es analytiques microlocales, Ast´erisque 95, 1982. [22] J. Sj¨ ostrand, Semi-exited states in nondegenerate potential wells, Asymptotic Anal., Vol. 6, 1992, pp. 29–43. [23] J. Sj¨ ostrand, A trace formula and review of some estimates for resonances. In: L. Rodino (eds.) Microlocal analysis and spectral theory. Nato ASI Series C: Mathematical and Physical Sciences, 490, pp. 377–437: Kluwer Academic Publishers 1997 [24] J. Sj¨ ostrand and M. Zworski, Complex scaling and the distribution of scattering poles, Journal of AMS, Vol. 4(4), 1991, pp. 729–769. [25] Stefanov P., Quasimodes and resonances: Sharp lower bounds, Duke Math. J., 99, 1, 1999, pp. 75–92. [26] S.-H. Tang and M. Zworski, From quasimodes to resonances, Math. Res. Lett., 5, 1998, pp. 261–272.

Georgi Popov* D´epartement de Math´ematiques UMR 6629 Universit´e de Nantes - CNRS B.P. 92208 F-44322 Nantes-Cedex 03, France e-mail: [email protected] *Author partially supported by grant MM-706/97 with MES, Bulgaria Communicated by J. Bellissard submitted 15/09/98, accepted 06/01/99

Ann. Henri Poincar´ e 1 (2000) 281 – 306 c Birkh¨

auser Verlag, Basel, 2000 1424-0637/00/020281-26 $ 1.50+0.20/0

Annales Henri Poincar´ e

On the Asymptotic Exactness of Thomas-Fermi Theory in the Thermodynamic Limit P.B. Matesanz and J.P. Solovej Abstract. In this paper we obtain a new version of stability of matter, which in particular shows that Thomas-Fermi theory is asymptotically correct in the limit of large nuclear charges uniformly in the number of nuclei. As a consequence we give a new lower bound on the volume of matter with an improved dependence on the nuclear charges.

I Introduction One of the most celebrated results in mathematical physics is the Theorem on Stability of Matter. This result, which was originally proved by Dyson and Lenard [5], states that the binding energy per particle, for a a system of charged quantum particles, where either the positive or negatively charged particles are fermions, is bounded independently of the number of particles. Another and maybe more intuitively understandable formulation of this result is that the volume occupied by the particles increases at least linearly in the number of particles. Since the original work of Dyson and Lenard there has been numerous improvements and generalizations. In particular the work of Lieb and Thirring [21] (see also the review [14]) established the connection between stability of matter and the semi-classical Thomas-Fermi theory and greatly improved the numerical constants found by Dyson and Lenard. Among other proofs of stability of matter we can mention the work of Federbush [6] and the recent proof of Graf [10], which we shall use in the present work. The result on stability of matter has been generalized to include relativistic effects [3, 9, 22, 20], classical magnetic fields [8, 17], and quantized fields [7]. In the present work we are concerned with another type of generalization of stability of matter. We are interested in the correct dependence in the physical parameters. One of the remarkable features of macroscopic matter is that the mean atomic spacing is nearly independent of the type of atoms that the matter consists of. Or put differently the mean atomic spacing is nearly independent of the nuclear charges. In Thomas-Fermi theory distances scale as the −1/3 power of the nuclear charge, i.e., Z −1/3 . One would therefore naively expect that in macroscopic matter the volume per particle would behave as Z −1 . This is however in stark contrast to the near independence of Z which is found experimentally.

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The Lieb-Thirring proof of stability of matter implies (see [21]) a lower bound on the volume per atom of the form Z −1 for large Z. Our main goal here is to show that this is indeed not optimal. We prove that there exists δ 0 > 0 such that 0 the volume per atom is bounded below by Z −1+δ . Based on the experimental evidence one would hope to prove δ 0 = 1, we are, however, very far from such a result (see Theorem 2). Our proof is based on first showing that Thomas-Fermi theory not only gives a bound on the energy, but that this bound is indeed asymptotically correct in the limit of large nuclear charges. The bound of volume of matter is arrived at by a careful study of Thomas-Fermi theory. The fact that Thomas-Fermi theory is asymptotically correct in the limit of large nuclear charge is a classical result due to Lieb and Simon [19]. The new feature is that we establish this asymptotics uniformly in the number of nuclei allowing us to use the Thomas-Fermi approximation independently of the number of nuclei. We consider matter formed by M nuclei of charges Zj ≥ 1, j = 1, . . . , M located at positions Rj ∈ R3 . We denote R = (R1 , . . . , RM ) and Z = (Z1 , . . . , ZM ). We consider these nuclei as static and consider the non-relativistic Hamiltonian of N electrons moving in the electric potential of the nuclei. The Hamiltonian is   N  M X X X X Zj  1 Zi Zj HR,Z,N = −∆i − + + .   |x − R | |x − x | |R j i j i − Rj | i=1 j=1 i 1≤i<j≤N

1≤i<j≤M

(1) Our improvement on stability of matter is as follows. PM 1 Theorem 1 (stability of matter) Let Z = M i=1 Zi . Assume there are 0 < a ≤ A < ∞ such that the charges satisfy aZ ≤ Zj ≤ AZ for all j and that Z ≥ 1. Then there is δ > 0 universal and C,c > 0 finite constants only depending on a,A such that inf

Ψ∈H,kΨk=1

hΨ,HR,Z,N ΨiH ≥ −CTF

M X

7/3

Zj

− CM Z

7/3−δ

j=1

+cZ

M 7/3 X

1/3 Γ Z δ(Rj ) .

(2)

j=1 N V Here the Hilbert space where the Hamiltonian acts is H = L2 (R3 ;C2 ) (Fermion space), Γ(t) = min{t−1 ,t−7 } and δ(Rj ) = mini6=j |Ri − Rj | is the distance of Rj to its nearest neighbor.

In Theorem 1, the constant CTF is the corresponding Thomas-Fermi constant (see Sect. IV below for the definition) in the case Z = 1, M = 1. The last term

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283

1/3 7/3 PM cZ δ(Rj ) is positive and is an estimate on the energy it takes j=1 Γ Z to bring the nuclei close, i.e., the pressure. A similar effect holds in the context of Thomas-Fermi theory (we refer again to Sect. IV). It is worth noting that the kind of estimate given in this theorem is essentially optimal. In fact, by taking the nuclei very far apart and using the Theorem of Lieb and Simon [19] on the exactness of Thomas-Fermi theory for one nucleus we obtain inf

inf

R:#R=M Ψ∈H,kΨk=1

hΨ,HR,Z,N ΨiH ≤ −CTF

M X

7/3

Zj

+ CM Z

7/3−δ

.

(3)

j=1

As an application of Theorem 1 we shall derive our results on the total volume and density of matter. This is the content of the next theorem. Theorem 2 (the volume of matter) With the conditions in Theorem 1, assume moreover that for the configuration R inf

hΨ,HR,Z,N ΨiH ≤ −CTF

M X

7/3

Zj

+ CM Z

7/3−δ

.

(4)

Then we have that R should satisfy, for 0 < δ1 < δ/7 and k finite n o −1/3+δ1 −(δ−7δ1 ) # j : δ(Rj ) ≤ Z = o(M ), Z 7→ ∞. ≤ kM Z

(5)

Ψ∈H,kΨk=1

j=1

Another interpretation of this result is as follows: Let us call M X 1 3 volume = B Rj , 2 δ(Rj ) ; B(x,r) = y ∈ R : |x − y| ≤ r . j=1

(Above, and hereafter, |G| stands for Lebesgue measure of G). Then we have the estimate volume ≥ k0 M Z

−1+3δ1

(6)

where k0 > 0 is a universal constant. We shall prove these theorems using a tiling of space into simplices and an electrostatic inequality developed by Graf [10] and Graf-Schenker [11] which enables us to localize the electrostatic interactions into the tiles and to ignore the electrostatic interactions between these tiles up to some error which can be controlled. This reduces our problem to bounded regions, and for these we shall use Thomas-Fermi theory in order to estimate the maximal number of nuclei which is allowed in order to have non-positive energy. We ignore the tiles with positive energy, in the tiles which may have non-positive energy, we show how to reduce the asymptotic estimates to that of the whole space.

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II Graf & Schenker (GS) electrostatic inequality The setting we shall use for the GS inequality is as follows: Let Q = [0,1]3 be the unit cube in R3 . The cube Q can be written as a union of 24 congruent tetrahedra. (To see this first note that there are 6 pyramids with top at the center of the cube and base equal to one of the faces of the cube. Each pyramid consists of 4 congruent tetrahedra.) Denote by D0 the open interior of one of these tetrahedra. We then have a “tiling” TS = {Dα }α∈N , i.e, a collection of disjoint tetrahedra all congruent to D0 such that α∈N Dα = R3 . We shall also need to consider the tiling −1/3+δ

Tl = {lDα }α∈N of scale l > 0, (which will be chosen later to be Z ). Given a rotation R ∈ SO(3) and a y ∈ Q we denote by R(Tl + ly) the tiling {R(lDα + ly)}. We are now ready to state the result of Graf and Schenker. Given points x1 ,... ,xK ∈ R3 . We then consider the function 1 if x,x0 belong to the same tetrahedron of R(Tl + ly) δR,y (x,x0 ) = 0 otherwise for R ∈ SO(3) and y ∈ Q. Theorem 3 (GS inequality) There is a C > 0 such that for all K ∈ N, all (x1 ,... ,xK ) ∈ R3K , all (z1 ,... ,zK ) ∈ CK , and any l > 0 we have X 1 ≤ i,j ≤ K i 6= j

zi zj ≥ |xi − xj |

*

X 1 ≤ i,j ≤ K i 6= j

+ K zi zj CX 2 δR,y (xi ,xj ) − |zi | . |xi − xj | l i=1

(7)

For any function f on SO(3)×Q we have here defined its average over translations and rotations Z f (R,y)dµ(R)dy (8) hf i := SO(3)×Q

where dµ(R) stands for Haar measure on SO(3). We refer to [10] and [11] for a proof of this inequality.

III Localizing the Hamiltonian into the tiles The inequality of Graf and Schenker allows us to localize the potential energy into tiles. We shall also localize the kinetic energy. Since we are asking for estimates from below it is natural to do this by Neumann-bracketing.

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Lemma 1 (localization estimate) Corresponding to a tiling R(Tl + ly) with R ∈ SO(3), y ∈ Q we define the Neumann Laplacians −∆α for the tile lDα (R,y) := R(lDα + ly), α ∈ N. and in terms of this the Hamiltonians

Hα,N 0

  N0  M X X Zj χα (Rj )  + := −∆iα −  |xi − Rj |  i=1

+

j=1

X 1≤i<j≤M

X 1≤i<j≤N

1 |x − xj | i 0

(9)

Zi Zj χα (Ri )χα (Rj ) |Ri − Rj |

N V0 acting on L2 (lDα (R,y);C2 ). Here χα is the characteristic function of lDα (R,y). Then we have the following decoupling inequality:

( ) M E X XD C inf specHR,Z,N ≥ Pinf Zj2 ). inf specHα,Nα − (N + l Nα : α Nα =N α j=1

(10)

Proof. Given α ˜ = (α1 ,... ,αN ) ∈ NN . We consider the subset Aα˜ (R,y) of R3N given by

Aα˜ (R,y) = {(x1 ,... ,xN ) ∈ R3N : xj ∈ lDαj (R,y), j = 1,...,N }. We let Nα , α ∈ N denote the number of j such that αj = α. The sets Aα˜ (R,y) corresponding to different α ˜ are disjoint and we can write R3N as the union

R3N =

[

Aα˜ (R,y).

α ˜

We denote by Ψα˜ = ΨχAα˜ . We first investigate the kinetic energy

(Ψ,

N X j=1

−∆j Ψ) =

N Z XX ˜ (R,y) α ˜ j=1 Aα

|∇j Ψ| = 2

∞ XX α ˜ α=1

*

X

j:αj =α

Ψα˜ ,−∆jα Ψα˜

+ .

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The left side is independent of R and y and we may therefore average over rotated and translated tiles. From the Graf-Schenker inequality we have M X N h X Ψ, −

Zj + |xi − Rj | j=1 i=1

X 1≤i<j≤N

1 + |xi − xj |

h X M X N Zj δR,y (xi ,Rj ) ≥ Ψ, − + |xi − Rj | j=1 i=1 X

+

1≤i<j≤M

=

X 1≤i<j≤M

X 1≤i<j≤N

δR,y (xi ,xj ) |xi − xj |

M X C Zi Zj δR,y (Ri ,Rj ) i Ψ − (N + Zj2 ) |Ri − Rj | l j=1

M X N ∞ h X XX Zj χα (xi )χα (Rj ) Ψα˜ , − + |xi − Rj | α=1 j=1 i=1 α ˜

X

+

1≤i<j≤M

Zi Zj i Ψ |Ri − Rj |

Zi Zj χα (Ri )χα (Rj ) i Ψα˜ |Ri − Rj |

−

X 1≤i<j≤N

χα (xi )χα (xj ) |xi − xj |

M X C (N + Zj2 ). l j=1

Note that, although Ψα˜ is not antisymmetric in all variables, it is antisymmetric in the variables belonging to the same tile. Hence M ∞ XX X C 2 Zj2 ). (Ψ,HR,Z,N Ψ) ≥ inf specHα,Nα kΨα˜ k − (N + l α=1 j=1 α ˜

Noting that

P

2 ˜k =1 α ˜ kΨα

we conclude (10).

IV Some results about Thomas-Fermi theory In this section we shall prove a result purely about Thomas-Fermi theory. It is closely related to the results (and ideas) of Brezis and Lieb in [2] about the asymptotic behavior of many-body potentials in Thomas-Fermi theory. In this section we shall assume that we deal with neutral systems. The effect of screening in neutral systems is that long-range interactions are much smaller than for non-neutral systems. Suppose we have some configuration of M nuclei of charges Z = (Z1 ,... ,ZM ) and positions R = (R1 ,... ,RM ). The Thomas-Fermi model for this problem is defined by the following functional on positive densities ρ ∈ L1 (R3 ) ∩ L5/3 (R3 ). Z Z β 5/3 ρ(x) dx − V (x)ρ(x)dx ETF,R,Z (ρ) = β R3

1 + 2

ZZ

R3

X Zl Zs ρ(x)ρ(y) dxdy + |Rl − Rs | R3 ×R3 |x − y| M

l<s

(11)

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On the Asymptotic Exactness of Thomas-Fermi Theory

where the potential is V (x) =

PM

Zi i=1 |x−Ri |

287

and β > 0 is a parameter. In our units

the physical value of this parameter is βphys = 35 (6π 2 )2/3 . We shall however also use Thomas-Fermi theory for other values of this parameter and we therefore allow it to be arbitrary for the moment. The Thomas-Fermi energy is defined by β := ETF,R,Z

inf

0≤ρ∈L1 (R3 )∩L5/3 (R3 )

β ETF,R,Z (ρ).

(12)

β simply as ETF,R,Z . If M = 1 and When β = βphys we shall write ETF,R,Z β β Z = 1 we shall write CTF instead of |ETF,R,Z | and if β = βphys simply CTF (see also Theorems 1–2). The energy satisfies the scaling properties β λβ β ETF,R,Z = λETF,λR,Z = λ7 ETF,λR,λ −3 Z

(13)

for any λ > 0. It is known that there is a unique function ρ which minimizes the functional (12). This function fulfills the Thomas-Fermi equations 5 2/3 (x) 3 βρ

= φ(x) 1 ∗ ρ(x) φ(x) = V (x) − |y|

(14)

and is moreover the unique non-negative solution ρ of (14). This function has R P the property that ρ = M j=1 Zj , i.e., the density minimizing the functional (11) corresponds to a neutral system. For future use we shall denote by φ(R,Z,x) the unique solution φ of (14), and φat (R,Z,x) the one corresponding to a single “atom” of charge Z located at R ∈ R3 . We refer to the original paper [19] or the review [15] for the proofs of these statements and for further results on Thomas-Fermi theory. We consider the following function : β β f β (R,Z) = ETF,R,Z + CTF

M X

7/3

Zj .

(15)

J=1

Using the scaling properties of Thomas-Fermi theory we have f β (R,Z) = Z

7/3 β

f

1/3 −1 Z R,Z Z .

(16)

In [2] the asymptotics of λ7 f β (λR,Z) for large λ was studied in the case of a finite number of nuclei. Our goal here is to show an estimate on f β (R,Z) similar to the asymptotics of [2] which holds uniformly in the number of nuclei.

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Lemma 2 (pressure in Thomas-Fermi theory) Assume there is a constant a > 0 such that aZ ≤ Zj for all j. Then there is a universal c > 0 (in particular independent of M ) such that for each R ∈ R3M we have f β (R,Z) ≥ cβ −1 (aZ)7/3

M X Γ (aZ)1/3 β −1 δ(Rj )

(17)

j=1

where Γ(t) = min{t−1 ,t−7 } and δ(Rj ) = mini6=j |Ri − Rj | is as before the nearest neighbor distance. Moreover, f β is a non-decreasing function of all the Zi variables. Proof. By the scaling property (13) it is enough to prove the lemma for Zj ≥ 1 and β = βphys and in this case we omit the superscript. Then we have the following (a Feynman-Hellman type result) ∂ f (R,Z) = lim [φ(R,Z,x) − φat (Ri ,Zi ,x)]. x7→Ri ∂Zi

(18)

The right side of (18) is non-negative because of Teller’s Lemma (see [15] Theorem 3.4) , which implies that f (R,Z) is non-decreasing in any of the Zi arguments. Therefore we have that f (R,Z) ≥ f (R,(1... ,1)).

(19)

H(λ) = f (R,(λ,...,λ)).

(20)

Let H(λ) be defined by

Applying Feynman-Hellman’s formula again we obtain M X d lim [φ(R,(λ,...,λ),x) − φat (Ri ,λ,x))]. H(λ) = x7→Ri dλ i=1

(21)

For any of the points Ri we choose a nearest neighbor, which shall be denoted Ri0 , i.e., |Ri − Ri0 | = δ(Ri ). Then, by Teller’s Lemma once again, we have that M X d H(λ) ≥ lim [φ((Ri ,Ri0 ),(λ,λ),x) − φat (Ri ,λ,x))] x7→Ri dλ i=1

(22)

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Since H(0) = 0, the estimate (22) implies that Z

1

H(1) ≥

dλ 0

=

1 2

Z

M X i=1

1

dλ 0

lim [φ((Ri ,Ri0 ),(λ,λ),x) − φat (Ri ,λ,x))]

x7→Ri

M X i=1

lim [φ((Ri ,Ri0 ),(λ,λ),x) − φat (Ri ,λ,x))]

x7→Ri

+ lim 0 [φ((Ri ,Ri0 ),(λ,λ),x) − φat (Ri0 ,λ,x))] x7→Ri =

1X f ((Ri ,Ri0 ),(1,1)) 2 i=1

=

1X f ((0,Ri − Ri0 ),(1,1)). 2 i=1

(23)

M

M

In order to finish the proof, it is hence enough to show that there is a c > 0 such that cΓ(R) ≤ f ((0,R),(1,1))

(24)

If |R| ≥ 1, (24) follows from the analysis in [2]. On the other hand, if |R| ≤ 1 we will use the Thomas-Fermi equation directly to analyze the behavior. We have by the Thomas-Fermi scaling φ((0,R),(λ,λ),x) − φat (0,λ,x) = λ4/3 φ((0,λ1/3 R),(1,1),λ1/3 x) −φat (0,1,λ1/3 x)) . By the Thomas-Fermi (14) equation and the fact (see [15] Corollary 3.6) that φ((0,r),(1,1),x) ≤ φat (0,1,x) + φat (r,1,x) we have lim φ((0,r),(1,1),x) − φat (0,1,x) |x|→0

= ≥ ≥

Z

Z (3φat (0,1,y))3/2 (3φ((0,r),(1,1),y))3/2 dy − dy 3/2 (5βphys ) |y| (5βphys )3/2 |y| √ Z (3φ((0,r),(1,1),y))3/2 √ Z (3φ(y − r))3/2 1 + (1 − 2) dy − 2 dy |r| (5βphys )3/2 |y| (5βphys )3/2 |y| 1 −C |r| 1 + |r|

for r < 1. The estimate (24) then follows from the Feynman-Hellmann identity (18).

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V Estimating the kinetic energy Now, in order to compare the localized Hamiltonians with their localized ThomasFermi counterparts, we need to establish some inequalities like those of LiebThirring. More specifically, we ask for estimates on Tr(−∆0 − V )α − where ∆0 is the Neumann-Laplacian on the tile lD0 and V is a real-valued function. We are interested mainly in the cases α = 0,1. We point out that if we were dealing with Dirichlet Laplacians, the problem can be automatically reduced to that of the whole space just by extension. In the case at hand, this is not allowed, and because of that we shall formulate and prove first a version of these inequalities in the case of a cube. Finally we are dealing with tiles rather than cubes, but the problem for the tiles can be reduced to that of the cubes as explained in the appendix below. Theorem 4 (Lieb-Thirring type estimate) Let Q be the unit cube in R3 and 0 ≤ α. If V is a real-valued function in L1loc (Q) for which [V ]− ∈ L3/2+α (Q), then the following trace estimate holds: Z α Z 3/2+α α Tr(−∆Q − V )− ≤ Cα [V ]− + Cα [V ]− (25) Q

Q

Here −∆Q is the Neumann-Laplacian in Q. The constant Cα depends only on α. The same result (possibly with a different constant Cα ) holds if the cube Q is replaced by the tile D0 . We did not find a reference for this theorem and we therefore include a proof in the appendix. Using the theorem above we can provide an estimate for the kinetic energy of N antisymmetric particles on a tile (of any scale l > 0) in terms of the 1-particle density ρψ (x) which is defined as Z ρψ (x) := N kψ(x,x2 ,... ,xN )k2C2N dx2 ,... ,dxN (26) (lD0 )N −1

N V for ψ ∈ (L2 (lD0 ;C2 )) normalized. This function is an analogue of a charge distribution, and has the property R ρ =N. lD0 ψ

Theorem 5 (kinetic energy estimate for fermions on a tile) Let N V ψ ∈ L2 (lD0 ;C2 ) normalized and define ρψ as before. Then, if H0 =

N X (−∆i0 ) i=1

we have

Z hψ,H0 ψi ≥ KLT lD0

ρψ (x)5/3 dx − K1 |lD0 |−1

Z ρψ (x)2/3 dx lD0

(27)

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where KLT and K1 are positive and finite absolute constants. Proof. By a simple rescaling it is enough to consider the case l = 1. LetP us define the N potential V (x) = −ρψ (x)2/3 and the N-particle Hamiltonian HN := i=1 (−∆i0 + λV (xi )) where λ > 0 is a parameter to be fixed later. This Hamiltonian acts on N V (L2 (lD0 ;C2 )), and it can be diagonalized by the eigenfunctions of HV = −∆0 +V on L2 (lD0 ;C2 ). In fact, the lowest eigenvalue of HN is just the sum of the first N negative eigenvalues of HV so we have from Theorem 4 that Z Z C1 λ 5/2 5/3 ρψ (x) dx − ρψ (x)2/3 dx inf specHN = −Tr(HV )− ≥ −CLT λ |D0 | D0 D0 From the variational principle Z λ ρψ (x)5/3 dx = hψ,HN ψi ≥ inf specHN hψ,H0 ψi − |D0 | D0 and thus we get Z hψ,H0 ψi ≥ (λ − CLT λ5/3 )

D0

ρψ (x)5/3 dx −

C1 λ |D0 |

Z D0

ρψ (x)2/3 dx

(28)

The theorem follows by choosing λ appropriately. Theorem 5 will be one of the main tools to provide a link between Quantum Theory and Thomas-Fermi theory. This is the subject of the next section.

VI Estimating the energy of the tiles with too many nuclei The principal task we will pursue in this section is to prove that the energy of the localized pieces of the Hamiltonian on tiles is positive if the tile contains a large enough number of nuclei, depending only on the average charge Z of these nuclei and the scale l of the tiles. We shall use the estimate for the kinetic energy given by Theorem 5, but we also need an estimate which allow us to compare the electron-electron repulsion term of the localized quantum Hamiltonians with the corresponding term in the Thomas-Fermi functional (11). Such an estimate is provided by the Lieb-Oxford inequality (see [18]). N

Lemma 3 (Lieb-Oxford inequality) Let ψ ∈ L2 (R3N ;C2 ) normalized and ρψ (x) the corresponding 1-particle density function. Then * N + ZZ Z X 1 1 ρψ (x)ρψ (y) ψ, ψ ≥ dxdy − 1.68 ρψ (x)4/3 dx. |x − x | 2 |x − y| i j i<j

(29)

We are now in a position to prove a lower bound on the localized Hamiltonians in terms of the neutral Thomas-Fermi theory.

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Lemma 4 (lower bound in terms of TF theory) Let Hα,N 0 be the operator defined N V0 in (9) acting acting on L2 (lDα (R,y);C2 ). Then there exists c > 0 such that β0 inf specHα,N 0 ≥ ETF,R − cN 0 (l−2 + 1) α ,Zα

(30)

where β0 = KLT /2 and Rα and Zα are the coordinates and charges of the nuclei in the tile lDα (R,y). Before proving this lemma we shall state the following corollary, which is a simple consequence of it: β0 Corollary 1 Assume that ETF,R ≥ 0 and that Zj ≥ aZ for all charges. Then α ,Zα 0 there is c,c > 0, depending only on a and β0 , such that

inf specHα,N 0

≥ −CTF

M X

Zj χα (Rj ) + c0 Z 7/3

M 7/3 X

j=1 0

− cN (l

−2

1/3 Γ Z δα (Rj ) χα (Rj )

j=1

+ 1)

(31)

where δα is the nearest neighbor distance among the nuclei in the collection Rα . Proof. In our case at hand we have that 0 < β0 ≤ βphys . Call κ = βphys /β0 ≥ 1. We β0 − cN 0 (l−2 + 1) and in the case have, by Lemma 4 that inf specHα,N 0 ≥ ETF,R α ,Zα β0 ETF,Rα ,Zα ≥ 0 we have, using the scaling properties of Thomas-Fermi theory (13), β0 ETF,R = κETF,κRα ,Zα ≥ ETF,κRα ,Zα . Finally, by Lemma 2 we get α ,Zα ETF,κRα ,Zα

≥ −CTF

M X

7/3

Zj χα (Rj ) + f (κRα ,Zα )

(32)

j=1

≥ −CTF

M X j=1

7/3

Zj χα (Rj ) + c(a,κ)Z

M 7/3 X

1/3 Γ Z δα (Rj ) χα (Rj )

j=1

where we have used the fact that Γ(tx) ≥ t−7 Γ(x); x > 0, t ≥ 1.

Proof of Lemma 4 From the kinetic energy estimate (27) and the Lieb-Oxford N V0 2 inequality (29) we get, for ψ ∈ L (lDα (R,y);C2 ) normalized that Z Z 5/3 hψ,Hα,N 0 ψi ≥ KLT ρψ − Vα (x)ρψ (x)dx ZZ ρψ (x)ρψ (y) 1 dxdy + Uα (33) + 2 |x − y| Z Z 4/3 2/3 −C ρψ + l−3 ρψ

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where Vα (x) :=

M X

M X Zj Zi Zj χα (Rj ) and Uα := χα (Ri )χα (Rj ) |x − Rj | |Ri − Rj | j=1 i<j

(34)

Using H¨ older’s inequality and the fact that ρψ is supported in lDα (R,y) we have 1/2

Z

Z 4/3

5/3

ρψ ≤ Hence KLT 2

ρψ

(N 0 )1/2

Z

Z 5/3

ρψ − C

Z

ρψ + l−3 4/3

ρψ ≤ c(N 0 )2/3 l ≤ N 0 l. 2/3

and

Z 2/3

ρψ

≥ −cN 0 (l−2 + 1)

(35)

We shall see now that if a tile contains too many nuclei then the Thomasβ0 is positive. If on the other hand the tile contains so few Fermi energy ETF,R α ,Zα nuclei that the Thomas-Fermi energy is negative we shall in the next section prove an estimate similar to (30) but with the correct physical constant βphys rather than β0 . Lemma 5 Consider a tile lDα (R,y) and let Zα and Rα denote the charges and positions of nuclei in this tile. Assume as before that all the nuclear charges are bounded below by aZ and above by AZ. Let Mα denote the number of nuclei in the tile. There is then a constant c > 0 depending only on a, and A such that for all β β > 0 ETF,R ≥ 0 if Mα ≥ max{2,cβ −3 l3 Z}. α ,Zα Proof. By the scaling property (13) of Thomas-Fermi theory we need only consider the case β = βphys and l = 1. From Lemma 2 we have ETF,Rα ,Zα ≥ −CTF Mα (AZ)7/3 + c(aZ)7/3 λ−1 M 0

(36)

where M 0 is the number of nuclei in the tile for which the distance to the nearest other nuclei in the tile is less than or equal to λ(aZ)−1/3 for some 0 < λ < 1. We can place Mα − M 0 disjoint balls of radius (aZ)−1/3 λ/2 centered at all the remaining nuclei. Since Mα ≥ 2 it is clear that either Mα − M 0 = 0 or this radius is universally bounded. Thus we see that these balls cover a region of universally bounded volume. We conclude that (Mα − M 0 ) ≤ caZλ−3 . Hence 7/3 ETF,Rα ,Zα ≥ Mα Z −CTF A7/3 + ca7/3 λ−1 − c(aZ)10/3 λ−4 and the lemma follows if we choose λ small enough.

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VII Proving the main result We shall now use a semiclassical approximation to show that if the number of nuclei in a tile is small enough then one may use the correct physical constant βphys instead of β0 in the lower bound in (30) . The semiclassical approximation will be done using the method of coherent states in a manner very similar to the argument given in [15]. Lemma 6 (lower bound in terms of physical TF theory) Let Hα,N 0 be the operaN V0 tor defined in (9) acting acting on L2 (lDα (R,y);C2 ). We assume as before that β0 ≤ 0, where aZ ≤ Zj ≤ AZ for all j = 1,... ,M. Assume moreover that ETF,R α ,Zα β0 = KLT /2 and Rα and Zα are the coordinates and charges of the nuclei in the tile lDα (R,y). Then there exist constants c,C > 0 depending only on a,A such that if we choose δ = 2/87 and l = Z

− 13 +δ

then 4

inf specHα,N 0 ≥ ETF,Rα ,Zα − cN 0 Z 3

−δ

7

− CMα Z 3

−δ

,

(37)

where Mα is the number of nuclei in the tile. β0 Proof. Lemma 5 shows that the condition ETF,R ≤ 0 implies that α ,Zα Mα ≤ max{2,Cl3 Z}. We shall choose l such that l3 Z ≥ 1 (see the end of the proof) we may therefore assume that Mα ≤ Cl3 Z. N V0 2 For ψ ∈ L (Dα (R,y);C2 ) normalized we consider again the 1-particle density function ρψ (x) defined in (26). Using the Lieb-Oxford inequality (29) and the positivity of the Coulomb kernel,

D(f,g) :=

1 2

ZZ R3 ×R3

f (x)g(y) dxdy |x − y|

we find that for any 0 ≤ ρ˜ ∈ L1 (R3 ) ∩ L5/3 (R3 ) and 0 < < 1 we have * hψ,Hα,N 0 ψi ≥ ψ,

0

N X

+ hi ψ − D(˜ ρ, ρ˜) − (1.68)

Z ρψ (x)4/3 dx + hψ,H0 ψi + Uα (38)

i=1

where we have introduced the one-particle operator h := −(1 − )∆α − Vα (x) + | · |−1 ∗ ρ˜(x) and used Vα and Uα defined in (34). In equation (38) we have kept part of the PN 0 kinetic energy hψ,H0 ψi = i=1 hψ,−∆iα ψi in order to later use it to control errors. We choose ρ˜ to be the density that minimizes the Thomas-Fermi problem β β with parameter β equal to (1 − )βphys , i.e., ETF,R (˜ ρ) = ETF,R . α ,Zα α ,Zα

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It is not convenient to work with Neumann Laplacians, so our next goal is to replace them by Dirichlet Laplacians. In order to do this we take a partition of unity, with the following properties, on lDα (R,y) 0 ≤ Θt ,Ξt ≤ 1 Θ2t + Ξ2t = 1 Θt (x) = 1 if dist(x,∂lDα ) ≥ t t Θt (x) = 0 if dist(x,∂lDα ) ≤ 2 C |∇Θt |∞ + |∇Ξt |∞ ≤ t C |∆Θt |∞ + |∆Ξt |∞ ≤ 2 . t

(39) (40) (41) (42) (43) (44)

For any φ ∈ L2 (lDα (R,y);C2 ) we then have (omitting the parameters R,y) Z Z Z 2 2 2 |∇φ| = Θt |∇φ| + Ξ2t |∇φ|2 lDα lDα lDα Z Z Z Z 2 2 2 = |∇(φΘt )| + |φ| Θt ∆Θt + |∇(φΞt )| + |φ|2 Ξt ∆Ξt lDα lDα lDα lDα Z Z Z C ≥ |∇(φΘt )|2 + |∇(φΞt )|2 − 2 |φ|2 . (45) t lDα lDα lDα Let γ be the one-particle density matrix on L2 (lDα ;C2 ) defined as the operator with integral kernel Z γ(x,y) := N 0 hψ(y,x2 ,... ,xN 0 ),ψ(x,x2 ,... ,xN 0 )idx2 ...dxN 0 (46) (lDα (R,y))N 0 −1

N0

where h , i here denotes the inner product (antilinear in the first variable) in C2 . Because of the antisymmetry of ψ, the one-particle density matrix γ satisfies the fundamental operator inequalities 0 ≤ γ ≤ 1; 0 ≤ Trγ ≤ N 0 .

(47)

2 3 2 To any positive definite trace class operator γ 0 on L R (R ;C ) we have a non1 3 0 negative density ργ 0 ∈ L (R ), defined by Tr(γ f ) = ργ 0 (x)f (x)dx for any f ∈ L∞ (R3 ) identified as a multiplication operator on L2 (R3 ;C2 ) (it is a simple exercise in measure theory to see that this defines an L1 function). In the special case (46) (assuming that ψ is in the operator we have ργ (x) = γ(x,x) = ρψ (x). D Moreover E PN 0 domain of H0 ) we have that ψ, i=1 hi ψ = Tr(hγ) so in terms of γ we can rewrite (38) as Z hψ,Hα,N 0 ψi ≥ Tr(hγ) − D(˜ ρ, ρ˜) − (1.68) ρψ (x)4/3 dx + hψ,H0 ψi + Uα . (48) R3

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By (47) γ can be written as X γ= λi hui ,·iui ; 0 ≤ λi ≤ 1; {ui } orthonormal set

(49)

i

where the functions ui belong to the Sobolev space H 2 (lDα (R,y);C2 ). using this spectral representation together with (45) yields Tr(hγ) ≥ Tr(hΘt γΘt ) + Tr(hΞt γΞt ) −

C 0 N t2

(50)

where in (50) both Θt and Ξt are regarded as multiplication operators. (1) (2) (1) (2) Let γt = Θt γΘt and γt = Ξt γΞt . Then γt and γt fulfill the bounds (47) if γ does. We now notice that (1) ˜ t γΘt Tr hγt = Tr(hΘt γΘt ) = Tr hΘ (51) ˜ is defined as h but with Dirichlet boundary conditions instead of Neumann where h boundary conditions. Now (48), (50) and (51) imply that Z ˜ (1) − D(˜ hψ,Hα,N 0 ψi ≥ Tr hγ − (1.68) ρψ (x)4/3 dx ρ , ρ ˜ ) + U α t R3

C (2) +hψ,H0 ψi + Tr hγt − 2 N 0. t

(52)

Now we shall use coherent states given asR follows: pick some function g ∈ H 1 (R3 ) which is spherically symmetric and with R3 |g|2 = 1. Introduce a parameter r > 0 and the family of functions gr := r−3/2 g(r−1 ·). The coherent states we shall use are then given by fp,s;r (x) = gr (x − s)eip·x ; p,s ∈ R3 and let us introduce the projections

πp,s;r = hfp,s;r ,·ifp,s;r ⊗ I; I =

1 0 0 1

(53) .

Then, for any m ∈ L2 (R3 ;C2 ) a computation gives Z kmk2 = (2π)−3 hm,πp,s;r midpds R3 ×R3 Z Z |∇m(x)|2 dx = (2π)−3 |p|2 hm,πp,s;r midpds R3 R3 ×R3 Z 2 |∇gr (x)|2 dx −kmk R3 Z Z 2˜ −3 ˜ |m(x)| φr (x)dx = (2π) φ(s)hm,π p,s;r midpds R3

R3 ×R3

(54)

(55)

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˜ where φ˜r (x) = |gr |2 ∗ φ(x) and φ˜ is any function on R3 . We shall now use this for ˜ the specific choice φ(x) = Vα (x) − | · |−1 ∗ ρ˜(x), i.e., the potential appearing in the ˜ Introducing the operator h ˜ r = −(1− )∆i − φ˜r we split the first operators h and h. term in (52) as follows ˜ (1) = Tr h ˜ r γ (1) + ar γ (1) . Tr hγ (56) t t t Here

Z (1) ar (γt ) = −

R3

Vα (x) − |gr |2 ∗ Vα (x) − (Ψ(x) − |gr |2 ∗ Ψ(x)) ργ (1) (x)dx t

where Ψ = | · |−1 ∗ ρ˜ and ργ (1) (x) is the density corresponding to the operator γt . t We may then write Z i h −3 ˜ r γ (1) ˜ Tr h M (p,s)dpds = (2π) (1 − )|p|2 − φ(s) t R3 ×R3 Z (1) |∇gr (x)|2 dx (57) −Trγt (1)

R3

(1) (1) where M (p,s) = Tr γt πp,s;r . We have here considered γt as an operator on all of L2 (R3 ;C2 ). The function M (p,s) satisfies 0 ≤ M (p,s) ≤ 2.

(58)

If we minimize (57) over the functions with the property (58) we find that the minimum is given by the “bath-tub” principle, which tell us that the minimizer actually is the function M (p,s) = 2χ{(p,s):φ(s)−(1−)|p| 2 ≥0} . A straightforward ˜ calculation then shows that β ˜ r γ (1) − D(˜ Tr h − CN 0 r−2 . (59) ρ, ρ˜) + Uα ≥ ETF,R t α ,Zα Returning to the estimate of the error term ar we note that Ψ − |gr |2 ∗ Ψ ≥ 0 since (being the convolution of | · |−1 against some integrable function) ψ is superharmonic and |gr |2 is spherically symmetric and of integral one. Therefore Z (1) ar (γt ) ≥ − Vα − |gr |2 ∗ Vα ργ (1) t R3

2 ≥ − Vα − |gr | ∗ Vα 5/2 ργ (1) (60) t

5/3

by H¨ older’s inequality. We will use the term hψ,H0 ψi in (52) to control the errors R 4/3 (1) ar (γt ) and −C ρψ through the use of the Lieb-Thirring estimate Theorem 5. If we write Z (1) (61) br := hψ,H0 ψi − (1.68) ρψ (x)4/3 dx − ar (γt )

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we find from Theorem 5 and H¨older’s inequality (see also (35) )that

5/2 br ≥ −C−3/2 Vα − |gr |2 ∗ Vα 5/2 − C(−1 + l−2 )N 0

(62)

where we have used the obvious fact that ργ (1) ≤ ρψ . From Minkowski’s inequality t we have that

Vα − |gr |2 ∗ Vα

5/2

≤

Mα X

Zj | · −Rj |−1 − |gr |2 ∗ | · −Rj |−1 5/2

j=1

≤ AZ

Mα X

| · −Rj |−1 − |gr |2 ∗ | · −Rj |−1

5/2

j=1

but a computation gives

| · −Rj |−1 − |gr |2 ∗ | · −Rj |−1 = Cr1/5 5/2

(63)

so we get

Vα − |gr |2 ∗ Vα 5/2 ≤ Cr1/2 (ZMα )5/2 ≤ CMα r1/2 l9/2 Z 4 5/2

(64)

(2) It remains to estimate the term Tr hγt in (52). First we note that (2) (2) = Tr ht γt Tr hγt

with ht = −(1 − )∆α − Vα (x)χ{x:dist(x,∂(lDα ))≤t} . (65)

Now we will apply the Lieb-Thirring inequality from Theorem 4 (2) (2) Tr ht γt ≥ −Tr(ht )− ≥ −Tr ht γt − Z ≥ −CLT Vα (x)5/2 χ{x:dist(x,∂(lDα ))≤t} dx lDα Z −3 Vα (x)χ{x:dist(x,∂(lDα ))≤t} dx −C1 l

(66)

lDα

(2)

where in (66) we have used that 0 ≤ γt ≤ 1. Again using Minkowski’s inequality we conclude that Z 5/2 1/2 4 Vα (x)5/2 χ{x:dist(x,∂(lDα ))≤t} dx ≤ C Mα Z t ≤ CMα t1/2 l9/2 Z (67) lDα

l−3

Z lDα

Vα (x)χ{x:dist(x,∂(lDα ))≤t} dx ≤ CMα l−2 tZ.

(68)

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Finally we must deal with the fact that it is β = (1 − )βphys and not βphys that β0 ≤ appears in (59). We first note that it follows from the assumption that ETF,R α ,Zα 7/3

0 that ETF,Rα ,Zα ≤ CMα Z . To see this note first that if we use the minimizer β0 /2 β0 as a trial density in the functional ETF,R we ρβ0 for the functional ETF,R α ,Zα α ,Zα conclude that Z Z β0 β0 7/3 5/3 β0 /2 5/3 β0 0 ≥ ETF,Rα ,Zα ≥ ρβ0 + ETF,Rα ,Zα ≥ ρβ0 − CMα Z 2 2 where the last estimate follows from the no-binding Theorem of Thomas-Fermi theory, i.e., the positivity of the function f β0 /2 (see Lemma 2). Now using ρβ0 as a trial density in the physical Thomas-Fermi functional ETF,Rα ,Zα gives Z 5/3 β0 ETF,Rα ,Zα ≤ ETF,Rα ,Zα (ρβ0 ) = (βphys − β0 ) ρβ0 + ETF,R α ,Zα Z 7/3 5/3 ≤ (βphys − β0 ) ρβ0 ≤ 2(βphys − β0 )β0−1 CMα Z . β in terms of the physical We are now in a position to estimate ETF,R α ,Zα Thomas-Fermi energy ETF,Rα ,Zα . Indeed using the scaling (13) we see that β ETF,R α ,Zα

6

= (1 − ) ETF,Rα ,(1−)−3 Zα   M   X 7/3 6 −7 = (1 − ) f (Rα ,(1 − )−3 Zα ) − CTF (1 − ) Zj χα (Rj )   j=1   M   X 7/3 6 −7 ≥ (1 − ) f (Rα ,Zα ) − CTF (1 − ) Zj χα (Rj )   j=1   M   iX h 7/3 6 −7 Zj χα (Rj ) = (1 − ) ETF,Rα ,Zα − CTF (1 − ) − 1   j=1

≥ ETF,Rα ,Zα − CMα Z

7/3

[2 − (1 − ) − (1 − )−7 ] 6

(69)

where in the third line of (69) we have used that f is a nondecreasing function of the nuclear charges. together with the observation that (1− )−3 > 1 and in the 7/3

6

bottom line the fact that ETF,Rα ,Zα ≤ CMα Z and (1 − ) < 1. The lemma now follows if we combine all the estimates (52), (56), (59), (61), (62), (64), (66), (67), (68), and (69) and choose = Z −51δ/2

−δ

, r=Z

−57δ/2

, l=Z

−1/3+δ

,

, where δ = 2/87. The following Lemma is of purely geometrical content. It claims that if two nuclei are nearest neighbors and are close enough, then the set of (R,y) ∈ SO(3)×Q for which the two nuclei belong to the same tile of the tiling {lDα (R,y)} has positive measure. t=Z

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Lemma 7 Let B(x0 ,r0 ) ⊂ D0 be the maximal ball contained in D0 and assume that x,x0 ∈ R3 satisfy |x − x0 | ≤ lr0 /2. Then, if we call Ax,x0 = {(R,y) ∈ SO(3) × Q : x,x0 belong to the same tile of {lDα (R,y)}} we have the estimate π µ(Ax,x0 ) ≥ r03 , 6

µ = measure on SO(3) × Q.

(70)

Proof. It is fairly easy to see that the claim is translationally invariant. We may therefore assume that x = x0 = 0. Then x,x0 ∈ B(0,lr0 /2). For |y| ≤ r0 /2 and all R ∈ SO(3) we have R−1 (B(0,r0 /2)) − y = B(0,r0 /2) − y ⊂ D0 . Hence x,x0 ∈ B(0,lr0 /2) ⊂ lR(D0 + y) = lDα (R,y) and the result follows.

We are now ready to give a proof of our main result, Theorem 1 : Proof of Theorem 1. We make the tiling localization as before with a tiling of scale −1/3+δ β0 l=Z . Let A = {α : ETF,R ≥ 0}. We then find from (10) together with α ,Zα (30) and (37) that * inf specHR,Z,N ≥

X

+ * β0 ETF,R α ,Zα

+

+

X

ETF,Rα ,Zα

α6∈A

α∈A

7/3−δ

− CN Z

7/3−δ

− CN Z

− CM Z

4/3−δ

.

(71)

+ * X 1/3 Γ Z δα (Rj ) χα (Rj )

(72)

Using Corollary 1 and Lemma 2 we get inf specHR,Z,N

≥ −CTF

M X

7/3

Zj

j=1 0

+c Z

M 7/3 X j=1

− CM Z

4/3−δ

α

where c0 > 0 and δα (Rj ) denotes the nearest neighbor distance among nuclei in the tile lDα (R,y). If δ(Rj ) ≤ lr0 /2 we use Lemma 7 to conclude that + * 1/3 X 7/3 1/3 7/3 Z Γ Z δα (Rj ) χα (Rj ) ≥ c00 Z Γ Z δ(Rj ) . α

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For a nucleus Rj such that δ(Rj ) ≥ lr0 /2 we cannot argue as above, but in 7/3

1/3

7/3−7δ

7/3−δ

this case we have that Z Γ(Z δ(Rj )) ≤ Z Z and this contribution can therefore be included at the expense of increasing the constant in front of the error term. It is a result of Lieb [16] that if N ≥ N0 , where N0 is the greater integer P smaller than or equal M + 2 M j=1 Zj then inf specHR,Z,N ≥ inf specHR,Z,N0 . We may therefore without loss of generality assume that N ≤ c(Z +1)M and we hence conclude that M M 1/3 X 7/3−δ 7/3 X 7/3 inf specHR,Z,N ≥ −CTF Zj − CM Z + cZ Γ Z δ(Rj ) . j=1

j=1

This concludes the proof of Theorem 1.

The proof of Theorem 2 follows inmediately from Theorem 1 noticing that M 1/3 n o X −7δ1 −1/3+δ1 Γ Z δ(Rj ) ≥ Z # Rj ∈ R : δ(Rj ) ≤ Z . j=1

A Appendix: the Lieb-Thirring inequality (Th. 4) Proof of Theorem 4. We shall look first at the case α = 0. In the case of the Laplacian on the whole space the corresponding inequality, which holds without the last term, is the celebrated CLR estimate, proved independently by Cwickel [4], Lieb [13], and Rozenblum [23]. We are however not aware of a reference in the Neumann case. For completeness we therefore include a proof. The method of Lieb uses the Feynman-Kac formula and gives by far the best constant. In order not to introduce the Feynman-Kac formula on the cube we here appeal to Rozenblum’s method. The appearance of the constant term is due to the fact that even if V = 0, 0 is an eigenvalue corresponding to a constant eigenfunction. On the orthogonal complement of the constants the Neumann Laplacian satisfies a Sobolev inequality similar to the Dirichlet Laplacian (see e.g. [1]). More ˜ ⊂ R3 with ecprecisely, there exists CS > 0 such that for all rectangular boxes Q centricity (the ratio between R the longest and the shortest side) bounded by 2 we ˜ with ˜ φ = 0 that have for all φ ∈ L2 (Q) Q Z

Z ˜ Q

Thus if

|∇φ|2 ≥ CS

˜ Q

1/3 |φ|6

.

Z 3/2

˜ Q

[V ]− ≤ (CS )3/2 ≡ A

(73)

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P.B. Matesanz and J.P. Solovej

we have for all φ with

˜ φ=0 Q

Z

Z |∇φ| + 2

˜ Q

R

˜ Q

that

Z V |φ|

2

≥

Ann. Henri Poincar´e

Z |∇φ| − 2

˜ Q

˜ Q

Z

≥ CS

[V ]− |φ|2

1/3 |φ|

˜ Q

Hence −∆Q˜ + V ≥ 0 in {χQ˜ }⊥ if

Z −

6

2/3 Z [V

˜ Q

3/2 ]−

1/3 |φ|

6

˜ Q

≥0

R

3/2 3/2 ≡ A. ˜ [V ]− ≤ (CS ) Q

Now the idea, which goes back to Rozenblum [23], is to try to cover the whole original unit cube by cubes of eccentricity bounded by 2 (which from now on will be called bricks) such that in each of them the condition (73) is satisfied. In order to do this we shall employ the following special case of a general covering lemma given by M. de Guzm´ an [12]. Lemma 8 (covering lemma) Let J(G) be a real-valued function defined over the class of Borel subsets of a d-dimensional cube Q which is lower-semiadditive (i.e J(G1 ∪ G2 ) ≥ J(G1 ) + J(G2 ) if G1 ∩ G2 = ∅) and continuous in measure (i.e if Gt is a nested family of sets continuous in measure, t → J(Gt ) is continuous). Then, ˜ ⊂ Q such that the for every integer n ≥ 1, there is a covering Ξ of Q by bricks Q number of bricks #Ξ ≤ n, each point of the cube belongs to at most d bricks and ˜ ∈Ξ for any Q ˜ ≤ J(Q)

2d+1 J(Q). n

(74)

We will use this Lemma 8 for the function Z 3/2

J(G) = G

[V ]−

(75)

together with the following, which is a consequence of the min-max principle for the eigenvalues of self-adjoint operators: n− (−∆Q + V ) = min{codimV; V ⊂ L2 (Q) : −∆Q + V > 0 on V}

(76)

where n− (−∆Q + V ) is the number of non-positive eigenvalues of −∆Q + V . Now R 3/2 there are two cases: if Q˜ [V ]− ≤ A, then we have exactly one non-positive eigenR 3/2 value. If, on the other hand, this is not satisfied, we choose n such that 16 n Q [3V ]− ˜ 1 ,... , Q˜n ≤ A and also being the smallest integer with this property. Moreover, let Q be the covering given by the Guzm´ an Lemma and let V be sp{χQ˜1 ,... ,χQ˜n }⊥ . Since

Vol. 1, 2000

1≤

On the Asymptotic Exactness of Thomas-Fermi Theory

Pn

˜j j=1 χQ

303

≤ 3 we have, for φ ∈ V

Z

Z

Z

|∇φ|2 + Q

V |φ|2

|∇φ|2

Q

Q

1X 3 j=1 n

≥

1X χ˜ − 3 j=1 Qj n

≥

(Z

Z [V ]− |φ|2

)

Z

|∇φ| −

χQ˜ j

j=1

2

˜j Q

n X

[3V ]− |φ|

2

˜j Q

≥ 0 which follows since

Z

16 3/2 3 J(Q) ≤ A n

3/2

˜ Q

˜ ≤ [3V ]− = 33/2 J(Q)

by our choice of n and this proves the α = 0 case. Hence Z 33/2 16 3/2 [V ]− + 1. n− (−∆Q + V ) ≤ codim V = n ≤ A Q

(77)

In order to deal with other α values, we need a bound on the bottom of the spectrum. This is provided by the following: write φ ∈ H 1 (Q) as φ = φ1 + φ2 where R φ1 = ( Q φ)χQ . We have |φ|2 ≤ 2|φ1 |2 +2|φ2 |2 and then, assuming also φ normalized we get since |φ1 |2 ≤ kφk2 χQ (recall that Q is a unit cube) Z |∇φ| + Q

Z

Z V |φ|

2

2

1/3

≥ CS

Q

|φ2 |

6

Z

Q 3 2−2s

−2 ( ≥

Q

2−2s Z 3

[V ]−

t>0

Q

Z ≥ −C(s) Q

3 2−2s

2(1−s) 3s

Z −2

1−s

t

[V ]− Q

)

2−2s 3

[V ]−

3 2−2s

[V ]−

1−s 3

Q

Z

inf CS t − 2

|φ2 |

6

Z −2

[V ]− Q

Z −2

[V ]−

(78)

Q

where we have picked some 0 < s < 1 in (78). Now we can end the proof noticing that Z ∞ Tr(−∆Q + V )α = α n−λ (−∆Q + V )λα−1 dλ (79) − 0

(here n−λ (−∆Q + V ) is the number of eigenvalues less than or equal to λ ) and moreover n−λ (−∆Q + V ) ≤ n− (−∆Q − [V + λ]− )

(80)

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which is an easy consequence of the minimax principle. Now (79), (80) together imply Z |inf spec(−∆Q +V )| α Tr(−∆Q + V )− ≤ α n− (−∆Q − [V + λ]− )λα−1 dλ. (81) 0

Now we choose s =

2α 3+2α

Z Tr(−∆Q + V )α − ≤α

and from (78), (79), (80) we get

|inf spec(−∆Q +V )|

0

Z

Q

Z

≤ C0 α

[V ]−

dx Q

0

Z

3

= C0 α Z

Q

≤ Cα

3/2

[V + λ]− λα−1 dλ + |inf spec(−∆Q + V )|α

+α

2 dx[V ]−

[V Q

Z 3/2 λα−1 dλ C0 [V + λ]− dx + 1

Z

1

(1 − λ)3/2 λα−1 dλ + |inf spec(−∆Q + V )|α 0

3/2+α ]− + 2α

Z

α [V ]−

(82)

Q

where in the last inequality we have used the elementary inequalities (x + y)α ≤ xα + y α ; x,y ≥ 0, 0 < α ≤ 1 or (x + y)α ≤ 2α−1 (xα + y α ); x,y ≥ 0, 1 < α and this completes the proof for the unit cube. In order to prove the Theorem in the case of the tetrahedra described in the beginning of Sect. II we shall show that the Neumann eigenfunctions in a tetrahedra can be extended to Neumann eigenfunctions in the unit cube. Note that we can get all 24 tetrahedra making up the unit cube by repeated reflections (through faces) of one of the tetrahedra, say, D0 . Moreover, it can be easily seen that it always takes an even number of reflections to return to a given tetrahedron. Since an even number of reflections leaving D0 invariant is the identity we see that any function on D0 can be extended consistently to the whole unit cube by reflections. If φ is an eigenfunction of −∆D0 − V , then φ ∈ H 2 (D0 ) (the Sobolev space of order 2) with ∂N φ|∂D0 = 0. If we therefore define φ˜ as the extension by reflection of φ to the whole unit cube Q then φ ∈ H 2 (Q) with ∂N φ|∂Q = 0. Moreover, if V˜ is the reflected extension of V , then φ˜ is a Neumann eigenfunction of −∆Q − V˜ with the same eigenvalue. Thus Tr(−∆D0 − V )α −

≤ Tr(−∆Q − V˜ )α − R α R 3/2+α ˜ ≤ Cα Q [V ]− + Cα Q [V˜ ]− R α R 3/2+α = 24Cα D0 [V ]− + Cα 24 D0 [V ]−

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On the Asymptotic Exactness of Thomas-Fermi Theory

305

References [1] H. Brezis, Analyse Fonctionelle. Masson, Paris, (1983) [2] H. Brezis & E. Lieb, Long range atomic potentials on the Thomas-Fermi Theory. Commun.Math.Phys., 65, 231–246 (1979) [3] G. Conlon, The ground state energy of a classical gas. Commun.Math.Phys., 94, 439–458 (1984) [4] M. Cwikel, Weak type estimates for singular values and the number of bound states of Schr¨ odinger operators. Ann. of Math. 106, 93–100 (1977) [5] F. Dyson & A. L´enard, Stability of Matter I and II, J.Math.Phys., 8, 423–434 (1967); ibid 9, 698-711 (1968) [6] P. Federbush, A new approach to the stability of matter problem. I, II. J.Math.Phys., 16, 347–351 (1975); ibid. 16, 706–709 (1975) [7] C. Fefferman, J. Fr¨ ohlich, & G-M. Graf, Stability of ultraviolet-cutoff quantum electrodynamics with non-relativistic matter Commun.Math.Phys., 190 309–330 (1997) [8] C. Fefferman, Stability of Coulomb systems in a magnetic field, Proc.Nat.Acad.Sci. U.S.A., 92, 5006–5007 (1995) [9] C. Fefferman & R. de la Llave, Relativistic stability of matter. I. Rev.Mat.Iberoamericana, 2, 119–213 (1986) [10] G.M. Graf, Stability of matter through an electrostatic inequality. Helv.Phys.Acta, 70, 72–79 (1997) [11] G.M. Graf & D. Schenker: On the Molecular Limit of Coulomb gases. Commun.Math.Phys., 174, 215–227 (1995) [12] M. de Guzm´an: A covering lemma with applications to differentiability of measures and singular integral operators, Studia Math., 34, 299–317 1970 [13] E.H. Lieb, Bounds on the eigenvalues of the Laplace and Schr¨ odinger operators. Bull. Amer. Math. Soc.,82 751–753 (1976) [14] E.H. Lieb, The stability of matter. Rev.Mod.Phys., 48, 553–569 (1981) [15] E.H. Lieb, Thomas-Fermi and related theories of atoms and molecules. Rev.Mod.Phys., 53, 603–641 (1981) [16] E.H. Lieb, Bound of the maximum negative ionization of atoms and molecules, Phys. Rev. A, 29, 3018–3028 (1984)

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[17] E.H. Lieb, M. Loss, & J.P. Solovej, Stability of matter in magnetic fields. Phys.Rev.Lett., 75, 985–989 (1995) [18] E.H. Lieb & S. Oxford, An improved lower bound of the indirect Coulomb energy. Int.J.Quamtum.Chem., 19, 427–439 (1981) [19] E.H. Lieb & B. Simon, The Thomas-Fermi theory of atoms, molecules and solids. Advances in Math., 23, 22–116, (1977) [20] E.H. Lieb, H. Siedentop, & J.P. Solovej, Stability and instability of relativistic electrons in classical electromagnetic fields. Jour.Stat.Phys., 89, 37–59 (1997) [21] E.H. Lieb & W.E. Thirring, Bound for the kinetic energy of fermions which proves the stability of matter. Phys.Rev.Lett., 35, 687–689 (1975) [22] E.H. Lieb & H.T. Yau, The stability and instability of relativistic matter. Commun.Math.Phys., 118, 177–213 (1988) [23] G. Rozenblum. Distribution of the discrete spectrum of singular differential operators. Doklady Akademii Nauk SSSR, 202 5, 1012–1015 (1972) This work was supported in parts by the EU TMR-grant FMRX-CT 960001. The authors are grateful to Aarhus University where this work was initiated. J.P. Solovej is also supported by MaPhySto – Centre for Mathematical Physics and Stochastics, funded by a grant from The Danish National Research Foundation and by a grant from the Danish Natural Science Research Council

Pedro Balodis Matesanz and Jan Philip Solovej Department of Mathematics University of Copenhagen Universitetsparken 5 DK-2100 Copenhagen e-mail : [email protected] e-mail : [email protected] Communicated by J. Bellissard submitted 05/10/98 ; revised 27/10/98 ; accepted 23/12/98

Ann. Henri Poincar´ e 1 (2000) 307 – 340 c Birkh¨

auser Verlag, Basel, 2000 1424-0637/00/020307-34 $ 1.50+0.20/0

Annales Henri Poincar´ e

Global Solutions to Maxwell Equations in a Ferromagnetic Medium ∗ J.L. Joly, G. M´etivier and J. Rauch

Abstract. We study the Cauchy problem for the Landau-Lifschitz model in ferromagnetism without exchange energy. Once existence of global finite energy solutions is obtained, we study additional uniqueness and regularity properties of these solutions.

I Introduction Recently the Cauchy problem for some nonlinear models in electromagnetism have been shown to possess large finite energy solutions, by which is meant solutions satisfying the fundamental physical energy estimates, such as the electromagnetic energy and some additional conservative or dissipative estimates on new components of the electromagnetic field such as the polarization P or the magnetization M of the medium. The physical estimates hinted at are 0-order with respect to derivatives. Hence, from a mathematical point of view, the above mentioned problems should be called weak Cauchy problems if one recalls the general Cauchy nonlinear problem is well-posed under stronger regularity assumptions on data, including L2 control on derivatives up to an order that depends on the space dimension and is always greater than 0. Instead, due to special structure of the nonlinearities, the problems we mention obey strong continuity properties that mix the properties of the nonlinearities and some geometric properties of the differential operator. It allows a mathematical analysis of the solutions leading to existence, uniqueness and regularity properties, which recalls what identifies the so-called strong Cauchy problem. This is the reason why ”weak solution” is not to be found in the title of these papers. The papers mentioned above deal with nonlinear optics in nonmagnetic mediums where the relation D = E + P involves the polarization P of the electric medium modeling the interaction between medium and light which, due to large intensity of the electric field E, is nonlinear. The anharmonic oscillator model is examined in [JMR 1], [DR] being addressed to the Maxwell-Bloch quantic model ∗ Research partially supported by the U.S. National Science Foundation, U.S. Office of Naval Research, and the NSF-CNRS cooperation program under grants number NSF-DMS-9203413 and OD-G-N0014-92-J-1245 NSF-INT-9314095 respectively, and the CNRS through the Groupe de Recherche G1180 POAN.

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Ann. Henri Poincar´ e

which introduces a larger set of components, involving the density of exited electrons. In contrast the present work concerns magnetic mediums where, in a suitable system of units, D = E and the magnetic induction B = H + M involves the magnetization M which satisfies a differential equation with a source term which is nonlinear in M and H. The so-called Landau-Lifschitz model (see [LL 1] [LL 2]) for the propagation of the electromagnetic field in a ferromagnetic medium uses the nonlinear interaction term γ α F (m, h) := m∧h+ (m ∧ (m ∧ h)) , h, m ∈ R3 , (I.1) 2 1+α |m| where γ, the gyromagnetic constant, and α, the damping factor, are positive constants. In (1.1) the variables m and h need to be replaced by, respectively, the polarization M and the magnetic field H. The aim of this paper is to prove the existence of global finite energy solutions to the corresponding Maxwell system   ∂t E − curl H = 0 ∂ H + curl E = −∂t M (I.2)  t ∂t M = F (M, H) with the usual divergence free conditions for E and B divE = div(H + M ) = 0.

(I.3)

Note that the divergence free condition (1.3) is satisfied as soon as it is satisfied at t = 0, since (1.2) immediately implies that ∂t (divE(t)) = ∂t (div(H(t) + M (t))) = 0 . Functions E, H and M denote R3 -valued functions of the time-space variables (t, x) ∈ R1+3 . The electromagnetic field is (E, H) and M is the magnetization of the ferromagnetic medium. F is given by (1.1) or can be a more general interaction, see §2 below. We also study the uniqueness and regularity properties of the energy solutions. With suitable assumptions on the nonlinearity F , we show that if the Cauchy data are smooth, then the solution remains smooth for all time. Uniqueness is proved for solutions with (curlE, curlH) in L2 . The uniqueness of energy solutions remains an open problem. The precise results are presented in the next section. Before stating the main results let us say more about the model (see [JMR3] for more complete discussion and bibliography). The nonlinearity we have chosen in (1.1) is far from complete. A more complete version consists in replacing F (M, H) in the third equation of (1.2) by F (M, H + Hef f ) with Hef f = Hs + Ha + He ,

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Global Solutions to Maxwell Equations in a Ferromagnetic Medium

309

where Hs = Hs (x) is given independent of t, Ha = −k (p · M ) p, k ≥ 0, p a given unit vector in R3 and He = −k0 1Ω 4 M , k0 ≥ 0 where Ω is an open subset of R3 such that Ω = supportM . The case He 6= 0 is completely different [V], [CF]. In particular, the equations are no longer hyperbolic. In this paper we consider only the hyperbolic case, and for simplicity we assume that Hef f = 0. The terms Hs and Ha can be treated with minor modifications. Also note that the Cauchy problem for (1.2) is solved in space dimension one, in [JV1], [JV2].

II Statement of the main results First, we detail the properties required for the function F which appears in (1.2). Assumption 2.1 F is a C ∞ function on R3 × R3 with values in R3 , such that h 7→ F (m, h) is linear, m ∈ R3 F (m, h) · m = 0, m, h ∈ R3 F (m, h) · h ≤ 0, m ∈ R3

(II.1) (II.2) (II.3)

If F satisfies Assumption 2.1 there exists a function C(R) such that for all R > 0, |F (m0 , h) − F (m, h)| ≤ C(R) |m0 − m| |h|, |m| ≤ R, |m0 | ≤ R (II.4) |F (m, h)| ≤ C(R) |h|, |m| ≤ R . Assumption 2.2 F ∈ C ∞ (R3 \ {0} × R3 ; R3 ) satisfies (2.1) (2.2) (2.3) and the inequalities (2.4) hold for m, m0 6= 0. The function given by (1.1) with α = 0 or the function F (m, h) :=

α γ m∧h+ p 2 (m ∧ (m ∧ h)) , 1 + α2 δ + |m|2

h, m ∈ R3 , δ > 0

satisfy Assumption 2.1 whereas the function (1.1) with α > 0, which is homogeneous of degree one in m, is not C 1 at m = 0 but satisfies Assumption 2.2. Let us now state what we mean by finite energy solution. Definition 2.3 We say that U = (E, H, M ) is a finite energy solution in ΩT = [0, T ] × R3 if all components E, H, M belong to C 0 [0, T ]; L2 (R3 ) , M is in L∞ ([0, T ] × R3 ) and if U is a distributional solution to (1.2) (1.3). Proposition 2.4 Any finite energy solution satisfies Z Z 2 2 (|E(t, x)| + |H(t, x)| ) dx ≤ (|E(0, x)|2 + |H(0, x)|2 ) dx R3

R3

(II.5)

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Ann. Henri Poincar´ e

and |M (t, x)| = |M (0, x)| a.e.

(II.6)

Proof. Replacing ∂t M by F (M, H) in the second equation in (1.2) shows that a finite energy solution (E, H) is the solution of a linear first order symmetric hyperbolic system with source term in L1 [0, T ]; L2 (R3 ) hence it satisfies the usual energy identities. With (2.3) this implies that Z (|E(t, x)|2 + |H(t, x)|2 ) dx , 0 ≤ t ≤ T R3

decreases in time. Similarly, ∂t M = F (M, H) ∈ L1 [0, T ]; L2 (R3 ) and ∂t M ·M = 0 by (2.2). Therefore |M (t, x)| , 0 ≤ t ≤ T is time-invariant. Summing up, the quantity n0 (t) =

q kE(t)k22 + kH(t)k22 + kM (t)kL2 ∩L∞ .

(II.7)

satisfies n0 (t) ≤ n0 (0) ,

0≤t≤T

(II.8)

Remark. Condition (2.3) is not necessary for the validity of the theorems to be stated below. It only simplifies some of the estimates. It insures dissipativity as observed in (2.5). If (2.3) is suppressed in Assumptions 2.1 and 2.2, one has in place of (2.5) Z Z (|E(t, x)|2 + |H(t, x)|2 ) dx ≤ eCt (|E(0, x)|2 + |H(0, x)|2 ) dx R3

R3

with some constant C depending only on kM0 k∞ . Notation. L0 denotes the space of Cauchy data U0 = (E0 , H0 , M0 ) ∈ L2 (R3 ) such that div E0 = div (H0 + M0 ) = 0 and M0 ∈ L∞ (R3 ). Theorem 2.5 Suppose that F satisfies Assumption 2.2. Then the Cauchy problem for (1.2) with initial data U0 ∈ L0 has a finite energy solution on Ω∞ = [0, +∞[×R3 . Moreover, suppose that U0 is a bounded subset of L0 which is compact in (L2 (R3 )). Then the set U of finite energy solutions on Ω∞ , with Cauchy data in U0 is compact in (C 0 ([0, T ]; L2 (R3 )))3 for all T > 0.

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311

The solutions are constructed as limits of solutions of reguralized equations. For all λ > 1 define S λ = ϕ(λ−1 Dx ) (II.9) where ϕ ∈ C0∞ is a cut-off function supported by |ξ| ≤ 2, equal to 1 on |ξ| = 1 and such that 0 ≤ ϕ ≤ 1. Consider the following approximation of the Cauchy problem for (1.2).  λ λ = 0   ∂t E − curlH λ λ (II.10) = −S λ F (M λ , H λ ) ∂t H + curlE   λ λ λ = F (M , H ) ∂t M with initial conditions E0λ = S λ E0 , H0λ = S λ H0 , M0λ = M0 .

(II.11)

Theorem 2.6 Let F satisfy Assumption 2.2 and U0 ∈ L0 . Then, for each λ ≥ 1, the Cauchy problem (2.10), (2.11) has a unique global solution U λ which belongs to C 1 ([0, +∞[; H s × H s × L∞ ) for all s. Moreover U λ has a subsequence which converges in C 0 ([0, T ]; L2 ×L2 ×L∞ ) for all T > 0 to a global finite energy solution U ∞ of (1.2) with initial data U0 . The regularized system (2.10) has been chosen so that the two conservations laws (2.5) and (2.6) hold. This explains why there is no S λ in front of F in the right hand side of the third equation. Other regularizations having this property could be considered. Thus the family of solutions U λ is bounded in C 0 ([0, ∞[; L2 (R3 )) and M λ is bounded in L∞ . Therefore there are subsequences which converge weakly. The difficulty is to pass to the limit in the nonlinear term F (M λ , H λ ). The main point in the proof is to show that if a subsequence U λ converges weakly, then it converges strongly, and therefore the limit is a finite energy solution. This argument also accounts for the compactness result stated in Theorem 2.5. Note that Theorems 2.5 and 2.6 do not depend specifically on the space dimension 3. Analogous results could be proved in higher dimension. The proofs are given in sections 3 and 4. Next we study the smoothness and uniqueness of the finite energy solutions. The components of U do not behave all the same : some are propagated at speed 1, some are propagated at speed 0. Introduce the orthogonal decomposition of L2 (R3 ) L2 (R3 ) = L2k (R3 ) ⊕ L2⊥ (R3 ). (II.12) Functions in L2k satisfy curl = 0 whereas those in L2⊥ are such that div = 0. The corresponding projectors are denoted by Pk : L2 → L2k and P⊥ : L2 → L2⊥ , the same notation being used in Lp , 1 < p < +∞. They are Fourier multipliers

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with symbols 1 2 (ξ, · )ξ and − 1 2 ξ ∧ (ξ ∧ · ) respectively. If U = (E, H, M ) is |ξ| |ξ| a finite energy solution on ΩT (Definition 2.3), it follows from div E(t) = 0 and div (H(t) + M (t)) = 0 that Ek (t) = 0 and (H(t) + M (t))k = 0. Thus, E and H have the orthogonal decomposition E(t) = E⊥ (t),

H(t) = H⊥ (t) − Mk (t) .

(II.13)

The fields M and Hk are propagated at speed 0, while E⊥ = E and H⊥ satisfy a wave equation. More precisely one can extract from (1.2) a linear system for (E⊥ , H⊥ ) with source term and coefficients of the zero-th order term depending on M and Mk . This point of view is developed in section 5. A remarkable fact is that, for µ ≤ 1, H µ regularity for (E⊥ , H⊥ ) is propagated from the initial data, without assuming the same regularity for M . This is partly due to the fact that ∂t M ∈ L2 , while ∂t is not characteristic for the system satisfied by (E⊥ , H⊥ ). The actual proof relies on the use of Strichartz inequalities, for which the space dimension 3 is critical. Theorem 2.7 Let F satisfy Assumption 2.2. Let U0 ∈ L0 be such that E0 and H0⊥ belong to H µ (R3 ), where µ ∈]0, 1]. Consider on Ω∞ a finite energy solution U with initial data U0 . Then E and H⊥ belong to C 0 ([0, +∞[; H µ (R3 )). The next theorem completes Theorem 2.7 when µ = 1. It is proved in section 6. Theorem 2.8 Let F satisfy Assumption 2.2. Let U0 ∈ L0 be such that curlE0 and curlH0 belong to L2 (R3 ). Then there exists a unique finite energy solution on Ω∞ satisfying the Cauchy condition U|t=0 = U0 . Furthermore curlE et curlH belong to C 0 ([0, +∞[; L2 (R3 )). The propagation of the L2 regularity of curlE and curlH follows from Theorem 2.7. The uniqueness is a consequence of a stronger result about the L2 stability of such solutions. When H ∈ L∞ , uniqueness and L2 stability are trivial. For general finite energy solution, it seems difficult to get such an L∞ bound. First, the projector Pk does not act in L∞ , and all we can insure is that Hk ∈ Lp for all finite p. Second, H⊥ satisfies a wave equation, for which L∞ bounds are not an easy matter. However, when (E, H⊥ ) ∈ H 1 , one has H⊥ ∈ L1 ([0, T ]; L2 ), and estimating the L2 (L∞ ) norm of H⊥ by the L1 (L2 ) norm of H⊥ is just the forbidden limit case of Strichartz estimates in space dimension 3. Proposition 6.3 is a substitute for these estimates. It is proved in section 8. Finally, the conclusion is that H is “almost” L∞ and this is the key for Theorem 2.8. In section 7, we study the higher order regularity of solutions. Theorem 2.9 Let F satisfy Assumption 2.1 and let s ≥ 2. If U0 ∈ L0 ∩H s (R3 ) then the unique finite energy solution with initial data U0 belongs to C 0 ([0,+∞[;H s (R3 )).

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For the nonlinear function F to act in H 2 , it must be smooth enough. This explains why Assumption 2.1 is required there.

III Existence of energy solutions This section is devoted to the proof of Theorems 2.5 and 2.6. First we prove the existence and uniqueness of solutions to the approximate equations (2.10). Proposition 3.1 Let F satisfy Assumption 2.2 and Cauchy data U0 belong to L0 . Then, for all λ ≥ 1, the Cauchy problem (2.10) (2.11) has a unique global solution U λ which belongs to C 1 ([0, +∞[; H s × H s × L∞ ) for all s. Moreover there exist C > 0 independent of λ such that, for all λ ≥ 1, the following estimates hold |M λ (t, x)| kE

λ

(t)k2L2

+ kH

λ

(t)k2L2

= ≤

div E λ (t) = div (H λ (t) + S λ M λ (t)) =

|M0 (x)| , a.e. eCt kE0λ k2L2 + kH0λ k2L2

(III.1)

0 0

(III.3) (III.4)

(III.2)

Proof. For U := (E, H, M ) define Gλ (U ) := (curl H, −curl E − S λ F (M, H), F (M, H)),

(III.5)

so that the Cauchy problem (2.10), (2.11) reads d λ U (t) = Gλ (U λ (t)), dt

U (0) = (E0λ , H0λ , M0 ).

(III.6)

Let L2λ denote the closed linear subspace of L2 of functions u satisfying supp u ˆ⊂ {|ξ| ≤ 2λ}. One has L2λ ⊂ H s for all s. The space L2λ is equipped with the scalar product of L2 and satisfies L2λ ⊂ L∞ with a continuous injection kukL∞ ≤ (2λ)3/2 kukL2 , u ∈ L2λ . λ

(III.7)

L2λ ×L2λ ×L∞

into itself and is locally lipschitzean. It 1) We first show that G maps is true for the first component since curl maps L2λ linearly into itself with norm less than 2λ. The second component U 7→ −curl E − S λ F (M, H) maps L2λ × L2λ × L∞ into L2λ since kF (M, H)kL2 ≤ C(kM kL∞ )kHkL2 and S λ maps L2 into L2λ . Moreover, writing F (M, H) − F (M 0 , H 0 ) = F (M, H) − F (M 0 , H) + F (M 0 , H − H 0 ) and using (2.4) and the fact that S λ is norm one in L2 , we get kS λ F (M, H) − S λ F (M 0 , H 0 )kL2 ≤ C(R)kM − M 0 kL∞ + C(R)kH − H 0 kL2

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if kM kL∞ ≤ R, kM 0 kL∞ ≤ R, kHkL2 ≤ R. The third component belongs to L∞ since inequalities (2.4) and (3.7) imply that kF (M, H)kL∞ ≤ C(kM kL∞ )kHkL∞ ≤ (2λ)3/2 C(kM kL∞ )kHkL2 . Moreover

kF (M, H) − F (M 0 , H 0 )kL∞ ≤ C(R) kM − M 0 kL∞ kHkL∞ + kH − H 0 kL∞ ≤ (2λ)3/2 RC(R)kM − M 0 kL∞ + C(R)kH − H 0 kL2 . (III.8) for kM kL∞ ≤ R, kM 0 kL∞ ≤ R, kHkL2 ≤ R.

2) The usual theorem for ordinary differential equations in Banach spaces applies to (3.6). There exist T > 0 and U λ ∈ C 1 ([0, T [; L2λ × L2λ × L∞ ) such that U λ is the unique maximal solution of (3.6). 3) The identity (3.1) follows from (2.2), like (2.6). Let R be such that kM0 kL∞ ≤ R. Multiplying the first and second equations in (2.10) by E λ and H λ , we get, using (3.1) and (2.4), ∂t kE λ (t)k2L2 + ∂t kH λ (t)k2L2 ≤ C(R) kH λ (t)k2L2 , from which the second estimate (3.2) follows, with C = C(R). This proves that T = ∞ as claimed. 4) The first equation in (2.10) implies that ∂t div E λ = 0. The right hand side of the second equation is −∂t S λ M λ and therefore ∂t (div H λ + S λ M λ ) = 0. Since the initial conditions satisfy div E0 = div (H0 + M0 ) = 0, one has div E0λ = div (H0λ + S λ M0 ) = 0 and (3.3) (3.4) follow. This ends the proof of Proposition 3.1. We proceed now with the proof of Theorem 2.6. Because of (3.1), (3.2) the set (U λ )λ is weakly relatively compact in L2loc ([0, +∞[; L2 (R3 )). Extracting a subsequence, we may suppose that the family (U λ )λ converges weakly to U ∞ in L2loc ([0, +∞[; L2 (R3 )) thus in L2 ([0, T ]; L2 (R3 )), for every T > 0. The proof aims at showing that U ∞ is a global finite energy solution. This involves getting continuity properties of the nonlinear term. A key step in this process is a weighted L2 estimate on differences M λ − M µ with a weight that depends on the solution U ∞ . Plugging F (M λ , H λ ) − F (M µ , H µ ) = F (M λ , H ∞ ) − F (M µ , H ∞ ) +F (M λ , H λ − H ∞ ) − F (M µ , H µ − H ∞ ), into the difference of the equations for M λ and M µ in (2.10) and using (2.4) and (3.1) yields the pointwise estimate

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− M µ |2 ≤ C(R)|H ∞ ||M λ − M µ |2 + F (M λ , H λ − H ∞ ) − F (M µ , H µ − H ∞ ) · (M λ − M µ ), 1 λ 2∂ t |M

315

(III.9)

for some R such that |M0 (x)| ≤ R. The weight e−2a(t,x) absorbs the first term in the right hand side of (3.9) if ∂t a(t, x) = C(R)|H ∞ (t, x)|. The precise choice Z a(t, x) = |x|2 +

t

C(R)|H ∞ (s, x)|ds

(III.10)

0

provides a positive and almost everywhere finite function such that e−2a(t) ∈ Lp (R3 ) for all 1 ≤ p ≤ ∞ which will be used later. With a defined by (3.10), (3.9) yields 1 ∂t (e−2a |M λ −M µ |2 ) ≤ e−2a F (M λ , H λ −H ∞ )−F (M µ , H µ −H ∞ ) ·(M λ −M µ ). 2 (III.11) Proposition 3.2 There is a constant C(R, T ) such that for all δ > 0 there exist N (δ) such that for all λ ≥ N (δ) and µ ≥ N (δ) and all t ∈ [0, T ], ke

−a(t)

(M (t) − M λ

µ

(t))k2L2

Z t ≤C δ+ ke−a(s) (M λ (s) − M ∞ (s))k2L2 ds . 0

(III.12)

We postpone the proof of Proposition 3.2 until next section and finish the proof of Theorem 2.6. We fix T > 0 and prove that U ∞ is a finite energy solution on ΩT . 1) We first show that for all t ∈ [0, T ], M µ (t) converges weakly in L2 to ∞ M (t). This result being independent of Proposition 3.2 can be used for its proof. The family M λ is bounded in L∞ and (E λ , H λ ) is bounded in C 0 ([0, T ]; L2 ), as a consequence of (3.1) (3.2). Therefore, the third equation in (2.10) and the estimate (2.4) imply that there is a constant K such that for all µ and t ≤ t0 in [0, T ] kM µ (t) − M µ (t0 )kL2 ≤ K |t − t0 |. Thus {M µ }µ is equicontinuous in time with value in L2 . RAscoli’s Theorem implies that for all test function ϕ, the family of functions t 7→ M µ (t, x)ϕ(x) dx, which is bounded and equicontinuous, has subsequences which converge in C 0 ([0, T ]). R λ ∞ µ (t, x) dt dx converges RSince U → ∞U weakly, it follows that R ψ(t)ϕ(x)M R to µ ψ(t)ϕ(x)M (t, x) dt dx. This shows that M (t, x)ϕ(x) dx converges to ϕ(x)

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M ∞ (t, x) dx uniformly in t. Therefore, for all t ∈ [0, T ], M µ (t) converges weakly in L2 to M ∞ (t). As a consequence, for all t ∈ [0, T ], we have ke−a(t) (M λ − M ∞ )(t)k2L2 ≤ C lim inf ke−a(t) (M λ − M µ )(t)k2L2 . µ

Letting µ tends to infinity in (3.12) implies that that for all λ ≥ N (δ) −a(t)

ke

(M − M λ

∞

)(t)k2L2

Z ≤C δ+

t

ke−a(s) (M λ − M ∞ )(s)k2L2 ds . (III.13)

0

Gronwall’s Lemma and (3.13) imply that M λ converge to M ∞ in L2 (ΩT , e−a(t,x) dtdx)). Since a is finite almost everywhere, e−a 6= 0 almost everywhere. Thus, from any subsequence of M λ we can extract a subsequence converging pointwise almost everywhere and thus in L2 (ΩT , dtdx) thanks to the pointwise estimate (3.1) and Lebesgue’s Dominated Convergence Theorem. The limit is M ∞ , and the convergence holds for the full sequence. Thus t 7→ kM λ (t) − M µ (t)kL2 converges to 0 in L2 ([0, T ]). Since the sequence M λ is equicontinuous in t with value in L2 the above convergence holds in C 0 ([0, T ]). Thus M λ → M ∞ in C 0 ([0, T ]; L2 ). 2) We show that S λ (F (M λ , H λ )) converges in L1 ([0, T ]; L2 ) to F (M ∞ , H ∞ ) and (E λ , H λ ) converges to (E ∞ , H ∞ ) in C 0 ([0, T ]; L2 × L2 ). The Maxwell system for the difference (E λ − E ∞ , H λ − H ∞ ) involves the source term δF = S λ (F (M λ , H λ )) − F (M ∞ , H ∞ ) which we write (III.14) δF = S λ (Aλ ) + S λ (B λ ) + (I − S λ ) F (M ∞ , H ∞ ) . with

Aλ = F (M λ , H λ − H ∞ ) ,

B λ = F (M λ , H ∞ ) − F (M ∞ , H ∞ ).

The L2 estimate for the Maxwell system implies that Z t k(δE(t), δH(t))kL2 ≤ C k(δE(0), δH(0))kL2 + kδF (s)kL2 , ds .

(III.15)

0

We need to estimate the L1 ([0, T ]; L2 ) norm of the 3 terms of the decomposition (3.14). The uniform estimate of M λ and (2.4) yield Z

Z

t

kS λ (Aλ )(s)kL2 ds ≤ C 0

t

kδH(s)kL2 ds.

(III.16)

0

Using (2.4), one has the pointwise estimate |B λ (s, x)| ≤ C(R)|M λ (s, x) − M ∞ (s, x)| |H ∞ (s, x)| ≤ 2RC(R)|H ∞ (s, x)|, (III.17)

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which proves that B λ is dominated by a function that belongs to L1 ([0, T ]; L2 ). Moreover, since M λ converges to M ∞ in L2 (ΩT ), (3.17) also implies that B λ (s, x) → 0 in L1 (ΩT ). Lebesgue’s Theorem implies that B λ (s, x) → 0 in L2 (ΩT ) thus in L1 ([0, T ]; L2 ). The same result holds for the third term since S λ → I pointwise in L2 . Since (δE(0), δH(0)) converges to 0 in L2 , estimate (3.15) and Gronwall’s Lemma imply that (δE(t), δH(t)) converges to 0 in L2 , uniformly for t ∈ [0, T ]. 3) We have proved that U λ converges to U ∞ in C 0 ([0, T ]; L2 ) for all T > 0. Thus F (M λ , H λ ) and S λ (F (M λ , H λ ) converge to F (M ∞ , H ∞ ) and U ∞ = (E ∞ , H ∞ , M ∞ ) is a solution to (1.2) in the distribution sense satisfying U ∞ (0) = U0 . The estimates (3.1) imply that M ∞ ∈ L∞ (Ω∞ ). Equalities div E ∞ = div (H ∞ + M ∞ ) = 0 follow from (3.3), (3.4) letting λ → ∞. This achieves the proof that U ∞ is a finite energy solution of with initial data U0 . The proof of Theorem 2.6 is now complete. It implies the first part of Theorem 2.5. To end the proof of Theorem 2.5, consider a bounded sequence in L0 of Cauchy data U0n , such that U0n converges to U0∞ in L2 . Denote by U n a finite energy n = U0n . We need to show there exists a subsequence, still solution such that U|t=0 n denoted by U , which converges strongly to a finite energy solution U ∞ with initial ∞ data U|t=0 = U0∞ . The proof is quite parallel to the proof of Theorem 2.6. The a-priori estimates (2.5) (2.6) show that U n is bounded in C 0 ([0, +∞[; L2 ) and M n is bounded in L∞ (Ω∞ ). Therefore, extracting a subsequence, one can assume that U n converges weakly to U ∞ in L2 (ΩT ) for all T > 0. With a given by (3.10), the 0 inequality (3.11) holds for U n and U n and (3.12) is to be replaced by 0

0

ke−a(t) (M n (t) − M n (t))k2L2 ≤ ke−a(0) (M0n − M0n )k2L2 Z t +C δ+ ke−a(s) (M n (s) − M ∞ (s))k2L2 ds .

(III.18)

0

Using this estimate and the equicontinuity of M n (t), one deduces the strong convergence M n → M ∞ in C 0 ([0, T ]; L2 ) for all T > 0 as before. The strong convergence (E n , H n ) → (E ∞ , H ∞ ) follows from the energy estimate (3.15), with the simplification that there is no S λ in the analogues of (3.14-16). The strong convergence imply that F (M n , H n ) → F (M ∞ , H ∞ ) which proves that U ∞ is a finite energy solution with initial data U0 .

IV Proof of Proposition 3.2. Note that M λ and M µ have the same initial data M0 . Integrating (3.11) in t, x yields

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ke−a(t) (M λ (t) − M µ (t))k2L2 ≤ Z tZ 2 e−2a(s) F M λ (s), H λ (s) − H ∞ (s) · (M λ (s) − M µ (s))ds dx 0 Z tZ −2 . e−2a(s) F (M µ (s), H µ (s) − H ∞ (s) · (M λ (s) − M µ (s)) dsdx(IV.1) 0

We then write each term F (M ρ , H ρ − H ∞ ) in the right hand side of (4.1), where ρ stands for λ or µ as the sum of two terms, using the linearity of F with ρ ∞ respect to H and the decomposition (2.12), H ρ = H⊥ + Hkρ and H ∞ = H⊥ + Hk∞ . ρ ρ ρ The equation (3.4) implies that Hk = −S Pk M . Taking weak limits in (3.4) implies that div (H ∞ + M ∞ ) = 0 and thus Hk∞ = −Pk M ∞ . Accordingly, the two terms in the right hand side of (4.1) are equal to Z tZ 0

Z tZ − 0

Z tZ 0

ρ ∞ e−2a(s) F (M ρ (s), H⊥ (s) − H⊥ (s)) · (M λ (s) − M µ (s)) dxds

(IV.2)

e−2a(s) F M ρ (s), S ρ Pk (M ρ (s) − M ∞ (s)) · (M λ (s) − M µ (s)) dxds

(IV.3) e−2a(s) F M ρ (s), (I − S ρ )Pk (M ∞ (s)) · (M λ (s) − M µ (s)) dxds. (IV.4)

The linearity of F with respect to H implies that (4.3) is the sum of −

Z tZ F M ρ (s), S ρ Pk (e−a(s) (M ρ (s) − M ∞ (s))) · e−a(s) (M λ (s) − M µ (s)) dxds 0

(IV.5) and Z tZ F M ρ (s), [e−a(s) , S ρ Pk ](M ρ (s)−M ∞ (s)) ·e−a(s) (M λ (s)−M µ (s)) dxds − 0

(IV.6) where [ A, B ] = AB − BA. The main step in the proof of Proposition 3.2 is to show that (4.2) and (4.6) are small when λ and µ are large as asserted by the following two propositions to be proved later. Proposition 4.1 For all δ > 0 there exists N (δ) > 0 such that for all λ, µ ≥ N (δ) and for all 0 ≤ t ≤ T , Z tZ ρ ∞ (s) − H⊥ (s)) · (M λ (s) − M µ (s)) dsdx ≤ δ. (IV.7) e−2a(s) F (M ρ (s), H⊥ 0

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Proposition 4.2 For all δ > 0 there exists N (T, δ) such that for all λ, µ ≥ N and all 0 ≤ t ≤ T , Z tZ F M ρ (t), [e−a(t) , S ρ Pk ] (M ∞ (t) − M ρ (t)) · (M λ (t) − M µ (t)) dxdt ≤ δ . 0

(IV.8) Since Pk M ∞ ∈ L2 (ΩT ), (I − S ρ )Pk M ∞ converges to 0 in L2 . Together with the uniform bounds for M λ , this implies that Z tZ e−2a(s) F M ρ (s), S ρ (I − S ρ )Pk M ∞ (s)) · (M λ (s) − M µ (s)) dxds ≤ δ 0

(IV.9) when λ and µ are large enough. Using (2.4) and that S ρ Pk is bounded in L2 , the term in (4.5) is estimated by Z t Z C(R) t −a(s) ke (M ρ (s) − M ∞ (s))k2L2 ds + ke−a(s) (M λ (s) − M µ (s))k2L2 ds . 2 0 0 (IV.10) When ρ = µ, we substitute (M µ − M λ ) + (M λ − M ∞ ) in place of M ρ − M ∞ . Therefore, the sum of the two terms (4.10) is less than or equal to Z t Z t 2C(R) ke−a(s) (M λ (s) − M ∞ (s))k2L2 ds + ke−a(s) (M λ (s) − M µ (s))k2L2 ds . 0

0

(IV.11) The estimates above show that for all δ > 0, there is N (δ, T ) such that for all λ ≥ N (δ, T ), µ ≥ N (δ, T ) and t ∈ [0, T ], the right hand side of (4.1) is less than or equal to δ plus twice the term in (4.11). Using Gronwall’s Lemma, the estimate (3.12) follows Proof of Proposition 4.1. The linearity of F yields ρ ρ ∞ ∞ F (M ρ , H⊥ − H⊥ ) · (M λ − M µ ) = G(M λ , M µ ) · (H⊥ − H⊥ ),

where G is lipschitzean so that the function Aλ,µ (s, x) to be integrated in (4.7) ρ ∞ reads Aλ,µ = e−2a G(M λ , M µ ) · (H⊥ − H⊥ ) and is the product of e−2a G(M λ , M µ ) ∞ by H⊥ −H⊥ . We now study the regularity properties of each factor of this product. The Lipschitz property of G, (3.1), M0 ∈ L2 ∩ L∞ and (3.10) imply that kG(M λ , M µ )kC 0 ([0,T ];L2 ) ≤ C ,

k∂t G(M λ , M µ )kL2 (ΩT ) ≤ C.

(IV.12)

Here we have used that the Lipschitz regularity of G is sufficient to differentiate G(M λ , M µ ) with respect to t. From (2.10), we get ρ H⊥ = −∂t2 M ρ = −∂t P⊥ S ρ F (M ρ , H ρ ) .

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ρ ∞ Thus H⊥ is bounded in H −1 (ΩT ) and H⊥ ∈ H −1 (ΩT ). Moreover, ∞ kC 0 ([0,T ];L2 ) ≤ C , kH⊥ − H⊥

∞ k ((H⊥ − H⊥ ))kH −1 (ΩT ) ≤ C.

(IV.13)

In (4.12), (4.13), C is a constant which only depends on the Cauchy data and ϕ. The a-priori estimates (4.13) implies that the microlocal defect measures of H ρ − H ∞ is contained in the characteristic variety of , that is C = {τ 2 = |ξ|2 }\{0}. Similarly, the microlocal defect measures of G(M λ , M µ ) is contained in C∂t = {τ = 0}\{0}. Since 0 ∈ / C + C∂t , this implies that the product G(M λ , Gµ )(H ρ − H ∞ ) 1 tends to 0 in Lloc (ΩT ), see [Ta], [G´e]. This result can also be obtained as a consequence of theorems of multiplications of distributions with microlocal additional smoothness. In addition, (4.12) and (4.13) show that G(M λ , Gµ )(H ρ − H ∞ ) is bounded 0 in C ([0, T ], L1 ). Since the weight e−2a tends to zero as |x| → ∞, we conclude that G(M λ , Gµ )(H ρ − H ∞ ) tends to 0 in L1 (ΩT ) and Proposition 4.1 is proved. The proof of Proposition 4.2 relies on the following lemma. Lemma 4.3 For all 0 ≤ t ≤ T , lim k[e−a(t) , S ρ Pk ] (M ∞ (t) − M ρ (t))kL2 = 0

λ→∞

(IV.14)

Proof. Fix t ∈ [0, T ]. Write [e−a(t) , S ρ Pk ] = [e−a(t) − b, S ρ Pk ] + [b, S ρ Pk ], where b ∈ C0∞ . Since S ρ Pk define a bonded family of continuous operators on L4 and L2 , there exists a constant C such that k[e−a(t) − b, S ρ Pk ](M ∞ (t) − M ρ (t))kL2 ≤ Cke−a(t) − bkL4 kM ∞ (t) − M ρ (t)kL4 . (IV.15) Fix δ > 0. Estimate (3.1) implies that the sequence M ∞ (t)−M ρ (t) is bounded in L and L2 hence in L4 . We choose b ∈ C0∞ (R3 ) such that ke−a(t) − bkL4 is so small that the left hand side of (4.15) is less than δ/3 for ρ ≥ 1. Choose now ψ1 and ψ2 in C0∞ such that bψ1 = b and ψ1 ψ2 = ψ1 . The commutators [b, S ρ Pk ] form a bounded family of pseudodifferential operators of degree −1. Since (M ρ −M ∞ )(t) converges weakly to zero in L2 , as remarked in the first step of the proof of Theorem 2.6, this implies that ψ2 [b, S ρ Pk ](M ρ (t)−M ∞ (t) converges strongly to 0 in L2 . On the other hand b(M ρ (t) − M ∞ (t) converge strongly to 0 in H −1 and (1 − ψ2 )[b, S ρ Pk ] = (1 − ψ2 )S ρ Pk ψ1 b(M ρ (t) − M ∞ (t) converge strongly to 0 in L2 since (1− ψ2 )S ρ Pk ψ1 is a bounded family of operators of degree -1. Remark. Consider m(x) ∈ L∞ and p(D) of order 0. Then [m, p(D)] is not, in general, a compact operator on L2 . What Lemma 4.3 shows is that this commutator is compact when restricted to bounded subsets of Lp ∩ L2 and when m belongs to ∞

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Lq with 1/p + 1/q = 1/2, p > 2. This is the main reason why we introduced the term |x|2 in the definition (3. 10) of the weight a. Proof of Proposition 4.2. From (3.1), (2.4) and Cauchy-Schwarz inequality, it fol lows that the family t 7→ F M ρ (t), [e−a(t) , S ρ Pk ] (M ∞ (t) − M ρ (t)) · (M λ (t) − M µ (t)) is bounded in L∞ ([0, T ]; L1 ). From Lemma 4.3 it also follows that t 7→ kF M ρ (t), [e−a(t) , S ρ Pk ] (M ∞ (t) − M ρ (t)) · (M λ (t) − M µ (t))k1 converges pointwise to 0. Lebesgue’s Dominated Convergence Theorem implies that Z TZ |F M ρ (t), [e−a(t) , S ρ Pk ] (M ∞ (t) − M ρ (t)) · (M λ (t) − M µ (t))| dxdt → 0 0

as λ, µ → ∞, thus achieving the proof of Proposition 4.3.

V Curl estimates for the electromagnetic field In this section we prove Theorem 2.7. We first consider the case µ < 1 and next use the H 1/2 estimate to prove the H 1 regularity. Consider a finite energy solution U = (E, H, M ) on Ω∞ . We show that if the initial data E0 and H0 have curl in the Sobolev space H µ−1 then the same property holds for all time. We use the projectors Pk and P⊥ introduced in (2.12). Apply P⊥ to the first two equations in (1.2). Using (2.13) and the identity E = E⊥ , curlH = curlH⊥ , yields ∂t E⊥ − curlH⊥ = 0 (V.1) ∂t H⊥ + curlE⊥ = P⊥ (A H⊥ ) + P⊥ g where A is such that F (M, H) = −AH and g := AMk . We consider (5.1) as a linear system for u = (E⊥ , H⊥ ), Lu := ∂t u + Λ(∂x )u = P (au) + P g ,

(V.2)

with a given coefficient a and a given source term P g. P = P (Dx ) is a Fourier multiplier with P (ξ) a projector in C6 which is C ∞ and homogeneous of degree 0. Λ(ξ) commutes to P (ξ) and Λ(ξ)P (ξ) has eigenvalues of constant multiplicity ±|ξ|. We know that M ∈ L∞ (Ω∞ ) and M and H are continuous and bounded in time with values in L2 . Therefore ∂t M = F (M, H) ∈ L∞ ([0, ∞[; L2 ) ∩ C 0 ([0, +∞[; Lp ) for all p < ∞. The same regularity holds for any Lipschitz function of M and thus a and g satisfy a ∈ L∞ (Ω∞ ) ∩ C 0 ([0, +∞[; L2 ) ,

∂t a ∈ L∞ ([0, +∞[; L2 ) ,

(V.3)

g ∈ C 0 ([0, +∞[; Lσ ) for all σ < ∞, (V.4) ∂t g ∈ L∞ ([0, +∞[; Lq ) for all q < 2 . Note that L is symmetric hyperbolic. Therefore the Cauchy problem for (5.2), with initial data in u0 ∈ L2 such that P u0 = u0 has a unique solution u ∈ C 0 ([0, +∞[; L2 ) which satisfies P u = u . When µ < 1, Theorem 2.7 follows from the next proposition.

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Proposition 5.1 Suppose that a and g satisfy (5.3) (5.4), µ ∈]0, 1[ and u0 ∈ H µ satisfies P u0 = u0 . Then the unique solution u to (5.2) with initial data u0 belongs to C 0 ([0, +∞[; H µ ). For T > 0, introduce the space 0 Y µ (T ) = C 0 ([0, T ], H µ ) ∩ C 1 ([0, T ]; H µ−1 ) ∩ Lr ([0, T ]; Bp,2 )

(V.5)

s 3 where 2/p = 1 − µ, 2/r = µ and Bp,p 0 denotes the scale of Besov spaces in R µ (the definition is recalled below). Introduce next Z (T ), the space of functions f on ΩT which admits the decomposition f = f1 + f2 with

f1 ∈ L1 ([0, T ], H µ ) ∩ C 0 ([0, T ]; L2 ), and

−1 f2 ∈ C 0 ([0, T ]; L2 ) ∩ Lr ([0, T ]; Bp,2 ), 0 ) ∂t f2 ∈ L1 ([0, T ], H µ−1 ) + Ls ([0, T ]; Bq,2

(V.6)

(V.7)

with 2/q = 2 − µ and 2/s = 1 + µ. These spaces are equipped with the obvious norms. The main step in the proof of Proposition 5.1 is to prove a local existence theorem in Y µ (T ), with a control on T . This relies on two estimates. Lemma 5.2 For T ∈]0, 1], f ∈ Z µ (T ) and u0 ∈ H µ such that P u0 = u0 , the solution u of Lu = P f , ut=0 = u0 (V.8) belongs to Y µ (T ) and satisfies kukC 0 ([0,T ];L2 ) ≤ ku0 kL2 + 2 T kf kC 0 ([0,T ];L2 ) ,

(V.9)

kukY µ (T ) ≤ C ku0 kH µ + kf kZ µ (T ) ,

(V.10)

where C only depends on µ. Lemma 5.3 There is a constant C which only depends on kakL∞ , k∂t akL∞ (L2 ) and µ, such, that for all T ∈]0, 1] and u ∈ Y µ (T ), the product au belongs to Z µ and kaukZ µ (T ) ≤ C T µ/2 ku0 kH µ + C kukC 0 ([0,T ];L2 ) .

(V.11)

Let K denote the mapping w 7→ u, where u solves (5.8) with f = au and u0 = 0. The estimates in the two lemmas above show that there is T1 , which only depends on the norms of a, such that K 2 is a contraction in Y µ (T ) if T ≤ T1 . −1 0 The estimates (5.4) for g and the embedding Lp ⊂ Bp,2 and Lq ⊂ Bq,2 for q ≤ 2 −1 0 2 r s 0 (see [Tr]), show that g ∈ C (L ), g ∈ L (Bp,2 ) and ∂t g ∈ L (Bq,2 ). Therefore

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g ∈ Z µ (T ) for all T . Thus, the problem (5.2) with initial data u0 ∈ H µ such that P u0 = u0 has a solution u ∈ Y µ (T1 ), which satisfies P u = u and ku(T1 )kH µ ≤ C (ku0 kH µ + kgkZ µ (T1 ) ) .

(V.12)

Since T1 only depends on the norms of a, the solution can be continued to 2T1 and, by induction, to all time. So, to finish the proof of Proposition 5.1, it remains to prove the two lemmas above. Proof of Lemma 5.2. The linear problem (5.8) has a unique solution which is smooth when the data are smooth. Thus it is sufficient to prove the estimates (5.9) and (5.10) for smooth solutions. The first one is the standard energy estimate in L2 for symmetric hyperbolic systems. The main ingredient to prove (5.10) are Strichartz estimates. 1) First we recall the definition of Besov spaces. Introduce ϕ ∈ C0∞ (R3 ), 0 ≤ ϕ ≤ 1, supported in {|ξ| ≤ 2} and equal to 1 on {|ξ| ≤ 1}. Introduce next ϕk (ξ) := ϕ(2−k ξ) for k ≥ 0 (V.13) ψk (ξ) := ϕ(2−k ξ) − ϕ(2−k+1 ξ) for k ∈ Z . Denote by Sk [resp ∆k ] the Fourier multipliers with symbols ϕk [resp ψk ]. s Recall that Bp,2 is the space of temperate distributions u such that kS0 uk2Lp +

+∞ X

2−2ks k∆k uk2Lp < ∞ .

(V.14)

k=1 µ s Also recall that H µ = B2,2 ([Tr]). The homogeneous spaces B˙ p,2 have a similar definition with +∞ X 2−2ks k∆k uk2Lp < ∞ . k=−∞

in place of (5.14). 2) Introduce the groups of operators G± (t) of Fourier multipliers e±it|ξ| . Since the eigenvalues of Λ(ξ)P (ξ) are ±|ξ| and have constant multiplicity, the fundamental solution of L, for data in the kernel of I − P , is G+ (t) P+ + G− (t) P− ,

(V.15)

where P± are Fourier multipliers with smooth symbols P± (ξ) which are orthogonal s projectors, with P = P+ + P− . The operators P± act in Lσ and in Bσ,2 for all s and all σ ∈]1, +∞[. For G± , we use the Strichartz estimates proved in [GV] (see also [LS]). For v0 ∈ H˙ µ and f ∈ Ls1 ([0, T ], B˙ qσ1 ,2 ), v(t) = G± (t)v0 and w(t) = Rt G± (t − t0 )f (t0 )dt0 belong to Lr1 ([0, T ], B˙ pρ1 ,2 ) and 0 ( kvkLr1 ([0,T ],B˙ σ ) ≤ C kv0 kH˙ µ p1 ,2 (V.16) kwkLr1 ([0,T ],B˙ σ ) ≤ C kf kLs1 ([0,T ],B˙ σ ) p1 ,2

q1 ,2

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provided that

µ = ρ + 1 − 2/p1 = σ + 1 − 2/q1 , 1/r1 + 1/p1 = 1/2 , 2 ≤ p1 < ∞ , 1/s1 + 1/q1 = 3/2 ,

1 < q1 ≤ 2 .

In the case ρ = µ, p1 = 2, r1 = ∞, v and w belong C 0 ([0, T ], H˙ µ ). 3) To prove (5.10), we use the linearity of (5.8) and split u and the data into several pieces. First, we note that the low frequencies of u are controlled by the L2 norm. Thus the estimate (5.10) for S0 u immediately follows from (5.9). 4) Consider the solution of (5.8) with f = 0 and initial data (1 − S0 )u0 = 0. Then, thanks to the form (5.13) of the fundamental solution, the estimates (5.16) imply that 0 ) ≤ C ku0 kH µ . kukC 0 ([0,T ];H µ ) + kukLr ([0,T ];Bp,2 We have replaced the homogeneous spaces by the inhomogeneous ones, using that the spectrum of u is contained in |ξ| ≥ 1. The same remark holds below when we use again (5.16). Moreover, since Lu = 0, one has k∂t ukC 0 ([0,T ];H µ−1 ) ≤ C kukC 0 ([0,T ];H µ ) , and (5.10) is satisfied. 5) Split f into f1 +f2 such that (5.6) (5.7) hold. Consider the solution of (5.8) with right hand side (1−S0 )f1 and vanishing initial data. Then (5.16) implies that 0 ) ≤ C kf1 kL1 ([0,T ];H µ ) kukC 0 ([0,T ];H µ ) + kukLr ([0,T ];Bp,2

To estimate ∂t u one uses the equation and the inequality kf1 kC 0 ([0,T ];H µ−1 ) ≤ kf1 kC 0 ([0,T ];L2 ) . This implies (5.10). 6) With f2 satisfying (5.7), consider the solution u of (5.8) with right hand side (1 − S0 )f2 and vanishing initial data. Using the fundamental solution (5.15), one gets that u = u+ + u− , whose spatial Fourier transforms are given by Z t ei(t−s)|ξ| fb± (s, ξ) ds , f± := (1 − S0 )P± f2 . u b± (t, ξ) = 0

Integrating by parts shows that u+ = v + w with Z t i vb(t, ξ) = ei(t−s)|ξ| gb(s, ξ) ds , gb(s, ξ) := − ∂t fb± (s, ξ) |ξ| 0 and w(t) = − G+ (t)h(0) + h(t) ,

i b b h(t, ξ) := f± (s, ξ) . |ξ|

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The assumption (5.7) implies that ∂t f2 is a sum of two terms f20 + f200 with the 0 00 indicated regularity. Accordingly, one has f+ = f+ +f+ , g = g 0 +g 00 and v = v 0 +v 00 . The estimates (5.16) imply that 0 0 0 0 ) ≤ C kg kL1 ([0,T ];H µ ) ≤ C kf kL1 ([0,T ];H µ−1 ) kv 0 kC 0 ([0,T ];H µ )∩Lr ([0,T ];Bp,2 2

and 00 0 00 0 ) ≤ C kg kLs ([0,T ];B 1 ) ≤ C kf kLs ([0,T ];B 0 ) . kv 00 kC 0 ([0,T ];H µ )∩Lr ([0,T ];Bp,2 2 q,2 q,2

For the boundary term w, we have 0 ) kwkC 0 ([0,T ];H µ )∩Lr ([0,T ];Bp,2

0 ) ≤ 2 khkC 0 ([0,T ];H µ )∩Lr ([0,T ];Bp,2 ≤ Ckf2 kC 0 ([0,T ];L2 )∩Lr ([0,T ];B −1 ) . p,2

Adding up the different estimates above and using the equation to estimate ∂t u+ yields ku+ kY µ (T ) ≤ C kf kZ µ (T ) . The estimate for u− is similar and thus (5.10) is satisfied. This finishes the proof of Lemma 5.2. Proof of Lemma 5.3. To simplify notations, we assume that a and u are scalar functions. For smooth functions, the product au can be split into two pieces (see [Bo]) e u) + Π(u, a) := a u = Π(a,

+∞ X

Sk+2 a ∆k u +

k=0

+∞ X

Sk−3 u∆k a .

(V.17)

k=3

Here the notations are slightly different from those used in (5.13). From now on, by e u) and Π(u, a) extend as bilinear operators definition, ∆0 = S0 . We prove that Π(a, acting on functions a which satisfy (5.3) and u ∈ Y µ , so that they satisfy (5.6) and (5.7) respectively. 1) For fixed t and σ ≥ 0, let us prove that e u)(t)kH σ ≤ C ka(t)kL∞ ku(t)kH σ . kΠ(a,

(V.18)

For σ > 0, (5.18) follows from the estimate kSk+2 a ∆k ukL2 ≤ kSk+2 akL∞ k∆k ukL2 and the fact that the spectrum of Sk+2 a ∆k u is contained in the ball {|ξ| ≤ 2k+4 }. For σ = 0 the proof is much more delicate. It is a classical result in harmonic analysis which can be found for instance in [CM].

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For all ρ < ∞, (5.3) implies P that a ∈ C 0 ([0, T ], Lρ ). This implies that for 0 µ all u ∈ C ([0, T ], H ), the series k Sk+2 a ∆k u converges in C 0 ([0, T ], H σ ) for all σ < µ and the partial sums are bounded in C 0 ([0, T ], H µ ). Therefore the sum e u) extends to the belongs to C 0 ([0, T ], L2 ) ∩ L∞ ([0, T ], H µ ). This shows that Π(a, 0 µ e a’s which satisfy (5.3) and u ∈ C ([0, T ], H ). Moreover, Π(a, u) ∈ C 0 ([0, T ], L2 ) ∩ L1 ([0, T ], H µ ) and (5.18) implies that e u)kC 0 ([0,T ],L2 )∩L1 ([0,T ],H µ ) ≤ C kakL∞ T kukC 0 ([0,T ];H µ ) + kukC 0 ([0,T ];L2 ) . kΠ(a, (V.19) 2) For fixed t and σ ≤ 0, let us show that kΠ(u, a)(t)kH σ ≤ C ka(t)kL∞ ku(t)kH σ .

(V.20)

P The proof for σ < 0 is easy, using that kSk−3 u ∆k akL2 ≤ j≤k−3 k∆j ukL2 k∆k akL∞ and the fact that the spectrum of Sk+2 a ∆k u is contained in the annulus {2k−2 ≤ |ξ| ≤ 2k+2 }. The result for σ = 0 is proved in [CM]. It is equivalent to (5.18) with σ = 0 since the product au has an obvious estimate in L2 . In addi0 tion, since a ∈ C 0 ([0, T ], Lρ ) for all ρ < ∞ and u ∈ C 0 ([0, T ], H µ ) ⊂ C 0 ([0, T ], Lρ ) e u) exfor some ρ0 > 2, it follows that au ∈ C 0 ([0, T ], L2 ). Thus Π(u, a) = au− Π(a, 0 µ 0 tends to a satisfying (5.3) and u ∈ C ([0, T ], H ) so that Π(u, a) ∈ C ([0, T ], L2 ). Moreover, (5.20) implies that kΠ(u, a)kC 0 ([0,T ],L2 ) ≤ C kakL∞ kukC 0 ([0,T ],L2 ) .

(V.21)

For p ≥ 2, one has k∆j ukLp ≤ C23j(1/2−1/p) k∆j ukLp . Thus 1 − 2/p = µ yields X X k∆j ukLp k∆k akL∞ ≤ C kakL∞ 23jµ/2 k∆j ukL2 . kSk−3 u ∆k akLp ≤ j≤k−3

j≤k−3

Since the spectrum of Sk+2 a ∆k u is contained in {2k−2 ≤ |ξ| ≤ 2k+2 }, it follows that kΠ(u, a)(t)kB −µ/2 ≤ C ka(t)kL∞ ku(t)kH µ . p,2

Therefore 1/r = µ/2 yields kΠ(u, a)kLr ([0,T ];B −1 ) ≤ C T µ/2 kakL∞ kukC 0 ([0,T ];H µ ) . p,2

(V.22)

This estimate also holds for the extended definition of Π(u, a), since the space in the left hand side is a dual and (5.22) provides uniform estimates for approximations of a and u. −1 3) So far, we have proved that Π(u, a) ∈ C 0 ([0, T ]; L2 ) ∩ Lr ([0, T ]; Bp,2 ). We now study ∂t Π(u, a). For smooth a and u, one has

∂t Π(a u) = Π(∂t u, a) + Π(u, ∂t a) .

(V.23)

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The estimate (5.20) shows that kΠ(∂t u, a)kL∞ ([0,T ];H µ−1 ) ≤ C kakL∞ k∂t ukC 0 ([0,T ];H µ−1 ) .

(V.24)

For the second term, use the relation 1/p + 1/2 = 1/q to find kSk−3 u ∆k ∂t akLq ≤ kSk−3 ukLp k∆k ∂t akL2 . 0 Since p ≥ 2, Bp,2 ⊂ Lp (see e.g. [Tr]). Thus kSk u(t)kLp ≤ Cku(t)kLp ≤ 0 k−2 0 . Since the spectrum of Sk−3 u ∆k a is contained in {2 ≤ |ξ| ≤ C ku(t)kBp,2

2k+2 }, this shows that

0 0 ≤ C k∂t a(t)kL2 ku(t)kBp,2 kΠ(u, ∂t a)(t)kBq,2

and, with 1/s − 1/r = 1/2, that 1/2 0 ) ≤ C T 0 ). k∂t akL∞ ([0,T ],L2 ) kukLr ([0,T ];Bp,2 kΠ(u, ∂t a)kLs ([0,T ];Bq,2

(V.25)

The spaces in the left hand sides of (5.24) and (5.25) are dual spaces. Thus the bilinear operators Π(∂t u, a) and Π(u, ∂t a) extend to the spaces which occur in the right hand sides. This shows that for a satisfying (5.3) and u ∈ Y µ (T ), one gets 0 that ∂t Π(u, a) ∈ L1 ([0, T ], H µ−1 ) + Ls ([0, T ], Bq,2 ) and 0 ) ≤ C T kakL∞ k∂t ukC 0 ([0,T ];H µ−1 ) k∂t Π(u, a)kL1 ([0,T ],H µ−1 )+Ls ([0,T ];Bq,2 0 ). + C T 1/2 k∂t akL∞ ([0,T ],L2 ) kukLr ([0,T ];Bp,2

(V.26)

Together with (5.19), (5.21) and (5.22), this finishes the proof of Lemma 5.3.

Proof of Theorem 2.7, when µ = 1. Consider a solution U of (1.2) (1.3) with Cauchy data E0 ∈ H 1 and H0⊥ ∈ H 1 . Theorem 2.7 with µ = 1/2 implies that H⊥ ∈ C 0 ([0, +∞[; H 1/2 ). The Sobolev embedding H 1/2 ⊂ L3 implies that 3 ∂t M = F (M, H) = F (M, H⊥ ) − F (M, Mk ) ∈ L∞ loc ([0, +∞[; L ) .

(V.27)

Therefore, the coefficient a and the source term g in (5.1) or (5.2) satisfy, in addition to (5.3) (5.4), ∂t a ∈ L∞ ([0, T ]; L3 ),

∂t g ∈ L∞ ([0, T ]; L2 )

(V.28)

for all T > 0. Again, we consider u = (E⊥ , H⊥ ) as the unique solution to the linear problem (5.2) and prove that if the initial data belong to H 1 , then there is a solution in C 0 ([0, T ], H 1 ) for all T > 0.

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1) For f ∈ L1 ([0, T ], L2 ), the solution to (5.8) satisfies Z t ku(t)kL2 ≤ ku(0)kL2 + 2 kf (s)kL2 ds . 0

If f ∈ C 0 ([0, T ], L2 ) and ∂t f ∈ L1 ([0, T ], L2 ), then, using the fundamental solution (5.13) and integrating by parts as in the proof of Lemma 5.2, one obtains Z t k∂x u(t)kL2 ≤ k∂x u(0)kL2 + C k∂t f (s)kL2 ds + kf (0)kL2 + kf (t)kL2 . 0

Moreover k∂t u(t)kL2 ≤ C k∂x u(t)kL2 + ku(t)kL2 . 2) One has ka(t)u(t)kL2 ≤ kakL∞ ku(t)kL2 . Using the Sobolev inequality kukL6 ≤ Ck∂x ukL2 , one gets k∂t (au)(t)kL2 ≤ kakL∞ k∂t u(t)kL2 + k∂t akL2 k∂x u(t)kL2 . 3) Let K denote the operator v 7→ u, where u is the solution of (5.8) with source term f = av and vanishing initial data. The estimates above show that K maps C 0 ([0, T ]; H 1 ) ∩ C 1 ([0, T ], L2 ) into itself and Rt kKv(t)kL2 ≤ C 0 kv(s)kL2 ds , Rt (V.29) k∂t,x Kv(t)kL2 ≤ C 0 k∂t,x v(s)kL2 ds + kv(t)kL2 + kv(0)kL2 . where C only depends on a. 4) The first two steps imply that the solution to (5.8), with source term g and initial data u0 , belongs to C 0 ([0, T ]; H 1 ) ∩ C 1 ([0, T ], L2 ). The third step implies that Picard’s iterates converge in C 0 ([0, T ]; H 1 ) ∩ C 1 ([0, T ], L2 ), proving that the unique solution to (5.2) also belongs to C 0 ([0, T ]; H 1 ) ∩ C 1 ([0, T ], L2 ). The proof of Theorem 2.7 is now complete.

VI Uniqueness and L2 -stability of the Hcurl solution In this section we prove Theorem 2.8. It follows from a stronger result on the stabil0 ity of finite energy solutions U such that curl E and curl H belong to C [0, +∞[; 2 3 L (R ) . Before stating the result, we make a few remarks. Consider two finite energy solutions U and U on ΩT . Then δU = U − U satisfies   0 LδU :=  −δF  (VI.1) δF

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where L denotes the first-order system in (1.2) and δF = F (M, H) − F ( M , H) = F (M, H − H) + F (M, H) − F ( M , H). Suppose that k M (0)kL∞ ≤ R and kM (0)kL∞ ≤ R. Then, k M kL∞ ≤ R and kM kL∞ ≤ R. Suppose in addition that (VI.2) k HkL∞ (ΩT ) < ∞. Then |δF | ≤ C|δU |, where C depends on R and k HkL∞ (ΩT ) . The standard energy estimate for (6.1) implies that kδU (t)kL2 ≤ eCt kδU (0)kL2

(VI.3)

proving that U is the unique finite energy solution with initial data U (0). The uniqueness of bounded solutions of semilinear equations is well known. Here, we have a-priori bounds of the L∞ norms of M and M . The interesting point is that (6.2) involves only H. The main goal of this section is to weaken condition (6.2). The price is the lost of the Lipschitz dependence of U (t) on U (0). Theorem 6.1 Let U be a finite energy solution on ΩT such that k M (0)kL∞ ≤ R, curl E and curl H belong to C 0 [0, T ]; L2 (R3 ) . Then there exist constants C > 0, c > 0 and ρ > 0, such that for all finite energy solution U on ΩT which satisfies kM (0)kL∞ ≤ R and kU (0) − U (0)kL2 ≤ c, one has for all t ∈ [0, T ] γ(t)

kU (t) − U (t)kL2 ≤ C kU (0) − U (0)kL2 ,

(VI.4)

−ρt

with γ(t) := e

The main ingredient is a substitute to the L∞ estimate (6.2). Lemma 6.2 There is a constant C such that for all λ ≥ e, there is H λ ∈ L∞ (ΩT ) and functions αλ ∈ L2 ([0, T ]), βλ ∈ L∞ ([0, T ]) such that for all t ∈ [0, T ] k H λ (t)kL∞ ≤ αλ (t) + βλ (t) , k( H − H λ )(t)kL2 ≤ C/λ , √ kαλ kL2 ([0,T ]) ≤ C ln λ , kαλ kL∞ ([0,T ]) ≤ C ln λ .

(VI.5) (VI.6)

Proof. Using (2.13) we write H = H ⊥ − M k and study each term separately. 1) The operator Pk maps Lp in Lp for all finite p, with norm less or equal to C0 p, with C0 independent of p (see [St] for instance). Therefore, for all p ∈ [2, +∞[, k M k (t)kp ≤ C0 p k M (t)kL2 ∩L∞ ≤ C0 p k M (0)kL2 ∩L∞ . Define M λk (t, x) = M k (t, x) when | M k (t, x)| ≤ C ln λ and M λk (t, x) = 0 otherwise. Then 1 (C1 p)p k( M k − M λk )(t)k2L2 ≤ k M k (t)kpLp ≤ p−2 (C ln λ) (C ln λ)p−2 with C1 = C0 k M (0)kL2 ∩L∞ . Choose C = 2eC1 and p = 2 ln λ ≥ 2. Then the right hand side is less than (C ln λ)2 e−p = (C ln λ)2 λ−2 ≤ C 0 λ−1 .

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2) Applying P⊥ ∂t to the second equation in (1.2) yields H ⊥ = −P⊥ ∂t F ( M , H).

(VI.7)

Since F is Lipschitzean and (∂t M , ∂t H) ∈ L2 , one can differentiate F ( M , H) with respect to t. Using (2.4), one obtains that |∂t F ( M , H)| ≤ C (|∂t H| + | H||∂t M |) ≤ C (|∂t H| + | H|2 ). We know that ∂t H = −curl E − F ( M , H) ∈ L1 ([0, T ]; L2 ). Moreover, M k (t) is bounded in L4 and H ⊥ (t) is bounded in H 1 thus in L4 . Therefore H(t) = H ⊥ (t) − M k (t) is bounded in L4 and the right hands side of (6.7) belongs to L1 ([0, T ]; L2 ). In addition, the initial values of H ⊥ satisfy H ⊥ (0) ∈ H 1 ,

∂t H ⊥ (0) = −curl E(0) − F ( M (0), H(0)) ∈ L2 .

One would like to use the Stichartz estimates to control the L2 (L∞ ) norm of H ⊥ by the L1 (L2 ) norm of H ⊥ . This corresponds to the forbidden limit case p1 = ∞ in (5.16) for which the inequality is known to be false (see [L] , [KM]). Nevertheless we persist in following this idea. The Strichartz inequality in the limit case p1 = ∞ holds for functions whose Fourier transform is supported in a ball and it is possible to give a sharp estimate of the constant involved in term of the radius of the ball. Precisely, recalling the definition of S λ in (2.9), we have Proposition 6.3 There exists a constant c such that for all λ > 0, all T > 0 and all u ∈ C 0 [0, ∞[; H 2 (R3 ) , p kS λ ukL2 ([0,T ]; L∞ (R3 )) ≤ c log(1 + λT ) k∂t,x u(0)kL2 (R3 ) + k ukL1 ([0,T ]; L2 (R3 )) .

The proof is delayed until section 8. A similar idea would be to estimate the constant C in (5.16) as p1 → ∞. It would lead to similar results. This proposition applies to H ⊥ . Therefore the function αλ (t) := kS λ H ⊥ (t)kL∞ belongs to L2 ([0, T ]) and p √ kαλ kL2 ([0,T ]) ≤ C ln(1 + λT ) ≤ C 0 ln λ (VI.8) for ln λ ≥ 1. Next, we note that H ⊥ − S λ H ⊥ satisfies k(I − S λ ) H ⊥ (t)kL2 ≤

1 k H ⊥ (t)kH 1 ≤ C/λ . 2λ

3) The Lemma 6.2 follows, with H λ = S λ H ⊥ − M λk , αλ (t) := kS λ H ⊥ (t)kL∞ and βλ (t) := k M λk (t)kL∞ .

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Proof of Theorem 6.1. Using the inequalities (2.4) for the function F and the L∞ bounds (2.6) for M and M , one obtains that the right hand side δF = F (M, δH)+ F (M, H) − F ( M , H) in (6.1) satisfies |δF | ≤ C(R) (|δH| + | H||δM |).

(VI.9)

For all λ ≥ e, choose H λ as indicated in Lemma 6.2. Then, (6.9) implies that 1 kδF (t)kL2 ≤ C kδH(t)kL2 + (αλ (t) + βλ (t)) kδM (t)kL2 + kδM (t)k∞ . λ With the obvious estimate |δM | ≤ |M | + | M | ≤ 2R, this implies that 1 kδF (t)kL2 ≤ C (1 + αλ (t) + βλ (t)) kδU (t)kL2 + . (VI.10) λ . for some C that depends only on k M (0)kL2 ∩L∞ , R and kcurl Hk 0 1 3 C

[0,T ];H (R )

Introduce δ(t) = kδU (t)kL2 . The energy estimate for (6.1) together with (6.10) yields Z t 1 1 + αλ (s) + βλ (s) δ(s) + δ(t) ≤ δ(0) + 2C ds. λ 0 Gronwall’s Lemma implies that for all λ ≥ e and all t ∈ [0, T ] δ(t) ≤ (δ(0) + where

Z Aλ (t) := 2C

t

Ct Aλ (t) )e λ

(VI.11)

1 + αλ (s) + βλ (s) ds.

0

Lemma 6.2 implies that for all λ ≥ e and 0 ≤ t ≤ T , √ Aλ (t) ≤ C t ln λ + t ln λ ≤ C1 (1 + t ln λ)

(VI.12)

Suppose that λ > 1/δ(0). Then (6.11) and (6.12) imply δ(t) ≤ δ(0)(1 + Ct) eC1 (1+t ln λ) .

(VI.13)

Suppose that δ(0) < 1/e. For t ≤ T1 := 1/2C1 , one can let λ → 1/δ(0) to find δ(t) ≤ eC1 (1 + CT1 ) δ(0)1−tC1 . Introduce ρ = C1 ln 4. Then 1 − tC1 ≥ γ(t) := e−ρt for t ≤ 1/2C1 . Summing up, we have shown that there are constant T1 > 0, C2 and ρ such that, if δ(0) ≤ 1/e, then for t ≤ T1 : δ(t) ≤ C2 δ(0)γ(t) . (VI.14) If C2 δ(0)γ(T1 ) ≤ 1/e, one can apply (6.14) to the Cauchy problem with initial time T1 and prove that (6.14) with another constant C2 holds on [T1 , 2T1 ]. By induction this implies that for δ(0) small enough, the estimate (6.4) follows.

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VII Global smooth solutions In this section, we prove Theorem 2.9. We therefore assume that F satisfies Assumption 2.1 and in particular is infinitely smooth. Classical results for the semilinear Cauchy problem in R1+3 with initial data U0 in H s , s ≥ 2, say that there exists a unique maximal solution U ∈ C 0 ([0, T [; H s ) of (1.2) satisfying U (0) = U0 . Moreover, if T < ∞, then kU (t)kL∞ → ∞, and hence kU (t)kH 2 → ∞ as t → T . Therefore, Theorem 2.9 is a consequence of the following a priori estimate. Proposition 7.1 Let F satisfy Assumption 2.1. For all T > 0, any finite energy solution U in ΩT belonging to C 0 ([0, T [; H 2 (R3 )), satisfies sup kU (t)kH 2 (R3 ) ≤ C

0≤t
where C depends only on T and kU (0)kH 2 (R3 ) . Let ∂U denote any x-derivative of U . Differentiating twice the equations for U leads to a system for U , ∂U and ∂ 2 U which reads LU = f0 ,

L∂U = f1 ,

L∂ 2 U = f2 .

(VII.1)

The nonlinear part of fi , i = 1, 2 involve the first and second order derivatives of F with respect to m. Assumption 2.1, the chain rule and the uniform bound (2.6) for M immediately yield the following pointwise estimates which hold almost everywhere on ΩT : |f0 | ≤ C|H|, |f1 | ≤ C |∂H| + |H| |∂M | , (VII.2) |f2 | ≤ C |∂ 2 H| + |∂H| |∂M | + |H| |∂ 2 M | + |H| |∂M |2 , where C is some constant that depends on kM (0)k∞ = R. Introduce n2 (t) = sup kU (s)kH 2 .

(VII.3)

0≤s≤t

In order to apply the energy estimate for the large system (7.1) we need to bound from above the L2 -norms of the fi (t) by some functions of n2 (t). The linear terms in (7.2) are trivially bounded by n2 (t). Thus it remains to control the quadratic and cubic terms. The key is to obtain bounds for the L∞ norm of H. For Hk = −Mk , we use a Judovic-type inequality. Lemma 7.2 There exists C depending only on kM (0)kL2 ∩L∞ such that for all t ∈ [0, T [ kMk (t)k∞ ≤ C 1 + ln+ (kM (t)kH 2 ) . (VII.4) where ln+ denotes the positive part of ln.

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P∞ Proof. Introduce a Littlewood-Paley decomposition 1 = ϕ(ξ) + k=1 ψ1 (2−k ξ) as in (5.13) and use the notations Sk := ϕ(Dx ), ∆k := ψk (Dx ). We first prove that for all k ≥ 1, 3 (VII.5) kSk Mk kL∞ ≤ 2 2 kM kL2 + c kkM kL∞ , where c only depends on ϕ. One has 3

3

kS0 Mk kL∞ ≤ 2 2 kS0 Mk kL2 ≤ 2 2 kM kL2 .

(VII.6)

since Pk is an orthogonal projector on L2 and ϕ is supported in |ξ| ≤ 2. Denote by pk (ξ) = |ξ|1 2 (ξ, ·)ξ the symbol of Pk . Since pk ψ1 ∈ C0∞ , its Fourier transform χ(x) belongs to the Schwartz class S. Since pk is homogeneous of degree 0, it follows that, for all k ≥ 1, Pk ◦ ψ1 (2−k Dx ) is the convolution operator with 23k χ(2k x), whose L1 -norm is independent of k. Thus, for all k ≥ 1, P

k∆k Mk kL∞ ≤ kχkL1 kM kL∞

(VII.7)

Since Sk = S0 + j≤k ∆j , (7.6) and (7.7) imply (7.5). To end the proof of the lemma, note that in R3 k(I − Sk )(u)kL∞ ≤ c 2−k/2 k∂ 2 ukL2

(VII.8)

Since Pk is orthogonal and commute with ∂ 2 , this implies kMk − Sk Mk kL∞ ≤ c 2−k/2 kM kH 2 . (VII.9) √ Let λ := kM (t)kH 2 /kM (t)kL∞ , noticing that λ ≥ 8π. Choose k such that 2k/2 ≤ λ ≤ 2(k+1)/2 . Then (7.5), (7.9) and the inequality kM (t)kL∞ ∩L2 ≤ kM (0)kL∞ ∩L2 imply (7.4). Lemma 7.3 There are a universal constant c and a constant C1 which depends only on kM (0)kL∞ ∩L2 , kcurl(E, H)(0)kL2 and T , such√that for all λ ≥ e, there is a function αλ ∈ L2 ([0, T ]) such that kαλ kL2 ([0,T ]) ≤ C1 ln λ and for all t ∈ [0, T [, c kH⊥ (t)kL∞ ≤ αλ (t) + √ kH(t)kH 2 . (VII.10) λ Proof. By Theorems 2.7 and 2.8, we know that U extends as a finite energy solution on Ω∞ , with (E, H⊥ ) ∈ C 0 ([0, ∞[; H 1 ). Thus the norm of (E, H⊥ ) in C 0 ([0, T ]; H 1 ) is bounded by a constant C1 which depends only on kM (0)kL∞ ∩L2 , kcurl(E, H)(0)kL2 and T . As in the proof of Lemma 6.2, this implies that H⊥ ∈ L1 ([0, T ]; L2 ), H⊥ (0) ∈ H 1 and ∂t H⊥ (0) ∈ L2 , with norms less than a similar conλ stant C1 . Therefore Proposition 6.3 implies that √ the function t 7→ kS (H⊥ (t))kL∞ 2 belongs to L ([0, T ]) with norm less than C1 ln λ. To prove (7.10) write kH⊥ (t)kL∞ ≤ kS λ (H⊥ (t))kL∞ + k(I − S λ )(H⊥ (s))kL∞ and recall (7.8), which implies that k(I − S λ )(H⊥ (t))kL∞ ≤ c kH⊥ (t)kH 2 ≤ c kH(t)kH 2 .

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Proof of Proposition 7.1. Let U ∈ C 0 ([0, T [; H 2 (R3 )) be a finite energy solution in ΩT . We estimate the L2 norms of the fi (t), using (7.2). Recall the CagliardoNirenberg inequality k |∂u|2 kL2 ≤ 4 kukL∞ k∂ 2 ukL2 . It implies that 1/2

1/2

k |∂H(t)| |∂M (t)| kL2 ≤ 4 kH(t)kL∞ kM (t)kL∞ kU (t)kH 2 and k |H(t)| |∂M (t)|2 kL2 ≤ 4 kH(t)kL∞ kM (t)kL∞ kM (t)kH 2 . Therefore, (7.2) implies that kfi (t) kL2 ≤ C (1 + kH(t)kL∞ ) kU (t)kH 2 . where C only depends on kM0 kL2 ∩L∞ . Next write H = H⊥ − Mk and use Lemmas 7.2 and 7.3 to find kU (t)kH 2 √ kfi (t) kL2 ≤ C 1 + αλ (t) + + ln+ (kU (t)kH 2 ) kU (t)kH 2 . (VII.11) λ Therefore, the energy estimates for the large system (7.1) implies that kU (t)kH 2 ≤

kU (0)kH 2 + + ln+ (kU (s)kH 2 ) kU (s)kH 2 ds. (VII.12) 2 + n √ is positive and increasing in n Since the function n 7→ n + αλ n + n ln (n) + λ the same inequality is true for n2 (t) = sup0≤s≤t kU (t)kH 2 , that is Rt C 0 1 + αλ (s) +

n2 (t) ≤ n2 (0) + C

kU (s)kH 2 √ λ

Z t n2 (s) 1 + αλ (s) + √ + ln+ (n2 (s)) n2 (s) ds. λ 0

(VII.13)

This family of inequalities hold for all t ∈ [0, T [ and λ ≥ e. The constant C only depends on kM0 kL2 ∩L∞ and the family of functions αλ satisfy √ kαλ kL2 ([0,T ]) ≤ C1 ln λ, (VII.14) where C1 only depends on kM (0)kL∞ ∩L2 , kcurl(E, H)(0)kL2 and T .

that

To complete the proof of Proposition 7.1 consider T1 < T and choose λ such √ λ = n2 (T1 ) so that inequality (7.14) implies that, for 0 ≤ t ≤ T1 ,

Z t n2 (t) ≤ ν(t) := n2 (0) + C (2 + αλ (s))n2 (s) + n2 (s) ln+ n2 (s) ds. (VII.15) 0

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The function ν(t) is absolutely continuous on [0, T1 ] and its derivative is ν 0 (t) = C 2 + αλ (t) + ln+ n2 (t) n2 (t) ≤ C 2 + αλ (t) + ln+ ν(t) ν(t). (VII.16) In (7.13) and (7.15) one can increase n2 (0) and therefore assume that n2 (0) > 1, so that ν(t) > 1. Then the function ln ν is absolutely continuous and its derivative is ν 0 (t) (ln ν)0 (t) = (VII.17) ≤ C 2 + αλ (t) + ln ν(t) . ν(t) Therefore, for t ≤ T1 , one gets ln ν(t) ≤ e

Ct

Z

t

et−s (2 + αλ (s))ds.

ln ν(0) +

(VII.18)

0

Take t = T1 in this estimate. Using (7.14) and recalling the choice λ = n2 (T1 )2 , (7.15) and (7.17) imply p (VII.19) ln n2 (T1 ) ≤ ln ν(T1 ) ≤ eCT ln n2 (0) + eCT C1 2T ln n2 (T1 ). Therefore ln n2 (T1 ) ≤ 2 eCT ln n2 (0) + 2T e2CT C12

(VII.20)

and Proposition 7.1 follows.

VIII Limit Strichartz-type estimates In this section we prove Proposition 6.3. Recall that the space dimension is equal to 3. The proof follows p the methods in [GV] or [LS], but we give the details to obtain the sharp bound ln(1 + λT ). Consider v ∈ C ∞ (R; S(R3 )) such that v = g,

v|t=0 = 0 ,

∂t v|t=0 = 0,

(VIII.1)

Lemma 8.1 Suppose that the support of the spatial Fourier transform gˆ(t, ξ) of g is contained in the ball {|ξ| ≤ λ}. Then for all t ≥ 0, Rt k v(t) k2L2 ≤ 2π5 2 ln(1 + λt) 0 kg(s)k2L1 dt, Rt (VIII.2) k∂t v(t)k2H˙ −1 ≤ 2π5 2 ln(1 + λt) 0 kg(s)k2L1 dt. Proof. We consider first the case λ = 1. The general case follows using dilations. 1) Suppose that support gˆ(t, ξ) ⊂ {|ξ| ≤ 1}. (VIII.3) The solution to (8.1) satisfies Z

t

vˆ(t, ξ) = 0

ds sin (t − s)|ξ| gˆ(s, ξ) , |ξ|

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and ∂t vˆ(t, ξ) = |ξ| Thus

Z

t

0

Ann. Henri Poincar´ e

ds cos (t − s)|ξ| gˆ(s, ξ) . |ξ|

Z tZ tZ

k v(t) k2L2 =

0 0 R3 ×R3

K− (τ, σ, x − y) g(σ, y)g(τ, x) dx dy dτ dσ

where, taking the support condition (8.3) into account, Z dξ sin (t − σ)|ξ| sin (t − τ )|ξ| e−i z·ξ . (2π)3 K− (τ, σ, z) := |ξ|2 |ξ|≤1

(VIII.4)

(VIII.5)

Similarly, ∂t v(t) 2 kL2 = k |ξ|

Z tZ tZ 0 0 R3 ×R3

with

K+ (τ, σ, x − y) g(σ, y)g(τ, x) dx dy dτ dσ

(VIII.6)

Z

(2π)3 K+ (τ, σ, z) :=

dξ cos (t − σ)|ξ| cos (t − τ )|ξ| e−i z·ξ 2 . (VIII.7) |ξ| |ξ|≤1 Z

Introduce M (λ, z) =

cos(λ|ξ|) e−i z·ξ

|ξ|≤1

dξ . |ξ|2

(VIII.8)

It follows that (2π)3 K± (τ, σ, z) =

1 M (τ − σ, z) ± M (2t − σ − τ, z) . 2

(VIII.9)

In order to apply Schur’s Lemma to (8.4) and (8.6), we need sharp bounds for supz |K± (τ, σ, z)|, hence, in view of (8.9), of supz |M (λ, z)|. From (8.8) one gets first that, for all λ, z, |M (λ, z)| ≤ 4π. (VIII.10) Note that M is real and rotation invariant in z. Taking polar coordinates for ξ, one obtains that Z 1Z 1 Z 1 sin(|z|ω + λ) sin(|z|ω − λ) M (λ, z) = 2π + dω. cos(λr)ei |z|rω drdω = π |z|ω + λ |z|ω − λ 0 −1 −1 (VIII.11) Writing Z |z|±λ Z 1 1 sin(|z|ω ± λ) sin(a) dω = da, |z|ω ± λ |z| −|z|±λ a −1 it follows that |M (λ, z)| ≤

4πSi(π) 8π < |z| |z|

(VIII.12)

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since the function Si(x) = (8.11) one also gets

Rx

sin(a) a

0

337

da satisfies |Si(x)| ≤ Si(π) < 2, x ∈ R. From 8π |λ|

sup |M (λ, z)| ≤

|z|≤ |λ| 2

(VIII.13)

which, with (8.12) and (8.10), yields sup |M (λ, z)| ≤ z

20π . 1 + |λ|

To estimate (8.4) and (8.6) we note that RRR tt σ, x − y) g(σ, y)g(τ, x) dx dy dτ dσ ≤ 0 0 R3 ×R3 K± (τ, Rt Rt supτ 0 supz |K± (τ, σ, z)|dσ 0 kg(s)k2L1 ds.

(VIII.14)

(VIII.15)

Using (8.9) and (8.14), one has (2π)3 sup |K± (τ, σ, z)| ≤ z

10π 10π + . 1 + |τ − σ| 1 + |2t − τ − σ|

Hence Z (2π)3

t

sup |K± (τ, σ, z)| ≤ 10π ln(1 + τ ) + ln(1 + 2t − τ ) z

0

and

Z τ

t

sup |K± (τ, σ, z)|dσ ≤

sup 0

z

5 log(1 + t). 2π 2

Substituting this estimate in (8.15), equalities (8.4) and (8.6) yield Z t 5 2 k∂t v(t)kH˙ −1 ≤ 2 ln(1 + t) k g(s) k2L1 ds, 2π 0 and kv(t)k2L2 ≤

5 ln(1 + t) 2π 2

Z

(VIII.16)

t

k g(s) k2L1 ds.

(VIII.17)

0

2) Suppose next that gˆ(t, ξ) is supported in {|ξ| ≤ λ}. Introduce gλ (t, x) :=

1 t x g( , ) , λ2 λ λ

t x vλ (t, x) := v( , ) . λ λ

Then vλ = gλ ,

vλ|t=0 = 0 ,

∂t vλ|t=0 = 0 ,

and gˆλ (t, ξ) = λ gˆ(t/λ, λξ) is supported in {|ξ| ≤ 1}. Thus (8.16) (8.17) apply to vλ and gλ and (8.2) follows.

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Proof of Proposition 6.3. Consider f ∈ C ∞ (R; S(R3 )), u0 ∈ S(R3 ) and u1 ∈ + 2 3 S(R3 ). Let u ∈ L∞ loc (R ; L (R ) denote the solution to u =f,

u|t=0 = u0 ,

∂t u|t=0 = u1 .

(VIII.18)

Recall from (2.9) that S λ = ϕ(λ−1 Dx ) where ϕ ∈ C0∞ (R3 )) is real, equal to 1 on |ξ| ≤ 1 and is supported in |ξ| ≤ 2. We show now there is a constant C, depending on ϕ but not on T and u, such that s Z T Z T p kS λ (u)(t)k2∞ dt ≤ C log(1 + 2λT ) ku0 kH˙ 1 + ku1 k2 + kf (t)kL2 dt . 0

0

(VIII.19) Note that the left hand side of (8.20) is well defined. One has Z sup S λ (u) g dxdt , kS λ (u)(t)kL2 ([0,T ];L∞ ) = kgkL2 ([0,T ];L1 ) ≤1

(VIII.20)

ΩT

where ΩT = [0, T ] × R3 and the functions g are supposed smooth. To any such g corresponds a unique v solution on {t ≤ T } × R3 to v = g, Since ϕ is real, we get

v|t=T = 0 ,

Z

(VIII.21)

Z λ

u S λ (g) dxdt.

S (u) g dxdt = ΩT

Commuting S λ and

∂t v|t=T = 0.

ΩT

yields

Sλv = Sλg ,

S λ v|t=T = 0 ,

∂t S λ v|t=T = 0.

Thus, integrating by part, one obtains R ΩT S λ (u) g dxdt ≤ kf kL1 ([0,T ];L2 ) kS λ (v)kL∞ ([0,T ];L2 ) + ku0 kH˙ 1 k∂t S λ (v)(0)kH˙ −1 + ku1 kL2 k∂t S λ (v)(0)kL2 .

(VIII.22)

Using (8.2) for S λ (v) whose spectrum belongs to |ξ| ≤ 2λ, it follows that R ΩT S λ (u) g dxdt ≤ q kS λ (g)kL2 ([0,T ];L1 ) 2π5 2 log(1 + 2λT ) ku0 kH˙ 1 + ku1 kL2 + kf kL1 ([0,T ];L2 ) . (VIII.23) To conclude note that the operators S λ are uniformly bounded from L1 (R3 ) to L (R3 ). Thus kS λ (g)kL2 ([0,T ];L1 ) ≤ Ck(g)kL2 ([0,T ];L1 ) ≤ C for some C that only depends on ϕ. With (8.21), (8.24) implies (8.20) and Proposition 6.3. 1

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[16] Lindblad H, Sogge C.D., On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal., 130, 1995, pp 357–426. [17] Stein, E. Singular integrals and differentiability properties of functions, Princeton University Press, 1970. [18] Tartar L., H-measures, a new approach for studying homogeneization, oscillations and concentrations effects in partial differential equations, Poc. Roy. Soc. Edinburgh, 115 (A), 1990, pp193–230. [19] Triebel H., Theory of functions spaces, Birkh¨ auser, Basel, 1983. [20] Visintin, A. On Landau-Lifshitz equations for ferromagnetism, Japan J. of Appl. Math., 2, No. 1, 1985, 69–84.

J.L. Joly MAB, Universit´e Bordeaux I F-33405 Talence, France e-mail : [email protected] G. M´etivier IRMAR, Universit´e Rennes I F-35042 Rennes, France e-mail : [email protected] J. Rauch Department of Mathematics University of Michigan Ann Arbor MI 48109, USA

Communicated by J. Bellissard submitted 21/01/98 ; accepted 10/04/98

Ann. Henri Poincar´ e 1 (2000) 341 – 384 c Birkh¨

auser Verlag, Basel, 2000 1424-0637/00/020341-44 $ 1.50+0.20/0

Annales Henri Poincar´ e

On the Initial Value Problem for the Ishimori System C.E. Kenig, Gustavo Ponce and Luis Vega

I Introduction In this paper we study Ishimori system  2 2  ∂t S = S ∧ (∂x S ± ∂y S) + b(∂x φ∂y S + ∂y φ∂x S), t ∈ R, x, y ∈ R   2 2 ∂x φ ∓ ∂y φ = ∓2S · (∂x S ∧ ∂y S),

(I.1)

where S(·, t) : R2 → R3 with kSk = 1, S → (0, 0, 1) as k(x, y)k → ∞, and ∧ denotes the wedge product in R3 . This model was proposed by Y. Ishimori in [Is] as a two dimensional generalization of the Heisenberg equation in ferromagnetism, which corresponds to the case b = 0 and signs (−, +, +) in (1.1) and was studied in [SuSuBa]. For b = 1 the system (1.1) is completely integrable by inverse scattering, (see [AbHa],[BeCo],[KoMa],[Sn],[ZaKu] and references therein). Using the stereographic variable u : R2 → C one can get rid of the constraint kSk = 1. Thus, for u=

S1 + iS2 1 , S = (S1 , S2 , S3 ) = (u + u ¯, −i(u − u ¯), 1 − |u|2 ), 1 + S3 1 + |u|2

(I.2)

the initial value problem (IVP) for (1.1) can be written as   i∂t u + ∂x2 u ± ∂y2 u =    

2¯ u 2 1+|u|2 ((∂x u)

∂ u∂y u ¯+∂x u ¯∂y u  ∂x2 φ ∓ ∂y2 φ = 4i x (1+|u| , 2 )2    u(x, y, 0) = u (x, y),

− (∂y u)2 ) +ib(∂x φ∂y u + ∂y φ∂x u),

(I.3)

0

with the condition u(x, y, t) → 0 as k(x, y)k → ∞. The case (−, +) in (1.3), i.e. − in the first equation and + in the second, was studied by A. Souyer [So]. He obtained local well posedness and global existence of solution for small data in an appropriate Sobolev space. It was remarked in [So] that the arguments there do not extend to the case (+, −) in (1.3). The case, (+, −) in (1.3), was first studied by Hayashi-Saut [HySa]. They consider the problem in a class of analytic functions which allowed them obtain

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local and global existence for small analytic data, thus overcoming the so called “loss of derivatives” introduced by the nonlinearity. In [Hy], N. Hayashi removed the analyticity assumptions in [HySa] by establishing the local well posedness of the IVP (1.3), for the case (+, −), with small data u0 in the weighted Sobolev space H 4 (R2 ) ∩ L2 ((x2 + y 2 )4 dxdy). Our main result here, Theorem 1.1, removes the smallness assumptions in [Hy]. In particular we show the local well-posedness of the IVP (1.3) with (+, −) sign, and data of arbitrary size in a weighted Sobolev space. Before stating our results we shall discuss the problem in a more general context. By inverting the operator ∂x2 ∓ ∂y2 one can rewrite the system in (1.3) as an scalar equation of Schr¨ odinger type i∂t u + ∂x2 u ∓ ∂y2 u = F (u, ∇x u, u, ∇x u, Ku, ...)

(I.4)

where F (·) represents the nonlinearity and K = ∂∂(−∆)−1 for the + sign and K an operator of “order one” for the − sign. The IVP for the equation in (1.4), without the operator K in the nonlinearity F (·) and in arbitrary dimension, i.e. ( ∂t u = iLu + F (u, ∇x u, u, ∇x u), x ∈ Rn (I.5) u(x, 0) = u0 (x), where ∇x = (∂x1 , .., ∂xn ), L is a non-degenerate constant coefficient, second order operator X X L= ∂x2j − ∂x2j , for some k ∈ {1, .., n}, (I.6) j≤k

j>k

and F (·) is a polynomial, having no constant or linear terms, has been studied in recents works. In [KePoVe1] we proved that (1.5) is locally well posed for “small” data, in some weighted Sobolev spaces. The proof in [KePoVe1] applies to the general form of L in (1.6). In [KePoVe1] the key estimates were  RT R 1/2 itL 1/2 itL 2 1/2  (i) |||D e u0 |||T ≡ supµ∈Zn ( 0 Qµ |D e u0 | dxdt) (I.7) ≤ cku0 k2 ,  R t i(t−t0 )L  0 0 0 (ii) |||∇x 0 e F (t )dt |||T ≤ c|||F |||T , where {Qµ }µ∈Zn is a family cubes of side one with disjoint interiors covering Rn , and D = (−∆)1/2 . The local smoothing effect in (i), known as Kato smoothing effects, see [Kt], was proven by Constantin-Saut [CnSa], Sj¨ olin [Sj], and Vega [Ve]. We proved the inhomogeneous version (ii) in [KePoVe1]. It is essential the gain of one derivative in (1.7) (ii). This allows to use the contraction principle in (1.5) and avoid the “loss of derivatives”. However, the

Vol. 1, 2000

On the Initial Value Problem for the Ishimori System

||| · |||T norm forces the use the following X (Sup[0,T ] SupQµ |G(x, t)|. kGklµ1 (L∞ (Qµ ×[0,T ])) =

343

(I.8)

µ∈Zn

This factor cannot be made small by taking T small, except if G(t) is small at t = 0. It is here where the restriction on the size of the data appears. In [HyOz] for the one dimensional case n = 1, Hayashi-Ozawa removed the smallness assumption on the size of the data in [KePoVe1]. By introducing a change of variables they reduced the problem to a new one which can be treated by standard energy methods. This technique is similar to that used by A. Souyer in [So] in his study of (1.3) with signs (−, +). In [Ch] for the elliptic case L = ∆, H. Chihara was able to remove the size restriction on the data in [KePoVe1] in any dimension. Finally in[KePoVe2] we showed how to remove the smallness assumptions in [KePoVe1] for the general dispersive operator L in (1.5), see (1.6). The arguments in [Ch], [KePoVe2] are based in techniques involving ψ.d.o’s. However, in some cases it is not clear how to extend them to treat specific models arising in both mathematics and physics. For example, consider the IVP for the Davey-Stewartson (D-S) system which arises in water waves problems, see [DS],[DjRe],[ZaSc], and inverse scattering see [AbHa],[BeCo],[KoMa],  2 2 2  i∂t v + c0 ∂x v + ∂y v = c1 |v| v + c2 u∂x ϕ, (I.9) ∂x2 ϕ + c3 ∂y2 ϕ = ∂x |v|2   v(x, y, 0) = v0 (x, y) where c0 , .., c3 are real parameters. In [GhSa], Ghidaglia-Saut studied the existence problem for solutions of the IVP (1.9). They classified the system as elliptic-elliptic, elliptic-hyperbolic, hyperbolic-elliptic and hyperbolic-hyperbolic according to the respective sign of (c0 , c3 ): (+, +), (+, −), (−, +) and (−, −). In [LiPo], Linares-Ponce adapted the results in [KePoVe1] to show that in this hyperbolic-hyperbolic case the IVP (1.9) is locally well posed for small data in weighted Sobolev spaces, (see also [HySa]). However, this smallness assumptions have yet to be removed. For the ellipticelliptic, elliptic-hyperbolic, and hyperbolic-elliptic cases, where a more complete set of results are available, we refer to [GhSa],[Hy],[HySa],[LiPo], and references therein. The necessity of the decay assumption on the data can be justified by the following result due to S. Mizohata [Mz]. Consider the linear IVP ( ∂t v = i∆v + b(x) · ∇x v + f (x, t), t ∈ R, x ∈ Rn , (I.10) v(x, 0) = v0 (x) ∈ L2 (Rn ), with b(·) and f (·) smooth enough functions. In [Mz] it was shown that the

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following condition is necessary for the L2 -local solvability of (1.8) Z R sup | Im b(x + rω) · ωdr| < ∞. x∈Rn ,ω∈Sn−1 ,R>0

(I.11)

0

We observe that the condition (1.11) may fail for b ∈ H s (Rn ). It holds for b ∈ W s,1 (Rn ), s large, however this class is not preserve by the associated linear group {eit∆ : t ∈ R}. Thus the use of weighted L2 spaces seems natural. Following [Hy] one extends the classification in for the DS system in (1.9) given in [GhSa] to the Ishimori system by considering its generalized form  u ¯ ((∂x u)2 +c2 (∂y u)2 )  i∂t u + ∂x2 u + c0 ∂y2 u = c1  1+|u|2    +c3 (∂x φ∂y u + ∂y φ∂x u), ¯+∂x u ¯ ∂y u ∂x u ∂y u 2 2  ∂x φ + c4 ∂y φ = c5 ,  (1+|u|2 )2   u(x, y, 0) = u (x, y),

(I.12)

0

where c0 , c4 ∈ R − {0}, and c1 , c2 , c3 , c5 ∈ C. As was remarked in [Hy] the local existence results for arbitrary size data in [So] applies to the cases (c0 , c2 , c4 ) = (c, c, b) in (1.12) with c ∈ R − {0}, and b > 0. The small data results in [Hy] corresponds to the elliptic-hyperbolic case, i.e. c0 > 0, c4 < 0, and as was mentioned there it does not extend to the hyperbolichyperbolic case i.e. c0 < 0, c4 < 0. In fact for this case no existence results are known besides those in [HySa] for “small” analytic data. In [Sn], L. Y. Sung using the gauge equivalence between the integrable case of the Ishimori system, i.e. b = 1 in (1.3), and the DS system in (1.6) with (c0 , c1 , c2 , c3 ) = (−1, 2, −1, 1) proved global existence of solution of (1.3), with b = 1 with “small” data. Our main result is the following Theorem. Theorem 1.1 Given N ≥ 1 there exist s, m ∈ Z+ such that for any u0 ∈ H s ∩ L2 (|x|m dx) the IVP (1.12) has a unique solution u(·) defined in the time interval [0, T ] satisfying that u ∈ C([0, T ] : H s (R2 ) ∩ L2 (R2 : |x|m dx)),

(I.13)

kλN (x)Jxs+1/2 ukL2x,y,T + kλN (y)Jys+1/2 ukL2x,y,T < ∞,

(I.14)

λN (x) = (1 + x2 )−N/2 , λN (y) = (1 + y 2 )−N/2 .

(I.15)

and

where

Moreover, the map data → solution from H s (R2 ) ∩ L2 (R2 : |x|m dx) into the class in (1.13)-(1.14) is locally continuous.

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On the Initial Value Problem for the Ishimori System

345

If s0 > s , then the above results hold, with s0 instead of s, in the same time interval [0, T ]. To explain our method of proof we assume without loss of generality c0 = 1 , c4 = −1 , c2 = 1. Then we rotate coordinates in the xy-plane and rewrite the equations in (1.12) as  u ¯   i∂t u + ∆u = c1 1+|u|2 ∂x u∂y u + c3 (∂x φ∂x u + ∂y φ∂y u),  

(I.16) ∂x ∂y φ = c5

¯−∂x u ¯∂y u ∂x u∂y u , (1+|u|2 )2

and assuming, without loss of generality, a trivial radiation condition at infinity, we write the IVP (1.16) as an scalar equation  u ¯    i∂t u + ∆u = c1 1+|u|2 ∂x u∂y u (I.17)    +c6 ∂x u∂y−1 ∂x u∂y u¯−∂2x u¯2∂y u − c7 ∂y u∂x−1 ∂x u∂y u¯+∂2x u¯2∂y u , (1+|u| ) (1+|u| ) where ∂x−1 f (x, y) =

Z

x

−∞

f (x0 , y)dx0 ,

(resp. ∂y−1 ).

(I.18)

First we observe that ∂x−1 is not a ψ.d.o. However by adding some decay from the coefficients we get that for large M 1 1 −1 ∂˜x−1 f (x, y) = ∂ f (·, y) (I.19) (1 + x2 )M x (1 + x2 )M defines a ψ.d.o. of order −1 in the x-variable. However ∂˜x−1 is not a ψ.d.o. in both variables. Thus the techniques in [Ch],[KePoVe2] and in recent related works [CrKaSt], [Do] can not be carried out. One has to work in each variable separated and when results in both variables are required one uses operator valued version of some of the techniques. For example, to establish the local smoothing effect in its homogeneous and inhomogeneous versions, see (1.7), we shall use the operator valued version of the sharp G˚ arding inequality, see [Ho]. Another feature of our approach is that for the linearized system associated to (1.17) the coefficients of the first order terms do not decay in both variables. More precisely, in a simplified setting, our linearized IVP is as that in (1.8) with b(x) · ∇x = b1 (x, y)∂x + b2 (x, y)∂y

(I.20)

where b1 is a smooth function with decay in x, uniformly in y, and b2 is a smooth function with decay in y, uniformly in x. Under these assumptions is clear that Mizohata’s condition in (1.11) for the IVP (1.10) holds. However, we consider

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the operator ∂x2 − ∂y2 under the above decay assumptions on the coefficient b(·) then the IVP (1.10) is, in general, ill-posed since in this case Mizohata’s condition reads Z sup x∈R2 ,ω∈S1 ,R>0

| Im

R

b(x + rω) · ω ˜ dr| < ∞,

ω ˜ = (ω2 , −ω1 ).

(I.21)

0

This may explain why for the hyperbolic-hyperbolic Ishimori system no existence results are available besides those in [HySa] for “small” analytic data. In fact, our main step in the proof of Theorem 1.1 is the following regularity result for the linearized IVP associated to (1.17)  ∂t z = i∆z + r1 ∂x z + r2 ∂y z + ϕ1 ∂x ∂y−1 ϕ2 z + ϕ3 ∂y ∂x−1 ϕ4 z     +ϕ5 ∂x ∂y−1 ϕ6 z¯ + ϕ7 ∂y ∂x−1 ϕ8 z¯ + p1 ∂x f1 + p2 ∂y f2  +φ1 ∂x ∂y−1 φ2 f3 + φ3 ∂y ∂x−1 φ4 f4 + f5 ,    z(x, y, 0) = z0 (x, y),

(I.22)

where rj = rj (x, y), j = 1, 2 are smooth functions, r1 with decay in x, uniformly in y, r2 with decay in y, uniformly in x, ϕj = ϕj (x, y), j = 1, .., 8 are smooth with decay in both variables, p1 = p1 (x, y, t), p2 = p2 (x, y, t) behave like r1 , r2 respectively uniformly in t ∈ [0, T ], and φj , j = 1, .., 4 are like the ϕj ’s uniformly in t ∈ [0, T ].

Theorem 1.2 Under the above hypothesis on the coefficients given N > 1 there exist M > 0, k ∈ Z+ and T > 0 small enough such that the solution of the IVP (1.22) with c0 > 0, c4 < 0 satisfies u ∈ C([0, T ] : L2 (R2 )) with

sup 0≤t≤T

1/2

1/2

kz(t)k2L2 + kλN (x)Jx zk2L2 x,y

x,y,T

≤ ckz0 k2L2 + cA

P4 j=1

x,y

1/2 +cA(kλN (x)Jx f1 k2L2

x,y,T

1/2

+cA(kλN (x)Jx f3 k2L2

x,y,T

+ kλN (y)Jy zk2L2

x,y,T

sup0≤t≤T kfj k2L2

+

x,y

1/2 kλN (y)Jy f2 k2L2

)

x,y,T

1/2

+ kλN (y)Jy f4 k2L2

x,y,T

+cT 1/2 kf5 k2L2

x,y,T

,

)

(I.23)

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On the Initial Value Problem for the Ishimori System

347

where c = c(N ) and A≤T +T

P2 j=1

P4 j=1

kpj k2L2 C k (R2 T

b

x,y )

kφ˜j k4L2 C k (R2 b

x,y )

+

P2 j=1

2 kλ−2 N (xj )pj kL∞ x,y,T

+ kφ˜1 k2L2 L∞ k∂x φ˜2 k2L2 L∞ T

x,y

T

x,y

+kφ˜3 k2L2 L∞ k∂y φ˜4 k2L2 L∞ T

x,y

T

x,y

(I.24) ˜ 2 +kλ−1 N (x)φ1 kL∞ x,y,T

˜ 2 kλ−1 N (x)φ2 kL∞ x,y,T

˜ 2 ˜ 2 +kλ−1 kλ−1 , N (y)φ3 kL∞ N (y)φ4 kL∞ x,y,T x,y,T where x1 = x, x2 = y, φ˜j (x, y, t) = (1 + y 2 )M/2 φj (x, y, t), j = 1, 2, (I.25) φ˜j (x, y, t) = (1 + x2 )M/2 φj (x, y, t), j = 3, 4. Moreover there is a continuous dependence of the solution with respect to the coefficients in the norms appearing in (1.24). We observe that the result of Theorem 1.2 holds for solutions of the IVP (1.22) with i∆ + ∆ instead of i∆, uniformly for ∈ (0, 1], (see Corollary 4.1 at the end of Section 4). This provides a slightly weaker version of Theorem 1.1. Theorem 1.3 Given N > 1 here exist s, m ∈ Z+ such that for any u0 ∈ H s ∩ L2 (|x|m dx) the IVP (1.12)with c0 > 0, c4 < 0 has a unique solution u(·) defined in the time interval [0, T ] satisfying that u ∈ C([0, T ] : H s−1 ∩ L2 (|x|m−1 dx)) ∩ L∞ ([0, T ] : H s ∩ L2 (|x|m dx)),

(I.26)

and kλN (x)Jxs+1/2 ukL2x,y,T + kλN (y)Jys+1/2 ukL2x,y,T < ∞,

(I.27)

where λN (x) = (1 + x2 )−N/2 ,

λN (y) = (1 + y 2 )−N/2 .

(I.28)

If s0 > s , then the above results hold, with s0 instead of s, in the same time interval [0, T ].Once Theorem 1.3 has been established the proof of Theorem 1.1 follows by combining Theorem 1.3 and Theorem 1.2 and their proofs.

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II Preliminary estimates This section contains some estimates to be used in the coming sections. We start by recalling some results on ψ.d.o’s Definition 2.1 The symbol class S k (R2n ) consists of the set of a ∈ C ∞ (R2n ) such that |∂xα ∂ξβ a(x, ξ)| ≤ cα,β (1 + |ξ|)k−|β| ,

x, ξ ∈ Rn ,

(II.1)

for all multi-indices α, β ∈ Nn . k We say that a ∈ SM (R2n ) if it is of class C M (R2n ) and (2.1) holds for |α|, |β| ≤ M . 0 Theorem 2.2 Let A, B ∈ SM (R2n ) with M C(x, D), where

c(x, ξ) = a(x, ξ)b(x, ξ) +

large. Then A(x, D)B(x, D) =

X Z

Z Z qγ,θ (x, ξ) = Os

qγ,θ (x, ξ)dθ,

(II.2)

0

|γ|=1

with

1

e−iy·η ∂ξ a(x, ξ + θη)∂x(γ) b(x + y, ξ)dydη. (γ)

(II.3)

−1 (R2n ) seminorms of Qγ,θ are bounded by products of semiMoreover, the SM (γ) (γ) norms of ∂ξ a, ∂x b, uniformly in θ ∈ [0, 1]. Also A∗ (x, D) has symbol

a∗ (x, ξ) = a(x, ξ) +

X Z |γ|=1

where ∗ qγ,θ (x, ξ)

Moreover, the (γ) (γ) ∂ξ ∂x a,

Z Z = Os

1

∗ qγ,θ (x, ξ)dθ,

e−iy·η ∂ξ ∂x(γ) a(x + y, ξ + θη)dydη. (γ)

−1 (R2n ) seminorms of SM

(II.4)

0

Q∗γ,θ

(II.5)

are bounded by seminorms of

uniformly in θ ∈ [0, 1].

Proposition 2.3 Given M > 0 there exists N > 0 such that 1 1 −1 ∂ f = a(x, D)f = ∂˜x−1 f (1 + |x|2 )N x (1 + |x0 |2 ))N −1 with a ∈ SM (R × R).

(II.6)

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349

Remark. This proposition allows to rewrite (1.22) substituting, after changing the φ’s and ϕ’s, ∂x−1 by ∂˜x−1 , ∂y−1 by ∂˜y−1 everywhere. Proof of Proposition 2.3. We use that Z x Z g(x0 )dx0 = ∂x−1 g(x) = −∞

−∞

to obtain 1 (1+|x|2 )N

= =

1 (1+|x|2 )N

1 (1+|x|2 )N

∞

R∞

R∞

∂x−1

χ[0,∞) (x − x0 )g(x0 )dx0 ,

f (x0 ) (1+|x0 |2 )N

0

−∞

(II.7)

f (x ) 0 χ[0,∞) (x − x0 ) (1+|x 0 |2 )N dx 0

χ[0,∞) (x−x ) (1+|x−x0 |2 )N −∞ (1+|x−x0 |2 )N (1+|x0 |2 )N

(II.8)

f (x0 )dx0 .

Now we observe that 1 1 (1+|x|2 )N (1+|x0 |2 )N

= =

PN

PN P j=0

(1 + |x − x0 |2 )N

|x−x0 |2(N −j) j=0 cN,j (1+|x|2 )N (1+|x0 |2 )N 2a 02b

x x a,b≥0 cN,j,a,b (1+|x|2 )(1+|x0 |2 )N a+b=N−j

(II.9) .

Thus since x0 (1 + |x0 |2 )N 2b

x2a (1 + |x|2 )N

,

(II.10)

are bounded functions, together with all their derivatives, to establish the claim we just need to show that if χ[0,∞) (x) , (1 + |x|2 )N

(II.11)

b N (ξ) = a(ξ) ∈ S −1 . K M

(II.12)

KN (x) = then

To prove (2.12) we write Z a(ξ) = 0

∞

eixξ

1 dx. (1 + |x|2 )N

Clearly a ∈ L∞ (R). Next by integrating by parts it follows that Z 1 N ∞ ixξ 2x a(ξ) = − − e dx, iξ iξ 0 (1 + |x|2 )N+1

(II.13)

(II.14)

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which shows that |a(ξ)| ≤

c , |ξ|

for large ξ.

(II.15)

Also, (2.14) and integration by parts lead to R∞ a0 (ξ) = iξ12 + iξN2 0 eixξ (1+|x|2x2 )N +1 dx R ∞ ixξ 2x2 − iN iξ 0 e (1+|x|2 )N +1 dx R∞ = ξc2 + iξN2 0 eixξ (1+|x|2x2 )N +1 dx R ∞ ixξ ∂ iN 2ix2 + (iξ) e ∂x (1+|x| dx 2 2 )N +1 0

(II.16)

which shows that |a0 (ξ)| ≤

c , |ξ|2

for large |ξ|.

(II.17)

The proof for the higher derivatives is similar. Thus we have established the claim (2.12) and completed the proof of Proposition 2.2. 0 Let us now consider the action of ψ.d.o. in SM (R2n ), on weighted L2 spaces. We recall the notation λN (x) = hxi−N =

1 , (1 + |x|2 )N/2

x ∈ Rn .

(II.18)

Lemma 2.4 Given N ≥ 0, there exists M = M (n, N ) > 0 such that, if a ∈ 0 SM (R2n ), then a(x, D) : L2 (Rn : λN (x)dx) → L2 (Rn : λN (x)dx),

(II.19)

with norm depending only on n, N, cα,β , |α|, |β| ≤ M . Proof. (see [KePoVe2], Lemma 2.3). Next we recall some fact of the theory of vector valued ψ.d.o’s of classical type, (as reference see [Ho], vol. 3, section 18.1, in particular Remark 2, page 79). Let H = L2 (R : dy), and consider operators of the form Z (II.20) Bf (x, y) = eix·ξ1 b(x, ξ1 )fˆx (ξ1 , −)dξ1 , where for each (x, ξ1 ) ∈ R2 , b(x, ξ1 ) is the symbol of an operator in H. k In this case the class SM is defined by the inequality |||∂xα ∂ξβ1 b(x, ξ1 )||| ≤ cα,β (1 + |ξ1 |)k−|β| , for |α|, |β| ≤ M , where |||∂xα ∂ξβ1 b(x, ξ1 )||| denotes the operator norm in H.

(II.21)

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Thus the calculus of ψ.d.o, L2 -boundedness, etc., have corresponding version in this context. We also need the operator valued version of the sharp G˚ arding inequality: Theorem 2.5 Let b(x, ξ1 ) be the symbol of the operator B defined in (2.20). Assume 1 that b(x, ξ1 ) satisfies (2.21) with k = 1 for |α| + |β| ≤ M , M large, (i.e., B ∈ SM ). If h(b(x, ξ1 )) + b(x, ξ1 )∗ )h, hiH ≥ 0,

∀h ∈ H,

(II.22)

RehBf, f i ≥ −ckf k2L2 (H) = −ckf k2L2 (R2 :dxdy) ,

(II.23)

for x ∈ R, and |ξ1 | ≥ N , then

where

Z Z hBf, f i =

Bf (x, y)f (x, y)dy dx =

Z hBf, f iH dx.

(II.24)

Clearly a corresponding theory holds if we interchange x and y.

III Proof of Theorem 1.2 (Diagonalization) We split the proof of Theorem 1.2 in two steps. This section contains the first step, i.e. the diagonalization reduction. By possibly changing ϕ0 s and φ0 s we can rewrite the IVP (1.22) replacing −1 ∂x , ∂y−1 by ∂ex−1 , ∂ey−1 respectively, (see the remark after the statement of Proposition 2.3). Also we introduce the following notations: R = r1 ∂x + r2 ∂y

;

¯ = r¯1 ∂x + r¯2 ∂y , R

(III.1)

L1 = ϕ1 ∂x ∂ey−1 ϕ2 + ϕ3 ∂y ∂ex−1 ϕ4

;

L1 = ϕ1 ∂x ∂ey−1 ϕ2 + ϕ3 ∂y ∂ex−1 ϕ4 ,

(III.2)

L2 = ϕ5 ∂x ∂ey−1 ϕ6 + ϕ7 ∂y ∂ex−1 ϕ8

;

L2 = ϕ5 ∂x ∂ey−1 ϕ6 + ϕ7 ∂y ∂ex−1 ϕ8 ,

(III.3)

;

(III.4)

F1 = p1 ∂x f1 + p2 ∂y f2

F 1 = p1 ∂x f1 + p2 ∂y f2 ,

F2 = φ1 ∂x ∂ey−1 φ2 f3 + φ3 ∂y ∂ex−1 φ4 f4 (III.5) F2 = φ1 ∂x ∂ey−1 φ2 f 3 + φ3 ∂y ∂ex−1 φ4 f 4 .

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Using (3.1)–(3.5) we rewrite the equation (1.22) as a system in w ~ = (z z¯)T , with the notations ! ! ∆ 0 R 0 H = 0 −∆ , B= 0 R (III.6) ¯

β1 =

L1 0

0 L¯1

! , β2 =

L2 0

0 L¯2

! , F~ =

F1 + F2 + f5 F¯1 + F¯2 + f5

! (III.7)

as ~ = iH w ~ + Bw ~ + β1 w ~ + β2 w ~ + F~ . ∂t w

(III.8)

Our goal in this step is to “eliminate β2 ” by accepting “semilinear errors.” We introduce the operator ! 0 S1 (III.9) Λ = I − S, S = S 0 , 2 where Si , i = 1, 2 are to be determined, and write the system for ~v = Λw. ~ We shall see that modulo “semilinear terms”, i.e. bounded L2 -terms, one has ! 0 −S1 ∆ 0 ∆S1 − S ∆ ΛH − HΛ = HS − SH = 0 2 −∆S2 0 ! 0 ∆S1 + S1 ∆ (III.10) = −S ∆ − ∆S 0 2 2 similarly, modulo bdd-L2 , term we have ΛB − BΛ = BS − SB =

Λβ1 − β1 Λ =

Λβ2 =

0 L¯1 S2 − S2 L1

0 L¯2

and ~− ΛF~ = F

! ¯ RS1 − S1 R , 0

0 ¯ 2 − S2 R RS

L2 0

! −

L1 S1 − S1 L¯1 0

S1 L¯2 0

0 S2 L2

(III.11)

! ,

(III.12)

!

! S1 (F¯1 + F¯2 + f¯5 ) S1 (F1 + F2 + f5 ) .

,

(III.13)

(III.14)

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Indeed we shall show the following four statements (3.15(i))-(3.15(iv)): S can be chosen such that Λ is invertible in various spaces,

(III.15(i))

i.e. for ~v = Λw ~ sup kw(t)k ~ v (t)kH s , H s ≤ c sup k~

|t|≤T

(III.16)

|t|≤T

and for N > 0 kλN (x)Jxs+1/2 wk ~ L2x,y,t + kλN (y)Jys+1/2 wk ~ L2x,y,t (III.17) ≤

c{kλN (x)Jxs+1/2~v kL2x,y,t

0 −i(∆S2 + S2 ∆)

Λβ1 − β1 Λ ,

+

! i(∆S1 + S1 ∆) + 0

S1 L¯2 0

0 S2 L2

kλN (y)Jys+1/2~v kL2x,y,t },

0 L¯2

L2 0

! is L2 -bounded, (III.15(ii))

! , ΛB − BΛ are L2 -bounded,

(III.15(iii))

! S1 (F¯1 + F¯2 + f¯5 ) S2 (F1 + F2 + f5 ) has “semilinear control,”

(III.15(iv))

i.e. kS1 (F¯1 + F¯2 + f¯5 )kL2x,y,t + kS2 (F1 + F2 + f5 )kL2x,y,t ≤c

4 4 X X 2 2 2 2 (kpj kL2T L∞ + k∇p k + kφ k + k∇φ k ) · kf4 kL∞ ∞ ∞ ∞ j j j LT Lxy LT Lxy LT Lxy T Lxy xy j=1

k=1 2 X

kpj kL2T L∞ + ckf5 kL2x,y,t ≡ Φ. xy

(III.18)

j=1

Our choice for S is given as follows. Let θ ∈ C∞ (R), even, and ( 1, |x| ≥ 2, θ(x) = 0, |x| ≤ 1,

(III.19)

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C.E. Kenig, Gustavo Ponce and Luis Vega

and ∧ (∆−1 R f ) (ξ1 , ξ2 ) = −

1 θ ξ12 + ξ22

|(ξ1 , ξ2 )| R

Ann. Henri Poincar´ e

fˆ(ξ1 , ξ2 ),

(III.20)

and define S1 =

1 L2 ∆−1 R 2i

,

S2 =

1 ¯ −1 L2 ∆R , 2i

(III.21)

where R is going to be chosen sufficiently large.

Verification of the Properties (3.15(i))–(3.15(iv)) We shall start with (3.15(iv)), estimating Si (Fi ) and obtain the “semilinear control” for it, see (3.18). For S1 (F¯1 ) = S1 (¯ p1 ∂x f¯1 + p¯2 ∂y f¯2 ) we consider first 1 p1 ∂x f¯1 ) = − L2 ∆−1 p1 ∂x f¯1 ) S1 (¯ R (¯ 2i 1 1 = − (L2 ∆−1 p1 f¯1 ) + L2 ∆−1 ¯1 )f¯1 ). R ∂x )(¯ R ((∂x p 2i 2i

(III.22)

For the first term we write −1 −1 e−1 e−1 L2 ∆−1 R ∂x = ϕ5 ∂x ∂y ϕ6 ∆R ∂x + ϕ7 ∂y ∂x ϕ8 ∆R ∂x = I+II.

(III.23)

We claim that both I and II are L2 -bounded. For I we use that 2 −1 e−1 I = ϕ5 ∂ey−1 [ϕ6 ; ∂x ]∆−1 R ∂x + ϕ5 ∂y ϕ6 ∂x ∆R .

(III.24)

e−1 are L2 -bdd, the first term in (3.24) is L2 -bounded. Since [ϕ6 ; ∂x ] , ∆−1 R ∂x , ∂y 2 For the second term we observe that ∂x2 ∆−1 R is L -bdd. 2 For II in (3.23) we proceed similarly using that ∂y ∂x ∆−1 R is L -bdd. Arguing in a similar manner for S1 (¯ p2 ∂y f¯2 ) we see that kS1 (F¯1 )kL2T L2xy ≤ c

2 X 2 , (kpj kL2T L∞ + k∇pj kL2T L∞ ) kfj kL∞ T Lxy xy xy

(III.25)

j=1

which is the desired “semilinear estimate”, see (3.18)). We next estimate S1 (F¯2 ) S1 (F¯2 ) = S1 (φ¯1 ∂x ∂ey−1 φ¯2 f¯3 ) ¯ e−1 ¯ ¯ = ϕ5 ∂x ∂ey−1 ϕ6 ∆−1 R (φ1 ∂x ∂y φ2 f3 ) ¯ e−1 ¯ ¯ +ϕ7 ∂y ∂ex−1 ϕ8 ∆−1 R (φ3 ∂x ∂y φ4 f3 ).

(III.26)

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Since ∂ey−1 is bounded in L2 , we get the same bound as in (3.25) for S1 (F¯1 ). The bound for S1 (F¯2 ) and S1 (f5 ) are similar. Next we want to verify the property in (3.15(ii)). We look at 1 1 −1 L2 + i(∆S1 + S1 ∆) = L2 − ∆L2 ∆−1 R − L2 ∆R ∆, 2 2 and observe that ∆−1 R ∆ has multiplier |(ξ1 , ξ2 )| θ = 1 − ψR (ξ1 , ξ2 ). R

(III.27)

(III.28)

Since L2 = ϕ5 ∂x ∂ey−1 ϕ6 +ϕ7 ∂y ∂ex−1 ϕ8 we get that L2 ψR is L2 -bounded (ϕ6 ψR , ϕ8 ψR are S −∞ in both variables). Thus 1 1 2 − L2 ∆−1 R ∆ = − L2 + L -bdd. 2 2

(III.29)

1 1 1 −1 L2 − ∆L2 ∆−1 R = (L2 − ∆L2 ∆R ). 2 2 2

(III.30)

Now we consider

We have that ∆L2 = (∂x2 + ∂y2 )L2 = (∂x2 + ∂y2 )(ϕ5 ∂x ∂ey−1 ϕ6 + ϕ7 ∂y ∂ex−1 ϕ8 ).

(III.31)

The first term in the r.h.s. above can be written as ∂x2 (ϕ5 ∂x ∂ey−1 ϕ6 ) = (∂x2 ϕ5 )∂x ∂ey−1 ϕ6 + ϕ5 ∂x ∂ey−1 (∂x2 ϕ6 ) +2ϕ5 ∂x ∂ey−1 (∂x ϕ6 )∂x + ϕ5 ∂x ∂ey−1 ϕ6 ∂x2 + 2(∂x ϕ5 )∂x2 ∂ey−1 ϕ6 .

(III.32)

When we compose on the right with ∆−1 R , all terms except the next to last give bounded operators in L2 . Similarly for the second term in the r.h.s. of (3.31) 2 ∂x2 (ϕ7 ∂y ∂ex−1 ϕ8 ) = ϕ7 ∂y ∂ex−1 ϕ8 ∂x2 + o.w.c.r. ∆−1 R − L -bdd,

(III.33)

2 where o.w.c.r. ∆−1 R − L bdd means operators which composed on the right with 2 ∆−1 are L -bdd. R Then, one sees that −1 2 2 ∆L2 ∆−1 R = L2 ∆∆R + L -bdd = L2 + L -bdd,

(III.34)

i(∆S1 + S1 ∆) + L2 = L2 -bdd,

(III.35)

and thus

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and similarly −i(∆S2 + S2 ∆) + L¯2 = L2 -bdd.

(III.36)

This proves (3.15(ii)). Next we shall verify (3.15(iii)). First we shall check that Λβ1 −β1 Λ is L2 -bdd, by proving that L1 S1 , S1 L¯1 , L¯1 S2 and S2 L1 are L2 -bdd. We recall that ! 0 L1 S1 − S1 L¯1 (III.37) Λβ1 − β1 Λ = L¯ S − S L 0 1 2 2 1 with L1 = ϕ1 ∂x ∂ey−1 ϕ2 + ϕ3 ∂y ∂ex−1 ϕ4 , L2 = ϕ5 ∂x ∂ey−1 ϕ6 + ϕ7 ∂y ∂ex−1 ϕ8 , and S1 =

−1 −1 L2 ∆−1 R , with ∆R defined in (3.20). 2i

(III.38)

We first consider L1 S1 , −2iL1 S1 = (ϕ1 ∂x ∂ey−1 ϕ2 + ϕ3 ∂y ∂ex−1 ϕ4 )(ϕ5 ∂x ∂ey−1 ϕ6 + ϕ7 ∂y ∂ex−1 ϕ8 )∆−1 R = ϕ1 ∂x ∂e−1 ϕ2 ϕ5 ∂x ∂e−1 ϕ6 ∆−1 + ϕ1 ∂x ∂e−1 ϕ2 ϕ7 ∂y ∂e−1 ϕ8 ∆−1 y

y

R −1 −1 −1 e e +ϕ3 ∂y ∂x ϕ4 ϕ5 ∂x ∂y ϕ6 ∆R

y

+

x

R −1 −1 −1 e e ϕ3 ∂y ∂x ϕ4 ϕ7 ∂y ∂x ϕ8 ∆R .

(III.39)

We take the first term on the r.h.s. of (3.39). −1 e−1 e−1 ϕ1 ∂x ∂ey−1 ϕ2 ϕ5 ∂x ∂ey−1 ϕ6 ∆−1 R = ϕ1 ∂y (∂x ϕ2 )ϕ5 ∂x ∂y ϕ6 ∆R −1 2 e−1 e−1 +ϕ1 ∂ey−1 ϕ2 (∂x ϕ5 )∂x ∂ey−1 ϕ6 ∆−1 R + ϕ1 ∂y ϕ2 ϕ5 ∂x ∂y ϕ6 ∆R −1 e−1 e−1 = ϕ1 ∂ey−1 (∂x ϕ2 )ϕ5 ∂ey−1 (∂x ϕ6 )∆−1 R + ϕ1 ∂y (∂x ϕ2 )ϕ5 ∂y ϕ6 ∂x ∆R

(III.40)

−1 e−1 e−1 +ϕ1 ∂ey−1 ϕ2 (∂x ϕ5 )∂ey−1 (∂x ϕ6 )∆−1 R + ϕ1 ∂y ϕ2 (∂x ϕ5 )∂y ϕ6 ∂x ∆R

+ϕ1 ∂ey−1 ϕ2 ϕ5 ∂x2 ∂ey−1 ϕ6 ∆−1 R . The first four terms in the right hand side of (3.40) are clearly L2 -bdd. For the fifth one we write −1 e−1 e−1 2 ϕ1 ∂ey−1 ϕ2 ϕ5 ∂x2 ∂ey−1 ϕ6 ∆−1 R = ϕ1 ∂y ϕ2 ϕ5 ∂y (∂x ϕ6 )∆R 2 −1 e−1 e−1 +2ϕ1 ∂ey−1 ϕ2 ϕ5 ∂ey−1 (∂x ϕ6 )∂x ∆−1 R + ϕ1 ∂y ϕ2 ϕ5 ∂y ϕ6 ∂x ∆R ,

(III.41)

which are all L2 -bdd. The second term in the right-hand side of (3.39) is slightly better because −1 e ∂x ∂x is L2 -bdd. The third one is like the second one and the fourth like the first one. Thus collecting this information we find that L1 S1 is L2 -bdd. The proof of the L2 -boundedness of S1 L¯1 , L¯1 S2 and S2 L1 is similar.

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Thus we have completed the proof of the first part of (3.15(iii)) i.e., Λβ1 −β1 Λ is L2 bounded. The proofs of the L2 -boundedness of S1 L¯2 and S2 L2 are similar. Thus we study ! 0 RS1 − S1 R ΛB − BΛ = RS . (III.42) ¯ 2 − S2 R 0 ¯ RS ¯ 2 , S2 R are L2 -bdd. Thus since It will be shown that RS1 , S1 R, 2iRS1

= r1 ∂x ((ϕ5 ∂x ∂ey−1 ϕ6 + ϕ7 ∂y ∂ex−1 ∂8 )∆−1 R ) +r2 ∂y ((ϕ5 ∂x ∂ey−1 ϕ6 + ϕ7 ∂y ∂ex−1 ϕ3 )∆−1 R ),

(III.43)

¯ the previous argument provides the result. A similar conclusion applies to S1 R, ¯ RS2 and S2 R. Finally we shall prove (3.15(i)), i.e., the invertibility of Λ. First we shall see that S1 , S2 have operator norm on L2 , which tends to zero as R ↑ ∞. Thus, we consider S1 =

1 (ϕ5 ∂x ∂ey−1 ϕ6 + ϕ7 ∂y ∂ex−1 ϕ8 )∆−1 R . 2i

(III.44)

We take the first term in the r.h.s. of (3.44) and remark that the proof for the second one is similar. Then −1 −1 e−1 e−1 ϕ5 ∂x ∂ey−1 ϕ6 ∆−1 R = ϕ5 ∂y (∂x ϕ6 )∆R + ϕ5 ∂y ϕ6 ∂x ∆R .

(III.45)

−1 2 Now ∆−1 R and ∂x ∆R have norms on L which tend to zero as R ↑ ∞, see 2 (3.20). This proves the invertibility in L of Λ = I − S. Next we shall show that for N > 1, and s ≥ 0 the operator norm of

λN (x)Jxs Sj Jx−s

and λN (y)Jys Sj Jy−s ,

j = 1, 2,

(III.46)

in L2 ([0, T ] × Rx × Ry ) tend to zero as R ↑ ∞. We start out with −1 −s −s s e−1 Jxs ϕ5 ∂x ∂ey−1 ϕ6 ∆−1 R Jx = [Jx ; ϕ5 ]∂x ∂y ϕ6 ∆R Jx −s +ϕ5 Jxs ∂x ∂ey−1 ϕ6 ∆−1 R Jx = I+II .

(III.47)

For I we observe that [Jxs ; ϕ5 ]∂x = L1 is an operator of order s in x, uniformly in y so L1 = (L1 Jx−s )Jxs , where L1 Jx−s is an operator of order zero in x uniformly in y, i.e., (L1 Jx−s )f (x, y) = Ty f (·, y)(x),

(III.48)

where Ty is a classical zero order ψ.d.o. in x with seminorms bounded uniformly in y.

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Returning to (3.47), by the continuity of the ψ.d.o. of order zero in L2 (λN (x) dx) (see Lemma 2.4) we have that kI(f )kL2 (R2 ×[0,T ]:λN (x)dxdydt) = k(L1 J −s )(J s ∂e−1 ϕ6 ∆−1 J −s )f kL2 (λ ≤ ≤ ≤

(III.49) x x y N (x)dxdydt) R x −1 −s −1 s e−1 −1 s −s Ck(Jx ∂y ϕ6 ∆R Jx )f kL2 (λN (x)) = Ck∂ey Jx ϕ6 ∆R Jx (f )kL2 (λN (x)) −s s −s −1 CkJxs ϕ6 ∆−1 R Jx (f )kL2 (λN (x)) = CkJx ϕ6 Jx ∆R (f )kL2 (λN (x)) Ck∆−1 R (f )kL2 (λN (x)) ,

since ∂ey−1 is bdd in L2 (dy) and Jxs ϕ6 Jx−s is a ψ.d.o. of order 0 in x, uniformly in y. We are now going to prove that for N ≥ 1 2 2 ∆−1 R : L (λN (x)dx dy dt) → L (λN (x)dx dy dt)

(III.50)

is bounded with norm tending to zero as R ↑ ∞ uniformly in N ≤ N0 . It suffices to see that for f = f (x, y) 1 ∆−1 ((1 + x2 )f ) is bdd in L2 (dx dy), 1 + x2 R

(III.51)

with norm tending to 0 as R ↑ ∞ (the proof for general N ∈ Z+ is similar). Taking Fourier transform it follows that 1 ∆−1 ((1 + x2 )f ) 1 + x2 R ZZ θ |(ξ1R,ξ2 )| 1 ((1 + x2 )f )∧ (ξ1 , ξ2 )dξ1 dξ2 = ei(xξ1 +yξ2 ) 1 + x2 ξ12 + ξ22 ZZ θ |(ξ1R,ξ2 )| 1 = (I − ∂ξ21 )fˆ(ξ1 , ξ2 )dξ1 dξ2 ei(xξ1 +yξ2 ) 1 + x2 ξ12 + ξ22   |(ξ1 ,ξ2 )| ZZ   θ R 1 2 i(xξ1 +yξ2 ) = (I − ∂ ) e fˆ(ξ1 , ξ2 )dξ1 dξ2 ξ 1  1 + x2 ξ12 + ξ22    ZZ  θ |(ξ1R,ξ2 )|  1 = ei(xξ1 +yξ2 ) (I − ∂ξ21 ) fˆ(ξ1 , ξ2 )dξ1 dξ2  ξ12 + ξ22  1 + x2 |(ξ1 ,ξ2 )| ZZ 2 θ R x + ei(xξ1 +yξ2 ) fˆ(ξ1 , ξ2 )dξ1 dξ2 1 + x2 ξ12 + ξ22   ZZ  θ |(ξ1R,ξ2 )|  2ix fˆ(ξ1 , ξ2 )dξ1 dξ2 , − ei(xξ1 +yξ2 ) ∂ξ1  ξ12 + ξ22  1 + x2

(III.52)

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from which the result follows. Now for II in (3.47) we have −s II(f ) = ϕ5 Jxs ∂x ∂ey−1 ϕ6 ∆−1 R Jx (f ) s −s −1 e−1 = ϕ5 ∂ey−1 [Jxs ∂x ; ϕ6 ]Jx−s ∆−1 R (f ) + ϕ5 ∂y ϕ6 Jx ∂x Js ∆R (f ) = II1 (f ) + II2 (f ).

(III.53)

In II1 (·), [Jxs ∂x ; ϕ6 ]Jx−s is a ψ.d.o. of order 0 in x, uniformly in y and we −1 proceed as for I(·). For II2 (·), we combine that Jxs ∂x Jx−s ∆−1 and that R = ∂x ∆R −1 2 ∂x ∆R is L (λN (x)dx dy dt) is bounded with norm tending to 0 as R ↑ ∞. The proof of this last fact is similar to that in (3.52). The other piece of S1 in (3.44) corresponds to −s Jxs ϕ7 ∂y ∂ex−1 ϕ8 ∆−1 R J = [J s ; ϕ7 ]∂y ∂e−1 ϕ8 ∆−1 J −s + ϕ7 J s ∂y ∂e−1 ϕ8 ∆−1 J −s x

x x R x −s = ([Jxs ; ϕ7 ]Jx−s )Jxs ∂y ∂ex−1 ϕ8 ∆−1 J R x 0 0

x

+

R x −s ϕ7 Jxs ∂y ∂ex−1 ϕ8 ∆−1 R Jx

(III.54)

= I + II .

Now for I0 we write I0 = [Jxs ; ϕ7 ]Jx−s ∂y Jxs ∂ex−1 ϕ8 Jx−s ∆−1 R s −s s e−1 −s −1 = (|J ; ϕ7 ]J )J ∂ (∂y ϕ8 )J ∆

(III.55)

x x x x x R −1 s −s s e−1 −s +([Jx ; ϕ7 ]Jx )Jx ∂x ϕ8 Jx ∂y ∆R .

Since [Jxs ; ϕ7 ]∂ex−1 (∂y ϕ8 )Jx−s is a ψ.d.o. of order 0 (in fact, of order −2) in x, uniformly in y we can handle the bound of the first term in the r.h.s. of (3.55) as that for I in the previous case. For the second term in the r.h.s. of (3.55) we see that [Jxs ; ϕ7 ]∂ex−1 ϕ8 Jx−s is a ψ.d.o of order 0 (in fact, of order −2) in x uniformly in y and ∂y ∆−1 is L2 (λN (x)dxdydt)-bounded with norm tending to zero as R ↑ ∞. R Finally we look at II0 = ϕ7 Jxs ∂y ∂ex−1 ϕ8 Jx−s ∆−1 R = ϕ7 J s ∂e−1 (∂y ϕ8 )J −s ∆−1 + ϕ7 J s ∂e−1 ϕ8 J −s ∂y ∆−1 . x x

x

R

x x

x

R

(III.56)

For the first term in the right hand side of (3.56) we use that ϕ7 Jxs ∂ex−1 (∂y ϕ8 ) is of order 0 in x (in fact, or order −1) uniformly in y, and for the second one we use that ϕ7 Jxs ∂ex−1 ϕ8 J −s is of order 0 in x uniformly in y, together with a previous argument. By symmetry we have finished the proof of (3.15(i))–(3.15(iv)) and concluded the diagonalization. Thus we have that ~v = Λw ~ = Λ(z z¯)T verifies the system

Jx−s

~ ∂t~v = iH~v + B~v + β1~v + C1~v + G,

(III.57)

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with H, B, β1 as in (3.6)–(3.7), C an operator in the (x, y)-variables, which is L2 -bounded (indeed H 2 -bdd) and ~ = F~ + E, ~ G

(III.58)

~ L2 kEk ≤ Φ defined in (3.18), x,y,t

(III.59)

where

and where F1 = p1 ∂x f1 + p2 ∂y f2 , F2 = φ1 ∂x ∂ey−1 φ2 f3 + φ3 ∂y ∂ex−1 φ4 f4 ,

(III.60)

and F~ =

F1 + F2 + f5 F¯1 + F¯2 + f¯5

! .

(III.61)

Thus we are reduced to proving the estimates for ~v , which solves the “diagonal system” (3.57).

IV Proof of Theorem 1.2 (Conclusion) In this section we complete the proof of Theorem 1.2. From the results in the previous section for all practical purpose to work with “diagonal system” (3.57) is equivalent to work with the single equation ∂t z = i∆z + r1 ∂x z + r2 ∂y z + ϕ1 ∂x ∂ey−1 ϕ2 z + ϕ3 ∂y ∂ex−1 ϕ4 z +c1 z + φ1 ∂x f1 + φ2 ∂y f2 + φ3 ∂x ∂ey−1 φ4 f3 + φ5 ∂y ∂ex−1 φ6 f4 + f5

(IV.1)

= i∆z + ~r · ∇z + ϕ1 ∂x ∂ey−1 ϕ2 z + ϕ3 ∂y ∂ex−1 ϕ4 z + C1 z + Γ, with C1 bounded in L2 , f5 ∈ L2T L2xy and z(0) = z0 (x). We introduce classical ψ.d.o. in each variable. First we have Cx (x, Dx ), whose symbol is Z −M x 2 ξ1 2 ξ1 CM,R (x, ξ1 ) = exp µ (s)ds θ (IV.2) 2 |ξ1 | R 0 with θ(·) defined in (3.19), and µ(·) ∈ C ∞ , an even function, µ ∈ L2 ([0, ∞)), with a decay at infinity to be determined. Clearly Cx ∈ S 0 (R2 ). Similarly we define Cy (y, Dy ). We observe that the symbol of ∂x2 Cx = σ(∂x2 Cx ) is σ(∂x2 Cx ) = −ξ12 CM,R (x, ξ1 ) + 2iξ1 ∂x CM,R (x, ξ1 ) + ∂x2 CM,R (x, ξ1 ),

(IV.3)

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On the Initial Value Problem for the Ishimori System

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and σ(Cx ∂x2 ) = −ξ12 CM,R (x, ξ1 ).

(IV.4)

Therefore σ(i[Cx ∂x2 − ∂x2 Cx ]) = 2ξ1 ∂x CM,R (x, ξ1 ) + L2 -bdd = −M µ2 (x)|ξ1 |θ2 ξR1 CM,R (x, ξ1 ) + L2 -bdd.

(IV.5)

Define the (self adjoint) operator C as C = (Cx Cy )∗ Cx Cy = Cx∗ Cx Cy∗ Cy ,

(IV.6)

A = C 2 = C ∗ C = Cx∗ Cx Cx∗ Cx Cy∗ Cy Cy∗ Cy = Ax Ay .

(IV.7)

and

We shall compute ∂t hAz, zi = hA∂t z, zi + hAz, ∂t zi = hiA∆z, zi + hA~r · ∇z, zi + hA(ϕ1 ∂x ∂ey−1 ϕ2 )z, zi +hAϕ3 ∂y ∂ex−1 ϕ4 z, zi + hAC1 z, zi + hAΓ, zi +hAz, i∆zi + hAz, ~r · ∇zi + hAz, ϕ1 ∂x ∂e−1 ϕ2 zi

(IV.8)

y

+hAz, ϕ3 ∂y ∂ex−1 ϕ4 zi + hAz, C1 zi + hAz, Γi. We shall use that hiA∆z, zi + hAz, i∆zi = hi[A∆ − ∆A]z, zi,

(IV.9)

i[A∆ − ∆A] = i(A∂x2 − ∂x2 A) + i(A∂y2 − ∂y2 A),

(IV.10)

σ(i(A∂x2

−

∂x2 A))

=

−4M µ (x)|ξ1 |θ 2

2

σ(i(A∂y2

−

∂y2 A))

=

ξ1 R

−4M µ (y)|ξ2 |θ 2

2

ξ2 R

ax ay + bdd-L2 ,

(IV.11)

ax ay + bdd-L2 ,

( σ(Ax ) = ax (x, ξ1 ) = C4M,R (x, ξ1 ) + Sx−1 , σ(Ay ) = ay (y, ξ2 ) = C4M,R (y, ξ2 ) + Sy−1 ,

(IV.12)

(IV.13)

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Ann. Henri Poincar´ e

with Sx−1 , Sy−1 ∈ S −1 (R2 ) (see Theorem 2.2). Also hAz, zi = hC ∗ Cz, zi = hCz, Czi,

(IV.14)

hA~r · ∇z, zi + hAz, ~r · ∇zi = hA~r · ∇z, zi + hz, A~r · ∇zi = 2 RehA~r · ∇z, zi.

(IV.15)

and

The sum of the third and ninth terms in the right hand side of (4.8) gives 2 RehAϕ1 ∂x ∂ey−1 ϕ2 z, zi, the sum of the 4th and 10th gives 2 RehAϕ3 ∂y ∂ex−1 ϕ4 z, zi, the sum of the 5th and 11th gives 2 RehAC1 z, zi, and finally the sum of 6th and 12th gives 2 RehAΓ, zi. In order to apply the vector value sharp G˚ arding inequality (Theorem 2.5) we write out operators in a vector valued form. Thus from (4.11)-(4.13) ξ1 σ(i[A∂x2 − ∂x2 A]) = −4M µ2 (x)|ξ1 |θ2 (IV.16) ax ay + bdd-L2 , R and −4M µ (x)|ξ1 |θ 2

2

ξ1 R

ax = −4M µ (x)|ξ1 |θ 2

2

ξ1 R

C4M,R (x, ξ1 ) + L0 , (IV.17)

where L0 = L0 (x, ξ1 ) is the symbol of a bdd operator in L2 (dx). Thus, modulo L2 -bdd operator (L0 ) we have

=

R

i[A∂x2 − ∂x2 A]f (x, y) (IV.18) ixξ1 2 2 ξ1 ∧x e (−4M µ (x)|ξ1 |θ R C4M,R (x, ξ1 )(Ay f (x, −)) (ξ1 ))dξ1 .

Now (Ay f (x, −))∧x (ξ1 ) =

R

e−ixξ1 Ay f (x, −)(y)dx R −ixξ 1 e f (x, −)dx (y). = Ay

(IV.19)

Thus, i[A∂x2 − ∂x2 A] has vector valued symbol (modulo L2xy -bdd operator) −4M µ (x)|ξ1 |θ 2

2

ξ1 R

C4M,R (x, ξ1 )Ay .

(IV.20)

Next we look at the term hAr1 ∂x z, zi = hAx Ay r1 ∂x z, zi. We recall that r1 = r1 (x, y) decays in x, uniformly in y, then we write r1 (x, y) = λ2N (x)λ−2 r1 (x, y), N (x)r1 (x, y) = λN (x)˜

(IV.21)

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and r1 ∂x = Ax λ2N (x)Ay r˜1 ∂x Ax Ay r1 ∂x = Ax Ay λ2N (x)˜ = Ax λ2N (x)∂x Ay r˜1 − Ax λ2N (x)Ay (∂x r˜1 ) = λ2N (x)∂x Ax Ay r˜1 + L2xy -bdd.

(IV.22)

Similarly Aϕ1 ∂x ∂ey−1 ϕ2 = Ax Ay ϕ1 ∂x ∂ey−1 ϕ2 = Ax Ay λ2N (x)ϕ e1 ∂x ∂ey−1 ϕ2 = Ax λ2N (x)Ay ϕ e1 ∂x ∂ey−1 ϕ2 = Ax λ2N (x)∂x Ay ϕ e1 ∂ey−1 ϕ2 + L2 -bdd =

λ2N (x)∂x Ax Ay ϕ e1 ∂ey−1 ϕ2

(IV.23)

2

+ L -bdd.

Thus modulo L2xy -bdd operators we have the vector value symbols σ(Ar1 ∂x ) = λ2N (x)(iξ1 )C4M,R (x, ξ1 )Ay r˜1 + L2 -bdd,

(IV.24)

and e1 ∂ey−1 ϕ2 + L2 -bdd. σ(Aϕ1 ∂x ∂ey−1 ϕ2 ) = λ2N (x)(iξ1 )C4M,R (x, ξ1 )Ay ϕ

(IV.25)

Note that all these operators are of order 1. In order to apply the vector valued sharp G˚ arding inequality (Theorem 2.5) we make the following claims : Claim 1 We can choose M, R, µ so that for |ξ1 | large −M µ2 (x)|ξ1 |θ2 ξR1 C4M,R (x, ξ1 ){Ay + A∗y } r1 A∗y , ≤ λ2N (x)iξ1 C4M,R (x, ξ1 )Ay r˜1 − λ2N (x)iξ1 C4M,R (x, ξ1 )˜

(IV.26)

as operators on L2 (R : dy), and −M µ2 (x)|ξ1 |θ2

ξ1 R

C4M,R (x, ξ1 ){Ay + A∗y }

e1 ∂ey−1 ϕ2 ≤ iξ1 λ2N (x)C4M,R (x, ξ1 )Ay ϕ −iξ1 λ2 (x)C4M,R (x, ξ1 ){ϕ¯2 (∂e−1 )∗ ϕ e1 A∗ }, N

y

(IV.27)

y

as operators on L2 (R : dy). Claim 2 With M, R, µ chosen as in Claim 1, we can choose R even larger so that Cx is invertible in L2 (dx). Proof of Claim 1. Since Ay = Cy∗ Cy Cy∗ Cy then A∗y = Ay .

(IV.28)

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Thus, we take f ∈ L2 (R : dy) and want to show that for ξ1 large −2M µ2 (x)|ξ1 |θ2 Rξ C4M,R (x, ξ1 )hAy f, f i ≤ 2 Reh−λ2N (x)iξ1 C4M,R (x, ξ1 )Ay r˜1 f, f i = 2 Re{−λ2N (x)iξ1 C4M,R (x, ξ1 )hAy r˜1 f, f i}.

(IV.29)

It suffices to show that for ξ1 large 2 λ (x)|ξ1 |C4M,R (x, ξ1 )hAy r˜1 f, f i N ≤ M µ2 (x)|ξ1 |θ2 ξR1 C4M,R (x, ξ1 )hAy f, f i or

λ2N (x)|hAy r˜1 f, f i| ≤ M µ2 (x)θ2

ξ1 R

(IV.30)

hAy f, f i

(IV.31)

for ξ1 large. Thus, for ξ1 such that |ξ1 | ≥ 2R (4.31) becomes (see (3.19)) λ2N (x)|hAy r˜1 f, f i| ≤ M µ2 (x)hAy f, f i.

(IV.32)

Choosing µ2 (x) = λ2N (x),

N > 1,

(IV.33)

|hAy r˜1 f, f i| ≤ M hAy f, f i.

(IV.34)

we reduce the proof of (4.26) to see that

We recall that Ay = Cy∗ Cy Cy∗ Cy , and so hAy f, f i = hCy∗ Cy f, Cy∗ Cy f i = kCy∗ Cy f k2L2y .

(IV.35)

Now |hAy r˜1 f, f i| = |hCy∗ Cy r˜1 f, Cy∗ Cy f i| ≤ kCy∗ Cy r˜1 f kL2y kCy∗ Cy f kL2y .

(IV.36)

Thus we just need to establish the inequality kCy∗ Cy r˜1 f kL2y ≤ M kCy∗ Cy f kL2y ,

uniformly in x.

(IV.37)

Letting g = Cy∗ Cy f , it suffices to show that there exist M, R so that kCy∗ Cy r˜1 (Cy∗ Cy )−1 gkL2y ≤ M kgkL2y ,

(IV.38)

uniformly in x. We recall that σ(Cy ) = CM,R (y, ξ2 ) = exp

−M 2

Z

y

µ2 (s)ds 0

ξ2 2 θ |ξ2 |

ξ2 R

.

(IV.39)

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365

We first choose M such that |˜ r1 (x, y)| ≤

M . 10

(IV.40)

With M so chosen, we will choose R. First we compute Cy∗ Cy . From Theorem 2.2 it follows that σ(Cy∗ ) = CM,R (y, ξ2 ) + q1 (y, ξ2 ),

(IV.41)

Ry 2 µ (s)ds |ξξ22 | θ2 ξR2 ∂ξ2 ∂y CM,R (y, ξ2 ) = ∂ξ2 ∂y exp −M 2 0 n o CM,R (y, ξ2 ) = −M µ2 (y)∂ξ2 |ξξ22 | θ2 ξR2 = −2M µ2 (y) |ξξ22 | R1 θ ξR2 θ0 ξR2 CM,R (y, ξ2 ).

(IV.42)

where q1 (y, ξ2 ) involves

Note that the S 0 seminorms of CM,R (y, ξ2 ) are uniformly bounded, depending only on M for R ≥ 1. Thus the S 0 seminorms of q1 (y, ξ2 ) are cM /R, for R ≥ 1. Hence Cy∗ = Cy + E0 ,

with

|||E0 ||| ≤

cM , R

(IV.43)

with ||| · ||| denoting the operator norm in L2 (R : dy). Thus Cy∗ Cy = Cy Cy + ECy = Cy Cy + E00 ,

(IV.44)

where E00 = E0 Cy inherits the boundedness property of E0 in (4.43). Now we compute Cy Cy σ(Cy Cy ) = C2M,R (y, ξ2 ) + q2 (y, ξ2 ),

(IV.45)

where q2 (·, ·) depends on ∂ξ2 CM,R and ∂y CM,R , (see Theorem 2.2). A computation similar to that in (4.42) gives |∂y CM,R (y, ξ2 )| ≤ cM ,

|∂ξ2 CM,R (y, ξ2 )| ≤ cM /R,

for R ≥ 1. Therefore combining (4.44)-(4.46) it follows that Z y ξ2 2 ξ2 ∗ 2 µ (s)ds σ(Cy Cy ) = exp −M θ + e(y, ξ2 ), |ξ2 | R 0

(IV.46)

(IV.47)

where the operator E(y, Dy ) with symbol e(y, ξ2 ) has operator norm ||| · ||| satisfying |||E||| ≤

CM R

in L2 (dy).

(IV.48)

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Ann. Henri Poincar´ e

Now we define the operator S = S(y, Dy ) by its symbol s(y, ξ2 ) as Z y ξ2 2 ξ2 2 s(y, ξ2 ) = exp M µ (s)ds θ . |ξ2 | R 0

(IV.49)

As before one sees that its operator norm |||S||| in L2 (dy) satisfies |||S||| ≤ CM , for R > 1.

(IV.50)

The same argument gives that Cy∗ Cy S = I + E1 , and SCy∗ Cy = I + E2 ,

(IV.51)

with |||Ej ||| ≤

CM , R

j = 1, 2, and R > 1.

(IV.52)

For M fixed we choose R large enough such that CM ≤ 1/2, R

(IV.53)

and get that for T1 = (I + E1 )−1 ,

T2 = (I + E2 )−1 ,

(IV.54)

one has |||Tj ||| ≤ 2, j = 1, 2,

T2 S = ST1 = (Cy∗ Cy )−1 .

(IV.55)

Also T1 = I + E3 , with |||E3 ||| <

CM , R

(IV.56)

and ST1 = S + SE3 with |||E30 ||| ≤

CM . R

(IV.57)

Thus (Cy∗ Cy )−1 = S + E30 .

(IV.58)

Now we proceed to estimate the norm of Cy∗ Cy r˜1 (Cy∗ Cy )−1 = Cy∗ Cy r˜1 S + Cy∗ Cy r˜1 E30 .

(IV.59)

We notice that |||Cy∗ Cy r˜1 E30 ||| ≤

CM , as operator in L2 (dy), uniformly in x, R

(IV.60)

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and

Z

y

ξ2 2 µ (s)ds θ |ξ2 |

2

σ(˜ r1 S) = r˜1 (x, y) exp M 0

ξ2 R

.

(IV.61)

Since modulo L2 -bdd operator with norm bounded by CM /R, Cy∗ Cy r˜1 S is the product of the symbols CM,2R (y, ξ2 ) and r˜1 (x, y)CM,−2R (y, ξ2 ) we get Cy∗ Cy r˜1 (Cy∗ Cy )−1 = r˜1 + Cy∗ Cy r˜1 E30 + E4 ,

(IV.62)

where the seminorms of the symbol of E4 are controlled by products of the one of ∂ξ2 C2M,R and those of ∂y (C−2M,R (·)˜ r1 (x, −)), uniformly in x. Hence |||E4 ||| ≤

CM , R

as operator in L2 (dy) uniformly in x,

(IV.63)

and consequently form (4.60), (4.63) Cy∗ Cy r˜1 (Cy∗ Cy )−1 = r˜1 + E5 ,

(IV.64)

as operator in L2 (dy), uniformly in x.

(IV.65)

with |||E5 ||| ≤

CM , R

Since |˜ r1 (x, y)| ≤

M , 10

(IV.66)

we obtain (4.38) by fixing R such that CM /R ≤ M/10. This proves the first part of Claim1, i.e. the inequality (4.26). We turn to the proof of (4.27), the second part of Claim 1, i.e. for f ∈ L2 (R : dy) Re{−M µ2 (x)|ξ1 |θ2 ξR1 C4M,R (x, ξ1 )hAy f, f i} e1 ∂ey−1 ϕ2 f, f i}. ≤ Re{iξ1 λ2N (x)C4M,R hAy ϕ

(IV.67)

We proceed as in (4.28)–(4.34). By taking ξ1 such that |ξ1 | ≥ 2R we need to show that |hAy ϕ e1 ∂ey−1 ϕ2 f, f i| ≤ M hAy f, f i

(IV.68)

e1 ∂ey−1 ϕ2 f, Cy∗ Cy f i| ≤ M kCy∗ Cy f k2L2y . |hCy∗ Cy ϕ

(IV.69)

or

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Thus, it suffices to show that e1 ∂ey−1 ϕ2 (Cy∗ Cy )−1 ||| ≤ M, in L2 (dy) uniformly in x. |||Cy∗ Cy ϕ

(IV.70)

We saw in (4.56)-(4.58) that (Cy∗ Cy )−1 = S + E30 , with |||E30 ||| ≤ CM /R,

(IV.71)

so we shall show that M can be chosen so that for R large enough e1 ∂ey−1 ϕ2 S||| ≤ M/2, as operator in L2 (dy), uniformly in x. |||Cy∗ Cy ϕ From (4.47) one has that Z ∗ σ(Cy Cy ) = exp −M 0

y

ξ2 2 µ (s)ds θ |ξ2 |

2

ξ2 R

(IV.72)

+ e(y, ξ2 ),

(IV.73)

with σ(E(y, Dy )) = e(y, ξ2 ) and |||E||| ≤ CM /R. We notice that ϕ2 S = Sϕ2 + E6 , and

with |||E6 ||| ≤ CM /R,

Ry e1 ) = ϕ e1 (x, y) exp −M 0 µ2 (x)ds σ(Cy∗ Cy ϕ

ξ2 2 |ξ2 | θ

(IV.74)

ξ2 R

+ e7 (x, y, ξ2 )

=ϕ e1 (x, y)C2M,R (y, ξ2 ) + e7 (x, y, ξ2 ),

(IV.75)

with σ(E7x (y, Dy )) = e7 (x, y, ξ2 ) satisfying that |||E7 ||| ≤

CM , as operator in L2 (dy), uniformly in x. R

(IV.76)

Thus the problem has been reduced to show that |||ϕ e1 C2M,R (y, Dy )∂ey−1 C−2M,R (y, Dy )ϕ2 ||| ≤ M/4,

(IV.77)

as operator in L2 (dy) uniformly in x. But we can choose M > 1 such that kϕ e1 kL∞ (R2 ) , kϕ2 ||L∞ (R2 ) ≤

M 1/4 . 10

(IV.78)

Thus we see that it suffices to show that M 1/4 |||C−2M,R (y, Dy )∂ey−1 C−2M,R (y, Dy )||| ≤ , 10

(IV.79)

with M large and R chosen after M . We recall that σ(∂ey−1 ) =

N0 X j=0

ψj1 (·)m(ξ2 )ψj2 (·),

(IV.80)

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where ψj1 , ψj2 ’s are multiplication operators by bounded smooth functions and in −1 m(·) is a multiplier in SM . So we can reduce ourselves to show that |||C2M,R (y, Dy )m(Dy )C−2M,R (y, Dy )||| ≤

M 1/4 . 10

(IV.81)

From Theorem 2.2 we have σ(m(Dy )C−2M,R (y, Dy )) = a(y, ξ2 ) + q(y, ξ2 ), where

Z

y

0

ξ2 2 µ (s)ds θ |ξ2 |

Z

1

a(y, ξ2 ) = m(ξ2 ) exp −M

2

ξ2 R

(IV.82)

,

(IV.83)

and q(y, ξ2 ) =

qδ (y, ξ2 )dθ,

(IV.84)

0

with

ZZ qδ (y, ξ2 ) =

e−izη m0 (ξ2 + δη)∂y C−2M,R (y + z, ξ2 )dz dη.

Now ξ2 2 θ ∂y C−2M,R (y, ξ2 ) = C−2M,R (y, ξ2 )2M µ (y) |ξ2 | 2

therefore qδ (y, ξ2 ) = =

ξ2 2 |ξ2 | θ ξ2 2 |ξ2 | θ

RR ξ2 R

ξ2 R

ξ2 R

(IV.85)

,

(IV.86)

e−izη m0 (ξ2 + δη)b1 (y + z, ξ2 )dz dη

q1,δ (y, ξ2 ),

(IV.87)

with b1 (y, ξ2 ) = M µ2 (y)C−2M,R (y, ξ2 ).

(IV.88)

Note that m0 ∈ S −2 , and b1 ∈ S 0 with semi-norms depending only on M and not on R ≥ 1. Thus q1,δ ∈ S −2 with bounds depending only on M . But then the S 0 seminorms of qδ are CM /R2 , uniformly in δ. Hence m(Dy )C−2M,R (y, Dy ) = C−2M,R (y, Dy )m(Dy ) + E8 ,

(IV.89)

with |||E8 ||| ≤ CM /R, as operator in L2 (dy).

(IV.90)

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Ann. Henri Poincar´ e

Finally C2M,R (y, Dy )C−2M,R (y, Dy )m(Dy ) = m(Dy ) + E9 , |||E9 ||| ≤ CM /R. (IV.91) We now take M so large that kmkL∞ (R) ≤

M 1/4 , 10010

(IV.92)

and then choose R large, and (4.27), and consequently Claim 1, has been proved. Note that Claim 2, and the symmetric of Claim 1 (a),(b) follow in the same manner. We now fix λ, M and R as in all the claims, to obtain using the vector valued sharp G˚ arding inequality that 2 Rehi[A∂x2 − A∂x2 ]z, zi + 2 RehAr1 ∂x z, zi + 2 RehAϕ1 ∂x ∂ey−1 ϕ2 z, zi ξ1 Ax Ay z, zi + c(M,R) kzk2L2 , ≤ −M hµ2 (x)|ξ1 |θ2 R

(IV.93)

2 Rehi[A∂y2 − A∂y2 ]z, zi + 2 RehAr2 ∂x z, zi + 2RehAϕ3 ∂y ∂ex−1 ϕ4 z, zi ξ2 2 2 Ax Ay z, zi + c(M,R) kzk2L2 . ≤ −M hµ (y)|ξ2 |θ R

(IV.94)

and

Thus upon integration between 0 and T in (4.8) we get RT kCz(t)k2L2 ≤ kCz0 k2L2 − M 0 hµ2 (x)|ξ1 |θ2 ξR1 Ax Ay z, zidt RT −M 0 hµ2 (y)|ξ2 |θ2 ξR2 Ax Ay z, zidt (IV.95) R t +c(M,R) T sup0≤t≤T kz(t)k2L2 + 2 Re 0 hAΓ, zi(t0 )dt0 . Hence if we denote by Qx = Q(x, D1 ) (resp. Qy = Q(y, D2 )) the operator defined by its symbol ξ1 ξ2 µ2 (x)|ξ1 |θ2 , resp. µ2 (y)|ξ2 |θ2 , (IV.96) R R it follows that Z T M (hQx Ax Ay z, zi + hQy Ax Ay z, zi)dt + cM sup kz(t)k2L2 0≤t≤T

0

≤

C(M )kz0 k2L2

+ T c(M,R) sup kz(t)k2L2 0≤t≤T

Z t + sup 2 Re hAΓ, zidt0 . 0≤t≤T 0

(IV.97)

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We now work with the term hQx Ax Ay z, zi = hQx Ax Cy∗ Cy Cy∗ Cy z, zi RR = hQx Ax Cy∗ Cy z, Cy∗ Cy zi ≡ Qx Ax Cy∗ Cy zCy∗ Cy z dx dy.

(IV.98)

We remark that Qx Ax is a ψ.d.o., in the x variable, of order 1, and that the symbol of Ax ≥ c(M,R) modulo terms of order −1. By the sharp G˚ arding inequality in the x variable, we have, uniformly in y that Z R Qx Ax Cy∗ Cy zCy∗ Cy z dx ≥ c(M,R) Qx Cy∗ Cy zCy∗ Cy z dx R (IV.99) −c(M,R) |z(x, y)|2 dx. Thus from (4.98)-(4.99), upon y-integration, it follows that hQx Ax Ay z, zi ≥ c(M,R) hQx Cy∗ Cy z, Cy∗ Cy zi − c(M,R) kzk2L2x,y .

(IV.100)

We remark that µ2 (x)|ξ1 |θ2 ξR1 Ax is the symbol of a ψ.d.o., in the xvariable, of order 1, and that the symbol of Ax ≥ c(M,R) modulo terms of order −1. By the sharp G˚ arding inequality in the x variable, we have, uniformly in y that R 2 µ (x)|ξ1 |θ2 ξR1 Ax Cy∗ Cy zCy∗ Cy z dx (IV.101) R 2 R ≥ c(M,R) µ (x)|ξ1 |θ2 ξR1 Cy∗ Cy zCy∗ Cy z dx − c(M,R) x |z(x, y)|2 dx. Thus upon y-integration it follows that hµ2 (x)|ξ1 |θ2 ξR1 Ax Ay z, zi ≥ c(M,R) hµ2 (x)|ξ1 |θ2 ξR1 Cy∗ Cy z, Cy∗ Cy zi − c(M,R) kzk2L2 . Next we observe that ξ1 2 2 Qx = Os µ (x)|ξ1 |θ = (µ(x)Jx1/2 )(µ(x)Jx1/2 ) + L0 , R

(IV.102)

(IV.103)

where L0 is L2 -bdd. Hence, from (4.100)-(4.103) and (4.33) we have hQx Ax Ay z, zi ≥ c(M,R) kµ(x)Jx Cy∗ Cy zk2L2 − c(M,R) kzk2L2 1/2

= c(M,R) kCy∗ Cy µ(x)Jx zk22 − c(M,R) kzk2L2 1/2

≥

1/2 c(M,R) kλN (x)Jx zk22

− c(M,R) kzk2L2 ,

(IV.104)

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C.E. Kenig, Gustavo Ponce and Luis Vega

Ann. Henri Poincar´ e

since Cy∗ Cy is invertible in L2x,y . Gathering this information we end up with RT 0

1/2

1/2

(kλN (x)Jx zk2L2 + kλN (y)Jy zk2L2 )dt + sup0≤t≤T kz(t)k2L2 ≤ c(M,R) kz0 k22 + c(M,R) T sup0≤t≤T kz(t)k2L2 (IV.105) R t +c(M,R) sup0≤t≤T 0 hAΓ, zidt0 .

It remains to study hAΓ, zi = hΓ, Azi. We recall that from (4.1) Γ = φ1 ∂x f1 + φ2 ∂y f2 + φ3 ∂x ∂ey−1 φ4 f3 + φ5 ∂y ∂ex−1 φ6 f4 + f5 .

(IV.106)

First we have for t ∈ [0, T ] Z t R R hf5 , Azidt ≤ T kf5 kL2 kAzkL2 dt ≤ C T kf5 kL2 kzkL2 dt 0 0 xy xy xy xy 0

≤ CT 1/2 kf5 kL2T,x,y sup0≤t≤T kz(t)kL2xy ≤

1 2c(M,R)

sup0≤t≤T kz(t)k2L2 xy

+

(IV.107)

c(M,R) T kf5 k2L2 . T,x,y

Next we consider Z t Z t hφ1 ∂x f1 , Azidt = hφ1 ∂x f1 , Ax Ay zidt . 0

(IV.108)

0

1/2 1/2

Writing ∂x = Rx Jx Jx

with Rx a ψ.d.o. of order zero in x, one sees that 1/2 1/2

hφ1 ∂x f1 , Ax Ay zi = hφ1 Rx Jx Jx f1 , Ax Ay zi 1/2

1/2

1/2

1/2

= h[φ1 Rx ; Jx ]Jx f1 , Ax Ay zi + hJx φ1 Rx Jx f1 , Ax Ay zi =

1/2 1/2 h[φ1 Rx ; Jx ]Jx f1 , Ax Ay zi

+

(IV.109)

1/2 1/2 hJx [Rx ; φ1 ]Jx f1 , Ax Ay zi

1/2 1/2 +hJx Rx φ1 Jx f1 ; Ax Ay zi. 1/2

1/2

1/2

1/2

Since [φ1 Rx ; Jx ]Jx , and Jx [Rx ; φ1 ]Jx are ψ.d.o’s of order zero in x, uniformly in y and t, the first and second term in the r.h.s. of (4.109) are bounded, after integration in time, by C ≤ CT

1/2

RT 0

kzkL2xy kφ1 kCbN (R2x,y ) kf1 kL2xy dt

(IV.110)

kφ1 kL2T (Cbk (R2x,y )) sup0≤t≤T kz(t)kL2xy sup0≤t≤T kf1 kL2xy .

For the third term in the r.h.s. of (4.109) we have that hJx Rx φ1 Jx f1 , Ax Ay zi = hφ1 Jx f1 , Rx∗ Jx Ax Ay zi 1/2

1/2

1/2

−1/2

= hφ1 Jx f1 , Rx∗ Jx Ax Jx 1/2

1/2

1/2

1/2

Ay Jx zi.

(IV.111)

Vol. 1, 2000

On the Initial Value Problem for the Ishimori System −1/2

Using that P = Rx∗ Jx Ax Jx (2.19), Lemma 2.4, that 1/2

−1/2

373

is a ψ.d.o. of order zero in x we get from

Ay Jx z, φ1 Jx f1 i| |hRx∗ Jx Ax Jx RR 1/2 1/2 = PAy Jx zφ1 Jx f dx dy 1/2 R 1/2 R R 1/2 1/2 |φ1 Jx f1 |2 λ2dx(x) ≤ |PAy Jx z|2 λ2N (x)dx dy N 1/2 R 1/2 R R 1/2 1/2 |Ay Jx z|2 λ2N (x)dx |φ1 Jx f1 |2 λ2dx(x) dy ≤C y N RR RR 1/2 1/2 dy 1 ≤ c(M,R) |Ay Jx z|2 λ2N (x)dx dy + c(M,R) |φ1 Jx f1 |2 λdx 2 (x) N RR 1/2 2 2 1 ≤ c(M,R) |Jx z| λN (x)dx dy

2

RR 1/2 2

|Jx f1 | λN (x)dx dy · φλ12(·,t) +c(M,R) . (x) ∞ 1/2

1/2

1/2

(IV.112)

Lx,y

N

Hence fixing t ∈ [0, T ] one gets R t c(M,R) 0 hφ1 ∂x f1 , Azidt R T RR 1/2 2 2 ≤ 12 0 |Jx z| λN (x)dx dy dt +

1 2

sup0≤t≤T kz(t)k2L2

x,y

+c(M,R) T sup0≤t≤T kf1 (t)k2L2 kφ1 k2L2 (C N (R2 )) x,y x,y T b

2

φ1 1/2 2 +c(M,R) λ2 (x) ∞ kλ(x)Jx f1 kL2 . N

Lx,y,T

The bound for the term

(IV.113)

T,x,y

Z t hφ2 ∂y f2 , Azidt

(IV.114)

0

is similar. We next turn to the estimate for hφ3 ∂x ∂ey−1 φ4 f3 , Ax Ay zi = hφ3 ∂ey−1 (∂x φ4 )f3 , Ax Ay zi + hφ3 ∂ey−1 φ4 ∂x f3 , Ax Ay zi.

(IV.115)

For the first term in the r.h.s. of (4.115) one has, after integration in time, the bound Rt c(M,R) 0 kzkL2x,y kφ3 kL∞ k∂x φ4 kL∞ kf3 kL2x,y dt (IV.116) x,y x,y ≤

1 4

sup0≤t≤T kz(t)k2L2 + c(M,R) kf3 k2L∞ L2 kφ3 k2L2 L∞ k∂x φ4 k2L2 L∞ x,y

T

x,y

T

1/2

x,y

1/2

For the second term in (4.115) we write ∂x = Rx Jx Jx

T

x,y

to have

hφ3 ∂ey−1 φ4 ∂x f3 , Ax Ay zi = hφ3 ∂ey−1 Rx Jx φ4 Jx f3 , Ax Ay zi 1/2

1/2

1/2 1/2 +hφ3 ∂ey−1 [φ4 ; Rx Jx ]Jx f3 , Ax Ay zi

(IV.117)

374

C.E. Kenig, Gustavo Ponce and Luis Vega 1/2

Ann. Henri Poincar´ e

1/2

where [φ4 ; Rx Jx ]Jx is a ψ.d.o. of order zero in x, uniform in y. Thus, the second term in the r.h.s. in (4.117) can be estimated as in (4.112). For the first one in we write 1/2 1/2 hφ3 ∂ey−1 Rx Jx φ4 Jx f3 , Ax Ay zi

= hRx Jx φ3 ∂ey−1 φ4 Jx f3 , Ax Ay zi 1/2

1/2

(IV.118)

+h[φ3 ; Rx Jx ]∂ey−1 φ4 Jx f3 , Ax Ay zi. 1/2

1/2

Now [φ3 ; Rx Jx ]∂ey−1 φ4 Jx 1/2

1/2

1/2

1/2 −1/2 e−1 1/2 ∂y φ4 Jx

= [φ3 ; Rx Jx ]Jx Jx

1/2 1/2 −1/2 1/2 = [φ3 ; Rx Jx ]Jx ∂ey−1 Jx φ4 Jx . 1/2

−1/2

1/2

(IV.119)

1/2

φ4 Jx are ψ.d.o. of order zero in x uniformly in Since [φ3 ; Rx Jx ]Jx and Jx y the bound of the second term in (4.118) follows the argument in (4.112). For the 1/2 −1/2 first term in (4.118) using that Jx Rx∗ Ax Jx is a ψ.d.o. of order zero in x, and Ay 1/2 −1/2 is a ψ.d.o. of order zero in y we obtain, using the notation A = Jx Rx∗ Ax Jx Ay and Theorem 2.4 that 1/2 1/2 hRx Jx φ3 ∂ey−1 φ4 Jx f3 , Ax Ay zi 1/2 1/2 = hφ3 ∂ey−1 φ4 Jx f3 , Jx Rx∗ Ax Ay zi 1/2 1/2 −1/2 1/2 Ay Jx zi = hφ3 ∂ey−1 φ4 Jx f3 , Jx Rx∗ Ax Jx 1/2 1/2 (IV.120) = hφ3 ∂ey−1 φ4 Jx f3 , AJx zi RR 1/2 RR 1/2 1/2 1/2 dy |AJx z|2 λ2N (x)dx dy |φ3 ∂ey−1 φ4 Jx f3 |2 λdx ≤ 2 (x) N RR 1/2 RR 1/2 1/2 1/2 dy |Jx z|2 λ2N (x)dx dy |φ3 ∂ey−1 φ4 J4 f3 |2 λdx ≤c . 2 (x) N

Thus after integrating in time we get the bound 1/2 1 2 4 kλN (x)Jx zkLxyT

3 2

4 2 1/2 +c(M,R) λNφ(x)

∞ λNφ(x)

∞ kλN (x)Jx f3 k2L2 LT xy

LT xy

(IV.121) .

x,y,T

Gathering this information we get the desired bound for the term in (4.115). In the same approach shows that the term Z 0

T

hφ5 ∂y ∂ex−1 φ6 f4 , Ax Ay zidt,

(IV.122)

Vol. 1, 2000

On the Initial Value Problem for the Ishimori System

375

is bounded by 1 4

1/2

sup0≤t≤T kz(t)k2L2 + 14 kλN (y)Jy zk2L2 x,y

x,y

+c(M,R) kf4 k2L∞ L2 kφ5 k2L2 L∞ k∂y φ6 k2L2 L∞ x,y T T x,y T x,y

2

2

φ5

φ6 2

1/2 +c(M,R) λN (y) ∞ λN (y) ∞ λN (y)Jy f4 2 Lx,y,T

Lx,y,T

(IV.123) .

Lx,y,T

Finally, collecting the information in (4.105)-(4.123) we complete the proof of Theorem 1.2. Corollary 4.1 Under the hypothesis of Theorem 1.2 the same results hold for solutions of the IVP (1.22) with i∆ + ∆ instead of i∆, uniformly for ∈ (0, 1]. Proof of Corollary 4.1. With the notation in (4.7) it suffices to see that (see (4.8)) hA∆z, zi + hAz, ∆zi = 2RehA∆z, zi ≤ ckzk2L2x,y , with c independent of ∈ (0, 1]. Thus, we write hA∂x2 z, zi = hAy Ax ∂x2 z, zi = hAy ∂x Ax ∂x z, zi + hAy [Ax ; ∂x ]∂x z, zi. Since Cx∗ Cx , Cy∗ Cy are invertible (see Claim 2 after (4.27)) it follows that hAy ∂x Ax ∂x z, zi = −hAy Ax ∂x z, ∂x zi = −kCx∗ Cx Cy∗ Cy ∂x zk2L2 ≤ −ck∂x zk2L2 , x,y

x,y

(IV.124)

which combined with |hAy [Ax ; ∂x ]∂x z, zi| ≤ ck∂x zkL2x,y kzkL2x,y

(IV.125)

yields the result.

V Proof of Theorem 1.3 We split the proof in three steps. STEP 1. Existence of a local solution u of (5.1) in a time interval [0, T ]. The proof of Theorem 1.3 is based on the viscosity method. Thus, for ∈ (0, 1] we consider the IVP ( ∂t u − i∆u − ∆u = G(u, ∇x u, u ¯, ∇x u ¯), (V.1) u(x, y, 0) = u0 (x, y),

376

C.E. Kenig, Gustavo Ponce and Luis Vega

Ann. Henri Poincar´ e

where u ¯ ¯, ∇x u ¯) = c1 1+|u| G(u, ∇x u, u 2 ∂x u∂y u

+c6 ∂x u∂y−1

∂x u∂y u ¯−∂x u ¯∂y u (1+|u|2 )2

+ c7 ∂y u∂x−1

∂x u∂y u ¯−∂x u ¯∂y u (1+|u|2 )2

(V.2)

.

We write (5.1) in the intergal equation form Z t 0 e(+i)(t−t )∆ G(u, ∇x u, u ¯, ∇x u ¯)(t0 )dt0 , u(t) = e(+i)t∆ u0 +

(V.3)

0

and defines the operator Φ = Φu0 as Z t 0 e(+i)(t−t )∆ G(v, ∇x v, v¯, ∇x v¯)(t0 )dt0 , Φ(v)(t) = e(+i)t∆ u0 +

(V.4)

0

for T v ∈ Xs,a = {v ∈ C([0, T ] : H s ) :

sup ku(t)kH s ≤ a}.

(V.5)

0≤t≤T

We shall use that if f ∈ H s , s ∈ R, then for ∈ (0, 1] ( ke(+i)t∆ f kH s ≤ kf kH s , k∇x e(+i)t∆ f kH s ≤ c√s t kf kH s−1 ,

(V.6)

where ∇x = (∂x , ∂y ). Using that H s (R2 ), with s > 1, is an algebra respect to the pointwise product of function, it follows that for s ≥ 3 and T > 0 sup kG(v, ..)(t)kH s−1 ≤ cs sup kv(t)k3H s .

0≤t≤T

(V.7)

0≤t≤T

Thus, inserting (5.6)-(5.7) in (5.4) we get that for s ≥ 3 sup0≤t≤T kΦ(v)(tkH s ≤ cs ku0 kH s + cs −1/2 ≤ cs ku0 kH s +

RT

√ 1 0 kG(v, ..)(t0 )kH s−1 dt0 t−t cs −1/2 T 1/2 sup0≤t≤T kv(t)k3H s . 0

(V.8)

Therefore, fixing a = 2cs ku0 kH s

and

T = (20 c42 −1/4 ku0 k2H s )−4 ,

(V.9)

T T ) ⊂ Xs,a . A similar argument shows that in (5.5) it follows that Φ(Xs,a

sup kΦ(v) − Φ(w)kH s ≤ 0≤t≤T

1 2

sup kv − wkH s , 0≤t≤T

(V.10)

Vol. 1, 2000

On the Initial Value Problem for the Ishimori System

377

T for any v, w ∈ Xs,a . Hence, (5.3), and consequently (5.1) has a unique solution s u ∈ C([0, T ] : H ).

STEP 2. A priori estimates for the u ’s in C([0, T ] : H s ∩ L2 ((x2 + y ) dxdy)), with T > 0 independent of ∈ (0, 1]. In this step, one of the key in the proof, we will use Theorem 1.2. From (5.3) one has that 2 m/2

sup ku (t)kL2 ≤ c0 ku0 kL2 + c0 T 1/2 sup ku (t)k3H 3 .

0≤t≤T

(V.11)

0≤t≤T

Next, we apply the operator ∂xs to the equation in (5.1) and write the result using the notation v1 = ∂xs u,

(V.12)

to get that   ∂t v1 − i∆v1 − ∆v1 = r1 (u0 )∂x v1 + r2 (u0 )∂y v1    −1 −1   v1  +ϕ1 (u0 )∂x ∂y ϕ2 (u0 )v1 + ϕ5 (u0 )∂x ∂y ϕ6 (u0 )¯ −1 +p1 ∂x f1 + p2 ∂y f2 + φ1,1 ∂x ∂y φ1,2 f1 + φ2,1 ∂x ∂y−1 φ2,2 f1    +φ5,1 ∂x ∂y−1 φ5,2 f2 + φ6,1 ∂x ∂y−1 φ6,2 f2 + f5 ,     v(x, y, 0) = ∂ s u (x, y), x 0

(V.13)

where r1 (u0 ) = r1,1 (u0 ) + r1,2 (u0 ) =

c1 u ¯0 1+|u0 |2 ∂y u0

+ c6 ∂y−1

r2 (u0 ) = r2,1 (u0 ) + r2,2 (u0 ) =

c1 u ¯0 1+|u0 |2 ∂x u0

+ c7 ∂x−1

ϕ1 (u0 ) = c6 ∂x u0 ,

ϕ5 (u0 ) = −c6 ∂x u0 ,

∂x u0 ∂y u ¯0 −∂x u ¯0 ∂y u0 (1+|u0 |2 )2

ϕ6 (u0 ) =

p1 = r1 (u (t)) − r1 (u0 ), φ1,1 = ϕ1 (u (t)) − ϕ1 (u0 ),

,

(V.14)

,

(V.15)

∂y u ¯0 (1 + |u0 |2 )2

(V.16)

∂x u0 ∂y u ¯0 −∂x u ¯0 ∂y u0 (1+|u0 |2 )2

ϕ2 (u0 ) =

∂y u0 (1 + |u0 |2 )2

p2 = r2 (u (t)) − r2 (u0 ), φ1,2 = ϕ2 ((u (t)),

(V.17)

(V.18) (V.19)

378

C.E. Kenig, Gustavo Ponce and Luis Vega

φ2,1 = ϕ1 (u0 ),

φ2,2 = ϕ2 (u (t)) − ϕ2 (u0 ),

φ5,1 = ϕ5 (u (t)) − ϕ5 (u0 ), φ6,1 = ϕ5 (u0 ), f1 = ∂xs u = v1 ,

Ann. Henri Poincar´ e

(V.20)

φ5,2 = ϕ2 ((u (t)),

(V.21)

φ6,2 = ϕ2 (u (t)) − ϕ2 (u0 ),

(V.22)

f2 = ∂¯xs u = v¯1 ,

(V.23)

and f5 contains all the lower order term and satisfies sup kf5 kL2x,y ≤ cs sup ku k3H s .

0≤t≤T

(V.24)

0≤t≤T

To obtain the desired a priori estimate we apply Theorem 1.2 to the IVP (5.13). Thus, we need to show that the constant A in (1.24) can be made as small as we please by taking T sufficiently small, uniformly in ∈ (0, 1]. We shall only be concerned with the term in (1.24) not having the factor T on it, i.e. the second, sixth and seventh. We observe from (5.14)-(5.22) that in each of these factor there is a term of the form Z t d g(u (t0 ))dt0 . (V.25) g(u (t)) − g(u0 ) ≡ 0 dt Thus, using the equation (5.13) we obtain an appropriate bound with a factor T on it. Thus, for the second terms in (1.24) we have Rt d 2 0 0 2 kλ−2 = kλ−2 N (x)p1 kL∞ N (x) 0 dt r1 (u (·, t )dt kL∞ x,y,T x,y,T R t d c1 u¯ ∂y u ¯ −∂x u ¯ ∂y u −2 −1 ∂x u ∂y u = kλN (x) 0 dt ( 1+|u |2 + c6 ∂y ( ))dt0 k2L∞ (V.26) (1+|u |2 )2 x,y,T P −2 2 6 α 2 ≤ CT sup0≤t≤T (ku (t)kH 3 + ku (t)kH 3 ) |α|≤4 kλN (x)∂ u kL∞ L2 ). T

x,y

Next, we have the estimate for the sixth term in (1.24) −1 2 ˜ 2 = kλ−1 kλ−1 N (x)φ1 kL∞ N (x)λM (y)φ1,1 kL∞ x,y,T x,y,T −1 2 = kλ−1 N (x)λM (y)(ϕ1 (u (t)) − ϕ1 (u0 ))kL∞ x,y,T Rt d −1 0 0 2 = kλ−1 (x)λ (y) ϕ (u (t )dt k ∞ Lx,y,T N M 0 dt 1

(V.27)

−1 2 ≤ cT kλ−1 N (x)λM (y)∂x ∂t u (t)kL∞ x,y,T P −1 4 α 2 ≤ cT (1 + sup0≤t≤T ku (t)kH 4 ) |α|≤6 kλ−1 N (x)λM (y)∂ u kL∞ L2 . T

x,y

The other terms in (1.24) can be bounded in a similar manner. Thus to close these estimates we need to bound terms of the form X −1 α 2 kλ−1 (V.28) 2 L (x)λL (y)∂ u kL∞ T Lx,y |α|≤6

Vol. 1, 2000

On the Initial Value Problem for the Ishimori System

379

with L = max{N, M }. To achieve this we use the operators Γx = x + 2it∂x ,

Γy = y + 2it∂y ,

(V.29)

and the following commutative relations and identities    [Γx ; ∂t − i∆] = [Γy ; ∂t − i∆] = 0, [Γx ; ∆] = −2∂x , [Γy ; ∆] = −2∂y ,   [Γx , ∂x ] = [Γy ; ∂y ] = −1, Γ(f g) = f Γ(g) + 2itg∂f.

(V.30)

We shall estimate Γu , Γ∂u , Γ2 u , .., (Γβ ∂ α u )|β|≤2L, |α|≤6 , in this order. For the first step we apply w1 = Γx u to get  u ¯   (∂t − i∆ − ∆)w1 = c1 1+|u|2 ∂y u∂x w1  

Γx

(V.31)

in (5.1) and use the notation

(V.32) ∂ u∂y u ¯−∂x u ¯∂y u +c6 ∂x w1 ∂y−1 ( x (1+|u| ) 2 )2

+

∂ u∂y u ¯−∂x u ¯∂y u c7 ∂y w1 ∂x−1 u( x (1+|u| ) 2 )2

+ f5,1 ,

where f5,1 satisfies an estimate similar to that (5.24). To apply Theorem 1.2 we rewrite the IVP (5.31) using the notation in (5.14)-(5.22) as ( (∂t − i∆ − ∆)w1 = r1 (u0 )∂x w1 + r2,2 (u0 )∂y w1 (V.33) +p1 ∂x f1 + (r2,2 (u (t)) − r2,2 (u0 ))∂y f1 + f5,1 , with f1 = w1 . We observe that the coefficients in the equation in (5.33) are basically the same as those in (5.13). In fact, this is the case for all the equation for the terms in (5.31) except that in each case kf5,· kL2 can be bounded using the previous terms. Hence, defining for T > 0 2 2 kukT ≡ sup ku(t)kH s + kλ−1 (V.34) m (x + y )u(t)kL2 , [0,T ]

with m ≥ L, s > 2m and using that x = Γx − 2it∂x ; x2 = Γ2x + 4itΓx ∂x + 4t2 ∂x2 + 2it; x3 = Γ3 + 2t(...),

(V.35)

from the above argument we get that 2 2 ku kT ≤ c(ku0 kH s + kλ−1 m (x + y )u0 kL2 ) +cT (1 + T 2 ) ku k3T + ku k6T .

(V.36)

380

C.E. Kenig, Gustavo Ponce and Luis Vega

Ann. Henri Poincar´ e

2 2 Then we conclude that there exists T0 = T0 (ku0 kH s +kλ−1 m (x +y )u0 kL2 ) > 0 such that the solution u ’s can be extended to the interval [0, T ] such that u ∈ C([0, T ] : H s ∩ L2 ((x2 + y 2 )m dxdy)), with 2 2 ku kT0 ≤ 2cs,m ku0 kH s + kλ−1 (V.37) m (x + y )u0 kL2 ≡ δ.

STEP 3. Convergence of u ’s in L∞ ([0, T ] : L2 )-norm as ↓ 0. In this step we shall use again Theorem 1.2. 0 0 For > 0 > 0 we define ω = ω , = u − u which satisfies the IVP   ∂t ω − i∆ω − 0 ∆ω − ( − 0 )∆u   0 0   u ¯ u ¯  = c1 1+|u  0 |2 ∂y u ∂x ω + c1 1+|u0 |2 ∂x u ∂y ω   0 ω∂y u ¯ −¯ ω ∂y u +r1,2 (u (t))∂x ω + c6 ∂x u ∂y−1 ∂x 2 2 | )  (1+|u   ω∂x u ¯ −¯ ω∂x u 0 −1  (u (t))∂ ω + c ∂ u ∂ ∂ +r  2,2 y 7 x y x (1+|u |2 )2    0  +Q(∂ α u|α|≤1 , ∂ β u|β|≤1 , ω)(t),

(V.38)

where (see (5.37)) 0

kQ(∂ α u|α|≤1 , ∂ β u|β|≤1 , ω)(t)kL2x,y ≤ cδkω(t)kL2x,y .

(V.39)

To apply Theorem 1.2 we rewrite (5.38) using the notation in (5.14)-(5.22) as   ∂t ω − i∆ω − 0 ∆ω − ( − 0 )∆u     = r1 (u0 )∂x ω + r2 (u0 )∂y ω      +ϕ1 (u0 )∂x ∂y−1 ϕ2 (u0 )ω + ϕ3 (u0 )∂y ∂x−1 ϕ4 (u0 )ω     +ϕ (u )∂ ∂ −1 ϕ (u )¯ −1 ω 5 0 x y 6 0 ω + ϕ7 (u0 )∂y ∂x ϕ8 (u0 )¯  +p1,1 ∂x f + p2,1 ∂y f      +φ1,1 ∂x ∂y−1 φ1,2 f1 + φ2,1 ∂x ∂y−1 φ2,2 f1 + φ3,1 ∂y ∂x−1 φ3,2 f1     +φ4,1 ∂y ∂x−1 φ4,2 f1 + φ5,1 ∂x ∂y−1 φ5,2 f2 + φ6,1 ∂x ∂y−1 φ6,2 f2    0  +φ7,1 ∂y ∂x−1 φ7,2 f2 + φ8,1 ∂y ∂x−1 φ8,2 f2 + Q(∂ α u|α|≤1 , ∂ β u|β|≤1 , ω)(t),

(V.40)

where ϕ1 , ϕ2 , ϕ5 , ϕ6 were defined in (5.16)-(5.17) and ϕ3 (u0 ) = −c7 ∂y u0 ,

ϕ3 (u0 ) = c7 ∂y u0 ,

ϕ4 (u0 ) =

ϕ4 (u0 ) =

∂x u ¯0 , (1 + |u0 |2 )2 ∂x u0 , (1 + |u0 |2 )2

(V.41)

(V.42)

Vol. 1, 2000

On the Initial Value Problem for the Ishimori System 0

!

p1,1 = c1

u0 ∂y u0 u ∂y u 0 2 2 − (1 + |u | ) (1 + |u0 |2 )2

p2,1 = c1

u0 ∂x u0 u ∂x u 0 2 2 − (1 + |u | ) (1 + |u0 |2 )2

0

0

φ1,1 = ∂x u − ∂x u0 ,

φ2,1 = ∂x u0 ,

φ2,2 =

(V.44)

∂y u ¯ , (1 + |u |2 )2

(V.45)

∂x u ¯ , (1 + |u |2 )2

φ6,2 = −

φ5,2 =

0

φ7,1 = ∂y u − ∂y u0 ,

φ8,2 =

∂y u ¯ (1 + |u |2 )2

∂x u ∂x u ¯ ¯0 − 2 2 (1 + |u | ) (1 + |u0 |2 )2 φ7,2 =

∂x u ¯ (1 + |u |2 )2

f2 = ω ¯.

(V.48)

(V.49)

∂x u ∂x u ¯ ¯0 − , (1 + |u |2 )2 (1 + |u0 |2 )2

f1 = ω,

(V.46)

(V.47)

∂x u ∂x u ¯ ¯0 − , 2 2 (1 + |u | ) (1 + |u0 |2 )2

φ8,1 = ∂y u0 ,

+ (r2,2 (u (t)) − r2,2 (u0 ))

!

φ3,2 =

φ5,1 = −(∂x u − ∂x u0 ),

φ6,1 = −∂x u0 ,

(V.43)

∂y u ∂y u ¯ ¯0 − , (1 + |u |2 )2 (1 + |u0 |2 )2

0

φ4,2 =

+ (r1,2 (u (t)) − r1,2 (u0 )),

φ1,2 =

φ3,1 = ∂y u − ∂y u0 ,

φ4,1 = ∂y u0 ,

381

(V.50)

(V.51)

(V.52)

(V.53)

To apply Theorem 1.2 to the IVP (5.40) we observe that as in the previous cases all the terms in (1.24) involved a factor T or a factor of the form described in (5.25), which can be bounded with a bound having a factor T on it. Thus, for T sufficiently small we get 0

lim sup kω, (t)kL2x,y = 0.

,0 →0 [0,T ]

(V.54)

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Ann. Henri Poincar´ e

This proves the convergence of the u ’s to a function u. By interpolation we get that the u ’s converges to u in C([0, T ] : H s−1 ∩ L2 (|x|m−1 dx)). Using weak∗-compactness and Fatou’s lemma it follows that u ∈ ∩L∞ ([0, T ] : H s ), and u ∈ ∩L∞ ([0, T ] : L2 (|x|m dx)) respectively. It is clear that in the time interval [0, T ] u is a solution of the IVP (5.1). Finally we remark that the proof of the uniqueness of the solution u in its class is similar to the argument described in (5.38)-(5.53), therefore it will be omitted.

References [AbHa]

Ablowitz, M. J., and Haberman, R., Nonlinear evolution equations in two and three dimensions, Phys. Rev. Lett. 35 1185–1188 (1975).

[BeCo]

Beals, R., and Coifmann, R. R., The spectral problem for the DaveyStewartson and Ishimori hierarchies, Proc. Conf. on Nonlinear Evolution Equations: Integrability and Spectral Methods, Manchester, U.K. (1988)

[Ch]

Chihara, H., The initial value problem semilinear Schr¨odinger equations, Publ. RIMS. Kyoto Univ. 32, 445–471 (1996).

[CnSa]

Constantin, P. and Saut, J. C., Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1, 413–446 (1989).

[CrKaSt] Craig, W., Kappeler T., and Strauss, W., Microlocal dispersive smoothing for the Schr¨ odinger equation, Comm. Pure Appl. Math. 48, 769–860 (1995). [DS]

Davey, A., and Stewartson, K., On three dimensional packets of surface waves, Proc. R. Soc. A 338, 101–110 (1974).

[DjRe]

Djordjevic, V. D., and Redekopp, L. P., On two-dimensional packets of capillarity-gravity waves, J. Fluid Mech. 79, 703–714 (1977).

[Do]

Doi, S., Remarks on the Cauchy problem for Schr¨ odinger type equations, Comm. P.D.E. 21, 163–178 (1996).

[GhSa]

Ghidaglia, J. M., and Saut, J. C., On the initial value problem for the Davey- Stewartson System, Nonlinearity 3, 475–506 (1990).

[Hy]

Hayashi, N., Local existence in time of small solutions to the Ishimori system, preprint

[HyOz]

Hayashi, N., and Ozawa, T., Remarks on nonlinear Schr¨ odinger equations in one space dimension, Diff. Integral Eqs 2, 453–461 (1994).

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On the Initial Value Problem for the Ishimori System

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[HySa]

Hayashi, N., and Saut, J-C., Global existence of small solutions to the Davey-Stewartson and the Ishimori systems, Diff. Integral Eqs 8, 1657– 1675 (1995).

[Ho]

H¨ ormander, L., The Analysis of Linear Partial Differential Operators, III, Springer-Verlag (1985).

[Is]

Ishimori, Y., Multi vortex solutions of a two dimensional nonlinear wave equation, Progr. Theor. Phys. 72, 33–37 (1984).

[Kt]

Kato, T., On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Advances in Math. Supp. Studies, Studies in Applied Math. 8, 93–128 (1983).

[KePoVe1] Kenig, C. E., Ponce, G., and Vega, L., Small solutions to nonlinear Schr¨ odinger equations, Annales de l’I.H.P. 10, 255–288 (1993). [KePoVe2] Kenig, C. E., Ponce, G. and Vega, L., Smoothing effects and local existence theory for the generalized nonlinear Schr¨ odinger equations, to appear in Inventiones Math [KoMa]

Konopelchenko, B. G., and Matkarimov, B. T., Inverse spectral transform for the nonlinear evolution equation generating the DaveyStewartson and Ishimori equations, Stud. Appl. Math. 82, 319–359 (1990).

[LiPo]

Linares, F., and Ponce, G., On the Davey-Stewartson systems, Annales de l’I.H.P. Analyse non lin´eaire 10, 523–548 (1993).

[Mz]

Mizohata, S., On the Cauchy problem, Notes and Reports in Math. in Science and Engineering, Science Press & Academic Press 3 (1985).

[Sj]

Sj¨ olin, P., Regularity of solutions to the Schr¨ odinger equations, Duke Math. J. 55, 699–715 (1987).

[So]

Souyer, A., The Cauchy problem for the Ishimori equations, J. Funct. Anal. 105, 233–255 (1992).

[Sn]

Sung, L.-Y., The Cauchy problem for the Ishimori equation, J. Funct. Anal. 139, 29–67 (1996).

[SuSuBa] Sulem, P.L., Sulem, C., and Bardos, C., On the continuous limit for a system of classical spins, Comm. Math. Phys. 107, 431–454 (1986). [Ve]

Vega, L., The Schr¨odinger equation: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 , 874–878 (1988).

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Ann. Henri Poincar´ e

[ZaKu]

Zakharov, V. E., and Kuznetson, E. A., Multi-scale expansions in the theory of systems integrable by the inverse scattering method, Physica D 18, 455–463 (1986).

[ZaSc]

Zakharov, V. E., and Schulman, E. I., Degenerated dispersion laws, motion invariant and kinetic equations, Physica 1D, 185–250 (1980).

C.E. Kenig Dept. of Mathematics Univ. of Chicago Chicago, IL 60637 e-mail:[email protected]

G. Ponce Dept. of Mathematics Univ. of California Santa Barbara, CA 93106

Communicated by J. Bellissard submitted 25/09/98, accepted 11/01/99

L. Vega Dep. de Matematicas Univ. del Pais Vasco Apartado 644

Ann. Henri Poincar´ e 1 (2000) 385 – 404 c Birkh¨

auser Verlag, Basel, 2000 1424-0637/00/020385-20 $ 1.50+0.20/0

Annales Henri Poincar´ e

Syst`eme de Yang-Mills-Vlasov pour des Particules avec Densit´e de Charge de Jauge Non-Ab´elienne sur un Espace-Temps Courbe N. Noutchegueme and P. Noundjeu Abstract. We prove local in time existence theorems of solutions of the Cauchy problem for the Yang-Mills system in temporal gauge, with current generated by a distribution function that satisfies a Vlasov equation, and an unknown non-abelian charge density subject to a conservation equation. R´esum´e. Nous d´ emontrons des th´ eor` emes d’existence locale dans le temps d’une solution du probl`eme de Cauchy pour le syst`eme de Yang-Mills en jauge temporelle, dont le courant est engendr´e par une fonction de distribution satisfaisant `a une ´equation de Vlasov, et une charge de jauge non-abelienne de densit´e inconnue soumise ` a une ´equation de conservation.

Introduction Un plasma est un train de particules ´evoluant a` tr`es grande vitesse et sous l’effet des forces qu’elles cr´eent. Pour des ´electrons, dans le cas ab´elien,les forces cr´e´ees sont des forces ´electromagn´etiques qui a` leur tour r´eifluencent ces particules, et, lorsqu’il n’y a pas collision, ce ph´enom`ene auto-entretenu est gouvern´e par le syst`eme de Maxwell-Vlasov, qui a ´et´e largement ´etudi´e ces derni`eres ann´ees. Nous consid´erons ici le cas de plasmas plus g´en´eraux o` u les particules ont une charge non ab´elienne, par exemple le champ de quarks que l’on rencontre en chromodynamique; le champ ´electromagn´etique est remplac´e par le champ de Yang-Mills, par exemple le champ des gluons de cette chromodynamique. Le plasma obtenu, appel´e ”plasma quarksgluons”, est sens´e exister `a tr`es haute temp´erature. Dans le cas sans collisions, ce ph´enom`ene est gouvern´e par le syst`eme coupl´e de Yang-Mills Vlasov, dont les inconnues sont: les potentiels de Yang-Mills, d’o` u d´erive le champ de YangMills et la fonction de distribution des particules Yang-Mills. Nous prenons les particules en jauge temporelle, nous les couplons avec le champ de Yang-Mills comme s’ils ´etaient, a` priori ,ind´ependants, et nous supposons que le courant de Yang-Mills, source du champ de Yang-Mills, est engendr´e d’une part par la fonction de distribution des particules, qui v´erifie l’´equation de Vlasov et d’autre part par une charge de jauge non-ab´elienne de densit´e inconnue, soumise a` une ´equation de conservation, cons´equence de la conservation du courant. Ceci nous permet d’une part de g´en´eraliser et d’´etendre aux espaces-temps courbes les r´esultats connus sur le syst`eme de Maxwell-Vlasov sur l’espace-temps plat de Minkowski [2], [3], 5], [8]

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d’autre part d’´etendre les r´esultats de [4] qui ´etudie le cas o` u la densit´e de charge est nulle. Nous ´etudions le probl`eme de Cauchy pour ce syst`eme coupl´e, sur un espacetemps courbe et nous incluons le cas o` u les particules ont une masse propre au repos nulle. Nous utilisons des majorations avec poids et nous d´emontrons l’existence de solutions locales dans le temps et globales dans l’espace, dans des espaces fontionnels qui ne leur imposent pas de d´ecroissance `a l’infini spatial.

I Hypoth`eses et notations Tout indice grec varie de 0 `a 3 et tout indice latin de 1 a` 3,sauf mention contraire.On adopte la convention de sommation d’Einstein. • Vari´et´e riemannienne de base :(V, g), C ∞ , de dimension 4, orient´ee dans le temps o` u; • La m´etrique g est de signature hyperbolique (+, −, −, −), V = R × S de coordonn´ees locales (xα ) avec x0 ou t sur R (xi ), i = 1, 2, 3 sur S; St = {t}×S spatial, R × {x} temporel, x ∈ S. Nous prenons g sous la forme g = α2 dt2 − g˜ij dxi dxj

(1.1)

o` u g˜ij = −gij ; (gij ) d´efinie n´egative. • Les m´etriques gt = (˜ gij ) induites par g sur S sont proprement riemanniennes et uniform´ement ´equivalentes a` une m´etrique g¯ sur S de rayon d’injectivit´e δ > 0, donc compl`ete. On suppose qu’il existe : A1 , A2 , B1 , B2 > 0 tels que dans les boules g´eod´esiques de rayon δ pour gt sur St on ait 3 3 X X i 2 i j (ξ ) ≤ g˜ij ξ ξ ≤ A2 (ξ i )2 ; B1 ≤ α ≤ B2 . A1 i=1

(1.2)

i=1

• Pour d´efinir les normes de tenseur, on associe a` g, la m´etrique elliptique γ sur V , d’expression locale : γ = α2 dt2 + g˜ij dxi dxj

(1.3)

• On suppose que les d´eriv´ees covariantes du tenseur de courbure R de g [resp. R de g¯] sont born´es. • A repr´esente un potentiel de Yang-Mills; c’est une 1-forme sur V `a valeurs dans G alg`ebre de Lie d’un groupe de Lie G, muni d’un produit scalaire Ad-invariant, not´e · i.e. tel que a · [b, c] = [a, b] · c ∀ a, b, c ∈ G

(1.4)

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o` u [, ] d´esigne le crochet de Lie de G.G est consid´er´e comme un espace vectoriel de dimension N sur R, dont on fixera une base orthonorm´ee (εa ), a = 1, 2, 3, ·, N. On ´ecrira dans les coordonn´ees locales xµ sur V et dans la base (εa ) de G : A = (Aµ ) = (Aaµ ) = Aaµ dxµ ⊗ εa

(1.5)

1 • F = dA + [A, A] est la courbure de A, ou le champ de Yang-Mills associ´e `a 2 A. F est une 2-forme antisym´etrique sur V `a valeurs dans G. On a localement a a F = (Fλµ ) = (Fλµ ); Fλµ = ∇λ Aaµ − ∇µ Aaλ + [Aλ , Aµ ]a

(1.6)

o` u ∇ est la d´eriv´ee covariante dans g. On a a b c [Aλ , Aµ ]a = Cbc Aλ Aµ

(1.7)

a a a o` u les Cbc sont les constantes de structure de G. On a Cbc = −Ccb et (1.4) a a entraˆıne que les Cbc on une trace nulle Cba = 0. Le potentiel A est pris en jauge temporelle i.e.

A0 = 0.

(1.8)

• On suppose que les particules ont une masse propre au repos m ≥ 0. • La charge de jauge non abelienne, appel´ee encore ” charge de couleur” et que nous appelons simplement : charge de Yang-Mills, est repr´esent´ee par une fonction C ∞ donn´ee q : V → G, q = qa εa , de grandeur donn´ee e. on d´esigne par ϑ la sh`ere de G d´efinie par ϑ:

q · q = |q|2 = e2

(1.9)

• On suppose que la charge de Yang-Mills q a une densit´e physique inconnue ρ, fonction r´eelle positive sur V ; ρ : V → R+ .

II Equations et probl`emes de Cauchy La trajectoire d’une particule dans un champ de Yang-Mills sur (V, g) est contenue dans l’espace de phase des particules, P = T (V ) × G de coordonn´ees locales (xα , pα , q a ) o` u x = (xα ) represente la position de la particule, p = (pα ) son impulsion et q = (qa ) sa charge; q = (q a ) repr´esentera aussi le point courant dans G. Cette trajectoire v´erifie le syst`eme diff´erentiel: a dpα dxα λ µ β α dq a α b c p p + p q · F ; p Aα q = pα ; = −Γα = −Cbc λµ β ds ds ds

(2.1)

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Ann. Henri Poincar´e

o` u les Γβλµ sont les coefficients de connexion de g. (2.1) exprime que dans P, le vecteur tangent a` la trajectoire est : Y = (p, P, Q) o` u λ µ β α a a α b c P α = −Γα λµ p p + p q · Fβ , Q = −Cbc p Aα q ; Mais les trajectoires de ces particules de masse propre m et de charge de grandeur e sont en fait contenues dans le sous-fibr´e Pm,e et de P d’´equations: gαβ pα pβ = m2 ; |q|2 = e2

(2.2)

d’o` u l’on d´eduit, vus (1.1) et (1.2). p0 = α−1 (m2 + g˜ij pi pj )1/2

(2.3)

Ce choix qui entraˆıne p0 ≥ 0 exprime le fait que les particules s’´ejectent vers le futur. Les coordonn´ees locales sur Pm,e ≡ R × T (S) × ϑ sont: x0 = t, xi , pi , q A o` u vu (2.3), A = 1, 2, ...N − 1. En un point de Pm,e , le vecteur Y lui est tangent. f : P → R+ repr´esente la fonction de distribution des particules. La conservation du nombre des particules dans le cas sans collision o` u nous nous pla¸cons est exprim´ee par l’´equation: LY f = pα

∂f ∂f ∂f + P α α + Qa a = 0 α ∂x ∂p ∂q

(2.4)

dite ´equation de Vlasov. f induit sur Pm,e une fonction not´ee encore f et qui v´erifie: LY f = pα

∂f ∂f ∂f + P i i + QA A = 0 ∂xα ∂p ∂q

(2.5)

J : V → G repr´esente le courant de Yang-Mills, engendr´e par les particules de masse m, de fonction de distribution f, de charge Yang-Mills q qui a une grandeur donn´ee e et une densit´e physique inconnue ρ. On a localement J = (J β ) = (J β,A ) o´ u J β,A est d´efinie au point de x de V par: Z J β,A (x) = pβ q A f (x, p, q)ωp ωq − ρ(x)U β (x)q A (x) (2.6) R3 ×ϑ

dp1 dp2 dp3 ; ωq est l’´el´ement de volume canonique de ϑ; U est le p0 vecteur unitaire temporel orient´e vers le futur et tangent aux g´eod´esiques dans (V, g) : U α ∇α U β = 0; gαβ U α U β = 1; U 0 > 0 (2.70 )

o` u ωp = |g|1/2

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Les ´equations de Yang-Mills s’´ecrivent : b α F αβ = J β ∇

(2.7)

ˆ α est la d´eriv´ee covariante de jauge, d´efinie sur les fonctions de V dans G par o` u∇ b α = ∇α + [Aα , ·] ∇

(2.8)

b α Fλµ + ∇ b λ Fµα + ∇ b µ Fαλ = 0 ∇

(2.9)

b α∇ b β F αβ ≡ 0 ∇

(2.10)

On a les identit´es de Bianchi

et les identit´es

(2.11) entraˆıne , vu (2.8) que J = (J β ) doit v´erifier la loi de conservation b βJβ = 0 ∇ a En utilisant Cba = o, on montre le r´esultat suivant Lemme 2.1 Soit Z β,A K (x) = pβ q A f (x, p, q)ωp ωp

(2.11)

(2.12)

R3 ×ϑ

Alors 1. Si f est de classe C 1 , ` a support compact et v´erifie l’´equation de Vlasov avec A et F de classe C 2 on a b β K β,A = 0 ∇

(2.13)

2. Si de plus ρ v´erifie l’´equation de conservation

∇α (ρU α ) = 0

(2.14)

alors le courant J = (J α ) v´erifie la loi de conservation (2.12). La jauge temporelle (1.8) entraˆıne, vu (1.6) que A v´erifie ∂0 Ai = F0i

(2.15)

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Ann. Henri Poincar´e

les ´equations de Yang-Mills-Vlasov sur (V, g) sont form´ees des ´equations: (2.6), (2.8), (2.16) en f, F, A, auxquelles on adjoint l’´equation de conservation de la densit´e de charge (2.15) en ρ, impos´ee par la conservation du courant de YangMills J = (J α ). (2.15) est, vu, (2.3) ´equivalente a` l’´equation de conservation de la charge de Yang-Mills b α (ρU α q) = 0 ∇ (2.150 ) Nous ´etudierons ce syst`eme int´egro-diff´erentiel non lin´eaire et nous montrerons que sa solution v´erifie effectivement la condition de conservation (2.12). Les ´equations de Yang-Mills (2.8) sont seulement 4 ´equations pour les 6 inconnues F 0i et F ij . On leur ajoute les identit´es de Bianchi (2.10) consid´er´ees comme ´equations en F. ceci nous emm`ene a` prendre la ”partie ´el´ectrique” F 0i sous forme contravariante et la partie ”magn´etique” Fij sous forme covariante. On choisit parmi ces 10 ´equations, les trois ´equations (2.10) d’indices α = 0, λ = i, µ = j, et les trois ´equations (2.8) d’indice β = i. On obtient le syst`eme suivant en F, A, f, ρ, `a ´etudier :  b α F αi = J i  ∇      b 0 Fij + ∇ b i Fj0 + ∇ b j F0i = 0  ∇    ∂0 Ai = F0i (Σ)     ∇α (ρ U α ) = 0       LY f = pα ∂f + P i ∂f + QA ∂f = 0 ∂xα ∂pi ∂q A

(2.80 ) (2.90 ) (2.16) (2.15) (2.6)

Les donn´ees de Cauchy sur S d’´equations locales x0 = 0 sont : • Pour le potentiel A, la donn´ee d’un potentiel de Yang-Mills a : S → G, o` u a = i∗ A, et i est l’immersion S → V. Localement a = (ai ) = (abi ) ; b = 1, 2, · · · N. • Pour le champ F, les donn´ees : i) Pour la partie ´el´ectrique, d’un vecteur E : S → G, localement E = (E i ) = (E i,a ), a = 1, 2, · · · , N ii) Pour la partie magn´etique, d’une 2-forme antisym´etrique Φ0 : S → G ; localement, Φ0 = (Φa0,i,j ) a = 1, 2, · · · , N . on prend , vu (1.6) Φ0 = b da

(2.16)

o` ub d est la diff´erentielle de jauge sur S, d´efinie par b d = d¯+ [a, ·], d¯ ´etant la diff´erentielle ext´erieure dans (S, g¯).

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• Pour la fonction de distribution f, la donn´ee d’une fonction num´erique posb itive ϕ sur Sb ≡ T (S) × θ telle que : ϕ = f |S. • Pour la densit´e de charge ρ, la donn´ee d’une fonction num´erique positive ρ0 sur S telle que ρ0 = ρ|S. Contraintes. Les donn´ees de Cauchy sont soumises a` des contraintes provenant d’une part des ´equations (2.10) en F , non trait´ees i.e. celles d’indices α = i, λ = j, µ = k et d’autre part de l’´equation (2.8) non trait´ee i.e. celle d’indice β = 0. (2.10) impose que l’on doit avoir b dΦ0 = 0. Mais puisque l’on a de mani`ere 2 d = 0, il en sera toujours ainsi si Φ analogue `a la diff´erentielle ext´erieure usuelle b 0

satisfait la contrainte (2.17). (2.8) d’indice β = 0 donne la contraine sur S : Z ˆ divE + p0 q ϕ ωp ωq − ρ0 U 0 q = 0

(2.17)

R3 ×ϑ

ˆ b E i = ∇ E i + [a , E i ] est la divergence covariante de jauge sur = ∇ o` u divE i i i S, associ´ee `a la d´eriv´ee covariante ∇ dans (S, g¯. On sait r´esoudre (2.18) en E, moyennant certains hypoth`eses sur a, ϕ et ρ0 [1]. Si ces contraintes sont r´esolues, les seules donn´ees ind´ependantes du probl`eme sont a, ϕ et ρ0 et on montre le r´esultat suivant. Lemme 2.2Toute solution (F 0i , Fij ), Ai , ρ au moins C 2 et f au moins C 1 du b α J α = 0, et qui prend les probl`eme de Cauchy pour le syst`eme (Σ) telle que ∇ donn´ees de Cauchy satisfaisant aux contraintes (2.17) et (2.18) : 1 – est telle que F est la 2-forme de courbure de A 2 – v´erifie les ´equations non trait´ees b α F α0 = J 0 ∇

(2.800 )

b i Fjk + ∇ b j Fki + ∇ b k Fij = 0 ∇

(2.100 )

3 – est solution du syst`eme complet de Yang-Mills-Vlasov coupl´e avec l’´equation de conservation de la charge de Yang-Mills. (2.8”) signifie que la contrainte (2.18) se conserve pendant l’´evolution et la courbure F de A v´erifie toujours les identit´es de Bianchi et donc (2.10’). D’apr`es ce lemme, la r´esolution du syst`eme (2.6), (2.8),(2.15), (2.16) se ram`enent `a celle de (Σ).

III Syst`eme d’´evolution lin´eaire associ´e ˜ F˜ , et on consid`ere le syst`eme lin´eaire suivant aux On suppose donn´es f˜, ρ˜, A, inconnues F, A, f, ρ d´eduit de (Σ)

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 R  ∇0 F 0i + ∇j F ji = −[A˜j , F˜ ji ] + R3 ×ϑ pi q f˜ ωp ωq − ρ˜U i q     ∇0 Fij + ∇i Fj0 + ∇j F0i = −[A˜j , F˜0i ] + [A˜i , F˜0j ]     ∂ A = F˜    0 i α 0i ∇ α (ρ U ) = 0 (Σ0 ) L  ˜f = 0 Y    o` u     ˜ = (p, P˜ , Q) ˜ avec  Y    Pei = −Γi pλ pµ + pβ q · Fei ; Q ebα q c e A = C A pα A λµ β bc

Ann. Henri Poincar´e

(3.1) (3.2) (3.3) (2.15) (3.4)

(3.40 )

Les donn´ees de Cauchy en x0 = 0 sont respectivement Φ = (E, Φ0 ), a, ρ, ϕ. On ´etudie (Σ0 ) sur VT = [0, T ] × S et VˆT = [0, T ] × T (S) × ϑ o` u T > 0. Si E ⊂ F on d´esigne par C0∞ (E) l’ensemble des restrictions a` E des fonctions ou tenseurs C ∞ et `a support compact dans F, o` u on prendra : F = V et E = VT ; F = P et E = VˆT ; E = F = S. Notons tout de suite que si Fe ∈ C0∞ (VT ) alors l’´equation (3.3) admet une solution unique A dans C0∞ (VT ) donn´ee localement par: Z Ai = ai +

t

Feoi dτ

(3.5)

0

On a : Proposition 3.1 e ∈ C0∞ (VT ) alors l’´equation (3.4) en f admet une solution unique 1. Si Fe et A ∞ ˆ ˆ f ∈ C0 (VT ) prenant la donn´ee de Cauchy ϕ ∈ C0∞ (S) a) quel que soit T > 0 si m > 0 gij pi pj ≥ C > 0}, si m = 0. b) Pour T assez petit et suppϕ ⊂ C0 ≡ {¯ e Fe ∈ C0∞ (VT ) alors le syst`eme (3.1)-(3.2) admet une 2. Si fe ∈ C ∞ (VˆT ), ρe, A, solution unique F ∈ C0∞ (VT ) prenant la donn´ee de Cauchy Φ ∈ C0∞ (S). 3. L’´equation (2.15) admet une solution unique ρ ∈ C0∞ (VT ) prenant la donn´ee de Cauchy ρ0 ∈ C0∞ (S). Preuve. 1) Consid´erons l’´equation (3.4) en f sur [−T, T ] × T S × ϑ ≡ WT ∂f a) Si m > 0, elle s’´ecrit vu (2.1): = 0, ´equation diff´erentielle sur les ∂τ trajectoires du champ Ye qui, d’apr`es les r´esultats sur les syst`emes diff´erentiels u si τ → (t + existent sur [−T, T ] car , d’apr`es (2.4) m > 0 ⇒ p0 > 0. D’o` τ, y(τ, t, yt )) est la trajectoire de Ye passant par le point fix´e (t, yt ) = (t, xi , pi , q A ) de Sˆt = T St × ϑ pour τ = 0 et qui coupe Sˆ pour t + τ = 0 i.e. pour τ = −t alors l’unique solution f de (3.4) dans C0∞ (WT ) qui prend la valeur ϕ sur Sˆ est : f (t, xi , pi , q A ) = ϕ[y(−t, t, xi , pi , q A )].

(3.6)

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Sa restriction a` VˆT est la solution cherch´ee. b) Si m = 0, la continuit´e de (xα , pi ) → p0 montre que l’on a p0 > 0 dans un voisinage de tout point X = (X0 , p0 , q0 ) de P = T (V ) × G o` u p00 > 0 et, d’apr`es a) les trajectoires de Ye sont d´efinies dans un tel voisinage. Pour X ∈ C0 , on a p00 > 0 et les trajectoires de Ye issues de C0 existent donc dans un voisinage de C0 dans P, donc dans VˆT o` u T > 0 est petit. L’hypoth`ese supp ϕ ⊂ C0 montre donc que (3.4) admet encore une solution f donn´ee par (3.6) pour 0 ≤ t < T, T petit. 2) Provient du fait que (3.1)-(3.2) forme un syst`eme x0 -hyperbolique sym´etrique du 1er ordre au sens de Friedrichs. 3) Le syst`eme diff´erentiel caract´eristique associ´e montre que le probl`eme de Cauchy pour (2.15) avec la donn´ee ρ0 sur S admet la solution unique, sur VT Z t ∇α U α ρ(t, x ¯) = ρ0 (¯ x)exp − dx ¯∈S (3.7) 0 ,x U0 0

IV Espaces fonctionnels et estimations a priori Soit Bδ une boule g´eod´esique de rayon δ de (S, g¯) et Ωt ⊂ R × S d´efini par:  !1/2  3   X T δ 2 Ωt = 0 ≤ x0 ≤ t, K(T − x0 ) ≥ xi ;0 ≤ t ≤ ;T ≤ (4.1)   2 K i=1

On pose: ˆ τ = Ωτ × R3 × ϑ; 0 ≤ τ ≤ t ˆ τ = ωτ × R3 × ϑ; Ω ωτ = Ωt ∩ Sτ ; ω

(4.2)

et quand les int´egrales existent pour f, fonction sur VˆT , U = A ou F : VT → G, tenseur et ρ : VT → R fonction :  1/2 Z X ˆ ˘ yβ,f (τ ) o` u yβ,f (τ ) = (p0 )2k+2(β+β)+1 | Dβ f |2 θτ (4.3) kf kτs,2,k =  ω ˆτ

|β|≤s

kU kτs,2

 1/2 Z X = zl,U (τ ) o` u zl,U (τ ) = l≤s

kρkτs,2

| ∇l U |2 µτ avec U = A ou F (4.4)

ωτ

 1/2 Z X   = zl,ρ (τ ) o` u zl,ρ (τ ) = l≤s

U 0 | ∇l ρ |2 µτ

(4.40 )

ωτ

avec | ∇l U |2 = γλ1 σ1 · · · γλl σl γα1 β1 · · · γαk βk ∇λ1 · · · ∇λl U α1 ···αk · ∇σ1 · · · ∇σl U β1 ···βk (4.5)

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| ∇l ρ |2 = γλ1 σ1 · · · γλl σl ∇λ1 · · · ∇λl ρ∇σ1 · · · ∇σl ρ

(4.50 )

o` u ˘ est le nombre de d´eriv´ees en pi (resp. q A ) dans le multi Dans (4.3), βˆ (resp. β) + indice β et k ∈ R ; po est donn´e par (2.4); s ∈ N. θτ = µτ ∧ ωp ∧ ωq o` u µτ est l’´el´ement de volume induit sur ωτ par l’´el´ement de volume µ = µ(g) associ´e `a g. Si τ = 0, µ0 = µ(¯ g ), rappelons que ω0 ⊂ Bδ ⊂ S. Dans (4.4) et (4.4’), l, s ∈ N et dans(4.5) γ est d´efini par (1.3) et le point d´esigne le produit scalaire dans G. Noter que, vues les propri´etes de g et g¯, les normes (4.4) et (4.4’) sur ωτ sont ´equivalentes a` celles obtenues en y rempla¸cant ∇ par D et avec des constantes ne d´ependant que de g. [si τ = 0, on remplace dans (4.5) et ¯ ou D]. ¯ D’apr`es les propri´et´es de g, (2.7’) entraˆıne qu’il (4.5’) γ par g¯ et ∇ par ∇ existe C1 > 0, C2 > 0., tels que C1 ≤ U 0 ≤ C2 ; la norme kρkτs,2 d´efinie par (4.4’) est donc ´equivalente a` la norme d´efinie sans le facteur U 0 . On d´esigne ensuite par : s ˆ t ), le compl´et´e de C0∞ (Ω ˆ t ) dans la norme: 1) E2,k (Ω kf kE s

b

2,k (Ωl )

= Sup kf kτs,2,k

(4.6)

0≤τ ≤t

2) E2s (Ωt ), le compl´et´e de C0∞ (Ωt ) dans la norme: kZkE2s (Ωt ) = Sup kZkτs,2

(4.7)

0≤τ ≤t

o` u d’apr`es ce qui pr´ec`ede, Z d´esigne A, F ou ρ. s ˆ t) (Ω Eks (Ωt ) = (E2s (Ωt ))3 × E2,k

3) L, l’op´erateur de Maxwell d´efini par le membre de gauche de (3.1)-(3.2) et e Fe, fe, ρe) ou d = d(A, F, f, ρ) le membre de droite de ce syst`eme. d = d(A, b t ), s ∈ N, Proposition 4.1 Soient A, F, ρ ∈ C0∞ (Ωt ), f, b ∈ C0∞ (Ω τ ∈ [0, t], C > 0. 1) Si LY f = b o` u Y = Y (F, A), on a pour s ≤ S, S > 4 : Z τ Z τ 0 0 0 kf kτs,2,k ≤ C kf k0s,2,k + kbkτs,2,k dτ 0 exp C 1 + kAkτs,2 + kF kτs,2 dτ 0 0

0

(4.8) 2) Si LF = d on a kF kτs,2

Z 0 ≤ C kF ks,2 +

τ

0 kdkτs,2

dτ

0

(4.9)

0

3) Si ∂0 Ai = F0i , on a

Z kAkτs,2 ≤ C kAk0s,2 + 0

τ

0

kF kτs,2 dτ 0

(4.10)

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4) Si ∇α (ρ U α ) = 0 on a kρkτs,2 ≤ Ckρk0s,2

(4.11)

et si ρ ∈ E2s (Ωt ) on a kJkτs,2 ≤ C kf kτs,2,k + kρkτs,2

(4.12)

s b t ), avec k > 5) Si f ∈ E2,k (Ω

3 2

Si m = 0, on suppose en outre au 1) et 5) que supp f ⊂ {p0 ≥ k0 > 0}. Preuve. 1), 2), 3), 4): on d´erive les ´equations et on forme des produits que l’on b t . En prenant K suffisamment grand, la surface lat´erale de Ωt int`egre sur Ωt et Ω est spatiale. On applique la formule de Stokes et le lemme de Gronwall. Pour 1) l’hypoth`ese S > 4 permet d’utiliser les th´eor`emes d’injection de Sobolev et on applique le th´eor`eme de multiplication de Sobolev pour s´eparer les produits Af et F f. Les espaces fonctionnels ont en fait ´et´e construits a` partir de ces in´egalit´es qui expliquent la pr´esence des poids en p0 dans (4.3) et en U 0 dans (4.4’). 5) s’´etablit en prenant d’abord f et ρ dans C0∞ . On utilise la d´efinition par s b t ) et E2s (Ωt ); l’hypoth`ese k > 3 assure la convergence a` (Ω compl´etion de E2,k 2 l’infini des int´egrales en (pi ) ∈ R3 et si m = 0, l’hypoth`ese sur f dans 1) et 5) assure la convergence de ces int´egrales en (pi ) = 0.

V Solution locale dans le temps et dans l’espace du syst`eme non lin´eaire e2s (Ωt ) et E e s (Ω b t ) les sous espaces vectoriels de Soit s ≥ 1. on d´esigne par E 2,k s−1 s−1 b E2 (Ωt ) et E2,k (Ωt ) form´es des (classes de ) fonctions admettant des d´eriv´ees au sens des distributions qui sont des fonctions telles que kU kEe s (Ωt ) = kU kE s−1 (Ωt ) + EssSup (zs,U (τ ))1/2 < +∞ 2

2

 kf kEe s

b

2,k (Ωt )

= kf kE s−1 (Ωb t ) + EssSup  2,k

(5.1)

0<τ
0<τ
X

1/2 yβ,f (τ )

< +∞

(5.2)

kβk=s

o` u dans (5.1), U = F, A ou ρ; les quantit´es ci dessus ´etant d´efinies au paragraphe IV. On pose 3 s s e2s (Ωt ) × E e2,k b t) (Ωt ) = E (Ω Ee2,k 3

Hks (ω0 ) = (H s (ω0 )) × Hks (ˆ ω0 )

(5.3)

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o` u Hks (ˆ ω0 ) est le coml´et´e de C0∞ (ˆ ω0 ) dans la norme  kϕkHks (ˆω0 ) = 

X Z

|β|≤s

1/2 2 ˆ ˘ (p0 )2k+2(β+β)+1 Dβ ϕ θ0 

(5.4)

ω ˆ0

et H s (ω0 ) est l’espace de sobolev usuel sur la vari´et´e riemannienne (ω0 , g¯). Theor`eme 5.1[Th´eor`eme d’existence et d’unicit´e]. Soit U0 = (Φ, a, ρ0 , ϕ) ∈ Hks (ω0 ) satisfaisant les contraintes. Alors si s > N2 + 5 et k > 32 : Il existe un nombre t0 > 0 ne d´ependant que de la norme de U0 dans Hks (ω0 ), fonction d´ecroissante de cette norme, tel que, le probl`eme de Cauchy pour le syst`eme de Yang-MillsVlasov coupl´e avec l’´equation de conservation de la charge de Yang-Mills admet s une solution unique U = (F, A, ρ, f ) dans Ee2,k (Ωt ) prenant la donn´ee de Cauchy U0 pour t = 0. Si m = 0 on suppose en outre que supp ϕ ⊂ C0 ≡ {¯ gij pi pj ≥ C > 0}. La preuve utilisera le Lemme suivant: o` u Y = Y (F, A), Yi = Y (Fi , Ai ), i = 1, 2 : b t ), F1 , F2 ∈ C0∞ (Ωl ), s > 3, alors Lemme 5.1 1) Si f ∈ C0∞ (Ω kLY1 −Y2 f kE s (Ωb t ) ≤ C kF1 − F2 kE2s (Ωt ) + kA1 − A2 kE2s (Ωt ) kf kE s+1 (Ωt ) 2,k

2,k

(5.5)

s b t ), F1 , F2 , A1 , A2 ∈ E2s (Ωt ), avec s > N + 4, on a si 0 ≤ τ ≤ t, (Ω 2) Si f ∈ E2,k 2 τ τ kLY1 −Y2 f k0,2,k ≤ C kF1 − F2 k0,2 + kA1 − A2 kτ0,2 kf kτs,2,k (5.6)

3) Si F ∈ E2s (Ωt ), d ∈ E2s−1 (Ωt ), o` u s > 3, et si LF = d, on a ∀τ ∈ [0, t] : Z τ τ 0 τ0 0 kdk0,2 dτ kF k0,2 ≤ C kF k0,2 +

(5.7)

0

u s > 3, et si ∂0 Ai = F0i , on a ∀τ ∈ [0, t] :, 4) Si A, F ∈ E2s (Ωt ), o` Z τ τ 0 τ0 0 kF k0,2 dτ kAk0,2 ≤ C kAk0,2 +

(5.8)

0

u s > 3, et si ∇α (ρ U α ) = 0 on a ∀τ ∈ [0, t] :, 5) Si ρ ∈ E2s (Ωt ), o` kρkτ0,2 ≤ Ckρk00,2

(5.9)

s b t ), b ∈ E s−1 (Ω b t ), A, F ∈ E2s (Ωt ), o` 6) Si f ∈ E2,k (Ω u s > N2 + 4 et si LY f = b, si 2,k τ ∈ [0, t] : Z τ Z τ 0 0 kbkτ0,2,k dτ 0 exp C (1 + kF kτs,2 ) dτ 0 (5.10) kf kτ0,2,k ≤ Ckf k00,2,k + 0

0

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Preuve du lemme 5.1. Pour 1) et 2) on utilise le th´eor`eme de multiplication de Sobolev. Le d´ecompte des d´eriv´ees en q A pour ´etablir (5.5) justifie la pr´esente de β˘ dans (4.3). Pour 3), 4), 5) l’hypoth`ese s > 3 entraine, vus les th´eor`emes d’injection de Sobolev, que F, A, ρ ∈ C01 (Ωt ) car Ωt est born´e; on utilise la formule de Stokes et (5.7), (5.8), (5.9) s’´etablisent respectivement comme (4.9), (4.10), (4.11) pour b t , ceci d’apr`es s = 0. Pour 6) S > N2 + 4 ⇒ f est born´ee et de classe C 1 sur Ω 3 b les th´eor`emes d’inclusion de Sobolev. Mais Ωt = Ωt × R × ϑ n’est pas born´e. On utilise alors une suite tronquante qui permet d’appliquer la formule de Stokes et d’obtenir (5.10) par passage a` la limite. Preuve du th´eor`eme. Y ´etant d´efini par (2.2) et L et d au paragraphe IV on posera dn = d(An , Fn , fn , ρn ); Yn = Y (Fn , An ), b t ); t > 0; ω0 ); C0∞ (t) = (C0∞ (Ωt ))3 × C0∞ (Ω C0∞ (0) = (C0∞ (ω0 ))3 × C0∞ (b U0 = (Φ, a, ρ0 , ϕ), Un = (Fn , An , ρn , fn ) n ≥ 1, b t ), t > 0. D0 (t) = (D0 (Ωt ))3 × D0 (Ω On suppose toujours supp ϕ ⊂ C0 si m = 0. La preuve du th´eor`eme (5.1) se fera en six ´etapes. 1) Construction et convergence des it´er´ees (s > 4) On prend d’abord U0 ∈ C0∞ (0); on prend ensuite U1 ∈ C0∞ (t); on construit la suite (Un )n≥2 par it´eration des solutions du syst`eme lin´eaire associ´e (Σ0 ) du paragraphe III i.e on y fait A˜ = An , F˜ = Fn , f˜ = fn , ρ˜ = ρn n ≥ 1, et on d´efinit Un+1 = (Fn+1 , An+1 , ρn+1 , fn+1 ) comme solution du probl`eme de Cauchy ( LFn+1 = dn ; ∂0 An+1,i = Fn,0i ; ∇α (ρn+1 U α ) = 0 ; LYn fn+1 = 0 Fn+1 = Φ ; An+1 = a ; ρn+1 = ρ0 ; fn+1 = ϕ en t = o

(5.11)

En prenant U1 = [(x0 )s+1 + 1]U0 on d´eduit de (5.11) que kFn k0s,2 , kAn k0s,2 , kfn k0s,2,k , se majorent uniquement par kU0 kHsk (ω0 ) .

kρn k0s,2 ,

Si m = 0, supp ϕ ⊂ C0 ⇒ supp fn ⊂ {p0 > k0 > 0}; vu (3.6). On d´eduit alors des majorations §IV, Proposition 4.1 du fait que pour s > 32 , s E2 (Ωt ) est une alg`ebre, ce qui permet vu (1.7) de s´eparer le produit An · Fn dans dn , que la suite (Un ) est born´ee dans Eks (Ωt ) pour t > 0, petit, et que t est une

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raison inverse de kU0 kHsk (ω0 ) . (5.11) donne ensuite :   L(Fn+1 − Fn ) = dn+1 − dn ; ∂0 (An+1,i − An,i ) = Fn,0i − Fn−1,0i ; ∇α [(ρn+1 − ρn )( U α )] = 0 ; LYn (fn+1 − fn ) = −LYn −Yn−1 (fn )   Fn+1 − Fn = 0 ; An+1 − An = 0 ; ρn+1 − ρn = 0 ; fn+1 − fn = 0 en t = o (5.12) On d´eduit de (5.12) en utilisant encore les estimations §IV, Proposition 4.1 et (5.5) pour l’´equation en f, et le fait que kFn kE2s (Ωt ) , kAn kE2s (Ωt ) , kfn kE s (Ωˆ t ) 2,k sont born´ees pour t petit, que la suite (Un ) est de Cauchy dans l’espace de Banach Eks (Ωt ) et y converge vers U = (F, A, ρ, f ) ∈ Eks (Ωt ). 2) Solution dans Eks (Ωt ) du probl`eme de Cauchy non lin´eaire avec donn´ees dans C0∞ (0) pour le syst`eme (Σ) § II (s > 4). U = (F, A, ρ, f ) ´etant d´efini ci-dessus, on d´eduit de la convergence de Un vers U dans Eks (Ωt ) que LFn+1 → LF et dn → d dans E20 (Ωt ), ∂0 An+1,,i → ∂0 Ai et Fn,0i → F0i dans D0 (Ωt ) faible; (ρn ) est constante et converge vers l’unique solution b t ) o` ρ ∈ C0∞ (Ωt ) car ρ0 ∈ C0∞ (ω0 ), LYn fn+1 → LY f dans L2loc (Ω u Y = Y (F, A) et 0 si m = 0 : supp ϕ ⊂ C0 ⇒ supp f ⊂ {p ≥ k0 > 0}; Fn → F et An → A dans L2 (ω0 ); fn → f, dans Hk0 (ˆ ω0 ). (5.11) entraˆıne alors ( LF = d ; ∂0 Ai = F0i ; ∇α (ρ U α ) = 0 ; LY f = 0 ; F = Φ ; A = a ; ρ = ρ0 ; f = ϕ en t = o

(5.13)

avec dans (5.13) d = d(F, A, ρ, f ); Y = Y (F, A). Et U = (F, A, , ρ, f ) ∈ Eks (Ωt ) est solution du probl`eme de Cauchy non lin´eaire pour (Σ). 3) Unicit´e de la solution du probl`eme de Cauchy non lin´eaire avec donn´ees C0∞ (s > n2 + 4) Soient Ui = (Fi , Ai , ρi , fi ), i = 1, 2 deux solutions du probl`eme de Cauchy dans Eks (Ωti ) pour la mˆeme donn´ee U0 = (Φ, a, ρ0 , ϕ) ∈ C0∞ (0). Soit t0 = inf (t1 , t2 ) > 0. Montrons que U1 = U2 sur [0, t0 ]. On sait que ρ1 = ρ2 ; (5.13) donne :   L(F1 − F2 ) = d1 − d2 ; ∂0 (A1,i − A2,i ) = F1,0i − F2,0i ; LY1 (f1 − f2 ) = −LY1 −Y2 f2 ;   F1 − F2 = 0 ; A1 − A2 = 0 ; f1 − f2 = 0 en t = o

(5.14)

On applique a` (5.14); i) pour F : (5.7) puis (4.12) pour s = 0 et les in´egalit´es de Sobolev pour traiter [A1 , F1 ] − [A2 , F2 ] = [A1 − A2 , F1 ] + [A2 , F1 − F2 ] qui intervient dans d1 − d2 ;

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ii) pour A : (5.8); iii) pour f : (5.10) puis (5.6), on obtient en faisant la somme des in´egalit´es obtenues, ∀ τ ∈ [0, t0 ].  Rτ 0 0  y(τ ) ≤ C 0 y(τ ) dτ (5.15) o` u   τ τ τ y(τ ) = kF1 − F2 k0,2 + kA1 − A2 k0,2 + kf1 − f2 k0,2,k o` u C est une constante ne d´ependant que des normes des Ui dans Eks (Ωti ) i = 1, 2. (5.15) donne, vu le lemme de Gronwall, U1 = U2 dans Ek0 (Ωt0 ); or cet u l’unicit´e dans Eks (Ωt ) pour t petit. espace contient Eks (Ωt0 ) car s > 4. D’o` b αJ α = 0 J 4) La solution U = (F, A, ρ, f ) dans Eks (t) est C ∞ et v´erifie ∇ d´efini par (2.7) a) Dans ce cas des donn´ees dans C0∞ (0), on trouve donc U ∈ Eks (Ωt ), ∀ s ∈ N. Les in´egalit´es de Sobolev entraˆınent que ∀ j ∈ N si on prend s > N2 + 4 + j, alors U est solution de classe C j . D’o` u U est C ∞ . b) Un = (Fn , An , ρn , fn ) n ≥ 1, est solution dans C0∞ (t) ⊂ C01 (t) de (5.11). b β Knβ,A = 0 o` u Knβ,A est d´efini par (2.13) D’apr`es § II, lemme 2.1, Un v´erifie ∇ α b α Jnα = 0, vue l’´equivalence de avec f = fn . Puisque ∇α (ρn U ) = 0, Un v´erifie ∇ (2.15) et (2.15’) et o` u Jn est d´efini par (2.17) avec f = fn et ρ = ρn . On en d´eduit b α J α = 0 en utilisant Un → U dans E s (Ωt ), donc dans D0 (t) faible, o` que ∇ u la k b d´erivation covariante de jauge ∇α est continue. 5) Solution dans ξ˜ks (Ωl ) du probl`eme de Cauchy non lin´eaire avec donn´ees dans Hks (ω0 ) o` u s > N2 + 5 On prend U0 = (Φ, a, ρ, ϕ) ∈ Hks (ω0 ). Soit Vn = (Φn , an , ρ0,n , ϕn ) ∈ C0∞ (0) telle que Vn tend vers U0 dans Hks (ω0 ). Si m = 0, on prend ϕn telle que supp(ϕ)⊂ C0 . On construit comme au 1) par it´eration sur ν, une suite (Un,ν )ν∈N? telle que Un,ν = (Fn,ν , An,ν , ρn,ν , ϕn,ν ) ∈ C0∞ (t) v´erifie ( LFn,ν+1 = dn,ν ; ∂0 (An,ν,i ) = Fn,ν−1,0i ; ∇α (ρn,ν U α ) = 0 ; LYn,ν fn,ν+1 = 0 Fn,ν+1 = Φn ; An,ν = an ; ρn,ν = ρ0,n ; fn,ν+1 = ϕn en t = o (5.16) On prend Un,1 = [(x0 )s+1 + 1]Vn . La suite convergente (Vn ) ´etant born´ee, on montre comme au 1) que la suite double (Un,ν )n,ν est born´ee et que la suite (Un,ν )ν est de Cauchy et converge vers Un = (Fn , An , ρn , fn ) dans Eks−1 (Ωl ), o u ´ s − 1 > N2 + 4, pour t suffisamment petit et ne d´ependant que de kU0 kHsk (ω0 ) en raison inverse de cette norme. Maintenant (5.16) donne :

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 α  L(Fn+1,ν − Fn,ν ) = dn+1,ν−1 − dn,ν−1 ; ∇α [(ρn+1,ν − ρn,ν )U ] = 0 ∂0 (An+1,ν,i − An,ν,i ) = Fn+1,ν−1,0i − Fn,ν−1,0i ;   LYn+1,ν−1 (fn+1,ν − fn,ν ) = LYn+1,ν−1 −Yn,ν−1 (fn,ν )

(5.17)

o´ u

Fn+1,ν − Fn,ν = Φn+1 − Φn ; An+1,ν − An,ν = an+1 − an, ; ρn+1,ν − ρn,ν = ρ0,n+1 − ρ0,n ; fn+1,ν − fn,ν = ϕn+1 − ϕn en t = 0

(5.170 )

On applique les estimations a` priori du §IV, prop. 4.1 a` (5.17). En utilisant (5.17’) et le fait que la suite Vn = (Φn , an , ρ0,n , ϕn ) est de Cauchy dans Hks (ω0 ); on passe la limite dans les in´egalit´es obtenues pour ν −→ ∞. On d´eduit que Un = (Fn , An , ρn , fn ) est une suite de Cauchy dans Eks−1 (Ωt ), et qu’elle y converge vers U = (F, A, ρ, f ) ∈ Eks−1 (Ωt ). On montre alors, comme au 2) et 3) et utilisant la convergence de Vn vers U0 dans Hks (ω0 ) que U est la solution unique du probl`eme de Cauchy dans Eks−1 (Ωt ) avec la donn´ee U0 dans Hks (ω0 ). Maintenant, l’espace ξ˜ks (Ωt ) est le dual d’un espace de Banach; ses boules ferm´ees sont faiblement compactes, donc faiblement ferm´ees. On en d´eduit, en extrayant des sous suites de (Un,ν ) et (Un ) qui convergent faiblement dans ξ˜ks (Ωt ) et donc `a fortiori, dans D0 (t) faible, que U est dans le sous espace ξ˜ks (Ωt ) de Eks−1 (Ωt ). 6) Solution du syst`eme complet de Yang-Mills-Vlasov coupl´e avec l’´equation de conservation de la charge de Yang-Mills b α Jnα = 0, o` u D’apr`es 4) la suite Un = (Fn , An , ρn , fn ) d´efinie au 5) v´erifie ∇ Jn est d´efinie par (2.7) avec f = fn et ρ = ρn . En utilisant la convergence de (Un ) b α J α = 0, o` vers U = (F, A, ρ, f ) dans Eks−1 (Ωt ) on a ∇ u J est d´efini par (2.7) avec N ces fonctions f et ρ; comme s > 2 + 5, dans U = (F, A, ρ, f, ) F, A, ρ sont de classe C 2 et f de classe C 1 , et U est solution du probl`eme de Cauchy pour (Σ) avec des donn´ees de Cauchy qui satisfont aux contraintes. D’apr`es le lemme 2.2, §II, U est solution du syst`eme complet de Yang-Mills-Vlasov coupl´e avec l’´equation de conservation de la charge de Yang-Mills. De plus la solution F est la 2-forme de courbure de la solution A. D’o` u le th´eor`eme 5.1.

VI Solution locale dans le temps et globale dans l’espace du syst`eme non lin´eaire a) Espaces fonctionnels des donn´ees sur S et Sˆ = T S × ϑ. On d´efinit sur Sˆ de coordonn´ees locales xi , pi , q A `a partir de la m´etrique g¯ de S la m´etrique proprement riemannienne d’expression locale : ¯ i Dp ¯ j + ds20 b g¯ = g¯ij dxi dxj + (p0 )−2 g¯ij Dp

(6.1)

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i

¯ i = dpi + Γhk ph dxk et ds20 est la m´etrique induite sur la sph`ere ϑ de G o` u Dp ¯ connexion de g¯). d´efinie par (1.9), par la m´etrique euclidienne de G (Γ b ¯ ¯ On d´esigne par ∇ et par ∇ la d´eriv´ee covariante au sens des distributions ˆ b sur (S, g¯) (resp; sur (S, g¯). On pose pour v = Φ ou a, tenseurs de S dans G, ˆ ρ0 : S −→ R, ϕ : S −→ R des fonctions : ¯ l v |2 = ∇ ¯ i1 · · · ∇ ¯ il v h1 ···hp · ∇ ¯ i1 · · · ∇ ¯ i vh1 ···hp |∇ l ¯ l ρ0 |2 = ∇ ¯i ···∇ ¯ i1 · · · ∇ ¯ il ρ0 ∇ ¯ i ρ0 |∇ 1

l

i1

l∈N

l

il

ˆ¯ · · · ∇ ˆ¯ ϕ∇ ˆ¯ · · · ∇ ˆ¯ ϕ ˆ¯ ϕ |2 = ∇ |∇ i1 il

(6.2) (6.3) (6.4)

O` u dans (6.2) le point d´esigne le produit scalaire dans G et dans (6.2), (6.3) (6.4) on monte et on desend les indices avec g¯ et gˆ¯. On consid`ere un recouvrement Bδ ˆ = B × R3 × ϑ forment un recouvrement de S, par des boules B; le boules B ˆ de S; chaque boule est une boule g´eod´esique de rayon δ de (S, g¯) [δ > 0 est le l,u ˆ rayon d’injectivit´e introduit au §I]. On d´esigne par Hsl,u (S) [resp. Hk,s (S), ] s ∈ N, l’espace des tenseurs de S dans G ou des fonctions de S dans R, w = Φ, a, ρ [resp. b j ϕ |2 ] ˆ mesurables, tels que | ∇l w |2 [resp. (p0 )2k+2(ˆj+˘j)+1 | ∇ des fonctions ϕ sur S] ˆ par rapport aux ´el´ements de volume µ0 = µ(¯ soit int´egrable sur B (resp. B) g ) sur ˆ et d’int´egrales uniform´ement born´ees. S (resp. θ = µ0 ∧ ωp ∧ ωq sur S] On munit Hsl,u (S) de la norme : kwkHsl,u (S) = sup kwkH s (B) o` u kwkH s (B)

1/2  XZ l = | ∇ w |2 µ0  l≤s

(6.40 )

B

et o` u le supremum est pris sur l’ensembe des B dans Bδ . Ce qui en fait un espace de Banach. Si w = ρ0 , la relation 0 < C1 ≤ U 0 ≤ C2 sur S montre que la norme (6.4’) o` u B est remplac´e par ω0 est ´equivalente a` (4.4’) pour τ = 0. On munit l,u ˆ (S) de la norme : Hk,s kϕkH l,u (S) ˆ ˆ = sup kϕkH s (B) k,s

k

o` u kφkH s (B) ˆ k

1/2  XZ ˆ ˘ ˆ |2 θ  = (p0 )2k+2(j+j)+1 | ∇ϕ 0 j≤s

(6.5)

ˆ B

ˆ et o` o` u le supremum est pris sur l’ensemble des boules B, u, vues les propri´et´es de ˆ dans (6.5) est en rempla¸cant B ˆ par ω g¯ et la d´efinition (6.1) de gˆ¯ la norme sur B ˆ0 ´equivalente a` (5.4), avec des constantes ne d´ependant que de g; d’o` u la notation l,u ˆ analogue adopt´ee. Hk,s (S) est un espace de Banach. On pose l,u l,u ˆ Hk,s (S) = (Hsl,u (S))3 × Hk,s (S)

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ˆ si U0 ∈ Hl,u (S) alors i∗ω U0 ∈ Noter que ∀B ∈ Bδ , si ω0 ⊂ B et donc ω ˆ0 ⊂ B k,s 0 s Hk (ω0 ) [def (5.3)] o` u iω0 : ω0 → S et on a ki∗ω0 U0 kHsk (ω0 ) ≤ CkU0 kkHl,u (ω0 )

(6.6)

k,s

o` u C ne d´epend que de de g. b) Espaces fonctionnels pour le probl`eme d’´evolution Nous allons construire un espace fonctionnel adapt´e au probl`eme. D’apr`es le th´eor`eme d’unicit´e, le probl`eme de Cauchy a` donn´ee nulle U0 = 0 sur ω0 n’admet que la solution nulle sur Ωt . Consid´erons le probl`eme de Cauchy a` donn´ee U0 sur l,u ˆ S tout entier. Soient s > N2 + 5; k > 32 , U0 = (Φ, a, ρ0 , ϕ) ∈ Hk,s (S), U0 6= 0, donc kU0 kHl,u (S) 6= 0. k,s

∀B ∈ Bδ , ∃xi ∈ S tel que BS= B(xi , δ). Notons des indices i S Iˆ l’ensemble ˆi = Bi × R3 × ϑ. et Bi = B(xi , δ). On a donc S ⊂ i∈I Bi et Sˆ ⊂ i∈I B uB i o` Soit ti > 0 le nombre fourni par le th´eor`eme (5.1) (d’existence) et Ωiti ⊂ R × Bi , le domaine de VT d´efini par (4.1) de ”base” ω0,i = Ωiti ∩ S, tel que le probl`eme de Cauchy pour le syst`eme non lin´eaire de Yang-Mills-Vlasov, coupl´e avec l’´equation de conservation de la charge de Yang-Mills, avec les donn´ees de Cauchy jω∗ 0,i U0 o` u jω0,i : ω0,i → S, admet une solution unique Ui = (Fi , Ai , ρi , fi ) dans ξ˜ks (Ωiti ). Le nombre ti ´etant une fonction d´ecroissante de kjω∗ 0,i U0 kHls , la famille (ti )i∈I k (ω) est vu (6.6), minor´ee par un nombre t0 > 0 ne d´ependant que de la norme de l,u (S). On a Ωit0 ⊂ Ωiti car ti ≥ t0 , ∀i ∈ I et Ui ∈ ξ˜ks (Ωit0 ), ∀i. On a U0 dans Hk,s S S ˆ it ). Vt0 = [0, t0 ] × S = i∈I Ωit0 ; Vˆt0 = i∈I Ω 0 ˆ On d´efinit sur Vt0 , de coordonn´ees locales x0 , xi , pi , q A , `a partir de la m´etrique γ [def.(1.3)] la m´etrique proprement riemannienne d’expression locale γˆ = γαβ dxα dxβ + (p0 )−2 Dpi Dpj + ds20

(6.7)

o` u Dpi = dpi + Γihk ph dxk

( Γ connexion de g)

ˆ ) la d´eriv´ee covariante au sens des distributions sur On d´esigne par ∇ (resp ∇ ˆ (Vt0 , g) [resp. (Vt0 , γˆ ) ] On pose, pour une fonction f sur Vˆt0 : ˆ j f |2 = ∇ ˆ A1 · · · ∇ ˆ Aj f ∇ ˆ A1 · · · ∇ ˆ Aj f |∇

j∈N

o` u on monte et desend les indices avec γ. Noter que vues les hypoth`eses sur g donc sur γ et vue la d´efinition (6.7) de γˆ , la norme kf kτs,2,k d´efinie par (4.3) est ´equivalente a`  1/2 XZ ˆ ˘ ˆ j f |2 θτ   (p0 )2k+2(j+j)+1 | ∇ j≤s

ω ˆr

(6.8)

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avec des constantes ne d´ependant que de g. On d´efinit donc une norme ´equivalente en rempla¸cant dans (4.6) kf kτs,2,k par (6.8). ˜ s (Vt ) et E ˜s ˆ On d´efinit les espaces locaux E 2,loc 2,k,loc (Vt ) comme suit: s ˜2,loc ˜2s (Ωjt ); U ∈E (Vt0 ) ⇔ i∗j U ∈ E 0 s s ˜2,k,loc ˜2,k ˆ jt ); f ∈E (Vˆt ) ⇔ f |Ωˆ j ) ∈ E (Ω 0 t0

Ωjt0

∀j ∈ I, o` u ij : → Vt0 et U = F, A tenseurs de Vt0 dans G, ou U = ρ, fonction de Vt0 dans R et f fonction de Vˆt0 dans R. ˜ s (Vt ) [resp. E ˜s ˆ La topologie de E efinie par la famille des 0 2,loc 2,k,loc (Vt0 ) ] est d´ semi normes pj,s (U ) = kU kE˜ s (Ωj ) [resp; pˆj,s (f ) = kf kE˜ s (Ωˆ j ) ] j ∈ I. qui en font t0 t0 2 2,k des espaces localement convexes s´epar´es. On pose s s s ˜2,loc ˜2,k,loc ξ˜2,k,loc (Vt0 ) = (E (Vt0 ))3 × E (Vˆt0 ).

c) Le th´eor`eme d’existence Th´eor`eme [solution globale dans l’espace et locale dans le temps] Si s > N2 + 5 et l,u si U0 = (Φ, a, ρ0 , ϕ) ∈ Hk,s (S) et satisfait les contraintes, alors il existe un nombre l,u (S), tel que le syst`eme de t0 > 0, ne d´ependant que de la norme de U0 dans Hk,s Yang-Mills-Vlasov coupl´e avec l’´equation de conservation de la charge de Yangs Mills ait une solution et une seule U = (F, A, ρ, f ) dans l’espace ξ˜2,k,loc (Vt0 ) prenant la donn´e de Cauchy U0 sur S. Si m = 0, on suppose en outre que supp(ϕ) ⊂ {¯ gi,j pi pj ≥ C > 0}. Preuve. On d´efinit U = (F, A, ρ, f ) sur (Vt0 )3 × Vˆt0 par recollement des solutions ˆ it en posant U = Ui sur Ui = (Fi , Ai , ρi , fi ) du syst`eme sur Ωi (t0 ) = (Ωit0 )3 × Ω 0 i Ω (t0 ). Pour i 6= j, on a la mˆeme donn´ee de Cauchy sur ω0,i ∩ ω0,j et le th´eor`eme d’unicit´e assure que Ui = Uj dans Ωi (t0 ) ∩ Ωj (t0 ). Ensuite, puisque ∀i ∈ I, Ui ∈ s s s (Ωit0 ) on a U ∈ ξ˜2,k,loc (Vt0 ). Enfin l’unicit´e de la solution U dans ξ˜2,k,loc (Vt0 ) ξ˜2,k 0 ˜ s’obtient a` partir de son unicit´e dans l’espace localement convexe s´epar´e ξ2,k,loc (Vt0 ) que l’on prouve comme pour le th´eor`eme (5.1) en utilisant prop (5.1), §V et les semi normes pi,0 et pˆi,o . Remarque Dans le cas m = 0, on peut en prenant V = R × S 3 , en d´eduire par transformation conforme, l’existense globale dans le temps, sur l’espace-temps de Minkwowki. Conclusion Le probl`eme de l’existence globale dans le temps reste ouvert. Nous allons pour terminer indiquer quelques applications, parmi les plus r´ecentes de ces ´equations de Yang-Mills que nous venons d‘´etudier. On peut citer: 1) L’´etude des excitations collectives et des ph´enom`enes de transport de plasma de Yang-Mills `a haute temp´erature [7]

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2) La reproduction par ces ´equations, dans certains r´egimes, des r´esulats de la th´eorie quantique des champs [6] 3) L’obtention des solutions num´eriques des ´equations de Yang-Mills-Vlasov pour le probl`eme de la Baryogen`ese dans l’univers primordial [9], ou pour la physique hadronique ”`a petit x” dans les collisions d’ions lourds ultrarelativistes, r´esultats obtenus r´ecemment par W. Poeschl et B. M¨ uller.

References [1] Y. Choquet-Bruhat et D. Christodoulou, ”Existence of global solutions of Yang-Mills-Higgs and Spinor fields, Ann. Ec. Norm. Sup. 4`eme s`erie, t.14(1981), pp. 481–500. [2] R. Glassey, W. Strauss, ”Singularity formation in a collisionless plasma could occur only at high velocities. Arc. Rational Mech. Anal. 92(1986) pp. 59–90 [3] S. Wolman, Local existence and uniqqueness theory of the Vlasox-Maxwell system”,J. Math.Anal. appl. 127(1987), pp. 103–121 [4] Y. Choquet-Bruhat et N. Noutchegueme: syst`eme de Yang-Mills-Vlasov en jauge temporelle.Ann. Inst. Henri Poincar´e, Vol. 55, N 3, 1991, pp. 759–787 [5] R. Glassey, I. Scheffer : The relativistic Vlasov-Maxwell equations in low dimension, Pitman research Notes in Mathematics series, MKV Murthy abd Spagnolo, (1992) [6] J.P. Blaizot, E. Iancu, Physical Review Letters 70 (1993) 3376, and Nuclear Physics B 417 (1994) 608 [7] P.F. Kelly, Q. Liu, C. Lucchesi, C. Manuel, Physical Review Letters 72 (1994) 3461, and Physical Review D 50 (1994) 4209. [8] R. Balean, T. Bartnick, The null-time boundary problem for Maxwell equations in Minkowski-space,Proc. R. Soc. London, (1998),pp 2041–2057 [9] G.D. Moore, C. Hu, B. Muller, Physical Review D58 (1998) 04001. N. Noutchegueme and P. Noundjeu Universit´e de Yaound´e I Facult´e des Sciences D´epartement de Mathematiques B.P. 812 Yaound´e, Cameroun e-mail : [email protected] Communicated by V. Rivasseau submitted 16/04/99, revised 3/07/99, accepted 1/07/99

Ann. Henri Poincar´ e 1 (2000) 405 – 442 c Birkh¨

auser Verlag, Basel, 2000 1424-0637/00/030405-38 $ 1.50+0.20/0

Annales Henri Poincar´ e

On the Extended Nature of Edge States of Quantum Hall Hamiltonians J. Fr¨ ohlich, G. M. Graf and J. Walcher

Abstract. Properties of eigenstates of one-particle Quantum Hall Hamiltonians localized near the boundary of a two-dimensional electron gas - so-called edge states - are studied. For finite samples it is shown that edge states with energy in an appropriate range between Landau levels remain extended along the boundary in the presence of a small amount of disorder, in the sense that they carry a non-zero chiral edge current. For a two-dimensional electron gas confined to a half-plane, or to a domain in the plane satisfying a certain geometric condition, the Mourre theory of positive commutators is applied to prove absolute continuity of the energy spectrum well in between Landau levels, corresponding to edge states.

1 Introduction and summary of results In this paper, we study two-dimensional electron gases in a uniform magnetic field perpendicular to the plane, in the presence of a small amount of disorder. The integer quantum Hall effect, discovered by von Klitzing [1], is the phenomenon that when the Fermi energy of the electron gas is well in between two Landau levels, the Hall conductance is equal to an integer multiple of e2 /h. Under the assumption of negligibly small electron-electron interactions, the integer quantum Hall effect can be derived from a simple one-electron picture. For an appropriate choice of sample geometry, described by a potential confining the electrons to the sample, and for a small amount of disorder, one can analyze, qualitatively, the energy spectrum of the corresponding one-particle Hamiltonian. In particular, as we show in this paper, eigenenergies well in between Landau levels correspond to eigenstates localized near, but extended along, the boundary of the sample, so called edge states. Those edge states carry a non-zero chiral edge current. Given a small voltage drop between two parallel components of the boundary, the edge states corresponding to the two boundary components will be filled somewhat asymmetrically with electrons. The result is a net Hall current parallel to the boundary and proportional to the voltage drop. The proportionality factor is the Hall conductivity. If the Fermi energy of the electron gas is well in between two Landau levels, and if the voltage drop is small compared to the energy gap between two adjacent Landau levels and to the Zeeman energy of the magnetic moment of an electron, the spectral properties of the Hamiltonian yield a Hall conductivity equal to e2 /h times the number of Landau levels below the Fermi energy, which is an integer. An argument of this sort, based on a clever use

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of gauge invariance, was first given by Laughlin [2] and subsequently refined by many other people (see e.g. [3], [4]). The idea that the Hall current is supported by edge states first appeared in a paper of Halperin [3]. The fundamental role of edge currents in the integer and the fractional quantum Hall effect was later understood in terms of a gauge anomaly cancellation mechanism in [5] and [6]. In this paper, we provide a rigorous analysis of one important detail underlying Halperin’s argument, namely of the question whether, and in what sense, the edge states are indeed extended states. Because we neglect electron-electron interactions, the magnetic moment of the electron turns out to be essentially irrelevant in our analysis, and we thus neglect electron spin. The one-electron Hamiltonian is therefore given by H=

1 ~ 2+V. (~ p − eA) 2m

(1.1)

~ is an electromagnetic In (1.1), m is the mass of an electron, e is its charge, A ~ = curl A, ~ and V = vector potential corresponding to a constant magnetic field B V0 + gVd is an external potential consisting of an edge potential, V0 , that confines the electron to the sample, and a disorder potential, gVd , corresponding to the presence of random impurities. The factor g, a “coupling constant”, is a measure for the strength of the disorder. The potential V0 can be replaced by appropriate ~ 2 , which boundary conditions in the definition of the covariant Laplacian, (~ p − eA) prevent an electron from leaving the sample; see, for example, [17], and section 6 of the present paper. The location of the energy spectrum of the one-particle Hamiltonian (1.1) is indicated in figure 1. This spectrum consists of a part corresponding to “bulk states” and a part corresponding to “edge states”. The former is located near the Landau levels, which are broadened by the disorder potential. Most of the bulk states are localized, but close to each Landau level, there are eigenvalues corresponding to extended bulk states. It is well known that in order to observe quantum Hall plateaux, one needs to have localized bulk states. The energy spectrum corresponding to edge states is located in the intervals between the broadened Landau levels. For a sample covering the entire plane, the intervals between the broadened Landau levels would be spectral gaps. The edge states of clean samples (g = 0) are well understood. For a bounded sample and weak disorder, one may use analytic perturbation theory in the disorder potential, gVd , in order to analyze the edge states. Unfortunately, as the sample size increases, the spacing between eigenvalues of H corresponding to edge states becomes smaller and smaller, and, as a consequence, the convergence radius of the perturbation series in g becomes smaller and smaller. Perturbation theory cannot be used in the limit of an infinitely large sample. The relation between quantization of the Hall conductivity and the extended nature of edge states is reviewed in section 2. Our definition of extended edge states for bounded samples is that they carry a non-vanishing chiral current. It is this property that plays an essential role in our analysis of the quantization of

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edge spectrum Landau levels bulk spectrum

E min

11111 00000 00000 11111 00000 11111 1 2

h ωc

11111 00000 00000 11111 00000 11111 3 2

h ωc

11111 00000 00000 11111 00000 11111 5 2

h ωc

11111 00000 00000 11111 00000 11111 7 2

E

h ωc

Figure 1: Bulk and edge spectrum of H. The energy scale is the cyclotron frequency ~ωc = eB/m.

the Hall conductivity. In sections 4, 5, and 6 the extended nature of edge states is established on the basis of arguments which are valid for sufficiently weak disorder, i.e. for |g| < g∗ , but our bounds on g∗ are uniform in the sample size. In sections 2 and 4, our sample has the original Laughlin cylinder geometry, but our proofs can be adapted for other sample shapes, such as the Corbino disc geometry used by Halperin [3], which we treat in an appendix. For infinite samples, the natural definition of “extended states” is that they correspond to absolutely continuous spectrum. In section 5, we consider the case of a two-dimensional electron gas confined to a half-plane by a smooth but steep edge potential, and prove that the energy spectrum well in between Landau levels is absolutely continuous for weak disorder. It turns out that our bound on the allowed strength of the disorder becomes smaller as the edge is made steeper. In section 6, we treat an “infinitely steep” edge directly by introducing Dirichlet boundary conditions in the definition of the covariant Laplacian. We show that for weak disorder, the edge states are again extended states. Our proofs can be extended to more general domains than the half-plane, provided they satisfy a certain geometric condition. The proofs in sections 5 and 6 are based on an application of the Mourre theory of positive commutators [7], which is briefly presented in section 3. For both finite and infinite samples, the extended nature of the edge states is analyzed with the help of the so-called “guiding center” of cyclotron motion. The commutator of the coordinate of the guiding center along the edge with the Hamiltonian is given by the derivative of the potential in the direction perpen-

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dicular to the edge∗ . The proofs are reduced to showing that this commutator is positive on states with energy well in between Landau levels. Instead of the guiding center, one can also use the coordinate of the particle itself along the edge as conjugate operator in the sense of Mourre theory. For the problem with Dirichlet boundary conditions, De Bi`evre and Pul´e [20] have shown that this allows to relax the assumptions on the disorder potential. It turns out that a Mourre estimate for one commutator is equivalent to a Mourre estimate for the other with the same lower bound on the commutator, but the techniques used to prove the estimate are different. In section 6, combining the two ideas, we use the coordinate of the particle itself as conjugate operator in the sense of Mourre theory, but prove the positivity of the commutator by considering the coordinate of the guiding center. Recently, Macris, Martin, and Pul´e (see [19]) have studied the half-plane case by a somewhat different method. They rule out the existence of eigenvalues between the broadened Landau levels by showing that the expectation value of the derivative of the potential in the direction perpendicular to the edge would be positive in an assumed eigenstate with energy between the broadened Landau levels. This would contradict the fact that the expectation value of a commutator with the Hamiltonian in an energy eigenstate must vanish by the virial theorem. To prove the positivity of the commutator in an assumed eigenstate, for weak disorder, they estimate the decay of edge state eigenfunctions into the edge with the help of Brownian motion techniques. Our use of the conjugate operator method allows us to exclude not only point spectrum, but also singular continuous spectrum. Furthermore, whereas the estimates for smooth potentials tend to fail in the limit of an infinitely steep edge, we also treat the problem with Dirichlet boundary conditions, and for more general domains than the half-plane.

2 The Laughlin argument revisited In this section, we review the argument leading to the integer quantization of the Hall conductivity, motivating our interest for the extended nature of edge states. In order to keep our analysis as simple as possible, we consider the cylinder geometry used in the original Laughlin argument [2]. The Hall current flows along the circumference and the Hall voltage is measured between two edge circles (see figure 2). In addition to the homogeneous magnetic field perpendicular to the surface of the cylinder, there is a “magnetic flux tube”, Φ, at the axis of the cylinder. The cylinder is characterized by two length scales, the radius, R, and the distance, L, between the two edges. Both lengths play a role in the mathematical analysis. Increasing R reduces the spacing between edge state eigenvalues of the Hamiltonian, and thus limits the applicability of perturbation theory to analyze the edge states. On the other hand, L influences the tunneling probability between two edges. Physically, we expect that for weak disorder, the tunneling probability ∗ This

is in the case of an edge potential, for Dirichlet boundary conditions, see section 6.

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L VH

φ

Iϕ

y Φ

B

R

Figure 2: Cylinder geometry

per edge length for states with energy well in between Landau levels is suppressed exponentially in L/lc , where lc is the cyclotron length. In this paper, we shall not provide rigorous bounds on those tunneling rates, but only deal with the problem connected to the spacing of eigenvalues. Possible tunneling between edges will be avoided by considering only one edge, or, equivalently, taking one edge to infinity. In the Corbino disc or annulus geometry introduced by Halperin [3], one cannot completely eliminate the tunneling problem, because, even if one considers only the outer edge of the annulus, the flux tube at the center is comparable to having a second edge, in the sense that for generic Φ, there are eigenvalues between Landau levels. Without precise estimates on tunneling probabilities between inner and outer edge, we can only show that edge states are extended for small |Φ|. The argument for the Corbino disc is carried out in appendix A. The coordinate along the axis of the cylinder will be denoted by y, and the coordinate perpendicular to it will be x = Rϕ, where 0 < ϕ ≤ 2π. The magnetic field is pointing radially outward, and the vector potential is chosen in ϕ-direction Aϕ = −By + Φ/2πR. Of course, the magnetic field can only be homogeneous on the two-dimensional surface of the cylinder, since otherwise the Maxwell equations would be violated. In these coordinates, the Hamiltonian is 2 ! 1 1 Φ H= −∂y2 + ∂ϕ − e −By + + V (y, ϕ), (2.1) 2m iR 2πR in units where ~ = 1. We start with V = 0, that is, with a cylinder infinite in y-direction and without disorder. The states can be labeled by the angular momentum quantum number l and the Landau band index n. The energy depends

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only on n through En,l = (n + 1/2)ωc . In the y-direction, the eigenfunctions are harmonic oscillator wave functions, localized near y0 (l − eΦ/2π) = (−l + eΦ/2π)/eBR. Changing Φ does not affect the energy of the states, but only their position along the cylinder. In particular, a change of Φ by 2π/e maps states with angular momentum l on those with angular momentum l − 1. This “spectral flow” produced by the change in Φ plays an important role in the following arguments. For a symmetric confining potential V0 = V0 (y), it is possible to continue labeling the states by l and n and to qualitatively discuss the dependence of the energy En,l (Φ) on l and Φ. The one-dimensional Hamiltonian for the motion in y-direction that results after separating the angular momentum, is analytic in the parameter l − eΦ/2π. Therefore, En,l (Φ) = En (l − eΦ/2π) are analytic functions, and the spectral flow of eigenstates with changing Φ is preserved by a symmetric V0 , En,l (Φ + 2π/e) = En,l−1 (Φ).

(2.2)

Furthermore, all eigenstates are well localized in the y-direction and the localization position, y0 (l − eΦ/2π), is also an analytic function, which is monotonically decreasing as can be seen by inspection of the Hamiltonian (2.1). En,l (Φ) can lie between Landau levels only if y0 (l − eΦ/2π) comes close to an edge of the cylinder, that is, only for edge states. For each n, it is possible to identify those l which correspond to states at the left and right edge. Large positive l correspond to the left edge, and large negative l to the right edge. Consider the current carried by an (n, l)-state in ϕ-direction, Iϕ,n,l = −

dEn,l (Φ) . dΦ

(2.3)

Under the assumption that V0 is monotonically increasing as one leaves the sample on either edge of the cylinder, so that it correctly describes the confining of the electron gas to the sample, it is easy to see that Iϕ,n,l has a definite sign for states localized at either edge. Edge states carry a chiral edge current. The main goal of our present work is to show that the edge states remain extended in the sense that they carry a chiral edge current if the sample contains a small amount of disorder. To motivate our interest for edge states, we now show that the chirality of our edge states implies the integer quantization of the Hall conductivity. Our argument is of a very general character and can be applied independently of a labeling of states by angular momentum and Landau band index. It is only to identify the integer ν = σH /(e2 /h) as the number of Landau levels below the Fermi energy that one must consider a situation where the Landau band index is a good quantum number. In our general argument, edge states are labeled by an index α, with corresponding energies Eα (Φ). We are interested in calculating the Hall conductance, σH , when the Fermi energy, EF , of the electron gas on the surface of the cylinder lies well in between two Landau levels. We assume that the disorder is sufficiently small so that Landau

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bands are still well defined in the bulk, and that there is no bulk spectrum in the vicinity of the Fermi energy. Experimentally, the Quantum Hall effect is observed in macroscopic samples as plateaux in the Hall conductance as a function of the magnetic field or carrier density. Therefore, the Fermi energy has to remain in between Landau levels, such that σH is quantized, for a sufficiently wide range of magnetic field or carrier density. This can only be achieved if there exist localized bulk states at EF . This fact is very well known, but we shall not make the attempt to solve the analytical problems connected with localized bulk states. For the rest of the argument, assume that the only occupied energy levels near the Fermi energy correspond to edge states. This edge spectrum will then be discrete. A small voltage drop, 0 < eVH = µr − µl ωc , between the left and right edge of the sample is taken into account by assuming that states localized at the right edge are occupied up to µr , and states localized the left edge up to µl . The Hall current is the current induced by this asymmetrical filling of edge states, in excess of the current carried by the electron gas in the ground state. If, for simplicity, we assume that µl = EF , the Hall current is carried by electrons on the right edge only. We denote by I the set of labels α of occupied edge states with µl ≤ Eα (0) ≤ µr . The Hall current can then be written as Iϕ =

X

−

α∈I

dEα (Φ) |Φ=0 . dΦ

(2.4)

As mentioned above, we shall neglect effects due to tunneling between the edges. On physical grounds, we expect that the tunneling rates are suppressed exponentially in L/lc , and tunneling will play no role when describing measurements performed on laboratory scales. We therefore take the limit L → ∞ in (2.4), and consider only the right edge. The rigorous justification of this restriction to a sample with only one edge is a rather difficult problem in localization theory which is not considered here. The next step in our argument is then to replace the expression (2.4), where dE/dΦ is evaluated at Φ = 0 by the average over a range of 2π/e,

Iϕ =

X I

e − 2π

2π/e Z

dΦ

dEα (Φ) . dΦ

(2.5)

0

It is this step which would fail in a sample with two edges at a finite distance L from each other, because resonances would necessarily occur at intermediate values of Φ. On physical grounds, expression (2.5) is a good approximation to (2.4) (in the limit L → ∞) provided the number of contributing states, |I|, is large, which is equivalent to a small spacing between successive eigenvalues, or to a large radius R. Because the spectrum of H is invariant under a change of Φ by 2π/e, there

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is a bijective map α 7→ β(α) with Eα (2π/e) = Eβ(α) (0). Then X e X Iϕ = − Eα (0) − Eα (0) . 2π β(I)\I

Ann. Henri Poincar´e

(2.6)

I\β(I)

Taking (2.6) as our expression for the Hall current, we now exploit the chirality of the edge states. For samples with only one edge, our analysis in subsequent sections implies that states corresponding to an energy in between Landau levels are localized near the edge of the sample and that the current they carry satisfies an estimate of the form C dEα (Φ) C0 ≥ ≥ , R dΦ R

(2.7)

where C and C 0 are non-zero constants whose common sign depends upon whether we consider a right or a left edge, the other being at infinity, and R is the size of the sample. It is easy to see that (2.7) is satisfied in the situation of a clean sample (g = 0) described above. These inequalities imply that C/R ≥ Eβ(α) (0) − Eα (0) e/2π ≥ C 0 /R > 0, if α corresponds to an edge state at the right edge. Therefore, ( µr + O(1/R) if α ∈ β(I) \ I (2.8) Eα (0) = µl + O(1/R) if α ∈ I \ β(I) . Because β is bijective, |I \ β(I)| = |β(I) \ I|, and recalling that µr − µl = eVH , we obtain that σH = |β(I) \ I| (1 + O(1/R)) (2.9) (e2 /h) is an integer, where σH = Iϕ /VH is the Hall conductance. This argument goes back essentially to ideas of Laughlin [2] and Halperin [3]. In the form presented above, it completely clarifies the universal character of the integer quantum Hall effect. The effect is not related to any particular geometry or symmetry of the sample. Our argument rests on the identification of the Hall current, as given by (2.5) or (2.6), and on the estimate (2.7), which we prove under various hypotheses in subsequent sections. Let us summarize the approximations made in the derivation: Taking the limit L → ∞ is justified by the exponential suppression of tunneling rates, and making this rigorous requires localization theory techniques. Taking the “thermodynamic limit” plays an important role also in the argument where the Hall conductance is identified with a charge index (see [18]). Indeed, the charge index vanishes in the generic situation with two edges. In our argument, the expression for the Hall current simplifies when we average over Φ, and this averaging is still possible, if more subtle, in a situation with two edges (I will in general depend on Φ). Precise estimates of the tunneling rates would make the transition from (2.4)

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to (2.5) rigorous. After having taken the limit L → ∞, the averaging over Φ and the replacement of Eα (0) with µr or µl for α ∈ β(I) \ I or α ∈ I \ β(I), respectively, become increasingly good approximations as the spacing between edge state eigenvalues decreases with increasing sample size, R. Here, finite size corrections are in principle easier to estimate than the tunneling rates. The definition of extended states as states whose energy increases (or decreases) monotonously when Φ is varied is consistent with the fact that an eigenfunction of H whose support does not surround the flux will, up to a phase factor, not be affected by a change of Φ, since, in a simply connected region, the influence of Φ can always be gauged away. Before closing this section, we explain how the integer ν = σH /(e2 /h) can be identified as the number of Landau bands below the Fermi energy, if n and l are good quantum numbers. For clean samples, this follows immediately from (2.2) and (2.6). We now show that this identification is still possible after inclusion of a small amount of disorder, gVd . In the generic situation with only one edge, the eigenvalues of the clean sample, En,l (Φ), will be non-degenerate for all Φ. One may then appeal to analytic perturbation theory to continue labeling the states by n and l in the presence of disorder, with energies En,l (Φ, g) which are analytic functions in g, for |g| small enough. By assumption, En,l (Φ, 0) = En,l (Φ). Because the eigenfunctions are also analytic in g, labels (n, l) that correspond to edge states for g = 0 will also correspond to states localized at the edge for |g| non-zero, but small. Thus, we propose to show the analog of equation (2.2), En,l (Φ + 2π/e, g) = En,l−1 (Φ, g),

(2.10)

which will imply the extended nature of the edge states and thereby the integer quantum Hall effect, with ν equal to the number of Landau bands below the Fermi energy (see figure 3). The shift in energy due to the disorder, En,l (Φ, g) − En,l (Φ, 0), is of the order of |g|, by the Feynman-Hellmann theorem, and (2.2) then immediately implies that En,l (Φ+2π/e, g)−En,l−1 (Φ, g) is also of the order of |g|. A change of Φ by 2π/e can be compensated by a gauge transformation and does not change the spectrum at all. Therefore, there must be an n0 and an l0 with En,l (Φ + 2π/e, g) = En0 ,l0 (Φ, g). But if, without disorder, the energies are sufficiently far apart and non-degenerate, this is only possible for n0 = n and l0 = l − 1, for small |g|, i.e., for weak disorder. This implies (2.10). The perturbative argument for the spectral flow may break down when the radius of the cylinder becomes very large, because the spacing between eigenvalues at g = 0 decreases with increasing R. But, as we have already noted, finite size corrections become small at least inversely proportional to R. It is therefore desirable to have an argument that establishes the spectral flow with a bound on the allowed strength of the disorder that is uniform in R. Such an argument will be provided in section 4. The rigorous proofs establishing the bounds in (2.7) are contained in section 5.

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E µr

EF

µl −(l − eΦ/2π) Figure 3: The spectral flow of edge states under variation of Φ. Each line represents a Landau band with eigenvalues symbolized by a dot. µl and µr are the chemical potentials on the left and right edge of the cylinder, respectively.

3 Methods and tools 3.1 The guiding center It is well known that the classical cyclotron orbit of a charged particle in a homogeneous magnetic field drifts under the influence of an electrostatic potential. This can be seen most simply by considering the “guiding center” of the motion, which is the center of a circle with cyclotron radius r = v/ωc = v/(eB/m) that passes through the position of the particle. The velocity, v, of the particle is given by ~v =

1 ~ (~ p − eA). m

(3.1)

This yields for the guiding center ˆ ~ = ~r + (~ ~ × B, Z p − eA) B

(3.2)

ˆ is a unit vector in B-direction perpendicular to the plane. where B ~ in a potential V is It is easily established that the equation of motion of Z ˆ ~˙ = B × grad V. Z B

(3.3)

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The separation of the motion of the guiding center from the cyclotronic motion will be used in section 4 and appendix A to get an estimable expression for the azimuthal current carried by any eigenstate of the Hamiltonian. In sections 5 and 6, the coordinate of the guiding center along the edge is used to establish estimates needed in Mourre theory, to which we turn now.

3.2 Positive commutators and absolutely continuous spectrum If in a classical Hamiltonian system, one can show that for some orbit the Poisson bracket of some coordinate K with the Hamiltonian H remains bounded away from zero, K˙ = {K, H} ≥ α > 0, for all times, one can conclude that the motion is extended along this coordinate. The quantum mechanical counterpart of this simple statement is the following: Assume there exists a “conjugate operator”, A, such that the commutator of A with the Hamiltonian is positive on some energy interval ∆, E∆ (H) [H, iA] E∆ (H) ≥ αE∆ (H),

(3.4)

with α > 0, and where E∆ (H) denotes the spectral projector of H on ∆. Noting that if ψ is an eigenstate of H, we have (ψ, [H, iA] ψ) = 0 by the virial theorem, we can conclude from (3.4) that H can not have an eigenvalue in the interval ∆. It was first proved by Mourre [7] that under additional regularity assumptions on H and its commutators with A, the spectrum of H is actually purely absolutely continuous on ∆. Equation (3.4) is termed a Mourre estimate. The assumptions on H have been subsequently relaxed considerably ([8, 9, 10], see also [11]). For the treatment of the problem with a smooth, steep edge potential in section 5, the original assumptions of Mourre can be verified. Those are (see [7]): (i) H and A are self-adjoint operators with domains D(H) and D(A). D(H) ∩ D(A) is a core for H. (ii) The unitary group eiAa generated by A leaves D(H) invariant and for all ψ ∈ D(H)

sup HeiAa ψ < ∞. |a|<1

(iii) The quadratic form [H, iA] which is defined on D(H) ∩ D(A), is bounded below and closable; the associated self-adjoint operator admits a domain containing D(H). 2 (iv) The quadratic form [H, iA], iA is form-bounded by |H| . Following Mourre’s work, the main efforts have gone in the direction of reducing the regularity assumptions described in (iii) and (iv). A very complete treatment of necessary and sufficient conditions for applicability of Mourre theory can be found in [10].

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It is also possible to apply Mourre theory with a non self-adjoint conjugate operator. Such situations are, for instance, treated in references [12], [13], and, implicitly, also in [14]. However, a general treatment of Mourre theory with non self-adjoint conjugate operator, in particular as regards the regularity assumptions, is to our knowledge still missing. In the case of Dirichlet boundary conditions, the coordinate of the guiding center is non self-adjoint. Still, it is possible to use it as a conjugate operator to prove absolute continuity of the spectrum [15, 16]. In section 6, we shall circumvent those technical difficulties, using ideas suggested in [20].

3.3 Decay of edge state eigenfunctions In the case of bounded samples, the decay of edge state eigenfunctions into the bulk can be proved with the help of the equation ψ = (E − H0 − Vd )−1 V0 ψ,

(3.5)

where ψ is an eigenfunction of H with energy E in the gaps of the bulk spectrum, ~ 2 /2m, H = H0 + V0 + Vd . and H0 = (~ p − eA) The free resolvent (E −H0 )−1 can be calculated explicitly in coordinate space representation, and it has Gaussian decay. Using this decay, equation (3.5), and the fact that V0 ψ is supported at the edge, one can show that the eigenfunction decays exponentially into the bulk of the sample. More technical details can be found in appendix A.

4 Extended edge states in large cylindrical samples In section 2, the notion of an “extended edge state” for arbitrarily large, bounded samples was defined as one that carries a chiral edge current. The central estimate needed for the derivation of the integer quantum Hall effect is (2.7). In the present section, we show that if ψ is an eigenstate of the Hamiltonian H, with an energy between Landau levels, and if the disorder is weak, the current carried by ψ is non-vanishing and has a definite sign. We assume the cylinder geometry described in section 2, with an edge potential confining the electron to the region y < 0. The current in ϕ-direction can be written as 2 ∂ϕ Φ dE Iϕ = − = ψ, − + By ψ . (4.1) dΦ 2πR iR 2πR Here and from now on, units are chosen in which m = 1/2 and e = 1. Because ψ is an eigenstate of H, the expectation value of [H, ipy ] = −B

∂

ϕ

iR

−

Φ + By − ∂yV 2πR

(4.2)

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in ψ vanishes by the virial theorem. This yields the equation Iϕ = −

1 ψ, ∂yV ψ . 2πBR

(4.3)

Equation (4.3) can be interpreted as saying that the current arises solely from the motion of the guiding center, whereas the cyclotronic motion does not contribute to the current when averaged over the whole cylinder. It is now intuitively clear why the current is chiral and non-vanishing. ψ, ∂yV ψ gets a positive contribution from the edge potential, and a contribution of indefinite sign from the disorder. If the disorder is weak, and because an edge state is localized near the edge, the contribution from the edge potential dominates the one from the disorder potential, so that the claim follows. The calculations needed to rigorously establish the lower bound on ψ, ∂yV ψ can be found in section 5 (proof of Theorem 1). On the other hand (∂ϕ /iR − Φ/2πR + By)ψ is bounded in norm by the Hamiltonian. Equations (4.1) and (4.3) therefore imply the estimate (2.7).

5 Steep edge potentials In the one particle picture that is the basis for our arguments in section 2, the edge of the sample can be naturally modelled by a smooth, but steep edge potential. In this section, we shall consider a half-plane geometry. The magnetic field B points in the z-direction, the sample is infinite in the x-direction, and the electron gas is confined by a wall to the region y < 0. The half-plane can be viewed as the limiting case R → ∞ of the cylinder geometry with one edge. The spectrum of the Hamiltonian is not purely discrete anymore and it is not possible to induce a spectral flow by changing a flux Φ. The definition of extended states is that the corresponding spectrum is absolutely continuous. The edge potential, V0 , is assumed to vanish for y < 0 and to rapidly increase for y > 0. The total potential is V = V0 + Vd , and the Hamiltonian is ~ 2 + V = H0 + V = H0 + V0 + Vd , H = (~ p − A)

(5.1)

where we have again chosen units with e = 1 and m = 1/2. The vector potential is taken in the Landau gauge, Ax = −By, Ay = Az = 0. We show absolute continuity for parts of the spectrum of H located between Landau levels, using Mourre theory with the x-coordinate of the guiding center as conjugate operator. The case V = V0 , that is without disorder potential, is standard. In the Landau gauge, the y-coordinate of the guiding center Zy = −px /B is a cyclic coordinate. After a Fourier transformation in the x-direction, the problem splits into one-dimensional Hamiltonians Hk indexed by the constant of motion k = −BZy . Those have spectrum En (k), where n is the Landau band index. For k → ∞, one can easily establish En (k) → (2n + 1)B, while for k → −∞, we have En (k) → ∞.

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Also En (k) is analytic as a function of k, so this implies that the spectrum of the full Hamiltonian is absolutely continuous (see [23], Theorem XIII.16). After introducing a disorder potential, it is worthwhile to first estimate the changes in the location of the spectrum. If the disorder is a random potential satisfying certain reasonable assumptions, it is known that the almost sure spectrum of the full Hamiltonian contains the spectrum of the clean Hamiltonian as a subset [24]. More details about this will be found in appendix B. The situation is of course more complicated for an arbitrary deterministic potential, but the results about continuity of the spectrum do not depend upon existence of spectrum. In the chosen gauge, our conjugate operator is Π = BZx = py + Bx. The commutator with H is [H, iΠ] = −∂yV , so that one rather has a “negative commutator” than a positive one, but this does obviously not hinder the application of Mourre theory. In addition to establishing a Mourre estimate, we need to assume that the edge potential allows the verification of conditions (i) to (iv) from section 3.2. We shall not make the attempt to present the optimal conditions on V0 , but simply note that (iii) and (iv) are valid, for example, if the potential is an upper bound for its own derivatives. Assumption (i) is trivially valid since the C ∞ function with compact support, Cc∞ , form a core for both H and Π. As for (ii), note that up to a phase factor, the group generated by Π are the translations in y-direction. If the edge potential V0 = V0 (y) does not increase too fast, for example subexponentially, so that an estimate of the form V0 (y+α) ≤ CV0 (y) holds uniformly in y, the domain of V0 is invariant under those translations, and with it also the domain of H, since the domain of the kinetic energy is trivially invariant. As noted above, extensions of the Mourre theory allow the treatment of much more singular potentials. For the following theorem, we assume an unbounded edge potential, vanishing for y < 0, with V00 (y) ≥ 0 for all y and inf {V00 (y); y ≥ b} > 0 for all b > 0. We discuss afterwards how the assumption that V0 is unbounded can be avoided. Theorem 1 (Mourre estimate). Assume E ∈ / σ(H0 ) = {(2n + 1)B, n ∈ N0 }. Then there is a constant δ, such that if the disorder potential satisfies |Vd | ≤ δ, there is an open interval ∆ 3 E and a positive constant α with − E∆ (H) [H, iΠ] E∆ (H) ≥ αE∆ (H).

(5.2)

The strategy for the proof is clear: Since − [H, iΠ] = ∂yV0 + ∂yVd , one first establishes the estimate considering only ∂yV0 , but including Vd in the Hamiltonian. This is the content of proposition 1, which yields a bound E∆ (H)∂yV0 E∆ (H) ≥ α ˜ E∆ (H). Then one estimates |∂yVd | on ∆ by E and δ, and Theorem 1 follows if |E∆ ∂yVd E∆ | < α. ˜ It is also possible to introduce another constant δ 0 to control the derivative |∂yVd | ≤ δ 0 separately, so that (5.2) follows for δ 0 < α ˜ . This allows a somewhat more generous choice of δ. We point out again that the proof of Theorem 1 can be transferred, without any changes, to the cylinder geometry to prove that ψ, ∂y V ψ is positive if ψ is an energy eigenstate with energy well in between Landau levels.

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Proposition 1. Assume E ∈ / σ(H0 ). Then there is a constant δ, such that if the disorder satisfies |Vd | ≤ δ, there is an open interval ∆ 3 E and a positive constant α ˜ with E∆ (H) ∂yV0 E∆ (H) ≥ α ˜ E∆ (H).

(5.3)

This proposition is at the heart of the matter, and its proof will be presented in some detail. 2 We have to show that ψ, V00 ψ ≥α ˜ kψk with α ˜ > 0 holds for all ψ with † 0 ψ = E∆(H)ψ . Obviously ψ, V0ψ ≥ 0 is non-negative. The intuition is that if ψ, V00 ψ goes to 0, then ψ, V0 ψ is also small, whence ψ is supported in the bulk and cannot be an edge state, so that ψ = E∆ (H)ψ is impossible. The problem is to estimate ψ, V00 ψ in terms of ψ, V0 ψ . Proof of proposition 1. Let η = dist(E, σ(H0 )), so that for all φ ∈ D(H0 ) in the domain of H0 , k(E − H0 )φk ≥ η kφk holds. The condition we put on δ is η > δ. Then E lies in the gaps of the bulk Hamiltonian H0 + Vd . Choose an > 0 with η > > δ and a smooth “cutoff” function j = j(y) with 1 ≥ j ≥ 0, j(y) = 1 for y ≤ b for some b > 0 and sup(|j(y)V (x, y)|) ≤ . This is possible because of the assumptions on V0 and because of |Vd | ≤ δ < . (see figure 4).

V0 j

Vd ≤ δ

0 Figure 4: The cutoff function j † From

now on, a

0

will denote a derivative with respect to y.

b

a

y

420

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We control j and its derivatives by introducing the following finite constants: C1 = sup{1/V00 (y)} : y ∈ supp(1 − j)} , C2 = sup{(j 00 (y))2 /V00 (y) : y ∈ supp(1 − j)} ,

(5.4)

C3 = sup{(j 0 (y))2 /V00 (y) : y ∈ supp(1 − j)} , C4 = sup{|j 0 (y)|}

Keeping track of these different constants will later allow to determine the dependence of the estimates on the steepness of the potential. Now let ∆ 3 E be an interval around E, and ψ = E∆ (H)ψ. Then ψ˜ = jψ ∈ D(H0 ) and the assumption on E yields the estimate η kjψk ≤ k(E − H0 )jψk ≤ k(E − H)jψk + kV jψk | {z } ≤ kψk .

(5.5)

A bound of kjψk in the other direction is obtained from k(1 − j)ψk2 = ψ, (1 − j)2 ψ ≤ C1 ψ, V00 ψ and is 1/2

kjψk ≥ kψk − k(1 − j)ψk ≥ kψk − C1

ψ, V00 ψ

1/2

(5.6)

.

(5.7)

Hj = jH − 2i(py − Ay )j 0 + j 00 yields k(E − H)jψk ≤ kj(E − H)ψk + 2 k(py − Ay )j 0 ψk + kj 00 ψk .

(5.8) The terms on the right hand side can be controlled in terms of |∆| and ψ, V00 ψ as follows: 2 kj 00 ψk = ψ, (j 00 )2 ψ ≤ C2 ψ, V00 ψ , (5.9) kj(E − H)ψk ≤ |∆| kψk , 2 k(py − Ay )j 0 ψk = ψ, j 0 (py − Ay )2 j 0 ψ (5.10) ≤ ψ, j 0 Hj 0 ψ + δ ψ, (j 0 )2 ψ because V + δ, (px − Ax )2 ≥ 0. Using j 0 Hj 0 = (j 0 H + Hj 0 )/2 + (j 00 )2 , the first term on the right hand side of (5.10) can be further estimated as 2

2

1 1 ψ, j 0 Hj 0 ψ = j 0 ψ, j 0 Hψ + j 0 Hψ, j 0 ψ + ψ, (j 00 )2 ψ 2 2 ≤ kj 0 ψk kj 0 Hψk + ψ, (j 00 )2 ψ .

(5.11)

Since

2 kj 0 ψk ≤ C3 ψ, V00 ψ ,

kj 0 Hψk ≤ kj 0 (E − H)ψk + E kj 0 ψk ≤ C4 |∆| kψk +

1/2 EC3

ψ, V00 ψ

1/2

(5.12) ,

(5.13)

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equations (5.10) and (5.11) imply 1/2 1/2 2 C3 C4 |∆| kψk , k(py − Ay )j 0 ψk ≤ ψ, V00 ψ [EC3 + C2 + δC3 ] + ψ, V00 ψ and thus k(py − Ay )j 0 ψk ≤ ψ, V00 ψ

1/2

[EC3 + C2 + δC3 ]1/2 1/4 1/4 1/2 1/2 + ψ, V00 ψ C3 C4 |∆| .

(5.14)

We now combine (5.5), (5.7), (5.8), (5.9) and the last inequality to: 1/2 1/2 η kψk − C1 ψ, V00 ψ ≤ η kjψk ≤ k(E − H)jψk + kψk

1/2 1/2 + ≤ |∆| kψk + kψk + C2 ψ, V00 ψ 1/2 1/2 0 + 2 ψ, V0 ψ (E + δ)C3 + C2 1/4 1/4 1/2 + 2 ψ, V00 ψ C3 C4 |∆|1/2 kψk1/2 (5.15) 1/2 1/2 1/4 1/2 1/2 Abbreviating D1 = C2 + 2 (E + δ)C3 + C2 , D2 = 2C3 C4 and D3 = C1 , the conclusion is that for all ψ = E∆ (H)ψ: (η − |∆| − ) kψk ≤ ψ, V00 ψ

1/2

≤ 2 ψ, V00 ψ

(D1 + ηD3 ) + D2 ψ, V00 ψ

1/2

1/4

1/2

|∆|

1/2

kψk

(D1 + ηD3 ) + (λ − 1) |∆| kψk , (5.16)

where λ − 1 = D22 /4(D1 + ηD3 ). Since η − > 0, and by making |∆| small enough, one finally gets: ψ, V00 ψ

η − λ |∆| − ≥ 2(D1 + ηD3 )

2 kψk2 =: α ˜ kψk2 .

(5.17)

Proof of Theorem 1. The missing piece is the estimate of ∂yVd = −[Vd , ipy ] on the energy interval ∆. ψ, [Vd , ipy ]ψ = ψ, [Vd , i(py − Ay )]ψ (5.18) = Vd ψ, (py − Ay )ψ − (py − Ay )ψ, Vd ψ ≤ 2δ kψk k(py − Ay )ψk .

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For ψ = E∆ (H)ψ with ∆ 3 E, 2 k(py − Ay )ψk = ψ, (py − Ay )2 ψ ≤ ψ, (H + δ)ψ ≤ (E + |∆| + δ) kψk

(5.19)

2

so that ψ, [Vd , ipy ]ψ ≤ 2δ(E + |∆| + δ)1/2 kψk2 .

(5.20)

With the additional condition on δ, ˜, 2δ(E + |∆| + δ)1/2 < α

(5.21)

the proof is complete. We now discuss the dependence of the estimate on the assumptions about the disorder and edge potentials. As mentioned above, one can relax the constraints on δ by introducing another constant δ 0 and imposing |Vd0 | ≤ δ 0 < α ˜ with α ˜ as in (5.17). It is actually enough if Vd0 is small near the edge, as one can easily show in a manner similar to the above by introducing a partition of unity separating the regions where Vd is smooth from those where √ it is rougher. The length scale is of course set by the = 1/ B: If Vd varies strongly on a scale of lc , it is better to cyclotron length l c √ use δ E as in (5.21) to control |Vd0 |. If Vd is smooth on this scale, the use of a δ 0 is more appropriate. In an alternative approach, using the x-coordinate of the particle itself as conjugate operator, similarly to section 6, it is possible to avoid assumptions on the derivative of the potential altogether. We will not make this explicit here. We now turn to the dependence of the estimates on the edge potential. α ˜ as defined in (5.17) depends not only on the disorder potential via δ, but also on V0 via the constants in the denominator, which are constrained by |jV | < : Assume V0 increases from 0 to on a length scale a. Then j must go from 1 to 0 on this scale (see figure 4), and for small a, the constants vary as: C1 ∼

a ,

C2 ∼

1 , a3

C3 ∼

1 , a

D12 ∼

1 , a3

D32 ∼

a .

(5.22)

Together with (5.17), small a or a steep potential implies α ˜ ∼ (η − )2 a3 . A steeper edge thus seems to allow less disorder. This problem is unexpected, since in the classical case also a hard wall leads to extended edge states. Using dimensional 0 analysis, one can argue that any direct estimate of ψ, V ψ in terms of η, which 0 is roughly the same as ψ, V0 ψ , will have a dependence on a that makes it fail when a tends to zero. Before we return to this problem in the next section by analyzing the problem with Dirichlet boundary conditions, we indicate what must be changed in the case of a bounded edge potential.

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V0 E0

j1

j2

j3

0

y

Figure 5: Partition of unity for bounded V0

We assume that the edge potential levels off above some height E0 , but still with V00 ≥ 0. Let η = min(dist(E, σ(H0 )), E0 − E), and choose δ < < η as above. Introduce a partition of unity according to figure 5, satisfying sup(|j1 V |) ≤ and V0 ≥ E0 > E on supp j3 . The condition inf{V00 (y) : y ≥ b} > 0 for all b > 0 is replaced by the condition inf{V00 (y) : y ∈ supp(j2 )} > 0. From the proof of proposition 1, we know that for small |∆|, η kj1 ψk ≤ k(E − H0 )j1 ψk ≤ λ |∆| kψk + kj1 ψk + C ψ, V00 ψ

1/2

(5.23) .

The constants C and λ might change subsequently, but are independent of ψ, δ, and . Since η − ≤ V0 − E + Vd on supp(j3 ), 2

(η − ) kj3 ψk = (η − ) ψ, j32 ψ

≤ ψ, j3 (H − E)j3 ψ

≤ kj3 ψk kj3 (H − E)ψk + ψ, (j30 )2 ψ 1/2 ≤ kj3 ψk |∆| kψk + kj3 ψk C ψ, V00 ψ .

(5.24)

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Therefore, η kj3 ψk ≤ |∆| kψk + kj3 ψk + C ψ, V00 ψ

1/2

.

(5.25)

Together with kψk ≤ kj1 ψk + kj2 ψk + kj3 ψk , 1/2 , kj2 ψk ≤ C ψ, V00 ψ

(5.26) (5.27)

equations (5.23) and (5.25) yield η kψk ≤ kψk + λ |∆| kψk + C ψ, V00 ψ so that one gets an estimate of ψ, V00 ψ as above.

1/2

(5.28)

6 Dirichlet boundary conditions In this section, we analyze the problem of extended edge states with the smooth edge potential replaced with Dirichlet boundary conditions. The fact that the conjugate operator that was used in section 5 is not self-adjoint in this situation makes special care necessary on the technical side. It is very likely that Mourre theory can be extended in some generality to the case when the conjugate operator is not self-adjoint. Elements of this theory can be found in [12, 13, 14]. That Mourre theory can be used to show absolute continuity of the edge spectrum with the x-coordinate of the guiding center, Zx , as conjugate operator, has been demonstrated in the preprint version of this paper [16], see also [15], which contains the first proof of absolute continuity of the spectrum as well as the first analysis of Dirichlet boundary conditions. Here, we shall resolve the problem as follows, using ideas suggested in reference [20]. We apply Mourre theory with the x-coordinate of the particle itself along the edge as conjugate operator, which is manifestly selfadjoint. The positivity of the commutator with the Hamiltonian, [H, ix] is reduced to the positivity of the commutator [H, iZx ]. Classically, only Zx is monotonically increasing in time, but quantum mechanically, the positivity of either of the two commutators is equivalent to the positivity of the other. In comparison with section 5, the assumptions on the disorder potential can be relaxed in that the boundedness of the derivative is not necessary. Furthermore, the proof of the positivity of the commutator i[H, Zy ] does not depend on having the geometry of a half-plane, but can be extended to more general domains in the plane satisfying a certain geometric condition which we state below. This section is organized as follows: We first give precise definitions of the occuring spaces and operators, and state the main theorems. We then prove the theorems for the half-plane, since the ideas can be made more transparent in this situation. We finally give the proofs for more general domains.

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6.1 Definitions and Theorems Let Ω be a domain in R2 . Ω may be the half-plane R2− = R × R− or a more general domain satisying a certain geometric assumption (GA) stated in section 6.3. To convey an idea of the domains under consideration, we present three examples in figure 6. (GA) allows for the domains (a, b), but not for (c).

Ω Ω β

(a)

Ω

(b): β = −π

(c): β = π

Figure 6: Examples for domains. (a,b) satisfy (GA), (c) does not. The Hamiltonian in this section will be again H0 +Vd , where the unperturbed Hamiltonian is H0 = (−i∇ − A)2

(6.1)

on L2 (Ω) with Dirichlet boundary conditions, i.e., with domain D(H0 ) = W2 (Ω, A) ∩W01 (Ω, A), where the magnetic Sobolev spaces are defined using covariant derivatives, e.g., W1 (Ω, A) = {ψ ∈ L2 (Ω) | (−i∂i − Ai )ψ ∈ L2 (Ω), i = x, y} .

(6.2)

Note that we do not specify the gauge at this point. We shall be working in Landau gauge for the half-plane, but for more general domains this is not appropriate for obvious geometrical reasons. Theorem 2. Let E/B ∈ / 2N0 + 1. For sufficiently small kVd k∞ /B, the spectrum of H on the half–plane R2− is purely absolutely continuous near E. Here, and below, ‘sufficiently small’ is meant depending only on quantities explicitly mentioned. In particular, for Theorem 2, the bound is uniform in B. In a classical picture, absolute continuity of the spectrum corresponds to the guiding center of the electron jumping in a definite direction along the boundary, ∂Ω, of Ω = R2− , each time the electron hits the wall. If the boundary is not a straight line, then at each collision the guiding center moves forward in the direction of the tangent vector to ∂Ω at the collision point. Yet, this may be a backward motion with respect to the tangent vector at the next collision point,

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with the result that a classical edge trajectory can get trapped. No trapping is possible, however, if the cyclotron radius of the electron is small compared to the radius of curvature of ∂Ω (more precisely: compared to the radius of injectivity of the tubular map associated with ∂Ω). Then the bouncing motion will result in an effective progression along ∂Ω. The generalization of Theorem 2 to more general domains reads as follows. Theorem 3. Assume Ω satisfies (GA). Fix E/B ∈ / 2N0 + 1 and let kVd k∞ /B be small enough. Then, the spectrum of H is purely absolutely continuous near E, provided B is large enough. Finally, no trapping is possible if Ω lies on one side of the graph of a function f, Ω = {(x, y) ∈ R2 | y < f (x)} ,

(6.3)

with f 0 ∈ C2unif (R). Theorem 4. Let E/B ∈ / 2N0 + 1, and B0 > 0. For sufficiently small kVd k∞ /B, the spectrum of H on the domain (6.3) is purely absolutely continuous near E, provided B ≥ B0 . Under the scaling x → λx, B → λ−2 B, V → λ−2 V , we have H → λ−2 H. Without loss of generality we may thus set B=1 in Theorem 2, and the same is true for Theorem 3 if its last sentence is replaced by: Then, the spectrum of H on λΩ is purely absolutely continuous near E, provided λ > 0 is large enough. Similarly the conclusion of Theorem 4 is ... the spectrum of H on the domain defined by fλ (x) = λf (x/λ) is purely absolutely continuous near 1/2 E, provided λ ≥ λ0 = B0 . The Theorems will be proved in this form.

6.2 The half–plane We begin the proof of Theorem 2 with some preliminaries (1.–3.): 1. We shall work in Landau gauge A = (−y, 0) .

(6.4)

2. Let p = −i∇. As mentioned above, the component py = −i∂y cannot be defined self-adjointly on L2 (Ω), a fact related to the integration by parts identity (py ω, ρ) − (ω, py ρ) = ihω, ρi ,

(ω, ρ ∈ W1 (Ω, A)) ,

(6.5)

where h·, ·i denotes the inner product on L2 (∂Ω). Strictly speaking one should write hLω, Lρi instead of hω, ρi, where L is the boundary trace operator, which is bounded as a map L : W1 (Ω, A) → L2 (∂Ω). Below we shall need another, similar statement (see e.g. [25], Theorem 8.3 or [26], Section 7.50),

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Theorem A. The map L : W2 (Ω, A) → W3/2 (∂Ω) × W1/2 (∂Ω) ψ 7→ (ψ ∂Ω, ∂y ψ ∂Ω)

(6.6)

is continuous, has ker L = W02 (Ω, A), and admits a continuous right inverse R: LR = 1. Proof. The statement in [25, 26] refers to A = 0; to extend it to the present situation, pick a function j ∈ C∞ 0 (−∞, 0] with j(y) = 1 for y near 0. Then L and R can be replaced by Lj, and jR respectively. The claim now follows, since the map ψ 7→ jψ is bounded as a map W2 (Ω, A) → W2 (Ω, 0), resp. W2 (Ω, 0) → W2 (Ω, A), for the special gauge (6.4). 3. The proof of Theorem 2 is based again on Mourre theory [7, 8, 9, 10]. We will here refer to the formulation given in [9], using the following pieces of notation: B(H) is the algebra of bounded operators on the Hilbert space H, ρ(H) 3 z is the resolvent set of H, on which R(z) := (H − z)−1 , and E∆ (H) is the spectral projection for H onto ∆ ⊂ R. Theorem B. Let H, A be self–adjoint operators on H, and E ∈ R. Assume: (i) There is z ∈ ρ(H) such that the map g : R → B(H) ,

s 7→ eisA R(z)e−isA

(6.7)

is of class C1+ε with 0+ ≤ ε ≤ 1 in the norm topology (with ε = 0+ meaning that g 0 (s) is Dini continuous); (ii) There is an open interval ∆ 3 E and an α > 0 such that E∆ (H)i[H, A]E∆ (H) ≥ αE∆ (H) .

(6.8)

Then the spectrum of H is purely absolutely continuous near E. According to Theorem 6.2.10 in [27], g ∈ C1 implies that [H, A], defined as a quadratic form on D(H) ∩ D(A), extends to a bounded operator [H, A] : D(H) → D(H)∗ . In particular, the left hand side of (6.8) is a bounded operator on H. We now compute the commutator of the non self-adjoint operator Π = −py − x with H0 , in the sense of quadratic forms. Lemma 1. For ϕ, ψ ∈ D(H0 ) ∩ D(x) we have i[(H0 ϕ, Πψ) − (Πϕ, H0 ψ)] = hpy ϕ, py ψi .

(6.9)

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Proof. By density we may assume that ϕ, ψ are of compact support. The separate contributions from the two terms in H0 = p2y + (px + y)2 are i[(p2y ϕ, Πψ) − (Πϕ, p2y ψ)] = hpy ϕ, py ψi , i[((px + y) ϕ, Πψ) − (Πϕ, (px + y) ψ)] = 0 . 2

2

(6.10) (6.11)

In fact, as (6.10) is concerned, the contribution of x to the commutator is, using (6.5), i[(py ϕ, py xψ) − (py xϕ, py ψ)] = 0 ,

(6.12)

whereas that of py is i[(p2y ϕ, py ψ) − (py ϕ, p2y ψ)] = −hpy ϕ, py ψi .

(6.13)

To verify (6.11), we note that [px + y, py + x]ψ = 0. Hence (px + y)2 ϕ, (py + x)ψ = = (px + y)ϕ, (px + y)(py + x)ψ

= (px + y)ϕ, (py + x)(px + y)ψ = (py + x)(px + y)ϕ, (px + y)ψ = (px + y)(py + x)ϕ, (px + y)ψ = (py + x)ϕ, (px + y)2 ψ ,

(6.14)

where, in the third equality, we used (6.5) with (px + y)ϕ ∂Ω = 0. We now turn to the proof of the basic positivity estimate. Lemma 2. Fix E ∈ / 2N0 + 1. Then there is an open interval ∆ 3 E and an α > 0 such that |||py E∆ (H0 )ψ|||2 ≥ αkE∆ (H0 )ψk2 ,

(6.15)

where we set ||| · |||2 = h·, ·i. Proof. We note that RLD(H0 ) ⊂ D(H0 ), and write for ψ ∈ D(H0 ) (H0 − E)ψ = (H0 − E)(ψ − RLψ) + (H0 − E)RLψ .

(6.16)

Since L(ψ−RLψ) = 0, the function ψ−RLψ ∈ W02 (Ω, A) can be extended to R2 by 0, and the Dirichlet boundary condition dropped. Hence k(H0 − E)(ψ − RLψ)k ≥ dist(E, 2N0 + 1)kψ − RLψk. Denoting the r-th Sobolev norm by k · kr , we obtain from k(H0 − E)RLψk ≤ const kRLψk2 ≤ const (|||ψ|||3/2 + |||∂1 ψ|||1/2 ) (Theorem A) and ψ ∂Ω = 0 that k(H0 − E)ψk ≥ dist(E, 2N + 1)kψk − const |||∂1 ψ|||1/2 .

(6.17)

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For any ε > 0 we have |||∂1 ψ|||1/2 ≤ const (ε−1 |||∂1 ψ|||0 + ε|||∂1 ψ|||1 ) , 5/4

with |||∂1 ψ|||1 ≤ const kψk5/2 ≤ const kH0 ψk. Here we used the boundedness of the map W5/2 (Ω, A) → W1 (∂Ω), ψ 7→ ∂1 ψ ∂Ω (see [25], Theorem 9.4 or [26], Theorem 7.58). We insert these estimates in equation (6.17) and use 5/4 kH0 E∆ (H0 )ψk ≤ (|E| + |∆|)5/4 kψk and k(H0 − E)E∆ (H0 )ψk ≤ |∆|kψk. For sufficiently small ε and |∆| we obtain the claim. Following [20], we deduce absolute continuity of the spectrum using A = −x as conjugate operator in the sense of Mourre theory (Theorem B), which in contrast to Π is self–adjoint on its natural domain. The right hand side in H(τ ) = eiτ x He−iτ x = p2y + (px − τ + y)2 + Vd extends from τ ∈ R to an entire analytic family of type A. Hence assumption (i) of Theorem B is fulfilled. Assumption (ii) holds because of the following lemma. Lemma 3. Let E ∈ / 2N0 + 1. Then there is an open interval ∆ 3 E and an α > 0 such that −E∆ (H)i[H, x]E∆ (H) ≥ αE∆ (H) , provided kV k∞ is small enough. Proof. We shall prove the statement in the case Vd = 0 first. Because of Lemmas 1 and 2, it suffices to show that the difference Π−(−x) = −py contributes arbitrarily little to the commutator. In a state ψ = E∆ (H0 )ψ, this contribution is estimated as |(H0 ψ, py ψ) − (py ψ, H0 ψ)| = | (H0 − E)ψ, py ψ − py ψ, (H0 − E)ψ | ≤ 2k(H0 − E)ψkkpy ψk ≤ 2|∆|(|E| + |∆|)1/2 kψk2 , where the equality is justified by (6.5) and ψ ∈ D(H0 ). The right hand side can be made arbitrarily small by letting |∆| → 0. We next extend the result to H and note that H 0 (0) = −i[H, x] = −i[H0 , x], which is relatively bounded with respect to H0 , is unchanged. Let α > 0 and ∆ 3 E be as given by Lemma 2. We may assume ∆ to be centered on E. For e 3 E, we split ∆ E∆ e (H) = E∆ (H0 )E∆ e (H) + (1 − E∆ (H0 ))E∆ e (H) , and obtain 0 0 E∆ e (H)(H (0) − α)E∆ e (H) ≥ E∆ e (H)E∆ (H0 )(H (0) − α)E∆ (H0 )E∆ e (H) 0 − 2k(H (0) − α)(1 − E∆ (H0 ))E∆ e (H)k .

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The first term on the right hand side is non-negative, and the second line can be estimated by a constant times kH0 (1 − E∆ (H0 ))(H0 − E)−1 kk(H0 − E)E∆ e (H)k e + kV k∞ ) . ≤ (1 + 2|E||∆|−1 )(|∆| Hence, 0 E∆ e (H)(H (0) − α)E∆ e (H) ≥ −α/2 ,

e + kV k∞ small enough, and therefore E e (H)H 0 (0)E e (H) ≥ (α/2)E e (H). for |∆| ∆ ∆ ∆ Proof of Theorem 2. Application of Theorem B.

6.3 General domains We now turn to the proof of Theorem 3. We first state precisely the geometrical assumption made in Theorem 3. Assumption. (GA) Let Ω ⊂ R2 be an open set. (i) Ω has the uniform C3 –property in the sense of [26]; (ii) Let ∂Ω consist of finitely many connected components γ, each parametrized by its arclength s (with the induced orientation). Let there be a function s ∈ C2 (Ω) extending arclength from ∂Ω to Ω, i.e., s(γ(s0 )) = s0 for s0 ∈ R, satisfying k∂i sk∞ < ∞ ,

k∂i ∂j sk∞ < ∞ .

(6.18)

We note that, by (i), a bounded component of ∂Ω would be a closed curve, but by (ii) no such curve is allowed. Hence Ω is simply connected and unbounded. Also, (GA) is not affected by the ambiguity s 7→ s − s0 implicit in the definition of arclength. We illustrate (GA) with an example. Let Ω ⊂ R2 be a simply connected open set with oriented boundary ∂Ω consisting of a single unbounded smooth curve γ ∈ C3unif (R), parametrized by arclength s ∈ R. For simplicity, we shall assume that γ is asymptotically straight, i.e., γ¨ ∈ L1 (ds) , with ˙ = d/ds. The overall bending of γ is Z∞ γ¨ (s)ds ,

β= −∞

and takes values in [−π, π].

(6.19)

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Example 1. If β 6= π ,

(6.20)

then Ω as described satisfies (GA). In particular, the domains (a, b) in Figure 6 are allowed, but not (c). Proof. Only part (ii) of (GA) requires proof. Because of γ ∈ C3unif (R) and of (6.19) we also have lims→±∞ γ¨ (s) = 0. Elementary geometric considerations show that, for large enough r > 0, the equation |γ(s)| = r has exactly two solutions s = s± (r), with s± (r) → ±∞, (r → ∞). We define functions ϕ± (r) through γ(s± (r)) = r(cos ϕ± (r), sin ϕ± (r)) in such a way that ϕ± (r) are continuous and that ϕ− (r) − ϕ+ (r) ∈ (0, 2π). Their limiting values ϕ∞ ± = limr→∞ ϕ± (r) exist and ∞ satisfy β + (ϕ∞ − ϕ ) = π. Condition (6.20) implies − + ∞ ϕ∞ − − ϕ+ > 0 .

(6.21)

Under our assumptions, ds± → ±1 , dr

d2 s± →0, dr2

r

dϕ± →0, dr

r

d2 ϕ± →0, dr2

(6.22)

as r → ∞. For R > 0 large enough we define s(x, y) on Ω ∩ {(x, y) | |(x, y)| > R} = {(x, y) = r(cos ϕ, sin ϕ) | r > R, ϕ+ (r) ≤ ϕ ≤ ϕ− (r)} by linear interpolation along arcs, |(x, y)| = r, i.e., by s(x, y) = s− (r)

ϕ − ϕ+ (r) ϕ− (r) − ϕ + s+ (r) ; ϕ− (r) − ϕ+ (r) ϕ− (r) − ϕ+ (r)

we then smoothly extend s further to the compact complement Ω ∩ {x | |x| ≤ R}. Now (6.18) follows from (6.21, 6.22). To prove Theorem 3, we will just address the changes required for the generalization from the proof of Theorem 2. For notational simplicity in intermediate results, we first consider the Hamiltonian on Ω. The required scaling Ω → λΩ (see the end of Section 6.1) will be done later. We define the components of the velocity as πi = −i∂i − Ai (i = x, y) with domain W1 (Ω, A); see equation (6.2). We introduce the matrix ε with entries εxy = −εyx = −1, εxx = εyy = 0. which represents a rotation by π/2, and define the outer unit normal n = n(s) = −εγ(s), ˙ with components ni . For ω, ρ ∈ W1 (Ω, A) we then have the integration by parts identity (πi ω, ρ) − (ω, πi ρ) = ihω, ni ρi , Also, H0 ψ = (πx2 + πy2 )ψ for ψ ∈ D(H0 ). We next extend the trace theorem (Theorem A) to the present setting.

(6.23)

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Theorem C. The map ˙ × W1/2 (∂Ω, A · γ) ˙ L : W2 (Ω, A) → W3/2 (∂Ω, A · γ) ψ 7→ (ψ ∂Ω, π · nψ ∂Ω)

(6.24) (6.25)

is continuous, has ker L = W02 (Ω, A), and admits a continuous right inverse R: LR = 1. The norms of L and R depend only on the C3 –regularity data of Ω. Proof. The uniform C3 –property is associated with a cover {Uj } of ∂Ω and corresponding diffeomorphisms (of class C3 ) Ψj : B = {(x, y) ∈ R2 | |(x, y)| < 1} → Uj , such that Ψj (B ∩{y < 0}) = Ω∩Uj ; see [26], Section 4.6 for details. The statement for A = 0 is [26], Theorem 7.53 (see also Section 4.29 allowing for unbounded ∂Ω; or [27], Theorem 5.9). The proof makes use of a partition of unity for ∂Ω subordinate to {Uj }, with the effect of reducing the statement to the analogous restriction property from B− := B ∩ {y < 0} to B0 := B ∩ {y = 0}. Similarly, in our case (A 6= 0) matters are reduced to the same statement, (6.25), with the replacements Ω → B− , ∂Ω → B0 , A → A˜ = Ψ∗j A and A · γ˙ → A˜x , where A˜ is the pull–back of A under Ψj , in the sense of 1–forms. The claim so left to prove is gauge covariant, and we may choose the special gauge Zy ˜ A(x, y) = − ω ˜ (x, ξ)dξ, 0 , 0

where ω ˜ (x, y)dx ∧ dy = Ψ∗j (dx ∧ dy) is the pull–back of the area 2–form on Uj . We note that ω ˜ , A˜ ∈ C2 (B− ), with bounded derivatives. Moreover, A˜ = 0 on B0 . Now the claim again follows from the case A˜ = 0, since the norms of W2 (B− ) and of ˜ are equivalent. W2 (B− , A) Remark. As in the proof of Lemma 2, we shall also need the boundedness of the map W5/2 (Ω, A) → W1 (∂Ω, A · γ), ˙ ψ 7→ π · nψ ∂Ω. The same argument applies and, in fact, it is only here that a uniform C2 –property does not suffice. In the context of Theorem 3, a formal candidate for a conjugate operator is Π = s(Z), where s is arclength as given by (GA), and Z is the guiding center, which ~ = ~x − ~π . This definition is unsuitable, because p = −i∇ and we can write as Z hence Z are not self–adjoint (vector) operators, and because the two components ~ − ~x and set of Z do not commute. Instead, we formally linearize s(Z) in Z Π = s(~x) − ∇s · ε~π = s(~x) − ε~π · ∇s ,

(6.26)

which is a well–defined (non self–adjoint) operator. The second expression follows from εij [πj , ∂i s] = −iεij ∂j ∂i s = 0. Here, and in the following, the summation convention is understood.

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Lemma 4. Let s satisfy (6.18). For ϕ, ψ ∈ D(H0 ) ∩ D(s) we have i[(H0 ϕ, Πψ) − (Πϕ, H0 ψ)] = (ϕ, πi (mij + mji )πj ψ) + hπi ϕ, (γ˙ j ∂j s)πi ψi , (6.27) where mij (~x) = εjk ∂k ∂i s. The last term in (6.27) can also be written as h(π · n)ϕ, (γ˙ · ∇s)(π · n)ψi, since πi ψ = ni (π · n)ψ on γ for ψ ∈ D(H0 ). If, as in (GA), s is equal to arclength on γ we have γ˙ · ∇s = 1 by definition. Proof. We first claim that for ϕ ∈ D(H0 ) ∩ D(s), ρ ∈ W1 (Ω, A) ∩ D(s) we have i[(πi ϕ, Πρ) − (Πϕ, πi ρ)] = (πj ϕ, mij ρ) .

(6.28)

Indeed, the contribution from s in (6.26) is i[(πi ϕ, sρ) − (sϕ, πi ρ)] = i(ϕ, [πi , s]ρ) = (ϕ, ∂i sρ) ,

(6.29)

by using (6.23) and ϕ ∈ D(H0 ), which makes the boundary term vanish. To compute the other contribution we note that i[(πi ϕ, πj ρ) − (πj ϕ, πi ρ)] = −i([πi , πj ]ϕ, ρ) + hπi ϕ, nj ρi − hπj ϕ, ni ρi = εij (ϕ, ρ) , since πi ϕ = ni (π · n)ϕ on ∂Ω. Hence, for fj = εkj ∂k s, i[(πi ϕ, fj πj ρ) − (fj πj ϕ, πi ρ)] = i[(πi ϕ, πj fj ρ) − (πj ϕ, πi fj ρ)] − (πi ϕ, ∂j fj ρ) + (πj ϕ, ∂i fj ρ) (6.30) = (ϕ, ∂i sρ) − (πj ϕ, mij ρ) , because of εij fj = ∂i s, ∂j fj = 0 and of ∂i fj = −mij . Subtracting (6.30) from (6.29) yields (6.28), which can also be written as i[(πi ω, Πψ) − (Πω, πi ψ)] = (ω, mij πj ψ) ,

(6.31)

for ω ∈ W1 (Ω, A) ∩ D(s). Moreover, (6.23) yields i[(Πω, ρ) − (ω, Πρ)] = hω, εji ∂j sni ρi = hω, γ˙ i ∂i sρi .

(6.32)

Writing the left hand side of (6.27) as i[(πi2 ϕ, Πψ) − (Ππi ϕ, πi ψ) + (Ππi ϕ, πi ψ) − (πi ϕ, Ππi ψ) + (πi ϕ, Ππi ψ) − (Πϕ, πi2 ψ)] . and using (6.31, 6.32, 6.28) with ω = πi ϕ and ρ = πi ψ, respectively, concludes the proof if ω, ρ ∈ D(s). By density this suffices.

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Proof of Theorem 3. Statement and proof of Lemma 2 hold true with the replacement py → π · n, with ∆, α depending only on the regularity data for Ω. These data are inherited by λΩ with λ ≥ 1. On the other hand λΩ satisfies part (ii) of (GA) with sλ (~x) = λs(~x/λ), and correspondingly mλ (~x) = λ−1 m(~x/λ) in equation (6.27). There, the first term on the right hand side is then estimated in absolute value for ϕ = ψ = E∆ (Hλ )ψ by a constant times λ−1 kπE∆ (Hλ )ψk2 ≤ λ−1 (|E| + |∆|)1/2 kψk2 . As a result, the positivity of the commutator (6.27) on E∆ (Hλ ) obtains for large λ > 0. As to the regularity assumption, the conjugate operator A = s(x) gives rise to the analytic family H(τ ) = eiτ s He−iτ s = H − τ (∇s · π + π · ∇s) + τ 2 (∇s)2 .

(6.33)

Statement and proof of Lemma 3 are changed accordingly, with Π−s(x) = −επ·∇s replacing py . In (6.33) we have suppressed the subscript λ. One should, however, notice that the relative bound of H 0 (0) = ∇s · π + π · ∇s with respect to H is independent of λ. We finally come to the Proof of Theorem 4. We set s(x, y) = −x, i.e., Π = −πy − x. Then, in (6.27), mij = 0 and γ˙ · ∇s = (1 + f 0 (x)2 )−1/2 ≥ δ for some δ > 0 and all x ∈ R. Upon scaling, fλ0 (x) = f 0 (x/λ) has C2unif (R)–norm which can be bounded independently of λ ≥ λ0 . In particular, bounds on the norms associated with (6.25), as well as δ, are independent of λ ≥ λ0 .

Acknowledgment We thank the referees for useful comments that helped clarify the argument in section 2, and improve the presentation in section 6.

A Corbino disc geometry In this appendix, we adapt the arguments of section 4 to the Corbino disc geometry (see figure 7). In polar coordinates and with a suitable gauge, the Hamiltonian is 2 ! 1 Φ 1 1 Br 2 H= ∂ϕ − e + −∂r − ∂r + + V (r, ϕ). (A.1) 2m r ir 2 2πr For V = 0, the spectrum and eigenfunctions of H can be obtained by elementary methods. For Φ = 0, the spectrum consists only of the Landau levels, with energy (n + 1/2)ωc . In contrast to the case of the cylinder, there is here a

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Φ

~ B

~ B

VH

~r E Iϕ

Figure 7: Corbino disc geometry

restriction on the angular p momentum, l ≥ −n. The states are localized in radial direction near r0 (l) = 2 |l| /eB. If Φ 6= 0, 0 < Φ ≤ 2π/e, the localization of the states is shifted from r0 (l) to r0 (l − eΦ/2π). The energy of the states with l − eΦ/2π ≥ 0 remains unchanged, but the energy of the states with l − eΦ/2π < 0 is changed to (n + eΦ/2π + 1/2)ωc . It is convenient to change the definition of the index n and to introduce functions En,l (Φ) in such a way that the spectral flow can be written as in equation (2.2), En,l (Φ + 2π/e) = En,l−1 (Φ).

(A.2)

The energies are En,l (Φ) = (n + 1/2)ωc if l − eΦ/2π ≥ 0 and En,l (Φ) = (n − l + eΦ/2π + 1/2)ωc if l − eΦ/2π < 0. With this definition of n, l is unrestricted, but the bands for fixed n are bent upwards when l − eΦ/2π is negative. Thus, the

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flux Φ has effects similar to an edge at the center of the disc. Besides considering only one edge, we also need to restrict the analysis to Φ = 0 in order to avoid resonances. The discussion with included edge potential, V0 , and weak disorder potential, gVd , is completely parallel to the case of the cylinder geometry and we do not repeat it here. We now derive the expression for the azimuthal current corresponding to equation (4.3) of section 4. In the Corbino disc geometry, the azimuthal current carried by a state, ψ, is ∂E 2 ∂ϕ Φ Br Iϕ = − = ψ, − − ψ ∂Φ 2πr ir 2πr 2 (A.3) ˆ × ~r B = ψ, [H, i~r]ψ . 2πr2 ˆ × ~r/r is a unit vector pointing in the azimuthal direction, and Note here that B that the commutator [H, i~r] gives the velocity (m = 1/2 and e = 1). Now replace ~ − (~ ~ × B/B, ˆ ~r in the commutator with the guiding center, ~r = Z p − A) and use the ~ ~ ˆ equation of motion for Z, [H, iZ] = B/B × grad V . This yields Iϕ = ψ,

ˆ × ~r 1 B ~ × B]ψ ˆ . ∂r V ψ − ψ, [H, i(~ p − A) 2 2πBr 2πBr

(A.4)

Using the fact that the expectation value of a commutator with H in an energy eigenstate vanishes, the second term in (A.4) is equal to ψ,

ˆ × ~r 1 B ~ × Bψ ˆ . H, i 2 (~ p − A) 2πB r

(A.5)

A straightforward calculation in polar coordinates then shows that the azimuthal current carried by an eigenstate of H can be written as 2 ~ ϕψ (~ p − A) 2πr 1 2 1 1 1 2 ~ 2ϕ ψ . = ψ, p − A) ∂r V ψ + ∂r ψ, ∂r ψ − ψ, 2 (~ 2πBr 2πB r r 2πB r

Iϕ = ψ,

(A.6)

This expression is similar to (4.3), with correction terms due to the circular geometry. The first term in (A.6) is positive for weak disorder, and decays inversely proportional to the size R of the sample. The second term is always positive, while the third term is negative, but bounded, and decays as 1/R2 . Thus Iϕ |Φ=0 is positive for an eigenstate of H with energy between Landau levels, if the sample is large, and the disorder weak. The rest of this appendix is devoted to making the estimates in this argument rigorous. To deal with the singularity at r = 0, it will be useful to know that edge states eigenfunctions are exponentially small near the origin. We start by

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considering the spatial decay of the free resolvent (E − H0 )−1 (for Φ = 0). This free resolvent for a homogeneous magnetic field Hamiltonian in two dimensions can be calculated using an expansion in eigenfunctions of H0 , which are known explicitly in terms of Laguerre polynomials. See [21] and [22] for more details. The result contains the confluent hypergeometric function called Ψ in the notation of [29], R0 (x, y; E) = (E − H0 )−1 (r1 eiϕ1 , r2 eiϕ2 ) 2 B B 1 2 = − eiπζ Γ(−ζ)eir1 r2 sin(ϕ1 −ϕ2 ) e− 4 |x−y| Ψ −ζ, 1; |x − y| . 4π 4 (A.7) Here, ζ is related to the energy by E = (2ζ + 1)B, so that R0 has singularities at Landau band energies. Ψ(−ζ, 1; z) has a logarithmic singularity at z = 0 and behaves for large |z| as |z|ζ . This implies a Gaussian decay for large |x − y|. We will, however, only use an estimate of the form |R0 (x, y; E)| ≤ Ce−|x−y|/ξ |ln(|x − y| /ξ)| ,

(A.8) √

with a decay length scale ξ on the order of the magnetic length B We now include the disorder potential and claim the following

−1

.

Lemma 5. Let V0 be a not too fast increasing edge potential that describes the Corbino disc with varying sample size R. Let further Vd be bounded by a constant sufficiently small compared to the magnetic field. Let finally ∆ be an energy interval in the spectral gaps of H0 + Vd . Whenever a < R, there are positive constants C and λ, such that an eigenfunction ψ of H = H0 + V0 + Vd with energy E ∈ ∆ satisfies |ψ(x)| ≤ Ce−(R−a)/λ kψk

(A.9)

for |x| ≤ a, uniformly in the sample size R and the energy E ∈ ∆. Proof. Use the equation ψ(x) =

Z

(E − H0 − Vd )−1 (x, y)V0 (y)ψ(y)dy

(A.10)

and expand the resolvent in a Neumann-series. (E − H0 − Vd )−1 =

∞ X

(E − H0 )−1 Vd

n

(E − H0 )−1 .

(A.11)

n=0

Consider a fixed n, and use the estimate (A.8) for each free resolvent. This results in the following integrals to be estimated: Z Y n C |ln(|zi−1 − zi | /ξ)| e−|zi−1 −zi |/ξ Vd (zi ) × i=1

C |ln(|zn − y| /ξ)| e−|zn −y|/ξ V0 (y) |ψ(y)| dz1 . . . dzn dy,

(A.12)

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with z0 = x, zn+1 = y. Taking out Vd out of the integrals, and splitting the exponentials in 3, this is estimated by

(C kVd k∞ )n Z

sup z1 ,...zn+1

−|w|/3ξ

e

exp(−

n+1 X

|zi−1 − zi | /3ξ) ×

i=1

Z

n

|ln(|w| /ξ)|

sup z1 ,...zn

exp(−

n+1 X

e−|zn −y|/3ξ |ln(|zn − y| /ξ)| × |zi−1 − zi | /3ξ)V0 (y) |ψ(y)| dy. (A.13)

i=1

P Now inf z1 ,...zn n+1 i=1 |zi−1 − zi | ≥ |x − y| and, applying the Schwarz inequality to the last integral, the bound moves to (C˜ kVd k∞ )n e−|x−R|/3ξ

Z

1/2 × e−2|w|/3ξ |ln(|w| /ξ)|2 dw 1/2 Z 2 . (A.14) e−2|x−y|/3ξ (V0 (y))2 |ψ(y)| dy

If the potential does not increase too fast, the integral containing V0 converges and is bounded by E kψk, and after summing over n, making the bound on the disorder small enough, the claim follows.

With the help of Lemma 5, we now prove the positivity of the current from the expression (A.6), 2 ~ ϕψ (~ p − A) 2πr 1 2 1 1 1 2 ~ 2ϕ ψ . = ψ, p − A) ∂r V ψ + ∂r ψ, ∂r ψ − ψ, 2 (~ 2πBr 2πB r r 2πB r

Iϕ = ψ,

(A.15)

The first step is to eliminate the singularity at r = 0 by replacing ψ with jψ, where j is a cutoff at radius a > 0 near the origin. We want to show that the error introduced by this replacement, 1 1 ψ, (p − A)ϕ ψ − jψ, (p − A)ϕ jψ = r r 1 1 (1 − j)ψ, (p − A)ϕ jψ + jψ, (p − A)ϕ (1 − j)ψ + r r 1 (1 − j)ψ, (p − A)ϕ (1 − j)ψ , r

(A.16)

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is small for large R. Consider bounding the term (1 − j)ψ, 1 (p − A)ϕ (1 − j)ψ ≤ k 1 ψk k 1 (p − A)ϕ ψk a r r3/4 a r1/2

1

(p − A)2ϕ ψ 1/2 ≤ Ce−R/λ a1/4 kψk k ψk1/2 a r a ≤ Ce−3R/2λ a1/2 kψk2 (A.17) where k · ka has its obvious meaning, and we have taken advantage of the fact the ~ 2ϕ ψk can be bounded by the energy‡ . r commutes with (p − A)ϕ and that k(~ p − A) The other terms in (A.16) similarly decay exponentially with the sample size. The replacement of ψ with jψ introduces additional terms, because in the derivation of (A.6), we used that ψ was an eigenstate of H, whereas jψ is not. Those additional terms are jψ,

ˆ × ~r ˆ × ~r 1 B 1 B ~ × B[H, ˆ ~ × Bjψ ˆ i 2 (~ i 2 (~ p − A) j]ψ + ψ, [H, j] p − A) 2πB r 2πB r 1 1 = jψ, (−∂r )[H, j]ψ + ψ, [H, j] (−∂r )jψ (A.18) 2πBr 2πBr

[H, j] = −2j 0 ∂r − j 00 − j 0 /r has support for r near a. Organize the various terms so that either no radial derivative is acting on ψ, or one radial derivative, or the expression 1r ∂r r∂r . Terms with one radial derivative are estimated, for example, as jψ, j 00 1 ∂r jψ ≤ kj 00 1 j 0 ψk k∂r ψk a r r (A.19) C −(R−a)/λ 2 kψk , ≤ 4e a where C contains the energy as bound for k∂r ψk. The terms with two radial derivatives, such as 1 1 j 0 j ψ, ∂r r∂r ψ , r r

(A.20)

are estimated similarly. Terms without radial derivative acting on ψ are even easier. Cutting off with j thus introduces error terms that decay exponentially with the sample size. From (A.6) with jψ instead of ψ, we now estimate

~ 2ϕ jψ ≤ k 1 jψk ~ 2ϕ ψ jψ, 1 (~ p − A) p − A) (A.21)

j(~

. 2 2 r r ‡ This Kato estimate for (~ ~ 2ϕ is not trivial, since radial and azimuthal part of the kinetic p − A) energy do not commute. The same kind of estimate in the half-plane geometry was needed in section 6.

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The second factor is bounded by the energy. The first is split into a part with the radial coordinate between a and R/2, so that it decays as e−R/2λ /a2 , and a part from R/2 to ∞ which decays as 1/R2 . The positivity of 1 jψ, ∂r V jψ r

(A.22)

is proved in a fashion very similar to the one in section 5 by introducing another cutoff function at the edge r = R§ . The lower bound for the expression decays as 1/R. If the sample is large and the disorder weak, the current is positive.

B Random potentials and almost sure spectrum Consider a Hamiltonian H1 on H = L2 (R2 ) which is invariant in x-direction. Let σ(H1 ) be the spectrum of H1 . Add to H1 a disorder in the form of a random potential Vd,ω , where ω ∈ Ω, and (Ω, P ) is a probability space. Assume three things about (Ω, P ): (i) For every ω ∈ Ω, Vd,ω is bounded by a constant δ which is independent of ω. (ii) The group G(x) of translations in x-direction acts measure-preserving and ergodically on (Ω, P ). This allows one to speak of an almost sure (a.s.) spectrum of Hω = H1 + Vd,ω , denoted by Σ(Hω ). (iii) For every measurable compact set Λ ⊂ R2 and every > 0 the probability P ω; |Vd,ω (x, y)| < ∀(x, y) ∈ Λ

(B.1)

is positive. Those assumptions are, for example, satisfied in an Anderson model for the disorder. Assumption (i) is added for consistency with the proofs in sections 5 and 6. As mentioned in section 3, it can very likely be replaced by boundedness of the variance of Vd,ω . Assumptions (ii) and (iii) allow the proof of the following: Lemma 6. Let notation be as introduced and assumptions (i) to (iii) be satisfied. Then σ(H1 ) ⊂ Σ(Hω ) ⊂ σ(H1 ) + [−δ, δ] .

(B.2)

For a proof, see [24]. § The cutoff at r = a does not significantly perturb the argument of section 5 because ψ is exponentially small near the center.

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[16] J. Fr¨ohlich, G.M. Graf, J. Walcher, On the extended nature of edge states of quantum Hall Hamiltonians, math-ph/9903014 [17] E. Akkermans, J. E. Avron, R. Narevich, R. Seiler, Boundary conditions for bulk and edge states in quantum Hall systems, SFB 288 Preprint No. 248, TU Berlin 1997, mp-arc 96--646 [18] J. E. Avron, R. Seiler, B. Simon, Charge deficiency, charge transport and comparison of dimensions, Commun. Math. Phys. 159, 399 (1994) [19] N. Macris, P. A. Martin, J. V. Pul´e, On edge states in semi-infinite quantum Hall systems, J. Phys. A 32, 1985–1996 (1998), cond-mat 9812367 [20] S. De Bi`evre and J. V. Pul´e, private communication; see also Propagating Edge States for a Magnetic Hamiltonian, Math. Phys. Electr. J. 5 (1999), math-ph/9903034 [21] R. Joynt and R. E. Prange, Conditions for the Quantum Hall effect, Phys. Rev. B 29, 3303 (1984) [22] D. Lehmann, A microscopic derivation of the critical magnetic field in a Superconductor, Commun. Math. Phys. 173, 155–174 (1995) [23] M. Reed and B. Simon, Methods of modern mathematical physics, vols 1–4, Academic Press 1972-1979 [24] H. Kunz, B. Souillard, Sur le spectre des op´erateurs aux diff´erences finies al´eatoires, Commun. Math. Phys. 78, 201–246 (1980) [25] J. L. Lions and E. Mag`enes, Non-homogeneous boundary value problems and applications, vol. 1, Springer 1972 [26] R.A. Adams, Sobolev spaces, Academic Press (1975). [27] V.I. Burenkov, Sobolev spaces on domains, Teubner (1998). [28] F. G. Tricomi, Fonctions hyperg´eometriques confluentes, M´emorial des sciences math´ematiques CXL (1960) [29] A. Erd´elyi et al., Higher Transcendental Functions, vols. I and II, McGrawHill 1953 J. Fr¨ohlich, G. M. Graf and J. Walcher Institut de Physique Th´eorique ETH-H¨ onggerberg CH-8093 Z¨ urich, Suisse Email : [email protected], [email protected], [email protected] Communicated by J. Bellissard submitted 02/04/99, revised 08/09/99, accepted 27/09/99

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auser Verlag, Basel, 2000 1424-0637/00/030443-17 $ 1.50+0.20/0

Annales Henri Poincar´ e

On the Existence of Ground States for Massless PauliFierz Hamiltonians C. G´erard

I Introduction We consider in this paper the problem of the existence of a ground state for a class of Hamiltonians used in physics to describe a confined quantum system (”matter”) interacting with a massless bosonic field. These Hamiltonians were called PauliFierz Hamiltonians in [DG]. Examples, like the spin-boson model or a simplified model of a confined atom interacting with a bosonic field are given in [DG, Sect. 3.3]. Pauli-Fierz Hamiltonians can be described as follows: Let K and K be respectively the Hilbert space and the Hamiltonian describing the matter. The assumption that the matter is confined is expressed mathematically by the fact that (K + i)−1 is compact on K. The bosonic field is described by the Fock space Γ(h) with the one-particle space h = L2 (IRd , dk), where IRd is the momentum space, and the free Hamiltonian Z dΓ(ω(k)) = ω(k)a∗ (k)a(k)dk. The positive function ω(k) is called the dispersion relation. The constant m := inf ω can be called the mass of the bosons, and we will consider here the case of massless bosons , ie we assume that m = 0. The interaction of the “matter” and the bosons is described by the operator Z V = v(k) ⊗ a∗ (k) + v ∗ (k) ⊗ a(k)dk, where IRd 3 k → v(k) is a function with values in operators on K. Thus, the system is described by the Hilbert space H := K ⊗ Γ(h) and the Hamiltonian H = K ⊗ 1l + 1l ⊗ dΓ(ω(k)) + gV,

(I.1)

g being a coupling constant. If K = C, the Hamiltonian H is solvable (see eg [Be, Sect. 7]) and H is defined as a selfadjoint operator if Z 1 |v(k)|2 dk < ∞, ω(k)

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and admits a ground state in H if and only if Z 1 |v(k)|2 dk < ∞. ω(k)2 In this paper we show that H admits a ground state in H for all values of the coupling constant under corresponding assumptions in the general case. The existence of a ground state of H in the Hilbert space H is an important physical property of the system described by H. For example it has the following consequence for the scattering theory of H: assume that ω ∈ C ∞ ({k|ω(k) > 0}) and ∇ω(k) 6= 0 in {k|ω(k) > 0}. Assume also that IRd 3 k 7→ kv(k)(K + 1)− 2 kB(K) 1

is locally in the Sobolev space H s in {k|ω(k > 0} for some s > 1 (a short-range condition on the interaction). Then under the conditions (H0), (H1), (I1) below, it is easy to prove the existence of the limits W ± (h) := s- lim eitH eiφ(ht ) e−itH t→±∞

− 12

for h ∈ h0 := {h ∈ h|ω h ∈ h} and ht = e−itω h. The operators W ± (h) are called asymptotic Weyl operators. They satisfy W ± (h)W ± (g) = e−i 2 Im(h|g) W (h + g), h, g ∈ h0 , 1

and

eitH W ± (h)e−itH = W ± (h−t ).

In particular they form two regular CCR representations over the preHilbert space h0 . It is easy to show that the space of bound states Hpp (H) of H is included into the space of vacua for these representations (see for example [DG]). Hence the existence of a ground state for H implies that the CCR representations defined by the asymptotic Weyl operators admit Fock subrepresentations. As a consequence wave operators can be defined. When the Hamiltonian H admits no ground state in the Hilbert space H, the ground state of H has to be interpreted as a state ω on some C ∗ −algebra of field observables. Similarly the scattering theory for H has to be significantly modified. These phenomena have been extensively studied by Froehlich [Fr]. In particular the arguments in the proof of Lemma IV.5 are inspired by [Fr, Sect. 2.3], where it is shown that the state ω is locally normal. Let us end the introduction by making some comments on related works. In [AH], the existence of a ground state is shown under rather similar conditions, if the coupling constant g is sufficiently small. In [Sp], the same problem is considered in the case the small system described by (K, K) is a confined atom, and the coupling function k 7→ v(k) is a real multiplication operator in the atomic variables (ie v ∗ (k) = v(−k) is a multiplication operator on K). Using functional integral

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methods and Perron-Frobenius arguments, the existence of a ground state is shown for all values of the coupling constant. Our result is hence a generalization of the results both of [AH] and [Sp]. If we drop the assumption that the small system is confined (mathematically this amounts to drop the hypothesis (H0) below), then the only result is the one of [BFS], where the existence of a ground state is shown for small coupling constant if K is an atomic Hamiltonian and assumptions similar to (I1), (I2) are made.

II Result II.1

Introduction

In this section we introduce the class of Hamiltonians that we will study in this paper. We have stated our result under rather general hypotheses, allowing for a mild UV divergency of the interaction. Clearly the behavior of the interaction for large momenta should not be important for the existence of a ground state, which essentially depends only on the low momentum behavior of the interaction. Therefore the reader wishing to avoid some technicalities can for example assume that the operator K is bounded and that the function IRd 3 k 7→ v(k) is compactly supported.

II.2

Hamiltonian

Let K be a separable Hilbert space representing the degrees of freedom of the atomic system. The Hamiltonian describing the atomic system is denoted by K. We assume that K is selfadjoint on D(K) ⊂ K and bounded below. Without loss of generality we can assume that K is positive. We assume (H0)

(K + i)−1 is compact.

The physical interpretation is that the atomic system is confined. Let h = L2 (IRd , dk) be the 1−particle Hilbert space in the momentum representation and let Γ(h) be the bosonic Fock space over h, representing the field degrees of freedom. We will denote by k the momentum operator of multiplication by k on L2 (IRd , dk), and by x = i∇k the position operator on L2 (IRd , dk). Let ω ∈ C(IRd , IR) be the boson dispersion relation. We assume   ∇ω ∈ L∞ (IRd ),    (H1) lim|k|→∞ ω(k) = +∞,     inf ω(k) = 0. To stay close to the usual physical situation, we will also assume that ω(0) = 0, ω(k) 6= 0 for k 6= 0, although the results below hold also in the general case. The typical example is of course the massless relativistic dispersion relation ω(k) = |k|.

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The Hamiltonian describing the field is equal to dΓ(ω). The Hilbert space of the interacting system is H := K ⊗ Γ(h). The Hamiltonian H0 := K ⊗ 1l + 1l ⊗ dΓ(ω) of the non-interacting system is associated with the quadratic form Z 1 1 2 2 Q0 (u, u) := (K ⊗ 1lu, K ⊗ 1lu) + ω(k)(1l ⊗ a(k)u, 1l ⊗ a(k)u)dk, 1

with D(Q0 ) = D((K + dΓ(ω)) 2 ). The interaction between the atom and the boson field is described with a coupling function v IRd 3 k 7→ v(k), 1

such that for a.e. k ∈ IRd , v(k) is a bounded operator from D(K 2 ) into K and 1 from K into D(K 2 )∗ . We associate to the coupling function v the quadratic form Z V (u, u) =

(1l ⊗ a(k)u, v(k) ⊗ 1lu) + (v(k) ⊗ 1lu, 1l ⊗ a(k)u)dk,

(II.1)

A rather minimal assumption under which the quadratic form Q = Q0 + V gives rise to a selfadjoint operator is for a.e. k ∈ IRd v(k)(K + 1)− 2 , (K + 1)− 2 v(k) ∈ B(K), 1

1

∀u1 , u2 ∈ K, k 7→ (u2 , v(k)(K + 1)− 2 u1 ), k 7→ (u2 , (K + 1)− 2 v(k)u1 ) 1

(I1)

1

are measurable, R 1 1 1 (kv(k)(K + R)− 2 k2 + k(K + R)− 2 v(k)k2 )dk < ∞, C(R) := ω(k) limR→+∞ C(R) = 0.

Note that it follows from the results quoted in the Appendix that the functions 1 1 k 7→ kv(k)(K + R)− 2 k, k 7→ k(K + R)− 2 v(k)k are measurable, and hence the last condition in (I1) has a meaning. Proposition II.1 Assume hypothesis (I1). Then the quadratic form V is Q0 −form bounded with relative bound 0. Consequently one can associate with the quadratic 1 form Q = Q0 + V a unique bounded below selfadjoint operator H with D(H 2 ) = 1

D(H02 ). The Hamiltonian H is called a Pauli-Fierz Hamiltonian. Proof. We apply the estimate (A.1) in Lemma A.1 with B = K, m = ω.

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Results

Under assumption (I1), one can associate a bounded below, selfadjoint Hamiltonian H to the quadratic form Q. Let us introduce the following assumption on the behavior of v(k) near {k|ω(k) = 0}: Z 1 1 (I2) kv(k)(K + 1)− 2 k2 dk < ∞. 2 ω(k) Theorem 1 Assume hypotheses (H0), (H1), (I1), (I2). Then inf spec(H) is an eigenvalue of H. In other words H admits a ground state in H.

III The cut-off Hamiltonians III.1

Operator bounds

Let us introduce the following assumption: R 1 1 1 )(kv(k)(K + R)− 2 k2 + k(K + 1)− 2 v(k)k2 )dk < ∞, C 0 (R) := (1 + ω(k) 0 (I1 ) limR→+∞ C 0 (R) = 0. Proposition III.1 Assume (I1), (I1’). Then the operator Z V = a∗ (v) + a(v) = v(k) ⊗ a∗ (k) + v ∗ (k) ⊗ a(k)d k is H0 −bounded with relative bound 0. Consequently H = H0 + V is a bounded below selfadjoint operator with D(H) = D(H0 ). Proof. We apply the estimates (A.2), (A.3) in Lemma A.1 with B = K, m = ω.2

III.2

Cut-off Hamiltonians

In the sequel we will need to introduce various cut-off Hamiltonians. For 0 < σ 1 an infrared cutoff parameter and τ 1 an ultraviolet cutoff parameter, we denote by Vσ , Vσ,τ the quadratic forms defined as V in (II.1) with the coupling function v replaced respectively by vσ , vσ,τ for vσ = 1l{σ≤ω} (k)v, vσ,τ = 1l{σ≤ω≤τ } (k)v. We denote by Hσ , Hσ,τ the selfadjoint operators associated with the quadratic forms Q0 + Vσ , Q0 + Vσ,τ . Note that since vσ,τ satisfies (I1’), we have D(Hσ,τ ) = D(H0 ). 1

1

Applying Lemma A.2 in the Appendix and the fact that D(H 2 ) = D(H02 ) we obtain limτ →+∞ (Hσ,τ − λ)−1 = (Hσ − λ)−1 , (III.1) limσ→0 (Hσ − λ)−1 = (H − λ)−1 ,

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for λ ∈ IR, λ −1, and 1 k (Hσ,τ − z)−1 − (Hσ − z)−1 (H0 + 1) 2 k ∈ o(1)|Imz|−1 τ → +∞, 1 k (Hσ − z)−1 − (H − z)−1 (H0 + 1) 2 k ∈ o(1)|Imz|−1 σ → 0,

(III.2)

for z ∈ C\IR.

III.3

Existence of ground states for the cut-off Hamiltonians

Let ω ˜ σ : IRd → IR be a dispersion relation satisfying   ∇˜ ωσ ∈ L∞ (IRd ),    ω ˜ σ (k) = ω(k) if ω(k) ≥ σ,     ω ˜ σ (k) ≥ σ/2.

(III.3)

1

˜ σ be the operator associated to the quadratic form kK 2 uk2 + Let H ka(k)uk2 dk + Vσ (u, u).

R

ω ˜ σ (k)

˜ σ admits a ground Lemma III.2 Hσ admits a ground state in H if and only if H state in H. 2 Proof. Let hσ := L2 ({k|ω(k) < σ}, dk), h⊥ σ = L ({k|ω(k) ≥ σ}, dk). Let U be the canonical unitary map U : Γ(h) → Γ(h⊥ σ ) ⊗ Γ(hσ )

(see for example [DG, Sect. 2.7 ]). Let us still denote by U the unitary map 1lK ⊗ U ∗ from H = K⊗Γ(h) into K⊗Γ(h⊥ σ )⊗Γ(hσ ). By [DG, Sect. 2.7], the operator U Hσ U is equal to 1lK⊗Γ(h⊥ ⊗ dΓ(ωσ,1 ) + Hσ2 ⊗ 1lΓ(hσ ) , σ) where ωσ,1 = ω|hσ and Hσ2 is the operator associated with the quadratic form R 1 ˜ σ U ∗ is equal to kK 2 uk2 + {ω(k)≥σ} ωσ (k)ka(k)uk2 dk + Vσ (u, u). Similarly U H ⊗ dΓ(˜ ωσ,1 ) + Hσ2 ⊗ 1lΓ(hσ ) , 1lK⊗Γ(h⊥ σ) ˜ σ U ∗ or U Hσ U ∗ where ω ˜ σ,1 = ω ˜ σ|hσ . Now Hσ2 has a ground state ψ if and only if U H have a ground state (equal to ψ ⊗ Ω, where Ω ∈ Γ(hσ ) is the vacuum vector). This proves the lemma. 2 The following result is essentially well known (see [AH], [BFS]) and rather easy to show. Proposition III.3 Assume hypotheses (H0), (H1), (I1). Then for any σ > 0 Hσ admits a ground state.

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˜ σ admits a ground state. Let for Proof. By Lemma III.2 it suffices to show that H 1 ˜ τR ∈ IN Hσ,τ be the Hamiltonian associated with the quadratic form kK 2 uk2 + ω ˜ σ (k)ka(k)uk2 dk + Vσ,τ (u, u). Let ˜σ,τ = inf spec(H ˜ σ,τ ), E ˜σ = inf spec(H ˜ σ ). E Applying Lemma A.2, we have for z ∈ C\IR ˜ σ )−1 = lim (z − H ˜ σ,n )−1 . (z − H n→+∞

(III.4)

˜ σ,τ ) = D(K + On the other hand applying the bounds in Lemma A.1 we have D(H ˜ dΓ(˜ ωσ )). The Hamiltonian Hσ,τ is very similar to the class of massive Pauli-Fierz Hamiltonians studied in [DG]. It is easy to see that the arguments of [DG] extend ˜ σ,τ . In particular, following the proofs of [DG, Lemma 3.4], [DG, Thm. 4.1], to H ˜σ,τ + σ/2[). Using (III.4) ˜ σ,τ ) is compact if χ ∈ C0∞ (] − ∞, E we obtain that χ(H ˜ ˜ and the fact that Eσ = limn→+∞ Eσ,τ , we obtain that χ(H˜σ ) is compact if ˜σ + σ/2[). This implies that H ˜ σ and hence Hσ admit a ground χ ∈ C0∞ (] − ∞, E state. 2

III.4

The pullthrough formula

As in [BFS], we shall use the pullthrough formula to get control on the ground states of Hσ . Since the domain Hσ is not explicitely known under assumption (I1), some care is needed to prove the pullthrough formula in our situation. Proposition III.4 As an identity on L2loc (IRd \{0}, dk; H), we have: (Hσ + ω(k) − z)−1 a(k)ψ = a(k)(Hσ − z)−1 ψ + (Hσ + ω(k) − z)−1 vσ (k)(Hσ − z)−1 ψ, ψ ∈ H. Proof. For u1 , u2 ∈ D(H0 ), the following identity makes sense as an identity on L2loc (IRd \{0}, dk): (a∗ (k)u1 , (Hσ,τ − z)u2 ) = ((Hσ,τ + ω(k) − z)u1 , a(k)u2 ) + (u1 , vσ,τ (k)u2 ). Setting u2 = (Hσ,τ − z)−1 v2 , we obtain that for v2 ∈ H, a(k)v2 ∈ L2loc (IRd \{0}, dk; D(H0 )∗ ) and a(k)v2 = (Hσ,τ + ω(k) − z)a(k)(Hσ,τ − z)−1 v2 + vσ,τ (k)(Hσ,τ − z)−1 v2 . Hence for ψ ∈ H, (Hσ + ω(k) − z)−1 a(k)ψ ∈ L2loc (IRd \{0}, dk; H) and (Hσ,τ + ω(k) − z)−1 a(k)ψ = a(k)(Hσ,τ − z)−1 ψ + (Hσ,τ + ω(k) − z)−1 vσ,τ (k)(Hσ,τ − z)−1 ψ,

(III.5)

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holds as an identity in L2loc (IRd \{0}, dk; H). 1 By (I1), (vσ,τ (k) − vσ (k))(H0 + 1)− 2 tends to 0 in L2 (IRd \{0}, dk; B(K)) when τ → +∞. Using also (III.2) and letting τ → +∞ we obtain (Hσ + ω(k) − z)−1 a(k)ψ = a(k)(Hσ − z)−1 ψ + (Hσ + ω(k) − z)−1 vσ (k)(Hσ − z)−1 ψ, 2

as claimed.

IV Proof of Thm. 1 Let Eσ := inf spec(Hσ ), E := inf spec(H). We denote by ψσ , σ > 0 a normalized ground state of Hσ . Applying the pullthrough formula to ψσ , we obtain easily the following identity on L2 (IRd , dk; H): a(k)ψσ = (Eσ − Hσ − ω(k))−1 vσ (k)ψσ .

(IV.1)

The first rather obvious bound on the family of ground states ψσ is the following. Lemma IV.1 Assume hypotheses (H0), (H1), (I1). Then (ψσ , H0 ψσ ) ≤ C, uniformly in σ > 0.

(IV.2)

The bound (IV.2) follows immediately from the fact that the quadratic forms Qσ are equivalent to Q0 , uniformly in σ. The following lemma is also well-known (see eg [BFS, Thm. II.5], [AH, Lemma 4.3]). We denote by N the number operator on Γ(h). Lemma IV.2 Assume hypotheses (H0), (H1), (I1), (I2). Then (ψσ , N ψσ ) ≤ C, uniformly in σ > 0.

(IV.3)

Proof. We have using (IV.1) R (ψσ , N ψσ ) = ka(k)ψσ k2 dk R = k(Eσ − Hσ (k) − ω(k))−1 vσ (k)ψσ k2 dk R 1 1 − 12 2 ≤ k(H0 + 1) 2 ψσ k2 ω(k) k dk 2 kvσ (k)(K + 1) ≤ C, uniformly in σ > 0 using (I2) and (IV.2).

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Lemma IV.3 Assume hypotheses (H0), (H1), (I1), (I2). Then E − Eσ ∈ o(σ).

(IV.4)

Proof. Let 0 < σ0 < σ. We have Eσ0 − Eσ ≤ (Qσ0 − Qσ )(ψσ , ψσ ) = (Vσ0 − Vσ )(ψσ , ψσ ), Eσ − Eσ0 ≤ (Qσ − Qσ0 )(ψσ0 , ψσ0 ) = (Vσ − Vσ0 )(ψσ0 , ψσ0 ),

(IV.5)

Applying (A.1) with m(k) = 1, we obtain |(Vσ0 − Vσ )(u, u)| ≤ C(σ 0 , σ)(u, N u) 2 (u, (K + 1)u) 2 , 1

for C(σ 0 , σ) =

Z

1

kv(k)(K + R)− 2 k2 dk 1

{σ0 <ω(k)≤σ}

(IV.6)

12

Using (IV.6) for u = ψσ or ψσ0 , the right hand side of (IV.5) is bounded by C0 C(σ 0 , σ), uniformly in σ, σ 0 , using (IV.2) and (IV.3). We note that by (III.1) E = limσ0 →0 Eσ0 . Hence letting σ 0 tend to 0 we get |E − Eσ | ≤ C0 C(0, σ) ∈ o(σ), using hypothesis (I2). 2 Proposition IV.4 Assume hypotheses (H0), (H1), (I1), (I2). Then a(k)ψσ − (E − H − ω(k))−1 v(k)ψσ → 0 when σ → 0 in L2 (IRd , dk; H). Proof. We have, using (IV.1) a(k)ψσ − (E − H − ω(k))−1 v(k)ψσ = (Eσ − Hσ − ω(k))−1 vσ (k)ψσ − (E − H − ω(k))−1 v(k)ψσ = −1l{ω(k)≤σ} (k)(E − H − ω(k))−1 v(k)ψσ +(E − H − ω(k))−1 (H − Hσ )(Eσ − Hσ − ω(k))−1 vσ (k)ψσ +(Eσ − E)(E − H − ω(k))−1 (Eσ − Hσ − ω(k))−1 vσ (k)ψσ =: Rσ,1 (k) + Rσ,2 (k) + Rσ,3 (k). We will estimate separately Rσ,i , 1 ≤ i ≤ 3. First kRσ,1 (k)kH ≤ 1l{ω(k)≤σ} (k)

1 1 1 kv(k)(K + 1)− 2 kB(K) k(K + 1) 2 ψσ kH , ω(k)

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which shows using hypothesis (I2) and (IV.2) that Rσ,1 ∈ o(σ 0 ) in L2 (IRd , dk; H).

(IV.7)

Let us next estimate Rσ,2 . Using the fact that (v − vσ )(k)(K + 1)− 2 belongs to L2 (IRd , dk; H), it is easy to verify that 1

(E − H − ω(k))−1 (H − Hσ )(Eσ − Hσ − ω(k))−1 vσ (k)ψσ = (E − H − ω(k))−1 (a∗ (v − vσ ) + a(v − vσ ))(Eσ − Hσ − ω(k))−1 vσ (k)ψσ . Note that it follows from functional calculus that k(E − H − ω(k))−1 (H + b) 2 k ≤ C sup(ω(k)−1 , ω(k)− 2 ). 1

1

(IV.8)

Using also the fact that (K + 1) 2 (H + b)− 2 is bounded, we have: 1

1

k(E − H − ω(k))−1 (a∗ (v − vσ ) + a(v − vσ ))(Eσ − Hσ − ω(k))−1 vσ (k)ψσ k ≤ C sup(ω(k)−1 , ω(k)− 2 )k(K + 1)− 2 (a∗ (v − vσ ) + a(v − vσ )) 1

1

(Eσ − Hσ − ω(k))−1 vσ (k)ψσ k ≤ C sup(ω(k)−1 , ω(k)− 2 ) 1

R

v(k)k2 dk

kv(k)(K + 1)− 2 k2 + k(K + 1)− 2 1

{ω(k)≤σ}

12

1

× k(N + 1) 2 (Eσ − Hσ − ω(k))−1 vσ (k)ψσ k, 1

applying the estimates (A.2), (A.3) in Lemma A.1 to B = 1l, m = 1, v(k) = 1 (K + 1)− 2 (v − vσ )(k). To bound k(N + 1) 2 (Eσ − Hσ − ω(k))−1 vσ (k)ψσ k, we write using again the pullthrough formula (IV.1): 1

a(k0 )(Eσ − Hσ − ω(k))−1 vσ (k)ψσ = (Eσ − Hσ − ω(k) − ω(k0 ))−1 a(k0 )vσ (k)ψσ +(Eσ − Hσ − ω(k0 ))−1 vσ (k0 )(Eσ − Hσ − ω(k))−1 vσ (k)ψσ = (Eσ − Hσ − ω(k) − ω(k0 ))−1 vσ (k)(Eσ − Hσ − ω(k0 ))−1 vσ (k0 )ψσ +(Eσ − Hσ − ω(k0 ))−1 vσ (k0 )(Eσ − Hσ − ω(k))−1 vσ (k)ψσ .

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This gives kN 2 (Eσ − Hσ − ω(k))−1 vσ (k)ψσ k2 R = ka(k0 )(Eσ − Hσ − ω(k))−1 vσ (k)ψσ k2 dk0 R ≤ 2 k(Eσ − Hσ − ω(k) − ω(k0 ))−1 vσ (k)(Eσ − Hσ − ω(k0 ))−1 vσ (k0 )ψσ k2 dk0 R +2 k(Eσ − Hσ − ω(k0 ))−1 vσ (k0 )(Eσ − Hσ − ω(k))−1 vσ (k)ψσ k2 dk0 R 1 1 − 12 2 ≤ C ω(k) k k(K + 1) 2 (Eσ − Hσ − ω(k0 ))−1 k2 × 2 kvσ (k)(K + 1) 1

kvσ (k0 )(K + 1)− 2 k2 k(K + 1) 2 ψσ k2 dk0 R 1 1 +C ω(k10 )2 kvσ (k0 )(K + 1)− 2 k2 k(K + 1) 2 (Eσ − Hσ − ω(k))−1 k2 × 1

1

kvσ (k)(K + 1)− 2 k2 k(K + 1) 2 ψσ k2 dk0 . 1

1

We use the bound (IV.8) and we obtain kN 2 (Eσ − Hσ − ω(k))−1 vσ (k)ψσ k2 1

≤ C(sup(ω(k)−1 , ω(k)− 2 ))2 kvσ (k)(K + 1)− 2 k2 × R 1 1 (sup(ω(k0 )−1 , ω(k0 )− 2 ))2 kvσ (k0 )(K + 1)− 2 k2 dk0 × 1

1

1

k(K + 1) 2 ψσ k2 ≤ C(sup(ω(k)−1 , ω(k)− 2 ))2 kvσ (k)(K + 1)− 2 k2 , 1

1

using (IV.2) and hypothesis (I2). Hence kRσ,2 (k)kH ≤ R 1 1 1 1 C(sup(ω(k)−1 , ω(k)− 2 ))2 kvσ (k)(K + 1)− 2 k( {ω(k)≤σ} k(K + 1)− 2 v(k)k2 dk) 2 . By (I2),

Z ( {ω(k)≤σ}

k(K + 1)− 2 v(k)k2 dk) 2 ∈ o(σ), 1

1

and since suppvσ ⊂ {ω(k) ≥ σ}, we obtain kRσ,2 (k)k ≤ o(σ 0 ) sup(ω(k)−1 , ω(k)− 2 )kv(k)(K + 1)− 2 k. 1

1

(IV.9)

Finally using Lemma IV.3, (IV.2) and the fact that suppvσ ⊂ {ω(k) ≥ σ}, we obtain kR3,σ (k)k ≤ o(σ 0 ) sup(ω(k)−1 , ω(k)− 2 )kv(k)(K + 1)− 2 k. 1

1

(IV.10)

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Combining (IV.7), (IV.9), (IV.10) and using (I2) we obtain the proposition. 2 As a consequence of Prop. IV.4, we have the following lemma, which is the main part of the proof of Thm. 1. We recall that x := i∇k is the position operator on L2 (IRd , dk). Lemma IV.5 Let F ∈ C0∞ (IR) be a cutoff function with 0 ≤ F ≤ 1, F (s) = 1 for |s| ≤ 12 , F (s) = 0 for |s| ≥ 1. Let FR (x) = F ( |x| R ). Then lim

σ→0,R→+∞

(ψσ , dΓ(1 − FR )ψσ ) = 0.

(IV.11)

Proof. Recall that if B is a bounded operator on h with distribution kernel b(k, k 0 ), we have Z Z 1 (u, dΓ(B)u) = b(k, k 0 )(a(k)u, a(k 0 )u)dkdk0 , u ∈ D(N 2 ). Using this identity, we obtain (ψσ , dΓ(1 − FR )ψσ ) = (a(·)ψσ , (1 − F (

|Dk | ))a(·)ψσ )L2 (IRd ,dk;H) . R

By Prop. IV.4, we have (ψσ , dΓ(1 − FR )ψσ ) = ((E − H − ω(·))−1 v(·)ψσ , (1 − F ( |DRk | ))(E − H − ω(·))−1 v(·)ψσ ) + o(σ 0 ), uniformly in R. This yields (ψσ , dΓ(1 − FR )ψσ ) ≤ k(E − H − ω(·))−1 v(·)kL2 (IRd ,dk,B(H)) × k(1 − F ( |DRk | ))(E − H − ω(·))−1 v(·)kL2 (IRd ,dk,B(H)) + o(σ 0 ). Now it follows from hypothesis (I2) and (IV.8) that (E − H − ω(·))−1 v(·) belongs to L2 (IRd , dk, B(H)), and hence k(1 − F (

|Dk | ))(E − H − ω(·))−1 v(·)kL2 (IRd ,dk,B(H)) ∈ o(R0 ). R

This proves (IV.11). 2 We can now prove Theorem 1. Proof of Theorem 1. Let us first recall the a priori bounds on the family of ground states {ψσ }. From (IV.2), (IV.3), we have 1

1

kN 2 ψσ k ≤ C, kH02 ψσ k ≤ C, uniformly in σ.

(IV.12)

Let also F be a cutoff function as in Lemma IV.5. Then it is easy to verify, using the fact that 0 ≤ F ≤ 1, that (1 − Γ(FR ))2 ≤ (1 − Γ(FR )) ≤ dΓ(1 − FR ).

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Using Lemma IV.5, we obtain lim

σ→0,R→∞

k1 − Γ(FR )ψσ k = 0.

(IV.13)

Let us denote by χ(s ≤ s0 ) a cutoff function supported in {|s| ≤ s0 }, equal to 1 in {|s| ≤ s0 /2}. Since the unit ball in H is compact for the weak topology, there exist a sequence σn → 0 and a vector ψ ∈ H such that ψσn tends weakly to ψ. By Lemma A.3 in the Appendix, it suffices to show that ψ 6= 0 to prove the theorem. Assume that ψ = 0. Note using hypotheses (H0), (H1), that for any λ, R the operator χ(N ≤ λ)χ(H0 ≤ λ)Γ(FR ) is compact on H. Then lim χ(N ≤ λ)χ(H0 ≤ λ)Γ(FR )ψσn = 0,

n→∞

(IV.14)

for any λ, R. By (IV.13), we can pick R large enough such that for n ≥ n0 k(1 − Γ(FR ))ψσn k ≤ 10−2 . −1

(IV.15)

1

Since (1 − χ(s ≤ s0 )) ≤ s0 2 s 2 , we can using (IV.12) pick λ large enough such that k(1 − χ(N ≤ λ))ψσn k ≤ 10−2 , k(1 − χ(H0 ≤ λ))ψσn k ≤ 10−2 .

(IV.16)

But (IV.15), (IV.16) and (IV.14) imply that for n large enough kψσn k ≤ 10−1 which is a contradiction. Hence ψ 6= 0 and the theorem is proved.

A Appendix We use the notations of Sect. II. The following lemma is well known if the coupling function v(k) is of the form vλ(k) for v a fixed linear operator on K and k 7→ λ(k) a scalar function. In our general setting it seems not to be in the literature. Let us first recall some terminology and results about measurability of vector and operator-valued functions. Let K be a Hilbert space. A map k 7→ ψ(k) ∈ K is said measurable if it is measurable if we equip K with the norm topology. Let now IRd 3 k 7→ T (k) ∈ B(K) be defined for a.e. k. The map k 7→ T (k) is said weakly measurable if for all ψ1 , ψ2 ∈ K the map k 7→ (ψ2 , T (k)ψ1 ) is measurable. If K is separable the following facts are true (see eg [Di, Chap. II §2]): i) the function k 7→ kT (k)k is measurable, ii) for any k 7→ ψ(k) ∈ K measurable, the function k 7→ T (k)ψ(k) is measurable. In particular for ψ ∈ K the function k 7→ T (k)ψ is measurable. These facts will be used in the proof of Lemma A.1 below. Lemma A.1 Let B ≥ 0 be a selfadjoint operator on the separable Hilbert space K, 1 v : IRd 3 k 7→ v(k) a function such that for a.e. k ∈ IRd , v(k)(B + 1)− 2 ∈ B(K),

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IRd 3 k 7→ v(k)(B + 1)− 2 ∈ B(K) is weakly measurable and m : IRd 3 k 7→ m(k) ∈ IR+ be a measurable function. Then Z 1 1 | (v(k)u, a(k)u)dk| ≤ C(R)(u, dΓ(m)u) 2 (u, (B + R)u) 2 , (A.1) 1

for

Z C(R) = (

1 1 1 kv(k)(B + R)− 2 k2 dk) 2 . m(k)

If moreover for a.e. k ∈ IRd , (B+1)− 2 v(k) ∈ B(K) and IRd 3 k 7→ (B+1)− 2 v(k) ∈ B(K) is weakly measurable, then Z 1 1 k v ∗ (k) ⊗ a(k)u dkk ≤ C1 (R)k(B + R) 2 ⊗ dΓ(m) 2 uk, (A.2) 1

for

Z C1 (R) = (

and

Z k

for

1

1 1 1 k(B + R)− 2 v(k)k2 dk) 2 , m(k)

v(k) ⊗ a∗ (k)u dkk ≤ C2 (R)k(B + R) 2 ⊗ dΓ(m) 2 uk + C3 (R)kuk, 1

1

(A.3)

Z C2 (R) = (

1 1 1 kv(k)(B + R)− 2 k2 dk) 2 , m(k) Z 1 1 C3 (R) = ( kv(k)(B + R)− 2 k2 dk) 2 .

Proof. The estimate (A.1) follows directly from Cauchy-Schwarz inequality. (We 1 1 use the fact that for u ∈ K ⊗ D(N 2 ) ∩ D(dΓ(m) 2 ) the map k 7→ a(k)u ∈ H is measurable). To prove (A.2), we consider the operator wR : K 3 u 7→ wR (k)u := m(k)− 2 (B + R)− 2 v(k)u ∈ L2 (IRd , dk; K) = K ⊗ h. 1

1

∗ kB(K⊗h) ≤ C1 (R)2 . This gives Clearly kwR kB(K,K⊗h) ≤ C1 (R) and hence kwR wR Z Z Z ∗ ∗ (k)ψ(k), wR (k0 )ψ(k 0 ))K dkdk0 | ≤ C1 (R)2 kψ(k)k2K dk, (A.4) | (wR

for ψ ∈ L2 (IRd , dk; K). The bound (A.4) still holds for ψ ∈ L2 (IRd , dk; H) if we replace the scalar product (., .)K by the scalar product (., .)H . We have: R ka(v)uk2 = k v ∗ (k)a(k)u dkk2 RR ∗ = (v (k)a(k)u, v ∗ (k0 )a(k 0 )u)H dkdk0 RR ∗ ∗ = (wR (k)ψ(k), wR (k0 )ψ(k 0 ))H dkdk,0

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1

for ψ(k) = m(k) 2 a(k)(B + R) 2 u. Using (A.4) we obtain R 1 ka(v)uk2 ≤ C1 (R)2 ω(k)ka(k)(B + R) 2 uk2 dk 1

1

= C1 (R)2 k(B + R) 2 ⊗ dΓ(m) 2 uk2 . This proves (A.2). Similarly, introducing the operator ˜R (k)u = m(k)− 2 v(k)(B + R)− 2 ∈ L2 (IRd , dk; K) = K ⊗ h, w ˜R : K 3 u 7→ w 1

1

∗ we have kw ˜R kB(K,K⊗h) ≤ C2 (R) and hence kw ˜R w ˜R kB(K) ≤ C2 (R)2 . This yields Z ∗ k w ˜R (k)w ˜R (k)dkkB(K) ≤ C2 (R)2 . (A.5)

(The integral in (A.5) should be considered in the weak sense on B(K), ie as a quadratic form on K). We have RR (v(k)a∗ (k)u, v(k 0 )a∗ (k0 )u)H dkdk0 ka∗ (v)uk2 = RR = (v(k)a(k 0 )u, v(k0 )a(k)u)H dkdk0 R + (v(k)u, v(k)u)dk. The second term in the r.h.s. is bounded by R 1 1 kv(k)(B + R)− 2 k2 k(B + R) 2 uk2 dk 1

≤ C32 (R)k(B + R) 2 uk2 . We write then the first term as RR (w ˜R (k)ψ(k 0 ), w ˜R (k0 )ψ(k))H dkdk0 RR ≤ kw ˜R (k)ψ(k 0 )k2H dkdk0 R ∗ R ≤k w ˜R (k)w ˜R (k)dkk kψ(k 0 )k2H dk0 1

1

≤ C2 (R)2 k(B + R) 2 ⊗ dΓ(m) 2 uk2 , 2

which proves (A.3).

Lemma A.2 Let Q be a closed, positive quadratic form, Qn be closed quadratic forms on D(Q) such that Qn converges to Q when n → +∞ in the topology of D(Q). Let H, Hn be the associated selfadjoint operators. Then for z in a bounded set U ⊂ C\IR , we have: k((H − z)−1 − (Hn − z)−1 )(H + 1)− 2 k ∈ o(1)|Imz|−1 , when n → +∞. 1

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and for λ ∈ IR, λ −1 k((H − λ)−1 − (Hn − λ)−1 )(H + 1)− 2 k ∈ o(1) when n → +∞. 1

Proof. Let for z ∈ C, u ∈ H, Rn (z) = (Hn − z)−1 , R(z) = (H − z)−1 , r = Rn (z)u − R(z)u. We have for v ∈ D(Q): (v, u)

= Q(v, R(z)u) − z(v, R(z)u) = Qn (v, Rn (z)u) − z(v, Rn (z)u).

Hence for v = r we obtain Q(r, R(z)u) − Qn (r, Rn (z)u) + zkrk2 = 0, or Q(r, r) − zkrk2 = (Q − Qn )(r, R(z)u).

(A.6)

If λ ∈ IR, λ −1, we deduce from (A.6) that 1

1

(Q + 1)(r, r) ∈ o(1)(Q + 1)(r, r) 2 (Q + 1)(R(λ)u, R(λ)u) 2 . This implies that (Q + 1)(r, r) is o(1)kuk, as claimed. Let now z ∈ U ⊂ C\IR. Taking the imaginary part of (A.6) we obtain krk2

∈ o(1)|Imz|−1 (Q + 1)(r, r) 2 (Q + 1)(R(z)u, R(z)u) 2 1

1

∈ o(1)|Imz|−2 (Q + 1)(r, r) 2 kuk2 , 1

since (Q + 1)(R(z)u, R(z)u) is bounded by |Imz|−2 kuk2 for z ∈ U . Taking then the real part of (A.6) we obtain |Q(r, r)|

∈ o(1)(Q + 1)(r, r) 2 (Q + 1)(R(z)u, R(z)u) 2 + o(1)|Imz|−2 1 (Q + 1)(r, r) 2 kuk2 1 ∈ o(1)|Imz|−2 (Q + 1) 2 (r, r)kuk2 . 1

1

This implies that (Q + 1)(r, r) 2 ∈ o(1)|Imz|−1 kuk as claimed. The following result is shown in [AH, Lemma 4.9] 1

2

Lemma A.3 Let H, Hn for n ∈ IN be selfadjoint operators on a Hilbert space H. Let ψn be a normalized eigenvector of Hn with eigenvalue En . Assume that i) Hn → H when n → ∞ in strong resolvent sense, ii) limn→∞ En = E, iii) w− limn→∞ ψn = ψ 6= 0. Then ψ is an eigenvector of H with eigenvalue E.

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Acknowledgement We would like to thank J.M. Bony, J. Fr¨ ohlich and G.M. Graf for useful discussions.

References [AH] Arai, A., Hirokawa, M.: On the existence and uniqueness of ground states of a generalized spin-boson model. J. Funct.Anal. 151 (1997) 455–503. [BFS] Bach, V., Fr¨ ohlich, J., Sigal, I.: Quantum electrodynamics of confined nonrelativistic particles, Sonderforschungsbereich 288 preprint, 1996. [Be]

Berezin, F.A.: The Method of Second Quantization, Academic Press New York San Francisco London, 1966.

[DG] Derezi´ nski, J., G´erard, C.: Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians, Rev. Math. Phys. 11 (4) (1999) p383–450. [Di]

Dixmier, J.: Les alg`ebres d’op´erateurs dans l’espace hilbertien, GauthierVillars, Paris, 1957.

[Fr]

Fr¨ohlich, J.: On the infrared problem in a model of scalar electrons and massless scalar bosons, Ann. Inst. Henri Poincar´e A19 (1973) 1–103.

[Sp]

Spohn, H.: Ground state of a quantum particle coupled to a scalar Bose field. Lett. Math. Phys. 44 (1998), no. 1, 9–16.

C. G´erard Centre de Math´ematiques URA 7640 CNRS Ecole Polytechnique 91128 Palaiseau Cedex FRANCE e-mail : [email protected] Communicated by Gian Michele Graf submitted 25/06/99, accepted 14/12/99

Ann. Henri Poincar´ e 1 (2000) 461 – 472 c Birkh¨

auser Verlag, Basel, 2000 1424-0637/00/030461-12 $ 1.50+0.20/0

Annales Henri Poincar´ e

Long Time Behavior of Solutions to Nernst-Planck and Debye-H¨ uckel Drift-Diffusion Systems Piotr Biler and Jean Dolbeault Abstract.We study the convergence rates of solutions to drift-diffusion systems (arising from plasma, semiconductors and electrolytes theories) to their self-similar or steady states. This analysis involves entropy-type Lyapunov functionals and logarithmic Sobolev inequalities.

1 Introduction We consider the long time asymptotics of solutions to drift-diffusion systems ut = ∇ · (∇u + u∇φ) ,

(1.1)

vt = ∇ · (∇v − v∇φ) ,

(1.2)

∆φ = v − u ,

(1.3)

where u, v denote densities of negatively, respectively positively, charged particles. The Poisson equation (1.3) defines the electric potential φ coupling the equations (1.1)-(1.2) for the temporal evolution of charge distributions. The system (1.1)(1.3) was formulated by W. Nernst and M. Planck at the end of the nineteenth century as a basic model for electrodiffusion of ions in electrolytes filling the whole space IR3 . Note that the case of multicharged particles is also covered by (1.1)-(1.3) since u and v denote the charge densities. Supplemented with the no-flux boundary conditions ∂φ ∂u +u =0, ∂ν ∂ν

(1.4)

∂φ ∂v −v =0 ∂ν ∂ν

(1.5)

on the boundary of a bounded domain Ω ⊂ IRd , d ≤ 3, and either φ = 0 on ∂Ω ,

(1.6)

φ = Ed ∗ (v − u) ,

(1.7)

or where Ed is the fundamental solution of the Laplacian in IRd , the system (1.1)(1.3) was also studied by P. Debye and E. H¨ uckel in the 1920’s. (1.6) signifies a

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conducting boundary of the container, while in the case of a bounded domain the “free” boundary condition (1.7) corresponds to a container immersed in a medium with the same dielectric constant as the solute. These equations, together with their generalizations including e.g. an exterior potential, known as drift-diffusion Poisson systems, also appear in plasma physics and (supplemented with some mixed linear boundary conditions instead of (1.4)(1.5)) in semiconductor device modelling. To determine completely the evolution, the initial conditions u(x,0) = u0 (x) ,

v(x,0) = v0 (x)

(1.8)

are added. Obviously, positivity of u0 ≥ 0, v0 ≥ 0 is conserved: u(x,t) ≥ 0, v(x,t) ≥ 0, as well as the total charges Z Z Z Z (1.9) Mu = u0 (x) dx = u(x,t) dx , Mv = v0 (x) dx = v(x,t) dx . Here Mu , Mv are not necessarily the same, i.e. the electroneutrality condition Mu = Mv

(1.10)

is not, in general, required. Condition (1.10) must be satisfied in the case of the homogeneous Neumann boundary conditions ∂φ ∂ν = 0 (i.e. an isolated wall of the container) leading together with (1.4)-(1.5) to ∂u ∂v ∂φ = = =0. ∂ν ∂ν ∂ν Our results (Theorem 1.2 below) are valid in that case, with even a simpler proof. The asymptotic properties of solutions to (1.1)-(1.3), (1.7) have been studied recently in [1]. The authors proved that (for d ≥ 3, Mu = Mv = 1 and u0 and v0 regular enough) u, v tend to their self-similar asymptotic states at an algebraic rate. We improve these results by relaxing assumptions on the initial data and showing a stronger (still algebraic) decay rate, which we expect to be optimal (see Theorem 1.1 below). In the case of a bounded domain, the convergence (with no specific speed) in the L1 -norm of u and v solving (1.1)-(1.5) to their corresponding steady states has been proved in [5] (as well as the L∞ -convergence for more regular u0 , v0 ). Here we prove the exponential convergence towards the steady states with a decay rate depending on Ω ⊂ IRd , d ≥ 2, and the initial value of the entropy functional only (see Theorem 1.2 below). Notation. The Lp -norm in IRd or Ω ⊂⊂ IRd is denoted by | · |p , and inessential constants (which may vary from line to line) are denoted generically by C.

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Define the asymptotic states in IRd by Mu |x|2 exp − , uas (x,t) = 2(2t + 1) (2π(2t + 1))d/2 Mv |x|2 exp − , vas (x,t) = 2(2t + 1) (2π(2t + 1))d/2

(1.11)

(1.12)

where the charges of the solution hu,vi of (1.1)-(1.2) are given by (1.9), and the entropy functional by Z Z u(x,t) v(x,t) 1 L(t) = u(x,t)log dx + v(x,t)log dx + |∇φ(t)|22 . uas (x,t) vas (x,t) 2 (1.13) Theorem 1.1 There exists a constant C = C(d,Mu ,Mv ,L0 ) such that for each solution hu,vi of (1.1)-(1.3), (1.7)-(1.8) in IRd , d ≥ 3, if L(0) = L0 , then for all t ≥ 0, L(t) ≤ C H(t)

(1.14)

|u(t) − uas (t)|21 + |v(t) − vas (t)|21 + |∇φ(t)|22 ≤ C H(t) ,

(1.15)

and    

where H(t) =

  

(2t+ 1)−1/2 , −1 (2t + 1) log(2t + 1) + 1 ,

d=4,

(2t + 1)−1 ,

d>4.

−1

Moreover if Mu = Mv , then H(t) = (2t + 1)

d=3,

for any d ≥ 3.

In the case of a bounded domain, define the entropy functional Z Z W (t) = u(x,t)logu(x,t) dx − U (x)logU (x) dx Z Z + v(x,t)logv(x,t) dx − V (x)logV (x) dx Z Z 1 1 (u − v)φ dx − (U − V )Φ dx , + 2 2

(1.16)

for the solution hu,v,φi of (1.1)-(1.5), (1.6) or (1.7), (1.8) and the unique steady state hU,V,Φi of the Debye-H¨ uckel system with Z Z Mu = U (x) dx , Mv = V (x) dx . (1.17) Note that for the condition (1.6) the fifth and the sixth terms in W (t) take the form 12 |∇φ|22 − 12 |∇Φ|22 .

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Theorem 1.2 If d ≥ 2, then there exist two constants λ = λ(Ω) > 0 and C = C(Mu ,Mv ,W0 ) such that for each solution hu,v,φi of (1.1)-(1.6), (1.8) in a bounded uniformly convex domain Ω, if W (0) = W0 , then for all t ≥ 0, W (t) ≤ W (0) e−λt ,

(1.18)

|u(t) − U |21 + |v(t) − V |21 + |∇(φ − Φ)|22 ≤ C e−λt .

(1.19)

and

2 Proof of Theorem 1.1 We begin with a rescaling of the system (1.1)-(1.3) which will lead to a system with a quadratic confinement potential, and therefore (eliminating the dispersion) to the expected exponential convergence to the steady states. This idea was applied in [8] and [7], as well as in [1], to a variety of problems ranging from kinetic equations to porous media equations. Let x ¯ ∈ IRd , τ > 0, be the new variables defined by x ¯=

x , R(t)

τ = logR(t) ,

R(t) = (2t + 1)1/2 ,

(2.1)

and consider the rescaled functions u ¯, v¯, φ¯ such that 1 u ¯ (¯ x,τ ) , Rd (t) 1 x,τ ) , v(x,t) = d v¯ (¯ R (t) ¯ x,τ ) . φ(x,t) = φ(¯ u(x,t) =

(2.2)

This whole section will deal with the rescaled system, so omitting the bars over x, u, v, φ will not lead to confusions with the original system, which now takes, after rescaling, the form uτ = ∇ · (∇u + ux + u∇φ) , (2.3) vτ = ∇ · (∇v + vx − v∇φ) ,

(2.4)

∆φ = e−τ (d−2) (v − u) .

(2.5)

The scaling (2.2) preserves the L1 -norms, so the rescaled initial data u0 , v0 still satisfy Z Z Z Z Mu = u0 (x) dx = u(x,τ ) dx , Mv = v0 (x) dx = v(x,τ ) dx . (2.6)

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Denote by hu∞ ,v∞ i the steady state of (2.3)-(2.4), that is Mu |x|2 u∞ (x) = , exp − 2 (2π)d/2 Mv |x|2 v∞ (x) = exp − . 2 (2π)d/2

465

(2.7) (2.8)

Of course, going back to the original variables x, t, hu∞ ,v∞ i corresponds to the asymptotic state huas ,vas i defined by (1.11)-(1.12). Writing φ = βψ with β = β(τ ) = e−τ (d−2) → 0 as τ → +∞, we introduce the relative entropy Z Z v β u dx + v log dx + |∇ψ|22 (2.9) W (τ ) = ulog u∞ v∞ 2 corresponding to the original entropy functional L in (1.13). The evolution of W is given by Z Z v 2 d u 2 dW = − u ∇ log − 1 β|∇ψ|22 , (2.10) dx − v ∇ log dx − dτ U V 2 with U , V denoting the local Maxwellians exp − 21 |x|2 − φ(x,τ ) , exp − 12 |y|2 − φ(y,τ ) dy exp − 21 |x|2 + φ(x,τ ) R , V (x,τ ) = Mv exp − 12 |y|2 + φ(y,τ ) dy

U (x,τ ) = Mu R

so that ∇U /U = −(x + ∇φ), ∇V /V = −(x − ∇φ). Using the notation 2 2 Z Z ∇u ∇v 1 1 u + x dx + v + x dx , J= 2 u 2 v

(2.11)

(2.12)

(2.13)

(2.10) can be rewritten as Z Z dW = −2J − 2 (∇u − ∇v) · ∇φ dx − 2 (u − v) x · ∇φ dx dτ Z d − (u + v)|∇φ|2 dx − (2.14) − 1 β|∇ψ|22 2 Z d = −2J − β 2 (u + v)|∇ψ|2 dx − 2β|u − v|22 + − 1 β|∇ψ|22 . 2 The quantity J in (2.13) can be estimated from below using the Gross logarithmic Sobolev inequality Z Z 1 a |∇f |2 f dx (2.15) dx + d 1 + log(2πa) |f |1 ≤ f log |f |1 2 2 f

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valid for each a > 0, see e.g. [11] or a thorough discussion of different versions of logarithmic Sobolev inequalities in [2]. (2.15) becomes an equality if and only if f (x) = C exp −|x|2 /(2a) (up to a translation). Taking a = 1 in (2.15), the relation (2.14) leads to d dW + 2W ≥ 2β|u − v|22 − β |∇ψ|22 ≥ −Cβ(Mu + Mv )2 (2.16) − dτ 2 (d−2)/2 d/2 Σ , because by the Hardy-Littlewoodwith a constant C = C(d) = d2 d−2 4 Sobolev inequality and an interpolation 4/d

2−4/d

|∇ψ|22 ≤ Σ|u − v|22d/(d+2) ≤ Σ|u − v|1 |u − v|2

4 ≤ |u − v|22 + C|u − v|21 . d

Clearly, (2.16) implies d 2τ e W (τ ) ≤ C(Mu + Mv )2 eτ (4−d) dτ and, after one integration, we obtain W (τ ) ≤ W (0)e−τ + C(Mu + Mv )2 e−τ in the case d = 3,

(2.17)

W (τ ) ≤ W (0) + C(Mu + Mv ) τ e−2τ

(2.18)

if d = 4, and finally for all d > 4 W (τ ) ≤ W (0) + C(Mu + Mv )2 e−2τ .

(2.19)

2

Since from the Csisz´ ar-Kullback inequality (cf. (1.9) in [2], App. D in [7], [6] or [10]) W (τ ) controls the L1 -norm of u − u∞ and v − v∞ , we get the same decay rates as in (2.17)-(2.19) for |u(τ ) − u∞ |21 + |v(τ ) − v∞ |21 + β|∇ψ(τ )|22 ≤ 2 max(Mu ,Mv ) + 1 W (τ ) . (2.20) Returning to the original variables x, t, this implies, of course, the estimates (1.14)(1.15) of Theorem 1.1 in the general case. In the electroneutrality case (1.10): Mu = Mv , since u∞ = v∞ , so for d = 3, |u − v|21 = |u − u∞ + v∞ − v|21 ≤ Ce−τ . Next, a modification of (2.16) reads d e2τ W (τ ) ≤ Ce2τ β|u − v|21 ≤ C , dτ

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and this leads to W (τ ) ≤ C(1 + τ )e−2τ . Inserting this into (2.20) and (2.16) once again implies d e2τ W (τ ) ≤ C(1 + τ )e−τ , dτ so that W (τ ) ≤ Ce−2τ . If d = 4, the same reasoning once again applies providing also the same improved decay rate. Remark. 2.1 Note that the constant C in (1.15) depends on d, Mu , Mv and L(0) only, and is independent of e.g. |u0 |r , |v0 |r with some r > d/2 — as it was in fact in [1]. Conditions like |u0 |r + |v0 |r < ∞ are sufficient for (local in time) existence of solutions to the considered systems (cf. Theorem 2 in [5]), but they can be relaxed — as it was done for a related parabolic-elliptic system describing the gravitational interaction of particles in [4]. Thus, compared to [1], Theorem 1.1 gives not only an improvement of the exponents but also gets rid of the unnecessary dependence on quantities other than L(0), Mu , Mv . We do not know if the exponents in Theorem 1.1 are optimal, but such a conjecture is supported by the calculations in the proof of the following Proposition p 2.2 There exists aconstant λ > 0 depending only on d with λ ≥ λ(d) = (d − 1)2 + 3 − (d − 1) , such that (d − 2) ≤

W (τ ) and hence L(t)

W (0) e−λτ

(2.21)

L(0) (2t + 1)−λ/2

≤

for each solution hu,vi to the Nernst-Planck system. Remark. 2.3 The interest of this proposition is that the constants controlling the convergence of W (t), L(t), and hence |u − uas |1 , |v − vas |1 in (1.15), depend on the initial values of W (0), L(0) onlyR(and not onR |u|1 = Mu , |v|1 = Mv , which are quantities not comparable with, say, ulogu dx, v logv dx in the whole IRd space case). However, the exponent λ — which is evaluated explicitly — is not as good as the one in Theorem 1.1. Proof of Proposition 2.2. Using (2.9), (2.13), (2.14), we may write for any positive λ Z Z dW v u − − v log + λW = λ J − ulog dτ u∞ v∞ +(2 − λ)J + B + 2E − µF , (2.22) where

Z B =β

2

(u + v)|∇ψ|2 dx ,

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E = β|u − v|22 , d F= − 1 β|∇ψ|22 , 2 λ . µ=1+ d−2 Observe that if we define Z ∇u + x · ∇φ dx , G1 = u u Z

then G1 − G2 =

Z G2 =

v

∇v + x · ∇φ dx , v

Z ∇(u − v) · ∇φ dx +

(u − v) (x · ∇φ) dx = E − F.

Define now √ ∇u +x , f1 = 2 − λ· u u

√ g1 = u∇φ ,

√ √ ∇v f2 = 2 − λ · v +x , v

√ g2 = v∇φ ,

√

a1 = |f1 |2 ,

b1 = |g1 |2 ,

a2 = |f2 |2 ,

b2 = |g2 |2 .

By the Cauchy-Schwarz inequality we have (2 − λ)1/2 |E − F | = (2 − λ)1/2 |G1 − G2 | Z = (f1 g1 − f2 g2 ) dx ≤ a1 b1 + a2 b2 . But 0 ≤ (a1 b2 − a2 b1 )2 = (a21 + a22 )(b21 + b22 ) − (a1 b1 + a2 b2 )2 , q √ q a1 b1 + a2 b2 ≤ 2 (a21 + a22 )/2 b21 + b22 1 1 2 ≤√ (a1 + a22 ) + (b21 + b22 ) 2 2 1 = √ (2 − λ)J + B , 2 and thus

1 (2 − λ)1/2 |E − F | ≤ √ (2 − λ)J + B . 2

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Using (2.22) we get p dW + λW ≥ 2(2 − λ)|E − F | + 2E − µF − dτ p =F · 2(2 − λ)|X − 1| + 2X − µ

469

(2.23)

with X = E/F ≥ 0. For either d ≥ 4 and λ ≤ 2, or d = 3 and λ ≤ 1, we have µ ≤ 2. The right hand side of (2.23) (positive for X ≥ µ/2) equals (for X ≤ µ/2 ≤ 1) p p p 2(2 − λ)(1 − X) + 2X − µ = 2 − 2(2 − λ) X + 2(2 − λ)− µ , so that

p 2(2 − λ) ≥ µ

(2.24)

equivalent guarantees dW dτ + λW ≤ 0, which implies (2.21). The condition (2.24) is √ 7 − 2 < 1, to λ ≤ λ(d). In particular, λ(d) is an increasing function of d, λ(3) = √ λ(4) = 4 3 − 6 and limd→+∞ λ(d) = 32 . Remark. 2.4 In the case of one species of particles, i.e. v ≡ 0 as was in [3] and [4], the result of Proposition 2.2 still holds. Finally, we remark that there is, in general, no hope to have λ > 2 in nontrivial cases. This can be inferred from the formula (2.22), where for each χ > 1, J − R R χ ulog uu∞ dx + v log vv∞ dx could be negative and dominate the other terms (for instance, in the limit Mu , Mv → 0+ ).

3 Proof of Theorem 1.2 First, we recall that steady states hU,V,Φi of (1.1)-(1.3) satisfy the relations ∇ · (e−Φ ∇(eΦ U )) = 0 , hence U = Mu R

e−Φ , e−Φ dx

∇ · (eΦ ∇(e−Φ V )) = 0 , V = Mv R

eΦ . eΦ dx

(3.1)

Together with (1.3) this leads to the Poisson-Boltzmann equation ∆Φ = Mv R

eΦ e−Φ R − M . u eΦ dx e−Φ dx

(3.2)

This equation, supplemented with the Dirichlet boundary condition (1.6) or the free condition (1.7), for every Mu ,Mv ≥ 0, has a unique (weak) solution Φ, see [9] or Proposition 2 in [5] (and this solution is classical whenever ∂Ω is of class C 1+ for some > 0).

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The evolution of the Lyapunov functional defined by (1.16) in the case of the Dirichlet boundary condition (1.6) or in the case (1.7) is given by Z Z dW 2 (3.3) = − u|∇(logu + φ)| dx − v|∇(logv − φ)|2 dx , dt cf. (35) in [5], where the above relation is obtained for weak solutions to the Debye-H¨ uckel system. Concerning the global in time existence of solutions to the Debye-H¨ uckel system with nonlinear boundary conditions (1.4)-(1.5), we note that this was proved for d = 2 only in Theorem 3 of [5]. Thus, in higher dimensions d ≥ 3, we assume that hu(t),v(t)i exists for all t ≥ 0. If equations (1.1)-(1.3) are supplemented with linear type boundary conditions (as it is the case in semiconductor modelling), the assumption u0 , v0 ∈ Lr (Ω) with an exponent r > d/2 (cf. Theorem 2 (ii) in [5] and [1] for the case of the whole space IRd ) guarantees the existence of hu(t),v(t)i for all t ≥ 0. First, we represent the entropy production terms in (3.3) as Z Z Z −φ φ 2 φ φ 2 R e u ∇ log(ue ) dx = ue ∇ log(ue ) dx · e−φ dx, e−φ dx

(3.4)

with an obvious modification for the second term. Note that (1.4)-(1.5) read ∂ ∂ φ = ∂ν ve−φ = 0. Then we recall Remark 3.7 of [2], where counterparts of ∂ν ue the logarithmic Sobolev inequality (2.15) (or Poincar´e-type inequalities) are discussed in the case of a bounded uniformly convex domain. We apply this remark to the domain Ω and the probability measure ρ0 (x) = R

e−φ e−φ dx

in the first entropy production term in (3.3) written as in (3.4). This implies the existence of a constant C(Ω) > 0 such that Z Z f |∇f |2 f R dx, dρ0 ≤ C(Ω) Ψ00 R Ψ R f dρ0 f dρ0 ( f dρ0 )2 where Ψ(s) = 1 − s + slogs and f = ueφ . Here we have Z Z |∇f |2 φ 2 00 R f R dx u ∇ log(ue ) dx = Mu Ψ f dρ0 ( f dρ0 )2 R R u since f dρ0 = ueφ dρ0 = R M . Thus we arrive at −φ e

Z

dx

2 Mu u ∇ log(ueφ ) dx ≥ C(Ω)

Z

R

f f f log R +1− R dρ0 , f dρ0 f dρ0 f dρ0

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Long Time Behavior of Solutions to Drift-Diffusion Systems

Z

2 u ∇ log(ueφ ) dx ≥

1 C(Ω)

Similarly, we have Z 2 v ∇ log(ve−φ ) dx ≥ Now we compute the expression  Z ueφ δ = ulog  Mu R

Z

1 C(Ω)

  dx +

e−φ dx

Z



 ulog 

Z

471

φ

ue R

Mu e−φ dx

 v log 

−φ

 dx.

(3.5)



ve  dx. R Mv

(3.6)

eφ dx



 −φ ve v log  Mv  dx. R eφ dx

If hU,V,Φi is the solution of the Poisson-Boltzmann equation (3.2) with the homogeneous Dirichlet boundary conditions for Φ, then it can be checked that δ = W + J[φ] − J[Φ], where

Z

Z

1 2

(3.7) Z

|∇φ|2 dx Z Z Z 1 |∇Φ|2 dx − U logU dx − V logV dx − 2

W =

is as in (1.16), and J[φ] =

1 2

ulogu dx +

v logv dx +

Z

Z |∇φ|2 dx + Mu log

Z e−φ dx + Mv log eφ dx

is a strictly convex functional reaching its minimum at Φ. dW dt

Now it is clear from (3.3), (3.5)-(3.6) and (3.7) that for some λ = λ(Ω) > 0 + λW ≤ 0, i.e. W (t) decays exponentially in t W (t) ≤ W (0) e−λt .

(3.8)

By the Csisz´ar-Kullback inequality (as was in Section 2), W (t) controls the L1 convergence to the unique steady state, so the conclusion (1.19) of Theorem 1.2 follows from (3.8). This improves (34) in Theorem 6 of [5] in two ways. First, there is an exponential decay rate. Second, (34) is proved under the assumption W (0) < ∞, which is much weaker than the assumption on the L2 -boundedness in time of the solution hu,vi in Theorem 6 of [5]. Evidently, this result is also valid for one species case (Mu or Mv equal to 0), so Theorem 2 in [3] is also improved. Acknowledgements. This research was partially supported by the grants POLONIUM 98111 and KBN 324/P03/97/12. The second author thanks the program on Charged Particle Kinetics at the Erwin Schr¨ odinger Institute and the TMR Kinetic Equations for partial support too.

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References [1] A. Arnold, P. Markowich, G. Toscani, On large time asymptotics for driftdiffusion Poisson systems, preprint ESI no. 655 (1999), 1–11. [2] A. Arnold, P. Markowich, G. Toscani, A. Unterreiter, On logarithmic Sobolev inequalities, Csisz´ ar-Kullback inequalities, and the rate of convergence to equilibrium for Fokker-Planck type equations, preprint TMR “Asymptotic Methods in Kinetic Theory” 12 (1998), 1–77. [3] P. Biler, Existence and asymptotics of solutions for a parabolic-elliptic system with nonlinear no-flux boundary conditions, Nonlinear Analysis T.M.A. 19 (1992), 1121–1136. [4] P. Biler, The Cauchy problem and self-similar solutions for a nonlinear parabolic equation, Studia Math. 114 (1995), 181–205. [5] P. Biler, W. Hebisch, T. Nadzieja, The Debye system: existence and large time behavior of solutions, Nonlinear Analysis T.M.A. 23 (1994), 1189–1209. [6] I. Csisz´ar, Information-type measures of difference of probability distributions and indirect observations, Studia Sci. Math. Hungar. 2 (1967), 299–318. [7] J. Dolbeault, M. del Pino, Generalized Sobolev inequalities and asymptotic behaviour in fast diffusion and porous medium problems, preprint Ceremade no. 9905 / ESI no. 704 (1999), 1–45. [8] J. Dolbeault, G. Rein, Time-dependent rescalings and Lyapunov functionals for the Vlasov-Poisson and Euler-Poisson systems, and for related models of kinetic equations, fluid dynamics and quantum physics, preprint Ceremade no. 9945 / TMR “Asymptotic Methods in Kinetic Theory” no. 102 (1999), 1–34, to appear in Math. Mod. Appl. Sci. [9] D. Gogny, P.-L. Lions, Sur les ´etats d’´equilibre pour les densit´es ´electroniques dans les plasmas, RAIRO Mod´el. Math. Anal. Num´er. 23 (1989), 137–153. [10] S. Kullback, A lower bound for discrimination information in terms of variation, IEEE Trans. Information Theory 13 (1967), 126–127. [11] G. Toscani, Sur l’in´egalit´e logarithmique de Sobolev,C. R. Acad. Sci. Paris S´er. I Math. 324 (1997), 689–694. Piotr Biler Mathematical Institute University of Wroclaw pl. Grunwaldzki 2/4 50-384 Wroclaw, Poland e-mail : [email protected] Communicated by J. Bellissard submitted 19/03/99 ; accepted 07/09/99

Jean Dolbeault Ceremade, U.M.R. C.N.R.S. no. 7534 Universit Paris IX-Dauphine Pl. du Marchal de Lattre de Tassigny F-75775 Paris Cedex 16, France e-mail : [email protected]

Ann. Henri Poincar´ e 1 (2000) 473 – 498 c Birkh¨

auser Verlag, Basel, 2000 1424-0637/00/030473-26 $ 1.50+0.20/0

Annales Henri Poincar´ e

Simple Examples of Lifschitz Tails in Gaussian Random Magnetic Fields Naomasa Ueki

Abstract. This is a first attempt to investigate the asymptotic behavior of the integrated density of states at the infimum of the spectrum for Schr¨ odinger operators with magnetic fields which are Gaussian random fields. In simple examples it is shown that the integrated density of states decays exponentially. These examples shall give a hint to consider in more general framework. R´esum´e Ceci est un premier essai pour rechercher le comportement asymptotique, a la borne inf´erieure du spectre, de la densit´ ` e d’´ etat int´egr´ ee pour l’op´ erateur de Schr¨ odinger avec champ magn´ etique qui est un champ al´ eatoire gaussiens. Sur des exemples simples nous allons montrer que la densit´ e d’´ etat int´egr´ ee d´ ecroˆıt exponentiellement. Ces exemples donneront une indication pour consid´erer des cas plus g´ en´ eraux.

1 Introduction This is a first attempt to consider the asymptotic behavior of the integrated density of states N (λ) of the Schr¨ odinger operator L(bω ) with the magnetic field dbω at the infimum of the spectrum, where

L(bω ) =

2 2 1X ∂ i j − bjω (x) 2 j=1 ∂x

(i =

√ −1)

(1.1)

P and bω = 2j=1 bjω (x)dxj (ω ∈ Ω, x ∈ R2 ) is a 1-form valued Gaussian random field on a probability space Ω that is stationary and ergodic with respect to the shift in the variable x ∈ R2 . We show that N (λ) decays exponentially in the following two cases: Case I. b1ω (x) = 0 and b2ω (x) = gω (x1 ) depends only on the first coordinate. Case II. bω (x) = d∗ fω (x), where fω (x) is a scalar valued Gaussian random field. The main results are stated in Theorems 3.1 and 4.1 below. These simple examples shall give a hint to consider in more general framework. The asymptotics of the integrated density of states has been studied mainly

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for the operator 1X 2 j=1 d

H(Vω ) = −

∂ ∂xj

2 + Vω (x)

(1.2)

on Rd , with a real valued random field Vω , as other spectral properties. For this aspect, we refer Carmona-Lacroix [4] and Pastur-Figotin [21]. In particular, the exponential decay of the integrated density of states conjectured by Lifschitz [16], which is governed by both the kinetic energy and the potential energy, was first proved by the large deviation theory of Donsker and Varadhan in the case that Z Vω (x) = u(x − y)µω (dy), (1.3) where µω (dy) is a Poisson random measure and u is a nonnegative integrable function on Rd such that u(x) = o(|x|−d−2 ) as |x| → ∞ [6], [20]. For the operator 2 d ∂ 1X j i − bω (x) + Vω (x), 2 j=1 ∂xj Matsumoto [18] and the author [28] studied simple cases where the result is not affected by the magnetic field. The most studied case is that dbω is a deterministic constant and Vω is as in 1.3 [2], [7], [10]. This case is related with the quantum Hall effect. Broderix, Hundertmark, Kirsch and Leschke [2] showed that the integrated density of states decays exponentially and determined the leading term which are governed solely by Vω , when u decays polynomially. Erd¨os [7] showed that the integrated density of states decays polynomially and determined the leading order which is governed also by the magnetic field when u is of compact support. Hupfer, Leschke and Warzel [10] studied the case that u decays exponentially and showed that the situation on the dependence of the magnetic field changes when u decays as exp(−C|x|2 ). Form these results the effect of the vector potential is weaker than that of the scalar potential. Therefore our problem in this paper is important mathematically to make clear the effect of the vector potential. On the other hand, the operator with random vector potential having the same form with 1.1 is studied also in the nonlinear electromagnetic field theory (see e.g. [1], [19] and the references therein). In that case, the field bω is a distribution valued random field and the analysis is much harder than ours. Our problem is equivalent, by the Tauberian theory, to investigate the asymptotic behavior as t ↑ ∞ of the Laplace-Stieltjes transform Z ∞ e (t) := N e−tλ dN (λ) 0

2 Z t X 1 w×b j j = E exp i bω (w(s)) ◦ dw (s) w(t) = 0 , 2πt j=1 0

(1.4)

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where E w×b is the expectation with respect to the 2-dimensional Wiener process w(s) = (w1 (s), w2 (s)), s ≥ 0, starting at 0 and the random field bω . The difficult points not appearing for the operator 1.2 are that the right hand side of 1.4 is an oscillatory integral and that the phase functional is given by a stochastic integral. Therefore our problem is one of the problems on the asymptotics of the stochastic oscillatory integrals (cf.[11],[12],[17]). However in Case I these difficulties disappear. In fact in this case the expectation with respect to w2 (s) in the right hand side of 1.4 can be computed easily and our problem is reduced to the same problem for the operator 1.2 on R with Vω (x) =

1 gω (x)2 . 2

(1.5)

We here recall a work by Figotin [8]: he determined the leading term of the asymptotics of the integrated density of states for the operator 1.2 with Z 1 Vω (x) = u(x − y)gω (x)2 dx, (1.6) 2 Rd where u is as in 1.3, by modifying the method by Donsker and Varadhan [6], [20]. By his method, we can obtain a lower estimate of the integrated density of states. However, to obtain an upper estimate, the techniques in [8] are not enough: in [8], the function u(x) plays an important role to make the potential Vω (x) good enough to apply the large deviation theory by Donsker-Varadhan for the occupation measure by the Wiener process in the weak topology (see also [5], [6], [20], [21], [29]). On the other hand, Kirsch and Martinelli give a general upper estimate of the integrated density of states for the operator 1.2 with Vω (x) = v(gω (x)),

(1.7)

where v(x) is a locally bounded positive real function on R, polynomially bounded at infinity, by a functional analytic approach [15]. From their estimate, we can obtain an upper estimate of the integrated density of states. For Case II, we can give some upper and lower bounds which are treated samely as in Case I. The organization of this paper is as follows. In Section 2, we follow Figotin [8] and Kirsch and Martinelli [15] to show that the integrated density of states for the operator H(Vω ) in 1.2 with the potential Vω as that in 1.5 decays exponentially at the infimum of the spectrum. In Section 3, we consider Case I and in Section 4, we consider Case II.

2 Square of Gaussian random fields In this section we follow Figotin [8] (see also Pastur-Figotin [21]) and Kirsch and Martinelli [15] to show that the integrated density of states N (λ) for the operator

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H(Vω ) in 1.2 with the potential 1X j g (x)2 2 j=1 ω q

Vω (x) =

(2.1)

decays exponentially at the infimum of the spectrum, where {(gωj (x))j=1,2,··· ,q , x ∈ Rd } is the Rq -valued stationary ergodic Gaussian random field with mean zero and the covariance βjk (x) = E[gωj (x)gωk (0)], j, k = 1, 2, · · · , q. In the following, we set Z c e−iξ·x βjk (x)dx, βjk (ξ) = Rd

we use the polar coordinate ξ = (ζ, θ), ζ > 0, θ ∈ S d−1 , and we denote the eigenvalues of any nonnegative q × q matrix A by µ1 (A) ≤ µ2 (A) ≤ · · · ≤ µq (A). We introduce the following conditions: for α > d and τ > 0, (P1, α, d) (i) supξ |ζ α βc jk (ξ)| < ∞; ◦

◦

d−1 and some β jk (θ) ∈ R; (ii) ζ α βc jk (ξ) → β jk (θ) as ζ → ∞ for any θ ∈ S

(iii) supx |x|m |βjk (x)| < ∞ for some m > d + 2 + 2d/(α − d); ◦

(iv) vol {θ ∈ S d−1 ; µq (β(θ)) 6= 0} is positive, where vol is the volume measure on S d−1 . (E1, τ, d) (i) There exist σ > 0, 1 ≤ q∗ ≤ q and functions Rj (ξ), 1 ≤ j ≤ q∗ such that b supθ |µj (β(ξ)) exp(σζ τ +Rj (ξ))−1| → 0 as ζ → ∞, where supθ |Rj (ξ)|ζ −τ → b is compact for 0 as ζ → ∞ for any 1 ≤ j ≤ q∗ and the support of µj (β(ξ)) any j > q∗ ; (ii) supx |x|m |βjk (x)| < ∞ for some m > d + 2. (C1, d) (i) The support of βc jk (ξ) is compact; (ii) supx |x|m |βjk (x)| < ∞ for some m > d + 2; (iii) βjj (0) > 0 for some j. Then the result is the following:

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Theorem 2.1 – (i) We assume the condition (P1, α, d) for some α > d. Then for small enough ε > 0, there exist c, c0 > 0 such that −1

−cλ−(α−d+2)/{2(αd

−1)}

≤ log N (λ) ≤ −c0 λ−d/2 log λ−1

for any 0 < λ < ε. (ii) We assume the condition (E1, τ, d) for some τ > 0. Then for small enough ε > 0, there exist c, c0 > 0 such that −cλ−d/2 log(d+τ )/τ λ−1 ≤ log N (λ) ≤ −c0 λ−d/2 log λ−1 for any 0 < λ < ε. (iii) We assume the condition (C1, d). Then for small enough ε > 0, there exist c, c0 > 0 such that −cλ−d/2 log λ−1 ≤ log N (λ) ≤ −c0 λ−d/2 log λ−1 for any 0 < λ < ε.

Remark 2.1 – In our methods, the leading term in Theorem 2.1 (iii) is not determined. In fact, if we assume d = q = 1, β(x) ≥ 0 and the condition (C1, 1), then our method gives √ λ log N (λ) 32 b =: −c1 (β) b lim | supp β| ≥ − log λ−1 25/2 π λ↓0 and

√ Z π λ log N (λ) b lim ≥ − √ β(0) |β(x)|dx =: −c2 (β). λ↓0 log λ−1 3 6 R

For these bounds it holds that b 1 (β) b : β satisfies (C1, 1) and β(x) ≥ 0} = sup{c2 (β)/c

2π √ < 1. 27 3

This theorem is obtained from the corollary of Theorem 2.2 and Theorem 2.3 below. For Theorem 2.2 and its corollary, we follow Figotin [8] (see also PasturFigotin [21]). To state the theorem, we introduce the following condition:

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(H1, d) b (i) supξ Tr[β(ξ)]

Qd

j=1 (1

+ |ξ j |1+γ ) < ∞ for some γ > 0;

(ii) supx |x|m |βjk (x)| < ∞ for some m > d + 2 + 2γ −1 ; (iii) limt→∞ Φ(ct)/Φ(t) = cκ for any c > 0 and some κ ∈ [0, 1), where 1 Φ(t) := (2π)d

Z

b dξ Tr[log(I + tβ(ξ)].

(2.2)

Rd

Let η(t), t > 0, be the solution of η

d/2

(t)Φ

t

η d/2 (t)

t . η(t)

=

(2.3)

Then we have the following: Theorem 2.2 – Under the condition (H1, d), we have η(t) e (t) > −∞. log N t→∞ t

(2.4)

lim

From this theorem, we have the following: Corollary – (i) Under the condition (P1, α, d) for some α > d, we have lim

e (t) log N

(α−d+2)/{α(d+2)/d−d} t→∞ t

> −∞.

(2.5)

(ii) Under the condition (E1, τ, d) for some τ > 0, we have e (t) log N

lim

t→∞ td/(d+2)

log{2(d+τ )}/{τ (d+2)} t

> −∞.

(2.6)

(iii) Under the condition (C1, d), we have lim

e (t) log N

t→∞ td/(d+2)

log2/(d+2) t

> −∞.

(2.7)

Theorem 2.3 is a special case of a theorem obtained by Kirsch and Martinelli [15]. To state the theorem, we introduce the following condition:

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(H2, d) (i) βjj (x) ∈ L1 (Rd ); (ii) βjj (x) is Riemannian approximable as lim ε↓0

X

Z εd βjj (εy) =

y∈Zd

Rd

βjj (x)dx.

Then we have the following: Theorem 2.3 – We assume the condition (H2, d). Then for any ε > 0 and 0 < s < 1, there exist positive constants c and λ0 independent of the random field gω such that r 1 n 21−d/2 π d dd/2 q min1≤j≤q βjj (0) o − ε log N (λ) ≤ cq exp − (2.8) λs B(d + 2)1+d/2 λd/2 for any 0 < λ < λ0 , where Z B = max

1≤j≤q

Rd

. |βjj (x)|dx βjj (0).

In particular we have π d dd/2 λd/2 log N (λ) ≤ − . λ↓0 log λ−1 B(d + 2)1+d/2 2d/2

lim

(2.9)

Proof of Theorem 2.3 – As in (21) in [15], we have N (λ) ≤ k(1 + α)d/2 λd/2 P (λ1 (−∆N Λ + Vω ) < 2λ), where k is a positive constant depending only on the dimension d, Λ = [−L/2, L/2]d , L = π(1+α)−1/2 (1+η)−1/2 (2λ)−1/2 , α is a positive constant specified later, η is an arbitrarily small positive constant and λ1 (−∆N Λ +Vω ) is the least eigenvalue of the operator −∆N + V with the Neumann condition on Λ. We dominate the ω Λ probability as P (λ1 (−∆N Λ + Vω ) < 2λ) ≤

q X

j 2 P (λ1 (−∆N Λ + qgω (x) ) < 2λ).

j=1

We estimate the each probability in the right hand side as in the proof of Theorem 6 in [15]. Then we have |Λ| j 2 P (λ1 (−∆N Gj (α, λ) , Λ + qgω (x) ) < 2λ) ≤ exp − Qj

480

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Ann. Henri Poincar´e

where |Λ| = Ld is the volume, Z |βjj (x)|dx βjj (0), Qj = Rd t t Gj (α, λ) = sup − log(exp P (qgωj (0)2 ≤ (2λ)s ) 1+α α t≥0 2tλ j 2 s + exp P (qgω (0) > (2λ) ) (2λ)s + 2αλ (2λ)s + 2αλ log P (qgωj (0)2 ≤ (2λ)s )−1 = (1 + α)(2λ)s α((2λ)s + 2αλ) + − α(2λ)1−s log P (qgωj (0)2 s (1 + α)(2λ) > (2λ)s )−1 + G0 (α, λ) and G0 (α, λ) is a function depending only on s such that lim λ↓0 G0 (α, λ) > −∞. We estimate as P (qgωj (0)2 > (2λ)s ) ≤ 1 and s 2(2λ)s . P (qgωj (0)2 ≤ (2λ)s ) ≤ qπβjj (0) Then for any ε > 0, there exist λ1 > 0 independent of gω such that r α qβjj (0) − ε log Gj (α, λ) ≥ 1+α λs for any 0 < λ < λ1 . Thus for any ε > 0, there exist λ2 > 0 independent of gω such that N (λ) ≤ k(1 + α)d/2 q exp g −

πd α − ε × B(1 + α)1+d/2 (1 + η)d/2 2d/2 r 1 q minj βjj (0) log λs λd/2

for any 0 < λ < λ2 . By taking the minimum with respect to α, we obtain 2.8. To prove Theorem 2.2, we prepare several lemmas: Lemma 2.1 – The function Φ(t) defined in 2.2 satisfies the following: (i) Φ(t) is well defined as a positive function of t > 0. (ii) Φ is concave and monotone increasing. (iii) Φ(t)/t is monotone decreasing. (iv) for any ε > 0, there exists C(ε) > 0 such that Φ(t) ≤ C(ε)t1/(1+γ)+ε for any t ≥ 1, where γ is the constant in the condition (H1, d) (i).

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Proof – (iv) is proven by (H1, d) (i) and the induction on the dimension d. Lemma 2.2 – There exists a positive function η(t), t > 0 , such that the equation 2.3 is satisfied identically. This function satisfies the following: (i) the functions η(t) and t/η d/2+1 (t) are monotone increasing and tend to ∞ as t → ∞. (ii) limt→∞ t/η δ (t) = 0 for any δ > d/2 + (1 + γ)/γ. (iii) limt→∞ (η(t)/t) log η(t) = 0. Proof – There exists a positive function T (t), t > 0, such that Φ(T (t))/T (t) = (T (t)/t)2/d by Lemma 2.1 (iii). Then the function η(t) := (t/T (t))2/d satisfies 2.3. The rest of this proposition is also proven by using Lemma 2.1. We next consider the functional Z X q p t η(t) g j 2 log E exp(− (g ( η(t)x)) φ(x)dx Φ(φ, t) = − t 2 Rd j=1 ω

(2.10)

in 0 ≤ φ ∈ L1 (Rd ) and t > 0, where E g is the expectation with respect to gω : Lemma 2.3 – (i) For any t > 0, let B(t) be an operator on L2 (Rd → Rq ) defined by Z X q p j βjk ( η(t)(x − y))ϕk (y)dy. {B(t)ϕ} (x) = Rd k=1

Then, for any 0 ≤ φ ∈ L1 (Rd ) and t > 0, Φ(φ, t) is represented as Φ(φ, t) =

p p η(t) Tr[log(I + t φB(t) φ)], 2t

(2.11)

where Tr[·] is the trace on the space L2 (Rd → Rq ). (ii) Φ(φ, t) is continuous in φ in the norm of L1 (Rd ). (iii) Φ(φ1 , t) ≤ Φ(φ2 , t) if φ1 ≤ φ2 . (iv) Φ(φ(· + a), t) = Φ(φ, t) for any a ∈ Rd . √ √ Proof – For (i), referring that φB(t) φ is a Hilbert-Schmidt operator, we take d q a complete orthonormal √ √ system {ϕn }n≥1 of L2 (R → R ) consisting of the eigenfunctions of φB(t) φ. Let {βn }n≥1 be the corresponding eigenvalues. Then we can write as N i h X p p √ η(t) log E g×X exp i Xn t(gω ( η(t)x) φ(x), ϕn (x)) , N→∞ t n=1

Φ(φ, t) = − lim

482

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Ann. Henri Poincar´e

where (·, ·) is the inner product of the space L2 (Rd → Rq ) and E g×X is the expectation with respect to gω and a system {Xn }n≥1 of independent copies of the Gaussian random variable with mean zero and covariance 1. We first calculate the expectation with respect to gω and then calculate the expectation with respect to {Xn }. Then we have Φ(φ, t) =

∞ p p η(t) η(t) X log(1 + tβn ) = Tr[log(I + t φB(t) φ)]. 2t n=1 2t

(ii) – (iv) are easily obtained from the expression 2.10 . To treat the expression as 2.11, we prepare the following: Lemma 2.4 – For any nonnegative operators B1 and B2 on a Hilbert space of trace class, we have the following: (i) Tr[log(I + B1 + B2 )] ≤ Tr[log(I + B1 )] + Tr[log(I + B2 )]. (ii) | Tr[log(I +B1 )]−Tr[log(I +B2 )]| ≤ kB1 −B2 k(Tr[log(I +B1 )]+Tr[log(I + B2 )]) if Tr[B1 ] = Tr[B2 ], where k · k is the operator norm. Proof – (i) Since the operators B1 and B2 are of trace class, it is enough to show this lemma on a finite dimensional Hilbert space. If we set ϕ(t) := Tr[log(I + B1 )] + Tr[log(I + tB2 )] − Tr[log(I + B1 + tB2 )], then we have ϕ0 (t) = Tr[{(I + tB2 )−1 − (I + B1 + tB2 )−1 }B2 ]. Since (I + tB2 )−1 ≥ (I + B1 + tB2 )−1 , we have ϕ0 (t) ≥ 0 and ϕ(t) ≥ 0. (ii) By the same method in the proof of (i), we have d Tr[log(I + B(t)) − B(t)] = Tr[(B1 − B2 )B(t)(I + B(t))−1 ], dt where B(t) := tB1 + (1 − t)B2 for 0 ≤ t ≤ 1. Then, by λ(1 + λ)−1 ≤ log(1 + λ) for any λ ≥ 0 and (i), we have | Tr[log(I + B1 ) − B1 ] − Tr[log(I + B2 ) − B2 ]| ≤kB1 − B2 k(Tr[log(I + B1 )] + Tr[log(I + B2 )]). To consider the asymptotics as t → ∞, we prepare the following:

(2.12)

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Lemma 2.5 – Let ΦR (t) := R−d Tr[log(I + tχΛ(R) BχΛ(R) )]

(2.13)

for R > 0, where Λ(R) = [−R/2, R/2]d and B is an operator on L2 (Rd → Rq ) defined by Z X q j {Bϕ} (x) = βjk (x − y)ϕk (y)dy. Rd k=1

Then we have lim ΦR (t) = Φ(t)

R→∞

for any t ≥ 0, where Φ(t) is the function defined in 2.2. Proof – For any ` ∈ N, we have lim R

−d

R→∞

1 Tr[(χΛ(R) BχΛ(R) ) ] = (2π)d

Z

`

b `] dξ Tr[β(ξ)

Rd

by the Lebesgue convergence theorem, the condition (H1, d) (ii) and the Fourier transform. Then by using also the condition (H1, d) (i), (ii), we see that the measure X δµ(j;R) (λ)λdλ mR (dλ) = R−d j

on [0, ∞) converges weakly to 1 m∞ (dλ) = (2π)d

Z dξ Rd

X

δµ(k;ξ) (λ)λdλ

k

as R → ∞, where, for each R > 0 and ξ ∈ Rd , {µ(j; R)}j and {µ(k; ξ)}k are b the eigenvalues of the operator χΛ(R) BχΛ(R) and the matrix β(ξ), respectively. By −1 taking a test function as λ log(1 + tλ), we can complete the proof. We now consider the asymptotics of Φ(φ, t) as t → ∞: Lemma 2.6 – For any c > 0 and ρ > 0, we have lim Φ(cχΛ(ρ) , t) ≤

t→∞

1 κ d c ρ . 2

(2.14)

Proof – By Lemma 2.3 (i) and 2.3, we have Φ(cχΛ(ρ) , t) =

1 Tr[log(I + ctχΛ(ρ) B(t)χΛ(ρ) )] Φ(ct/η d/2 (t)) . 2 η d/2 (t)Φ(ct/η d/2 (t)) Φ(t/η d/2 (t))

(2.15)

484

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Ann. Henri Poincar´e

By the condition (H1, d) (iii) and Lemma 2.2 (i), we have Φ(ct/ηd/2 (t)) = cκ . t→∞ Φ(t/η d/2 (t)) lim

Then it is enough to show that Tr[log(I + ctχΛ(ρ) B(t)χΛ(ρ) )] ≤ ρd . t→∞ η d/2 (t)Φ(ct/η d/2 (t)) lim

(2.16)

For any small ε > 0, we use Lemma 2.3 (iii) to estimate as (2.17) Tr[log(I + ctχΛ(n(ρ+ε)) B(t)χΛ(n(ρ+ε)) )] ≥ Tr[log(I + ctΣ + ctΣ0 )], P P where Σ = ` χρ,` B(t)χρ,` , Σ0 = `6=k χρ,` B(t)χρ,k and χρ,` (x) = χΛ(ρ) (x − (ρ + ε)`). By Lemma 2.4 (ii) and Tr[Σ0 ] = 0, the right hand side of 2.17 is dominated from below by Tr[log(I + ctΣ)] − ctkΣ0 k{Tr[log(I + ctΣ + ctΣ0 )] + Tr[log(I + ctΣ)]}. By the condition (H1, d) (ii), we have p |Σ0 (x, y)| ≤ c1 χ{|x−y|>ε} ( η(t)|x − y|)−m . Thus we have Tr[log(I + ctχΛ(n(ρ+ε)) B(t)χΛ(n(ρ+ε)) )] X 1 − cc2 t/η m/2 (t) Tr[log(I + ct χρ,` B(t)χρ,` )]. ≥ m/2 1 + cc2 t/η (t) `

(2.18)

p By the unitary transform φ(x) → η d/4 (t)φ( η(t)x) and Lemma 2.3 (i),(iv), we rewrite as h i ct Tr log I + d/2 χΛ(n(ρ+ε)√η(t)) BχΛ(n(ρ+ε)√η(t)) η (t) (2.19) m/2 1 − cc2 t/η (t) d Tr[log(I + ctχρ B(t)χρ )]n . ≥ 1 + cc2 t/ηm/2 (t) By using Lemma 2.5 (ii) to take the limit of n → ∞, we have Φ

ct 1 − cc2 t/ηm/2 (t) d d/2 (ρ + ε) η (t) ≥ Tr[log(I + ctχρ B(t)χρ )]. η d/2 (t) 1 + cc2 t/ηm/2 (t)

By Lemma 2.2 (ii), we obtain 2.16. Proof of Theorem 2.2 – It is well known that Z q 1 tX j 1 w×g 2 e (t) = E exp(− (g (w(s))) ds w(t) = 0 , N 2 0 j=1 ω (2πt)d/2

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where E w×g is the expectation with respect to the d-dimensional Wiener process w(s) and gωj (x) (cf.[4],[21]). We use the scaling property of the Wiener process to rewrite as Z q t 1 η(t) t/η(t) X j p w×g 2 e N (t) = exp(− E (gω ( η(t)w(s))) ds w =0 2 0 η(t) (2πt)d/2 j=1 i h 1 t H(η(t)) (0, 0) , = d/2 E g exp − η(t) η (t) where exp(−tH(η))(x, y), t > 0, x, y ∈ Rd , is the integral kernel of the heat semigroup generated by the operator H(η) =

q X 1 √ (gωj ( η·))2 . −∆+η 2 j=1

By Theorem (9.5) of [21] , we have e (t) ≥ N

t k∇φk2 1 2 exp − + Φ(φ , t) η(t) 2 η d/2 (t)| supp φ|

(2.20)

for any φ ∈ C0∞ (Rd ) such that kφk2 = 1, where k · k2 and (·, ·) are the norm and the inner product of L2 (Rd ), respectively. By Lemma 2.6 and Lemma 2.2 (iii), we have η(t) e (t) ≥ −1 (k∇φk22 + c2κ Rd ), log N 2 t→∞ t lim

(2.21)

where c, R > 0 are taken as |φ| ≤ c and supp φ ⊂ Λ(R). Corollary (i), (iii) of Theorem 2.2 are proven by Theorem 2.2 and the following: Lemma 2.7 – (i) If Φ(t) = CtA logB t(1 + o(1)) as t → ∞ for some B, C > 0 and 0 ≤ A < 1, then 0

tA η(t) = 0 (1 + o(1)) C 0 logB t

(2.22)

as t → ∞, where A0 =

B 2(1 − A) 2B 2 , B0 = and C 0 = C 2/(d(1−A)+2) . d(1 − A) + 2 d(1 − A) + 2 d(1 − A) + 2 0

(ii) If η(t) =

tA (1 + o(1)) C logB t

486

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Ann. Henri Poincar´e

and

η(t) e (t) = −D(1 + o(1)) log N t as t → ∞ for some B, C, D > 0 and 0 < A < 1, then e (t) = −CDt1−A logB t(1 + o(1)) log N

(2.23)

as t → ∞. This is easily proven by 2.3. For Corollary (ii), we use also the following: Lemma 2.8 – For arbitrarily fixed σ, τ , R > 0, let Z ∞ τ Φσ,R (t) := dζζ d−1 log(1 + te−σζ ),

t > 0.

(2.24)

R

Then, for any fixed K > 0, we have Φσ,R (Kt) = c log(d+τ )/τ t(1 + o(1))

(2.25)

as t → ∞, where c = τ /{(d + τ )dσ d/τ }. Theorem 2.1 is obtained from Corollary of Theorem 2.2, Theorem 2.3 and the following: Lemma 2.9 – If

e (t) log N ≥ −C B t→∞ tA log t for some B, C > 0 and 0 < A < 1, then we have lim

0

λA log N (λ) ≥ −C 0 , lim B0 λ↓0 log (1/λ) where A0 = A/(1 − A), B 0 = B/(1 − A) and C 0 = C 1/(1−A) (1 − A)−B/(1−A) .

3 Case I Let {gω (y), y ∈ R} be the stationary ergodic Gaussian random process with mean zero and the covariance β(y) = E[gω (y)gω (0)]. We consider the operator L(bω ) in 1.1 with b1ω (x) = 0 and b2ω (x) = gω (x1 ). For this operator, we introduce the integrated density of states N (λ) by the same methods in [2], [7], [10], [18] and [28]. We use the condition (P1, α, 1), (E1, τ, 1), (C1, 1), (H1, 1) and (H2, 1) introduced in the last section. Then the result is the following:

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Theorem 3.1 – (i) We assume the condition (P1, α, 1) for some α > 1. Then for small enough ε > 0, there exist c, c0 > 0 such that −cλ−(α+1)/{2(α−1)} ≤ log N (λ) ≤ −c0 λ−p/2 log λ−1 for any 0 < λ < ε. (ii) We assume the condition (E1, τ, 1) for some τ > 0. Then for small enough ε > 0, there exist c, c0 > 0 such that −cλ−1/2 log(1+τ )/τ λ−1 ≤ log N (λ) ≤ −c0 λ−1/2 log λ−1 for any 0 < λ < ε. (iii) We assume the condition (C1, 1). Then for small enough ε > 0, there exist c, c0 > 0 such that −cλ−1/2 log λ−1 ≤ log N (λ) ≤ −c0 λ−1/2 log λ−1 for any 0 < λ < ε.

Remark 3.1 – The constants c and c0 in Theorem 3.1 are same as in Theorem 2.1. Thus the leading term is not determined in Theorem 3.1 (iii) as stated in Remark 2.1.

Theorem 3.2 – Under the condition (H1, 1), we have η(t) e (t) > −∞. log N t→∞ t lim

(3.1)

Corollary – (i) Under the condition (P1, α, 1) for some α > 1, we have lim

e (t) log N

(α−1)/(3α−1) t→∞ t

> −∞.

(3.2)

(ii) Under the condition (E1, τ, p) for some τ > 0, we have e (t) log N

lim

t→∞ t1/3

log2(τ +1)/(3τ ) t

> −∞.

(3.3)

(iii) Under the condition (C1, p), we have lim

e (t) log N

t→∞ t1/3

log2/3 t

> −∞.

(3.4)

488

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Ann. Henri Poincar´e

Theorem 3.3 – Under the condition (H2, 1), we have π λ1/2 log N (λ) ≤ − 3/2 1/2 , −1 λ↓0 log λ B3 2

lim

Z

where B=

R

(3.5)

. |β(x)|dx β(0).

Theorem 3.1 is obtained from the corollary of Theorem 3.2 and Theorem 3.3. For the proof of Theorems 3.2 and 3.3, we prepare the following: Lemma 3.1 – (i) It holds that Z 1Z t 1 w×g 2 e N (t) = dξE (gω (w(s)) + ξ) ds w(t) = 0 , (3.6) exp − 2 0 (2π)3/2 t1/2 R where E w×g is the expectation with respect to the 1-dimensional Wiener process w and gω . (ii) It holds that i h e (t) = 1 E w×g exp − t V (gω , w, t) w(t) = 0 , (3.7) N 2πt 2 where

Z

t

V (gω , w, t) = 0

ds t

2 Z t dr gω (w(r)) gω (w(s)) − t 0

is the sample variance. (iii) It holds that 1Z t 1 w×g 2 e N (t) ≥ gω (w(s)) ds w(t) = 0 . exp − E 2πt 2 0

(3.8)

(iv) It holds that Z t 1/2 e (t) ≤ (1 + tβ(0)) E w×g exp − 1 N gω (w(s))2 ds w(t) = 0 . 2πt 2 0

(3.9)

Proof – (i) By Theorem 6.1 in Broderix, Hundertmark and Leschke [3], the integral kernel of the heat semigroup generated by the operator L(bω ) is represented as Z t −tL(bω ) w 1 1 2 (x, y) =E exp i gω (x + w (s))dw (s) x + w(t) = y e 0

×

|x − y|2 1 exp − , 2πt 2t

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where E w is the expectation with respect to the 2-dimensional Wiener process w(s) = (w1 (s), w2 (s)) (cf.[11], [25], [26], [27]). Then by using the Fourier transform we have Z Z Z t 1 2 2 w 1 2 (0, 0) = dξ dy exp(iξy )E g (w (s))dw (s) exp i ω (2π)2 t R 0 R (y 2 )2 1 w (t) = 0, w2 (t) = y 2 exp − , 2t Z Z t 1 w j 1 2 1 w exp i dξE (g (w (s)) + ξ)dw (s) (t) = 0 . ω (2π)3/2 t1/2 R 0

−tL(bω )

e

=

By calculating the expectation with respect to w2 (s) and taking the expectation with respect to bω , we obtain 3.6. (ii) and (iii) are easily obtained from (i) and (ii), respectively. (iv) We rewrite 3.6 as Z 1 e (t) = N dξE w [exp(−tΦ(µtw , t, ξ))|w(t) = 0], (3.10) (2π)3/2 t1/2 R where

Z 1 t (gω (x) + ξ)2 µ(dx) Φ(µ, t, ξ) = − log E exp(− t 2 R

(3.11)

for any probability measure µ on R and µtw is the empirical distribution of the Wiener process: {w(s) : 0 ≤ s ≤ t} 7→ µtw (B) := t−1 ( the measure of {s : 0 ≤ s ≤ t, w(s) ∈ B}) for any Borel set B in R. If we define an operator B(µ) on L2 (R, dµ) by Z {B(µ)ϕ}(x) =

R

β(x − y)ϕ(y)µ(dy)

for each probability measure µ on R, then we easily see that B(µ) is a HilbertSchmidt operator. Accordingly we take a complete orthonormal system {ϕn }n≥1 of L2 (R, dµ) consisting of the eigenfunctions of B(µ). Let {βn }n≥1 be the corresponding eigenvalues. Then by the same method as in the proof of Lemma 2.3 (i), we can write as Φ(µ, t, ξ) =

∞ ∞ 1 X 1 X αn2 log(1 + tβn ) + , 2t n=1 2 n=1 1 + tβn

(3.12)

where αn = (ξ, ϕn )µ and (·, ·)µ is the inner product of the space L2 (R, dµ). The first term is equal to Φ(µ, t, 0) and the second term is estimated as Φ(µ, t, ξ) − Φ(µ, t, 0) ≥

∞ X 1 |ξ|2 αn2 = 2(1 + tβ(0)) n=1 2(1 + tβ(0))

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Ann. Henri Poincar´e

since the operator norm of B(µ) is estimated as kB(µ)k ≤ β(0). By applying this to 3.10, we obtain 3.9. Now Theorem 3.2 and its corollary are easily obtained from Lemma 3.1 (iii), Theorem 2.2 and its corollary. To prove Theorem 3.3, we further use the following: Lemma 3.2 – (i) If e (t) log N ≤ −C t→∞ tA logB t lim

(3.13)

for some B, C > 0 and 0 < A < 1, then we have 0

λA log N (λ) ≤ −C 0 , 0 λ↓0 logB (1/λ)

lim

(3.14)

where A0 = A/(1 − A), B 0 = B/(1 − A) and C 0 = C 1/(1−A) AA/(1−A) (1 − A)1−B/(1−A) . (ii) Conversely if 3.14 holds, then 3.13 also holds. Proof – The proof of (i) is almost same with that of Fukushima [9] as in Lemma 2.9. Now if 3.14 holds, then for any ε > 0, there exists λ0 > 0 such that 0 0 N (λ) ≤ exp(−(C 0 − ε)λ−A logB λ−1 ) for any 0 < λ < λ0 . Accordingly we devide as Z ∞ Z λ0 −tλ f1 (t) + N f2 (t) e e dN (λ) + e−tλ dN (λ) =: N N (t) = λ0

0

and estimate as

. f2 (t) ≤ e−t(1−ε)λ0 N e (εt) ≤ e−t(1−ε)λ0 (2πεt) N

and f1 (t) ≤ t N

Z

λ0

e−g(λ) dλ + e−g(λ0 ) ,

0 0

0

where g(λ) := tλ + (C 0 − ε)λ−A logB λ−1 . For each t > 0, we see that min0≤λ≤λ0 g(λ) = g(λ(t)) where λ(t) is the solution of 0

0

0

0

0

λ(t) = (C 0 − ε)1/(A +1) t−1/(A +1) (B 0 logB −1 λ(t)−1 + A0 logB λ(t)−1 )1/(A +1) . By the iteration, we have min g(λ) ∼ tA logB tC(ε)

0≤λ≤λ0

0

0

as t → ∞, where C(ε) = (C 0 − ε)1/(A +1) (A0 + 1)−B ((A0 )1/(A +1) + (A0 )−A ). From this we obtain 3.13.

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4 Case II Let {fω (x), x ∈ R2 } be the real valued stationary ergodic Gaussian random field with mean zero and the covariance β(x) = E[fω (x)fω (0)]. Under the integrability b of β(x) and β(ξ)(1 + |ξ|2 ), {∇fω (x), x ∈ R2 } is the stationary ergodic Gaussian random field with mean zero and the covariance h ∂f ∂fω i ∂2β ω E (x) (0) = − (x) (4.1) ∂xj ∂xk ∂xj ∂xk for any j, k = 1, 2. Then, for the operator L(bω ) in 1.1 with b1ω (x) =

∂fω ∂fω (x) and b2ω (x) = − 1 (x), ∂x2 ∂x

(4.2)

we have the integrated density of states N (λ) as in the last section. We introduce the following conditions: for α > 4 and τ > 0, b = |ξ|−α for |ξ| ≥ R. (P2, α) βb ∈ C ∞ and there exists R > 0 such that β(ξ) ∞ b b (E2, τ ) β ∈ C and there exists R > 0 such that β(ξ) = exp(−|ξ|τ ) for |ξ| ≥ R. (C2) 0 6= βb ∈ C0∞ . Then the result is the following: Theorem 4.1 – (i) We assume the condition (P2, α) for some α > 4. Then for small enough ε > 0, there exist c, c0 > 0 such that −cλ−(α−2)/(α−4) ≤ log N (λ) ≤ −c0 λ−1/2 log λ−1 for any 0 < λ < ε. (ii) We assume the condition (E2, τ ) for some τ > 0. Then for small enough ε > 0, there exist c, c0 > 0 such that −cλ−1 log(τ +2)/τ λ−1 ≤ log N (λ) ≤ −c0 λ−1/2 log λ−1 for any 0 < λ < ε. (iii) We assume the condition (C2). Then for small enough ε > 0, there exist c, c0 > 0 such that −cλ−1 log λ−1 ≤ log N (λ) ≤ −c0 λ−1/2 log λ−1 for any 0 < λ < ε. As in Theorems 2.1 and 3.1, this theorem is obtained from the corollary of Theorem 4.2 and Therem 4.3 below. Therefore the conditions (P2, α), (E2, τ ) and (C2) can be weaken. To state Theorem 4.2, we introduce the following condition: (H3)

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b + |ξ|)γ+4 < ∞ for some γ > 0; (i) sup β(ξ)(1 ξ

∂ 2 β(x) (ii) β is C 2 and sup |x|n j k < ∞ for any j, k = 1, 2 and some ∂x ∂x x n > 4(γ −1 + 1); (iii) limt→∞ Φ(ct)/Φ(t) = cκ for any c > 0 and some κ ∈ [0, 1), where Φ(t) :=

1 (2π)2

Z

b dξ Tr[log(I + tM (ξ, β)],

b = (Mjk (ξ, β)) b 1≤j,k≤2 and M (ξ, β) Z b = e−iξ·x E[bjω (x)bkω (0)]dx Mjk (ξ, β) R2  X X ξ`ξm + ξ`ξm − = `>j,m>k

(4.3)

R2

`<j,m
X

X

ξ`ξm −

`>j,m
 b ξ ` ξ m β(ξ).

`<j,m>k

Let η(t), t > 0, be the solution of 2.3 where Φ(t) is replaced by that defined in 4.3. Then we have the following: Theorem 4.2 – Under the condition (H3), we have η(t) e (t) > −∞. log N t→∞ t lim

Corollary – (i) Under the condition (P2, α) for some α > 4, we have lim t→∞

e (t) log N t(α−2)/(2α−6)

> −∞.

(ii) Under the condition (E2, τ ) for some τ > 0, we have lim

t→∞ t1/2

e (t) log N log(τ +2)/(2τ ) t

> −∞.

(iii) Under the condition (C2), we have lim

e (t) log N

t→∞ t1/2

log1/2 t

> −∞.

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To state Theorem 4.3, we introduce the following condition: (H4) b (i) β(ξ) is C 4 ; Z ∞ ∂ k b θ) < ∞ for k = 0, 1, 2; β(ζ, dζ(1 + ζ)3 (ii) sup ∂ζ θ∈S 1 0 Z ∞ b θ) > 0, (iii) inf1 dζζ 3 β(ζ, θ∈S

0

where (ζ, θ), ζ ≥ 0, θ ∈ S 1 , is the polar coordinate of ξ. Then we have the following: Theorem 4.3 – Under the condition (H4), we have π λ1/2 log N (λ) ≤ − 3/2 , λ↓0 log λ−1 3 B

lim where

Z B = sup θ∈S 1

R

Z dx 1

0

∞

(4.4)

Z dζ cos(ζx )ζ β(ζ, θ) 1

3b

∞

b θ). dζζ 3 β(ζ,

0

To prove Theorem 4.3, we prepare the following: Lemma 4.1 – Under the condition (H4), it holds that Z fθ (t)dθ, e (t) ≤ 1 N N 2π S 1 where

Z t 1 eθ (t) = E w×g exp i gθ,ω (w1 (s))dw2 (s) w(t) = 0 N 2πt 0

(4.5)

(4.6)

is the Laplace-Stieltjes transform of the integrated density of states for the Schr¨ odinger operator L(bω ) in 1.1 with b1ω = 0 and b2ω = gθ,ω (x1 ), {gθ,ω (x1 ), x1 ∈ R} is the stationary ergodic Gaussian random field with mean zero and the covariance Z ∞ 1 1 b θ) cos(ζx1 ), βθ (x ) = dζζ 3 β(ζ, (4.7) 2π 0 and E w×g is the expectation with respect to w(s) = (w1 (s), w2 (s)) and gθ,ω (x1 ).

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Proof – By Theorem 6.1 in Broderix, Hundertmark and Leschke [3] and [28], we have e (t) = E w×f [eiΨ(w,ω) |w(t) = 0] N where Ψ(w, ω) =

1 , (2πt)d/2

(4.8)

Z t ∂fω ∂fω 1 2 (w(s))dw (s) − (w(s))dw (s) ∂x2 ∂x1 0

and E w×f is the expectation with respect to w(s) and fω . By calculating the expectation with respect to fω , we have i h e (t) = E w exp − 1 E f [Ψ(w, ω)2 ] w(t) = 0 1 , N (4.9) 2 2πt where E f is the expectation with respect to fω . In terms of the covariance β, we can write as Z t 2 Z 1 iξ·w(s) b , dξ β(ξ) e dξ ∧ w(s) E f [Ψ(w, ω)2 ] = 2 (2π) R2 0

(4.10)

where ξ ∧ w(s) = ξ 2 w1 (s) − ξ 1 w2 (s). In fact, if we take a division ∆ of the interval [0, t] given by 0 = t0 < t1 < · · · < tn = t, then we have I(∆) = J(∆), where I(∆) := E

J(∆) :=

f

n−1 X

1 (2π)2

2 ∂fω ∂fω 1 2 , (w(ta ))∆w (ta ) − (w(ta ))∆w (ta ) ∂x2 ∂x1 a=0 n−1 2 Z X iξ·w(t ) a b dξ β(ξ) e ∆ξ ∧ w(t ) a R2

a=0

and ∆w(ta ) = w(ta+1 ) − w(ta ). By the definition of the Itˆ o integral, we see that I(∆) and J(∆) converge the left and right hand sides of 4.10 respectively in L1 (P W (·|w(t) = 0)) as |∆| := maxa |ta+1 − ta | → 0 where P W is the Wiener measure. We now use Jensen’s inequality in 4.9 as follows: Z 1 e ˘θ (t)dθ, N (t) ≤ N (4.11) 2π S 1 where

Z t 2 1Z ∞ 1 iξ·w(s) b ˘ dζ · ζ β(ζ, θ) e dξ ∧ w(s) w(t) = 0 Nθ (t) := E exp − . 2 0 2πt 0

By the O(2)-invariance, the process ξ · w(s) ξ ∧ w(s) , ζ ζ s

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is a 2-dimensional Wiener process independent of ζ. Thus we have i h ˘θ (t) = E exp − 1 I(θ, w) w(t) = 0 1 , N 2 2πt where 1 I(θ, w) = 2π

Z

∞

0

2 Z t iζw1 (s) 2 dζ · ζ β(ζ, θ) e dw (s) . 3b

0

By the same method as in 4.10, we have E g [Gθ (w, ω)2 ] = I(θ, w) for any θ ∈ S 1 , where Z

t

gθ,ω (w1 (s))dw2 (s)

Gθ (w, ω) = 0

˘θ (t) = N fθ (t). and E g is the expectation with respect to gθ,ω . Now we see that N Proof of Theorem 4.3 – By Lemma 4.1 and 3.9, we have r 1 + tβθ (0) g e ˘ (t) N (t) ≤ sup N0,θ (t) =: N 1 2πt θ∈S g for any (j, k) ∈ S, where N 0,θ (t) is the Laplace-Stieltjes transform of the integrated density of states for the operator H(Vω ) in 1.2 on L2 (R) with Vω (x) = gθ,ω (x)2 /2. By 2.8 we see that, for any small ε > 0 and any 0 < s < 1, there exist c, λ0 > 0 such that r 21/2 π 1 β0 N0,θ (λ) ≤ c exp{− − ε 1/2 log λs B33/2 λ for any 0 < λ < λ0 , where N0,θ (λ) is the integrated density of states for the operator H(Vω ) in 1.2 on L2 (R) with Vω (x) = gθ,ω (x)2 /2 and β0 = supθ∈S 1 βθ (0). By the same method as in the proof of Lemma 3.2 (ii), we have lim

˘ (t) log N

t→∞ t1/3

log

2/3

π 2 1/3 1 ≤− . 2/3 2 (3B) t

From this and Lemma 3.2 (i), we obtain 4.4. Proof of Theorem 4.2 – We use the scaling property of the Wiener process to obtain p p t 1 e−tL(bω ) (0, 0) = exp − L( η(t)bω ( η(t)·)) (0, 0) , η(t) η(t)d/2

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where η(t) is the function defined in 2.3. As in 2.20, by Theorem (9.5) of [21], we have i h p p 1 t e (t) ≥ N E f exp − (L( η(t)bω ( η(t)·))φ, φ) η(t)| supp φ| η(t) for any φ ∈ C0∞ (R2 → C). By restricting the function φ to be of real valued, we have n o t k∇φk2 1 e (t) ≥ exp − + Φ(φ2 , t) N η(t)| supp φ| η(t) 2 for any φ ∈ C0∞ (R2 → R), where tZ p η(t) 2 f 2 2 Φ(φ , t) = − |bω ( η(t)x)| φ (x)dx . log E exp − t 2 R2 Thus we can complete the proof as in the proof of Theorem 2.2.

References [1] S. Albeverio and H.Tamura, On the propagator of a scalar field in the presence of confining nonlinear electromagnetic force, preprint. [2] K. Broderix, D. Hundertmark, W. Kirsch and H.Leschke, The fate of Lifschits tails in magnetic fields, J. Statist. Phys. 80 (1995), 1-22. [3] K. Broderix, D. Hundertmark and H.Leschke, Continuity properties of Schr¨ odinger semigroups with magnetic fields, preprint [4] R. Carmona and J. Lacroix, Spectral theory of random Schr¨ odinger operators, Birkh¨ auser, Boston, 1990. [5] J. -D. Deuschel and D. W. Stroock, Large deviations, Academic Press, Boston, 1989. [6] M. D. Donsker and S. R. S. Varadhan, Asymptotics for the Wiener sausage, Comm. Pure Appl. Math. 28 (1975), 525–565. [7] L. Erd¨os, Lifschitz tail in a magnetic field: the nonclassical regime, Probab. Theory Relat. Fields, 112 (1998), 321–371. [8] A. L. Figotin, Eigenvalue distribution of random Schr¨ odinger equations and asymptotic behavior of certain Wiener integrals, Dopovidi Akad. Nauk Ukr. SSR (6) (1981), 27–29. [9] M. Fukushima, On the spectral distribution of a disordered system and the range of a random walk, Osaka J. Math. 11 (1974), 73–85.

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[10] T. Hupfer, H. Leschke and S. Warzel, Poissonian obstacles with Gaussian walls discriminate between classical and quantum Lifshitz tailing in magnetic fields, preprint [11] N. Ikeda and S. Manabe, Asymptotic formulas for stochastic oscillatory integral, Asymptotic Problems in Probability Theory: Wiener Functionals and Asymptotics, Longman, New York,1993, 136–155. [12] N. Ikeda, I. Shigekawa and S. Taniguchi, The Malliavin calculus and long time asymptotics of certain Wiener integrals, Proc. Center for Math. Anal. Australian Nat. Univ. 9 Canberra, 1985, 46–113. [13] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, 2nd. ed., Kodansha/North-Holland, Tokyo/Amsterdam, 1989. [14] A. K. Klein, L. J. Landau and D. S. Shucker, Decoupling inequalities for stationary Gaussian processes, Ann. Probab., 10 (1982), 702–708. [15] W. Kirsch and F. Martinelli, Large deviations and Lifschitz singularity of the integrated density of states of random Hamiltonians, Commun. Math. Phys., 89 (1983), 27–40. [16] I. M. Lifschitz, Energy spectrum structure and quantum states of disordered condensed systems, Usp. Fiz. Nauk 83 (1964), 617–663. [17] P. Malliavin and S. Taniguchi, Analytic functions, Cauchy formula and stationary phase on a real abstract Wiener space, J. Funct. Anal. 143 (1997), 470–528. [18] H. Matsumoto, On the integrated density of states for the Schr¨odinger operators with certain random electromagnetic potentials, J. Math. Soc. Japan 45 (1993), 197–214. [19] K. Nakane, The Schr¨odinger operator with random vector potentials, Hokkaido Math. J. 25 (1996), 55–80. [20] S. Nakao, On the spectral distribution of the Schr¨ odinger operator with random potential, Japan J. Math. 3 (1977), 111–139. [21] L. A. Pastur and A. L. Figotin, Spectra of random and almost-periodic operators, Springer, Berlin, Heidelberg and New York, 1992. [22] M. Reed and B. Simon, Methods of modern mathematical physics, II : Fourier analysis and self-adjointness, Academic Press, New York, 1975. [23] M. Reed and B. Simon, Methods of modern mathematical physics, IV : Analysis of operators, Academic Press, New York, 1978.

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[24] D. Revuz and M. Yor, Continuous martingales and Brownian motion, Springer, Berlin, Heidelberg and New York, 1991. [25] B. Simon, Functional Integration and Quantum Physics, Academic Press, London, 1979. [26] B. Simon, Maximal and minimal Schr¨odinger operators and forms, J. Operator Theory Appl. 1 (1979), 37–47. [27] B. Simon, Schr¨odinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 447–526. [28] N. Ueki, On spectra of random Schr¨ odinger operators with magnetic fields, Osaka J. Math. 31 (1994), 177–187. [29] S. R. S. Varadhan, Large deviations and applications, Society for Industrial and Applied Mathematics, Philadelphia, 1984.

Naomasa Ueki Graduate School of Human and Environmental Studies Kyoto University Kyoto 606-8501, Japan Email : ueki@@math.h.kyoto-u.ac.jp

Communicated by J. Bellissard submitted 06/04/98, revised 21/04/99, accepted 07/10/99

Ann. Henri Poincar´ e 1 (2000) 499 – 541 c Birkh¨

auser Verlag, Basel, 2000 1424-0637/00/030499-43 $ 1.50+0.20/0

Annales Henri Poincar´ e

Sine-Gordon Revisited J. Dimock and T. R. Hurd Abstract. We study the sine-Gordon model in two dimensional space time in two different domains. For β > 8π and weak coupling, we introduce an ultraviolet cutoff and study the infrared behavior. A renormalization group analysis shows that the model is asymptotically free in the infrared. For β < 8π and weak coupling, we introduce an infrared cutoff and study the ultraviolet behavior. A renormalization group analysis shows that the model is asymptotically free in the ultraviolet.

I Introduction We are concerned with the two dimensional sine-Gordon model. The model is characterized by its partition function which is formally Z Y Z Z 1 2 Z = exp ζ cos(φ(x))dx − dφ(x) . (1) |∂φ(x)| dx 2β 2 x∈R

It is of interest both as a Euclidean quantum field theory and because it describes the classical statistical mechanics of a Coulomb gas with inverse temperature β and activity ζ/2. The expression for Z is ill-defined. To make sense of it we first replace the plane R2 by the torus ΛM = R2 /LM Z2 , where M is a non-negative integer and L is a fixed large positive constant. Then the quadratic term is combined with the non-existent Lebesgue measure to give a Gaussian measure. Introduction of a short M distance cutoff at scale L−N gives the Gaussian measure µβv−N with covariance M (x − y) = β|ΛM |−1 βv−N

X p∈Λ∗ M ,p6=0

eip(x−y) −p4 L−4N e p2

(2)

where Λ∗M = R(2πL)−M Z2 . Since p = 0 is excluded the measure is supported on fields φ with φ = 0. Furthermore the covariance is smooth and so the measure is supported on smooth functions. Thus the cutoff expression Z Z M (φ). Z = exp ζ cos(φ(x))dx dµβv−N (3) ΛM

is well-defined. We are interested in studying the limits N → ∞ (the UV problem) and M → ∞ (the IR problem).

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There are two distinct domains in which these problems are tractable. For β > 8π and ζ small it turns out that the long distance behavior differs only slightly from that of the free model (ζ = 0) and thus the IR problem can be controlled. For β < 8π and ζ small it turns out that the short distance behavior differs only slightly from free and thus the UV problem can be controlled. The purpose of this paper is to carry out the analysis in each case using a renormalization group (RG) method. Each of these problems have been previously studied by the authors in [13] and [15]. Unfortunately there is an error which occurs in both papers and spoils the proofs of the main results.1 In our present paper we are at last able to fix this error, and reinstate our earlier results. The fix requires some substantial modifications to the method, and so we give here reasonably self-contained proofs of the main technical lemmas. We first discuss the IR problem for β > 8π. We study the expression (3) with the ultraviolet cutoff N fixed: for simplicity we take N = 0. The RG method involves the introduction of a sum over scales. For any 0 ≤ j ≤ M we have v0M (x

− y) =

j−1 X

C M −k (L−k (x − y)) + v0M −j (L−j (x − y)) .

(4)

k=0

The slice covariances are defined by2 C M (x − y) = |ΛM |−1

X p∈Λ∗ M ,p6=0

4 4 eip(x−y) −p4 (e − e−L p ) . 2 p

(5)

The integral over µβv0M in the partition function can then be evaluated by successively taking convolutions with µβC and then scaling down by L. After j steps we have the expression Z Z = Zj (φ) dµβvM −j (φ) (6) 0

with successive densities Zj defined on ΛM −j and related by Z Zj+1 (φ) = (µβC ∗ Zj )(φL ) = Zj (φL + ζ)dµβC (ζ)

(7)

where φL (x) = φ(x/L) is the canonical rescaling of the field for d = 2. Equation (7) R is the RG map. We want to study the flow starting with Z0 (φ) = exp(ζ cos φ). 1 The problem is that for the homotopy property one needs κ small, but the limitation on κ cannot be made independently of L as was implicitly assumed. In fact one needs κ ≤ O(L−2 ) or smaller. Then the use of Sobolev inequalities require κ(h∗1 )2 ≥ O(1) and hence h∗1 ≥ O(L). This spoils the estimate above line (49) in [15]. There is a similar problem in [13]. 2 We have chosen to take e−p4 rather than say e−p2 in (2),(5) in order to have a smoother approach to infinite volume at p = 0.

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501

To track the flow we must isolate the fastest growing parts of Zj during each RG step. We extract a constant part and a gradient part and instead of (7) now define Zj+1 by ! Z δσj 2 = (µβC ∗ Zj ) (φL ) Zj+1 (φ) exp δEj |ΛM −j | − (∂φ) (8) 2β ΛM −j−1 with special choices of δEj , δσj . The quadratic factor is absorbed into the measure at each step and so instead of (6) we have for some constants Ej , σj Z Ej Zj (φ) dµβvM −j (σj ) (φ) , Z=e (9) 0

where v0M (σ; x − y) = |ΛM |−1

X p∈Λ∗ M ,p6=0

eip(x−y) p4 (e + σ)−1 . p2

(10)

The successive values of σj are given by σj+1 = σj + δσj and there is a similar formula for Ej+1 in terms of Ej , δEj and δσj . To state the main result we need one more ingredient. This is a local structure for the densities Zj . Following Brydges and Yau [11] densities are represented by polymer expansions Zj (φ) = Exp(2 + Kj )(Λj , φ) as we now explain. A closed polymer X is a union of closed unit squares centered on lattice points. A polymer activity is a function K(X, φ) depending on polymers X and fields φ with the property that the dependence on φ is localized in X. One can define a product on polymer activities and an associated exponential function (Exp(K))(X, φ). If 2 is the characteristic function of open unit cells then XY Exp(2 + K)(X, φ) = K(Xi , φ) (11) {Xi } i

where the sum is over collections of disjoint polymers {Xi } in X. For an exposition of polymers see [3]. Now we can state the IR result: Theorem 1 Let β > 8π, let > 0, let L be sufficiently large, and let |ζ| be sufficiently small. Then for j = 0, 1, 2, .. the partition function Z defined by (3) with N = 0 can be written Z Z = eEj Exp(2 + Kj )(ΛM −j , φ)dµβvM −j (σj ) (φ) (12) 0

where Ej /|ΛM | and σj are bounded and O(|ζ|) uniformly in M . The polymer activities Kj are even and 2π–periodic in φ. There is a norm k · k∞ such that kKj k∞ ≤ δ j |ζ|1− where δ = O(1) max{L−2 , L2−β/4π } < 1/4

(13)

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Here and throughout the paper O(1) means a constant which is independent of L, ζ, M, j. The norm kKj k∞ of Kj (X, φ) enforces conditions of growth and analyticity in φ and requires tree decay in X. A more precise version of the theorem will be stated later when we come to the proof. The point is that Kj shrinks uniformly in M so that the dominant contribution as j → M is from the Gaussian measure. The result gives a uniform bound on the energy density log Z/|ΛM | and it should be possible to also take the limit M → ∞. Everything should also be analytic in ζ in a complex neighborhood of the origin. The only difficult part here is working with complex measures; see [11] for a treatment of this problem for the closely related dipole gas. A modification of this theorem to include local perturbations should make it possible to study correlation functions for the model, proving the existence of the M → ∞ limit and showing that the long distance behavior of correlations is essentially the same as free. See [19] for results of this nature, and [14], [8] for the closely related dipole gas. Let us mention some earlier work on this model. It was first treated heuristically by Kosterlitz and Thouless [20]. Fr¨ ohlich and Spencer later gave a rigorous treatment for β large [17] by a special method (not the RG). The range of validity was extended to β > 8π by Marchetti and Klein [21]. Now we discuss the UV problem for β < 8π. We start with a fixed torus ΛM . 0 . We also For simplicity take the unit torus Λ0 so the starting covariance is v−N make a renormalization replacing cos(φ(x)) by the Wick ordered version 0 0 : cos(φ(x)) :βv−N = exp(βv−N (0)/2) cos(φ(x)) .

Thus we study the partition function Z Z 0 0 : cos(φ(x)) :βv−N dx dµβv−N (φ) . Z = exp ζ

(14)

(15)

Λ0

We scale up to get an expression for Z on ΛN . Absorbing the Wick ordering constant into the coupling constant one finds that Z Z Z = exp ζ−N cos(φ(x))dx dµβv0N (φ) (16) ΛN

where for any j ≤ 0 0 ζj = L−2|j| exp(βv−|j| (0)/2)ζ ≈ L−(2−β/4π)|j| ζ .

(17)

The UV problem of controlling the limit N → ∞ by an RG analysis looks very much like the IR problem. The main difference is that the coupling constants ζj start out ultra small at j = −N and grow to a small value ζ0 = ζ, instead of starting out small and then shrinking. A technical simplification is that the field strength extraction is no longer needed and we can take σj = 0.

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Sine-Gordon Revisited

We define

R

V (X, φ) =

∆

503

cos(φ(x))dx 0

X = ∆ = unit square |X| ≥ 2

(18)

and then the result is : Theorem 2 Let β < 8π, let > 0, let L be sufficiently large, and let |ζ| be sufficiently small. Then for j = −N, −N + 1, . . . , 0 the partition function given by (15) or (16) can be written Z Z = eEj Exp(2 + Kj )(Λ|j| , φ)dµβv|j| (φ) (19) 0

where Ej

=

j−1 X

δEk |Λ|k| | ,

k=−N

Kj

˜j . = ζj V + K

(20)

˜ j are analytic in ζ and satisfy ˜ j are even and 2π–periodic in φ, and δEj , K The K |δEj | ≤ |ζj |2− ˜ j k∞ ≤ |ζj |2− kK

(21)

for some norm k · k∞ . For β < 4π the theorem implies that Z is uniformly bounded and analytic in ζ. For 4π ≤ β < 8π it isolates the divergence in Z. One can also show that δEj , Kj have limits as N → ∞, and hence so does Z (for β < 4π). [15] A modification of this theorem to include local perturbations should make it possible to study correlation functions for β < 8π . (The potentially divergent factor Ej does not contribute to correlation functions). One should be able to take the N → ∞ limit and study the short distance behavior of correlations. See [15],[19] for results of this nature. Also see [12] for a proof that at β = 4π the theory is equivalent to a theory of massive free fermions. Earlier work on this problem can be found in [16], [2], [23] [22].

II Estimates on the RG map Our treatment of the RG map on polymer activities is similar to that used in previous papers [13],[15], [5], [6]. However there are essential modifications: references [5], [6], which we follow as much as possible, use open polymers while we have to use closed polymers as in [13],[15] (see the discussion in the next section ). Our norms are now simpler as well, a simplification we pay for with some harder proofs.

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In this chapter, we analyze a single RG map on a torus Λ = ΛM = Rd /LM Zd of arbitrary dimension d ≥ 2. We work with the fixed covariance X

C M (σ, x − y) = |ΛM |−1

p∈Λ∗ M ,p6=0

4 4 4 eipx [(ep + σ)−1 − (eL p + σ)−1 ] 2 p

(22)

and |σ| assumed small, although the results holds for a much larger class. We start by defining our norms. Then we consider separately the three pieces of the RG: fluctuation, extraction, and scaling. Finally we put them together in Theorem 10 to give an overall estimate on the RG map.

II.1

Norms

Let the Banach spaces C r (X), C˜s (X) of smooth fields φ(x) on a closed polymer X be defined respectively for fixed r, s ≥ 0 by the following norms: kφkX ≡ kφk∞,r,X

=

sup |α|≤r, x∈X



kφks,X ≡ kφk2,s,X

= 

X Z

|α|≤s

|∂ α φ(x)| 1/2 |∂ α φ(x)|2 dx

(23)

X

We assume s > d/2 + r to ensure a Sobolev inequality kφk∞,r,X ≤ O(1)kφk2,s,X and the corresponding dense embedding C˜s (X) ⊂ C r (X). Let K(X, φ) be a smooth function on C˜s (X). Thus we assume the existence of all derivatives Kn (X, φ). These are continuous symmetric multilinear functionals on C˜s (X). In fact we make a stronger assumption that these derivatives have continuous extensions to C r by demanding the finiteness of the following norm kKn (X, φ)k =

sup ˜s (X) fi ∈C kfi k∞,r,X ≤1

|Kn (X, φ; f1 , ..., fn )|.

(24)

A large field regulator is a functional of the form G(κ, X, φ) = G0 (κ, X, φ)δG(κ, ∂X, φ) where G0 (κ, X, φ)

X Z

= exp(κ

1≤|α|≤s

δG(κ, ∂X, φ)

= exp(κc

X Z

|α|=1

with constants κ, c ≤ 1 to be specified.

(25)

|∂ α φ|2 )

X

∂X

|∂ α φ|2 )

(26)

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A large set regulator Γ(X) has the form Γ(X) = A|X| Θ(X)

(27)

for a parameter A ≥ 1 and factor Θ(X) such that Θ(X)−1 has polynomial tree decay (see [11], [6] for the exact definition). For our present paper we fix A = Ld+3 , and also define Γp (X) = 2p|X| Γ(X) for any p = ±1, ±2, . . .. In terms of regulators G, Γ and a further parameter h ≥ 0 we define the norms : X kKkG,h,Γ = Γ(X)kK(X)kG,h , kK(X)kG,h

=

kKn (X)kG

=

X⊃∆ ∞ n X

h kKn (X)kG , n! n=0 sup φ∈C˜s (X)

kKn (X, φ)kG(X, φ)−1 .

(28)

The sum over X is independent of the unit block ∆ for translation invariant K. These norms are simpler than the norms in earlier versions of this formalism in which one first localizes the derivatives in unit blocks, then takes the supremum over the fields, and finally sums over blocks. The previous version (designed for models in d > 2) controls the fluctuation step in an elegant manner, but in d = 2 leads to unbounded growth in the parameter h in the scaling step. The present norms require a different treatment of fluctuation, but avoid growth in h. Another point concerns the boundary term δG(κ, ∂X, φ) in the large field regulator G(κ, X, φ). It is present to absorb the growth of G0 (κ, X, φ) a feature upon which we elaborate in the next section. However we also need G(X)G(Y ) ≤ G(X ∪ Y ) for disjoint polymers. The boundary term spoils this if the polymers are open since disjoint polymers may have pieces of their boundaries in common. This is the reason we have taken closed polymers.

II.2

Fluctuation

Given a localized density Exp(2+K) and the Gaussian measure µC we want to find new polymer activities FK such that µC ∗ Exp(2 + K) = Exp(2 + FK) and such that we more or less preserve control over size and localization. We accomplish this using the framework of Brydges and Yau [11] (see also Brydges and Kennedy [9]). Those authors actually give two constructions. The first is by solving a functional Hamilton-Jacobi equation for µtC ∗ Exp(2 + K). This is elegant and efficient, but with our new norms we cannot take advantage of it (since we can no longer keep the large set regulator Γ constant). Instead we use the second construction of Brydges and Yau, an explicit cluster expansion. Cluster expansions have a long history starting with [18]. We begin with the purely combinatoric part. Let F (s) be a continuously differentiable function of s = {sij } where 0 ≤ sij ≤ 1 and where ij runs over the

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distinct unordered pairs (bonds) from some finite index set. A graph G on this set is a collection of bonds , and it is called a forest if it has no closed loops. The set of all forests is denoted F. Finally we define σij (G, s) = inf{sb : b ∈ path joining ij in G}

(29)

with the convention that σij (G, s) = 0 if there is no path joining ij in G. Then for 1 = {1, 1, ..., 1} XZ Y Y F (1) = dsb ( ∂sb F )(σ(G, s)) (30) G∈F

b∈G

b∈G

where the G = ∅ term is interpreted as F (0). For the proof see Abdesselam and Rivasseau [1] or Brydges and Martin, Theorem VIII.2 [10]. Now for any X, Y define C(X, Y )(x, y) =

1 [χX (x)χY (y) + χY (x)χX (y)]C(x − y) 2

(31)

and let CX = C(X, X) be the restriction of C to X. Suppose that {Xi } is a collection of disjoint polymers whose union is X. Then the restriction CX can be written X CX = C(Xi , Xj ) (32) i,j

with the sum over ordered pairs. We weaken the coupling between Xi , Xj with parameters sij and define X CX (s) = C(Xi , Xj )sij (33) i,j

where sii = 1. Now while CX (s) is not necessarily positive definite, CX (σ(T, s)) is positive definite for any s and any tree T [9], [10]. Let ∆C be the functional Laplacian given formally by: Z δ δ 1 C(x, y) dxdy . (34) ∆C = 2 δφ(x) δφ(y) Lemma 3 [11], [1] For smooth polymer activities K µC ∗ Exp(2 + K) = Exp(2 + FK) with FK(X) =

X {Xi },T →X

Z dsT µCX (σ(T,s)) ∗

Y ij∈T

(−2∆C(Xi ,Xj ) )

(35) Y

K(Xi )

(36)

i

where the sum is over collections of disjoint polymers {Xi } whose union is X, and over tree graphs T on {Xi }. If {Xi } = {X} the term is interpreted as µC ∗ K(X).

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Proof. We start with µC ∗ Exp(2 + K)(X) =

X

µC ∗

Y

{Xi }

K(Xi ) .

(37)

i

Q

In the expression µC ∗ i K(Xi , φ) we regard the product as a function of fields φ on X only, and replace the covariance C by CX .3 If {Xi } = {X} has only one element we leave the expression alone. Otherwise there are two or more subsets and for each Q {Xi } weQ analyze F (1) = µCX ∗ i K(Xi ) by introducing the interpolation F (s) = µCX (s) ∗ i K(Xi ) with CX (s) given by (33), and then making the expansion (30). This gives the expression ! Y X XZ G G K(Xi ) . (38) ds ∂ µCX (s) ∗ {Xi } G

i

s=σ(G,s)

Now the graph G can be regarded as a union of trees {Tk }. Grouping together the polymers {Xi } linked by the trees yields new disjoint polymers {Yk }. The covariance CX (s) preserves the {Yk } since σij (G, s) = 0 for blocks Xi , Xj in different trees. We can write CX (s) = ⊕k CYk (s). Then the integrand above factors and we have   ! X XY Z Y  dsTk ∂ Tk µCY (s) ∗  . K(Xi ) (39) k i:Xi ⊂Yk

{Xi } {Tk } k

s=σ(Tk ,s)

Now we group together the terms in the sum by the {Yk } they determine and find XY FK(Yk ) = Exp(2 + FK) (40) {Yk } k

where X

FK(Y ) =

Z T

ds

∂ µCY (s) ∗ T

{Xi },T →Y

Y i

! K(Xi )

.

(41)

s=σ(T,s)

The result now follows since ∂CY (s)/∂sij = 2C(Xi , Xj ) and hence ∂/∂sij (µCY (s) ∗ F ) = µCY (s) ∗ (−2∆C(Xi ,Xj ) )F and hence Y Y Y K(Xi ) = µCY (s) ∗ (−2∆C(Xi ,Xj ) ) K(Xi ) . (42) ∂ T µCY (s) ∗ i

ij∈T

i

2 The behavior in φ of FK(X, φ) will turn out to be slightly worse than that for K(X, φ) which means we have to take a larger large field regulator. It is convenient 3 Our assumption that K(X, φ) has φ dependence localized in X means that the function is measurable with respect to ΣX , the σ-algebra generated by {φx }x∈X .

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to choose a regulator which is a scaling of the original. Let the field scaled up by ` > 1 be defined by φ` (x) = `−(d−2)/2 φ(x/`) (43) (our convention here is different from earlier papers). Then define G` (κ, X, φ) = G(κ, `−1 X, φ`−1 )  Z X Z X 2|α|−2 α 2  ` |∂ φ| + κc ` = exp κ X

1≤|α|≤s

|α|=1

 |∂ α φ|2 . (44)

∂X

For the applications we have in mind we need 1 < ` ≤ L: for definiteness we take ` = 2 in the following. For unit blocks ∆, ∆0 define C∗ (∆, ∆0 ) = kC(∆, ∆0 )kd(∆, ∆0 )2d θ(∆, ∆0 ) .

(45)

Here the norm is the C r norm in each variable, d(∆, ∆0 ) is Euclidean distance between block centres, and θ is the distance function built into the tree decay factor Θ in (27). Now define X kCk∗ = sup C∗ (∆, ∆0 ) . (46) ∆

∆0 6=∆

Theorem 4 Let κc−1 L2 be sufficiently small. Then there is a constant γ depending only on the dimension such that if 0 < δh < h and for some p δh2 ≥ 8γ 2 kCk∗ kKkG(κ),h,Γp+3

(47)

kFKkG` (κ),h−δh,Γp ≤ 2kKkG(κ),h,Γp+3 .

(48)

then

Remark. The linearization F1 K = µC ∗ K satisfies the same bound (or even the better bound with δh = 0). Proof. We adapt the analysis of [11] to our norms. In (36) change to a sum on disjoint ordered polymers (X1 , ..., XN ) and regard T as a tree on (1, ..., N ). Then we have FK(X) = µC ∗ K(X) +

∞ X N=2

1 N!

X

XZ

(49) dsT µCX (σ(T,s)) ∗

(X1 ,...,XN ) T

Y

∆C(Xi ,Xj )

ij∈T

We next introduce a sum over unit blocks: for b = {ij} X ∆C(Xi ,Xj ) = ∆C(∆bi ,∆bj ) ∆bi ∈Xi ,∆bj ∈Xj

N Y

K(Xi ) .

i=1

(50)

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Taking derivatives and norms yields ∞ X X X X 1 N! N=2 (X1 ,...,XN ) T {∆bi ,∆bj } " # Z N Y Y dsT kµCX (σ(T,s)) ∗ ∆C(∆bi ,∆bj ) Kni (Xi ) k . (51)

k(FK)n (X)k ≤ kµC ∗ Kn (X)k + X n1 ,...,nN

n! n1 !...nN !

i=1

b∈T

In Lemma 5 to follow we show that for κc−1 L2 sufficiently small µCX (σ(T,s)) ∗ G(κ, X) ≤ G` (κ, X)2|X|

(52)

which makes Q it possible to estimate the above convolutions. Using this and G(κ, X) = i G(κ, Xi ) (since the Xi are disjoint) we find (see [6] for more details) k(FK)n (X)kG` (κ) ≤ kKn kG(κ) +

∞ X 1 N!

N=2

X n1 ,...,nN

X

X

(X1 ,...,XN ) T

X {∆bi ,∆bj }

Y n! kC(∆bi , ∆bj )k kKni +di (Xi )kG(κ) 2|Xi | . n1 !...nN ! i=1 N Y

(53)

b∈T

Here di is the incidence number for the ith vertex in the graph T . Now multiply by (h − δh)n /n! and sum over n to obtain kFK(X)kG` (κ),h−δh

≤ kKn (X)kG(κ),h +

∞ X 1 N!

N=2

Y

kC(∆bi , ∆bj )k

N Y i=1

b∈T

X

X

(X1 ,...,XN ) T

X {∆bi ,∆bj }

d di kK(Xi )kG(κ),h−δh 2|Xi | .(54) dh

A Cauchy bound yields (

−di d di di !kK(Xi )kG(κ),h . ) kK(Xi )kG(κ),h−δh ≤ δh dh

It is proved in Lemma 6 to follow that for any i we have Y di ! ≤ γ di d(∆bi , ∆bj )d

(55)

(56)

b3i

P for some constant γ. Taking into account that i di = 2N − 2 this gives Y Y di ! ≤ γ 2N−2 d(∆bi , ∆bj )2d i

b

(57)

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and so kFK(X)kG` (κ),h−δh ≤ kK(X)kG(κ),h +

∞ X 1 N!

N=2

(γ δh−1 )2N−2

Y

kC(∆bi , ∆bj )k∞ d(∆bi , ∆bj )2d

X

X

(X1 ,...,XN ) T N Y

X {∆bi ,∆bj }

kK(Xi )kG(κ),h 2|Xi | . (58)

i=1

b∈T

Now multiply by Γp (X) ≤

Y

Γp (Xi )

Y

i

θ(∆bi , ∆bj )

(59)

b

Q and identify b C∗ (∆bi , ∆bj ). Next sum over X ⊃ ∆ and dominate the expression by a sum over i0 and a sum over unrestricted disjoint (X1 , ..., XN ) such that Xi0 ⊃ ∆. To estimate this sum and the sum over {∆bi , ∆bj }, we start at the twigs of the tree and work inward leaving to the last the set Xi0 which is pinned. Suppose that when we come to a vertex i we have gained a factor |Xi |di −1 from the previous estimates. If b = {ij} is the remaining inward bond at this vertex and ∆ = ∆bi , ∆0 = ∆bj , then we have X X C∗ (∆, ∆0 )kK(Xi )kG(κ),h Γp+1 (Xi )|Xi |di −1 Xi ∆∈Xi ,∆0 ∈Xj

≤

X

X

C∗ (∆, ∆0 )kK(Xi )kG(κ),h Γp+3 (Xi )(di − 1)!

Xi ∆∈Xi ,∆0 ∈Xj

≤

X

C∗ (∆, ∆0 )kKkG(κ),h,Γp+3 (di − 1)!

∆0 ∈Xj ,∆

≤ kCk∗ kKkG(κ),h,Γp+3 |Xj |(di − 1)! .

(60)

This gives a factor |Xj | for the j vertex. The case for i = i0 is special and we have di0 ! ≤ (N − 1)(di0 − 1)!) . There is also a factor N for the sum over i0 and combining all the above yields kFKkG` (κ),h−δh,Γp ≤ kKkG(κ),h,Γp+3 (1 +

∞ X N=2

αN−1 X Y (di − 1)!) (N − 2)! i=1 N

(61)

T

where α = γ 2 δh−2 kCk∗ kKkG(κ),h,Γ . But the number of trees with given p+3 Q incidence numbers di is (N −2)!/ i (di −1)! by Cayley’s theorem, and the number the sum over T is bounded by of choices of di is bounded by 22N−2 = 4N−1 . Thus P N−1 (N − 2)!4N−1 . Then the sum over N is bounded by ∞ and this is less N=2 (4α) that 1 since our basic assumption is 4α ≤ 1/2 2 This completes the proof of the theorem, except for the following two results which we skipped.

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Lemma 5 Let κc−1 L2 be sufficiently small. Then µCX (σ(T,s)) ∗ G(κ, X) ≤ G` (κ, X)2|X| .

(62)

Proof. (see [3] for more details) Consider for 0 ≤ t ≤ 1 the family of large field regulators t 1−t Gt (κ, X) = 2t|X| [G` (κ, X)] [G(κ, X)] . (63) We prove for 0 ≤ t ≤ 1 that µtCX (σ(T,s)) ∗ G0 (κ, X) ≤ Gt (κ, X) .

(64)

The result we want comes at t = 1. We have Gt (X) = exp(U (t, X)) where (with ` = 2) Z X Z |∂ α φ|2 · (22|α|−2 t + (1 − t)) + κc U (t, X) = t log(2)|X| + κ X

1≤|α|≤s

|∂φ|2 (1 + t)

∂X

(65) The bound (64) is implied by 1 ∆CX (σ(T,s)) U + CX (σ(T, s)) 2

∂U ∂U , ∂φ ∂φ

≤

∂U . ∂t

(66)

Showing (66) is a somewhat lengthy computation in which every term on the left is bounded by corresponding terms on the right for κ sufficiently small. The terms with |α| = 1 are special since there is no corresponding term on the right. Instead one integrates by parts. This adds derivatives and boundary terms both of which can be bounded. The condition on κ turns out to be that the following quantities be sufficiently small : κ

sup |(∂xα ∂yβ CX (σ(T, s)))(x, x)|

sup

1≤|α|,|β|≤s x∈X

κc−1

sup

0≤|α|,|β|≤s x∈X

κc−1

sup

Z |(∂xα ∂yβ CX (σ(T, s)))(x, y)|dy

sup X

Z

|(∂xα ∂yβ CX (σ(T, s)))(x, y)|dy .

sup

0≤|α|,|β|≤s x∈X

(67)

∂X

These quantities are bounded by the corresponding quantities with σ = 1. Note from Lemma 22 in the appendix, (∂xα ∂yβ C)(x, x) is bounded by O(1). The second and third quantities are bounded by same expressions with X = Λ and X = the d − 1 dimensional “checkerboard” in Λ. For both these integrals, we use Lemma 22 again and find the worst bound is κc−1 L2 . Hence the result follows. 2

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Lemma 6 Let ∆ and ∆1 , ...∆n be distinct unit blocks. Then there is a constant γ depending only on the dimension d such that n! ≤ γ n

n Y

d(∆, ∆j )d .

(68)

j=1

Remark. Bounds of this type were introduced in [18]. Proof. Let mr be the number of unit blocks intersecting a ball of radius r centered on a lattice point, and select γ so mr ≤ γrd for all r > 1. Order the blocks so that d(∆, ∆1 ) ≤ ... ≤ d(∆, ∆n ) .

(69)

Then the ball of radius rk = d(∆, ∆k ) around the center of ∆ intersects mrk unit blocks and mrk ≥ k. Then k ≤ mrk ≤ γrkd and we have n! =

n Y k=1

k≤

n Y

γrkd = γ n

k=1

n Y

d(∆, ∆k )d .

(70)

k=1

2

II.3

Extraction

In the extraction step we remove a polymer activity F from the general activity K. Usually F is some low order terms in K but we do not assume this at first. The extraction is defined so that ! X Exp(2 + K)(Λ, φ) = exp F (X, φ) Exp(2 + E(K, F ))(Λ, φ) (71) X⊂Λ

with new polymer activities E(K, F ). To specify E(K, F ) we define ˜ K(X) = K(X) − (eF − 1)+ (X) , X Y (eF − 1)+ (Y ) = (eF (Yj ) − 1) {Yj }→Y

(72)

j

where the sum is over collections {Yj } of distinct polymers which are overlap connected and whose union is Y . Then formula (71) holds with E(K, F ) given by Y X Y ˜ i ) (e−F (Yj ) − 1) K(X (73) E(K, F )(Z) = {Xi },{Yj }→Z

i

j

where the sum is over collections of disjoint subsets {Xi } and collections of distinct subsets {Yj } each intersecting some Xi , so that the {Xi }, {Yj } are overlap connected and their union is Z. This version of extraction is taken from [14], to which

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we refer for a proof. The linearization of E(K, F ) in K and F is E1 (K, F ) = K − F : this is the sense in which F has been removed from K. To obtain estimates on E(K, F ) we will need estimates like G(X) ≤ G(Z) when X ⊂ Z. For this to be true we have to be able to dominate δG by G0 so we can “dissolve” the pieces of ∂X which do not contribute to ∂Z. This means that the constant c in δG has to be sufficiently small. Let cs be the Sobolev R defined P constant so that for x ∈ ∆, the closed unit block, we have |∂φ(x)|2 ≤ cs 1≤|α|≤s ∆ |∂ α φ|2 . Lemma 7 For X ⊂ Z, κ > 0 and c < (2d cs )−1 we have G(κ, X) ≤ G(κ, Z) .

(74)

If c < (4d cs )−1 the same bound holds with G replaced by G` , ` = 2. Proof. Let f be a face (d − 1 cell) in ∂X which does not contribute to ∂Z. Any such face f must be also be a face for some ∆ in Z − X. Then we can “dissolve” the boundary by using the Sobolev inequality and the bound on c to obtain δG(κ, f ) ≤ G0 (κ/2d, ∆) .

(75)

Each ∆ arises from at most 2d faces and so δG(κ, ∂X − ∂Z) ≤ G0 (κ, Z − X) .

(76)

G(κ, X) = G0 (κ, X)δG(κ, ∂X − ∂Z)δG(κ, ∂Z ∩ ∂X) ≤ G0 (κ, Z)G(κ, ∂Z ∩ ∂X) .

(77)

Thus we have

Since δG(κ, ∂Z ∩ ∂X) ≤ δG(κ, ∂Z) the result follows.

2

We now assume F satisfies the following localization property: F (X, φ) has the decomposition X F (X, ∆, φ) (78) F (X, φ) = ∆⊂X

where ∆ is summed over unit blocks, and F (X, ∆, φ) has the φ dependence localized in ∆. We also need stability conditions on the perturbation F . Let f (X) be a collection of constants. We say that F is stable for (G, h, f (X)) if for complex z(X) ( ) X sup k exp z(X)F (X, ∆) kG,h ≤ 2 . (79) |z(X)|f (X)≤1

X⊃∆

For a method to verify the stability hypothesis see the appendix.

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Theorem 8 Let c < (2d cs )−1 . Suppose that F is stable for (G(κ), h, f (X)) and for (G0 (δκ), h, δf (X)) and that kf kΓp+4 ,kδf kΓp+2 and kKkG(κ),h,Γp+2 are sufficiently small. Then there is a constant O(1) such that kE(K, F )kG(κ+δκ),h,Γp ≤ O(1)(kKkG(κ),h,Γp+2 + kf kΓp+4 ) .

(80)

For c < (4d cs )−1 the same bound holds with each G replaced by G` , ` = 2. Proof. The proof is similar to [6] where however the extraction is not global. We start with (73) which can be written X Y 1 Z Y dzj ˜ i) E(K, F )(Z) = K(X exp {−zj F (Yj )} . (81) 2πi z (z j j − 1) i j {Xi },{Yj }→Z

The integral is over the circles |zj |δf (Yj ) = 1. Inserting F (Y ) = we can rewrite this as

P ∆⊂Y

F (Y, ∆)

E(K, F )(Z) X =

Y

{Xi },{Yj }→Z

i

    Y 1 Z X Y dzj ˜ i) K(X exp − zj F (Yj , ∆) (82)   2πi zj (zj − 1) j j ∆⊂Z

Now we note Y

G(κ, Xi )

i

Y

G0 (δκ, ∆) ≤ G(κ + δκ, Z) .

(83)

∆⊂Z

Q This follows from i G(κ, Xi ) = G(κ, ∪i Xi ) ≤ G(κ, Z) (by the lemma) and from Q 0 0 ∆⊂Z G (δκ, ∆) = G (δκ, Z) ≤ G(δκ, Z). Using this estimate and the multiplicative property of the norm we obtain X Y Y ˜ i )kG(κ),h kE(K, F )(Z)kG(κ+δκ),h ≤ kK(X O(1)δf (Yj ) {Xi },{Yj }→Z

sup

i

j

   X  Y k exp − zj F (Yj , ∆) kG0 (δκ),h .  

|zj |δf (Yj )≤1 ∆⊂Z

(84)

j

By our second stability assumption the last factor is bounded by 2|Z| . Now we write X 1 X X = N !M ! {Xi },{Yj }

N,M

(X1 ,...,XN ),(Y1 ,...,YM )

where the sum is over ordered sets, but otherwise the restrictions apply. Q We mul|Z| tiply by Γ (Z), identify 2 Γ (Z) = Γ (Z) and use Γ (Z) ≤ p p p+1 p+1 i Γp+1 (Xi ) Q j Γp+1 (Yj ) which follows from the overlap connectedness. Then sum over Z

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with a pin, and use a spanning tree argument and the small norm hypotheses to obtain X (N + M )! M ˜ N kE(K, F )kG(κ+δκ),h,Γp ≤ (O(1))N+M kKk G(κ),h,Γp+2 kδf kΓp+2 N !M ! N≥1,M ≥0

˜ G(κ+δκ),h,Γ ≤ O(1)kKk . p+2 (In the last step use (N + M )!/N !M ! ≤ 2N+M . ) ˜ = K − (eF − 1)+ . We write Recall that K XY 1 Z dzj exp {zj F (Yj )} (eF − 1)+ (Y ) = 2πi zj (zj − 1) j

(85)

(86)

{Yj }

now with the integral over |zj |f (Yj ) = 1. Proceeding as above and using the first stability assumption we have XY O(1)f (Yj ) (87) k(eF − 1)+ (Y )kG(κ),h ≤ 2|Y | {Yj } j

and hence k(eF − 1)+ kG(κ),h,Γp+2 ≤

∞ X

(O(1))N kf kN Γp+4 ≤ O(1)kf kΓp+4 .

(88)

N=1

This gives the result. 2

II.4

Scaling

In the scaling step we define new polymer activities S(K) so that Exp(2 + K)(Λ, φL ) = Exp(2 + S(K))(L−1 Λ, φ) .

(89)

Here the scaled field is φL (x) = L−α φ(x/L) with α = dim φ = (d − 2)/2. After a rearrangement one finds X Y K(Yi , φL ) (90) S(K)(X, φ) = {Yi }→LX

i

where the Yi are disjoint but the L-closures Y¯iL overlap and fill LX. Theorem 9 Let c < (2d Ld/2 cs )−1 and define hL = L−α h. For any positive p, q there is a constant O(1) such that kS(K)kG(κ),h,Γp ≤ O(1)Ld kKkGL (κ),hL ,Γp−q provided kKkGL (κ),hL ,Γp−q is sufficiently small.

(91)

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Proof. Let Y = ∪i Yi . Since L−1 Y ⊂ X we have by a generalization of Lemma 7 and the bound c < (2d Ld/2 cs )−1 Y G(κ, L−1 Yi ) = G(κ, L−1 Y ) ≤ G(κ, X) . (92) i

The point here is that we need the Sobolev inequality on the L−1 scale which means that we must replace cs by the larger Ld/2 cs . In the definition of S(K) we write K(Yi , φL ) = KL−1 (L−1 Yi , φ) and by (92) and the multiplicative property of the norm we have X Y kS(K)(X)kG(κ),h ≤ kKL−1 (L−1 Yi )kG(κ),h . (93) {Yi }→LX

i

However kKL−1 (L−1 Y )kG(κ),h ≤ kK(Y )kGL (κ),hL and so X

Y

{Yi }→LX

i

kS(K)(X)kG(κ),h ≤

kK(Yi )kGL (κ),hL .

Now multiply by Γp (X). By the connectedness we have Γp (X) ≤ Γp (L−1 Y¯iL ). Furthermore we have the bound [11] for some constant O(1) : Γp (L−1 Y¯ L ) ≤ O(1)Γp−q (Y )

(94) Q i

(95)

Summing over X with a pin and using a spanning tree argument we obtain kS(K)kG(κ),h,Γp ≤

∞ X

(O(1)Ld kKkGL (κ),hL ,Γp−q )N .

(96)

N=1

This gives the result. 2 Remark.

The linearization given by S1 (K)(X, φ) =

X

K(Y, φL )

(97)

Y¯ L =LX

also satisfies the same bound.

II.5

Summary

We combine the three steps into one theorem which tells how the polymer activity changes under a single RG step. Our assumptions on the polymer activity K, the extraction F , and parameters κ, δκ, h, δh are as follows: 1. kKkG(κ),h,Γ is sufficiently small.

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2. The constants κ, c in G(κ) satisfy c ≤ (2d Ld/2 cs )−1 and κc−1 L2 is sufficiently small. 3. The inequality (δh)2 ≥ 8γ 2 kCk∗ kKkG(κ),h,Γ holds. 4. The extraction F is stable for (G` (κ), h − δh, f (X)) and for (G0` (δκ), h − δh, δf (X)) with constants f (X), δf (X) such that kf kΓ−1 , kδf kΓ−3 are sufficiently small and such that kf kΓ−1 ≤ O(1)kKkG(κ),h,Γ . Theorem 10 Under the above assumptions (µC ∗ Exp(2 + K)(Λ)) (φL ) = exp

X

!

F (X, φL )

Exp(2 + R(K, F ))(L−1 Λ, φ)

X⊂Λ

(98) where R(K, F ) = S(E(F(K), F )) .

(99)

kR(K, F )kG(κ+δκ),h−δh,Γ ≤ O(1)Ld kKkG(κ),h,Γ .

(100)

In addition Proof. If K # = F(K) then by conditions 2,3, Theorem 4 is applicable and so (µC ∗ Exp(2 + K)(Λ)) (φ) = Exp(2 + K # )(Λ, φ)

(101)

kK # kG` (κ),h−δh,Γ−3 ≤ 2kKkG(κ),h,Γ .

(102)

and Then we extract F and we find Exp(2 + K )(Λ, φ) = exp #

X

! F (X, φ) Exp(2 + K ∗ )(Λ, φ)

(103)

X⊂Λ

where K ∗ = E(K # , F ). The hypotheses of Theorem 8 hold for K # and p = −5: one has that kK # kG` (κ),h−δh,Γ−3 is sufficiently small by assumption 1 and (102). Therefore kK ∗ kG` (κ+δκ),h−δh,Γ−5 ≤ O(1)(kK # kG` (κ),h−δh,Γ−3 + kf kΓ−1 ) ≤ O(1)kKkG(κ),h,Γ (104) Finally we scale and find by Theorem 9 that Exp(2 + K ∗ )(Λ, φL ) = Exp(2 + K 0 )(L−1 Λ, φ)

(105)

where K 0 = S(K ∗ ) = R(K, F ), and since kK ∗ kG` (κ+δk),h−δh,Γ−5 is sufficiently small we have kK 0 kG(κ+δk),h−δh,Γ

≤ O(1)Ld kK ∗ kGL (κ+δk),(h−δh)L ,Γ−5 ≤ O(1)Ld kK ∗ kG` (κ+δk),h−δh,Γ−5 ≤ O(1)Ld kKkG(κ),h,Γ .

(106)

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This completes the proof. 2 Remark. The linearization R1 (K, F ) = S1 E1 (F1 K, F ) satisfies the same bound.

III More estimates The last theorem exhibits the obstruction to iterating the RG, namely the Ld growth factor. The aim in what follows is to exhibit special cases where one can beat this growth factor. There are three mechanisms which are more or less model independent: higher order terms, large sets, and scaling for small sets with extractions. A fourth mechanism is estimates on the fluctuation integral for small sets and charged polymers and is special to the two dimensional sine-Gordon model. We discuss each of these in turn.

III.1

Higher order terms

We show that if K, F are small enough then the higher order terms in R(K, F ) are even smaller. This fact, which follows from the next proposition with D = O(Ld ), will allow us to restrict attention to the linearized RG. Lemma 11 Suppose that K, F are small enough so that sK, sF satisfy the hypotheses of Theorem 10 for all complex s in the disc |s| ≤ D for some D ≥ 2. Then R(K, F ) = R1 (K, F ) + R≥2 (K, F )

(107)

where R1 (K, F ) is the linearization and kR≥2 (K, F )kG(κ+δκ),h−δh,Γ ≤ O(1)D−1 Ld kKkG(κ),h,Γ .

(108)

Proof. By Theorem 10 we have that R(sK, sF ) is well-defined for |s| ≤ D and satisfies (109) kR(sK, sF )kG(κ+δκ),h−δh,Γ ≤ O(1)DLd kKkG(κ),h,Γ . Furthermore it is not difficult to see that R(sK, sF ) is analytic in s . Expand around s = 0 and evaluate at s = 1 and obtain (107) with the remainder given by I 1 R(sK, sF ) ds . (110) R≥2 (K, F ) = 2πi |s|=D s2 (s − 1) Using the bound (109) and picking up an extra factor |s−2 | = D−2 we have the result. 2

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Large sets

We next study the linearization R1 (K, F ) on large sets, that is on large polymers. A polymer X is called small if it is connected and has |X| ≤ 2d . Otherwise it is a large polymer. The following gives favourable bounds for large sets : Lemma 12 Let K be supported on large sets. Then for any p, q > 0 kS1 (K)kG,h,Γp ≤ O(1)L−2 kKkGL ,hL ,Γp−q .

(111)

Under the hypotheses of theorem 10 : kS1 F1 KkG(κ+δκ),h−δh,Γ ≤ O(1)L−2 kKkG(κ),h,Γ .

(112)

Proof. The first bound follows by following the proof of Theorem 9 for the linear terms only, but replacing (95) by the stronger inequality ¯ L ) ≤ O(1)L−d−2 Γp−q (X) Γp (L−1 X

(113)

which is valid for large sets X. This inequality is proved in [11] and [6], Lemma 1. For the second bound we note that if K is supported on large sets then so is F1 K. Thus we can use the first bound followed by our bound on F1 . 2 Remark. The second bound gives a good bound on R1 (K, F ) = S1 E1 (F1 K, F ) since we will use it in a situation where E1 (K, F ) = K and hence R1 (K, F ) = S1 F1 K.

III.3

Small sets

For small sets the usual strategy would be to extract the fastest growing terms (the relevant variables) and get good bounds on the remainder. This generally works when the canonical scaling dimension of the field is positive. However in d=2 the field has dimension zero and any polynomial in the field is relevant, rendering the strategy intractable. For sine-Gordon we use the fact that the interaction is periodic under translations φ → φ + 2π in field space. This allows a Fourier analysis in this translation variable and a new contraction mechanism for the nonzero Fourier modes. The remaining zero modes depend only on ∂φ which has a positive dimension and thus these terms can be handled by extraction. We now give the details. Let K be a polymer activity which satisfies K(X, φ+2π) = K(X, φ). Expand K(X, Φ + φ) in a Fourier series in the real variable Φ X K(X, Φ + φ) = k0 (X, φ) + eiqΦ kq (X, φ) (114) q6=0

520

where

J. Dimock and T. R. Hurd

1 kq (X, φ) = 2π

Z

π

e−iqΦ K(X, Φ + φ)dΦ.

Ann. Henri Poincar´ e

(115)

−π

Then K(X, φ) = k0 (X, φ) +

X

kq (X, φ).

(116)

q6=0

The terms with q 6= 0 are called the charged terms and the q = 0 term is called the neutral term. The terminology is consistent with the Coulomb gas interpretation ¯ of the model. We sometimes also use the notation K(X, φ) = k0 (X, φ). Note that for a constant shift c of the field kq (X, φ + c) = eiqc kq (X, φ).

(117)

Also using G(κ, X, φ) = G(κ, X, φ + Φ) one can show kkq kG(κ),h,Γ ≤ kKkG(κ),h,Γ .

(118)

III.3.1 Charged sector Now we show how in dimension two only, the charged terms exhibit significantly improved behaviour under the fluctuation step. Lemma 13 Let K(X, φ) be supported on small sets, and be periodic in φ with Fourier coefficients kq (X, φ) as above. Then for q 6= 0 kµC ∗ kq kG` (κ),h,Γ−1 ≤ mq kkq kG(κ),h+NC ,Γ

(119)

inf kC(· − x) − C(0)kX sup small x∈X = exp[−(|q| − 1/2)C(0)].

(120)

where NC

=

X

mq

Remark. The right side of (119)Pcan also be bounded by mq kKkG(κ),h+NC ,Γ . Then if k0 = 0 so that K(X, φ) = q6=0 kq (X, φ) we have X kµC ∗ KkG` (κ),h,Γ−1 ≤ ( mq ) kKkG(κ),h+NC ,Γ ≤ O(1)e−C(0)/2 kKkG(κ),h+NC ,Γ . q6=0

(121) In d = 2, C(0) = O(log L), giving a significant decay factor for L large. In d > 2, C(0) = O(1), and the decay factor is not significant. Proof. We have

Z (µC ∗ kq )(X, φ) =

kq (X, φ + ζ)dµC (ζ) .

(122)

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Now let f be any function and shift the integral by ζ → ζ + iσq f where σq is the sign of q. We find our expression is Z −1 −1 e(f,C f )/2 e−iσq (ζ,C f ) kq (X, φ + ζ + iσq f ) dµC (ζ) . (123) Taking f (y) = C(y − x) where x is an arbitrary point of X gives Z C(0)/2 e−iσq ζ(x) kq (X, φ + ζ + iσq C(· − x))dµC (ζ) . e

(124)

Now use (117) with c = iσq C(0) to obtain Z (µC ∗ kq )(X, φ) = mq e−iσq ζ(x) kq,x (X, φ + ζ)dµC (ζ)

(125)

where kq,x (X, φ) = kq (X, φ + iσq (C(· − x) − C(0))) is a translation of kq . Taking derivatives and norms : k(µC ∗ kq )n (X, φ)k ≤ mq

(126)

Z k(kq,x )n (X, φ + ζ)kdµC (ζ) .

(127)

By Lemma 5, µC ∗ G(κ, X) ≤ G` (κ, X)2|X| and hence k(µC ∗ kq )(X)kG` ,h ≤ mq kkq,x (X)kG,h 2|X|

(128)

(still for any x ∈ X). Now in general we can estimate translations by kK(X, · + f )kG,h ≤ kK(X)kG,h+kf kX

(129)

where kf kX is defined in (23). This can be seen by making a power series expansion in f . We apply this to kq,x and choose x ∈ X to minimize kC(· − x) − C(0)kX , and find kkq,x (X)kG,h ≤ kkq (X)kG,h+NC . (130) Combining (128) and (130) gives the result. 2 Remark. The price we have paid for the strong contraction factor is a slight loss in the region of analyticity h + NC → h or h → h − NC . Iterating this is a problem in d = 2 since we do not recover analyticity in the scaling step. For the UV problem this could be overcome by taking h very large at the start. However for the IR problem we just have to do better. Lemma 14 Let the hypotheses of Lemma 13 hold. For 0 ≤ η ≤ 1 , and any p, r ≥ 0, kS1 kq kG(κ),h,Γp ≤ O(1)Ld eηhL |q| kkq kGL (κ),hL (1−η/2),Γp−r .

(131)

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Remark. Suppose d = 2 so that hL = h. The point of the lemma is that we have traded a slightly worse bound (the factor e|q|ηh ) for better analyticity (the improvement from h(1 − η/2) to h). If we combine Lemma 13 and Lemma 14 with the choice η = 2h−1 NC (assumed less than 1) we find kS1 F1 kq kG(κ),h,Γ

= kS1 (µC ∗ kq )kG(κ),h,Γ ≤ O(1)L2 eηh|q| k(µC ∗ kq )kGL (κ),h(1−η/2),Γ−1 ≤ O(1)L2 e2NC |q| mq kkq kG(κ),h,Γ .

(132)

Since C(0) = O(log L) and NC = O(1) the factor mq = exp(−(|q| − 1/2)C(0)) is stronger than the factor e2NC |q| . Hence we have accomplished the goal of finding a strong contraction factor without losing analyticity. (Of course we still have to see if it is strong enough to beat the factor L2 .) Before embarking on the proof of the lemma we note a preliminary result which exhibits improved scaling behavior when a function vanishes at a point. Lemma 15 Let Y be a small set in LX and suppose fL (y) = L−α f (x/L) vanishes at some point y∗ ∈ Y . Then kfL kY ≤ O(1)L−1−α kf kX .

(133)

Proof. First observe that ∂ β (fL (x)) = L−|β|−α (∂ β f )(L−1 x), so we need only look at the nonderivative term in the norm. Now note that for any y ∈ Y the length of the shortest rectilinear path within Y from y to y∗ , is less than O(1). Therefore since fL vanishes at y∗ |fL (y)| = |fL (y) − fL (y∗ )| ≤ O(1) sup |∂ β fL (z)| z∈Y,|β|=1

≤ O(1)L−1−α kf kX . Proof (of Lemma 14). With kq0 = S1 kq we have kq0 (X, φ) =

X

kq (Y, φL )

(134) 2

(135)

¯ L =LX Y :Y

where the sum is over small sets. For each term of (135) we use (117) to shift φL by a constant ηφL (y∗ ) where y∗ is an arbitrary point of Y . Then we have kq0 (X, φ) =

X Y¯ L =LX

eiqηφL (y∗ ) kq (Y, (1 − η)φL + η φ˜L ) .

(136)

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Here we have defined f˜(x) = f (x) − f (y ∗ /L) so that f˜L (y) = fL (y) − fL (y ∗ ). Lemma 15 implies k(1 − η)fL + η f˜L kY ≤ L−α [(1 − η) + (O(1)/L)η] ≤ L−α [1 − η/2]

(137)

whenever kf kX ≤ 1 and so when computing derivatives we obtain k(kq0 )n (X, φ)k ≤

X

X

¯ L =LX a+b=n Y

n! −nα L (|q|η)a (1−η/2)b k(kq )b (Y, (1−η)φL +η φ˜L )k a!b! (138)

We also have by (92) GL (κ, Y, (1 − η)φL + η φ˜L ) = G(κ, L−1 Y, φ) ≤ G(κ, X, φ)

(139)

and so k(kq0 )n (X)kG ≤

X

X

¯ L =LX a+b=n Y

and so

n! −nα L (|q|η)a (1 − η/2)b k(kq )b (Y )kGL a!b!

kkq0 (X)kG,h ≤ eηhL |q|

X

kkq (Y )kGL ,hL (1−η/2) .

(140)

(141)

Y¯ L =LX

The rest of the proof follows as in theorem 9. 2 III.3.2 Neutral sector Improved bounds can be arranged for general activities defined on small sets by extracting a finite number of terms characterised by low “scaling dimension”. As in [6] we define the scaling dimension dim K of any polymer activity K by dim(Kn ) = rn + n dim φ; dim(K) = inf dim(Kn ) n

(142)

where the infimum is taken over n such that Kn (X, 0) 6= 0. Here rn is defined to be the largest integer satisfying rn ≤ r and Kn (X, φ = 0; p×n ) = 0 whenever p×n = (p1 , . . . , pn ) is an n–tuple of polynomials of total degree less than rn . One can interpret rn as the number of derivatives present in the φn part of K (up to a maximum r). For comparison purposes we quote the following result from [6]: Theorem 16 Suppose d ≥ 3,, K(X, φ) is supported on small sets, and κh2 ≥ O(1). Then for any p, q ≥ 0 there is a constant O(1) such that kS1 (K)kG,h,Γp ≤ O(1)Ld−dim(K) kKkGL ,h,Γp−q .

(143)

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The proof needs dim φ > 0 and fails for d = 2. However we can obtain a similar result for d = 2 if we restrict to the neutral sector. Lemma 17 Suppose d = 2, K(X, φ) is supported on small sets and satisfies the neutrality condition K(X, φ + c) = K(X, φ) for any real c, and that κh2 ≥ O(1). Then for any p, q ≥ 0 there is a constant O(1) such that kS1 (K)kG,h,Γp ≤ O(1)L2−dim(K) kKkGL ,h,Γp−q .

(144)

Remark. The neutrality condition implies Kn (X, φ; f1 , ..., fn ) vanishes if any fi is a constant. Hence dim Kn = rn ≥ n for n < r and dim Kn = rn = r for n ≥ r. Proof. Starting from the definition (97) we have X

(S1 K)n (X, φ) =

(KL−1 )n (L−1 Y, φ) .

(145)

|Kn (Y, φL ; f1,L , ..., fn,L )| .

(146)

¯ L =LX Y :Y

Thus we need to estimate k(KL−1 )n (L−1 Y, φ)k =

sup kfi kX ≤1

By the remark above the supremum can be taken over fields fi such that fi,L vanishes at a point in Y . For such fields Lemma 15 applies again giving kfi,L kY ≤ O(1)L−1 kfi kX and it follows that k(S1 K)n (X, φ)k ≤

X

kKn (Y, φL )k(O(1)L−1 )n .

(147)

Y

We proceed as in the proof of Theorem 9, first summing only over n ≥ dim(K) so we can gain a factor L− dim(K) . With dim(K) = k we have X

hn /n!k(S1 K)n (X)kG ≤ O(1)L−k

X

kK(Y )kGL ,h .

(148)

Y

n≥k

We do something different for derivatives Kn with n < k. We have the representation k−1 X

1 ×(m−n) Km (Y, 0; φL × fL×n ) (m − n)! m=n Z 1 (1 − t)k−n−1 ×(k−n) + dt Kk (Y, tφL ; φL × fL×n ) .(149) (k − n − 1)! 0

Kn (Y, φL ; fL×n ) =

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Again by the neutrality condition we can replace φL by φ˜L (y) = φL (y) − φL (y∗ ) for some y∗ ∈ Y , and similarly for fL . Now in [6], Lemma 15, it is proved that |Kn (Y, 0; fL×n )| ≤ (O(1))n L− dim Kn kKn (Y, 0)k

n Y

kfj kX .

(150)

j=1

Use this bound on the terms in the sum. The remainder is estimated using kφ˜L kY ≤ ˜ X from Lemma 15. We find O(1)L−1 kφk X k(S1 K)n (X, φ)k ≤ O(1)L−k ( k−1 X

Y

Z

˜ m−n + kKm (Y, 0)k kφk X

m=n

0

)

1

˜ k−n (151) dt (1 − t)k−n−1 kKk (Y, tφL )k kφk X

Now multiply by G(κ, X, φ)−1 . For the remainder term we use G(κ, X, φ)−1

= G(κt2 , X, φ)−1 G(κ(1 − t2 ), X, φ)−1 ≤ GL (κt2 , Y, φL )−1 G(κ(1 − t2 ), X, φ)−1

(152)

where we have used (92) again. We next use ˜ a G(κ(1 − t2 ), X, φ)−1 ≤ O(1)(κ(1 − t2 ))−a/2 . sup kφk X

(153)

φ

This is a Sobolev inequality on derivatives of order up to r and needs s > d/2 + r. For the zeroeth derivative we dominate φ˜ by a first derivative and then use the Sobolev inequality. Here we use the fact that X is necessarily small and so has diameter O(1). Now the integral over t can be estimated by O(1)kKk (Y )kGL κ−(k−n)/2 . The terms in the sum over m are treated similarly and we end up with k XX kKm (Y )kGL κ−(m−n)/2 . (154) k(S1 K)n (X)kG ≤ O(1)L−k Y −1/2

Since κ

m=n

≤ O(1)h, this leads for n < k to X hn k(S1 K)n (X)kG ≤ O(1)L−k kK(Y )kGL ,h . n!

(155)

Y

Combining this with (148) we find k(S1 K)(X)kG,h ≤ O(1)L−k

X

kK(Y )kGL ,h

(156)

Y

and the result follows as before. 2

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IV The infrared problem We return to the sine-Gordon model in d = 2. The infrared problem for β > 8π is to study the partition function Z Z Z = exp ζ cos(φ(x))dx dµβv0M (φ). (157) ΛM

in the limit M → ∞. In particular we want to prove Theorem 1. We shall use a family of polymer activity norms defined for j = 0, 1, 2, ... by kKkj = k · kG(κj ),hj ,Γ

(158)

where the underlying φ–norms in (23) are taken with r = 4, s = 6. The large field regulator is G(κj ) defined by (25) with ! j X −k κj = κ0 2 . (159) k=0 −1

−1

2

We choose c = (8Lcs ) and κ0 c L sufficiently small that Lemma 5 holds (thus κ0 ≤ O(L−3 )). Note that κj increases slowly in j. The domain of analyticity is defined by   ∞ X (160) hj = h∞ 1 + 2−k  k=j+1 −1/2 (so with h∞ = κ0 1/2 1/2 κj hj 0 ≥ κ0 h∞ =

h∞ ≥ O(L )). Note that hj decreases slowly in j and that 1. Finally Γ is defined as in (27). We restate Theorem 1 as 3/2

follows : Theorem 18 Let β be chosen from a compact subset of (8π, ∞), let 0 < < 1, and let L be chosen sufficiently large. Then there is a number ζ¯ such that for all ζ real with |ζ| ≤ ζ¯ and any 0 ≤ j ≤ M the partition function has the form Z Ej Z=e Exp(2 + Kj )(ΛM −j , φ)dµβvM −j (σj ) (φ) (161) 0

where the polymer activities Kj are translation invariant, and even and 2π–periodic in φ. They satisfy the bounds kKj kj ≤ δ j |ζ|1− −2

where δ = O(1) max{L , L the field strength have the form

2−β/4π

Ej

(162)

} < 1/4. Furthermore the energy density and

=

j−1 X

δEk

k=0

σj

=

j−1 X k=0

δσk

(163)

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and satisfy the bounds |δEk | ≤ O(1) δ k |ζ|1− |ΛM −k | k 1− . |δσk | ≤ O(1) h−2 ∞ δ |ζ|

(164)

Remark. Since kKj k∞ ≤ kKj kj the version stated in Theorem 1 follows as well. Proof. The proof is by induction on j. For j = 0 we write the interaction as a sum over unit blocks, make a Mayer expansion, and then group together into connected components to obtain X XY exp( ζV (∆)) = (eζV (∆i ) − 1) = Exp(2 + K0 )(ΛM ) . (165) {∆i } i

∆⊂ΛM

Here K0 is supported on connected polymers and is given by Y K0 (X) = (eζV (∆) − 1) .

(166)

∆⊂X

However by Lemma 20 in the appendix we have the estimate for |ζ| sufficiently small (167) keζV (∆) − 1k1,h0 ≤ |ζ|1−/2 . Hence kK0 (X)k1,h0 ≤ (|ζ|1−/2 )|X| and it follows by a standard bound [13] that kK0 k0 ≤ |ζ|1− . Thus the representation and the bound hold for j = 0. Before proceeding to the general step of the induction we specify the extractions we want to make. For an expression Exp(2 + K)(Λ, φ),Rthe extracted part π ¯ F = F (K) is taken from the neutral sector K(X, φ) = (2π)−1 −π K(X, Φ + φ)dΦ ¯ −F ) for small sets. It is chosen satisfying F (X, φ+c) = F (X, φ) and so that dim(K ¯ is larger than zero. In fact we want to choose F so that dim(K − F ) ≥ 4 ( this is why we need r = 4). These conditions are more than sufficient to beat the factor L2 in the scaling step. As noted earlier the neutrality condition implies ¯ n ) ≥ min(n, 4) , and hence we may take Fn = 0 for n ≥ 4. Also note dim(K ¯ n (X, 0) = 0 for n odd, and hence we may take F1 , F3 = 0. The remaining that K conditions are for small sets X : ¯ − F )0 (X, 0) = 0 (K ¯ (K − F )2 (X, 0; xµ , xν ) = 0 ¯ (K − F )2 (X, 0; xµ , xν xρ ) = 0 . (168) P If we define the extracted part by F (X) = ∆ F (X, ∆) and Z Z X X (2) (2) 2 F (X, ∆, φ) = α(0) (X) + αµ,ν (X) (∂µ φ)(∂ν φ) + αµ,νρ (X) (∂µ φ)(∂νρ φ) µ,ν

∆

µ,νρ

∆

(169)

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then the conditions (168) determine ¯ 0 (X, 0) 1S (X) α(0) (X) = |X|−1 K (2) −1 ¯ αµ,ν (X) = (2|X|) K2 (X, 0; xµ , xν ) 1S (X) (2) ¯ 2 (X, 0; xµ , xν xρ ) 1S (X) αµ,νρ (X) = |X|−1 K

(170)

where 1S is the characteristic function of small sets. The last two equations define F = F (K). Now we continue with the induction, supposing the theorem is true for j and proving it for j + 1. The RG applied to Exp(2 + Kj )(ΛM −j , φ) starts with a fluctuation integral with the measure µβCj where Cj (x − y) = v0M −j (σj , x − y) − v0M −j−1 (σj , (x − y)/L) .

(171)

Let Fj be the map on polymer activities associated with this operation, so the new activities are Kj# = Fj (Kj ). Next we extract Fj = F (Kj# ) with coefficients αj as specified above. Finally we scale to the volume ΛM −j−1 . Thus as in Theorem 10 :

µβCj ∗ Exp(2 + Kj )(ΛM −j ) (φL )   X = exp  Fj (X, φL ) Exp(2 + Kj+1 )(ΛM −j−1 , φ)

(172)

X⊂ΛM −j

where Kj+1 = Rj (Kj ) ≡ S(E(Kj# , F (Kj# )) .

(173)

Using the lattice invariances one can prove that X

(0)

αj (X) = δEj

X⊃∆

X

αj,µ,ν (X) = −(2β)−1 δµν δσj (2)

X⊃∆

X

(2)

αj,µ,νρ (X) = 0

(174)

X⊃∆

for some constants δEj , δσj . Now (172) becomes (175) (µβCj ∗ Exp(2 + Kj )(ΛM −j ))(φL ) ! Z δσj (∂φ)2 Exp(2 + Kj+1 )(ΛM −j−1 , φ). (176) = exp δEj |ΛM −j | − 2β ΛM −j−1

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The partition function Z is the integral of this with respect to µβvM −j−1 (σj ) . Ab0 R sorbing the (∂φ)2 term into this measure changes v(σj ) to v(σj+1 ) with σj+1 = σj + δσj and we have Z = eEj+1

(177)

Z Exp(2 + Kj+1 )(ΛM −j−1 , φ)dµβvM −j−1 (σj+1 ) (φ) 0

where

"Z

Ej+1 = Ej + δEj |ΛM −j | + log

exp

−δσj 2β

(178)

!

Z 2

(∂φ) ΛM −j−1

# dµβvM −j−1 (σj ) (φ) 0

(179) This establishes the required form (161) for j + 1. Theorem 10 will be used to obtain a crude bound on kKj+1 kj+1 . With δhj = hj − hj+1 = 2−j−1 h∞ and δκj = κj+1 − κj = 2−j−1 κ0 we check the hypotheses of this theorem. 1. This is true by the inductive assumption on Kj for ζ¯ sufficiently small. 2. True by our choice of κ0 , c. 3. First note from Lemma 23 in the appendix that kβCj k∗ is bounded uniformly in j. Also δ j (δhj )−2 is bounded uniformly in j for L sufficiently large, and therefore kKj kj ≤ δ j |ζ|1− ≤ (8γ 2 kβCj k∗ )−1 (δhj )2 (180) holds for all j provided ζ¯ is small enough. 4. The stability conditions will be verified by using Lemma 21 in the appendix which involves X X (2) (2) kα(X)ka = |α(0) (X)| + a2 |αµν (X)| + a2 |(αµνρ (X)| . (181) µν

µνρ

By this lemma Fj is stable for (G0` (κj ), hj+1 , fj (X)) if we take the definition fj (X) = O(1)kαj (X)khj+1 . We need kfj kΓ−1 small and kfj kΓ−1 ≤ O(1)kKj kj and it suffices to show the latter. Now in the definition of αj (X) replace x by x−x∗ where x∗ is some point in X. Then we obtain the estimates for n = 0, 2 (n)

¯ n# (X, 0)k ≤ O(1)kKn# (X, 0)k ≤ O(1)kKn# (X)kG (κ ) . |αj (X)| ≤ O(1)kK j ` (182) It follows that kαj (X)khj+1 ≤ O(1)kK # (X)kG` (κj ),hj+1 .

(183)

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and hence kfj kΓ−3 ≤ O(1)kK # kG` (κj ),hj+1 ,Γ−3 ≤ O(1)kKj kj

(184)

Since f is supported on small sets the same bound holds for kfj kΓ−1 . Lemma 21 also says that F is stable for (G` (δκj ), hj+1 , δfj (X)) if we define δfj (X) = O(1)kαj (X)kδκ−1/2 . We must show that kδfj kΓ−3 is sufficiently j

−2 j+1 small under our hypotheses. We have that 1 ≤ δκ−1 and hence j hj+1 ≤ 2 j |δfj (X)| ≤ O(1)2 |fj (X)|. Therefore

|δfj |Γ−3 ≤ O(1)2j |fj |Γ−3 ≤ O(1)2j kKj kj ≤ O(1)(2δ)j |ζ|1−

(185)

which is small for ζ small. This verifies the hypotheses of Theorem 10 and we conclude kKj+1 kj+1 = kRj (Kj )kj+1 ≤ O(1)L2 kKj kj .

(186)

It remains to improve the crude bound on Kj+1 to kKj+1 kj+1 ≤ δkKj kj so we get the required kKj+1 kj+1 ≤ δ j+1 |ζ|1− . To accomplish this let 1S (respectively P 1S¯) be the characteristic function of small (large) sets, write Kj = q kq as in (116), and make the decomposition X Kj+1 = R≥2 (Kj ) + R1 (Kj 1S¯) + R1 ( kq 1S ) + R1 (k0 1S ) . (187) q6=0

We will show that each of the four terms on the right can be bounded by (δ/4)kKj kj . 1. As above one can check that Theorem 10 holds for sKj , sFj with |s| ≤ L4 . Then by Lemma 11 with D = L4 kR≥2 (Kj )kj+1 ≤ O(1)L−2 kKj kj ≤

δ kKj kj . 4

(188)

2. The extraction is zero on large sets and so by Lemma 12 kR1 (Kj 1S¯)kj+1 = kS1 F1 (Kj 1S¯)kj+1 ≤ O(1)L−2 kKj kj ≤

δ kKj kj . (189) 4

3. There is no extraction in R1 (kq 1S ) since the P extraction is based on F1 (kq 1S ) = F1 (k¯q 1S ) = 0. Hence the third term is q6=0 S1 F1 (kq 1S ) which we bound by putting together Lemmas 13, 14. As in (132) we have kS1 F1 (kq 1S )kj+1 ≤ O(1)L2 e2NβCj |q| e−(|q|−1/2)βCj (0) kkq kj .

(190)

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However by estimates on Cj in Lemma 23 in the Appendix we have NβCj ≤ βk∂Cj k∞ ≤ O(1) and Cj (0) =

(191)

M −j−1 log L /2 ). + O(e−L 2π(1 + σj )

(192)

Using also kkq kj ≤ kKj kj and the bound on σj we have for L sufficiently large :   X X −|q|(βC (0)−2N )+βC (0)/2 2 j j βCj   kq 1S kj+1 ≤ O(1)L e kKj kj kR1 q6=0

q6=0

≤ O(1)L δ ≤ kKj kj . 4

2−β/4π

kKj kj (193)

4. This term has the desired bound because of the extraction. Let K † = F1 (k0 1S ). Then we have R1 (k0 1S ) = S1 (K † − F (K † )). The extraction F ¯ † − F (K † )) ≥ 4, but we have K ¯ † = K † (since the is defined so that dim(K † † same is true of k0 ) and hence dim(K −F (K )) ≥ 4. Then Lemma 17 applies (note κj+1 h2j+1 ≥ 1) and gives kR1 (k0 1S )kj+1 ≤ O(1)L−2 kK † − F (K † )kG` (κj+1 ),hj+1 ,Γ−3 .

(194)

Now kK † kG` (κj+1 ),hj+1 ,Γ−3 ≤ O(1)kKj kj . Furthermore the same bound holds for F (K † ). To see this extend the argument of Lemma 21 in the appendix. If α† is defined from K † we argue as in (239) and (183) and find k(F (K † ))(X)kG` (κj+1 ),hj+1

≤ O(1) kα† (X)khj+1 ≤ O(1)kK † (X)kG` (κj+1 ),hj+1

(195)

which is enough. Thus kR1 (k0 1S )kj+1 ≤ O(1)L−2 kKj kj ≤

δ kKj kj . 4

(196)

This completes the bound on kKj+1 kj+1 . The last step is to establish the bounds (164). Using (182) we have |δEj | ≤ O(1)kK0# kG` (κj ),Γ−3 ≤ O(1)kKj kj ≤ O(1)δ j |ζ|1− −2 j 1− |δσj | ≤ O(1)βkK2# kG` (κj ),Γ−3 ≤ O(1)h−2 (197) j+1 kKj kj ≤ O(1)h∞ δ |ζ|

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We also need to bound δEj . Let v = v0M −j−1 (σj ) and let T = v 1/2 ∆v 1/2 , a positive self-adjoint operator. Doing the integral in (179) we find δEj = δEj |ΛM −j | + log det(1 + δσj T )−1/2 1 = δEj |ΛM −j | − tr (log(1 + δσj T )) . 2

(198)

But kT k ≤ 2 and |δσj | is small so the spectrum of δσj T is confined to a small neighborhood of the origin. Hence | log(1 + λ)| ≤ O(1)|λ| for any eigenvalue λ and hence |tr (log(1 + δσj T )) | ≤ O(1)tr(|δσj T |) = O(1)|δσj |tr(T ) ≤ O(1)|δσj ||ΛM −j−1 | (199) where the last step is an explicit computation. Now the bounds on δEj and δσj yield the bound |δEj | ≤ O(1)δ j |ζ|1− |ΛM −j |. This completes the proof of the infrared theorem. 2

V The ultraviolet problem The ultraviolet problem on the unit torus Λ0 for β < 8π is equivalent to a scaling limit for unit cutoff theories. Thus we study the N → ∞ limit of the partition function Z Z Z=

exp ζ−N ΛN

cos(φ(x))dx dµβv0N (φ).

(200)

After a number of RG transformations the RG index will increase from −N to a value j ≤ 0 and we will be on a volume Λ|j| with a coupling constant which will have grown from the ultra small ζ−N to |j|

ζj = L−2|j| eβv0

(0)/2

ζ.

(201)

At this point polymer activities are estimated with a norm essentially the same as for the IR problem, but with relaxed smoothness in φ characterized by r = 2, s = 4 in (23). We take k · kj = k · kG(κ),hj ,Γ (202) with c = (8Lcs )−1 , κ = O(L−3 ) sufficiently small so that Lemma 5 holds, and |j| X −k 2 hj = h0 1 + k=1

(which decreases in j) with h0 = κ−1/2 = O(L3/2 ), and Γ as in (27). Our aim is now to prove the following refinement of Theorem 2.

(203)

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Theorem 19 Let β be chosen from a compact subset of (0, 8π), let 0 < < 1/4, and let L be chosen sufficiently large. Then there is a number ζ¯ such that for all ζ complex with |ζ| ≤ ζ¯ and any −N ≤ j ≤ 0 the partition function has the form Z Ej Z=e (204) Exp(2 + Kj )(Λ|j| , φ)dµβv|j| (φ) . 0

The polymer activities Kj are translation invariant, even and 2π–periodic in φ, analytic in ζ and have the form ˜j Kj = ζj V + K

(205)

where V is given by (18). We have the estimates kζj V kj ˜ j kj kK

≤ |ζj |1− ≤ |ζj |2−4 .

(206)

Furthermore, the energy density has the form Ej =

j−1 X

δEk |Λ|k| |

(207)

k=−N

where |δEk | ≤ O(1)|ζk |2−4 .

(208)

Proof. The proof is by induction on j = −N, ... − 1. For j = −N the initial density can be written just as in the IR problem as Z exp(ζ−N cos φ) = Exp(2 + K−N )(ΛN , φ) (209) ΛN

where K−N is supported on connected polymers and given by Y K−N (X) = (eζ−N V (∆) − 1) .

(210)

∆⊂X

˜ −N (∆) where If X = ∆ we write K−N (∆) = ζ−N V (∆) + K ˜ −N (∆) = eζ−N V (∆) − ζ−N V (∆) − 1 . K

(211)

2− ˜ −N (∆)k1,h now follows from Lemma 20 in the apThe bound kK −N ≤ |ζ−N | ˜ pendix. Also for |X| ≥ 2 we have kK−N (X)k1,h−N = kK−N (X)k1,h−N ≤ O(1)(|ζ−N |1− )|X| . From these two bounds we can deduce that for j = −N

˜ −N (X)kG(κ),h ,Γ = |ζ−N |2−2 . kK −N

(212)

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Thus the theorem is established for j = −N . Next we specify the extractions F (K) from a polymer activity K in the general step. Again the extraction is from the neutral part on small sets, but now ¯ − F (K)) ≥ 2. Thus we extract only the constant we only need dim(K ¯ (F (K))(X) = α(X)|X| = K(X, 0) 1S (X) .

(213)

Now we suppose the theorem is true for j and prove it for j + 1. Starting with Exp(2 + Kj )(Λ|j| , φ) we do a fluctuation integral with the measure µβCj where |j|

|j+1|

Cj (x − y) = v0 (0, x − y) − v0

(0, (x − y)/L).

(214)

This produces new polymer activities Kj# = F(Kj ). Then we extract Fj (X) = ¯ # (X, 0)1S as above. Finally we scale down to the volume Λ|j+1| . αj (X)|X| = K j Thus we have as in Theorem 10 (µβC ∗ Exp(2 + Kj )(Λ|j| ))(φL )   X Fj (X) Exp(2 + Kj+1 )(Λ|j+1| , φ) = exp  X⊂Λ|j|

= exp δEj |Λ|j| | Exp(2 + Kj+1 )(Λ|j+1| , φ)

(215)

where =

δEj

X

αj (X)

X⊃∆

= R(Kj ) = S(E(Kj# , F (Kj# ))) .

Kj+1

(216)

The partition function is obtained from (215) by multiplying by eEj and integrating with respect to µβv|j+1| and has the required form 0

Z = eEj+1

Z Exp(2 + Kj+1 )(Λ|j+1| , φ)dµβv|j+1| (φ)

(217)

0

where Ej+1 = Ej + δEj |Λ|j| |.

(218)

Next we check the hypotheses of Theorem 10 with δhj = hj −hj+1 and δκ = 0. It is easier than before since only constants are extracted. A degenerate version of Lemma 21 with α(2) = 0 implies that Fj is stable for (G` (κ), hj+1 , f (X)) and (1, hj+1 , δf (X)) with f (X) = δf (X) = O(1)|αj (X)| . Since |αj |Γ−3 ≤ O(1)kKj kj is certainly small enough, the stability assumption of Theorem 10 holds. The other assumptions are easily checked and we conclude kKj+1 kj+1 ≤ O(1)L2 kKj kj .

(219)

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The leading behaviour of the RG is given by noting that R1 (ζj V ) = ζj+1 V .

(220)

Indeed simple computations give F1 (V ) = e−βC(0)/2 V and E1 (V, F (V )) = V − F (V ) = V and S1 V = L2 V . Thus R1 (ζj V ) = L2 e−βC(0)/2 ζj V and since L2 e−βC(0)/2 ζj = ζj+1 the claim is verified. Because of this we now have :

˜j = If we expand K ˜ j+1 K

P

˜ j+1 = R1 (K ˜ j ) + R≥2 (Kj ) . K

kq on small sets as before this can be written X ˜ j 1S¯) + R1 ( = R≥2 (Kj ) + R1 (K kq 1S ) + R1 (k0 1S ) .

(221)

q

(222)

q6=0

˜ j+1 kj+1 ≤ |ζj+1 |2−4 we show that each of the four terms on the To show kK right of (222) can be bounded by |ζj+1 |2−4 /4. 1. One checks that Theorem 10 holds for sKj , sFj with s ≤ |ζj |−1+2 . Then by Lemma 11 with D = |ζj |−1+2 we have kR≥2 (Kj )kj+1 ≤ O(1)L2 |ζj |1−2 kKj kj ≤ |ζj |2−4 /4 .

(223)

˜ j 1S¯) = S1 F1 (K ˜ j 1S¯). There2. No extractions are taken from large sets so R1 (K fore we can use Lemma 12 and find ˜ j 1S¯)kj+1 ≤ O(1)L−2 kK ˜ j 1S¯kj ≤ |ζj |2−4 /4 ≤ |ζj+1 |2−4 /4 . kR1 (K

(224)

3. The third term is bounded using Lemmas 13,14 just as in the infrared section, and we gain a factor e−βC(0)/2+2NβC = O(1)L−β/4π . We have   X X ˜ j kj kR1  kq  1S kj+1 ≤ O(1)L2 e−|q|(βC(0)−2NβC )+βC(0)/2 kK q6=0

q6=0

≤ O(1)L

2−β/4π

|ζj |2−4

≤ O(1)L(2−β/4π)(1−(2−4)) |ζj+1 |2−4 ≤ |ζj+1 |2−4 /4 .

(225)

Here we have used |ζj | ≤ O(1)L−(2−β/4π) |ζj+1 |. 4. Letting K † = F1 (k0 1S ) we have R1 (k0 1S ) = S1 (K † −F (K † )). The extraction F is now defined so that dim(K † − F (K † )) ≥ 2 and Lemma 17 gives kR1 (k0 1S )kj+1 ≤ O(1)kK † − F (K † )kG` ,hj+1 ,Γ−3 .

(226)

˜ j kj and thus This is bounded by O(1) kK † kG` ,hj+1 ,Γ−3 ≤ O(1)kK kR1 (k0 1S )kj+1 ≤ O(1)|ζj |2− ≤ |ζj+1 |2− /4 .

(227)

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˜ j+1 kj+1 is complete. Next we need the bound on δEj . Now the bound on kK We have as before |δEj | ≤ O(1)kKj# kG` ,hj+1 ,Γ−3 ≤ O(1)kKj kj ≤ O(1)|ζj |1− .

(228)

But we are claiming more, namely that the bound is actually O(1)|ζj |2−4 . To see ¯ # where K # = F(Kj ) = F1 (Kj ) + the improvement note that δEj depends on K j j ¯ j ) and since V¯ = 0 this term only depends of K ˜j. F≥2 (Kj ) Since F1 (Kj ) = F1 (K Thus both terms are O(|ζj |2−4 ). We omit the details of this estimate. The analyticity of Kj (X, φ) in ζ follows by observing that K−N (X, φ) is analytic for complex |ζ| ≤ ζ¯ and that each RG transformation preserves this property. This completes the proof of the ultraviolet theorem. 2

A Appendix A.1 Estimates on potentials Lemma 20 Let V (∆, φ) = ζ.

R

∆

cos(φ(x))dx for a unit block ∆. Then for any complex

kV (∆)kG=1,h keζV (∆) kG=1,h

≤ eh ≤ 2 exp(|ζ|e2h ) .

(229)

Furthermore for 0 < < 1 and |ζ| sufficiently small (depending on h, ) keζV (∆) − 1kG=1,h keζV (∆) − ζV (∆) − 1kG=1,h

≤ |ζ|1− ≤ |ζ|2− .

(230)

Proof. A computation shows that kVn (∆, φ)k ≤ 1 and the first bound follows. For the second bound we compute the nth derivative and resum as in [5] and find ! ∞ X (2h)n (2h)n ζV (∆) )n (φ)k ≤ exp (231) k(e |ζ|kVn (∆, φ)k . n! n! n=0 Again we use kVn (∆, φ)k ≤ 1 and then take the supremum over φ to obtain (2h)n k(eζV (∆) )n kG=1 ≤ exp |ζ|e2h . n! Now multiply by 2−n and sum over n to get the result.

(232)

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For the third bound we write ζV (∆)

e

1 −1= 2πi

Z

ezζV (∆) dz z(z − 1)

(233)

where the contour is the circle |z| = |ζ|−1+/2 ≥ 2. Since kezζV (∆) k1,h ≤ O(1) for |ζ| small by the second bound we get a bound O(1)|ζ|1−/2 ≤ |ζ|1− . The fourth bound is similar. This completes the proof. 2 The next lemma is useful in verifying the stability hypothesis. Fix a unit square ∆ and consider a family of quadratic polynomials F (X, ∆) defined for small sets X ⊃ ∆ which have the form Z X (2) αab (X) ∂ a φ(x)∂ b φ(x)dx (234) F (X, ∆) = α(0) (X) + ∆

1≤|a|,|b|≤r

where a, b are multi-indices. (We could as well include a term linear in ∂φ.) We also define X (2) kα(X)ka = |α(0) (X)| + a2 |α(2) (X)| ≡ |α(0) (X)| + a2 |αab (X)| . (235) ab

Lemma 21 Let α(X) be supported on small sets and let a = max{κ−1/2 , h} for κ ≤ 1 and h ≥ 1. Also let k = O(1) be the number of small sets containing a unit block ∆. Then for all complex z(X) satisfying 40k|z(X)|kα(X)ka ≤ 1 we have k exp

X

(236)

! z(X)F (X, ∆) kG0 (κ),h ≤ 2 .

(237)

X⊃∆

Thus F is stable for (G0 (κ), h, 40kkα(X)ka ), Remark. Similarly F is stable for (G0` (κ), h, O(1)kα(X)ka ) with a larger constant O(1). Proof. We have as above P (3h)n X⊃∆ z(X)F (X, ∆)) n (φ)k ≤ n! k exp( P P2 (3h)n |z(X)| kF (X, ∆, φ)k . exp n X⊃∆ n=0 n!

(238)

Now compute the derivatives and estimate them by |F0 (X, ∆, φ)|

≤ |α(0) (X)| + |α(2) (X)|k∂φk2s,∆

kF1 (X, ∆, φ)k

≤ 2|α(2) (X)|k∂φks,∆

kF2 (X, ∆, φ)k

≤ 2|α(2) (X)| .

(239)

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Then estimate 2 X (3h)n kFn (X, ∆, φ)k ≤ |α(0) (X)| + k∂φk2s,∆ + 6hk∂φks,∆ + 9h2 |α(2) (X)| n! n=0 ≤ |α(0) (X)| + 10k∂φk2s,∆ + 10h2 |α(2) (X)|

≤ |α(0) (X)| + 10a2 (1 + κk∂φk2s,∆ )|α(2) (X)| ≤ 40(1/4 + κk∂φk2s,∆ )kα(X)ka .

(240)

Now since 40|z(X)|kα(X)ka ≤ k−1 we find X X⊃∆

|z(X)|

2 X (3h)n kFn (X, ∆, φ)k ≤ 1/4 + κk∂φk2∆,s . n! n=0

(241)

Using this in (238) yields ! X 1 (3h)n z(X)F (X, ∆)) kG0 (κ) ≤ e 4 . k exp( n! X⊃∆

(242)

n

Now multiply by 3−n and sum over n to obtain the result. 2

A.2 Estimates on covariances Let C∞ (σ, x) be the covariance on Rd , d ≥ 2 defined by C∞ (σ, x) = (2π)−d

Z dp Rd

4 4 4 eipx [(ep + σ)−1 − (eL p + σ)−1 ] . 2 p

(243)

Lemma 22 1. There is σ0 = O(1) such that for |σ| ≤ σ0 and any multi-index β there are constants c1 , c2 such that Z

|∂ β C∞ (σ, x)| |∂ β C∞ (σ, x)|dx

≤ c1 exp(−|x|/L) ≤ c2 .

(244)

The constant c1 = O(1) log L for d = 2, β = 0, but may be chosen indepenRL dent of L otherwise. We also have c2 ≤ O(1) 1 s1−|β| ds. 2. In d = 2, C∞ (σ, 0) =

log L . 2π(1 + σ)

(245)

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Proof. We rewrite the covariance and its derivatives as Z Z L 4 4 eipx ∂ ∂ β C∞ (σ, x) = (2π)−d ds dp 2 (ip)β (− )(es p + σ)−1 p ∂s 1 Rd Z L Z ipx 3 4 s4 p4 e (4s p e ) = (2π)−d ds dp 2 (ip)β s4 p4 2 p (e + σ) d 1 R " # Z L Z 4 ds ep −d is−1 px β 2 = 4(2π) dp e (ip) p (246) d−1+|β| (ep4 + σ)2 1 s Rd The function in brackets is analytic, bounded and integrable in the strip |Im(p)| ≤ 1 around the real axis when |σ| is small. Therefore we can shift the p integral one unit in an imaginary direction and exhibit the exponential decay in x. We find Z L −1 ds |∂ β C∞ (σ, x)| ≤ O(1) e−s |x| (247) d−1+|β| s 1 and the bounds (244) follow. In d = 2 we compute Z ∞ Z L 4 r2 er ds −2 2πrdr r4 (248) C∞ (σ, 0) = π s 0 (e + σ)2 1 log L . (249) = 2π(1 + σ) This completes the proof. 2 Now let C M (σ, x) be the covariance on ΛM as defined in (22). Lemma 23 Let |σ| ≤ σ0 . 1. For any multi-index β and |x| ≤ LM /2 Z

|∂ β C M (σ, x)| ≤ O(1)c1 exp(−|x|/L) |∂ β C M (σ, x)|dx ≤ O(1)c2 .

(250)

2. In d = 2, C M (σ, 0) =

M −1 log L /2 . + O(1)e−L 2π(1 + σ)

Proof. We have the representation C M (σ, x) =

X

C∞ (σ, x + nLM ) .

(251)

(252)

n∈Z2

This follows since both sides are doubly periodic with period LM , and they have 4 4 4 the same Fourier coefficients, namely p−2 ((ep + σ)−1 − (eL p + σ)−1 ) for p 6= 0 and 0 for p = 0. The terms in the sum are estimated by the previous lemma and we obtain all the stated results. 2

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Acknowledgement We thank David Brydges for his encouragement and many helpful conversations.

References [1] A. Abdesselam, V. Rivasseau, Trees, forests, and jungles: a botanical garden for cluster expansions, in V. Rivasseau, editor, Constructive Physics, 311–326, Berlin, 1995, Springer. [2] G. Benfatto, G. Gallavotti, F. Nicolo, On the massive sine-Gordon equation in the first few regions of collapse. Commun. Math. Phys. 83, 387–410 (1982) [3] D.Brydges, J.Dimock, T.R. Hurd, Weak perturbations of Gaussian measures. In J.Feldman, R.Froese, and L.Rosen, editors, Mathematical Quantum Theory I: Field Theory and Many–body Theory. AMS, 1994. [4] D.Brydges, J.Dimock, T.R. Hurd, Applications of the renormalization group. In J.Feldman, R.Froese, and L.Rosen, editors, Mathematical Quantum Theory I: Field Theory and Many–body Theory. AMS, 1994. [5] D.C. Brydges, J.Dimock, T.R. Hurd, The short distance behavior of φ43 . Commun. Math. Phys., 172, 143–186, 1995. [6] D.C. Brydges, J.Dimock, T.R. Hurd, Estimates on renormalization group transformations, Can. J. Math. 50, 756–793, (1998) [7] D.C. Brydges, J.Dimock, T.R. Hurd,, A non-Gaussian fixed point for φ4 in 4 − dimensions, Commun. Math. Phys. 198, 111–156, (1998) [8] D. Brydges and G. Keller, Correlation functions of general observables in dipole-type systems, Helv. Phys. Acta67, 43–116 (1994). [9] D. Brydges, T. Kennedy, Mayer expansions and the Hamilton-Jacobi equation, J. Stat. Phys. 48, 19 (1987). [10] D. Brydges, P. Martin, Coulomb systems at low density, cond-mat/9904122 [11] D.C. Brydges, H.T. Yau, Grad φ perturbations of massless Gaussian fields. Commun. Math. Phys., 129, 351–392, (1990). [12] J. Dimock, Bosonization of Massive Fermions Commun. Math. Phys. 198, 247–281, (1998). [13] J.Dimock, T.R. Hurd, A renormalization group analysis of the KosterlitzThouless phase. Commun. Math. Phys., 137, 263–287, (1991).

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[14] J. Dimock, T. R. Hurd, A renormalization group analysis of correlation functions for the dipole gas. J. Stat. Phys., 66, 1277–1318, (1992). [15] J.Dimock, T.R.Hurd, Construction of the two-dimensional sine-Gordon model for β < 8π. Commun. Math. Phys., 156, 547–580, (1993). [16] J.Fr¨ ohlich, E.Seiler, The massive Thirring-Schwinger model (QED2 ): convergence of perturbation theory and particle structure. Helv. Phys. Acta, 49, 889–, (1976). [17] J. Fr¨ohlich, T. Spencer, The Kosterlitz-Thouless transition in two-dimensional Abelian spin systems and Coulomb gas, Commun. Math. Phys. 81, 527–602, (1981). [18] J. Glimm, A. Jaffe, T. Spencer, The particle structure of the weakly coupled P (φ)2 model and other applications of high temperature expansions. in G. Velo., A. Wightman, eds, Constructive Quantum field theory, Springer-Verlag, New York, 1973 [19] T.R. Hurd, Charge correlations for the two-dimensional Coulomb gas. in V. Rivasseau, editor, Constructive Physics, 311-326, Berlin, 1995, Springer. [20] J.M. Kosterlitz, D.J. Thouless: Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C 6, 1181–1203 (1973) [21] D.H.U. Marchetti, A. Klein, Power-law falloff in two- dimensional Coulomb gases at inverse temperature β > 8π. J. Stat. Phys. 64, 135–162 (1991) [22] F. Nicolo, P. Perfetti, The sine-Gordon field theory model at α2 = 8π, the non-superrenormalizable theory. Commun. Math. Phys. 123, 425–452 (1989) [23] F. Nicolo, J. Renn, A. Steinmann: On the massive sine-Gordon equation in all regions of collapse. Commun. Math. Phys. 105, 291–326 (1986) J. Dimock Research supported by NSF Grant PHY9722045 Dept. of Mathematics SUNY at Buffalo Buffalo, N.Y. 14214

T.R. Hurd Research supported by Natural Sciences and the Engineering Research Council of Canada Dept. of Mathematics and Statistics McMaster University Hamilton, Ontario, L8S 4K1 e-mail : [email protected]

Communicated by V. Rivasseau submitted 08/09/99, accepted 01/12/99

Ann. Henri Poincar´ e 1 (2000) 543 – 567 c Birkh¨

auser Verlag, Basel, 2000 1424-0637/00/030543-25 $ 1.50+0.20/0

Annales Henri Poincar´ e

Maximum Principles for Null Hypersurfaces and Null Splitting Theorems Gregory J. Galloway

I Introduction The geometric maximum principle for smooth (spacelike) hypersurfaces, which is a consequence of Alexandrov’s [1] strong maximum for second order quasi-linear elliptic operators, is a basic tool in Riemannian and Lorentzian geometry. In [2], extending earlier work of Eschenburg [7], a version of the geometric maximum principle in the Lorentzian setting was obtained for rough (C 0 ) spacelike hypersurfaces which obey mean curvature inequalities in the sense of support hypersurfaces. In the present paper we establish an analogous result for null hypersurfaces (Theorem III.4) and consider some applications. For the applications, it is important to have a version of the maximum principle for null hypersurfaces which does not require smoothness. The reason for this, which is described in more detail in Section 3, is that the null hypersurfaces which arise most naturally in spacetime geometry and general relativity, such as black hole event horizons, are in general C 0 but not C 1 . To establish our basic approach, we first prove a maximum principle for smooth null hypersurfaces (Theorem II.1), and then proceed to the C 0 case. The general C 0 version is then applied to study some rigidity properties of spacetimes which contain null lines (inextendible globally maximal null geodesics). The standard Lorentzian splitting theorem, which is the Lorentzian analogue of the Cheeger-Gromoll splitting theorem of Riemannian geometry, describes the rigidity of spacetimes which contain timelike lines (inextendible globally maximal timelike geodesics); see [3, Chapter 14] for a nice exposition. Here we show how the maximum principle for rough null hypersurfaces can be used to obtain a general “splitting theorem” for spacetimes with null lines (Theorem IV.1). We then consider an application of this null splitting theorem to asymptotically flat spacetimes. We prove that a nonsingular asymptotically flat (in the sense of Penrose [17]) vacuum (i.e., Ricci flat) spacetime which contains a null line is isometric to Minkowski space (Theorem IV.3). In Section 2 we review the relevant aspects of the geometry of null hypersurfaces and present the maximum principle for smooth null hypersurfaces. In Section 3 we present the maximum principle for C 0 null hypersurfaces. In Section 4 we consider the aforementioned applications. For basic notions used below from Lorentzian geometry and causal theory, we refer the reader to the excellent references, [3], [13], [16] and [18].

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II The maximum principle for smooth null hypersurfaces II.1

The geometry of null hypersurfaces

Here we review some aspects of the geometry of null hypersurfaces. For further details, see e.g. [14] which is written from a similar point of view. Let M be a spacetime, i.e., a smooth time-oriented Lorentzian manifold, of dimension n ≥ 3. We denote the Lorentzian metric on M by g or h , i. A (smooth) null hypersurface in M is a smooth co-dimension one embedded submanifold S of M such that the pullback of the metric g to S is degenerate. Because of the Lorentz signature of g, the null space of the pullback is one dimensional at each point of S. Hence, every null hypersurface S admits a smooth nonvanishing future directed null vector field K ∈ ΓT S such that the normal space of K at p ∈ S coincides with the tangent space of S at p, i.e., Kp⊥ = Tp S for all p ∈ S. It follows, in particular, that tangent vectors to S not parallel to K are spacelike. It is well-known that the integral curves of K, when suitably parameterized, are null geodesics. These integral curves are called the null geodesic generators of S. We note that the vector field K is unique up to a positive (pointwise) scale factor. Since K is orthogonal to S we can introduce the null Weingarten map and null second fundamental form of S with respect K in a manner roughly analogous to what is done for spacelike hypersurfaces or hypersurfaces in a Riemannian manifold. We introduce the following equivalence relation on tangent vectors : For X, X 0 ∈ Tp S, X 0 = X mod K if and only if X 0 −X = λK for some λ ∈ R. Let X denote the equivalence class of X. Simple computations show that if X 0 = X mod K and Y 0 = Y mod K then hX 0 , Y 0 i = hX, Y i and h∇X 0 K, Y 0 i = h∇X K, Y i, where ∇ is the Levi-Civita connection of M . Hence, for various quantities of interest, components along K are not of interest. For this reason one works with the tangent space of S modded out by K, i.e., let Tp S/K = {X : X ∈ Tp S} and T S/K = ∪p∈S Tp S/K. T S/K is a rank n − 2 vector bundle over S. This vector bundle does not depend on the particular choice of null vector field K. There is a natural positive definite metric h in T S/K induced from h , i: For each p ∈ S, define h : Tp S/K × Tp S/K → R by h(X, Y ) = hX, Y i. From remarks above, h is well-defined. The null Weingarten map b = bK of S with respect to K is, for each point p ∈ S, a linear map b : Tp S/K → Tp S/K defined by b(X) = ∇X K. It is easily e = f K, f ∈ C ∞ (S), is any other future verified that b is well-defined. Note if K e = f ∇X K mod K. It follows directed null vector field tangent to S, then ∇X K that the Weingarten map b of S is unique up to positive scale factor and that b at a given point p ∈ S depends only on the value of K at p. A standard computation shows, h(b(X), Y ) = h∇X K, Y i = hX, ∇Y Ki = h(X, b(Y )). Hence b is self-adjoint with respect to h. The null second fundamental form B = BK of S with respect to K is the bilinear form associated to b via h: For each p ∈ S, B : Tp S/K × Tp S/K → R is defined by B(X, Y ) = h(b(X), Y ) =

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h∇X K, Y i. Since b is self-adjoint, B is symmetric. We say that S is totally geodesic iff B ≡ 0. This has the usual geometric consequence: A geodesic in M starting tangent to a totally geodesic null hypersurface S remains in S. Null hyperplanes in Minkowski space are totally geodesic, as is the event horizon in Shwarzschild spacetime. The null mean curvature of S with respect to K is the smooth scalar field θ ∈ C ∞ (S) defined by θ = tr b. Let e1 , e2 , ...en−2 be n−2 orthonormal spacelike vectors (with respect to h, i) tangent to S at p. Then {e1 , e2 , ...en−2 } is an orthonormal basis (with respect to h) of Tp S/K. Hence at p, θ

= tr b =

n−2 X

h(b(ei ), ei )

i=1

=

n−2 X

h∇ei K, ei i.

(II.1)

i=1

Let Σ be the properly transverse intersection of a hypersurface P in M with S. By properly transverse we mean that K is not tangent to P at any point of Σ. Then Σ is a smooth (n − 2)-dimensional spacelike submanifold of M contained in S which meets K orthogonally. From Equation II.1, θ|Σ = divΣ K, and hence the null mean curvature gives a measure of the divergence of the null generators of e = f K then θe = f θ. Thus the null mean curvature inequalities S. Note that if K θ ≥ 0, θ ≤ 0, are invariant under positive rescaling of K. In Minkowski space, a future null cone S = ∂I + (p) \ {p} (resp., past null cone S = ∂I − (p) \ {p}) has positive null mean curvature, θ > 0 (resp., negative null mean curvature, θ < 0). The null second fundamental form of a null hypersurface obeys a well-defined comparison theory roughly similar to the comparison theory satisfied by the second fundamental forms of a family of parallel spacelike hypersurfaces (cf., Eschenburg [6], which we follow in spirit). Let η : (a, b) → M , s → η(s), be a future directed affinely parameterized null geodesic generator of S. For each s ∈ (a, b), let b(s) = bη0 (s) : Tη(s) S/η 0 (s) → Tη(s) S/η 0 (s) be the Weingarten map based at η(s) with respect to the null vector K = η 0 (s). The one parameter family of Weingarten maps s → b(s), obeys the following Ricatti equation, b0 + b2 + R = 0.

(II.2)

Here 0 denotes covariant differentiation in the direction η 0 (s); if X = X(s) is a vector field along η tangent to S, we define, b0 (X) = b(X)0 − b(X 0 ).

(II.3)

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R : Tη(s) S/η 0 (s) → Tη(s) S/η 0 (s) is the curvature endomorphism defined by R(X) = R(X, η 0 (s))η 0 (s), where (X, Y, Z) → R(X, Y )Z is the Riemann curvature tensor of M , R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z. We indicate the proof of Equation II.2. Fix a point p = η(s0 ), s0 ∈ (a, b), on η. On a neighborhood U of p in S we can scale the null vector field K so that K is a geodesic vector field, ∇K K = 0, and so that K, restricted to η, is the velocity vector field to η, i.e., for each s near s0 , Kη(s) = η 0 (s). Let X ∈ Tp M . Shrinking U if necessary, we can extend X to a smooth vector field on U so that [X, K] = ∇X K − ∇K X = 0. Then, R(X, K)K = ∇X ∇K K − ∇K ∇X K − ∇[X,K] K = −∇K ∇K X. Hence along η we have, X 00 = −R(X, η 0 )η 0 (which implies that X, restricted to η, is a Jacobi field along η). Thus, from Equation II.3, at the point p we have, b0 (X) = ∇X K 0 − b(∇K X) = ∇K X 0 − b(∇X K) = X 00 − b(b(X)) = −R(X, η 0 )η 0 − b2 (X) = −R(X) − b2 (X), which establishes Equation II.2. By taking the trace of II.2 we obtain the following formula for the derivative of the null mean curvature θ = θ(s) along η, θ0 = −Ric(η 0 , η 0 ) − σ2 −

1 θ2 , n−2

(II.4)

where σ, the shear scalar, is the trace of the square of the trace free part of b. Equation II.4 is the well-known Raychaudhuri equation (for an irrotational null geodesic congruence) of relativity. This equation shows how the Ricci curvature of spacetime influences the null mean curvature of a null hypersurface.

II.2

The maximum principle for smooth null hypersurfaces

The aim here is to prove the geometric maximum principle for smooth null hypersurfaces. Because of its natural invariance, we restrict attention to the zero mean curvature case. In the statement we make use of the following intuitive terminology. Let S1 and S2 be null hypersurfaces that meet at a point p. We say that S2 lies to the future (resp., past) side of S1 near p provided for some neighborhood U of p in M in which S1 is closed and achronal, S2 ∩ U ⊂ J + (S1 ∩ U, U ) (resp., S2 ∩ U ⊂ J − (S1 ∩ U, U )). Theorem II.1 Let S1 and S2 be smooth null hypersurfaces in a spacetime M . Suppose, (1) S1 and S2 meet at p ∈ M and S2 lies to the future side of S1 near p, and (2) the null mean curvature scalars θ1 of S1 , and θ2 of S2 , satisfy, θ1 ≤ 0 ≤ θ2 .

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Then S1 and S2 coincide near p and this common null hypersurface has null mean curvature θ = 0. The proof is an application of Alexandrov’s [1] strong maximum principle for second order quasi-linear elliptic operators. It will be convenient to state the precise form of this result needed here. For connected open sets Ω ⊂ Rn and U ⊂ Rn × R × Rn , we say u ∈ C 2 (Ω) is U -admissible provided (x, u(x), ∂u(x)) ∈ U for all x = (x1 , x2 , ..., xn ) ∈ Ω, where ∂u ∂u = (∂1 u, ∂2 u, ..., ∂n u), and ∂i u = ∂x i. Let Q = Q(u) be a second order quasi-linear operator, i.e., for U -admissible u ∈ C 2 (Ω), consider Q(u) =

n X

aij (x, u, ∂u)∂ij u + b(x, u, ∂u),

(II.5)

i,j=1 2

where aij , b ∈ C 1 (U ), aij = aji , and ∂ij = ∂u∂j ∂ui . The operator Q is elliptic provided for each (x, r, p) ∈ U , and for all ξ = (ξ 1 , ...ξ n ) ∈ Rn , ξ 6= 0, n X

aij (x, r, p)ξ i ξ j > 0.

i,j=1

We now state the form of the strong maximum principle for second order quasilinear elliptic operators most suitable for our purposes. Theorem II.2 Let Q = Q(u) be a second order quasi-linear elliptic operator as described above. Suppose the U -admissible functions u, v ∈ C 2 (Ω) satisfy, (1) u ≤ v on Ω and u(x0 ) = v(x0 ) for some x0 ∈ Ω, and (2) Q(v) ≤ Q(u) on Ω. Then u = v on Ω. The idea of the proof of Theorem II.1 is to intersect, in a properly transverse manner, the null hypersurfaces S1 and S2 with a timelike (i.e., Lorentzian in the induced metric) hypersurface through the point p, and to show that the spacelike intersections agree. Analytically, intersecting the null hypersurfaces in this manner reduces the problem to a nondegenerate elliptic one. In order to apply Theorem II.2 we need a suitable analytic expression for the null mean curvature, which we now derive. Let p be a point in a spacetime M , and let P be a timelike hypersurface passing through p. Let V be a connected spacelike hypersurface in P (and hence a co-dimension two spacelike submanifold of M ) passing through p. Via the normal exponential map along V in P , we can assume, by shrinking P if necessary, that P can be expressed as, P = (−a, a) × V,

(II.6)

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and that the induce metric on P takes the form,

ds2 = −dt2 +

n−2 X

gij (t, x)dxi dxj ,

(II.7)

i=1

where x = (x1 , ..., xn−2 ) are coordinates in V centered on p. Let S be a null hypersurface which meets P properly transversely in a spacelike hypersurface Σ in P . By adjusting the size of P and S if necessary, we may assume that Σ can be expressed as a graph over V . Thus, there exists u ∈ C ∞ (V ) such that Σ = graph u = {(u(x), x) ∈ P : x ∈ V }. Let H(u) denote the mean curvature of Σ = graph u. (By our sign conventions the mean curvature of Σ is + the divergence of the future pointing normal along Σ.) To describe H(u) we introduce the following notation. Let h be the Riemannian metric on V whose components are given by hij (x) = gij (u(x), x), and let hij be the i, jth entry of the inverse [hij ]−1 . Let P matrix P∇u denote the gradient of u. In terms of coordinates, j ∇u = j u ∂j , where uj = i hij ∂i u. Finally, introduce the quantity, 1 . ν=p 1 − |∇u|2 The positivity of ν is equivalent to Σ = graph u being spacelike. With respect to these quantities, we have (cf., [2]),

H(u) =

n−2 X

aij (x, u, ∂u)∂ij u + b(x, u, ∂u),

(II.8)

i,j=1

where aij = νhij + ν 3 ui uj and b is a polynomial expression in ∂i u, hij , hij , ∂t gij (u(x), x) and ν. From the form of aij , it is clear that H = H(u) is a second order quasi-linear elliptic operator. Let K be a future directed null vector field on S. Since K is orthogonal to Σ, by rescaling we may assume that along Σ , K = Z + N , where Z is the future directed unit normal vector field to Σ in P and N is one of the two unit spacelike normal vector fields to P in M . Let θ be the null mean curvature of S with respect to K. We obtain an expression for θ along Σ. Let BP denote the second fundamental form of P with respect to N , and let BΣ denote the second fundamental form of Σ in P with respect to Z. Then for q ∈ Σ and vectors X, Y ∈ Tq Σ, BP (X, Y ) = h∇X N, Y i, and BΣ (X, Y ) = h∇X Z, Y i = h∇X Z, Y i, where ∇ is the induced Levi-Civita connection on P .

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Now let {e1 , e2 , ..., en−2 } be an orthonormal basis for Tq Σ. Then the value of θ at q is given by, θ

=

=

n−2 X

n−2 X

n−2 X

i=1 n−2 X

i=1 n−2 X

i=1

h∇ei K, ei i = BΣ (ei , ei ) +

i=1

h∇ei Z, ei i +

h∇ei N, ei i

BP (ei , ei )

i=1

= HΣ + BP (Z, Z) + HP ,

(II.9)

where HΣ is the mean curvature of Σ and HP is the mean curvature of P . In the notation introduced above, Z

= ν(∂0 + ∇u) n−2 X = νui ∂i ,

(II.10)

i=0

where ∂0 =

∂ ∂t ,

u0 = 1, and as above, ui =

Pn−2 j=1

n−2 X

BP (Z, Z) = BP (

i=0

=

n−2 X

hij ∂j u, i = 1, ..., n − 2. Hence,

νui ∂i ,

n−2 X

νui ∂i )

i=0

ν 2 βij (u)ui uj ,

(II.11)

i,j=1

where for x ∈ V , βij (u)(x) = BP (∂i , ∂j )|(u(x),x) . Now let θ(u) denote the null mean curvature of S along Σ = graph u. Equations II.9 and II.11 give, θ(u) = H(u) +

n−2 X

ν 2 βij (u)ui uj + α(u),

i,j=1

where α(u) is the function on V defined by α(u)(x) = HP (u(x), x). Thus, by II.8, we finally arrive at, θ(u) =

n−2 X

aij (x, u, ∂u)∂ij u + b1 (x, u, ∂u)

(II.12)

i,j=1

where, b1 (x, u, ∂u) = b(x, u, ∂u) +

n−2 X i,j=1

ν 2 βij (u)ui uj + α(u),

(II.13)

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and where aij and b are as in Equation II.8. In particular, θ = θ(u) is a second order quasi-linear elliptic operator with the same symbol as the mean curvature operator for spacelike hypersurfaces in P . Proof of Theorem 2.1: Let P be a timelike hypersurface passing through p, as described by equations II.6 and II.7. S1 and S2 have a common null tangent at p. Choose P so that it is transverse to this tangent. Then, by choosing P small enough the intersections Σ1 = S1 ∩ P and Σ2 = S2 ∩ P will be properly transverse, and hence Σ1 and Σ2 will be spacelike hypersurfaces in P . Let Ki , i = 1, 2, be the null vector field on Si with respect to which the null mean curvature function θi is defined. Let N be the unit normal to P pointing to the same side of P as K1 |Σ1 and K2 |Σ2 . By rescaling we can assume Ki |Σi = Zi + N |Σi , where Zi is the future directed unit normal to Σi in P . Let ui = ui (x), i = 1, 2, be the smooth function on V such that Σi = graph ui . From the hypotheses of Theorem II.1 we know, (1) u1 ≤ u2 on V and u1 (p) = u2 (p), and (2) θ(u2 ) ≤ θ(u1 ) on V . By Theorem II.2, we conclude that u1 = u2 on V , i.e. Σ1 = Σ2 . Now, Si , i = 1, 2, is obtained locally by exponentiating out from Σi in the orthogonal direction Ki |Σi = Zi + N |Σi . It follows that S1 and S2 agree near p, i.e., there is a spacetime neighborhood O of p such that S1 ∩ O = S2 ∩ O = S, and S has null mean curvature θ = 0. 2

III The maximum principle for C 0 null hypersurfaces III.1

C 0 null hypersurfaces

The usefulness of the maximum principle for smooth null hypersurfaces obtained in the previous section is limited by the fact that the most interesting null hypersurfaces arising in general relativity, such as black hole event horizons and Cauchy horizons, are rough, i.e., are C 0 but in general not C 1 . The aim of this section is to present a maximum principle for rough null hypersurfaces, similar in spirit to the maximum principle for rough spacelike hypersurfaces obtained in [2]. Horizons and other null hypersurfaces commonly occurring in relativity arise essentially as the null portions of achronal boundaries which are sets of the form ∂I ± (A), A ⊂ M . Achronal boundaries are always C 0 hypersurfaces, but simple examples illustrate that they (and their null portions) may fail to be differentiable at certain points. Consider, for example, the set S = ∂I − (A)\ A, where A consists of two disjoint closed disks in the t = 0 slice of Minkowski 3-space. This surface, which represents the merger of two truncated cones, has a “crease”, i.e., a curve of nondifferentiable points (corresponding to the intersection of the two cones) but which otherwise is a smooth null hypersurface.

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An important feature of the null portion of an achronal boundary is that it is ruled by null geodesics which are either past or future inextendible within the hypersurface. This is illustrated in the example above. S is ruled by null geodesics which are future inextendible in S, but which are in general not past inextendible in S. Null geodesics in S that meet the crease when extended toward the past leave S when extended further, and hence have past end points on S. We now formulate a general definition of C 0 null hypersurface which captures the essential features of these examples. A set A ⊂ M is said to be achronal if no two points of A can be joined by a timelike curve. A ⊂ M is locally achronal if for each p ∈ A there is a neighborhood U of p such that A ∩ U is achronal in U . A C 0 nontimelike hypersurface in M is a topological hypersurface S in M which is locally achronal. We remark that for each p ∈ S, there exist arbitrarily small neighborhoods U of p such that S ∩ U is closed and achronal in U , and for each q ∈ U \ S, either q ∈ I + (S ∩ U, U ) or q ∈ I − (S ∩ U, U ). Definition III.1 A C 0 future null hypersurface in M is a nontimelike hypersurface S in M such that for each p ∈ S and any neighborhood U of p in which S is achronal, there exists a point q ∈ S, q 6= p, such that q ∈ J + (p, U ). Since q ∈ J + (p, U ) \ I + (p, U ), there is a null geodesic segment η from p to q. The segment η must be contained in S, for otherwise at some point η would enter either I + (S, U ) or I − (S, U ), which would contradict the achronality of S in U . The geodesic η can be extended further to the future in S: Choose r in S ∩ J + (q, U ), r 6= q. The null geodesic from q to r in S must smoothly extend the one from p to q, otherwise there would be an achronality violation of S in U . Thus, for each point p ∈ S there is a future directed null geodesic in S starting at, or passing through, p which is future inextendible in S, i.e., which does not have a future end point in S. These null geodesics are called the null geodesic generators of S. They may or may not have past end points in S. Summarizing, a C 0 future null hypersurface is a locally achronal topological hypersurface S of M which is ruled by future inextendible null geodesics. A C 0 past null hypersurface is defined in a time-dual manner. Let S be a C 0 future null hypersurface. Adopting the terminology introduced in [4] for event horizons, a semi-tangent of S is a future directed null vector K which is tangent to a null generator of S. We do not want to distinguish between semi-tangents based at the same point and pointing in the same null direction, so we assume the semi-tangents of S have been uniformly normalized in some manner, e.g., by requiring each semi-tangent to have unit length with respect to some auxilliary Riemannian metric on M . Then note that the local achronality of S implies that at each interior point (non-past end point) of a null generator of S there is a unique semi-tangent at that point. Techniques from [4] can be adapted to prove the following. Lemma III.1 Let S be a C 0 future null hypersurface in a spacetime M .

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1. If pn → p in S and Xn → X in T M , where, for each n, Xn is a semi-tangent of S at pn then X is a semi-tangent of S. 2. Suppose p is an interior point of a null generator of S, and let X be the unique semi-tangent of S at p. Then semi-tangents of S at points near p must be close to X, i.e., if Xn is any semi-tangent of S at pn , and pn → p then Xn → X. The proof of the maximum principle for C 0 null hypersurfaces will proceed in a fashion similar to the smooth case. Thus we will need to consider the intersection of a C 0 null hypersurface S with a smooth timelike hypersurface P . Lemma III.2 Let S be a C 0 future null hypersurface and let p ∈ S be an interior point of a null generator η of S. Let P be a smooth timelike hypersurface passing through p transverse to η. Then there exists a neighborhood O of p in P such that Σ = S ∩ P is a partial Cauchy surface in O, i.e., Σ is a closed acausal C 0 hypersurface in O. Proof. The proof uses the edge concept, in particular the result that an achronal set is a closed C 0 hypersurface if and only if it is edgeless; see e.g., Corollary 26, p. 414 in [16]. Let U be a neighborhood of p in M in which S is achronal and edgeless. Then V = U ∩ P is a neighborhood of p in P in which Σ is achronal and edgeless in P . Hence, Σ is a closed achronal C 0 hypersurface in V , and it remains to show that it is actually acausal in a perhaps smaller neighborhood. Suppose there exists a sequence of neighborhoods Vn ⊂ V of p, which shrink to p, such that Σ is not acausal in Vn for each n. Then, for each n, there exists a pair of points pn , qn ∈ Σ ∩ Vn such that pn → p and qn ∈ J + (pn , Vn ). Hence for each n, there exists a P -null geodesic ηn from pn to qn . Now, ηn is a causal curve in U , and, in fact, must be a null geodesic in U , since otherwise we would have qn ∈ I + (pn , U ), which would violate the achronality of S in U . Hence ηn ⊂ S, and the initial tangent Xn to ηn , when suitably normalized, is a semi-tangent of S at pn . By the second part of Lemma III.1, Xn → X, where X is tangent to η at p. But X is tangent to P , since each Xn is, which contradicts the assumption that P is transverse to η at p. We now extend the meaning of mean curvature inequalities to C 0 null hypersurfaces. The idea, motivated by previous work involving spacelike hypersurfaces ([7], [2]) is to use smooth null support hypersurfaces. Henceforth we set the scale for all null vectors on M by requiring that they have unit length with respect to a fixed Riemannian metric on M . Definition III.2 Let S be a C 0 future null hypersurface in M . We say that S has null mean curvature θ ≥ 0 in the sense of support hypersurfaces provided for each p ∈ S and for each > 0 there exists a smooth (at least C 2 ) null hypersurface Sp, such that,

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(1) Sp, is a past support hypersurface for S at p, i.e., Sp, passes through p and lies to the past side of S near p, and (2) the null mean curvature of Sp, at p satisfies θp, ≥ −. For example, if p is a point in Minkowski space, the future null cone ∂I + (p) has null mean curvature θ ≥ 0 in the sense of support hypersurfaces. One can use null hyperplanes, even at the vertex, as support hypersurfaces. Another, less trivial example is that of a black hole event horizon H = ∂I − (I + ) in an asymptotically flat black hole spacetime M . Here I + refers to future null infinity; see Section 4. Assuming the generators of H are future complete and M obeys the null energy condition, Ric(X, X) ≥ 0, for all null vectors X, it follows from Lemma IV.2 in Section 4 that H has null mean curvature θ ≥ 0 in the sense of support hypersurfaces. This observation and other consequences of the existence of smooth null support hypersurfaces provided the initial impetus for the development of a proof of the black hole area theorem under natural regularity conditions, i.e. the regularity implicit in the fact that H is an achronal boundary, cf. [5]. With the notation as in Definition III.2, let Bp, denote the null second fundamental form of Sp, at p. We say that the collection of null second fundamental forms {Bp, : p ∈ S, > 0} is locally bounded from below provided that for all p ∈ S there is a neighborhood W of p in S and a constant k > 0 such that Bq, ≥ −khq,

for all q ∈ W and > 0,

(III.14)

where hq, is the Riemannian metric on Tq Sq, /Kq, , as defined in Section 2.1. This technical condition arises in the statement of the maximum principle for C 0 null hypersurfaces, and is satisfied in many natural geometric situations for essentially a priori reasons. Lemma III.3 Let S be a C 0 future null hypersurface and let W be a smooth null hypersurface which is a past support hypersurface for S at p. If K ∈ Tp W is the future directed (normalized) null tangent of W at p then K is a semi-tangent of S at p. Proof. Let U be a neighborhood of p such that S ∩ U is closed and achronal in U and W ∩ U ⊂ J − (S ∩ U, U ). For simplicity, we may assume that S ⊂ U and W ⊂ J − (S, U ). Let η ⊂ U be an initial segment in U of the future directed null generator of W starting at p with initial tangent K. Then η ⊂ J − (S, U )∩J + (S, U ). By the remark before Definition III.1, if η leaves S at some point, it will enter either I − (S, U ) or I + (S, U ). Either case leads to an achronality violation. Hence, η must be a null generator of S, which implies that K is a semi-tangent of S. If S is a C 0 past null hypersurface, one defines in a time-dual fashion what it means for S to have null mean curvature θ ≤ 0 in the sense of support hypersurfaces. In this case one uses smooth null hypersurfaces which lie locally to the future of S. Although, in principle, one can also consider future null hypersurfaces

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with nonpositive null mean curvature, this appears to be a less useful notion, as future support hypersurfaces cannot typically be constructed at past end points of generators.

III.2

The maximum principle for C 0 null hypersurfaces

The aim now is to present a proof of the geometric maximum principle stated below. Unless otherwise stated, we continue to assume that all null vectors are normalized to unit length with respect to a fixed background Riemannian metric. Theorem III.4 Let S1 be a C 0 future null hypersurface and let S2 be a C 0 past null hypersurface in a spacetime M . Suppose, (1) S1 and S2 meet at p ∈ M and S2 lies to the future side of S1 near p, (2) S1 has null mean curvature θ1 ≥ 0 in the sense of support hypersurfaces, with null second fundamental forms {Bp, : p ∈ S1 , > 0} locally bounded from below, and (3) S2 has null mean curvature θ2 ≤ 0 in the sense of support hypersurfaces. Then S1 and S2 coincide near p, i.e., there is a neighborhood Oof p such that S1 ∩ O = S2 ∩ O. Moreover, S1 ∩ O = S2 ∩ O is a smooth null hypersurface with null mean curvature θ = 0. The proof proceeds in a similar fashion to the proof of Theorem II.1. Instead of Theorem II.2, we use the strong maximum principle for weak (in the sense of support functions) sub and super solutions of second order quasi-linear elliptic equations obtained in [2]. We will state here a restricted form of this result, adapted to our purposes. Let Ω be a domain in Rn and let U be an open set in Rn × R × Rn of the form U = Ω × I × B, where I is an open interval and B is an open ball in Rn . Consider the second order quasi-linear elliptic operator Q = Q(u) as defined in Equation II.5 for U -admissible functions u ∈ C 2 (Ω), where now we assume aij , b ∈ C ∞ (U ). We now Pn also ijassume Qi =j Q(u) is uniformly elliptic, by which we mean (1) the quantity i,j=1 a (x, r, p)ξ ξ is uniformly positive and bounded away from infinity for all (x, r, p) ∈ U and all unit vectors ξ = (ξ1 , ..., ξn ), and (2) aij , b and their first order partial derivatives are bounded on U . We need the notion of a support function. Given u ∈ C 0 (Ω) and x0 ∈ Ω, φ is an upper (resp., lower) support function for u at x0 provided φ(x0 ) = u(x0 ) and φ ≥ u (resp., φ ≤ u) on some neighborhood of x0 . We say that a function u ∈ C 0 (Ω) satisfies Q(u) ≥ 0 in the sense of support functions iff for all > 0 and all x ∈ Ω there is a U -admissible lower support function φx, , which is C 2 in a neighborhood of x, such that Q(φx, )(x) ≥ −. We say that u satisfies Q(u) ≥ 0 in the sense of support functions with Hessians locally bounded from below iff, in addition, there is a constant k > 0, independent of and x, such that Hess(φx, )(x) ≥ −kI, where

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I is the identity matrix. For u ∈ C 0 (Ω), we define Q(u) ≤ 0 in the sense of support functions in an analogous fashion. The following theorem is a special case of Theorem 2.4 in [2] Theorem III.5 Let Q = Q(u) be a second order quasi-linear uniformly elliptic operator as described above. Suppose u, v ∈ C 0 (Ω) satisfy, (1) u ≤ v on Ω and u(x0 ) = v(x0 ) for some x0 ∈ Ω, (2) Q(u) ≥ 0 in the sense of support functions with Hessians locally bounded from below, and (3) Q(v) ≤ 0 in the sense of support functions. Then u = v on Ω and u = v ∈ C ∞ (Ω). We now proceed to the proof of the maximum principle for C 0 null hypersurfaces. Proof of Theorem III.4: We first observe that p is an interior point of null generators for both S1 and S2 , and that these two null generators agree near p. To show this, let ηi be a null generator of Si starting at p, i = 1, 2; η1 is future directed and η2 is past directed. Let U be a neighborhood of p in which S1 is closed achronal , such that S2 ∩ U ⊂ J + (S1 ∩ U, U ). We may assume η1 and η2 are contained in U . Since η2 is past pointing, η2 ⊂ J − (S1 ∩ U, U ) ∩ J + (S1 ∩ U, U ). Then, as in Lemma III.3, the achronality of S1 in U forces η2 ⊂ S1 . To avoid an achronality violation, η = −η2 ∪ η1 must be an unbroken null geodesic, and hence a null generator of S1 passing through p. Similarly, −η = −η1 ∪ η2 is a null generator of S2 passing through p. Let P be a timelike hypersurface passing through p transverse to η. Let Kp be the normalized semi-tangent of S1 at p; Kp is tangent to η. Let N be the spacelike unit normal vector field to P that points to the same side of P as Kp . By making a homothetic change in the background Riemannian metric we may assume hKp , Np i = 1. Hence, Kp is of the form, Kp = Zp + Np , where Zp ∈ Tp P is a future directed unit timelike vector. As in the proof of Theorem II.1, P in the induced metric can be expressed as in Equations II.6 and II.7. Moreover, V can be constructed so that Zp is perpendicular to V . Then Kp = (∂0 + N )p . By Lemma III.2, provided P is taken small enough, Σ1 = S1 ∩P , and Σ2 = S2 ∩P will be partial Cauchy surfaces in P passing through p, with Σ2 to the future of Σ1 . Thus, shrinking P further if necessary, there exist functions ui ∈ C 0 (V ), i = 1, 2, such that Σi = graph(ui ) and (1) u1 ≤ u2 on V and u1 (p) = u2 (p) = 0. Let {Sq, } be the family of smooth null lower support hypersurfaces for S1 . Restrict attention to those members of the family for which q ∈ Σ1 . Let Bq, be the null second fundametal form of Sq, at q with respect to the null vector

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Kq, . By Lemmas III.1 and III.3, the collection of null vectors {Kq, } can be made arbitrarily close to Kp by taking P sufficiently small. This has several implications. It implies, in particular, for P sufficiently small, that Kq, is transverse to P . Hence, by shrinking Sq, about q, if necessary, Sq, meets P in a properly transverse manner, and thus Σq, = Sq, ∩ P is a smooth spacelike hypersurface in P . Thus for each > 0 and q ∈ Σ1 , there exists φq, ∈ C 2 (Wq, ), Wq, ⊂ V , such that Σq, = graph(φq, ). We now consider the null mean curvature operator θ = θ(u), as described in equations II.12 and II.13, on the set U = V × (−a, a) × B, where B is an open ball in Rn−2 centered at the origin. By choosing V , a and B sufficiently small, θ = θ(u) will be uniformly elliptic on U , in the sense described above. Since the vectors {Kq, } can be made uniformly close to Kp , the inner products hKq, , Nq i can be made uniformly close to the value one. Hence, we can rescale the vectors Kq, so that hKq, , Nq i = 1 without altering the validity of the assumed null mean curvature inequality θ1 ≥ 0 in the sense of support hypersurfaces, at points of S1 in P . Then Kq, can be expressed as, Kq, = Zq, + Nq , where Zq, ∈ Tq P is a future directed unit timelike vector. Moreover, the vectors Zq, can be made uniformly close to Zp = ∂0 |p by taking P small enough. Equation II.10 then implies that the Euclidean vectors ∂φq, (q) = (∂1 φq, (q), ..., ∂n−2 φq, (q)) can be made to lie in the ball B. We conclude that by taking P sufficiently small, each function φq, is U admissible. Now, φq, is a C 2 lower support function for Σ1 at q. By assumption, the null mean curvature of Sq, at q satisfies, θq, (q) ≥ −, which, in the analytic setting, translates into, θ(φq, )(q) ≥ −. Thus, u1 satisfies, θ(u1 ) ≥ 0 in the sense of support functions. For each q ∈ Σ1 , Zq, is the future directed timelike unit normal to Σq, . Let βq, be the second fundamental form of Σq, ⊂ P at q with respect to the normal Zq, . Let BP,q be the second fundamental form of P at q with respect to N . The second fundamental forms Bq, , βq, , and BP,q are related by Bq, (X, X) = βq, (X, X) + BP,q (X, X),

(III.15)

for all unit vectors X ∈ Tq Σq, = [Zq, ]⊥ ⊂ Tq P . In a sufficiently small relatively compact neighborhood P0 of p in P , the collection of vectors {Zq, : q ∈ Σ1 ∩ P0 } has compact closure in T P . It follows that the collection of vectors X = {Xq ∈ Tq Σq, : q ∈ Σ1 ∩ P0 , |Xq | = 1} has compact closure in T P , as well. Hence the set of numbers {BP,q (Xq , Xq ) : Xq ∈ X } is bounded. Coupled with the assumption that the second fundamenal forms {Bq, } are locally bounded from below, we conclude, by shrinking P further if necessary, that the second fundamental forms {βq, : q ∈ Σ1 } are locally bounded from below, i.e., for each q0 ∈ Σ1 there is a neighborhood W of q0 in Σ1 and a constant k such that βq, ≥ −kgq,

for all q ∈ W and > 0,

(III.16)

where gq, is the induced metric on Σq, at q. For P sufficiently small, the induced metrics gq, will be close to the metric of V at p. Using the relationship between

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βq, and Hess φq, , worked out, for example, in Section 3.1 in [2], inequality III.16 translates into the analytic statement that for each q0 ∈ Σ1 there is a neighborhood W of q0 in Σ1 and a constant k1 such that Hess φq, (q) ≥ −k1 I for all q ∈ W and > 0. Thus, we finally conclude that, (2) u1 satisfies θ(u1 ) ≥ 0 in the sense of support functions with Hessians locally bounded from below. By similar reasoning, adjusting the size of P as necessary, we have that (3) u2 satisfies θ(u2 ) ≤ 0 in the sense of support functions. In view of (1), (2), and (3), Theorem III.5 applied to the operator θ = θ(u) implies that u1 = u2 on V and u1 = u2 is C ∞ . Hence, Σ1 and Σ2 are smooth spacelike hypersurfaces in P which coincide near p. Then near p, S1 and S2 are obtained by exponentiating normally out along a common smooth null orthogonal vector field 2 along Σ1 = Σ2 . The conclusion to Theorem III.4 now follows. For simplicity we have restricted attention to the null mean curvature inequalities θ1 ≤ 0 ≤ θ2 . However, Theorem III.4, with an obvious modification of Definition III.2, remains valid under the null mean curvature inequalities θ1 ≤ a ≤ θ2 , for any a ∈ R.

IV The null splitting theorem We now consider some consequences of Theorem III.4. The proof of many global results in general relativity, such as the classical Hawking-Penrose singularity theorems and more recent results such as those concerning topological censorship (see e.g., [11, 12]) involve the construction of a timelike line or a null line in spacetime. A timelike geodesic segment is maximal if it is longest among all causal curves joining its end points, or equivalently, if it realizes the Lorentzian distance between its end points. A timelike line is an inextendible timelike geodesic each segment of which is maximal. Similarly, a null line is an inextendible null geodesic each segment of which is maximal. But since each segment of a null geodesic has zero length, it follows that an inextendible null geodesic is a null line iff it is achronal, i.e., iff no two points of it can be joined by a timelike curve. In particular, null lines, like timelike lines, must be free of conjugate points. The standard Lorentzian splitting theorem [3, Chapter 14], considers the rigidity of spacetimes which contain timelike lines. Recall, it asserts that a timelike geodesically complete spacetime (M, g) which obeys the strong energy condition, Ric(X, X) ≥ 0 for all timelike vectors X, and which contains a timelike line splits along the line, i.e., is isometric to (R × V, −dt2 ⊕ h), where (V, h) is a complete Riemannian manifold. We now consider the analogous problem for spacetimes with null lines. The theorem stated below establishes the rigidity of spacetime in this null case. Unless otherwise stated, we continue to assume that all null vectors are normalized to unit length with respect to a fixed background Riemannian metric.

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Theorem IV.1 Let M be a null geodesically complete spacetime which obeys the null energy condition, Ric(X, X) ≥ 0 for all null vectors X and contains a null line η. Then η is contained in a smooth closed achronal totally geodesic (B ≡ 0) null hypersurface S. The simplest illustration of Theorem IV.1 is Minkowski space: Each null geodesic in Minkowski space is contained in a null hyperplane. De Sitter space, anti-de Sitter, and gravitational plane wave solutions furnish other illustrations. Remark IV.1 Theorem IV.1 may be viewed as a “splitting” theorem of sorts, where the splitting takes place in S. Let K be the unique (up to scale) future pointing null vector field of S. The vanishing of the null second fundamental form B of S means that the standard kinematical quantities associated with K, i.e., the expansion θ, and shear σ (as well as vorticity) of K vanish. In a similar vein, the vanishing of B implies that the metric h on T S/K defined in Section 2.1 is invariant under the flow generated by K; see [14] for a precise statement and proof. To take this a step further, suppose, in the setting of Theorem IV.1, that M is also globally hyperbolic. Let V be a smooth Cauchy surface for M . Since V is Cauchy, and the generators of S are inextendible in S, each null generator of S meets V exactly once, and this intersection is properly transverse. Hence, Σ = S ∩ V is a smooth codimension two spacelike submanifold of M , and the map Φ : R × Σ → S, defined by, Φ(s, x) = expx sK(x), is a diffeomorphism. The invariance of the metric ∂ h with respect to the flow generated by K, or equivalently, by Φ∗ ( ∂s ), implies that ∗ ∗ (i ◦ Φ) g = π g0 , where i : S ,→ M is inclusion, π : R× Σ → Σ is projection, and g0 is the induced metric on Σ. In more heuristic terms, S ≈ R×Σ, and i∗ g = 0dt2 +g0 . It is in this sense that one may view Theorem IV.1 as a splitting theorem. + (η) = ∂I − (η), \ \ Remark IV.2 The proof of Theorem IV.1 shows that S = ∂I ± ± \ where ∂I (η) is the component of ∂I (η) containing η. In more heuristic terms S is obtained as a limit of future null cones ∂I + (η(t)) (resp., past null cones ∂I − (η(t))) as t → −∞ (resp., t → ∞). The proof also shows that the assumption of null completeness can be weakened. It is sufficient to require that the generators of ∂I − (η) be future geodesically complete and the generators of ∂I + (η) be past geodesically complete. (As discussed in the proof, the generators of ∂I − (η) (resp., ∂I + (η)) are in general future (resp., past) inextendible in M .)

The proof of Theorem IV.1 is an application of Theorem III.4, and relies on the following basic lemma. Lemma IV.2 Let M be a spacetime which satisfies the null energy condition. Suppose S is an achronal C 0 future null hypersurface whose null generators are future geodesically complete. Then S has null mean curvature θ ≥ 0 in the sense of support hypersurfaces, with null second fundamental forms locally bounded from below.

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Proof. The basic idea for constructing past support hypersurfaces for S is to consider the “past null cones” of points on generators of S formed by past null geodesics emanating from these points. For the purpose of establishing certain properties about these null hypersurfaces, it is useful to relate them to achronal boundaries of the form ∂J − (q), q ∈ S, which are defined purely in terms of the causal structure of M . For this reason we assume initially that M is globally hyperbolic. At the end we will indicate how to remove this assumption. For each p ∈ S and normalized semi-tangent K at p, let η : [0, ∞) → M , η(s) = expp sK, be the affinely parameterized null geodesic generator of S starting at p with initial tangent K. Since S is achronal each such generator is a null ray, i.e., a maximal null half-geodesic. Since η is maximal, no point on η|[0,r) is conjugate to η(r). For each r > 0, consider the achronal boundary ∂J − (η(r)), which is a C 0 hypersurface containing η|[0,r] . By standard properties of the null cut locus (see especially, Theorems 9.15 and 9.35 in [3], which assume global hyperbolicity) there is a neighborhood U of η|[0,r) such that Sp,K,r = ∂J − (η(r)) ∩ U is a smooth null hypersurface diffeomorphic under the exponential map based at η(r) to a neighborhood of the line segment {−τ η 0 (r) : 0 < τ ≤ 1} in the past null cone Λ− η(r) ⊂ Tη(r) M . From the achronality of S we observe, ∂J − (η(r)) ∩ I + (S) = ∅. This implies that Sp,K,r is a past support hypersurface for S at p. Let θp,K,r denote the null mean curvature of Sp,K,r at p with respect to K. We use Equation II.4 to obtain the lower bound θp,K,r ≥ −

n−2 . r

(IV.17)

The argument is standard. In the notation of Section 2.1, let θ(s), s ∈ [0, r), be the null mean curvature of Sp,K,r at η(s) with respect to η 0 (s). Equation II.4 and the energy condition imply, dθ 1 ≤− θ2 . ds n−2

(IV.18)

Without loss of generality we may assume θ(0) = θp,K,r < 0. Then θ = θ(s) is strictly negative on [0, r), and we can devide IV.18 by θ2 to obtain, 1 d −1 θ ≥ . ds n−2

(IV.19)

Integrating IV.19 from 0 to r−δ, and letting δ → 0 we obtain the lower bound IV.17. Thus, since r can be taken arbitrarily large, we have shown, in the globally hyperbolic case, that S has null mean curvature θ ≥ 0 in the sense of support hypersurfaces with respect to the collection {Sp,K,r } of smooth null hypersurfaces. By Lemma III.3, K is tangent to Sp,K,r at p. Let Bp,K,r denote the null second fundamental form of {Sp,K,r } at p with respect to K. We now argue that the collection of null second fundamental forms {Bp,K,r : r ≥ r0 }, for some r0 > 0,

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is locally bounded from below. Recall, “locally”, means “locally in the point p”; the lower bound must hold for all r > r0 , cf., inequality III.14. This lower bound follows from a continuity argument and an elementary monotonicity result, as we now discuss. Fix p ∈ S. Let U be a convex normal neighborhood of p. Thus, for each q ∈ U , U is the diffeomorphic image under the exponential map based at q of a neighborhood of the origin in Tq M . Provided U is small enough, we have that for −1 − each q ∈ U , ∂J − (q) ∩ U = ∂(J − (q) ∩ U ) = expq (Λ− q ∩ expq (U )), where Λq is the − past null cone in Tq M . In particular, for each q ∈ U , W (q) = ∂J (q) ∩ U \ {q} is a smooth null hypersurface in U such that if qn → q in U , W (qn ) converges smoothly to W (q). There exists a neighborhood V of p, V ⊂ U , and r0 > 0 such that for each q ∈ V , and each normalized null vector K ∈ Tq M , the null geodesic segment s → η(s) = expq sK, s ∈ [0, r0 ] is contained in U . Let B(q, K) be the null second fundamental form of W (expq r0 K) at q with respect to K. Since, as (qn , Kn ) → (q, K), W (expqn r0 Kn ) converges smoothly to W (expq r0 K), we have that B(qn , Kn ) → B(q, K) smoothly. Returning to the original family of support hypersurfaces, {Sp,K,r }, with associated family of null second fundamental forms {Bp,K,r }, observe that when q ∈ S ∩ V , Sq,K,r0 agrees with W (expq r0 K) near q. Hence, if (qn , Kn ) → (q, K) in S ∩ V , Bqn ,Kn ,r0 → Bq,K,r0 smoothly. It follows that the collection of null second fundamental forms {Bp,K,r0 } is locally bounded from below. Consider as in the beginning, for p ∈ S and K a normalized semi-tangent at p, the null geodesic generator, η : [0, ∞) → M , η(s) = expp sK, s ≥ 0. For 0 < r < t < ∞, J − (η(r)) ⊂ J − (η(t)). Then, since ∂J − (η(t)) is achronal, ∂J − (η(r)) cannot enter I + (∂J − (η(t)). It follows that for r < t, Sp,K,r lies to the past side of Sp,K,t near p. By an elementary comparison of null second fundamental forms at p we obtain the monotonicity property, Bp,K,t ≥ Bp,K,r

for all 0 < r < t < ∞.

(IV.20)

This monotonicity now implies that the entire family of null second fundamental forms {Bp,K,r : r ≥ r0 } is locally bounded from below. This concludes the proof of Lemma IV.2 under the assumption that M is globally hyperbolic. We now describe how to handle the general case. In general, M may have bad causal properties. In particular the generators of S could be closed. Nevertheless, the past support hypersurfaces are formed in the same manner, as the “past null cones” of points on generators of S formed by past null geodesics emanating from these points. But as an intermediary step, to take advantage of the arguments in the globally hyperbolic case, we pull back each generator to a spacetime having good causal properties. Again, consider, for p ∈ S and K a normalized semi-tangent at p, the null geodesic generator, η : [0, ∞) → M , η(s) = expp sK, s > 0. Restrict attention to the finite segment η|[0,r] . Roughly speaking, we introduce Fermi type coordinates

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near η|[0,r] . Let {e1 , e2 , ...en−1 } be an orthonormal frame of spacelike vectors in Tη(0) M . Parallel translate these vectors along η to obtain spacelike orthonormal ¯ ⊂ vector fields ei = ei (s), 1 ≤ i ≤ n − 2 along η|[0,r] . Consider the map Φ : M P n−1 ¯ is an Rn → M defined by Φ(s, x1 , x2 , ..., xn−1 ) = expη(s) ( i=1 xi ei ). Here M open set containing the line segment {(s, 0, 0, ..., 0) : 0 ≤ s ≤ r}. By the inverse ¯ so that Φ is a local diffeomorphism. Equip function theorem we can choose M ¯ with the pullback metric g¯ = Φ∗ g, where g is the Lorentzian metric on M , M ¯ Lorentzian and Φ a local isometry. Let t ∈ C ∞ (M ¯ ) be the 0th thereby making M coordinate function, t(s, x1 , x2 , ..., xn−1 ) = s. Since the slices t = s are spacelike, ¯ . Thus, M ¯ is a strongly causal ∇t is timelike, and hence t is a time function on M spacetime. ¯ , η¯(s) = (s, 0, 0, ..., 0), defined for δ > 0 sufficiently The curve η¯ : [0, r¯+δ] → M ¯ . Let K be a compact neighborhood of η¯ in small, is a maximal null geodesic in M ¯ M . Then by Corollary 7.7 in [18], any two causally related points in K can be joined by a longest causal curve γ in K, and each segment of γ contained in the interior of K is a maximal causal geodesic. This property is sufficient to push through, with only minor modifications, all relevant results concerning the null cut locus of η¯(0) ¯ of η¯|[0,r) such on η¯. In view of the above discussion, there is a neighborhood U − ¯ ¯ that Sp,K,r = ∂J (η(r))∩ U is a smooth null hypersurface diffeomorphic under the exponential map based at η¯(r) to a neighborhood of the line segment {−τ η¯0 (r) : ¯ ¯ ¯ 0 < τ ≤ 1} in the past null cone Λ− η¯(r) ⊂ Tη¯(r) M . Let V ⊂ U be a neighborhood of η¯(0) on which Φ is an isometry onto its image, and let Sp,K,r = Φ(S¯p,K,r ∩ V¯ ). Then {Sp,K,r } is the desired collection of past support hypersurfaces for S, having all the requisite properties. In particular, the mean curvature inequality IV.17 and the monotonicity property IV.20 hold for the family {S¯p,K,r }, and hence the family {Sp,K,r }, by just the same arguments as in the globally hyperbolic case. Proof of Theorem IV.1: Since η is achronal, it follows that η ⊂ ∂I − (η), and hence ∂I − (η) 6= ∅. Then, by standard properties of achronal boundaries, ∂I − (η) is a closed achronal C 0 hypersurface in M . We claim that ∂I − (η) is a C 0 future null hypersurface. By standard results on achronal boundaries, e.g., [18, Theorem 3.20], ∂I − (η) \ η¯ (where η¯ is the closure of η as a subset of M ) is ruled by future inextendible null geodesics. However, since we do not assume M is strongly causal, it is possible, in the worst case scenario, that η¯ = ∂I − (η), in which case [18, Theorem 3.20] gives no information. To show that ∂I − (η) is ruled by future inextendible null geodesics we apply instead the more general [18, Lemma 3.19]. Let N ⊂ U be a convex normal neighborhood of p. N as a spacetime in its own right is strongly causal. Let K be a compact neighborhood of p contained in N . Then for each t ∈ R, η|[t,∞) cannot remain in K if it ever meets it. Thus there exists a sequence of pi = η(ti ), ti % ∞, pi ∈ / K. It follows that for each x ∈ K ∩ I − (η), there exists a future directed timelike curve from x to a point on η not in K. We may then apply [18, Lemma 3.19] to conclude that p is the past end point of a future directed null geodesic segment contained in ∂I − (η). Hence, according to Definition III.1, ∂I − (η) is a C 0 future null hypersurface. Moreover, because ∂I − (η) is closed, the

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null generators of ∂I − (η) are future inextendible in M , and hence future complete. Let S− be the component of ∂I − (η) containing η. From the above, S− is an achronal C 0 future null hypersurface whose null generators are future geodesically complete. Thus, by Lemma IV.2, S− has null mean curvature θ− ≥ 0 in the sense of support hypersurfaces, with null second fundamental forms locally bounded from below. Let S+ be the component of ∂I + (η) containing η. Arguing in a time-dual fashion, S+ is an achronal C 0 past null hypersurface whose null generators are past geodesically complete. By the time-dual of Lemma IV.2, S+ has null mean curvature θ+ ≤ 0 in the sense of support hypersurfaces. Moreover at any point p on η, S+ lies to the future side of S− near p. Theorem III.4 then implies that S− = S+ = S is a smooth null hypersurface containing η with vanishing null mean curvature, θ ≡ 0. Equation II.4 and the null energy condition then imply that the shear scalar vanishes along each generator, which in turn implies that the null second fundamental form of S vanishes. 2 We conclude the paper with an application of Theorem IV.1. The application we consider is concerned with asymptotically simple (e.g., asymptotically flat and nonsingular) spacetimes as defined by Penrose [17] in terms of the notion of conformal infinity. Consider a 4-dimensional connected chronological spacetime M with metric g which can be conformally included into a spacetime-with-boundary M 0 with metric g 0 such that M is the interior of M 0 , M = M 0 \ ∂M 0 . Regarding the conformal factor, it is assumed that there exists a smooth function Ω on M 0 which satisfies, (i) Ω > 0 and g 0 = Ω2 g on M , and (ii) Ω = 0 and dΩ 6= 0 along ∂M 0 . The boundary I := ∂M 0 is assumed to consist of two components, I + and I − , future and past null infinity, respectively, which are smooth null hypersurfaces. I + (respectively, I − ) consists of points of I which are future (resp., past) endpoints of causal curves in M . A spacetime M satisfying the above is said to be asymptotically flat at null infinity. If, in addition, M satisfies, (iii) Every inextendible null geodesic in M has a past end point on I − and a future end point on I + then M is said to asymptotically simple with null conformal boundary. Condition (iii) is imposed to ensure that I includes all of the null infinity of M . It also implies that M is free of singularities which would prevent some null geodesics from reaching infinity. The notion of asymptotic simplicity was introduced by Penrose in order to facilitate the study of the asymptotic behavior of isolated gravitating systems. When restricting to vacuum (i.e., Ricci flat) spacetimes, asymptotic simplicity provides an elegant and rigorous setting for the study of gravitational radiation far from the radiating source. See the papers [9, 10] of Friedrich for discussion of

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the problem of global existence of asymptotically simple vacuum spacetimes. Here we prove the following rigidity result. Theorem IV.3 Suppose M is an asymptotically simple vacuum spacetime which contains a null line. Then M is isometric to Minkowski space. Proof. The first and main step of the proof is to show that M is flat (i.e., has vanishing Riemann curvature). Then certain global arguments will show that M is isometric to Minkowski space. For technical reasons, it is useful to extend M 0 slightly beyond its boundary. In fact, by smoothly attaching a collar to I + and to I − , we can extend M 0 to a spacetime P without boundary such that M 0 is a retract of P and both I + and I − separate P . It follows that I + and I − are globally achronal null hypersurfaces in P . Let M − = M ∪ I − . A straight forward limit curve argument, using the asymptotic simplicity of M 0 , shows that M − is causally simple. This means that the sets of the form J ± (x, M − ), x ∈ M − , are closed subsets of M − . (The limit curve lemma, in the form of Lemma 14.2 in [3], for example, is valid in P .) Let η be a null line in M which has past end point p ∈ I − and future end point q ∈ I + . Consider the “future null cone” at p, Np := ∂I + (p, M − ). From the causal simplicity of M − it follows that ∂I + (p, M − ) = J + (p, M − ) \ I + (p, M − ). Hence each point in Np can be joined to p by a null geodesic segment in M − . In particular Np is connected. ¿From the simple equality I + (p, M − ) = I + (η, M − ), it follows that, Np = ∂I + (η, M − ) = ∂I + (η, M ) ∪ γp = S ∪ γp , where S = ∂I + (η, M ) and γp is the future directed null generator of I − starting at p. Since asymptotically simple spacetimes are null geodesically complete, Theorem IV.1 implies that S = ∂I + (η, M ) is a smooth null hypersurface in M which is totally geodesic with respect to g. Since it is smooth and closed the generators of S never cross and never leave S in M − to the future. Moreover, since Np = ∂I + (p, M − ) is achronal, there are no conjugate points to p along the generators of Np . It follows that Np \ {p} is the diffeomorphic image under the expo−1 − + nential map expp : Tp P → P of (Λ+ p \ {0}) ∩ expp (M ), where Λp is the future null cone in Tp P . We are now fully justified in referring to Np as the future null cone in M − at p. Given a smooth null hypersurface, with smooth future pointing null normal vector field K, the shear tensor σab is the trace free part of the null second fundamental form Bab , σab = Bab − θ2 hab . Since S is totally geodesic in the physical 1 metric g and the shear scalar σ = (σab σab ) 2 is a conformal invariant, it follows by continuity that the shear tensor of Np \ {p} (with respect to an appropriately 0 chosen g 0 - null normal K 0 ) vanishes in the metric g 0 , σab = 0. The trace free part of 0 equation II.2 then implies that the components Ca0b0 (with respect to an appropriately chosen pseudo-orthonormal frame in which e0 = K 0 , cf., Section 4.2 in [13])

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of the conformal tensor of g 0 vanish on Np \ {p}. An argument of Friedrich in [8], which makes use of the Bianchi identities and, in the present case, the vacuum field equations expressed in terms of his regular conformal field equations, then shows that the so-called rescaled conformal tensor, and hence, the conformal tensor of the physical metric g must vanish on D+ (Np , M − ) ∩ M . Since M is Ricci flat, we conclude that M is flat on D+ (Np , M − ) ∩ M . In a precisely time-dual fashion M is flat on D− (Nq , M + ) ∩ M , where Nq is the past null cone of M + = M ∪ I + at q. To conclude that M is everywhere flat we show that M ⊂ D+ (Np , M − ) ∪ − D (Nq , M + ). Consider the set V = I + (S, M ) ∪ S ∪ I − (S, M ). It is clear from the fact that S is an achronal boundary that V is open in M . As S = ∂I + (η, M ), it follows that I + (S, M ) ⊂ J + (p, M − ), and time-dually, that I − (S, M ) ⊂ J − (q, M + ). Using the fact that J + (p, M − ) and J − (q, M + ) are closed subsets of M − and M + , respectively, it follows that V is closed in M . Hence, M = I + (S, M ) ∪ S ∪ I − (S, M ). We show that each term in this union is a subset of D+ (Np , M − ) ∪ D− (Nq , M + ). Trivially, S ⊂ Np ⊂ D+ (Np , M − ). Consider I + (S, M ) ⊂ J + (p, P ) ∩ M = J + (Np , P ) ∩ M . We claim that J + (Np , P ) ∩ M ⊂ D+ (Np , P ) ∩ M . If not, then H + (Np , P ) ∩ M 6= ∅. Choose a point x ∈ H + (Np , P ) ∩ M , and let ν be a null generator of H + (Np , P ) with future end point x. Since Np is edgeless, ν remains in H + (Np , P ) as it is extended into the past. By asymptotic simplicity, ν must meet I − . In fact, since no portion of ν can coincide with a generator of I − , ν must meet I − transversely and then enter I − (I − , P ). But this means that ν has left J + (Np , P ), which is a contradiction. Since D+ (Np , P ) ∩ M = D+ (Np , M − ) ∩ M , we have shown that I + (S, M ) ⊂ D+ (Np , M − ). By the timedual argument, I − (S, M ) ⊂ D− (Nq , M + ). Thus, M is globally flat. Also, as an asymptotically simple spacetime, M is null geodesically complete, simply connected and globally hyperbolic; cf., [13], [15]. We use these properties to show that M is geodesically complete. It then follows from the uniqueness of simply connected space forms that M is isometric to Minkowski space. We first observe that there exists a time orientation preserving local isometry φ : M → L, where L is Minkowski space. To obtain φ, first construct by a standard procedure a frame {e0 , e1 , ..., en−1 } of orthonormal parallel vector fields on M . Then solve dxi = hei , i, i = 0, ..., n − 1, for functions xi ∈ C ∞ (M ), and set φ = (x0 , x1 , ..., xn−1 ). From the fact that φ is a local isometry and M is null geodesically complete, it follows that any null geodesic, or broken null geodesic, in L can be lifted via φ to M . In particular it follows that φ is onto: If φ(M ) is not all of L then we can find a null geodesic η¯ in L that meets φ(M ) but is not entirely contained in φ(M ). Choose p ∈ M such that φ(p) is on η¯. Then the lift of η¯ through p is incomplete, contradicting the null geodesic completeness of M . We now show that M is timelike geodesically complete. If it is not, then, without loss of generality, there exists a future inextendible unit speed timelike geodesic γ : [0, a) → M , t → γ(t), with a < ∞. Let γ¯ = φ ◦ γ; γ¯ can be extended to a complete timelike geodesic in L which we still refer to as γ¯ . Let η¯ be a future

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directed broken null geodesic extending from p¯ = γ¯ (0) to q¯ = γ¯ (a). Let η be the lift of η¯ starting at p = γ(0); η extends to a point q ∈ I + (p), with φ(q) = q¯. Since M is globally hyperbolic there exists a maximal timelike geodesic segment µ from p to q. Then µ ¯ = φ ◦ µ is a timelike geodesic segment in L from p¯ to q¯. Hence, µ ¯ = γ¯ |[0,a] . It follows that µ extends γ to q, which is a contradiction. Finally, we show that M is spacelike geodesically complete. If it is not, then there exists an inextendible unit speed spacelike geodesic γ : [0, a) → M , t → γ(t), with a < ∞. Let γ¯ = φ ◦ γ; γ¯ can be extended to a complete spacelike geodesic in L which we still refer to as γ¯ . We now use the fact that timelike geodesics, and broken timelike geodesics, in L can be lifted via φ to M . Let α ¯ =α ¯1 + α ¯ 2 be a two-segment broken timelike geodesic extending from p¯ = γ¯ (0) to q¯ = γ¯ (a), with α ¯ 1 starting at p¯ and past pointing. Similarly, let β¯ = β¯1 + β¯2 be a two-segment broken timelike geodesic extending from p¯ to q¯, with β¯1 starting at p¯ and future pointing. Let x ¯ be the point at the corner of α, ¯ and let y¯ be the point at the corner ¯ Note that γ¯ |[0,a] ⊂ I + (¯ of β. x) ∩ I − (¯ y ). Let α be the lift of α ¯ starting at p = γ(0); α extends to a point q with φ(q) = q¯. Similarly, let β be the lift of β¯ starting at p. Let x be the point at the corner of α, and let y be the point at the corner of β. Note that an initial segment of γ is contained in J + (x) ∩ J − (y). We claim that γ is entirely contained in J + (x) ∩ J − (y). If not then γ either leaves J + (x) or J − (y). Suppose it leaves J + (x) at z ∈ γ ∩ ∂J + (x). Since M is globally hyperbolic, there exists a null geodesic segment η from x to z. Then η¯ = φ ◦ η is a null geodesic in L from x ¯ to z¯ ∈ γ¯ |[0,a] . But since, by construction, γ¯|[0,a] ⊂ I + (¯ x), no such null geodesic exists. Thus, γ ⊂ J + (x) ∩ J − (y). We show that γ extends to q, thereby obtaining a contradiction. Consider a sequence γ(tn ), tn → a. Since J + (x) ∩ J − (y) is compact, there exists a subsequence γ(tnk ) which converges to a point q 0 ∈ J + (x) ∩ J − (y). Since γ¯ (tn ) → q¯, it follows by continuity that φ(q 0 ) = q¯. Let µ be a causal geodesic segment from x to q0 . Then µ ¯ = φ ◦ µ is a causal geodesic from x ¯ to q¯ in L. Thus, µ ¯ = α ¯ 2 , and hence µ = α2 . Since α2 has future end point q, we conclude that q0 = q. Hence, since every sequence γ(tn ), tn → a, has a subsequence converging to q, it follows that limt→a γ(t) = q, and so γ extends to q. This concludes the proof that M is geodesically complete and hence, as noted above, isometric to Minkowski space. We remark in closing that Theorem IV.3 can be generalized in various directions. For example, it is possible to formulate a version of Theorem IV.3 for asymptotically flat spacetimes which contain singularities, black holes, etc., by imposing suitable conditions on the domain of outer communications D = I − (I + )∩I + (I − ), the conclusion then being that D is flat. Also, it appears that Theorem IV.3 can be extended to vacuum spacetimes with positive cosmological constant, Λ > 0, thereby yielding a characterization of de Sitter space. Details of this will appear elsewhere.

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Acknowledgement The author is indebted to Helmut Friedrich for many helpful discussions concerning the proof of Theorem IV.3. The author would also like to thank Lars Andersson and Piotr Chru´sciel for many valuable comments and suggestions. Part of the work on this paper was carried out during a visit to the Royal Institute of Technology in Stockholm, Sweden in 1999. The author wishes to thank the Institute for its hospitality and support.

References [1] A. D. Alexandrov, Some theorems on partial differential equations of second order, Vestnik Leningrad. Univ. Ser. Mat. Fiz. Him. 9 (1954), no. 8, 3–17. [2] L. Andersson, G.J. Galloway, and R. Howard, A strong maximum principle for weak solutions of quasi-linear elliptic equations with applications to Lorentzian and Riemannian geometry, Comm. Pure Appl. Math. 51 (1998), 581–624. [3] J. K. Beem, P. E. Ehrlich, and K. L.Easley, Global Lorentzian geometry, 2 ed., Pure and Applied Mathematics, vol. 202, Marcel Dekker, New York, 1996. [4] P.T. Chru´sciel and G.J. Galloway, Horizons non–differentiable on dense sets, Commun. Math. Phys. 193 (1998), 449–470, gr-qc/9611032. [5] P. T. Chru´sciel, E. Delay, G. J. Galloway, and R. Howard, The area theorem, preprint, 1999. [6] J.-H. Eschenburg, Comparison theorems and hypersurfaces, Manuscripta Math. 59 (1987), no. 3, 295–323. , Maximum principle for hypersurfaces, Manuscripta Math. 64 (1989),

[7] 55–75.

[8] H. Friedrich, Existence and structure of past asymptotically simple solutions of Einstein’s field eqautions with positive cosmological constant, J. Geom. Phys. 3 (1986) 101–117. [9]

, Einstein equations and conformal structure, in The Geometric universe: Science, geometry and the work of Roger Penrose, eds. S. A. Huggett et al., Oxford University Press, Oxford.

[10]

, Einstein’s equations and geometric asymptotics, in Gravitation and relativity: At the turn of the millennium. Proceedings of the GR-15 Conference at Pune, India, in June 1997, eds. N. Dadich and J. Narlikar, IUCAA, Pune, India.

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[11] J.L. Friedman, K. Schleich, and D.M. Witt, Toplogical censorship, Phys. Rev. Lett. 71 (1993) 1486–1489. [12] G.J. Galloway and E. Woolgar, The cosmic censor forbids naked topology, Classical and Quantum Grav. 14 (1997) L1–L7. [13] S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time, Cambridge University Press, Cambridge, 1973. [14] D.N. Kupeli, On null submanifolds in spacetimes, Geom. Dedicata 23 (1987), 33–51. [15] R.P.A.C. Newman, The global structure of simple space-times, Commun. Math. Phys. 123 (1989), 17–52. [16] B. O’Neill, Semi–Riemannian geometry, Academic Press, New York, 1983. [17] R. Penrose, Zero rest-mass fields including gravitation: asymptotic behavior, Proc. Roy. Soc. Lond. A 284 159–203. [18]

, Techniques of differential topology in relativity, SIAM, Philadelphia, 1972, (Regional Conf. Series in Appl. Math., vol. 7).

Gregory J. Galloway Supported in part by NSF grant # DMS-9803566. University of Miami, Department of Mathematics and Computer Science, Coral Gables, FL, 33124 email :[email protected]

Communicated by S. Klainerman submitted 13/09/99; accepted 25/09/99

Ann. Henri Poincar´ e 1 (2000) 569 – 605 c Birkh¨

auser Verlag, Basel, 2000 1424-0637/00/030569-37 $ 1.50+0.20/0

Annales Henri Poincar´ e

Towards F=ma in a General Setting for Lagrangian Mechanics Andrew D. Lewis Abstract. By using a suitably general definition of a force, one may geometrically cast the Euler-Lagrange equations in a “force balance” form. The key ingredient in such a construction is the Euler-Lagrange 2-force which is a bundle map from the bundle of two-jets into the first contact system. This 2-force can be used as the basis for a geometric presentation of Lagrangian mechanics with external forces and constraints. Also described is the precise correspondence between this 2-force and the Poincar´ e-Cartan two-form.

I Introduction In making a differential geometric presentation of Lagrangian mechanics, one of the cumbersome aspects has always been that the Euler-Lagrange equations are not themselves the components of a tensor field, and so one cannot easily assign to the equations a geometric object. One way of circumventing this has been to use the symplectic-like formalism of the Poincar´e-Cartan two-form. This is done in the time-independent case, e.g., by Abraham and Marsden [1] and Libermann and Marle [14], by pulling-back the symplectic form on the cotangent bundle via the Legendre transformation. In the time-dependent case one has to modify this construction to use the natural almost tangent-like structure on the bundle of one-jets [7]. In either case, the Euler-Lagrange equations themselves are somewhat obscured “inside” the two-form. We offer here an alternative to these two-form based geometric formulations of Lagrangian mechanics by making a general notion of a force, and then, with this general notion in hand, assigning to a Lagrangian function a force which may be thought of as a generalisation of the “inertial force” ma (i.e., mass×acceleration) in Newtonian mechanics. We call this force the Euler-Lagrange 2-force. Using this generalised inertial force, we can provide easy, intuitive characterisations of the Euler-Lagrange equations. For example, if the Lagrangian is regular then we can define the Euler-Lagrange vector field in a straightforward manner. Our presentation includes external forces, and a general class of constraints. The geometric object we define here is a fairly natural one to consider. However, it is interesting to see exactly how it is related to the standard constructions in Lagrangian mechanics. In particular, the exact relationship between the Euler-Lagrange 2force and the Poincar´e-Cartan two-form requires some work to make precise (see Theorem VI.1).

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Another aspect of our presentation is that we use infinite-dimensional manifolds throughout. Besides increasing the scope of the work, this avoids the preponderance of indices that results from a strictly finite-dimensional presentation. Binz, de Le´ on and Socolescu [3] provide a formulation for nonholonomic mechanics on Hilbert manifolds. The standard formalism of jet bundles, which is an appropriate general setting for Lagrangian mechanics, is readily adapted to infinite-dimensional manifolds, and the basic ideas and local representations are given in Section II. Using jet bundles as the starting point, we provide some basic background to Lagrangian mechanics in Section III. We investigate the geometry which can be associated with a Lagrangian function, as well as provide suitably general definitions of a force and a constraint. It is when defining forces that we provide the framework which allows for the definition of the “Euler-Lagrange 2-force” in Section IV. In finite-dimensions, this is essentially a force whose coefficients are the components of the Euler-Lagrange equations. It is in this way that we give geometric meaning to the Euler-Lagrange equations themselves, and not through other devices such as the two-form formalism. When one mimics the Hamiltonian formulation using the Poincar´e-Cartan two-form, the way one defines the Euler-Lagrange vector field, in the case when the Lagrangian is regular, follows just as it does in the Hamiltonian case (unsurprisingly). In Section V we indicate how to use the Euler-Lagrange 2force to define the Euler-Lagrange vector field in cases when L is regular. We also indicate how to handle the forced and constrained cases within our “force balance” framework. It should not be surprising that the Poincar´e-Cartan two-form and the Euler-Lagrange 2-force which we define are, when suitably interpreted, equivalent. The exact form of this equivalence is given in Section VI. It is somewhat non-trivial to derive the Poincar´e-Cartan two-form from the Euler-Lagrange 2-force.

II Jet bundle geometry In this section we provide a review of the geometric tools we will use in the paper. We shall for the most part adopt the notations and conventions of Abraham, Marsden and Ratiu [2]. In particular, we shall work within the category of C ∞ reflexive Banach manifolds. However, where clarity is assisted, the finite-dimensional coordinate formulas are provided. Here is a list of notation we use, in roughly alphabetical order. We shall define many objects upon their first usage, but all terminology should be found in this list in any case. (a, b) : the open interval in R with endpoints a and b with a < b α·e : the natural pairing of α ∈ E ∗ with e ∈ E X,Y : X is defined to be equal to Y V |M : the restriction to M ⊂ B of a vector bundle π : V → B ann(F ) : the annihilator in E ∗ of F ⊂ E Br,e : the open ball of radius r > 0 in E centred at e

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c0 (t) coann(Σ) Dk f (u)

: T c(t) · 1, where c : [t1 , t2 ] → M is a curve : the elements in E annihilated by Σ ⊂ E ∗ : the kth derivative at u of a map f : U → F with U an open subset of a Banach space E and F a Banach space Dk f (u1 , . . . , um ) : the kth partial derivative at (u1 , . . . , um ) of a map f : U1 × · · · × Um → F where Ui is an open subset of a Banach space Ei , i = 1, . . . , m, and F is a Banach space dk f : the k-jet derivative of f : J k Q → R E, F : typical Banach spaces ∞ : the sections of a vector bundle with total space V Γ (V ) V (V ) : the bundle of exterior k-forms for a vector bundle V k L(E; F ) : the set of continuous linear maps between Banach spaces E and F Tf : TM → TN : the derivative of a mapping f : M → N between manifolds M and N τM : T M → M : the tangent bundle projection τk,l : J k Q → J l Q : the natural projection for k > l (we denote τk = τk,0 ) U : the image of U under φ for a manifold chart (U, φ) : the restriction of τQ to V Q νQ : V Q → Q η : the pull-back of dt to J 1 Q Our notation ann(F ) and coann(Σ) is non-standard—normally one sees the notation F 0 and Σ⊥ . Also, we shall sometimes write A · e for the evaluation of A ∈ L(E; F ) on e ∈ E, especially if A itself depends on other arguments. This avoids awkward double parentheses like A(x)(e).

II.1

Jet bundles

We shall work in the strictly time-dependent formulation. Thus we consider a locally trivial fibre bundle π : Q → R which is generally not provided with a global trivialisation. A special case, of course, is that when Q = R × Q where Q is the configuration manifold and π is projection onto the first factor. In the general case, we call π : Q → R the configuration bundle. This point of view of using a non-trivial bundle is taken, for example, by Giachetta [9] and de Le´ on, Marrero and de Diego [7]. Various authors [11, 16] use the jet bundle formalism, but consider trivial bundles. Employing non-trivial bundles offers no practical advantages, but often simplifies the exposition by disallowing certain confusing identifications which can be made in the trivial case. Also, the employment of fibre bundles perhaps makes easier any future generalisations to Lagrangian field theory. We shall frequently work in an adapted chart for π : Q → R. Since the base space R is equipped with a natural chart, we shall always assume our adapted charts, which we denote (U, φ), are chosen so that U , image(φ) = (a, b)× U 0 , and so that the induced chart on the base space is the identity chart ((a, b), id(a,b) ).

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Here U 0 is an open subset of a Banach space we will usually denote by E. We shall write coordinates in an adapted chart as (t, u) ∈ (a, b) × U 0 . When we write finite-dimensional coordinate formulas, we denote coordinates by (t, q i ). In the finite-dimensional case, we shall suppose that dim(Q) = n + 1 where n ≥ 1. The vertical subbundle of the fibration π : Q → R we denote by V Q and recall that it is the kernel of the projection T π : T Q → T R. We denote the projection from V Q to Q by νQ . Associated with the configuration bundle π : Q → R are its jet bundles [19]. By convention define J 0 Q = Q. Fix t ∈ R and q ∈ π −1 (t). Two local sections c1 and c2 are equivalent to order one at q if c1 (t) = c2 (t) = q and c01 (t) = c02 (t). We write the equivalence class to order one at q containing c by [c(t)]1 . The set of all such equivalence classes we denote by J 1 Q which is the bundle of onejets. We denote by τ1 : J 1 Q → Q the natural projection which forgets first-order equivalence. If c : [t1 , t2 ] → Q is a local section, we define a local section j 1 c of π ◦ τ1 : J 1 Q → R by assigning to t ∈ [t1 , t2 ] the equivalence class [c(t)]1 . Now let t ∈ R and v ∈ (π ◦ τ1 )−1 (t). Two local sections c1 and c2 are equivalent to order two at v ∈ J 1 Q if they are equivalent to order one with [c1 (t)]1 = [c2 (t)]1 = v, and if (j 1 c1 )0 (t) = (j 1 c2 )0 (t). We denote the equivalence class to order two at q containing c by [c(t)]2 . The set of these equivalence classes we denote by J 2 Q and call the bundle of two-jets. The map τ2,1 : J 2 Q → J 1 Q is the natural projection which forgets equivalence to order two, but remembers equivalence to order one. Given a local section c : [t1 , t2 ] → Q we define a local section j 2 c of J 2 Q over R by assigning to t ∈ [t1 , t2 ] the equivalence class [c0 (t)]2 . One may inductively proceed in this way, defining the higher order jet bundles τk,k−1 : J k Q → J k−1 Q. We shall denote by τk,l : J k Q → J l Q the natural projection for l < k, and we shall adopt the convention that τk = τk,0 . It may be shown that the fibre bundle τk,k−1 : J k Q → J k−1 Q is an ∗ ∗ affine bundle modelled on the pull-back vector bundle τk−1 νQ : τk−1 V Q → J k−1 Q. ∗ ∗ k It is also true that the pull-back bundle τk νQ : τk V Q → J Q is naturally isomorphic to ker(T τk,k−1 ), and so is a subbundle of T (J k Q). If (U, φ) is an adapted chart for π : Q → R, we have induced natural charts for T Q and the jet bundles J k Q which we denote by (T U, T φ) and (J k U, j k φ). If φ is a bijection from U ⊂ Q to U = (a, b) × U 0 ⊂ R × E, then T φ takes its values in U × R × E, and j k φ takes its values in U × E × · · · × E (where there are k of the factors E). We shall write coordinates for T Q as ((t, u), (τ, v)), and coordinates for J k Q as (t, u, u1 , . . . , uk ). If we wish to express finite-dimensional coordinate formulas we write coordinates for T Q as ((t, q i ), (τ, v j )) and coordinates for J 2 Q as (t, q i , v j , ak ) (we shall not be employing anything higher than twojets). The adapted chart (U, φ) also induces a natural chart (T ∗ U, T ∗ φ) for T ∗ Q, and we denote coordinates here by ((t, u), (λ, α)) in infinite-dimensions, and by ((t, q i ), (λ, pi )) in finite-dimensions. There are natural inclusions of J k Q in T (J k−1 Q) for k ≥ 1. In natural coor-

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dinates, for k = 1, 2 the inclusions are given by (t, u, u1 ) 7→ ((t, u), (1, u1 )) (t, u, u1 , u2 ) 7→ ((t, u, u1 ), (1, u1 , u2 )). We also note that the local form of a vector in the vertical subbundle V Q is ((t, u), (0, e)).

II.2

The first contact system

Associated with the jet bundles are the contact systems. The kth contact system is a subbundle of T ∗ (J k Q). We shall only use the first contact system which we now describe using the characterisation of Gardner and Shadwick [8]. We define the fibre of C1 (Q), the first contact system, at j 1 c(t) ∈ J 1 Q by ∗ C1j 1 c(t) (Q) = (Tj 1 c(t) τ1 )∗ β − (Tt c ◦ Tj 1 c(t) (π ◦ τ1 ))∗ β | β ∈ Tc(t) Q . (1) If one works through this definition, it may be seen that C1 (Q) consists in a natural chart of those elements of T ∗ (J 1 Q) of the form ((t, u, u1 ), (−α · u1 , α, 0))

(2)

for some α ∈ E ∗ (here α · u1 means the natural pairing of α ∈ E ∗ with u1 ∈ E). In turn, one readily sees that this corresponds to the usual local basis {dq 1 − v 1 dt, . . . , dqn − v n dt} in finite-dimensions. Gardner and Shadwick [8] show that the definition (1) gives C1 (Q) the property that a local section σ : [t1 , t2 ] → J 1 Q has the property that σ 0 (t) is annihilated by C1 (Q) if and only if σ = j 1 c for a local section c : [t1 , t2 ] → Q. The map which in local coordinates is given by ((t, u, u1 ), (−α · u1 , α, 0)) 7→ ((t, u, u1 ), α) is a vector bundle isomorphism of C1 (Q) with τ1∗ V ∗ Q where V ∗ Q is the dual bundle to V Q. Note that V ∗ Q is not naturally a subbundle of T ∗ Q. We shall make frequent use of this natural identification of τ1∗ V ∗ Q with C1 (Q). Roughly speaking, it is often more convenient to represent objects in a chart by using τ1∗ V ∗ Q, but more natural intrinsically to use C1 (Q). In finite-dimensions, when we regard C1 (Q) as the dual bundle to τ1∗ V Q ' ker(T τ1 ), the local basis {dq1 − v 1 dt, . . . , dqn − v n dt} for C1 (Q) is dual to the local basis { ∂v∂ 1 , . . . , ∂v∂n } for τ1∗ V Q.

II.3

The jet derivatives

Lewis [12] introduced the idea of the “acceleration derivative” of a function on J 2 Q as a generalisation of the fibre derivative of a function on a vector bundle (also see [10]). Here we extend this to the k-jet derivative of a function f defined on J k Q. The definition goes as follows. Fix ξ ∈ J k−1 Q and let fξ denote the restriction of f to the fibre of J k Q over ξ. Since this fibre is an affine space modelled on the

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vector space Vτk−1 (ξ) Q, its derivative dfξ may be regarded as taking its values in Vτ∗k−1 (ξ) Q. But as we just saw, Vτ∗k−1 (ξ) Q is naturally isomorphic to C1τk−1,1 (ξ) (Q), and so in this way we construct a map dk f : J k Q → C1 (Q) so that the diagram dk f

J k QE

EE EE τk,1 EEE "

J 1Q

/ C1 (Q) w ww w ww {w w

commutes. In a natural chart we have dk f (t, u, u1 , . . . , uk ) = ((t, u, u1 ), D k+2 f (t, u, u1 , . . . , uk )) where D k+2 denotes the (k + 2)nd partial derivative.

II.4

The almost tangent-like structure on J 1 Q

Recall that on the tangent bundle of the manifold Q there is a natural almost tangent structure which we denote by SQ . Explicitly, SQ is the (1, 1) tensor field on T Q given by SQ (vq )(X) = verliftvq (Tvq τQ (X)) where vq ∈ Tq Q, X ∈ Tvq T Q, τQ : T Q → Q is the tangent bundle projection, and verliftvq is the vertical lift defined by d (vq + tuq ) verliftvq (uq ) = dt t=0 for uq ∈ Tq Q.1 It turns out that SQ |J 1 Q leaves T (J 1 Q) ⊂ T T Q invariant and so defines a (1, 1) tensor field on J 1 Q which we denote by S˜Q . In natural coordinates one may determine that S˜Q (t, u, u1 ) · (τ, e1 , e2 ) = (0, 0, e1 − τ u1 ). ∗ of T ∗ (J 1 Q) has local representative The dual endomorphism S˜Q ∗ S˜Q (t, u, u1 ) · (λ, α1 , α2 ) = (−α2 · u1 , α2 , 0).

(3)

∗ is a surjective map Given our local representation (2) of C1 (Q), this shows that S˜Q 1 onto C (Q). In finite-dimensions we have

∂ S˜Q = i ⊗ (dq i − v i dt). ∂v 1 Note that this definition of S has nothing to do with Q being the total space of a locally Q trivial fibre bundle over R.

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Second-order vector fields

Since J 2 Q is naturally a subset of T (J 1 Q), it makes sense to define a second-order vector field to be a vector field on J 1 Q which takes its values in J 2 Q. Thus, in a natural chart (J 1 U, j 1 φ), a second-order vector field has representative (t, u, u1 ) 7→ ((t, u, u1 ), (1, u1 , X(t, u, u1 ))) for some X : U × E → E. Note that every second-order vector field is annihilated by C1 (Q), and that if we add a section of τ1∗ V Q ' ker(T τ1 ) to a second-order vector field, we get another second-order vector field. If M is a submanifold of J 1 Q, a second-order vector field on M is a vector field on M taking values in (J 2 Q|M ) ∩ T M . Of course, for a general submanifold M , it is possible that there will be no second-order vector fields on M .

III The components of Lagrangian mechanics In this section we review some common terminology from Lagrangian mechanics. We wish to give some properties of Lagrangian functions, as well as present general definitions of forces and constraints.

III.1

Lagrangians

A Lagrangian is a R-valued function L on J 1 Q. For X ∈ Γ∞ (V Q) define a function LX on J 1 Q by LX (vq ) = hd1 L(vq ); X(q)i. Since X depends only on Q and since the derivative d1 is taken only with respect to the fibre in J 1 Q, d1 LX (vq ) depends only on the value of X at q, and not on the derivative of X. As a consequence we may define a symmetric (0, 2) tensor gL on the pull-back bundle τ1∗ V Q as follows: gL (vq )(X(q), Y (q)) = hd1 LX (vq ); Y (q)i where Y is a another vertical vector field on Q. In a natural chart we have gL (t, u, u1 ) = D23 L(t, u, u1 ). In finite-dimensions this yields the familiar formula gL =

∂2L (dq i − v i dt) ⊗ (dq j − v j dt) ∂v i ∂v j

[ where we identify τ1∗ V ∗ Q with C1 (Q). Associated to gL is a bundle map gL : ∗ 1 1 → τ1 V QC (Q) over the identity on J Q. L is weakly regular if gL (v) is a weakly nondegenerate form (i.e., gL (v)(X, Y ) = 0 for all Y implies X = 0) for each [ v ∈ J 1 Q, and regular if gL is a vector bundle isomorphism from τ1∗ V Q to τ1∗ V ∗ Q ' 1 C (Q). L is positive-definite (resp. negative-definite) if gL (v) is a positive-definite (resp. negative-definite) form for each v ∈ J 1 Q. If L is regular, we denote the ] [ by gL : C1 (Q) → τ1∗ V Q. inverse of gL

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Forces

We provide a definition of a force which allows dependence on time, configuration, and any finite number of derivatives of configuration with respect to time. One should think of a force as a mechanism for inhibiting motion in certain directions—thus it is intuitive to regard a force as taking values in τ1∗ V ∗ Q ' C1 (Q). Precisely, a k-force is a smooth map Φ : J k Q → C1 (Q) so that the diagram Φ

J k QE

EE EE τk,1 EEE "

J 1Q

/ C1 (Q) w ww ww w {w w

commutes. In a natural chart a k-force is represented by Φ(t, u, u1 , . . . , uk ) = ((t, u, u1 ), Φ(t, u, u1 , . . . , uk )) for some map Φ : U × E × · · · × E → E ∗ . Here, for the sake of making a shorter formula, we have identified C1 (Q) with τ1∗ V ∗ Q. Most forces one encounters are 1-forces (i.e., they depend on time, configuration, and velocity), but the main new idea of this paper, presented in Section IV, is that of a 2-force which we associate with a Lagrangian. Note that a 1-force is simply a C1 (Q)-valued one-form on J 1 Q. If c : [t1 , t2 ] → Q is a local section of π : Q → R, we may define a force along c to be a map Φ : [t1 , t2 ] → C1 (Q) so that the diagram

ww {ww

C1 (Q)

[a, b]

w

Φ www

AA AAc AA AA /Q

commutes.

III.3

Constraints

We also wish to consider constraints in our formulation of mechanics. We shall consider a fairly general notion of a constraint, but one which is nonetheless convenient for proving an existence result for solutions of the corresponding constrained Euler-Lagrange equations. It is necessary to devote some effort to providing local descriptions for the constraints we consider, so a significant portion of this section will be devoted to this task. The dividends will be reaped in the proof of Theorem V.5 where the notation we introduce shortly will prove useful. The reader will find our notation applied to a simple example in Section VII. A constraint is a pair (M, Λ) where M is a submanifold of J 1 Q and Λ is a subbundle of T ∗ (J 1 Q). We shall see that all that is actually required is the restriction,

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Λ|M , of Λ to M . Any extension of Λ off M will suffice since only algebraic constructions are performed with the sections of Λ. A local section c : [t1 , t2 ] → Q satisfies the constraint (M, Λ) if j 1 c(t) ∈ M and if (j 1 c)0 (t) Λj 1 c(t) = 0 for t ∈ [t1 , t2 ]. Given a constraint (M, Λ), we associate to it a subset F(M,Λ) of C1 (Q)|M defined ∗ ∗ (Λ|M ). This makes sense since S˜Q is C1 (Q)-valued by (2) and (3). by F(M,Λ) = S˜Q A constraint force is defined to be a force taking values in F(M,Λ) .2 The above general notion of a constraint is too unstructured. With this definition it is possible, for example, that constrained problems have no solutions. To enable us to state a general result concerning existence of solutions for constrained systems, we need to consider a class of constraints which is rather more restrictive. Even though it is restrictive, it contains the types of constraints most often considered—for example, as we indicate below, constraints which are affine in velocity are of the form we now introduce. Let us define a subset ΛM of T ∗ M by defining its fibre at v ∈ M to be ΛM,v = Tv∗ iM (ΛiM (v) ) where iM : M → J 1 Q is the inclusion. For a general submanifold M , ΛM cannot be expected to be a subbundle. The constraint (M, Λ) is called ideal if the following conditions hold: IC1. F(M,Λ) is a subbundle of C1 (Q)|M ; ∗ |M ) ∩ Λ|M is a subbundle of Λ|M ; IC2. ker(S˜Q

IC3. J 2 Q|M ∩ coann(ΛM ) is a non-trivial affine subbundle of J 2 Q|M modelled on the vector subbundle coann(F(M,Λ) ) ∩ (τ1 |M )∗ V Q of (τ1 |M )∗ V Q. The condition IC1 is a natural one as it asks that the set in which the constraint forces take their values be a subbundle. Condition IC2 will allow us to make a ∗ local decomposition of Λ|M which renders the map S˜Q |M in a particularly simple form. This in turn is helpful in the existence proof for solutions to the constrained problem. The final condition, IC3, provides a reasonable target set—an affine subbundle—for a second-order vector field which describes the constrained motion. As we shall see in Theorem V.5, this general notion of an ideal constraint is sufficient to establish the existence of a second-order vector field on M whose integral curves are solutions of the constrained equations of motion. These conditions are not, however, necessary in order for the constrained system to have solutions. Note that in the event that Q is modelled on a Hilbert space rather than a Banach space, the assumptions that various subsets be subbundles bear only on their having constant rank (or the infinite-dimensional equivalent). However, for Banach manifolds there is the additional hypothesis that the subspaces involved be split. 2 Just why one should define a constraint force in this way is not a simple matter to justify—it is really the key ingredient to the nature of the Euler-Lagrange equations for systems with nonlinear constraints. This matter is discussed by Chetaev [5].

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It will be important for us to have explicit local representations for a constraint (M, Λ), so let us introduce our notation for this. Let (U, φ) be an adapted chart for π : Q → R. We suppose that U = φ(U ) ⊂ R × E for some Banach space E. This induces natural charts (J k U, j k φ) for J k Q, k = 1, 2, and a natural chart (T ∗ (J 1 U ), T ∗ (j 1 φ)) for T ∗ (J 1 Q). Since Λ is a subbundle of T ∗ (J 1 Q), Λv is split in Tv∗ (J 1 Q) for each v ∈ J 1 Q. Therefore, by choosing U sufficiently small, there exists a vector bundle chart (T ∗ (J 1 U ), ψ) for T ∗ (J 1 Q) which is adapted to this subbundle. This means that the following hold: 1. ψ is a bijection from T ∗ (J 1 U ) onto U × E × F1∗ × F2∗ for Banach spaces F1 and F2 ; 2. ψ(Λ)(t,u,u1 ) = {(t, u, u1 )} × F1∗ × {0}; 3. the overlap map from T ∗ (j 1 φ)(T ∗ (J 1 U )) = U × E × R∗ × E ∗ × E ∗ to ψ(T ∗ (J 1 U )) = U × E × F1∗ × F2∗ has the form h : ((t, u, u1 ), (λ, α1 , α2 )) 7→ ((t, u, u1 ), (A10 (t, u, u1 ) · λ + A11 (t, u, u1 ) · α1 + A12 (t, u, u1 ) · α2 , A20 (t, u, u1 ) · λ + A21 (t, u, u1 ) · α1 + A22 (t, u, u1 ) · α2 )) for maps Aj0 : U × E → L(R∗ ; Fj∗ ) and Aij : U × E → L(E ∗ ; Fj∗ ), i, j = 1, 2. Let us denote the inverse of the overlap map by h−1 : ((t, u, u1 ), (ν 1 , ν 2 )) 7→ ((t, u, u1 ), (B01 (t, u, u1 ) · ν 1 + B02 (t, u, u1 ) · ν 2 , B11 (t, u, u1 ) · ν 1 + B12 (t, u, u1 ) · ν 2 , B21 (t, u, u1 ) · ν 1 , B22 (t, u, u1 ) · ν 2 )) for maps B0j : U × E → L(Fj∗ ; R∗ ) and Bij : U × E → L(Fj∗ ; E ∗ ), i, j = 1, 2. Thus the constraint codistribution Λ is locally given in the natural chart (T ∗ (J 1 U ), T ∗ (j 1 φ)) by Λ(t,u,u1 ) = {((t,u,u1 ),(B01 (t,u,u1 ) · ν 1 ,B11 (t,u,u1 ) · ν 1 ,B21 (t,u,u1 ) · ν 1 ))|ν 1 ∈ F1∗ }. ∗ ∗ Using the local form (3) of S˜Q , we see that the local form of S˜Q |Λ is given by

((t, u, u1 ), (B01 (t, u, u1 ) · ν 1 , B11 (t, u, u1 ) · ν 1 , B21 (t, u, u1 ) · ν 1 )) 7→ ((t, u, u1 ), B21 (t, u, u1 ) · ν 1 ) (4) where we think of C1 (Q) ' τ1∗ V ∗ Q. The local decomposition we have just presented is not really sufficient for our purposes. The problem arises because B21 (t, u, u1 ) may not be injective, corre∗ sponding to the fact that S˜Q |Λ may not be injective. To overcome this difficulty, we

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∗ make a further refinement of Λ which is appropriately adapted to the mapping S˜Q . 1 ∗ Let us suppose that F(M,Λ) is a subbundle of C (Q)|M and that ker(S˜Q |M ) ∩ Λ|M is a subbundle of Λ|M (as, for example, when (M, Λ) is ideal). In this case the map (4) is a local vector bundle mapping whose image and kernel are subbundles. Thus (see [2, Proposition 3.4.18]) we may further refine our local representation of Λ|M . Indeed, supposing that M ∩ J 1 U 6= ∅, we may choose a vector bundle chart (T ∗ (J 1 U ), ψ0 ) for T ∗ (J 1 Q) with the following properties: ∗ ∗ 1. ψ0 is a bijection from T ∗ (J 1 Q) onto U × E × F11 × F12 × F2∗ for Banach spaces F11 , F12 , and F2 ; ∗ ∗ 2. ψ0 (Λ)(t,u,u1 ) = {(t, u, u1 )} × F11 × F12 × {0}; ∗ ∗ 3. ψ0 (ker(S˜Q |M ) ∩ Λ|M ) = {(t, u, u1 )} × {0} × F12 × {0} if φ−1 (t, u, u1 ) ∈ M ;

4. the overlap map from T ∗ j 1 φ(T ∗ (J 1 U )) = U × E × R∗ × E ∗ × E ∗ ∗ ∗ to ψ0 (T ∗ (J 1 U )) = U × E × F11 × F12 × F2∗ has the form

h0 : ((t, u, u1 ), (λ, α1 , α2 )) 7→ ((t, u, u1 ), (A110 (t, u, u1 ) · λ + A111 (t, u, u1 ) · α1 + A112 (t, u, u1 ) · α2 , (A120 (t, u, u1 ) · λ + A121 (t, u, u1 ) · α1 + A122 (t, u, u1 ) · α2 , A20 (t, u, u1 ) · λ + A21 (t, u, u1 ) · α1 + A22 (t, u, u1 ) · α2 )) ∗ for maps A1j0 : U × E → L(R∗ ; F1j ), A1ij : U × E → L(E ∗ ; F1i ),

A20 : U × E → L(R∗ ; F2∗ ) and A2j : U × E → L(E ∗ ; Fj∗ ), i, j = 1, 2; 5. if the inverse of h0 is 0

h −1 : ((t, u, u1 ), (ν 11 , ν 12 , ν 2 )) 7→ ((t, u, u1 ), (B011 (t, u, u1 ) · ν 11 + B012 (t, u, u1 ) · ν 12 + B02 (t, u, u1 ) · ν 2 , B111 (t, u, u1 ) · ν 11 + B112 (t, u, u1 ) · ν 12 + B12 (t, u, u1 ) · ν 2 , B211 (t, u, u1 ) · ν 11 + B212 (t, u, u1 ) · ν 12 + B22 (t, u, u1 ) · ν 2 )) ∗ ∗ for maps B01j : U × E → L(F1j ; R∗ ), Bi1j : U × E → L(F1j ; E ∗ ), ∗ ∗ ∗ ∗ B02 : U × E → L(F2 ; R ), and Bj2 : U × E → L(F2 ; E ), i, j = 1, 2, then, when φ−1 (t, u, u1 ) ∈ M , the following hold:

(a) B211 (t, u, u1 ) is injective with split image; (b) B212 (t, u, u1 ) = 0; (c) the map ν 12 7→ B012 (t, u, u1 ) · ν 12 + B112 (t, u, u1 ) · ν 12 is injective with split image.

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With this refined splitting we locally have ∗ F(M,Λ) = ((t,u,u1 ),B211 (t,u,u1 ) · ν 11 ) ν 11 ∈ F11 , (t,u,u1 ) ∈ j 1 φ(M ∩ J 1 U ) and ∗ |M ) ∩ Λ|M = ((t, u, u1 ), (B012 (t, u, u1 ) · ν 12 , ker(S˜Q

∗ , (t, u, u1 ) ∈ j 1 φ(M ∩ J 1 U ) . B112 (t, u, u1 ) · ν 12 , 0)) | ν 12 ∈ F12

∗ |Λ, let us explicitly state To emphasise how this refined splitting is adapted to S˜Q that the local form of this vector bundle map is

((t, u, u1 ), (B011 (t, u, u1 ) · ν 11 + B012 (t, u, u1 ) · ν 12 , B111 (t, u, u1 ) · ν 11 + B112 (t, u, u1 ) · ν 12 B211 (t, u, u1 ) · ν 11 + B212 (t, u, u1 ) · ν 12 )) 7→ ((t, u, u1 ), B211 (t, u, u1 ) · ν 11 ). This describes one half of a constraint (M, Λ). To see how the submanifold M may be locally added to the mix, suppose that the chart (U, φ) is chosen so that ˜ , χ) for M . We V , M ∩ J 1 U is open in M and forms the domain of a chart (U ˜ ˜ ˜ →U ×E shall suppose that χ is E-valued for some Banach space E. Let iM : U be the local representative of iM which we write as iM (˜ u) = (C0 (˜ u), C1 (˜ u), C2 (˜ u)) ˜ → U ∩ R, C1 : U ˜ → U ∩ E, and C2 : U ˜ → E. for maps C0 : U In finite-dimensions, the above constructions may be described in terms of local bases for the various subbundles. We choose a local basis {β 1 , . . . , β 2n+1 } for T ∗ (J 1 Q) with the property that β 1 , . . . , β m generate Λ. We shall write the forms β a , a = 1, . . . , m, as β a = β0a dt + βia dq i + βˆia dv i .

(5)

One readily sees that the one-forms βˆia (dq i − v i dt),

a = 1, . . . , m

when restricted to M , locally generate F(M,Λ) . However, they will not in general be linearly independent. If F(M,Λ) is a subbundle, then we may choose the one-forms β a , a = 1, . . . , m, in such a way that the one-forms βˆia (dq i − v i dt),

a = 1, . . . , m, ˜

m ˜ ≤ m,

˜ + form a basis for F(M,Λ) when restricted to M , and the one-forms β a , a = m ∗ 1 ˜ 1, . . . , m form a basis for ker(SQ |M ) ∩ Λ|M when restricted to M . If (x , . . . , xr ) are coordinates for M , we may write the inclusion iM locally as (x1 , . . . , xr ) 7→ (C0 (x), C1i (x), C2j (x)),

i, j = 1, . . . , n.

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In this case, a local section given in coordinates by t 7→ (t, q i (t)) satisfies the constraint if and only if C0 (x(t)) = t,

C1i (x(t)) = q i (t),

C2j (x(t)) = q˙j (t),

i, j = 1, . . . , n

for some curve t 7→ (xα (t)) in M , and βˆia q¨i + βia q˙i + β 0 = 0,

a = 1, . . . , m.

Let us now employ our local notation to characterise ideal constraints. The local description of ΛM ⊂ T ∗ M is u, DC0 (˜ u)∗ · (B011 (t, u, u1 ) · ν 11 + B012 (t, u, u1 ) · ν 12 ) + ΛM = (˜ DC1 (˜ u)∗ ·(B111 (t, u, u1 )·ν 11 +B112 (t, u, u1 )·ν 12 )+DC2 (˜ u)∗ ·B211 (t, u, u1 )·ν 11 ) | ∗ ∗ , ν 12 ∈ F12 , (t, u, u1 ) = iM (˜ u) . (6) ν 11 ∈ F11 Therefore, J 2 Q|M ∩ coann(ΛM ) = (˜ u, e˜) | DC0 (˜ u) · e˜ = 1, DC1 (˜ u) · e˜ = u1 , ∗ ∗ ∗ u) · e˜ = 0, B011 (t, u, u1 ) · 1 + B111 (t, u, u1 ) · u1 + B211 (t, u, u1 ) ◦ DC2 (˜ ∗ ∗ B012 (t, u, u1 ) · 1 + B112 (t, u, u1 ) · u1 = 0, (t, u, u1 ) = iM (˜ u) . (7) We also have

∗ (t, u, u1 ) · e = 0 . coann(F(M,Λ) ) ∩ (τ1 |M )∗ V Q = (t, u, u1 ), (0, 0, e) | B112

(8)

If J 2 Q∩coann(ΛM ) is an affine subbundle modelled on coann(F(M,Λ) )∩(τ1 |M )∗ V Q we must have DC2 (˜ u) · e˜ + e ∈ image(DC2 (˜ u))

(9)

for each e which satisfies the relation in (8). This shows that for ideal constraints we have coann(F(M,Λ) ) ∩ (τ1 |M )∗ V Q ⊂ T M. Remarks III.1 1. The case of constraints for which M = J 1 Q is taken up by ∗ Giachetta [9]. In this case Giachetta calls a constraint “ideal” when S˜Q |Λ is 1 a vector bundle monomorphism. One then readily checks that (M = J Q, Λ) is ideal in our sense. 2. de Le´ on, Marrero and de Diego [6] provide a notion of an “admissible” con∗ straint as a pair (M, D) (here D is a distribution on M ) for which S˜Q | ann(D) is a vector bundle monomorphism. Taking Λ so that Λ|M = coann(D), this implies, but is not equivalent to, the conditions IC2 and IC1. This notion of “admissible” is not adequate to ensure solutions for the constrained dynamics; these are provided as separate conditions by de Le´ on, Marrero and de Diego.

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An important class of constraints is affine constraints which are defined by a codistribution Λ0 on Q. A local section c : [t1 , t2 ] → Q satisfies the affine constraint Λ0 if c0 (t) Λ0,c(t) = 0 for t ∈ [t1 , t2 ]. Let us see how we can construct a general constraint from an affine constraint. First of all, we define the submanifold M to be coann(Λ0 ) ∩ J 1 Q. This set will not always be non-empty. Lewis [13] shows that if dt ∧ Λ0 6= 0 then M defined in this way is non-empty, and is further an affine subbundle of J 1 Q. Let us say Λ0 is compatible with π if dt ∧ Λ0 6= 0. Also, one can lift Λ0 to a subbundle j 1 Λ0 of T ∗ (J 1 Q) as follows. For a section β of Λ0 define a function fβ on T Q by fβ (vq ) = dβ(q) · vq . We then define a subset ΛT0 of T ∗ T Q whose fibre over vq ∈ J 1 Q is ∞

ΛT0,vq = { dfβ (vq ) | β ∈ Γ (Λ0 )} . Lewis [13] shows that ΛT0 is a subbundle of T ∗ T Q. If ιQ : J 1 Q → T Q is the inclusion, Lewis further shows that if Λ0 is compatible with π then the map Tv∗ ιQ : Tv∗ T Q → Tv∗ (J 1 Q) restricted to ΛT0,v is an injection. Thus, if Λ0 is compatible with π then ΛT0 restricts to a well-defined subbundle of T ∗ (J 1 Q) which we denote by j 1 Λ0 . In this way, given an affine constraint Λ0 which is compatible with π we can define a constraint of general type given by (coann(Λ0 ) ∩ J 1 Q, j 1 Λ0 ). In finite-dimensions we have rank(j 1 Λ0 ) = 2 rank(Λ0 ). If we have a local basis for Λ0 given by one-forms β a = β0a dt + βia dqi ,

a = 1, . . . , m,

then Λ0 is compatible with π if and only if the matrix βia , a = 1, . . . , m, i = 1, . . . , n, has maximal rank (i.e., rank m). In this case the one-forms

a ∂β0 ∂βia i ∂β0a ∂βia i + j v dqj + βia dv i , + v dt + ∂t ∂t ∂q j ∂q

a = 1, . . . , m,

(10)

together with the local basis for Λ0 , form a local basis for j 1 Λ0 . An infinitedimensional version of this is given by Lewis [13], where it is also shown that (coann(Λ0 ) ∩ J 1 Q, j 1 Λ0 ) is ideal. A coordinate definition of j 1 Λ0 is given by de Le´ on, Marrero and de Diego [7]. We also give an example of a system, treated in our framework, with affine constraints, and we refer to Section VII for some further remarks on these systems.

IV The Euler-Lagrange 2-force Recall the unforced, unconstrained Euler-Lagrange equations in their classical form: ∂L d ∂L − i = 0, i = 1, . . . , n. dt ∂ q˙i ∂q

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To motivate our definition of the Euler-Lagrange 2-force, it is convenient to expand the Euler-Lagrange equations to ∂2L ∂2L ∂L ∂2L j q¨ + j i q˙j + − i = 0, j i i ∂ q˙ ∂ q˙ ∂q ∂ q˙ ∂t∂ q˙ ∂q

i = 1, . . . , n.

With this form of the Euler-Lagrange equations in front of us, it is natural to define the Euler-Lagrange 2-force to be the 2-force ΦL on Q which is given in a natural chart for J 2 Q by ΦL (t, u, u1 , u2 ) = ((t, u, u1 ), D 23 L(t, u, u1 ) · u2 + D 2 D3 L(t, u, u1 ) · (u1 , ·) + D1 D 3 L(t, u, u1 ) · (1, ·) − D1 L(t, u, u1 )), regarding ΦL as τ1∗ V ∗ Q-valued. We shall find the “place-holder” notation convenient. Thus, for example, D 2 D3 L(t, u, u1 ) · (u1 , ·) is the element of E ∗ defined by e 7→ D 2 D3 L(t, u, u1 ) · (u1 , e). Also note that we match the arguments with the partial derivatives in the same order—that is, the leftmost partial derivative takes the first argument, and so on. A partial derivative with respect to “t” will be supposed to be evaluated on the vector “1” unless otherwise indicated. This definition for ΦL needs to be shown to be independent of natural chart. It is well-known that the Euler-Lagrange equations are independent of coordinates in the sense that a solution in one set of coordinates will still be a solution when we make a change of coordinates. But we can do better than this with ΦL . Proposition IV.1 ΦL obeys the transformation property of a 2-force. Proof. This is a straightforward but tedious exercise in differential calculus, and ˜ be adapted charts we shall only outline the main points. Let (U, φ) and (U, φ) 0 ˜ ) = (a, b) × for Q with the same domain U . We let φ(U ) = (a, b) × U and φ(U 0 2 2 2 2˜ ˜ U . These charts induce natural charts (J U, j φ) and (J U, j φ) for J 2 Q. Given our assumption on the form of adapted charts for π : Q → R, the overlap map ˜ 0 . Let us φ˜ ◦ φ−1 is given by (t, u) 7→ (t, ψ(t, u)) for some map ψ : (a, b) × U 0 → U 2 2 denote coordinates in the chart (J U, j φ) by (t, u, u1 , u2 ) and coordinates in the ˜ by (t˜, u ˜, u ˜1 , u chart (J 2 U, j 2 φ) ˜2 ). The transformation property for the overlap map 2 ˜ ◦ 2 −1 j φ (j φ) (i.e., the transformation law for 2-jets) is given by u ˜1 = D1 ψ(t, u) + D2 ψ(t, u) · u1 , u ˜2 =

D21 ψ(t, u)

+ 2D2 D1 ψ(t, u) · u1 + D22 ψ(t, u) · (u1 , u1 ) + D 2 ψ(t, u) · u2 .

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Ann. Henri Poincar´e

For brevity let us denote χ = j 1 φ˜ ◦ (j 1 φ)−1 . We compute D1 (L ◦ χ)(t, u, u1 ) = D1 L(χ(t, u, u1 )) + D2 L(χ(t, u, u1 )) ◦ D1 ψ(t, u) + D3 L(χ(t, u, u1 )) ◦ D1 D 2 ψ(t, u) · u1 + D 3 L(χ(t, u, u1 )) ◦ D 21 ψ(t, u), and D 2 (L ◦ χ)(t, u, u1 ) = D2 L(χ(t, u, u1 )) ◦ D2 ψ(t, u) + D3 L(χ(t, u, u1 )) ◦ D22 ψ(t, u) · u1 + D3 L(χ(t, u, u1 )) ◦ D1 D 2 χ(t, u), and D3 (L ◦ χ)(t, u, u1 ) = D 3 L(χ(t, u, u1 )) ◦ D 2 ψ(t, u). From these we readily compute D 1 D3 (L ◦ χ)(t, u, u1 ) = D 1 D3 L(χ(t, u, u1 )) ◦ D2 ψ(t, u) + D 2 D3 L(χ(t, u, u1 )) · (D 1 ψ(t, u), D 2 ψ(t, u) · (·)) + D 3 L(χ(t, u, u1 )) ◦ D 1 D2 ψ(t, u) + D 23 L(χ(t, u, u1 )) · (D 1 D2 ψ(t, u) · u1 , D 2 ψ(t, u) · (·)) + D 23 L(χ(t, u, u1 )) · (D 21 ψ(t, u), D 2 ψ(t, u) · (·)), and D2 D 3 (L ◦ χ)(t, u, u1 ) = D2 D3 L(χ(t, u, u1 )) ◦ (D 2 ψ(t, u) × D 2 ψ(t, u)) + D23 L(χ(t, u, u1 )) ◦ (D 22 ψ(t, u) · u1 × D 2 ψ(t, u)) + D3 L(χ(t, u, u1 )) ◦ D22 ψ(t, u) D23 L(χ(t, u, u1 )) ◦ (D 1 D 2 ψ(t, u) × D 2 ψ(t, u)), and D23 (L ◦ χ)(t, u, u1 ) = D 23 L(χ(t, u, u1 )) ◦ (D2 ψ(t, u) × D2 ψ(t, u)), where, for example, D23 L(χ(t, u, u1 )) · (D21 ψ(t, u), D 2 ψ(t, u) · (·)) · e = D23 L(χ(t, u, u1 )) · (D 21 ψ(t, u), D 2 ψ(t, u) · e), and D23 L(χ(t, u, u1 )) ◦ (D 2 ψ(t, u) × D2 ψ(t, u)) · (e1 , e2 ) = D23 L(χ(t, u, u1 )) · (D 2 ψ(t, u) · e1 , D2 ψ(t, u) · e2 ).

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Collecting all this together, and using the transformation rule for J 2 Q gives D23 (L ◦ χ)(t, u, u1 ) · u2 + D 2 D 3 (L ◦ χ)(t, u, u1 ) · u1 + D1 D3 (L ◦ χ)(t, u, u1 ) − D2 (L ◦ χ)(t, u, u1 ) = (D 2 ψ(t, u))∗ D 23 L(χ(t, u, u1 )) · u ˜2 + (D 2 ψ(t, u))∗ D2 D 3 L(χ(t, u, u1 )) · u ˜1 + ∗ ∗ (D 2 ψ(t, u)) D 1 D 3 L(χ(t, u, u1 )) − (D 2 ψ(t, u)) D2 L(χ(t, u, u1 )). That is to say ˜, u ˜1 , u ˜2 ) = (D 2 ψ(t, u))∗ (ΦL (t, u, u1 , u2 )). ΦL (t˜, u It remains to show that this is how a 2-force should transform. Since a 2-force is C1 (Q)-valued, we need to see how sections of C1 (Q) transform under the change of coordinates χ. Recall from (2) that elements of C1 (Q) are locally of the form ((t, u, u1 ), (−α · u1 , α, 0)) for some α ∈ E ∗ . For (τ, e1 , e2 ) ∈ R × E × E we compute (−α · u ˜1 , α, 0) · (Dχ(t, u, u1 ) · (τ, e1 , e2 )) = (−((D 2 ψ(t, u))∗ · α) · u1 , (D 2 ψ(t, u))∗ · α, 0) · (τ, e1 , e2 ). This shows that ΦL transforms as do sections of C1 (Q) which completes the proof. Remarks IV.2

1. In finite-dimensions we have 2 ∂2L j ∂2L ∂L ∂ L j ΦL = a + i j v + i − i (dq i − v i dt). ∂v i ∂v j ∂v ∂q ∂v ∂t ∂q

2. In some sense, all the work of the above result is unnecessary as we shall see an intrinsic way of defining the Euler-Lagrange 2-force in Section VI. However, if we are to enable the Euler-Lagrange 2-force to stand on its own two feet, so to speak, then it is necessary to ensure that its definition is coordinate independent. 3. Of course, the Euler-Lagrange 2-force has been conceived of in various guises in the past. That is to say, it is possible to think intrinsically of the EulerLagrange equations in a manner different from what we do here. For example, Tulczyjew [20] provides a discussion of the “Lagrange differential.” With the Euler-Lagrange 2-force, it is a simple matter to provide an intuitive characterisation of solutions to the Euler-Lagrange equations. Since we are considering external forces and constraints, let us be precise about this. A Lagrangian system on Q is a triple (L, Φ, (M, Λ)) where L is a Lagrangian, Φ is a 1-force, and (M, Λ) is a constraint. A local section c : [t1 , t2 ] → Q is a solution for (L, Φ, (M, Λ))

586

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Ann. Henri Poincar´e

if c satisfies the constraint (M, Λ) and if there exists a constraint force λ along c so that ΦL (j 2 c(t)) = Φ(j 1 c(t)) + λ(t) for t ∈ [t1 , t2 ]. In finite-dimensions, let β a = β0a dt + βia dqi + βˆia dv i ,

a = 1, . . . , m,

be a local basis for Λ and suppose the inclusion of M into J 1 Q has the form (x1 , . . . , xr ) 7→ (C0 (x), C1i (x), C2j (x)). If a section c has the local form t 7→ (t, q i (t)), then c is a solution for (L, Φ, (M, Λ)) if and only if 1. there exists a curve t 7→ (xα (t)) in M so that C0 (x(t)) = t,

C1i (x(t)) = q i (t),

C2j (x(t)) = q˙i (t),

for i, j = 1, . . . , n, 2. the relations βˆia q¨i + βia q˙i + β0a = 0 hold for a = 1, . . . , m, and 3. the equality 2 ∂2L j ∂2L ∂ L j ∂L q¨ + i j q˙ + i − i (dq i − q˙i dt) = (Φi + λi )(dqi − q˙i dt) ∂v i ∂v j ∂v ∂q ∂v ∂t ∂q holds for some 1-force λ along c. Matching the coefficients of the basis for C1 (Q) in this last expression gives the usual Lagrange multiplier form of the Euler-Lagrange equations. Of course, our discussion here of solutions for a Lagrangian system does nothing to assert the existence of such. Let us now deal with exactly this question.

V The Euler-Lagrange vector field In Section IV we wrote down the forced, constrained Euler-Lagrange equations associated with a Lagrangian system (L, Φ, (M, Λ)) on the total space of a locally trivial fibre bundle π : Q → R. Now we wish to assert, under restrictions on the Lagrangian system, that the Euler-Lagrange equations have unique solutions. One way to do this is to construct a second-order vector field on M ⊂ J 1 Q whose integral curves are solutions of the Euler-Lagrange equations. This is indeed the route we, along with many others, choose when characterising Lagrangian systems which possess unique solutions. However, our characterisation explicitly uses the Euler-Lagrange 2-force rather than a two-form formalism.

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V.1 The unforced, unconstrained case We investigate the unforced, unconstrained case first. That is, we consider a Lagrangian system of the form (L, 0, (J 1 Q, {0})). The following result is, of course, well-known in that it asserts the existence of the Euler-Lagrange vector field when the Lagrangian is regular. Theorem V.1 If L is a regular Lagrangian then there exists a unique second-order vector field XL on J 1 Q with the property that ΦL ◦ XL = 0. We call XL the Euler-Lagrange vector field for the regular Lagrangian L. Proof. We work locally with an adapted chart (U, φ) for π : Q → R. We take φ as being R × E-valued for a Banach space E. Throughout we regard J 2 Q as a subset of T (J 1 Q), and we regard forces as being τ1∗ V ∗ Q-valued. It will be convenient to write ΦL ((t, u, u1 ), (1, u1 , u2 )) = ((t, u, u1 ), A−1 L (t, u, u1 ) · u2 + ξL (t, u, u1 )) where AL (t, u, u1 ) : E ∗ → E is the inverse of the map e 7→ D23 L(t, u, u1 ) · e (this inverse exists since we are assuming L regular) and ξL (t, u, u1 ) = D2 D 3 L(t, u, u1 ) · (u1 , ·) + D 1 D 3 L(t, u, u1 ) · (1, ·) − D2 L(t, u, u1 ). Suppose a second-order vector field X on J 1 Q has local representative (t, u, u1 ) 7→ ((t, u, u1 ), (1, u1 , X(t, u, u1 ))). Then ΦL ◦ X = 0 if and only if X(t, u, u1 ) = −AL (t, u, u1 ) · ξL (t, u, u1 ). Thus we choose XL to have local representative (t, u, u1 ) 7→ ((t, u, u1 ), (1, u1 , −AL (t, u, u1 ) · ξL (t, u, u1 ))) which shows that XL exists. That XL is the unique second-order vector field with the stated property is a consequence of all of the above statements being “if and only if.” The proof immediately yields the following local form of XL . Corollary V.2 Let L be a regular Lagrangian with XL the Euler-Lagrange vector field, and let (U, φ) be an adapted chart for π : Q → R. The local representative of XL is (t, u, u1 ) 7→ ((t, u, u1 ), (1, u1 , −AL (t, u, u1 ) · D 2 D 3 L(t, u(t), u1 ) · u1 − AL (t, u, u1 ) · D 1 D3 L(t, u(t), u1 ) + AL (t, u, u1 ) · D 2 L(t, u(t), u1 ))) where AL (t, u, u1 ) is the inverse of the map e 7→ D 23 L(t, u, u1 ) · e.

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In finite-dimensions we have XL =

∂ ∂2L ∂L ∂2L ∂ ∂ ij + − k j vk − + v i i + gL j j ∂t ∂q ∂q ∂v ∂t∂v ∂q ∂v i

ij where the matrix with components gL , i, j = 1, . . . , n, is the inverse of the matrix with components

∂2L , ∂v i ∂v j

i, j = 1, . . . , n.

(11)

V.2 The forced, unconstrained case Let us now consider the addition of a 1-force Φ to the problem data. That is, we consider Lagrangian systems of the form (L, Φ, (J 1 Q, {0})). As in the unforced case, we can formulate a purely geometric result which defines for us the required second-order vector field. The proof of the following result is a simple adaptation of that of Theorem V.1. Proposition V.3 Let L be a regular Lagrangian on Q and let Φ be a 1-force on Q. There exists a unique second-order vector field XL,Φ on J 1 Q with the property that ΦL ◦ XL,Φ = Φ. We call XL,Φ the forced Euler-Lagrange vector field for the regular Lagrangian L and the 1-force Φ. It is now natural to ask whether XL,Φ is related to XL , and if so in what way. Answering this is the following result. Proposition V.4 If L is a regular Lagrangian and Φ is a 1-force then XL,Φ = ] ◦ Φ. XL + gL ] ◦ Φ, Proof. This is a simple matter of examining the local representatives of XL , gL and XL,Φ which are (t, u, u1 ) 7→ ((t, u, u1 ), (1, u1 , −AL (t, u, u1 ) · ξL (t, u, u1 ))), (t, u, u1 ) 7→ ((t, u, u1 ), (0, 0, AL (t, u, u1 ) · Φ(t, u, u1 ))), (t, u, u1 ) 7→ ((t, u, u1 ), (1, u1 , −AL (t, u, u1 ) · ξL (t, u, u1 ) +AL (t, u, u1 ) · Φ(t, u, u1 ))), respectively. The result follows directly.

] ◦ Φ takes its values in τ ∗ V Q and so when we add it Note that the vector field gL 1 to a second-order vector field, the result will be another second-order vector field.

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V.3 The forced, constrained case We now consider the general situation when we have a full Lagrangian system (L, Φ, (M, Λ)). The construction of a vector field describing nonlinearly constrained dynamics is dealt with, for example, by Marle [15, Proposition 2.15] in the Hamiltonian context and by de Le´ on, Marrero and de Diego [6, Proposition 3.4] in the Lagrangian. Neither of these papers allow external forces, although it would not be difficult in either case to work out how this could be done. In our case, we expect that we will require at least a regular Lagrangian. The following result asserts that if, in addition, we ask that L be definite and that the constraints be ideal, then we may establish the existence of a vector field having certain properties. In Proposition V.6 below, we show that integral curves of this vector field are in 1–1 correspondence with solutions of (L, Φ, (M, Λ)). Recall from Section III the definition of the subbundle ΛM of T ∗ M . Theorem V.5 Let (L, Φ, (M, Λ)) be a Lagrangian system with L a regular Lagrangian and (M, Λ) a constraint which is ideal. Further assume that gL |(coann(F(M,Λ) ) ∩ τ1∗ V Q) is strongly nondegenerate. Then there exists a unique second-order vector field XL,Φ,(M,Λ) on M having the following two properties: (i) XL,Φ,(M,Λ) (M ) ⊂ (coann(ΛM ) ∩ J 2 Q|M ); (ii) (ΦL ◦ XL − Φ)(J 1 Q) ⊂ F(M,Λ) . In particular, if L is definite (i.e., positive or negative-definite) then XL,Φ,(M,Λ) exists and is uniquely determined by (i) and (ii). Proof. We work locally, and borrow the notation of Section III.3 and Theorem V.1. Thus suppose (U, φ) to be an adapted chart for π : Q → R taking values in R × E ˜ = M ∩ J 1 U is the domain for a chart for a Banach space E. We assume that U ˜ , χ) for M with χ taking values in E. ˜ Throughout the proof, we write a typical (U ˜ ) as (t, u, u1 ). We shall always be ˜ as u point in U ˜ and a typical point in iM (U considering points (t, u, u1 ) lying in the image of iM . If we write u ˜ and (t, u, u1 ) in the same equation, it will always be the case that (t, u, u1 ) = iM (˜ u). The local representative of ΦL will be taken to have the form ΦL (t, u, u1 , u2 ) = ((t, u, u1 ), A−1 L (t, u, u1 ) · u2 + ξL (t, u, u1 )) and Φ has local representative (t, u, u1 ) 7→ ((t, u, u1 ), Φ(t, u, u1 )). As in (7), the local model for coann(ΛM ) ∩ J 2 Q is ((t,u,u1 ),(DC0 (˜ u) · e˜,DC1 (˜ u) · e˜,DC2 (˜ u) · e˜)) | DC0 (˜ u) · e˜ = 1, DC1 (˜ u) · e˜ = u1 , ∗ ∗ ∗ ◦ ◦ ◦ u) + B111 (t, u, u1 ) DC1 (˜ u) + B211 (t, u, u1 ) DC2 (˜ u)) · e˜ = 0, (B011 (t, u, u1 ) DC0 (˜ ∗ ∗ ◦ ◦ (B012 (t, u, u1 ) DC0 (˜ u) + B112 (t, u, u1 ) DC1 (˜ u)) · e˜ = 0

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∗ for Banach spaces F11 and F12 , and maps B01j : U × E → L(F1j ; R∗ ), Bi1j : U × ∗ ∗ ∗ ∗ E → L(F1j ; E ), B02 : U × E → L(F2 ; R ), and Bj2 : U × E → L(F2∗ ; E ∗ ), i, j = ˜ R) DCi : U ˜ E), i = 1, 2. Also, the local model ˜ → L(E; ˜ → L(E; 1, 2, and DC0 : U for F(M,Λ) is ∗ . ((t, u, u1 ), B211 (t, u, u1 ) · ν 11 ) ν 11 ∈ F11

˜. ˜∈U Recall that, by definition, B211 (t, u, u1 ) is injective with split image for each u Let X be a second-order vector field on M with local representative u ˜ 7→ (˜ u, X(˜ u)). ˜ X satisfies (i) if and only if for each u ˜∈U DC0 (˜ u) · X(˜ u) = 1, u) · X(˜ u) = u 1 , DC1 (˜ ∗ ∗ ∗ (t, u, u1 ) ◦ DC2 (˜ u) · X(˜ u) = 0, B011 (t, u, u1 ) · 1 + B111 (t, u, u1 ) · u1 + B211 ∗ ∗ B011 (t, u, u1 ) · 1 + B111 (t, u, u1 ) · u1 = 0.

(12)

Since (M, Λ) is ideal, we know that such an X exists. X satisfies (ii) if and only ˜ u, u1 ) ∈ F ∗ with the if, for each (t, u, u1 ) ∈ image(iM ), there exists some λ(t, 11 property that ◦ A−1 u) · X(t, u, u1 ) + ξL (t, u, u1 ) − Φ(t, u, u1 ) = L (t, u, u1 ) DC2 (˜ ˜ u, u1 ). (13) B211 (t, u, u1 ) · λ(t,

Note that this implies that if X satisfies (ii) then it is uniquely determined by (13) ◦ as A−1 u) is injective. Furthermore, the same equation uniquely L (t, u, u1 ) DC2 (˜ ˜ u, u1 ) since B211 (t, u, u1 ) is injective. specifies λ(t, ˜ Fix To establish existence of XL,Φ,(M,Λ) we will explicitly determine λ. (t, u, u1 ) = iM (˜ u). By our assumption that gL |(coann(F(M,Λ) ) ∩ τ1∗ V Q) is strongly nondegenerate, the map ∗ ∗ B211 (t, u, u1 ) ◦ AL (t, u, u1 ) ◦ B211 (t, u, u1 ) ∈ L(F11 ; F11 )

(14)

is a Banach isomorphism. It is then a straightforward to check that if we choose ˜ u, u1 ) ∈ F ∗ to be the unique solution of λ(t, 11 ∗ ˜ u, u1 ) = (t, u, u1 ) ◦ AL (t, u, u1 ) ◦ B211 (t, u, u1 ) · λ(t, B211 ∗ ∗ (t, u, u1 ) ◦ AL (t, u, u1 ) · ξL (t, u, u1 ) − B211 (t, u, u1 ) ◦ AL (t, u, u1 ) · Φ(t, u, u1 ) − B211 ∗ ∗ (t, u, u1 ) · u1 − B011 (t, u, u1 ) · 1 B111

then X chosen as in (13) satisfies (12). Therefore, choosing XL,Φ,(M,Λ) = X in this way, we see that XL,Φ,(M,Λ) satisfies (i). Moreover, XL,Φ,(M,Λ) was constructed by

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requiring it to satisfy (ii). This establishes existence of XL,Φ,(M,Λ) and so completes the proof of the first assertion as uniqueness has already been proved. To prove the final assertion, we need to show that if gL is definite then (14) defines a Banach isomorphism. But if gL is positive-definite then the pairing

ν 11 ) (ν 11 , ν˜11 ) 7→ AL (t, u, u1 ) ◦ B211 (t, u, u1 )(ν 11 ); B211 (t, u, u1 )(˜ ∗ defines an inner product on F11 (it is symmetric since A∗L (t, u, u1 ) = AL (t, u, u1 )). If gL is negative definite we may obtain an inner product by multiplying by −1. Therefore, by the Riesz Representation Theorem, the map which sends ν 11 to the element

ν 11 ); B211 (t, u, u1 )(ν 11 ) ν˜11 7→ AL (t, u, u1 ) ◦ B211 (t, u, u1 )(˜

of F11 is a Banach isomorphism. In other words, the map (14) is a Banach isomorphism. For linear constraints in finite-dimensions, the computations we perform in the proof of the theorem are standard [17, Section 6.1.2]. Note that the requirement that gL |(coann(F(M,Λ) ) ∩ τ1∗ V Q) be strongly nondegenerate can fail in finite-dimensions, even when L is regular, as can be seen with a simple example. We let Q = R×Q with Q = R2 . Denote natural coordinates for J 1 Q with respect to Cartesian coordinates (x, y) on R2 by (t, x, y, vx , vy ). The Lagrangian L=

1 2 (v − vy2 ) 2 x

is regular, and the constraint (M = J 1 Q, Λ) where Λ is generated by the one-form β = dvx − dvy is ideal. In this case F(M,Λ) is generated by the one-form dx−vx dt−(dy−vy dt) and so coann(F(M,Λ) ) ∩ τ1∗ V Q is spanned by ∂v∂x + ∂v∂y . Therefore gL |(coann(F(M,Λ) ) ∩ τ1∗ V Q) is zero. The following characterisation of XL,Φ,(M,Λ) establishes that the integral curves of XL,Φ,(M,Λ) are indeed solutions for (L, Φ, (M, Λ)). Proposition V.6 If (L, Φ, (M, Λ)) is a Lagrangian system satisfying the hypotheses of Theorem V.5, then XL,Φ,(M,Λ) is the unique second-order vector field on M with the property that a local section t 7→ j 1 c(t) ∈ M is an integral curve of XL,Φ,(M,Λ) if and only if t 7→ c(t) is a solution of (L, Φ, (M, Λ)). Proof. We work locally with the notation of Theorem V.5. We take a curve on M which is locally given by t 7→ u ˜(t). This is a solution of (L, Φ, (M, Λ)) if and only if

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1. C0 (˜ u(t)) = t, C1 (˜ u(t)) = u(t), and C2 (˜ u(t)) = u(t) ˙ (which defines a curve t 7→ (t, u(t)) in U ), ∗ ∗ ∗ 2. B211 (t, u(t), u(t)) ˙ ·u ¨(t) + B111 (t, u(t), u(t)) ˙ · u(t) ˙ + B011 (t, u(t), u(t)) ˙ = 0 and ∗ ∗ B112 (t, u(t), u(t)) ˙ · u(t) ˙ + B012 (t, u(t), u(t)) ˙ = 0, and

3. A−1 ˙ u ¨(t)+ξL (t, u(t), u(t))−Φ(t, ˙ u(t), u(t)) ˙ = B211 (t, u(t), u(t))· ˙ L (t, u(t), u(t))· ∗ ˜ ˜ λ(t) for some t 7→ λ(t) ∈ F11 . ˜ is We now proceed exactly as in the proof of Theorem V.5 and ascertain that λ(t) uniquely defined by its being the unique solution of ∗ ˜ = ◦ AL (t, u(t), u(t)) ◦ B211 (t, u(t), u(t)) (t, u(t), u(t)) ˙ ˙ ˙ · λ(t) B211 ∗ ◦ AL (t, u(t), u(t)) B211 (t, u(t), u(t)) ˙ ˙ · ξL (t, u(t), u(t)) ˙ − ∗ ◦ AL (t, u(t), u(t)) ˙ ˙ · Φ(t, u(t), u(t)) ˙ − B211 (t, u(t), u(t)) ∗ ∗ (t, u(t), u(t)) ˙ · u˙ − B011 (t, u(t), u(t)) ˙ · 1. B111

This completes the proof.

Note that under the hypotheses of Theorem V.5, the constraint forces have more structure than that of forces along a curve—they are determined by a welldefined 1-force on M . In finite-dimensions, we may be explicit about writing this constraint 1-force. Take a local basis β0a dt + βia dqi + βˆia dv i ,

a = 1, . . . , m,

for Λ with the property that, when restricted to M , the forms βˆia (dq i − v i dt),

a = 1, . . . , m, ˜

form a basis for F(M,Λ) . Then a 1-force which is a constraint force for Λ will have ˜ a (dq i − v i dt) for some functions λ ˜ a , a = 1, . . . , m. ˜ For the constraint the form βˆia λ 1-force these functions are given by 2 ∂ L k ∂2L ∂L b i b ˜ a = Cab βˆb (t, u, v)g ij λ v + − − Φ − β v − β j i i 0 , (15) L ∂q k ∂v j ∂t∂v j ∂q j a = 1, . . . , m, ˜ where Cab , a, b = 1, . . . , m ˜ is the matrix whose components are ij formed by the inverse of the matrix with components βˆia βˆjb gL , a, b = 1, . . . , m, ˜ ij and, as usual, gL , i, j = 1, . . . , n, are the components of the inverse of the matrix whose components are given by (11).

VI The relationship of ΦL to the Poincar´e-Cartan two-form In Sections IV and V we formulated the Euler-Lagrange equations and derived the Euler-Lagrange vector field (when possible) using as our primary tool the EulerLagrange 2-force ΦL . This is not necessarily the standard way to accomplish these

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tasks. One more common way to do this [11, 7] is to construct a two-form where solutions are then defined by their making the two-form vanish upon taking interior products (we will be precise about this shortly). This approach may be seen as borrowing from the Hamiltonian formulation. In this section we make precise the relationship between this standard approach and our approach as presented in Section V.

VI.1

The unforced, unconstrained case

To avoid confusion as to where the one-form dt lives, let us define η = (π ◦ τ1 )∗ dt as its pull-back to J 1 Q. For a Lagrangian L on Q define the Poincar´e-Cartan one-form on J 1 Q by ∗ ΘL = Lη + S˜Q (dL)

and define the Poincar´e-Cartan two-form on J 1 Q by ΩL = −dΘL . The one-form ΘL was first introduced by Cartan [4]. In infinite-dimensions, one uses the definition of the exterior derivative in terms of the Lie derivative by Palais [18]. This gives the local representations for the Poincar´e-Cartan forms as ΘL (t, u, u1 ) = ((t, u, u1 ), (L − D3 L(t, u, u1 ) · u1 , D3 L(t, u, u1 ), 0)),

(16)

and ΩL (t, u, u1 ) · ((τ1 , e1 , f1 ), (τ2 , e2 , f2 )) = τ1 D2 L · e2 − τ2 D2 L · e1 − τ1 D2 D 3 L · (e2 , u1 ) + τ2 D 2 D3 L · (e1 , u1 ) − τ1 D23 L · (u1 , f2 ) + τ2 D 23 L · (u1 , f1 ) + D1 D 3 L · (τ2 , e1 ) − D1 D 3 L · (τ1 , e2 ) + D2 D 3 L · (e2 , e1 ) − D2 D 3 L · (e1 , e2 ) + D23 L · (e1 , f2 ) − D23 L · (e2 , f1 ) (17) where all derivatives in (17) are evaluated at (t, u, u1 ). In finite-dimensions these read ΘL = Ldt + and

∂L (dq i − v i dt) ∂v i

∂L ∂L − i dt ∧ (dq i − v i dt). ΩL = − d ∂v i ∂q

It is our goal to relate ΩL with ΦL . ToVdo this requires the following exterior algebraic construction. We denote by k (T (J 1 Q)) the bundle of exterior k-forms on J 1 Q.VAssociated with the subbundle C1 (Q) of T ∗ (J 1 Q) is the subbundle (C1 (Q))k of k (T (J 1 Q)) whose fibre at v ∈ J 1 Q is V (C1 (Q))kv = α ∈ k (Tv (J 1 Q)) | α(X1 , . . . , Xk ) = 0 for all X1 , . . . , Xk ∈ coann(C1v (Q)) .

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V Thus the sections of ⊕k≥1 (C1 (Q))k form the algebraic ideal in Γ∞ ( (J 1 Q)) corresponding to the subbundle C1 (Q) of T ∗ (J 1 Q). We can now establish an exact correspondence between ΦL and ΩL . Theorem VI.1 Let L be a Lagrangian on Q. The following statements hold. (i) There exists a unique two-form Ω on J 1 Q with the following properties: (a) Ω is closed; (b) Ω ∈ Γ∞ ((C1 (Q))2 ); (c) for every second-order vector field X on J 1 Q, ΦL ◦ X = −X

Ω.

Furthermore, this unique two-form is exactly ΩL . (ii) Conversely, there exists a unique 2-force Φ on Q for which Φ ◦ X = −X ΩL , and this 2-force is exactly ΦL . Proof. (i) Let (U, φ) be an adapted chart for π : Q → R with (J 1 U, j 1 φ) the corresponding natural chart for J 1 Q. Take φ to be R × E-valued, and suppose that U = (a, b) × Br,0 for some a, b ∈ R and r > 0 (here Br,0 is the open ball of radius r centred at 0). In this case, U and U × E are contractible. Suppose Ω is a closed two-form on U × E. By the Poincar´e lemma, Ω is exact, so suppose Ω = dΘ and that Θ is given by Θ(t, u, u1 ) = ((t, u, u1 ), (A0 (t, u, u1 ), A1 (t, u, u1 ), A2 (t, u, u1 ))) for A0 : U × E → R∗ and Ai : U × E → E ∗ , i = 1, 2. A computation gives (τ, e1 , e2 ) dΘ(t, u, u1 ) = ((t, u, u1 ), (D 2 A0 · (e1 , ·) − D 1 A1 · (·, e1 ) + D 3 A0 · (e2 , ·) − D1 A2 · (·, e2 ), −D2 A0 · (·, τ ) + D1 A1 · (τ, ·) + D2 A1 · (e1 , ·) − D2 A1 · (·, e1 ) + D3 A1 · (e2 , ·) − D2 A2 (·, e2 ), − D 3 A0 · (·, τ ) + D 1 A2 · (τ, ·) − D3 A1 · (·, e1 ) + D 2 A2 · (e1 , ·) + D3 A2 · (e2 , ·) − D3 A2 (·, e2 ))). Now let us require (i b) to hold. Elements of coann(C1 (Q)) at (t, u, u1 ) have the form (τ, τ u1 , e) ∈ R × E × E. An element of C1 (Q) at (t, u, u1 ) has the form (−α · u1 , α, 0) for some α ∈ E ∗ . Thus Ω(t, u, u1 ) satisfies (i a) and (i b) if and only if τ D2 A0 · (u1 , ·) − τ D1 A1 · (·, u1 ) + D3 A0 · (e, ·) − D1 A2 · (·, e) = −α · u1 −D2 A0 · (·, τ ) + D 1 A1 · (τ, ·) + τ D 2 A1 · (u1 , ·) − τ D2 A1 · (·, u1 ) + D3 A1 · (e, ·) − D2 A2 (·, e) = α −D3 A0 · (·, τ ) + D 1 A2 · (τ, ·) − τ D 3 A1 · (·, u1 ) + τ D2 A2 · (u1 , ·) + D 3 A2 · (e, ·) − D 3 A2 (·, e) = 0

(18)

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for some α ∈ E ∗ and for every (τ, e) ∈ R × E. Here all derivatives have been evaluated at some (t, u, u1 ) ∈ U × E. Note that α is allowed to vary as a function of τ and e, as well as of (t, u, u1 ). In what follows, we shall write α(τ, e) and take the (t, u, u1 ) dependence for granted. Now let X be a second order vector field on U × E defined by (t, u, u1 ) 7→ ((t, u, u1 ), (1, u1 , e)) for some e ∈ E which varies with (t, u, u1 ). Then ΦL ◦ X(t, u, u1 ) = ((t, u, u1 ), (−D23 L(t, u, u1 ) · (e, u1 ) − ξL (t, u, u1 ) · u1 , D 23 L(t, u, u1 ) · e + ξL (t, u, u1 ), 0)) where ξL (t, u, u1 ) = D1 D 3 L(t, u, u1 ) · (1, ·) + D2 D3 L(t, u, u1 ) · (u1 , ·) − D2 L. Thus Ω satisfies (i a) and (i c) if and only if D 2 A0 · (u1 , ·) − D1 A1 · (·, u1 ) + D 3 A0 · (e, ·) − D1 A2 · (·, e) = D23 L(t, u, u1 ) · (u1 , e) + ξL (t, u, u1 ) · u1 −D2 A0 · (·, 1) + D 1 A1 · (1, ·) + D 2 A1 · (u1 , ·) − D 2 A1 · (·, u1 ) + D 3 A1 · (e, ·) − D 2 A2 (·, e) = −D23 L(t, u, u1 ) · e − ξL (t, u, u1 )

(19)

−D3 A0 · (·, 1) + D 1 A2 · (1, ·) − D 3 A1 · (·, u1 ) + D 2 A2 · (u1 , ·) + D 3 A2 · (e, ·) − D 3 A2 (·, e) = 0 for every e ∈ E. Let us take the third equation from (18) with τ = 0: D 3 A2 · (e, ·) − D 3 A2 · (·, e) = 0 for every e ∈ E. This means that for each fixed (t, u) ∈ U the one-form on E defined by u1 7→ A2 (t, u, u1 ) is closed, and so by the Poincar´e lemma exact. Therefore there exists a function F : U × E → R so that A2 (t, u, u1 ) = D3 F (t, u, u1 ).

(20)

If we take the second of equations (18) with τ = 1 and subtract from it the second of equations (19), we get α(1, e) = −D23 L(t, u, u1 ) · e − ξL (t, u, u1 ). From the second of equations (18) we see that α(τ, e) + D2 A2 · (e, ·) − D3 A1 · (·, e) is a linear function of τ and is independent of e; let us write α(τ, e) = D3 A1 · (·, e) − D2 A2 · (e, ·) + τ β

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where β depends only on (t, u, u1 ). We then have −D23 L(t, u, u1 ) · e − ξL (t, u, u1 ) = D3 A1 · (·, e) − D2 A2 · (e, ·) + β for every e ∈ E. Therefore D3 A1 · (·, e) − D2 A2 · (e, ·) = −D 23 L(t, u, u1 ) · e

(21)

and β = −ξL (t, u, u1 ) and so α(τ, e) = −D23 L(t, u, u1 ) · e − τ ξL (t, u, u1 ). Now we substitute (20) into (21) to get D3 A1 · (·, e) = D 2 D3 F · (e, ·) − D 23 L(t, u, u1 ) · e. Therefore we must have A1 (t, u, u1 ) = D 2 F (t, u, u1 ) − D 3 L(t, u, u1 ) + G(t, u)

(22)

where F is as above and for some G : U → E ∗ . Now we substitute (20) and (22) into the third of equations (19) to get −D3 A0 · (·, 1) + D1 D 3 F · (1, ·) + D 23 L(t, u, u1 ) · u1 = 0. Note that D23 L(t, u, u1 ) · u1 = D3 (D 3 L(t, u, u1 ) · u1 ) − D 3 L(t, u, u1 ). This implies that A0 (t, u, u1 ) = D1 F (t, u, u1 ) · 1 + D3 L(t, u, u1 ) · u1 − L(t, u, u1 ) + H(t, u)

(23)

for some H : U → R. Now substitute (20), (22), and (23) into the first of equations (19) with e = 0 to get (D2 H − D 1 G · 1) · u1 = 0. Since G and H are independent of u1 and u1 ∈ E is arbitrary we get D 1 G · 1 = D2 H.

(24)

Next we substitute (20), (22), (23), and (24) into the second of equations (19) with e = 0 to get D2 G · (u1 , ·) − D2 G · (·, u1 ) = 0.

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Since G is independent of u1 and u1 ∈ E is arbitrary, this implies that for each fixed t ∈ (a, b) the one-form u 7→ G(t, u) is closed. By the Poincar´e lemma we may then write G(t, u) = D2 R for some function R on U . In the expression (22) for A1 , we can simply absorb R into F and write A1 (t, u, u1 ) = D 2 F − D 3 L for some function F on U . By (24) we have D 2 H(t, u) = D2 D 1 R(t, u) which means that H(t, u) = D 1 R(t, u) + S(t) for some function S : (a, b) → R. As a result of the above computations we have Θ = Θ1 + Θ2 where Θ1 (t, u, u1 ) = ((t, u, u1 ), (D 3 L(t, u, u1 ) · u1 − L, −D3 L(t, u, u1 ), 0)), Θ2 (t, u, u1 ) = ((t, u, u1 ), (D 1 F (t, u) · 1 + D 1 R(t, u) + S(t), D 2 F (t, u), D 3 F )). By (16), Θ1 is exactly the local representative of −ΘL , and it is a straightforward computation to check that Θ2 is closed. Therefore, we have shown that the conditions (i a), (i b), and (i c) imply that Ω = −dΘL . One may further check that Ω so defined does indeed satisfy the conditions (i a), (i b), and (i c). (ii) As above, let X be the second order vector field on U × E taking the value (1, u1 , e) at (t, u, u1 ) for e ∈ E. Now consider a local 2-force Φ whose value at ((t, u, u1 ), (1, u1 , u2 )) is (−α(t, u, u1 , u2 ) · u1 , α(t, u, u1 , u2 ), 0) for α(t, u, u1 , u2 ) ∈ E ∗ . Given the coordinate form for ΩL in (17) we compute − (1, u1 , e)

ΩL (t, u, u1 ) = (D 2 L · u1 − D 2 D3 L · (u1 , u1 ) − D23 L · (u1 , e) −

D1 D 3 L · (1, u1 ), D 1 D 3 L · (1, ·) + D 2 D3 L(u1 , ·) + D 23 L · e − D2 L, 0). (25) We also compute Φ ◦ X(t, u, u1 ) = (−α(t, u, u1 , e) · u1 , α(t, u, u1 , e), 0).

(26)

Thus Φ ◦ X = −X Ω if and only if all components of (25) and (26) match. In particular, matching the second component gives α(t, u, u1 , e) = D 1 D3 L · (1, ·) + D2 D 3 L(u1 , ·) + D23 L · e − D 2 L for every e ∈ E. But this α makes Φ exactly the local representative of ΦL . One may easily see that this α also makes the first components of (25) and (26) match. Remarks VI.2 1. Thus we have an exact correspondence between ΦL and ΩL in that given ΦL we can compute ΩL using the geometry of J 1 Q and the condition that ΩL be closed, and given ΩL we can directly determine ΦL . As the proof of Theorem VI.1 suggests, determining ΩL from ΦL is not altogether trivial, even though the statement of the correspondence is quite benign. 2. One can use Theorem VI.1(ii) as an intrinsic definition of ΦL .

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3. None of the three conditions in Theorem VI.1(i) can be omitted if the result is to be true. Hermann [11] gives a characterisation of ΩL , but his conditions are not enough to determine it uniquely. 4. An obvious consequence of Theorem VI.1 is that a local section c : [t1 , t2 ] → Q is a solution for (L, 0, (J 1 Q, {0})) if and only if (j 1 c)0 (t) ΩL (j 1 c(t)) = 0 for every t ∈ [t1 , t2 ]. 5. When L is regular, the characteristic distribution of ΩL , whose fibre at v ∈ J 1 Q is D(ΩL )v = X ∈ Tv (J 1 Q) X ΩL (v) = 0 , is a subbundle of rank 1, and is generated by XL . In particular, if Q is finite-dimensional then ΩL defines a contact structure on J 1 Q.

VI.2

The forced, unconstrained case

It is a fairly simple matter to add an external force to the formulation above. Recall that a 1-force Φ may be regarded as a C1 (Q)-valued one-form on J 1 Q. The forced Poincar´e-Cartan two-form is the two-form on J 1 Q given by ΩL,Φ = ΩL − Φ ∧ η. In finite-dimensions one readily computes ∂L ∂L ΩL,Φ (t, q, v) = − d − i dt − Φi dt ∧ (dq i − v i dt). ∂v i ∂q Given Theorem VI.1, the following result is natural. Its proof follows very much along the lines of that of the theorem. Proposition VI.3 Let L be a Lagrangian and let Φ be a 1-force on Q. There exists a unique two-form Ω on J 1 Q with the properties: (i) Ω + Φ ∧ η is closed; (ii) Ω ∈ Γ∞ ((C1 (Q))2 ); (iii) ΦL ◦ X − Φ = −X Ω for every second-order vector field X on J 1 Q. Furthermore, this unique two-form is precisely ΩL,Φ . Remarks VI.4 1. It certainly need not be the case that ΩL,Φ be closed as is ΩL . In particular, ΩL,Φ will not generally be exact. These issues are discussed by Hermann [11, §13]. 2. By Proposition V.3 and property (iii) of Proposition VI.3, a local section c : [t1 , t2 ] → Q is a solution for (L, Φ, (J 1 Q, {0})) if and only if (j 1 c)0 (t) ΩL,Φ (j 1 c(t)) = 0 for t ∈ [t1 , t2 ].

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3. When L is regular, the characteristic distribution of ΩL,Φ is a subbundle of T (J 1 Q) of rank 1, and is generated by XL,Φ . Indeed, XL,Φ is the unique second-order vector field for which XL,Φ ΩL,Φ = 0.

VI.3

The forced, constrained case

Now let us use the Poincar´e-Cartan two-form to develop conditions for solutions of (L, Φ, (M, Λ)) when (M, Λ) is possibly non-trivial. What we do here is reminiscent of the development of de Le´ on, Marrero and de Diego [7]. As is to be expected, when one adds constraints, the way in which one characterises solutions changes somewhat. We do this in the following result. Proposition VI.5 A local section c : [t1 , t2 ] → Q is a solution for the Lagrangian system (L, Φ, (M, Λ)) if and only if c satisfies the constraint (M, Λ) and (j 1 c)0 (t) ΩL,Φ |M (j 1 c(t)) ∈ F(M,Λ) for t ∈ [t1 , t2 ]. Proof. By definition, c is a solution if and only if it satisfies the constraint and ΦL ◦ j 2 c(t) − Φ ◦ j 1 c(t) ∈ F(M,Λ) for t ∈ [t1 , t2 ]. Since (j 1 c)0 is a second-order vector field along j 1 c, by Proposition VI.3(iii) this is equivalent to c satisfying the constraint and (j 1 c)0 (t) ΩL,Φ |M (j 1 c(t)) ∈ F(M,Λ) . In the case when we are assured of solutions to (M, Λ) we have the following assertion which follows from Theorem V.5 and Proposition VI.3(iii). Proposition VI.6 If (L, Φ, (M, Λ)) is a Lagrangian system with L a definite Lagrangian, and (M, Λ) an ideal constraint, then there exists a unique second-order vector field X on M with the properties (i) X(M ) ⊂ (coann(ΛM ) ∩ J 2 Q|M ); (ii) X

ΩL,Φ |M ∈ F(M,Λ) .

Furthermore, this vector field is precisely XL,Φ,(M,Λ) .

VII An example In order to illustrate the methodology of the paper, let us look at an example. The system we look at is a simple one with constraints linear in velocity, and our intention is to illustrate the concepts of the paper, in particular our general constructions with constraints in Section III.3. We look at a system with a trivial configuration bundle Q = R × Q where Q = SE(2) × SO(2). Here SE(2) denotes the group of proper isometries of the twodimensional Euclidean plane E 2 , and SO(2) denotes the special orthogonal group in two-dimensions. To coordinatise SE(2) we fix a point O ∈ E 2 and attach to O an orthonormal frame {e1 , e2 }. An element Ψ ∈ SE(2) will map the orthonormal frame {e1 , e2 } to an orthonormal frame {f 1 , f 2 } which is attached to a point P ∈ E 2 . To Ψ we associated the coordinates (x, y, θ) where (x, y) = P − O ∈ R2

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and θ ∈ R has the property that f i = R(θ)ei , i = 1, 2, where R(θ) is rotation by θ mod 2π. We use φ to coordinatise SO(2) in the usual manner, and this gives coordinates (x, y, θ, φ) for Q. Of course, these coordinates are not globally defined, but that will not bother us here. On J 1 Q ' R × T Q we consider the Lagrangian L(t, x, y, θ, φ, vx , vy , vθ , vφ ) =

1 1 1 m(vx2 + vy2 ) + I1 vθ2 + I2 vφ2 2 2 2

where m, I1 , and I2 are positive real constants. Let us now specify a constraint (M, Λ) for the system. We take M = { (t, x, y, θ, φ, vx , vy , vθ , vφ ) | vx = r cos θvφ , vy = r sin θvφ } for some constant r > 0. This defines the subset of J 1 Q within which motion will be constrained, and we use as coordinates for M the coordinates (t, x, y, θ, φ, vθ , vφ ). This defines the inclusion of M into J 1 Q by iM (t, x, y, θ, φ, vθ , vφ ) = (t, v, y, θ, φ, r cos θvφ , r sin θvφ , vθ , vφ ). Therefore, using notation from Section III.3, we have DC0 = 1 0 0 0 0 0 0 ,   0 1 0 0 0 0 0 0 0 1 0 0 0 0  DC1 =  0 0 0 1 0 0 0 , 0 0 0 0 1 0 0   0 0 0 −r sin θvφ 0 0 r cos θ 0 0 0 r cos θvφ 0 0 r sin θ  . DC2 =  0 0 0 0 0 1 0  0 0 0 0 0 0 1 The constraints here are linear in velocity and so are defined by a codistri˜ 0 on Q which pulls back to a codistribution Λ0 on Q. We take Λ ˜ 0 to be bution Λ the codistribution generated by the one-forms dx − r cos θdφ,

dy − r sin θdφ.

Of course, these are also the generators of Λ0 , but thought of as forms on Q rather that Q. Using (10), one then computes the generators of Λ = j 1 Λ0 to be dx − r cos θdφ, dy − r sin θdφ r sin θvφ dθ + dvx − r cos θdvφ , −r cos θvφ dθ + dvy − r sin θdvφ . This then gives the constraint (M, Λ) which we will consider in this section.

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Remark VII.1 With this linear constraint, the system models a disk rolling upright on the plane. The coordinates (x, y) locate the point of contact of the disk with the plane, θ is the “heading angle” of the disk, and φ is the angle of rotation of the disk. The parameter m is the mass of the disk, I1 is the disk’s moment of inertia about the axis normal to the surface, I2 is the moment of inertia of the disk about its centre of rotation, and r is the radius of the disk. Now let us compute the various subbundles associated with the constraint. To do so, we select a basis of one-forms on T ∗ (J 1 Q) as outlined in Section III.3. Let us choose a basis β 1 = r sin θvφ dθ + dvx − r cos θdvφ β 2 = − r cos θvφ dθ + dvy − r sin θdvφ β 3 = dx − r cos θdφ β 4 = dy − r sin θdφ τ = dt ; α1 = dvθ ; α2 = dvφ ; α3 = dθ ; α4 = dφ. ∗ |M ) ∩ Λ|M and so This basis is chosen so that {β 3 , β 4 } form a basis for ker(S˜Q that, in the notation of (5), the one-forms

{βˆi1 (dxi − v i dt), βˆi2 (dxi − v i dt)} form a basis for F(M,Λ) . From the expressions for our adapted basis one-forms, we may ascertain that, in the notation of Section III.3, we have B011 = 0 0 ,  0  0 B111 =  r sin θvφ 0  0 0 0 0 B12 =  0 0 0 0  B211

B012 = 0 0 , B02 = 1 0   0 1  0  0  , B112 =   0 −r cos θvφ  −r cos θ 0   0 0 0 0 0 0 0 0 0 0 0 0  , B22 =  0 1 0 0 1 0 0 0 1 0 0 1   1 0 0  0  0 1  , B212 =  =  0 0 0  −r cos θ −r sin θ 0

From (6) we compute generators for ΛM to be dx − r cos θdφ,

dy − r sin θdφ.

0 0 0 ,  0 1  , 0  −r sin θ  0 0 0 0 , 0 0 0 0  0 0 . 0 0

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From (8) we compute generators for coann(F(M,Λ) ) ∩ (τ1 |M )∗ V Q to be ∂ , ∂vθ

r cos θ

∂ ∂ ∂ + r sin θ + . ∂vx ∂vy ∂vφ

(27)

One readily ascertains that these vector fields are tangent to M . Restricted to M , and using the coordinates for M , these vectors are exactly ∂ , ∂vθ

∂ . ∂vφ

(28)

There is a notational confusion here because the vectors in equation (27) live on J 1 Q whereas the vectors in equation (28) live on M . Confusion arises since we are naming the coordinates for M the same as a subset of the coordinates for J 1 Q. In any event, using (7) we also derive J 2 Q|M ∩ coann(ΛM ) = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂t + r cos θvφ ∂x + r sin θvφ ∂y + vθ ∂θ + vφ ∂φ + a ∂vθ + b ∂vφ | a, b ∈ R . With all these calculations, one readily checks that the conditions IC1, IC2, and IC3 are satisfied by the constraint (M, Λ). This is to be expected given the general fact that linear constraints are ideal. Remark VII.2 For this example, we observe the following facts: ∗ 1. F(M,Λ) = ΛM = ker(S˜Q |M ) ∩ (Λ|M );

2. the generators for these codistributions on M are “the same” as those for Λ0 (keeping in mind that the forms live on different spaces); 3. coann(F(M,Λ) ) ∩ (τ1 |M )∗ V Q = T M ∩ (τ1 |M )∗ V Q; 4. in local coordinates, [J 2 Q|M ∩ coann(ΛM )]X = (1, v) + TX M ∩ (τ1 |M )∗ V Q for X = ((t, q), (1, v)) ∈ M . These observations will generally hold for systems with compatible affine constraints. The Euler-Lagrange 2-force is easily computed: ΦL = max (dx − vx dt) + may (dy − vy dt) + I1 vθ (dθ − vθ dt) + I2 vφ (dφ − vφ dt) where (ax , ay , aθ , aφ ) are the fibre coordinates for J 2 Q over J 1 Q. As a general constraint force has the form λ1 (dx − vx dt − r cos θ(dφ − vφ dt)) + λ2 (dy − vy dt − r sin θ(dφ − vφ dt))

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for some functions λ1 and λ2 , we ascertain that the equations of motion subject to the external 1-force Φ = Φx (dx − vx dt) + Φy (dy − vy dt) + Φθ (dθ − vθ dt) + Φφ (dφ − vφ dt) are m¨ x = Φx + λ1 m¨ y = Φy + λ2 I1 θ¨ = Φθ I2 φ¨ = Φφ − r cos θλ1 − r sin θλ2 which are algebro-differential equations when combined with the constraint equations ˙ x˙ = r cos θφ,

˙ y˙ = r sin θφ.

We may also explicitly write the vector field XL,Φ,(M,Λ) . First we compute the Lagrange multipliers using (15). The matrix with components denoted Cab in equation (15) is computed to be # " m(I2 +mr2 sin2 θ) m2 r2 sin θ cos θ − 2 2 I2 +mr I2 +mr . C= 2 2 m(I2 +mr2 cos2 θ) sin θ cos θ − m rI2 +mr 2 I2 +mr2 We thus compute

λ1 =

1 {(mr2 (cos 2θ − 1) − 2I2 )Φx + mr2 sin 2θΦy 2(I2 + mr2 ) +2mr cos θΦφ − 2mr(I2 + mr2 ) sin θvθ vφ }

λ2 =

1 {mr2 sin 2θΦx − (mr2 (cos 2θ + 1) + 2I2 )Φy 2(I2 + mr2 ) +2mr sin θΦφ + 2mr(I2 + mr2 ) cos θvθ vφ }.

Substituting these expressions for λ1 and λ2 into the vector field Φy + λ2 ∂ ∂ ∂ ∂ ∂ ∂ Φx + λ1 ∂ + + + vx + vy + vθ + vφ + ∂t ∂x ∂y ∂θ ∂φ m ∂vx m ∂vy Φφ − r cos θλ1 − r sin θλ2 ∂ Φθ ∂ + I1 ∂vθ I2 ∂vφ gives a vector defined on all of J 1 Q. We are, of course, only in the restriction of this vector field to M . First of all, one may readily check that the vector field does

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in fact restrict to M . In the coordinates (t, x, y, θ, φ, vθ , vφ ) for M this restricted vector field is given by XL,Φ,(M,Λ) =

∂ ∂ ∂ ∂ ∂ + r cos θvφ + r sin θvφ + vθ + vφ + ∂t ∂x ∂y ∂θ ∂φ r cos θΦx + r sin θΦy + Φφ ∂ Φθ ∂ + . I1 ∂vθ I2 + mr2 ∂vφ

Finally, for the sake of completeness, let us write the two-form ΩL,Φ : ΩL,Φ = (mdvx − Φx dt) ∧ (dx − vx dt) + (mdvy − Φy dt) ∧ (dy − vy dt) + (I1 dvθ − Φθ dt) ∧ (dθ − vθ dt) + (I2 dvφ − Φφ dt) ∧ (dφ − vφ dt).

References [1] Ralph Abraham and Jerrold E. Marsden. Foundations of Mechanics. Addison Wesley, Reading, MA, second edition, 1978. [2] Ralph Abraham, Jerrold E. Marsden, and Tudor S. Ratiu. Manifolds, Tensor Analysis, and Applications. Number 75 in Applied Mathematical Sciences. Springer-Verlag, second edition, 1988. [3] Ernst Binz, Manuel de Le´ on, and Dan Socolescu. Infinite dimensional Lagrangian systems with nonholonomic constraints. Preprint, 1998. ´ [4] Elie Cartan. Le¸cons sur les invariants int´egraux. Hermann, Paris, 1971. Reprint of 1922 second printing. [5] Nikolai Gurevich Chetaev. Theoretical Mechanics. Springer-Verlag, New York-Heidelberg-Berlin, 1989. Translated from the Russian by Irene Aleksanova. [6] Manuel de Le´ on, Juan C. Marrero, and David Mart´ın de Diego. Mechanical systems with nonlinear constraints. Int. Jour. Theor. Phys., 36(4), 979–995, 1997. [7] Manuel de Le´ on, Juan C. Marrero, and David Mart´ın de Diego. Nonholonomic Lagrangian systems in jet manifolds. Jour. of Phys. A. Mathematical and General,30, 1167–1190, 1997. [8] Robert B. Gardner and William F. Shadwick. A simple characterization of the contact system on J k (E). The Rocky Mountain Jour. of Math., 17(1), 19–21, 1987. [9] Giovanni Giachetta. Jet methods in nonholonomic mechanics. Jour. Math. Phys., 33(5), 1652–1665, 1992. [10] Xavier Gr` acia. The fibre derivatives and some of its applications. Memorias de la Real Academia de Ciencias Exactas, F´ı sicas y Naturales de Madrid. Serie de Ciencias Exactas, 32, 43–58, 1998.

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[11] Robert Hermann. The differential geometric structure of general mechanical systems from the Lagrangian point of view. Jour. Math. Phys., 23(11), 2077– 2089, 1982. [12] Andrew D. Lewis. The geometry of the Gibbs-Appell equations and Gauss’s Principle of Least Constraint. Rep. on Math. Phys., 38(1), 11–28, 1996. [13] Andrew D. Lewis. Lifting distributions to tangent and jet bundles. Preprint, 1998. [14] Paulette Libermann and Charles-Michel Marle. Symplectic Geometry and Analytical Mechanics. Mathematics and its Applications. D. Reidel Publishing Company, Dordrecht/Boston/Lancaster/Tokyo, 1987. [15] Charles-Michel Marle. Reduction of constrained mechanical systems and stability of relative equilibria. Commun. Math. Phys., 174(2), 295–318, 1997. [16] Enrico Massa and Enrico Pagani. Jet bundle geometry, dynamical connections, and the inverse problem of Lagrangian mechanics. Ann. de l’Inst. Henri Poincar´e. Physique Th´eorique, 61(1), 17–62, 1994. [17] Richard M. Murray, Zexiang Li, and S. Shankar Sastry. A Mathematical Introduction to Robotic Manipulation. CRC Press, 2000 Corporate Blvd., N.W., Boca Raton, Florida 33431, 1994. [18] Richard S. Palais. Definition of the exterior derivative in terms of the Lie derivative. Proc. of the American Mathematical Society, 5, 902–908, 1954. [19] D. J. Saunders. The Geometry of Jet Bundles. Number 142 in London Mathematical Society Lecture Note Series. Cambridge University Press, New York/Port Chester/Melbourne/Sydney, 1989. [20] W. M. Tulczyjew. The Lagrange differential. Bulletin de l’Acad´emie Polonaise des Sciences. S´erie des Sciences Math´ematiques, Astronomiques et Physiques, 24(12), 1089–1096, 1976.

Andrew D. Lewis Department of Mathematics & Statistics Queen’s University Kingston, ON K7L 3N6, Canada Email :[email protected]

Communicated by J. Bellissard submitted 23/03/99, revised 27/08/99, accepted 15/09/99

Ann. Henri Poincar´ e 1 (2000) 607 – 623 c Birkh¨ auser Verlag, Basel, 2000 1424-0637/00/040607-17 $ 1.50+0.20/0

Annales Henri Poincar´ e

Algebraic Holography K.-H. Rehren Abstract. A rigorous (and simple) proof is given that there is a one-to-one correspondence between causal anti-deSitter covariant quantum ﬁeld theories on anti-deSitter space and causal conformally covariant quantum ﬁeld theories on its conformal boundary. The correspondence is given by the explicit identiﬁcation of observables localized in wedge regions in anti-deSitter space and observables localized in double-cone regions in its boundary. It takes vacuum states into vacuum states, and positive-energy representations into positive-energy representations.

I Introduction and results The conjectured correspondence (so-called “holography”) [14, 20] between quantum ﬁeld theories on 1+s-dimensional anti-deSitter space-time AdS1,s (the “bulk space”) and conformal quantum ﬁeld theories on its conformal boundary CM1,s−1 which is a compactiﬁcation of Minkowski space R1,s−1 , has recently raised enthusiastic interest. If anti-deSitter space is considered as an approximation to the space-time geometry near certain gravitational horizons (extremal black holes), then the correspondence lends support to the informal idea of reduction of degrees of freedom due to the thermodynamic properties of black holes [10, 18]. Thus, holography is expected to give an important clue for the understanding of quantum theory in strong gravitational ﬁelds and, ultimately, of quantum gravity. While the original conjecture [14] was based on “stringy” pictures, it was soon formulated [20] in terms of (Euclidean) conventional quantum ﬁeld theory, and a speciﬁc relation between generating functionals was conjectured. These conjectures have since been exposed with success to many structural and group theoretical tests, yet a rigorous proof has not been given. The problem is, of course, that the “holographic” transition from anti-deSitter space to its boundary and back, is by no means a point transformation, thus preventing a simple (pointwise) operator identiﬁcation between bulk ﬁelds and boundary ﬁelds. In the present note, we show that in contrast, an identiﬁcation between the algebras generated by the respective local bulk and boundary ﬁelds is indeed possible in a very transparent manner. These algebraic data are completely suﬃcient to reconstruct the respective theories. We want to remind the reader of the point of view due to Haag and Kastler (see [8] for a standard textbook reference) which emphasizes that, while any choice of particular ﬁelds in a quantum ﬁeld theory may be a matter of convenience without aﬀecting the physical content of the theory (comparable to the choice of coor-

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dinates in geometry), the algebras they generate and their algebraic interrelations, notably causal commutativity, supply all the relevant physical information in an invariant manner. The interested reader will ﬁnd in [2] a review of the (far from obvious, indeed) equivalence between quantum ﬁeld theory in terms of ﬁelds and quantum ﬁeld theory in terms of algebras, notably on the strategies available to extract physically relevant information, such as the particle spectrum, superselection charges, and scattering amplitudes, from the net of algebras without knowing the ﬁelds. It is crucial in the algebraic approach, however, to keep track of the localization of the observables. Indeed, the physical interpretation of a theory is coded in the structure of a “causal net” of algebras [8] which means the speciﬁcation of the sets of observables B(X) which are localized in any given space-time region X.1 The assignment X → B(X) is subject to the conditions of isotony (an observable localized in a region X is localized in any larger region Y ⊃ X, thus B(Y ) ⊃ B(X)), causal commutativity (two observables localized at space-like distance commute with each other), and covariance (the Poincar´e transform of an observable localized in X is localized in the transformed region gX; in the context at hand replace “Poincar´e” by “anti-deSitter”). Each B(X) should in fact be an algebra of operators (with the observables its selfadjoint elements), and to have suﬃcient control of limits and convergence in order to compute physical quantities of interest, it is convenient to let B(X) be von Neumann algebras.2 For most purposes it is convenient to consider as typical compact regions the “double-cones”, that is, intersections of a future directed and a past directed light-cone, and to think of point-like localization in terms of very small doublecones. On the other hand, certain aspects of the theory are better captured by “wedge” regions which extend to space-like inﬁnity. A space-like wedge (for short: wedge) in Minkowski space is a region of the form {x : x1 > |x0 |}, or any Poincar´e transform thereof. The corresponding regions in anti-deSitter space turn out to be intersections of AdS1,s with suitable ﬂat space wedges in the ambient space R2,s , see below. In conformally covariant theories there is no distinction between double-cones and wedges since conformal transformations map the former onto the latter. It will become apparent in the sequel that to understand the issue of “holography”, the algebraic framework proves to be most appropriate. The basis for the holography conjectures is, of course, the coincidence between the anti-deSitter group SO0 (2, s) and the conformal group SO0 (2, s). (SO0 (n, m) is the identity component of the group SO(n, m), that is the proper orthochronous subgroup distinguished by the invariant condition that the determinants of the 1 The assignment X → B(X) is a “net” in the mathematical sense: a generalized sequence with a partially ordered index set (namely the set of regions X). 2 A von Neumann algebra is an algebra of bounded operators on a Hilbert space which is closed in the weak topology of matrix elements. E.g., if φ is a hermitean ﬁeld and φ(f ) a ﬁeld operator smeared over a region X containing the support of f , then operators like exp iφ(f ) belong to B(X).

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time-like n × n and of the space-like m × m sub-matrices are both positive.) The former group acts on AdS1,s (as a “deformation” from the ﬂat space Poincar´e group in 1+s dimensions, SO0 (1, s) R1,s ), and the latter group acts on the conformal boundary CM1,s−1 of AdS1,s (as an extension of the Poincar´e group in 1+(s − 1) dimensions, SO0 (1, s − 1) R1,s−1 ) by restriction of the former group action on the bulk. The representation theoretical aspect of this coincidence has been elaborated (in Euclidean metric) in [6]. In terms of covariant nets of algebras of local observables (“local algebras”), it is thus suﬃcient to identify one suitable algebra in anti-deSitter space with another suitable algebra in the conformal boundary space, and then to let SO0 (2, s) act to provide the remaining identiﬁcations. As any double-cone region in conformal space determines a subgroup of the conformal group SO0 (2, s) which preserves this double-cone, it is natural to identify its algebra with the algebra of a region in anti-deSitter space which is preserved by the same subgroup of the anti-deSitter group SO0 (2, s). It turns out that this region is a space-like wedge region which intersects the boundary in the given double-cone. For a typical bulk observable localized in a wedge region, the reader is invited to think of a ﬁeld operator for a Mandelstam string which stretches to space-like inﬁnity. Its holographic localization on the boundary has ﬁnite size, but it becomes sharper and sharper as the string is “pulled to inﬁnity”. We shall see that one may be forced to take into consideration theories which possess only wedge-localized, but no compactly localized observables. Our main algebraic result rests on the following geometric Lemma.3 Lemma Between the set of space-like wedge regions in anti-deSitter space, W ⊂ AdS1,s , and the set of double-cones in its conformal boundary space, I ⊂ CM1,s−1 , there is a canonical bijection α : W → I = α(W ) preserving inclusions and causal complements, and intertwining the actions of the anti-deSitter group SO0 (2, s) and of the conformal group SO0 (2, s) α(g(W )) = g(α(W ˙ )),

α−1 (g(I)) ˙ = g(α−1 (I))

where g˙ is the restriction of the action of g to the boundary. The double-cone I = α(W ) associated with a wedge W is the intersection of W with the boundary. Given the Lemma, the main algebraic result states that bulk observables localized in wedge regions are identiﬁed with boundary observables localized in double-cone regions: Corollary 1 The identiﬁcation of local observables B(W ) := A(α(W )),

A(I) := B(α−1 (I))

3 For details, see Sect. 2. We denote double-cones in the boundary by the symbol I, because (i) we prefer to reserve the “standard” symbol O for double-cones in the bulk space, and because (ii) in 1+1 dimensions the “double-cones” on the boundary are in fact open intervals on the circle S1.

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gives rise to a 1:1 correspondence between isotonous causal conformally covariant nets of algebras I → A(I) on CM1,s−1 and isotonous causal anti-deSitter covariant nets of algebras W → B(W ) on AdS1,s . An observable localized in a double-cone O in anti-deSitter space is localized in any wedge containing O, hence the local algebra B(O) should be contained in all B(W ), W ⊃ O. We shall deﬁne B(O) as the intersection of all these wedge algebras. These intersections do no longer correspond to simple geometric regions in CM1,s−1 (so points in the bulk have a complicated geometry in the boundary), as will be discussed in more detail in 1+1 dimensions below. The following result also identiﬁes states and representations of the corresponding theories: Corollary 2 Under the identiﬁcation of Corollary 1, a vacuum state on the net A corresponds to a vacuum state on the net B. Positive-energy representations of the net A correspond to positive-energy representations of the net B. The net A satisﬁes essential Haag duality if and only if the net B does. The modular group and modular conjugation (in the sense of Tomita-Takesaki) of a wedge algebra B(W ) in a vacuum state act geometrically (by a subgroup of SO0 (2, s) which preserves W and by a CPT reﬂection, respectively) if and only if the same holds for the double-cone algebras A(I). Essential Haag duality means that the algebras associated with causally complementary wedges not only commute as required by locality, but either algebra is in fact the maximal algebra commuting with the other one. The last statement in the Corollary refers to the modular theory of von Neumann algebras which states that every (normal and cyclic) state on a von Neumann algebra is a thermal equilibrium state with respect to a unique adapted “time” evolution (one-parameter group of automorphisms = modular group) of the algebra. In quantum ﬁeld theories in Minkowski space, whose local algebras are generated by smeared Wightman ﬁelds, the modular groups have been computed for wedge algebras in the vacuum state [5] and were found to coincide with the boost subgroup of the Lorentz group which preserves the wedge (geometric action). In conformal theories, mapping wedges onto double-cones by suitable conformal transformations, the same result also applies to double-cones [9]. This result is an algebraic explanation of the Unruh eﬀect according to which a uniformly accelerated observer attributes a temperature to the vacuum state, and provides also an explanation of Hawking radiation if the wedge region is replaced by the space-time region outside the horizon of a Schwarzschild black hole [17]. The modular theory also provides a “modular conjugation” which maps the algebra onto its commutant. For Minkowski space Wightman ﬁeld theories in the vacuum state as before, the modular conjugation of a wedge algebra turns out to act geometrically as a CPT-type reﬂection (CPT up to a rotation) along the “ridge” of the wedge which maps the wedge onto its causal complement. This entails essential duality for Minkowski space [5] as well as conformally covariant [9] Wightman theories.

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The statement in Corollary 2 on the modular group thus implies that, if the boundary theory is a Wightman theory, then the boundary and the bulk theory both satisfy essential Haag duality, and also in anti-deSitter space a vacuum state of B in restriction to a wedge algebra B(W ) is a thermal equilibrium state with respect to the associated one-parameter boost subgroup of the anti-deSitter group which preserves W , i.e., the Unruh eﬀect takes place for a uniformly accelerated observer. Furthermore, the CPT theorem holds for the theory on anti-deSitter space. On the other hand, essential Haag duality and geometric modular action for quantum ﬁeld theories on AdS1,s were established under much more general assumptions [3], implying the same properties for the associated boundary theory even when it is not a Wightman theory (see below). 1 We emphasize that the Hamiltonians R M0,d on AdS1,s and P 0 on CM1,s−1 are not identiﬁed under the identiﬁcation of the anti-deSitter group and the conformal group. Instead, M0,d is (in suitable coordinates) identiﬁed with the combination 12 (P 0 + K 0 ) of translations and special conformal transformations in the 0-direction of CM1,s−1 . This is diﬀerent from the Euclidean picture [20] where the anti-deSitter Hamiltonian is identiﬁed with the dilatation subgroup of the conformal group. In Lorentzian metric, the dilatations correspond to a space-like “translation” subgroup of the anti-deSitter group. This must have been expected since the generator of dilatations does not have a one-sided spectrum as is required for the real-time Hamiltonian. The subgroup generated by 12 (P 0 + K 0 ) is well-known to be periodic and to satisfy the spectrum condition in every positiveenergy representation. (Periodicity in bulk time of course implies a mass gap for the underlying bulk theory. This is not in conﬂict with the boundary theory being massless since the respective subgroups of time evolution cannot be identiﬁed.)

Diﬀerent Hamiltonians give rise to diﬀerent counting of degrees of freedom, since entropy is deﬁned via the partition function. Thus, the “holographic” reduction of degrees of freedom [10, 18] can be viewed as a consequence of the choice of the Hamiltonian: The anti-deSitter Hamiltonian M0,d = 12 (P 0 + K 0 ) has discrete spectrum and has a chance (at least in 1+1 dimensions) to yield a ﬁnite partition function. One the other hand, the partition function with respect to the boundary Hamiltonian P 0 exhibits the usual infrared divergence due to inﬁnite volume and continuous spectrum. A crucial aspect of the present analysis is the identiﬁcation of compact regions in the boundary with wedge regions in the bulk. With a little hindsight, this aspect is indeed also present in the proposal for the identiﬁcation of generating functionals [20]. While the latter is given in the Euclidean approach, it should refer in real time to a hyperbolic diﬀerential equation with initial values given in a double-cone on the boundary which determine its solution in a wedge region of bulk space. We also show that in 1+1 dimensions there are suﬃciently many observables localized in arbitrarily small compact regions in the bulk space to ensure that compactly localized observables generate the wedge algebras. This property is crucial if we want to think of local algebras as being generated by local ﬁelds:

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Proposition Assume that the boundary theory A on S 1 is weakly additive (i.e., A(I) is generated by A(Jn ) whenever the interval I is covered by a family of intervals Jn ). If a wedge W in AdS1,1 is covered by a family of double-cones On ⊂ W , then the algebra B(W ) is generated by the observables localized in On : B(W ) =

B(On ).

n

In order to establish this result, we explicitly determine the observables localized in a double-cone region on AdS1,1 . Their algebra B(O) turns out to be non-trivial: it is the intersection of two interval algebras A(Ii ) on the boundary S 1 where the intersection of the two intervals Ii is a union of two disjoint intervals Ji . B(O) contains therefore at least A(J1 ) and A(J2 ). In fact, it is even larger than that, containing also observables corresponding to a “charge transport” [8], that is, operators which annihilate a superselection charge in J1 and create the same charge in J2 . The inclusion A(I1 ) ∨ A(J2 ) ⊂ B(O) therefore carries (complete) algebraic information about the superselection structure of the chiral boundary theory [11]. As the double-cone on AdS1,1 shrinks, the size of the intervals Ji also shrinks but not their distance, so points in 1+1-dimensional anti-deSitter space are related to pairs of points in conformal space. But we see that sharply localized bulk observables involve boundary observables localized in large intervals: the above charge transporters. This result provides an algebraic interpretation of the obstruction against a point transformation between bulk and boundary. The issue of compactly localized observables in anti-deSitter space is more complicated in more than two dimensions, and deserves a separate careful analysis. Some preliminary results will be presented in Section 2.3. They show that if the bulk theory possesses observables localized in double-cones, then the corresponding boundary theory violates an additivity property which is characteristic for Wightman ﬁeld theories, while its violation is expected for non-abelian gauge theories due to the presence of gauge-invariant Wilson loop operators. Conversely, if the boundary theory satisﬁes this additivity property, then the observables of the corresponding bulk theory are always attached to inﬁnity, as in topological (Chern-Simons) theories. Let us point out that the conjectures in [14, 20] suggest a much more ambitious interpretation, namely that the correspondence pertains to bulk theories involving quantum gravity, while the anti-deSitter space and its boundary are understood in some asymptotic (semi-classical) sense. Indeed, the algebraic approach is no more able to describe proper quantum gravity as any other mathematically unambiguous framework up to now. Most arguments given in the literature in favour of the conjectures refer to gravity as perturbative gravity on a background space-time. Likewise, our present results concern the semi-classical version of the conjectures, treating gravity like any other quantum ﬁeld theory as a theory of observables on a classical background geometry. In fact, the presence or absence

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of gravity in the bulk theory plays no particular role. This is only apparently in conﬂict with the original arguments for a holographic reduction of degrees of freedom of a bulk theory in the vicinity of a gravitational horizon [10, 18] in which gravity is essential. Namely, our statement can be interpreted in the sense that once the anti-deSitter geometry is given for whatever reason (e.g., the presence of a gravitational horizon), then it can support only the degrees of freedom of a boundary theory.

II

Identiﬁcation of observables

We denote by H1,s the d=1+s-dimensional hypersurface deﬁned through its embedding into ambient R2,s , x20 − x21 − . . . − x2s + x2d = R2 with Lorentzian metric induced from the 2+s-dimensional metric ds2 = dx20 − dx21 − . . . − dx2s + dx2d . Its group of isometries is the Lorentz group O(2, s) of the ambient space in which the reﬂection x → −x is central. Anti-deSitter space is the quotient manifold AdS1,s = H1,s /Z2 (with the same Lorentzian metric locally). We denote by p the projection H1,s → AdS1,s . Two open regions in anti-deSitter space are called “causally disjoint” if none of their points can be connected by a time-like geodesic. The largest open region causally disjoint from a given region is called the causal complement. In a causal quantum ﬁeld theory on the quotient space AdS1,s , observables and hence algebras associated with causally disjoint regions commute with each other. The reader should be worried about this deﬁnition, since causal independence of observables should be linked to causal connectedness by time-like curves rather than geodesics. But on anti-deSitter space, any two points can be connected by a time-like curve, so they are indeed causally connected, and the requirement that causally disconnected observables commute is empty. Yet, as our Corollary 1 shows, if the boundary theory is causal, then the associated bulk theory is indeed causal in the present (geodesic) sense. We refer also to [3] where it is shown that vacuum expectation values of commutators of observables with causally disjoint localization have to vanish whenever the vacuum state has reasonable properties (invariance and thermodynamic passivity), but without any a priori assumptions on causal commutation relations (neither in bulk nor on the boundary). Thus in the theories on anti-deSitter space we consider in this paper, observables localized in causally disjoint but causally connected regions commute; see [3] for a discussion of the ensuing physical constraints on the nature of interactions on anti-deSitter space. The causal structure of AdS1,s is determined by its metric modulo conformal transformations which preserve angles and geodesics. As a causal manifold,

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AdS1,s has a boundary (the “asymptotic directions” of geodesics). The boundary inherits the causal structure of the bulk space AdS1,s , and the anti-deSitter group SO0 (2, s) acts on this space. It is well known that this boundary is a compactiﬁcation CM1,s−1 = (S 1 × S s−1 )/Z2 of Minkowski space R1,s−1 , and SO0 (2, s) acts on it like the conformal group. The notions of causal disjoint and causal complements on CM1,s−1 coincide, up to conformal transformations, with those on Minkowski space [12]. In d =1+1 dimensions, s = 1, the conformal space is S 1 , and the causal complement of an interval I is I c = S 1 \I. Both anti-deSitter space and its causal boundary have a “global time-arrow”, that is, the distinction between the future and past light-cone in the tangent spaces (which are ordinary Minkowski spaces) at each point x can be globally chosen continuous in x (and consistent with the reﬂection x → −x). The time orientation on the bulk space induces the time orientation on the boundary. The time arrow is crucial in order to distinguish representations of positive energy. 2.1 Proof of the Lemma Any ordered pair of light-like vectors (e, f ) in the ambient space R2,s such that e·f < 0 deﬁnes an open subspace of the hypersurface H1,s given by (e, f ) = {x ∈ R2,s : x2 = R2 , e·x > 0, f ·x > 0}. W This space has two connected components. Namely, the tangent vector at each (e, f ) under the boost in the e-f -plane, δe,f x = (f·x)e−(e·x)f , is either point x ∈ W a future or a past directed time-like vector, since (δe,f x)2 = −2(e·f )(e·x)(f·x) > 0. − (e, f ) the connected components of W (e, f ) in + (e, f ) and W We denote by W + (f, e) = which δe,f x is future and past directed, respectively. By this deﬁnition, W W− (e, f ), and W+ (−e, −f ) = −W+ (e, f ). ± (e, f ) as specThe wedge regions in the hypersurface H1,s are the regions W iﬁed. The wedge regions in anti-deSitter space are their quotients W± (e, f ) = ± (e, f ). One has W+ (e, f ) = W− (f, e) = W+ (−e, −f ), and W+ (e, f ) and pW W− (e, f ) are each other’s causal complements. For an illustration in 1+1 dimensions, cf. Figure 1. We claim that the projected wedges W± (e, f ) intersect the boundary of AdS1,s in regions I± (e, f ) which are double-cones of Minkowski space R1,s−1 or images thereof under some conformal transformation. Note that any two double-cones in R1,s−1 are connected by a conformal transformation, and among their conformal transforms are also the past and future light-cones and space-like wedges in R1,s−1 .

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x0 xd

BOUNDARY

xs

~ W

_~ W_

+

arrow of time

I+ (

(

x

(

(

arrow of time

_x I_

~

W_

BULK

+ (e, f ) and W − (e, f ) in 1+1 dimensions, and their Figure 1. Wedge regions W − intersections with the boundary. The light-like vectors e and f are tangent to −W in its apex x. In anti-deSitter space, W− is identiﬁed with −W− , and W± = pW± are causal complements of each other. We claim also that the causal complement W− (e, f ) of the wedge W+ (e, f ) intersects the boundary in the causal complement I− (e, f ) = I+ (e, f )c of I+ (e, f ). It would be suﬃcient to compute the intersections of any single pair of wedges ± (e, f ) with the boundary and see that it is a pair of causally complementary W conformal double-cones in CM1,s−1 , since the claim then follows for any other pair of wedges by covariance. For illustrative reason we shall compute two such examples. We ﬁx the “arrow of time” by declaring the tangent vector of the rotation in the 0-d-plane, δt x = (−xd , 0, . . . , 0, x0 ), to be future directed. In stereographic coordinates (y0 , !y , x− ) of the hypersurface x2 = R2 , where x− = xd − xs and (y0 , !y) = (x0 , !x)/x− , !x = (x1 , . . . , xs−1 ), the boundary is given by |x− | = ∞. Thus, in the limit of inﬁnite x− one obtains a chart yµ = (y0 , !y ) of CM1,s−1 . The induced conformal structure is that of Minkowski space, dy 2 = f (y)2 (dy02 − d!y2 ). Our ﬁrst example is the one underlying Figure 1: we choose eµ = (0, . . . , 0, 1, 1) (e, f ) read xd − xs > 0 and and fµ = (0, . . . , 0, 1, −1). The conditions for x ∈ W 2 2 2 xd + xs < 0, implying xd − xs < 0 and hence x0 − !x2 > R2 . The tangent vector δe,f x has d-component δe,f xd = −2xs > 0. Hence, it is future directed if x0 > 0, and past directed if x0 < 0: + (e, f ) = {x : x2 = R2 , xs < −|xd |, x0 > 0}. W After dividing (x0 , !x) by x− = xd − xs ∞, we obtain the boundary region I+ (e, f ) = {y = (y0 , !y ) : y02 − !y 2 > 0, y0 > 0},

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that is, the future light-cone in Minkowski space; similarly, I− (e, f ) is the past light-cone, and I± (e, f ) are each other’s causal complements in CM1,s−1 . Next, we choose eµ = (1, 1, 0, . . . , 0) and fµ = (−1, 1, 0, . . . , 0). The conditions (e, f ) read x1 < −|x0 |, implying x20 −!x2 < 0 and hence x2 −x2s > R2 > 0. for x ∈ W d The tangent vector δe,f x has 0-component δe,f xd = −2x1 > 0. Hence, it is future directed if xd < 0 hence xd − xs < 0, and past directed if xd > 0 hence xd − xs > 0: + (e, f ) = {x : x2 = R2 , x1 < −|x0 |, x− < 0}. W After dividing (x0 , !x) by x− = xd − xs −∞, we obtain the boundary region I+ (e, f ) = {y = (y0 , !y) : y1 > |y0 |}, that is, a space-like wedge region in Minkowski space; similarly, I− (e, f ) is the opposite wedge y1 < −|y0 |, which is again the causal complement of I+ (e, f ) in CM1,s−1 . Both light-cones and wedge regions in Minkowski space are well known to be conformal transforms of double-cones, and hence they are double-cones on CM1,s−1 . The pairs of regions computed above are indeed causally complementary pairs. We now consider the map α : W+ (e, f ) → I+ (e, f ). Since the action of the conformal group on the boundary is induced by the action of the anti-deSitter ± (ge, gf ) = g(W ± (e, f )) and I± (ge, gf ) = group on the bulk, we see that W g(I ˙ ± (e, f )), hence α intertwines the actions of the anti-deSitter and the conformal group. It is clear that α preserves inclusions, and we have seen that it preserves causal complements for one, and hence for all wedges. Since SO0 (2, s) acts transitively on the set of double-cones of CM1,s−1 , the map α is surjective. Finally, since W+ (e, f ) and I+ (e, f ) have the same stabilizer subgroup of SO0 (2, s), it is also injective. This completes the proof of the Lemma. 2.2 Proof of the Corollaries We identify wedge algebras on AdS1,s and double-cone algebras on CM1,s−1 by B(W± (e, f )) = A(I± (e, f )), that is, B(W ) = A(α(W )). The Lemma implies that if A is given as an isotonous, causal and conformally covariant net of algebras on CM1,s−1 , then B(W ) deﬁned by this identiﬁcation constitute an isotonous, causal and anti-deSitter covariant net of algebras on AdS1,s , and vice versa. Namely, the identiﬁcation is just a relabelling of the index set of the net which preserves inclusions and causal complements and intertwines the action of SO0 (2, s). Thus we have established Corollary 1. As for Corollary 2, we note that, as the algebras are identiﬁed, states and representations of the nets A and B are also identiﬁed. Since the identiﬁcation intertwines the action of the anti-dSitter group and of the conformal group, an anti-deSitter invariant state on B corresponds to a conformally invariant state on A. The generator of time translations in the anti-deSitter

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group corresponds to the generator 12 (P 0 + K 0 ) in the conformal group which is known to be positive if and only if P 0 is positive (note that K 0 is conformally conjugate to P 0 ). Hence the conditions for positivity of the respective generators of time-translations are equivalent. By the identiﬁcation of states and algebras, also the modular groups are identiﬁed. The modular group and modular conjugation for double-cone algebras in a vacuum state of conformally covariant quantum ﬁeld theories are conformally conjugate to the modular group and modular conjugation of a Minkowski space wedge algebra, which in turn are given by the Lorentz boosts in the wedge direction and the reﬂection along the ridge of the wedge [5, 9]. It follows that the modular group for a wedge algebra on anti-deSitter space is given by the corresponding subgroup of the anti-deSitter group which preserves the wedge (for a wedge W+ (e, f ), this is the subgroup of boosts in the e-f -plane), and the modular conjugation is a CPT transformation which maps W+ onto W− . These remarks suﬃce to complete the proof of Corollary 2. Let us mention that the correspondence given in Corollary 1 holds also for “weakly local” nets both on the bulk and on the boundary. In a weakly local net, the vacuum expectation value of the commutator of two causally disjoint observables vanishes, but not necessarily the commutator itself. Weak locality for quantum ﬁeld theories on anti-deSitter space follows [3] from very conservative assumptions on the vacuum state without any commutation relations assumed. Thus, also the boundary theory will always be weakly local. 2.3 Compact localization in anti-deSitter space Let us ﬁrst note that as the ridge of a wedge is shifted into the interior of the wedge, the double-cone on the boundary shrinks. Thus, sharply localized boundary observables correspond to bulk observables at space-like inﬁnity [1]. We now show that sharply localized bulk observables do not correspond to a simple geometry on the boundary, but must be determined algebraically. An observable localized in a double-cone O of anti-deSitter space must be contained in every wedge algebra B(W ) such that O ⊂ W . The algebra B(O) is thus at most the intersection of all B(W ) such that O ⊂ W . We may deﬁne it as this intersection, thereby ensuring isotony, causal commutativity and covariance for the net of double-cone algebras in an obvious manner. Double-cone algebras on anti-deSitter space are thus delicate intersections of algebras of double-cones and their conformal images on the boundary, and might turn out trivial. In 1+1 dimensions, the geometry is particularly simple since a double-cone is an intersection of only two wedges. We show that the corresponding intersection of algebras is non-trivial, and shall turn to d > 1 + 1 below. Let us write (in 1+1 dimensions) the relation B(O) = B(W1 ) ∩ B(W2 ) ≡ A(I1 ) ∩ A(I2 )

whenever

O = W1 ∩ W2 ,

where Wi are any pair of wedge regions in AdS1,1 and Ii = α(Wi ) their intersections with the boundary, that is, open intervals on S 1 .

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The intersection W1 ∩ W2 might not be a double-cone. It might be empty, or it might be another wedge region. Before discussing the above relation as a deﬁnition for the double-cone algebra B(O) if O = W1 ∩ W2 is a double-cone, we shall ﬁrst convince ourselves that it is consistent also in these other cases. If W1 contains W2 , or vice versa, then O equals the larger wedge, and the relation holds by isotony. If W1 and W2 are disjoint, then the intersections with the boundary are also disjoint, and B(∅) = A(I1 ) ∩ A(I2 ) is trivial if the boundary net A on S 1 is irreducible (that is, disjoint intervals have no nontrivial observables in common). Next, it might happen that W1 and W2 have a nontrivial intersection without the apex of one wedge lying inside the other wedge. In this case, the intersection is again a wedge, say W3 . Namely, any wedge in AdS1,1 is of the form W+ (e, f ) where e and f are a future and a past directed light-like tangent vector in the apex x (the unique point in AdS1,1 solving e·x = f ·x = 0). The condition e·f < 0 implies that both tangent vectors point in the same (positive or negative) 1-direction. The wedge itself is the surface between the two light-rays emanating from x in the directions −e and −f (cf. Figure 1). The present situation arises if the future directed light-ray of W1 intersects the past directed light-ray of W2 (or vice versa) in a point x3 without the other pair of light-rays intersecting each other. The intersection of the two wedges is the surface between the two intersecting lightrays travelling on from the point x3 , which is another wedge region W3 with apex x3 . It follows that the intersection of the intersections Ii of Wi with the boundary equals the intersection I3 of W3 with the boundary. Hence consistency of the above relation is guaranteed by A(I1 ) ∩ A(I2 ) = A(I3 ) where I1 and I2 are two intervals on S 1 whose intersection I3 is again an interval. Now we come to the case that W1 ∩ W2 is a double-cone O in the proper sense. This is the case if the closure of the causal complement of W1 is contained in W2 . It follows that the closure of the causal complement of I1 is contained in I2 , hence the intersection of I1 and I2 is the union of two disjoint intervals J1 and J2 . The latter are the two light-like geodesic “shadows”, cast by O onto the boundary. Thus, the observables localized in a double-cone in anti-deSitter space AdS1,1 are given by the intersection of two interval algebras A(Ii ) on the boundary for intervals Ii with disconnected intersections (or equivalently, by essential duality, the joint commutant of two interval algebras for disjoint intervals). Such algebras have received much attention in the literature [16, 21, 11], notably within the context of superselection sectors. Namely, if I1 ∩I2 = J1 ∪J2 consists of two disjoint intervals, then the intersection of algebras A(I1 ) ∩ A(I2 ) is larger than the algebra A(J1 ) ∨ A(J2 ). The excess can be attributed to the existence of superselection sectors [11], the extra operators being intertwiners which transport a superselection charge from one of the intervals Ji to the other. We conclude that (certain) compactly localized observables on anti-deSitter space are strongly delocalized observables (charge transporters) of the boundary theory. Yet there is no obstruction against both theories being Wightman theories generated by local Wightman ﬁelds, as the following simple example shows.

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In suitable coordinates xµ = R · (cos t, cos x, sin t)/ sin x, the bulk is the strip (t, x) ∈ R × (0, π) with points (t, x) ∼ (t + π, π − x) identiﬁed, while the boundary are the points (0, u), u ∈ R mod 2π. The metric is a multiple of dt2 − dx2 , thus the light rays emanating from the bulk point (t, x) hit the boundary at the points u± = t ± x mod 2π. We see that, as the double-cone O shrinks to a point (t, x) in bulk, the two intervals Ji on the boundary also shrink to points (namely u± ) while their distance remains ﬁnite. Now, we consider the abelian current ﬁeld j(u) on the boundary, and determine the associated ﬁelds on anti-deSitter space. First, for (t, x) in the strip, both j(t ± x) are localized at (t, x) and give rise to a conserved vector current j µ with components j 0 (t, x) = j(t + x) + j(t − x), j 1 (t, x) = −j(t + x) + j(t − x). Fur t+x thermore, the ﬁelds φα (t, x) = exp iα t−x j(u)du (suitably regularized, of course), α ∈ R, are also localized at (t, x). Namely, since the charge operator S 1 j(u)du is a number q in each irreducible representation, φα (t, x) may as well be represented as t−x eiαq exp −iα t+x−2π j(u)du and hence is localized in both complementary boundary intervals [t − x, t + x] and [t + x − 2π, t − x] which overlap in the points u+ and u− , as required. Indeed, the ﬁelds φα can be obtained from bounded Weyl operators with ﬁnite localization as follows. A(I) is generated by boundary observables of the Weyl form W (f ) = exp ij(f ) where f is a smearing function on S 1 which is constant outside the interval I. Adding a constant to f is immaterial forthe localization since the commutation relations are given by the symplectic form f g du. A Weyl operator W (f ) is localized in both intervals I1 , I2 (notation as before) if f has constant values on both gaps between J1 , J2 , but it is not a product of Weyl operators in J1 and in J2 whenever these values are diﬀerent. As a bulk observable, W (f ) is localized in the double-cone O = W1 ∩ W2 , and operators of this form generate B(O). Suitably regularized limits of W (f ) yield the point-like local ﬁelds φα (t, x). For the more expert reader, we mention that our identiﬁcation of double-cone algebras in bulk with two-interval algebras on the boundary also shows how the notorious diﬃculty to compute the modular group for two-interval algebras [16] is related to the diﬃculty to compute the modular group of double-cone algebras in massive theories. (We discuss below that in a scaling limit the massive anti-deSitter theory approaches a conformal ﬂat space theory. In this limit, the modular group can again be computed.) We now prove the Proposition of Sect. 1. It asserts that the algebras B(On ) generate B(W ) whenever a family of double-cones On ⊂ W covers the wedge W ⊂ AdS1,1 . Each B(On ) is of the form A(In1 ) ∩ A(In2 ) where In1 ⊂ I = α(W ) and In1 ∩In2 = Jn1 ∪Jn2 is a union of two disjoint intervals. By deﬁnition, the assertion is equivalent to A(I) = A(In1 ) ∩ A(In2 ), n

where the inclusion “⊃” holds since each A(In1 ) is contained in A(I). On the other

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hand, the algebras on the right hand side are larger than A(Jn1 ) ∨ A(Jn2 ). If On cover the wedge W , then the intervals Jn1 and Jn2 , as n runs, cover the interval I = α(W ). So the claim follows from weak additivity of the boundary theory. In d ≥ 2 + 1 dimensions, the situation is drastically diﬀerent. Namely, if a family of small boundary double-cones Ii covers the space-like basis of a large double-cone I, and Wi and W denote the associated anti-deSitter wedge regions, then – unlike in 1+1 dimensions – W will contain a bulk double-cone O which is space-like to all Wi . Consequently, B(O) ⊂ B(W ) = A(I) must commute with the algebra i A(Ii ) generated by all B(Wi ) = A(Ii ). But in theories based on gauge-invariant Wightman ﬁelds (with the localization of operators determined in terms of smearing functions), the latter algebra coincides with A( i Ii ). This algebra in turn coincides with A(I) whenever the dynamics is generated by a Hamiltonian which is an integral over a local density, because then the observables in a neighbourhood of the space-like basis of I determine the observables in all of I. Thus B(O) must belong to the center of A(I) which is commutative (classical). Hence, a Wightman boundary theory is associated with a bulk theory without compactly localized quantum observables. Conversely, if there are double-cone localized bulk observables (e.g., if the bulk theory is itself described by a Wightman ﬁeld [7]), then the nontriviality of B(O) requires A( i Ii ) = A(I) to be strictly larger than i A(Ii ). This violation of additivity seems to be characteristic of non-abelian gauge theories where Wilson loop operators are not generated by point-like gauge invariant ﬁelds (cf. also the discussion in [19]). These issues certainly deserve a more detailed and careful analysis. For the moment, we conclude that the holographic correspondence necessarily relates, in more than 1+1 dimensions, Wightman type boundary theories to bulk theories without compactly localized observables (topological theories), in agreement with a remark on Chern-Simons theories in [20], and, conversely, bulk theories with point-like ﬁelds to boundary theories which share properties of non-abelian gauge theories, in agreement with the occurrence of Yang-Mills theories in [14].

III

Speculations

It is an interesting side-aspect of the last remark in the previous section that the holographic correspondence in both directions relates gauge theories to Wightman theories. It might therefore provide a new constructive scheme giving access to gauge theories. If one is interested in quantum ﬁeld theories on Minkowski rather than antideSitter space, one may consider the ﬂat space limit in which the curvature radius R of anti-deSitter space tends to inﬁnity, or equivalently consider a region of anti-deSitter space which is much smaller than the curvature radius. The regime |x| << R asymptotically becomes ﬂat Minkowski space, and the anti-deSitter group contracts to the Poincar´e group. Thus, one obtains a Minkowski space theory

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on R1,s from a conformal theory on CM1,s−1 through a scaling limit [2, 4] of the associated theory on AdS1,s . For d = 1 + 1, this can be done quite explicitly. The double-cone algebras B(O) are certain extensions of the algebras A(J1 ) ∨ A(J2 ), as discussed before. Now in the ﬂat regime the intervals Ji become small of order |O|/R. Thus for a substantial portion of Minkowski space, the relevant intervals Ji are all contained in a suitable but ﬁxed pair of non-overlapping intervals Ki . Let us now assume that the conformal net has the split property (an algebraic property valid in any chiral quantum ﬁeld theory for which Tr exp −βL0 exists), which ensures that states can be independently prepared on causally disjoint regions with a ﬁnite distance [2]. Then A(K1 ) ∨ A(K2 ) is unitarily isomorphic to A(K1 ) ⊗ A(K2 ), and the isomorphism is inherited by all its subalgebras A(J1 ) ∨ A(J2 ) A(J1 ) ⊗ A(J2 ). Under this isomorphism, the larger algebra B(O) is identiﬁed [11] with the standard construction [13] of 1+1-dimensional conformal Minkowski space observables from a given chiral conformal net (which corresponds to the diagonal modular invariant and is sometimes quoted as the Longo-Rehren net): B(O) BLR (J1 × J2 ) if O corresponds to I1 ∩ I2 = J1 ∪ J2 ⊂ K1 ∪ K2 . The unitary isomorphism of algebras, however, does not take the vacuum state on B to the vacuum state on the LR net. Thus, the ﬂat space limit of the anti-deSitter space theory in 1+1 dimensions associated with a given chiral conformal theory, is given by the LR net associated with that same chiral theory. Note that the LR net has 1+1-dimensional conformal symmetry, but of course the anti-deSitter net is not conformally invariant due to the presence of the curvature scale R. It would be interesting to get an analogous understanding of the ﬂat space limit of the anti-deSitter space theory in higher dimensions in terms of the associated conformal theory. One might speculate whether one can “iterate holography”, and use the ﬂat space limit of the bulk theory on AdS1,s as a boundary input for a new bulk theory on AdS1,s+1 . Here, however, a warning is in order. Namely, the limiting ﬂat space theory on Minkowski space R1,s will, like the LR net, in general not be extendible to the conformal compactiﬁcation CM1,s but rather to a covering thereof. One might therefore endeavour to extend the present analysis to theories on covering spaces both of anti-deSitter space and of its boundary. There is an independent and physically motivated reason to study quantum ﬁeld theories on a covering of anti-deSitter space. Namely, it has been observed (see above, [3]) that the local commutativity for causally disjoint but not causally disconnected observables leads to severe constraints on the possible interactions on anti-deSitter space proper. These constraints will disappear on the universal covering space. This “anti-deSitter causality paradox” parallels very much the old “conformal causality paradox” that causal commutativity on CM1,s proper excludes most conformal theories of interest; it was solved [15] by the recognition that conformal ﬁelds naturally live on a covering space. Holography tells us that both problems

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are the two sides of the same coin. Extending the present analysis to covering spaces seems a dubious task for d = 1+1 since the boundary of the covering of two-dimensional anti-deSitter space has two connected components. In higher dimensions, however, we do not expect serious obstacles.

Acknowledgment Previous versions of this paper have been improved in several respects on the basis of discussions with D. Buchholz, R. Verch (who also made the Figure), B. Schroer, R. Helling and many others, as well as questions raised by the referees. My thanks are due to all of them.

References [1] M. Bertola, J. Bros, U. Moschella, R. Schaeﬀer: AdS/CFT correspondence for n-point functions, hep-th/9908140. [2] D. Buchholz: Current trends in axiomatic quantum ﬁeld theory, G¨ ottingen preprint (1998), hep-th/9811233. [3] D. Buchholz, M. Florig, S.J. Summers: The second law of thermodynamics, TCP and Einstein causality in anti-de Sitter space-time, Class-Qu-Grav. 17(2000) L31–37. [4] D. Buchholz, R. Verch: Scaling algebras and renormalization group in algebraic quantum ﬁeld theory, Rev. Math. Phys. 7 (1995) 1195–1240. [5] J.J. Bisognano, E.H. Wichmann: On the duality condition for a hermitean scalar ﬁeld, J. Math. Phys. 16 (1975) 985–1007. [6] V. Dobrev: Intertwining operator realization of the AdS/CFT correspondence, Nucl. Phys. B 553 [PM] (1999) 559–582. [7] C. Fronsdal: Elementary particles in a curved space. II, Phys. Rev. D 10 (1974) 589–598. [8] R. Haag: Local Quantum Physics, Springer 1992. [9] P.D. Hislop, R. Longo: Modular structure of the local algebras associated with the free massless scalar ﬁeld theory, Commun. Math. Phys. 84 (1982) 71–85. [10] G. ’t Hooft: Dimensional reduction in quantum gravity, in: Salam-Festschrift, A. Ali et al. eds., World Scientiﬁc 1993, pp. 284–296. [11] Y. Kawahigashi, R. Longo, M. M¨ uger: Multi-interval subfactors and modularity of representations in conformal ﬁeld theory, Tokyo-Rome preprint (1999), math.OA/9903104. [12] M. L¨ uscher, G. Mack: Global conformal invariance in quantum ﬁeld theory, Commun. Math. Phys. 41 (1975) 203–234. [13] R. Longo, K.-H. Rehren: Nets of subfactors, Rev. Math. Phys. 7 (1995) 567–597. [14] J. Maldacena: The large N limit of superconformal ﬁeld theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231–252. [15] B. Schroer, J.A. Swieca: Conformal transformations for quantized ﬁelds, Phys. Rev. D 10 (1974) 480–485. [16] B. Schroer, H.-W. Wiesbrock: Modular theory and geometry, Rev. Math. Phys. 12 (2000) 139–158. [17] G.L. Sewell: Quantum ﬁelds on manifolds: PCT and gravitationally induced thermal states, Ann. Phys. (N.Y.) 141 (1982) 201–224. [18] L. Susskind: The world as a hologram, J. Math. Phys. 36 (1995) 6377–6396. [19] L. Susskind, N. Toumbas: Wilson loops as precursors, Phys. Rev. D 61 (2000) 044001.

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[20] E. Witten: Anti de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253–291. [21] Xu Feng: Jones-Wassermann subfactors for disconnected intervals, UCLA preprint (1997), q-alg/9704003.

K.-H. Rehren Institut f¨ ur Theoretische Physik Universit¨ at G¨ ottingen 37073 G¨ ottingen, Germany Email : [email protected]

Communicated by D. Buchholz submitted 26/05/99, revised 07/10/99, accepted 08/10/99

Ann. Henri Poincar´ e 1 (2000) 625 – 653 c Birkh¨ auser Verlag, Basel, 2000 1424-0637/00/040625-29 $ 1.50+0.20/0

Annales Henri Poincar´ e

Adiabatic Limit for the Maxwell–Lorentz Equations M. Kunze and H. Spohn Abstract. We consider the Abraham model of a rigid charge distribution coupled to the electromagnetic ﬁeld and subject to, on the scale of the charge diameter, slowly varying external potentials. We prove that in the adiabatic limit the motion of the charged particle is governed by a eﬀective Hamiltonian. To next order one has to add as a small correction the relativistically covariant radiation reaction. This third order equation has a repulsive center manifold and the true solution is well approximated by a trajectory on the center manifold. We also prove that in the adiabatic limit the ﬁelds are derived from the Li´enard–Wiechert potentials of a moving point charge and that the radiated energy is given by Larmor’s formula.

I Introduction The Maxwell–Lorentz equations describe the motion of a charged particle coupled to its own electromagnetic ﬁeld and possibly subject to external ﬁelds. We assume the particle to be relativistic, characterized by its bare mass mb and charge e. To have the dynamics well deﬁned one has to smoothen out the charge. For simplicity we adopt here the Abraham model of a rigid, velocity independent charge distribution [1, 15]. Taking units of space–time such that the speed of light c = 1 and unit of charge such that ε0 = 1 = µ0 , Maxwell’s equations read ∂ B(x, t) = −∇ ∧ E(x, t), ∂t

∂ E(x, t) = ∇ ∧ B(x, t) − eρ(x − q(t))v(t) ∂t

(1)

together with the constraints ∇ · E(x, t) = eρ(x − q(t)),

∇ · B(x, t) = 0.

(2)

They are coupled to the Lorentz equation, d mb γ(v(t))v(t) = m0 (v(t))v(t) ˙ dt =e

d3 x ρ(x − q(t)) (E(x, t) + Eex (x)) + v(t) ∧ (B(x, t) + Bex (x)) , −1/2

(3)

where v(t) = q(t). ˙ Here x ∈ IR3 , t ∈ IR, γ(v) = (1 − v 2 ) , E, B are the electric and magnetic ﬁelds, and q(t) ∈ IR3 and v(t) ∈ V = {v : |v| < 1} are the position and the velocity of the charged particle, respectively. The velocity dependent (bare)

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mass m0 (v) is deﬁned as a 3 × 3 matrix through m0 (v)v˙ = mb (γ v˙ + γ 3 (v · v)v). ˙ The prescribed external ﬁelds, Eex , Bex , are assumed to be time-independent and are derived from external potentials as Eex = −∇φex ,

Bex = ∇ ∧ Aex .

These potentials can be fairly arbitrary, precise conditions will be stated below in (P ). Finally, ρ is the charge distribution. It is normalized, d3 xρ(x) = 1, (4) and we require ρ(x) to be smooth, radial, and compactly supported, i.e., ρ ∈ C0∞ (IR3 ) ,

ρ(x) = ρr (|x|) ,

ρ(x) = 0 for |x| ≥ Rρ .

(C)

For an electron, Rρ would be the classical electron radius which is deﬁned by equating the rest energy with the electrostatic energy, i.e. mc2 = e2 /Rρ . In numbers one has Rρ = 3 × 10−13 cm, even smaller by a factor of 1836 for protons. External ﬁelds which can be generated in the laboratory vary slowly on that scale. Thus it seems natural, and physically mandatory, to study the motion of the charged particle under the condition that the external potentials φex , Aex vary slowly on the scale of Rρ . This is the adiabatic limit. Formally we introduce a small dimensionless parameter ε, ε 1, and require the potentials to be slowly varying as φex (εx),

Aex (εx).

(5)

In the Lorentz equation (3) the external ﬁelds then have to be substituted by εEex (εx), εBex (εx). Let us ﬁrst develop a qualitative picture for the motion of the charged particle, many details of which will be proved below. On a time scale O(Rρ ), i.e., O(Rρ /c) with the speed of light c = 1, the forces are of order ε. Thus the particle moves on an essentially straight line trajectory with constant velocity. The electromagnetic ﬁelds adapt themselves correspondingly and in approximation one has a solution to (1) - (3) with Eex = 0 = Bex travelling at constant velocity v. In analogy to nonlinear wave equations we call this solution a charged soliton. Depending on v it carries an energy Es (v) and a momentum Ps (v), both of which are the sum of a mechanical and a ﬁeld contribution. The interaction with the ﬁeld modiﬁes the bare energy–momentum relation, E(p) = (m2b + p2 )1/2 , to an eﬀective energy– momentum relation Eeﬀ (p) = Es (v(p)), where v(Ps ) is the inverse function to Ps (v). This relation will be computed in Section 2. We deﬁne the microscopic space–time scale by q = O(Rρ ), t = O(Rρ ). On the macroscopic space–time scale, q = O(ε−1 Rρ ), t = O(ε−1 Rρ ), the charge responds to the external forces through the eﬀective energy–momentum relation. Thus on this scale we have a Hamiltonian motion with a velocity–dependent

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mass m(v) = m0 (v)+ mf (v), where mf (v) is the contribution from the ﬁeld carried along with the charge. In a relativistic theory one would expect that mf (v) = (me /mb )m0 (v),

(6)

where me is the electrostatic energy of the charge distribution, 1 1 2 ρ(k)| k−2 . me = e2 d3 xd3 y (4π|x − y|)−1 ρ(x)ρ(y) = e2 d3 k |ˆ 2 2 However the Abraham model is nonrelativistic and therefore mf (v) has a velocity dependence diﬀerent from (6), cf. (33) below for the explicit expression. The eﬀective Hamiltonian motion is then governed by d r(t) = u(t), dt

(m0 (u) + mf (u))

d (7) u(t) = e Eex (r(t)) + u(t) ∧ Bex (r(t)) dt

with (r, u) the position and velocity of the charge on the macroscopic scale. Accelerated charges radiate at a rate roughly proportional to v(t) ˙ 2 , which cort 2 ∼ ˙ in energy over the time span t. We have |v(t)| ˙ responds to a loss 0 ds v(s) = O(ε) −1 and therefore on the macroscopic scale, t = O(ε ), the energy loss is proportional to ε. Thus there must be dissipative corrections of order ε to the Hamiltonian motion (7). The structure of this radiation reaction was ﬁrst guessed by Abraham [1, p. 123, Eq. (85)] in 1904 and derived more systematically in a famous paper by Dirac [4]. In our notation it reads (m0 (u) + mf (u))u˙ = e (Eex (r) + u ∧ Bex (r)) 2 + ε(e2 /6π) γ 4 (u · u ¨)u + 3γ 6 (u · u) ˙ u + 3γ 4 (u · u) ˙ u˙ + γ 2 u ¨

(8)

with u = r. ˙ The radiative reaction contains terms proportional to u ¨(t). Thus (8) has undesirable solutions, the runaway solutions, whose acceleration increases without limit. Dirac [4], reemphasized by Haag [7], proposed to eliminate such unphysical solutions by adding the asymptotic condition u(t) ˙ → 0 as t → ∞. Of course, one would hope that the asymptotic condition is in fact a consequence of the equations of motion. More seriously, the instability of the solution ﬂow to (8) renders it rather impractical in the sense that, unless one has an explicit solution, it is diﬃcult to sort physical solutions from runaways. As recognized in [5, 6, 11], equations of the form (8) have a repulsive center manifold Iε . For initial conditions away from Iε the solution grows without bound, whereas on Iε the acceleration u(t) ˙ remains bounded and the velocity u(t) stays bounded away from 1. The main goal of our paper is to unravel an identical structure for the (nonrelativistic) Lorentz–Dirac equation (8) and to establish that for small ε the true solution of the Maxwell–Lorentz equations (1)-(3) is tracked with high precision by a solution of (8) on its center manifold. We will prove

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that the energy of the true solution and of the comparison solution on the center manifold of (8) diﬀer only by order 2 on the macroscopic scale, where energy refers to the eﬀective hamiltonian for (7). While this singles out the dissipative correction to (8), it leaves open whether the usual distance between these two solutions is of order 2 . Using results from geometric singular perturbation theory, the center manifold can be expanded in powers of ε. Up to order ε one obtains a second order diﬀerential equation, cf. Eq. (24) below, which has only stable solutions and describes the physically correct long–time behavior. Thus in the adiabatic limit the motion of the charged particle is governed by an eﬀective second order diﬀerential equation and the associated electromagnetic ﬁelds are the Li´enard–Wiechert ﬁelds of a point charge. We outline the contents of our paper. In Section II we discuss the charged solitons and their energy and momentum. In Section III we state precise assumptions on the external potentials, explain the structure of the center manifold, and list our main results. Their proof uses a strategy which is similar to [10] but more involved. Thus one has to show that on the microscopic....time scale the derivatives ... are bounded as |q(t)| ˙ < 1, |¨ q (t)| ≤ Cε, | q (t)| ≤ Cε2 , | q (t)| ≤ Cε3 uniformly in t. This is the most technical part of our paper and deferred to an appendix, Section VIII. With these a priori bounds we are allowed to Taylor expand the self–force (Section IV) up to a uniformly bounded error of order ε3 on the microscopic scale. The center manifold is discussed in Section V. Finally in Sections VI and VII we determine the adiabatic limit of the ﬁelds and the energy radiated per unit time, which is in agreement with Larmor’s formula, of course. There is one result that came as a surprise to us and thus seems worth to be pointed out. As explained, since the Abraham model of a rigid charge is nonrelativistic, also the eﬀective energy–momentum relation Eeﬀ (p) is nonrelativistic. However, the radiation reaction is relativistic and agrees with the result formally derived by Dirac. In covariant notation the radiation reaction term in the second line of (8) reads, reintroducing the velocity of light, c, (e2 /6πc3 )(¨ uµ − c−2 u˙ λ u˙ λ uµ ), compare with [12]. The radiation reaction apparently only knows about the total charge, but no ﬁner details of the charge distribution.

II Energy-momentum relation for a charged soliton Throughout this section we consider the Maxwell–Lorentz equations with Eex = 0 = Bex . Since then the equations are invariant under spatial translations, one expects to have solutions travelling at constant velocity. In fact, for every v ∈ V there is a unique solution given by Sq,v (t) = (Ev (x − q − vt), Bv (x − q − vt), q + vt, v),

q ∈ IR3 , |v| < 1, t ∈ IR, (9)

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where Ev (x) = −∇φv (x) + v(v · ∇)φv (x),

Bv (x) = −v ∧ ∇φv (x),

(10)

ρ(k)/[k2 − (v · k)2 ], see [10]; the hat denotes Fourier transform. We and φˆv (k) = eˆ call Sq,v = Sq,v (0) the charged soliton with center q ∈ IR3 and velocity v ∈ V. If there are no external forces, the set of solitons is invariant under our dynamics. The Maxwell–Lorentz equations can be rewritten in Hamiltonian form. Since the system is translation invariant, there is a conserved energy and momentum. In our variables these are given by 1 H0 (E, B, q, v) = mb γ(v) + d3 x [E 2 (x) + B 2 (x)], 2 P0 (E, B, q, v) = mb γ(v)v + d3 x [E(x) ∧ B(x)]. In particular, the charged soliton Sq,v carries the energy Es (v) = H0 (Sq,v ) and the momentum Ps (v) = P0 (Sq,v ). They are explicitly given by 1 1 + |v| Es (v) = mb γ(v) + me log −1 , (11) |v| 1 − |v| and −2

Ps (v) = mb γ(v)v + me |v|

1 + v2 1 + |v| log − 1 v. 2|v| 1 − |v|

(12)

We note that the soliton Sq,v is also characterized as the minimizer of H0 at ﬁxed total momentum P0 = Ps (v) under the constraints ∇ · E = eρ(· − q) and ∇ · B = 0. The map Ps : V → IR3 is one–to–one with inverse v(Ps ). Thus the energy– momentum relation of the charged solitons is deﬁned by Eeﬀ (p) = Es (v(p)).

(13)

Eeﬀ (p) determines the eﬀective equations of motion (7) on the macroscopic scale. In the canonical variables q, p they are of Hamiltonian form with Hamiltonian H(q, p) = Eeﬀ (p − eAex (q)) + eφex (q).

(14)

To check this, one only has to recall that m(v) = (dPs /dv)(v) and v(dPs /dv) = (dEs /dv), as can be veriﬁed from the explicit expressions (11) and (12). If one expands (11), (12) for small velocities, then 1 Es (v) = mb + 2me + (mb + (4me /3)) v 2 + O(v 4 ), 2 Ps (v) = (mb + (4me /3))v + O(|v|3 ).

(15)

Therefore in the small velocity limit the eﬀective mass of the charged particle equals mb + 4me /3.

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III Main results We ﬁrst transform (1)–(3) with external potentials (5) to the macroscopic space– time scale through t = ε−1 t , x = ε−1 x , E(x, t) = ε3/2 E (x , t ), B(x, t) = ε3/2 B (x , t ), and q(t) = ε−1 q (t ). The amplitudes of the transformed ﬁelds have been chosen such that the ﬁeld energy remains invariant. There is little risk in omitting the primes, for simplicity. However, we indicate explicitly the ε–dependence of q (t ) and denote it by q ε (t) with velocity v ε (t) = q˙ε (t). Then √ ∂ ∂ B(x, t) = −∇ ∧ E(x, t), E(x, t) = ∇ ∧ B(x, t) − ε eρε (x − q ε (t))v ε (t), ∂t ∂t q˙ε (t) = v ε (t), m0 (v ε (t))v˙ ε (t) = [e(Eex + v ε (t) ∧ Bex ) ∗ ρε ](q ε (t)) √ + ε e d3 x ρε (x − q ε (t))[E(x, t) + v ε (t) ∧ B(x, t)] (16) with the constraints ∇ · E(x, t) =

√ ε eρε (x − q ε (t)),

∇ · B(x, t) = 0,

and ρε (x) = ε−3 ρ(ε−1 x). The energy function on the macroscopic scale is deﬁned by 1 Hmac (E, B, q, v) = mb γ(v) + [eφex ∗ ρε ](q) + d3 x [E 2 (x) + B 2 (x)]. 2 For the external potentials we require φex ∈ C 3 (IR3 ), Aex ∈ C 3 (IR3 , IR3 ), and |∇k φex |∞ ≤ C (k = 0, 1, 2, 3),

|∇k Aex |∞ ≤ C, (k = 1, 2, 3).

(P )

We could equally well treat the conﬁning case, i.e., φex (q) → ∞ as |q| → ∞, and |∇k φex |∞ ≤ C only for k = 1, 2, 3; cf. [11]. We note that in the rescaled coordinates, the Maxwell-Lorentz equations retain their form. Only the support√of the charge distribution now shrinks propotional to and the total charge is εe. Of course, on the microscopic scale ε is a parameter appearing in the external potentials. The physical charge and electrostatic energy of the charge distribution remain ﬁxed. In the more traditional point charge limit the charge distribution tends to a δ-function, which means that the electrostatic energy diverges. The particle and its comoving ﬁeld can no longer respond to external forces. The only way to compensate is the to let the bare charge tend to −∞ [4], which however results in instabilities [2]. Next, (16) has to be supplemented with initial data (E 0 , B 0 , q 0 , v 0 ). If they have ﬁnite energy, then (16) has a unique (weak) solution global in time and the

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energy does not change along the solution trajectory. We refer to [10] and to [2], where also the smoothness of solutions in dependence on the smoothness of the initial data is discussed. For the adiabatic limit it is natural to assume initially a charged soliton centered at q 0 with velocity v 0 . With some extra eﬀort we could deal with a more general class of initial data, but this issue will not be persued further. To deﬁne the initial data for (16) we only have to transform (10) to the macroscopic scale, which yields q ε (0) = q 0 , v ε (0) = v 0 , E 0 (x) = −∇φεv0 (x − q 0 ) + (v 0 · ∇φεv0 (x − q 0 ))v 0 , B 0 (x) = −v 0 ∧ ∇φεv0 (x − q 0 ),

(17)

where φˆεv (k) =

√ −1 εe ρˆ(εk)[k2 − (k · v)2 ] .

Our ﬁrst main result is an a priori estimate on the derivatives of q ε (t) uniformly in ε and t. It is proved in Appendix 8 and states Lemma 3.1 For any solution of (16) with initial conditions given by (17) we have sup |v ε (t)| ≤ v¯ < 1.

t∈IR

In addition, there exist constants C > 0 and δ0 > 0 such that ...ε

v ε (t)| + sup | v (t)| ≤ C sup |v˙ ε (t)| + sup |¨ t∈IR

t∈IR

t∈IR

provided e ≤ δ0 . Here v¯, C, and δ0 depend only on bounds for the initial data, but not on ε. With this result we are allowed to Taylor expand q ε (t) and v ε (t) in the self–force, √ (18) Fsε (t) = ε e d3 x ρε (x − q ε (t))[E(x, t) + v ε (t) ∧ B(x, t)], where E(x, t) and B(x, t) are to be inserted from the solution of the inhomogeneous Maxwell equations. Fsε (t) will have then a contribution from the initial ﬁelds, which vanishes for t ≥ εtρ := 2εRρ /(1−¯ v ) using the compact support of ρ. The remaining contribution comes from the retarded ﬁelds and has the form Fsε (t)

= −mf (v ε (t))v˙ ε (t) + ε(e2 /6π) γ 4 (v ε (t) · v¨ε (t))v ε (t) + 3γ 6 (v ε (t) · v˙ ε (t))2 v ε (t) +3γ 4 (v ε (t) · v˙ ε (t))v˙ ε (t) + γ 2 v¨ε (t) + O(ε2 )

(19)

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−1/2

for small ε, where γ = (1 − v ε (t)2 ) and the 3 × 3-matrix mf (v) is deﬁned in (33) below. Because m(v) = (dPs /dv)(v) = m0 (v) + mf (v), we obtain from (16) and (19) m(v ε )v˙ ε

= [e(Eex + v ε ∧ Bex ) ∗ ρε ](q ε ) + ε(e2 /6π) γ 4 (v ε · v¨ε )v ε + 3γ 6 (v ε · v˙ ε )2 v ε +3γ 4 (v ε · v˙ ε )v˙ ε + γ 2 v¨ε + ε2 f ε (t).

(20)

Since ρε → δ0 , the natural comparison dynamics to (20) is given by m(u)u˙ = e(Eex (r) + u ∧ Bex (r)) 2 ¨)u + 3γ 6 (u · u) ˙ u + 3γ 4 (u · u) ˙ u˙ + γ 2 u ¨ , (21) + ε(e2 /6π) γ 4 (u · u −1/2

with γ = (1 − u2 ) . Eqn. (21) can be rewritten as a singular perturbation problem. For this, we introduce x = (r, u) ∈ IR3 × V , y = u˙ ∈ IR3 , f (x, y) = (x2 , y) ∈ V × IR3 , and g(x, y, ε) = a(x2 )−1 (6π/e2 )[m(x2 )y − Fex (x)] − ε [3γ 6 (x2 · y)2 x2 + 3γ 4 (x2 · y)y] , −1/2

, Fex (x) = e(Eex (r) + u ∧ Bex (r)), and a(u)z = γ 4 (u · z)u + where γ = (1 − x22 ) 3 2 γ z, z ∈ IR , with matrix inverse a(u)−1 z = γ −2 [z − (u · z)u]. Thus (21) becomes x˙ = f (x, y),

εy˙ = g(x, y, ε).

(22)

For ε = 0, (22) has the invariant manifold ˙ : m(u)u˙ = Fex (r, u)} I0 = {(x, y) : g(x, y, 0) = 0} = {(r, u, u) = {(x, h(x)) : x ∈ IR3 × V},

(23)

where h(x) = m(x2 )−1 Fex (x). To ﬁnd a global invariant manifold Iε for (22) with ε > 0 small we apply the geometric singular perturbation theory, cf. [13, 8], which however requires to modify the functions a(u) and m(u) for |u| near 1, due to the singularity at |u| = 1; see [11] for a similar approach. In fact, for the motion on Iε such modiﬁcation will play no role according to Lemma 3.1. Let us explain in more detail the structure of the solution ﬂow to (21). In ¯ v ) > 0 satisfying some estimates; hence δ¯ is (38) below, we will ﬁx a small δ¯ = δ(¯ determined only through bounds for the initial data. We deﬁne ¯ K1−δ¯ = IR3 × {u ∈ IR3 : |u| ≤ 1 − δ}

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and continue a(u) and m(u) with their values at |u| = 1 − δ¯ to the missing inﬁnite strip 1 − δ¯ < |u| < 1. With these modiﬁcations the basic assumptions (I), (II) from [13, p. 45] are satisﬁed, since I0 is normally hyperbolic (here: repulsive) at a ¯ > 0 and a rate independent of ε; cf. Lemma 5.1. Therefore there exist ε0 = ε0 (δ) 3 3 1 C -function h(x, ε) = hε (x) : IR × V×]0, ε0 ] → IR such that for ε ≤ ε0 , Iε = {(x, hε (x)) : x ∈ IR3 × V} is forward invariant for the ﬂow of (21) with the modiﬁed functions a(u) and m(u). Since the modiﬁed equation agrees with (21) in the interior of K1−δ¯, it follows that Iε is locally invariant for the ﬂow of (21), i.e. the solution of the modiﬁed equation is the solution to the original equation as long as it does not reach the boundary ¯ The ﬂow for ε = 0 is governed set {(x, hε (x)) = (r, u, hε (r, u)) : |u| = 1 − δ}. by x˙ = f (x, h(x)), which agrees with (7). It then perturbs to x˙ = f (x, hε (x)) for ε ≤ ε0 , i.e. r˙ = u,

−1

u˙ = hε (r, u) = m(u)

Fex (r, u) + h1,ε (r, u),

(24)

cf. [13, Thm. 2.1, Thm. 2.9], where ¯ and |h1,ε (r, u)| ≤ c(δ)ε ¯ |hε (r, u)| ≤ c(δ)

(25)

for (r, u) ∈ IR3 × V and ε ∈]0, ε0 ]. The function h1,ε (r, u) can be expanded in ε. Then, up to errors of order ε2 , the motion on the center manifold is given by r˙ = u,

m(u)u˙ = e(Eex (r) + u ∧ Bex (r)) + ε(e2 /6π) a(u)[u · ∇r + h · ∇u + 3γ 4 (u · h)]h

(26)

with h = h(r, u) = em(u)−1 (Eex (r) + u ∧ Bex (r)). Note that the error in passing from (21) to (26) is of the same order as the one neglected already in (21). Thus Eqn. (26) should be regarded as the true eﬀective dynamics we have been searching for, including the correction due to radiation. We emphasize that (26) has no runaway solutions and can easily be implemented numerically. Having then established all the assertions of Sections IV and V, we obtain the following main result. Theorem 3.2 Assume (P ) for the potentials, and let the initial data (E 0 (x), B 0 (x), q 0 , p0 ) be given by (17). Let the charge e and let ε ≤ ε1 be suﬃciently small, and introduce the center manifolds Iε for the comparison dynamics (21) as explained above. At time εtρ = 2εRρ /(1 − v¯) we match the initial values, r(εtρ ) = q ε (εtρ ), u(εtρ ) = v ε (εtρ ), for the motion on the center manifold, i.e., the initial data for the comparison dynamics are (q ε (εtρ ), v ε (εtρ ), hε (q ε (εtρ ), v ε (εtρ ))) ∈ Iε .

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Then for every τ > 0 there exists c(τ ) > 0 such that for all t ∈ [εtρ , εtρ + τ ] we have the bounds |q ε (t) − r(t)| ≤ c(τ )ε, and

|v ε (t) − u(t)| ≤ c(τ )ε,

|v˙ ε (t) − u(t)| ˙ ≤ c(τ )ε,

[Es (v ε (t)) + eφex (q ε (t))] − [Es (u(t)) + eφex (r(t))] ≤ c(τ )ε2

with Es from (11). Furthermore for t = O(ε), i.e. t ∈ [εtρ , εtρ + ετ ], the bound |q ε (t) − r(t)| ≤ c(τ )ε3

and

|v ε (t) − u(t)| ≤ c(τ )ε2

is satisﬁed. Proof . We refer to [11, Section 5]. There the analogue of Theorem 3.2 is proved for the case of a scalar wave ﬁeld, starting from a diﬀerential equation which has the same qualitative properties as (20). ✷ The smallness of the charge e, needed in the proof of Lemma 3.1, is presumably an artifact. However, the upper bound for e is determined only through bounds for the data in (17) and does not depend on other properties of the solutions. As a further weakness of Theorem 3.2, the radiation reaction correction of order ε in the comparison dynamics is required only in the short time bound and in the energy estimate. One would hope that Theorem 3.2 can be improved in the sense that the eﬀective dynamics is ε2 close to the true solution over some compact time interval. Up to now we only discussed the adiabatic limit for the motion of the particle. Since lim q ε (t) = r(t)

ε→0

and

lim v ε (t) = u(t)

ε→0

exist by Theorem 3.2, we can in addition infer the adiabatic limit for the electric and magnetic ﬁelds. We introduce tret , the retarded time depending on x and t, as the unique solution of tret = t − |x − r(tret )| and denote n ˆ (x, t) = (x − r(tret ))/|x − r(tret )|. Theorem 3.3 For the ﬁelds E(t, x), B(t, x) from (16) we have for x = r(t) the pointwise limits 1 ¯ t) and lim √1 B(x, t) = B(x, ¯ t), lim √ E(x, t) = E(x, ε→0 ε→0 ε ε where e ¯ t) = E(x, ˆ (x, t) · u(tret )]−3 4π|x−r(tret )| [1 − n

(1−u2 (tret )) ˙ ret )) + |x−r(tret )| [ˆ n(x, t) − u(tret )] (27) n ˆ (x, t) ∧ ([ˆ n(x, t) − u(tret )] ∧ u(t

¯ t) = n ¯ t). E, ¯ B ¯ are the Li´enard-Wiechert ﬁelds of a point and B(x, ˆ (x, t) ∧ E(x, charge [14].

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IV Representation of the self-force In this section we show that (19) holds. The calculations will be carried out on the original time scale of (1)-(3) to keep notation simpler. On this scale the self-force (18) reads Fs (t) = e d3 x ρ(x − q(t))[E(x, t) + v(t) ∧ B(x, t)]. (28) From Maxwell’s equations in (1), (2), cf. [10] and the arguments in the proof of Lemma 8.1 below, the ﬁelds are a sum of the initial and the retarded ﬁelds, E(x, t) = E(0) (x, t) + E(r) (x, t)

and B(x, t) = B(0) (x, t) + B(r) (x, t),

(29)

where ˆ(0) (k, t) = cos |k|t E(k, ˆ 0) − i sin |k|t k ∧ B(k, ˆ 0), E |k| ˆ 0), ˆ 0) + i sin |k|t k ∧ E(k, ˆ(0) (k, t) = cos |k|t B(k, B |k| t t sin |k|(t − s) ˆ(r) (k, t) = − E eˆ ρ(k, s)k, (30) ds cos |k|(t − s) ˆj(k, s) + i ds |k| 0 0 t sin |k|(t − s) ˆ(r) (k, t) = −i B ds k ∧ ˆj(k, s), (31) |k| 0 ρ(k)v(t)eik·q(t) . According to (29), we with ρˆ(k, t) = ρˆ(k)eik·q(t) and ˆj(k, t) = eˆ decompose Fs (t) from (28) as Fs (t) = e d3 x ρ(x − q(t))[E(0) (x, t) + v(t) ∧ B(0) (x, t)] + e d3 x ρ(x − q(t))[E(r) (x, t) + v(t) ∧ B(r) (x, t)] = F(0) (t) + F(r) (t). The following lemma states that only F(r) (t) contributes to the self-force. Lemma 4.1 For t ≥ tρ = 2Rρ /(1 − v¯) we have F(0) (t) = 0. Proof . Let G(t) denote the group generated by the free wave equation in D1,2 (IR3 ) ⊕L2 (IR3 ), with D1,2 (IR3 ) = {φ ∈ L2 (IR3 ) : ∇φ ∈ L2 (IR3 , IR3 )}. Then by (17) and Fourier transform,

E(0) (x, t) E˙ (0) (x, t)

= −e

0

−∞

ds [G(t − s)ΦE (· − q 0 − v 0 s)](x) ΦE (x) =

ρ(x)v 0 ∇ρ(x)

,

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Ann. Henri Poincar´e

and an analogous representation of (B(0) (x, t), B˙ (0) (x, t)), with ΦE replaced through ΦB (x) = (0, v 0 ∧ ∇ρ(x)); observe E(0) (x, 0) = E 0 (x) and E˙ (0) (x, 0) = ∇ ∧ B 0 (x) as well as B(0) (x, 0) = B 0 (x) and B˙ (0) (x, 0) = −∇ ∧ E 0 (x). Hence both ΦE and ΦB have their supports in B Rρ (0). Therefore by Kirchhoﬀ’s formula applied to [. . . ](x) it can be seen that E(0) (x, t) = 0 = B(0) (x, t) for |x − q 0 | ≤ t − Rρ . Substituting into the deﬁnition of F(0) (t) yields the claim, since |x − q(t)| ≤ Rρ and |x − q 0 | ≥ t − Rρ implies t ≤ tρ by Lemma 3.1. ✷

Lemma 4.2 For t ≥ tρ , = −mf (v)v˙ + (e2 /6π) γ 4 (v · v¨)v + 3γ 6 (v · v) ˙ 2v + 3γ 4 (v · v) ˙ v˙ + γ 2 v¨ + O(ε3 )

F(r) (t)

(32)

with the 3 × 3-matrix

2 γ 3 + v2 2 mf (v)z = me (3 − v ) − arth|v| (z · v)v v4 |v|5

1 + v2 arth|v| z , + − |v|−2 + |v|3 denoting arth|v| =

1 2

(33)

1+|v| log 1−|v| .

Proof . Using Lemma 3.1 we are allowed to expand as v(s) eik·[q(s)−q(t)] v(s)

1 = v − vτ ˙ + v¨τ 2 + O(ε3 ) 2 1 −i(k·v)τ ˙ + v¨ τ 2 = e v − vτ ˙ + i(k · v)v 2

i 1 3 2 4 ˙ v˙ τ − (k · v) ˙ vτ + O(ε3 ), − (k · v¨)v + 3(k · v) 6 8

with τ = t − s and v = v(t), etc. Introducing Ip =

dτ 0

Jp = 0

T

T

sin(|k|τ ) −i(k·v)τ p e τ , |k|

dτ cos(|k|τ )e−i(k·v)τ τ p ,

p = 0, . . . , 4,

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we obtain in Fourier transformed form by the deﬁnition of F(r) (t) and by (30), (31), 1 2 2 3 F(r) (t) = − e ˙ + v¨ J2 − ρ(k)| vJ0 − vJ ˙ 1 + i(k · v)v d k |ˆ 2

i 1 (k · v¨)v + 3(k · v) ˙ v˙ J3 − (k · v) ˙ 2 vJ4 6 8 ρ(k)|2 i (1 − v 2 )k + (k · v)v I0 + i (v · v)k ˙ − (k · v)v˙ I1 + e2 d3 k |ˆ 1 2 ˙ − (k · v)(k · v)v ˙ − i(v · v¨)k + i(k · v)¨ v I2 (v − 1)(k · v)k 2 1 + (1 − v 2 )(k · v¨)k + (k · v)(k · v¨)v − 3(v · v)(k ˙ · v)k ˙ 6 +

+3(k · v)(k · v) ˙ v˙ I3

i 2 + (k · v) ˙ (v 2 − 1)k − (k · v)v I4 + O(ε3 ). 8

(34)

This formula holds for arbitrary t, T ≥ tρ , since analogously to [9, Lemma 5.1] or [10, Lemma 3.2] it may be shown that for such t, T due to the compact support of t t ρ the 0 ds(. . . )-integrals can be changed to t−T ds(. . . )-integrals when (30) and (31) are inserted into F(r) (t). To evaluate the terms in (34) as T → ∞, we abbreviate the terms containing Jp as Jp and Ip as Ip , respectively. We ﬁrst consider the Ip and then turn to the Jp . ρ(k)|2 kI0 → 0 as T → ∞ by [9, Appendix A], we have I0 -limit: Since d3 k |ˆ I0 → 0. I1 -limit: As a consequence of the relation ∇v Ip = −ikIp+1 we obtain

2 2 3 3 ˙ v ρ(k)| I0 + v˙ (v · ∇v ) d k |ˆ ρ(k)| I0 . d k |ˆ I1 = −(v · v)∇ Then we may evaluate as T → ∞ e2 d3 k |ˆ ρ(k)|2 I0 → e2 d3 k hence

e I1 → 2me 2

|ˆ ρ(k)|2 = 2me |v|−1 arth|v|, k2 − (k · v)2

γ2 −3 2 −1 − |v| arth|v| v + γ − |v| arth|v| v˙ . − (v · v) ˙ v2

I2 -limit: From [9, Appendix A] and [10, Lemma 3.3] we know 1 2 2 2 3 d k |ˆ ρ(k)| (k · v)kI ˙ 2 → −µ(v)v, ˙ d3 k |ˆ ρ(k)| I1 → γ 2 /4π, e 2

638

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as T → ∞. Here 2 γ4 γ −3 2 −5 µ(v)z = me − |v| arth|v| z + 4 (5v − 3) + 3|v| arth|v| (v · z)v v2 v for |v| < 1. Thus e2 I2 → γ −2 µ(v)v˙ + v (v · µ(v)v) ˙ + 12 e2 (v · v¨)∇v (γ 2 /4π) − 12 e2 v¨ (v · ∇v )(γ 2 /4π) 4 2 = me γv4 (−v 4 + 6v 2 − 3) + 3−v arth|v| (v · v)v ˙ 5 |v| −2 + |v|−2 − γ|v|3 arth|v| v˙ + (e2 /4π)γ 4 (v · v¨)v − v 2 v¨ . I3 -limit: Making use of the relation (ξ · ∇v )∇v I1 = −k(k · ξ)I3 , we arrive at

3 3 1 2 2 ∇v )∇v d k |ˆ ρ(k)| I1 − 6 v (v · ∇v )(¨ v · ∇v ) d k |ˆ ρ(k)| I1 I3 =

3 3 1 1 2 2 ˙ v˙ · ∇v )∇v d k |ˆ ρ(k)| I1 − 2 v˙ (v · ∇v )(v˙ · ∇v ) d k |ˆ ρ(k)| I1 + 2 (v · v)( 1 1 → ( 4π ) − 13 γ 4 (1 + 4γ 2 )(v · v¨) + 4γ 6 (v · v) ˙ 2 v − ( π1 )γ 6 v 2 (v · v) ˙ v˙ − ( 12π )γ 2 v¨.

− 16 γ −2 (¨ v·

I4 -limit: Observing (ξ · ∇v )2 ∇v I1 = ik(k · ξ)2 I4 it follows that I4

1 2 2 3 2 = − (1 − v )(v˙ · ∇v ) ∇v ρ(k)| I1 d k |ˆ 8

1 2 3 2 ρ(k)| I1 d k |ˆ − v (v · ∇v )(v˙ · ∇v ) 8 → −(1/4π) γ 6 v˙ 2 + 2γ 6 (v · v) ˙ 2 + 6γ 8 (v · v) ˙ 2 v − (1/2π)γ 4 (v · v) ˙ v. ˙

J0 -limit: As in [9, Appendix A] it may be shown through direct calculation that as T → ∞ |ˆ ρ(k)|2 ρ(k)|2 J0 → i d3 k 2 (k · v) = 0, (35) d3 k |ˆ k − (k · v)2 the latter due to the symmetry of ρ. Hence J0 → 0. J1 -limit: Analogously to (35) one ﬁnds here that

ρ(k)|2 J1 → − d3 k |ˆ

Therefore J1 → −2e−2 me γ 2 v. ˙

d3 k |ˆ ρ(k)|2

k2 + (k · v)2 = −2e−2 me γ 2 . [k2 − (k · v)2 ]2

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Adiabatic Limit for the Maxwell–Lorentz Equations

(J2 + J3 + J4 )-limit: By Lemma 4.3 below, T → ∞. From this we obtain

639

d3 k |ˆ ρ(k)|2 J2 → −γ 4 /2π as

J2 + J3 + J4

1 1 3 2 3 2 = d k |ˆ ρ(k)| J1 − v¨ ρ(k)| J2 v (v˙ · ∇v ) d k |ˆ 2 2

1 d3 k |ˆ ρ(k)|2 J2 v · ∇v ) − v (¨ 6

1 1 3 2 2 3 2 d k |ˆ ρ(k)| J2 − v (v˙ · ∇v ) ρ(k)| J2 d k |ˆ − v˙ (v˙ · ∇v ) 2 8 4 ˙ + (1/4π) γ 6 (v · v¨) + γ 6 v˙ 2 + 6γ 8 (v · v) ˙ 2 v → −2e−2 me γ 4 (v · v)v 3 ˙ v˙ + (1/4π)γ 4 v¨. +(1/π)γ 6 (v · v) ✷

Collecting all the terms together we get (32). It remains to show Lemma 4.3 We have ∞

2

dt t

ρ(k)|2 cos |k|t e−i(k·v)t = −γ 4 /2π. d3 k|ˆ

0

Proof . Since through integration by parts, Jp = and since

sin |k|T −i(k·v)T p T − pIp−1 − (v · ∇v )Ip−1 , e |k|

p ≥ 1,

ρ(k)|2 I1 → γ 2 /4π, we only need to prove d3 k |ˆ sin |k|T −i(k·v)T 2 d3 k |ˆ ρ(k)|2 T → 0, T → ∞, e |k|

for ﬁxed |v| < 1. After some transformations this expression may be written as T ρ(k)|2 3 |ˆ −3/2 T sin(|k|T ) sin(|k||v|T ) = (2π) d k d3 x ρ(x) |v| k2 |v| (36) d3 y ρ(y)ψ(x − y), ˆ where ψ(k) = k−2 sin(|k|T ) sin(|k||v|T ). We claim that the integral in (36) vanishes for T > 2Rρ /(1 − |v|). For this, according to (C) it suﬃces to show ψ(x) = 0 for |x| ≤ 2Rρ and T > 2Rρ /(1 − |v|). Explicitly we have through transformation to polar coordinates 4π ∞ dR sin(R|x|) sin(RT ) sin(R|v|T ). (2π)3/2 ψ(x) = |x| 0 R

640

M. Kunze and H. Spohn

With Si(x) = 0

a

x 0

sin t t

Ann. Henri Poincar´e

dt,

dR 1 sin(R|x|) sin(RT ) sin(R|v|T ) = Si(a[|x| − (1 − |v|)T ]) R 4 1 + Si(a[|x| + (1 − |v|)T ]) 4 1 − Si(a[|x| + (1 + |v|)T ]) 4 1 − Si(a[|x| − (1 + |v|)T ]), 4

and the latter tends to zero as a → ∞, for |x| ≤ 2Rρ and T > 2Rρ /(1 − |v|), so we are done. ✷

V Dynamics on the center manifold In this section we study the asymptotic properties of solutions to (21), or equivalently (22), on Iε . The results are quite similar to those in [11, Section 4], so we do not repeat all details. ¯ Since a(u) and m(u) have been changed to a constant outside {u : |u| ≤ 1−δ}, the following lemma shows that Iε is repulsive in normal direction at some εindependent rate. Lemma 5.1 The eigenvalues of Dy g(x, y, 0) = (6π/e2 )a(x2 )−1 m(x2 ) are bounded below by a positive constant, uniformly in x = (r, u) with r ∈ IR3 and |u| ≤ 1 − δ, for all given δ ∈]0, 1]. Proof . Analogously to [11, Lemma 4.1] one obtains the lower bound (6π/e2 )(1 − u2 ). ✷ By means of a suitable Lyapunov function we can deduce much information on the asymptotic behaviour of solutions. The following result is straightforward. Lemma 5.2 Let Gε (r, u, u) ˙ = Es (u) + eφex (r) − ε(e2 /6π) γ 4 (u · u), ˙ with Es from (11). Then d Gε (r, u, u) ˙ = −ε(e2 /6π) [γ 4 u˙ 2 + γ 6 (u · u) ˙ 2] dt along solution trajectories (r(t), u(t), u(t)) ˙ of (21).

(37)

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The term on the right-hand side of (37) accounts for the energy radiated to inﬁnity; see Section VII below. Since inf q∈IR3 φex (q) > −∞ by (P ), Lemma 5.2 implies through integration of (37) over [0, ∞[, analogously to [11, Thm. 4.3], the following Theorem 5.3 Let (P ) hold for the potentials, and let a global solution (r(t), u(t)) of (21) be given with supt≥0 |u(t)| ≤ u ¯(ε) < 1 and supt≥0 |u(t)| ˙ ≤ c(ε), for possibly ε-dependent constants u ¯(ε) and c(ε). Then u(t) ˙ → 0,

u ¨(t) → 0,

and

Fex (r(t), u(t)) → 0

t → ∞.

as

If also supt≥0 |r(t)| ≤ c(ε) is known, then additionally u(t) → 0

and

∇φex (r(t)) → 0

as

t → ∞.

The latter implies that the trajectory approaches a distinguished critical point of the electrostatic potential φex in case the set of critical points is discrete. In Section III we modiﬁed the functions a(u) and m(u) to be constant outside ¯ with some δ¯ = δ(¯ ¯ v ) > 0 small; we now give the precise deﬁnition {u : |u| ≤ 1 − δ} ¯ Let of δ. δ¯ := min{(1 − v¯)/2, (1 − s0 )/2},

(38)

where, as a consequence of Es (u) → ∞ for |u| → 1, r, u ˜) : Es (˜ u) + eφex (˜ r) ≤ c0 + 1} for some r ∈ IR3 < 1, s0 = sup |u| : (r, u) ∈ {(˜ with c0 chosen such that Kv¯ ⊂ {(˜ r, u ˜) : Es (˜ u) + eφex (˜ r) ≤ c0 }, as is possible, since φex is bounded. The next theorem summarizes properties of solutions of (21) resp. (22) on Iε , with the modiﬁed functions a(u) and m(u). Theorem 5.4 Assume (P ) for the potentials. Then there exists ε1 > 0 depending only on the bound v¯ from Lemma 3.1 such that for ε ∈]0, ε1 ] the following holds. (a) All solutions starting in points (r, u, hε (r, u)) ∈ Iε , |u| ≤ v¯, stay away from ¯ for all future times. the boundary {(r, u, hε (r, u)) : |u| = 1 − δ} (b) Solutions as in (a) exist globally, are solutions to (21), and remain on Iε for all future times. (c) Solutions as in (a) satisfy sup{|u(t)| : t ∈ IR, ε ∈]0, ε1 ]} ≤ 1 − 2δ¯ < 1 , and ¯ sup{|u(t)| ˙ : t ∈ IR, ε ∈]0, ε1 ]} + sup{|¨ u(t)| : t ∈ IR, ε ∈]0, ε1 ]} ≤ c(δ), and hence also u(t) ˙ → 0,

u ¨(t) → 0,

and

Fex (r(t), u(t)) → 0

as

t → ∞.

(39)

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Ann. Henri Poincar´e

Proof . Analogously to Theorem 4.4 and Corollary 4.5 in [11]; the constant ca has to be replaced by (e2 /6π) sup|u|≤¯v γ 4 . For the estimates in (39), it is important to use (25). ✷ Having seen that Iε is a global forward invariant manifold for (21), we now address the converse problem and look for conditions which guarantee that some solution of (21) already has to lie on Iε . The following theorem rests on the fact that according to the normal hyperbolicity of I0 , cf. Lemma 5.1, no trajectory can remain arbitrarily close to Iε without being contained in Iε . Theorem 5.5 Suppose a family (rε (t), uε (t)), ε ∈]0, ε2 ], of solutions to (21) is given such that sup{|uε (t)| : t ∈ IR, ε ∈]0, ε2 ]} ≤ u ¯ < 1. Additionally, assume that one of the following two conditions is satisﬁed. (a) The bound sup{|u˙ ε (t)| : t ∈ IR, ε ∈]0, ε2 ]} + sup{|¨ uε (t)| : t ∈ IR, ε ∈]0, ε2 ]} ≤ c1 holds, or (b) the asymptotic condition of Dirac and Haag, u˙ ε (t) → 0 as t → ∞, holds for each ε ∈]0, ε2 ]. Then the solutions have to lie on Iε , for suﬃciently small ε. ✷

Proof . We refer to [11, Propositions 4.6 and 4.7].

VI Adiabatic limit of the ﬁelds We prove Theorem 3.3. First we need to derive suitable representation formulas for ¨(0) = ∆E(0) , E(x, t) and B(x, t) from (16). Since E(0) solves the wave equation E 0 0 ˙ E(0) (x, 0) = E (x), E(0) (x, 0) = ∇ ∧ B (x), we ﬁnd from (17) that 1 √ E(0) (x, t) ε

= −e

0

−∞ 0

ds [G(t − s)ΦE,ε (· − q 0 − v 0 s)]1 (x)

1 = −e ds 4π(t − s)2 −∞ −e

0

ds −∞

1 4π(t − s)

|y−x|=(t−s)

|y−x|=(t−s)

d2 y [((y − x) · ∇)jε0 (y, s) +jε0 (y, s)]

d2 y ∇ρ0ε (y, s),

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with ΦE,ε (x) = (ρε (x)v 0 , ∇ρε (x)), jε0 (y, s) = ρε (y − q 0 − v 0 s)v 0 , and ρ0ε (y, s) = ρε (y − q 0 − v 0 s), see also the proof of Lemma 4.1; [. . . ]1 is the ﬁrst component of [. . . ]. Similarly, 0 1 1 √ B(0) (x, t) = e ds d2 y ∇ ∧ jε0 (y, s). 4π(t − s) |y−x|=(t−s) ε −∞ Extending q ε (t) resp. v ε (t) to times t ≤ 0 by q 0 + v 0 t resp. v 0 , we hence obtain through inverse Fourier transform of (30), with jε (y, s) = ρε (y − q ε (s))v ε (s) and ρε (y, s) = ρε (y − q ε (s)), 1 √ E(x, t) = ε t 1 = −e −∞ ds 4π(t−s) d2 y [((y − x) · ∇)jε (y, s) + jε (y, s)] 2 |y−x|=(t−s) t 1 −e −∞ ds 4π(t−s) d2 y ∇ρε (y, s) |y−x|=(t−s) 1 ε ε = −e d3 y 4π|x−y| 2 v (t − |x − y|) ((y − x) · ∇)ρε (y − q (t − |x − y|)) +ρε (y − q ε (t − |x − y|))v ε (t − |x − y|)

1 ε (40) + 4π|x−y| ∇ρε (y − q (t − |x − y|)) . In the same manner, 1 1 √ B(x, t) = −e d3 y v ε (t − |x − y|) ∧ ∇ρε (y − q ε (t − |x − y|)). (41) 4π|x − y| ε Now in both integrals (40) and (41) we introduce the transformation z(y) = −1 y − q ε (t − |x − y|) with det(dy/dz) = [1 − v ε (t − |x − y|) · (x − y)/|x − y|] and integrate by parts to remove the ∇’s from the ρε (z). Then we calculate the respective derivatives with respect to z and pass to the limit ε → 0, using ρε (z) = ε−3 ρ(ε−1 z) → δ0 in the sense of distributions, according to (4). Moreover, in the limit ε → 0 the equation 0 = y − r(t − |x − y|) for z = 0 is solved by ˆ (x, t) · u(tret )]−1 . Proceeding as iny = r(tret ); in particular, det(dy/dz) → [1 − n ¯ t) and dicated, a somewhat lengthy calculation then yields ε−1/2 E(x, t) → E(x, ¯ t) with E(x, ¯ t) from (27) and B(x, ¯ t) = n ¯ t). ✷ ε−1/2 B(x, t) → B(x, ˆ (x, t) ∧ E(x,

VII Radiated energy In this section we determine the energy radiated to inﬁnity. For this, let HR,qε (t) (t + R)

= mb γ(v ε (t + R)) + [eφex ∗ ρε ](q ε (t + R)) 1 d3 x [E 2 (x, t + R) + B 2 (x, t + R)] + 2 |x−qε (t)|≤R

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be the energy of (16) in a ball of radius R around the particle, at time t + R. Utilizing (16), this energy changes as d dt

HR,qε (t) (t + R) = 12 |x−qε (t)|=R d2 x E 2 (x, t + R) + B 2 (t, x + R) (n(x) · v ε (t)) + |x−qε (t)|=R d2 x E(x, t + R) · [n(x) ∧ B(x, t + R)]

=

R2 2

d2 ω E 2 (q ε (t) + Rω, t + R) + B 2 (q ε (t) + Rω, t + R) (ω · v ε (t)) |ω|=1 +R2 |ω|=1 d2 ω E(q ε (t) + Rω, t + R) · [ω ∧ B(q ε (t) + Rω, t + R)],

where n(x) is the outer normal. In this calculation we have used that |x−qε (t)|>R

d3 x ρε (x − q ε (t + R))E(x, t + R) · v ε (t + R) = 0

for R large, since |x − q ε (t + R)| ≥ |x − q ε (t)| − |q ε (t + R) − q ε (t)| ≥ (1 − v¯)R ≥ εRρ according to Lemma 3.1. Thus by Theorem 3.3, 1 d ε dt

HR,qε (t) (t + R) −→ 2 R2 ¯ 2 (r(t) + Rω, t + R) (ω · u(t)) ¯ (r(t) + Rω, t + R) + B d2 ω E 2 |ω|=1 2 ¯ ¯ +R d2 ω E(r(t) + Rω, t + R) · [ω ∧ B(r(t) + Rω, t + R)] |ω|=1

as ε → 0. In order to determine the amount of energy radiated from the particle to inﬁnity, we take the limit R → ∞ of the latter expression. For this, ﬁrst observe that with x = r(t) + Rω and τ = t + R it follows that tret = tret (x, τ ) = t, since ¯ τ ) = B(x, ¯ τ) then τ − tret = |x − r(tret )|. Hence also n ˆ (x, τ ) = ω, and n ˆ (x, τ ) ∧ E(x, thus yields 1 ¯2 ¯ 2 (x, τ )](ω · u(t)) + E(x, ¯ τ ) · [ω ∧ B(x, ¯ τ )] [E (x, τ ) + B 2 ¯ τ ))2 = −(1 − ω · u(t))E¯ 2 (x, τ ) + (1 − (ω · u(t))/2)(ω · E(x, = −(1 − ω · u(t))E¯ 2 (x, τ ) + O(R−4 ). In the last equation we have used that by (27) ¯ τ) = e n ˆ (x, τ ) · E(x,

(1 − u2 (tret )) [1 − n ˆ (x, t) · u(tret )]−2 = O(R−2 ). 4π|x − r(tret )|2

(42)

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Therefore (42) and (27) yield

1 d lim lim HR,qε (t) (t + R) R→∞ ε→0 ε dt e2 2 d2 ω − (1 − u2 (t))[1 − ω · u(t)]−5 (ω · u(t)) ˙ = − 2 16 π |ω|=1 +2 [1 − ω · u(t)]

−4

(ω · u(t))(u(t) ˙ · u(t)) ˙ + [1 − ω · u(t)]

u˙ (t)

−3 2

2 2 ˙ ] = −(e2 /6π)γ 6 [u˙ 2 (t) − (u(t) ∧ u(t)) ˙ ] = −(e2 /6π)[γ 4 u˙ 2 (t) + γ 6 (u(t) · u(t))

in agreement with (37) and Larmor’s formula [14]. We remark that the same result is obtained if the limits ε → 0 and R → ∞ are interchanged. This can be shown as in [11, Section 7].

VIII Appendix We prove Lemma 3.1, and for this it will be more convenient to switch back to the original time scale of (1)-(3). Then Lemma 3.1 reads as Lemma 3.1 For any solution of (3) we have sup |v(t)| ≤ v¯ < 1. t∈IR

In addition, there exist constants C > 0 and δ0 > 0 such that ...

sup |v(t)| ˙ ≤ Cε,

sup |¨ v (t)| ≤ Cε2 ,

sup | v (t)| ≤ Cε3

t∈IR

t∈IR

t∈IR

in case that e ≤ δ0 and that the trajectory starts on the soliton manifold. Here v¯, C, and δ0 do depend only on bounds for the initial data, but not on ε. We split the proof into four subsections.

VIII.1

Bounding |v(t)|

By (P ) we obtain

H(E 0 (x), B 0 (x), q 0 , v 0 ) ≥ mb γ(v(t)) + e

inf 3 φex (q) ,

q∈IR

since H(E(x, t), B(x, t), q(t), v(t)) = [φex (ε·) ∗ eρ](q(t)) + H0 (E(x, t), B(x, t), q(t), v(t)) is constant along solutions of (1)-(3); see [10]. Hence sup |v(t)| ≤ v¯ < 1,

(43)

t∈IR

with v¯ depending only on the data. For this estimate it was neither necessary that e be small nor that the trajectory started on the soliton manifold.

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Bounding |v(t)| ˙

By (3) and since Sq,v (t) is a solution with ε = 0, v˙ = em0 (v)−1 ε d3 x ρ(x − q)[Eex (εx) + v ∧ Bex (εx)] + d3 x ρ(x − q) [E(x) − Ev (x − q)] + v ∧ [B(x) − Bv (x − q)] ,

(44)

−1

where m0 (v) z = mb −1 γ −1 (z − (v · z)v), z ∈ IR3 , is the matrix inverse of m0 (v). To estimate the second integral term, deﬁne

E(x, t) − Ev(t) (x − q(t)) Z(x, t) = . (45) B(x, t) − Bv(t) (x − q(t)) Then Maxwell equations and the relations (v · ∇)Ev (x) = −∇ ∧ Bv (x) + eρ(x)v, (v · ∇)Bv (x) = ∇ ∧ Ev (x) imply

0 ∇∧ ˙ Z(t) = AZ(t) − f (t) , with A = −∇∧ 0 and f (x, t) =

(v(t) ˙ · ∇v )Ev (x − q(t)) (v(t) ˙ · ∇v )Bv (x − q(t))

.

(46)

The Maxwell operator A is the generator of a C 0 -group U (t), t ∈ IR, of isometries in L2 (IR3 )3 ⊕ L2 (IR3 )3 , cf. [3, p. 435; (H2)]. Therefore we have the mild solution representation t ds [U (t − s)f (·, s)](x) . (47) Z(x, t) = [U (t)Z(·, 0)](x) − 0

Since the solution is supposed to start on the soliton manifold, Z(0) = 0, so here the ﬁrst term drops out. In the following lemma we state an explicit representation for U (t − s)f (·, s). Lemma 8.1 For given f = (f1 , f2 ) with ∇ · f1 = 0 and ∇ · f2 = 0 we have for W (t, s, x) = (W1 (t, s, x), W2 (t, s, x)) = [U (t − s)f (·, s)](x) 1 W1 (t, s, x) = 4π(t − s)2 |y−x|=(t−s) d2 y (t − s)∇ ∧ f2 (y, s) + f1 (y, s) + ((y − x) · ∇)f1 (y, s) , (48) 1 W2 (t, s, x) = 4π(t − s)2 |y−x|=(t−s) d2 y − (t − s)∇ ∧ f1 (y, s) + f2 (y, s) + ((y − x) · ∇)f2 (y, s) .

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Proof . We follow [10, Section A.1] and introduce the complex C(x, t) = W1 (t, s, x) 3 ˙ ˆ i.e., C(k, ˆ t) = +iW2 (t, s, x) ∈ C . Then C˙ = −i ∇∧C, thus Cˆ = −k ∧ Cˆ =: m(k)C, 2 2 3 2 ˆ exp([t − s]m(k))C(k, s). Since m (k) = −k Id + k ⊗ k and m (k) = −k m(k), etc., the exponential can be evaluated explicitly as exp(tm(k)) = cos |k|t Id + |k|−2 (1 − ˆ s) = fˆ1 (k, s) + ifˆ2 (k, s) and cos |k|t)(k ⊗ k) + |k|−1 sin |k|t m(k). Because C(k, ˆ s) = (k · C(k, ˆ s))k = 0, whence ∇ · f1 = 0 = ∇ · f2 , we obtain (k ⊗ k)C(k, −1 ˆ ˆ C(k, t) = [cos |k|(t − s) Id + |k| sin |k|(t − s) m(k)]C(k, s). Taking into account that a general f (x) = |y−x|=τ d2 y g(y) has fˆ(k) = 4πτ |k|−1 sin |k|τ gˆ(k), and ∇ ∧ f (k) = −i k ∧ fˆ(k), this yields the claim. ✷

Corollary 8.2 In the setting of Lemma 8.1, if in addition |f1 (y, s)| + |f2 (y, s)| ≤ C

α(s) n (1 + |y − q(s)| )

and

(49) |∇f1 (y, s)| + |∇f2 (y, s)| ≤ C

α(s) (1 + |y − q(s)|n+1 )

,

then |W1 (t, s, x + q(t))| + |W2 (t, s, x + q(t))| ≤ C

α(s) , 1 + (t − s)n

|x| ≤ Rρ .

Proof . We deal with e.g. the ﬁrst component W1 (t, s, x + q(t)) for |x| ≤ Rρ . Then for y ∈ IR3 with |y − (x + q(t))| = (t − s) we ﬁnd |y − q(s)| ≥ (t − s) − Rρ − |q(t) − q(s)| ≥ (1 − v¯)(t − s) − Rρ by (43). Hence we can distinguish cases e.g. (t − s) ≥ 1 + 2(1 − v¯)−1 Rρ and (t − s) ≤ . . . , and then use (49) in (48) for (t − s) ≥ . . . to −1 ✷ get |W1 (t, s, x + q(t))| ≤ C(1 + (t − s)n ) α(s) for |x| ≤ Rρ . In particular, both the lemma and the corollary (with n = 2 and α(s) = e|v(s)|) ˙ apply to f deﬁned in (46). To see this, observe ∇ · Bv = 0, and ∇ · Ev = eρ is independent of v, hence ∇ · f1 = 0 = ∇ · f2 . Concerning (49), we have |∇v Ev (x)| + |∇v Bv (x)| ≤ C(|∇φv (x)| + |∇v ∇φv (x)|) and |∇v ∇Ev (x)| + |∇v ∇Bv (x)| ≤ C(|∇2 φv (x)|+|∇v ∇2 φv (x)|). Through tedious, but straightforward calculation one can derive the following estimates on the derivatives of ∇φv , cf. [11]. These will also be used frequently in the sequel. We have |∇φv (x)| + |∇v ∇φv (x)| + |∇2v ∇φv (x)| + |∇3v ∇φv (x)| ≤ Ce(1 + |x|)−2 , |∇2 φv (x)| + |∇v ∇2 φv (x)| + |∇2v ∇2 φv (x)| + |∇3v ∇2 φv (x)| ≤ Ce(1 + |x|)−3 , |∇3 φv (x)| + |∇v ∇3 φv (x)| + |∇2v ∇3 φv (x)| + |∇3v ∇3 φv (x)| ≤ Ce(1 + |x|)−4 , |∇4 φv (x)| + |∇v ∇4 φv (x)| + |∇2v ∇4 φv (x)| + |∇3v ∇4 φv (x)| ≤ Ce(1 + |x|)−5 , (50)

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for x ∈ IR3 and |v| ≤ v¯ < 1, with ∇ = ∇x . This implies that (49) holds as well. Consequently, since U (t) is unitary, it follows from (47) that t |v(s)| ˙ Z(· + q(t), t)L2 (B R (0))3 ⊕L2 (B R (0))3 ≤ Ce ds . (51) ρ ρ 1 + (t − s)2 0 Therefore by (44) and assumptions (P ) and (C), since |m0 (v)−1 | ≤ C,

t |v(s)| ˙ . ds |v(t)| ˙ ≤ Ce ε + e 1 + (t − s)2 0 Choosing e suﬃciently small, this implies ˙ ≤ Cε. sup |v(t)|

(52)

t∈IR

VIII.3

Bounding |¨ v (t)|

Let L(t)φ = (v(t) · ∇x )φ + ∂t φ for functions φ = φ(x, t). We also introduce the notation ϕ(x, t) = E(x, t) − Ev(t) (x − q(t)) ,

ψ(x, t) = B(x, t) − Bv(t) (x − q(t)) ,

i.e. Z = (ϕ, ψ), cf. (45). Using the explicit form of m0 (v)−1 it follows from (52) that d m0 (v(t))−1 ≤ C|v(t)| ˙ ≤ Cε. dt Thus we obtain by diﬀerentiation of (44), invoking also the last step and (P ),

˙ ∧ d3 x ρ(x)ψ(x + q(t), t) + |M (t)| , |¨ v (t)| ≤ C ε2 + v(t) where

M (t) =

d3 x ρ(x) (L(t)ϕ)(x + q(t)) + v(t) ∧ (L(t)ψ)(x + q(t)) .

(53)

Therefore by (52) and (51), |¨ v (t)| ≤ C ε2 + |M (t)| ;

(54)

whence it remains to derive a bound for M (t). For this, we introduce Σ(x, t) = (L(t)Z)(x, t) so that M (t) = d3 x ρ(x)[Σ1 (x + q(t), t) + v(t) ∧ Σ2 (x + q(t), t)].

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d Since generally dt [L(t)φ] = L(t)φ˙ + (v˙ · ∇)φ, it follows from (46) that Σ˙ = AΣ − L(t)f + (v˙ · ∇)Z. Therefore

Σ(x + q(t), t) = [U (t)Σ(0)](x + q(t)) + t ds [U (t − s)((v(s) ˙ · ∇)Z(·, s) − L(s)f (·, s))](x + q(t)),

(55)

0

the ﬁrst term being zero as a consequence of Z(0) = 0. We estimate both parts of the integral term separately. d What concerns the ﬁrst, observe dt (∇Z) = A(∇Z) − ∇f by (46), thus

t

ds [U (t − s)((v(s) ˙ · ∇)Z(·, s))](x + q(t)) = s t ds dτ [U (t − τ )((v(s) ˙ · ∇)f (·, τ ))](x + q(t)), − 0

0

(56)

0

since ∇Z(0) = 0. Now ∇ · (∇f ) = ∇(∇ · f ) = 0, hence we can use the estimates in (50) to see that Corollary 8.2 (with n = 3) applies as above to yield by (52) the bound Cε2 on the right-hand side of (56), for |x| ≤ Rρ ; observe that f has an additional v˙ and that it was not necessary here to keep the e from the estimates in (50), since the bound Cε2 is already ﬁne. Returning to the second part of the integral term in (55), we have [L(s)f (·, s)](x) = (¨ v · ∇v )Φv (x − q) + (v˙ · ∇v )2 Φv (x − q) with q = q(s), etc., and Φv := (Ev , Bv ). Since ∇ · Ev = eρ and ∇ · Bv = 0, we 2 obtain ∇ · (L(s)f ) = 0. Thus by (50), Corollary 8.2 (with n = 2 and α(s) = |v(s)| ˙ resp. α(s) = e|¨ v (s)|), and (52), t

t |¨ v (s)| 2 ds [U (t − s)(L(s)f (·, s))](x + q(t)) ≤ C ε + e ds , 1 + (t − s)2 0 0 |x| ≤ Rρ .

(57)

Collecting together all the parts of M (t), we ﬁnally arrive at t ds |M (t)| ≤ C ε2 + e 0

|¨ v (s)| . 1 + (t − s)2

Hence by (54) for e suﬃciently small, sup |¨ v (t)| ≤ Cε2 .

t∈IR

(58)

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VIII.4

Ann. Henri Poincar´e

...

Bounding | v (t)| ...

The estimate supt∈IR | v (t)| ≤ Cε3 will be derived by diﬀerentiating (44) twice. Since we have 2 d 2 −1 m (v(t)) + |¨ v (t)| ≤ Cε2 , ≤ C | v(t)| ˙ 0 dt2 we can use the previous steps and (P ) to ﬁnd

... | v | ≤ C ε3 + v¨ ∧ d3 xρ(x)ψ(x + q,t) + v˙ ∧ d3 x[L(t)ψ](x + q) + |M˙ (t)| . By (52), (58), and the estimates of the two preceding sections, also 3 v ∧ d x ρ(x)ψ(x + q, t) + v ˙ ∧ d3 x [L(t)ψ](x + q) ≤ Cε3 . ¨ To derive the same bound for M˙ (t), with M (t) from (53), observe M˙ (t) = v˙ ∧ d3 x [L(t)ψ](x + q) + d3 x ρ(x) (L(t)ϕ)(x + q(t)) + v(t) ∧ (L(t)ψ)(x + q(t)) ,

(59)

where L(t)φ = (v(t) ˙ · ∇)φ + (v(t) · ∇)2 φ + 2(v(t) · ∇)φ˙ + φ¨ d [(L(t)φ)(x+q(t))] = (L(t)φ)(x+q(t)). for a general φ = φ(x, t); (59) results from dt The ﬁrst term on the right-hand side of (59) is again bounded by Cε3 , so we have d to analyze the second. Since Z˙ = AZ − f , we ﬁnd the relation dt (L(t)Z) = ˙ A(L(t)Z) − L(t)f + (¨ v · ∇)Z + 2[(v · ∇)(v˙ · ∇)Z + (v˙ · ∇)Z], and thus, since L(0)Z(0) = 0, t [L(t)Z](x + q(t)) = ds U (t − s) − L(s)f (·, s) + (¨ v (s) · ∇)Z(·, s) 0 ˙ s)] (x + q(t)) + 2[(v(s) · ∇)(v(s) ˙ · ∇)Z(·, s) + (v(s) ˙ · ∇)Z(·,

=: T1 + T2 + T3 . ˙ replaced by Estimate of T2 : This is the same term as (56), only with v(s) v¨(s). Hence |T2 | ≤ Cε3 for |x| ≤ Rρ by (58). Estimate of T1 : By a direct calculation it may be shown that ...

v · ∇v )Φv (x − q) [L(s)f (·, s)](x) = ( v ·∇v )Φv (x − q) + 3(v˙ · ∇v )(¨ +(v˙ · ∇v )3 Φv (x − q),

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where q = q(s), etc., and Φv = (Ev , Bv ). Therefore in particular ∇ · (L(s)f ) = 0, whence the estimates in (50), Corollary 8.2 with n = 2, (52), and (58) yield

t ... | v (s)| |T1 | ≤ C ε3 + e ds , |x| ≤ Rρ . 1 + (t − s)2 0 Estimate of T3 : Let P (t)φ = (v(t) · ∇)∇φ + ∇φ˙ for a general φ = φ(x, t). d Then dt (P (t)Z) = A(P (t)Z) − P (t)f + (v˙ · ∇)∇Z gives, since P (0)Z(0) = 0, T3

=

t

2 0 t

=

2

ds v(s) ˙ · [U (t − s)(P (s)Z(·, s))](x + q(t)) s ds v(s) ˙ · dτ U (t − τ )[−P (τ )f (·, τ ) + (v(τ ˙ ) · ∇)∇Z(·, τ )]

0

0

(x + q(t)) =: T3,1 + T3,2 . What concerns T3,1 , we have [P (τ )f (·, τ )](x) = (¨ v · ∇v )∇Φv (x − q) + (v˙ · ∇v )2 ∇Φv (x − q) . Hence the standard argument (n = 3 in Corollary 8.2) implies |T3,1 | ≤ Cε3 for ˙ in the integral. Thus it remains to bound |x| ≤ Rρ ; observe the additional v(s) d T3,2 . As dt (∇2 Z) = A(∇2 Z) − ∇2 f , it follows from ∇2 Z(0) = 0 that T3,2 = 2

t

ds 0

s

τ

dσ [U (t − σ)(v(s) ˙ · ∇)(v(τ ˙ ) · ∇)f (·, σ)](x + q(t)) .

dτ 0

0

Since ∇ · (∇2 f ) = 0, again Corollary 8.2 applies, this time with n = 4, and we obtain for |x| ≤ Rρ |T3,2 | ≤ Cε

3

t

ds 0

s

dτ 0

τ

dσ 0

1 ≤ Cε3 . 1 + (t − σ)4

Summarizing all the estimates in this section results in

t ... | v (s)| ... 3 | v (t)| ≤ C ε + e . ds 1 + (t − s)2 0 Thus for e suﬃciently small we get ...

sup | v (t)| ≤ Cε3 , t∈IR

and the proof of Lemma 3.1 is complete.

✷

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References [1] Abraham M., Theorie der Elektrizit¨ at, Band II: Elektromagnetische Theorie der Strahlung, Teubner, Leipzig 1905 [2] Bauer G. & D¨ urr D., The Maxwell-Lorentz system of a rigid charge distribution, preprint [3] Dautray R. & Lions J.-L., Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5: Evolution Problems I, Springer, BerlinHeidelberg-New York 1992 [4] Dirac P.A.M., Classical theory of radiating electrons, Proc. Royal Soc. London A 167, 148-169 (1938) [5] Carati A. & Galgani L., Asymptotic character of the series of classical electrodynamics and an application to bremsstrahlung, Nonlinearity 6, 905-914 (1993) [6] Carati A., Delzanno P., Galgani L. & Sassarini J., Nonuniqueness properties of the physical solutions of the Lorentz-Dirac equation, Nonlinearity 8, 65-79 (1995) [7] R. Haag, Die Selbstwechselwirkung des Elektrons, Z. Naturforsch. 10a, 752761 (1955) [8] Jones Ch., Geometric singular perturbation theory, in Dynamical Systems, Proceedings, Montecatini Terme 1994, Ed. Johnson R., LNM 1609, Springer, Berlin-New York 1995, pp. 44-118 [9] Komech A., Kunze M. & Spohn H., Eﬀective dynamics for a mechanical particle coupled to a wave ﬁeld, Comm. Math. Phys. 203, 1–19 (1999). [10] Komech A. & Spohn H., Long-time asymptotics for the coupled MaxwellLorentz equations, Comm. Partial Diﬀerential Equations25(3) (2000). [11] Kunze M. & Spohn H., Radiation reaction and center manifolds,to appear in SIAM J. Math. Anal. [12] Rohrlich F., Classical Charged Particles, 2nd edition, Addison-Wesley, Reading, MA, 1990 [13] Sakamoto K., Invariant manifolds in singular perturbation problems for ordinary diﬀerential equations, Proc. Roy. Soc. Edinburgh Sect. A 116, 45-78 (1990) [14] Scharf G., From Electrostatics to Optics, Springer, Heidelberg 1994

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[15] Yaghjian A.D., Relativistic Dynamics of a Charged Sphere, Lecture Notes in Physics m 11, Springer, Berlin-New York 1992

Markus Kunze Mathematisches Institut der Universit¨ at K¨ oln Weyertal 86 D-50931 K¨ oln, Germany Email : [email protected]

Herbert Spohn Zentrum Mathematik and Physik Department, TU M¨ unchen D-80290 M¨ unchen, Germany Email : [email protected]

Communicated by J. Bellissard submitted 07/04/99, revised 16/07/99 ; accepted 28/10/99

Ann. Henri Poincar´ e 1 (2000) 655 – 684 c Birkh¨ auser Verlag, Basel, 2000 1424-0637/00/040655-30 $ 1.50+0.20/0

Annales Henri Poincar´ e

The Palais-Smale Condition and Ma˜ n´e’s Critical Values G. Contreras, R. Iturriaga, G. P. Paternain, M. Paternain Abstract. Let L be a convex superlinear autonomous Lagrangian on a closed connected manifold N . We consider critical values of Lagrangians as deﬁned by R. Ma˜ n´ e in [23]. We deﬁne energy levels satisfying the Palais-Smale condition and we show that the critical value of the lift of L to any covering of N equals the inﬁmum of the values of k such that the energy level t satisﬁes the Palais-Smale condition for every t > k provided that the Peierls barrier is ﬁnite. When the static set is not empty, the Peierls barrier is always ﬁnite and thus we obtain a characterization of the critical value of L in terms of the Palais-Smale condition. We also show that if an energy level without conjugate points has energy strictly bigger than cu (L) (the critical value of the lift of L to the universal covering of N ), then two diﬀerent points in the universal covering can be joined by a unique solution of the Euler-Lagrange equation that lives in the given energy level. Conversely, if the latter property holds, then the energy of the energy level is greater than or equal to cu (L). In this way, we obtain a characterization of the energy levels where an analogue of the Hadamard theorem holds. We conclude the paper showing other applications such as the existence of minimizing periodic orbits in every non-trivial homotopy class with energy greater than cu (L) and homologically trivial periodic orbits such that the action of L + k is negative if cu (L) < k < ca (L), where ca (L) is the critical value of the lift of L the abelian covering of N . We also prove that given an Anosov energy level, there exists in each non-trivial free homotopy class a unique closed orbit of the Euler-Lagrange ﬂow in the given energy level.

I Introduction In this paper we study geometric and dynamical properties of convex and superlinear Lagrangians, and it can be considered as a continuation of our previous paper [9]. This time we study the action functional from the viewpoint of Morse theory and we show, among other results, that for a compact manifold the critical value as deﬁned by R. Ma˜ n´e in [23] can be characterized by the Palais-Smale condition. This will follow from more general results to be precisely stated below. It is well known that the action functional of the Lagrangian L(x, v) =

1 2 |v| 2 x

arising from a Riemannian metric satisﬁes the Palais-Smale condition (cf. [29]). This condition ensures that the minimax principle holds and from the latter many standard properties of geodesics easily follow, namely, The Hopf-Rinow theorem, the Hadamard theorem and the existence of closed geodesics in each homotopy class. As we explain below, these and other properties as well as the approach in [29]

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and [31] hold for energy levels of convex and superlinear autonomous Lagrangians if the energy is greater than the critical value. In order to describe precisely our results let us recall some preliminaries. Let N be a closed connected smooth manifold and let L : T N → R be a smooth convex superlinear Lagrangian. This means that L restricted to each Tx N has positive deﬁnite Hessian and that for some Riemannian metric we have that lim

|v|→∞

L(x, v) = ∞, |v|

uniformly on x ∈ N. Since N is compact, the extremals of L give rise to a complete ﬂow φt : T N → T N called the Euler-Lagrange ﬂow of the Lagrangian. Recall that the energy EL : T N → R is deﬁned by EL (x, v) =

∂L (x, v).v − L(x, v). ∂v

Since L is autonomous, EL is a ﬁrst integral of the ﬂow φt . Recall also that the action of the Lagrangian L on an absolutely continuous curve γ : [a, b] → N is deﬁned by b SL (γ) = L(γ(t), γ(t)) ˙ dt. a

Given two points, q1 and q2 in N and T > 0 denote by C(q1 , q2 ; T ) the set of absolutely continuous curves γ : [0, T ] → N , with γ(0) = q1 and γ(T ) = q2 . For each k ∈ R we deﬁne the action potential Φk : N × N → R by Φk (q1 , q2 ) = inf{SL+k (γ) : γ ∈ ∪T >0 C(q1 , q2 ; T )}. The critical value of L, which was introduced by Ma˜ n´e in [23], is the real number c(L) deﬁned as the inﬁmum of k ∈ R such that for some q ∈ N , Φk (q, q) > −∞. Since L is convex and superlinear and N is compact such a number exists and it has various important properties [23, 6]. We brieﬂy mention a few of them since we shall need them below. For any k ≥ c(L), the action potential Φk is a Lipschitz function that satisﬁes a triangle inequality. In general the action potential is not symmetric but if we deﬁne dk : N × N → R by setting dk (q1 , q2 ) = Φk (q1 , q2 ) + Φk (q2 , q1 ), then dk is a distance function for all k > c(L) and a pseudo-distance for k = c(L). In [23, 6] the critical value is characterized in other ways relating it to minimizing measures or to the existence of Tonelli minimizers with ﬁxed energy between two points. We can also consider the critical value of the lift of the Lagrangian L to a covering of the compact manifold N . Suppose that p : M → N is a covering space

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and consider the Lagrangian L : T M → R given by L := L ◦ dp. For each k ∈ R we can deﬁne an action potential Φk in M × M just as above and similarly we obtain a critical value c(L) for L. It can be easily checked that if M1 and M2 are coverings of N such that M1 covers M2 , then c(L1 ) ≤ c(L2 ),

(1)

where L1 and L2 denote the lifts of the Lagrangian L to M1 and M2 respectively. Among all possible coverings of N there are two distinguished ones; the uni˜ , and the abelian covering which we versal covering which we shall denote by N shall denote by N . The latter is deﬁned as the covering of N whose fundamental group is the kernel of the Hurewicz homomorphism π1 (N ) → H1 (N, R). When ˜ is a ﬁnite covering of N . π1 (N ) is abelian, N The universal covering of N gives rise to the critical value def ˜ ), cu (L) = c(lift of L to N

and the abelian covering of N gives rise to the critical value def

ca (L) = c(lift of L to N ). From inequality (1) it follows that cu (L) ≤ ca (L), but in general the inequality may be strict as it was shown in [30]. The critical values have another important feature: they single out those energy levels in which relevant globally minimizing objects (orbits or measures) live [10, 23, 6]. The study of these globally minimizing objects has a long history that goes back to M. Morse [27] and G.A. Hedlund [19]. Recent work on this subject has been done by V. Bangert [3, 4], M.J. Dias Carneiro [10], A. Fathi [14, 15, 16, 17], R. Ma˜ n´e [23, 24] and J. Mather [25, 26]. We refer to [8, 18] for comprehensive accounts of the theory. Static and semistatic curves are the paradigms of what we mean by globally minimizing orbits and since they will play an important role in our results we give now their deﬁnition (cf. [23, 26]). Set c = c(L). We say that x : [a, b] → M is a semistatic curve if it is absolutely continuous and: SL+c x|[t0 ,t1 ] = Φc (x(t0 ), x(t1 )) , (2) for all a < t0 ≤ t1 < b; and that it is a static curve if SL+c x|[t0 ,t1 ] = −Φc (x(t1 ), x(t0 )) for all a < t0 ≤ t1 < b. Observe that since Φc (x(t0 ), x(t1 )) + Φc (x(t1 ), x(t0 )) = dc (x(t0 ), x(t1 )) ≥ 0

(3)

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a static curve is a semistatic curve for which dc (x(t0 ), x(t1 )) = 0 for all a < t0 ≤ t1 < b. Let Σ(L) be the set of vectors v ∈ T M such that the solutions γ : R → M of the Euler-Lagrange equation satisfying γ(0) ˙ = v are static. We call Σ(L) the static set. As we mentioned before, one of our aims in this paper is to relate the Morse theory of the action functional to the critical values. Let H 1 (Rk ) be the set of absolutely continuous curves x : [0, 1] → Rk such that

1 2 |x(t)| ˙ dt < ∞.

0

It is well known that H 1 (Rk ) is a Hilbert space with the inner product deﬁned by 1 x, y1 = x(0), y(0)Rk + x(t), ˙ y(t) ˙ Rk dt, 0

where · , · Rk is the standard inner product of Rk . Given any Riemannian metric of M we may assume, on account of the Nash embedding theorem, that M is isometrically embedded in some Rk . Take q1 and q2 in M and let Ω(q1 , q2 ) be the set of elements of H 1 (Rk ) such that x([0, 1]) ⊂ M , x(0) = q1 and x(1) = q2 . It follows from the arguments in [29] that Ω(q1 , q2 ) inherits a Hilbert manifold structure compatible with the Riemannian metric on M . We now deﬁne another action AL closely related to the action SL we deﬁned before. Given the Lagrangian L : T M → R deﬁne AL : R+ × Ω(q1 , q2 ) → R by AL (b, x) =

1

b L(x(t), x(t)/b) ˙ dt. 0

Observe that AL (b, x) = SL (y), where y(t) = x(t/b). We now recall the deﬁnition of the Palais-Smale condition. In fact, this is a rather stronger version of the condition in [29] and [31] that we borrow from [20] and [22]. Deﬁnition 1 Let f : X → R be a C 1 map where X is an open set of a Hilbert manifold. We say that f satisﬁes the Palais-Smale condition if every sequence {xn } such that {f (xn )} is bounded and ||dxn f || → 0 as n → ∞ has a converging subsequence.

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We remark that the manifold R+ × Ω(q1 , q2 ) is not complete, however if q1 = q2 then the set {AL+k ≤ a} is complete when k is strictly bigger that the critical value (see Lemma 15). This is important when we apply the minimax principle (see Proposition 21). In order to show that the action functional AL+k is C 2 and satisﬁes the Palais-Smale condition we need the Lagrangian to be quadratic at inﬁnity: Deﬁnition 2 We say that L : T N → R is quadratic at inﬁnity if there exists a Riemannian metric on N and R > 0 such that for each x ∈ N and |v|x > R, L(x, v) has the form L(x, v) =

1 2 |v| + θx (v) − V (x), 2 x

where θ is a smooth 1-form on N and V : N → R a smooth function. A lifted Lagrangian L : T M → R is said to be quadratic at inﬁnity if it is the lift of a Lagrangian quadratic at inﬁnity on N . In Section 3 (cf. Proposition 18) we shall show that given a Lagrangian L and k ∈ R there is a Lagrangian L0 quadratic at inﬁnity such that L(x, v) = L0 (x, v) for all (x, v) with E(x, v) ≤ k + 1. We shall also show (cf. Lemma 19) that given two Lagrangians L and L0 which agree for any (x, v) with E(x, v) ≤ c(L) + 1, then c(L) = c(L0 ). These properties motivate the following deﬁnition: Deﬁnition 3 We say that the energy level E −1 (k) of a convex and superlinear Lagrangian L satisﬁes the Palais-Smale condition if there is a Lagrangian L0 quadratic at inﬁnity such that L and L0 agree for any (x, v) with E(x, v) ≤ k + 1 and the action functional AL0 +k satisﬁes the Palais-Smale condition on R+ × Ω(q1 , q2 ) for every q1 = q2 . In Section 3 we shall prove: Theorem A. Let M be any covering of the closed manifold N and let L : T M → R be the lifted Lagrangian. If the static set Σ(L) is not empty, then c(L) = inf t ∈ R : EL−1 (k) satisﬁes the Palais-Smale condition for every k > t . When M is compact the set Σ(L) is not empty [6, 23] and therefore we obtain: Corollary. Suppose that M is compact. Then c(L) = inf t ∈ R : EL−1 (k) satisﬁes the Palais-Smale condition for every k > t . S. Bolotin in [5] explores ideas which are similar to the ones we develop here. Using a somewhat diﬀerent language he also notes that the Palais-Smale

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condition holds for the action AL+k for large values of k but he does not give a characterization of c(L) as before. Theorem A will follow from Theorems B and C below. To state these theorems we need two more deﬁnitions. For k ∈ R, let Φk (q1 , q2 ; T ) := hk (q1 , q2 ) :=

inf

γ∈C(q1 ,q2 ;T )

SL+k (γ)

lim inf Φk (q1 , q2 ; T ).

T →+∞

The function hc is known as the Peierls barrier [15, 26]. Theorem B. Assume that L is quadratic at inﬁnity and hk (q1 , q2 ) = +∞. Then the action functional AL+k : R+ × Ω(q1 , q2 ) → R satisﬁes the Palais-Smale condition provided q1 = q2 . Theorem C. Assume that L is quadratic at inﬁnity and that for some pair (q1 , q2 ), hc (q1 , q2 ) < +∞. Then the action functional AL+c : R+ × Ω(q1 , q2 ) → R does not satisfy the Palais-Smale condition. At the end of Section 3 we explain in detail how to derive Theorem A from Theorems B and C. Two extra ingredients needed for this derivation are given by Corollary 12 in Section 3, which states that the Peierls barrier is ﬁnite for every pair (q1 , q2 ) in M when the static set Σ(L) is not empty and Lemma 10 which states that hk (q1 , q2 ) = +∞ for every pair (q1 , q2 ) provided that k > c(L). We remark that if hc (q1 , q2 ) is ﬁnite for some pair (q1 , q2 ) then it is ﬁnite for all pairs (q1 , q2 ). In the appendix we give an example of a Lagrangian on R2 for which the static set is empty and the Peierls barrier hc is inﬁnite (and hence the PalaisSmale condition holds at critical energy). Even though this Lagrangian is not the lift of a Lagrangian on a compact manifold, it shows that most likely Theorems A, B and C are optimal. It is unknown whether the energy level EL−1 (k) satisﬁes the Palais-Smale condition for k < c(L) (some authors have assumed that the Palais-Smale condition holds at subcritical energies for magnetic Lagrangians and this gap has been pointed out by S. Bolotin, see [33] for a discussion of the problem). In Section 4 we prove an analogue of the Hadamard theorem on ﬁxed energy levels. A pair of points (x1 , v1 ), (x2 , v2 ) in T M are said to be conjugate if (x2 , v2 ) = φt (x1 , v1 ) for t = 0 and dφt (V (x1 , v1 )) intersects V (x2 , v2 ) non-trivially. Here, V (x, v) is the vertical ﬁbre at (x, v) deﬁned as usual as the kernel of dπ(x,v) : T(x,v) T N → Tx N where π : T N → N is the canonical projection. We have:

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Theorem D. Let L : T N → R be a convex and superlinear Lagrangian. Assume that there is q0 ∈ N such that q0 has no conjugate points in EL−1 (k). Let q˜0 be a ˜ , the universal covering of N and let L be the lift of L to N ˜ . Denote lift of q0 to N the following statement by (H): ˜ there is a unique solution of the Euler-Lagrange equation (H) For every q˜ in N of L with energy k, joining q˜0 to q˜. Then, k > cu (L) (H)

=⇒ (H) =⇒ k ≥ cu (L).

We note that when k = cu (L) there are examples in [7] where (H) does not hold. Also there are examples in [7] of multivalued Lagrangians, that become honests Lagrangians in the universal covering for which (H) does hold. In Section 5 we recall some results on Morse theory that we will use in the last section. In Section 6 we give applications such as the existence of minimizing periodic orbits in every non-trivial free homotopy class with energy greater than cu (L) and homologically trivial periodic orbits such that the action of L + k is negative if cu (L) < k < ca (L), where ca (L) is the critical value of the lift of L to the abelian covering of N . These results should be compared with the work of S.P. Novikov, I. Taimanov and A. Bahri and I. Taimanov on the existence of closed orbits for magnetic Lagrangians. See [2, 28, 33, 34] and the extensive references therein. We also prove in Section 6 that given an Anosov energy level, there exists in each non-trivial free homotopy class a unique closed orbit of the Euler-Lagrange ﬂow in the given energy level.

II First and Second Variations In this section we calculate the ﬁrst and second variations of the action functional AL+k . These computations do not need any assumptions on the Lagrangian. However, if we want them to be the ﬁrst and second derivative of the action functional AL+k we need the Lagrangian to be quadratic at inﬁnity. Take a curve s → (bs , xs ) ∈ R+ × Ω(q1 , q2 ) and set b := b0 , x := x0 , dbs s ξ(t) := ∂x ∂s |s=0 (t), α := ds |s=0 and g(s) := AL+k (bs , xs ). A straightforward calculation in local coordinates gives: Lemma 4

d(b,x) AL+k (α, ξ) := g (0) = α + 0

1

k − EL (x(t), x(t)/b) ˙ dt

0 1

˙ bLx (x(t), x(t)/b)ξ(t) ˙ + Lv (x(t), x(t)/b) ˙ ξ(t) dt.

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Remark 5 If (b, x) is a critical point of AL+k , then y : [0, b] → M given by y(t) = x(t/b) is a solution of the Euler-Lagrange equation of L with energy k (see [1] and [6]). Indeed, the second term in the last equation is equal to b d Lx (y, y) ˙ − dt Lv (y, y) ˙ η dt , 0

where η(t) := ξ(t/b). Since it is zero for all variations η, then y satisﬁes the EulerLagrange equation. Since the ﬁrst term is zero for α = 1, then EL (y, y) ˙ ≡ k. Remark 6

1 ∂AL+k dAL+k (bs , x) k − EL (x(t), x(t)/b) = α ˙ dt, (α) := ∂b ds 0 s=0 α b k − EL (y(s), y(s)) = ˙ ds, b 0

where y(s) = x(s/b). Lemma 7

∂ 2 AL+k d2 AL+k (b, xs ) (ξ, ξ) := ∂2x ds2 s=0 b = {ηLxx (y, y)η ˙ + ηLxv (y, y) ˙ η˙ + ηL ˙ vx (y, y)η ˙ + ηL ˙ vv (y, y) ˙ η} ˙ dt, 0

where η(t) = ξ(t/b) and y(t) = x(t/b). Proof . Calculate g (0) where g(s) = AL+k (b, xs ).

The following lemma is an immediate consequence of the Morse Index Theorem for convex Lagrangians (cf. [11]). Lemma 8 Let y, (b, x) be as in the previous lemma. If in addition y is a solution 2 of the Euler-Lagrange equation with no conjugate points, then ∂ ∂A2L+k is positive x deﬁnite. The following theorem is a particular case of a theorem due to Smale [31]. Theorem 9 If a Lagrangian L is quadratic at inﬁnity then the corresponding action functional AL+k : R+ ×Ω(q1 , q2 ) → R is a C 2 function with respect to the canonical Hilbert structure of R+ × Ω(q1 , q2 ). Moreover, the diﬀerential of AL+k evaluated at the tangent vector (α, ξ) is given precisely by the number d(b,x) AL+k (α, ξ) deﬁned in Lemma 4.

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Lemma 8 motivates the following: Question: Is it true that (b, x) is a local minimum of AL+k provided that x(t/b) is a solution of the Euler-Lagrange equation with no conjugate points? The next example shows that the answer to this question is negative. This example was motivated by the referee who pointed out a mistake in a previous version of the mansucript. We thank him or her for this observation. On the other hand we shall show in Lemma 30 that the answer to the question is aﬃrmative in the case of Anosov energy levels and the space of closed paths with a ﬁxed non-trivial homotopy class. Let L be the Lagrangian on T R2 given by: ˙ L(x, y, x, ˙ y) ˙ = 12 (x˙ 2 + y˙ 2 ) − xy. We take k = 1/2, q1 = (1, 0) and q2 = (−1, 0). The orbits of L with energy 1/2 are circles of radius one oriented counterclockwise. Let c0 be the half circle of radius one connecting q1 to q2 . Let (−ε,√ε) s → cs be a small variation of c0 by circles with center at (0, s) and radius 1 + s2 . We parametrize these circles by √ arc length, hence cs connects q1 to q2 in time b(s) = 1 + s2 (π + 2 a(s)), where a(s) is the angle in (−π/2, π/2) whose tangent is s. By observing that ˙ y) ˙ = 1 − x y, ˙ (L + 12 )(x, y, x, when x˙ 2 + y˙ 2 = 1 one obtains: A(s) := AL+1/2 (bs , xs ) = SL+1/2 (cs ) = b(s) − (1 + s2 )

π 2

+ a(s) +

1 2

sin(2a(s)) .

A somewhat tedious computation shows that: A (0) = 0,

A (0) = 0,

A (0) = −2.

Hence s → A(s) does not have a local minumum at s = 0. On the other hand the piece of orbit [0, π] s → c0 (s) does not have conjugate points [8, Example A.3]. Finally we observe that if we consider a ball in R2 with radius, let us say, four, then L restricted to this ball can be embedded into a convex superlinear Lagrangian on a closed surface.

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III The Palais-Smale condition for Lagrangians quadratic at inﬁnity III.1

The Peierls barrier

For k ∈ R, let Φk (q1 , q2 ; T ) := hk (q1 , q2 ) :=

inf

γ∈C(q1 ,q2 ,T )

SL+k (γ)

lim inf Φk (q1 , q2 ; T ).

T →+∞

The function hc is called the Peierls barrier [15]. Lemma 10 Set c := c(L). If k > c then hk (q1 , q2 ) = +∞ for all q1 , q2 ∈ M . If k < c then hk (q1 , q2 ) = −∞ for all q1 , q2 ∈ M . Proof . If k > c, we have that Φk (q1 , q2 ; T ) ≥ Φc (q1 , q2 ) + (k − c) T. Hence hk (q1 , q2 ) = +∞. If k < c, since Φk (q2 , q2 ) = −∞, there is a curve γ ∈ C(q2 , q2 ; T ) with T > 0 and SL+k (γ) < 0. Then n

Φk (q1 , q2 ; 1 + n T ) ≤ Φk (q1 , q2 ; 1) + n SL+k (γ) −→ −∞. Thus hk (q1 , q2 ) = −∞. Proposition 11 hc (p, p) = 0 iﬀ p ∈ π Σ(L). Proof . First take p ∈ π Σ(L) and let γ : R → M be a static curve with γ(0) = p. Let ε > 0 be given. For any t > 0 there is a curve γt : [0, Tt ] → M such that γt (0) = γ(t), γt (Tt ) = γ(0) and SL+c (γt ) ≤ Φc (γ(t), p) + ε. Then 0 ≤ hc (p, p) ≤ lim inf SL+c (γ|[0,t] ∗ γt ) t→+∞ ≤ lim inf SL+c (γ|[0,t] ) + SL+c (γt ) t→+∞

≤ lim inf (Φc (p, γ(t)) + Φc (γ(t), p)) + ε t→∞

= ε,

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where the last equality holds because γ is a static curve. Now assume that hc (p, p) = 0. Then there is Tn → ∞ and Tonelli minimizers γn : [0, Tn ] → M with γn (0) = γn (Tn ) = p and SL+c (γn ) → 0. By Lemma 16 the speed of the Tonelli minimizers γn is bounded and hence there is a subsequence such that γ˙ n (0) → v. Let γ be the solution of the Euler-Lagrange equation with initial conditions (p, v). We are going to show that γ is static. Take 0 < t1 < t2 and ε > 0. Since Φc is continuous we can take n so big that Tn > t2 , SL+c (γn ) < ε, Φc (γ(t1 ), γ(t2 )) < Φc (γn (t1 ), γn (t2 )) + ε and Φc (γ(t2 ), γ(t1 )) < Φc (γn (t2 ), γn (t1 )) + ε. Next observe that Φc (γn (t1 ), γn (t2 )) ≤ SL+c (γn |[t1 ,t2 ] ) and Φc (γn (t2 ), γn (t1 )) ≤ SL+c (γn |[t2 ,Tn ] ) + SL+c (γn |[0,t1 ] ). Hence Φc (γ(t1 ), γ(t2 )) + Φc (γ(t2 ), γ(t1 )) ≤ 3ε, and since ε was arbitrary we deduce that γ|(0,∞) is static. By Proposition 3.5.5 in [8], γ : R → M is a static curve as desired. Corollary 12 If the static set Σ(L) is not empty, then for any pair (q1 , q2 ) we have hc (q1 , q2 ) ≤ Φc (q1 , p) + Φc (p, q2 ), for any p ∈ π Σ(L). Proof . From the deﬁnition of the Peierls barrier hc we have: hc (q1 , q2 ) ≤ Φc (q1 , p) + hc (p, q) + Φc (q, q2 ). Set p = q ∈ π Σ(L) and use Proposition 11. Even though we do not need the next result for the proof of the theorems in the introduction, we include it here because it gives an interesting characterization of the Peierls barrier in terms of the action potential. This result should be compared with Fathi’s results in [15], in which he characterizes the Peierls barrier in terms of conjugate weak KAM solutions.

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Proposition 13 If M is compact then hc (x, y) =

inf

p∈π(Σ(L))

{Φc (x, p) + Φc (p, y)} .

Proof . Recall that when M is compact the set Σ(L) is not empty [6, 23]. Using Corollary 12 we get that hc (x, y) ≤ inf Φc (x, p) + Φc (p, y) . p∈πΣ(L)

Now let γn ∈ C(x, y; Tn ) be Tonelli minimizers with Tn → +∞ and SL+c (γn ) → hc (x, y) < +∞. Then T1n SL+c (γn ) → 0. Observe that by Lemma 16 the speed of the Tonelli minimizers γn is bounded. Let µ be a weak limit of a subsequence of the probability measures µγn supported on the piece of orbits [0, Tn ] t → (γn (t), γ˙ n (t)).

Then µ is a minimizing measure (see [6, 23]). Let q ∈ π supp(µ) and qn ∈ γn ([0, Tn ]) be such that limn qn = q. Then, Φc (x, q) + Φc (q, y) ≤ Φc (x, qn ) + Φc (qn , y) + Φc (qn , q) + Φc (q, qn ) ≤ SL+c (γn ) + Φc (qn , q) + Φc (q, qn ). Letting n → ∞, we get that Φc (x, q) + Φc (q, y) ≤ hc (x, y).

III.2

Proof of Theorem B

We begin with the following lemma: Lemma 14 Suppose that L is quadratic at inﬁnity. Given any coordinate chart there are positive numbers δ, B, C and D such that in the given chart we have: C|v − w|2 ≤ (Lv (x, v) − Lv (y, w)) · (v − w) + [B (|v| + |w|) + D] |v − w| d(x, y) provided that d(x, y) < δ. Proof . Take δ such that if d(x, y) < δ then x, y belong to the same coordinate chart of M . Next we work in local coordinates as if L were deﬁned in R2n . 1 (v − w) · Lvv t (x, v) + (1 − t) (y, w) · (v − w) dt = 0 = Lv (x, v) − Lv (y, w) · (v − w) 1 − (v − w) · Lxv (x, v)t + (y, w)(1 − t) · (x − y) dt. 0

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Now, since L is quadratic at inﬁnity |Lxv (x, v)| ≤ B1 |v|x + D1 for some positive constants B1 and D1 . On the other hand, since Lvv is positive deﬁnite, there is C > 0 such that u Lvv u ≥ C |u|2 for every u. If we now take into account the equivalence between d and the euclidean metric in the given coordinate chart we can easily obtain the statement of the lemma for appropriate constants B and D.

The next lemma will ensure that if q1 = q2 then the set {AL+k ≤ a} is complete when hk (q1 , q2 ) = ∞. This observation is needed when we apply the minimax principle given by Proposition 21. Lemma 15 If q1 = q2 and AL+k (bn , xn ) ≤ D then bn is bounded away from zero. Proof . Since our Lagrangian is quadratic at inﬁnity, there are positive constants D1 and D2 such that L(x, v) ≥ D1 |v|2 − D2 , consequently D ≥ AL+k (bn , xn ) ≥

D1 bn

1

|x˙n |2 dt + (k − D2 ) bn , 0

and thus there are positive numbers D3 and D4 such that for all n, 1 |x˙ n |2 dt ≤ D3 bn + D4 b2n .

(4)

0

Observe that if bn → 0 then ||xn ||1 → 0 and then the length of xn goes to 0 which is absurd provided q1 = q2 . Let us begin now with the proof of Theorem B. Take {(bn , xn )} such that {AL+k (bn , xn )} is bounded and ||d(bn ,xn ) AL+k ||1 → 0. Let yn (t) = xn (t/bn ). Then {bn } is bounded, for if not, we may assume that bn → +∞ and then n

AL+k (bn , xn ) = SL+k (yn ) ≥ Φk (q1 , q2 ; bn ) −→ +∞. So, we can assume that {bn } converges. Let b = limn bn . Let

yn (s) if s ≤ bn , wn (s) = q2 if bn ≤ s ≤ b + 1.

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Then wn ∈ C(q1 , q2 ; b + 1) and SL (wn ) = SL (yn ) + L(q2 , 0) (b + 1 − bn ) ≤ AL+k (bn , xn ) − k bn + 2 L(q2 , 0), if bn ≥ b − 1. By the same arguments in the proof of Tonelli’s Theorem and by the Arzela-Ascoli Theorem there is a convergent subsequence of wn in the C 0 topology (cf. [8, 25]). This implies that also {yn } and {xn } have convergent subsequences in the C 0 topology. For the sequel we work with a convergent subsequence of {xn }. We shall assume without loss of generality that the limit point of the sequence {xn } is contained in a coordinate chart, for if not we can cover it with a ﬁnite number of charts and we do the argument below on each chart. Hence for n large enough xn has its image contained in the same local chart as the limit point. By Lemma 15 we may assume that bn → b = 0. Write zn = xn /bn . Also by Lemma 15 (see inequality (4)) there is K > 0 such that ||xn ||1 ≤ K

and

||zn ||1 ≤ K

(recall that || . ||1 is the norm in H 1 (Rk )). Now we follow the lines of Lemma 7.1 in [31]. Since ||d(bn ,xn ) AL+k ||1 → 0 , given ε there is N such that ||d(bn ,xn ) AL+k (η) − d(bm ,xm ) AL+k (η)||1 < ε for every n, m ≥ N and ||η||1 ≤ 2K. We can take in particular η = xn − xm and therefore using Lemma 4 we have, 1 {bn Lx (xn , z˙n ) − bm Lx (xm , z˙m )} (xn − xm ) dt + 0 1 {Lv (xn , z˙n ) − Lv (xm , z˙m )} (x˙ n − x˙ m ) dt < ε + 0

for m, n > N . Since our Lagrangian is quadratic at inﬁnity, then there are positive constants a and c such that ||Lx || < a |v|2x + c. Using (4) in the proof of Lemma 15 we have that the ﬁrst term is bounded by (2 a D3 +(bn +bm )(a D4 + c))||xn −xm ||∞ . Consequently the second integral is small for big m and n. Now we apply Lemma 14 to obtain C 0

1

z˙n − z˙m 2 dt ≤

1

{ Lv (xn , z˙n ) − Lv (xm , z˙m ) } · z˙n − z˙m dt +

0

1

+B

0 1

+D 0

|z˙n | + |z˙m |) z˙n − z˙m |xn − xm | dt

z˙n − z˙m |xn − xm | dt.

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Now

1

|z˙n | + |z˙m |) z˙n − z˙m dt ≤ ||zn ||1 + ||zm ||1 zn − zm 1

0

by the Cauchy-Shwartz inequality. Therefore 2 C z˙n − z˙m L2 ≤

1

{ Lv (xn , z˙n ) − Lv (xm , z˙m ) } · z˙n − z˙m dt +

0

+ (4 K 2 B + 2KD) ||xn − xm ||∞ . Since xn converges in the C 0 topology and the integral is small, we conclude that zn converges in the H 1 -norm. Again, since b = 0, xn also converges in the H 1 -norm, ﬁnishing the proof of the theorem.

III.3

Proof of Theorem C

Lemma 16 ([25]) For B > 0 there exists C = C(B) > 0 such that if x, y ∈ M and γ ∈ C(x, y; T ) is a solution of the Euler-Lagrange equation with AL (γ) ≤ B T , then |γ(t)| ˙ < C for all t ∈ [0, T ]. Proof . By the superlinearity there is D > 0 such that L(x, v) ≥ |v| − D for all (x, v) ∈ T M . Since SL (γ) ≤ B T , the mean value theorem implies that there is t0 ∈ (0, T ) such that |γ(t ˙ 0 )| ≤ D + B. The conservation of the energy implies that there is C = C(B) > 0 such that |γ| ˙ ≤ C. Lemma 17 For all x, y ∈ M and ε > 0, the function t → Φk (x, y; t) is Lipschitz on ε < t < +∞. Proof . Fix ε > 0. If T > ε, let γ ∈ C(x, y; T ) be a Tonelli minimizer. Let τ : [0, T ] → M be a geodesic with speed d(x, y)/T < d(x, y)/ε connecting x to y. Let B=

max

{(x,v): |v|≤d(x,y)/ε}

L(x, v).

Then since γ is a Tonelli minimizer we have SL (γ) ≤ SL (τ ) ≤ B T . On account of Lemma 16 there exists C = C(ε) > 0 such that |E(γ, γ) ˙ − k| ≤ C(ε) + |k|. Denote h(s) := Φk (x, y; s). If γs (t) := γ(T t/s) with t ∈ [0, s], then h(s) ≤ SL+k (γs ) =:

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B(s). Using Remark 6 we have that h(T + δ) − h(T ) δ δ→0 T 1 ≤ B (T ) = k − E(γ, γ) ˙ dt T 0 ≤ C(ε) + |k|.

f (T ) : = lim sup

If S, T > ε we have that S Φk (x, y; S) ≤ Φk (x, y; T ) + f (t) dt T ≤ Φk (x, y; T ) + C(ε) + |k| |T − S|. Since we can reverse the roles of S and T , this implies the Lipschitz condition for T → Φk (x, y; T ). We begin now with the proof of Theorem C. Let f (t) := Φc (x, y; t). Case 1: Suppose ﬁrst that there is T0 > 0 such that f is monotonous on [T0 , +∞). Since by Lemma 17 f is Lipschitz on [T0 , +∞) by Rademacher’s theorem [13], f is diﬀerentiable almost everywhere and

t

f (t) − f (T0 ) =

f (s) ds.

(5)

T0

Since f is monotonous f ≥ 0 or f ≤ 0 for all t ≥ T0 and

lim f (t) =

t→+∞

lim inf f (t) = hc (x, y) < +∞. This implies that there is a sequence of diﬀerent→+∞

tiability points tn → +∞ such that f (tn ) → 0, for otherwise there would exist K > 0 and R > 0 such that f (s) ≥ K > 0 for s ≥ R or f (s) ≤ −K for s ≥ R. Consequently equation (5) would imply that limt→+∞ f (t) is inﬁnite. Let γn be a Tonelli minimizer in C(x, y; tn ) and ηs (t) = γn ( tsn t). Then SL+k (ηs ) ≥ f (s) for s in an open interval contaning tn . This implies that

f (tn ) =

d ds tn SL+k (ηs )

1 = tn

tn

k − E(γn , γ˙ n ) dt,

0

where the second equality follows from Remark 6. Since γn is a solution of the Euler-Lagrange equation, by Remark 5 and Remark 6, if xn (s) = γn (s tn ) dAL+k (tn , xn )(ξ, α) = α f (tn ) → 0. Observe also that AL+k (tn , xn ) → hc (x, y) < +∞. On the other hand (tn , xn ) does not converge.

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Case 2: Suppose that the set of local minima of f is unbounded. We claim that there exists a sequence tn → +∞ of local minima of f such that limn f (tn ) = lim inf f (t). For let sn be an increasing sequence of local minima of f such that t→+∞

sn → +∞. There exists a sequence rn > sn such that limn f (rn ) = lim inf f (t). t→+∞

Excluding some sn ’s if necessary we can assume that sn < rn < sn+1 . Minimizing f on the interval [sn , sn+1 ], we obtain a local minimum tn ∈ [sn , sn+1 ] such that limn tn = limn sn = +∞ and f (tn ) ≤ f (rn ) so that limn f (tn ) = lim inf f (t). t→+∞

Let γn be a Tonelli minimizer in C(x, y; tn ) and ηs (t) = γn ( tsn t). Then SL+k (ηs ) ≥ f (tn ) for s in a neighbourhood of tn . In particular, tn is also a local minimum of s → SL+k (ηs ). Since s → SL+k (ηs ) is diﬀerentiable, dds |tn AL+k (ηs ) = 0. By Remark 5 and Remark 6, if xn (s) = γn (s tn ) dAL+k (tn , xn )(ξ, α) = α

d ds |tn AL+k (ηs )

= 0.

Observe also that AL+k (tn , xn ) → hc (x, y) < +∞. On the other hand (tn , xn ) does not converge.

III.4

Proof of Theorem A

Proposition 18 Given a convex and superlinear Lagrangian L : T N → R and k ∈ R there is a Lagrangian L0 , convex and quadratic at inﬁnity such that L0 (x, v) = L(x, v) for every (x, v) such that EL (x, v) ≤ k + 1. Proof . Without loss of generality we may assume that L ≥ 0. Choose R > 0 such that EL (x, v) ≤ k + 1 Deﬁne ψ : R → R by

implies

|v|x ≤ R.

ψ(s) = max L(x, sv). x,|v|=1

Let ϕ1 : R → R be an even smooth strictly convex function such that ϕ1 (2R) = ϕ1 (−2R) = 0 and such that ϕ1 (s) > ψ(s) for |s| > R1 where R1 is a suﬃciently large positive number bigger than 2R. Deﬁne L1 (x, v) := ϕ1 (|v|x ) and L2 := max{L, L1 }. Then L2 coincides with L for those (x, v) with |v|x ≤ 2R and coincides with L1 for those (x, v) with |v|x > R1 . The Lagrangian L2 is strictly convex and may be approximated by a smooth strictly convex Lagrangian L3 such that L3 coincides with L for those (x, v) with |v|x ≤ R and coincides with L1 for those (x, v) with |v|x > R1 . We brieﬂy explain how to achieve this approximation given a strictly convex function f : Rn → R. This approximation can be done on each tangent space Tx N . The idea is to smooth out f using a convolution. Let

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η ∈ C ∞ (Rn ) be the function that equals C exp |x|21−1 if |x| < 1 and 0 if |x| ≥ 1. The constant C is selected so that Rn η dx = 1. For each ε > 0 set 1 x ηε (x) := n η . ε ε The functions ηε are C ∞ , their integrals equal one and their support is inside the ball of radius ε around the origin. The function η is called the standard molliﬁer. We deﬁne the molliﬁcation of f by f ε := ηε ∗ f . That is, f ε (x) = ηε (y − x)f (y) dy = ηε (y)f (x + y) dy. Rn

Rn

It is straightforward to verify that if f is strictly convex then f ε is also strictly convex. Moreover f ε is C ∞ and approximates f as ε → 0 uniformly on compact subsets [12, Appendix C]. It follows that the Hessian of f ε is positive deﬁnite. Suppose in addition that f is C ∞ in some open set U of the form, U = {x ∈ Rn : |x| < a + δ and |x| > b − δ}, with 0 < a < b and δ very small. Now let r1 and r2 be positive numbers such that r1 < a r2 . Now let hε := (1 − α)f ε + αf = f ε + α(f − f ε ). We have: 1. hε is C ∞ ; 2. hε coincides with f for |x| < r1 and |x| > r2 ; 3. for any ε suﬃciently small the function hε is strictly convex since the derivatives of f ε approximate the derivatives of f uniformly on the set r1 ≤ |x| ≤ a and b ≤ |x| ≤ r2 as ε → 0. Then hε gives the desired approximation of f . Let us complete now the proof of the proposition. Let ϕ2 : R → R be a smooth strictly convex function such that • ϕ2 (s) = ϕ1 (s) for |s| ≤ R1 + 1; • ϕ2 (s) = s2 for |s| > R2 , where R2 is a suﬃciently large positive number bigger than R1 + 1. Finally let L0 be the Lagrangian which coincides with L3 for those (x, v) with |v|x ≤ R1 and coincides with ϕ2 (|v|x ) for those (x, v) with |v|x ≥ R1 . Then, L0 is smooth, strictly convex, quadratic at inﬁnity and coincides with L for those (x, v) with |v|x ≤ R.

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Lemma 19 Assume that L and L0 agree on the set of those (x, v) satisfying EL (x, v) ≤ c(L) + 1. Then c(L0 ) = c(L). Proof . We use the following characterization of the critical value taken from [9] c(L) =

inf

sup H(x, dx u),

u∈C ∞ (M,R) x∈M

(6)

where H : T ∗ M → R is the Hamiltonian associated to L. Let 0 < ε < 1. Observe that H(x, p) < c(L) + ε implies H0 (x, p) = H(x, p) < c(L) + ε, where H0 is the Hamiltonian associated to L0 . By (6) there is u ∈ C ∞ (M, R) such that H(x, dx u) < c(L) + ε and hence H0 (x, dx u) = H(x, dx u) < c(L) + ε. Since ε is arbitrary, we obtain c(L0 ) ≤ c(L). Changing the roles of H0 , H, c(L0 ), c(L) we obtain c(L) ≤ c(L0 ), concluding the proof. Corollary 20 Given a convex superlinear Lagrangian L : T M → R and k ≥ c(L) there is a Lagrangian L0 convex and quadratic at inﬁnity, such that L and L0 agree on the set of those (x, v) satisfying EL (x, v) ≤ k + 1 and c(L0 ) = c(L). Proof . It follows from Proposition 18 and Lemma 19. Let us prove Theorem A. Take L0 such that c(L0 ) = c(L) according to the preceding corollary. Since L0 and L agree on a neighbourhood of E −1 (c(L)), then L and L0 have the same static set since the latter must be contained in E −1 (c(L)). Now if k > c(L0 ) = c(L), the barrier hk of L0 is +∞, and then AL0 +k satisﬁes the Palais-Smale condition by Theorem B. This means that EL−1 (k) is a Palais-Smale level. On the other hand, since the static set Σ(L) is not empty, hc < +∞, and then Theorem C completes the proof.

IV Proof of Theorem D The statement k > cu (L) ⇒ (H) was proved in [7] and could also be obtained using the corollary of Theorem A in [9]: we reparametrize the energy level in the universal covering to obtain a complete Finsler metric to which we apply Morse theory which is known to hold for Finsler geometry. ˜ . Recall Now we prove that (H) =⇒ k ≥ cu (L). Let L be the lift of L to T N that the Hamiltonian H associated to L is given by ˜ }, H(x, p) = sup {p(v) − L(x, v) | v ∈ Tx N

(7)

and the supremum is achieved at v such that p = Lv (x, v). Thus H(x, Lv (x, v)) = E(x, v). ˜ , let (bq , xq ) be the unique critical point of AL+k in R+ ×Ω(˜ Given q˜ ∈ N q0 , q˜). ˜ Write yq˜(t) := xq (t/bq˜) and deﬁne f : N → R by f (˜ q ) = AL+k (bq˜, xq˜) = SL+k (yq˜).

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The uniqueness of yq˜ implies that f is of class C 1 because it is a composition of the action functional with an analogue of the exponential map expq˜ on EL−1 (k). ˜. ˜ be a smooth curve such that α(0) = q˜ and α(0) Let α : (−ε, ε) → N ˙ = w ∈ Tq˜N If we diﬀerentiate f (α(s)) at s = 0 using the ﬁrst variation given by Lemma 4 and integration by parts we obtain: dq˜f (w) = Lv (˜ q , y˙ q˜(bq˜)) · w. This implies that q , y˙ q˜(bq˜)) = k. H(˜ q , dq˜f ) = E(˜ In [9] we showed that supx∈N˜ H(x, dx f ) . cu (L) = inf f ∈C ∞ (N,R) ˜ However the same proof in [9] shows that one can replace in the above equality ˜ , R) by C k (N ˜ , R) for any k ≥ 1 and we obtain the same critical value. We C ∞ (N ˜ see right away that k ≥ cu (L).

V Some results on Morse theory In this section we recall some results on Morse theory (cf. [29, 31]) that we shall use in the next section. Let X be an open set in a Hilbert manifold and f : X → R is a C 2 map. The following version of the minimax principle (Proposition 21 below) is a modiﬁcation of that of [20] (see also [32]). The only (minor) diﬀerence with the usual minimax principle is that our manifold X is not necessarily complete, but instead each set [f ≤ b] ⊆ X is complete. Observe that if the vector ﬁeld Y = −∇f is not globally Lipschitz, the gradient ﬂow ψt of −f is a priori only a local ﬂow. Given p ∈ X, t > 0 deﬁne α(p) := sup{ a > 0 | s → ψs (p) is deﬁned on s ∈ [0, a] }. We say that a function τ : X → [0, +∞) is an admisible time if τ is diﬀerentiable and 0 ≤ τ (x) < α(x) for all x ∈ X. Given and admisible time τ , and a subset F ⊂ X. deﬁne Fτ := { ψτ (p) (p) | p ∈ F }. Let F be a family of subsets F ⊂ X. We say that F is forward invariant if Fτ ∈ F for all F ∈ F and any admisible time τ . Deﬁne c(f, F) = inf sup f (p). F ∈F p∈F

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Proposition 21 Let f be a C 1 function satisfying the Palais-Smale condition. Assume also that F is forward invariant under the gradient ﬂow of −f . Suppose that there is b such that −∞ < c(f, F) < b < +∞ and such that the subset [f ≤ b] ⊆ X is complete. Then c(f, F) is a critical value of f . Proof . We borrow the following lemma from [31], Lemma 22 Suppose that f : X → R is C 1 , ψt is the gradient ﬂow of −f and the subset [a ≤ f ≤ b] ⊂ X is complete. Then the ﬂow ψt is relatively complete on [a ≤ f ≤ b], that is, if a ≤ f (p) ≤ b, then either α(p) = +∞ or f (ψβ (p)) ≤ a for some 0 ≤ β < α(p). Proof . Let ψt be the ﬂow of Y = −∇f . Then t2 f (ψt1 (p)) − f (ψt2 (p)) = − ∇f (ψs (p)) · Y (ψs (p)) ds = t1

t2

Y (ψs (p))2 ds.

t1

(8) Moreover, using the Cauchy-Schwartz inequality, we have that t2 t2 2 Y (ψs (p)) ds ≤ |t2 − t1 | Y (ψs (p))2 ds. d(ψt1 (p), ψt2 (p))2 ≤ t1

t1

Thus d(ψt1 (p), ψt2 (p))2 ≤ |t2 − t1 | | f (ψt1 (p)) − f (ψt2 (p)) |.

(9)

Let I = [0, α[ a maximal interval of deﬁnition of t → ψt (p). Suppose that a ≤ f (ψt (p)) ≤ b for 0 ≤ t < α < ∞. Let tn ↑ α. By inequality (9), n → ψtn (p) is a Cauchy sequence on [a ≤ f ≤ b]. Since [a ≤ f ≤ b] is complete, it has a limit q = limn ψtn (p) = ψα (p). Since f is C 1 , we can extend the solution t → ψt (p) at t = α. This contradicts the deﬁnition of α. Write c = c(f, F). We shall prove that for all ε > 0 there is a critical value cε such that c − ε < cε < c + ε. Then, using ε = n1 , the Palais-Smale condition implies that c is a critical value. Suppose that there are no critical points on A(ε) := [c − ε ≤ f ≤ c + ε]. The Palais-Smale condition implies that there is δ > 0 such that ∇f (p) > δ for all p ∈ A(ε). If f (p) ≤ c + ε, let τ (p) := inf{ t > 0 | s → ψs (p) is deﬁned on [0, t] and f (ψt (p)) ≤ c − ε }. Since s → f (ψs (p)) is decreasing, by Lemma 22, either τ (p) = +∞ or τ (p) < α(p) and f (ψτ (p) (p)) = c − ε. Since ∇f > δ on A(ε), by equation (8), c − ε ≤ f (ψt (p)) ≤ f (p) − t δ 2 ≤ c + ε − t δ 2

for 0 ≤ t ≤ τ (p).

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Thus τ (p) ≤ 2ε/δ 2 for all p ∈ A(ε); in particular this shows that τ (p) cannot be +∞. Observe that the implicit function theorem applied to the function F (x, t) = ∂ f (ψt (x)) implies that p → τ (p) is diﬀerentiable because ∂t F (x, t) = ∇f · X = − ∇f < −δ. By the deﬁnition of c(f, F) there exists F ∈ F such that sup f (x) ≤ c + ε.

x∈F

Then sup f (x) = sup f (ψτ (p) (p)) ≤ c − ε.

x∈Fτ

x∈F

Since τ is an admisible time, this contradicts the deﬁnition of c(f, F). From Proposition 21 we derive, taking F to be the family of sets of the form {p} with p ∈ X, the following Corollary 23 Let f : X → R be a C 2 function for which [f ≤ b] is complete for every b. Suppose that f is bounded from below and satisﬁes the Palais-Smale condition. Then f has a global minimum. It is convenient to obtain a further reﬁnement of the corollary above which will be useful in the next section. Corollary 24 Let X be a connected manifold. Let f : X → R be a C 2 function for which [f ≤ b] is complete for every b, satisfying the Palais-Smale condition. Suppose that p1 is a relative minimizer of f and suppose that f admits a second relative minimizer p2 = p1 . Then, 1. either there exists a critical point p of f which is not a relative minimum or 2. p1 and p2 can be connected in any neighborhood of the set of relative minimizers p of f with f (p) = f (p1 ). Necessarily then f (p1 ) = f (p2 ). Proof . A detailed proof can be found in [32, Theorem 10.3]. The idea is to apply again the minimax principle; this time F is the family of subsets of the form x([0, 1]) where x is a curve joining p1 to p2 . We conclude this section with the following suggestive remark. Let F be the family of all subsets F of T ∗ M of the form F = {(x, dx u) : x ∈ M } where u ∈ C ∞ (M, R). Then c(H, F) = inf

sup H(x, p)

F ∈F (x,p)∈F

=

inf

sup H(x, dx u)

u∈C ∞ (M,R) x∈M

= c(L), where the last equality is proved in [9]. Hence Ma˜ n´e’s critical value resembles a critical value of H as a smooth function even though in general it is not a critical

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value of H as a smooth function. This resemblance shows that the name “critical value” for c(L) is appropriate and explains the (intentional) similarities in our notation.

VI Applications to the reduced action functional VI.1

Periodic Orbits

In this subsection we show the existence of periodic orbits in every nontrivial free homotopy class and every energy level above cu (L), where L : T N → R is a convex Lagrangian quadratic at inﬁnity and the manifold N is compact. Let σ be a nontrivial free homotopy class of closed loops in a compact manifold N . As before, deﬁne Ωσ as the set of elements of R+ × H 1 (Rk ) of the form (b, x) where x([0, 1]) ⊂ N , x(0) = x(1) and x ∈ σ. Let L : T N → R be a convex and superlinear Lagrangian. The action AL : R+ × Ωσ → R is deﬁned by 1 AL (b, x) = b(L(x(t), x(t)/b)) ˙ dt. 0

The previous discussion about the Palais-Smale condition translates to the case of free loops in a non trivial free homotopy class with only minor changes. In particular if L is quadratic at inﬁnity and k > cu (L), then AL+k satisﬁes the Palais-Smale condition on R+ × Ωσ . However one has to use the compactness of N in the following lemma. Lemma 25 Let k > cu (L) and (bn , xn ) ∈ R+ × Ωσ such that AL+k (bn , xn ) is bounded. Then {(bn , xn )} has a converging subsequence in the C 0 topology. Let us prove ﬁrst: Lemma 26 If k ≥ cu (L), then inf AL+k > −∞.

R+ ×Ωσ

Proof . Fix x0 ∈ σ and let R be twice the diameter of N . Take C > 0 such that SL+k (z) ≤ C for all curves z : [0, R] → N such that |z| ˙ ≤ 1. Let (b, x) be an arbitrary element of R+ × Ωσ . Let x1 (t) := x(t/b). Then there is a curve z, parametrized by arc length, −1 with length not greater than R such that γ = x−1 is homotopic to 0 ∗ z ∗ x1 ∗ z zero. Since γ lifts as a closed curve we have SL+k (γ) ≥ 0.

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Then SL+k (x1 ) ≥ −SL+k (x0 ) − 2C. Since SL+k (x1 ) = AL+k (x), we are done. Proof of Lemma 25. Observe that if k > c := cu (L) then, AL+k (bn , xn ) = AL+c (bn , xn ) + (k − c)bn , hence if AL+k (bn , xn ) is bounded it follows that {bn } is also bounded. Since the manifold N is compact by the same arguments in the proof of Tonelli’s Theorem in Ωσ and by the Arzela-Ascoli Theorem there is a convergent subsequence of xn in the C 0 topology (cf. [8, 6, 25]) and hence Lemma 25 follows. Using Corollary 23 of the previous section we can obtain right away the following theorem: Theorem 27 Let N be a closed manifold and let L : T N → R be a convex Lagrangian quadratic at inﬁnity. Then for every k > cu (L) and every nontrivial free homotopy class σ there is a periodic orbit of L in σ having energy k that minimizes the action of AL+k on R+ × Ωσ . We conclude this subsection by showing: Proposition 28 Let cu (L) < k < ca (L). Then there is a periodic orbit γ of L which is homologically trivial and also has SL+k (γ) < 0. Proof . Since k < ca (L) there is a closed curve α such that SL+k (α) < 0. Such a curve cannot be homotopically trivial, otherwise, we lift it to the universal covering as a closed curve having negative action, contradicting the condition cu (L) < k. Let σ be the (non trivial) homotopy class of α. By the previous theorem AL+k : R+ × Ωσ → R has a global minimum (b, x) which is a periodic orbit with energy k. If we set y(t) := x(t/b), then AL+k (b, x) = SL+k (y) ≤ SL+k (α) < 0, as desired.

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Periodic orbits of Anosov energy levels

In this last subsection we show the following theorem. It was proved for geodesic ﬂows by W. Klingenberg [21]. Theorem 29 Let N be a closed manifold and let L : T N → R be a convex superlinear Lagrangian. Suppose that the Euler-Lagrange ﬂow of L restricted to the regular energy level E−1 (k) is Anosov. Then in any non-trivial free homotopy class there is a unique closed orbit of L with energy k. Proof . We proved in [9] that if the Euler-Lagrange ﬂow of L restricted to the regular energy level E−1 (k) is Anosov then k > cu (L). Let σ be a non-trivial free homotopy class. Without loss of generality we can assume that L is quadratic at inﬁnity and hence AL+k satisﬁes the Palais-Smale condition on R+ × Ωσ . By Theorem 27 we know that there exists a closed orbit of L with energy k that minimizes AL+k on R+ × Ωσ . The next lemma shows in fact that every closed orbit with energy k in the homotopy class σ has this minimizing property. We will postpone its proof until we complete the proof of the theorem. Lemma 30 Every closed orbit of L with energy k in the homotopy class σ is a minimum of AL+k on R+ × Ωσ . Now suppose that we have two geometrically diﬀerent closed orbits γ1 and γ2 of L with energy k in the free homotopy class σ. By Lemma 30 all the critical points of AL+k on R+ × Ωσ are minimizers and hence in Corollary 24 the second alternative holds. This contradicts that γ1 (or γ2 ) is hyperbolic. Proof of Lemma 30. Let γ be a closed orbit of L with energy k in the free homotopy class σ. Let W s (γ) be the weak stable leaf of γ for the corresponding Hamiltonian s (γ) be its lift to the universal covering. We proved in [9] that ﬂow and let W s →R W (γ) is the graph of an exact 1-form. This means that there exists u : N } and since W s (γ) is contained in the energy s (γ) = {(x, dx u) : x ∈ N such that W level k we have that H(x, dx u) = k for all x ∈ N . By the relation between H and L we have L(x, v) − dx u(v) + k ≥ 0,

(10)

and equality holds if and only if Lv (x, v) = dx u, i.e. when (x, v) belongs to the s (γ) under the Legendre transform which is the same as the inverse image of W lift of the weak stable leaf for the Euler-Lagrange ﬂow on T N . be a fundamental domain for the action of π1 (N ). Let γ Let D ⊂ N be a lift with initial point in D. Let η : [0, T1 ] → N of the closed curve γ : [0, T ] → N to N with initial be a closed curve in the free homotopy class σ and let η be a lift to N point in D. Let a : [0, 1] → N be a curve such that a(0) = γ (0) and a(1) = η(0).

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Ann. Henri Poincar´e

→ N be the covering transformation that takes γ Let ϕ : N (0) to γ (T ). Let be the curve ϕn a−1 . Using (10) we get: bn : [0, 1] → N L − du + k ≥ 0 = L − du + k. a∗ η ∗···∗ϕn−1 η ∗bn

γ ∗···∗ϕn−1 γ

Observe that the curves a ∗ η ∗ · · · ∗ ϕn−1 η ∗ bn and γ ∗ · · · ∗ ϕn−1 γ have the same end points, hence (L + k) + (L + k) + n (L + k) ≥ n (L + k) a

Since

bn

η

bn

γ

(L + k) is independent of n, diving by n and letting n → ∞ we obtain: SL+k (η) ≥ SL+k (γ).

VII Appendix: an example of a Lagrangian with hc = +∞. Let L : T R2 → R be L(x, v) = 12 |v|2 + ψ(x), where | · | is the euclidean metric on 1 R2 and ψ(x) is a smooth function with ψ(x) = |x| for |x| ≥ 2, ψ > 0 and ψ(x) = 2 for 0 ≤ |x| ≤ 1. Then c(L) = − inf ψ = 0, because if γn : [0, Tn ] → R2 is a smooth closed curve with length ?(γn ) = 1, |γn (t)| ≥ n for t ∈ [0, Tn ] and energy E(γn ) = 12 |γ˙ n |2 − ψ(γn ) ≡ 0, then c(L) ≥ − inf AL (γn ) ≥ − n>0

=− 0

Tn

0

Tn 1 2

|γ˙ n |2 + ψ(γn )

|γ˙ n |2 ≥ − n2 −→ 0.

On the other hand, c(L) = − inf {AL (γ) | γ closed } ≤ 0, because L > 0. Observe that since L > 0 and on compact subsets of R2 , L > a > 0, then we have that dc (x, y) = Φc (x, y) > 0 for all x, y ∈ R2 with x = y. Hence Σ(L) = ø.

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Suppose that hc (0, 0) < +∞. For every x ∈ R2 there exists a vector v ∈ Tx R2 such that the solution of the Euler-Lagrange equation with initial conditions (x, v) is semistatic in forward time [9]. Let v be such a vector in T0 R2 and write xv (t) = (r(t), θ(t)) in polar coordinates about the origin 0 ∈ R2 . Then lim inf t→+∞ r(t) = +∞ because otherwise the orbit of v would lie on a compact subset of E −1 (0) and then ø = ω − limit(v) ⊆ Σ(L) = ø (see [8] for a proof of the fact that the ω − limit set of a forward semistatic orbit is contained in the static set). Note that for any t with r(t) ≥ 2 we have: |x˙ v (t)| =

and L(φt v) = |x˙ v (t)|2 =

2 r(t)

2 r(t)

|x˙ v (t)|.

Let Tn → +∞ be such that r(Tn ) → +∞. Hence there is n0 such that for all n ≥ n0 , r(Tn ) ≥ 2. Since L + c = L > 0, then +∞ hc (0, 0) ≥ (L(φt (v)) + c) dt 0

≥ lim sup Tn

Tn0

≥ lim sup Tn

Tn

Tn0

≥ lim sup Tn

Tn

Tn

Tn0

r(Tn )

= lim sup n

Tn0

2 r(t)

|x˙ v (t)| dt

2 r(t)

|r| ˙ dt

2 r

r˙ dt

2 r

dr = +∞.

Acknowledgments M. Paternain thanks the CIMAT, Guanajuato, for hospitality while this work was in progress. We thank the referee for pointing out a mistake in a previous version of this manuscript and for various other corrections and suggestions for improvement.

References [1] R. Abraham, J. Marsden, Foundations of Mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced book program, Reading Mass., 1978. [2] A. Bahri, I. Taimanov, Periodic orbits in magnetic ﬁelds and Ricci curvature of Lagrangian systems, Trans. Amer. Math. Soc. 350 (1998) 2679–2717. [3] V. Bangert, Mather sets for twists maps and geodesics on tori, Dynamics reported, Vol. 1, 1–56, Teubner, Stuttgart, 1988.

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[4] V. Bangert, Minimal geodesics, Ergod. Th. and Dynam. Sys. 10 (1989) 263– 286. [5] S. Bolotin, Homoclinic orbits in invariant tori of Hamiltonian systems, Dynamical systems in classical mechanics, V.V. Kozlov ed, Amer. Math. Soc. Transl. Ser. 2, 168, Amer. Math. Soc., Providence, RI, 1995, 21–90. [6] G. Contreras, J. Delgado, R. Iturriaga, Lagrangian ﬂows: the dynamics of globally minimizing orbits II, Bol. Soc. Bras. Mat. Vol. 28, N.2, (1997) 155– 196. [7] G. Contreras, R. Iturriaga, Convex Hamiltonians without conjugate points, Ergod. Th and Dynam. Sys. 19 (1999) 901–952. [8] G. Contreras, R. Iturriaga, Global minimizers of autonomous Lagrangians, 22 Col´ oquio Bras. de Mat. IMPA. Rio de Janeiro, 1999. [9] G. Contreras, R. Iturriaga, G.P. Paternain, M. Paternain, Lagrangian graphs, minimizing measures, and Ma˜ n´e’s critical values, Geom. and Funct. Anal. 8 (1998) 788–809. [10] M.J. Dias Carneiro, On minimizing measures of the action of autonomous Lagrangians, Nonlinearity 8 (1995) 1077–1085. [11] J.J. Duistermaat, On the Morse index in variational calculus, Advances in Math. 21 (1976) 173–195. [12] L.C. Evans, Partial diﬀerential equations, Graduate Studies in Mathematics 19, American Mathematical Society, Providence, Rhode Island, 1998. [13] L.C. Evans, R.F. Gariepy, Measure theory and ﬁne properties of functions, Studies in Advanced Mathematics, CRC Press, 1992. [14] A. Fathi, Th´eor`eme KAM faible et Th´eorie de Mather sur les systems Lagrangiens, C.R. Acad. Sci. Paris, t. 324, S´erie I (1997) 1043–1046. [15] A. Fathi, Solutions KAM faibles conjugu´ees et barri`eres de Peierls, C.R. Acad. Sci. Paris, t. 325, S´erie I (1997) 649–652. [16] A. Fathi, Orbites h´et´eroclines et ensemble de Peierls, C. R. Acad. Sci. Paris, t. 326, S´erie I (1998) 1213–1216. [17] A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris, t. 327, S´erie I (1998) 267–270. [18] A. Fathi, Dynamique Lagrangiene, Lecture Notes, ENS Lyon, 1998. [19] G.A. Hedlund, Geodesics on a two-dimensional Riemannian manifold with periodic coeﬃcients, Ann. of Math. 33 (1932) 719–739.

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[20] H. Hofer, E. Zhender, Symplectic invariants and Hamiltonian dynamics, Birkh¨auser, 1994. [21] W. Klingenberg, Riemannian manifolds with geodesic ﬂow of Anosov type, Ann. of Math. 99 (1974) 1–13. [22] R. Ma˜ n´e. Global Variational Methods in Conservative Dynamics. 18o Coloquio Bras. de Mat. IMPA. Rio de Janeiro, 1991. [23] R. Ma˜ n´e, Lagrangian ﬂows: the dynamics of globally minimizing orbits, International Congress on Dynamical Systems in Montevideo (a tribute to Ricardo Ma˜ n´e), F. Ledrappier, J. Lewowicz, S. Newhouse eds, Pitman Research Notes in Math. 362 (1996) 120–131. Reprinted in Bol. Soc. Bras. Mat. Vol 28, N. 2, (1997) 141-153. [24] R. Ma˜ n´e, Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity 9 (1996) 273–310. [25] J. Mather, Action minimizing measures for positive deﬁnite Lagrangian systems, Math. Z. 207 (1991) 169–207. [26] J. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier 43 (1993) 1349–1386. [27] M. Morse, A fundamental class of geodesics on any closed surface of genus greater than one, Trans. Amer. Math. Soc. 26 (1924) 25–60. [28] S.P. Novikov, Hamiltonian formalism and a multivalued analogue of Morse theory, Russian Math. Surveys 37 (1982) 1–56. [29] R. Palais, Morse theory on Hilbert manifolds, Topology 2 (1963) 299–340. [30] G. P. Paternain, M. Paternain, Critical values of autonomous Lagrangian systems, Comment. Math. Helvetici 72 (1997) 481–499. [31] S. Smale, Morse theory and a non-linear generalization of the Dirichlet problem, Ann. of Math. 80 (1964) 382–396. [32] M. Struwe, Variational methods. Applications to nonlinear partial diﬀerential equations and Hamiltonian systems, 2nd rev. and exp. ed. (English) Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 34. Berlin: Springer, 1996. [33] I. Taimanov, Non-self-intersecting closed extremals of multivalued or not everywhere positive functional, Math. USSR-Izv., 38 (1992) 359–374. [34] I. Taimanov, Closed extremals on two-dimensional manifolds, Russian Math. Surveys, 47 (1992) 163–211.

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G. Contreras, R. Iturriaga, G.P. Paternain CIMAT A.P. 402, 36000 Guanajuato. Gto. M´exico email : [email protected] [email protected] [email protected]

Ann. Henri Poincar´e

M. Paternain Centro de Matem´atica Facultad de Ciencias Igu` a 4225 11400 Montevideo Uruguay [email protected]

Gonzalo Contreras was partially supported by a CONACYT-M´exico grant # 28489-E and by CNPq-Brazil Renato Iturriaga was partially supported by a CONACYT-M´exico grant # 28489E Miguel Paternain was partially supported by a CSIC-Uruguay grant Gabriel Paternain is on leave from Centro de Matem´ atica, Facultad de Ciencias, Igu´ a 4225, 11400 Montevideo, Uruguay. Communicated by Jean Pierre Eckmann submitted 25/06/99, accepted 23/02/00

Ann. Henri Poincar´ e 1 (2000) 685 – 714 c Birkh¨ auser Verlag, Basel, 2000 1424-0637/00/040685-30 $ 1.50+0.20/0

Annales Henri Poincar´ e

The Wigner Function for General Lie Groups and the Wavelet Transform S. Twareque Ali, Natig M. Atakishiyev, Sergey M. Chumakov, Kurt Bernardo Wolf Abstract. We build Wigner maps, functions and operators on general phase spaces arising from a class of Lie groups, including non-unimodular groups (such as the aﬃne group). The phase spaces are coadjoint orbits in the dual of the Lie algebra of these groups and they come equipped with natural symplectic structures and Liouville-type invariant measures. When the group admits square-integrable representations, we present a very general construction of a Wigner function which enjoys all the desirable properties, including full covariance and reconstruction formulae. We study in detail the case of the aﬃne group on the line. In particular, we put into focus the close connection between the well-known wavelet transform and the Wigner function on such groups.

I Introduction: the original Wigner function The notion of a quantum-mechanical phase space, where the evolution of a state can be described by a (quasi-)probability distribution function over classical (‘cnumber’) coordinates of position and momentum (q, p) ∈ R2 , hinges on the function introduced by Wigner [25] in 1932. Given two wavefunctions φ(x) and ψ(x), of the space coordinate x ∈ R, their Wigner function on phase space (q, p) is 1 QM W (φ, ψ|q, p; h) = dx φ(q − 12 x) e−2πi xp/h ψ(q + 12 x). (1) h R Here h is the Planck constant, which ﬁxes the scale (and units) of the two classical coordinates, incorporating the uncertainty principle. When φ = ψ, we write W QM (ψ|q, p; h) = W QM (ψ, ψ|q, p; h) and speak of the Wigner function of the state ψ on phase space. While phase-space probability functions for classical systems are non-negative but otherwise arbitrary, a Wigner function is much more restrictive and (except for Gaussian functions) does have (usually small) regions where its values are negative. For this reason it is properly called a quasi - probability distribution function and named the Wigner function for short. Nevertheless, issues of quantum measurement can be discussed adequately in terms of the Wigner function, and it serves well in formulating a picture of quantum mechanics at par with the Schr¨ odinger, Heisenberg and Feynman formulations [16]. More recently, it has served as a ﬁne tool for quantum optics [1], [19], since using it density matrices (i.e., ‘impure states’, or positive trace-class Hilbert-Schmidt operators) can also be eﬀectively plotted in terms of position-momentum or action-angle phase spaces.

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Ann. Henri Poincar´ e

The analysis of signals in time and frequency is crucial for radar technology, so quite early analogues of the Wigner function, such as the Woodward self-ambiguity function [28], were extensively studied and applied [23], [8]; in this case, the unit of phase space area h in (1), is set equal to one. Also in the context of monochromatic, paraxial wave optics, the Wigner function is easily used for the analysis of onedimensional signals such as a code bar [6], since a simple lens arrangement will produce on a screen an intensity ﬁeld which is closely related to its Wigner function [18]. The two canonically conjugate coordinates in this case are position on the screen and spatial frequency (optical momentum), with the fundamental unit of phase space area being now λ, the wavelength of the light. Recently, the grouptheoretical study of various optical and quantum mechanical models, where the observable quantities of a system are the eigenvalues of the generators of a Lie group [27], [21], [5], has suggested a substantial generalization of the concepts of Wigner functions and of phase space. In this article we develop a fairly rigorous group-theoretic foundation for the study of Wigner functions for a class of Lie groups which admit square-integrable, unitary irreducible representations. Typically these groups are related to the underlying symmetry and dynamics of the physical system, and incorporate its speciﬁc geometry. The group has a natural action on the dual space of its Lie algebra, called the coadjoint action, and the orbits of this action are possible phase spaces of the system. These coadjoint orbits carry natural symplectic structures which make them geometrically similar to classical phase spaces. They carry invariant measures under the group action, which are analogues of the well-known Liouville measure on ordinary phase space. The original Wigner function (1) arises from a square-integrable representation of the Heisenberg-Weyl group —the group of the canonical commutation relations. This has pointed the direction to follow toward a generalization of the notion of the Wigner function to other Lie groups [27]. As an application of the general theory, we treat in detail the aﬃne group, whose Wigner function is shown to be closely related to the well-known wavelet transform. This two-parameter group is non-unimodular, i.e., its left- and rightinvariant Haar measures are distinct. The aﬃne group, being non-unimodular, makes it imperative that the mathematical properties of square-integrable representations be used in order to arrive at an adequate generalization of the Wigner phase space formalism. In Section II we present a ﬁrst approximation to the generalization of Wigner’s phase space formalism, developed in Refs. [27], [21], and [5], to display the desirable properties of a phase space function. To make this paper self contained, the following three Sections organize the required mathematical fundamentals and thus the notation. Section III recalls the exponential map between a Lie algebra and group, to deﬁne the coadjoint action of the group on the dual of the algebra, providing the ‘c-number’ arguments of the Wigner function, such as those on the right-hand side of Eq. (1). Section IV looks at various natural representations of the group, associated to the adjoint and coadjoint actions. Finally, Section V discusses the class of group representations for which Wigner functions can be built and presents the

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objects whose Wigner functions are to be determined; these are Hilbert-Schmidt operators (‘density matrices’), including ‘pure state’ wavefunctions. In Section VI we deﬁne the Wigner map and function of Hilbert-Schmidt operators on the Hilbert space carrying the group representation, and look at their properties. In particular, we obtain the formula for the reconstruction of the density matrix from its Wigner function. In Section VII we explain the relationship between the Wigner transform, coherent states of the group and the generalized wavelet transform. Finally, some of the key expressions are written down in coordinate terms, for comparison with known formulae. The speciﬁc case of the two-dimensional aﬃne group is examined in Section VIII and its Wigner transform is calculated there as an explicit example and application. In Section IX, we display for the aﬃne group the exact relation between the Wigner map (the Wigner function with one ﬁxed ‘mother’ wavelet) and the wavelet transform. Section X is devoted to showing how the original Wigner function in (1) can be derived using the general theory as applied to the Heisenberg-Weyl group. We conclude in Section XI with some general comments and indicate some directions for further research. The Wigner function does not contain more information than the original signal or wavefunction. But, in the same way that a musical score shows through notes on the pentagram the essence of a tune better than a complete pressure-wave register of a performance, the information is presented in a form kinder to human comprehension.

II The Wigner operator: a ﬁrst generalization In Reference [27] it was proposed that the Wigner function (1) of φ, ψ, or of the density matrix ρ = |ψφ|, be written as the matrix element of a ‘Wigner operator’ W, which is a (measurable, operator-valued) function on phase space, W (φ, ψ | X ∗ ) = φ | W(X ∗ ) | ψ,

W (ρ | X ∗ ) = Tr[ρ W(X ∗ )],

(2)

where X ∗ is an element of the dual of the Lie algebra whose components in a chosen basis provide the ‘c-number’ arguments of the Wigner function.

II.1

The Wigner operator

The Wigner operator is, roughly speaking, the Fourier transform of the (Hilbert space representation of) the group elements g ∈ G, previously written in terms of the coordinates ξ = {ξi }ni=1 of X ∗ as ), dµ(g [x ]) e−iξ· x g [x ], g [x ] = exp(ix · X (3) W(ξ ) = RG

= {Xn }N where X n=1 is a basis of generators (consisting of Hilbert space operators) for the Lie algebra and the square brackets indicate that the arguments are the

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polar coordinates of the group, with domain x ∈ RG ⊂ RN . The measure dµ(g ) is the left invariant Haar measure (also the right invariant Haar measure, since in this expression the group had been assumed to be unimodular). For the HeisenbergWeyl algebra [26], Eqs. (2) and (3) yield the Wigner function (1), with the common = {Q, P, Λ} and Λ = h1, in the irreducible representation Schr¨ odinger operators X labelled by h ∈ R − {0}; the corresponding c-number coordinates are ξ = {q, p, h}. With abuse of notation [24] one may think of this operator as (2π)3 δ(Q − q) δ(P − p) δ(Λ − h). The following subsections highlight the basic properties of the construction (2)–(3) which show that it is a proper generalization of the original Wigner function of quantum mechanics [16], [14].

II.2

Sesquilinearity and reality

In a unitary representation of the group, the Wigner operator (3) is (formally) selfadjoint. Hence the Wigner functions (1) and (2) are sesquilinear in their Hilbert space arguments, i.e., W (φ, aψ1 + bψ2 | X ∗ ) = aW (φ, ψ1 | X ∗ ) + bW (φ, ψ2 | X ∗ ), W (ψ, φ | X ∗ ) = W (φ, ψ | X ∗ ).

(4) (5)

It follows that the Wigner function of a single wave function is real, W (ψ | X ∗ ) = W (ψ, ψ | X ∗ ) ∈ R. For a sum of functions (such as Schr¨ odinger-cat states), W (φ + ψ | X ∗ ) = W (φ | X ∗ ) + W (ψ | X ∗ ) + 2 Re W (φ, ψ | X ∗ ).

(6)

There is holographic information in the interference cross-term.

II.3

Covariance

Translations and linear transformations of classical phase space X ∗ = (q, p) → T(q, p), are equivalent to the canonical transforms T of the wavefunctions, for the original Wigner function [12]. More generally [5], under an automorphism group of the Lie algebra, the form of (3) implies the covariance property W (ψ | TX ∗ ) = W (T ψ | X ∗ ).

(7)

This formula is important because it ensures the correspondence between classical and quantum phase spaces.

II.4

Overlap and reconstruction

From the Wigner function one can recover the wavefunction (up to an overall phase) or the density matrix by exploiting the overlap and reconstruction relations.

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For the traditional case of the Heisenberg-Weyl algebra (1) the overlap condition is dq dp W (φ, ψ | q, p) W (υ, ω | q, p) = φ | ω υ | ψ. (8) R2

This formula is important because it shows the passage from the Wigner to the Schr¨ odinger or Heisenberg formalisms of quantum mechanics, and holds correspondingly between the Wigner function and wavefunctions on Lie groups. From Eq. (8) also follow various properties of marginal distributions, or projections over some of the phase-space coordinates, and of moments [16], e.g., dp W (φ, ψ | q, p) = φ(q) ψ(q). (9) R

III Lie groups, Lie algebras and their duals In this and the next two sections we collect some relevant notions and results on Lie groups and some of their representations. While most of the concepts and results presented here are known, especially to people working in representation theory, they may not be generally familiar —at least not in the setting in which we need them. Moreover, since the terminology for analogous concepts in quantum optics is often quite diﬀerent, we go into some detail to explain the mathematical notions and terminology, particularly since the Wigner function has such wide applications in wave and quantum optics.

III.1

Exponential map; adjoint action

Let G be a Lie group generated by a Lie algebra g. The Lie algebra is a linear vector space, and the exponential map from g to G, exp X = g,

X ∈ N0 ⊂ g,

g ∈ Ve ⊂ G,

(10)

is a topological homeomorphism between an open set N0 around the origin 0 in g, and an open set Ve around the identity element e in G. We denote the inverse map from G to g by log g = X,

X ∈ N0 ⊂ g,

g ∈ Ve ⊂ G.

(11)

It is well known [15] that the neighbourhood Ve in G can be chosen to be symmetric, i.e., if g ∈ Ve , then also g −1 ∈ Ve . In this paper we shall also assume that Ve can be chosen to be an open dense set in G, such that its complement has (Haar) measure zero. (A group such as Sp(2n, R) does not have this property.) There is a natural action of the group G on its Lie algebra g, called the adjoint action, Ad. For g, go ∈ Ve ⊂ G such that go g go−1 ∈ Ve , this action maps X ∈ g to Y = Adgo X ∈ g through exp(Adgo X) = exp Y = go (exp X)go−1 .

(12)

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If G is a matrix group, so that g also consists of matrices, then Adgo X = go X go−1 .

III.2

(13)

Dual space, coadjoint action

The Lie algebra g is a vector space, and hence has a dual denoted g∗ , also a vector space, consisting of all linear functionals on g. Let X ∗ ; X denote the dual paring between X ∗ ∈ g∗ and X ∈ g. In other words, X ∗ ; X is simply the value X ∗ (X) of the functional X ∗ , computed on the vector X. If we introduce a basis {Xi }ni=1 in g (n being its dimension), and the dual basis {X ∗i }ni=1 in g∗ (i.e., X ∗i ; Xj = δji ), n n then for X = i=1 xi Xi and X ∗ = i=1 ξi X ∗i , with xi and ξi real numbers, we get n X ∗ ; X = xi ξi = x · ξ. (14) i=1

Also in these coordinates, the (translation invariant) Lebesgue measures on g and g∗ assume respectively the forms dX → dx = dx1 ∧ dx2 ∧ . . . ∧ dxn ,

dX ∗ → dξ = dξ1 ∧ dξ2 ∧ . . . ∧ dξn .

(15)

The coadjoint action Adgo of a group element go ∈ Ve ⊂ g on a vector X ∈ g∗ is deﬁned by the relation ∗

Adgo X ∗ ; X = X ∗ ; Adgo−1 X.

(16)

This is the dual action on g∗ induced by the adjoint action. In terms of the bases introduced in g and g∗ as above, if the adjoint action Adgo has a matrix representation, then the representation of the coadjoint action Adgo is given by the inverse transposed matrix.

III.3

Coadjoint orbits and invariant measures

If X0∗ is a ﬁxed element of g∗ , the set of all elements of g∗ of the type Adg X0∗ , g ∈ G, is its orbit under G, denoted O∗ . Orbits under the coadjoint action are symplectic manifolds, i.e., smooth surfaces in g∗ , generally of lower dimension than g∗ , which have a structure similar to classical phase spaces. Moreover, these orbits O∗ come naturally equipped with measures dΩ(X ∗ ) (analogues of the Liouville measure on ordinary phase space), which are invariant under the coadjoint action of G [17]. Namely, (17) dΩ(X ∗ ) = dΩ(Adg X ∗ ), X ∗ ∈ O∗ , for all g ∈ G. An orbit O∗ of this type is called a coadjoint orbit. Clearly, two such orbits are either distinct or else they coincide entirely. The collection of all coadjoint orbits exhausts g∗ and we may write Oλ∗ = g∗ , (18) λ∈J

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where λ could be a discrete or continuous parameter, or set of parameters that characterize the orbits, and J is the appropriate index set. We denote the invariant measure on the coadjoint orbit Oλ∗ by dΩλ . In view of (18), any element X ∗ in the vector space g∗ belongs to some orbit ∗ Oλ and hence may be written as Xλ∗ , to display the orbit dependence explicitly. We shall assume the following decomposition of the Lebesgue measure on g∗ : dX ∗ = dκ(λ) σλ (Xλ∗ ) dΩλ (Xλ∗ ),

Xλ∗ ∈ Oλ∗ ,

(19)

where κ is the appropriate measure on the parameter space J and σλ a positive, non-vanishing function on the orbit Oλ∗ . Depending on the nature of J, the measure dκ can be discrete or continuous, or can have both a discrete and a continuous part. Such a decomposition will hold for all the cases of interest to us, while more generally, it is a statement of a certain regularity condition [11].

IV Invariant Haar measures, the regular and coadjoint representations Every Lie group carries a left- and a right-invariant Haar measure, dµ and dµr respectively. These satisfy dµ (g) = dµ (go g),

dµr (g) = dµr (ggo ), g ∈ G,

(20)

for ﬁxed but arbitrary g0 ∈ G. In general the left and right Haar measures are diﬀerent (though equivalent in the sense of measures). However, for unimodular groups (including compact, abelian, certain semidirect products, etc.) they turn out to be the same, i.e., dµ (g) = dµr (g). Note that Na˘ımark groups, including solvable groups such as the two-parameter aﬃne group of translations and dilatations, are not unimodular.

IV.1

Modular function; left- and right-regular representations

Since generally, the left and right Haar measures are measure-theoretically equivalent, they are related through a modular function ∆(g). This is a positive and real-valued measurable function on G, satisfying dµ (g) = ∆(g) dµr (g) = dµr (g −1 ).

(21)

The modular function is also a group homomorphism: ∆(g1 g2 ) = ∆(g1 )∆(g2 ), for all g1 , g2 ∈ G and ∆(e) = 1. In what follows, we shall only need the left Haar measure dµ and so write it simply as dµ. (All conclusions can be formulated equivalently in terms of the right Haar measure.) Using the Haar measure dµ one can build two unitary representations of G: the left- and right-regular representations. Consider the Hilbert space L2 (G, dµ),

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of all measurable, complex-valued functions f on G, satisfying |f (g)|2 dµ(g) < ∞. G

On this Hilbert space we deﬁne a representation of G by unitary operators U (g), g ∈ G, such that (U (g)f )(g ) = f (g −1 g ),

f ∈ L2 (G, dµ),

(22)

holding for almost all g ∈ G with respect to the measure dµ. This is called the left-regular representation; its unitarity is trivially checked using the invariance properties of the measure dµ. Similarly, we deﬁne the right-regular representation Ur (g) on the same Hilbert space L2 (G, dµ), where it is also unitary, and given by (Ur (g)f )(g ) = [∆(g)] 2 f (g g), 1

f ∈ L2 (G, dµ),

(23)

again for almost all g ∈ G with respect to dµ. It is easy to verify that the leftand right-regular operators U (g) and Ur (g) commute. Moreover, the two representations are unitarily equivalent by the map M on f ∈ L2 (G, dµ) given by 1 (M f )(g) = [∆(g)]− 2 f (g −1 ). This map is unitary and a straightforward computation using (21) and the homomorphism properties of the modular function, shows that M U (g)M −1 = Ur (g).

IV.2

Adjoint and coadjoint representations

The adjoint and the coadjoint actions of the group give rise to two interesting unitary representations connected by an integral, generalized Fourier transform relation. Consider the two Hilbert spaces L2 (g, dX) and L2 (g∗ , dX ∗ ) of Lebesguemeasurable complex-valued functions on the Lie algebra and its dual, respectively, and which are square-integrable with respect to the corresponding Lebesgue measures. On L2 (g, dX) one deﬁnes the adjoint representation V (g) of g ∈ G by the operators (V (g)F )(X) = Adg − 2 F (Adg−1 X), 1

F ∈ L2 (g, dX),

(24)

where Adg is the determinant of the linear transformation Adg on g. The operators V (g) form a unitary representation of G. Similarly, on L2 (g∗ , dX ∗ ) one deﬁnes the coadjoint representation by the operators V (g), g ∈ G, given by 1 (V (g)F)(X ∗ ) = Adg − 2 F(Adg−1 X ∗ ),

F ∈ L2 (g∗ , dX ∗ ),

(25)

where Adg is now the determinant of the linear transformation Adg on g∗ . Again, the operators V (g) form a unitary representation of G.

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The dual representations V (g) and V (g) are unitarily equivalent. They are related by the Fourier transform F : L2 (g, dX) → L2 (g∗ , dX ∗ ), ∗ 1 ∗ ∗ e−iX ; X F (X) dX, (26) (FF )(X ) = F (X ) = n 2 (2π) g which is a unitary map. The unitary equivalence of the two representations: FV (g)F −1 = V (g),

g ∈ G,

(27)

is then established using (16) and the facts that Adg = Adg −1 and d(Adg X) = Adg dX.

IV.3

Covariant coadjoint representation

We now construct a unitary representation of G on each coadjoint orbit Oλ∗ . Going back to (17)–(18), for each λ ∈ J we deﬁne a Hilbert space Hλ = L2 (Oλ∗ , dΩλ ), consisting of all complex-valued dΩλ -measurable functions Fλ on the orbit Oλ∗ , which are square-integrable, |Fλ (X ∗ )|2 dΩλ (X ∗ ) < ∞. ∗ Oλ

We represent g ∈ G on the Hilbert space Hλ by the operator Uλ (g), where, (Uλ (g)Fλ )(X ∗ ) = Fλ (Adg−1 X ∗ ),

X ∗ ∈ Oλ∗ .

(28)

Because of the invariance of dΩλ under the coadjoint action Adg , this representation is unitary. Comparing V in (25) with Uλ in (28), we see that although both representations are built on the dual of the Lie algebra g∗ , Uλ appears to be more covariant 1 —in the sense that it has no ‘scaling factor’ Adg − 2 in it. This is because the measure dΩλ is actually invariant under the coadjoint action, while the measure dX ∗ is (in general) not. On the other hand, the representation Uλ is restricted to functions on a single orbit Oλ only (λ is ﬁxed), while V is deﬁned on functions on the entire dual space g∗ . However, as we now proceed to show, using the factored form (19) of the Lebesgue measure, it is possible to combine all the representations Uλ , λ ∈ J, into a single representation U , which is unitarily equivalent to V , and which is deﬁned on functions on all of g∗ . We deﬁne the covariant coadjoint representation U on the set of all the Hilbert spaces Hλ that carry the representations Uλ , combined into a single direct of the component spaces Hλ , integral Hilbert space H ⊕ Hλ dκ(λ). (29) H= J

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consist of collections of vectors The elements of H Φ = {Fλ }λ∈J ,

Fλ ∈ Hλ .

(30)

When Φ = {Fλ } and Φ = {Fλ } are two such collections, and α, β are complex numbers, then we may also deﬁne their linear combination, αΦ + βΦ = {αFλ + β Fλ }. In this way we provide a linear vector space structure on the set of all such collections of vectors Φ. Next, using the measure dκ appearing in (19), we retain only those vectors Φ which satisfy 2 2 ∗ 2 ∗ Φ = Fλ dκ(λ) = |Fλ (X )| dΩλ (X ) dκ(λ) < ∞. (31) J

J

∗ Oλ

on which Eq. (31) deﬁnes a The set of all such vectors forms the Hilbert space H norm. All the coadjoint representations Uλ in (28) can now be collected into the in the manner: one covariant coadjoint representation U on H U (g)Φ = {Uλ (g)Fλ }λ∈J ,

g ∈ G,

(32)

which is unitary by construction. Note that this is indeed deﬁned on all of g∗ , since for any Xλ∗ ∈ g∗ coming from the coadjoint orbit Oλ∗ , we have (U (g)Φ)(Xλ∗ ) = (Uλ (g)Fλ )(Xλ∗ ) = Fλ (Adg−1 Xλ∗ ),

(33)

by (28).

IV.4

Unitary equivalence of representations

We end this section by showing that the coadjoint representation V in (25) and the covariant coadjoint representation U above, are unitarily equivalent; this will complete the proof of the equivalence of all three representations: the adjoint V , the coadjoint V and covariant coadjoint U . Consider the relation (19) between the Lebesgue measure dX ∗ on the dual ∗ g of the Lie algebra, and the invariant measures dΩλ on the coadjoint orbits Oλ∗ . Again, since d(Adg X ∗ ) = Adg dX ∗ , and dΩλ (Adg Xλ∗ ) = dΩλ (X ∗ ), it follows that (34) Adg σλ (Xλ∗ ) = σλ (Adg Xλ∗ ). intertwining the : L2 (g∗ , dX ∗ ) → H Thus we introduce ﬁnally the linear map N coadjoint and the covariant coadjoint representations, F = {G λ }λ∈J , N

λ (Xλ ) = [σλ (X ∗ )] 12 F(Xλ∗ ), Xλ∗ ∈ Oλ∗ . where, G

(35)

is a unitary map Using (19) and (34) it is then straightforward to check that N for which −1 = U (g). V (g)N (36) N

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With these preliminaries attended to, we turn our attention in the next two Sections to the actual construction of the Wigner map, the Wigner function and the Wigner operator.

V Square-integrable representations The existence of the Wigner map hinges on the existence of a class of representations for certain types of groups. These are the so-called square-integrable [2] or discrete-series representations which enjoy an intertwining property with the left-regular representation U , characterized in Eq. (22).

V.1 Admissible vectors Let U be an unitary irreducible representation of G on a Hilbert space H. We recall that U is square-integrable if there exists a non-zero vector η ∈ H, called an admissible vector, such that c(η) = |U (g)η|η|2 dµ(g) < ∞. (37) G

It is easy to see that if η is admissible, then so is also U (g)η for any g ∈ G. In other words, the set A of all admissible vectors is invariant under U ; then, the irreducibility of U implies that either A is dense in H, or else A = {0}; in the latter case, U is not square-integrable. When G is a unimodular group, the square integrability of U implies that A = H, i.e., every vector in H is admissible (see, e.g., [13]).

V.2 Orthogonality relations The matrix elements U (g)ψ|φ of a square- integrable representation U satisfy certain useful orthogonality relations. Indeed, every square-integrable representation determines a unique positive invertible operator C on H, whose domain coincides with the set A of all admissible vectors. Furthermore, for all vectors η1 , η2 ∈ A and arbitrary vectors φ1 , φ2 ∈ H, the following orthogonality relation holds: U (g)η2 |φ2 U (g)η1 |φ1 dµ(g) = Cη1 |Cη2 φ2 |φ1 . (38) G

When G is a unimodular group, then C is a positive multiple of the identity, i.e., C = λI, for some λ > 0. For non-unimodular groups, C is an unbounded operator and its domain A is only dense in H. This form of the orthogonality relations is well-known; however, for our purposes it will be convenient to use an extended version of these relations [3]. Let B2 (H) denote the space of all Hilbert-Schmidt operators on H. This is the Hilbert space obtained by taking all ﬁnite complex combinations of rank-one

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operators on H of the type ρ = |ψφ|, ψ, φ ∈ H, and closing the resulting set in 1 the norm ρB = {Tr[ρ∗ ρ]} 2 , which arises from the scalar product ρ2 |ρ1 B = Tr [ρ2 ∗ ρ1 ].

(39)

The orthogonality relations (38) can now be extended to hold between pairs of elements in the Hilbert space B2 (H). This is done using the Wigner transform, as we show below.

V.3 The Wigner transform Let η ∈ H be an admissible vector and consider the vector ψ = Cη, which is in the range of C (i.e., in the domain D of C −1 , dense in H). Using such vectors we : H ⊗ D(C −1 ) → L2 (G, dµ), deﬁne the Wigner transform as the linear map W (Wρ)(g) = U (g)C −1 ψ|φ = Tr[U (g)∗ ρC −1 ],

(40)

where ρ = |φ ψ| ∈ B2 (H) and the star denotes the adjoint. Then for any two such ρi ∈ B2 (H), i = 1, 2 and ρi = |φi ψi |, the orthogonality relations (38) may be reexpressed as: 2 )(g)(Wρ 1 )(g) dµ(g) = Tr [ρ2 ∗ ρ1 ] = ρ2 |ρ1 B . (Wρ (41) G

from H ⊗ D(C −1 ) [the The relation (40) deﬁnes a linear transform map W dense subspace of B2 (H) generated by vectors of the form |φ ψ|, φ ∈ H, and ψ in the domain of the operator C −1 ] into L2 (G, dµ). Foundations of this map for the Heisenberg-Weyl algebra can be seen in [20], where it is known as the characteristic function, and in [14] as the Wigner transform. Our construction ﬁnds more basic is the (generalized) the Wigner map W deﬁned in the next Section, and of which W Fourier transform. preserves the scalar product, hence it is an The Wigner transform map W isometry; it may be therefore extended by continuity to an isometry valid on all of B2 (H). We use the same notation for this extended Wigner transform map, now : B2 (H) → L2 (G, dµ). It associates to each Hilbert-Schmidt operator ρ a squareW integrable function fρ (g) on the group. On a dense set of elements ρ ∈ B2 (H), we can recover the original function fρ by the trace formula: = Tr[U (g)∗ ρC −1 ]. fρ (g) = (Wρ)(g)

(42)

The unitary representation U (g) acting on the Hilbert space H can be lifted immediately to a unitary representation U on the Hilbert space B2 (H). Indeed, we simply deﬁne its action on a vector ρ ∈ B2 (H) by ordinary operator product from the left, U (g)ρ = U (g)ρ. (43)

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Now, for any g ∈ G, the operator C satisﬁes the covariance conditions U (g)∗ CU (g) = [∆(g)]− 2 C,

CU (g)C −1 = [∆(g)]− 2 U (g).

1

1

(44)

intertwines U (g) With this, it is easily veriﬁed that the Wigner transform map W on B2 (H) with the left-regular representation U on L2 (G, dµ) which was deﬁned in (22), namely U (g) = U (g)W, W g ∈ G. (45)

VI The Wigner map and function We now introduce the Wigner map W which is essentially the Fourier transform in Subsect. V.3. We shall assume as in Subsect. III.1 that the of the map W exponential map (10) relates the open neighbourhood N0 of g to an open dense set Ve in the connected part of the identity of G and such that the complement of Ve has measure zero. We can then use the exponential map g = eX to introduce local over the set Ve in G; in the basis Xi , i = 1, 2, . . . , n we write X = n coordinates i x = (x1 , x2 , . . . , xn ) ∈ Rn . i=1 x Xi ∈ g, so the group element g ∈ Ve will map to In these coordinates, the left invariant Haar measure on the group will become dµ(g) → m(X) dX,

(46)

where m is a positive Lebesgue-measurable function on N0 , and the relations (41) assume the form 2 )(eX )(Wρ 1 )(eX ) m(X) dX = Tr [ρ2 ∗ ρ1 ] = ρ2 |ρ1 B . (Wρ (47) N0

VI.1

The Wigner map

We deﬁne a linear map W from the space of Hilbert-Schmidt operators ρ ∈ B2 (H) to functions of Xλ∗ ∈ Oλ∗ on the coadjoint orbits, by the Fourier transform-type integral [σλ (Xλ∗ )] 2 n (2π) 2 1

(Wρ)(Xλ∗ ) =

∗ 1 X e−iXλ ; X (Wρ)(e )[m(X)] 2 dX,

(48)

N0

where σλ is the density function in (19). Using this, Eq. (47) and standard properties of the Fourier transform, we immediately establish that W maps any Hilbert deﬁned in Schmidt operator ρ to an element of the direct integral Hilbert space H (29)–(31), and that this map is a linear isometry. We call this map W : B2 (H) → H, the Wigner map.

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The Wigner operator and matrix

In the previous literature [27], [21], [5], it was found useful to deﬁne a formal operator, called the Wigner operator , which acts in the Wigner map (48) extended to the Hilbert space H of this Section. It is 1 ∗ 1 [σ(X ∗ )] 2 ∗ W(X ) = e−iX ; X C −1 U (e−X ) [m(X)] 2 dX. (49) n (2π) 2 N0 Formally, this deﬁnes an operator on H for almost all X ∗ ∈ g∗ with respect to the Lebesgue measure. For any pair of functions φ, ψ ∈ H, or any Hilbert-Schmidt operator ρ, the Wigner function is W (φ, ψ | X ∗ ) = φ | W(X ∗ ) | ψH ,

W (ρ | X ∗ ) = Tr [ρW(X ∗ )].

(50)

In the group ISO(2) studied in [21] and SU(2) in [5] and [9], the Hilbert space H has a denumerable basis φλm , m ∈ Jλ an integer and a ﬁnite number, respectively. Then, it is convenient to deﬁne the (inﬁnite) Wigner matrix W(X ∗ ) with diagonal blocks Wλ (X ∗ ), whose matrix elements are λ ∗ λ ∗ λ Wm,m (X ) = φm | W(X ) | φm H ,

(51)

and which are reduced to integrals of special functions to be computed.

VI.3

The Wigner function

Introducing the positive Lebesgue-measurable function σ on X ∗ ∈ g∗ , which assumes the value σλ (X ∗ ) for all X ∗ ∈ Oλ∗ , we can write Eq. (48) on the whole of g∗ , as W (ρ | X ∗ ) = (Wρ)(X ∗ ) (52) ∗ 1 1 e−iX ; X Tr[U (e−X )ρC −1 ] [σ(X ∗ ) m(X)] 2 dX. = n (2π) 2 N0 We call this the Wigner function of the Hilbert-Schmidt operator ρ, on the dual g∗ of the the Lie algebra of G (or, more accurately, on its coadjoint orbits Oλ∗ , λ ∈ J). The basic properties of the Wigner map and the Wigner function can immediately be read oﬀ their deﬁnitions (48) and (52) and compared with the properties listed in Section II, as we now proceed to show.

VI.4

Reality/sesquilinearity

First note that the Wigner map (48) of elements ρ ∈ B2 (H) is linear. Now, let ρ∗ be the adjoint of the operator ρ; then, since N0 is invariant under the interchange X → −X, and since by virtue of (21) ∆(eX )m(−X) = m(X),

(53)

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replacing X by −X in the integral in (52) and using (44), we obtain W (ρ | X ∗ ) = W (ρ∗ | X ∗ ).

(54)

Hence, if ρ is self-adjoint then its Wigner function W (ρ | X ∗ ) is real. For elements of the type ρ = |φψ|, the map W can be looked upon as i.e., linear in φ, antilinear in ψ and nona sesquilinear map from H × H into H, degenerate, in the sense that W(|ψψ|) = 0 if and only if ψ = 0. The corresponding Wigner functions, written W (ψ, φ | X ∗ ), hence satisfy Eqs. (5) and (6), and are real (cf. Subsect. II.2).

VI.5

Covariance

Here we verify that the covariance property of Subsect. II.3 holds in our new generalized setting. In order to do this, consider the representation Ub of G, on the Hilbert space B2 (H) of Hilbert-Schmidt operators, Ub (g)ρ = U (g)ρU (g)∗ ,

g ∈ G,

(55)

where U is the square-integrable representation introduced in Section V. The representation Ub in (55) is clearly unitary. Now, since dµ(g0 gg0−1 ) = −1 ∆(g0 ) dµ(g), we easily derive that m(Adg X) =

m(X) , Adg ∆(g)

X ∈ g,

g ∈ G.

(56)

Using this relation, Adg = Adg −1 and Eq. (34), we ﬁnd after some routine computations that the Wigner map W intertwines the representation Ub with the covariant coadjoint representation U , deﬁned in (33), i.e., WUb (g) = U (g)W,

g ∈ G.

(57)

In terms of the Wigner function [cf. Eq. (7)], this is W (U (g)ρU (g)∗ | X ∗ ) = W (ρ | Adg−1 X ∗ ),

VI.6

g ∈ G,

X ∗ ∈ g∗ .

(58)

Overlap and reconstruction formulae

The Wigner map is an isometry since it preserves scalar products: for any two Hilbert-Schmidt operators, ρ1 and ρ2 , we have Wρ1 | Wρ2 H = ρ1 | ρ2 B . This can be written alternatively as the overlap condition [cf. Eq. (8)], W (ρ1 |X ∗ ) W (ρ2 |X ∗ ) [σ(X ∗ )]−1 dX ∗ = Tr[ρ∗1 ρ2 ]. g∗

(59)

(60)

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It is easy to invert the Wigner map using the overlap formula (60). Indeed, taking ρ2 = ρ and ρ1 = |ψφ|, with ψ in the domain of C −1 , we obtain W (|φψ| |X ∗ ) W (ρ|X ∗ ) [σ(X ∗ )]−1 dX ∗ = φ|ρ ψ. (61) g∗ Using (52) and noting that φ and ψ are arbitrary, yields

1 1 m(X) 2 −iX ∗ ; X ∗ X −1 ρ= e W (ρ|X )U (e )C dX dX ∗ . n σ(X ∗ ) (2π) 2 g∗ N0 (62) We shall refer to this relation as the reconstruction formula.

VII The Wigner function and wavelet transform There is a very interesting relation between the Wigner function introduced in the previous Section and the generalized wavelet transform.

VII.1

The wavelet transform

In the square-integrable representation U , any admissible vector η ∈ A, with c(η) as in (37), can be used to deﬁne the (generalized) wavelet transform fη,φ of an arbitrary φ ∈ H: 1 g ∈ G. (63) fη,φ (g) = 1 U (g)η|φ, [c(η)] 2 The wavelet transform is a square-integrable function on G – an element of the Hilbert space L2 (G, dµ). In fact, the map φ → fη,φ in (63) may be shown to be an isometry [13], i.e., |fη,φ (g)|2 dµ(g) = φ2 . (64) G

Note that the standard wavelet transform discussed in the literature [10] is a special case of the transform (63) when the group G is the aﬃne group of the real line.

VII.2

Coherent states

It is worthwhile to mention at this point the role of coherent states, where the standard or canonical coherent states belong to the special case of the HeisenbergWeyl group. The orthogonality relations (38) imply the resolution of the identity, 1 |ηg ηg | dµ(g) = I, ηg = U (g)η, (65) c(η) G and then the vectors [c(η)]− 2 η are called coherent states of the group G in the unitary irreducible representation U (see, e.g., [3], [22]). Thus, the generalized wavelet transform (63) may be called also the coherent state transform. 1

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The Wigner Function for General Lie Groups and the Wavelet Transform 701

The Wigner-wavelet relations

To give explicitly the relation between the Wigner function W (ρ|X ∗ ) in (2) and the wavelet transform fη,φ (g) of φ ∈ H in (63), for ﬁxed (admissible) η ∈ A and arbitrary φ ∈ H, consider the Hilbert-Schmidt operators of the form ρη,φ =

1 1

[c(η)] 2

|φη|C.

Comparing now (63) with (52), we conclude that ∗ 1 1 W (ρη,φ |X ∗ ) = e−iX ; X fη,φ (eX ) [σ(X ∗ ) m(X)] 2 dX. n (2π) 2 N0

(66)

(67)

This relation is easily inverted, yielding the wavelet transform in terms of the Wigner function, namely ∗ 1 1 X eiX ; X W (ρη,φ |X ∗ ) [σ(X ∗ ) m(X)]− 2 dX ∗ . (68) fη,φ (e ) = n (2π) 2 g∗

VII.4

Bases and coordinates in the Lie algebra

As indicated in Section III, one can introduce bases in the Lie algebra g and its dual g∗ , in terms of which X ∈ g has the coordinate representation x = (x1 , x2 , . . . , xn ), while X ∗ ∈ g∗ has the coordinates ξ = (ξ1 , ξ2 , . . . , ξn ); their Lebesgue measures 0 be the image, in these coordinates, of the set have the forms given in (15). So let N N0 ⊂ g (the domain of the exponential map (10), the range Ve of which is assumed i to be dense in G with its completment having zero Haar measure). Denote by X the Hilbert space operators that represent the basis elements Xi ∈ g, i.e., the operators on H such that U (eXi ) = e−iXi . (69) 1 , X 2 , . . . , X n . the vector operator with components X Finally, denote by X In these terms the Wigner function (52) and its inverse (62) can be written

1 1 Tr ei(X−ξ )· x ρC −1 [σ(ξ ) m(x)] 2 dx, (70) W (ρ | ξ ) = n (2π) 2 N 0 1 e−i(X−ξ )· x W (ρ|ξ ) ρ = n (2π) 2 g∗ N 0 12  m(x) × C −1 dx dξ. (71) σ(ξ ) Similarly, all other coordinate-free expressions appearing earlier can be written in these coordinates, which are most useful for computational purposes. In particular,

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the overlap condition (60), for ρ1 = |φ1 ψ1 | and ρ2 = |φ2 ψ2 | becomes W (ψ1 , φ1 |ξ ) W (ψ2 , φ2 |ξ ) [σ(ξ )]−1 dξ = φ1 |φ2 ψ2 |ψ1 . g∗

(72)

In these coordinates, the covariance condition (7) or (58) assumes the form: W (U (g)ψ, U (g)φ|ξ ) = W (ψ, φ|M T (g)ξ ),

g ∈ G,

φ, ψ ∈ H,

(73)

where M (g −1 )T is the matrix of the coadjoint map Adg .

VIII Wigner functions for the the two-dimensional aﬃne group In this section we apply the theory presented above to the important particular case of the aﬃne group of the line, Gaﬀ , consisting of all transformations of the form x → ax + b, x ∈ R, with a > 0, b ∈ R. A group element is thus given by + a pair (a, b) ∈ R+ ∗ × R. (Note that R∗ denotes the positive real line without the origin.) Group multiplication replicates matrix multiplication when we represent group elements by the matrices

a b . (74) g= 0 1 Wigner functions for this group have been obtained earlier in [7], using diﬀerent methods. Our analysis reproduces the same results.

VIII.1

Aﬃne algebra and group matrices

The Lie algebra gaﬀ of Gaﬀ is generated by the two elements

1 0 0 1 , X2 = , X1 = 0 0 0 0 so that the one-parameter subgroups of Gaﬀ are

a 0 1 b (log a)X1 bX2 e = , e = . 0 1 0 1 Thus, for a general element in the Lie algebra

1 x X = x1 X1 + x2 X2 = 0

x2 0

(75)

(76)

,

the group element obtained from the exponential map is

1 x2 1 x a b − 1) ex x1 (e g = eX = = . 0 1 0 1

(77)

(78)

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From here follows the inverse map from the group to the algebra, X = log g = x1 X1 + x2 X2 ,

x1 = log a,

x2 =

b log a . a−1

(79)

Since every X ∈ gaﬀ is mapped to an element g ∈ Gaﬀ by the exponential map 0 with the full real plane and use x = (x1 , x2 ) ∈ R2 (78), we identify its domain N as the coordinates for the elements of the Lie algebra.

VIII.2

Haar measures

The left- and right-invariant measures on Gaﬀ are easily computed in the polar coordinates (78), da db 1 − e−x = dx1 dx2 , 2 a x1 1 ex − 1 1 2 da db = dx dx , a x1 1

dµ(g) = dµ (g) = dµr (g) = and the modular function is

∆(g) =

1 . a

(80) (81)

(82)

Writing as in (46), we ﬁnd da db = m(x) dx, a2

VIII.3

1 − e−x . x1 1

m(x1 , x2 ) =

(83)

Adjoint and coadjoint action

The adjoint action of the group Gaﬀ on an element (77) of the Lie algebra is easily computed to be

1 x −bx1 + ax2 . (84) Adg X = gXg −1 = 0 0 The matrix of this transformation which acts on (column) vectors x = (x1 , x2 ) ∈ R2 is

1 0 . (85) M (g) = −b a On the dual of the Lie algebra, X ∗ ∈ g∗ has coordinates ξ = (ξ1 , ξ2 ) ; on this column vector, the coadjoint action is represented by the inverse transpose matrix,

1 ba−1 −1 M (g) = M (g ) = . (86) 0 a−1 The determinants of these matrices are Adg = a = Adg −1 .

(87)

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The coadjoint representation of this group in Eq. (25), V , is carried by the Hilbert space L2 (R2 , dξ ) and has the form 1 1 (V (g)F)(ξ ) = a 2 F(M (g) ξ ) = a 2 F(ξ1 − bξ2 , aξ2 ).

VIII.4

(88)

Coadjoint orbits of the aﬃne group

The coadjoint orbits of the aﬃne group are found from the action of the matrices (86) on ﬁxed vectors ξ ∈ R2 . The following orbit structure emerges: 1. The orbit obtained by acting with the matrices M (g −1 ) on the column vector (0, 1) , ∗ = {ξ+ = (ξ1 , ξ2 ) ∈ R2 |ξ2 > 0} = R × R+ O+ ∗.

(89)

2. The orbit obtained by acting on (0, −1) with the same matrices, ∗ = {ξ− = (ξ1 , ξ2 ) ∈ R2 |ξ2 < 0} = R × R− O− ∗.

(90)

3. Applying the matrices to the column vector (α, 0) , for each α ∈ R we obtain an orbit that consists of the single point (α, 0); this we denote by Oα∗ . We may thus characterize the foliation of the dual of the Lie algebra g∗aﬀ , by the set J = {+, −, R}. This we identify with the real plane, R2 = Oλ∗ . (91) λ∈J

Note that in this classiﬁcation, the last set of orbits Oα∗ are a set of Lebesgue ∗ measure zero in R2 , while the other two orbits O± are open sets in R2 and their union is dense. Under the coadjoint action, the latter two orbits carry the invariant measures dξ1 dξ2 ∗ . (92) , ξ ∈ O± dΩ± (ξ ) = 1 (2π) 2 |ξ2 | Comparing with (19), we now deﬁne a measure dκ(λ) on the Borel sets of the set J = {+, −, R} as follows: κ({±}) = 1,

κ({E}) = 0,

(93)

of Eq. (29) for for any open set E ⊂ R. Thus the direct integral Hilbert space H the covariant coadjoint representation is now just an orthogonal sum, = H+ ⊕ H − , H

∗ where H± = L2 (O± , dΩ± ).

consist of pairs of functions, F = (F+ , F− ), with F± ∈ H± . Elements in H

(94)

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The Wigner Function for General Lie Groups and the Wavelet Transform 705

Covariant coadjoint representation

as we deﬁned it in The covariant coadjoint representation of Gaﬀ is carried by H; Subsect. IV.3, it has the form (U (g)F)(ξ ) = F(M (g) ξ ) = F(ξ1 − bξ2 , aξ2 ),

F ∈ H,

(95)

where ξ1 is the translation parameter in the aﬃne group while ξ2 is the scale parameter. In order to compute the explicit form of the Wigner function for the aﬃne group, it is necessary to use its unitary irreducible representations; there are only two such representations. To examine them, consider the representation U (g) on the Hilbert space L2 (R, dt), t − b 1 (U (g)φ)(t) = a− 2 φ , a

φ ∈ L2 (R, dt),

g = (a, b) ∈ Gaﬀ .

(96)

This representation is unitary but not irreducible. To isolate its irreducible components we go over to the Fourier-transformed Hilbert space L2 (R, dω), where the representation is 1 −ibω (g)φ)(ω) , (U = a 2 φ(aω)e

φ ∈ L2 (R, dω),

g = (a, b) ∈ Gaﬀ .

(97)

It is now clear that each of the two subspaces of functions deﬁned on the intervals (0, ∞) and (−∞, 0), H± = L2 (R± , dω), (98) (g), and in fact are irreducible subspaces are stable under the action of the U to these two subspaces by under this action. We shall denote the restrictions of U ± U , respectively. The two subrepresentations are then inequivalent, but both are square-integrable in the sense of Section V.

VIII.6

(g)+ Wigner functions for U

We shall now derive Wigner functions for the unitary irreducible representation + ; analogous results hold in an obvious way for the representation U − as well. U + A vector η ∈ H is admissible in the sense of (37), if and only if it satisﬁes the condition (see, e.g., [10]), ∞ 2π | η (ω)|2 dω < ∞. (99) ω 0 This means that η must lie in the domain of the positive unbounded operator C, whose action is 12 2π (C η)(ω) = η(ω), ω ≥ 0. (100) ω

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Using (92), we identify the density σ appearing in the Wigner function (70) to be ∗ ∗ ξ ∈ O+ ∪ O− .

σ(ξ ) = |ξ2 |,

(101)

Let ψ ∈ L2 (R+ , dω) be any vector in the domain of the operator C −1 (i.e., ψ = C η for some admissible vector η), and let φ be an arbitrary element in L2 (R+ , dω). Then, combining (70) with (83), (97), (99) and (101), after some computation we obtain φ | ξ1 , ξ2 ) = W (ψ,

1 1 (2π) 2

∞

ψ

−∞

x

ξ2 e 2 sinch x2

x ξ2 e−iξ1 x ξ2 e− 2 dx, φ sinch x2 sinch x2

(102)

which is valid for all ξ2 > 0, and where sinch (u) = (sinh u)/u. The above Wigner function was obtained using the irreducible representation ∗ + , and it lead to Wigner functions which are supported on the orbit O+ . Had we U − ∗ we would have obtained an analogous function supported on O− used U . When + − we use the reducible representation U = U ⊕ U given in (97), for arbitrary φ ∈ L2 (R, dω) and ψ ∈ L2 (R, dω) satisfying

∞

−∞

|ω| |ψ(ω)|2 dω < ∞, 2π

(103)

we ﬁnd the Wigner function φ | ξ1 , ξ2 ) = W (ψ,

1 1 (2π) 2

∞

ψ

−∞

x

ξ2 e 2 sinch x2

x |ξ2 | e−iξ1 x ξ2 e− 2 dx, φ sinch x2 sinch x2

(104)

which is now valid for all ξ ∈ R2 .

VIII.7

Aﬃne covariance

It is easily veriﬁed that the Wigner function (102) satisﬁes the correct covariance condition (7)–(73)–(86), U φ | ξ1 − bξ2 , aξ2 ). (g)ψ, (g)φ | ξ ) = W (ψ, W (U

(105)

Comparing with (57) and (95), we see that the Wigner map intertwines the repre (g)ρU (g)∗ with the covariant coadjoint representation U (g) sentation Ub (g)ρ = U in (95). The overlap condition (72) is also straightforward to verify.

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The Wigner Function for General Lie Groups and the Wavelet Transform 707

Marginality relations of the aﬃne Wigner function

Although the Wigner functions obtained above are deﬁned on all of R2 , they should ∗ ∗ be regarded as functions deﬁned on the orbits O+ ∪ O− . This is because we would like to think of the Wigner functions as phase space distributions, and these orbits have the structure of symplectic manifolds with invariant measures, and hence are classical phase spaces. The interpretation of the Wigner function as a distribution ∗ ∗ over the phase space O+ ∪ O− is further supported by the following observations. Consider the aﬃne Wigner function (104) of one wavefunction, W (ψ | ξ ) = ψ | ξ ). Integrating this over the coadjoint orbits with respect to the invariant W (ψ, phase space measure (92), we obtain the full projection to a positive number, dξ1 dξ2 W (ψ | ξ ) dΩ± (ξ ) = W (ψ | ξ ) 1 ∗ ∪O ∗ ∗ ∪O ∗ (2π) 2 |ξ2 | O+ O+ − − ∞ 2 |ψ(ω)| dω = ψ. (106) = −∞

For an arbitrary density matrix ρ, the result is W (ρ | ξ ) dΩ± (ξ ) = Tr ρ. ∗ ∪O ∗ O+ −

(107)

Therefore, though the Wigner function, even for a pure state, is not in general positive, its phase space integral has the proper measurement-theoretic interpretation as the squared norm of the state. The well-known projection or marginality properties satisﬁed by the original Wigner function discussed in Section I, cannot be expected to hold in the aﬃne case. We do have however, a similar relation when we project (integrate) over the translation parameter ξ1 of the aﬃne group, to ﬁnd the marginal distribution over the scale parameter ξ1 , namely ∞ 1 dξ1 2 )|2 . = |ψ(ξ W (ψ | ξ ) (108) 1 |ξ | 2 (2π) −∞ 2 In the scale parameter ξ2 however, the marginality relation has a more complicated form. Indeed, a straightforward manipulation of integrals leads to the relation ∞ ∞ ∞ 1 dξ2 x − x2 ) dx dω. 2 )ψ(ωe = W (ψ | ξ ) e−iξ1 x ψ(ωe (109) 1 |ξ2 | (2π) 2 −∞ −∞ −∞ On the other hand, the choice of phase space coordinates (ξ1 , ξ2 ), which we have adopted here, is not the only possible one and a diﬀerent choice could lead to a simpler form for this marginality condition. Unfortunately, there does not seem to exist an obvious “natural” choice of coordinates for general phase spaces.

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IX Wavelet transform and Wigner function in the aﬃne group It is worthwhile to display in detail the relationship between the Wigner functions of the aﬃne group and the wavelet transform, since both the wavelet transform and the Wigner function are used extensively in image reconstruction computations. As pointed out in the general setting in Section VI, Eqs. (67) and (68), there is an intimate connection between the two.

IX.1

Coherent states of the aﬃne group

Consider the (doubly reducible) representation of the aﬃne group given in (97). A mother wavelet is any vector (a signal) η in the carrier Hilbert space L2 (R, dω) of the representation, which satisﬁes the admissibility condition [10] of Eqs. (99) and (99), ∞ 2π (110) | η (ω)|2 dω < ∞. −∞ |ω| This implies in particular that η(ω) must vanish at the origin. Now choose a particular mother wavelet, normalized by (37) so that ∞ ∞ da (a, b) db |U η | η |2 = 1. (111) c( η) = a2 −∞ 0 Using this mother wavelet we deﬁne a family of wavelets, or equivalently coherent states of the aﬃne group, (a, b) ηa,b = U η,

(a, b) ∈ Gaﬀ .

(112)

In view of (96), these are simply functions in the inverse Fourier domain which are scaled and translated versions of the mother wavelet, and with the same normalization (111). The resolution of the identity (65) on the Hilbert space L2 (R, dω) now assumes the form ∞ ∞ da db | ηa,b ηa,b | = I. (113) a2 −∞ 0

IX.2

Wigner-wavelet relations

Consider an arbitrary signal φ ∈ L2 (R, dω) and its wavelet transform in the translation and scale parameters a, b of the wavelet family, fη,φ (a, b) = ηa,b |φ.

(114)

Next, for the chosen mother wavelet η, note that the function (C η)(ω) =

2π 1

|ω| 2

η(ω),

ω ∈ R,

(115)

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is admissible and hence vanishes at the origin ω = 0. Then, specializing the integral in (67) to the aﬃne group and using (78), we ﬁnd the relation between the wavelet transform and the Wigner function given by 1 W (C η, φ | ξ ) = 2π

−i x·ξ

e R2

1 1 x1 fη,φ ex , x2 e 2 sinch x2

12

|ξ2 | e

x1 2

1

sinch x2

dx.

(116) In this expression the choice of the mother wavelet η is ﬁxed, and the equation refers to φ only. It is a routine matter now to invert this relation and to write the wavelet transform in terms of the Wigner function, 1 12 dξ dξ x1 1 1 1 1 2 x 2 x2 x1 ei x·ξ W (C η, φ | ξ ) e 2 sinch x2 fη,φ e , x e sinch 2 = 1 2π R2 |ξ2 | 2 (117)

X The standard Wigner function revisited We ﬁnally go back to the well-known Wigner function in (1) at the beginning of this paper and see how it ﬁts into the same theoretical considerations. (It will actually be necessary to do a somewhat diﬀerent constuction in this case, since the representation in question will not be square integrable with respect to the entire group.) As mentioned in the Introduction, the Wigner function has its origin in the Heisenberg-Weyl group GHW (of the canonical commutation relations). This group is the central extension of the abelian group of R2 and is topologically isomorphic to R × R2 . Denoting a generic element in GHW by g = (θ, ξ, η), the multiplication rule is, (θ1 , ξ1 , η1 ) (θ2 , ξ2 , η2 ) = (θ1 + θ2 +

1 2

[η1 ξ2 − η2 ξ1 ], ξ1 + ξ2 , η1 + η2 ).

(118)

The corresponding Lie algebra gHW is generated by the three elements X0 , X1 , X2 , satisfying the Lie bracket relations [X1 , X2 ] = X0 and [Xi , X0 ] = 0, i = 1, 2. The central element X0 generates the phase subgroup Θ, consisting of group elements of the type (θ, 0, 0). We shall actually use the three elements X0 , X1 and −X2 as basis vectors for the Lie algebra, and write its general element as X = x0 X0 + x1 X1 − x2 X2 . From the relation g0 gg0−1 = (θ + η0 ξ − ηξ0 , ξ, η), we readily derive the matrices of the adjoint and coadjoint They are    1 1 η ξ M (θ, ξ, η) =  0 1 0  and M (θ, ξ, η) =  −η −ξ 0 0 1

(119) actions in this basis.  0 0 1 0, 0 1

(120)

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respectively. The coadjoint action on a vector γ = (γ0 , γ1 , γ2 )T is,   γ0 M (θ, ξ, η)γ =  γ1 − γ0 η  . γ2 − γ0 ξ

(121)

From this we see that the coadjoint orbits of GHW are of the following types: 1. The planes

Oλ∗ = {(λ, x)T | x ∈ R2 },

(122)

one for each λ = 0 and generated from the vector (λ, 0, 0) . T

2. The singletons

O λ∗ = {(0, λ)T },

(123)

one for each λ ∈ R2 and generated from the vector (0, λ)T . The invariant measures on the orbits Oλ∗ are simply the Lebesgue measures dx on the planes. Corresponding to each one of the non-degenerate orbits Oλ∗ , there is a unitary irreducible representation U λ of GHW carried by the Hilbert space H = L2 (R, dx): ξ

(U λ (θ, ξ, η)φ)(x) = eiλθ eiλη(x− 2 ) φ(x − ξ).

(124)

Since we may also write, U λ (θ, ξ, η) = U λ (eX ) = eiλ(θI+ηQ−ξP ) ,

X = x0 X0 + x1 X1 − x2 X2 ,

(125)

with

i ∂φ(x) , (126) λ ∂x the Hilbert space generators corresponding to X0 , X1 and X2 are seen to be I, Q and P , respectively, with the further identiﬁcation, x0 = −θ, x1 = −η and x2 = −ξ. Let us consider the case λ = 1 (equivalent to setting = 1) and simply write U for the corresponding representation. Also, we shall write U (0, ξ, η) = U (ξ, η). This representation is not square-integrable with respect to the whole group GHW , but only with respect to the homogeneous space GHW /Θ Oλ∗ R2 [2, 4]. However, it is possible to adapt the theory of square integrable representations, as outlined in Section V, to this situation [4, 14]. Basically, we work with the operators U (ξ, η), which give a multiplier representation of R2 and which admit the following orthogonality relations: 1 U (ξ, η)φ1 |ψ1 U (ξ, η)φ2 |ψ2 dξ dη = φ1 |φ2 ψ1 |ψ2 , (127) 2π R2 (Qφ)(x) = xφ(x)

and (P φ)(x) = −

for arbitrary vectors φ1 , φ2 , ψ1 and ψ2 in the Hilbert space. The operator C in (38) 1 is in this case (2π) 2 I. Since the phase subgroup Θ has now been factored out, the

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Wigner transform has to be deﬁned using the remaining two generators Q and P ∗ and will be a function on the coadjoint orbit Oλ=1 R2 . (These considerations will be made more rigorous in a subsequent publication, where we intend to deal with Wigner functions obtained from group representations which are square integrable only with respect to a homogeneous space. It will turn out that we shall have to extend the general theory to include certain types of reducible square integrable representations.) It is also clear that both the densities σ and m, appearing in the expression for the Wigner function in (70), are constants in this case and we set them equal to unity. The Wigner function, for arbitrary Hilbert-Schmidt operators ρ, now assumes the form, 1 W (ρ | γ1 , γ2 ) = ei(γ1 η−γ2 ξ) Tr[e−i(Qη−P ξ) ρ] dξ dη. (128) 2π R2 Taking ρ = |ψφ|, for wave functions φ, ψ ∈ H, writing (q, p) for (γ1 , γ2 ) and simplifying the resulting expression, we easily obtain 1 W (φ, ψ | q, p) = e−ixp φ(q − 12 x) ψ(q + 12 x) dx, (129) 2π R which is exactly the same expression as in (1).

XI Conclusion The object which is crucial to our construction of the Wigner function is a squareintegrable group representation [2]. Such a representation belongs to the discrete series of the group, and not every group has a representation in this series. The groups studied thus far, the Heisenberg-Weyl group [27], the Euclidean group [21], and the spin group [5], have these representations, are unimodular, and enjoy several other simplifying properties, such having global polar coordinates. The aﬃne group is the simplest example where one of these properties — unimodularity— is transcended. We have reﬁned the deﬁnitions of the Wigner operator and function given in the previous literature so that the aﬃne case is included cogently, and we have compared the results on wavelets richly contained in the literature. We have found that indeed there is a close relation between the Wigner function and the wavelet transform: they are essentially Fourier transforms of each other. This has been noted before in the case of the Heisenberg-Weyl group, where the Wigner and the radar Woodward ambiguity functions [28] are also Fourier transforms [23]. The important advantage of the Wigner function (52) is that it is deﬁned on a coadjoint orbit. This ensures its interpretation as a (quasi-)distribution on a phase space. The fact that coherent states which satisfy a resolution of the identity of the type (65) are also associated to square integrable representations, gives the link between generalized wavelet transforms and generalized Wigner functions. In a

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following publication we shall indicate how the construction of this paper can be extended to certain other types of representations that are not square-integrable, such as the continuous series of Sp(2,R) representations.

Acknowledgements This work was supported by project dgapa–unam IN104198, Optica Matem´ atica. One of us (STA) would like to acknowledge grants from the NSERC, Canada and FCAR, Qu´ebec, and the hospitality of the Centro Internacional de Ciencias, Cuernavaca.

References [1] G.S. Agarwal, Entropy, the Wigner distribution function and the approach to equilibrium of systems of coupled harmonic oscillators, Phys. Rev. A3 (1971) 828–831. [2] S.T. Ali, J-P. Antoine and J-P. Gazeau, Square integrability of group representations on homogeneous spaces. I. Reproducing triples and frames, Ann. Inst. H. Poincar´e 55 (1991) 829–855; II. Coherent and quasi-coherent states. The case of the Poincar´e group, Ann. Inst. H. Poincar´e 55 (1991) 857-890. [3] S.T. Ali, J-P. Antoine, J-P. Gazeau and U.A. Mueller, Coherent states and their generalizations: A mathematical overview, Reviews Math. Phys. 7 (1995) 1013–1104; also, S.T. Ali, J.-P. Antoine and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations, (Springer-Verlag, New York, 2000). [4] S.T. Ali, A general theorem on square-integrability: Vector coherent states, J. Math. Phys. 39 (1998) 3954-3964. [5] N.M. Atakishiyev, S.M. Chumakov, and K.B. Wolf, Wigner distribution function for ﬁnite systems, J. Math. Phys. 139 (1998) 6244–6261. [6] M.J. Bastiaans, The Wigner distribution function applied to optical signals and systems, Opt. Comm. 25 (1978) 26–30; M.J. Bastiaans, Wigner distribution functions and its application to ﬁrst-order optics, J. Opt. Soc. Am. 69 (1979) 1710–1716. [7] J. Bertrand and P. Bertrand, Repr´esentations temps-fr´equence des signaux, C.R. Acad. Sc. Paris 299, S´erie I (1984) 635-638; A class of Wigner functions with extended covariance properties, J. Math. Phys. 33 (1992) 2515–2527. [8] J. Bertrand and P. Bertrand, Microwave imaging of time-varying radar targets, Inverse Problems, 13 (1997) 621–645.

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[9] S.M. Chumakov, A. Frank and K.B. Wolf, Finite Kerr medium: macroscopic quantum superposition states and Wigner functions on the sphere, Physical Review A60 (1999) 1817–1823. [10] I. Daubechies, Ten Lectures on Wavelets, (SIAM, Philadelphia, 1992). [11] H. F¨ uhr and M. Mayer, Continuous wavelet transforms from cyclic representations: A general approach using Plancherel measure, Univ. M¨ unchen preprint (July 1998). [12] G. Garc´ıa-Calder´ on and M. Moshinsky, Wigner distribution functions and the representation of canonical transformations in quantum mechanics, J. Phys. A 13 (1980) L185–L188. [13] A. Grossmann, J. Morlet and T. Paul, Integral transforms associated to square integrable representations. I. General results, J. Math. Phys 26 (1985) 2473– 2479. [14] D.M. Healy Jr. and F.E. Schroeck Jr., On informational completeness of covariant localization observables and Wigner coeﬃcients, J. Math. Phys. 36 (1995) 453–507; Franklin E. Schroeck, Jr., Quantum Mechanics on Phase Space (Kluwer Academic Publ., Boston, 1996). [15] S. Helgason, Diﬀerential Geometry, Lie Groups, and Symmetric Spaces, (Academic Press, Inc, San Diego, Ca, 1978). [16] M. Hillery, R.F. O’Connel, M.O. Scully, and E.P. Wigner, Distribution functions in physics: fundamentals, Phys. Rep. 106 (1984) 121–167; Y.S. Kim and M.E. Noz, Phase Space Picture of Quantum Mechanics, (World Scientiﬁc, Singapore, 1991); H.-W. Lee, Theory and application of the quantum phase-space distribution functions, Phys. Rep. 259 (1995) 147–211; Vladimir I. Man’ko, Conventional quantum mechanics without wave function and density matrix. In: Proceedings of the 1998 Latin American School of Physics (American Institute of Physics, New York, 1999). [17] A.A. Kirillov, Elements of the Theory of Representations, (Springer-Verlag, Berlin, 1976). [18] A. Lohmann, The Wigner function and its optical production, Opt. Comm. 42 (1980) 32–37. [19] S. Mancini, V.I. Man’ko, and P. Tombesi, Reconstructing the density operator by using generalized ﬁeld quadratures, Quantum Semiclass. Opt. 7 (1995) 615; S. Mancini, V.I. Man’ko and P. Tombesi, Phys. Lett. A 213 (1996) 1–6; S. Mancini, V.I. Man’ko, and P. Tombesi, Found. Phys. 27 (1997) 81. [20] J.E. Moyal, Quantum mechanics as a statistical theory, Proc. Cambridge Phil. Soc. 45 (1949) 99–124.

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[21] L.M. Nieto, N.M. Atakishiyev, S.M. Chumakov and K.B. Wolf, Wigner distribution function for Euclidean systems, J. Phys. A 31 (1998) 3875–3895. [22] A. Perelomov, Generalized Coherent States and their Applications, (SpringerVerlag, Heidelberg, 1986). [23] W. Schempp, Analog radar signal design and digital signal processing – a Heisenberg nilpotent Lie group approach. In Lie Methods in Optics. Ed. by J. S´ anchez-Mondrag´on and K.B. Wolf. Springer Lecture Notes in Physics, Vol. 250 (Springer Verlag, Heidelberg, 1986); W. Schempp, Harmonic Analysis on the Heisenberg Nilpotent Lie Group, with Applications to Signal Theory, Pitman Research Notes in Mathematics, Vol. 147 (Longman Scientiﬁc and Technical, London, 1986); W. Schempp, Magnetic Resonance Imaging: Mathematical Foundations and Applications, (John wiley & Sons, New York, 1997). [24] Remark by R. Simon (Institute of Mathematical Science, Madras, India). [25] E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932) 749–759. [26] K.B. Wolf, The Heisenberg–Weyl ring in quantum mechanics. In Group Theory and its Applications, III, Ed. by E.M. Loebl (Academic Press, New York, 1975), pp. 189–247. [27] K.B. Wolf, Wigner distribution function for paraxial polychromatic optics, Opt. Comm. 132 (1996) 343–352. [28] P.M. Woodward, Probability and Information Theory, with Applications to Radar , (Artech House, Dedham MA, 1980). S. Twareque Ali Department of Mathematics and Statistics Concordia University Montr´eal, Qu´ebec, Canada H4B 1R6

Natig M. Atakishiyev Instituto de Matem´ aticas UNAM Cuernavaca, M´exico

Sergey M. Chumakov Departamento de Ciencias B´asicas Universidad de Guadalajara M´exico

Kurt Bernardo Wolf Centro de Ciencias F´ısicas Universidad Nacional Aut´ onoma de M´exico Apartado Postal 48–3 62251 Cuernavaca, Morelos, M´exico

Communicated by Gian Michele Graf submitted 05/07/99, accepted 24/01/2000

Ann. Henri Poincar´ e 1 (2000) 715 – 752 c Birkh¨ auser Verlag, Basel, 2000 1424-0637/00/040715-38 $ 1.50+0.20/0

Annales Henri Poincar´ e

Topological Entropy and ε-Entropy for Damped Hyperbolic Equations P. Collet, J.-P. Eckmann

Abstract. We study damped hyperbolic equations on the inﬁnite line. We show that on the global attracting set G the topological entropy per unit length exists in the topology of W 1,∞ . We also show that the topological entropy of G exists. These results are shown using two main techniques: Bounds in bounded domains in position space and for large momenta, and a novel submultiplicativity argument in W 1,∞ .

1 Introduction This paper is an extension of our earlier papers [CE1, CE2] to mixed parabolichyperbolic equations in the inﬁnite domain R : η 2 ∂t2 u(x, t) + ∂t u(x, t) = ∂x2 u(x, t) + U u(x, t) ,

(1.1)

where U (s) = s2 /2 − s4 /4. The particular choice of the potential U is in fact not very important, but we will deal only with this one. This problem can be written as a system: ∂t u(x, t) = v(x, t) ,

η ∂t v(x, t) = −v(x, t) + ∂x2 u(x, t) + U u(x, t) . 2

(1.2)

The functions u will be real-valued, but the extension to vector-valued functions is easy and is left to the reader, since it only complicates notation. The question we ask is about the nature of the attracting set for this problem, its complexity, and in particular its ε-entropy. We have developed this subject for parabolic problems in the two papers described above and we study now the complexity in this parabolic-hyperbolic setting. The diﬀerence with the parabolic case is the absence of regularization. In the parabolic case, the dispersion law is, written in Fourier space for the linearized equation ∂t u ˜(k, t) = (1 − k2 )˜ u(k, t) ,

(1.3)

when U (s) = s2 /2 + O(s3 ) near s = 0. In the case we consider now, the problem is rather a system of the form

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∂t u ˜(k, t) = v˜(k, t) , η ∂t v˜(k, t) = −˜ v (k, t) + (1 − k2 )˜ u(k, t) . 2

Ann. Henri Poincar´ e

(1.4)

Thus, as is well known, (1.3) regularizes derivatives because |k| exp (1 − k2 )t is bounded in k when t > 0, while the real part of the eigenvalue of the system (1.4) −2 is, for large |k|, only as negative as −O(η ), and therefore the exponential is −2 only bounded like |k| exp −Cη t for some C > 0. This diverges with |k|, but converges (non-uniformly) to 0 as t → ∞. One can ask whether this reduced form of regularization manifests itself in an increased complexity of either the attracting set, or some forward invariant set of bounded initial data. The conclusion of our paper is that the complexity of the problem (1.4) is of the same order as that of (1.3). Since we work on the inﬁnite line, we need local topologies. This will be achieved by choosing a cutoﬀ function h: hδ (x) =

1 . (1 + δ 2 x2 )2

We could take other functions with suﬃciently strong polynomial decay, but the nice ideas of Mielke [M1, M2] using exponentially decaying cutoﬀ functions do not seem to work here. We then consider local Sobolev norms of the form (u, v)2hδ ,2 = dx hδ (x) u2 + 2(u )2 + (u )2 + η 2 v 2 + (v )2 (x) , 2 with the norm and then local spaces Hδ,loc

(u, v)δ,loc,2 = sup (u, v)hδ,ξ ,2 , ξ∈R

where hδ,ξ (x) = hδ (x − ξ). Note that this norm, and many others used in this paper, has one more derivative in u than in v. Such norms are typical when one writes equations such as (1.1) with two components as in (1.2). We will also need bounds on the third derivative: (u, v)2hδ ,3 = dx hδ (x) u2 +2(u )2 +2(u )2 +(u )2 +η 2 v 2 +2(v )2 +(v )2 (x) , 3 . and the corresponding space Hδ,loc Our study will be based on a subtle interplay between several norms. We will show that every initial condition with ﬁnite (u, v)δ,loc,3 will end up after some ﬁnite time in a bounded set in this norm. We call this bounded set the attracting set G. The attractor G∞ is then deﬁned as G∞ = Φt (G) , t>0

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where t → Φt is the ﬂow deﬁned by (1.1). We will not only study the complexity of G∞ , but we can also make statements about functions which have “evolved for long enough,” namely functions in GT ≡ T >t>0 Φt (G) for some large T . Remark. Our deﬁnitions are similar to, but not quite the same as those in [F], 2 where the author considers Hδ,loc and makes statements about G∞ in that topology. Probably, with a little more work, our estimates would yield again the results of [F]. Deﬁnition. Given some interval [−L, L] in R, and T ∈ R+ , we denote by NL,T (ε) the number of balls in Hδ,loc,2 needed to cover GT when restricted to [−L, L] in the variable x. In fact, we use a slightly diﬀerent topology, see (8.1). We show in (9.3) that there are three constants A, C, and τ∗ such that NL,T +τ∗ (ε) ≤ NL+A/ε,T (2ε)C L ,

(1.5)

(Proposition 8.4 and (9.20)). It is at this point that we use the deﬁnition of GT , the fact that high-momentum parts of the solution are damped with an exponential rate of about η −2 , and that the low momentum parts are Fourier transforms of analytic functions, which can be ﬁnitely sampled by the Cartwright formula (8.10). Iterating the inequality (1.5) and observing that G0 = G has ﬁnite radius K in Hδ,loc,2 , we see that for L > A/ε log2 ((1 + K)/ε) and for T > τ∗ log2 ((1 + K)/ε) one ﬁnds NL,T (ε) ≤ C (L+A/ε) log2 ((1+K)/ε) . We then change topology to W 1,∞ (functions in L∞ whose derivatives are 2 also in L∞ ) and show that the results obtained for Hδ,loc give bounds in W 1,∞ . We introduce a new type of submultiplicativity bound in Section 9.2. Indeed, we show in Corollary 9.2 (which is an easy consequence of the Theorem 9.1) that if a bounded set of functions in C 2 can be covered by N1 balls of radius ε in WI1,∞ 1 and by N2 on WI1,∞ , where I1 and I2 are disjoint intervals, then it can be covered 2 by C(ε)N1 N2

(1.6)

balls of radius ε in WI1,∞ . The point here is that C only depends on ε (and the 1 ∪I2 bound on the functions) and that the balls on I1 ∪ I2 have the same radius as the original balls. Indeed, if one allows a doubling of the radius, the corresponding inequality is trivial, but insuﬃcient for taking the thermodynamic limit in the entropy. Thus, our bound is an essential tool for showing the existence of inﬁnite volume limits in topologies where the “matching” of functions needs some care. Once all these tools are in place, we can easily repeat the proofs of the existence of the topological entropy using the methods developed in [CE1] and [CE2]. The paper is structured as follows. In Section 2 we bound the ﬂow in time, using localized versions of coercive functionals as introduced by Feireisl [F]. The

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main result is Theorem 2.6 and its corollary (2.22) and (2.23) which show that the 3 solution to (1.2) is well behaved in Hδ,loc . In Section 3 and Section 4 we study the linear part of (2.1) localized in coordinate and momentum space. We next study the decay of the high momentum part in Section 5 and Section 6. This allows, in Section 7 to study the time evolution of diﬀerences of two solutions of (2.1), in 2 other words, we control now the deformation of balls (in the topology of Hδ,loc ). In Section 8 (Proposition 8.4 and (9.20)) we show how to cover GT with balls as explained in (1.6) above. In Section 9.2 we deal with the technically delicate submultiplicativity bound mentioned before. Finally, in Section 9.3 and Section 9.4 we show without eﬀort the bound Theorem 9.5 on the ε-entropy and the Theorem 9.7 which shows the existence of the topological entropy per unit length.

2 Coercive functionals In this section, we study some functionals which control the ﬂow in time. The ﬁrst part of this material is an adaptation from the work of Feireisl[F]. We consider here the problem (1.1) in the form u˙ = v , η v˙ = −v + u + U (u) , 2

2

(2.1)

4

where we take U (s) = s2 − s4 , but many other choices are of course possible. To simplify things, we assume throughout that 0 < η < 1, and in fact, in subsequent sections we will assume η < η0 for some small η0 . We shall use throughout a localization function hα which depends on a small parameter α, to be determined later on. The constant α will only depend on the coeﬃcients of (2.1) (but not on η < 1). We set 1 hα (x) = . (2.2) (1 + α2 x2 )2 Note that α dx hα is independent of α. Remark. We will only use values of 0 < α ≤ 12 and this will be tacitly assumed in all the estimates. Using hα , we introduce the norm (u, v)2hα ,1 = dx hα (x) η 2 v 2 + u2 + (u )2 (x) . (2.3) We also need a translation invariant topology on (u, v). Let hα,ξ (x) = hα (x − ξ). Deﬁnition 2.1 We deﬁne the norm (u, v)α,loc,1 = sup (u, v)hα,ξ ,1 , ξ∈R

(2.4)

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1 and the space Hα,loc is deﬁned by 1 Hα,loc = (u, v)

(u, v)α,loc,1 < ∞ .

719

(2.5)

The norm introduced above is not very convenient for estimates, and thus we introduce as in Feireisl[F] the quantity F0 (which is not a norm) by 2 2 2 2 F0 (u, v) = α dx hα (x) η v (x) + (u (x)) + V (u(x)) + η u(x)v(x) . (2.6) Here, V is chosen such that U (x) + 2V (x) − η 2 x = 0 ,

(2.7)

with V (0) = 0. Note that U (x) → −∞ as |x| → ∞ at a rate O(x4 ), and therefore V (x) → +∞ at a rate O(x4 ). In particular, V (x) ≥ x2 for suﬃciently large |x| .

(2.8)

The following bound can be found in Feireisl[F]. Lemma 2.2 There are constants a0 > 0 and b0 > 0 (independent of η for 0 < η < 1) for which one has the inequality ∂t F0 (ut , vt ) ≤ −a0 F0 (ut , vt ) + b0 ,

(2.9)

where ut (x) = u(x, t), vt (x) = v(x, t) is the solution of (2.1). This bound can be used to bound (u, v)hα ,1 . Recall that V diverges like |x|4 . Using the bound η2 v2 η 2 u2 η 2 |uv| ≤ + , 2 2 this implies η2 v2 + u2 − C0 , V (u) + η 2 uv ≥ − 2 for some constant C0 . Therefore, one has the inequality (u, v)2hα ,1 ≤ 2F0 (u, v) + C1 . Using (2.9) we conclude Lemma 2.3 There is a constant C2 (independent of 0 < η < 1) for which the following holds. Assume that F0 (u0 , v0 ) < ∞. Then, for all t > 0 one has (ut , vt )hα ,1 < ∞ and there is a T = T (u0 , v0 ) for which the solution (ut , vt ) of (2.1) with initial data (u0 , v0 ) satisﬁes for all t > T : (ut , vt )hα ,1 ≤ C2 .

(2.10)

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1 We can extend this result to the topology of Hα,loc . Let u0,ξ (x) = u0 (x − ξ) and similarly for v0 .

Proposition 2.4 There is a constant C3 (independent of 0 < η < 1) for which the following holds. Assume that supξ∈R F0 (u0,ξ , v0,ξ ) < ∞. Then there is a T = T (u0 , v0 ) for which the solution (ut , vt ) of (2.1) with initial data (u0 , v0 ) satisﬁes for all t > T : (ut , vt )α,loc,1 ≤ C2 ,

(2.11)

and ut ∞ ≤ C3 . Proof. Consider the quantities F0,ξ deﬁned by replacing hα (x) by its translate hα (x + ξ) in Eq. (2.6). Then, F0 (ut,ξ , vt,ξ ) = F0,ξ (ut , vt ). Clearly, for every ξ we have ∂t F0,ξ (ut , vt ) ≤ −a0 F0,ξ (ut , vt ) + b0 , since (2.1) does not depend explicitly on x. It follows from the above that if supξ F (u0,ξ , v0,ξ ) < ∞ there is a ﬁnite time T after which (ut , vt )α,loc,1 ≤ C2 .

(2.12)

This proves (2.11). To conclude that u is bounded we need the following easy Lemma 2.5 There is a constant C4 = C4 (δ) such that sup |f (x)| ≤ C4 f hδ ,1 .

(2.13)

x∈[−1,1]

Proof. From the explicit form of hδ we conclude that there is a constant C = (1 + δ 2 )2 such that 1 2 2 2 dx |f (x)| ≤ (1 + δ ) dx hδ |f (x)|2 , −1

and similarly

1

−1

dx |f (x)|2 ≤ (1 + δ 2 )2

dx hδ |f (x)|2 .

The result follows from the standard Sobolev inequality. The proof of Lemma 2.5 is complete. Using this lemma, and observing that the · α,loc,1 norm is translation invariant we conclude immediately from (2.12) that there is a constant C3 for which sup |u(x, t)| ≤ C3 , x

(2.14)

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for all t > T . The proof of Proposition 2.4 is complete. We next deal with the slightly more complicated bounds on the spatial derivatives of u and v. Let w = u and let z = v . They satisfy the equations w˙ t = zt , η z˙t = −zt + wt + U (ut )wt , 2

(2.15)

where U (s) = 1 − 3s2 , for the U we have taken above. We consider initial data (w0 , z0 ) which will be bounded later and assume (in view of Proposition 2.4) that (u0 , v0 )α,loc,1 ≤ C2 which implies supx |u(x, t)| ≤ C3 for all t > 0. We are going to bound the growth of (w, z) as a function of time. We introduce a positive constant µ (which we ﬁx later) and set F1 (w, z) = α dx hα (x) η 2 z 2 (x) + (w (x))2 + η 2 µw(x)z(x) . (2.16) When no confusion is possible, we henceforth write (wt , zt ) satisfying (2.15) :

f for dx f (x). One ﬁnds for

η2 µ η2 µ w˙ t zt + wt z˙t hα η 2 zt z˙t + wt w˙ t + 2 2 η2 µ 2 z = α hα − zt2 + zt wt + U (ut )zt wt + wt zt + 2 t µ µ µ − wt zt + wt wt + U (ut )wt2 2 2 2 2 η µ ) + U (ut )zt wt = α hα − zt2 (1 − 2 µ µ 2 µ 2 − wt zt − (wt ) + U (ut )wt 2 2 2 µ − α hα zt wt + wt wt . 2

1 2 ∂t F1 (wt , zt )

= α

Note now that by the deﬁnition (2.2) of hα and the restriction α ≤ ﬁnd that the quotient |hα /hα | is bounded by 2α ≤ 1. Using this, we get |hα zt wt | ≤ αhα zt2 + (wt )2 , and

(2.17)

1 2,

we

|µhα wt wt | ≤ αµhα wt2 + (wt )2 .

We will also use the bound |wt zt | ≤ α−1 wt2 + αzt2 . Finally note that there is a constant C5 for which sup |U (s)| ≤ C5 . |s|≤C3

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Combining these bounds with the last equality of (2.17), we get for times t > 0: η2 µ 2 1 1 − ∂ F (w , z ) ≤ −α h z − α − µ/2 − αC5 t 1 t t α t 2 2 2 (2.18) − α hα (wt ) µ/2 − αµ/2 − α + αµ + µ/2 + µC5 /2 + α−1 C5 α hα wt2 . It is clear that if we choose α and µ suﬃciently small (but independent of η for small enough η), then we get, for some (large) constant C6 , µ 2 2 2 1 + C ∂ F (w , z ) ≤ α h + (w ) w z − . (2.19) t t α 6 t t 2 t 1 4 t Note now that for small µ > 0 and η > 0, η 2 µwz ≤ (1 − η 2 )z 2 +

η2 µ w2 , 4(1 − η 2 )

which is equivalent to −z 2 ≤ −η 2 z 2 − η 2 µwz +

η2 µ w2 . 4(1 − η 2 )

Therefore, (2.19) leads to 1 2 ∂t F1 (wt , zt )

≤ −

αµ 4

hα η 2 zt2 + (wt )2 + η 2 µwt zt + C7

hα wt2 .

The last term is bounded by (2.11), and therefore we ﬁnd: ∂t F1 (wt , zt ) ≤ −a1 F1 (wt , zt ) + b1 , for some ﬁnite positive a1 and b1 . Using again the methods leading to Proposition 2.4, we obtain Theorem 2.6 There are constants C8 , C9 and C10 (independent of η < 1) for which the following holds. Assume supξ∈R F0 (u0,ξ , v0,ξ ) < C8 and supξ∈R F1 (u0,ξ , v0,ξ ) < C8 . Then the solution (ut , vt ) of (2.1) with initial data (u0 , v0 ) satisﬁes for all t > 0: (ut , vt )α,loc,2 ≤ C9 ,

(2.20)

sup |u(x, t)| + |u (x, t)| + |v(x, t)| ≤ C10 .

(2.21)

and x∈R

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Remark. The technique used above can be extended to show that any derivative of u(x, t) and v(x, t) is eventually bounded (if the potential U is suﬃciently differentiable and the initial data are suﬃciently regular). The details are left to the reader. We will in fact use bounds on the second derivative at some later point in the argument, i.e., bounds of the form (ut , vt )α,loc,3 ≤ C9 ,

(2.22)

and sup |u(x, t)| + |u (x, t)| + |u (x, t)| + |v(x, t)| + |v (x, t)| ≤ C10 .

(2.23)

x∈R

3 The linearized evolution In this section, we study the linear part of the solution. By this we mean solutions of the equation u˙ = v , η v = −v + u . 2˙

(3.1)

It will be useful to rewrite this system of equations as u u˙ = L . v˙ v

(3.2)

Next, we go through a second round of estimates, similar to the ones in Section 2, to see how fast (u, v) can grow. We use again the cutoﬀ function hδ (x) =

1 , (1 + δ 2 x2 )2

with a parameter δ diﬀerent from α which will be ﬁxed in Section 6. We consider the functional (3.3) H = hδ η 2 v2 + u2 + (u )2 = (u, v)hδ ,1 , and proceed to bound it. One gets for the solution (ut , vt ) of (3.2) : 1 ∂ H(u , v ) = dx hδ η 2 vt v˙ t + ut u˙ t + ut u˙ t t t 2 t = dx hδ −v2t + vt ut + vt ut + ut vt = dx hδ −v2t + vt ut − hδ vt ut .

(3.4)

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Observe that by construction, |hδ /hδ | ≤ 1 , when δ <

1 2

(which we assume throughout). We use now (since u and v are real): |vt | |ut | + |hδ ut /hδ | ≤ 12 v2t + u2t + (ut )2 .

We can use this inequality to bound the mixed terms in the last equality of (3.4), and compensate the term 12 (v2t ) with the negative term −v2t . Therefore, we have shown that there is a constant C11 independent of η < 1 for which ∂t H(ut , vt ) ≤ C11 H(ut , vt ) .

(3.5)

We next deﬁne a space in which both the functions and their derivatives are controlled. This is the space in which our ﬁnal bounds will be spelled out. Using still the cutoﬀ function 1 hδ (x) = , (1 + δ 2 x2 )2 we deﬁne the norm

(u, v)2hδ ,2 =

hδ η 2 v 2 + (v )2 + u2 + 2(u )2 + (u )2 .

Deﬁnition 3.1 The Hilbert space Hh2 δ is deﬁned by

2

Hhδ = (u, v) (u, v)hδ ,2 < ∞ .

(3.6)

(3.7)

We also need the translates of hδ to deﬁne a translation invariant topology. Let hδ,ξ (x) = hδ (x − ξ). 2 Deﬁnition 3.2 The space Hδ,loc is deﬁned by

2 = (u, v)

sup (u, v)hδ,ξ ,2 < ∞ . Hδ,loc

(3.8)

ξ

Remark. Note that (u, v)2hδ ,2 = (u, v)2hδ ,1 + (u , v )2hδ ,1 . Next, we observe that L and ∂x commute. Therefore, the bounds on (ut , vt )hδ ,1 can be extended immediately to similar bounds on (ut , vt )hδ ,2 and we get from (3.5) : Theorem 3.3 There is a constant C11 such that if (ut , vt ) solves (3.1) then (ut , vt )2hδ ,2 ≤ eC11t (u0 , v0 )2hδ ,2 .

(3.9)

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4 Momentum localization Let m be a bounded positive function of x which decays faster than any inverse power as |x| → ∞. We deﬁne the convolution operator Ma by Ma f (x) = dy m a(x − y) f (y) . (4.1) Let again hδ (x) =

1 . (1 + δ 2 x2 )2

(4.2)

Lemma 4.1 There is a constant C12 such that if δ > 0 and a > 0, then the operator Ma is bounded on L2 (hδ dx ), with norm bounded by 2

Ma δ ≤ C12

1 + aδ 2 . a

(4.3)

ˆ a with integral kernel Proof. We will prove the result by bounding the operator M −1/2 1/2 hδ (x) m a(x − y) hδ (y) (4.4) on L2 (dx). Writing 1 = χ(2|x| > |y|) + χ(2|x| < |y|), with characteristic functions χ, and multiplying the kernel (4.4) with them, we induce a decomposition of this operator as a sum of two pieces. Fix |x|. We consider ﬁrst the integration region |y| < 2|x|. In that region, we have a bound 1/2 h (x) 0 < δ1/2 ≤ 4. hδ (y) Therefore, in this region, the integral kernel is bounded by 4|m(a(x − y))|. Since m decreases like an arbitrary power we get for every + > 0 a bound 1/2 hδ (x)|m a(x − y) | 1 dy |f (y)| ≤ 4K dy |f (y)| . 1/2 hδ (y) |y|<2|x| |y|<2|x| (1 + a|x − y|) When + > 1, then a/(1 + a|x|) is bounded in L1 (uniformly in a) and therefore, ˆ a f is bounded in L2 (dx), with norm Young’s inequality shows that this piece of M −1 less that C13 a f 2 , and C13 independent of δ and a. In the region |y| ≥ 2|x|, we use |x − y| > |y|/2 and |x − y| > |x|. Therefore, using hδ (x) ≤ 1, we ﬁnd for + > 3, 1/2 hδ (x)|m a(x − y) | dy |f (y)| ≤ hδ (y)1/2 |y|≥2|x| (+2) 1 + δ2 y2 2 2 K dy −2 +2 |f (y)| . |y|≥2|x| (1 + a|x|) 2 (1 + a|y|) 2

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1 + δ2 y2 δ2 ≤ 1 + . (1 + a|y|)2 a2

Using the Schwarz inequality yields a bound (1 + δ 2 /a2 )/a on the second piece of Ma δ . Combining the two pieces completes the proof of Lemma 4.1. We need later the following variant of this result: Let ϑ = ϑ(k) be a smooth characteristic function which equals 1 for |k| ≤ 1 and 0 for |k| > 2. Let qa (x) = ˜ aϑ(ax), with ϑ˜ the inverse Fourier transform of ϑ. For a > 0, let Qa be the convolution operator deﬁned by Qa f (x) = dy qa (x − y)f (y) . (4.5) This operator is a substitute for a projection onto momenta less than a. Setting ˜ m(x) = ϑ(x), we get from Lemma 4.1: Corollary 4.2 There is a constant C12 such that if δ > 0 and a > 0 then the operator Qa is bounded on L2 (hδ dx ), with norm bounded by δ2 Qa δ ≤ C12 1 + 2 . a

(4.6)

5 High momentum bounds We consider again the function hδ as deﬁned in (4.2), and we study functions u for which dx hδ (x) |u(x)|2 + |u (x)|2 < ∞. Such functions have a Fourier transform u ˜ in the sense of tempered distributions, and we deﬁne now

2 2

Ka = u dx hδ (x) |u(x)| + |u (x)| < ∞ and supp u ˜ ∈ R \ (−a, a) . Thus, apart from not being deﬁned as a function, the Fourier transform u ˜ of a 2 u ∈ Ka has support at momenta larger than a. If h (x) ≡ 1 and u ∈ L (dx), then, δ obviously, for u ∈ Ka , one has |u|2 ≤ a−2 |u |2 . The following proposition whose elegant proof was kindly provided by H. Epstein, shows that the cutoﬀ function hδ does only moderately change this property. Proposition 5.1 Assume that a > 0 and δ > 0. There is a constant ν(a, δ) < ∞ such that for all u ∈ Ka one has the inequality 2 (5.1) dx hδ (x)|u(x)| ≤ ν(a, δ) dx hδ (x)|u (x)|2 . There is a constant C14 > 0 such that one can choose 2

(1 + aδ 2 )2 . ν(a, δ) = C14 a2

(5.2)

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Remark. We will need the result only for δ < a, so that we can use the simpler bound ν(a, δ) ≤

C15 . a2

Proof. Let ϑ be a smooth characteristic function which equals 1 for |k| ≤

(5.3) 1 2

and 0

for |k| > 1. Let u ∈ Ka . Since the distribution u˜ has support in the complement u(k), we of the interval (−a, a), and the Fourier transform u of the derivative is ik˜ see that 1 − ϑ(k/a) u (k) . u ˜(k) = ik Deﬁne next 1 − ϑ(k) . m(k) ˜ ≡ ik The (inverse) Fourier transform, m, of m ˜ decreases faster than any power of |x| at inﬁnity. If we let ma (x) = m(ax), then m ˜ a (k) = m(k/a)/a. ˜ Thus, it follows with the notation of Section 4 that u(x) = Ma (u ) (x) . By Lemma 4.1, we conclude that δ2 2 2 2 2 2 2 (1 + a2 ) uδ = hδ |u| = hδ |Ma (u )| = Ma (u )δ ≤ C12 hδ |u |2 , a2 and the claim (5.2) follows.

6 The linear high frequency part We begin by deﬁning the projection onto high frequencies, on a space with weight hδ (x) = (1 + δ 2 x2 )−2 . We ﬁrst recall the notion of projection onto low frequencies from Section 4. Denote by ϑ a smooth characteristic function, equal to 1 for |k| ≤ 1 and vanishing for |k| > 2. We ﬁx now a (large) cutoﬀ scale k∗ and we deﬁne as before qk∗ (x) = k∗ ϑ(k∗ x) , and Qk∗ f (x) =

dy qk∗ (x − y)f (y) .

(6.1)

In Corollary 4.2, we showed that on L2 (hδ dx ), the operator Qk∗ is bounded by 2 C12 1 + kδ 2 . Therefore, the projection onto high momenta ∗

Pk∗ = 1 − Qk∗ ,

(6.2)

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is also bounded on that space. Henceforth, we shall assume δ < k∗ , and thus we get immediately the bound. Lemma 6.1 There is a constant C16 such that if k∗ > δ > 0, then the operator Pk∗ satisﬁes Pk∗ δ ≤ C16 ,

(6.3)

as a map on L2 (hδ dx ). Lemma 6.2 There is a constant C17 such that for k∗ > δ > 0 the operator Pk∗ ⊕Pk∗ 2 is bounded in norm by C17 as a map from Hδ,loc to itself. Proof. We have already checked in Lemma 6.1 that Pk∗ is bounded on L2 (hδ dx ). Note that Pk∗ is a convolution operator and so Pk∗ and ∂x commute, and the 2 extension of the result to Hδ,loc (as deﬁned in Deﬁnition 3.2) follows at once. So far, we have argued that Pk∗ is bounded. We will now use the high momentum bound of Section 5 with a = k∗ , and k∗ ≤ η −1 to show that the semi-group generated by the free evolution (see below) is a (strong) contraction. In fact, we will show that the contraction rate is O(k∗2 ) as long as k∗ < η −1 , O(η −2 ) for any cutoﬀ k∗ ≥ η −1 . This behavior is typical of the mixed parabolic-hyperbolic problems we consider here, since the linearized evolution, written in momentum space, has the generator 0 1 −η −2 k2 −η −2 with eigenvalues λ± =

−1 ±

1 − 4k2 η −2 . 2η 2

One can see from the expression for the eigenvalues that the real part never gets more negative than −O(η −2 ). We study now the properties of the operator L deﬁned as in Eqs.(3.1) and (3.2) by u v . L = η −2 −v + u v We introduce parameters γ > 0, and δ > 0 which will be ﬁxed later and we consider the functional J: J = Jhδ ,γ (u, v) = (u, v)2hδ ,2 + η 2 γ dx hδ (x) u(x)v(x) + u (x)v (x) , (6.4)

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where the norm (u, v)hδ ,2 was deﬁned in Eq. (3.6) : 2 2 2 2 2 2 2 (u, v)hδ ,2 = hδ η v + (v ) + u + 2(u ) + (u ) . Consider the solution (ut , vt ) of (3.2). Then, writing J for Jhδ ,γ , we ﬁnd 2 1 hδ η vt v˙ t + vt v˙ t + ut u˙ t + 2ut u˙ t + ut u˙ t 2 ∂t J(ut , vt ) = 1 2 + 2 η γ hδ ut v˙ t + vt u˙ t + ut v˙ t + vt u˙ t = hδ − v2t + vt ut − (vt )2 + vt u + u v + 2u v + u v t t t t t t t 2 2 + 12 γ hδ −ut vt + ut ut + η 2 v2t − ut vt + ut u t + η (vt ) 2 2 1 2 1 = hδ − vt + (vt ) (1 − 2 η γ) + ut vt + ut vt (1 − 2 γ) 2 2 1 − 2 γ hδ (ut ) + (ut ) − hδ vt ut + vt ut + 12 γ ut ut + ut ut . (6.5) By construction, we have |hδ /hδ | ≤ C18 δ, and therefore the last integral in (6.5) can be bounded (in modulus) by 2 2 2 2 2 2 2 1 C18 δ hδ vt + (ut ) + (vt ) + (ut ) + 2 γ ut + 2(ut ) + (ut ) . Thus, we ﬁnd

1 2 ∂t J(ut , vt )

hδ v2t + (vt )2 1 − 12 η 2 γ − C18 δ C18 δ 1 − 2 γ hδ (ut )2 + (ut )2 1 − − 2C18 δ γ + 12 C18 δγ hδ u2t + hδ ut vt + ut vt (1 − 12 γ) .

≤−

(6.6)

Recall that η > 0 is given, and that we want to prove results for all η < η0 , where η0 is our (only) small parameter. We rewrite the last integral in (6.6) as (6.7) − 18 η 2 γ 2 hδ ut vt + ut vt + hδ ut vt + ut vt (1 + 18 η 2 γ 2 − 12 γ) .

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We introduce now the ﬁrst restrictions on η and k∗ : Fix 1 η0 ≤ √ , 40

(6.8)

and k0 ≥

40C15 .

(6.9)

These bounds will be made more stringent below. We shall always require 0 < η < η0 ,

and ∞ > k∗ > k0 .

We next deﬁne γ ≡ min(η −2 , k∗2 /C15 )/320 , and we choose the space-cutoﬀ parameter δ as δ = min 1/2, 1/(40C18 ) .

(6.10)

(6.11)

Note that γ is essentially the inverse of the dispersion law as explained at the beginning of this section. With the above requirements we ﬁnd γ > 2 and |1 + 18 η 2 γ 2 − 12 γ| ≤ γ . We polarize the second integral in (6.7) (but not the ﬁrst) and bound it (in modulus) by |1 + 18 η 2 γ 2 − 12 γ| hδ (|ut vt | + |ut vt |) ≤ γ hδ (|ut vt | + |ut vt |) (6.12) 1 2 2 2 2 2 . ≤ hδ 8γ ut + (ut ) + 8 vt + (vt ) Combining (6.6) with the decomposition (6.7) and the bound (6.12), we ﬁnd 1 ∂ J(u , v ) ≤ − hδ v2t + (vt )2 1 − 12 η 2 γ − C18 δ − 18 t t 2 t C18 δ − 12 γ hδ u2t + 2(ut )2 + (ut )2 1 − − 2C18 δ γ 2 C18 δ − 2C18 δ) + hδ ut + (ut )2 (8γ 2 + 12 C18 δγ) + 12 γ(1 − γ − 18 η 2 γ 2 hδ ut vt + ut vt . (6.13)

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The bizarre decomposition of the terms involving (ut )2 will become clear soon. Note that by our choice of constants, (6.13) can be simpliﬁed to the slightly less good bound 1 1 hδ v2t + (vt )2 2 ∂t J(ut , vt ) ≤ − 2 − 14 γ hδ u2t + 2(ut )2 + (ut )2 (6.14) 2 + 16γ hδ u2t + (ut )2 − 18 η 2 γ 2 hδ ut vt + ut vt . We project onto high momenta, and exploit the contraction properties : We assume from now on that Qk∗ u0 = 0 and Qk∗ v0 = 0. Note that if this property holds at time zero, it holds for all times for the evolution deﬁned by L, because L commutes with Qk∗ ⊕ Qk∗ . Using the bounds of Section 5, we have hδ u2t ≤ ν hδ (ut )2 , (6.15) 2 2 hδ (ut ) ≤ ν hδ (ut ) , where

ν = C15 k∗−2 .

Thus, (6.14) can be improved to 1 2 ∂t J

≤−

1 2

hδ v2t + (vt )2

hδ u2t + 2(ut )2 + (ut )2 2 + 16νγ hδ (ut )2 + (ut )2 − 18 η 2 γ 2 hδ ut vt + ut vt .

− 14 γ

This leads to a bound of the form 1 2 ∂t J

≤ − 12 η −2 hδ η 2 v2t + (vt )2 − 18 γ hδ u2t + 2(ut )2 + (ut )2 − 18 η 2 γ 2 hδ ut vt + ut vt .

(6.16)

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Taking the least good bounds above, we ﬁnally get the decay of the high frequency part (since η −2 ≥ γ): 2 2 2 2 2 2 2 1 1 hδ η vt + (vt ) + ut + 2(ut ) + (ut ) + η γ ut vt + ut vt 2 ∂t J ≡ 2 ∂t 2 2 2 2 2 2 2 1 ≤ − 8 γ hδ η vt + (vt ) + ut + 2(ut ) + (ut ) + η γ ut vt + ut vt 1 min(η −2 , k∗2 /C15 )J . = − 18 γJ = − 2560

(6.17) Thus we have shown the Proposition 6.3 There is a (small) η0 > 0 such that for all η < η0 the following holds for the functional 2 2 2 2 2 2 2 Jhδ ,γ (ut , vt ) = hδ (x) η vt +(vt ) +ut +2(ut ) +(ut ) +η γ ut vt +ut vt ) (x) . Let (ut , vt ) = eLt (u0 , v0 ), and assume (u0 , v0 ) ∈ Kk∗ ⊕ Kk∗ . Then Jhδ ,γ (ut , vt ) ≤ exp −γt/80 · Jhδ ,γ (u0 , v0 ) ,

(6.18)

where γ = min(η −2 , k∗2 /C15 )/320 .

(6.19)

We come now back to the deﬁnition (6.18) of J, and compare it to the norm ·hδ ,2 deﬁned in Eq. (3.6). These two quantities deﬁne equivalent topologies when considered on Kk∗ ⊕ Kk∗ . Lemma 6.4 On Kk∗ ⊕ Kk∗ one has the bound

η 2 γ

hδ uv + u v

≤ hδ 12 (u )2 + (u )2 + 18 v2 + (v )2 . Remark. This lemma eliminates the somewhat arbitrary quantity γ from the topology, see Theorem 6.5 below. Proof. This is a combination of earlier estimates. Indeed, we have already seen in Eq. (6.12) that the mixed terms in Eq. (6.7) can be bounded by

≡ X . η 2 γ

hδ uv + u v

≤ η 2 hδ 8γ 2 u2 + (u )2 + 18 v2 + (v )2 Furthermore, by (6.15) and the choice of k∗ , we get X ≤ η 2 C15 k∗−2 hδ 8γ 2 (u )2 + (u )2 + 18 v2 + (v )2 .

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Since we have also chosen γ = min(η −2 , k∗2 /C15 )/320, we get ﬁnally , X ≤ hδ 12 (u )2 + (u )2 + 18 v2 + (v )2 which is what we asserted. Recall the deﬁnition (6.2) of the projection Pk∗ onto momenta larger than k∗ . From Lemma 6.4 and Proposition 6.3 we have immediately, with the notation 2 of (3.6) and (6.4) in the topology of Hδ,loc (which does not depend on δ0 ) : Theorem 6.5 Assume η0 and k∗ satisfy (6.8) and (6.9), and assume δ ≤ 1/(40C18 ). For all η satisfying 0 < η < η0 the following holds: If (u0 , v0 )hδ ,2 < ∞ and (ut , vt ) = eLt (u0 , v0 ) then one has the bounds Jhδ ,γ (Pk∗ ut , Pk∗ vt )/2 ≤ (Pk∗ ut , Pk∗ vt )2hδ ,2 ≤ 2Jhδ ,γ (Pk∗ ut , Pk∗ vt ) ,

(6.20)

and (Pk∗ ut , Pk∗ vt )hδ ,2 ≤ 4 exp(−γt/80) · (Pk∗ u0 , Pk∗ v0 )hδ ,2 ,

(6.21)

where γ = min(η −2 , k∗2 C15 )/320.

7 The evolution of diﬀerences In this section, we combine the results of Sections 3 and 6 into bounds on the evolution of the diﬀerence of solutions to (2.1). We will ﬁrst treat the general case, and show a bound which diverges exponentially with time, and then we will treat the high frequency case where we have decay. We consider two initial conditions, and their respective evolutions under the semi-ﬂow deﬁned by (2.1). We call these functions (u1 , v1 ) and (u2 , v2 ), respectively. The evolution for the diﬀerence (u, v) = (u1 − u2 , v1 − v2 ) takes now the form u˙ = v , η v = −v + u + M (u1 , u2 )u , 2˙

(7.1)

where M (u1 , u2 ) is deﬁned by M (u1 , u2 )(u1 − u2 ) = U (u1 ) − U (u2 ). It will be useful to rewrite this system of equations as ut 0 u˙ t = L + . (7.2) v˙ t vt Mu1,t ,u2,t ut Note that M (u1 , u2 ) is really a space-time dependent coeﬃcient of the linear problem (3.2). The important observation is now that on the attracting set, i.e., for all

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suﬃciently large t (depending on the initial conditions u1 , u2 , v1 , v2 ) we have, by Theorem 2.6, a universal bound sup |M (u1,t (x), u2,t (x))| + sup |∂x M (u1,t (x), u2,t (x))| ≤ M∗ .

x∈R

(7.3)

x∈R

Since we already know bounds on the solution, we can write it as follows for ut (x) = u(x, t) and vt (x) = v(x, t): t ut u0 0 tL (t−s)L ds e = e + . (7.4) vt v0 Mu1,s ,u2,s us 0 Proposition 7.1 Assume (u1,0 , v1,0 ) and (u2,0 , v2,0 ) are in G. Let ut = u1,t − u2,t and let vt = v1,t − v2,t . There are constants C19 and C20 such that for all t > 0, (ut , vt )hδ ,2 ≤ C19 eC20 t (u0 , v0 )hδ ,2 .

(7.5)

Proof. We have already seen in (7.3) that |M (u1,t , u2,t )| and its derivative are bounded and then the result follows at once from the representation (7.4) and the bound of Theorem 3.3. The handling of the high frequency part Pk∗ (ut , vt ) is similar. Instead of (7.2), we get ∂t Pk∗ ut Pk∗ ut 0 = L + . (7.6) Pk∗ Mu1,t ,u2,t ut ∂t Pk∗ vt Pk∗ vt The solution of this problem is t Pk∗ u0 0 Pk∗ ut tL (t−s)L ds e = e + . Pk∗ Mu1,s ,u2,s us Pk∗ vt Pk∗ v0 0

(7.7)

What is important here is that in both terms the operator L acts on functions with high frequencies. Proposition 7.2 Assume (u1,0 , v1,0 ) and (u2,0 , v2,0 ) are in G. Let ut = u1,t − u2,t and let vt = v1,t − v2,t . There are constants C21 , C22 , and C23 such that for all t > 0, eC23 t (Pk∗ ut , Pk∗ vt )hδ ,2 ≤ C21 e−γt/80 + C22 (7.8) (u0 , v0 )hδ ,2 , γ where γ = min(η −2 , k∗2 /C15 )/320. Remark. In fact, one can choose C23 = C20 . Proof. We use again (7.3) to bound M and ∂x M . Furthermore, Pk∗ is bounded and then the result follows at once from the representation (7.7) and the bound (6.21) of Theorem 6.5 for the ﬁrst term of (7.8) and additionally the bound (7.5) of Proposition 7.1 for the second.

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8 Covering the set of long-lived functions GT We deﬁne a new norm by (u, v)δ,L,2 =

(u, v)hδ,ξ ,2 ,

sup

(8.1)

ξ∈[−L,L]

where hδ,ξ (x) =

1 , (1 + δ 2 (x − ξ)2 )2

and (u, v)hδ ,2 was deﬁned in (3.6). This norm looks at a “window” of size 2L in 2 Hδ,loc . We next deﬁne a natural unit of time, τ∗ . We recall the deﬁnition (6.19) of γ: γ = min(η −2 , k∗2 /C15 )/320. We deﬁne τ∗ =

b log γ , γ

(8.2)

where the (small positive) constant b is chosen such that the factor in (7.8) is minimal and when γ is large (say, γ > γ0 ), we get C21 M M 9e−γτ∗ /80 + C22

eC23 τ∗ γ −κ ≤ , γ 4

(8.3)

for some κ > 0. We will use this bound in the sequel. For ε > 0 we deﬁne NL,T (ε) as the minimum number of balls of radius ε (in the norm · δ,L,2 ), needed to cover the set GT . Theorem 8.1 There exist ﬁnite constants A, and C24 such that for all ε with 0 < ε < 1 and all L > A/ε one has L NL−A/ε,T +τ∗ (ε) ≤ C24 NL,T (2ε) .

(8.4)

Proof. We denote t → Φt the ﬂow deﬁned by (2.1). Let B be a ﬁnite collection of balls of radius 2ε in the topology deﬁned by · δ,L,2 which cover GT . Since GT +τ∗ ⊂ Φτ∗ GT , we see that GT +τ∗ ⊂ Φτ∗

B∈B

B ∩ GT

⊂

Φτ∗ B .

B∈B

Consider now any of the B in B. We are going to cover Φτ∗ (B) by balls of radius ε. Let ϕ0 and ψ0 be two elements of the ball B and assume furthermore ϕ0 and ψ0 are in GT . This implies ϕ0 − ψ0 δ,L,2 ≤ 4ε ,

(8.5)

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and, since ϕ0 and ψ0 are in the global attracting set G, we also have ϕ0 − ψ0 hδ ,2 ≤ C25 ,

(8.6)

for some constant C25 . With τ∗ as in 8.2, we let ϕ = Φτ∗ (ϕ0 ) ,

ψ = Φτ∗ (ψ0 ) .

We then rewrite ϕ − ψ as ϕ − ψ = Pk∗ (ϕ − ψ) + Qk∗ (ϕ − ψ) ,

(8.7)

where (the direct sums of) Pk∗ and Qk∗ are the high- and low-momentum projections introduced earlier (in (6.1) and (6.2)). Our aim is to bound this diﬀerence in the norm · δ,L−A/ε,2 , where A is a large constant to be determined later. We begin with Pk∗ (ϕ − ψ). By our choice of τ∗ in (8.2), we have, by (8.3) and Proposition 7.2, Pk∗ ϕ − ψ Pk∗ Φτ∗ (ϕ0 ) − Φτ∗ (ψ0 ) ≤ ≤ γ −κ ε . δ,L−A/ε,2 δ,L,2 We now ﬁx η0 > 0 so small and k0 so large (and at least satisfying Eqs.(6.8) and (6.9)) such that for all η < η0 and all k∗ > k0 one has −κ 1 , ≤ γ −κ = min(η −2 , k∗2 /C15 )/320 8

(8.8)

and also γ > γ0 , see (8.3). Summarizing the bounds for this piece, we get Pk∗ (ϕ − ψ)δ,L−A/ε,2 ≤

ε . 8

(8.9)

We bound next Qk∗ (ϕ−ψ) by decomposing it into a ﬁnite sum plus a remainder. We will work with the two components of Qk∗ ϕ or Qk∗ ψ separately. Since the norm on the ﬁrst component has 2 derivatives and the norm on the second only 1, we will deal only with the ﬁrst case and leave the other case to the reader. We will work with the notion of Bernstein class Bσ (K), deﬁned by

Bσ (K) = h

|h(x + iy)| ≤ Keσ|y| for all x, y ∈ R . If h ∈ Bσ (K), it can be represented by the Cartwright interpolation formula [KT, Eq. (191)] (or [B] for a proof) with σ = π/2 and ω = π/4) as h(x) = where the xj = because of

jπ 2σ

∞ sin( σx − πj ) sin(2σx) (−1)j σx 2 πj 4 h(xj ) , 4 ( 2 − 4 )2 j=−∞

(8.10)

are discrete sampling points. This class is useful in our context

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Lemma 8.2 There is a constant C26 such that if u ∈ L∞ , then Qk∗ u ∈ B2k∗ (C26 k∗ u∞ ) .

(8.11)

Proof. This amounts to saying that a function with frequency support in [−2k∗ , 2k∗ ] is in the Bernstein class. This is almost obvious, except for the smooth cutoﬀ. In fact, with the function ϑ as deﬁned in Section 5, we consider ik(x+iy) dk e ϑ(k/k∗ ) = k∗ d+ eik∗ (x+iy) ϑ(+) , (8.12) which is in L1 ( dx ) for any y ∈ R. And the L1 ( dx ) norm is bounded by O(1) k∗ + k∗−1 e2k∗ |y| . Therefore, the convolution operator deﬁned by (8.12) maps u to B2k∗ (O(k∗ )u∞ ). We next bound the functions appearing in (8.10) in our favorite topology: Lemma 8.3 Let σ > 2 and let fj be deﬁned by πj πj σx σx sin 4( σx j sin 2σx sin( 2 − 2 − 4 ) sin( 2 − 4 ) fj (x) ≡ = (−1) πj 2 πj 2 4( σx 4( σx 2 − 4 ) 2 − 4 )

πj 4 )

.

There is a constant C27 independent of j and ξ, such that for all j and ξ one has: 2 2 σ4 C27 dx hδ (x − ξ) fj2 (x) + 2 fj (x) + fj (x) . (8.13) ≤ 1 + (2σξ − πj)4 Remark. The numerical coeﬃcient C27 depends on δ, but δ has been ﬁxed in Eq. (6.11) : δ = 1/(40C18 ). Proof. The function fj can be bounded as

sin 4( σx − πj ) sin( σx −

2 4 2 |fj (x)| ≤

πj 2

4( σx 2 − 4 )

πj

4 )

C28

≤ σx

1+( 2 −

πj 2 4 )

,

since the numerator vanishes simultaneously with the denominator (and to order 2). Similarly, the derivative is bounded by

πj πj

σx σx C29 (σ/2)

sin 4( 2 − 4 sin( 2 − 4 )

, + = 1, 2 , (8.14)

∂x

≤ πj πj 2 σx 2

4( 2 − 4 ) 1 + 4( σx 2 − 4 ) since σ > 2 by assumption. It follows that 1 1 · . dx hδ,ξ (x)|fj (x)|2 ≤ C30 dx (1 + δ 2 (x − ξ)2 )2 (1 + (2σx − πj)2 )2

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Setting ρ = min(δ, 2σ), we ﬁnd that this is bounded by C31 1 dx hδ,ξ (x)|fj (x)|2 ≤ ρ 1 + ρ2 (ξ −

Ann. Henri Poincar´ e

πj 2 2 2σ )

.

In view of (8.14) one gets a similar bound for the derivatives, and thus (8.13) follows. Consider the element (u, v) ∈ G. We know that (u, v)δ,loc,2 ≤ C9 . For the ﬁrst component, u, this means sup dx hδ (x − ξ) |u(x)|2 + 2|u (x)|2 + |u (x)|2 ≤ C92 . ξ∈R

From this, we conclude using the Sobolev inequality in the form of Lemma 2.5 that u∞ ≤ C32 for some ﬁnite C32 . By Lemma 8.2 we then get that Qk∗ u∞ ≤ C26 k∗ C32 and furthermore, Qk∗ u ∈ B2k∗ (C26 k∗ C32 ). Thus, we can apply the Cartwright formula to h = Qk∗ u, with σ = 2k∗ . Throughout, Lk∗ has to be suﬃciently large. We deﬁne SL (h) =

where xj =

jπ 4k∗

sin(4k∗ x) 4

(−1)j

sin(k∗ x −

|j|≤2Lk∗

πj 4 ) h(xj ) πj 2 ) 4

(k∗ x −

,

(8.15)

are the discrete sampling points. We decompose Qk∗ u =

h − SL (h) + SL h .

(8.16)

The ﬁrst term in (8.16) will be small because h − SL (h) is the remainder of the converging sum in (8.10), and for the second one we will use a covering argument. We ﬁrst show that XL ≡ h − SL (h) is small when L is large. The diﬀerence can be written as

h − SL (h) (x) =

(−1)j

|j|>2Lk∗

sin(4k∗ x) sin(k∗ x − 4(k∗ x −

πj 2 4 )

πj 4 )

h(

jπ ). 4k∗

Using (8.13), we get as a bound for XL when |ξ| ≤ L:

dx hδ (x − ξ) |XL (x)|2 + 2|XL (x)|2 + |XL (x)|2 , ≤ C27

|j|≥2Lk∗

1 1 + (k∗ ξ −

πj 2 4 )

≤

1/2 C

33

.

1 + L − |ξ|

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This argument can be repeated for the second component. Since in the deﬁnition (8.1) of · δ,L−A/ε,2 we have |ξ| ≤ L − A/ε, we ﬁnd a bound on the exterior part of Qk∗ (ϕ − ψ): Qk∗ (ϕ − ψ) − SL Qk∗ (ϕ − ψ)

δ,L−A/ε,2

≤

C34 ε C34 ε sup ϕ − ψhδ,ξ ,2 ≤ C25 , A ξ∈R A (8.17)

using (8.6). Clearly, if A is suﬃciently large (but independent of ε and L), we get the bound ε ≤ . (8.18) Qk∗ (ϕ − ψ) − SL Qk∗ (ϕ − ψ) 8 δ,L−A/ε,2 We ﬁnally deal with the central part, namely SL (Qk∗ (ϕ − ψ)). This is described in Proposition 8.4 There is a constant C35 such that the following holds. Let B be a ball of radius ε in the topology deﬁned by · δ,L,2 . Then the set SL (B ∩ G) can be covered by no more than L C35 balls of radius ε/8. Proof. Since ϕ, ψ ∈ G, Lemma 8.2 implies Qk∗ (ϕ − ψ) ∈ B2k∗ (X), where X = diamL∞ (G)C26 k∗ . Moreover, from Corollary 4.2 we deduce Qk∗ (ϕ − ψ)δ,L,2 ≤ C36 ε . Using the Sobolev inequality from Lemma 2.5, this implies

sup Qk∗ (ϕ − ψ) (x) ≤ C37 ε . x∈[−L,L]

We use next the bounds

Qk∗ (ϕ − ψ) (xj ) ≤ C37 ε , for |j| < 2Lk∗ . We let n be a large integer which will be ﬁxed at the end of the proof. The set of values of each of the 2 components of Qk∗ (ϕ − ψ) (xj ) can be covered by 8nC37 balls of radius ε/(4n), for each of the 2(2Lk ∗ ) + 1 possible values of j. We bound now in detail the sum in SL Qk∗ (ϕ − ψ) as deﬁned in (8.15). 4(2Lk∗ )+1 We ﬁx one of the (8nC ) grid points for the components of Qk∗ (ϕ− 37 ψ) (xj ). For each component, we get a set of 2(2Lk∗ ) + 1 numbers q , with |+| ≤

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2Lk∗ . We pick numbers r satisfying |r − q | < ε/(4n) for all + and we want to show that the function ∆(x) =

sin(4k∗ x) 4

|j|≤2Lk∗

(−1)j

sin(k∗ x − (k∗ x −

πj 4 ) (rj πj 2 4 )

− qj )

has a · δ,L,2 norm less than ε/8. This will clearly suﬃce to show Proposition 8.4. By Lemma 8.3, we get ∆δ,L,2 ≤ C38 sup

|ξ|≤L

|j|≤2Lk∗

1 ε ε ≤ C39 . 1 + (4k∗ ξ − πj)2 4n n

L We choose n = 8C39 , and we see that, all in all, one needs (8nC37 )4(2Lk∗ )+1 ≤ C35 balls of radius ε/8 to cover SL (B ∩ G). (Note that C38 and C39 depend on k∗ . In fact they are bounded by O(k∗4 ).)

Proof of Theorem 8.1. We combine now the various estimates to prove (8.4). Let B be one of the NL,T (2ε) balls of radius 2ε needed to cover GT and let ϕ0 ∈ B ∩ GT . Let ϕ = Φτ∗ ϕ0 . All we need to show is that the set of all ψ ∈ Φτ∗ (B ∩ GT ) can be covered by C2 4L balls of radius ε in the topology of the norm · δ,L−A/ε,2 . We decompose ϕ − ψ according to (8.7) and then Qk∗ (ϕ − ψ) according to (8.16), so that we have three terms. The ﬁrst is bounded by ε/8 using (8.9), and this bound uses the fact that the solution is older by τ∗ than the elements of GT . The second L is bounded by (8.18). Since SL (B ∩ GT ) can be covered by C35 balls of radius ε/8 in the norm · δ,L,2 it can also be covered by the same number of balls in the L norm · δ,L−A/ε,2 . Thus the sum of the three contributions can be covered by C35 balls of radius 3ε/8 < ε. The proof of Theorem 8.1 is complete.

9 The ε-entropy and the topological entropy 9.1 Introduction In this section, we exploit the results obtained so far to show that the ε-entropy and the topological entropy per unit length can be deﬁned for the Eq. (2.1). The reasoning here is very close to the one used in [CE2], and so there is no need to repeat it here. What needs however some special attention is the choice of topology for which the entropy per unit length can be deﬁned. We basically need a topology which has a submultiplicativity property which we deﬁne below. The most simple example of such a topology was used in [CE2], namely L∞ . The property which we used there is that if a set S of functions is deﬁned on the union of 2 adjacent intervals, say I1 ∪ I2 , then the following is true: If S restricted to I1 can be covered by NI1 balls of radius ε in L∞ (I1 ), and S|I2 can be covered by NI2 balls in L∞ (I2 ), then S|I1 ∪I2 can be covered by NI1 · NI2 balls in L∞ (I1 ∪ I2 ) (all of radius ε). In

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L∞ , this property is obvious: Let B1,i , with i = 1, . . . , NI1 be the balls covering S|I1 and B2,j , with j = 1, . . . , NI2 those covering S|I2 . Then one can just take the set Si,j of functions

Si,j = f

f |I1 ∈ B1,i and f |I2 ∈ Bj,2 , and this is a ball of radius ε in L∞ (I1 ∪ I2 ). The diﬃculty with topologies which are ﬁner than L∞ is that we have to patch the functions on I1 and I2 together in such a way that the patched function is an element of a ball in the topology on I1 ∪ I2 . We do not know how to do this in the topologies used in the earlier sections, and therefore we go to a new topology in which the submultiplicativity property holds in the sense that there is a constant C = C(ε) independent of I1 and I2 such that the functions on the union of I1 and I2 can be covered by NI1 (ε) · NI2 (ε) · C(ε)

(9.1)

balls of radius ε. It is well known from the literature on statistical mechanics (see e.g., Ruelle [R]) and easy to see that this weaker form of submultiplicativity suﬃces to prove the existence of limits (of the logarithms) per unit length. The topology we will use is W 1,∞ , deﬁned by f W 1,∞ ≡ max sup |f (x)|, sup |f (y)| . (9.2) x∈R

y∈R

This is a “good” topology for our problem, because we can control the evolution of functions in W 1,∞ . However, it is obvious that the submultiplicativity property is not immediate, since the matching of functions has to be continuous and once diﬀerentiable.

9.2 Submultiplicativity in W 1,∞ We develop here the estimates leading to Eq. (9.1) for balls in W 1,∞ . Our main result will be Corollary 9.2. We let R > 5 be a large constant which will be determined in Eq. (9.8) below. Notation. Throughout, we will use the notation |g|I = sup |g(x)| . x∈I

We let WI1,∞ be the space of continuously diﬀerentiable functions g : I → R, equipped with the norm gI = max |g|I , |g |I .

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1,∞ (Thus, comparing with (9.2) we have gW 1,∞ = gR .) Assume gL ∈ W[−R,0] and 1,∞ and let gR ∈ W[0,R]

Eε,G,gL ,gR =

u ∈ C 2 ([−R, R]) : |u |[−R,R] ≤ G ,

u − gL [−R,0] ≤ ε , u − gR [0,R] ≤ ε .

Theorem 9.1 There are a K (depending only on ε and G), and functions 1,∞ g1 , . . . , gN ∈ W[−R,R] satisfying gi (−R) = gL (−R) ,

gi (R) = gR (R) ,

(9.3)

for i = 1, . . . , N , such that the following holds: For every u ∈ Eε,G,gL ,gR , there is a j = j(u) ∈ {1, . . . , N } such that u − gj [−R,R] ≤ ε . Deﬁnition. We say a set {g1 , . . . , gK } of functions gi ∈ W 1,∞ ε-covers a set F of W 1,∞ functions on I if for every g ∈ F there is a k ∈ {1, . . . , K} for which |g − gk |I ≤ ε ,

and |g − gk |I ≤ ε .

Corollary 9.2 Assume that a collection F of C 2 functions is given on [−L, L ] and assume that each f ∈ F satisﬁes |f |[−L,L ] ≤ α, |f |[−L,L ] ≤ β, and |f |[−L,L ] ≤ γ. There are constants R, ε0 > 0 and a family of constants Kε (depending only on α, β, and γ) such that the following holds for any L, L > R and any ε ≤ ε0 : 1,∞ and If F|[−L,0] and F|[0,L ] can be ε-covered by S, (resp. S ) functions in W[−L,0] 1,∞ W[0,L ] respectively, then F|[−L,L ] can be ε-covered by no more than S · S · Kε 1,∞ functions in W[−L,L ].

Proof of Theorem 9.1. We will ﬁrst ﬁnd ﬁnite constants a, b, c (> 1) with the following property: Fix gL and gR and assume E ≡ Eε,G,gL ,gR = ∅ (that is, there is a connecting function in an ε-neighborhood of gL and gR ). We claim one can construct a W 2,∞ function g for which the following inequalities hold : g − gL [−R,0] ≤ aε ,

g − gR [0,R] ≤ bε ,

|g |[−R,R] ≤ c + G .

(9.4)

Furthermore, g will satisfy g(−R) = gL (−R) ,

g(R) = gR (R) .

(9.5)

In other words, this is in principle a good approximation, which in addition matches exactly at the boundary, but the bound has deteriorated to aε and bε and a and b might be larger than 1. The point of Theorem 9.1 and Corollary 9.2 is

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that a (and b) can be pushed down to 1 by increasing the number of connecting functions to a number of functions which does not depend on gL and gR . Fix an arbitrary function u0 ∈ E. We construct a function g which interpolates between gL and gR , using u0 as a bridge. Let ψ be a C ∞ function, 0 ≤ ψ(x) ≤ 1 satisfying ψ(x) = 0 for x < R − 3 and ψ(x) = 1 for x ≥ R. We deﬁne g by g(x) = u0 (x) − ψ(x) · u0 (R) − gR (R) − ψ(−x) · u0 (−R) − gL (−R) . (9.6) This function is clearly continuously diﬀerentiable since u0 is continuously diﬀerentiable. Let I = [0, R]. From (9.6) we ﬁnd for x ∈ I: g(x) − gR (x) = u0 (x) − gR (x) − ψ(x) · u0 (R) − gR (R) , and therefore, |g − gR |I ≤ |u0 − gR |I + |u0 − gR |I · |ψ|I ≤ 2ε , |g − gR |I ≤ |u0 − gR |I + |u0 − gR |I · |ψ |I ≤ ε 1 + |ψ |I . The negative x are handled in the same way. Finally, the last inequality of Eq. (9.4) follows at once from Eq. (9.6). We note that by the construction in Eq. (9.6), the boundary condition (9.5) is fulﬁlled. Deﬁnition. We denote by Fε,A,B,C the set of C 2 functions deﬁned by Fε,A,B,C = {f : |f (±R)| ≤ ε , |f |[−R,R] ≤ A , |f |[−R,R] ≤ B , |f |[−R,R] ≤ C} . We shall need later the sets 0 Fδ,A,B,C,ξ = {f : |f (0)| ≤ δ , |f |[0,ξ] ≤ A , |f |[0,ξ] ≤ B , |f |[0,ξ] ≤ C} .

Let u ∈ Eε,G,gL ,gR and deﬁne g as in Eq. (9.6). If we let f = u−g, then by Eq. (9.4), and the construction of g, we see that f ∈ Fε,A,B,C , with A = B = (a + b)ε and C = c + G. We will now use the following bound on Fε,A,B,C . Proposition 9.3 Fix 0 < ε ≤ A, B ≥ 0, and C ≥ 1. There is a ﬁnite set H of W 1,∞ functions H = {h1 , . . . , hN }, which ε-covers Fε,A,B,C on [−R, R] and which furthermore satisﬁes hj (±R) = 0 ,

(9.7)

for j = 1, . . . , N . Using Proposition 9.3 we can complete the proof of Theorem 9.1. Given gL and gR as above, we construct ﬁrst a g as in Eq. (9.6). When u ∈ E, then f = u−g is in Fε,A,B,C by the bounds Eq. (9.4) and the equality (9.5). Thus, by Proposition

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9.3 the f are ε-covered by the N functions {h1 , . . . , hN }. Deﬁne now ui = hi + g, i = 1, . . . , N , and then the set Eε,G,gL ,gR of functions u is ε-covered by the ui , since u − ui = (u − g) − (ui − g) = f − hi , and we have just stated that the f are ε-covered by a ﬁnite number of ui . Furthermore, the hi vanish at the boundary of [−R, R]. Thus, we have interpolated between gL and gR , with N functions in W 1,∞ which ε-cover the original set. The proof of Theorem 9.1 is complete. The corollary then follows at once since the factor N does not depend on the choice of gL and gR (except that the bound is too pessimistic in case Eε,G,gL ,gR happens to be empty). Remark. The diﬃculty in proving Proposition 9.3 lies in the fact that the hj vanish at the endpoints while the functions f in Fε,A,B,C may be as large as ε near the boundary, |f (±R)| = ε, so there is no space near ±R with which just to construct an open cover. The main ingredient to the proof of Proposition 9.3 is the following local lemma. Before we formulate it, we assume, without loss of generality, that C > 1. Since we are interested in small ε, we shall also assume ε < 1. We introduce two fundamental scales ξ and τ in our analysis: ξ =

ε , 10C

and τ =

ε . 10

We will ﬁrst consider a (small) interval J whose left endpoint is the origin. Deﬁnition of R. We can now ﬁx R by setting it to 41 ≥ R ≥ 40,

R = m∗ ξ ,

(9.8)

where m∗ is an integer. This choice is only good for ε ≤ 10C and we leave the trivial modiﬁcations for arbitrary ε to the reader. Lemma 9.4 Let J = [0, ξ]. There is a ﬁnite set of linear functions of the form 0 , for every δ ∈ [0, ε]. One has in fact better gj (x) = jτ x, which ε-covers Fδ,A,B,C,ξ 0 bounds: there is for every f ∈ Fδ,A,B,C,ξ a j with |j| ≤ B τ + 2 for which |f − gj |J ≤ max(δ, ε2 ν) ,

and

|f − gj |J ≤ ε

and furthermore, at the right endpoint, one has |f (ξ) − gj (ξ)| ≤ max δ − µε2 , ε2 ν ,

3 , 10

(9.9)

where ν =

1 , 40C

µ =

1 . 200C

(9.10)

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Proof of Lemma 9.4. This is just a construction of the “right” j, followed by some 0 veriﬁcations. Note ﬁrst that if f ∈ Fδ,A,B,C,ξ , then we have f (x) = f (0) + xf (0) + x2 v(x) , f (x) = f (0) + xw(x) , with |v|[0,ξ] ≤ C/2 and |w|[0,ξ] ≤ C. We will pursue the proof for the case when f (0) ≥ 0, the other case is handled by symmetry. We deﬁne f (0) f (0) j = +2 = +1+ρ , (9.11) τ τ with ρ ∈ (0, 1]. Here [x] = inf n∈Z,n≥x n is the integer part of x. Now set g(x) = cx, with c = jτ : c = f (0) + τ + τ ρ . Clearly, g equals one of the gj of Lemma 9.4 if we take the ﬁnite set of j to contain |j| ≤ B τ + 2. Next, we estimate the quality of the approximation. First we have f (x) − g (x) = f (x) − c = f (0) + xw(x) − f (0) − τ − τ ρ . This leads, for x ∈ [0, ξ], to ε , 10 2ε ε f (x) − c ≥ −Cξ − 2τ ≥ − − . 10 10 f (x) − c ≤

Cξ − τ ≤ Cξ ≤

We conclude that |f − gj |J ≤

3 ε. 10

We consider next f (x) − cx. We ﬁnd f (x) − cx = f (0) + xf (0) + x2 v(x) − xf (0) − τ x − ρτ x . This leads to the bounds C 10 C ε C 2 x − τ x ≤ δ + x2 − τ x = δ − x(1 − x · ) , (9.12) 2 2 10 2 ε C 2 C 2 f (x) − cx ≥ f (0) − x − 2τ x ≥ − x − 2τ x . (9.13) 2 2

f (x) − cx ≤ f (0) +

Since we consider only x ∈ [0, ξ], we ﬁnd that 1 − x 10C 2ε ≥ 1 − therefore f (x) − cx ≤ δ .

ε 10C 10C 2ε

=

1 2

and

(9.14)

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Recall that we deal with the case f (0) ≥ 0. Thus, we also get for x ∈ [0, ξ], f (x) − cx ≥ −

C ε2 2ε2 ε2 4ε2 ε2 − = − − = − ≡ −ε2 ν . 2 2 2 10 C 100C 200C 200C 40C (9.15)

Thus, we conclude that |f − gj |J ≤ ε, provided ε ≤ 40C. (This un-intuitive bound comes from having chosen ξ = ε/(10C) which is unreasonable when C " 1.) We next show that the bound on f (x) − cx is tighter than what we got so far when x = ξ. Indeed, we get in this case from Eq.(9.12), f (ξ) − cξ ≤ δ −

1 ε2 ≡ δ − µε2 . 2 100C

(9.16)

The assertion Eq. (9.9) follows by combining Eq. (9.16) with Eq. (9.15). It remains to see that the set of possible j is ﬁnite. Considering Eq. (9.11) 0 and the fact that f ∈ Fδ,A,B,C,ξ we see that j can take at most 2(B/τ + 2) + 1 possible values. The proof of Lemma 9.4 is complete. Proof of Proposition 9.3. This proof is a repeated application of Lemma 9.4. We ε2 retain the assumptions and notations from that proof. Let η = ξτ = 100C . We consider the grid (in the (x, y)-plane): mξ, nη : m = −m∗ , −m∗ + 1, . . . , m∗ ; n = −n∗ , . . . , n∗ , where m∗ = R/ξ ,

n∗ = [A/η] + 1 ,

recalling that R/ξ is an integer. In other words, we cover the range of possible arguments (in [−R, R]) and values (in [−A, A]) of f ∈ Fε,A,B,C by a ﬁne grid. Consider now the set of all continuous, piecewise linear functions h(x), connecting linearly successive lattice points (mξ, nη) with ((m + 1)ξ, n η), with −m∗ ≤ m < m∗ , |n| ≤ n∗ and |n | ≤ n∗ . Furthermore, we require that h(−R) = h(R) = 0. There are a ﬁnite number of such functions, namely at most (2n∗ + 1)2m∗ −1 . Note that η has been chosen in such a way that the slopes of the straight pieces of h are integer multiples of τ . We show next that every f ∈ Fε,A,B,C is, together with its derivative, ε-close to one of the h. We begin by constructing the piecewise linear approximation h. We start at the point x = −R, y = 0, and shift the origin to this point by deﬁning: f0 (x) = f (x + R) . 0 0 ⊃ Fδ,A,B,C,ξ , with δ = ε, and by Lemma 9.4, f0 is Then f0 is in Fε,A,B,C,2R approximated by one of the linear functions, say n0 τ x, with n0 = [2 + f0 (0)/τ ] on the interval [0, ξ] (when f (0) > 0). Note that we also have (when ε is small) |f0 (ξ) − n0 τ ξ| ≤ ε − µε2 . We deﬁne

h(x) = n0 τ (x + R) , for x ∈ [−R, −R + ξ] .

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Next shift the origin of the (x, y)-plane to (−R + ξ, n0 η) = −R + ξ, h(ξ) , and deﬁne f1 (x) = f (x + R − ξ) − h(x + R − ξ) = f (x + R − ξ) − n0 η . The deﬁnition of the ﬁrst segment of h and the bounds on f0 show that 0 f1 ∈ Fε−µε 2 ,A+|n |η,B,C,2R−ξ . 0 0 We now apply Lemma 9.4 to f1 . Note that f1 is not in Fδ,A,B,C,ξ but in a space with a worse bound on the absolute value. However, the value of A does not enter the construction of the proof of Lemma 9.4 and hence is irrelevant for our inductive construction of h. Applying Lemma 9.4 to f1 , we ﬁnd the second linear piece of the function h, and get a piecewise linear, continuous approximation of f on [−R, −R+2ξ]. The ﬁnal point of the approximation by h is now (−R+2ξ, n1 η), and we construct f2 by translating the origin to that point. Assuming that ε − 2µε2 > νε2 , we conclude that 0 f2 ∈ Fε−2µε 2 ,A+|n +n |η,B,C,2R−2ξ . 0 1

Note that the construction can not “drift away” in the y-direction, since we assumed from the outset that |f |[−R,R] ≤ A, and hence the y-translates never exceed A by more than ε (since h is an approximation to f ). We continue the construction in the same way as before, until x = 0 is reached. At this point we have achieved the following: The original function is approximated by the piecewise linear function h on J = [−R, 0] with the bound |f − h|J ≤ ε ,

|f − h |J ≤ ε .

Furthermore, at the point x = ξ the approximation is really “good:” Consider the deﬁnition (9.8) of R. The number of steps from −R to 0 is m∗ ≥ 40 · 1ξ − 1 = 40·10C − 1 and in each step we gain a constant µε2 , as long as δ > ε2 ν. Therefore, ε (9.17) |f (0) − h(0)| ≤ max ε2 ν, ε − m∗ ε2 µ = ε2 ν , where the last equality follows from 400C 400C 1 − 2 = ε2 · −2 ≥ ε2 µm∗ ≥ ε2 µ(m∗ − 1) ≥ ε2 µ ε 200C ε 2ε2 ≥ε, 2ε − 200C

(9.18)

when ε ≤ 100C. We repeat the same construction from the right endpoint, (with m∗ −1 steps, which is also covered by (9.18)) obtaining the piecewise linear function h on the set J = [ξ, R], and again a bound, using (9.10) : |f (ξ) − h(ξ)| ≤ max ε2 ν, ε − (m∗ − 1)ε2 µ ≤ ε2 ν . (9.19)

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We complete the deﬁnition of h by connecting 0, h(0) linearly with ξ, h(ξ) . Note that it is necessarily a line segment connecting two of the grid points and so h is one of the functions we counted earlier. We need to verify the bounds on J = [0, ξ]. It is here that the Eqs.(9.17) and (9.19) are relevant. We write f (x) = f (0) · (1 −

x x ) + f (ξ) + r(x) , ξ ξ

and then by the bounds on the second derivative of f we get |r|J ≤ Cξ 2 /8, and |r |J ≤ Cξ/2. Since x x h(x) = h(0) · (1 − ) + h(ξ) , ξ ξ we ﬁnd for ε ≤ 800C/21, C ε2 ε ε = ε( + ) ≤ ε, 2 2 8 C 10 40C 8 · 102 C 2 · 10C 2 Cε 11 2 Cξ 1 = ·ε · + = ε ≤ ε. |f − h |J ≤ ε2 ν + ξ 2 ε 40C 2 · 10C 20 |f − h|J ≤ ε2 ν +

Thus, we have shown the required bound on all of [−R, R]. The piecewise linear, continuous function obtained in this way will be called hf (x). It is clearly one of the functions we constructed. It approximates f and f on all of [−R, R]. We have thus found a ﬁnite family of piecewise linear functions which ε-covers Fε,A,B,C . The proof of Proposition 9.3 is complete.

9.3 The ε-entropy of Kolmogorov and Tikhomirov We proceed as in [CE1], but with a change of topology as explained above. We have deﬁned in Section 8 the minimum number NL,T (ε) of balls in the norm · δ,L,2 needed to cover the set GT . We also showed in Theorem 8.1 that NL−A/ε,T +τ∗ (ε) ≤ C2 4L NL,T (2ε) ,

(9.20)

with some constants A and C2 4 depending only on the coeﬃcients of the problem (2.1). If we iterate Eq. (9.20) m times, we get L+A/ε

NL,T (ε) ≤ C24

L+2A/ε

C24

L+mA/ε

· · · C24

NL+A/ε+2A/ε+···+mA/ε,T −mτ∗ (2m ε) , (9.21)

provided both T − mτ∗ ≥ 0. In (2.20) we have shown that there is a constant C9 which bounds the radius of G in Hδ,loc,2 . (The bound C9 in (2.20) was for Hα,loc,2 .) Therefore, one ball of radius C9 suﬃces to cover G|[−L,L] . Choosing m = m(ε) in

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such a way that 2m ε > C9 , we conclude that Gm(ε)τ∗ |[−L,L] can be covered by a ﬁnite number of balls in Hδ,loc,2 :1 m(ε)(L+A/ε)

NL,m(ε)τ∗ (ε) ≤ C24

,

with m(ε) = O(log(1/ε)). We deﬁne similarly for any interval I, the minimal number MI,T (ε) of balls needed to cover GT in the topology · W 1,∞ . By the I Sobolev inequality from Lemma 2.5, we see that uW 1,∞

[−L,L]

≤ C4 uδ,L,2 .

Therefore, if GT can be covered by NL,T (ε/C4 ) balls of radius ε/C4 in the norm · δ,L,2 , it can obviously be covered by the same number of balls of radius ε in the norm · W 1,∞ . Thus we have [−L,L]

M[−L,L],T (ε) ≤ NL,T (ε/C4 ) .

(9.22)

We now apply Corollary 9.2. We ﬁrst note that by Eqs.(2.22) and (2.23) it is adequate to consider functions with bounded second derivative. (In fact this is the only place where these higher derivatives are needed.) Thus, we can apply Corollary 9.2 and we conclude that for two intervals I1 and I2 , one has MI1 ∪I2 ,T (ε) ≤ MI1 ,T (ε)MI2 ,T (ε)Kε .

(9.23)

Thus, we have established submultiplicativity (in I) and ﬁniteness of MI,T (ε). Furthermore, from the construction of m in (9.21) with 2m ε > C9 , we ﬁnd by choosing the minimal such m = m(ε) = O(log(1/ε)) and for T > m(ε)τ∗ , m(ε)(C40 |I|+C41 A/ε)

MI,T (ε) ≤ C24

.

Using this bound and (9.23), we get convergence and a bound on the ε-entropy Hε (G∞ ): Theorem 9.5 The ε-entropy per unit length of G∞ in W 1,∞ exists and is bounded by 1 log M[−L,L],m(ε)τ∗ (ε) ≤ C42 log(1/ε) . L→∞ L

Hε (G∞ ) ≤ lim

(9.24)

9.4 Existence of the topological entropy per unit length This material is taken from [CE2], and we introduce it without proofs, just to show what follows from the bounds of the previous sections. 1 The argument used here is more elegant than the one used in [CE1]. We thank Y. Colin de Verdi` ere for suggesting it.

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For any ε > 0 and any interval I in R, we deﬁne WIε as the set of all ﬁnite covers of G by open sets in WI1,∞ of diameter at most ε. Note that by the argument of Section 9.3, such a ﬁnite cover exists. Note also that elements of WI1,∞ are pairs of functions (u, v) and that the topology is WI1,∞ on the u-component and L∞ (I) of the v-component. Let τ > 0 be a ﬁxed time step, and let T = nτ with n ∈ Z. Deﬁnitions. Let U ∈ WIε . We say that two elements A1 and A2 in G are (U, T )separated if there is at least one k ∈ {0, . . . , n} for which the points Φkτ (A1 ) and Φkτ (A2 ) do not belong to the same atom of U. We deﬁne NT,τ (U) to be the largest number of elements which are pairwise (U, T )-separated (and considered with time-step τ .) Note that this number is ﬁnite since it is at most (Card U)2T /τ . Finally, we deﬁne NI,T,τ,ε = inf ε NT,τ (U) . U∈WI

Lemma 9.6 (Lemma 2.1. of [CE2]). Let I1 and I2 be two disjoint intervals (perhaps with common boundary) and let I = I1 ∪ I2 . The functions NI,T,τ,ε satisfy the following bounds: There is a constant C = C(ε) such that: i) NI,T,τ,ε is non-increasing in ε. ii) NI,T1 +T2 ,τ,ε ≤ NI,T1 ,τ,ε NI,T2 ,τ,ε . iii) NI1 ∪I2 ,T,τ,ε ≤ C NI1 ,T,τ,ε NI2 ,T,τ,ε . Remark. It is important here that C(ε) does not depend on the lengths of I1 and I2 . Proof. The properties i) and ii) are shown exactly as in [CE2]. However, the proof and the statement of iii) are now modiﬁed since we consider the topology of W 1,∞ . In order to prove iii), we consider U1 ∈ WIε1 and U2 ∈ WIε2 . Since we are using the W 1,∞ norm we have NT,τ (U1 ∩ U2 ) ≤ NT,τ (U1 ) NT,τ (U2 ) . We also have easily WIε1 ∩ WIε2 ⊂ WIε1 ∪I2 . The claim iii) now follows easily. Remark. Henceforth, we shall work with domains which are intervals IL = [−L, L].

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Theorem 9.7 The following limit exists h = lim lim

ε→0 L→∞

1 1 lim log NIL ,T,τ,ε . L T →∞ T

(9.25)

Moreover, h does not depend on τ . It is called the topological entropy per unit volume of the system. Proof. The proof is given in [CE2]. Remark. It also follows from Section 9.3 that h is bounded.

9.5 Sampling The results we describe in this section are, on the surface, the same as those obtained in [CE2]. This means that by discrete sampling of the signal in a spacetime region [−L − A log(1/ε), L + A log(1/ε)] × [0, τ∗ log(1/ε)] ,

(9.26)

the function observed can be determined to precision ε everywhere on the interval [−L, L] at time τ∗ log(1/ε). In the current context this result can be worked out in detail in the following sense: Assume that two solutions u1 and u2 and their ﬁrst and second space derivatives (as well as ∂t u1 and ∂t u2 and their ﬁrst derivatives) coincide to within ε in the region (9.26) on a space-time grid with mesh O(1/k∗ ) × O(τ∗ ). Then one can conclude that u1 (τ∗ log(1/ε), ·) − u2 (τ∗ log(1/ε), ·)W 1,∞

[−L,L]

+∂t u1 (τ∗ log(1/ε), ·) − ∂t u2 (τ∗ log(1/ε), ·)L∞ ≤ C43 ε , [−L,L] for some universal constant C43 . This result allows, in principle, to reconstruct the K2 -entropy. In our view, the result sketched above is somewhat unsatisfactory, and its clariﬁcation needs further work. Namely, we would like to be able to make positive statements based on sampling only function values, and not their derivatives, in particular, not the second derivative. (They are needed to bound the diﬀerence in W 1,∞ .) Indeed, a quick inspection of properties of the Bernstein class shows that we have no reasonable bound on SL (Qk∗ f ) − SL (f ) in W 1,∞ if we only have information about the function and not about its derivatives.

Acknowledgment We have proﬁted from useful discussions with Th. Gallay and J. Rougemont. This work was partially supported by the Fonds National Suisse. We thank the referee for pinpointing some misleading deﬁnitions.

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References [B]

R.P.Boas, Entire Functions, New York, Academic Press 1954.

[CE1] P. Collet, and J.-P. Eckmann, Extensive properties of extended systems, Commun.Math.Phys. 200 699–722 (1999). [CE2] P. Collet, and J.-P. Eckmann, Topological entropy per unit volume in parabolic PDE’s, Nonlinearity 12 451–473 (1999). [F]

E. Feireisl, Bounded, locally compact global attractors for semilinear damped wave equations on RN , Diﬀer. Integral Equ. 9, 1147–1156 (1996).

[KT] A.N. Kolmogorv and V.M. Tikhomirov, ε-entropy and ε-capacity of sets in functional spaces.2 In selected works of A.N. Kolmogorov, Vol. III, A.N. Shirayayev, Ed. Dordrecht, Kluver (1993) [M1]

A. Mielke, The complex Ginzburg-Landau equation on large and unbounded domains : sharper bounds and attractors. Nonlinearity 10, 199–222 (1997).

[M2]

A. Mielke, private communication.

[MS]

A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains - existence and comparison, Nonlinearity 8, 743–768 (1995).

[R]

D. Ruelle, Statistical Mechanics, New York, Benjamin (1963).

P. Collet Centre de Physique Th´eorique UMR 7644, CNRS Ecole Polytechnique F-91128 Palaiseau Cedex e-mail : [email protected] J.-P. Eckmann D´epartement de Physique Th´eorique Universit´e de Gen`eve CH-1211 Gen`eve 4, Switzerland Section de Math´ematiques Universit´e de Gen`eve e-mail : [email protected] CH-1211 Gen`eve 4, Switzerland Communicated by Rafael D. Benguria submitted 7/09/99, accepted 23/02/2000 2 The version in this collection is more complete than the original paper of Uspekhi Mat. Nauk, 14, 3–86 (1959).

Ann. Henri Poincar´ e 1 (2000) 753 – 800 c Birkh¨ auser Verlag, Basel, 2000 1424-0637/00/040753-48 $ 1.50+0.20/0

Annales Henri Poincar´ e

Long Range Scattering and Modiﬁed Wave Operators for some Hartree Type Equations II∗ J. Ginibre, G. Velo

Abstract. We study the theory of scattering for a class of Hartree type equations with long range interactions in space dimension n ≥ 3, including Hartree equations with potential V (x) = λ|x|−γ . For 0 < γ ≤ 1 we prove the existence of modiﬁed wave operators with no size restriction on the data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators, thereby extending the results of a previous paper which covered the range 1/2 < γ < 1.

1 Introduction This is the second paper where we study the theory of scattering and more precisely the existence of modiﬁed wave operators for a class of long range Hartree type equations 1 (1.1) i∂t u + ∆u = g(|u|2 )u 2 where u is a complex function deﬁned in space time IRn+1 , ∆ is the Laplacian in IRn , and g(|u|2 ) = λtµ−γ ωµ−n |u|2 (1.2) with ω = (−∆)1/2 , λ ∈ IR, 0 < γ ≤ 1 and 0 < µ < n. The operator ω µ−n can also be represented by the convolution in x ωµ−n f = Cn,µ |x|−µ ∗ f

(1.3)

so that (1.2) is a Hartree type interaction with potential V (x) = C|x|−µ . The more standard Hartree equation corresponds to the case γ = µ. In that case, the nonlinearity g(|u|2 ) becomes g(|u|2 ) = V ∗ |u|2 = λ|x|−γ ∗ |u|2

(1.4)

with a suitable redeﬁnition of λ. A large amount of work has been devoted to the theory of scattering for the Hartree equation (1.1) with nonlinearity (1.4) as well as with similar nonlinearities with more general potentials. As in the case of the linear Schr¨ odinger equation, one must distinguish the short range case, corresponding to γ > 1, from the long range ∗ Work

supported in part by NATO Collaborative Research Grant 972231

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case corresponding to γ ≤ 1. In the short range case, it is known that the (ordinary) wave operators exist in suitable function spaces for γ > 1 [11]. Furthermore for repulsive interactions, namely for λ ≥ 0, it is known that all solutions in suitable spaces admit asymptotic states in L2 for γ > 1, and that asymptotic completeness holds for γ > 4/3 [10]. In the long range case γ ≤ 1, the ordinary wave operators are known not to exist in any reasonable sense [10], and should be replaced by modiﬁed wave operators including a suitable phase in their deﬁnition, as is the case for the linear Schr¨odinger equation. A well developed theory of long range scattering exists for the latter. See for instance [1] for a recent treatment and for an extensive bibliography. In contrast with that situation, only partial results are available for the Hartree equation. On the one hand, the existence of modiﬁed wave operators has been proved in the critical case γ = 1 for small solutions [2]. On the other hand, it has been shown, ﬁrst in the critical case γ = 1 [6, 9] and then in the whole range 0 < γ ≤ 1 [5, 7, 8] that the global solutions of the Hartree equation (1.1) (1.3) with small initial data exhibit an asymptotic behaviour as t → ±∞ of the expected scattering type characterized by scattering states u± and including suitable phase factors that are typical of long range scattering. In particular, in the framework of scattering theory, the results of [5, 7, 8] are closely related to the property of asymptotic completeness for small data. In a previous paper with the same title [4], hereafter referred to as I, we proved the existence of modiﬁed wave operators for the equation (1.1) (1.2), and we gave a description of the asymptotic behaviour in time of solutions in the ranges of those operators, with no size restriction on the data, in suitable spaces and for γ in the range 1/2 < γ < 1. The method is an extension of the energy method used in [5, 7, 8], and uses in particular the equations introduced in [7] to study the asymptotic behaviour of small solutions. The spaces of initial data, namely in the present case of asymptotic states, are Sobolev spaces of ﬁnite order similar to those used in [8]. The present paper is devoted to the extension of the previous results to the whole range 0 < γ ≤ 1. The methods used here are natural extensions of those used in I. They require in particular the same restrictions on µ and n, in particular µ ≤ n − 2 and n ≥ 3. We refer to the introduction of I for a discussion of those conditions. The construction of the modiﬁed wave operators is too complicated to allow for a more precise statement of results at this stage, and will be described in Section 2 below, which is a summary and continuation of Section 2 of I. That construction involves the study of the same auxiliary system of equations as in I, for a new function w and a phase ϕ instead of the original function u, and relies as a preliminary step on the construction of local wave operators in a neighborhood of inﬁnity for that system. That step requires the deﬁnition of a modiﬁed asymptotic dynamics which is signiﬁcantly more complicated than that used in I. We now give a brief outline of the contents of this paper. A more detailed description of the technical parts will be given at the end of Section 2. After collecting some notation and preliminary estimates in Section 3 and recalling from I some preliminary results on the auxiliary system in Section 4, we deﬁne and study

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the asymptotic dynamics in Section 5. We then study the asymptotic behaviour of solutions for the auxiliary system in Section 6. In particular we essentially construct local wave operators at inﬁnity for that system. We then come back from the auxiliary system to the original equation (1.1) for u and construct the wave operators for the latter in Section 7, where the ﬁnal result will be stated in Proposition 7.5. We have tried to make this paper as self-contained as possible and at the same time to keep duplication with I to a minimum. Duplication occurs in the beginning of Section 3 and in Section 4 where we recall estimates and results from I. On the other hand, Sections 6 and 7 follow the same pattern as Sections 5, 6 and 7 of I, with the appropriate changes needed to handle the more general situation. We conclude this section with some general notation which will be used freely throughout this paper. We denote by · r the norm in Lr ≡ Lr (IRn ). For any interval I and any Banach space X, we denote by C(I, X) the space of strongly continuous functions from I to X and by L∞ (I, X) (resp. L∞ loc (I, X)) the space of measurable essentially bounded (resp. locally essentially bounded) functions from I to X. For real numbers a and b, we use the notation a ∨ b = Max(a, b), a ∧ b = Min(a, b) and [a] = integral part of a. In the estimates of solutions of the relevant equations, we shall use the letter C to denote constants, possibly diﬀerent from an estimate to the next, depending on various parameters such as γ, but not on the solutions themselves or on their initial data. Those constants will be bounded in γ for γ away from zero. We shall use the notation A(a1 , a2 , · · · ) for estimating functions, also possibly diﬀerent from an estimate to the next, depending in addition on suitable norms a1 , a2 , · · · of the solutions or of their initial data. Finally Item (p.q) of I will be referred to as Item (I.p.q). Additional notation will be given at the beginning of Section 3. In all this paper, we assume that n ≥ 3, 0 < µ ≤ n − 2 and 0 < γ ≤ 1.

2 Heuristics In this section, we discuss in heuristic terms the construction of the modiﬁed wave operators for the equation (1.1), as it will be performed in this paper. That construction is an extension of that performed in I in the special case γ > 1/2, and we refer to Section I.2 for a more detailed introduction and for general background. The problem that we want to address is that of classifying the possible asymptotic behaviours of the solutions of (1.1) by relating them to a set of model functions V = {v = v(u+ )} parametrized by some data u+ and with suitably chosen and preferably simple asymptotic behaviour in time. For each v ∈ V, one tries to construct a solution u of (1.1) such that u(t) behaves as v(t) when t → ∞ in a suitable sense. The map Ω : u+ → u thereby obtained classiﬁes the asymptotic behaviours of solutions of (1.1) and is a preliminary version of the wave operator for positive time. A similar question can be asked for t → −∞. From now on we restrict our attention to positive time.

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In the short range case corresponding to γ > 1 in (1.1), the previous scheme can be implemented by taking for V the set V = {v = U (t)u+ } of solutions of the equation 1 i∂t v + ∆v = 0 , (2.1) 2 with U (t) being the unitary group U (t) = exp (i(t/2)∆)

.

(2.2)

The initial data u+ for v is called the asymptotic state for u. In the long range case corresponding to γ ≤ 1 in (1.1) (1.2), the previous set is known to be inadequate and has to be replaced by a better set of model functions obtained by modifying the previous ones by a suitable phase. The modiﬁcation that we use requires additional structure of U (t). In fact U (t) can be written as U (t) = M (t) D(t) F M (t)

(2.3)

where M (t) is the operator of multiplication by the function , M (t) = exp ix2 /2t

(2.4)

F is the Fourier transform and D(t) is the dilation operator deﬁned by (D(t) f ) (x) = (it)−n/2 f (x/t)

.

(2.5)

Let now ϕ(0) = ϕ(0) (x, t) be a real function of space time and let z (0) (x, t) = exp(−iϕ(0) (x, t)). We replace v(t) = U (t)u+ by the modiﬁed free evolution [12] [13] v(t) = M (t) D(t) z (0) (t) w+ (2.6) where w+ = F u+ . In order to allow for easy comparison of u with v, it is then convenient to represent u in terms of a phase factor z(t) = exp(−iϕ(t)) and of an amplitude w(t) in such a way that asymptotically ϕ(t) behaves as ϕ(0) (t) and w(t) tends to w+ . This is done by writing u in the form [7] [8] u(t) = M (t) D(t) z(t) w(t) ≡ (Λ(w, ϕ)) (t) .

(2.7)

In I, we introduced three possible modiﬁed free evolutions vi (t) i = 1, 2, 3 and correspondingly three parametrizations of u(t) by (wi (t), ϕi (t)), i = 1, 2, 3. The choice (2.6) (2.7) corresponds to i = 2. We shall work exclusively with that choice throughout this paper, and the subscript 2 is therefore consistently omitted. In I we used mostly the choice i = 3 and dropped the subscript 3, so that (w, ϕ) in I means (w3 , ϕ3 ) as opposed to (w2 , ϕ2 ) in this paper. This should be kept in mind when comparing results from I and from this paper. The construction of the wave operators for u proceeds by ﬁrst constructing the wave operators for the pair (w, ϕ) and then recovering the wave operators for

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u therefrom by the use of (2.7). The evolution equation for (w, ϕ) is obtained by substituting (2.7) into the equation (1.1). One obtains the equation i∂t + (2t2 )−1 ∆ − D∗ gD zw = 0 (2.8) for zw, with

g ≡ g |u|2 = g |Dw|2

,

or equivalently, by expanding the derivatives in (2.8), i∂t + (2t2 )−1 ∆ − i(2t2 )−1 (2∇ϕ · ∇ + (∆ϕ)) w + ∂t ϕ − (2t2 )−1 |∇ϕ|2 − D∗ gD w = 0 .

(2.9)

(2.10)

We are now in the situation of a gauge theory. The equation (2.8) or (2.10) is invariant under the gauge transformation (w, ϕ) → (w exp(iσ), ϕ + σ), where σ is an arbitrary function of space time, and the original gauge invariant equation is not suﬃcient to provide evolution equations for the two gauge dependent quantities w and ϕ. At this point we arbitrarily add the Hamilton-Jacobi equation as a gauge condition. This yields a system of evolution equations for (w, ϕ), namely   ∂t w = i(2t2 )−1 ∆w + (2t2 )−1 (2∇ϕ · ∇ + (∆ϕ)) w

(2.11)



(2.12)

∂t ϕ = (2t2 )−1 |∇ϕ|2 + t−γ g0 (w, w)

where we have deﬁned g0 (w1 , w2 ) = λ Re ω µ−n w1 w ¯2

(2.13)

and rewritten the nonlinear interaction term in (2.10) as D∗ g |Dw|2 D = t−γ g0 (w, w) . The gauge freedom in (2.11) (2.12) is now reduced to that given by an arbitrary function of space only. It can be shown, actually it has been shown in I, that the Cauchy problem for the system (2.11) (2.12) is locally wellposed in a neighborhood of inﬁnity in time. The solutions thereby obtained behave asymptotically as w(t) = O(1) and ϕ(t) = O(t1−γ ) as t → ∞, a behaviour that is immediately seen to be compatible with (2.11) (2.12). We next study the asymptotic behaviour of the solutions of the auxiliary system (2.11) (2.12) in more detail and try to construct wave operators for that system. For that purpose, we need to choose a set of model functions playing the role of v, in the spirit of (2.6). In the simple case γ > 1/2 considered in I, that set of model functions was taken to consist of solutions of the system   ∂t w(0) = 0 (2.14)  . ∂t ϕ(0) = t−γ g0 w(0) , w(0)

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The general solution of (2.14) is  (0) w (t) = w+       ϕ(0) (t) = ψ+ +

t

dt1 t−γ 1

(2.15) g0 (w+ , w+ ) ≡ ψ+ + ϕ0 (t)

1

and leads to (2.6) with ϕ(0) = ψ+ + ϕ0 . The asymptotic states for (w, ϕ) then consist of pairs (w+ , ψ+ ). The choice (2.14) (2.15) is adequate for γ > 1/2 because comparison of (2.11) (2.12) with (2.15) yields ∂t (ϕ − ϕ0 ) = O(t−2γ ) which is integrable at inﬁnity for γ > 1/2, thereby allowing for imposing an initial condition at t = ∞ for ψ1 = ϕ − ϕ0 . For γ ≤ 1/2 however, the choice (2.14) (2.15) is not suﬃcient and one needs to construct more accurate asymptotic functions. There are several ways to do that. The one we choose can be motivated heuristically as follows. Let p ≥ 0 be an integer. We write   wm + qp+1 ≡ Wp + qp+1 (2.16) w=     0≤m≤p    ϕ = ϕm + ψp+1 ≡ φp + ψp+1  

(2.17)

0≤m≤p

with the understanding that asymptotically in t wm (t) = O t−mγ , qp+1 (t) = o t−pγ

ϕm (t) = O t1−(m+1)γ

,

,

ψp+1 (t) = o t1−(p+1)γ

(2.18) .

(2.19)

Substituting (2.16) (2.17) into (2.11) (2.12) and identifying the various powers of t−γ yields the following system of equations for (wm , ϕm ) :  −1  (2∇ϕj · ∇ + (∆ϕj )) wm−j ∂t wm+1 = 2t2     0≤j≤m 2 −1    ∇ϕj · ∇ϕm−j + t−γ   ∂t ϕm+1 = 2t 0≤j≤m

(2.20)

g0 (wj , wm+1−j ) (2.21)

0≤j≤m+1

for m + 1 ≥ 0. Here it is understood that wj = 0 and ϕj = 0 for j < 0, so that the case m = −1 of (2.20) (2.21) reduces to (2.14) with w(0) = w0 and ϕ(0) = ϕ0 . We supplement that system with the initial conditions  (2.22)  w0 (∞) = w+ , wm (∞) = 0 for m ≥ 1 

ϕm (1) = 0

for 0 ≤ m ≤ p .

(2.23)

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The system (2.20) (2.21) with the initial conditions (2.22) (2.23) can be solved by successive integrations : knowing (wj , ϕj ) for 0 ≤ j ≤ m, one constructs successively wm+1 by integrating (2.20) between t and ∞, and then ϕm+1 by integrating (2.21) between 1 and t. If (p + 1)γ < 1, that method of resolution reproduces the asymptotic behaviour in time (2.18) (2.19) which was used in the ﬁrst place to provide a heuristic derivation of the system (2.20) (2.21). One can however consider that system and solve it by the same method for any integer p. If (p + 1)γ > 1, the asymptotic behaviour saturates at wm = O(t−1 ) for mγ > 1 and ϕm = O(1) for (m + 1)γ > 1. If γ −1 is an integer, (m + 1)γ = 1 for some m, then ϕm (t) = O(Log t) and wm+1 = O(t−1 Log t). We now argue that for suﬃciently large p, φp is a suﬃciently good approximation for ϕ to ensure that ψp+1 has a limit as t → ∞. In fact by comparing the system (2.20) (2.21) with (2.11) (2.12), one ﬁnds that ∂t ψp+1 is of the same order in t as ∂t ϕp+1 , namely ∂t ψp+1 = O(t−(p+2)γ ), which is integrable at inﬁnity for (p + 2)γ > 1. In this way every solution (w, ϕ) of the system (2.11) (2.12) as obtained previously has asymptotic states consisting of w+ = lim w(t) and t→∞

ψ+ = lim ψp+1 (t). t→∞

Conversely, under the condition (p + 2)γ > 1, we shall be able to solve the system (2.11) (2.12) by looking for solutions in the form (2.16) (2.17) with the additional initial condition ψp+1 (∞) = ψ+ , thereby getting a solution which is asymptotic to (Wp , φp + ψ+ ) with

, ϕ − φp − ψ+ = O t1−(p+2)γ . (2.24) w − Wp = O t−(p+1)γ This allows to deﬁne a map Ω0 : (w+ , ψ+ ) → (w, ϕ) which is essentially the wave operator for (w, ϕ). It is an unfortunate feature of the methods used in this paper that both the construction of the asymptotic states (w+ , ψ+ ) of a given solution (w, ϕ) and the construction of (w, ϕ) from given asymptotic states (w+ , ψ+ ) suﬀer from a loss of regularity of roughly p + 1 derivatives, which prevents the two constructions to be inverse of each other in a strict sense. We next discuss the gauge covariance properties of Ω0 . Two solutions (w, ϕ) and (w , ϕ ) of the system (2.11) (2.12) will be said to be gauge equivalent if they give rise to the same u through (2.7), namely if w exp(−iϕ) = w exp(−iϕ ). If (w, ϕ) and (w , ϕ ) are two gauge equivalent solutions, one can show easily that the diﬀerence ϕ− = ϕ −ϕ has a limit σ when t → ∞ and that w+ = w+ exp(iσ). Under that condition, it turns out that the phases {ϕj } and φp (but not the amplitudes) obtained by solving (2.20) (2.21) are gauge invariant, namely ϕm = ϕm for 0 ≤ m ≤ p and therefore φp = φp , so that ψ+ = ψ+ + σ. It is then natural to deﬁne gauge equivalence of asymptotic states (w+ , ψ+ ) and (w+ , ψ+ ) by the condition w+ exp(−iψ+ ) = w+ exp(−iψ+ ) and the previous result can be rephrased as the statement that gauge equivalent solutions of (2.11) (2.12) in R(Ω0 ) have gauge

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equivalent asymptotic states. Conversely, we are interested in showing that gauge equivalent asymptotic states have gauge equivalent images under Ω0 . Here however we meet with a technical problem coming from the construction of Ω0 itself. For given (w+ , ψ+ ) we construct (w, ϕ) in practice as follows. We take a (large) ﬁnite time t0 and we deﬁne a solution (wt0 , ϕt0 ) of the system (2.11) (2.12) by imposing a suitable initial condition at t0 , depending on (w+ , ψ+ ), and using the known results for the Cauchy problem with ﬁnite initial time. We then let t0 tend to inﬁnity and obtain (w, ϕ) as the limit of (wt0 , ϕt0 ). The simplest way to prove the gauge equivalence of two solutions (w, ϕ) and (w , ϕ ) obtained in this way from gauge equivalent (w+ , ψ+ ) and (w+ , ψ+ ) consists in using an initial condition at t0 which already ensures that (wt0 , ϕt0 ) and (wt 0 , ϕt0 ) are gauge equivalent. Unfortunately the natural choice (wt0 (t0 ), ϕt0 (t0 )) = (Wp (t0 ), φp (t0 ) + ψ+ ) does not satisfy that requirement as soon as p ≥ 1 because φp (t0 ) is gauge invariant while Wp (t0 ) exp(−ψ+ ) is not. In order to overcome that diﬃculty, we introduce a new amplitude V and a new phase χ deﬁned by solving the transport equations  (2.25)  ∂t V = (2t2 )−1 (2∇φp−1 · ∇ + (∆φp−1 )) V 

∂t χ = t−2 ∇φp−1 · ∇χ

(2.26)

with initial condition V (∞) = w+

,

χ(∞) = ψ+

.

(2.27)

It follows from (2.25) (2.26) that V exp(−iχ) satisﬁes the same transport equation as V , now with gauge invariant initial condition (V exp(−iχ))(∞) = w+ exp(−iψ+ ), and is therefore gauge invariant. Furthermore, (V, χ) is a suﬃciently good approximation of (Wp , ψ+ ) in the sense that

V (t) − Wp (t) = O t−(p+1)γ , χ(t) − ψ+ = O(t−γ ) . (2.28) One then takes (wt0 (t0 ), ϕt0 (t0 )) = (V (t0 ), φp (t0 )+χ(t0 )) as an initial condition at time t0 , thereby ensuring that (wt0 , ϕt0 ) and (wt 0 , ϕt0 ) are gauge equivalent. That equivalence is easily seen to be preserved in the limit t0 → ∞. Furthermore, the estimates (2.28) ensure that the asymptotic properties (2.24) are preserved by the modiﬁed construction. As a consequence of the previous discussion, the map Ω0 is gauge covariant, namely induces an injective map of gauge equivalence classes of asymptotic states (w+ , ψ+ ) to gauge equivalence classes of solutions (w, ϕ) of the system (2.11) (2.12). The wave operator for u is obtained from Ω0 just deﬁned and from Λ deﬁned by (2.7). From the previous discussion it follows that the map Λ ◦ Ω0 : (w+ , ψ+ ) → u is injective from gauge equivalence classes of asymptotic states (w+ , ψ+ ) to solutions of (1.1). In order to deﬁne a wave operator for u involving only the asymptotic state u+ but not an arbitrary phase ψ+ , we choose a representative in each equivalence class (w+ , ψ+ ), namely we deﬁne the wave operator for u as the

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map Ω : u+ → u = (Λ ◦ Ω0 )(F u+ , 0). Since each equivalence class of asymptotic states contains at most one element with ψ+ = 0, the map Ω is again injective. We shall prove in addition that R(Ω) = R(Λ ◦ Ω0 ) if p ≤ 2. (This need not be the case if p ≥ 3, because derivative losses in the construction generate a mismatch between the regularity properties required on w+ and ψ+ , so that gauge equivalence classes of asymptotic states need not contain an element with ψ+ = 0 in that case). The previous heuristic discussion was based in part on a number of asymptotic estimates in terms of negative powers of t. However if γ −1 is an integer some of these estimates have to be replaced or supplemented by logarithms. In order to treat all values of γ ∈ (0, 1] in a uniﬁed way, we shall introduce a number of estimating functions of time deﬁned by integral representations. Those functions are smooth in γ, in particular at integer values of γ −1 . They generate the logarithms automatically whenever needed, and they recombine nicely between themselves in the derivation of the main estimates. The simplest example thereof is h0 (t) deﬁned by (3.19) below. In the same way as in I, the system (2.11) (2.12) can be rewritten as a system of equations for w and for s = ∇ϕ, from which ϕ can then be recovered by (2.12), thereby leading to a slightly more general theory since the system for (w, s) can be studied without even assuming that s is a gradient. In I, we ﬁrst studied the system for (w, s) and then deduced therefrom the relevant results for (w, ϕ). Here for simplicity we shall use exclusively the variables (w, ϕ). The same remark applies to the system (2.20) (2.21). We are now in a position to describe in more detail the contents of the technical parts of this paper, namely Sections 3-7. In Section 3, we introduce some notation, we deﬁne the relevant function spaces needed to study the system (2.11) (2.12), we recall from I a number of Sobolev and energy estimates, we then introduce the estimating functions of time mentioned above and we derive a number of estimates for them. In Section 4, we recall from I some preliminary results on the Cauchy problem for the auxiliary system (2.11) (2.12) and on the asymptotic behaviour of its solutions. In Section 5 we study the systems (2.20) (2.21) and (2.25) (2.26) deﬁning the asymptotic dynamics. We ﬁrst derive a number of properties and estimates for the solutions of the system (2.20) (2.21), deﬁned inductively (Proposition 5.1). We then prove the existence and some properties of the solutions of the transport equations (2.25) and (2.26) in a slightly more general setting (Propositions 5.2 and 5.3 respectively). We ﬁnally specialize those results to the case at hand and compare V with Wp deﬁned by (2.16) (Proposition 5.4). In Section 6 we study in detail the asymptotic behaviour in time of solutions of the auxiliary system (2.11) (2.12). We ﬁrst derive asymptotic estimates on the approximation of the available solutions (w, ϕ) of that system by the asymptotic functions (Wm , φm ) deﬁned by (2.16) (2.17), and in particular we complete the proof of existence of asymptotic states for those solutions (Proposition 6.1). We then turn to the construction of local wave operators at inﬁnity. For a given solution (V, χ) of the system (2.25) (2.26) and a given (large) t0 , we construct a solution (wt0 , ϕt0 ) of the system (2.11) (2.12) which coincides with (V, φp + χ) at t0 and

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we estimate it uniformly in t0 (Proposition 6.2). We then prove that when t0 → ∞, (wt0 , ϕt0 ) has a limit (w, ϕ) which is asymptotic both to (V, φp + χ) and to (Wp , φp + ψ+ ) (Proposition 6.3). Finally in Section 7, we exploit the results of Section 6 to construct the wave operators for the equation (1.1) and to describe the asymptotic behaviour of solutions in their range. We ﬁrst prove that the local wave operator at inﬁnity for the system (2.11) (2.12) deﬁned through Proposition 6.3 in Deﬁnition 7.1 is gauge covariant in the sense of Deﬁnitions 7.2 and 7.3 in the best form that can be expected with the available regularity (Propositions 7.2 and 7.3). With the help of some information on the Cauchy problem for (1.1) at ﬁnite time (Proposition 7.1), we then deﬁne the wave operator Ω : u+ → u (Deﬁnition 7.4), we prove that it is injective and under suitable restrictions, that it has the expected range (Proposition 7.4). We then collect all the available information on Ω and on solutions of (1.1) in its range in Proposition 7.5, which contains the main results of this paper.

3 Notation and preliminary estimates In this section, we deﬁne the function spaces where we shall study the auxiliary system (2.11) (2.12) and we recall from I a number of Sobolev and energy type estimates which hold in those spaces. We then introduce a number of estimating functions of time and we derive a number of relations and estimates for them. We shall use Sobolev spaces of integer order Hrk deﬁned for 1 ≤ r ≤ ∞ by     Hrk = u : u; Hrk ≡

∂ j u r < ∞   0≤j≤k

and the associated homogeneous spaces H˙ rk with norm

u; H˙ rk = ∂ k u r where

∂ j u r =

∂ α u r

.

α:|α|=j

The subscript r will be omitted if r = 2. Let 30 = [n/2] and deﬁne r0 by δ(r0 ) = 30 so that r0 = 2n for odd n and r0 = ∞ for even n. Let k and 3 be nonnegative integers with 3 ≥ 30 − 1. We shall k k look for w as a complex valued function in spaces L∞ loc (I, H ) or C(I, H ) and for ∞ ϕ as a real valued function in spaces Lloc (I, Y ) or C(I, Y ) where Y = L∞ ∩ H˙ r10 ∩ H˙ 0 +1 ∩ H˙ +2

.

(3.1)

The spaces Y are easily seen to be duals of Banach spaces and satisfy the embed ding Y ⊂ Y for 3 ≥ 3. We shall use systematically the notation |w|k = w; H k

,

|ϕ| = ϕ; Y

(3.2)

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and the meaning of the symbol |a|b will be made unambiguous by the fact that the pair (a, b) contains either the pair (w, k) or the pair (ϕ, 3). Note that the second notation in (3.2) is diﬀerent from, although closely related to, the similar notation in I which was used for s = ∇ϕ. We recall the following result from I (see Lemma I.3.5). Lemma 3.1. Let ϕ be a real function with ∇ϕ ∈ L∞ ∩ H˙ for some 3 > n/2 and let k ≤ 3 + 1. Then the following estimate holds :

k |exp(−iϕ)w|k ≤ C 1+ ∇ϕ; L∞ ∩ H˙

|w|k . (3.3) Let in addition ϕ ∈ L∞ . Then the following estimate holds : |(exp(−iϕ) − 1) w|k ≤

k−1 C ϕ ∞ + ∇ϕ; L∞ ∩ H˙ 1+ ∇ϕ; L∞ ∩ H˙

|w|k .

(3.4)

In order to state the estimates that are relevant for the study of the system (2.11) (2.12), it is useful to give the following deﬁnition (see Deﬁnition I.3.1). Deﬁnition 3.1. Let 0 < µ ≤ n − 2. A pair of nonnegative integers (k, 3) will be called admissible if it satisﬁes k ≤ 3, 3 > n/2 and 3 + 2 + µ ≤ (n/2 + 2k) ∧ (n + k)

(3.5)

and in addition k > n/2 if 3 + 2 + µ = n + k and n/2 + 3 + µ < (n/2 + 2k) ∧ (n + k) if n is even. For µ = n − 2, admissible pairs are pairs (k, 3) such that k = 3 > n/2. If (k, 3) is admissible, so is (k + j, 3 + j) for any positive integer j. Admissible pairs always have k ≥ 2. For n = 3, µ = 1, the pair (2,2) is admissible. The following Sobolev like inequalities will be essential to study the system (2.11) (2.12). Lemma 3.2. Let 3 > n/2 and k ≤ 3. Then the following estimates hold : |(2∇ϕ · ∇ + (∆ϕ)) w|k−1 ≤ C|ϕ|−1 |w|k |∇ϕ1 · ∇ϕ2 |−1 ≤ C|ϕ1 | |ϕ2 |

.

,

(3.6) (3.7)

Assume in addition that (k, 3) is admissible. Let g0 be deﬁned by (2.13). Then |g0 (w1 w2 )| ≤ C|w1 |k |w2 |k |g0 (w1 w2 )|−1 ≤ C|w1 |k |w2 |k−1

,

(3.8) .

(3.9)

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Sketch of proof. (3.6) follows from Lemma I.3.4 by the same estimates as in Lemma I.3.9. The estimate (3.7) essentially follows from Lemma I.3.3. The estimates (3.8) and (3.9) follow from Corollary I.3.1. In addition to the previous estimates, we shall need energy type estimates for solutions of the following transport equations ∂t w = (2t2 )−1 iθ∆w + (2∇φ · ∇ + (∆φ)) w + R1 , (3.10) ∂t ϕ = (2t2 )−1 θ|∇ϕ|2 + 2∇φ · ∇ϕ + R2

(3.11)

where θ is a real constant and φ, R1 , R2 are given functions of space time. Those estimates will be stated in diﬀerential form for brevity, although they should be understood in integrated from. They hold for functions that are suﬃciently regular in time, for instance locally bounded in the relevant norms. Lemma 3.3. Let 3 > n/2 and k ≤ 3. (1) Let w satisfy (3.10). Then the following estimate holds : |∂t |w|k | ≤ C t−2 |φ| |w|k + |R1 |k

.

(3.12)

(2) Let ϕ satisfy (3.11). Then the following estimates hold : |∂t |ϕ| | ≤ C t−2 |ϕ| (|θ| |ϕ| + |φ|+1 ) + |R2 |

,

|∂t |ϕ|−1 | ≤ C t−2 |ϕ|−1 (|θ| |ϕ| + |φ| ) + |R2 |−1

(3.13) .

(3.14)

Sketch of proof. (3.12) follows from Lemmas I.3.2 and I.3.4 by the same estimates as in Lemma I.3.7. (3.13) and (3.14) follow from Lemmas I.3.2 and I.3.3 by the same estimates as in Lemmas I.3.7 and I.3.9. Lemma 3.4. Let 3 > n/2 and k ≤ 3. Let w and ϕ satisfy (3.10) and (3.11) respectively, with θ = 0, R1 = 0 and R2 = 0. Then the following estimates hold : |∂t |w|k+1 | ≤ C t−2 (|φ| |w|k+1 + |φ|+1 |w|k ) |∂t |ϕ|+1 | ≤ C t−2 (|φ| |ϕ|+1 + |φ|+2 |ϕ| )

, .

(3.15) (3.16)

Sketch of proof. (3.15) and (3.16) follow from Lemmas I.3.2, I.3.3 and I.3.4 by the same estimates as in Lemma I.3.8.

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Lemma 3.5. Let 3 > n/2 and k ≤ 3. Let w1 , w2 and ϕ1 , ϕ2 satisfy (3.10) and (3.11) with φ = φ1 and φ = φ2 respectively, and with θ = 0, R1 = 0 and R2 = 0. Let w− = w1 − w2 , ϕ− = ϕ1 − ϕ2 and φ− = φ1 − φ2 . Then the following estimates hold :

, (3.17) |∂t |w− |k | ≤ C t−2 |φ2 | |w− |k + |φ− | |w1 |k + ∇φ− ∞ |w1 |k+1

. (3.18) |∂t |ϕ− | | ≤ C t−2 |φ2 |+1 |ϕ− | + |φ− |+1 |ϕ1 | + ∇φ− ∞ |ϕ1 |+1 Sketch of proof. (3.17) and (3.18) follow from Lemmas I.3.2, I.3.3 and I.3.4 by the same estimates as in Lemma I.3.10. We now introduce a number of estimating functions of time and derive a number of estimates and relations for them. We start with

t h0 (t) = dt1 t−γ (3.19) 1 1

so that h0 (t) =

  (1 − γ)−1 (t1−γ − 1)

for γ = 1



for γ = 1 .

(3.20) Log t

The basic building block for the subsequent functions is the function h deﬁned by

∞ −1 h(t) = dt1 t−γ , (3.21) 1 (t ∨ t1 ) 1

which can also be written as h(t) = t−1 h0 (t) + γ −1 t−γ =

∞

dt1 t−2 1 h0 (t1 )

(3.22)

t

and is explicitly computed as  −1  γ (1 − γ)−1 (t−γ − γt−1 ) h(t) =  −1 t (1 + Log t)

for γ = 1 (3.23) for γ = 1 .

It follows from (3.21) that t h(t) is increasing in t and from (3.20) (3.23) that t h(t) h0 (t)−1 is decreasing in t. The function h satisﬁes the estimates . (3.24) γ −1 t−γ ∨ t−1 ≤ h(t) ≤ |1 − γ|−1 γ −1 t−γ ∨ t−1 The ﬁrst inequality in (3.24) follows in part from (3.22) and in part from the monotony of t h(t), while the second inequality follows from (3.23) and holds only for γ = 1.

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In some cases where we shall need to indicate the dependence of h0 and h on γ, we shall write h0 (γ, t) and h(γ, t) for h0 (t) and h(t). We next deﬁne for any m ≥ 0

t

Nm (t) =

dt1 t−γ hm (t1 ) 1

,

(3.25)

1

∞

Qm (t) =

−1 m dt1 t−γ h (t1 ) , 1 (t ∨ t1 )

(3.26)

1

so that N0 = h0 and Q0 = h. Those functions are smooth in γ. Clearly Nm is increasing and Qm is decreasing in t, while t Qm (t) is increasing in t, so that Qm (t) ≥ Qm (1) t−1 . From the fact that h is decreasing, it follows that Ni+j (t) ≤ γ −j Ni (t) ≤ γ −(i+j) h0 (t)

,

(3.27)

Qi+j (t) ≤ γ −j Qi (t) ≤ γ −(i+j) h(t)

(3.28)

for all i ≥ 0, j ≥ 0. It follows from (3.24) that Nm and Qm satisfy the lower and upper bounds γ −m h0 ((m + 1)γ, t) ≤ Nm (t) ≤ (1 − γ)−m γ −m h0 ((m + 1)γ, t)

(3.29)

γ −m h ((m + 1)γ, t) ≤ Qm (t) ≤ (1 − γ)−m γ −m h ((m + 1)γ, t)

(3.30)

where the lower bounds hold for all γ > 0 and the upper bounds for 0 < γ < 1 if m ≥ 1. From (3.20) and (3.23), it follows that Nm (t) and Qm (t) behave as t1−(m+1)γ and t−(m+1)γ respectively as t → ∞ if (m + 1)γ < 1. If (m + 1)γ = 1, Nm (t) and Qm (t) produce logarithms and behave as Log t and t−1 Log t respectively as t → ∞. If (m + 1)γ > 1, Nm (t) and Qm (t) saturate respectively as Constant and t−1 when t → ∞. For m ≥ 1, the upper bounds in (3.29) and (3.30) blow up when γ tends to one, but the same conclusions still hold. For (m + 2)γ > 1, we ﬁnally deﬁne

∞ dt1 t−γ h(t ∨ t1 ) hm (t1 ) , (3.31) Pm (t) = 1 1

Rm (t) =

∞

dt1 t−2 1 Pm (t1 ) .

(3.32)

t

In particular P0 is explicitly computed as P0 (t) = h0 (t) h(t) + γ −1 t−γ + 2γ −1 (2γ − 1)−1 t1−2γ

.

(3.33)

Clearly Pm (t) and Rm (t) are decreasing in t, while Pm (t) h(t)−1 is increasing in t, so that Pm (t) ≥ Pm (1)γh(t). It follows from (3.24) that Pm satisﬁes the lower and upper bounds

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≥1 ≤ (1 − γ)−(m+1)

×γ −(m+1) t−γ h0 ((m + 1)γ, t) + ((m + 2)γ − 1)−1 t1−(m+2)γ

Pm (t)

.

(3.34)

From (3.20) it follows that Pm (t) behaves as t1−(m+2)γ as t → ∞ if (m + 1)γ < 1. If (m + 1)γ = 1, Pm (t) behaves as t−γ Log t. If (m + 1)γ > 1, Pm (t) saturates at t−γ as long as γ < 1. We now collect a number of relations and estimates satisﬁed by the previous estimating functions. Lemma 3.6. Let i, j and m be nonnegative integers. Let 1 ≤ a ≤ b and t ≥ 1. Then the following identities and estimates hold :

∞ dt1 t−2 (3.35) 1 Nm (t1 ) = Qm (t) t

t

dt1 t−2 1 h0 (t1 ) Nm (t1 ) = Nm+1 (t) − h(t) Nm (t) ≤ Nm+1 (t)

1

∞

dt1 t−2 1 h0 (t1 ) Nm (t1 ) = Pm (t)

for (m + 2)γ > 1

(3.36) (3.37)

t

∞

dt1 t−γ 1

Ni (t) Nj (t) ≤ h0 (t) Ni+j (t)

(3.38)

Ni (t) Qj (t) ≤ h(t) Ni+j (t) ≤ Ni+j+1 (t)

(3.39)

(3.40) Qi (t) Qj (t) ≤ h(t) Qi+j (t)

∞ h(t1 ) Qm−1 (t1 ) ≤ dt1 t−γ for m ≥ 1 , (m+2)γ > 1 . 1 Qm (t1 )

t

t

∞

(3.41) dt1 t−γ 1

t

t

1

Qm (t1 ) ≤ Pm (t)

for (m + 2)γ > 1 .

dt1 t−γ h(t1 ) Qm−1 (t1 ) ≤ Nm+1 (t) 1

t

(3.42) (3.43)

dt1 t−γ Qm (t1 ) ≤ Nm+1 (t) 1

(3.44)

dt t−γ Qm (t) ≤ Qm (a) (h0 (b) − h0 (a))

(3.45)

dt t−γ h(t) Qm−1 (t) ≤ 2Qm (a) (h0 (b) − h0 (a))

(3.46)

1 b

a

b

a

Rm (t) ≤ Cm h(t) Qm (t)

for (m + 2)γ > 1

(3.47)

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J. Ginibre, G. Velo

where

−1

Cm = (2m + 3)γ ((m + 2)γ − 1) Proof. (3.35). By the deﬁnition of Nm and Qm

∞

∞

−2 dt1 t−2 N (t ) = dt t m 1 1 1 1 t

∞

=

dt2 t−γ hm (t2 ) 2

t

∞

dt1 t−2 1 =

t∨t2

1

t1

Ann. Henri Poincar´ e

.

dt2 t−γ hm (t2 ) 2

1 ∞

−1 dt2 t−γ hm (t2 ) = Qm (t) 2 (t ∨ t2 )

.

1

(3.36). By the deﬁnition of Nm and integration by parts

t

t dt1 t−2 h (t ) N (t ) = − dt1 h (t1 ) Nm (t1 ) 0 1 m 1 1 1

1

t

= −h(t) Nm (t) +

h(t) Nm (t) = Nm+1 (t) − h(t) Nm (t) .

1

(3.37). By the deﬁnitions of Nm and Pm and integration by parts

∞

∞ −2 dt1 t1 h0 (t1 ) Nm (t1 ) = h(t) Nm (t) + dt1 t−γ hm+1 (t1 ) 1 t

t ∞

=

dt1 t−γ h (t ∨ t1 ) hm (t1 ) = Pm (t) 1

.

1

(3.38). By the deﬁnition of Nm

t

t i Ni (t) Nj (t) = dt1 t−γ h (t ) dt2 t−γ hj (t2 ) . 1 1 2 1

1

For ﬁxed i + j, the integral is logarithmically convex in i (or j) and therefore estimated by the maximum of its values for i = 0 and j = 0, which are equal by symmetry and equal to the RHS of (3.38). (3.39). By the deﬁnition of Nm and Qm

t

∞ −1 −γ i Ni (t) Qj (t) = dt1 t1 h (t1 ) dt2 t−γ hj (t2 ) . 2 (t ∨ t2 ) 1

1

We split the integral over t2 into the subregions t2 ≤ t and t2 ≥ t. In the region t2 ≤ t, by logarithmic convexity and symmetry, we estimate the integral by replacing hi (t1 ) hj (t2 ) by hi+j (t1 ). In the region t2 ≥ t, we make the same replacement because t2 ≥ t ≥ t1 and h is decreasing in t. We obtain

t ··· ≤ dt1 t−γ hi+j (t1 ) h(t) = h(t) Ni+j (t) 1 1

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which yields the ﬁrst inequality in (3.39). Using in addition the fact that h(t) ≤ h(t1 ) for t1 ≤ t yields the second inequality. (3.40). By the deﬁnition of Qm , the LHS of (3.40) is logarithmically convex in i or j for ﬁxed i + j, and symmetric in i and j, and is therefore estimated by its end point values, namely with i, j replaced by 0 and i + j.

(3.41) and (3.42). By the deﬁnition of Qm

∞

∞ ∞ −1 −γ −γ dt1 t1 h(t1 ) Qm−1 (t1 ) = dt1 t1 h(t1 ) dt2 t−γ hm−1 (t2 ) . 2 (t1 ∨ t2 )

t

t

1

(3.48) We estimate the last integral by replacing h(t1 ) by h(t2 ), by logarithmic convexity and symmetry in the region t2 ≥ t and by monotony of h in the region t2 ≤ t(≤ t1 ), thereby continuing (3.48) by

∞

∞ −1 −γ dt1 t1 dt2 t−γ hm (t2 ) ··· ≤ 2 (t1 ∨ t2 ) t

1

which is the RHS of (3.41) and the LHS of (3.42),

∞

∞ −1 −γ m dt2 t2 h (t2 ) dt1 t−γ ··· = 1 (t1 ∨ t2 )

.

t

1

We estimate the last integral by h(t∨t2 ) by ﬁrst replacing t1 ∨t2 by t1 ∨t∨t2 , since t1 ≥ t, and then extending the integration over t1 to [1, ∞), thereby obtaining

∞ ··· ≤ dt2 t−γ hm (t2 ) h(t ∨ t2 ) = Pm (t) . 2 1

(3.43) and (3.44). By the deﬁnition of Qm

t dt1 t−γ 1 (h(t1 )Qm−1 (t1 ) or Qm (t1 )) 1

t

=

dt1 t−γ 1

1

≤

∞

−1 h(t1 ) hm−1 (t2 ) or hm (t2 ) dt2 t−γ 2 (t1 ∨ t2 )

1 t

dt1 t−γ 1

m

h (t1 )

1

∞

dt2 t−γ 2 (t1 ∨ t2 )

−1

= Nm+1 (t)

1

by logarithmic convexity and symmetry in the region t2 ≤ t and by monotony of h in the region t2 ≥ t(≥ t1 ). (3.45) follows immediately from the fact that Qm is decreasing in t. (3.46). We ﬁrst prove that h(t) Qm−1 (t) ≤ 2Qm (t)

.

(3.49)

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J. Ginibre, G. Velo

In fact

∞

Qm−1 (t) h(t) =

−1

dt1 t−γ 1 (t ∨ t1 )

1

∞

=

−1 dt1 t−γ 1 (t ∨ t1 )

1

∞

≤2

dt1 t−γ 1 (t ∨ t1 )

−1

∞

Ann. Henri Poincar´ e

−1

dt2 t−γ 2 (t ∨ t2 )

hm−1 (t1 )

1 ∞

−1 m−1 h dt2 t−γ (t1 ) + hm−1 (t2 ) 2 (t ∨ t2 )

t1

∞

dt2 t−γ 2 (t ∨ t1 ∨ t2 )

−1

hm−1 (t1 ) ≤ 2Qm (t) (3.50)

t1

1

since h is decreasing in t and

∞ −1 dt2 t−γ ≤ h (t ∨ t1 ) ≤ h(t1 ) . 2 (t ∨ t1 ∨ t2 ) t1

Now (3.46) follows from (3.49) and (3.45). (3.47). We ﬁrst deﬁne for future use

t −γ m + −1 Qm = Q− + Q = t dt t h (t ) + 1 1 m m 1 − + + Pm = h(t) Pm = Pm

1 t

dt1 t−γ hm (t1 ) + 1

− and we estimate Rm and − = Rm

∞

t

∞

≤ h(t)

∞

t + Rm

dt1 t−1−γ hm (t1 ) , 1

(3.51)

dt1 t−γ hm+1 (t1 ) , 1

(3.52)

t ∞

t

1

− + Rm = Rm + Rm =

∞

− dt1 t−2 1 Pm (t1 ) +

∞

+ dt1 t−2 1 Pm (t1 )

(3.53)

t

separately. We ﬁrst estimate

t1 dt1 t−2 h(t ) dt2 t−γ hm (t2 ) 1 1 2 1

−1 dt2 t−γ hm (t2 ) = h(t) Qm (t) 2 (t ∨ t2 )

(3.54)

1

by the monotony of h and after performing the integral over t1 . We next use the diﬀerential equation γh + t h = t−1 (3.55) + satisﬁed by h to rewrite Pm as follows

∞

+ m (t) = − dt1 t1−γ h (t ) h (t ) + γPm 1 1 1 t

∞

dt1 t1−1−γ hm (t1 ) .

t

Integrating by parts in the ﬁrst integral and using (3.51), we obtain + + (m + 1)γ Pm (t) = t1−γ hm+1 (t) + (1 − γ)Pm (t) + (m + 1) Q+ m (t)

namely + ((m + 2)γ − 1) Pm (t) = t1−γ hm+1 (t) + (m + 1) Q+ m (t)

Substituting that result into the deﬁnition of

+ Rm (t),

we obtain

.

Vol. 1, 2000

Scattering and Wave Operators for Hartree Type Equations II

+ ((m + 2)γ − 1) Rm (t) =

∞

−1−γ m+1 dt1 t1 h (t1 ) + (m + 1) t

∞

771

dt1 t−1−γ hm (t1 ) t−1 − t−1 1 1

t

+ (m + 1)t−1 Q+ m (t) + −1 Qm (t) ≤ h(t) + (m + 1)t

≤

Q+ m+1 (t)

by the monotony of h, ≤ (1 + (m + 1)γ) h Q+ m (t)

(3.56)

by (3.24). Collecting (3.54) and (3.56) yields (3.47).

4 Cauchy problem and preliminary asymptotics for the auxiliary system In this section, we collect a number of results from I on the Cauchy problem and on the asymptotic behaviour of solutions for the auxiliary system  −1 −1  ∆w + 2t2 (2∇ϕ · ∇ + (∆ϕ)) w  ∂t w = i 2t2

(2.11) ≡ (4.1)

  ∂ ϕ = 2t2 −1 |∇ϕ|2 + t−γ g (w, w) . t 0

(2.12) ≡ (4.2)

Those results are immediate extensions of results contained in I. The main diﬀerences are that (i) the results are stated here in terms of ϕ whereas they are stated in I in terms of s = ∇ϕ, and (ii) here we use systematically the estimating functions of time h0 and h introduced in Section 3, thereby covering the whole interval 0 < γ ≤ 1. The proofs will be sketched brieﬂy or omitted. We ﬁrst recall the results on the local Cauchy problem with ﬁnite initial time (see Proposition I.4.1). Proposition 4.1. Let (k, 3) be an admissible pair. Let t0 > 0. Then for any (w0 , ϕ0 ) ∈ H k ⊕ Y , there exist T± with 0 ≤ T− < t0 < T+ ≤ ∞ such that : (1) The system (4.1) (4.2) has a unique solution (w, ϕ) ∈ C(I, H k ⊕ Y ) with (w, ϕ)(t0 ) = (w0 , ϕ0 ), where I = (T− , T+ ). If T− > 0 (resp. T+ < ∞), then |w(t)|k + |ϕ(t)| → ∞ when t decreases to T− (resp. increases to T+ ). (2) If (w0 , ϕ0 ) ∈ H k ⊕ Y for some admissible pair (k , 3 ) with k ≥ k and k 3 ≥ 3, then (w, ϕ) ∈ C(I, H ⊕ Y ). (3) For any compact subinterval J ⊂⊂ I, the map (w0 , ϕ0 ) → (w, ϕ) is continuous from H k−1 ⊕ Y −1 to L∞ (J, H k−1 ⊕ Y −1 ) uniformly on the bounded sets of H k ⊕ Y , and is pointwise continuous from H k ⊕ Y to L∞ (J, H k ⊕ Y ). We next recall the results on the local Cauchy problem in a neighborhood of inﬁnity in time (see Proposition I.5.1).

772

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Ann. Henri Poincar´ e

Proposition 4.2. Let (k, 3) be an admissible pair. Let (w0 , ϕ 0 ) ∈ H k ⊕Y and deﬁne a = |w0 |k and b = |ϕ 0 | . Then there exists T0 < ∞, depending on a, b, such that for all t0 ≥ T0 , there exists T ≤ t0 , depending on a, b and t0 , such that the system 0 has a unique solution (4.1) (4.2) with initial data w(t0 ) = w0 , ϕ(t0 ) = h0 (t0 )ϕ (w, ϕ) in the interval [T, ∞) such that (w, h−1 ϕ) ∈ (C ∩ L∞ )([T, ∞), H k ⊗ Y ). 0 One can deﬁne T0 and T by C(b + a2 ) h (T0 ) = 1

(4.3)

T = h0 (t0 ) h (T0 )−1

(4.4)

and the solution (w, ϕ) is estimated for all t ≥ T by |w(t)|k ≤ C a |ϕ(t)| ≤ C(b + a2 ) h0 (t ∨ t0 )

(4.5) .

(4.6)

Sketch of proof. The proof is almost identical with that of Proposition I.5.1 and follows from a priori estimates of the maximal solution obtained from Proposition 4.1. Deﬁne y = |w|k and z = |ϕ| . By Lemmas 3.2 and 3.3, y and z satisfy   |∂t y| ≤ C t−2 y z (4.7)  |∂t z| ≤ C t−2 z 2 + C t−γ y 2 . For t ≥ t0 , we take t¯ > t0 , we deﬁne Y ≡ Y (t¯) = y; L∞ ([t0 , t¯]) and Z ≡ Z(t¯) =

h0 (t)−1 z; L∞ ([t0 , t¯]) , we substitute those deﬁnitions into (4.7), we integrate over t with the appropriate initial condition and we obtain   Y ≤ a + C Y Z h(t0 ) (4.8)  Z ≤ b + C Y 2 + C Z 2 h(t0 ) by (3.19) (3.22). For t ≤ t0 , we take t¯ < t0 , we deﬁne Y ≡ Y (t¯) = y; L∞ ([t¯, t0 ]) and Z ≡ Z(t¯) = z; L∞ ([t¯, t0 ]) , we substitute those deﬁnitions into (4.7), we integrate over t with the appropriate initial condition and we obtain   Y ≤ a + C t−1 Y Z (4.9)  Z ≤ b + C Y 2 h0 (t0 ) + C t−1 Z 2 . The proof then proceeds from (4.8) and (4.9) in the same way as that of Proposition I.5.1. For subsequent applications, we shall need the following lemma, which is essentially identical with Lemma I.5.1.

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Lemma 4.1. Let a > 0, b > 0, t0 > 1 and let y, z be nonnegative continuous functions satisfying y(t0 ) = y0 , z(t0 ) = z0 and   |∂t y| ≤ t−2 h0 (t) b y + t−2 a z (4.10)  |∂t z| ≤ t−2 h0 (t) b z + t−γ a y . Deﬁne y¯, z¯ by (y, z) = (¯ y , z¯) exp(b|h(t) − h(t0 )|) . Then for

γ(tγ0

(4.11)

∧ t ) ≥ 2a , the following estimates hold :   y¯ ≤ 2(y0 + a z0 t−1 0 ) γ

2

 for t ≥ t0 , and

(4.12)

z¯ ≤ z0 + 2a(y0 + a z0 t−1 0 ) h0 (t)

  y¯ ≤ y0 + 2a (z0 + a y0 h0 (t0 )) t−1 

(4.13) z¯ ≤ 2 (z0 + a y0 h0 (t0 ))

for 1 ≤ t ≤ t0 . As an easy consequence of Lemma 4.1, we obtain the following uniqueness result at inﬁnity for the system (4.1) (4.2) (see Proposition I.5.2). Proposition 4.3. Let (k, 3) be an admissible pair. Let (wi , ϕi ), i = 1, 2 be two ∞ k solutions of the system (4.1) (4.2) such that (wi , h−1 0 ϕi ) ∈ L ([T, ∞), H ⊕ Y ) for some T > 0 and such that |w1 (t) − w2 (t)|k−1 h0 (t) and |ϕ1 (t) − ϕ2 (t)|−1 tend to zero when t → ∞. Then (w1 , ϕ1 ) = (w2 , ϕ2 ). We ﬁnally recall the existence result for the limit of w(t) as t → ∞ for the solutions of the system (4.1) (4.2) obtained in Proposition 4.2 (see Proposition I.5.3). Proposition 4.4. Let (k, 3) satisfy k ≤ 3 + 1 and 3 > n/2. Let (w, ϕ) satisfy (4.1) ∞ k and be such that (w, h−1 0 ϕ) ∈ (C ∩ L )([T, ∞), H ⊕ Y ) for some T > 0. Let a = w; L∞ ([T, ∞), H k ) ,

∞ b = h−1 0 ϕ; L ([T, ∞), Y )

.

(4.14)

Then there exists w+ ∈ H k such that w(t) tends to w+ strongly in H k−1 and weakly in H k when t → ∞. Furthermore the following estimates hold |w+ |k ≤ a

(4.15)

|w(t 0 ) − w(t)| k−1 ≤ C a b h(t0 ∧ t)

(4.16)

|w(t) − w+ |k−1 ≤ C a b h(t)

(4.17)

for t0 , t suﬃciently large, namely bh(t0 ∧ t) ≤ C or bh(t) ≤ C, where w(t) = U (1/t)w(t).

774

J. Ginibre, G. Velo

Ann. Henri Poincar´ e

5 Existence and properties of the asymptotic dynamics In this section we derive the relevant properties of the solutions (wm , ϕm ) of the system (2.20) (2.21) with initial conditions (2.22) (2.23) and of the solutions (V, χ) of the transport equations (2.25) (2.26) with initial conditions (2.27). We use systematically the estimating functions of time Nm , Qm and Pm deﬁned by (3.25) (3.26) and (3.31). We begin with the system (2.20) (2.21), which is solved by successive integrations, as explained in Section 2. Proposition 5.1. Let (k, 3) be an admissible pair, let p ≥ 0 be an integer, let w+ ∈ H k+p and let a = |w+ |k+p . Let {w0 = w+ , wm+1 } and {ϕm }, 0 ≤ m ≤ p, be the solution of the system (2.20) (2.21) with initial conditions (2.22) (2.23). Then (1) wm+1 ∈ C([1, ∞), H k+p−m−1 ), ϕm ∈ C([1, ∞), Y +p−m ) and the following estimates hold for all t ≥ 1 : |wm+1 (t)|k+p−m−1 ≤ A(a) Qm (t)

(5.1)

|ϕm (t)|+p−m ≤ A(a) Nm (t)

(5.2)

for some estimating function A(a). If in addition (p + 2)γ > 1 and if we deﬁne ϕp+1 by (2.21) with initial condition ϕp+1 (∞) = 0, then ϕp+1 ∈ C([1, ∞), Y −1 ) and the following estimate holds : |ϕp+1 (t)|−1 ≤ A(a) Pp (t) . (5.3) (2) The functions {ϕm } are gauge invariant in the following sense. If w+ = w+ exp(iσ) for some real valued function σ and if w+ gives rise to {ϕm }, then ϕm = ϕm for 0 ≤ m ≤ p + 1. (3) The map w+ → {wm+1 , ϕm } is uniformly Lipschitz continuous on the bounded sets from the norm topology of w+ in H k+p to the norms ∞ k+p−m−1 −1

Q−1 ) and Nm ϕm ; L∞ ([1, ∞), Y +p−m ) , 0 ≤ m wm+1 ; L ([1, ∞), H m ≤ p. A similar continuity holds for ϕp+1 .

Proof. Part (1). The proof proceeds by induction on m. We assume the results to hold for (wj , ϕj ) for j ≤ m and we prove them for wm+1 and ϕm+1 . We ﬁrst consider wm+1 which is obtained from (2.20). From Lemma 3.2, especially (3.6) with (k, 3) replaced by (k + p − m, 3 + p − m) which is again an admissible pair and from the induction assumption, we obtain |∂t wm+1 |k+p−m−1 ≤ A(a) t−2 Nj (t) Qm−j−1 (t) + Nm (t) . (5.4) 0≤j≤m−1

Integrating (5.4) between t and ∞, using the initial condition wm+1 (∞) = 0 and using (3.39) (3.35) shows that wm+1 ∈ C([1, ∞), H k+p−m−1 ) and that wm+1 satisﬁes (5.1). We next consider ϕm+1 which is obtained from (2.21). From Lemma 3.2, especially (3.7) (3.9) with again (k, 3) replaced by (k + p − m, 3 + p − m), from the induction assumption and from the result for wm+1 , we obtain

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775

|∂t ϕm+1 |+p−m−1 ≤ Nj (t) Nm−j (t) + t−γ A(a) t−2 0≤j≤m

Qj (t) Qm−1−j (t) + Qm (t) .

0≤j≤m−1

(5.5) Integrating (5.5) between 1 and t, using the initial condition ϕm+1 (1) = 0, and using (3.38) (3.36) and (3.40) (3.43) (3.44) shows that ϕm+1 ∈ C([1, ∞), Y +p−m−1 ) and that ϕm+1 satisﬁes (5.2). We ﬁnally assume that (p+2)γ > 1 and estimate ∂t ϕp+1 by (5.5) with m = p. The last result and in particular the estimate (5.3) then follow by integration between t and ∞ and use of (3.38) (3.37) and (3.40) (3.41) (3.42) with m = p. Part (2). We deﬁne for 0 ≤ m ≤ p + 1 Bm = w ¯j wm−j 0≤j≤m

so that B0 = |w+ |2 and Bm is bounded in time and tends to zero at inﬁnity for m ≥ 1, for instance in H11 norm. The equation (2.21) for ϕm+1 can be rewritten as ∇ϕj · ∇ϕm−j + t−γ λ ω µ−n Bm+1 . (5.6) ∂t ϕm+1 = (2t2 )−1 0≤j≤m

We next compute ∂t Bm+1

= (2t2 )−1

2Re w ¯j

0≤j≤m

= t−2

(2∇ϕm−i−j · ∇ + (∆ ϕm−i−j ))wi

0≤i≤m−j

(∇ϕm−k · ∇ + (∆ ϕm−k ))Bk

.

0≤k≤m

(5.7) Using (5.6) and (5.7), we now show by induction on m that Bm and ϕm are gauge invariant. In fact assume that Bj and ϕj are gauge invariant for j ≤ m. Then ∂t Bm+1 is gauge invariant by (5.7) and therefore Bm+1 is gauge invariant because Bm+1 (∞) = 0. Substituting that result into (5.6) and using the induction assumption, we obtain from (5.6) that ∂t ϕm+1 is gauge invariant, and therefore ϕm+1 is gauge invariant since ϕm+1 (1) = 0 for m < p and ϕp+1 (∞) = 0. , ϕm } be the solutions of the system (2.20) (2.21) Part (3). Let {wm , ϕm } and {wm associated with w+ and w+ . From the fact that the RHS of (2.20) (2.21) are bilinear, it follows as in Part (1) by induction on m that the following estimates hold, with a = |w+ |k+p ∨ |w+ |k+p : wm+1 − wm+1 ≤ A(a) |w+ − w+ |k+p Qm (t) (5.8) k+p−m−1 |ϕm − ϕm |+p−m ≤ A(a) |w+ − w+ |k+p Nm (t)

(5.9)

776

J. Ginibre, G. Velo

for 0 ≤ m ≤ p, and if (p + 2)γ > 1, ϕp+1 − ϕp+1 ≤ A(a) |w+ − w+ |k+p Pp (t) −1

Ann. Henri Poincar´ e

.

(5.10)

The continuity as stated in Part (3) follows from those estimates. Remark 5.1. There is no upper bound on p in Proposition 5.1. However if (p+1)γ > 1, all the (wm , ϕm ) with (m + 1)γ > 1 have the same asymptotic behaviour in time and behave respectively as t−1 and Constant as t → ∞, because Qm and Nm saturate to those behaviours in that case. We deﬁne for future reference (see also (2.16) (2.17)) Wm = wj , φm = ϕj (5.11) 0≤j≤m

0≤j≤m

where wj , ϕj are obtained by Proposition 5.1. We now turn to the study of the transport equation ∂t V = (2t2 )−1 (2∇φ · ∇ + (∆φ)) V

(5.12)

which we shall use later with φ = φp−1 , as explained in Section 2 (see (2.25)). Proposition 5.2. Let 3 > n/2 and 1 ≤ k ≤ 3. Let T ≥ 1, I = [T, ∞), let φ ∈ C(I, Y ) ∞ k with h−1 0 φ ∈ L (I, Y ) and let w+ ∈ H . Then (1) The equation (5.12) has a solution V ∈ (C∩L∞ )(I, H k ) which is estimated by

V : L∞ (I, H k ) ≤ |w+ |k exp(C b γ −1 ) (5.13) where

∞ b = h−1 0 φ; L (I, Y ) ,

(5.14)

and which tends to w+ at inﬁnity in the sense that |V (t) − w+ |k−1 ≤ C b exp(C b γ −1 ) |w+ |k h(t)

.

(5.15)

(2) The solution V is unique in L∞ (I, L2 ) under the condition that

V (t) − w+ 2 tends to zero as t → ∞. (3) The map (w+ , φ) → V is uniformly Lipschitz continuous in w+ for the norm topology of H k and is continuous in φ for the topology of convergence in Y pointwise in t to the norm topology of L∞ (I, H k ) for h−1 0 φ in bounded sets of L∞ (I, Y ). Proof. Part (1). We ﬁrst take t0 ∈ I. Using a regularization (for instance parabolic), energy estimates as in Lemmas 3.2 and 3.3 (see especially (3.6) and (3.12)), and a limiting procedure, one obtains easily the existence of a solution Vt0 of the equation (5.12) with initial condition Vt0 (t0 ) = w+ , and such that Vt0 ∈ C(I, H k−1 ) ∩ (Cw ∩ L∞ )(I, H k ) .

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Using the same energy estimates, one then shows that ∂t |Vt0 (t)|k ≤ C b t−2 h0 (t) |Vt0 (t)|k −2 ∂t |Vt0 (t) − w+ |k−1 ≤ C b t h0 (t) Vt0 (t) − w+

k−1

(5.16) + |w+ |k

and for two solutions Vt0 and Vt1 associated with t0 and t1 ∂t |Vt0 (t) − Vt1 (t)|k−1 ≤ C b t−2 h0 (t)Vt0 (t) − Vt1 (t)

(5.17)

.

(5.18)

k−1

Integrating (5.16) (5.17) between t0 and t and integrating (5.18) between t1 and t, we obtain respectively |Vt0 (t)|k ≤ |w+ |k exp (C b|h(t) − h(t0 )|) ≤ |w+ |k exp(C b γ −1 ) , ≤ |w+ |k (exp (C b|h(t) − h(t0 )|) − 1) Vt0 (t) − w+ k−1

(5.20)

≤ |w+ |k C b exp C b γ −1 |h(t) − h(t0 )| , Vt0 (t) − Vt1 (t)

k−1

≤ Vt0 (t1 ) − w+

exp (C b|h(t) − h(t1 )|)

≤ Vt0 (t1 ) − w+

exp C b γ −1

k−1

(5.19)

(5.21) .

k−1

Substituting (5.20) into (5.21) yields |Vt0 (t) − Vt1 (t)|k−1 ≤ |w+ |k C b exp(2C bγ −1 ) |h(t1 ) − h(t0 )|

.

(5.22)

It follows from (5.22) that when t0 → ∞, Vt0 has a limit V ∈ (C ∩ L∞ )(I, H k−1 ) satisfying (5.15). One sees easily that V satisﬁes the equation (5.12). From the estimate (5.19) it follows by a standard compactness argument that V ∈ (Cw ∩ L∞ )(I, H k ) and that V satisﬁes the estimate (5.13). Furthermore V also satisﬁes (5.16) so that |V (t)|k is Lipschitz continuous in t, which together with weak continuity in H k implies strong continuity in H k . Part (2). If V1 and V2 are two solutions of (5.12) one obtains by the same energy estimates as above

V1 (t) − V2 (t) 2 ≤ V1 (t ) − V2 (t ) 2 exp (C b|h(t) − h(t )|)

.

(5.23)

Taking the limit t → ∞ shows that V1 = V2 . Part (3). Continuity of V with respect to w+ follows immediately from the linearity of the equation (5.12) and from the estimate (5.13). In order to prove continuity with respect to φ, we ﬁrst derive an estimate for the diﬀerence of two solutions

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V1 and V2 associated with φ1 and φ2 . We assume that φ1 ∈ C(I, Y +1 ) with ∞ +1 ∞ h−1 ), that φ2 ∈ C(I, Y ) with h−1 0 φ ∈ L (I, Y 0 φ2 ∈ L (I, Y ), that V1 ∈ (C ∩ ∞ k+1 ∞ k L )(I, H ) and that V2 ∈ (C ∩ L )(I, H ). Let V− = V1 − V2 and φ− = φ1 − φ2 . It follows from (5.12) that . (5.24) ∂t V− = (2t2 )−1 (2∇φ2 · ∇ + (∆φ2 )) V− + (2∇φ− · ∇ + (∆φ− )) V1 Let a = Max Vi ; L∞ (I, H k ) , i=1,2

∞ b = Max h−1 0 φi ; L (I, Y )

i=1,2

.

Estimating (5.24) by Lemma 3.5, we obtain ∂t |V− |k ≤ C t−2 |φ2 | |V− |k + |φ− | |V1 |k + |φ− |∗ |V1 |k+1 ≤ C t−2 b h0 |V− |k + a|φ− | + |φ− |∗ |V1 |k+1

(5.25)

where |f |∗ ≡ ∇f ∞ . On the other hand, by Lemma 3.4 we obtain ∂t |V1 |k+1 ≤ C t−2 |φ1 | |V1 |k+1 + |φ1 |+1 |V1 |k −2

≤C t

b h0 |V1 |k+1 + a|φ1 |+1 .

(5.26)

Integrating (5.26) between t0 and t and using the fact that |∂t y| ≤ C0 t−2 h0 y + z

(5.27)

implies t y(t) ≤ y(t0 ) exp (C0 |h(t) − h(t0 )|) + dt1 z(t1 ) exp (C0 |h(t) − h(t1 )|) t0

t −1 ≤ exp C0 γ y(t0 ) + dt1 z(t1 )

,

t0

(5.28) we obtain t |V1 (t)|k+1 ≤ C |V1 (t0 )|k+1 + dt1 t−2 |φ (t )| 1 1 +1 1

(5.29)

t0

where C depends on a, b. Substituting (5.29) into (5.25) and integrating between t0 and t yields similarly

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t t −2 + |V |V− (t)|k ≤ C |V− (t0 )|k + dt1 t−2 |φ (t )| (t )| dt t |φ (t )| − 1 1 0 k+1 1 1 − 1 ∗ 1 t0

t

+ dt1 t−2 |φ (t )| − 1 ∗ 1 t0

t0

t1

dt2 t−2 2

t0

|φ1 (t2 )|+1

.

(5.30)

In particular if V1 and V2 are solutions of (5.12) with φ1 and φ2 respectively and with initial data w+1 ∈ H k+1 and w+2 ∈ H k at time t0 , as obtained in Part (1), then the following estimate holds uniformly in t0 and t

∞ dt t−2 |φ− (t)| |V− (t)|k ≤ C |w+1 − w+2 |k +

+ |w+1 |k+1 +

∞

dt t−2 |φ1 (t)|+1

1

1

∞

dt t−2 |φ− (t)|∗

,

(5.31)

1

where all the integrals are convergent under the assumptions made on φ1 and φ2 . We can now prove the continuity with respect to φ. The proof proceeds as in Step 7 of that of Proposition I.4.1. We introduce a regularization deﬁned as follows. We choose a function ψ1 ∈ S(IRn ) such that dx ψ1 (x) = 1 and such that |ξ|−2 (ψ1 (ξ) − 1)|ξ=0 = 0. We deﬁne ψε (x) = ε−n ψ1 (x/ε), so that ψε (ξ) = ψ1 (εξ) and we deﬁne the regularization by f → fε = ψε ∗ f for all f ∈ S . An immediate computation yields

∂fε 2 ≤ ∂ψε 1 f 2 = ε−1 ∂ψ1 1 f 2

(5.32)

and

fε − f ∞ ≤ (ψε − 1)f 1 ≤ εθ |ξ|−n/2−θ (ψ1 (ξ) − 1) 2 f ; H˙ n/2+θ . (5.33) −1 ∞ Let now w+ ∈ H k and φ, φ ∈ C(I, Y ) with h−1 0 φ, h0 φ ∈ L (I, Y ) and such that −1 ∞ ∞

h−1 0 φ; L (I, Y ) ∨ h0 φ ; L (I, Y ) ≤ b . Let V and V be the solutions of the equation (5.12) with φ and φ respectively and with initial data w+ at t0 obtained in Part (1). We regularize w+ , φ, φ to w+ε , φε , φε , so that the following estimates hold : |w+ε |k+1 ≤ C ε−1 |w+ |k

,

|φε |+1 ≤ C ε−1 |φ|

and |φ − φε |∗ ≤ C ε3/2 |φ|

,

,

|φε |+1 ≤ C ε−1 |φ | (5.34)

|φ − φε |∗ ≤ C ε3/2 |φ |

.

(5.35)

The estimates (5.34) follow from (5.32) and the estimates (5.35) follow from (5.33) and from the deﬁnition of Y . Let Vε and Vε be the solutions of (5.12) obtained from (w+ε , φε ) and (w+ε , φε ). We estimate |V (t) − V (t)|k ≤ |V (t) − Vε (t)|k + |Vε (t) − Vε (t)|k + |Vε (t) − V (t)|k

. (5.36)

We estimate the three norms in the RHS of (5.36) by applying successively (5.31) with (V1 , V2 ) = (Vε , V ), (Vε , Vε ) and (Vε , V ). We obtain

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|V (t) − V (t)|k ≤

∞

C |w+ − w+ε |k + dt t−2 |φ(t) − φε (t)| + |φε (t) − φε (t)| + |φε (t) − φ (t)| +ε−1

1

∞

1

dt t−2 |φ(t) − φε (t)|∗ + |φε (t) − φε (t)|∗ + |φε (t) − φ (t)|∗

(5.37)

where we have used (5.34). Using the inequalities |φε − φε | ≤ |φ − φ | |φε − φε |∗ ≤ |φ − φ |∗ ≤ |φ − φ | |φε − φ | ≤ |φε − φ| + 2|φ − φ | and (5.35), we can continue (5.37) by

|V (t) − V (t)|k

≤ C |w+ − w+ε |k + + 1 + ε−1

∞

−2

dt t 1

∞

1

dt t−2 |φ(t) − φε (t)| + ε1/2

|φ(t) − φ (t)|

(5.38)

.

For ﬁxed φ, by the Lebesgue dominated convergence theorem, the ﬁrst integral in the RHS tends to zero when ε → 0, while the second integral tends to zero when φ → φ in Y pointwise in t. The RHS of (5.38) can then be made arbitrarily small by ﬁrst taking ε suﬃciently small and then letting φ tend to φ in the previous sense for ﬁxed ε. We next turn to the analogous transport equation ∂t χ = t−2 ∇φ · ∇χ

(5.39)

which we shall use together with (5.12), as explained in Section 2. Proposition 5.3. Let 3 > n/2. Let T ≥ 1, I = [T, ∞), let φ ∈ C(I, Y +1 ) with ∞ +1 h−1 ) and let ψ+ ∈ Y . Then 0 φ ∈ L (I, Y (1) The equation (5.39) has a solution χ ∈ (C ∩L∞ )(I, Y ) which is estimated by (5.40)

χ; L∞ (I, Y ) ≤ |ψ+ | exp(C b γ −1 ) where

∞ +1 b = h−1 ) , 0 φ; L (I, Y

(5.41)

and tends to ψ+ at inﬁnity in the sense that |χ(t) − ψ+ |−1 ≤ C b exp(C b γ −1 )|ψ+ | h(t)

.

(5.42)

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(2) The solution χ is unique in L∞ (I, L∞ ) under the condition that

χ(t) − ψ+ ∞ tends to zero as t → ∞. (3) The map (ψ+ , φ) → χ is uniformly Lipschitz continuous in ψ+ for the norm topology of Y and is continuous in φ for the topology of convergence in Y +1 pointwise in t to the norm topology of L∞ (I, Y ), for h−1 0 φ in bounded sets of L∞ (I, Y +1 ). (4) Let in addition w+ ∈ H k and let V be the solution of (5.12) obtained in Proposition 5.2. Then for ﬁxed φ, V exp(−iχ) is gauge invariant in the following sense : if (V, χ) and (V , χ ) are the solutions obtained from (w+ , ψ+ ) and (w+ , ψ+ ) and if w+ exp(−iψ+ ) = w+ exp(−iψ+ ), then V (t) exp(−iχ(t)) = V (t) exp(−iχ (t)) for all t ∈ I. Proof. Parts (1) (2) (3). The proof is the same as that of Proposition 5.2, starting from the estimates (3.13) (3.14) (3.16) (3.18) of Lemmas 3.3, 3.4 and 3.5. Part (4). It follows from (5.12) and (5.39) that V exp(−iχ) also satisﬁes (5.12), with gauge invariant initial condition V (∞) exp(−iχ(∞)) = w+ exp(−iψ+ ). The result then follows from the uniqueness statement of Proposition 5.2, part (2). In the subsequent applications, we shall use the solutions of the equations (5.12) and (5.39) associated with φ = φp−1 deﬁned by (5.11) (see (2.25) (2.26)). In particular we shall use V as a substitute for Wp , also deﬁned by (5.11), and we shall need the fact that V is a suﬃciently good approximation of Wp . We collect the relevant properties in the following proposition. Proposition 5.4. Let (k, 3) be an admissible pair. Let p ≥ 1 be an integer. Let w+ ∈ H k+p+1 , let a = |w+ |k+p+1 and let φ = φp−1 be deﬁned by (5.11) and ∞ +2 Proposition 5.1, so that h−1 ). Let V be the solution of 0 φ ∈ (C ∩ L )([1, ∞), Y (5.12) deﬁned by Proposition 5.2, so that V ∈ (C ∩ L∞ )([1, ∞), H k+2 ). (1) Let Wp be deﬁned by (5.11) and Proposition 5.1 so that Wp ∈ (C ∩ L∞ )([1, ∞), H k+1 ). Then |V (t) − Wp (t)|k ≤ A(a) Qp (t)

(5.43)

for some estimating function A(a). (2) Let ψ+ ∈ Y +1 and let χ be the solution of (5.39) deﬁned by Proposition 5.3, so that χ ∈ (C ∩ L∞ )([1, ∞), Y +1 ). Then |χ(t) − ψ+ | ≤ A(a) |ψ+ |+1 h(t)

(5.44)

and V (t) exp(−iχ(t)) is gauge invariant. (3) The map w+ → V is continuous from H k+p+1 to L∞ ([1, ∞), H k+2 ) and the map (w+ , ψ+ ) → χ is continuous from H k+p+1 ⊕ Y +1 to L∞ ([1, ∞), Y +1 ).

782

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Proof. Part (1). From (2.20) and (5.11) it follows that −1 (2∇φp−1 · ∇ + (∆φp−1 )) Wp − ∂t Wp = 2t2

Ann. Henri Poincar´ e

(2∇ϕi · ∇ + (∆ϕi )) wj

i≤p−1,j≤p i+j≥p

(5.45) so that

−1 (2∇φp−1 · ∇ + (∆φp−1 )) (V − Wp )+ ∂t (V − Wp ) = 2t2

(2∇ϕi · ∇ + (∆ϕi )) wj

.

i≤p−1,j≤p i+j≥p

From Lemma 3.3, esp. (3.12) and Lemma 3.2, esp. (3.6), we obtain (5.46) |ϕi | |wj |k+1 ∂t |V − Wp |k ≤ C t−2 h0 (t) b |V − Wp |k + t−2 i≤p−1,j≤p i+j≥p

where

∞ b = h−1 0 φp−1 ; L ([1, ∞), Y )

.

Integrating (5.46) between t and inﬁnity, using (5.1) (5.2) (5.27) (5.28), we obtain

∞ dt1 t−2 Ni (t1 ) Qj−1 (t1 ) |V (t) − Wp (t)|k ≤ A(a) 1 t

≤ A(a)

i≤p−1,j≤p i+j≥p

(5.47)

Qm (t) ≤ A(a) Qp (t)

p≤m≤2p−1

by (3.39) (3.35) and (3.28). Part (2) is a partial rewriting of Proposition 5.3 in the special case φ = φp−1 . Part (3). The continuity properties stated there follow by combining those of Propositions 5.1, part (3), 5.2 part (3) and 5.3, part (3). Remark 5.2. By keeping track of the orders of derivation more accurately, one sees easily that Proposition 5.4 holds with (k, 3) replaced everywhere by (k − 1, 3 − 1). We have stated Proposition 5.4 at the level of regularity which will be used in the subsequent applications.

6 Asymptotics and wave operators for the auxiliary system In this section we derive the main technical results of this paper. We prove that suﬃciently regular solutions (w, ϕ) of the auxiliary system (4.1) (4.2) have asymptotic states (w+ , ψ+ ), and conversely that suﬃciently regular asymptotic states (w+ , ψ+ ) generate solutions (w, ϕ) of the auxiliary system in the sense

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described in Section 2, thereby allowing for the deﬁnition of the wave operator Ω0 : (w+ , ψ+ ) → (w, ϕ). We ﬁrst prove the existence of asymptotic states of solutions (w, ϕ) of (4.1) (4.2). The existence of w+ is already established in Proposition 4.4 under rather general assumptions. However the existence of ψ+ requires a more complicated construction and stronger assumptions. Proposition 6.1. Let (k, 3) be an admissible pair. Let p ≥ 0 be an integer. Let T ≥ 2, I = [T, ∞), let (w, ϕ) be a solution of the system (4.1) (4.2) such that ∞ k+(p+1)∨2 (w, h−1 ⊕ Y +p ) and let 0 ϕ) ∈ (C ∩ L )(I, H a = w; L∞ ([T, ∞), H k+(p+1)∨2 ) ,

∞ +p b = h−1 )

0 ϕ; L ([T, ∞), Y

. (6.1) Let w+ = lim w(t) ∈ H k+p+1 be deﬁned by Proposition 4.4. Let {wm+1 , ϕm }, t→∞ 0 ≤ m ≤ p be deﬁned by Proposition 5.1, and let Wm , φm , 0 ≤ m ≤ p, be deﬁned by (5.11). Then the following estimates hold for all t ∈ I : ≤ A(a, b) Qm (t) (6.2) w(t) − Wm (t) k+p−m−1

ϕ(t) − φm (t)

+p−m−1

≤ A(a, b) Nm+1 (t)

(6.3)

for 0 ≤ m ≤ p, and for some estimating function A(a, b). If in addition (p + 2)γ > 1, then the following limit exists

lim ϕ(t) − φp (t) = ψ+

(6.4)

as a strong limit in Y −1 , and the following estimate holds ≤ A(a, b) Pp (t) . ϕ(t) − φp (t) − ψ+

(6.5)

t→∞

−1

Proof. The proof proceeds by induction on m. For 0 ≤ m ≤ p, we deﬁne qm+1 (t) = w(t) − Wm (t)

,

(6.6)

ψm+1 (t) = ϕ(t) − φm (t)

.

(6.7)

We also deﬁne q0 = w and ψ0 = ϕ. We assume that the estimates (6.2) (6.3) hold for (qj , ψj ), 0 ≤ j ≤ m and we derive them for (qm+1 , ψm+1 ). We substitute the decompositions w = Wm +qm+1 and ϕ = φm +ψm+1 in the LHS of (4.1) (4.2) and we partly substitute the decompositions w = Wm−1 + qm and ϕ = φm−1 + ψm in the RHS of the same equations. Using in addition (2.20) (2.21), we obtain ∂t qm+1 = (2t2 )−1 i∆w + (2∇ϕ · ∇ + (∆ϕ)) qm

784

J. Ginibre, G. Velo

+ (2∇ψm · ∇ + (∆ψm )) Wm−1 +

Ann. Henri Poincar´ e

(2∇ϕi · ∇ + (∆ϕi )) wj

,

0≤i,j≤m−1 i+j≥m

∂t ψm+1 = (2t2 )−1 (∇ϕ + ∇φm−1 ) · ∇ψm +

∇ϕi · ∇ϕj

0≤i,j≤m−1 i+j≥m

+t−γ g0 (qm , q1 ) + g0 (qm , Wm−1 − w0 ) + 2g0 (qm+1 , w0 ) +

(6.8)

g0 (wi , wj ) .

0≤i,j≤m−1 i+j≥m+1

(6.9) The equation (6.9) holds only for m ≥ 1 and the last bracket thereof has been obtained by using the fact that g0 (qm , w + Wm−1 ) − 2g0 (wm , w0 ) = g0 (qm , q1 ) + g0 (qm , Wm−1 − w0 ) +2g0 (qm+1 , w0 )

.

For m = 0, (6.9) should be replaced by ∂t ψ1 = (2t2 )−1 |∇ϕ|2 + t−γ (g0 (q1 , q1 ) + 2g0 (q1 , w0 )) .

(6.9)0

We estimate the RHS of (6.8) (6.9) by Lemma 3.2 with (k, 3) replaced by (k + p − m, 3+p−m), which is again an admissible pair. We use Proposition 5.1 to estimate Wm−1 , wj , φm−1 and ϕj , and we use the induction hypothesis to estimate qm , q1 , qm+1 and ψm . Note that Wm−1 − w0 , which occurs only for m ≥ 2, satisﬁes the same estimate as q1 . In the induction procedure, as in the proof of Proposition 5.1, one has ﬁrst to complete the estimation of qm+1 before estimating ψm+1 . One then obtains , ≤ A(a, b) t−2 1 + h0 Qm−1 + Nm + Ni Qj−1 ∂t qm+1 k+p−m−1

∂t ψm+1

m≤i+j≤2(m−1)

+p−m−1

≤ A(a, b) t−2 h0 Nm +

+t−γ h Qm−1 + Qm +

(6.10)

Ni Nj

m≤i+j≤2(m−1)

Qi−1 Qj−1

(6.11)

m+1≤i+j≤2(m−1)

for m ≥ 1, and

≤ A(a, b) t−2 (1 + h0 ) ∂t q1 k+p−1 ≤ A(a, b) t−2 h20 + t−γ h ∂t ψ1 +p−1

(6.10)0 .

(6.11)0

Integrating (6.10) between t and inﬁnity with the condition qm+1 (∞) = 0 which follows from the deﬁnition and using (3.39) (3.35) (3.28) and the fact that t−1 Qm (1) ≤ Qm (t) yields (6.2).

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Integrating (6.11) between T and t, using (3.38) (3.36) (3.27) for the ﬁrst bracket and (3.40) (3.43) (3.44) (3.28) for the second bracket yields ≤ C + A(a, b) Nm+1 (t) ψm+1 (t) +p−m−1 ≤ C Nm+1 (T )−1 + A(a, b) Nm+1 (t) C = ψm+1 (T )

with

, +p−m−1

where we have used the fact that Nm+1 is increasing in T , and we have assumed that T is bounded away from 1. This yields (6.3). We next turn to the proof of (6.5). In that case, the RHS of (6.11) with m = p is integrable in time, which proves the existence of the limit (6.4). Integrating (6.11) between t and inﬁnity and using (3.38) (3.37) for the ﬁrst bracket and (3.40) (3.41) (3.42) (3.28) for the second bracket yields (6.5). We now turn to the construction of solutions (w, ϕ) of the system (4.1) (4.2) with given asymptotic states (w+ , ψ+ ). For that purpose we ﬁrst take a (large) positive t0 and we construct the solution (wt0 , ϕt0 ) of (4.1) (4.2) with initial data (V (t0 ), φp (t0 ) + χ(t0 )) at t0 . The solution (w, ϕ) will then be obtained therefrom by taking the limit t0 → ∞, as explained in Section 2. Proposition 6.2. Let (k, 3) be an admissible pair and let p be an integer such that (p + 2)γ > 1. Let w+ ∈ H k+(p+1)∨2 and ψ+ ∈ Y +1 . Let φ = φp−1 be deﬁned by ∞ +2 (5.11) and Proposition 5.1, so that h−1 ). Let V and χ 0 φ ∈ (C ∩ L )([1, ∞), Y be the solutions of (5.12) and (5.39) respectively, obtained in Propositions 5.4, so that (V, χ) ∈ (C ∩ L∞ )([1, ∞), H k+2 ⊕ Y +1 ). Let a+ = |w+ |k+(p+1)∨2

,

b+ = |ψ+ |+1

.

(6.12)

Then there exist T0 and T , 1 ≤ T0 , T < ∞, depending only on (γ, p, a+ , b+ ) such that for all t0 ≥ T0 ∨ T , the system (4.1) (4.2) with initial data wt0 (t0 ) = V (t0 ), ϕt0 (t0 ) = φp (t0 ) + χ(t0 ) has a unique solution in the interval [T, ∞) such that ∞ k (wt0 , h−1 0 ϕt0 ) ∈ (C ∩L )([T, ∞), H ⊕ Y ). One can deﬁne T0 and T by conditions of the type A (a+ , b+ ) h(T0 ) = 1 (6.13) A (a+ , b+ ) ((p + 2)γ − 1)−1 h(T ) = 1 . The solution satisﬁes the estimates wt0 (t) − V (t) ∨ wt0 (t) − Wp (t) ≤ A (a+ , b+ ) Qp (t0 ) k

(6.14)

(6.15)

k

ϕt0 (t) − φp (t) − χ(t) ∨ ϕt0 (t) − φp (t) − ψ+ ≤ A (a+ , b+ ) Qp (t0 ) h0 (t) (6.16)

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for t ≥ t0 ,

Ann. Henri Poincar´ e

wt0 (t) − V (t) ∨ wt0 (t) − Wp (t) ≤ A (a+ , b+ ) Qp (t) k

(6.17)

k

ϕt0 (t) − φp (t) − χ(t) ∨ ϕt0 (t) − φp (t) − ψ+ ≤ A (a+ , b+ ) Pp (t)

(6.18)

for T ≤ t ≤ t0 , and |wt0 (t)|k ≤ A (a+ , b+ )

,

|ϕt0 (t)| ≤ A (a+ , b+ ) h0 (t)

(6.19)

for all t ≥ T . Proof. The result follows from Proposition 4.1 and standard globalisation arguments provided we can derive (6.15) (6.16) (6.17) (6.18) as a priori estimates under the assumptions of the proposition. Let (wt0 , ϕt0 ) be the maximal solution of (4.1) (4.2) with the appropriate initial condition at t0 . Deﬁne q = wt0 − V and ψ = ϕt0 − φp − χ. Comparing the equations (4.1) (4.2) and (5.12) (5.39), we obtain ∂t q = (2t2 )−1 i∆wt0 + (2∇ϕt0 · ∇ + (∆ϕt0 )) q + (2∇(ψ + ϕp + χ) · ∇ +(∆(ψ + ϕp + χ))) V

(6.20) ∇ϕi · ∇ϕj ∂t ψ = (2t2 )−1 |∇ψ|2 + 2∇ψ · ∇(φp + χ) + |∇χ|2 + 2∇χ · ∇ϕp + +t−γ g0 (q, q) + 2g0 (q, V ) + g0 (V − Wp , V + Wp ) +

0≤i,j≤p i+j≥p

g0 (wi wj )

(6.21)

0≤i,j≤p i+j≥p+1

where the last bracket is obtained by rewriting g0 (wt0 , wt0 ) − g0 (wi , wj ) . i+j≤p

We estimate (q, ψ) by Lemmas 3.2 and 3.3, especially (3.6) (3.12) for q and (3.7) (3.8) (3.13) for ψ and we obtain (6.22) ∂t |q|k ≤ C t−2 {|V |k+2 + |ϕt0 | |q|k + |ψ + ϕp + χ| |V |k+1 } ∂t |ψ| ≤ C t−2 |ψ|2 + |ψ| |φp + χ|+1 + |χ|2+1 +|ϕp |+1 |χ|+1 + |ϕi |+1 |ϕj |+1 0≤i,j≤p i+j≥p

+C t−γ |q|2k + |q|k |V |k + |V − Wp |k |V + Wp |k +

0≤i,j≤p i+j≥p+1

|wi |k |wj |k

. (6.23)

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From Propositions 5.1 and 5.4, it follows that there exist a and b depending on (a+ , b+ ), such that the following estimates hold. |V |k+2 ≤ a , |Wp |k ≤ a , |V − Wp |k ≤ a Qp |wj |k ≤ a Qj−1 |φp |+1 ≤ b h0 , |ϕj |+1 ≤ b Nj |χ|+1 ≤ b ≤ C b Np ≤ C b γ −p h0

, for 1 ≤ j ≤ p , for 0 ≤ j ≤ p , for t ≥ 2 .

We now deﬁne y = |q|k and z = |ψ| . Using the previous estimates, we obtain from (6.22) (6.23) |∂t y| ≤ C t−2 a + (z + b h0 )y + (z + b Np )a (6.24) Ni Nj + |∂t z| ≤ C t−2 (z + b h0 )z + b2 0≤i,j≤p i+j≥p

C t−γ y(y + a) + a2 Qp + a2

(6.25) .

Qi Qj

0≤i,j≤p−1 i+j≥p−1

In the last bracket in (6.25), the terms in a2 are absent for p = 0 since in that case V = W0 = w0 = w+ . We next estimate y and z from (6.24) (6.25), taking C = 1 for the rest of the proof. We distinguish two cases. Case t ≥ t0 . Let t¯ > t0 and deﬁne Y = Y (t¯) = y; L∞ ([t0 , t¯]) and Z = Z(t¯) = ∞ ¯ ¯

h−1 0 z; L ([t0 , t]) . Then for all t ∈ [t0 , t ] (6.26) ∂t y ≤ t−2 a + (Z + b)Y h0 + a Z h0 + a b Np Ni Nj ∂t z ≤ t−2 (Z + b)Z h20 + b2 0≤i,j≤p i+j≥p

+ t−γ Y (Y + a) + a2 Qp + a2

Qi Qj

(6.27) .

0≤i,j≤p−1 i+j≥p−1

Integrating (6.26) between t0 and t and using (3.35) we obtain y ≤ a t−1 0 + (Z + b)Y h(t0 ) + a Z h(t0 ) + a b Qp (t0 )

(6.28)

Y ≤ (Z + b)Y h(t0 ) + a Z h(t0 ) + a B1 (t0 )

(6.29)

−1 + b Qp (t0 ) . B1 (t0 ) = t−1 0 + b Qp (t0 ) ≤ Qp (1)

(6.30)

and therefore where

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Integrating (6.27) between t0 and t, we obtain similarly z ≤ (Z + b)Z h0 (t) h(t0 ) + Y (Y + a) h0 (t) + C(b2 + a2 ) h0 (t) Qp (t0 ) . (6.31) Here we have used the following relations

t

2 dt1 t−2 1 h0 (t1 ) ≤ h0 (t)(h(t0 ) − h(t))

t0

t

dt1 t−2 1 Ni (t1 ) Nj (t1 ) ≤

t0

t

dt1 t−2 1 h0 (t1 ) Ni+j (t1 )

t0

t

≤ h0 (t)

dt1 t−2 1 Ni+j (t1 ) = h0 (t) (Qi+j (t0 ) − Qi+j (t))

t0

by (3.38) (3.35),

t

dt1 t−γ Qp (t1 ) ≤ Qp (t0 ) (h0 (t) − h0 (t0 )) 1

t0

by (3.45), and

t

t0

dt1 t−γ Qi (t1 ) Qj (t1 ) ≤ 1

t

dt1 t−γ h(t1 ) Qi+j (t1 ) ≤ 2Qi+j+1 (t0 ) (h0 (t) − h0 (t0 )) 1

t0

by (3.40) and (3.46). From (6.31) we obtain Z ≤ (Z + b)Z h(t0 ) + Y (Y + a) + B2 (t0 )

(6.32)

B2 = C(b2 + a2 ) Qp (t0 ) .

(6.33)

with Now (6.29) (6.32) deﬁne a closed subset R of IR+ ×IR+ in the (Y, Z) variables, containing the point (0, 0), and (Y, Z) is a continuous function of t¯ starting from that point for t¯ = t0 . If we can ﬁnd an open region R1 of IR+ × IR+ containing (0, 0) and such that R ∩ R1 ⊂ R1 , then (Y, Z) will remain in R ∩ R1 for all time, because R ∩ R1 is both open and closed in R. We ﬁrst take t0 suﬃciently large so that (6.34) 4b h(t0 ) ≤ 1 , 16a2 h(t0 ) ≤ 1 , 4B1 (t0 ) ≤ 1 and we choose for R1 the region 4Z h(t0 ) < 1. From (6.29) (6.32) (6.34) it follows that in R ∩ R1

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  Y ≤ 2a (Z h(t0 ) + B1 (t0 )) ≤ a 

Z ≤ 4a Y + 2B2 (t0 )

and therefore   Y ≤ 4a B1 (t0 ) + 2B2 (t0 ) h(t0 ) 

(6.35)

Z ≤ 4 4a2 B1 (t0 ) + B2 (t0 )

so that the condition 4Zh(t0 ) < 1 is implied by 16 4a2 B1 (t0 ) + B2 (t0 ) h(t0 ) < 1 .

(6.36)

The estimates (6.15) (6.16) with V and χ now follow from (6.35), while the conditions (6.34) (6.36) reduce to the form (6.13). The estimates (6.15) (6.16) with Wp and ψ+ follow from the previous ones, from (5.43) (5.44) and from the fact that Qp (t) h0 (t) h(t)−1 = (t Qp (t)) h0 (t) t−1 h(t)−1 is an increasing function of t, so that h(t) ≤ h(t1 ) Qp (t1 )−1 h0 (t1 )−1 Qp (t0 ) h0 (t) for any (ﬁxed) t1 ≤ t0 . y; L∞ ([t¯, t0 ]) and Z = Case t ≤ t0 . Let t¯ < t0 and deﬁne Y = Y (t¯) = Q−1 p −1 ∞ ¯ ¯ Z(t) = Pp z; L ([t, t0 ]) . It then follows from (6.24) (6.25) that for all t ∈ [t¯, t0 ] |∂t y| ≤ t−2 a + Z Y Pp Qp + b Y h0 Qp + a Z Pp + a b Np

(6.37)

|∂t z| ≤ t−2 Z 2 Pp2 + b Z h0 Pp + b2 Ni Nj 0≤i,j≤p i+j≥p

+t−γ Y 2 Q2p + a Y Qp + a2 Qp + a2

Qi Qj

.

0≤i,j≤p−1 i+j≥p−1

Integrating (6.37) between t and t0 , using (3.32) (3.35) and

t0

dt1 t−2 1 Pp (t1 ) Qp (t1 ) ≤ Qp (t) (Rp (t) − Rp (t0 ))

t

t

t0

dt1 t−2 1 h0 (t1 ) Qp (t1 ) ≤ Qp (t) (h(t) − h(t0 ))

(6.38)

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we obtain y ≤ a t−1 + Z Y Rp (t) Qp (t) + b Y h(t) Qp (t) + a Z Rp (t) + a b Qp (t) (6.39) and therefore by (3.47) Y ≤ a c + Z Y Rp (t) + b Y h(t) + Cp a Z h(t) + a b

(6.40)

where c = Qp (1)−1 . We integrate similarly (6.38) between t and t0 . We use the relations

t0 2 dt1 t−2 1 Pp (t1 ) ≤ Pp (t) (Rp (t) − Rp (t0 )) ≤ Pp (t) Rp (t) t

t0 dt1 t−2 1 h0 (t1 ) Pp (t1 ) ≤ Pp (t) (h(t) − h(t0 )) t

t0

t0 dt1 t−2 N (t ) N (t ) ≤ dt1 t−2 i 1 j 1 1 1 h0 (t1 ) Ni+j (t1 ) t

t

≤ Pi+j (t) − Pi+j (t0 ) by (3.38) (3.37),

t0 2 dt1 t−γ Q (t ) ≤ Q (t) p p 1 1 t

by (3.42) and

t0

dt1 t−γ Qp (t1 ) ≤ Qp (t) Pp (t) 1

t

t0

dt1 t−γ Qi (t1 ) Qj (t1 ) ≤ Pi+j+1 (t) 1

t

by (3.40) (3.41) (3.42). We obtain z ≤ Z 2 Pp (t) Rp (t) + b Z Pp (t) h(t) + Y 2 Pp (t) Qp (t) + a Y Pp (t) + C(a2 + b2 )Pp (t) (6.41) and therefore Z ≤ Z 2 Rp (t) + b Z h(t) + Y 2 Qp (t) + a Y + C(a2 + b2 ) .

(6.42)

We now take t suﬃciently large so that bh(t) ≤ 1/4 and we proceed as in the case t ≥ t0 by taking for R1 the strip deﬁned by ZRp (t) < 1/4, Cp Zh(t) < b, thereby obtaining from (6.40) (6.42)   Y ≤ 2a(2b + c) (6.43)  Z ≤ 2C a2 + b2 + 4a2 (2b + c) (1 + 2(2b + c)Qp (1)) . The conditions bh ≤ 1/4, ZR < 1/4, Zh < b are then satisﬁed for t ≥ T with T deﬁned by a condition of the form (6.14), where the singular factor ((p+2)γ −1)−1

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comes from Cp . The estimates (6.17) (6.18) with V and χ follow from (6.43), and the analogous estimates with Wp and ψ+ follow therefrom and from (5.43) (5.44). Finally the estimates (6.19) follow from (6.15) (6.16) (6.17) (6.18) and from Proposition 5.1. We can now take the limit t0 → ∞ of the solution (wt0 , ϕt0 ) constructed in Proposition 6.2, for ﬁxed (w+ , ψ+ ). Proposition 6.3. Let (k, 3) be an admissible pair and let p be an integer such that (p + 2)γ > 1. Let w+ ∈ H k+(p+1)∨2 and ψ+ ∈ Y +1 . Let φ = φp−1 be deﬁned by ∞ +2 ). Let V and χ (5.11) and Proposition 5.1, so that h−1 0 φ ∈ (C ∩ L )([1, ∞), Y be the solutions of (5.12) and (5.39) respectively, obtained in Proposition 5.4 so that (V, χ) ∈ (C ∩ L∞ )([1, ∞), H k+2 ⊕ Y +1 ) and let a+ , b+ be deﬁned by (6.12). Then (1) There exists T , 1 ≤ T < ∞, depending only on (γ, p, a+ , b+ ) and there exists a unique solution (w, ϕ) of the system (4.1) (4.2) in the interval [T, ∞) such ∞ k that (w, h−1 0 ϕ) ∈ (C ∩ L )([T, ∞), H ⊕ Y ) and such that the following estimates hold for all t ≥ T . (6.44) |w(t) − V (t)|k ∨ w(t) − Wp (t) ≤ A (a+ , b+ ) Qp (t) k

ϕ(t) − φp (t) − χ(t) ∨ ϕ(t) − φp (t) − ψ+ ≤ A (a+ , b+ ) Pp (t)

(6.45)

|w(t)|k ≤ A (a+ , b+ )

,

|ϕ(t)| ≤ A (a+ , b+ ) h0 (t)

(6.46)

One can deﬁne T by a condition of the type (6.14). (2) Let (wt0 , ϕt0 ) be the solution of the system (4.1) (4.2) constructed in ∞ k Proposition 6.2 for t0 ≥ T0 ∨T and such that (wt0 , h−1 0 ϕt0 ) ∈ (C∩L )([T, ∞), H ⊕ Y ). Then (wt0 , ϕt0 ) converges to (w, ϕ) in norm in L∞ (J, H k−1 ⊕ Y −1 ) and in the weak-∗ sense in L∞ (J, H k ⊕Y ) for any compact J ⊂ [T, ∞), and in the weak-∗ sense in H k ⊕ Y pointwise in t. (3) The map (w+ , ψ+ ) → (w, ϕ) deﬁned in Part (1) is continuous on the bounded sets of H k+(p+1)∨2 ⊕ Y +1 from the norm topology of (w+ , ψ+ ) in H k+p−1 ⊕ Y −1 to the norm topology of (w, ϕ) in L∞ (J, H k−1 ⊕ Y −1 ) and to the weak-∗ topology in L∞ (J, H k ⊕ Y ) for any compact interval J ⊂ [T, ∞), and to the weak-∗ topology in H k ⊕ Y pointwise in t. Proof. Parts (1) and (2) will follow from the convergence of (wt0 , ϕt0 ) when t0 → ∞ in the topologies stated in Part (2). Let T0 ∨ T ≤ t0 ≤ t1 . From (6.17) (6.18) it follows that (6.47) wt1 (t0 ) − wt0 (t0 ) = wt1 (t0 ) − V (t0 ) ≤ A Qp (t0 ) k

k

ϕt1 (t0 ) − ϕt0 (t0 ) = ϕt1 (t0 ) − φp (t0 ) − χ(t0 ) ≤ A Pp (t0 )

(6.48)

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We now estimate (wt0 − wt1 , ϕt0 − ϕt1 ) in H k−1 ⊕ Y −1 for t ≤ t0 . Let , z = ϕt0 − ϕt1 . y = wt0 − wt1 k−1

(6.49)

−1

From (6.19) and Lemma 3.3, it follows that y and z satisfy the system (4.10). Integrating that system for t ≤ t0 with initial data at t0 , we obtain from Lemma 4.1   y(t) ≤ A y(t0 ) + t−1 (z(t0 ) + y(t0 )h0 (t0 )) (6.50)  z(t) ≤ A (z(t0 ) + y(t0 )h0 (t0 )) . From (6.47) (6.48) (6.50) and from the fact that Pp (t0 ) and Qp (t0 )h0 (t0 ) tend to zero when t0 → ∞, it follows that there exists (w, ϕ) ∈ C([T, ∞), H k−1 ⊕ Y −1 ) such that (wt0 , ϕt0 ) converges to (w, ϕ) in L∞ (J, H k−1 ⊕ Y −1 ) for all compact intervals J ⊂ [T, ∞). From that convergence, from (6.17) (6.18) (6.19) and standard ∞ k compactness arguments, it follows that (w, h−1 0 ϕ) ∈ (Cw∗ ∩ L )([T, ∞), H ⊕ Y ), that (w, ϕ) satisﬁes the estimates (6.44) (6.45) (6.46) for all t ≥ T , and that (wt0 , ϕt0 ) converges to (w, ϕ) in the other topologies considered in Part (2). Furthermore, (w, ϕ) satisﬁes the system (4.1) (4.2) and by Proposition 4.1, part (1), (w, ϕ) ∈ C([T, ∞), H k ⊕ Y ). Finally, uniqueness of (w, ϕ) under the conditions (6.44) (6.45) follows from Proposition 4.3 and from the fact that Pp (t) and Qp (t)h0 (t) tend to zero when t → ∞. Part (3). Let (w+ , ψ+ ) and (w+ , ψ+ ) belong to a ﬁxed bounded set of H k+(p+1)∨2 ⊕ +1 Y . Let (Wp , φp ) and (Wp , φp ) be the associated functions deﬁned by (5.11) and Proposition 5.1 and let (w, ϕ) and (w , ϕ ) be the associated solutions of the system (4.1) (4.2) deﬁned in Part (1). We assume that (w+ , ψ+ ) is close to (w+ , ψ+ ) in the sense that |w+ − w+ |k+p−1 ≤ ε (6.51) |ψ+ − ψ+ |−1 ≤ ε0

.

(6.52)

We now take t0 > T and we estimate (w − w , ϕ − ϕ ) in H k−1 ⊕ Y −1 for t ≤ t0 . Let y = |w − w |k−1 , z = |ϕ − ϕ |−1 . (6.53) From (4.1) (4.2) and Lemma 3.3, it follows that (y, z) satisfy the system (4.10). Integrating that system between t0 and t yields the estimate (6.50) for (y, z) deﬁned by (6.53). From (6.44) (6.45) we obtain   y(t ) ≤ A Q (t ) + (t ) − W (t ) W 0 p 0 p 0 0  p  k−1 (6.54)    z(t0 ) ≤ A Pp (t0 ) + ε0 + φp (t0 ) − φp (t0 ) . −1

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From estimates similar to (5.8) (5.9), and from (6.51) (6.52) it then follows that   y(t0 ) ≤ A (Qp (t0 ) + ε) (6.55)  z(t0 ) ≤ A (Pp (t0 ) + ε h0 (t0 )) + ε0 . We now choose t0 so that Qp (t0 ) = ε. Substituting (6.55) with that choice into (6.50) and using the asymptotic behaviour of Qp and Pp for large t, we obtain   y(t) ≤ A m(ε) + t−1 (ε0 + m(ε)) (6.56)  z(t) ≤ A (ε0 + m(ε)) where

 ((p+2)γ−1)/(p+1)γ ε      m(ε) = εγ Log ε      γ ε

for (p + 1)γ < 1 for(p + 1)γ = 1

(6.57)

for(p + 1)γ > 1 .

This implies the (uniform H¨ older) continuity of (w, ϕ) as a function of (w+ , ψ+ ) in the norm topology of L∞ (J, H k−1 ⊕ Y −1 ) for all compact intervals J ⊂ [T, ∞). The other continuities follow therefrom and from the boundedness of ∞ k (w, h−1 0 ϕ) in L ([T, ∞), H ⊕ Y ) by standard compactness arguments.

7 Asymptotics and wave operators for u In this section we complete the construction of the wave operators for the equation (1.1) and we derive asymptotic properties of solutions in their range. The construction relies in an essential way on those of Section 6, esp. Proposition 6.3, and will require a discussion of the gauge invariance of those constructions. We ﬁrst deﬁne the wave operator for the auxiliary system (4.1) (4.2). Deﬁnition 7.1. We deﬁne the wave operator Ω0 as the map Ω0 : (w+ , ψ+ ) → (w, ϕ)

(7.1)

from H k+(p+1)∨2 ⊕ Y +1 to the space of (w, ϕ) such that (w, h−1 0 ϕ) ∈ (C ∩ L∞ )([T, ∞), H k ⊕ Y ) for some T , 1 ≤ T < ∞, where (w, ϕ) is the solution of the system (4.1) (4.2) obtained in Proposition 6.3, part (1). Before deﬁning the wave operators for u, we now study the gauge invariance of Ω0 , which plays an important role in justifying that deﬁnition, as was explained in Section 2. For that purpose we need some information on the Cauchy problem for the equation (1.1) at ﬁnite times. In addition to the operators M = M (t) and D = D(t) deﬁned by (2.4) (2.5), we introduce the operator J = J(t) = x + it∇ ,

(7.2)

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the generator of Galilei transformations. The operators M , D, J satisfy the commutation relation iM D ∇=J M D . (7.3) For any interval I ⊂ [1, ∞) and any nonnegative integer k, we deﬁne the space X k (I) = u : D∗ M ∗ u ∈ C(I, H k ) (7.4) = u :< J(t) >k u ∈ C(I, L2 ) where < λ >= (1+λ2 )1/2 for any real number or self-adjoint operator λ and where the second equality follows from (7.3). Then (see Proposition I.7.1). Proposition 7.1. Let k be a positive integer and let 0 < µ < n∧2k. Then the Cauchy problem for the equation (1.1) with initial data u(t0 ) = u0 such that < J(t0 ) >k u0 ∈ L2 at some initial time t0 ≥ 1 is locally well posed in X k (·), namely (1) There exists T > 0 such that (1.1) has a unique solution with initial data u(t0 ) = u0 in X k ([1 ∨ (t0 − T ), t0 + T ]). (2) For any interval I, t0 ∈ I ⊂ [1, ∞), (1.1) with initial data u(t0 ) = u0 has at most one solution in X k (I). (3) The solution of Part (1) depends continuously on u0 in the norms considered there. We come back from the system (4.1) (4.2) to the equation (1.1) by reconstructing u from (w, ϕ) by (2.7) and accordingly we deﬁne the map Λ : (w, ϕ) → u = M D exp(−iϕ)w

.

(7.5)

It follows immediately from Lemma 3.1 that the map Λ satisﬁes the following property. Lemma 7.1. The map Λ deﬁned by (7.5) is bounded and continuous from C(I, H k ⊕ Y ) to X k (I) for any admissible pair (k, 3) and any interval I ⊂ [1, ∞). We now give the following deﬁnition. Deﬁnition 7.2. Let (k, 3) be an admissible pair and let (w, ϕ) and (w , ϕ ) be two solutions of the system (4.1) (4.2) in C(I, H k ⊕ Y ) for some interval I ⊂ [1, ∞). We say that (w, ϕ) and (w , ϕ ) are gauge equivalent if they give rise to the same u, namely if Λ(w, ϕ) = Λ(w , ϕ ), or equivalently if exp(−iϕ(t)) w(t) = exp(−iϕ (t)) w (t)

(7.6)

for all t ∈ I. A suﬃcient condition for gauge equivalence is given by the following Lemma. Lemma 7.2. Let (k, 3) be an admissible pair and let (w, ϕ) and (w , ϕ ) be two solutions of the system (4.1) (4.2) in C(I, H k ⊕ Y ). In order that (w, ϕ) and (w , ϕ ) be gauge equivalent, it is suﬃcient that (7.6) holds for one t ∈ I.

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Proof. An immediate consequence of Lemma 7.1, of Proposition 7.1, part (2), and of the fact that (k, 3) admissible implies k > 1 + µ/2. The gauge covariance properties of Ω0 will be expressed by the following two propositions. Proposition 7.2. Let (k, 3) be an admissible pair. Let (w, ϕ) and (w , ϕ ) be two solu−1 ∞ tions of the system (4.1) (4.2) such that (w, h−1 0 ϕ), (w , h0 ϕ ) ∈ (C ∩ L )([T, ∞), k H ⊕Y ) for some T ≥ 1, and assume that (w, ϕ) and (w , ϕ ) are gauge equivalent. Then (1) There exists σ ∈ Y −1 such that ϕ (t) − ϕ(t) converges to σ when t → ∞ strongly in Y −2 and in the weak-∗ sense in Y −1 . The following estimates hold :

ϕ (t) − ϕ(t) − σ; Y −2 ≤ A h(t)

(7.7)

−1 for some constant A depending on T and on the norms of h−1 0 ϕ, h0 ϕ ∞ in L (·, Y ), with the exception of the case n even, 3 = n/2 + 1 where the L∞ norm of ∇ϕ satisﬁes only

∇ϕ (t) − ∇ϕ(t) − ∇σ ∞ ≤ A h(t)1/2

.

(7.8)

be the limits of w(t) and w (t) as t → ∞, obtained in (2) Let w+ and w+ Proposition 4.4. Then w+ = w+ exp(−iσ). (3) Let p ≥ 0 be an integer. Assume in addition that w+ , w+ ∈ H k+p and let φp , φp be associated with w+ , w+ according to (5.11) and Proposition 5.1. Assume that the following limits exist

, lim ϕ (t) − φp (t) = ψ+ (7.9) lim ϕ(t) − φp (t) = ψ+ t→∞

t→∞

= ψ+ + σ. as strong limits in L∞ . Then ψ+

Proof. Part (1) is essentially identical with Proposition I.7.2, part (1). Part (2). We deﬁne ϕ− (t) = ϕ (t) − ϕ(t) and we estimate

w+ − w+ eiσ 2 ≤ w+ − w (t) 2 + w (t) − exp(iϕ− (t))w(t) 2

+ (exp(iϕ− (t)) − exp(iσ))w(t) 2 + exp(iσ)(w(t) − w+ ) 2 ≤ w+ − w (t) 2 + w(t) − w+ 2 + ϕ− (t) − σ ∞ w(t) 2

(7.10)

by gauge invariance. The last member of (7.10) tends to zero as t → ∞. Part (3). By gauge invariance, namely Proposition 5.1 part (2) and Part (2) of this proposition, φp = φp and therefore

− ψ+ = lim ϕ (t) − ϕ(t) = σ . (7.11) ψ+ t→∞

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Remark 7.1. The additional assumptions of Proposition 7.2, part (3) are satisﬁed either if (w, ϕ), (w , ϕ ) satisfy the assumptions of Proposition 6.1, or if (w, ϕ), (w , ϕ ) ∈ R(Ω0 ). We shall not consider the former case any further. In the latter case, it follows from (7.11) that actually σ ∈ Y +1 . Proposition 7.2 prompts us to make the following deﬁnition of gauge equivalence for asymptotic states. Deﬁnition 7.3. Two pairs (w+ , ψ+ ) and (w+ , ψ+ ) are gauge equivalent if w+ exp(−iψ+ ) = w+ exp(−iψ+ ). With this deﬁnition, Proposition 7.2 implies that two gauge equivalent solutions of the system (4.1) (4.2) in R(Ω0 ) are images of two gauge equivalent pairs of asymptotic states. The next proposition shows that conversely two gauge equivalent pairs of asymptotic states have gauge equivalent images under Ω0 . Proposition 7.3. Let (k, 3) be an admissible pair and let p be an integer such that (p + 2)γ > 1. Let (w+ , ψ+ ), (w+ , ψ+ ) ∈ H k+(p+1)∨2 ⊕ Y +1 be gauge equivalent, and let (w, ϕ), (w , ϕ ) be their images under Ω0 . Then (w, ϕ) and (w , ϕ ) are gauge equivalent. Proof. Let t0 be suﬃciently large and let (wt0 , ϕt0 ) and (wt 0 , ϕt0 ) be the solutions of the system (4.1) (4.2) constructed by Proposition 6.2. From the initial conditions wt0 (t0 ) = V (t0 )

,

ϕt0 (t0 ) = φp (t0 ) + χ(t0 ) ,

wt 0 (t0 ) = V (t0 ) , ϕt0 (t0 ) = φp (t0 ) + χ (t0 ) ,

from the fact that φp = φp by Proposition 5.1 part (2) and that V exp(−iχ) = V exp(−iχ ) by Proposition 5.4, part (2), it follows that wt0 (t0 ) exp(−iϕt0 (t0 )) = wt 0 (t0 ) exp(−iϕt0 (t0 )) and therefore by Lemma 7.2, (wt0 , ϕt0 ) and (wt 0 , ϕt0 ) are gauge equivalent, namely wt0 (t) exp(−iϕt0 (t)) = wt 0 (t) exp(−iϕt0 (t))

(7.12)

for all t for which both solutions are deﬁned. We now take the limit t0 → ∞ for ﬁxed t in (7.12). By Proposition 6.3 part (2), for ﬁxed t, (wt0 , ϕt0 ) and (wt 0 , ϕt0 ) converge respectively to (w, ϕ) and (w , ϕ ) in H k−1 ⊕ Y −1 . By Lemma 3.1, one can take the limit t0 → ∞ in (7.12), thereby obtaining (7.6), so that (w, ϕ) and (w , ϕ ) are gauge equivalent. We can now deﬁne the wave operators for u. We recall from the heuristic discussion of Section 2 that we want to exploit the operator Ω0 deﬁned in Deﬁnition 7.1, reconstruct u through the map Λ deﬁned by (7.5) and eliminate the arbitrariness in ψ+ by ﬁxing ψ+ = 0, thereby ensuring the injectivity of the wave operator for u. Deﬁnition 7.4. We deﬁne the wave operator Ω as the map Ω : u+ → u = (Λ ◦ Ω0 ) (F u+ , 0)

(7.13)

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from F H k+(p+1)∨2 to X k ([T, ∞)) for some T , 1 ≤ T < ∞, where k is the ﬁrst element of an admissible pair, and Ω0 and Λ are deﬁned by Deﬁnition 7.1 and by (7.5). The fact that Ω acts between the spaces indicated follows from Proposition 6.3 and Lemma 7.1. Proposition 7.4. (1) The map Ω is injective. (2) If 0 ≤ p ≤ 2, then R(Ω) = R(Λ ◦ Ω0 ). Proof. Part (1) follows from the fact that Ω0 is an injective map between gauge equivalence classes and that an equivalence class of asymptotic states contains at most one representative with ψ+ = 0. Part (2) follows from the fact that the gauge equivalence class of a given (w+ , ψ+ ) actually contains an element with ψ+ = 0, namely (w+ exp(−iψ+ ), 0), by Lemma 3.1. Remark 7.2. Part (2) of Proposition 7.4 does not extend to the case p ≥ 3 because in that case (w+ , ψ+ ) ∈ H k+p+1 ⊕ Y +1 does not imply that w+ exp(−iψ+ ) ∈ H k+p+1 , so that the gauge equivalence class of a given (w+ , ψ+ ) need not contain an element with ψ+ = 0. We now collect the information obtained for the solutions of the equation (1.1) so far constructed. The main result of this paper can be stated as follows. Proposition 7.5. Let n ≥ 3, 0 < µ ≤ n − 2 and 0 < γ ≤ 1. Let (k, 3) be an admissible pair. Let p ≥ 0 be an integer with (p + 2)γ > 1. Let u+ ∈ F H k+(p+1)∨2 and a+ = |F u+ |k+(p+1)∨2 . Let Wp and φp be deﬁned by (5.11) and Proposition 5.1 with w+ = F u+ . Then (1) There exists T , 1 ≤ T < ∞, and there exists a unique solution u ∈ X k ([T, ∞)) of the equation (1.1) which can be represented as u = M D exp(−iϕ)w where (w, ϕ) is a solution of the system (4.1) (4.2) such that (w, h−1 0 ϕ) ∈ (C ∩ L∞ )([T, ∞), H k ⊕ Y ) and such that h0 (t) → 0 (7.14) w(t) − F u+ k−1

ϕ(t) − φp (t)

−1

→0

(7.15)

when t → ∞, where h0 is deﬁned by (3.19). The time T can be deﬁned by (6.14) with b+ = 0. (2) The solution is obtained as u = Ω(u+ ) where the map Ω is deﬁned in Deﬁnition 7.4. The map Ω is injective.

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(3) The map Ω is continuous on the bounded sets of F H k+(p+1)∨2 from the norm topology in F H k+p−1 for u+ to the norm topology in X k−1 (I) and to the weak-∗ topology in X k (I) for u for any compact interval I ⊂ [T, ∞), and to the weak topology in M DH k pointwise in t. (4) The solution u satisﬁes the following estimates for t ≥ T

< J(t) >k exp(iφp (t, x/t))u(t) − M (t) D(t) F u+ 2 ≤ A(a+ ) Pp (t) (7.16) for some estimating function A(a+ ), where Pp (t) is deﬁned by (3.31). (5) Let r satisfy 0 ≤ δ(r) ≡ n/2 − n/r ≤ k ∧ n/2, δ(r) < n/2 if k = n/2. Then u satisﬁes the following estimate

u(t) − exp(−iφp (t, x/t))M (t) D(t) F u+ r ≤ A(a+ ) t−δ(r) Pp (t)

.

(7.17)

Proof. Parts (1) (2) (3) follow from Proposition 6.3, Proposition 4.3, from Deﬁnition 7.4, Proposition 7.4 part (1), and Lemma 7.1. Part (4). From the deﬁnition (7.2) of J(t), from the commutation relation (7.3) and from Lemma 3.1, it follows that the LHS of (7.16) is estimated by

· 2 = exp(i (φp − ϕ))w − F u+ ≤ |w − F u+ |k + (exp(i(φp − ϕ)) − 1) w k

≤ |w − F u+ |k + |φp − ϕ|−1 (1 + |φp − ϕ|−1 )k−1 |w|k

k

.

The result now follows from the estimates (4.17) and (6.45) (6.46). Part (5) follows from Part (4) and from the inequality

f r

= t−δ(r) D∗ M ∗ f r ≤ C t−δ(r) < ∇ >k D∗ M ∗ f 2 = C t−δ(r) < J(t) >k f 2

which follows from the commutation relation (7.3) and from Sobolev inequalities. Remark 7.3. In (7.16) and (7.17) one could replace M DF u+ by U (t)u+ since U (t)u+ − M DF u+ = O(t−1 ) in the relevant norms. One could also replace F u+ by Wp , but this would not produce any improvement in the ﬁnal estimates, since the main contribution of the difference between u and its asymptotic form is that of the phase. Finally, by combining Proposition 7.5 with the known results on the Cauchy problem for the equation (1.1) at ﬁnite times, one could extend the solutions u to arbitrary ﬁnite times and deﬁne more standard wave operators Ω1 : u+ → u(1) where u = Ωu+ . We refer to I for the details. Acknowledgements One of us (G. V.) is grateful to Professor J. C. Saut for the hospitality at the Laboratoire d’Analyse Num´erique et Equations aux D´eriv´ees Partielles and to Professor D. Schiﬀ for the hospitality at the Laboratoire de Physique Th´eorique.

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References [1] J. Derezinski, C. G´erard : Scattering theory of classical and quantum N particle systems, Springer, Berlin, 1997. [2] J. Ginibre, T. Ozawa : Long range scattering for nonlinear Schr¨ odinger and Hartree equations in space dimension n ≥ 2, Commun. Math. Phys. 151 (1993), 619–645. [3] J. Ginibre, G. Velo : On a class of nonlinear Schr¨ odinger equations with nonlocal interaction, Math. Z. 170 (1980), 109–136. [4] J. Ginibre, G. Velo : Long range scattering and modiﬁed wave operators for some Hartree type equations I, Rev. Math. Phys., 12 (2000), 361–429. [5] N. Hayashi, E. I. Kaikina, P. I. Naumkin : On the scattering theory for the cubic nonlinear Schr¨ odinger and Hartree type equations in one space dimension, Hokka¨ıdo Math. J., 27 (1998), 651–667. [6] N. Hayashi, P. I. Naumkin : Asymptotics for large time of solutions to the nonlinear Schr¨ odinger and Hartree equations, Am. J. Math. 120 (1998), 369– 389. [7] N. Hayashi, P. I. Naumkin : Scattering theory and large time asymptotics of solutions to Hartree type equations with a long range potential, preprint, 1997. [8] N. Hayashi, P. I. Naumkin : Remarks on scattering theory and large time asymptotics of solutions to Hartree type equations with a long range potential, SUT J. of Math. 34 (1998), 13–24. [9] N. Hayashi, P. I. Naumkin, T. Ozawa : Scattering theory for the Hartree equation, SIAM J. Math. Anal. 29 (1998), 1256–1267. [10] N. Hayashi, Y. Tsutsumi : Scattering theory for Hartree type equations, Ann. IHP (Phys. Th´eor.) 46 (1987), 187–213. [11] H. Nawa, T. Ozawa : Nonlinear scattering with nonlocal interaction,Commun. Math. Phys. 146 (1992), 259–275. [12] T. Ozawa : Long range scattering for nonlinear Schr¨ odinger equations in one space dimension, Commun. Math. Phys. 139 (1991), 479–493. [13] D. R. Yafaev : Wave operators for the Schr¨odinger equation, Theor. Mat. Phys. 45 (1980), 992–998.

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J. Ginibre Laboratoire de Physique Th´eorique* Universit´e de Paris XI, Bˆatiment 210 F-91405 Orsay Cedex, France. G. Velo Laboratoire d’Analyse Num´erique et E.D.P.† Universit´e de Paris XI, Bˆatiment 425 F-91405 Orsay Cedex, France. Permanent address : Dipartimento di Fisica Universit`a di Bologna and INFN, Sezione di Bologna, Italy. * Unit´e Mixte de Recherche (CNRS) UMR 8627 † Unit´e Mixte de Recherche (CNRS) UMR 8628. communicated by Rafael D. Benguria submitted 14/09/99, accepted 02/12/99

Ann. Henri Poincar´ e

Ann. Henri Poincar´ e 1 (2000) 801 – 821 c Birkh¨ auser Verlag, Basel, 2000 1424-0637/00/050801-21 $ 1.50+0.20/0

Annales Henri Poincar´ e

Strong Magnetic Field Asymptotics of the Integrated Density of States for a Random 3D Schr¨ odinger Operator W. Kirsch, G.D. Raikov Abstract. We consider the three-dimensional Schr¨ odinger operator with constant magnetic ﬁeld and bounded random electric potential. We investigate the asymptotic behaviour of the integrated density of states for this operator as the norm of the magnetic ﬁeld tends to inﬁnity. R´esum´e On consid`ere l’op´ erateur de Schr¨ odinger tridimensionnel avec un champ magn´ etique constant et un potentiel ´electrique al´ eatoire born´ e. On ´ etudie le comportement asymptotique de la densit´e d’´ etats pour cet op´erateur-ci lorsque la norme du champ magn´etique tend vers l’inﬁni.

1 Introduction Let b := (0, 0, b), b > 0, x = (x, y, z) ∈ R3 . Introduce the unperturbed selfadjoint Schr¨ odinger operator 2 2 2 b∧x ∂ ∂ ∂2 by bx H0 (b) := i∇ + ≡ i + i − 2, (1.1) − + 2 ∂x 2 ∂y 2 ∂z deﬁned originally on C0∞ (R3 ), and then closed in L2 (R3 ). It is well-known that for each b > 0 we have σ(H0 (b)) = [b, +∞) (1.2) where σ(H0 (b)) denotes the spectrum of the operator H0 (b) (see e.g. [A.H.S]). Let (Ω, F, P) be a probability space. Let Vω (x), ω ∈ Ω, x ∈ R3 , be a real random ﬁeld. We assume that Vω is G3 -ergodic with G = Z or G = R (see [K, Section 3.1] or [P.Fi, Section 1C]). In other words, there exists an ergodic group of measure preserving automorphisms Tk : Ω → Ω, k ∈ G3 , such that Vω (x + k) = VTk ω (x), x ∈ R3 , ω ∈ Ω, k ∈ G3 .

(1.3)

We recall that ergodicity of a group G of automorphisms of Ω means that the invariance of a given set A ∈ F under the action of G (i.e. gA = A for each g ∈ G) implies either P(A) = 1 or P(A) = 0. Let x ∈ R3 . We shall write occasionally x = (X, z) with X ∈ R2 and z ∈ R. We suppose that Vω is G-ergodic with G = Z or G = R in the direction of the magnetic ﬁeld (or, in brief, in the z-direction), i.e. that the subgroup {Tk |k = (0, 0, k), k ∈ G} is ergodic.

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Further, we assume that the realizations of Vω are almost surely uniformly bounded, i.e. we have c0 := ess − sup sup |Vω (x)| < ∞.

(1.4)

ω∈Ω x∈R3

Finally, for simplicity we suppose that the realizations of Vω are almost surely continuous. Examples : (i) Let αj : Ω → R, j ∈ Z3 , be independent identically distributed almost surely uniformly bounded random variables. Assume that v : R3 → R is a continuous function satisfying |v(x)| ≤ c(1 + |x|)−β , Set Vω (x) :=

c > 0,

β > 3.

αj (ω) v(x − j), ω ∈ Ω, x ∈ R3 .

j∈Z3

Then the random ﬁeld Vω is Z3 -ergodic (see [K, Model I, Section 3.3] or [P.Fi, Example 1.15a, p.23]), Z-ergodic in the direction of the magnetic ﬁeld (as a matter of fact, in all directions; see [E.K.Sch.S, Example 2, p.615]). Moreover, it is obvious that almost surely the realizations of Vω are uniformly bounded and continuous. (ii) Let ξω (x), ω ∈ Ω, x ∈ R3 , be a real-valued homogeneous Gaussian ﬁeld whose correlation function is continuous at the origin and decays at inﬁnity (see [P.Fi, Example 1.15c, p.26] and [E.K.Sch.S, Example 3, p.615]). Assume that F : R → R is a bounded continuous function. Set Vω (x) := F (ξω (x)), ω ∈ Ω, x ∈ R3 . Then the random ﬁeld Vω is R3 -ergodic, R-ergodic in the direction of the magnetic ﬁeld (and all other directions) whose realizations are almost surely uniformly bounded and continuous. On D(H0 (b)) deﬁne the perturbed Schr¨ odinger operator H(b, ω) := H0 (b) + Vω , b > 0, ω ∈ Ω. It follows from (1.2) and (1.4) that almost surely we have σ(H(b, ω)) ⊆ [b − c0 , +∞).

(1.5)

The aim of this paper is to study the asymptotic behaviour as b → ∞ of the integrated density of states (IDOS) for the operator H(b, ω). In order to recall the deﬁnition of the IDOS, we need several auxiliary concepts. Let ϕr , r ∈ R+ , and ϕ be non-decreasing functions deﬁned on a common domain I ⊆ R. We shall write v − lim ϕr = ϕ r→∞

if we have limr→∞ ϕr (t) = ϕ(t) at all continuity points t of the function ϕ. In this case the function ϕ is called the vague limit as r → ∞ of the family ϕr (cf. [K, p.313]).

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Further, let T = T ∗ be a selfadjoint operator in a Hilbert space. Denote by PI (T ) its spectral projection corresponding to the interval I ⊂ R. Set N (λ; T ) := rank P(−∞,λ) (T ), λ ∈ R, (1.6) n± (s; T ) := rank P(s,+∞) (±T ), s > 0. R 3 , R > 0, with Dirichlet boundary conditions On the Sobolev space H2 − R 2, 2 2 deﬁne the operator H0,R (b) := i∇ + b∧x . Put 2 Db (.) := v − lim R−3 N ( . ; H0,R (b) + Vω ). R→∞

(1.7)

Any non-decreasing function Db (µ), µ ∈ R, satisfying (1.7), is called IDOS for the operator H(b, ω). It is well-known that almost surely the vague limit (1.7) exists and the quantity Db (µ) is non-random (see e.g. [Bro.H.L], [Ma], [U], and the references cited there). Since Db is non-decreasing, the set of its eventual discontinuity points is not more than countable. Note that (1.5) implies that almost surely inf σ(H0,R (b) + Vω ) ≥ b − c0 for all R > 0. Therefore, (1.8) Db (µ) = 0, µ 0, R0 ∈ (0, R), hold with C independent of µ, R, and R0 . Then it follows easily from (1.7) that Db (µ − c0 ) ≤ Db (µ) ≤ Db (µ + c0 ), µ ∈ R.

(1.9)

In this paper we study the asymptotic behaviour as b → ∞ of Db (λ + b), the parameter λ ∈ R being ﬁxed.

2 Statement of the main result On H2 d2 − dz 2.

R R − 2 , 2 with Dirichlet boundary conditions deﬁne the operator h0,R :=

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Proposition 2.1. Let G = Z or G = R. Let fω (z), ω ∈ Ω, z ∈ R, be a real Gergodic random ﬁeld whose realizations are almost surely uniformly bounded and continuous. Then for each λ ∈ R the limit ,(λ; f ) := lim R−1 N (λ; h0,R + fω ) R→∞

(2.1)

exists almost surely. Moreover, the function ,(λ; f ) is non-random, and continuous with respect to λ ∈ R. The proof of the existence and the non-randomness of ,(λ; f ) for much more general ergodic ﬁelds fω can be found in [K, Chapter 7]. The continuity of ,(λ; f ) which is guaranteed by the fact that h0,R + fω is an ordinary diﬀerential operator, is discussed in [P.Fi, Chapter III]. Lemma 2.1. Assume that the hypotheses of Proposition 2.1 hold. Let T : Ω → Ω be a measure preserving automorphism. Then we have lim R−1 N (λ; h0,R + fT ω ) = lim R−1 N (λ; h0,R + fω ).

R→∞

R→∞

(2.2)

Proof. By [K, Theorem 6, Chapter 7] we have lim R−1 N (λ; h0,R + fω ) = sup R−1 E (N (λ; h0,R + fω )) ≡

R→∞

R>0

sup R−1 R>0

N (λ; h0,R + fω ) dP(ω) Ω

where E is used as the symbol of the mathematical expectation. Analogously, lim R−1 N (λ; h0,R + fT ω ) = sup R−1 N (λ; h0,R + fT ω ) dP(ω). R→∞

R>0

Ω

Since T is a measure preserving automorphism, we get sup R−1 N (λ; h0,R + fT ω ) dP(ω) = sup R−1 N (λ; h0,R + fω ) dP(ω) R>0

Ω

R>0

Ω

which yields (2.2).

Our assumptions concerning Vω guarantee that the random ﬁeld fω = Vω (X, .) depending on the parameter X ∈ R2 , satisﬁes the hypotheses of Proposition 2.1. Moreover, if G = Z, then the function ,(λ; V (X, .)) is Z2 -periodic with respect to X ∈ R2 . In order to see this, one may apply Lemma 2.1 for T = Tk0 (see (1.3)) with k0 = (K, 0), K ∈ Z2 , and conclude that ,(λ; V (X + K, .)) = ,(λ; V (X, .)), λ ∈ R, X ∈ R2 , K ∈ Z2 .

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Note that the continuity of Vω (x) with respect to x ∈ R3 , and the continuity of ,(λ; f ) with respect to λ ∈ R, imply the continuity of ,(λ; V (X, .)) with respect to X ∈ R2 . Taking into account also its periodicity, we ﬁnd that ,(λ; V (X, .)), 2 X ∈ R2 , is uniquely determined by its values for X ∈ − 12 , 12 . Similarly, if G = R, the quantity ,(λ; V (X, .)) is independent of X ∈ R2 . In order to see this, one may apply Lemma 2.1 for T = Tx0 (see (1.3)) with x0 = (X, 0), X ∈ R2 , and conclude that ,(λ; V (X, .)) = ,(λ; V (0, .)), λ ∈ R, X ∈ R2 . Finally, using the elementary estimate N (λ; h0,R + Vω (X, .)) ≤

R 1/2 (λ + c0 )+ , X∈ π

we get 1 1/2 , X∈ ,(λ; v(X, .)) ≤ (λ + c0 )+ π For λ ∈ R set

(− 12 , 12 )2 ,(λ, V (X, .)) dX k(λ) := ,(λ, V (0, .)) if G = R.

1 1 − , 2 2

1 1 − , 2 2

if

2 , R > 0,

2 .

G = Z,

Obviously, k(λ) is continuous with respect to λ. Theorem 2.1. Let G = Z or G = R. Let Vω be a real G3 -ergodic random ﬁeld whose realizations are almost surely uniformly bounded and continuous. Assume in addition that Vω is G-ergodic in the direction of the magnetic ﬁeld. Then we have 1 lim b−1 Db (λ + b) = k(λ), λ ∈ R. (2.3) b→∞ 2π Remark: For deﬁniteness, we shall prove Theorem 2.1 in the case G = Z. The proof in the case G = R is quite similar and only simpler. The asymptotics as b → ∞ of the IDOS for the two-dimensional Schr¨ odinger operator with constant magnetic ﬁeld has been extensively investigated during the last two decades (see e.g. [Br.G.I], [Bro.H.L], [M.Pu], [Pu.Sc], [W]). As far as the authors are informed, no results concerning the strong magnetic ﬁeld asymptotics of the IDOS for the three-dimensional Schr¨ odinger operator considered in this paper, are known. Besides the quantity Db (λ + b) whose main asymptotic term is obtained in (2.3), we could consider more general quantities Db (λ2 + /b) − Db (λ1 + /b) with λj ∈ R, j = 1, 2, λ1 < λ2 , and / ∈ R. Recall that the numbers {(2q − 1)b)}q≥1 are called Landau levels. For this reason we shall refer to the asymptotics as b → ∞ of

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Db (λ2 + (2q − 1)b) − Db (λ1 + (2q − 1)b), q ∈ Z, q ≥ 1, as the asymptotics of the IDOS near the kth Landau level. Analogously, if / > 1 is not an odd integer, we shall refer to the asymptotics as b → ∞ of Db (λ2 + /b) − Db (λ1 + /b) as the asymptotics of the IDOS far from the Landau levels. Note that the case / < 1 is trivial since (1.8) implies Db (λ + /b) = 0, λ ∈ R, / < 1, provided that b is large enough. Since (2.3) entails 1 (k(λ2 ) − k(λ1 )) , 2π we can say that Theorem 2.1 concerns the asymptotics of the IDOS near the ﬁrst Landau level. The problems of obtaining the ﬁrst asymptotic term of the IDOS near the higher Landau levels and far from the Landau levels remain open as far as the methods used in this paper are not directly applicable to them. We hope to solve these problems in a future work. Here we would like to note that lim b−1 (Db (λ2 + b) − Db (λ1 + b)) =

b→∞

lim b−1 (Db (λ2 + /b) − Db (λ1 + /b)) =

b→∞

1 1/2 1/2 (λ , ) − (λ ) 2 + 1 + 2π 2

(2.4)

if / > 1 is an odd integer, and lim b−1/2 (Db (λ2 + /b) − Db (λ1 + /b)) =

b→∞

1 (λ2 − λ1 ) 4π 2

(/− (2q − 1))−1/2 ,

1≤q<(+1)/2

(2.5) if / > 1 is not an odd integer. Combining (2.4) (respectively, (2.5)) with (1.9) we obtain generically the correct asymptotic order of the IDOS near the higher Landau levels (respectively, far from the Landau levels). Finally, note that (1.9) implies immediately lim b−3/2 Db (λ + /b) = D1 (/), λ ∈ R, / > 1,

b→∞

which however yields only the rough estimate Db (λ2 + /b) − Db (λ1 + /b) = o(b3/2 ),

b → ∞.

The methods we apply are relatively simple. First of all, we give an equivalent representation of Db (λ + b) which is more convenient for our purposes. Namely, on D(H0 (b)) we deﬁne the operator ˜ ω, λ, R) := H0 (b) − b + (Vω − λ − 1)1 R R 3 , b > 0, λ ∈ R, ω ∈ Ω, (2.6) H(b, (− , ) 2

2

(see (1.1) for the deﬁnition of H0 (b)), which has purely discrete negative spectrum, and show that almost surely we have ˜ ω, λ, R)), Db (λ + b) = lim R−3 N (−1; H(b, R→∞

provided that λ + b is a continuity point of Db (see Proposition 4.1 below).

(2.7)

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Moreover, we apply the Birman-Schwinger principle (see [B]), and similarly to [R 1 – 3] we employ the Kac-Murdock-Szeg¨o theorem in order to reduce the study of the asymptotics as b → ∞ of Db (λ + b) to the asymptotic analysis as R → ∞ and b → ∞ of the traces of the powers of certain trace-class operators tb,R (see (3.4) below). The Birkhoﬀ-Khintchine ergodic theorem plays a crucial rˆ ole in this analysis. The paper is organized as follows. In Section 3 we investigate the asymptotics of R → ∞ and b → ∞ of Tr tlb,R , l ≥ 1, and some related traces. Section 4 contains auxiliary results. In particular, we prove (2.7) as well as an analogous formula concerning ,(λ, V (X, .)) (see (2.1)). Finally, the proof of Theorem 2.1 can be found in Section 5.

3 Trace asymptotics ∂ Let H0 (b) := i ∂x −

by 2

2

∂ + i ∂y +

2

bx 2 2

be the selfadjoint operator deﬁned origi-

The spectrum of H0 (b) coincides with nally on C0∞ (R2 ) and then closed in L (R2 ). the set of the Landau levels, i.e. σ(H0 (b)) = ∞ q=1 {(2q − 1)b}, and the multiplicity of each eigenvalue (2q − 1)b, q ≥ 1, is inﬁnite. Denote by pb : L2 (R2 ) → L2 (R2 ) the orthogonal projection onto the eigenspace of H0 (b) associated with the ﬁrst Landau level b. In other words, pb w = w implies w ∈ D(H0 (b)) and H0 (b)w = bw. It is well-known that Pb (x, y; x , y )w(x , y ) dx dy , w ∈ L2 (R2 ), (pb w)(x, y) = R2

with b b 2 2 Pb (x, y; x , y ) := exp − (x − x ) + (y − y ) + 2i(xy − yx ) . 2π 4

Set

Pb :=

(3.1)

⊕

R

pb dz.

Evidently, Pb : L2 (R3 ) → L2 (R3 ) is an orthogonal projection, (Pb u)(x, y, z) =

R2

Pb (x, y; x , y )u(x , y , z) dx dy ,

and Pb commutes with H0 (b) and H0 (b)Pb u =

−

∂ ∂z .

u ∈ L2 (R3 ),

Moreover, we have

∂2 + b Pb u, ∂z 2

u ∈ D(H0 (b)),

(3.2)

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−1/2 ∂2 (see (1.1)). Deﬁne the operator r := − ∂z , bounded and selfadjoint in 2 + 1 L2 (R3 ). Evidently, 1 (r u)(x, y, z) = 2

e−|z−z | u(x, y, z ) dz ,

2

R

u ∈ L2 (R3 ).

Moreover, the operators Pb and r commute. Fix λ ∈ R and for brevity set Qω (x) ≡ Qω (x; λ) := Vω (x) − λ − 1,

x ∈ R3 ,

ω ∈ Ω.

Almost surely we have |Qω (x)| ≤ c1 , x ∈ R3 ,

(3.3)

with c1 := c0 + |λ + 1| (see (1.4)). Put Q˜ω ,R (x) := Qω (x)1(− R , R )3 (x), 2

2

x ∈ R3 .

Deﬁne the operator tb,R (Qω ) := Pb rQ˜ω ,R rPb , 2

(3.4)

3

compact and selfadjoint in L (R ). It is easy to check that we have bR3 ˜ ω,R |1/2 22 = b Pb r|Q c1 , |Qω (x)| dx ≤ 3 4π (− R2 , R2 ) 4π

(3.5)

where .2 denotes the Hilbert-Schmidt norm. Therefore, tb,R (Qω ) is a trace-class operator. Set b M1 (b) := Qω (X, z) dXdz . (3.6) E 4π (− 12 , 12 )3 Let l ≥ 2. Put b E 2π

Ml (b) :=

− 12 , 12

(

3

)

Qω (X1 , z1 )

R3(l−1)

Πls=2 Qω (X1 + b−1/2 Xs , z1 + zs )

ψl (z2 , . . . , zl )Ψl (X2 , . . . , Xl )dX2 . . . dXl dz2 . . . dzl dX1 dz1 ) , where

l−1 1 |zs+1 − zs | , ψl (z2 , . . . , zl ) := l exp −|z2 | − |zl | − 2 s=2

Ψl (X2 , . . . , Xl ) ≡ Ψl (x2 , y2 , . . . , xl , yl ) := and

(3.7)

1 exp {−Φl (x2 , y2 , . . . , xl , yl )}, (2π)l−1

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Φl (x2 , y2 , . . . , xl , yl ) := 1 2 x2 + y2 2 + xl 2 + yl 2 + 4 l−1 (xs+1 − xs )2 + (ys+1 − ys )2 + 2i(xs+1 ys − ys+1 xs )

;

s=2

if l = 2, then the sums with respect to s in the formulae deﬁning ψl and Φl , should be omitted. Note that ψl ∈ L1 (Rl−1 ), Ψl ∈ L1 (R2(l−1) ). Hence, (3.3) implies that the integral deﬁning Ml (b) is absolutely convergent. Proposition 3.1. Almost surely we have lim R−3 Tr tb,R (Qω )l = Ml (b),

R→∞

l ≥ 1,

(3.8)

the operator tb,R (Qω ) being deﬁned in (3.4). Proof. We shall prove (3.8) in the generic case l ≥ 2. l l It is easy to verify that Tr tb,R (Qω )l = Tr Pb r2 Q˜ω ,R , and that Tr Pb r2 Q˜ω ,R can be written in a standard way as an integral over R3l of the diagonal value of l the integral kernel of the operator Pb r2 Q˜ω ,R . Hence, we have Tr tb,R (Qω )l = 1 l −|zs+1 −zs | Pb (Xs+1 , Xs )e Πls=1 Q˜ω ,R (Xs , zs )dX1 . . . dXl dz1 . . . dzl Πs=1 2 R3l

where the notation Πls=1 means that in the product of l factors the variables Xl+1 and zl+1 , should be set equal respectively to X1 and z1 . Changing the variables X1 = X1 , Xs = X1 + b−1/2 Xs , s = 2, . . . , l, z1 = z1 , zs = z1 + zs , s = 2, . . . , l, we get Tr tb,R (Qω )l = b Qω (X1 , z1 ) Πls=2 Q˜ω ,R (X1 + b−1/2 Xs , z1 + zs ) 2π (− R2 , R2 )3 3(l−1) R

ψl (z2 , . . . , zl )Ψl (X2 , . . . , Xl )dX2 . . . dXl dz2 . . . dzl dX1 dz1 . (3.9) Our next step is to show that if we replace in (3.9) all the functions Q˜ω ,R by Qω , the error will be of order o(R3 ) as R → ∞. More precisely, we set ˜ l,R (b) := M

810

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b Q (X , z ) Πls=2 Qω (X1 + b−1/2 Xs , z1 + zs ) ω 1 1 2π (− R2 , R2 )3 R3(l−1) ψl (z2 , . . . , zl )Ψl (X2 , . . . , Xl )dX2 . . . dXl dz2 . . . dzl dX1 dz1 ,

l ≥ 2,

write ˜ l,R (b) + El (R, b, ω), l ≥ 2, Tr (tb,R (Qω ))l = M

(3.10)

and shall demonstrate that almost surely we have lim R−3 El (R, b, ω) = 0.

(3.11)

R→∞

Evidently, El (R, b, ω) admits the estimate

b l 3 c R 2π 1 (− 12 , 12 )3

|El (R, b, ω)| ≤

R3(l−1)

Πls=2 ER,b (X1 , . . . , Xl , z1 , . . . , zl )

ψl (z2 , . . . , zl )|Ψl (X2 , . . . , Xl )|dX2 . . . dXl dz2 . . . dzl dX1 dz1 where ER,b (X1 , . . . , Xl , z1 , . . . , zl ) := 1 − Πls=2 1(− R , R )3 (RX1 + b−1/2 Xs , Rz1 + zs ) . 2

Since ψl Ψl ∈ L (R 1

3(l−1)

2

), ER,b L∞

(− 12 , 12 )3 ×R3(l−1)

= 1 for every b > 0 and

R > 0, and lim ER,b (X1 , . . . , Xl , z1 , . . . , zl ) = 0

R→∞

3 for almost every (X1 , . . . , Xl , z1 , . . . , zl ) ∈ − 12 , 12 × R3(l−1) , the dominated convergence theorem yields (3.11). Set L = L(R) = ent R 2 where ent(x) denotes the integer part of x ∈ R. Obviously, ˜ l,R (b) = (2L + 1)−3 M ˜ l,2L+1 (b) + o(1), R → ∞. R−3 M

(3.12)

Let j = (J, j) ∈ Z3 , J ∈ Z2 , j ∈ Z. Introduce the random variables b γl,j (ω, b) := Qω (X1 +J, z1 +j) Πls=2 Qω (X1 +J+b−1/2 Xs , z1 +j+zs ) 2π (− 12 , 12 )3 R3(l−1) ψl (z2 , . . . , zl )Ψl (X2 , . . . , Xl )dX2 . . . dXl dz2 . . . dzl dX1 dz1 , l ≥ 2. Set

ΓL = j = (j1 , j2 , j3 ) ∈ Z3 ||js | ≤ L, s = 1, 2, 3 , L ∈ N, L ≥ 1.

It is easy to verify that ˜ l,2L+1 = M

j∈ΓL

γl,j (ω, b).

(3.13)

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On the other hand, it is obvious that for each k ∈ Z3 we have γl,j+k (ω, b) = γl,j (Tk ω, b), j ∈ Z3 , ω ∈ Ω, (see (1.3)). Hence, the sequence {γl,j }j∈Z3 is a Z3 -ergodic random ﬁeld. Therefore, we can apply the Birkhoﬀ-Khintchine ergodic theorem (see e.g. [K, Theorem 2, Section 3.2] or [P.Fi, Proposition 1.7, p.18]). As a result we get almost surely γl,j (ω, b) = E(γl,0 ) ≡ Ml (b), l ≥ 2. (3.14) lim (2L + 1)−3 L→∞

j∈ΓL

Now the combination of (3.10)-(3.14) yields (3.8) with l ≥ 2. The proof in the case l = 1 is similar but much simpler. 2 Fix X ∈ − 21 , 12 . Introduce the operator

−1/2 −1/2 d2 d2 Qω (X, z)1(− R , R ) (z) − 2 + 1 τR (Qω (X)) := − 2 + 1 2 2 dz dz

(3.15)

which is compact and selfadjoint in L2 (Rz ), and depends on the parameters X ∈ 1 1 2 − 2 , 2 and ω ∈ Ω (see (3.4) in order to compare τR (Qω (X)) with the operator tb,R (Qω )). It is easy to check that τR (Qω (X)) is a trace-class operator. 2 For X ∈ − 12 , 12 set 1 Qω (X, z) dz , µ1 (X) := E 2 (− 12 , 12 ) µl (X) :=

E

Qω (X, z1 ) (− 12 , 12 )

Πls=2 Qω (X, z1

Rl−1

+ zs )ψl (z2 , . . . , zl )dz2 . . . zl dz1

, l ≥ 2,

(see (3.6)–(3.7) in order to compare µl (X) with the quantities Ml (b), l ≥ 1). By analogy with Proposition 3.1 we can demonstrate the following Proposition 3.2. Almost surely we have lim R

R→∞

−1

Tr τR (Qω (X)) = µl (X), l ≥ 1, X ∈ l

1 1 − , 2 2

2 .

Set ml := (− 1 , 1 )2 µl (X)dX, l ≥ 1. 2 2 Proposition 3.3. We have lim b−1 Ml (b) =

b→∞

1 ml , l ≥ 1. 2π

(3.16)

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b Proof. Obviously M1 (b) = 2π m1 which yields immediately (3.16) with l = 1. Let l ≥ 2. Using the fact that p1 is an orthogonal projection, and taking into account the explicit form of its kernel P1 (see (3.1)), we get

R2(l−1)

Ψl (X2 , . . . , Xl )dX2 . . . dXl =

2π R2(l−1)

P1 (0; X2 )P1 (X2 ; X3 ) . . . P1 (Xl−1 ; Xl )P1 (Xl ; 0)dX2 . . . dXl = 2πP1 (0; 0) = 1.

(cf. [R 2, pp.16-17]). Hence, we have ml = E

(− 12 , 12 )

Qω (X1 , z1 ) 3

R3(l−1)

ψl (z2 , . . . , zl )Ψl (X2 , . . . , Xl )

Πls=2 Qω (X1 , z1 + zs )dX2 . . . dXl dz2 . . . dzl dX1 dz1 . Then, obviously, 1 1 E b−1 Ml (b) − ml = 2π 2π

(− 12 , 12 )3

Qω (X1 , z1 )

ψl (z2 , . . . , zl )Ψl (X2 , . . . , Xl )

R3(l−1)

Πls=2 Qω (X1 + b−1/2 Xs , z1 + zs ) − Πls=2 Qω (X1 , z1 + zs ) dX2 . . . dXl dz2 . . . dzl dX1 dz1 ) .

(3.17)

Since Qω is almost surely uniformly bounded and continuous, while ψl ∈ L1 (Rl−1 ) and Ψl ∈ L1 (R2(l−1) ), we ﬁnd that it follows from the dominated convergence theorem that (3.17) implies (3.16). Combining Propositions 3.1, 3.2, and 3.3, we get the following Corollary 3.1. For each l ≥ 1 almost surely the limits lim lim b−1 R−3 Tr tb,R (Qω )l

b→∞ R→∞

and

1 −1 Tr τR (Qω (X))l dX lim R 2π R→∞ (− 12 , 12 )2

exist, coincide, and are non-random.

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4 Auxiliary results ˜ ω, λ, R) be the operator deﬁned in (2.6). Let λ + b with Proposition 4.1. Let H(b, λ ∈ R and b > 0, be a continuity point of Db . Then (2.7) is valid. Proof. First, note that we have N (λ + b; H0,R (b) + Vω ) = N (−1; H0,R (b) − b + Vω − λ − 1). The minimax principle implies ˜ ω, λ, R)). N (−1; H0,R (b) − b + Vω − λ − 1) ≤ N (−1; H(b, Therefore, ˜ ω, λ, R)) ≥ lim inf R−3 N (λ + b; H0,R (b) + Vω ) = lim inf R−3 N (−1; H(b, R→∞

R→∞

lim R−3 N (λ + b; H0,R (b) + Vω ) = Db (λ + b).

(4.1)

R→∞

Further, ﬁx R > 0, R0 ∈ (0, R), put O1 = O1,R =

3 R R − , , 2 2

3 R − R0 R − R0 , O2 = O2,R,R0 = R3 \ − , 2 2

and pick two functions ϕ1 and ϕ2 satisfying the following properties : i) ϕj ∈ C ∞ (R3 ), j = 1, 2; ii) supp ϕj ⊆ Oj , j = 1, 2; iii) ϕ21 (x) + ϕ22 (x) = 1 for every x ∈ R3 ; iv) |∇ϕj (x)| ≤ c2 R0−1 for every x ∈ R3 with c2 > 0 which is independent of x, R, and R0 . Introduce the selfadjoint operator ˜ D (b, ω, λ, R, R0 ) := H

b∧x i∇ + 2

2 − b + (Vω − λ − 1)1(− R , R )3 2

2

whose quadratic form is deﬁned originally on C0∞ (O2 ), and then is closed in L2 (O2 ) (cf. (1.1) and (2.6)). Then the “magnetic” version of the so-called ISM localization formula (see [C.F.K.S, Section 3.1]) yields ˜ ω, λ, R)) ≤ N (−1; H0,R (b) − b + Vω − λ − 1 − N (−1; H(b, |∇ϕj |2 )+ j=1,2

˜ D (b, ω, λ, R, R0 ) − N (−1; H

j=1,2

|∇ϕj |2 ).

(4.2)

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Obviously, N (−1; H0,R (b) − b + Vω − λ − 1 −

|∇ϕj |2 ) ≤ N (λ + b + 2c22 R0−2 ; H0,R (b) + Vω ).

j=1,2

(4.3) Choose a sequence {εr }r≥1 such that εr > 0, r ≥ 1, limr→∞ εr = 0, and λ + b + εr , √ √ r ≥ 1, are continuity points of Db . Fix r ≥ 1 and set R0 = 2c2 / εr . Then we have lim sup R−3 N (λ + b + 2c22 R0−2 ; H0,R (b) + Vω ) = R→∞

lim R−3 N (λ + b + εr ; H0,R (b) + Vω ) = Db (λ + b + εr ), r ≥ 1.

R→∞

(4.4)

The combination of (4.3) and (4.4) yields lim sup R−3 N (−1; H0,R (b) − b + Vω − λ − 1 − R→∞

|∇ϕj |2 ) ≤ Db (λ + b + εr ), r ≥ 1.

j=1,2

(4.5) On the other hand, by the minimax principle we have ˜ D (b, ω, λ, R, R0 ) − N (−1; H

|∇ϕj |2 ) ≤ N (−1; H0 (b) − b + W )

(4.6)

j=1,2

where 

W = Wω,λ,R,R0 = (Vω − λ − 1)1(− R , R )3 − 2

2

 |∇ϕj |2  1O2 .

j=1,2

Arguing as in [R 1, Section 5], we deduce the estimate N (−1; H0 (b) − b + W ) ≤ c3

R3

|W |3/2 dx ≤ c4 R3 − (R − R0 )3

(4.7)

where the quantities c3 and c4 may depend on b, λ, and R0 , but are independent of R. The combination of (4.6) and (4.7) yields ˜ D (b, ω, λ, R, R0 ) − lim R−3 N (−1; H

R→∞

|∇ϕj |2 ) = 0.

j=1,2

Putting together (4.2), (4.5), and (4.8), we obtain ˜ ω, λ, R)) ≤ Db (λ + b + εr ), r ≥ 1. lim sup R−3 N (−1; H(b, R→∞

(4.8)

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Letting r → ∞ (hence, εr → 0), and bearing in mind that λ + b is a continuity point of Db , we get ˜ ω, λ, R)) ≤ Db (λ + b). lim sup R−3 N (−1; H(b,

(4.9)

R→∞

Now, the combination of (4.1) and (4.9) yields (2.7).

For b > 0, ω ∈ Ω, λ ∈ R, and R > 0, introduce the operator Tb,ω,λ,R := (H0 (b) − b + 1)−1/2 (Vω − λ − 1)1(− R , R )3 (H0 (b) − b + 1)−1/2 , (4.10) 2

2

compact and selfadjoint in L2 (R3 ). Corollary 4.1. Let λ + b be a continuity point of Db . Then almost surely we have Db (λ + b) = lim R−3 n− (1; Tb,ω,λ,R ), R→∞

the quantity n− (s; T ) being deﬁned in (1.6). Proof. It suﬃces to note that the Birman-Schwinger principle (see [B, Lemma 1.1]) ˜ ω, λ, R)) = n− (1; Tb,ω,λ,R ), and then to apply Proposition 4.1. implies N (−1; H(b, 1 1 3 Fix X ∈ − 2 , 2 and λ ∈ R. For s ∈ R, s = 0, set 1 λ+1 − 1; − V (X, .) ,˜λ (s; X) := −sign(s) , − s s (see (2.1) for the deﬁnition of ,(λ; f )). Since ,(λ, V (X, .)) is a continuous function with respect to λ ∈ R (see Proposition 2.1), and Vω is uniformly bounded (see (1.4)), we ﬁnd that ,˜λ (s; X) is a continuous function with respect to s ∈ R \ {0} for any ﬁxed λ ∈ R. Moreover, 2 1 1 . ,˜λ (−1, X) = ,(λ, V (X, .)), X ∈ − , 2 2 2 For R > 0, ω ∈ Ω, λ ∈ R, X ∈ − 12 , 12 , and s ∈ R \ {0}, set n− (−s; τR (Vω (X, .) − λ − 1)) if s < 0, νR,ω,X,λ (s) = −n+ (s; τR (Vω (X, .) − λ − 1)) if s > 0, the operator τR being deﬁned in (3.15). 2 Proposition 4.2. For every λ ∈ R, X ∈ − 12 , 12 , and s ∈ R \ {0}, we have ,˜λ (s; X) = lim R−1 νR,ω,X,λ (s) R→∞

almost surely.

(4.11)

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Proof. We shall prove (4.11) in the case s < 0. In this case, by Proposition 2.1 ,˜λ (s, X) = lim R−1 N (− R→∞

λ+1 1 − 1; h0,R − Vω (X, .)) = s s

1 lim R−1 N (−1; h0,R − (Vω (X, .) − λ − 1)). s

R→∞

(4.12)

On H2 (R) deﬁne the operator 2 ˜ s,ω,X,λ,R := − d − 1 (Vω (X, .) − λ − 1)1 R R , R > 0. h (− 2 , 2 ) dz 2 s

Applying the minimax principle, and bearing in mind that h0,R − 1s (Vω (X, .)−λ−1) ˜ s,ω,X,λ,R are second-order ordinary diﬀerential operators, we get and h ˜ s,ω,X,λ,R ) − N (−1; h0,R − 1 (Vω (X, .) − λ − 1)) ≤ 2. 0 ≤ N (−1; h s Therefore, (4.12) implies ˜ s,ω,X,λ,R ). ,˜λ (s, X) = lim R−1 N (−1; h R→∞

(4.13)

On the other hand, by the Birman-Schwinger principle we have ˜ s,ω,X,λ,R ) = n− (−s; τR (Vω (X, .) − λ − 1)) ≡ νR,ω,X,λ (s). N (−1; h

(4.14)

Putting together (4.13) and (4.14), we obtain (4.11) with s < 0. The proof in the case s > 0 is completely analogous. For λ ∈ R and s ∈ R \ {0} set ˜ ,˜λ (s; X)dX. kλ (s) := (− 12 , 12 )2

(4.15)

Obviously, k˜λ (s) is continuous with respect to s. Moreover, k˜λ (−1) = k(λ), λ ∈ R.

(4.16)

Remark. The function νR,ω,X,λ (s) of the variable s ∈ R \ {0} is non-negative on (−∞, 0), non-positive on (0, ∞), and non-decreasing on (−∞, 0) and (0, ∞). By (4.11) and (4.15), the functions ,˜λ (s; X) and k˜λ (s) have the same properties. Corollary 4.2. For each λ ∈ R almost surely we have 1 lim lim b−1 R−3 Tr tb,R (Vω − λ − 1)l = sl dk˜λ (s), l ≥ 1, b→∞ R→∞ 2π R the operator tb,R being deﬁned in (3.4).

(4.17)

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Proof. By Corollary 3.1, we have lim lim b−1 R−3 Tr tb,R (Vω − λ − 1)l =

b→∞ R→∞

1 −1 lim R Tr τR (Vω (X, .) − λ − 1)l dX = 2 2π R→∞ (− 12 , 12 ) 1 lim R−1 sl dνR,ω,X,λ (s) dX. 1 1 2 2π R→∞ (− 2 , 2 ) R

Proposition 4.2 easily implies that sl dνR,ω,X,λ (s) dX = lim R−1 2 R→∞ (− 12 , 12 ) R l s d˜ ,λ (s; X) dX = sl dk˜λ (s), R (− 12 , 12 )2 R

(4.18)

(4.19)

and the combination of (4.18) and (4.19) yields (4.17). Corollary 4.3. For each λ ∈ R and s < 0 we have lim lim b−1 R−3 n− (−s; tb,R (Vω − λ − 1)) =

b→∞ R→∞

1 ˜ kλ (s). 2π

(4.20)

Proof. We have tR,b (Vω − λ − 1) ≤ c1 (see (3.3)). Moreover, k˜λ (s) = 0 if |s| > c1 . Hence we can apply the Kac-Murdock-Szeg¨ o theorem (see [R 1, Section 3]) and, taking into account the continuity of k˜λ (.), to conclude that (4.20) follows from (4.17).

5 Proof of Theorem 2.1 In order to prove (2.3), it suﬃces to show that for each sequence {bj }j≥1 such that bj → ∞ as j → ∞, we have lim b−1 j Dbj (λ + bj ) =

j→∞

1 k(λ), λ ∈ R. 2π

(5.1)

− + ± Fix two sequences {λ± m }m≥1 such that λm < λ < λm , m ≥ 1, limm→∞ λm = λ, ± and λm +bj are continuity points of Dbj for all m ≥ 1 and j ≥ 1. Then by Corollary 4.1 we have −1 −3 lim sup b−1 n− (1; Tbj ,ω,λ+ ), j Dbj (λ + bj ) ≤ lim sup lim bj R m ,R j→∞

j→∞

R→∞

−1 −3 n− (1; Tbj ,ω,λ− ), lim inf b−1 j Dbj (λ + bj ) ≥ lim inf lim bj R m ,R j→∞

j→∞ R→∞

the operator Tb,ω,λ,R being deﬁned in (4.10). By the minimax principle

(5.2) (5.3)

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Ann. Henri Poincar´ e

−3 lim inf lim b−1 n− (1; Tbj ,ω,λ− )≥ j R m ,R j→∞ R→∞

−3 n− (1; Pbj Tbj ,ω,λ− Pbj ). lim inf lim inf b−1 j R m ,R j→∞

R→∞

(5.4)

Note that the operator Pb Tb,ω,λ,R Pb coincides with the operator tb,R (Vω − λ − 1) (see (4.10), (3.4), and (3.2)). Hence, Corollary 4.3 and (4.16) entail −3 lim inf lim inf b−1 n− (1; Pbj Tbj ,ω,λ− Pbj ) ≥ j R m ,R j→∞

lim

R→∞

lim b−1 R−3 n− (1; tR,b (Vω − λ− m − 1)) =

b→∞ R→∞

1 ˜ 1 k − (−1) = k(λ− m ). 2π λm 2π

(5.5)

The combination of (5.3)-(5.5) yields lim inf b−1 j Dbj (λ + bj ) ≥ j→∞

1 k(λ− m ). 2π

(5.6)

On the other hand, we have Tbj ,ω,λ+ = Pbj Tbj ,ω,λ+ Pbj + (Id − Pbj )Tbj ,ω,λ+ (Id − Pbj ) m ,R m ,R m ,R +2Re Pbj Tbj ,ω,λ+ (Id − Pbj ). m ,R

(5.7)

Set T˜b,ω,λ,R := (Id − Pb )(H0 (b) − b + 1)−1/2 |Vω − λ − 1|1(− R , R )3 (H0 (b) − b + 1)−1/2 (Id − Pb ). 2

2

Applying the elementary operator inequalities (Id − Pbj )Tbj ,ω,λ+ (Id − Pbj ) ≥ −T˜bj ,ω,λ+ , m ,R m ,R −2 ˜ Tbj ,ω,λ+ 2Re Pbj Tbj ,ω,λ+ (Id − Pbj ) ≥ −ε2 tbj ,R (|Vω − λ+ , ε > 0, m − 1|) − ε m ,R m ,R

we ﬁnd that (5.7) implies 2 + n− (1; Tbj ,ω,λ+ ) ≤ n− (1; tbj ,R (Vω − λ+ m − 1) − ε tbj ,R (|Vω − λm − 1|)) + m ,R

n+ (1; (1 + ε−2 )T˜bj ,ω,λ+ ), ε > 0. m ,R

(5.8)

It is easy to verify the estimate ≤ (bj + 1)−1 (c0 + |λ+ T˜bj ,ω,λ+ m + 1|), R > 0. m ,R

(5.9)

Fix ε > 0 and assume that bj is so large that (1 + ε−2 )(bj + 1)−1 (c0 + |λ+ m | + 1) < 1. Then (5.9) entails n+ (1; (1 + ε−2 )T˜bj ,ω,λ+ ) = 0. (5.10) m ,R

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By the Weyl inequalities for the eigenvalues of compact selfadjoint operators we have 2 + n− (1; tbj ,R (Vω − λ+ m − 1) − ε tbj ,R (|Vω − λm − 1|)) ≤ + n− (1 − ε; tbj ,R (Vω − λ+ m − 1)) + n+ (1; εtbj ,R (|Vω − λm − 1|)), ε ∈ (0, 1).

(5.11)

The estimate (3.5) implies 3 n+ (1; εtbj ,R (|Vω − λ+ m − 1|)) ≤ c5 εbj R

(5.12)

with c5 := (c0 + |λ+ m + 1|)/4π. Now, the combination of (5.8), (5.10), (5.11), and (5.12) yields −3 lim sup lim b−1 n− (1; Tbj ,ω,λ+ )≤ j R m ,R j→∞

R→∞

−3 n− (1 − ε; tbj ,R (Vω − λ+ lim sup lim sup b−1 m − 1)) + c5 ε, ε ∈ (0, 1). j R j→∞

(5.13)

R→∞

By Corollary 4.3 −3 n− (1 − ε; tbj ,R (Vω − λ+ lim sup lim sup b−1 m − 1)) = j R j→∞

R→∞

lim lim b−1 R−3 n− (1−ε; tb,R (Vω −λ+ m −1)) =

b→∞ R→∞

1 ˜ k + (−1+ε), ε ∈ (0, 1). (5.14) 2π λm

The combination of (5.2), (5.13), and (5.14) yields lim sup b−1 j Dbj (λ + bj ) ≤ j→∞

1 ˜ k + (−1 + ε) + c5 ε, ε ∈ (0, 1). 2π λm

Letting ε ↓ 0, we get lim sup b−1 j Dbj (λ + bj ) ≤ j→∞

1 ˜ 1 k + (−1) = k(λ+ m ). 2π λm 2π

(5.15)

Letting m → ∞ (hence, λ± m → λ) in (5.6) and (5.15), we arrive at (5.1). Acknowledgements This work was done during the second author’s visits to the Ruhr University in 1998 and 1999. Financial support of Sonderforschungsbereich 237 “Unordnung und grosse Fluktuationen”, and of Grant MM 612/96 of the Bulgarian Science Foundation, is gratefully acknowledged. The authors thank Prof. H. Leschke and Dr. E. Giere for several stimulating discussions. Acknowledgements are also due to the referee whose valuable remarks contributed to the improvement of the exposition.

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References [A.H.S]

J. Avron, I. Herbst, B. Simon, Schr¨ odinger operators with magnetic ﬁelds. I. General interactions, Duke. Math. J. 45 (1978), 847–883.

[B]

˘ Birman, On the spectrum of singular boundary value problems, M.S. Mat. Sbornik 55 (1961), 125–174 (Russian); Engl. transl. in Amer. Math. Soc. Transl., (2) 53 (1966), 23–80.

[Br.G.I]

E. Br´ezin, D.J. Gross, C. Itzykson, Density of states in the presence of a strong magnetic ﬁeld and random impurities, Nucl.Phys. B 235 (1984), 24–44.

[Bro.H.L]

K. Broderix, D. Hundertmark, H. Leschke, Self-averaging, decomposition and asymptotic properties of the density of states for random Schr¨ odinger operators with constant magnetic ﬁeld, In: Path Integrals from meV to MeV, Tutzing ’92, World Scientiﬁc, Singapore (1993), 98–107.

[CdV]

Y. Colin de Verdi`ere, L’asymptotique de Weyl pour les bouteilles magn´etiques, Commun.Math.Phys. 105 (1986), 327–335.

[C.F.K.S]

H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, Schr¨ odinger operators, with application to quantum mechanics and global geometry. Texts and Monographs in Physics. Springer-Verlag. Berlin etc.(1987).

[E.K.Sch.S]

H. Englisch, W. Kirsch, M. Schr¨ oder, B. Simon, Random Hamiltonians ergodic in all but one direction, Commun.Math.Phys. 128 (1990), 613-625.

[K]

W. Kirsch, Random Schr¨ odinger operators. In: Schroedinger operators, Proc. Nord. Summer Sch. Math., Sandbjerg Slot, Soenderborg/Denmark 1988, Lect. Notes Phys. 345, (1989) 264–370.

[M.Pu]

N. Macris, J.V. Pul´e, Density of states of random Schr¨ odinger operators with a uniform magnetic ﬁeld, Lett. Math. Phys. 24 (1992), 307–321.

[Ma]

H. Matsumoto, On the integrated density of states for the Schr¨ odinger operators with certain random electromagnetic potentials, J. Math. Soc. Japan 45 (1993), 197–214.

[P.Fi]

L. Pastur, A. Figotin, Spectra of Random and Almost-Periodic Operators. Grundlehren der Mathematischen Wissenschaften 297. Springer-Verlag, Berlin etc. (1992).

[Pu.Sc]

J.V. Pul´e, M. Scrowston, Inﬁnite degeneracy for a Landau Hamiltonian with Poisson impurities, J. Math. Phys. 38 (1997), 6304–6314.

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[R 1]

G.D. Raikov, Eigenvalue asymptotics for the Schr¨ odinger operator in strong constant magnetic ﬁelds, Commun. P.D.E. 23 (1998), 1583– 1620.

[R 2]

G.D. Raikov, Eigenvalue asymptotics for the Dirac operator in strong constant magnetic ﬁelds, Math.Phys.Electr.J., 5 (1999), No.2, 22 pp.

[R 3]

G.D. Raikov, Eigenvalue asymptotics for the Pauli operator in strong non-constant magnetic ﬁelds, Ann.Inst.Fourier, 49, (1999), 1603– 1636.

[U]

N. Ueki, On spectra of random Schr¨ odinger operators with magnetic ﬁelds, Osaka J. Math. 31 (1994), 177–187.

[W]

W.-M. Wang, Asymptotic expansion for the density of states of the magnetic Schr¨ odinger operator with a random potential, Commun. Math. Phys. 172 (1995), 401–425.

W.Kirsch Department of Mathematics Ruhr University Universit¨ atsstrasse, 150 44780 Bochum, Germany E-mail: [email protected] G.D.Raikov Institute of Mathematics and Informatics Bulgarian Academy of Sciences Acad.G.Bonchev Str., bl. 8 1113 Soﬁa, Bulgaria E-mail: [email protected] Communicated by Michael Aizenman submitted 19/09/99, accepted 28/02/2000

To access this journal online: http://www.birkhauser.ch

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Annales Henri Poincar´ e

Lifshitz Tail for 2D Discrete Schr¨ odinger Operator with Random Magnetic Field Shu Nakamura Abstract. Lifshitz tail for 2 dimensional discrete Schr¨ odinger operator with Andersontype random magnetic ﬁeld is proved. We ﬁrst prove local energy estimates for deterministic discrete magnetic Schr¨ odinger operators, and then follow the large deviation argument of Simon [6].

1 Introduction We ﬁrst deﬁne our Hamiltonian. Let F be the set of unit squares with the vertices in Z2 , and let E be the set of edges, i.e., F = [x1 , x1 + 1] × [x2 , x2 + 1] x1 , x2 ∈ Z , E = (x, y) x, y ∈ Z2 , |x − y| = 1 . For e = (x, y) ∈ E, we write e = (y, x). For f ∈ F, we denote the boundary of f by ∂f ⊂ E, i.e., if f = [x1 , x1 + 1] × [x2 , x2 + 1] then ∂f = (x, x + δ1 ), (x + δ1 , x + δ1 + δ2 ), (x + δ1 + δ2 , x + δ2 ), (x + δ2 , x) , where δ1 = (1, 0) and δ2 = (0, 1) ∈ Z2 . Let A(e) be a function of e ∈ E with values in R/(2πZ) such that A(e) = −A(e),

e ∈ E.

The discrete magnetic Schr¨ odinger operator H on Z2 with a vector potential A is deﬁned by ψ(x) − eiA((x,y)) ψ(y) , x ∈ Z2 , Hψ(x) = |x−y|=1

for ψ ∈ 2 (Z2 ). H is a bounded self-adjoint operator on 2 (Z2 ), and 0 ≤ H ≤ 8. For a given vector potential A, the magnetic ﬁeld is deﬁned by A(e), f ∈ F, B(f ) = e∈∂f

which is a function on F with values in R/(2πZ). It is well-known that the spectral properties of H depends only on B, not A. Namely, if A and A˜ induces the same

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˜ are unimagnetic ﬁeld B, then the corresponding Schr¨ odinger operators H and H tarily equivalent by a gauge transform (i.e., a unimodular multiplication operator on Z2 ). We suppose that B = Bw (f ), w ∈ Ω, is an identically distributed independent random variables (i.i.d.) with distribution dµ (on R/(2πZ)). We denote the probability space by Ω. Our principal example is the uniform distribution on R/(2πZ). Then H is an ergodic operator, and the spectrum is independent of w almost surely (a.s.). See, e.g., [2]. Moreover, the integrated density of states: k(E) = lim

L→∞

1 #{eigenvalues of H ΛL ≤ E} |ΛL |

exists a.s., where ΛL = x = (x1 , x2 ) ∈ Z2 |x1 |, |x2 | ≤ L , |ΛL | = (2L + 1)2 , and H ΛL is the Hamiltonian restricted on 2 (ΛL ). (See Appendix C for the proof.) We suppose the distribution dµ satisﬁes the following assumptions: Assumption A. (1) dµ is not point measure at 0, i.e., Bw is not identically zero. (2) There is C and a > 0 such that dµ([−ε, ε]) ≥ Cεa ,

0 ≤ ε ≤ π,

where we identify R/(2πZ) with [−π, π). Theorem 1. Let k(E) be the integrated density of states for H = Hw , and suppose Assumption A. Then lim log(− log k(E))/ log E = −1.

E↓0

Remark. (1) This is a natural analogue of the Lifshitz singularity of the integrated density of states for Schr¨ odinger operator with random potential (see, e.g., [6], [2], [4] and references therein). Lifshitz tail for magnetic Schr¨ odinger operator is recently studied by Ueki [8], where the random magnetic ﬁeld is supposed to be Gaussian random ﬁeld. Note that the space dimension d is 2 in our setting and the right hand side of the statement is −d/2. (2) Under our assumption, σ(Hw ) = [0, 8] a.s., and hence Theorem 1 describe the behavior of k(E) at the lower edge of the spectrum. Similar result holds at 8 = sup σ(H) with little modiﬁcations. In Section 2, we prove a simple local energy estimate for discrete magnetic Schr¨ odinger operators. It is an analogue of the Avron-Herbst-Simon estimate [1] for continuous magnetic Schr¨ odinger operators, and we also show that it is optimal. In the rest of the paper, we mimic the argument of Simon [6] to prove the Lifshitz tail. We deﬁne the Dirichlet-Neumann decoupling in Section 3 and prove Theorem 1 in

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Section 4. In Appendix A, we discuss a generalization of the local energy estimate of Section 2. It is not necessary in this paper, but is interesting in itself and maybe useful in the analysis of discrete magnetic Schr¨ odinger operators. In Appendix B, we give a proof of the spectral property of an example of Section 2. We give a proof of the existence of the integrated density of states in Appendix C. Notations. We denote the inner product of 2 (Z2 ) by ϕ(x)ψ(x), ϕ, ψ ∈ 2 (Z2 ).

ϕ|ψ = x∈Z2

P(·) and E(·) denote the probability and the expectation with respect to w ∈ Ω, respectively. For e ∈ E, i(e) and t(e) denote the initial point and the terminal point of e, respectively, i.e., i(e) = x,

t(e) = y,

if e = (x, y).

We write Z+ = {0, 1, 2 . . . }.

2 Local energy estimates for discrete magnetic Schr¨ odinger operators In this section, we consider deterministic magnetic Schr¨ odinger operators on Z2 . For a given magnetic ﬁeld B, we set WB (x) = (1 − cos(B(f )/4)), x ∈ Z2 . x∈f

Here we identify R/(2πZ) with [−π, π), and hence B(f )/4 ∈ [−π/4, π/4). Thus, in particular, √ 0 ≤ WB (x) ≤ 4(1 − 1/ 2), x ∈ Z2 . Theorem 2. Let H and WB as above. Then H ≥ WB , i.e.,

ψ|Hψ ≥ ψ|WB ψ = WB (x)|ψ(x)|2 , ψ ∈ 2 (Z2 ). x∈Z2

Remark. The motivation of this estimate comes from recent works by Higuchi and Shirai [3]. (See also Sunada [7] for related results on generalized Harper operators.) Proof. We note

ψ|Hψ =

|x−y|=1

=

|x−y|=1

ψ(x) ψ(x) − eiA((x,y)) ψ(y) ψ(y) ψ(y) − e−iA((x,y)) ψ(x)

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since A((y, x)) = −A((x, y)). We take the average of these expressions, and obtain

ψ|Hψ =

1 2

2 ψ(x) − eiA((x,y)) ψ(y) |x−y|=1

2 1 = ψ(i(e)) − eiA(e) ψ(t(e)) . 2 e∈E

Since each e ∈ E is an element of ∂f for only one f ∈ F, we may write

ψ|Hψ =

2 1 ψ(i(e)) − eiA(e) ψ(t(e)) . 2 f ∈F e∈∂f

Lemma 3. 2 B(f ) |ψ(x)|2 . ψ(i(e)) − eiA(e) ψ(t(e)) ≥ 2 1 − cos 4

e∈∂f

x∈f

Now Theorem 2 follows immediately. Proof of Lemma 3. Let f = [x1 , x1 + 1] × [x2 , x2 + 1],

x = (x1 , x2 ) ∈ Z2 ,

and we write u1 = ψ(x), u2 = ψ(x + δ1 ), u3 = ψ(x + δ1 + δ2 ), θ1 = A((x, x + δ1 )), θ2 = A((x + δ1 , x + δ1 + δ2 )), θ3 = A((x + δ1 + δ2 , x + δ2 )), θ4 = A((x + δ2 , x)).

u4 = ψ(x + δ2 ),

Then we have 2 ψ(i(e)) − eiA(e) ψ(t(e)) e∈∂f



2 −e−iθ1 = (u1 , u2 , u3 , u4 )   0 −eiθ4

−eiθ1 2 −e−iθ2 0

0 −eiθ2 2 −e−iθ3

  −e−iθ4 u1 u2  0     ≡ u∗ hθ u. −eiθ3  u3  u4 2

We set α1 = θ1 , α2 = θ1 + θ2 , α3 = θ1 + θ2 + θ3 , and   1   eiα1 . A= iα2   e iα3 e

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Then we have



2  −1 ∗ Ahθ A =   0 −eiB

827

 −1 0 −e−iB 2 −1 0   ≡ AB −1 2 −1  0 −1 2

since B = B(f ) = θ1 + θ2 + θ3 + θ4 . Note that AB is the Hamiltonian of the free discrete Schr¨odinger operator on the closed chain {0, 1, 2, 3} with the periodic boundary condition with an additional phase eiB . Thus the eigenvectors are given by vj = (1, eiµj , e2iµj , e3iµj ),

µj = (B + 2πj)/4,

j = 0, 1, 2, 3,

and the eigenvalues are λj = 2(1 − cos µj ). In particular, the lowest eigenvalue is λ0 = 2(1 − cos(B/4)) since B ∈ [−π, π). This implies AB ≥ 2(1 − cos(B/4)), and hence hθ ≥ 2(1 − cos(B/4)) to conclude the assertion. Example 1. Let b ∈ (−π, π) and let b, B(f ) = −b,

if x1 + x2 is even, if x1 + x2 is odd,

where f = [x1 , x1 + 1] × [x2 , x2 + 1]. In this case, WB (x) = 4(1 − cos(b/4)) for any x ∈ Z2 . H is solvable and we can show σ(H) = [4(1 − cos(b/4)), 4(1 + cos(b/4))]. This example shows that Theorem 1 is optimal. See Appendix B for the computation of this example. We will use the following (almost trivial) analogue of the Kato inequality: Lemma 4.

ψ|Hψ ≥ |ψ| | H0 |ψ|,

ψ ∈ 2 (Z2 ),

where H0 denotes the free discrete Schr¨ odinger operator on Z2 , i.e., H with A ≡ 0. Proof. We have

2 1 ψ(i(e)) − eiA(e) ψ(t(e)) 2 e 2 1 |ψ(i(e))| − |ψ(t(e))| = |ψ| | H0 |ψ|. ≥ 2 e

ψ|Hψ =

This implies, for example, 1 1

|ψ| | H0 |ψ| + ψ|WB ψ 2 2 1 = |ψ| |(H0 + WB )|ψ|. 2

ψ|Hψ ≥

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By the min-max principle, we obtain the following lemma. Lemma 5. inf σ(H) ≥

1 inf σ(H0 + WB ). 2

odinger operator. We will use (a Note that H0 + WB is a usual discrete Schr¨ modiﬁed version of) Lemma 5 in the proof of Theorem 1.

3 Dirichlet-Neumann bracketing In this section, we generalize the Dirichlet and Neumann decoupling of Simon [6, Section 2] to discrete magnetic Schr¨ odinger operators. Let Z2 = α Sα be a disjoint decomposition of Z2 , and let Σ = e ∈ E e ⊂ Sα for any Sα be the boundary set of the decomposition. Then for a given magnetic Schr¨ odinger operator H, we construct operators H Σ;N and H Σ;D such that H Σ;N ≤ H ≤ H Σ;D and they commute with the direct decomposition: 2 (Z2 ) = α 2 (Sα ), i.e., they act on each 2 (Sα ). For each e ∈ E, we set  1 iA(e)  ψ(x) − e ψ(t(e)) , if x = i(e),  2 1 −iA(e) Le ψ(x) = ψ(i(e)) , if x = t(e), 2 ψ(x) − e    0, otherwise. Le corresponds to 2 × 2-matrix 1 1 −eiA(ε) 1 2 −e−iA(e)

on 2 ({i(e), t(e)}),

and hence Le has eigenvalues 1 and 0. In particular, Le ≥ 0. We note Le . H= e∈E

On the other hand, we also set  1 iA(e)  ψ(x) + e ψ(t(e)) ,  2 Me ψ(x) = 1 ψ(x) + e−iA(e) ψ(i(e)) , 2    0,

if x = i(e), if x = t(e), otherwise.

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Then, similarly, we learn Me ≥ 0. Now we set Le = Le , H Σ;N = H − e∈Σ /

e∈Σ

H

Σ;D

=H+

Me .

e∈Σ

Then, clearly, we have 0 ≤ H Σ;N ≤ H ≤ H Σ;D , and H Σ;# has no oﬀ-diagonal elements corresponding to e ∈ Σ, where # = N or D. Hence H Σ;# acts on each component of the direct decomposition: 2 (Z2 ) = 2 α (Sα ). For L ∈ Z+ , we set Sα(L) = x ∈ Z2 Lαj ≤ xj < L(αj + 1), j = 1, 2 for α ∈ Z2 . (L)

Let H L;# be the operator H Σ;# restricted to 2 (S0 ). As in [6, Section 2], we set # kL (E) = L−2 E #{eigenvalues of H L;# ≤ E} with # = N or D. Then we have D N (E) ≤ k(E) ≤ kL (E), kL

E ∈ R,

N N (E) and kL (E) to obtain lower and upper for each L ∈ Z+ . We will estimate kD bounds of k(E), respectively.

4 Proof of Theorem 1 The proof of Theorem 1 is similar to [6], and we mainly discuss the necessary modiﬁcations.

4.1 Lower bound Recall that the lowest eigenvalue of H0L;D is given by eL;D = 2(1 − cos(π/L)) ([6, 0 Theorem 2.4]). For E > 0, we set L ∈ Z+ such that 4π 2 /E < L ≤ 4π 2 /E + 1 so that eL;D ≤ 2π 2 /L2 < E/2. If we suppose 0 |B(f )| ≤ E/8L

(L)

for any f ⊂ S0 ,

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then we can ﬁnd a vector potential A such that (L)

|A(e)| ≤ E/8 for any e ⊂ S0 . This implies H0L;D − H L;D ≤ E/2, and hence the lowest eigenvalue of H L;D is smaller than E. Thus we have (L) D (E) ≥ L−2 P |B(f )| ≤ E/8L for any f ⊂ S0 kL L2 2 2 = L−2 P |B(f )| ≤ E/8L ≥ L−2 C L (cE 3/2 )aL = exp −2 log L + L2 (log C + a log c + (3a/2) log E) This implies D lim inf log(− log kL (E))/ log E ≥ −1 E→0

since L ∼ 4π E 2

2

−1

as E → 0, and hence lim inf log(− log k(E))/ log E ≥ −1. E→0

4.2 Upper bound We ﬁrst note an analogue of Lemma 5. Lemma 6. Let WBL (x) =

B(f ) 1 − cos . 4 (L)

x∈f ⊂S0

Then inf σ(H L,N ) ≥

1 inf σ(H0L,N + WBL ). 2

The proof is almost the same, and we omit it. We ﬁx ε0 and f0 > 0 such that P(1 − cos(B(f )/4) ≥ ε0 ) = f0 Now the main lemmas of [6], i.e., Theorems 4.1 and 4.2 are rewritten as follows in our setting: Proposition 7. There exist L0 and α0 such that if L ≥ L0 and if (L) L−2 # x ∈ S0 WBL (x) ≥ ε0 ≥ f0 /3, then inf σ(H0L;N + WBL ) ≥ α0 L−2 .

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Proposition 8. 1 B(f ) (L) ≥ ε0 < f0 P (L − 1)−2 # f ∈ F f ⊂ S0 , 1 − cos 4 2 1 ≤ exp − f02 (L − 1)2 . 2 Proposition 8 implies 1 1 (L) P (L − 1)−2 # x ∈ S0 WBL (x) ≥ ε0 < f0 ≤ exp − f02 (L − 1)2 . 2 2 by the deﬁnition of WBL . Combining this with Proposition 7, we observe 1 P inf σ(H0L;N + WBL ) < α0 L−2 ≤ exp − f02 L2 3 if L > L0 . Hence we have N kL (E) ≤ P(inf σ(H L,N ) ≤ E) 1 ≤ P inf σ(H0L;N + WBL ) ≤ 2E ≤ exp − f02 L2 3

if we choose L so that 2E < α0 L−2 , i.e., if L < α0 /2E. We set L be the largest integer satisfying the above condition, so that L ∼ E −1/2 as E → 0. Then we learn α 1 2 0 N kL ∼ exp − f02 E −1 (E) ≤ exp − f02 α0 /2 · E −1/2 − 1 2 6

α0 /2·

as E → 0, and we conclude lim sup log(− log k(E))/ log E ≤ −1. E→0

Appendix A Generalization of local energy estimate Let m ∈ N and let Fm be the set of squares of size m with the vertices at Z2 , i.e., Fm = [x1 , x1 + m] × [x2 , x2 + m] x1 , x2 ∈ Z . Let A(e) be a given vector potential. Then for f ∈ Fm , we denote the magnetic ﬂux through f by B(f ) = A(e), e∈∂f

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where ∂f is the boundary of f . Here we again identify R/(2πZ) with [−π, π) and we always suppose B(f ) ∈ [−π, π). Then we set (m)

WB (x) =

1 m

B(f ) 1 − cos . 4m

x∈f ∈Fm

(m)

Theorem 9. Under the above notations, H ≥ WB

for any m ∈ N.

Proof. We note that each e ∈ E is an element of ∂f for m diﬀerent f ∈ Fm . Hence, as in the proof of Theorem 2, we have

ψ|Hψ =

2 1 ψ(i(e)) − eiA(e) ψ(t(e)) . 2m f ∈Fm

Then we use the following lemma: Lemma 10. For each f ∈ Fm , 2 B(f ) |ψ(x)|2 . ψ(i(e)) − eiA(e) ψ(t(e)) ≥ 2 1 − cos 4m

e∈∂f

x∈f

The lemma is proved similarly as Lemma 3, since the left hand side of the above formula is the energy function for the free discrete Schr¨ odinger operator on the closed chain of size 4m with the periodic boundary condition with the phase eiB(f ) . Now the theorem follows immediately. Theorem 9 may imply better estimate if B(f ) does not change the sign frequently. Let us consider the constant magnetic case to observe this. Example 2. Suppose B(f ) = b for all f ∈ F. Then for f ∈ Fm , B(f ) = m2 b (modulo 2πZ). We choose m ∈ N so that π π < |b| ≤ 2 (m + 1)2 m when |b| is small enough. Then m ∼ |π/b|1/2 and |m2 b| = π +O(|b|1/2 ) when b ∼ 0. By simple computations, we learn H ≥ WBm (x) =

π |b| + O(|b|3/2 ) 4

as b → 0. Probably this lower bound is not optimal, but it is better than the estimate which follows from Theorem 2 for small b.

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Appendix B Spectrum of Example 1 We set θ = b/2, θ, A((x, x + δ1 )) = −θ,

if x1 + x2 is even, if x1 + x2 is odd,

odinger and A((x, x + δ2 )) = 0 for any x ∈ Z2 . Then A deﬁnes a magnetic Schr¨ operator of Example 1. Note that this operator is invariant under the following change of coordinates: T1 ψ(x) = ψ(x + δ1 + δ2 ),

T2 ψ(x) = ψ(x + δ1 − δ2 )

where ψ ∈ 2 (Z2 ). Hence we can apply the Floquet-Bloch theory. Namely, we compute the (generalized) eigenfunction and eigenvalues of H under conditions T1 ψ = eiα ψ,

T2 ψ = eiβ ψ

with ﬁxed α, β ∈ R/(2πZ). These lead to a system of equations: Hψ(0) = 4ψ(0) − eiθ + ei(θ−α−β) + e−iα + e−iβ ψ(δ1 ), Hψ(δ1 ) = − e−iθ + ei(−θ+α+β) + eiα + eiβ ψ(0) + 4ψ(δ1 ).

4 λ with λ = Thus the eigenvalues are the characteristic roots of the matrix λ 4 −{eiθ + ei(θ−α−β) + e−iα + e−iβ }. They are 4 ± |λ|, and by simple computations we obtain |λ|2 = 4 1 + cos α cos β + cos θ(cos α + cos β) . It is then easy to see sup |λ|2 = 4(2 + 2 cos θ) = (4 cos(b/4))2 , α,β

and the range of |λ| (as a function of α and β) is [0, 4 cos(b/4)]. Thus the spectrum of H is given by σ(H) = 4 ± |λ| α, β ∈ R/(2πZ) = [4 − 4 cos(b/4), 4 + 4 cos(b/4)] and it is absolutely continuous.

Appendix C Existence of the integrated density of states Theorem 11. The integrated density of states k(E) = lim

L→∞

1 #{e.v. of H ΛL ≤ E} |ΛL |

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exists for E ∈ R almost surely. Moreover, D D (E) = sup kL (E) k(E) = lim kL L→∞

L

N N (E) = inf kL (E), = lim kL L→∞

where

D kL

and

N kL

L

are deﬁned in Section 3.

Proof. Let Λ ⊂ Z be a box in Z2 , and let H Λ;D and H Λ;N be deﬁned similarly as H L;D and H L;N in Section 3. Namely, they are deﬁned by 2 1

ψ|H Λ;N ψ = ψ(x) − eiA((x,y)) ψ(y) , 2 2

|x−y|=1 x,y∈Λ

ψ|H Λ;D ψ =

1 2

ψ(x) − eiA((x,y)) ψ(y)2 +

|x−y|=1 x,y∈Λ

|ψ(x)|2

|x−y|=1 x∈Λ,y∈Λc

for ψ ∈ 2 (Λ). We set ∗ (E) = kΛ

1 #{e.v. of H Λ;∗ ≤ E} |Λ|

∗ ∗ for E ∈ R with ∗ = D or N . Note that kΛ (E) = k2L+1 (E). We ﬁx E. Now it is L N D a standard procedure to see that kΛ (E) and kΛ (E) are subadditive process and superadditive process, respectively, in the sense of [2] Deﬁnition VI.1.6. Then by Theorem VI.1.7 of [2], we learn N N kN (E) = lim kL (E) = inf kL (E), L→∞

D

k (E) = lim

L→∞

L

D kL (E)

D = sup kL (E) L

exist. Since D (E) ≤ k2L+1

1 N #{e.v. of H ΛL ≤ E} ≤ k2L+1 (E), |Λ|

it remains only to show kN (E) = kD (E).

(N-D)

By the deﬁnition, H ΛL ;D −H ΛL ;N is an operator of rank #(∂ΛL ) = 4(2L+1). Hence, by the min-max principle ([5] Section XIII.1), we have 0 ≤ #{e.v. of H ΛL ;N ≤ E} − #{e.v. of H ΛL ;D ≤ E} ≤ 4(2L + 1). This implies N D 0 ≤ kΛ (E) − kΛ (E) ≤ L L

and (N-D) follows immediately from this.

4 2L + 1

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Acknowledgement. The author wishes to thank the referee for constructive comments. In particular, Appendix C is added following the referee’s suggestion.

References [1] Avron, J., Herbst, I., Simon, B.: Schr¨ odinger operators with magnetic ﬁelds I: General interactions. Duke Math. J. 45, 847–883 (1978). [2] Carmona, R., Lacroix, J.: Spectral Theory of Random Schr¨ odinger Operators. Birkh¨ auser 1990. [3] Higuchi, Yu., Shirai, T.: The spectrum of magnetic Schr¨ odinger operators on a graph with periodic structure. To appear in J. Funct. Anal.; Weak Bloch property of discrete magnetic Schr¨ odinger operators, Preprint. [4] Kirsch, W.: Random Schr¨ odinger operators. In Schr¨ odinger Operators (H. Holden, A. Jensen eds.), Springer Lecture Notes in Physics 345 (1989). [5] Reed, M., Simon, B.: Methods of Modern Mathematical Physiscs. Vol. IV. Analysis of Operators. Academic Press 1978. [6] Simon, B.: Lifshitz tails for the Anderson model. J. Stat. Phys. 38, 65–76 (1985). [7] Sunada, T.: A discrete analogue of periodic magnetic Schr¨ odinger operators. In Geometry of Spectrum (R. Brooks, C. Gordon, P. Perry eds.), AMS Contemporary Math. 173 (1994). [8] Ueki, N.: Simple examples of Lifschitz tails in Gaussian random magnetic ﬁelds, Ann. Henri Poincar´e 1 n◦ 3, 473–498 (2000). S. Nakamura Graduate School of Mathematical Sciences University of Tokyo 3-8-1, Komaba, Meguro-ku Tokyo, Japan 153-8914 E-mail : [email protected] Communicated by Gian Michele Graf submitted 29/11/99, revised 02/02/2000, accepted 10/02/2000

To access this journal online: http://www.birkhauser.ch

Ann. Henri Poincar´ e 1 (2000) 837 – 883 c Birkh¨ auser Verlag, Basel, 2000 1424-0637/00/050837-47 $ 1.50+0.20/0

Annales Henri Poincar´ e

Exponentially Accurate Semiclassical Dynamics: Propagation, Localization, Ehrenfest Times, Scattering, and More General States George A. Hagedorn, Alain Joye Abstract. We prove six theorems concerning exponentially accurate semiclassical quantum mechanics. Two of these theorems are known results, but have new proofs. Under appropriate hypotheses, they conclude that the exact and approximate dynamics of an initially localized wave packet agree up to exponentially small errors in for ﬁnite times and for Ehrenfest times. Two other theorems state that for such times the wave packets are localized near a classical orbit up to exponentially small errors. The ﬁfth theorem deals with inﬁnite times and states an exponentially accurate scattering result. The sixth theorem provides extensions of the other ﬁve by allowing more general initial conditions.

1 Introduction This paper is devoted to proving several theorems concerning exponentially accurate approximations to solutions of the time-dependent Schr¨ odinger equation i

∂ 2 Ψ(x, t, ) = − ∆ Ψ(x, t, ) + V (x) Ψ(x, t, ) ∂t 2

(1.1)

in the semiclassical limit → 0. The semiclassical approximation of quantum dynamics has been the object of several recent investigations from diﬀerent points of view. One approach uses coherent state initial conditions and approximates the evolved wave packet by suitable linear combinations of coherent states. Another approach considers the Heisenberg evolution of suitable bounded observables and approximates the corresponding operators by means of Egoroﬀ’s theorem. The goal of both approaches is to produce accurate, computable approximations as goes to zero, for as long a time interval as possible. In scattering situations the time interval is the whole real line. There are several results concerning the propagation of certain coherent states for ﬁnite time intervals. Early results [14, 7] constructed approximate solutions that were accurate up to O(1/2 ) errors. Later approximations were constructed with O(l/2 ) errors for any l [8, 9, 5, 16]. Very recently, approximations were constructed with errors of exponential order O(e−Γ/ ) with Γ > 0 in [11] (see also [22]). The validity of the corresponding approximations for time intervals of length O(ln(1/)), the so-called Ehrenfest time-scale, has been established up to O(l/2 )

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α

errors in [5], and up to O(e1/ ) errors with 0 < α < 1 in [11] . There is physical intuition and evidence that the Ehrenfest time scale is the natural limit for the validity of coherent state type approximations. This issue is studied in detail for the quantized Baker and Cat maps in [2]. Approximations have been constructed for inﬁnite times in the context of scattering theory for coherent states. Approximate solutions with errors of order O(1/2 ), uniformly in time, are produced in [7]. This yields approximations for the scattering matrix with errors that are also O(1/2 ). Related results for another class of states can be found in [20, 21]. Corresponding results for the approximation of observables in the Heisenberg picture can be found in [18] for approximations with O(l/2 ) errors for any l for ﬁnite times. Approximations with exponentially small errors both for ﬁnite times and for Ehrenfest times are constructed in [1] and [3]. The exponentially accurate results mentioned above, and those we present below, are obtained for Hamiltonians that satisfy certain analyticity conditions. The approximations are generated by optimal truncation of asymptotic series. For further information, we refer the reader to the review articles [4, 19, 15, 12]. The present paper is concerned with the propagation of coherent states in the spirit of the ﬁrst approach described above. We present a new construction of approximate solutions to the time dependent Schr¨ odinger equation that is an alternative to the one presented in [11]. The new expansion has several advantages. In addition to being exponentially accurate up to the Ehrenfest time scale (Theorems 3.1 and 3.3), it allows us to extend our previous results in four separate directions: 1. We get exponentially precise localization properties for both the approximation and the exact solution for both ﬁnite times and Ehrenfest times (Theorems 3.2 and 3.4). 2. We get exponentially accurate information on the semiclassical limit of the scattering matrix for suitable short range potentials (Theorem 3.5). 3. The new algorithm is superior for numerical computation. The work done to construct the approximate wave function for one value of is used for the construction for all smaller values of . This should be contrasted with the construction of [11] where every calculation must be redone for each value of . 4. The results in [11] concern the propagation of initial coherent states given by a linear combination of a ﬁnite number N of elementary coherent states. We can control the new approximation as a function of N , which also allows us to extend the validity of all previous results to a more general set of initial states (Theorem 3.6). In this case however, the algorithm requires the computation of diﬀerent quantities as varies. The technical diﬀerence between the present construction and the one in [11] is the following: In both papers, we use a suitable time dependent basis to convert the PDE (1.1) into an inﬁnite system of ODE’s for the expansion coeﬃcients in that basis of the solution to the Schr¨ odinger equation. In [11] we then construct

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the approximate solution by approximating this inﬁnite system by a ﬁnite system, which we solve exactly. In the new approach, we substitute an a priori expansion in powers of 1/2 into the original inﬁnite system of ODE’s. We construct our approximate solution by keeping a ﬁnite number of terms. This turns out to be quite eﬃcient. The new approximation also plays a vital role in the construction of an exponentially accurate time–dependent Born–Oppenheimer approximation [13].

2 Coherent States and Classical Dynamics We begin this section by recalling the deﬁnition of the coherent states φj (A, B, , a, η, x) described in detail in [10]. A more explicit, but more complicated deﬁnition is given in [9]. We adopt the standard multi-index notation. A multi-index j = (j1 , j2 , . . . , jd ) is a d-tuple of non-negative integers. We deﬁne |j| = dk=1 jk , xj = xj11 xj22 · · · xjdd , |j| j! = (j1 !)(j2 !) · · · (jd !), and Dj = (∂x1 )j1 (∂x∂2 )j2 ···(∂xd )jd . Throughout the paper we assume a ∈ IRd , η ∈ IRd and > 0. We also assume that A and B are d × d complex invertible matrices that satisfy At B − B t A = 0, A∗ B + B ∗ A = 2 I.

(2.1)

These conditions guarantee that both the real and imaginary parts of BA−1 are symmetric. −1 Furthermore, Re BA−1 is strictly positive deﬁnite, and Re BA−1 = A A∗ . Our deﬁnition of ϕj (A, B, , a, η, x) is based on the following raising operators that are deﬁned for m = 1, 2, . . . , d. d d 1 ∂ Am (A, B, , a, η) = √ B n m (xn − an ) − i An m (−i − ηn ) . ∂xn 2 n=1 n=1 ∗

The corresponding lowering operators Am (A, B, , a, η) are their formal adjoints. These operators satisfy commutation relations that lead to the properties of the φj (A, B, , a, η, x) that we list below. The raising operators Am (A, B, , a, η)∗ for m = 1, 2, . . . , d commute with one another, and the lowering operators Am (A, B, , a, η) commute with one another. However, Am (A, B, , a, η)∗ An (A, B, , a, η) − An (A, B, , a, η) Am (A, B, , a, η)∗ = −δm, n .

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Deﬁnition For the multi-index j = 0, we deﬁne the normalized complex Gaussian wave packet (modulo the sign of a square root) by φ0 (A, B, , a, η, x) = π −d/4 −d/4 (det(A))−1/2 × exp − (x − a), B A−1 (x − a) /(2) + i η, (x − a) / . Then, for any non-zero multi-index j, we deﬁne φj (A, B, , a, η, · ) =

1 √ ( A1 (A, B, , a, η)∗ )j1 ( A2 (A, B, , a, η)∗ )j2 · · · j! × ( Ad (A, B, , a, η)∗ )jd φ0 (A , B, , a, η, · ).

Properties 1. For A = B = I, = 1, and a = η = 0, the φj (A, B, , a, η, · ) are just the standard Harmonic oscillator eigenstates with energies |j| + d/2. 2. For each admissible A, B, , a, and η, the set { φj (A, B, , a, η, · ) } is an orthonormal basis for L2 (IRd ). 3. The raising operators can also be given by another formula that was omitted from [10] in the multi-dimensional case. If we set

∗ g(A, B, , a, x) = exp − (x − a), BA−1 (x − a) /(2) − i η, (x − a) / , then we have ( Am (A, B, , a, η)∗ ψ ) (x) d 1 ∂ = − An m ( g(A, B, , a, x) ψ(x) ) . 2 g(A, B, , a, x) n=1 ∂xn 4. In [9], the state φj (A, B, , a, η, x) is deﬁned as a normalization factor times Hj (A; −1/2 |A|−1 (x − a)) φ0 (A, B, , a, η, x). Here Hj (A; y) is a recursively deﬁned |j|th order polynomial in y that depends on A only through UA , where A = |A| UA is the polar decomposition of A. 5. By scaling out the |A| and dependence and using Remark 3 above, one can 2 show that Hj (A; y) e−y /2 is an (unnormalized) eigenstate of the usual Harmonic oscillator with energy |j| + d/2. 6. When the dimension d is 1, the position and momentum uncertainties of the φj (A, B, , a, η, · ) are (j + 1/2) |A| and (j + 1/2) |B|, respectively. In higher dimensions, they are bounded by (|j| + d/2)A and (|j| + d/2)B, respectively.

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7. When we approximately solve the Schr¨ odinger equation, the choice of the sign of the square root in the deﬁnition of φ0 (A, B, , a, η, · ) is determined by continuity in t after an arbitrary initial choice. 8. We prove below that the matrix elements of (x − a)m satisfy | φj (A, B, , a, η, x), (x − a)m φk (A, B, , a, η, x) | √ ≤ |m|/2 ( 2d)|m| A|m| (|k| + 1)(|k| + 2) · · · (|k| + |m|), and

φj (A, B, , a, η, x), (x − a)m φk (A, B, , a, η, x) = 0, if

|j| − |k|

> |m|. (2.2)

We now assume that the potential V : IRd → IR is smooth and bounded below. Our semiclassical approximations depend on solutions to the following classical equations of motion a(t) ˙ η(t) ˙ ˙ A(t) ˙ B(t) ˙ S(t)

= = = =

η(t), − ∇V (a(t)), i B(t), i V (2) (a(t)) A(t), η(t)2 − V (a(t)), = 2

(2.3)

where V (2) denotes the Hessian matrix for V , and the initial conditions A(0), B(0), a(0), η(0), and S(0) = 0 satisfy (2.1). The matrices A(t) and B(t) are related to the linearization of the classical ﬂow through the following identities: A(t) =

∂a(t) ∂a(t) A(0) + i B(0), ∂a(0) ∂η(0)

B(t)

∂η(t) ∂η(t) B(0) − i A(0). ∂η(0) ∂a(0)

=

Because V is smooth and bounded below, there exist global solutions to the ﬁrst two equations of the system (2.3) for any initial condition. From this, it follows immediately that the remaining three equations of the system (2.3) have global solutions. Furthermore, it is not diﬃcult [8, 9] to prove that conditions (2.1) are preserved by the ﬂow. The usefulness of our wave packets stems from the following important property [10]. If we decompose the potential as V (x) = Wa (x) + Za (x) ≡ Wa (x) + (V (x) − Wa (x)),

(2.4)

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where Wa (x) denotes the second order Taylor approximation (with the obvious abuse of notation) Wa (x) ≡ V (a) + V (1) (a) (x − a) + V (2) (a) (x − a)2 /2. then for all multi-indices j, ∂ iS(t)/ e φj (A(t), B(t), , a(t), η(t), x) ∂t 2 ∆ + Wa(t) (x) eiS(t)/ φj (A(t), B(t), , a(t), η(t), x) , (2.5) = − 2 i

if A(t), B(t), a(t), η(t), and S(t) satisfy (2.3). In other words, our semiclassical wave packets ϕj exactly take into account the kinetic energy and quadratic part Wa(t) (x) of the potential when propagated by means of the classical ﬂow and its linearization around the classical trajectory selected by the initial conditions. In the rest of the paper, whenever we write φj (A(t), B(t), , a(t), η(t), x), we tacitly assume that A(t), B(t), a(t), η(t), and S(t) are solutions to (2.3) with initial conditions satisfying (2.1).

3 The Main Results In this section, we list our results concerning the propagation of semiclassical wave packets. The ﬁrst is the construction of an approximate wave function that agrees with the exact wave function up to an exponentially small error. The construction is quite explicit. It depends on the somewhat arbitrary choice of a parameter g > 0. The precise result is summarized in the following theorem: Theorem 3.1. Suppose V (x) is real and bounded below for x ∈ IRd . Assume V extends to an analytic function in a neighborhood of the region Sδ = { z : | Im zj | ≤ δ } and satisﬁes |V (z)| ≤ M exp τ |z|2 for z ∈ Sδ and some positive constants M and τ . Fix T , choose a classical orbit a(t) for 0 ≤ t ≤ T , and consider an arbitrary normalized coherent state of the form cj (0) φj (A(0), B(0), , a(0), η(0), x). ψ(x, 0, ) = |j|≤J

There exists a number G > 0, such that for each choice of the parameter g ∈ (0, G), there exists an exact solution to the Schr¨ odinger equation, i

2 ∂Ψ = − ∆ Ψ + V Ψ, ∂t 2

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with Ψ(x, 0, ) = ψ(x, 0, ), that agrees with the approximate solution ψ(x, t, ) = eiS(t)/ cj (t, ) φj (A(t), B(t), , a(t), η(t), x), |j|≤J+3g/−3

up to an error whose L2 (IRd ) norm is bounded by C exp { −γg / }, with γg > 0. Furthermore, the complex coeﬃcients cj (t, ) are determined by an explicit procedure. The second result shows that the approximate wave function of Theorem 3.1 is concentrated within an arbitrarily small distance of the classical path up to an exponentially small error if g is chosen suﬃcently small. Theorem 3.2. Suppose that the hypotheses of Theorem 3.1 are satisﬁed and that b > 0 is given. For suﬃciently small values of the parameter g > 0, the wave packet ψ(x, t, ) is localized within a distance b of a(t), up to an error exp { −Γg / }, with Γg > 0, in the sense that 1/2

|ψ(x, t, )| dx 2

|x−a(t)|>b

≤ exp { −Γg / } .

Next, we turn to the validity of the approximation and its localization properties on the Ehrenfest time scale, i.e. when T is allowed to increase with as ln(1/). Theorem 3.3. Suppose the assumptions of Theorem 3.1 are satisﬁed except that the upper bound on V is replaced by |V (z)| ≤ M exp (τ |z|) for z ∈ Sδ and some positive constants M and τ . Further, assume the existence of a constant N > 0 and a positive Lyapunov exponent λ so that A(t) ≤ N exp(λt), for all t ≥ 0. Then, for suﬃciently small T > 0, there exist constants C > 0,γ > 0, σ ∈ (0, 1), and σ ∈ (0, 1), and an exact solution to the Schr¨ odinger equation that agrees with the approximation ψ(x, t, ) = eiS(t)/ cj (t, ) φj (A(t), B(t), , a(t), η(t), x), |j|≤J+3/σ −3

up to an error whose norm is bounded by C exp { −γ /σ }, whenever 0 ≤ t ≤ T ln(1/). 1 Moreover, if τ can be taken arbitrarily small, we can chose T = 6λ (1 − 2) where 2 is arbitrarily small. Remark. The semiclassical approximation of observables in the Heisenberg picture holds for any T < 2/(3λ), when τ << 1, as shown recently in [3]. That time interval is longer than those for which a localized coherent state can approximate the evolution of an initial coherent state, which is characterized by T < 1/(2λ). See [2] for a study of related issues on quantized hyperbolic maps on the torus.

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Theorem 3.4. Suppose the hypotheses of Theorem 3.3 are satisﬁed and that b > 0 is given. Then, for suﬃciently small T > 0, there exist Γ > 0, σ ∈ (0, 1), and σ ∈ (0, 1), such that the approximation of Theorem 3.3 satisﬁes 1/2

|ψ(x, t, )| dx 2

|x−a(t)|>b

≤ exp { −Γ /σ } ,

whenever 0 ≤ t ≤ T ln(1/). Moreover, if τ can be taken arbitrarily small, we can chose T = 2 is arbitrarily small.

1 6λ (1

− 2) where

We also explore the validity of the approximation in a scattering framework and its consequences on the corresponding semiclassical approximation of the scattering matrix S(). This requires assumptions on the decay of the potential and its derivatives at inﬁnity. For scattering theory, we assume V satisﬁes the following decay hypothesis. D: There exist β > 1, v0 > 0, and v1 > 0, such that for all x ∈ IRd and all multi-indices m ∈ Nd , |m| v0 v1 m! Dm V (x) ≤ , (3.1) x β+|m| √ where x = 1 + x2 . Theorem 1.2 of [7] shows that under the hypothesis D, the solution of the classical equations (2.3) satisﬁes the following asymptotic estimates: For any a− ∈ IRd , 0 = η− ∈ IRd such that (a− , η− ) ∈ IR2d \E, where E ⊆ {(a− , η− ) ∈ IR2d : η− = 0} is closed and of Lebesgue measure zero in IRd , there exists (a+ , η+ ) ∈ IR2d , η+ = 0, and S+ ∈ IR such that lim |a(t) − a± − η± t| = 0,

t→±∞

lim |η(t) − η± | = 0,

t→±∞

2 lim |S(t) − tη− /2| = 0,

t→−∞

2 /2| = 0. lim |S(t) − S+ − tη+

t→+∞

(3.2)

Moreover, for any d × d matrices (A− , B− ) satisfying condition (2.1), there exist matrices (A+ , B+ ) ∈ Md (I C )2 satisfying (2.1), such that lim A(t) − A± − iB± t = 0,

t→±∞

lim B(t) − B± = 0.

t→±∞

(3.3)

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Our assumption D implies that V is short range. It follows that if H0 () = 2 ∆, then the wave operators deﬁned by − 2 Ω∓ () = s − lim eiH()s/ e−iH0 ()s/ s→±∞

(3.4)

exist and have identical ranges equal to the absolutely continuous subspace of H(). As a result, the scattering matrix S() = Ω− ()∗ Ω+ ()

(3.5)

is unitary. Theorem 3.5. Suppose d ≥ 3 and assume hypothesis D. Let (a− , η− ) ∈ IR2d \E and (A− , B− ) ∈ Md (I C )2 satisfy condition (2.1). Let cj (−∞) ∈ C I , for j ∈ Nd , with 2d 2 |j| ≤ J, such that |j|≤J |cj (−∞)| = 1. Then, there exist (a+ , η+ ) ∈ IR , 2 (A+ , B+ ) ∈ Md (I C ) satisfying (2.1), S+ ∈ IR and explicit coeﬃcients cj (+∞, ) ∈ d C I , for all j ∈ N , with |j| ≤ J˜ with J˜ = J + 3g/ − 3 such that for some γ > 0, C > 0, g > 0 (depending on the the classical data), the states deﬁned by Φ− (A− , B− , , a− , η− , x) = cj (−∞)φj (A− , B− , , a− , η− , x) |j|≤J

Φ+ (A+ , B+ , , a+ , η+ , x)

iS+ /

= e

(3.6) cj (+∞, )φj (A+ , B+ , , a+ , η+ , x)

|j|≤J˜

satisfy S() Φ− (A− , B− , , a− , η− , ·) − Φ+ (A+ , B+ , , a+ , η+ , ·)L2 (IRd ) ≤ C e−γ/ , if is small enough. Finally, we address the question of the generalization of the initial coherent state, whose evolution can be controlled up to exponential accuracy in the diﬀerent settings considered above. For (a, η) ∈ IR2d , we deﬁne Λh (a, η) to be the operator √ (Λ (a, η)f )(x) = −d/2 eiη, (x−a) / f ((x − a)/ ). We deﬁne a dense set C in L2 (IRd ), that is contained in the set S of Schwartz functions, by   C = f (x) = cj φj (I I, I, I 1, 0, 0, x) ∈ S,  j   such that ∃ K > 0 with |cj |2 ≤ e−KJ , for large J . (3.7)  |j|>J

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Remark. It is easy to check that the inequality in (3.7) is equivalent to the requirement that the coeﬃcients of f satisfy |cj | ≤ e−K|j| , for large |j|. Another equivalent deﬁnition of C is C = ∪t>0 e−tHho S, where Hho = − ∆/2 + x2 /2 is the harmonic oscillator Hamiltonian. The set C is sometimes called the set of analytic vectors [17] for the harmonic oscillator Hamiltonian. Theorem 3.6. All theorems above remain true if the initial condition has the form ψ(x, 0, ) = (Λ (a, η)ϕ)(x), where ϕ ∈ C. Theorem 3.1 is proved in Section 6. Theorem 3.2 is proved in Section 7. Theorems 3.3 and 3.4 are proved in Section 8. Theorem 3.5 is proved in Section 9. Theorem 3.6 is proved in Section 10.

4 An Alternative Semiclassical Expansion In this section we derive an expansion in powers of 1/2 . In later sections we perform optimal truncation of this expansion to obtain exponentially accurate approximations. We wish approximately to solve the equation i

∂ψ 2 = − ∆ ψ + V (x) ψ, ∂t 2

with initial conditions of the form ψ(x, 0, ) = c0,j (0) φj (A(0), B(0), , a(0), η(0), x),

(4.1)

(4.2)

|j|≤J

where

2

| c0,j (0) |

= 1.

|j|≤J

We can write the exact solution to this equation in the basis of semiclassical wave packets, cj (t, ) φj (A(t), B(t), , a(t), η(t), x). (4.3) ψ(x, t, ) = eiS(t)/ j

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Note that the sum is over multi-indices j. The inﬁnite vector c whose entries are the coeﬃcients cj satisﬁes i c˙ = K(t, ) c, (4.4) where K(t, ) is an inﬁnite self–adjoint matrix. The matrix K(t, ) has an asymptotic expansion in powers of 1/2 . The cubic term in the expansion of V (x) around x = a(t) gives the leading non-zero term of order 3/2 . The quartic term in the expansion of V (x) gives the term of order 4/2 , etc. Thus, we can write K(t, ) ∼

∞

k/2 Kk (t),

(4.5)

k=3

with Kk (t) =

(Dm V )(a(t)) X(t)m , m!

(4.6)

|m|=k

where X(t)m is the inﬁnite matrix that represents −|m|/2 (x − a)m . Explicit formulas [10] show that entries of X(t)m and Kk (t) do not depend on . We formally expand the vector c as c(t, )

= c0 (t) + 1/2 c1 (t) + 2/2 c2 (t) + . . . = k/2 ck (t).

(4.7)

k

We denote the j th entry of ck (t) by ck,j (t). Note that k is a non-negative integer, and j is a multi-index. We substitute the two expansions (4.5) and (4.7) into (4.4) and divide by . We then equate terms of the same orders on the two sides of the resulting equation. Order 0. The zeroth order terms simply require i c˙0 = 0.

(4.8)

From (4.2), the solution is obviously c0,j (t) = c0,j (0). We note that c0,j (t) = 0 if |j| > J. Order 1. The ﬁrst order terms require i c˙1 = K3 (t) c0 (t).

(4.9)

We solve this by integrating. Because of (4.2), c1 (0) = 0. From the form of c0 (t), and c1,j = 0 whenever only ﬁnitely many of the entries of c1 (t) are non-zero, J +3+d non-zero |j| > J + 3. In d space dimensions, c1 (t) has at most d entries.

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Order 2. The second order terms require i c˙2 = K4 (t) c0 (t) + K3 (t) c1 (t).

(4.10)

Again, we can solve this by integrating with c2 (0) = 0. The only entries of c2 (t) that can be non-zero are c2,j (t) with |j| ≤ J + 6. In d dimensions, there are at J +6+d most non-zero entries. d Order n. In general, the nth order terms require n−1

i c˙n =

Kn+2−k (t) ck (t).

(4.11)

k=0

be non-zero only if To solve this, we simply integrate. We observe that cn,j (t) can J + 3n + d |j| ≤ J + 3n. In d dimensions, there are at most non-zero entries. d Our expansion is diﬀerent from the one constructed in [11], and it is different from the Dyson expansion used in [11]. All three of these expansions are asymptotic to the exact solution of the Schr¨ odinger equation. We note that the main construction in [11] yields a normalized wave function. The expansion derived above does not generate normalized wavepackets. To prove that this expansion is asymptotic, we apply Lemma 2.8 of [10]. To check the hypotheses of that lemma, we do the expansion above through order (l − 1) to obtain c0 (t), c1 (t), . . . , cl−1 (t). We substitute these into (4.7) with the sum cut oﬀ after k = l − 1. We then use the result in (4.3) and compute ∂ψ 2 (x, t, ) + ∆ ψ(x, t, ) − V (x) ψ(x, t, ) (4.12) ∂t 2 Because the ck (t) solve (4.8), (4.9), (4.10), etc., there are many cancellations. We obtain l−1 (l+1−k) k/2 Wa(t) (x) ξl (x, t, ) = eiS(t)/ ξl (x, t, ) = i

k=0

ck,j (t)φj (A(t), B(t), , a(t), η(t), x).

(4.13)

|j|≤J(l)

= J + 3l − 3, and for each q, W (q) (x) denotes the Taylor series error Here, J(l) a(t) (q)

Wa(t) (x)

= V (x) −

Dm V (a(t)) (x − a(t))m m!

|m|≤q

=

|m|=q+1

Dm V (ζm (x, a(t)) (x − a(t))m , m!

for some ζm (x, a(t)) = a(t) + θm,x,a(t) (x − a(t)), with θm,x,a(t) ∈ (0, 1).

(4.14)

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If V is C l+2 on some neighborhood of {a(t) : t ∈ [0, T ] }, then each Wa(t) (x) that occurs in (4.13) is bounded on a slightly smaller neighborhood of {a(t) : t ∈ [0, T ] }. Since (x − a(t))m φj (A(t), B(t), , a(t), η(t), x) has order |m|/2 , it follows that ξ(·, t, ) has order l+2 . Applying Lemma 2.8 of [10], we learn that the ψ(x, t, ) solves the Schr¨ odinger equation up to an error whose norm is bounded by Cl l/2 , when is suﬃciently small. Note that the argument above requires the insertion of cutoﬀs to handle the Gaussian tails or some other assumption, such as V ∈ C l+2 (IRd ) with |Dm V (x)| ≤ Mm exp(τ x2 ) for |m| ≤ l + 2.

5 Estimates of the Expansion Coeﬃcients In this section we study the behavior of ck (t). The ﬁrst step is to get a good estimate of the operator norm of the bounded operator (x − a)m P|j|≤n , where P|j|≤n denotes the projection onto the span of the φj with |j| ≤ n. Lemma 5.1. In d dimensions, (x − a)m P|j|≤n = P|j|≤n+|m| (x − a)m P|j|≤n , and (x − a) P|j|≤n ≤ m

√

|m| (n + |m|)! 1/2 2 d A . n!

(5.1)

(5.2)

Proof. Formula (2.22) of [10] states that (xi − ai ) = Ai p Ap (A, B, , a, η)∗ + Ai p Ap (A, B, , a, η) . 2 p p Note that the right hand side contains 2d terms. Suppose v is any vector in the range of P|j|≤n . Then using formulas (2.8) and (2.9) of [10], we easily deduce that √ n + 1 v , √ ≤ n v ,

Ap (A, B, , a, η)∗ v ≤ Ap (A, B, , a, η) v

and that both Ap (A, B, , a, η)∗ v and Ap (A, B, , a, η) v belong to the range of P|j|≤n+1 . It follows immediately that √ √ (xi − ai ) P|j|≤n ≤ 2 d A n + 1, and that (xi − ai ) P|j|≤n = P|j|≤n+1 (x − a) P|j|≤n . The lemma follows from these two results by a simple induction.

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The conclusion to the next lemma contains the binomial coeﬃcients k−1 . p−1 For k = 1 and p = 1 we deﬁne this to be 1. Lemma 5.2. Suppose V satisﬁes the hypotheses of Theorem 3.1. Fix T and choose a classical orbit a(t) for 0 ≤ t ≤ T . The hypotheses guarantee that n |n| | (D V )(a(t)) | (5.3) 1, sup δ D1 = max n! 0≤|n|, 0≤t≤T and D2 = max

1,

√ 2 d δ −1 A(t)

sup

! (5.4)

0≤t≤T

are ﬁnite.

We deﬁne D3 =

d+2 d−1

, which is the number of multi-indices m with

|m| = 3 in d dimensions. Suppose c0 (0) is a normalized vector with c0,j (0) non-zero only for |j| ≤ J, and suppose ck,j (0) = 0 for all j when k ≥ 1. Let ck,j (t) be the solution to (4.8), (4.9), . . . , (4.11), with these initial conditions. Then for t ∈ [0, T ], we have c0,j (t) = 0

|j| > J,

whenever

(5.5)

c0 (t) ≤ D1 , and for k ≥ 1, ck (t) =

k

(5.6)

[p]

ck (t),

(5.7)

p=1

where [p]

ck,j (t) = 0

|j| > J + k + 2p,

whenever

(5.8)

and [p] ck (t)

≤

k−1 p−1

D1p

D2k+2p

D3k

(J + k + 2p)! J!

1/2

tp . p!

(5.9)

Proof. The ﬁniteness of D1 and D2 is standard. The conclusions (5.5) and (5.6) are trivial. We assume t ∈ [0, T ], and let X(t) denote the formal vector whose entries Xi (t) denote the inﬁnite matrix that represents −1/2 (xi − ai (t)) in the basis { φj (A(t), B(t), , a(t), η(t), ·) }.

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From (4.9) we have i c˙1 (t) = K3 (t) c0 (t) =

(Dm V )(a(t)) X(t)m c0 (t). m!

|m|=3 [1]

We integrate to obtain c1 (t) = c1 (t). Lemma 5.1, (5.5), and (5.6) imply two conclusions: [1] c1,j (t) = 0 whenever |j| > J + 3, (5.10) and

1/2 (J + 3)! [1] c1 (t) ≤ D1 D23 D3 t, (5.11) J! d+2 where the factor is the number of multi-indices m with |m| = 3 in d d−1 dimensions. This proves (5.7), (5.8), and (5.9) for k = 1. For k = 2, we have from (4.10), i c˙2 (t)

= K4 (t) c0 (t) + K3 (t) c1 (t) =

(Dm V )(a(t)) (Dm V )(a(t)) X(t)m c0 (t) + X(t)m c1 (t). m! m!

|m|=4

|m|=3

[1]

The two terms on the right hand side of this equation produce two terms, c2 (t) [2] and c2 (t), when we integrate to obtain c2 (t). Using (5.5), (5.6), (5.10), (5.11), and [1] [2] two applications of Lemma 5.1 we learn that c2 (t) = c2 (t) + c2 (t), where [1]

c2,j (t) = 0

whenever

|j| > J + 4,

[2]

c2,j (t) = 0 whenever |j| > J + 6, 1/2 (J + 4)! d+3 [1] 4 c2 (t) ≤ D1 D2 t, d−1 J! and

[2] c2 (t)

d+3 d−1

≤

D12

D26

D32

(J + 6)! J!

1/2

t2 . 2!

(5.12) (5.13) (5.14)

(5.15)

in (5.14) is the number of multi-indices m with |m| = 4.) 2 d+3 d+2 = This implies (5.7), (5.8), and (5.9) for k = 2 because ≤ d−1 d−1 D32 . This combinatorial inequality follows because d ≥ 1 implies 2 −1 (d + 2)!2 4! (d − 1)! d+2 d+3 = d−1 d−1 3!2 (d − 1)!2 (d + 3)! 4d d + 2 d + 1 4 = ≥ ≥ 1. (5.16) d+3 3 2 1 + 3/d (The factor of

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From (4.11) with n = 3, we have i c˙3 (t)

= K5 (t) c0 (t) + K4 (t) c1 (t) + K3 (t) c2 (t) =

2

k=0 |m|=5−k

(Dm V )(a(t)) X(t)m ck (t). m!

Using (5.5), (5.6), (5.10), (5.11), (5.12), (5.13), (5.14), (5.15), and four applications [1] [2] [3] of Lemma 5.1 we learn that c3 (t) = c3 (t) + c3 (t) + c3 (t), where [1]

whenever

|j| > J + 5,

(5.17)

[2]

whenever

|j| > J + 7,

(5.18)

c3,j (t) = 0 c3,j (t) = 0 [3]

c3,j (t) = 0 whenever |j| > J + 9, 1/2 (J + 5)! d+4 [1] D1 D25 t, c3 (t) ≤ d−1 J! 1/2 2 (J + 7)! t d+3 [2] , c3 (t) ≤ 2 D12 D27 D3 d−1 J! 2! and

[3]

c3 (t) ≤ D13 D29 D33

(J + 9)! J!

1/2

t3 . 3!

(5.19) (5.20) (5.21)

(5.22)

This implies (5.7), (5.8), and (5.9) for k = 3 because of (5.16) and the similar inequality 3 d+4 d+2 = D33 . ≤ d−1 d−1 This inequality follows because d ≥ 1 implies " #2 −1 3 d+2 d+1 4 5 d+4 d+2 = ≥ 1. d−1 d−1 1 + 4/d 1 + 3/d 3 2 Now suppose inductively that the lemma is true for all k ≤ q, for some q ≥ 2. By integrating (4.11) with n = q + 1, we can decompose cq+1 (t) =

[1] cq+1 (t)

+

q n+1

d[q, n, p](t),

(5.23)

n=1 p=2

where

[1]

cq+1 (t) = − i

Kq+3 (s) c0 (s) ds,

d[q, n, p](t) = − i

(5.24)

0 t

Kq+3−n (s) c[p−1] (s) ds, n 0

for 1 ≤ n ≤ q and 2 ≤ p ≤ n + 1.

t

(5.25)

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We interchange the sums in (5.23) to obtain cq+1 (t) =

[1] cq+1 (t)

+

q+1

[p]

cq+1 (t),

(5.26)

p=2

where [p] cq+1 (t)

q

=

d[q, n, p](t),

(5.27)

n=p−1

for 2 ≤ p ≤ q + 1. This establishes (5.7) for k = q + 1. The induction hypotheses, formulas (4.6), (5.24), (5.25), (5.27), and Lemma 5.1 imply (5.8) for k = q + 1, as well as the two inequalities [1] cq+1 (t)

≤

d+q+2 d−1

D1 D2q+3

(J + q + 3)! J!

1/2 t,

(5.28)

and 1/2 (J + q + 1 + 2p)! d+q+2−n D1 D2q+3−n d−1 (J + n + 2p − 2! 1/2 p (J + n + 2p − 2)! t n−1 × D1p−1 D2n+2p−2 D3n p−2 J! p! n−1 d+q+2−n = D1p D2q+2p+1 D3n p−2 d−1

d[q, n, p](t)

≤

×

(J + q + 2p + 1)! J!

1/2

tp . p!

(5.29)

From these inequalities and (5.26), we obtain (5.9) for k = q + 1 as soon as we establish both the inequality q+1 n d+2 d+q+2−n d+2 ≤ (5.30) d−1 d−1 d−1 for 0 ≤ n ≤ q, and the identity q = p−1

q n−1 , p−2

(5.31)

n=p−1

for q ≥ 2 and 2 ≤ p ≤ q + 1. We set r = q − n and note that (5.30) is equivalent to −r−1 d+r+2 d+2 ≤ 1, d−1 d−1

(5.32)

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for 0 ≤ r ≤ q. However, −r−1 d+r+2 d+2 d−1 d−1 r+1 (d − 1)! 3! (d + 2 + r)! = (d − 1)! (r + 3)! (d + 2)! r+1 r+1 r+1 r+4 1 r+5 2 r+d+2 d−1 = ··· . 1 4 2 5 d−1 d+2 Inequality (5.32) follows if each of the factors in the square brackets is bounded by 1. Thus, we need only prove r+1 m+3 r+m+3 ≤ , m m for 1 ≤ m, which can be veriﬁed by using the binomial expansion: r+1 r+1 m+3 r+m+3 r+3 3 3 = . = 1+ ≤ 1+(r+1) + · · · = 1 + m m m m m This proves (5.32) and hence (5.30). The identity (5.31) is trivial for q = 2 and p = 2, 3. Assume inductively that it is true for all 2 ≤ q ≤ m and 2 ≤ p ≤ q + 1, where m ≥ 2. The identity (5.31) is trivial for q = m + 1 and p − 1 = m + 1, since m+1 m = 1 = . m+1 m Then for m + 1 > p − 1, we have (m + 1)! m+1 = p−1 (p − 1)! (m − p + 2)! (m − p + 2) (m!) (p − 1) (m!) + (p − 1)! (m − p + 2)! (p − 1)! (m − p + 2)! m m = + p−2 p−1 m m n−1 = + p−2 p−2

=

n=p−1

=

m+1 n=p−1

n−1 p−2

.

This proves (5.31) and completes the proof of the lemma.

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Corollary 5.3. Assume the hypotheses of Lemma 5.2. Then in addition to (5.5) and (5.6), we have the following for k ≥ 1: ck,j (t) = 0, and

ck (t) ≤

|j| > J + 3k,

(5.33)

k D2k D3k 1 + D1 D22 t . k!

(5.34)

whenever

(J + 3k)! J!

1/2

Proof. Since p ≤ k, (5.8) implies (5.33). ((J + k + 2p)!)1/2 is increasing in p. Thus, To prove (5.34), we note that p! (5.9) and p ≤ k imply [p] ck (t)

≤

k−1 p−1

D1p

D2k+2p

D3k

(J + 3k)! J!

1/2

tp . k!

Summing over p, we obtain ck (t) ≤

(J + 3k)! J!

1/2

k−1 D2k D3k 1 + D1 D22 t D1 D22 t. k!

This implies (5.34).

6 Optimal Truncation Estimates In this section we show that the error given by (4.12) and (4.13) is exponentially small if we choose l = [ g/ ] for an appropriate value of g (where [ · ] is the greatest integer less than or equal to x). The philosophy will be separately to estimate the error near the classical orbit and far from the orbit. To do so, we let b be any positive number and deﬁne χ1 (x, t) to be the characteristic function of { x : |x − a(t)| ≤ b }. We set χ2 (x, t) = 1 − χ1 (x, t). The following lemma controls errors near the classical path by estimating the Taylor series error. It is suﬃcient to combine estimates of the previous section carefully so we have enough control in l. Lemma 6.1. Assume V satisﬁes the hypotheses of Theorem 3.1. Deﬁne χ1 (x, t) as above and ξl (x, t, ) by (4.12). For ﬁxed T > 0 and b > 0, there exists G1 > 0, such that for each g ∈ (0, G1 ), there exist C1 and γ1 > 0, such that if l is chosen to depend on as l() = [ g/ ] , then −1

T

χ1 (·, t) ξl() (·, t, ) dt ≤ C1 exp { −γ1 / } . 0

(6.1)

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Proof. It is suﬃcient to prove the existence of α1 and β1 , such that −1

T

χ1 (·, t) ξl (·, t, ) dt ≤ α1 β1l ll/2 l/2 .

(6.2)

0

If this can be established, we choose G1 = β1−2 . Then 0 < g < G1 and l = [ g/ ] imply β12 g = e−ω , with ω > 0. Since α1 (β12 l )l/2 = α1 e−ωg/(2) , this implies the lemma with C1 = α1 and γ1 = ωg/2. To prove (6.2), we note ﬁrst that our hypotheses imply the ﬁniteness of D4 =

δ |n|

sup |n|≥0, 0≤t≤T, |x−a(t)|≤b

| (Dn V )(x) | . n!

(6.3)

We use this, (4.13), and (4.14) to see that χ1 (x, t) ξl (x, t, ) $ l−1 $ $ ≤ $ $

k/2 χ1 (x, t)

k=0 |m|=l+2−k

×

(Dm V )(ζm (x, a(t)) (x − a(t))m m!

|j|≤J(l)

l−1

≤

$ $ $ ck,j (t) φj (A(t), B(t), , a(t), η(t), x) $ $

k/2 D4 δ −l−2+k

k=0

×

|m|=l+2−k

≤ D4 l/2+1

l−1

$ $ $ $ (x − a(t))m $

|j|≤J(l)

δ −l−2+k

$ $ $ ck,j (t) φj (A(t), B(t), , a(t), η(t), x) $ $

X(t)m ck (t) ,

|m|=l+2−k

k=0

where X(t) is the inﬁnite matrix that represents −1/2 (x − a(t)) in the φj basis. Thus, −1

T

χ1 (·, t)ξl (·, t, )dt ≤ 0

D4 l/2

l−1 k=0

0

T

δ −l−2+k

X(t)m ck (t) dt.

(6.4)

|m|=l+2−k

We apply Lemmas 5.1 and 5.2 to estimate each integral on the right hand

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side of (6.4). For k = 0, we obtain 1/2 T (J + l + 2)! d+l+1 δ −l−2 X(t)m c0 (t) dt ≤ D1 D2l+2 T d−1 J! 0 |m|=l+2

≤

(J + 3l)! J!

1/2

D2l+2 D3l D1 D22 T. (l − 1)!

In the last step, we have used D2 ≥ 1, (5.30), and (J + l + 2)! ≤

(6.5)

(J + 3l)! , ((l − 1)!)2

which is true for l ≥ 1. For k ≥ 1, we write the integral on the right hand side of (6.4) as a sum of k terms by employing (5.7). By (5.8), (5.9), and Lemma 5.1, the pth integrand satisﬁes $ $ $ $ [p] δ −l−2+k $ X(t)m ck (t) $ |m|=l+2−k

≤ D2l+2−k

= ≤

k−1 p−1 k−1 p−1

1/2 (J + l + 2p + 2)! d+l+1−k d−1 (J + k + 2p)! 1/2 p (J + k + 2p)! t k−1 × D1p D2k+2p D3k p−1 J! p!

D1p D2l+2p+2 D3k

D1p D2l+2p+2 D3l

d+l+1−k d−1

(J + l + 2p + 2)! J!

(J + l + 2p + 2)! J!

1/2

tp . p!

1/2

tp p!

(6.6)

In the last step, we have again used (5.30). We now mimic the proof of Corollary 5.3 and then integrate to obtain the following estimate of the kth term in (6.4): 1/2 k+1 (J + 3l)! D3l D2l+2 1 + D1 D22 T −1 . 2 J! (l − 1)! (k + 1)D1 D2 We deﬁne D5 = 1 + D1 D22 T, (6.7) k+1 by D3l , and 1 + D1 D22 T − 1 by D5k+1 . We then

D3l (k + 1)D1 D22 sum over k in (6.4) to obtain the estimate 1/2 T (J + 3l)! D2l+2 D3l D5l+1 χ1 (·, t) ξl (·, t, ) dt ≤ D4 l/2 . −1 J! (l − 1)! D5 − 1 0 (6.8)

bound

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1 1 by l−1 , (l − 1)! ρ (l − 1)l−1 which holds for some constant ρ. After some algebra, this leads to the estimate (6.2). In this expression we bound (J +3l)! by (J +3l)J+3l and

Proving the analogous lemma with χ1 replaced by χ2 requires more work since controlling the error term in the Taylor series expansion of the potential now involves estimating products of gaussian wave packets and powers of x multiplied 2 by the potential which behaves as ex . The main diﬃculty lies in the fact that we need to control all estimates in l, the order of the approximation we consider. We proceed in two steps. Since we will use spherical coordinates for the variable y = −1/2 (x − a), we ﬁrst establish an estimate on the decay of radial part of the eigenfunctions of the harmonic oscillator in the classically forbidden region in Lemma 6.2. Then we use it in the proof of Lemma 6.3 to estimate the products of powers 2 of x and ex with functions φj (A(t), B(t), , 0, 0, x), which are such eigenstates, and appear in our constructions in (6.18). Once this quantity is estimated carefully as a function of all parameters, we proceed in a similar way as above to get a ﬁnal estimate in l that yields exponential decay once l 1/. In spherical coordinates when d ≥ 2, the operator −∆y + y 2 has the form −

L2 d−1 ∂ ∂2 + − + r2 . ∂r2 r ∂r r2

Here L2 is the Laplace–Beltrami operator on S d−1 . For d ≥ 3, it has eigenvalues λq = q(q + d − 2),

(6.9)

with multiplicities mq

= =

1 (q + 1)(q + 2) · · · (q + d − 3) (q + d − 2)(q + d − 1) − (q − 1)q (d − 1)! 1 (q + 1)(q + 2) · · · (q + d − 3)(d − 1)(2q + d − 2) (d − 2)!

≤ Cd eαd q ,

!

(6.10)

where q = 0, 1, . . .. We denote a corresponding orthonormal basis of eigenfunctions by Yq,m (ω) for 1 ≤ m ≤ mq . When d = 1, the analog of the Laplace–Beltrami operator is multiplication by λ = 0 on even functions and multiplication by λ = 1 on odd functions. The ∂2 ∂2 2 2 operator − ∂x 2 + x on IR just becomes the direct sum of two copies of − ∂r 2 + r on (0, ∞) with Neumann and Dirichlet boundary conditions at r = 0. When d = 2, (6.10) should be replaced with m0 = 1 and mq = 2 for q > 0, but the inequality mq ≤ Cd ead q still holds.

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The eigenvalues of −∆y + y 2 are E = 4n + 2q + d with normalized eigenfunctions % 2 2n! q+ d −1 rq Ln 2 (r2 ) e−r /2 Yq,m (ω). ψq,n,m (r, ω) = (6.11) d Γ(q + n + 2 ) Here Lβn (x)

=

n

m

(−1)

m=0

n+β n−m

xm m!

(6.12)

denotes the Laguerre polynomial that satisﬁes the diﬀerential equation x u (x) + (β − x + 1) u (x) + n u(x) = 0, and the normalization condition ∞ Lβn (x) Lβm (x) xβ e−x dx = 0

0 Γ(β+n+1) n!

if n = m , if n = m

(6.13)

(6.14)

for β > −1. The following lemma implies an estimate for |ψq,n,m (r, ω)| when r is in the region which is classically forbidden because of energy considerations. − 1 with q = 0, 1, . . ., the Laguerre polynomial Lβn (x) xn whenever x > 4n + 2β + 2 = 4n + 2q + d. in (6.11) satisﬁes |Lβn (x)| ≤ n!

Lemma 6.2. For β = q +

d 2

Proof. We mimic the proof of Lemma 3.1 of [11]. The ﬁrst step is to show that 2 g(r) = rβ Lβn (r2 ) e−r /2 cannot vanish in the classically forbidden region r2 > 4n + 2q + d. This function vanishes at inﬁnity and is a non-trivial solution to an equation of the form − g (r) + w(r) g(r) = 0, where w(r) > 0 for r2 > 4n + 2q + d. From this diﬀerential equation we conclude that g and g have the same sign in this region. By standard uniqueness theorems, g and g cannot both vanish at the same point. To obtain a contradition, suppose g has a zero at some point r1 with r12 > 4n + 2q + d. Since g vanishes at inﬁnity, the mean value theorem guarantees that g (r2 ) = 0 for some r2 > r1 . Without loss of generality, we may assume g(r2 ) > 0. This forces g (r2 ) > 0, so g is locally increasing. It follows that g is increasing for all r > r2 . Thus, g could not go to zero at inﬁnity. This contradiction shows that g could not have had a zero in the region. We now proceed by induction on n. Since Lβ0 (x) = 1, the lemma is true for n = 0. We now assume n ≥ 1 and that the lemma has been established for Lβn−1 (x).

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Our non-vanishing result and (6.12) imply Lβn (x) n! = 1 − Bβ, n (x), (−1)n xn where Bβ, n (x) = O(1/x) for large x, and Bβ, n (x) > −1, for x > 4n + 2β + 2. Using recurrence relation 8.971.3 of [6], we have d Bβ, n (x) = dx

x Lβn (x) − n Lβn (x) n! xn+1 (−1)n

= −

(n + β) Lβn−1 (x) n! . xn+1 (−1)n

By our induction hypothesis, Lβn−1 (x) has sign (−1)n−1 for x > 4n+2β −2, which includes the region of interest. Thus, Bβ, n (x) is increasing. Since it goes to zero at inﬁnity, it cannot be positive. This implies the lemma. Lemma 6.3. Assume V satisﬁes the hypotheses of Theorem 3.1. Deﬁne χ2 (x, t) as above and ξl (x, t, ) by (4.12). For ﬁxed T > 0 and b > 0, there exists G2 > 0, such that for each g ∈ (0, G2 ), there exist C2 and γ2 > 0, such that if l is chosen to depend on as l() = [ g/ ] , and is suﬃciently small, then −1

T

χ2 (·, t) ξl() (·, t, ) dt ≤ C2 exp { −γ2 / } .

(6.15)

0

Proof. We begin by using the analyticity of V to control Taylor series errors. We deﬁne Cδ (x) = {z ∈ C I d : zj = xj + δeiθj , θj ∈ [0, 2π), j = 1, 2, · · · , d}. If z ∈ Cδ (ζ(x, a)), then, for all j = 1, 2, · · · , d, |zj | ≤ δ + |ζj (x, a)| ≤ δ + |aj | + |xj − aj |. Using this and applying (b + c)2 ≤ 2(b2 + c2 ) several times, we see that z ∈ Cδ (ζ(x, a)) implies | V (z) | ≤ M exp(2τ (x − a)2 ) exp(4τ (δ 2 d + a2 )). 1 m D V (ζ(x, a)) as a d–dimensional Cauchy integral, we obtain Hence, writing m! the bound 1 exp(4τ (δ 2 d + a2 )) |Dp V (ζ(x, a))| ≤ M exp(2τ (x − a)2 ), p! δ |p|

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where ζ(x, a) is any value between x and a. Thus, for 0 ≤ t ≤ T , there exists a constant M1 , such that 1 M1 |(Dp V )(ζp (x, a(t)))| ≤ |p| exp(2τ (x − a(t))2 ). p! δ

(6.16)

We use this, (4.13), (4.14), and (5.33) to see that χ2 (x, t)ξl (x, t, ) $ l−1 $ $ ≤$ k/2 χ2 (x, t) $

|p|=l+2−k

k=0

(Dp V )(ζp (x, a(t)) (x − a(t))p p!

×

|j|≤J+3k

$ $ $ ck,j (t)φj (A(t), B(t), , a(t), η(t), x)$ $

$ $ $ ≤ k/2 M1 δ −l−2+k $χ2 (x, t) exp(2τ (x − a(t))2 )(x − a(t))p $ k=0 |p|=l+2−k $ $ $ × ck,j (t)φj (A(t), B(t), , a(t), η(t), x)$ $ l−1

|j|≤J+3k

$ $ 2 $ ≤ M1 δ −l−2 k/2 δ k |ck,j (t)| $χ2 (x, t)e2τ (x−a(t)) (x − a(t))p $ k=0 |j|≤J+3k |p|=l+2−k $ $ $ (6.17) ×φj (A(t), B(t), , a(t), η(t), x)$ . $ l−1

with |j| ≤ J + 3k. Note that (5.33) has been used to replace |j| ≤ J(l) The norm in the ﬁnal expression of (6.17) equals $ $ $ χ{ z: |z|>b } (x) exp(2τ x2 ) xp φj (A(t), B(t), , 0, 0, x) $ ,

(6.18)

where |p| = l + 2 − k. We assume that is suﬃciently small that 4τ |A(t)|2 < 2/3. Then the square of the quantity (6.18) equals 2−|j| (j!)−1 π −d/2 | det A(t)|−1 −d/2 2 −2 2 × x2p e4τ x |Hj (A; |A(t)|−1 −1/2 x)|2 e−|A(t)| x / dx |x|>b

≤

|p| A(t)2 2 2 |y|2|p| |Hj (A; y)|2 e(4τ |A(t)| −1)y dy. |j| d/2 2 (j!)π 1/2 ||A(t)|y|>b

862

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Ann. Henri Poincar´e

|p| A(t)2 2 |y|2|p| |Hj (A; y)|2 e−y 2/3 dy |j| d/2 2 (j!)π 1/2 ||A(t)|y|>b

≤

(6.19)

|p| A(t)2 2 ≤ e |y|2|p| |Hj (A; y)|2 e−y 1/2 dy. |j| d/2 2 (j!)π 1/2 ||A(t)|y|>b % 2 1 Hj (A; y) e−y /2 is a norBy formula (3.7) of [11], Ωj (y) = 2|j| j! π d/2 malized eigenfunction of −∆y + y 2 with eigenvalue 2|j| + d. Thus, in spherical coordinates, it can be written as dj,q,n,m ψq,n,m (r, ω), (6.20) Ωj (y) = −b2 /(6A(t)2 )

{q,n,m: 2n+q=|j|}

where

|dj,q,n,m |2 = 1.

{q,n,m: 2n+q=|j|}

= J + 3l − 3, We ultimately choose l = [ g/ ] , with 0 < g < G2 . Since J(l) ≤ C3 /. By choosing G2 suﬃciently there exists C3 , such that < 1 implies J(l) −2 2 1)/(2) for 0 ≤ t ≤ T and small . Thus, small, we also have J(l) < (A(t) b − the relevant values of j in (6.20) satisfy 2|j| + d < A(t)−1 b−1/2 .

r2n+q q+ d −1

whenever r2 > 4n + 2q + Lemma 6.2 shows that rq Ln 2 (r2 ) ≤ n! d = 2|j| + d. So, we see that (6.20) is bounded by |p| A(t)2|p| S d−1

r2|p| r>

b A1/2

2

2

dj,q,n,m ψq,n,m (r, ω)

er /2 rd−1 dr dω.

{q,n,m: 2n+q=|j|}

We interchange the sum and integrals and apply the Schwartz inequality to the sum. This shows that (6.20) is bounded by 2 2 2n! e−b /(6A(t) ) |p| A(t)2|p| Γ(q + d2 + n) {q,n,m:2n+q=|j|} ∞

2 2

q+ d −1

rd−1+2|p|+2q Ln 2 (r2 ) e−r /2 dr |Yq,m (ω)|2 dω × b A1/2

= e−b

2

S d−1

/(6A(t)2 ) |p|

A(t)2|p|

×

∞ b A1/2

2n! Γ(q + d2 + n) {q,n,m:2n+q=|j|}

2 2

q+ d −1

rd−1+2|p|+2q Ln 2 (r2 ) e−r /2 dr.

(6.21)

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By reducing the value of G2 if necessary, we can ensure that the hypotheses of Lemma 6.2 are satisﬁed in the integration region in the right hand side of (6.21). So, Lemma 6.2 shows that the integral satisﬁes ∞

2

2

q+ d −1 rd−1+2|p|+2q Ln 2 (r2 ) e−r /2 dr b A1/2

∞

≤

1 (n!)2

≤

22n+ 2 +q+|p| (n!)2

r4n+d−1+2|p|+2q e−r

d

∞

/2

dr

z 4n+d−1+2q+2|p| e−z dz 2

0

2n+ d 2 +q+|p|−1

=

2

b A1/2

2

Γ(2n +

(n!)2

d + q + |p|). 2

So, (6.21) is bounded by e−b

2

/(6A(t)2 ) |p|

A(t)2|p|

22n+ 2 +q+|p| Γ(2n + d2 + q + |p|) . n! Γ(q + d2 + n) d

{q,n,m: 2n+q=|j|}

We use (6.10) to estimate the sum over m ≤ mq and bound this by Cd e−b

2

/(6A(t)2 )

|p| A(t)2|p|

{n,q: q=|j|−2n}

= Cd e−b

2

/(6A(t)2 )

2|j|+ 2 +|p| Γ(|j| + d2 + |p|) n! Γ(q + d2 + n) d

eαd q

|p| A(t)2|p| 2|j|+ 2 +|p| eαd |j| Γ(|j| + d

×

n≤|j|/2

d + |p|) 2

e−2αd n . n! Γ(q + d2 + n)

Since e−2αd n ≤ 1, this is bounded by Cd e−b

2

/(6A(t)2 )

d |p| A(t)2|p| 2|j|+|p| eαd |j| Γ(|j| + + |p|) 2 1 . n! Γ(q + d2 + n) n≤|j|/2

(6.22)

d For n ≤ |j|/2 and d ﬁxed, there exists a constant C , such that Γ(|j| − n + ) ≥ 2 C (|j| − n)!. So, the sum over n in (6.22) is bounded by |j| |j| 1 1 1 2 ≤ . ≤ C n C n! (|j| − n)! C |j|! |j|! n≤|j|/2

n≤|j|/2

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Thus, (6.21) is bounded by C e−b

2

/(6A(t)2 )

|p| A(t)2|p| 2|p| eβd |j|

Γ(|j| + d2 + |p|) . |j|!

(6.23)

This quantity bounds (6.20), which, in turn, bounds the square of (6.18). Terms of the form (6.18) occur in (6.17). Putting this all together, we see that (6.17) is bounded by M1 C

l−1

1/2 −b2 /(12A(t)2 ) −l−2

e

δ

k/2 δ k

|j|≤J+3k

k=0

×

|ck,j (t)| βd |j|/2 e |j|!

|p|/2

|p|

A(t)

|p|/2

2

Γ(|j| +

|p|=l+2−k

d + |p|) . 2

(6.24)

The number of terms that occur in the ﬁnal sum of this expression is l−k+d+1 , and the terms in that sum are increasing. Thus, (6.24) is d−1 bounded by l−1

|ck,j (t)| βd |j|/2 e |j|! k=0 |j|≤J+3k d l−k+d+1 2 (l+2−k)/2 × 2A(t) Γ(|j| + + l + 2 − k) d−1 2

M1 C

1/2 −b2 /(12A(t)2 ) −l−2

= M1 C

1/2 −b2 /(12A(t)2 ) −l−2

×

e

δ

e

δ

l−k+d+1 d−1

k/2 δ k

l−1 l +1 −k/2 2A(t)2 2 δ k 2A(t)2 k=0

|ck,j (t)| βd |j|/2 e |j|! |j|≤J+3k

Γ(|j| +

d + l + 2 − k). 2

Applying the Schwartz inequality to the sum over j, we see that this expression is bounded by M1 C ×

1/2 −b2 /(12A(t)2 ) −l−2

e

l−k+d+1 d−1

δ

l−1 l +1 −k/2 2A(t)2 2 δ k 2 A(t)2

 ck (t) 

k=0

|j|≤J+3k

1/2 eβd |j| d Γ(|j| + + l + 2 − k) . |j|! 2

The number of terms that occur in the ﬁnal sum of this expression is J + 3k + d , and the terms in that sum are increasing. Thus, the expression d

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Exponentially Accurate Semiclassical Dynamics

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is bounded by M1 C

1/2 −b2 /(12A(t)2 ) −l−2

e

δ

l−1 l +1 −k/2 2A(t)2 2 δ k 2A(t)2 k=0

#1/2 " d l−k+d+1 J + 3k + d eβd (J+3k) Γ(J + 2k + + l + 2) × ck (t) . d−1 d (J + 3k)! 2 We now apply the estimate of ck (t) from Corollary 5.3 to bound this by M1 C

1/2 −b2 /(12A(t)2 ) −l−2

e

l−1

×

δ

−k/2 δ k 2A(t)2

l +1 2A(t)2 2

k=0 k k βd (J+3k)/2 D2 D3 (1

× e

l−k+d+1 d−1

+ D1 D22 |t|)k k!

J + 3k + d d

1/2

Γ(J + 2k + d2 + l + 2) J!

1/2

l +1 eβd J/2 √ 2A(t)2 2 J! 1/2 l−1 l−k+d+1 J + 3k + d k 2 −k/2 × δ 2A(t) d−1 d k=0 Dk Dk (1 + D1 D22 |t|)k d (6.25) × e3βd k/2 2 3 Γ(J + 2k + + l + 2). k! 2

≤ M1 C

1/2 −b2 /(12A(t)2 ) −l−2

e

δ

We now employ the following inequalities that hold for some numbers D5 and D6 :

l−k+d+1 l+d+1 ≤ , d−1 d−1 J + 3k + d J + 3l − 3 + d ≤ , d d e3βd k/4

D2k D3k (1 + D1 D22 |t|)k 1 k!

≤ e3βd (l−1)/4 , ≤ D5l−1 ,

d Γ(J + 2k + + l + 2) ≤ 2

l−1 k=0

−k/2 δ k 2 A(t)2

1 (l − 1)!

≤ D6l .

d Γ(J + 3l + ), 2

and

866

G. A. Hagedorn, A. Joye

We then see that (6.25) is bounded by M1 C ×

1/2 −b2 /(12A(t)2 ) −l−2

e

δ

J + 3l − 3 + d d

1/2

l +1 eβd J/2 √ 2A(t)2 2 J!

3βd (l−1)/4

e

D5l−1 D6l

1 (l − 1)!

We bound this expression by using the two inequalities l+d+1 ≤ (l + d + 1)d−1 d−1 J + 3l − 3 + d ≤ (J + 3l − 3 + d)d . d

Ann. Henri Poincar´e

l+d+1 d−1

d Γ(J + 3l + ). (6.26) 2

and

Note that the right hand sides of these inequalities grow polynomially with l. Since d is ﬁxed, we conclude that (6.26), and hence, (6.17) are bounded by a constant times γJ eγl d −b2 /(12A(t)2 ) e l/2 √ Γ(J + 3l + ), (6.27) e (l − 1)! 2 J! for some positive γ, and γ . With J ﬁxed, we apply Stirling’s formula to the factorial and Γ function to bound this by another constant times l 2 2 (3l)3l/2 γ 3/2 e−b /(12A(t) ) l/2 eγl = e 3 ( l)l/2+1 . ll By choosing l = [ g/ ] for some suﬃciently small g > 0, this is bounded by a constant times e−γ2 / . This implies the lemma. Theorem 3.1 follows immediately from Lemmas (6.1) and (6.3) with G = min { G1 , G2 }.

7 Localization Estimates for the Wave Packets In this section we show that our wave packets are localized near the classical path. Given any 2 > 0, we can choose the truncation parameter g > 0, such that our exponentially accurate wave packet is concentrated within { x : |x − a(t)| b }, and let ψ(x, t, ) be the result of our construction with the series truncated with l() = [ g/ ] . We must prove χ(·, t) ψ(·, t, ) ≤ exp { −Γ/ }, for some Γ > 0 when g > 0 is suﬃciently small.

(7.1)

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The left hand side of (7.1) is bounded by l−1

|ck,j (t)| χ(·, t) φj (A(t), B(t), , a(t), η(t), ·) .

(7.2)

k=0 |j|≤J+3k

The norm in this sum has the form (6.18), with n = 0 and τ = 0. Mimicking the estimation of (6.18), we obtain the estimate that corresponds to (6.24). We conclude that if g > 0 is suﬃciently small, then χ(·, t) φj (A(t), B(t), , a(t), η(t), ·) ≤ e−b

2

/(12A(t)2 )

˜

e|j|βd

whenever |j| ≤ J + 3l() − 3, for some β˜d . We use this and the Schwarz inequality to obtain, for some constants C0 , C1 , C2 and C3 l−1

k/2 |ck,j (t)|χ(·, t)φj (A(t), B(t), , a(t), η(t), ·)

k=0 |j|≤J+3k

≤

l−1

 k/2 ck (t)e−b

2

/(12A(t) )  2

1/2 ˜ |j|2β d

e

|j|≤J+3k

k=0

−b2 /(12A(t)2 )

≤e

l−1

D1 D22 t

k/2

k=0

(J + 3k)! J!

1/2

C0k D5k β˜d J e (J + 3k + d)d/2 k! (7.3)

≤ e−b

2

/(12A(t)2 )

C1

∞

(C2 k)

k/2

≤ C3 e−b

2

/(12A(t)2 )

,

(7.4)

k=0

provided g is small enough that C2 k ≤ C2 g < 1 is satisﬁed.

8 Ehrenfest Time Scale In this section we consider the accuracy of our construction when we allow T to grow as → 0. Since the results stated in Theorem 3.4 and the method of proof are basically equivalent to those of [11], we will be rather sketchy. Proof of Theorem 3.3. The ﬁrst point to notice is that the since potential is bounded from below, energy conservation implies that a(t) grows at most linearly with time. The exponential bound on the potential then implies the existence of 1 > 0 and v > 0 such that the quantity (5.3) is bounded by D 1 evτ T . D1 (T ) ≤ D

868

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2 Similarly, the existence of the Lyapunov exponent λ implies the existence of D such that the quantity (5.4) satisﬁes ˜ 1 eλT . D2 (T ) ≤ D It then remains for us to keep track of the time dependence in the proof of Theorem 3.1. In particular, the quantities (6.3) and (6.7) fulﬁll the following estimates, modulo a possible increase of v: D4 (T )

≤ D1 (T ),

D5 (T )

5 e(vτ +2λ)T . ≤ D

Using these bounds, we get the existence of constants C and D, independent of time, such that (6.2) can be replaced by T l/2 −1 χ1 (·, t)ξl (·, t, ) dt ≤ −1 D e(2vτ +3λ)T C l e(6λ+2vτ )T . (8.1) 0

Thus, if we choose l = g(T )/, then (8.1) is bounded by g(T )/(2) −1 D e(2vτ +3λ)T C g(T ) e(6λ+2vτ )T , so that we need g(T ) e(6λ+2vτ )T → 0 and g(T )/ → ∞. These demands are satisﬁed by the choices g(T ) = e−κT provided

and

T = T ln(1/),

(8.2)

6λ + 2vτ < κ < 1/T .

(8.3)

Note that the prefactor in (8.1) will be of order −ν1 , for some ﬁnite ν1 . It will thus play no role since it follows from these considerations that there exists γ1 > 0, such that T

−1

1−κT

χ1 (·, t)ξl (·, t, ) dt = O(−ν1 e−γ1 /

).

0

By a similar argument, we obtain the estimate corresponding to (6.15) with other time–independent constants D and C : T l/2 2 2λT χ2 (·, t)ξl (·, t, ) dt ≤ −1 e−b /(12e ) D C l e(8λ+2vτ )T . −1 0

(8.4) Inserting our choices (8.2) and constraints (8.3) in (8.4), it is elementary to see that there exists positive ν2 and γ2 , such that T 1−2λT −1 χ2 (·, t)ξl (·, t, ) dt = O(−ν2 e−γ2 / ), 0

which proves the Theorem.

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869

Proof of Theorem 3.4. Considerations similar to those in the second part of the proof of Theorem 3.3 show that there exists constants C0 , C1 independent of T such that χ2 (·, t)ψ(·, t, ) ≤

l−1



1/2

k/2 ck (t) 

χ2 (·, t) φj (A(t), B(t), , a(t), η(t), ·)2 

|j|≤J+3k

k=0

∞ k/2 2 2λT ke(2vτ +6λ)T C0 ≤ e−b /(12e ) C1k ,

g(T )/

≤ e−b

2

/(12e2λT )

k=0

k=0

where, by virtue of (8.2) and (8.3), we can take C1 < 1. So, the Theorem holds 2 (1−2λκT ) ) . with exponential decay of order e−b /(12

9 Scattering Theory In this section we show our approximations are valid up to exponentially small corrections in a scattering framework, provided the potential satisﬁes hypothesis D. Proof of Theorem 3.5. First note that equations (3.2) together with e−itH0 ()/ φj (A, B, , a, η, x) = eitη

2

/(2)

φj (A + tiB, B, , a + tη, η, x)

for any j ∈ Nd imply that as t → ±∞, eitH0 ()/ eiS(t)/ φj (A(t), B(t), , a(t), η(t), x) → eiS± / φj (A± , B± , , a± , η± , x) with S− = 0, for any j ∈ Nd . Moreover, using (3.2) and the property min(|v|, 1) t ≤ tv ≤ max(|v|, 1) t , √ for any v ∈ IRd and any t ∈ IR, with t = 1 + t2 , we get the existence of c˜0 > 0 and c˜1 > 0 depending on the asymptotic data (a± , η± ), such that

m |m|

D V (a(t))

≤ c˜0 c˜1

(9.1)

m! t β+|m| for large times. This estimate together with (3.3) and Lemma 5.1 yields the following estimate on the operator Kk (t) P|j|≤n deﬁned in (4.6): $ $ $ $ m $ $ D V (a(t)) m $ Kk (t) P|j|≤n = $ P X(t) |j|≤n $ $ m! $ $ |m|=k

≤

d−1+k d−1

(n + k)! c˜0 c˜k2 , n! t β

(9.2)

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Ann. Henri Poincar´e

where c˜2 depends on the asymptotic data (a± , η± , A± , B± ) and the binomial coeﬃcient gives the number of multi-indices of order k. At the possible cost of an increase in the constants, we may assume this estimate is valid for all t ∈ IR. This estimate shows in particular that Kk (t) P|j|≤n is integrable in time. From this, it is easy to check inductively that the solutions cn (t) to the equations (4.11) have limits as |t| → ∞. The asymptotic values of the coeﬃcients cn (t) at inﬁnity allow us to deﬁne the asymptotic states Φ± (A± , B± , , a± , η± , x) by (3.6) with initial conditions at −∞ characterized by arbitrary normalized coeﬃcients that satisfy c0,j (−∞) = 0, cn,j

= 0,

for |j| > J,

and

for n = 1, 2, · · · , and all j ∈ Nd .

Thus, our approximate solution ψ(x, t, ) = eiS(t)/

(9.3)

cj (t, ) φj (A(t), B(t), , a(t), η(t), x)

|j|≤J+3g/−3

has the asymptotic property as t → ±∞, eitH0 ()/ ψ(x, t, ) → Φ± (A± , B± , , a± , η± , x).

(9.4)

We prove below that ψ(x, t, ) − lim ei(t−s)H()/ eisH0 ()/ Φ− (A− , B− , , a− , η− , x) s→∞

= ψ(x, t, ) − eitH()/ Ω+ () Φ− (A− , B− , , a− , η− , x) = O(e−γ/ ), uniformly for t ∈ IR. Thus, making use of (9.4), we have lim eitH0 ()/ ψ(x, t, ) − eitH0 ()/ eitH()/ Ω+ ()Φ− (A− , B− , , a− , η− , x)

t→+∞

= Φ+ (A+ , B+ , , a+ , η+ , x) − Ω− ()∗ Ω+ ()Φ− (A− , B− , , a− , η− , x) = Φ+ (A+ , B+ , , a+ , η+ , x) − S()Φ− (A− , B− , , a− , η− , x) = O(e−γ/ ). Hence, we need only show that the estimate on ξl (x, t, ) corresponding to our approximation yields an exponentially small correction term after choosing l = g/ for suﬃciently small g, uniformly for t ∈ IR. We mimic Section 5 to get estimates on the coeﬃcients ck (t) =

k

[p]

ck (t)

(9.5)

p=1

starting with c0,j (t) = c0,j (−∞),

c0,j (−∞) = 0 if |j| > J,

and

c0 (−∞) = 1.

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We note that the number of components of the vectors ck (t) is the same as in (5.8) and that the combinatorics associated with the n and p dependence of the estimates is identical to that performed in Section 5. d+2 , at ﬁrst order we have Hence, with D3 = d−1 c1 (t) =

[1] c1 (t)

≤ D3

(J + 3)! c˜0 c˜32 J!

[1]

t

−∞

s −β ds.

[2]

At second order, we obtain c2 (t) = c2 (t) + c2 (t), where [1] c2 (t)

≤

D32

[2] c2 (t)

≤

D32

(J + 4)! c˜0 c˜42 J! (J + 6)! 2 6 c˜0 c˜2 J!

t

−∞

[1]

s −β ds

t

−∞

and

−β

ds1 s1

[2]

s1

−∞

ds2 s2 −β .

[3]

At third order, we obtain c3 (t) = c3 (t) + c3 (t) + c3 (t), where

[1] c3 (t)

[2]

c3 (t) [3]

c3 (t)

t (J + 5)! 5 ≤ s −β ds, c˜0 c˜2 J! −∞ s1 (J + 7)! 2 7 t 3 −β c˜0 c˜2 ≤ 2 D3 ds1 s1

ds2 s2 −β , and J! −∞ −∞ s1 s2 (J + 9)! 3 9 t ≤ D33 ds1 s1 −β ds2 s2 −β ds3 s3 −β . c˜0 c˜2 J! −∞ −∞ −∞ D33

Using the identity

t

−∞

−β

ds1 s1

s1

−∞

−β

ds2 s2

···

sn−1

−∞

−β

dsn sn

1 = n!

t

−∞

−β

s

n ds

,

we get estimates identical to (5.11), (5.14), (5.15), (5.20), (5.21), (5.22). It is easy to check that the induction can be carried out exactly as in Section 5 to give an analog of Corollary 5.3 that states Lemma 9.1. Assume the decay hypothesis D. The expansion coeﬃcients (9.5) satisfying (9.3) obey the following estimates: ck,j (t)

= 0,

[p]

ck (t)

≤

whenever k−1 p−1

D3k

|j| > J + 3k, (J + 3k)! p k+2p c˜0 c˜2 J!

* t

ds s −β −∞ k!

p

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for p ≤ k, and k−1 t t (J + 3k)! c˜k2 D3k 2 −β 2 ck (t) ≤ 1 + c˜0 c˜2 ds s

c˜0 c˜2 ds s −β . J! k! −∞ −∞ Our next task is to estimate the norm of ξl (x, t). We again consider separately the errors near the classical orbit and those far from the orbit. Let b(t) be a real valued function that satisﬁes a(t)

a(t)

≤ b(t) ≤ , 4 2

(9.6)

for all t ∈ IR. We deﬁne χ1 (x, t) to be the characteristic function of {x : |x−a(t)| ≤ b(t)} and χ2 (x, t) = 1−χ1 (x, t). Then, for some constants c˜3 and c˜4 and any t ∈ IR, we have

m |m|

D V (ζm (x, a(t))) v0 v1

≤

χ (x, t)) χ1 (x, t) 1

m! ζm (x, a(t))) β+|m| ≤

|m|

c˜3 c˜4 , t β+|m|

since for large times on the support of χ1 , |ζm (x, a(t)))| ≥ |a(t)|/4. Therefore, for some constants c˜5 and c˜6 , $ $ |m| m $ $ (n + |m|)! c˜5 c˜6 $ χ1 (x, t) D V (ζm (x, a(t))) X(t)m P|j|≤n $ ≤ , $ $ m! n! t β for any t ∈ IR. We now mimic the manipulations performed in Section 6 to get (9.7) χ1 (x, t)ξl (x, t, ) $ l−1 $ (Dm V )(ζm (x, a(t)) |m|/2 $ ≤ $ k/2 χ1 (x, t) (x − a(t))m −|m|/2 $ m! k=0 |m|=l+2−k $ $ $ ×P|j|≤J+3k ck,j (t)φj (A(t), B(t), , a(t), η(t), x)$ $ |j|≤J+3k

d−1+l+2−k ck (t) ≤ d−1 k=0 $ $ m $ $ $χ1 (x, t) D V (ζm (x, a(t))) X(t)m P|j|≤J+3k $ × max $ $ m! {m:|m|=l+2−k} l−1

(l+2)/2

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l−1

873

(J + 2k + l + 2)! c˜5 c˜l+2−k d−1+l+2−k 6 d−1 J! t β k−1 t t c˜k2 D3k 2 −β 2 1 + c˜0 c˜2 × ds s

c˜0 c˜2 ds s −β . k! −∞ −∞

k=0

(l+2)/2

Making use of (5.30), the deﬁnition of D3 , and introducing another constant ∞ c6 , where I = s −β ds, c˜7 = 1 + c˜0 c˜22 I c˜2 /˜ −∞

(9.7) is bounded by (J + 2k + l + 2)! k! l−1 (J + 3l)! c c˜ )2 I (l+2)/2 c˜5 c˜0 (˜ √ 6 2 (˜ c6 D3 )l c˜k7 β (l − 1)! J! t

k=0 (J + 3l)! c˜5 c˜0 (˜ c6 c˜2 )2 I √ (l+2)/2 (˜ c6 D3 c˜7 )l (l − 1)! (˜ c7 − 1) J! t β 2 l (J + 3l)! c6 c˜2 ) I c˜5 c˜0 (˜ (l+2)/2 2 √ D3 1 + c˜0 c˜2 I c˜2 . β (l − 1)! (˜ c7 − 1) J! t

l−1 c c˜ )2 I (l+2)/2 c˜5 c˜0 (˜ √ 6 2 (˜ c6 D3 )l c˜k7 J! t β k=0

≤ ≤ =

(9.8)

This estimate is integrable for t ∈ IR and yields a bound that implies exponential decay in by the optimal truncation technique. We now come to the estimate of χ2 (x, t)ξl (x, t), which is a little bit more elaborate. The diﬃculty stems from a lack of suﬃcient information on the position of ζm (x, a(t)). So, instead of the usual Taylor series error formula, we use the deﬁnition Dm V (a(t)) (q) (x − a(t))m . Wa(t) (x) = V (x) − m! |m|≤q

We ultimately use q = l + 1 − k, where k = 0, 1, · · · , l − 1. Our proof requires the space dimension to satisfy d ≥ 3 in order to obtain integrability in t. Consider the following integral 2 N = χ2 (x, t)2 |φj (A(t), B(t), , a(t), η(t), x)|2 V (x)2 dx IRd = |φj (A(t), B(t), , 0, 0, z)|2 V (z + a(t))2 dz. |z|≥b(t)

We use the formula −1

|φj (A(t), B(t), , 0, 0, z)| =

e−(|A| (t)z) /2 |Hj (A; |A(t)|−1 1/2 z)| , j! 2|j| π d/2 | det A(t)| d/2 2

874

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Ann. Henri Poincar´e

the asymptotic behavior (3.3), and the following estimate, which is valid on the support of χ2 , (|A(t)|−1 z)2 ≥ to obtain the bound ˜ N 2 ≤ e−b/

z2 b2 (t) a(t) 2 ≥ ≥ A(t)2 A(t)2 16 A(t)2

−1

|z|≥b(t)

e−(|A(t)| z) /2 |Hj (A; |A(t)|−1 1/2 z)|2 V (z + a(t))2 dz, j! 2|j| π d/2 | det A(t)| d/2 2

for some ﬁnite, positive ˜b. Note that this estimate has a uniform exponentially decreasing prefactor. As in Section 6, we use spherical coordinates and the decomposition (6.20) Ωj (y) = dj,q,n,m ψq,n,m (r, ω), {q,n,m: 2n+q=|j|}

%

where

|dj,q,n,m | = 1 and Ωj (y) = 2

{q,n,m: 2n+q=|j|}

2 1 Hj (A; y) e−y /2 . 2|j| j! π d/2

This leads to the estimate

˜

N 2 ≤ e−b/

{q,n,m:2n+q=|j|}

e+(|A(t)|

−1

|z|≥b(t)

z)2 /2

|ψq,n,m (rz , ωz )|2 V (z + a(t))2 dz, | det A(t)|d/2

where the spherical coordinates (rz , ωz ) describe the vector 1/2 |A(t)|−1 z. We choose p > 2, such that d/β 2 by 1/s + 1/p = 1/2. Applying H¨ older’s inequality, we get the bound

˜

N2 ≤

e−b/ | det A(t)|d/2

V 2p

{q,n,m:2n+q=|j|}

2/s

+(|A(t)|−1 z)2 s/4

e |z|≥b(t)

|ψq,n,m (rz , ωz )| dz s

.

We need to bound the integral in this expression. We √ change variables to y = |A(t)|−1 z/1/2 and use the estimate | |A(t)|y | ≤ b(t)/ , which is valid when √ |y| ≤ b(t)/(A(t) ). This yields −1 2 e+(|A(t)| z) s/4 |ψq,n,m (rz , ωz )|s dz |z|≥b(t)

≤

e+y b(t) √ |y|≥ A(t)

2

s/4

|ψq,n,m (r, ω)|s | det A(t)| d/2 dy,

(9.9)

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875

where the spherical coordinates (r, ω) now describe the vector y. Note that we have used det(|A|) = | det(A)|, which follows from A = UA |A|, where UA is unitary. ¯ Since √ b(t)/A(t) has a strictly positive inﬁmum b, and we ultimately choose l g/ , with g arbitrarily small, we can assume the integration in (9.9) is within the classically forbidden region where Lemma 4.2 applies, for all indices {q, n, m : 2n + q = |j|} of interest. Hence, manipulations similar to those performed in Section 6, show that (9.9) is bounded above by ∞ | det A(t)| d/2 2s/2 s d−1+sq+2sn −sr2 /4 dω |Y | e q,m √ dr r s/2 s/2 Γ(q + n + d/2) n! ¯ S d−1 b/ ∞ 2 | det A(t)| d/2 (2/s)d+s|j| 2s/2 s ≤ dω |Yq,m | dz z d−1+s|j| e−z s/2 s/2 Γ(q + n + d/2) n! 0 S d−1 d/2 d+s|j| s/2 | det A(t)| (2/s) d + s|j| 2 s /2. ≤ dω |Y | Γ q,m 2 Γ(q + n + d/2)s/2 n!s/2 S d−1 This implies the estimate ˜

N2 ≤

e−b/ V 2p (2/s)2(d/s+|j|) Γ ((d + s|j|)/2) 22/s−1 | det A(t)|1−2/s d/2−d/s  2/s * s dω |Y | q,m S d−1  . × Γ(q + n + d/2)s/2 n!s/2 2/s

{q,n,m: 2n+q=|j|}

We bound the integral in this expression by using the following crude lemma. Its proof is at the end of this section. Lemma 9.2. For some constants M0 and M1 , we have (9.10) | Yq,m (ω) | ≤ M0 M1q . |j| ≤ 2|j| to estimate We use this and the inequalities q ≤ |j| and n  2/s * s dω |Y | q,m d−1 S   Γ(q + n + d/2)s/2 n!s/2 {q,n,m: 2n+q=|j|}

M02 |S d−1 |2/s (2dM12 )|j| m|j|

2/s

≤

π d/2 M02 |S d−1 |2/s (2dM12 )|j| m|j| π d/2 C |j|!



|j|/2

n=0 2/s

≤





2/s 1  Γ(|j| − n + d/2)s/2 n!s/2

2/s |j| s/2   n |j|/2

n=0

876

G. A. Hagedorn, A. Joye

|j|2/s M02 |S d−1 |2/s (4dM12 )|j| m|j|

Ann. Henri Poincar´e

2/s

≤

π d/2 C |j|!

.

Hence, for some constants N0 and N1 , that depend on d and s only, ˜

N ≤ e−b/2

|j|

V p N0 N1 . | det A(t)|1/p d/p

By our choice of p, 1 1/ t d/p | det A(t)|1/p is integrable. For k ≤ l g/, with suﬃciently small g, this last estimate allows us to bound the corresponding term in ξl (x, t)χ2 (x, t) as follows (where the Ni , i = 0, 1, 2, 3 . . . are constants): $ $ $ $ l−1 $ $ k/2 $ ck,j (t) χ2 (x, t) V (x) φj (A(t), B(t), , a(t), η(t), x) $ $ $ $ |j|≤J+3k $ k=0   1/2 l−1 ≤ k/2 ck (t)  V (x) φj (A(t), B(t), , a(t), η(t), x)  |j|≤J+3k

k=0

≤

l−1

k/2

(J + 3k)! V p N0 ˜  e−b/2 J! k! | det A(t)|1/p d/p

1/2 2|j| 

N1

|j|≤J+3k

k=0

≤



N3k

−˜ b/2

e N4 | det A(t)|1/p d/p

l−1

(9.11)

kk/2 k/2 N5k

k=0

˜

e−b/2 N6 ≤ . | det A(t)|1/p d/p

(9.12)

It remains for us to control integrals of the form F 2 (p) = χ22 (x, t)|φj (A(t), B(t), , a(t), η(t), x)|2 |Dp V (a(t))(x − a(t))p |2 /(p!)2 dx IRd 2|p| c˜20 c˜1 (x − a(t))2|p| |φj (A(t), B(t), , a(t), η(t), x)|2 dx ≤ t 2(β+|p|) |x−a(t)|≥b(t) ≤

2|p| −˜ b/(2)

c˜20 c˜2

e t 2β

|p|

√ |y|≥¯ b/

y 2|p| | Hj (A; y)|2 ey

2

/2

/(2|j| j! π d/2 ) dy,

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877

where we used the same type of estimates as above. We bound the last integral in this expression by using spherical coordinates and noting that the integration region lies within the classically forbidden region, if g is suﬃciently small. The integral is thus bounded by {q,n,m : 2n+q=|j|}

22n+q+|p|+d/2−1 |p| Γ(2n + q + |p| + d/2) ≤ f0 f1 (|j| + |p|)!/|j|!. n! Γ(q + n + d/2)

So, for some other constants we have F (p)2 ≤

|j|

|p|

˜

f2 f3 f4 |p| e−b/(2) (|j| + |p|)! . t 2β |j|!

The corresponding sum in χ2 ξl is bounded by $ $ $ l−1 k/2 $ h ck,j (t) $ $ k=0 |j|≤J+3k

×

|p|≤l+1−k

≤

l−1

hk/2

k=0

Dp V (a(t)) (x − a(t))p χ2 (x, t) φj (A(t), B(t), , a(t), η(t), x) p! 

ck (t) 

|p|≤l+1−k

$ $ $ $ $ $

1/2 F 2 (p) 

|j|≤J+3k

(J + 3k)! k/2 k ≤ h f5 √ J! k! k=0  1/2 f2 f J+3k f |p| |p| e−˜b/(2) (J + 3k + |p|)! 3 4   t 2β (J + 3k)! l−1

|p|≤l+1−k

≤

|j|≤J+3k

l−1 ˜ 1/2 ¯ e−b/(4) f2 eβJ hk/2 f6k √ t β k! J! k=0

|p|

|p|/2 f7

(J + 3k + |p|)!, (9.13)

|p|≤l+1−k

where β¯ is independent of J. The last sum in this expression is bounded by l+1−k r=0

(f8 1/2 )r

(J + 3k + r)! =

l+1+2k

(f8 1/2 )s

(J + s)! (f8 1/2 )−3k .

s=3k

(9.14) Since s ≤ l + 1 + 2k ≤ 3l + 1 g/ and g is small, we have (f8 1/2 )s (J + s)! ≤ (f9 1/2 s1/2 )s ≤ α(g)s ,

878

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Ann. Henri Poincar´e

√ where α(g) = f10 g is smaller than one. Furthermore, the sum (9.14) is bounded by α(g)3k (f8 1/2 )−3k . 1 − α(g) From this we deduce that (9.13) is dominated by a constant times l−1 ˜ e−b/(4) hk/2 f6k 2 α(g)3k (f8 1/2 )−3k t β k! k=0

l−1 ˜ ˜ (g)k e−b/(4) α = 2 t β k k! k=0

≤ 2

−˜ b/(4)

e

t β

˜ eα(g)/

˜

≤

e−b/(8) , t β

(9.15)

provided g is small enough, since α(g) ˜ g 3/2 as g → 0, which is exponentially small and integrable in time. Finally, gathering estimates (9.8), (9.12) and (9.15), we get the existence of positive γ, H, G, and C, such that g < G, l() = g/, and < H imply ∞ dt ξl (x, t)/ ≤ C e−γ/ . −∞

Proof of Lemma 9.2. Let f ∈ S be the function that is given in spherical coordinates by % 2 2 rq e−r /2 Yq,m (ω). f (x) = d Γ(q + 2 ) For integers q > 0, the maximum absolute value of this function is % 2 q q/2 e−q/2 max | Yq,m (ω) |. ω Γ(q + d2 )

(9.16)

The function f is a normalized eigenfunction of −∆ + x2 with eigenvalue E = 2q + d. Its norm in the Sobolev space Hs for s > 0 satisﬁes f Hs ≤ C1 (s) f + (−∆)s/2 f ≤ C1 (s) 1 + (2q + d)s/2 , (9.17) for some C1 (s). older’s inequality, If s > d/2, then (1 + |k|2 )−s/2 is in L2 (IRd ). So, by H¨ $ $ $ $ $ $ $ $ | f (x) | ≤ (2π)−d/2 $ f+(k) (1 + |k|2 )s/2 $ $ (1 + |k|2 )−s/2 $ = C2 (s) f Hs .

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This and (9.17) imply that (9.16) is bounded by C3 (s)

1 + (2q + d)s/2 . Thus,

% max | Yq,m (ω) | ≤ C3 (s) ω

Γ(q + d2 ) 1 + (2q + d)s/2 q −q/2 eq/2 . 2

The lemma follows from this by an application of Stirling’s formula.

10 More General Coherent States In this section we extend all the previous theorems of the paper to allow initial conditions that are certain inﬁnite linear combinations of the φj . Proof of Theorem 3.6. The strategy is quite simple. Let ϕ ∈ C have expansion I I, I , 0, 0, x), and let ϕ = cj φj (I, ψ0 (x, 0, ) = (Λh (a, η)ϕ)(x) be our initial condition. By construction, cj φj (I, I I, I , a, η, x). ψ0 (x, 0, ) = d j∈N For J > 0, we deﬁne ψJ (x, 0, ) =

cj φj (I, I I, I , a, η, x),

|j|≤J

and denote the approximation that arises from this initial condition by ψJ(J,) (x, t, ) = cj (t, ) φj (A(t), B(t), , a(t), η(t), x). ˜ ˜ |j|≤J(J,)

We then have e−itH()/ ψ0 (0, ) = e−itH()/ (ψ0 (0, ) − ψJ (0, )) + e−itH()/ ψJ (0, )

(10.1)

= ψJ(J,) (t, ) + O(e−itH()/ ψJ (0, ) − ψJ(J,) (t, )) ˜ ˜ +O(ψ0 (0, ) − ψJ (0, ))). Thus, to make the error terms exponentially small in we need to consider values of the cutoﬀ J that grow to inﬁnity with in a suitable way, and we also need to control our approximation as a function of J.

880

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In the proofs of all previous theorems, the dependence of the approximation on l governs the estimates on the error terms. The dependence comes through the diﬀerent choices of l g/ or l g(T )/, with T ln(1/). The set C is chosen to give an exponentially small contribution as l → ∞ in the last term of (10.1) with the choice J = ν l, (10.2) for some ν > 0. We need only show that the basic estimates in the proofs above are unaltered by the replacement of J by ν g/, for g small enough. We can do this because we have been careful to make the J dependence explicit in all the key estimates, such as Corollary 5.3. In the contribution to the error term associated with χ1 given by (6.8) we adapt the last step by using the estimate (J(l) + 3l)! ((ν + 3)l)(ν+3)l ≤ c0 (ν) c1 (ν)l l3l , ≤ c0 (ν) J(l)! (νl)νl

(10.3)

for some constants c0 (ν) and c1 (ν). Hence, the remainder of the argument for Lemma 6.1 is the same, with updated constants. Since the constants are modiﬁed in a time independent way, the long time estimates are also unchanged. Consider now the contribution associated with χ2 in Lemma 6.3. We ﬁrst note ˜ = J(l)+3l−3 = (ν+3)l−3 that (10.2) implies (with a slight abuse of notation) J(l) ˜ g/. The arguments that rely on the smallness of g to so that we still have J(l) allow us to use of Lemma (6.2) remain in force. We thus arrive at (6.27). We deal with it by using (10.3), exactly as above, and obtain exponential decay again in case l = g/. The long time estimates are also valid as the time dependence of the constants is unaltered. This shows that Theorems 3.1 and 3.3 are true with our generalized initial coherent states. Theorem 3.2 also holds for these initial states provided we can control the sum in (7.3) with J(l) = νl. To do so, we ﬁrst note that the last two factors of (7.3) can be bounded by eβ J , for some β , so that they are of order eg/ . This is harmless if g is small enough because of the exponentially decreasing prefactor. Next, we use k ≤ l − 1 to obtain √ ((ν + 3)l)νl/2 (J + 3k)!/(J!) ≤ (J+3k)3k/2 (J+3k)J/2 / J! ≤ c(ν) (J+3k)3k/2 , (νl)νl/2 for some constant c(ν). Thus, the sum in (7.3) is bounded by c (ν)l

l−1

C2k k/2 (J + 3k)3k/2 /k!,

(10.4)

k=0

for another constant c (ν), and where C2 is proportional to C0 D5 in (7.3). Since (J + 3k)3k/2 ≤ ((ν + 3)l)3k/2 ≤ (((ν + 3)g)3/2 /3/2 )k ,

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we can bound (10.4) by

c (ν)g/ eC2 ((ν+3)g)

3/2

/

which, again, is harmless for suﬃciently small g. To prove the validity of Theorem 3.4, insert (8.2) and (8.3) in the estimates above and check that the conclusion still holds. This is straightforward. Finally, for Theorem 3.5 to hold, we ﬁrst must consider the contribution associated to χ1 ξl , which relies on (9.8). Here, (10.3) applies directly. Next, the ﬁrst contribution from χ2 ξl is (9.11). It has the form (7.3) and yields exponential decay in the same way, for g small enough. It remains for us to bound (9.13). With s ≤ l + 1 + 2k ≤ 3l, we use the estimate, (J + s)! ≤ (νl + s)νl/2 (νl + s)s/2 ≤ ((ν + 3)l)νl/2 ((ν + 3)g/)s/2 ¯ √ in (9.14). The ﬁrst factor when multiplied by eβJ / J! is of order ecg/ where c is independent of g. The ﬁnal factor allows us to repeat the argument that led to (9.15). Hence, for g small enough, we get an exponentially small contribution in and the result follows.

References [1] Bambusi, D., Graﬃ, S., and Paul, T.: Long Time Semiclassical Approximation of Quantum Flows: A Proof of Ehrenfest Time. 1998 preprint. [2] Bonechi, F. and De Bi`evre, S.: Exponential mixing and | ln()| time scales in quantized hyperbolic maps on the torus, 1999 preprint. [3] Bouzouina, A. and Robert D.: Uniform Semi-classical Estimates for the Propagation of Heisenberg Observables. 1999 preprint. [4] Combescure, M.: The eﬃciency of coherent states in various domains of semiclassical physics. C.R.A.S. s´erie 2b t. 325 pp. 635-648, (1997). [5] Combescure, M. and Robert, D.: Semiclassical Spreading of Quantum Wave Packets and Applications near Unstable Fixed Points of the Classical Flow. Asymptotic Anal. 14, pp. 377–404, (1997). [6] Gradsteyhn, I. S. and Ryzhik, I. M.: Table of Integrals, Series, and Products, Fifth Ed. New York: Academic Press 1994. [7] Hagedorn, G. A.: Semiclassical Quantum Mechanics I. The → 0 Limit for Coherent States. Commun. Math. Phys. 71, 77-93 (1980). [8] Hagedorn, G. A.: Semiclassical Quantum Mechanics III: The Large Order Asymptotics and More General States. Ann. Phys. 135, 58–70 (1981).

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[9] Hagedorn, G. A.: Semiclassical Quantum Mechanics IV: Large Order Asymptotics and More General States in More than One Dimension. Ann. Inst. H. Poincar´e Sect. A. 42, 363–374 (1985). [10] Hagedorn, G. A.: Raising and lowering operators for semiclassical wave packets. Ann. Phys. 269, 77–104 (1998). [11] Hagedorn, G. A. and Joye, A.: Semiclassical Dynamics with Exponentially Small Error Estimates. Commun. Math. Phys. 207, 439–465 (1999). [12] Hagedorn, G. A. and Joye, A.: Semiclassical Dynamics and Exponential Asymptotics. Proceedings of the 1999 UAB-GIT International Conference on Diﬀerential Equations and Mathematical Physics (to appear). [13] Hagedorn, G. A. and Joye, A.: A time Dependent Born-Oppenheimer Approximation with Exponentially Small Error Estimate, 2000 preprint. [14] Hepp, K.: The Classical Limit for Quantum Mechanical Correlation Functions. Commun. Math. Phys. 35, 265–277 (1974). [15] Paul, T.: Semi-Classical Methods with Emphasis on Coherent States in Quasiclassical Methods, J. Rauch and B. Simon eds, IMA Volumes in Mathematics and Applications 95, 51-97, Springer 1997. [16] Paul, T. and Uribe, A.: A Construction of Quasi-Modes using Coherent States. Ann. I.H.P. Sect. A., Physique Th´eorique 59, 357–381, (1993). [17] Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self-Adjointness. Academic Press, New York (1975). [18] Robert, D.: Autour de l’approximation semi-classique. Progress in Mathematics 68, Birkha¨ user, Boston (1987). [19] Robert, D.: Semi-Classical Approximation in Quantum Mechanics. A Survey of Old and Recent Mathematical Results. Mathematical results in quantum mechanics (Ascona, 1996). Helv. Phys. Acta 71, 44–116 (1998). [20] Yajima, K.: The Quasi–classical Limit of Quantum Scattering Theory. Commun. Math. Phys. 69, 101–130 (1979). [21] Yajima, K.: The Quasi–classical Limit of Quantum Scattering Theory, II. Long Range Scattering. Duke Math. J. 48, 1–21 (1981). [22] Yajima, K.: Gevrey Frequency Set and Semi-classical Behavior of Wave Packets, in Schr¨ odinger Operators, The Quantum Mechanical Many Body Problem, Lecture Notes in Physics, 403, ed. by E. Balslev. Berlin, Heidelberg, New York: Springer-Verlag, 1992, pp 248–264.

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George A. Hagedorn Department of Mathematics and Center for Statistical Mechanics and Mathematical Physics Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061-0123, U.S.A. Partially Supported by National Science Foundation Grant DMS–9703751. E-mail : [email protected] Alain Joye Institut Fourier Unit´e Mixte de Recherche CNRS-UJF 5582 Universit´e de Grenoble I BP 74 F–38402 Saint Martin d’H`eres Cedex, France E-mail : [email protected] Communicated by Bernard Helﬀer submitted 01/12/99, revised 15/02/2000, accepted 24/02/2000

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Autocorrelation Scaling and Fourier Transform of Non-Autonomous Systems C´esar R. de Oliveira Abstract. Upper bounds for the (strong) Fourier transform, of a rather general sequence of unitary operators, are related to the uniform α-H¨ older continuity of its autocorrelation measure. It is a natural generalization of the “Dynamical BombieriTaylor Conjecture.” Immediate applications include driven quantum systems, classical and quantum harmonic oscillators, and non-autonomous twisted generalized random walks in Hilbert spaces.

1 Introduction and Main Result We are interested in asymptotic properties of the time evolution of a class of quantum systems, particularly in the non-autonomous case, in which the Hamiltonian H depends on time; we shall present upper bounds for the growth of the strong Fourier transform of sequences of unitary operators in terms of the α-H¨older continuity of the corresponding autocorrelation measures. We use the autonomous case to motivate our main result, mentioning that some works on the asymptotic properties of systems with non-trivial time dependence include [1–18]. Let U (t, 0), t ∈ IR, be a strongly continuous one-parameter group of unitary operators on the (separable) Hilbert space H. Denote its inﬁnitesimal generator by H, i.e., H : dom H → H is a self-adjoint operator such that U (t, 0) = e−iHt , ∀t. With respect to the time evolution in quantum mechanics H represents the Hamiltonian operator and U (t, 0) is called propagator. The Mean Ergodic Theorem [19] states that for each ξ ∈ H, ω ∈ IR, 1 lim T →∞ T 2

2 T eiωt U (t, 0)ξdt = EH (ω)ξ2 = µξ ({ω}), 0

(1)

where EH (ω) is the orthogonal projector onto Ker(H − ωI) and µξ the spectral measure associated to the vector ξ. Thus, the left hand side of (1) can be used to recover the point spectrum of the Hamiltonian operator H. The mathematical formulation and adaptation of this result for a class of non-autonomous systems was discussed in [15]; now we brieﬂy recall its principal result. Let {Un }∞ n=1 be a sequence of unitary operators on H and Λ(n) = Un · · · U2 U1 .

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We set Λ(0) = I. This quantity can be seen as the time evolution operator of non-autonomous quantum systems for which the time-dependent law changes at every integer time. E.g., the time evolution associated to a family of kicked systems given by the Hamiltonian H(t) = H0 +

∞

Vj δ(t − j),

j=1

with {Vj } being a sequence of potentials; in this case Un = e−iVn e−iH0 . Another possibility is a time-dependent quantum system built upon a sequence of autonomous potentials Vj , with Vj acting on the time interval (j − 1, j]. A quantity that resembles the left hand side of the Mean Ergodic Theorem (1) for Λ was ﬁrst suggested about ten years ago in [2], motivated by a similar relation in the context of diﬀraction by aperiodic structures studied by Bombieri and Taylor [20, 21]; by borrowing a conjecture from Bombieri-Taylor, in [2] it was proposed that something like a point spectrum for Λ would be present if the limit 1 ϕN (ω, ξ) = 0 N→∞ N 2 lim

(2)

for some ω ∈ [0, 2π] , ξ ∈ H, where 2 N−1 iωj . e Λ(j)ξ ϕN (ω, ξ) = j=0

(3)

This idea was also employed in [3, 9]. The precise statement appeared in [15], also clarifying the meaning of the point spectrum for Λ, i.e., it was proven that lim sup N→∞

1 ϕN (ω, ξ) ≤ 2σξ ({ω}), N2

(4)

where σξ denotes the autocorrelation measure of {Λ(n)ξ}. Notice that the limit (2) is a kind of Fourier Transform of {Λ(n)ξ}. Recall that the autocorrelation measure σξ is a (ﬁnite) Borel positive measure on [0, 2π] (the unit circle), which is deﬁned by Bochner-Herglotz theorem—the second equality below—via the autocorrelation functions N−1 1 Λ(j + k)ξ|Λ(j)ξ = eiks dσξ (s). Cξ (k) = lim N→∞ N j=0 Recall also that the notion of autocorrelation measure is the natural generalization, for time-dependent systems, of spectral measures of propagators for autonomous quantum models. In this work the autocorrelation functions Cξ (k) are supposed to be well deﬁned.

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Relation (4) has been called the Dynamical Bombieri-Taylor conjecture, and it is a rigorous version of the Mean Ergodic Theorem (1) that holds for some systems with general time dependence. Notice, however, that there are examples [15] for which the left hand side of (4) vanishes while σξ is pure point; this happens because ϕN is too sensible to phases variations, at least when compared to autocorrelations functions. In summary, ϕN can be used, even numerically, to derive properties about the point components of the autocorrelation measures. Here it will be shown that ϕN can also be used to extract information on the continuous part of the autocorrelation measures; the point is to tune the growth rate of ϕN and consider ϕN /N 2−α , with α ≥ 0. Our main result asks also for an average on ω, so we introduce the following notation for positive integrable functions f : J → IR on a closed interval J ⊂ IR: 1 f (ω)dω f (ω) J = |J| J (| · | denotes Lebesgue measure). If |J| = 0 then f (ω) J ≡ 0. Deﬁnition 1. [22, 23] A σ-ﬁnite positive Borel measure µ, on subsets of IR, is uniformly α-H¨older continuous (UαH) on the interval J ⊂ IR if there is a positive constant C such that µ(J ) ≤ C|J |α , for any subinterval J ⊂ J with |J | < 1. Remark 1.1. UαH measures have been thought of as a kind of fractal measures among physicists. The most relevant property for quantum mechanics [22, 23] is that for each UαH measure µ in IR there exists a constant D < ∞ such that 1 T µ(t)|2 dt ≤ DT −α , for all T > 1 (ˆ µ denotes the Fourier Transform of µ). T 0 |ˆ Theorem 1. Suppose the autocorrelation functions exist for the sequence {Λ(n)ξ} in the Hilbert space H, with UαH autocorrelation measure σξ , 0 ≤ α ≤ 1, in the closed interval J ⊂ [0, 2π]. Then there exists a constant 0 ≤ K < ∞ such that lim sup N→∞

1 N 2−α

ϕN (ω, ξ) J ≤ K.

(5)

Remark 1.2. A clear corollary of this theorem is that if there is a subsequence of integers {Nr } such that 1

ϕNr (ω, ξ) J r→∞ Nr2−α lim

= ∞,

then σξ is not UβH, for α ≤ β, in the closed interval J. Remark 1.3. By combining this theorem with Fatou’s lemma, it follows that lim inf N→∞

ϕN (ω, ξ) < ∞, N 2−α

for ω in a set of full Lebesgue measure in J.

(6)

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Remark 1.4. Although inserted in the context of quantum mechanics, the above theorem holds for rather general sequences of unitary operators on Hilbert spaces; it has potential applications in the studies of general (classical and quantum) driven dynamical systems. Relations (5) and (6) can, in principle, be a theoretical and numerical source of information on the H¨ older properties of autocorrelation measures, which are in general very hard to be explicitly computed. Remark 1.5. For the case α = 0 there are more speciﬁc results, as discussed above, with a proper estimate for K—see (4). A measure may have a H¨ older exponent α that depends on the point in its support; in this case it is the smallest value of α in J that is relevant for thm. 1. Remark 1.6. If there is a unitary operator W such that Un = W, ∀n—the autonomous case—then the autocorrelation measures are replaced by spectral measures of W , and Hof [24] has got an equivalence in the analogous of thm. 1 (see [24] for details). I was not able to prove that relation (5) implies that σξ is UαH; based on the examples presented in [15] with strictly inequality in (4), we suspect that thm. 1 does not have a simple converse (see also the next two remarks). Remark 1.7. The wide generality of thm. 1 with respect to the time dependence of Λ justiﬁes, at least on intuitive grounds, the average over J in (5); it is a way to smear eventual wild oscillations of ϕN as function of ω. I do not know any non-trivial suﬃcient condition assuring that such average can be dropped out. Remark 1.8. The Mean Ergodic Theorem (1) is a kind of rigorous formulation of the physical concept of “energy representation” in quantum mechanics (for pure point Hamiltonians). In case of non-autonomous systems such representation is actually expected to fail in general; this is a physical reason for the inequality and average over frequencies in (5). Remark 1.9. The value α = 1 is related to measures absolutely continuous with respect to Lebesgue measure (with continuous density, at least), while α = 0 to point measures; but it is known that a UαH measure with 0 < α < 1 does not necessarily have a singular continuous component [23, 24]. The remaining of this paper is organized as follows. In Section 2 the proof of the above theorem is presented, and the critical growth exponent for a sequence of real numbers is deﬁned. In Section 3 thm. 1 is used to give upper bounds on the energy growth of a class of classical and quantum harmonic oscillators with general time dependence; that section ﬁnishes with a remark on non-autonomous twisted generalized random walks.

2 Proof of Theorem 1 In this Section we present the proof of thm. 1 and deﬁne the critical growth exponent for a sequence of positive real numbers.

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The case α = 0 was discussed in [15] and, since those results imply thm. 1 for this case, we suppose that α > 0 and that σξ is a continuous measure over J. Each function ϕN (·, ξ) is continuous and bounded by N 2 ξ2 , so it deﬁnes a measure dµN (ω) = ϕN (ω, ξ)/N dω absolutely continuous with respect to Lebesgue measure. Thm. 1 will follow, after additional manipulations, from the claim that, for each ξ ∈ H, the sequence {µN } converges, in the weak∗ topology, to the autocorrelation measure σξ . In fact, by expanding ϕN we get dµN (ω) =   N−1 1  = exp(i(n − j)ω)Λ(n)ξ|Λ(j)ξ  dω N n,j=0   N−1 N−1−j 1  ikω = e Λ(j + k)ξ|Λ(j)ξ  dω. N j=0 k=−j

Its Fourier transform at r ∈ ZZ is given by (for N > |r|) µ ˆN (r) = =

N−1 N−1−j 1 δk,r Λ(j + k)ξ|Λ(j)ξ N j=0 k=−j

=

1 N

N−1

Λ(j + r)ξ|Λ(j)ξ .

j=0

Therefore, for N → ∞ one gets that µ ˆN (r) → Cξ (r) = σ ˆξ (r), for any r ∈ ZZ, and so µN converges in the weak∗ topology to σξ . Since σξ is UαH on the bounded interval J, there exists 0 < C < ∞ such that σξ (J ) ≤ C|J |α for any interval J ⊂ J, and being σξ a regular and continuous measure we have limN→∞ µN (J ) = σξ (J ) (the border of any interval J has zero σξ measure) [25]. If |J| = 0 there is nothing to prove since thm. 1 becomes trivial, so we assume that |J| > 0. Pick 0 < ε < C|J|α ; by the above claim there exists M > 0 such that if N ≥ M then ϕN (ω, ξ)/N dω − ε ≤ σξ (J) ≤ C|J|α , J

and

1 1 ϕN (ω, ξ) J = N |J|

ϕN (ω, ξ)/N dω ≤ 2C|J|α−1 . J

Taking also M large enough so that |J| ≥ 1/M , it follows that, for N ≥ M , |J|α−1 ≤ N 1−α . Thus 1 ϕN (ω, ξ) J ≤ 2CN 1−α . N

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From this relation it follows that lim sup N→∞

1 N 2−α

ϕN (ω, ξ) J ≤ K,

with K = 2C and thm. 1 is proven.

Deﬁnition 2. Given a sequence u = {un }∞ n=1 of positive real numbers, its critical growth exponent β(u) is the unique real number such that un ∞ if γ < β(u) lim sup γ = 0 if γ > β(u). n→∞ n Thm. 1 can be expressed in terms of the critical exponent β: If the autocorrelation measure σξ is UαH, 0 ≤ α ≤ 1, in the closed interval J, then β(ϕN (ω, ξ) J ) ≤ (2 − α).

3 Driven Harmonic Oscillators Consider the Hamiltonian of a unidimensional harmonic oscillator, with natural frequency ωo , under a time-dependent force H(t) = H(ωo ) + qF (t)

(7)

with H(ωo ) = (p2 + ωo2 q 2 )/2, and F (t) being a piecewise continuous function. For example, given a sequence of continuous real functions Fn deﬁned on (0, 1] set F (t) = Fn (t − n),

t ∈ (n, n + 1].

(8)

Another interesting possibility is given by kicked oscillators with F (t) = ε

∞

ν(n)δ(t − n),

(9)

j=1

where {ν(n)} is a sequence (periodic, almost periodic or random) taking the values ±1, and ε is the kick intensity. Both, the classical and quantum dynamics of (7) with forces (8) and (9) are well deﬁned [6, 8]. We can apply thm. 1 to get Corollary 1. Let U (t, s) be the quantum propagator of (7) with forces (8) or (9) [8]. Set Un = s − lim U (t, s), Λ(n) = Un · · · U0 , t↑n,s↓(n−1)

and suppose the autocorrelation functions for {Λ(n)ξ} exist. If the corresponding autocorrelation measure σξ is UαH, 0 ≤ α ≤ 1, in the closed interval J ⊂ [0, 2π], then (5) holds in this case.

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An interesting point about the harmonic oscillator (7,9) is that a variation of ϕN is directly related to the unperturbed (classical and quantum) energy growth. Let N z(N, ωo ) ≡ einωo ν(n). n=1

For simplicity let’s suppose that initially the classical oscillator is at rest at the origin, i.e., p(0) = q(0) = 0; in this case the value of the unperturbed energy H(ωo ) is given by [6, 8] EC (N, ωo ) =

ε2 2 |z(N, ωo )| . 2

(10)

The time dependence of the energy for general initial conditions, as well as the expectation of the quantum unperturbed energy for ξ ∈ domH(ωo ), diﬀer from (10) by linear terms in z(N, ωo ). So, the long time behaviours of the classical and quantum unperturbed energies are essentially equivalent—see [6, 8, 26] for details, including applications to harmonic oscillators under perturbations modulated along random and substitution sequences. Corollary 2. Suppose the autocorrelation functions N 1 ν(n + r)ν(n) N→∞ N n=1

Cν (r) = lim

exist, and denote the corresponding autocorrelation measure by ην . If ην is UαH, 0 ≤ α ≤ 1, in the closed interval J, then there exists a constant 0 ≤ K < ∞ such that EC (N, ωo ) J ≤ Kε2 N 2−α

(11)

for any N ≥ 1, and the critical exponent β (EC (N, ωo ) J ) ≤ (2 − α) (here the average · J is with respect to ωo ). Proof. Since ν(n) takes values on ±1 it is a sequence of unitary operators on the ˜ ˜ Hilbert space IR. Set Λ(n) = ν(n), Λ(0) = I, and 2 N−1 ijω o˜ . Λ(j) e ϕ˜N (ωo , 1) = j=0

By noting that ϕ˜N (ωo , 1) = |z(N, ωo )|2 , corol. 2 is a simple consequence of thm. 1 and (10). Remark 3.1. General initial conditions are reﬂected only on the value of the constant K in (11); therefore only properties of the numerical sequence ν(n) are relevant for the exponent ruling the energy growth in this case. In [8] there are more

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speciﬁc results on random, Thue-Morse and Rudin-Shapiro sequences; here we need the average since thm. 1 holds for very general sequences {ν(n)}, and so it is expected to give weaker information than any speciﬁc analysis; e.g., the autocorrelation measures ην for Rudin-Shapiro sequence is Lebesgue measure, so (11) implies EC (N, ωo ) J ≤ Kε2 N , a result that follows directly from Saﬀari inequality [8, 27]. Remark 3.2. The upper bound on the average energy growth EC depends only on the behaviour of the autocorrelation measure ην of the perturbing sequence {ν(n)} near the natural frequency ωo ; it is a kind of resonance. In particular, if ην is a positive continuous function times Lebesgue measure in a neighbourhood V of ωo , and pure point outside V (indicating a highly correlated sequence), we still get a linear upper bound for the average energy growth around ωo . Remark 3.3. In the case of substitution sequences ν(n) with pure point autocorrelation measures, we can only infer from thm. 1 that, for any interval J, EC (N, ωo ) J ≤ Kε2 N 2 . That is the case, for instance, of Fibonacci, paper-folding and period doubling sequences [28, 29]. As a ﬁnal remark we comment upon twisted non-autonomous random walks in Hilbert spaces. They are built upon a sequence of unitary operators Un : H ←/ Λ(n) = Un · · · U1 , ω ∈ [0, 2π], and a vector ξ ∈ H; each walk is deﬁned by SN (ω, ξ) =

N

eijω Λ(j)ξ.

j=1

For theoretical and numerical investigations of similar random walks see [30, 24] and references there in. A fundamental question is about the asymptotic behaviour of the mean square displacement ϕN (ω, ξ) = SN (ω, ξ)2 . By thm. 1 it follows that if the autocorrelation measure σξ for {Λ(n)ξ} exists and is UαH in the closed interval J, then we have the following upper bound for the critical exponent of the average mean square displacement β(SN (ω, ξ)2 J ) ≤ (2 − α). Notice that average superdiﬀusive behaviour is not possible if σξ is absolutely continuous with respect to Lebesgue measure in J and with continuous density. Acknowledgements This work was partially supported by CNPq (Brazil).

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References [1] C.-A. Pillet, Commun. Math. Phys., Vol. 102, 1985, p. 237. [2] J.-M. Luck, H. Orland, and U. Smilansky, J. Stat. Phys., Vol. 53, 1988, p. 551. [3] R. Graham, Europhys. Lett., Vol. 8, 1989, p. 717. [4] N. F. de Godoy and R. Graham, Europhys. Lett., Vol. 16, 1991, p. 519. [5] H. R. Jauslin and J. L. Lebowitz, Chaos, Vol. 1, 1991, p. 114. [6] L. Bunimovich, H. Jauslin, J. Lebowitz, A. Pellegrinotti and P. Niebala, J. Stat. Phys., Vol. 62, 1991, p. 793. [7] P. M. Blekher, H. R. Jauslin and J. L. Lebowitz, J. Stat. Phys., Vol. 68, 1992, p. 271. [8] M. Combescure, Ann. Int. Henri Poincar´e, Vol. 57, 1992, p. 67. [9] C. R. de Oliveira, J. Phys. A: Math. Gen., Vol. 27, 1994, p. L847. [10] C. R. de Oliveira, J. Stat. Phys., Vol. 78, 1995, p. 1055. [11] C. R. de Oliveira, Europhys. Lett., Vol. 31, 1995, p. 63. [12] A. Joye, J. Stat. Phys., Vol. 75, 1996, p. 575. [13] W. F. Wreszinski, Helv. Phys. Acta, Vol. 70, 1997, p. 109. [14] J. M. Barbaroux and A. Joye, J. Stat. Phys., Vol. 90, 1998, p. 1225. [15] C. R. de Oliveira, J. Math. Phys., Vol. 39, 1998, p. 4335. [16] S. Tcheremchantsev, Commun. Math. Phys., Vol. 196, 1998, p. 105. [17] A. V. Zhukov, Phys. Lett. A, Vol. 256, 1999, p. 325. [18] J. C. A. Barata, Rev. Math. Phys., Vol. 12, 2000, p. 25. [19] W. O. Amrein, Non-Relativistic Quantum Dynamics, Reidel, 1981, ch. 4. [20] E. Bombieri and J. E. Taylor, J. Phys. Colloque C3, Vol. 47, 1986, p. 19. [21] E. Bombieri and J. E. Taylor, in The Legacy of S. Kovalevskaya, Contemporary Mathematics Vol. 64, American Mathematical Society, 1987. [22] R. S. Strichartz, J. Funct. Anal., Vol. 89, 1990, p. 154. [23] Y. Last, J. Funct. Anal., Vol. 142, 1996, p. 406. [24] A. Hof, Commun. Math. Phys., Vol. 184, p. 567.

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[25] P. Billingsley, Weak Convergence of Measures: Applications in Probability, SIAM, 1971. [26] G. A. Hagedorn, M. Loss and J. Slawny, J. Phys. A: Math. Gen., Vol. 19, 1986, p. 521. [27] B. Saﬀari, C. R. Acad. Sci. Paris, Vol. 304, 1987, p. 127. [28] M. Queﬀ´elec, Substitution Dynamical Systems – Spectral Analysis, LNM Vol. 1294, Springer, Berlin, 1987. [29] F. Axel and D. Gratias, (eds.), Beyond Quasicrystals, Les Editions de Physique/Springer, Berlin, 1995. [30] F. M. Dekking, Random and Automatic Walks, in ref. [29], p. 415. C´esar R. de Oliveira UFSCar – Departamento de Matematica Cx. Postal 676 Sao Carlos SP 13560-970 BRAZIL E-mail : [email protected] Communicated by Gian Michele Graf submitted 10/09/99, accepted 31/05/00

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Annales Henri Poincar´ e

Asymptotic properties of the electromagnetic ﬁeld in the external Schwarzschild spacetime Walter Inglese and Francesco Nicol` o Abstract. We study the asymptotic behaviour of the solutions to the vacuum Maxwell equations in the external Schwarzschild spacetime. The results are based on the extensive use of geometric considerations and the introduction of generalized energy estimates. We obtain the asymptotic behaviour along the null outgoing directions and we prove also some partial results concerning the behaviour along the timelike curves. Our techniques can be also used to control the asymptotic behaviour of the various derivatives of the Maxwell ﬁeld and to obtain the asymptotic behaviour of the Weyl tensor ﬁelds, solutions of the “spin 2” equations.

1 Introduction In this paper we study the asymptotic behaviour of the solutions of the vacuum Maxwell equations in the external Schwarzschild spacetime. The results are based on the extensive use of geometric considerations and the introduction of generalized energy estimates. These ideas and techniques have been introduced by D.Christodolou and S.Klainerman, [Ch-Kl1], in the case of the Minkowski spacetime for the Maxwell equations and for the linear spin 2 equations of the Weyl tensor ﬁeld, the Bianchi equations 1 . Their generalization has been subsequently used by the same authors to prove the much more complicated problem of the non linear stability of the Minkowski space, see [Ch-Kl2], [Kl-Ni] and [Ch-Kl-Ni]. To obtain the asymptotic behaviour of the solutions using geometric considerations is much more complicated in the Schwarzschild spacetime as, diﬀerently from the Minkowski spacetime, the conformal group is not anymore a group of isometries. The only Killing vectors are those associated to the rotation group and the one associated to time translations. Moreover there are not conformal Killing vectors. This is the reason why we have a complete control of the asymptotic behaviour only outside a cone of directions. In other words we are able to control the asymptotic behaviour along the null outgoing geodesics, while we have only partial results for the behaviour along a generic timelike curve. 1 In the ﬂat case one could also try to obtain the same results just looking at the fundamental solution of the equations, while this turns out much more complicated in the Schwarzschild background spacetime. In fact the “strong” Huygens principle is not true in the Schwarzschild spacetime, see [McL], [Fr]. This implies that the value of the solution at a generic point does not depend only on the values at the boundary of the domain of inﬂuence intersected with the hypersurface Σt=0 where the initial data are given.

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Apart from the geometric considerations, our results are based on the construction of a family of generalized energy-type norms and on the control of their boundedness. From it, using Sobolev estimates, we control the L∞ norms of the electromagnetic ﬁeld. Recalling the expression of the metric tensor, in t, r, θ, φ coordinates, ds2 = −(1 −

2m 2 2m −1 2 )dt + (1 − ) dr + r2 (dθ2 + sin2 θdφ2 ), r r

it turns out that the null ingoing geodesics tend asymptotically to the boundary ∂M = {p ∈ M|r(p) = 2m} of the external Schwarzschild spacetime, denoted hereafter by M. Viceversa the outgoing null geodesics starting near to the boundary remain for a “long time” at a small distance from it, the longer the nearer to the boundary the geodesic starts. This is the reason why, as we discuss in detail later on, the asymptotic behaviour of the Maxwell solutions, although with the same power decay as the one in the Minkowski case, it is not “uniform” with respect to the distance of the null curves from ∂M, at the time t = 0 2 . Using similar techniques we can also control the asymptotic behaviour of the various derivatives of the Maxwell solutions. This is possible, of course, if we assume suﬃciently regular initial data. These results will be carefully discussed in a subsequent paper. The central result of this paper is Theorem 3.6 : Let Mδ0 be the region of the spacetime outside the “cone” made by the null outgoing geodesics which at Σt=0 have r = 2m + δ0 . We denote α, α, ρ, σ the null decomposition of the Maxwell tensor ﬁeld with respect to a moving frame adapted to the null outgoing and ingoing “cones” of the Schwarzschild spacetime and assume the inital data suﬃciently regular; then there exists a positive function C2 depending on the initial data norms, the mass m and the distance δ0 such that 3 4 5

sup |r 2 α| ≤ C2 (m, δ0 )

Mδ0

3

sup |rτ−2 α| ≤ C2 (m, δ0 )

Mδ0

1

sup |r2 τ−2 (|ρ − ρ¯|, |σ − σ ¯ |)| ≤ C2 (m, δ0 ) .

Mδ0

2 Our results are obtained in a coordinate independent way. Nevertheless the behaviour near the boundary of the spacetime has a partial dependance on the choice of the moving frame, see the discussion in subsection 2.2. 1 3τ 2 − is the equivalent, in the Schwarzschild spacetime, of the function 1 + (t − r) 2 in the Minkowski case. 4ρ ¯ is the average of ρ over S, the two dimensional surface intersection of the outgoing and ingoing cones. As we discuss later on, r can be deﬁned in a coordinate independent way as 1 2 where |S| is the area of S. r ≡ |S| 4π

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Moreover, if the initial data satisfy also the following conditions q0 sup |r2 ρ| ≤ 4π Σt=0 h0 sup |r2 σ| ≤ 4π Σt=0 then there exists a function C3 depending on the initial data, m and δ0 such that sup |r2 (|ρ|, |σ|)| ≤ C3 (m, δ0 ) .

Mδ0

Finally the functions C2 (m, δ0 ) and C3 (m, δ0 ) diverge as δ0 → 0. Although in the seventies and in the eighties a considerable eﬀort has been done studying the behaviour of scalar, spin 1 and spin 2 wave equations in curved spacetimes and in particular in the Schwarzschild spacetime, see, for instance [Bar-Pre], [Por-St], [St-Sch], [St], [Pr1], [Pr2], nevertheless general results of this type are, in fact, absent. Moreover, although the asymptotic behaviour we ﬁnd can be considered the expected one as the asymptotic decay is the same as the one in the Minkowski case, nevertheless there are various aspects which is worthwhile to point out. a) This result seems in disagreement with the expectations associated to the Penrose compactiﬁcation method [Pe1], [Pe2]. b) Diﬀerently from the ﬂat case, the asymptotic estimates in the null directions are not uniform. We have a partial control of the non uniformity. q0 c) The initial conditions on ρ: supΣt=0 |r2 ρ| ≤ 4π can be interpreted as describing the electric charge inside the internal region of the extended Schwarzschild spacetime. Therefore this approach can be thought as a ﬁrst (linear) step toward the study of the Einstein equations coupled with the Maxwell equations in the presence of a charged “black hole” ([Haw-El]). It can be seen also as a counterpart of the Reissner-Norsdstrom model. d) A technical, but crucial aspect, discussed in detail along the paper, is the use of integral norms performed along the null outgoing and ingoing cones, instead of the more familiar ones, done over the constant time hypersurfaces. e) The same techniques can be applied to the spin 2 equations obtaining similar results. In this case they can be seen as a preliminary step toward the proof of the much more complicated problem of the global nonlinear stability of the spacetime outside the domain of dependance of a compact region at t = 0, see [Kl-Ni] and [Ch-Kl-Ni]. f) An extension of the work of D. Christodolou and S. Klainerman [Ch-Kl1] to general spin ﬁeld equations, always with the Minkowski spacetime as background spacetime, has been developed by Wei-Ton Shu [Shu]. In his ﬁnal remarks he also suggests an extension to the Schwarzschild spacetime proposing some of the modiﬁed pseudo Killing vector ﬁelds we use here. Nevertheless he does not seem to realize the serious diﬃculty arising from the fact that they are not anymore Killing and moreover, in the case of K0 , not even asymptotically Killing.

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2 General properties of the Schwarzschild spacetime and some analytic tools In the spherical coordinates the Schwarzschild metric has the form: ds2 = −(1 −

2m 2 r )dt

+ (1 −

2m −1 2 dr r )

+ r2 (dθ2 + sin2 θdφ2 )

(2.1)

where m is the gravitational mass (units with c = G = 1). This metric is singular for r = 0 and for r = 2m, therefore one has to cut out of the manifold, deﬁned by the coordinates (t, r, θ, φ), the regions r = 0 and r = 2m. The r > 2m region, denoted by M, is called the external Schwarzschild spacetime 5 . In this section we describe the causal structure of the Schwarzschild spacetime with its exact or approximate symmetries and some global Sobolev estimates which are needed to obtain the asymptotic behaviour.

2.1 The foliations of the Schwarzschild spacetime We denote C(u) the null outgoing hypersurfaces, we will call, hereafter, null outgoing cones. They are described by the equations u(p) = const where u is a solution of the eikonal equation g µν ∂µ w∂ν w = 0, satisfying initial conditions on the spacelike hypersurface Σ0 = {p ∈ M|t(p) = 0} such that the null geodesics generating it are outgoing ones. Their tangent vector ﬁeld is Lµ = −g µσ ∂σ u. Analogously C(u) are the null ingoing hypersurfaces, or ingoing cones, described by the equations u(p) = const and u is a solution of the eikonal equation satisfying initial conditions such that the null geodesics generating it are ingoing ones. Their tangent vector ﬁeld is Lµ = −g µσ ∂σ u. It is immediate to realize that in the Schwarzschild spacetime the functions u(p) and u(p) are u = t + r∗ , u = t − r∗ where r∗ ≡ r + 2m log( Deﬁning Φ2 = 1 −

r − 1) 2m

(2.2)

2m r , the null geodesic vector ﬁelds are ∂ ∂ ∂ ∂ ∂ ) = 2Φ−2 + = Φ−2 ( + ∂t ∂r ∂t ∂r∗ ∂u ∂ ∂ ∂ ∂ ∂ L = Φ−2 − ) = 2Φ−2 = Φ−2 ( − ∂t ∂r ∂t ∂r∗ ∂u

L = Φ−2

(2.3)

and g(L, L) = −2Φ−2 . From L and L we deﬁne the null vector ﬁelds ∂ ∂ ∂ + ) = 2Φ−1 ∂t ∂r∗ ∂u ∂ ∂ ∂ e3 = Φ−1 ( − ) = 2Φ−1 ∂t ∂r∗ ∂u

e4 = Φ−1 (

5 For

more details see [Haw-El] and [M-Th-W].

(2.4)

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forming a null pair satisfying g(e3 , e3 ) = g(e4 , e4 ) = 0 , g(e3 , e4 ) = −2 . The null pair {e3 , e4 } and the corresponding null frame is called “coordinate stationary observer frame”, see for instance [Pr1], [Pr2]. In fact it is obtained ∂ ∂ starting from the orthonormal vector ﬁelds Φ−1 ∂t and Φ ∂r associated to the t, r coordinates of a stationary observer. Later on we will discuss the null frames associated to moving observers. The null cones C(u) and C(u) foliate the Schwarzschild spacetime. From them it is possible to deﬁne the two dimensional spacelike surfaces S(u, u) = C(u) ∩ C(u)

(2.5)

which generate a two dimensional foliation. Another foliation we will use is the one made by the constant time hypersurfaces Σt = {p ∈ M|t(p) = t} and, obviously, S(u, u) = S(u, t) where S(u, t) = Σt ∩ C(u). Adding to e3 , e4 an orthonormal frame {ea }a=1,2 relative to the tangent space of the S(u, u) surfaces we obtain a null frame “adapted” to this foliation. For instance ∂ ∂ + ) = ΦL ∂t ∂r∗ ∂ ∂ e3 = Φ−1 ( − ) = ΦL ∂t ∂r∗ ∂ 1 ∂ 1 eθ = , eφ = r ∂θ r sin θ ∂φ e4 = Φ−1 (

(2.6)

The choice of the adapted null frame is not unique. In particular we can consider the following “scaling transformation” for the null pair e3 , e4 : e4 = ae4 , e3 = a−1 e3

(2.7)

where a is a scalar function on M. The vectors of the null pair ∂ ∂ ∂ ∂ + ) = Φ−1 + Φ ∂t ∂r∗ ∂t ∂r ∂ ∂ ∂ ∂ ) = Φ−1 − Φ e3 = Φ−1 ( − ∂t ∂r∗ ∂t ∂r e4 = Φ−1 (

(2.8)

are a combination of the Schwarzschild coordinate basis normalized vectors. They are interpreted as the null pair associated to a stationary observer. If, instead, we consider a “freely falling observer”, that is an observer moving along a radial

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geodesic, for instance from the spatial inﬁnity toward the “origin” of the Schwarzschild spacetime, we can associate to it a diﬀerent null pair e4 , e3 6 e4

=

1−β 1+β

12 e4 ,

e3

=

1+β 1−β

12 e3

(2.9)

12 is his speed at the points p with coordinate r(p) = r. This where β = − 2m r null pair is connected to the previous null pair by the scaling transformation of 12 . This allows to reinterpret the asymptotic decay as seen coeﬃcient a = 1−β 1+β by diﬀerent observer.

2.2 The symmetries of the Schwarzschild spacetime The Schwarzschild spacetime is static which means that the diﬀeomorphisms associated to the time translations are isometries, and is spherically symmetric, that is the group SO(3) is a group of isometries, the two spheres S(u, u) being their transitivity surfaces. A family of one parameter diﬀeomorphisms, Φt , generated by a vector ﬁeld X is a one parameter group of isometries if Φt ∗ g = g which implies LX g = 0, where LX g is the Lie derivative of the metric with respect to X. In this case X is called a Killing vector ﬁeld and LX g = 0 is equivalent to the equation 7 D(α Xβ) = 0. If the diﬀeomorphisms Φt are such that Φt ∗ g = Ω2t g, with Ωt a real, regular scalar pair e 4 , e3 is obtained in the following way: ﬁrst one proves that an observer starting at the “spatial” inﬁnity with zero speed moves radially toward the origin with radial velocity 1 2 . Next we recall that the components of the vector ﬁelds of the tangent space at β = − 2m r the generic point p of M can be interpreted as the normal coordinates associated to the point p of the manifold and, as they describe the local inertial frames of General Relativity, diﬀerent normal coordinates are connected through Lorentz transformations. The Lorentz transformation ∂ ∂ , eR = Φ ∂r into the new vectors associated to the previous β transforms eT = Φ−1 ∂t 6 The

eT = Φ−1 eT + Φ−1 βeR eR = −Φ−1 βeT + Φ−1 eR 1 recalling that Φ−1 = 1 − β 2 2 . Diﬀerently from the previous pair these vectors are not orthonormal, we form an orthonormal pair in the standard way obtaining 2β 1 − β2 eT − e = e , e T R = eR . 1 + β2 1 + β2 R √1 √1 The null pair e eT + e eT − e 4 = R , e3 = R satisﬁes the following relation with the 2 2 stationary observer’s one e 4 = 7D α

1−β 1+β

1 2

e4 , e 3 =

is the covariant derivative Dα ≡ D

symmetrisation symbol.

∂ ∂xα

1+β 1−β

1 2

e3 .

associated to the metric g and (α β) is the

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function on M diﬀerent from zero, then {Φt } describes a conformal isometry and, in this case, LX gαβ ≡(X) παβ = Dα Xβ + Dβ Xα = λX gαβ d where λX (p) = dt ΩX,t (p)t=0 is a regular function on the manifold M and X is a conformal Killing vector ﬁeld. The tensor (X) π is called the deformation tensor of X. As shown in [Ch-Kl1], see also [Kl1] and [Kl2], and discussed in the next subsection, the presence of isometries or conformal isometries is crucial to obtain generalized energy-type estimates. These are used to derive, via Sobolev type theorems, the asymptotic decays of the solutions of the linear equations. Therefore one has to ask which vector ﬁelds can be used in the Schwarzschild spacetime, as, in this case, there are not conformal Killing vector ﬁelds. We introduce the following vector ﬁelds which, although not conformal Killing vectors in M are such that their deformation tensors have nice properties. We call them “pseudo-Killing vectors” 8 9 : ∂ ∂ + r∗ ∂t ∂r∗ ∂ ∂ Ω(i,j) = xi j − xj i ∂x ∂x K0 = 2tS + (r∗2 − t2 )T0

T0 =

∂ , ∂t

S=t

(2.10)

With the exception of T0 , generator of the time translations, and of the Ω(i,j) ’s, generating the spatial rotations, the vector ﬁelds deﬁned in eqs. 2.11 are not Killing 8 In the eqs. 2.11, the coordinates xi are the usual Cartesian ones x1 = r sin θ cos φ , x2 = r sin θ sin φ , x3 = r cos θ. 9 Adding to the previous ones the vector ﬁelds ∂ ∂ ∂ Tr = , Ω(0,r) = −t + r∗ ∂r∗ ∂r∗ ∂t

Kr = 2r∗ S + (r∗2 − t2 )Tr the following commutation relations hold [T0 , Tr ] = 0 , [T0 , Ω(i,j) ] = 0 , [Tr , Ω(i,j) ] = 0 [T0 , Ω(0,r) ] = η00 Tr , [Ω(i,j) , S] = 0 [Tr , Ω(0,r) ] = −ηrr T0 , [S, Ω(0,r) ] = 0 [T0 , S] = T0 , [K0 , Ω(0,r) ] = η00 Kr , [K0 , S] = −K0 [Tr , S] = Tr , [Kr , Ω(0,r) ] = −ηrr K0 , [Kr , S] = Kr [K0 , Tr ] = 2Ω(0,r) , [K0 , T0 ] = 2η00 S [Kr , T0 ] = −2Ω(0,r) , [Kr , Tr ] = 2ηrr S , [Ω(0,r) , Ω(i,j) ] = 0 [Ω(i,j) , K0 ] = 0 , [Ω(i,j) , Kr ] = 0 , [K0 , Kr ] = −4r∗ (r∗2 − t2 )T0 [Ω(i,j) , Ω(l,k) ] = ηil Ω(k,j) − ηik Ω(l,j) + ηjl Ω(i,k) − ηjk Ω(i,l) . where ηµν is the metric of the ﬂat space. The whole algebra will be used in a following paper to obtain the asymptotic behaviour of the higher derivatives.

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nor conformal Killing and their deformation tensors satisfy the following lemma 10 : ˆ be the traceless part of the deformation tensor Lemma 2.1. Let (X) π Chosing as X the vectors S, K0 , the following expressions hold (X)

(X)

π ˆ αβ = µ(X) sign(α)gαβ

where sign(α) =

π = LX g. (2.11)

+1, if α : 0, r −1, if α : θ, φ

and µ(S) = 1 + r∗ (Φ∂r Φ −

Φ2 ) r

µ(K0 ) = 2tµ(S) Moreover, for

m r

(2.12)

small, the previous quantities are, approximately,

µ(S) = O(

m t r r log ) , µ(K0 ) = O(m log ) r 2m r 2m

(2.13)

showing that S and K0 , in the limit m → 0, are conformal Killing vector ﬁelds. ¯ = K0 + T0 in terms of the null It will be useful to express T0 , S, K0 and K pair {e4 , e3 }: Φ (e4 + e3 ) , S = 2 Φ K0 = (u2 e4 + u2 e3 ) , 2

T0 =

Φ (ue4 + ue3 ) 2 2 2 ¯ = Φ (τ+ K e4 + τ− e3 ) 2

(2.14)

2 ≡ l02 + (r∗ ± t)2 and l0 , chosen = 1 unless explicitely stated, where we deﬁne τ± has the dimension of a length. We will often use also the following vector ﬁelds:

1 ˜ = 1 (e4 − e3 ) = Φ ∂ ≡ ΦN T˜0 = (e4 + e3 ) = Φ−1 T0 , N 2 2 ∂r

(2.15)

2.3 The connection coeﬃcients of the external Schwarzschild spacetime The way the submanifolds S(u, u), see eq. 2.5, are embedded in M is determined by their null second fundamental forms χ, χ, two-covariant tensors on S(u, u), χ(X, Y ) = g(DX e4 , Y ), χ(X, Y ) = g(DX e3 , Y ) 10 The

µX functions for the ﬁelds, Kr , Ω(0,r) , introduced in footnote 9, are µ(Kr ) = 2r∗ + (r∗ 2 + t2 )(Φ∂r Φ −

and do not tend to zero for

m r

→ 0.

Φ2 Φ2 ) , µ(Ω(0,r) ) = t(Φ∂r Φ − ) r r

(2.16)

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where X, Y ∈ T S and D denotes the connection on (M, g). In the general case the 1-form ζ(X) = 12 g(DX e4 , e3 ), called the torsion of S, is also needed to describe the embedding. In the Schwarzschild spacetime, due to its rotational symmetry, ζ and the traceless parts of χ and χ are identically zero so that χab =

Φ Φ 1 1 δab trχ = δab , χab = δab trχ = −δab 2 r 2 r

(2.17)

The second fundamental forms χ, χ and the torsion ζ are a subset of the whole family of connection coeﬃcients which describe the geometric structure of the whole spacetime. The other ones diﬀerent from zero are 1 1 ω ≡ − D4 log Φ , ω ≡ − D3 log Φ 2 2

(2.18)

These coeﬃcients satisfy the null structure equations of the manifold.

2.4 Global Sobolev estimates We introduce on M the following Euclidean pointwise norm for the generic tensor ﬁeld U : |U |2

...ρm σ1 ...σm µ1 ν1 ≡ Uµρ11...µ Uν1 ...νn g¯ . . . g¯µn νn g¯ρ1 σ1 . . . g¯ρm σm n

m n

(2.19)

where g¯µν ≡ gµν +2(T˜0 )µ (T˜0 )ν . We denote D / 4, D / 3 the projections over the tangent space T S of D4 ≡ De4 and D3 ≡ De3 and ∇ / the Levi-Civita connection relative to the induced metric on S(u, u). The proofs of the following lemmas and propositions are in the Appendix. Lemma 2.2. Let G be a C ∞ tensor ﬁeld tangent to S(u, u), then the following Sobolev estimate holds

14 sup |G| ≤ cr− 2 1

S(u,u)

|G|4 + |r∇ / G|4

(2.20)

S(u,u)

Here and in the sequel c denotes a constant independent from the relevant parameters. Proposition 2.3. Let U be a C ∞ tensor ﬁeld tangent at each point to the corresponding S(u, u), let us denote C(u; [u0 , u]) the portion of the null outgoing cone where u varies in the interval [u0 , u] and introduce the analogous deﬁnition for C(u, [u0 , u]). The following estimates hold : 14 3 supS(u,u) (r 2 |U |) ≤ c S(u,u ) r4 |U |4 + r4 |r∇ / U |4 0 12 2 + C(u;[u ,u]) |U |2 + r2 |∇ / U |2 + r2 |D / 4 U |2 + r4 |∇ / U |2 + r4 |∇ /D / 4 U |2 0

(2.21)

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14 1 2 2 supS(u,u) (rτ−2 |U |) ≤ c S(u,u ) r2 τ− |U |4 + r2 τ− |r∇ / U |4 0 12 2 2 2 + C(u;[u ,u]) |U |2 + r2 |∇ / U |2 + τ− |D / 4 U |2 + r4 |∇ / U |2 + r2 τ− |∇ /D / 4 U |2 0

(2.22) where the integrals over the null cones C(u; [u0 , u]) and C(u; [u0 , u]) are deﬁned in the following way 11

u

F ≡ du ΦF

C(u;[u0 ,u])

u0 u

F ≡ u0

C(u;[u0 ,u])

du

S(u,u )

ΦF.

(2.23)

S(u ,u)

A similar result holds expressing the sup-norms in terms of integrals along the C(u, [u0 , u]) null cones: 14 3 / U |4 supS(u,u) (r 2 |U |) ≤ c S(u0 ,u) r4 |U |4 + r4 |r∇ 12 + C(u,[u0 ,u]) |U |2 + r2 |∇ / U |2 + r2 |D / 3 U |2 + r4 |∇ / 2 U |2 + r4 |∇ /D / 3 U |2 (2.24) 14 1 2 2 |U |4 + r2 τ− |r∇ / U |4 supS(u,u) (rτ−2 |U |) ≤ c S(u0 ,u) r2 τ− 12 2 2 + C(u,[u0 ,u]) |U |2 + r2 |∇ / U |2 + τ− |D / 3 U |2 + r4 |∇ / 2 U |2 + r2 τ− |∇ /D / 3 U |2 (2.25) Together with this proposition we will use another proposition, similar to the previous one, where the sup-norm of the function U is estimated in terms of L2 norms relative to a Σt hypersurface. Proposition 2.4. Let U be a C ∞ tensor ﬁeld tangent at each point p to the corresponding S(t, r)12 and satisfying lim r|U (r, ω)| = 0

r→∞

11 As C(u; [u , u]) and C(u; [u , u]) are null hypersurfaces, there is not a “canonical” deﬁnition 0 0 of their volume so that we are free to choose an appropriate deﬁnition for

and . C(u;[u0 ,u])

C(u;[u0 ,u])

The deﬁnition we use diﬀers by a factor Φ, from the one used in [Ch-Kl2], page 221. 12 S(t, r) = S(t(u, u), r(u, u)) = S(u, u).

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where with ω we indicate the angular coordinates, then the following estimates hold: Nondegenerate version: 3

sup (r 2 |U |) S(t,r)

≤c Σt

2

|U | + r |∇ / U | + r |D / N˜ U | + r |∇ / U | + r |∇ /D / N˜ U | 2

2

2

2

2

4

2

4

2

12 (2.26)

Degenerate version: 1

sup (rτ−2 |U |)

S(t,r)

2

≤c Σt

2 2 |U |2 + r2 |∇ / U |2 + τ− |D / N˜ U |2 + r4 |∇ / U |2 + r2 τ− |∇ /D / N˜ U |2

12

(2.27) where the integral over the spacelike hypersurface Σt is

∞

H≡ dr Φ−1 H S(t,r )

2m

Σt

13

An immediate corollary of this proposition is Corollary 2.5. Let U be a C ∞ tensor ﬁeld tangent at each point p to the corresponding S(t, r) and satisfying lim r|U (r, ω)| = 0

r→∞

where with ω we indicate the angular coordinates, then the following estimates hold: Nondegenerate version:

sup (r4 |U |4 + r4 |r∇ / U |4 ) r≥2m

S(t,r)

≤c Σt

|U |2 + r2 |∇ / U |2 + r2 |D / N˜ U |2 + r4 |∇ / 2 U |2 + r4 |∇ /D / N˜ U |2

12 (2.28)

13 In

this case 1

dV = θ(t) ∧θ(r) ∧θ(θ) ∧θ(φ) (T˜0 , ·, ·, ·) = T˜0ν ˜ναβγ dxα ∧dxβ ∧dxγ = Φ−1 |detg| 2 ijk dxi ∧dxj ∧dxk and

Σt

∞

dV =

dr r=2m

S(t,r)

Φ−1 r2 sinθdθdφ

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Degenerate version:

2 2 (r2 τ− |U |4 + r2 τ− |r∇ / U |4 ) sup r≥2m

S(t,r)

≤c

|U | + r |∇ / U| + 2

2

Σt

2

2 τ− |D / N˜ U |2

2

+ r |∇ / U| + 4

2

2 r2 τ− |∇ /D / N˜ U |2

12

(2.29)

3 The asymptotic behaviour of the solutions of the Maxwell equations 3.1 The null decomposition of the electromagnetic ﬁeld The electromagnetic ﬁeld is a two form F . We denote by ∗ F its Hodge dual, whose components are ∗ Fµν = 12 5˜µνρσ F ρσ where 5˜ = dV and, for a generic choice 1

1 2 3 4 of coordinates, 5˜µνρσ = (g) 2 δ[µ δν δρ δσ] . g denotes the absolute value |det{gµν }|. The vacuum Maxwell equations are 14 :

D[λ Fµν] = 0 , D[λ ∗ F µν] = 0

(3.30)

The following lemma, whose elementary proof is in [Ch-Kl1], will be used over and over: Lemma 3.1. The Maxwell equations are invariant under isometries and conformal isometries. In particular, if X is the Killing or conformal Killing vector ﬁeld associated to the isometry and F is a solution of the Maxwell equations, then also LX F is a solution. Given a vector ﬁeld X we deﬁne the one form iX F ≡ F (·, X). The one form iX ∗ F is deﬁned in the same way. iX F ,iX ∗ F completely determine the two form F at any point where X, X ≡ g(X, X) is diﬀerent from zero. If X = T˜0 the one forms E = iT˜0 F, H = iT˜0 ∗ F are called the electric and magnetic parts of F , respectively, and they are tangential to the hypersurfaces Σt . The null decomposition of the electromagnetic tensor F in terms of one forms and scalar functions on S is deﬁned in the following way 15 : αa ≡ α(F )(ea ) = F (ea , e4 ) 14 The

vacuum Maxwell equations can also be written as Dµ Fµν = 0 , Dµ∗ F µν = 0 .

as tensor ﬁelds on M can be written as 1 1 µν ν αµ = Πµλ Fλν eν3 , αµ = Πµλ Fλν eν4 , ρ = Fµν eµ ˜ Fµν 3 e4 , σ = 2 2 λ where ˜µν ≡ 12 ˜µνρσ eρ3 eσ 4 is the area form of the two spheres S(u, u) and Πµ is the projection µ ν 1 µ ν µν µν tensor over T S: Π = g + 2 (e3 e4 + e4 e3 ). 15 α, α, ρ, σ

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Asymptotics of the Electromagnetic Field in Schwarzschild Spacetime

αa ≡ α(F )(ea ) = F (ea , e3 ) 1 ρ ≡ ρ(F ) = F (e3 , e4 ) 2 σ ≡ σ(F ) = F (eθ , eφ ) ∗

F can be similarily decomposed in terms of ⊗ α , ⊗

α = ∗α ,

⊗

α = −∗ α ,

⊗

⊗

α,

ρ=σ ,

⊗

907

(3.31)

⊗

ρ,

⊗

σ with

σ = −ρ

(3.32)

and ∗ indicates the Hodge dual on S(u, u). Finally the null components can be expressed in terms of electric and magnetic parts of F : αa = F (ea , e4 ) = (Ea + 5ab Hb ) αa = F (ea , e3 ) = (Ea − 5ab Hb ) ˜ ) = −E⊥ ρ = F (T˜0 , N ∗ ˜ ) = −H⊥ σ = F (T˜0 , N

(3.33)

3.2 The Maxwell equations in the null decomposition With respect to the null decomposition, the Maxwell equations have the following form 16 : Φ )α + ∇ / ρ −∗ ∇ /σ = 0 r Φ /σ = 0 D / 3 α − (∂r Φ + )α − ∇ / ρ −∗ ∇ r Φ Φ D4 σ + 2 σ + curl / α = 0 , D4 ρ + 2 ρ − div / α=0 r r Φ Φ D3 σ − 2 σ + curl / α = 0 , D3 ρ − 2 ρ + div / α=0 r r D / 4 α + (∂r Φ +

(3.34)

which, written in terms of the null components, become Φ )α(ea ) + ∂ea ρ − 5ab ∂eb σ = 0 r Φ ∂e3 α(ea ) − (∂r Φ + )α(ea ) − ∂ea ρ − 5ab ∂eb σ = 0 r Φ cot θ ∂e4 σ + 2 σ + ∂eθ α(eφ ) − ∂eφ α(eθ ) + α(eφ ) = 0 r r Φ cot θ ∂e4 ρ + 2 ρ − ∂eθ α(eθ ) − ∂eφ α(eφ ) − α(eθ ) = 0 r r

∂e4 α(ea ) + (∂r Φ +

16 We

use the following deﬁnitions / f = ab ∇ / af b. div / f =∇ / a fa , curl

(3.35)

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Φ cot θ α(eφ ) = 0 ∂e3 σ − 2 σ + ∂eθ α(eφ ) − ∂eφ α(eθ ) + r r Φ cot θ ∂e3 ρ − 2 ρ + ∂eθ α(eθ ) + ∂eφ α(eφ ) + α(eθ ) = 0 r r

3.3 The energy-momentum tensor and the energy-type norms The control of the asymptotic behaviour of the solutions of Maxwell equations in the Schwarzschild spacetime is based, as discussed in the introduction, on two results: a) The existence of bounded “energy-type” norms. b) The existence of exact or, at least, approximated symmetries of the Schwarzschild spacetime. The starting ingredient to deﬁne these “energy-type” norms is the Maxwell energy-momentum tensor Q: Q(X, Y ) = iX F, iY F + iX ∗F , iY ∗F

(3.36)

for generic vector ﬁelds X, Y . Its components are Qµν = Fµρ Fν ρ + ∗Fµρ ∗Fν ρ = 2Fµρ Fν ρ − 12 gµν Fρσ F ρσ . Lemma 3.2. The energy-momentum tensor ﬁeld Q has the following properties: I) Q is symmetric, traceless and for any non spacelike future directed vector ﬁelds X, Y satisﬁes Q(X, Y ) ≥ 0. II) If F is a solution of the Maxwell equations then Q has vanishing divergence Dµ Qµν = 0. The proof of I) and II) is elementary, see [Ch-Kl1]. The following expressions hold: Q(e3 , e3 ) = 2|α|2 , Q(e4 , e4 ) = 2|α|2 Q(e3 , e4 ) = 2(ρ2 + σ2 )

(3.37)

¯ are timelike future directed vector ﬁelds, the following quantities are non As T0 , K negative 17 18 Q(T0 , e4 ) = Φ{|α|2 + (ρ2 + σ2 )} 2 2 ¯ e4 ) = Φ{τ+ Q(K, |α|2 + τ− (ρ2 + σ2 )} 2 2 Q(T0 , e3 ) = Φ{|α| + (ρ + σ2 )} 17 The

last two are in fact strictly positive. choice of T˜0 in the second argument of Q(·, ·) is due to the fact that T˜0 is the unit vector normal to Σt and plays the same role as e4 , e3 in the previous expressions. Viceversa in the ﬁrst ¯ approximates the corresponding conformal Killing argument, T0 is a Killing vector ﬁeld and K vector ﬁeld of the Minkowski spacetime. 18 The

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Asymptotics of the Electromagnetic Field in Schwarzschild Spacetime 2 2 2 ¯ e3 ) = Φ{τ− Q(K, |α|2 + τ+ (ρ + σ2 )} 2 2 Q(T0 , T˜0 ) = Φ/2{|α| + |α| + 2(ρ2 + σ2 )} 2

909

(3.38)

2

2 2 ¯ T˜0 ) = Φ/2{τ− 2 |α| + τ+ 2 |α| + (τ+ Q(K, + τ− )(ρ2 + σ2 )}

¯ T˜0 ) over the hypersurfaces Σt , the integrals The integrals of Q(T0 , T˜0 ), Q(K, ¯ ¯ e3 ) of Q(T0 , e4 ), Q(K, e4 ) along the null cones C(u) and those of Q(T0 , e3 ), Q(K, along the null cones C(u) are the energy-type norms we are going to use. Their relevance follows from the fact that the properties of the tensor Q and the asymptotically approximate symmetries of the Schwarzschild spacetime allow to prove their boundedness once they are bounded on the initial hypersurface. This result, crucial for proving the asymptotic behaviour, is thoroughly investigated in the next section. Here we assume that a family of integral norms is bounded and derive from this assumption the asymptotic behaviour for the solutions of the Maxwell equations. Let us consider the following integrals 19 :

Q(LaO F )(T0 , T˜0 ) , Q(LbT0 LaO F )(T0 , T˜0 ) (3.39) Σt

Σt

C(u;[u0 ,u])

¯ e4 ) Q(LaO F )(K,

, C(u;[u0 ,u])

C(u;[u0 ,u])

¯ e4 ) , Q(LS LaO F )(K,

where a ≥ 1, b ≥ 0,

C(u;[u0 ,u])

¯ e3 ) (3.40) Q(LS LaO F )(K,

C(u;[u0 ,u])

¯ e3 ) Q(LaO F )(K,

¯ e4 ) , Q(LbS LaO F )(K,

C(u;[u0 ,u])

¯ e3 ) Q(LbS LaO F )(K,

Q(LaO F )

Q(LbS LaO F )

is deﬁned in the following way

= Q(LbS LΩi1 j1 ...LΩia ja F )

(3.41)

i1 j1 ,..,ia ja

where Ω(ij) are the Killing vector ﬁelds associated to the rotation group and in the sum: il < jl . From these integrals we deﬁne the following norms:

O ¯ e4 ) Ik (u; [u0 , u]) = Q(LaO F )(K, 1≤a≤k+2

IkS (u; [u0 , u]) =

1≤a≤k+1

IO k (u; [u0 , u])

=

1≤a≤k+2

I Sk (u; [u0 , u])

=

1≤a≤k+1

C(u;[u0 ,u])

C(u;[u0 ,u])

¯ e4 ) Q(LS LaO F )(K,

C(u;[u0 ,u])

C(u;[u0 ,u])

¯ e3 ) Q(LaO F )(K,

(3.42)

¯ e3 ) Q(LS LaO F )(K,

¯ e4,3 ) and ¯ ˜ reason why we do not assume the integrals C,C Q(F )(K, Σ Q(F )(K, T0 ) bounded follows from the remark b) after Theorem 3.7. 19 The

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We introduce also similar quantities relative to Σt :

O ¯ T˜0 ) Ik (Σt ) = Q(LaO F )(K, 1≤a≤k+2

IkS (Σt )

=

Σt

Σt

1≤a≤k+1

and

¯ T˜0 ) Q(LS LaO F )(K,

(3.43)

20

EkO (Σt )

=

1≤a≤k+2

EkT0 (Σt )

=

1≤a≤k+1

Σt

Q(LaO F )(T0 , T˜0 )

Σt

Q(LT0 LaO F )(T0 , T˜0 )

(3.44)

Fixed (u, u) let t¯ ∈ [0, 12 (u + u)]; we denote u0 (t¯) and u0 (t¯) the values 21 of the functions u(p), u(p) at the intersections C(u) ∩ Σt¯ and C(u) ∩ Σt¯ respectively. Finally Σt (≥ r) is the portion of the hypersurface Σt made by points whose radial coordinates are greater or equal to r. The following proposition allows to estimate the sup-norms of the null components of the Maxwell ﬁelds in terms of the energy-type norms, eqs. 3.42, 3.43, relative to the null cones and to the Σt¯ hypersurface. Proposition 3.3. Let t = 12 (u + u) suﬃciently large. There exists t¯ ∈ [0, 12 (u + u)) such that every regular 22 solution of the vacuum Maxwell equations in the external Schwarzschild spacetime satisﬁes the following inequalities 1 5 r 2 |α(u, u)| ≤ cΦ(r(u, u0 (t¯)))−2 I0O (u; [u0 (t¯), u]) + I0S (u; [u0 (t¯), u]) 2 1 + I0O (Σt¯(≥ r(u, u0 (t¯))) + I0S (Σt¯(≥ r(u, u0 (t¯))) 2 3 12 S ¯ ¯ rτ−2 |α(u, u)| ≤ cΦ(r(u, u))−2 I O 0 (u; [u0 (t), u]) + I 0 (u; [u0 (t), u]) 1 + I0O (Σt¯(≥ r(u0 (t¯), u)) + I0S (Σt¯(≥ r(u0 (t¯), u)) 2 r2 (|ρ(u, u)|, |σ(u, u)|) ≤ c supΣt¯ |r2 (¯ ρ, σ ¯ )| 1 1 1 − ¯ 2 +cτ− 2 Φ(r(u, u))− 2 I O 0 (u; [u0 (t), u]) 1 + I0O (Σt¯(≥ r(u0 (t¯), u)) + I0S (Σt¯(≥ r(u0 (t¯), u)) 2

(3.45)

where c is a generic constant independent from u, u. 20 The E norms are introduced as they are used to obtain a weaker, but more general asymptotic result as discussed later on. 21 In fact u (t¯) and u (t¯) depend also on u and u respectively and should be written u (t¯, u) 0 0 0 and u0 (u, t¯), but we omit this dependence to simplify the notation. 22 The exact meaning of “regular” is discussed in Theorem 3.6.

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Proof. We sketch the proof for the null component α and leave in the Appendix the detailed proof for each component. We apply ﬁrst Proposition 2.3 with U = rα. ¯ ¯ Using the explicit expressions of the Q(L1,2 O F )(K, e4 ) and Q(LS LO F )(K, e4 ), 2 2 2 2 / α| , .... are see eqs. 3.41, it follows that the C(u, [u0 , u]) integrals of r |α| , r |r∇ bounded in terms of the norm integrals I0O (u; [u0 , u]) and I0S (u; [u0 , u]). It is here that the factor Φ(r(u, u0 (t¯))) appears. Next we use Corollary 2.5 to express the L4 (S(u, u0 (t¯))) norms of rα and r2 ∇ / α in terms of integrals 23 along the hyper¯ surface Σt¯(≥ r(u, u0 (t))) and, proceeding as before, these integrals are expressed in terms of the norms 3.43 producing, also in this case, the factor Φ(r(u, u0 (t¯))). Analogous arguments, described in the Appendix, apply for the other null components. We observe that in the case of α we are forced to use the norms I O 0 (u; [u0 , u]) and I S0 (u; [u0 , u]) and, in this case, the factor Φ(r(u0 (t¯), u)) appears. Remark. This Proposition is a preliminary step to the control of the asymptotic behaviour of the null components of the electromagnetic ﬁeld. Here the need for t¯ being large comes from the control of D / 4 α in terms of D / S α and D / 3 α, see eq. 5.129. The main result, still to prove, is to show that the energy-type norms in 3.45 are bounded in terms of (the norms of) the initial data. These estimates also require an appropriate choice of t¯. The next two propositions describe the central technical result of the paper and together with the previous one allow to prove the main theorem. They prove that the I norms appearing in Proposition 3.3, although not conserved, can be bounded in terms of the initial data. Let δ0 r∗ (δ0 ) = (2m + δ0 ) + 2m log (3.46) 2m and denote Mδ0 the region of the external Schwarzschild spacetime Mδ0 = {p ∈ M|t(p) ≥ 0, u(p) ≤ −r∗ (δ0 )}

(3.47)

We deﬁne V (u, u) the part of the domain of dependance of S(u, u) above the initial hypersurface V (u, u) = {p ∈ J − (S(u, u))|t(p) ≥ 0}

(3.48)

whose boundary is formed by the union of the portions of the null cones C(u) and C(u) lying in V (u, u) and a ﬁnite region of Σ0 . Moreover we decompose V (u, u) as V (u, u) = V≤t¯(u, u) ∪ V≥t¯(u, u)

(3.49)

where V≥t¯(u, u) denotes the part of V (u, u) above Σt¯. 23 We

have, nevertheless, to prove that the assumption lim r|U (r, ω)| = 0

r→∞

required in Proposition 2.4 is satisﬁed for the choice of the U tensor done here. These asymptotic spatial behaviour can be proved using this same Proposition 3.3 with a δ0 suﬃciently large to choose t¯ = 0 and the fact that, due to Proposition 3.5 the r.h.s. of eqs. 3.45 are bounded.

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Proposition 3.4. Let δ0 > 0 and assume t = 12 (u + u) suﬃciently large 24 . Given a positive constant C0 suﬃciently large, there exists a time 25 t¯0 = t¯(m, δ0 ) such that for any (u, u) ∈ Mδ0 , satisfying the assumption on t, and t¯ ∈ [t¯0 , t) the following estimates hold I0O (u; [u0 (t¯), u]) ≤ C0 I0O (Σt¯ ∩ V (u, u)) ≤ C0 I0O (Σt¯ ∩ Mδ0 ) I0S (u; [u0 (t¯), u]) ≤ C0 I0O (Σt¯ ∩ V (u, u)) + I0S (Σt¯ ∩ V (u, u)) ≤ C0 I0O (Σt¯ ∩ Mδ0 ) + I0S (Σt¯ ∩ Mδ0 ) O O ¯ IO 0 (u; [u0 (t), u]) ≤ C0 I0 (Σt¯ ∩ V (u, u)) ≤ C0 I0 (Σt¯ ∩ Mδ0 ) I S0 (u; [u0 (t¯), u]) ≤ C0 I0O (Σt¯ ∩ V (u, u)) + I0S (Σt¯ ∩ V (u, u)) ≤ C0 I0O (Σt¯ ∩ Mδ0 ) + I0S (Σt¯ ∩ Mδ0 )

(3.50)

where Σt¯ ∩ V (u, u) is the subset of Σt¯ with r(p) ∈ [r(u, u0 (t¯)), r(u0 (t¯), u)] and Σt¯ ∩ Mδ0 is the subset of Σt¯ with r(p) ≥ r(u = −r∗ (δ0 ), u0 (t¯)). From the previous equation it follows immediately that 26 sup (u,u)∈Mδ0 ∩V≥t¯(u,u)

sup (u,u)∈Mδ0 ∩V≥t¯(u,u)

sup (u,u)∈Mδ0 ∩V≥t¯(u,u)

sup (u,u)∈Mδ0 ∩V≥t¯(u,u)

I0O (u; [u0 (t¯), u]) ≤ C0 I0O (Σt¯ ∩ Mδ0 ) I0S (u; [u0 (t¯), u]) ≤ C0 I0O (Σt¯ ∩ Mδ0 ) + I0S (Σt¯ ∩ Mδ0 ) O ¯ IO 0 (u; [u0 (t), u]) ≤ C0 I0 (Σt¯ ∩ Mδ0 )

(3.51)

I S0 (u; [u0 (t¯), u]) ≤ C0 I0O (Σt¯ ∩ Mδ0 ) + I0S (Σt¯ ∩ Mδ0 )

Moreover the function t¯(m, δ0 ) diverges as δ0 → 0. Proposition 3.5. For a generic t¯ > 0 the following inequality holds O I0 (Σt¯ ∩ Mδ0 ) + I0S (Σt¯ ∩ Mδ0 ) ≤ C1 (m, δ0 ; t¯) I0O (Σ0 ) + I0S (Σ0 ) (3.52) C1 (m, δ0 ; t¯) is a positive function increasing in m and t¯ and decreasing in δ0 , such that, for m > 0 lim C1 (m, δ0 ; t¯) = ∞

δ0 →0

lim C1 (m, δ0 ; t¯) = ∞

t¯→∞

(3.53)

The proofs of Propositions 3.4, 3.5 are discussed in section 4. 24 The

meaning of “suﬃciently large” will be clear in the course of the proof. time t¯0 depends, obviously, also on C0 . O 26 It is worthwhile to observe that the estimates for sup ¯ (u,u)∈Mδ0 I 0 (u; [u0 (t), u]) and S ¯ sup(u,u)∈Mδ I 0 (u; [u0 (t), u]) are very rough. 25 The

0

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3.4 The asymptotic behaviour Using the results discussed in the previous propositions we can state the main theorems of this paper concerning the asymptotic behaviour of a class of solutions of the Maxwell equations. We consider the Cauchy problem for the Maxwell equations 3.30 in the external Schwarzschild spacetime where the initial data are given on the Σ0 hypersurface. We specify them in terms of the norms 3.43, 3.44 relative to the Σ0 hypersurface, with k = 0:

O ¯ T˜0 ) I0 (Σ0 ) = Q(LaO F )(K, 1≤a≤2

Σ0

1≤a≤1

Σ0

I0S (Σ0 ) =

E0O (Σ0 ) =

1≤a≤2

Σ0

1≤a≤1

Σ0

E0T0 (Σ0 ) =

¯ T˜0 ) Q(LS LaO F )(K,

(3.54)

Q(LaO F )(T0 , T˜0 ) Q(LT0 LaO F )(T0 , T˜0 )

(3.55)

Theorem 3.6. Let the initial data be such that I0O (Σ0 ) and I0S (Σ0 ) are bounded, let δ0 > 0 be ﬁxed, then there exists a positive function C2 depending on the initial data I-norms, m and δ0 such that 5

sup |r 2 α| ≤ C2 (m, δ0 )

Mδ0

3

sup |rτ−2 α| ≤ C2 (m, δ0 )

Mδ0

(3.56)

1

sup |r2 τ−2 (|ρ − ρ¯|, |σ − σ ¯ |)| ≤ C2 (m, δ0 ) .

Mδ0

Moreover, if the initial data satisfy also the following conditions sup |r2 ρ| ≤

q0 h0 , sup |r2 σ| ≤ 4π 4π Σ0

(3.57)

sup |r2 ρ| ≤

q0 h0 , sup |r2 σ| ≤ 4π 4π Σt

(3.58)

Σ0

it follows that

Σt

and there exists a constant C3 , depending on m, δ0 and the initial data, such that sup |r2 (|ρ|, |σ|)| ≤ C3 (m, δ0 )

Mδ0

(3.59)

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Finally the constants C2 (m, δ0 ) and C3 (m, δ0 ) are bounded by a positive function which, as δ0 → 0, diverges at most as:   δ0 3 |m log 2m | 2m  C2,3 (m, δ0 ) = O  (3.60) δ0 We recall that, for any δ0 > 0, the boundary of the region Mδ0 is the part of the null cone C(u = −r∗ (δ0 )) with t > 0. Therefore to control the asymptotic behaviour of the Maxwell solution in Mδ0 amounts to controlling the asymptotic behaviour along the null outgoing geodesics, that is moving on the null outgoing cones C(u) or, obviously, along the spacelike curves inside Mδ0 27 . Using the E norms, eq.3.44, it is easier to prove a weaker proposition analogous to Proposition 3.3 and obtain the following Theorem 3.7. Let the initial data of the Maxwell equations be such that E0O (Σ0 ), E0T0 (Σ0 ) are bounded, then, ﬁxed δ0 > 0, there exists a positive constant C4 depending on m, δ0 and the initial data E-norms such that for r ≥ 2m + δ0 ≡ r0 (δ0 ) 1

|r 2 α| ≤ C4 (m, δ0 )

sup r≥2m+δ0

1

|r 2 α| ≤ C4 (m, δ0 )

sup r≥2m+δ0

3

|r 2 (|ρ|, |σ|)| ≤ C5 (m, δ0 )

sup r≥2m+δ0 − 32

where C4 (m, δ0 ) ≤ CΦ(r0 (δ0 ))

− 12

and C5 (m, δ0 ) ≤ CΦ(r0 (δ0 ))

.

Remarks. a) The diﬀerence between the last theorem and the previous one is that, in this case, the curves along which we consider the asymptotic behaviour can go out from the region Mδ0 as, for instance, the time like curves r = const. The result is, nevertheless, much weaker and, probably, not optimal. b) The rational behind the assumption on ρ and σ on Σ0 is the following one: ˜ , T˜0 ) = if we assume that supΣ0 |r2 ρ| is bounded and diﬀerent from zero, from F (N −ρ and divE = 0 it follows that

q0 = ρ = 0. S(t=0,r=2m+δ0 )

This can be interpreted as the global electric charge contained in the internal part of the Schwarzschild spacetime. The requirement that supMδ0 |r2 ρ| is bounded can, therefore, be interpreted as the request that the global electric charge contained 27 In fact one can also build a timelike curve totally contained in M δ0 which asymptotically approximate a null geodesic. Along this curve the result of Theorem 3.6 holds.

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in the extended spacetime be ﬁnite. An analogous argument can be done for the assumption on σ, and physically S(t=0,r=2m+δ0 ) σ can be interpreted as the global magnetic charge contained in the extended spacetime. c) Proposition 3.4 has, in fact, a more general version allowing to bound S a larger family of norms containing IkO , IkS , I O k , I k for k ≥ 0 in terms of the corresponding quantities relative to the initial hypersurface. Using this general version we will be able to control the asymptotic behaviour of the derivatives of the solutions of the Maxwell equations in the region Mδ0 , if we control the analogous quantities on the initial hypersurface, that is provided that the initial data are suﬃciently regular. This will be discussed in a next paper where the full algebra of the pseudo Killing vector ﬁelds will be used. d) The main diﬀerence between the results proved here and the analogous ones proved in the ﬂat case, see [Ch-Kl1], is that the asymptotic behaviour here is not uniform. This is expressed by the fact that the constants C2 (m, δ0 ) and C3 (m, δ0 ) diverge as δ0 tends to zero 28 . Moreover we have to remark that the results obtained here are in disagreement with those expected using the compactiﬁcation arguments, see [Pe1], [Pe2], concerning the asymptotic behaviour of α. e) It is important to observe that if we choose a diﬀerent null pair, for instance the one associated to the “freely falling observer”, e4 = Λe4 , e3 = Λ−1 e3 where Λ = 1−β 1+β , the null components of the Maxwell tensor ﬁeld transform in the following way: αa = Λαa , αa = Λ−1 αa ρ = ρ , σ = σ

(3.61)

This remark shows that we cannot eliminate the non uniformity, for δ0 → 0, of the functions C2 (m, δ0 ) and C3 (m, δ0 ) just “changing the observer”.

4 The control of the energy-type norms In this section we prove that the energy-type norms introduced in the previous section are bounded in terms of analogous norms for the initial data. This is the content of Propositions 3.4 and 3.5 which are the main technical results of this work and allow to prove Theorems 3.6, 3.7 discussed in the previous section. We recall a Proposition, whose simple proof is in [Ch-Kl1] Proposition 4.1. Let Q(G) be the energy-momentum tensor ﬁeld of an antisymmetric two form G and let X be a vector ﬁeld. Deﬁne the covariant vector ﬁeld 28 One has also to recall that the null cones of the Schwarzschild spacetime diﬀer from the corresponding ones of the Minkowski spacetime and diverge from them asymptotically.

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P associated to X, Pα = Q(G)αβ X β , then, as Q is symmetric and traceless, it follows that29 1 (X) αβ β divP = (DivQ(G))β X + Qαβ π ˆ (4.62) 2 where the deformation tensor (X) π = LX g, measures how much the diﬀeomorphism generated by X diﬀers from an isometry or a conformal isometry. (X) π ˆ is its traceless part. Corollary 4.2. Let the tensor ﬁeld G satisfy the vacuum Maxwell equations and X be a Killing or conformal Killing vector ﬁeld, then divP = 0. Integrating divP in the region V≥t¯(u, u) and using Stokes theorem we obtain the following Lemma: Lemma 4.3. Let Pα = Q(G)αβ X β be deﬁned as in Proposition 4.1, then Stokes theorem implies

Φ−1 Q(G)(X, e3 ) + Φ−1 Q(G)(X, e4 ) C(u)∩V≥t¯(u,u)

−

Σt¯∩V (u,u)

Q(G)(X, T˜0 )

C(u)∩V≥t¯(u,u)

1 β αβ (X) =− π ˆ αβ (DivQ(G))β X + Q(G) 2 V≥t¯(u,u)

(4.63) Choosing as G, LΩij F and L2Ωij F , with F solution of the Maxwell equations, and observing that, due to the spherical symmetry of the spacetime, LnΩij F is also a solution for any ij and n ≥ 0, it follows that, posing X = T0 , the E norms deﬁned ¯ we have, see eq. 3.42, in eq. 3.44 are conserved 30 . Viceversa, posing X = K (O) O ¯ I0O (u; [u0 (t¯), u]) + I O (V≥t¯(u, u)) 0 (u; [u0 (t), u]) − I0 (Σt¯ ∩ V (u, u)) ≤ Err (4.64)

where Err

(O)

1 ¯ αβ (V≥t¯(u, u)) = |(K)π ˆ Q( LaO F )αβ | 2 V≥t¯(u,u)

(4.65)

1≤a≤2

The analogous inequality is more complicated for G = LS LΩij F as this form is not a solution of the Maxwell equations. We obtain I0S (u; [u0 (t¯), u]) + I S0 (u; [u0 (t¯), u]) − I0S (Σt¯ ∩ V (u, u)) ≤ Err (S) (V≥t¯(u, u)) (4.66) 29 divP 30 With

≡ Dα Pα . an obvious modiﬁcation of the region where the Stokes theorem applies.

Vol. 1, 2000

Asymptotics of the Electromagnetic Field in Schwarzschild Spacetime

where Err (S) (V≥t¯(u, u))

=

1 2

¯

|(K)π ˆ

αβ

917

Q(LS LO F )αβ |

V≥t¯(u,u)

¯ α| |(DivQ(LS LO F ))α K

+

(4.67)

V≥t¯(u,u)

Deﬁning HO (≥ t¯; u, u) ≡ HS (≥ t¯; u, u) ≡

sup (u ,u )∈V≥t¯(u,u)

sup

(u ,u )∈V≥t¯(u,u)

O ¯ I0 (u ; [u0 (t¯), u ]) + I O 0 (u ; [u0 (t), u ]) S I0 (u ; [u0 (t¯), u ]) + I S0 (u ; [u0 (t¯), u ]) (4.68)

H(≥ t¯; u, u) ≡ HO (≥ t¯; u, u) + HS (≥ t¯; u, u) H(Σt¯ ∩ V (u, u)) ≡ I0O (Σt¯ ∩ V (u, u)) + I0S (Σt¯ ∩ V (u, u))

(4.69)

the following inequality holds H(≥ t¯; u, u) − H(Σt¯ ∩ V (u, u)) ≤ Err(O) (V≥t¯(u, u)) + Err(S) (V≥t¯(u, u)) (4.70) In conclusion to bound H(≥ t¯; u, u) in terms of H(Σt¯ ∩ V (u, u)) we have to control Err(V≥t¯(u, u)) ≡ Err(O) (V≥t¯(u, u)) + Err (S) (V≥t¯(u, u))

(4.71)

From the inequality 4.70 it follows that we can control H(≥ t¯; u, u) in terms of H(Σt¯∩V (u, u)) if we are able to control the error term: Err(V≥t¯(u, u)). In a similar way to estimate H(Σt¯ ∩ V (u, u)) in terms of H(Σ0 ∩ V (u, u)) we have to control the corresponding error term relative, now, to the region V≤t¯. This is the core of the technical part. The control of these error terms allow to prove the following two Propositions which, at their turn, imply Propositions 3.4, 3.5. Proposition 4.4. Fixed m, δ0 > 0 let V (u, u) ⊂ Mδ0 with t = 12 (u + u) very large 31 , then a suﬃciently large t¯0 = t¯(m, δ0 ) < t exists, depending on m and δ0 , but independent from t, such that for any t¯ ≥ t¯0 the following inequality holds H(≥ t¯; u, u) ≤ C5 (m, δ0 )H(Σt¯ ∩ V (u, u))

(4.72)

where C5 (m, δ0 ) is a positive bounded function. Moreover, if δ0 is suﬃciently large 32 , it is possible to choose t¯(m, δ0 ) = 0. 31 As

m is the only intrinsic length unit, t suﬃciently large means t = Mm with M >> 1 much large is understood looking at the proof of the Proposition. In particular u0 (t¯ = 0) = r∗ (δ0 ) must be such that eqs. 4.82, 4.99 are satisﬁed. 32 How

918

W. Inglese, F. Nicol` o

Ann. Henri Poincar´ e

Next Proposition implies immediately Proposition 3.5: Proposition 4.5. For any given m, δ0 > 0 and a generic t¯ the following inequality holds H(Σt¯ ∩ Mδ0 (u, u)) ≤ C1 (m, δ0 , t¯)H(Σ0 )

(4.73)

where C1 (m, δ0 ; t¯) is a positive function increasing in m and t¯ and decreasing in δ0 , such that, for m > 0   δ0 3 |m log 2m | 2m  C1 (m, δ0 ) = O  (4.74) δ0 Remark. It is important to observe that we have chosen to estimate the energytype norms using the “ﬂux-norms” above Σt¯ and the energy-norms on the Σt ’s hypersurfaces, below Σt¯. The reason is that to prove Proposition 4.4 the use of the “ﬂux-norms” is required and the analogous result in terms of the Σt norms is false, see the remark at the end of the proof. The advantage of considering the ﬂux-norms only above Σt¯ lies in the fact that all the quantities Φ have, in this case, a lower bound strictly greater than zero and independent from u. Below Σt¯ there is no advantage in using the ﬂux-norms instead of the energy-type norms.

4.1 Proof of Proposition 4.4 The proof of Proposition 4.4 is divided in various parts. We start estimating HO (≥ t¯; u, u), see eq. 4.68, in terms of I0O (Σt¯ ∩ V (u, u)). Proposition 4.6. For any m, δ0 > 0, ﬁxed 50,1 small, it is possible to ﬁnd t¯ ≥ t¯0 such that for any V (u, u) ⊂ Mδ0 HO (≥ t¯; u, u) − HO (Σt¯ ∩ V (u, u)) ≤ 50,1 HO (≥ t¯; u, u)

(4.75)

Proof. We have to control, see eq. 4.64,

1 ¯ (O) Err (V≥t¯(u, u)) = |(K) π ˆ αβ Q( LaO F )αβ | . 2 V≥t¯(u,u) 1≤a≤2

As Q( LaO F )αβ is traceless and recalling the expression of ¯ (K) αβ

π ˆ

¯ (K)

π ˆ , eq. 2.11, we have

Q( LaO F )αβ = −4tµ(S) (r)Q( LaO F )(e3 , e4 )

where, see eq. 2.12, µ(S) (r) = 1 + r∗ (Φ∂r Φ −

m m r Φ2 )= 3−2 1−3 log( − 1) . r r r 2m

(4.76)

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Asymptotics of the Electromagnetic Field in Schwarzschild Spacetime

919

From eq. 3.38, as Q(e4 , e3 ), Q(e3 , e3 ) and Q(e4 , e4 ) are non negative, Q( LaO F )(e3 , e4 ) ≤

2 1 a ¯ 2 Q( LO F )(K, e3 ) Φ τ+

and, therefore,

¯ αβ |(K)π Q( LaO F )αβ | ≤ 8 V≥t¯(u,u)

≤c

m Φ(r(u, u0 (t¯)))

u

u0 (t¯)

du 2 τ+

1 1 a ¯ 2 Q( LO F )(K, e3 ) Φ τ+ r t 1 + | log( 2m − 1)| ¯ e3 ) Q( LaO F )(K, r

tµ(S) (r)

V≥t¯(u,u)

C(u ;[u0 (t¯),u])

(4.77) where c is a generic constant. To estimate the factor lemma easy to prove.

t r

in 4.77 we use the following

Lemma 4.7. Fixed m, δ0 > 0, deﬁning δ0 | , for δ0 < 2m t¯0 = 2|r∗ (δ0 )| = 2|(2m + δ0 ) + 2m log 2m t¯0 = 0 , for δ0 ≥ 2m

(4.78)

then, for t > t¯0 , on any C(u) ⊂ Mδ0 , the following inequality holds c2 ≤

t ≤ c1 r

(4.79)

where c1 , c2 are generic constants, moreover there exist constants c3 , c4 , independent from m, δ0 , such that c3 r ≤ r∗ ≤ c4 r. Substituting inequality 4.79 in the r.h.s. of 4.77 we obtain

¯ αβ |(K)π ˆ Q( LaO F )αβ | V≥t¯(u,u)

≤c

m Φ(r(u, u0 (t¯)))

m ≤c Φ(r(u, u0 (t¯)))

u

du

u 2

C(u ;[u0 (t¯),u])

¯ e3 ) Q( LaO F )(K,

sup

(u ,u )∈V≥t¯(u,u)

· 33 Recall

,

(t¯),u ) 1 + | log( r(u02m − 1)|

u0 (t¯)

33

∞

u0 (t¯)

du

C(u ;[u0 (t¯),u ])

¯ e3 ) Q( LaO F )(K,

(t¯),u ) 1 + | log( r(u02m − 1)| u 2

that the generic constant c can be diﬀerent in diﬀerent inequalities.

·

920

W. Inglese, F. Nicol` o

m ≤c Φ(r(u, u0 (t¯)))

Ann. Henri Poincar´ e

sup

(u ,u )∈V≥t¯(u,u)

·

∞

du

¯ IO 0 (u ; [u0 (t), u ])

·

(t¯),u ) 1 + | log( r(u02m − 1)| u 2

u0 (t¯)

(4.80)

and from it

 t¯),u0 (t¯)) m 1 + | log( r(u0 (2m − 1)|  HO (≥ t¯; u, u) Err(O) (V≥t¯(u, u)) ≤ c  Φ(r(u, u0 (t¯)))u0 (t¯) 

(4.81) Therefore, ﬁxed m, δ0 , given 50,1 small, it is possible to ﬁnd t¯ and, consequently, u0 (t¯) such that r m 1 + | log( 2m − 1)| c c˜0 (t¯, r0 ) ≤ 50,1 Φ(r(u, u0 (t¯)))u0 (t¯) proving the proposition. The estimate of HS (≥ t¯; u, u) is provided by the following Proposition 4.8. For any m, δ0 > 0, ﬁxed 50,2 small, it is possible to ﬁnd t¯ ≥ t¯0 such that for any V (u, u) ⊂ Mδ0 HS (≥ t¯; u, u) − HO (Σt¯ ∩ V (u, u)) + HS (Σt¯ ∩ V (u, u)) (4.82) ≤ 50,2 HO (≥ t¯; u, u) + HS (≥ t¯; u, u) Proof. We have to control the various integrals in

1 ¯ αβ |(K)π ˆ Q(LS LO F )αβ | Err(S) (V≥t¯(u, u)) = 2 V≥t¯(u,u)

¯ α| + |(DivQ(LS LO F ))α K

(4.83)

V≥t¯(u,u)

The ﬁrst term is controlled as in the previous case obtaining, for t¯ suﬃciently large,

1 ¯ αβ |(K)π ˆ Q(LS LO F )αβ | 2 V≥t¯(u,u)

50,2 ¯ e3 ) ≤ Q(LS LO F )(K, sup 2 u ∈[u0 ,u] C(u )∩V≥t¯(u,u) 50,2 50,2 S ≤ sup H (≥ t¯; u, u) I S0 (u , u ) ≤ (4.84) 2 2 (u ,u )∈V≥t (u,u)

Vol. 1, 2000

Asymptotics of the Electromagnetic Field in Schwarzschild Spacetime

921

The control of the second integral in 4.83 is more delicate. We need the explicit expression of DivQ(LS LO F ) which is provided by the following lemma, whose proof is in the Appendix, Lemma 4.9. Let F be a solution of the Maxwell equations, then, denoting F˜ ≡ ˜ µν ≡ Q(LS LΩ F )µν , the following relation holds LS LΩij F and Q ij ˜ µν = 2(Jρ(1) + Jρ(2) + Jρ(3) )F˜νρ Dµ Q (4.85) 3

−J (l) (e3 )eσ4 − J (l) (e4 )eσ3 + 2J (l) (eθ )eσθ + 2J (l) (eφ )eσφ (LS LΩij F )νσ = l=1

Jρ(1) ≡ Jρ(1) (LΩij F ) =(S) π ˆ µσ Dσ LΩij Fµρ

where

Jρ(2) ≡ Jρ(2) (LΩij F ) = Jρ(3)

≡

Jρ(3) (LΩij F )

=

(S) (S)

Γλ LΩij F λρ Γσρ λ LΩij F

(4.86) σλ

1 gλσ (sign(λ)) δµr + gλµ (sign(λ)) δσr − gσµ (sign(σ)) δλr 2 (S) Γλ = g σρ (S) Γσρ λ (4.87) (S)

with

Γσρ λ =

1 1 (sign(λ)) = + = ∂r ((S) µ + tr(S) π) = O(m) 2 , if λ ∈ {0, r} 4 r 1 log r (sign(λ)) = − = ∂r (−(S) µ + tr(S) π) = O(m) 2 , if λ ∈ {θ, φ} 4 r (4.88)

Moreover it is simple to obtain the following expression ¯ν = (DivQ(LS LO F ))ν K

3

[−J (l) (e3 )eσ4 − J (l) (e4 )eσ3 l=1

+ =

2J

(l)

3

(eθ )eσθ

+ 2J

(l)

34

(eφ )eσφ ](LS

2 ν 2 ν LO F )νσ (Φ/2)(τ+ e4 + τ− e3 )

2 Φ τ+ [J (l) (e4 )ρ(LS LO F ) − I (l) · α(LS LO F )]

l=1

−

2 [J (l) (e3 )ρ(LS LO F ) − I (l) · α(LS LO F )] τ−

where I (l) · α(LS LO F ) ≡ for I (l) · α(LS LO F ).

a

(4.89)

J (l) (ea ) · α(LS LO F )(ea ). A similar expression holds

34 This expression is written in a slightly simbolical way. For instance, with J (l) (e )ρ(L L F ) 4 S O we mean

(l) Jν (LΩij F )eν4 ρ(LS LΩij F ). J (l) (e4 )ρ(LS LO F ) = i<j

Hereafter with LO F we mean LΩij F for a generic i, j.

922

W. Inglese, F. Nicol` o

Ann. Henri Poincar´ e

¯ ν | we decompose the currents J (l) To estimate V≥t¯(u,u) |(DivQ(LS LO F ))ν K in terms of the null components of the Maxwell tensor ﬁeld. The explicit computation is in the Appendix and the result is Φ J (1) (e4 ) = µ(S) (r)[4 ρ − 2div / α] r Φ / α] J (1) (e3 ) = µ(S) (r)[4 ρ − 2div r Φ J (1) (eθ ) = µ(S) (r)[2∂eφ σ + (α(eθ ) − α(eθ ))] r Φ (1) J (eφ ) = µ(S) (r)[−2∂eθ σ + (α(eφ ) − α(eφ ))] r

(4.90)

where, here, α, α, ρ, σ denote α( LO F ), α( LO F ), ρ( LO F ), σ( LO F ). The estimates of the various components of these currents are given in the next Lemma. They use the fact that, due to the symmetries of the Schwarzschild spacetime, α(LΩij F ) = LΩij α(F ) and / α|2 + |α|2 , | LO α|2 = r2 |∇ / α|2 + |α|2 , | LO (ρ, σ)|2 = r2 |∇ / (ρ, σ)|2 . | LO α|2 = r2 |∇ Lemma 4.10. Let F be a solution of the Maxwell equations, let us denote 35 with LaO F a tensor LΩi1 j1 ...LΩia ja F , for an arbitrary choice of the indices i1 ...ia , j1 ...ja , then, using the explicit expression of (S) π ˆ µν and the Maxwell equations, the various (1) (2) components of the current J , J , J (3) associated to LaO F have the following estimates µ(S) (r) / ρ( LaO F )|) (|r∇ / α( LaO F )| + |r∇ r µ(S) (r) (|r∇ / α( LaO F )| + |r∇ c / ρ( LaO F )|) r µ(S) (r) (|r∇ / σ( LaO F )| + |α( LaO F )| + |α( LaO F )| c r / α( LaO F )|) |r∇ / α( LaO F )| + |r∇ µ(S) (r) c (|r∇ / σ( LaO F )| + |α( LaO F )| + |α( LaO F )| r |r∇ / α( LaO F )| + |r∇ / α( LaO F )|)

|J (1) (e3 )| ≤ c |J (1) (e4 )| ≤ |J (1) (eθ )| ≤ + |J (1) (eφ )| ≤ +

(4.91)

and, for the generic component of J (2,3) , |J (2,3) | ≤ c 35 Remark

µ(S) (r) (|α( LaO F )| + |α( LaO F )| + |ρ( LaO F )|) r

that this deﬁnition is slightly diﬀerent from the one in eq. 3.41.

(4.92)

Vol. 1, 2000

Asymptotics of the Electromagnetic Field in Schwarzschild Spacetime

923

Applying this Lemma to DivQ(LS LO F ) and recalling the relations between the Lie derivatives and the covariant derivatives, the following estimates hold µ(S) (r) |J (1) (e3 )| ≤ c / 2 α(F )| + |r∇ / ρ(F )| |r∇ / α(F )| + |r2 ∇ r µ (r) (S) 2 / α(F )| + |r∇ / ρ(F )|) (4.93) (|r∇ / α(F )| + |r2 ∇ |J (1) (e4 )| ≤ c r µ(S) (r) 2 2 (|r ∇ / σ(F )| + |α(F )| + |α(F )| + |r∇ / α(F )| + |r∇ / α(F )|) |J (1) (eθ )| ≤ c r µ(S) (r) 2 2 / σ(F )| + |α(F )| + |α(F )| + |r∇ / α(F )| + |r∇ / α(F )|) (|r ∇ |J (1) (eφ )| ≤ c r |J (2,3) | ≤ c

µ(S) (r) (|α(F )| + |r∇ / α(F )| + |α(F )| + |r∇ / α(F )| + |r∇ / ρ(F )|) r (4.94)

Using these estimates we obtain the following bound

¯ α| ≤ |(DivQ(LS LO F ))α K V≥t¯(u,u)

(4.95)

µ(S) (r) 2 2 / α(F )| + |r2 ∇ / α(F )| + |r∇ / ρ(F )| |ρ(LS LO F )| τ+ |r∇ r V≥t¯(u,u) / 2 σ(F )| + |α(F )| + |α(F )| + |r∇ / α(F )| + |r∇ / α(F )| |α(LS LO F )| + |r2 ∇ 2 2 + τ− |r∇ / α(F )| + |r2 ∇ / α(F )| + |r∇ / ρ(F )| |ρ(LS LO F )| + |r2 ∇ / 2 σ(F )| + |α(F )| + |α(F )| + |r∇ / α(F )| + |r∇ / α(F )| |α(LS LO F )| ¯ α | we have to control the large family To estimate V≥t¯(u,u) |(DivQ(LS LO F ))α K of integrals on V≥t¯(u, u) composing it. We divide them in two sets whose integrals have to be estimated in a diﬀerent way.

µ(S) (r) set (A): τ+ |r2 ∇ / 2 α(F )|τ+ |ρ(LS LO F )| r V≥t¯(u,u)

µ(S) (r) / α(F )|τ+ |ρ(LS LO F )| τ+ |r∇ r V≥t¯(u,u)

µ(S) (r) τ+ |r∇ / ρ(F )|τ+ |ρ(LS LO F )| r V≥t¯(u,u)

µ(S) (r) τ+ |α(F )|τ+ |α(LS LO F )| r V≥t¯(u,u)

µ(S) (r) / α(F )|τ+ |α(LS LO F )| τ+ |r∇ r V≥t¯(u,u) c

924

W. Inglese, F. Nicol` o

Ann. Henri Poincar´ e

V≥t¯(u,u)

µ(S) (r) / 2 σ(F )|τ+ |α(LS LO F )| τ+ |r2 ∇ r

V≥t¯(u,u)

µ(S) (r) τ− |r2 ∇ / 2 α(F )|τ− |ρ(LS LO F )| r

V≥t¯(u,u)

µ(S) (r) / ρ(F )|τ− |ρ(LS LO F )| τ− |r∇ r

V≥t¯(u,u)

µ(S) (r) τ− |α(F )|τ− |α(LS LO F )| r

V≥t¯(u,u)

µ(S) (r) τ− |r∇ / α(F )|τ− |α(LS LO F )| r

V≥t¯(u,u)

µ(S) (r) τ− |α(F )|τ− |α(LS LO F )| r

V≥t¯(u,u)

µ(S) (r) / α(F )|τ− |ρ(LS LO F )| τ− |r∇ r

V≥t¯(u,u)

µ(S) (r) τ− |r∇ / α(F )|τ− |α(LS LO F )| r

V≥t¯(u,u)

µ(S) (r) 2 / σ(F )|τ− |α(LS LO F )| τ− |r2 ∇ r

V≥t¯(u,u)

µ(S) (r) τ+ |α(F )|τ+ |α(LS LO F )| r

V≥t¯(u,u)

µ(S) (r) τ+ |r∇ / α(F )|τ+ |α(LS LO F )| r

set (B):

(4.96)

(4.97)

Every integral of the ﬁrst group satisﬁes the following inequality   

u

µ(S) (r(u, u )) ¯ e3 )  [(A)] ≤  Q( LaO F )(K, ) r(u, u ¯ u0 (t) C(u )∩V≥t¯(u,u) 1≤a≤2

 12

µ(S) (r(u , u)) ¯ e4 )  + Q( LaO F )(K, r(u , u) )∩V ¯(u,u) u0 (t¯) C(u ≥t 1≤a≤2

u µ(S) (r(u, u )) ¯ e3 ) · Q(LS LO F )(K, r(u, u ) u0 (t¯) C(u )∩V≥t¯(u,u)

12

u µ(S) (r(u , u)) ¯ e4 ) Q(LS LO F )(K, + r(u , u) u0 (t¯) C(u )∩V≥t¯(u,u)



u

Vol. 1, 2000

Asymptotics of the Electromagnetic Field in Schwarzschild Spacetime

≤ cm

t¯) 1 + | log( r( 2m − 1)| r(u, u0 (t¯))

sup

(u ,u )∈V≥t (u,u)

925

I0O (≥ t¯; u , u )+

S ¯ ¯ t¯; u , u ) + I O 0 (≥ t; u , u ) + I 0 (≥ t; u , u ) t¯) 1 + | log( r( − 1)| 2m ≤ cm H(≥ t¯; u, u) r(u, u0 (t¯))

I0S (≥

Again we can choose t¯ ≥ t¯0 suﬃciently large such that t¯) 1 + | log( r( 2m − 1)| 50,2 cm ≤ ¯ r(u, u0 (t)) 28

(4.98)

(4.99)

obtaining [(A)] ≤

50,2 50,2 O H (≥ t¯; u, u) + HS (≥ t¯; u, u) ≤ H(≥ t¯; u, u) 2 2

(4.100)

We estimate the ﬁrst integral integral of the group (B)

µ(S) (r) τ+ |α(F )|τ+ |α(LS LO F )| . r V≥t¯(u,u) The estimate of the other one is done exactly in the same way and we omit it. We ¯ e4 ) and Q(K, ¯ e3 ), see equations 3.38, that we recall from the expressions of Q(K, can control the integrals of α only along the null cones C(u) and those of α along the C(u) ones. Using Lemma 4.7 we bound τr+ in V≥t¯(u, u) for t¯ suﬃciently large, with a constant c independent from m, δ0 . Therefore

µ(S) (r) τ+ |α(F )|τ+ |α(LS LO F )| r V≥t¯(u,u)

12 2 τ− r(u , u ) 2 · − 1)| 1 + | log( ≤ cm 2 |α(F )| 2m τ+ V≥t¯(u,u)

12 2 τ+ r(u , u ) 2 − 1)| · 1 + | log( 2 |α(LS LO F )| 2m τ− V≥t¯(u,u)

12

τ+ u 1 + | log( − 1)| 2m ¯ e3 ) ≤ cm du Q( LO F )(K, · 2 τ+ u0 (t¯) C(u )∩V≥t¯(u,u)  12 

u 1 + | log( r(u2m,u) − 1)|

¯ e4 ) du Q(LS LO F )(K, · 2 τ− u0 (t¯) C(u )∩V≥t¯(u,u)

926

W. Inglese, F. Nicol` o

Ann. Henri Poincar´ e

 12 12  u r(u ,u) τ+ − 1)| 1 + | log( 1 + | log( 2m − 1)| 2m   . ≤ cm du du 2 2 τ+ τ− u0 (t¯) u0 (t¯)

¯ sup Q( LO F )(K, e3 )

u

C(u )∩V≥t¯(u,u)

[u0 ,u]×[u0 ,u]

12

C(u )∩V≥t¯(u,u)

¯ e4 ) Q(LS LO F )(K,

(4.101)

with a generic constant c independent from u, u, t¯. For t¯ suﬃciently large the ﬁrst integral of the factor 36  12

12  u r(u ,u) τ+ u 1 + | log( − 1)| 1 + | log( 2m − 1)| 2m   du du 2 2 τ τ u0 (t¯) u0 (t¯) + − ¯

which is O( logu u(0t¯()t) ), can be made suﬃciently small so that

37

0

u

du

1+

u0 (t¯)

τ+ | log( 2m 2 τ+

 12 1 + | log( r(u2m,u) − 1)|  ≤ 50,2 du 2 τ 4 ¯ u0 (t) −

12 

− 1)| 

u

(4.102) and the following inequality holds

µ(S) (r) τ+ |α(F )|τ+ |α(LS LO F )| r V≥t¯(u,u) 12

50,2 ¯ e3 ) sup Q( LO F )(K, · ≤ 4 u ∈[u0 ,u] C(u )∩V≥t¯(u,u) 12

¯ · sup Q(LS LO F )(K, e4 ) u ∈[u0 ,u]

≤

C(u )∩V≥t¯(u,u)

50,2 O H (≥ t¯; u, u) + HS (≥ t¯; u, u) 4

(4.103)

In conclusion [(B)] ≤ 36 This 37 The

50,2 50,2 O H (≥ t¯; u, u) + HS (≥ t¯; u, u) ≤ H(≥ t¯; u, u) 2 2

t¯ can be larger than the previous ones. second factor

u u0 (t¯)

du

1+| log(

r(u ,u) −1)| 2m 2 τ−

small if u = −r∗ (δ0 ) with δ0 suﬃciently small.

(4.104)

is bounded by c log

r(u,u) , m

not necessarily

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927

Collecting all these estimates Proposition 4.8 is proved, which together with Proposition 4.6 completes the proof of Proposition 4.4. Remark. It is due to the presence of the integrals of group (B) that the Σt energytype norms cannot be bounded in terms of the initial data. In fact the analogous of the inequality 4.104 cannot be obtained using the Σt energy-type norms. An estimate in terms of these norms is, anyway, needed to prove Proposition 4.5. This is the reason the function C1 (m, δ0 ; t¯) depends on t¯. In Proposition 4.5 this is not a problem, as t¯, although large, is ﬁxed independently from u and u.

4.2 Proof of Proposition 4.5 The proof is similar to the proof of Proposition 4.4, but much simpler. Deﬁning H(t) ≡ H(Σt (r ≥ r(u, u0 (t))) it follows that, see eq. 4.69, H(t) ≥ H(Σt ∩ V (u, u)) = I0O (Σt ∩ V (u, u)) + I0S (Σt ∩ V (u, u))

(4.105)

Recalling Proposition 4.1 and Corollary 4.2, applying again Stokes theorem, see Lemma 4.3, we obtain Lemma 4.11. Let Pα = Q(G)αβ X β be deﬁned as in Proposition 4.1 then the Stokes theorem implies

Q(G)(X, T˜0 ) − Q(G)(X, T˜0 ) Σt+δ (r≥r(u,u0 (t+δ)))

+

Φ−1 Q(G)(X, e4 )

C(u,[u0 (t),u0 (t+δ)])

1 (DivQ(G))β X β + Q(G)αβ (X)π αβ 2 V ([t,t+δ];u)

= −

Σt (r≥r(u,u0 (t)))

(4.106)

where V ([t, t + δ]; u) is the volume whose boundaries are: Σt+δ (r ≥ r(u, u0 (t + δ))), Σt (r ≥ r(u, u0 (t))) and C(u, [u0 (t), u0 (t + δ)]). As C(u,[u (t),u (t+δ)]) Φ−1 Q(G)(X, e4 ) is non negative 0 0 

 ¯ αβ H(t + δ) − H(t) ≤ |(K)π Q( LaO F )αβ |+ (4.107) V ([t,t+δ];u) 1≤a≤2 % ¯ αβ (K) α ¯ | π Q(LS LO F )αβ | + |(DivQ(LS LO F ))α K | and, taking the limit δ → 0, the following diﬀerential inequality holds: 

 dH ¯ αβ |(K)π Q( LaO F )αβ |+ ≤ dt Σt (r≥r(u,u0 (t))  1≤a≤2

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+|(K)π

αβ

Ann. Henri Poincar´ e

% ¯ α| Q(LS LO F )αβ | + |(DivQ(LS LO F ))α K

(4.108)

which we use to estimate H for t ∈ [0, t¯]. To achieve it we express the right hand side of 4.108 in terms of H. The following lemma is proved in the Appendix: Lemma 4.12. For t ≤ t¯0 = 2|r∗ (δ0 )| we have the following estimate  

¯ αβ ¯ αβ  |(K)π Q( LaO F )αβ | + |(K)π Q(LS LO F )αβ | Σt (r≥r(u,u0 (t))

1≤a≤2

(1 + (r∗ (δ0 ))2 ) ≤ c 1 + (r∗ (δ0 ))3 1 + 1 + t2

H(t)

(4.109)

¯ α | ≤ c 1 + (r∗ (δ0 ))2 H(t) |(DivQ(LS LO F ))α K

(4.110)

Σt (r≥r(u,u0 (t))

Using this Lemma we rewrite the diﬀerential inequality 4.108 for t ≤ t¯0 : dH (1 + (r∗ (δ0 ))2 ) 3 ≤ c 1 + (r∗ (δ0 )) H(t) 1+ dt 1 + t2

(4.111)

and from it we conclude H(t) ≤ H(0) exp{(1 + (r∗ (δ0 )) )t} ≤ 3

2m δ0

δ0 3 |m log 2m |

H(0)

(4.112)

5 Appendix 5.1 Proof of Proposition 2.3 Proposition 2.3 is a direct consequence of Lemma 2.2 and of the following one: Lemma 5.1. Let G be a C ∞ tensor ﬁeld tangent at each point to the corresponding S(u, u), then the following Sobolev inequalities hold

r4 |G|4 ≤ r4 |G|4 S(u,u)

S(u0 ,u)

|G|2 + r2 |∇ / G|2 + r2 |D / 3 G|2

+c C(u;[u0 ,u])

S(u,u)

2

2 r2 τ− |G|4 ≤

S(u0 ,u)

+c C(u;[u0 ,u])

2 r2 τ− |G|4

2 |G|2 + r2 |∇ / G|2 + τ− |D / 3 G|2

2

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Asymptotics of the Electromagnetic Field in Schwarzschild Spacetime

929

r |G| ≤ 4

r4 |G|4

4

S(u,u)

(5.113)

S(u,u0 )

2 |G| + r |∇ / G| + r |D / 3 G| 2

+c

2

2

2

2

C(u;[u0 ,u])

S(u,u)

2 r2 τ− |G|4 ≤

S(u,u0 )

+c

C(u;[u0 ,u])

2 r2 τ− |G|4

2

2 |G|2 + r2 |∇ / G|2 + τ− |D / 3 G|2 1

1

Substituting in Lemma 2.2 G with rU or with r 2 τ−2 U and using in the r.h.s. the inequalities provided by Lemma 5.1 we obtain the estimates 2.21...2.25, proving the Proposition 2.3. Lemma 2.2 is discussed in [Ch-Kl2], page 64. We discuss here the proof of Lemma 5.1. We write the integral S(u,u) r4 |G|4 in the following way

u

∂ r4 |G|4 = r4 |G|4 + du r4 |G|4 ∂u u0 S(u,u) S(u0 ,u) S(u ,u)

u

∂r 4 4 5 4 6 ∂ 4 = r |G| + du dω 6r |G| + r |G| ∂u ∂u u0 S(u0 ,u) S1

u

r4 |G|4 + 2 du r4 Φ|G|2 (G · D / 3 G) = S(u0 ,u)

u

du

+6 u0

∂r 3 r ∂u

S(u ,u)

u0

S(u ,u)

|G|4

(5.114)

∂ where dω is the angular part of the measure on S(u, u), ∂u = Φ2 D3 on the scalar functions, |G| is the norm of the tensor G with respect to the induced metric on S and (G · D / 3 G) is the scalar product between two tensors tangent to S made with respect to the same metric. As

∂r ∂r ∂r∗ 1 = = − Φ2 ≤ 0 ∂u ∂r∗ ∂u 2 the last term is non positive and we neglect it obtaining the following upper bound

4 4 4 4 r |G| ≤ r |G| + 2 r4 |G|2 |(G · D / 3 G)| S(u,u)

S(u0 ,u)

C(u;[u0 ,u])

(5.115) Applying the Schwartz inequality we obtain

r |G| ≤ 4

S(u,u)

r |G| + 2

4

4

S(u0 ,u)

12

r |G|

4

6

C(u;[u0 ,u])

12 r |D / 3 G|

6

2

2

C(u;[u0 ,u])

(5.116)

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Moreover using the isoperimetric inequality 38 it follows

6 6 4 4 2 2 2 r |G| ≤ c r |G| |G| + r |∇ / G| S(u ,u)

S(u ,u)

S(u ,u)

(5.117) and from it

r |G| ≤ c 6

C(u;[u0 ,u])

6

sup u ∈[u0 ,u]

S(u ,u)

r |G| 4

|G| + r |∇ / G|

4

2

2

2

C(u;[u0 ,u])

(5.118) Substituting this inequality in the previous one we obtain

4 4 4 4 sup r |G| ≤ r |G| + c sup u ∈[u0 ,u]

S(u ,u)

u ∈[u0 ,u]

S(u0 ,u)

12

S(u ,u)

·

4

12

12

|G|2 + r2 |∇ / G|2

·

r |G| 4

r2 |D / 3 G|2

C(u;[u0 ,u])

C(u;[u0 ,u])

(5.119) and ﬁnally

sup u ∈[u0 ,u]

S(u ,u)

r4 |G|4 ≤

r4 |G|4

S(u0 ,u)

2 |G| + r |∇ / G| + r |D / 3 G| 2

+c

2

2

2

2

(5.120)

C(u;[u0 ,u])

proving the ﬁrst line of 5.113. in proving the second line is that The only2 diﬀerence |G|4 and instead of 5.117 we derive, from we start with the integral S(u,u) r2 τ− the isoperimetric inequality,

4 2 6 2 2 4 2 2 2 r τ− |G| ≤ c r τ− |G| |G| + r |∇ / G| S(u ,u)

S(u ,u)

S(u ,u)

(5.121) so that we obtain

2 sup r2 τ− |G|4 ≤ u ∈[u0 ,u]

S(u ,u)

S(u0 ,u)

|G| + r |∇ / G| + C(u;[u0 ,u])

[Ch-Kl2], page 58.

2 2

+c 38 See

2 r2 τ− |G|4

2

2

2 τ− |D / 3 G|2

(5.122)

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931

To prove the last two inequalities of 5.113 all the previous computations have to be done again to express the sup-norms in terms of integrals along the uC(u; [u0 , u]) cones. The only diﬀerence is that the term, corresponding to 6 u0 du S(u ,u) ∂r 4 r3 ∂u |G| ,

u

6

du

r3 S(u,u )

u0

∂r |G|4 , ∂u

is non negative and, therefore, cannot be omitted. We bound it as

u 3 ∂r 4 du r |G| ≤ 3 r3 |G|4 6 ∂u u0 S(u,u ) C(u;[u0 ,u])

12

12 ≤3

r6 |G|6

|G|2

C(u;[u0 ,u])

(5.123)

C(u;[u0 ,u])

and add this term to the other one. The ﬁnal result is of the same type:

r4 |G|4 ≤ r4 |G|4 S(u,u )

S(u,u0 )

|G|2 + r2 |∇ / G|2 + r2 |D / 4 G|2

+c C(u;[u0 ,u])

S(u,u )

2

2 r2 τ− |G|4 ≤

S(u,u0 )

2 r2 τ− |G|4

(5.124) 2

|G| + r |∇ / G| + 2

+c

2

2

C(u;[u0 ,u])

5.2

2 τ− |D / 4 G|2

Proof of Proposition 2.4

To prove Proposition 2.4 we write r4 |G|4 in the following way

∞ ∂ 4 4 (r |G| )(r, ω) = − dr (r4 |G|4 ) ∂r r

39

: (5.125)

which is true, provided the assumption limr→∞ r4 |G|4 = 0 is satisﬁed. Then integrating both sides on S(t, r) we obtain

∞

∂ 4 4 dσ(r |G| )(r, ω) = − dr dσ (r4 |G|4 ) ∂r r S(t,r) S(t,r) reason for not starting in this proof directly from S(t,r) dσ r4 |G|4 is due to the fact that we do not want to make the assumption

lim dσ r4 |G|4 = 0 39 The

r→∞ S(t,r)

as this limit is not true for the solutions of the Maxwell equations.

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r2 ∂ 4 4 Φ(r ) (r |G| ) r2 ∂r r S(t,r ) 2

r ∂ = − dµ 2 Φ(r ) (r4 |G|4 ) r ∂r Σ ([r,∞))

t ≤ −4 dµ r2 r Φ(r )|G|4 Σt ([r,∞))

+4 dµ r2 r2 |G|2 |(G · DN˜ G)| Σt ([r,∞))

≤ 4 dµ r4 |G|2 |(G · DN˜ G)| (5.126)

= −

∞

dr

dσΦ(r )−1

Σt ([r,∞))

and the last inequality is nearly the same as in eq. 5.124, the main diﬀerence being the presence of DN˜ = ΦD ∂ instead of D / 4 or D / 3 . The remaining of the proof goes ∂r just mimicking all the previous steps of Proposition 2.3.

5.3 Proof of Proposition 3.3 The proof relies on the estimates of Proposition 2.3, where C(u; [u0 , u]) and C(u; [u0 , u]) are now the portions of the null cones above Σt¯. The strategy is to estimate the various integrals in the r.h.s. of Proposition 2.3, with U equal to the null components of the Maxwell ﬁeld, in terms of the integrals, on the same ¯ e(3,4) ) functions with (a, b) ∈ {(1, 0), (2, 0), (1, 1)}. cones, of the Q(LbS LaO F )(K, We divide the proof in various parts, each one relative to some null components in which the Maxwell tensor F is decomposed. 5.3.1 α and α 5

Choosing in eq. 2.21 U = rα(F ) it follows that to control supS(u,u) |r 2 α(F )| we have to control the following integrals

r2 |α(F )|2 , r4 |∇ / α(F )|2 C(u;[u0 ,u]) C(u;[u0 ,u])

2 r4 |D / 4 α(F )|2 , r6 |∇ / α(F )|2 C(u;[u0 ,u]) C(u;[u0 ,u])

6 2 r |∇ /D / 4 α(F )| (5.127) C(u;[u0 ,u])

To control the ﬁrst two integrals we observe that, due to the symmetries of the Schwarzschild spacetime, LΩij commutes with the null decomposition: α(LΩij F ) = LΩij α(F ). Moreover it is easy to prove by direct computation that | LO α|2 = r2 |∇ / α|2 + |α|2

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Asymptotics of the Electromagnetic Field in Schwarzschild Spacetime

where | LO α|2 ≡

i<j

C(u;[u0 ,u])

C(u;[u0 ,u])

933

|LΩ(ij) α|2 . Therefore

r2 |α(F )|2 ≤ Φ−1 (r(u, u0 ))

C(u;[u0 ,u])

r4 |∇ / α(F )|2 ≤ Φ−1 (r(u, u0 ))

¯ e4 ) Q(LO F )(K,

C(u;[u0 ,u])

¯ e4 ) Q(LO F )(K, (5.128)

In the same way it is easy to prove that

r6 |∇ / 2 α(F )|2 ≤ Φ−1 (r(u, u0 )) C(u;[u0 ,u])

C(u;[u0 ,u])

¯ e4 ) Q(L2O F )(K,

In order to estimate the fourth integral we recall that, as S =

Φ 2 (ue4

+ ue3 ),

2 / 4 α|2 ≤ c Φ−2 |D / S α|2 + τ− |D / 3 α|2 r2 |D so that we are reduced to control the integrals

r2 Φ−2 |D / S α(F )|2 , C(u;[u0 ,u])

C(u;[u0 ,u])

(5.129)

2 r2 τ− |D / 3 α(F )|2 .

The ﬁrst integral can be easily bounded in the following way 40

2 −2 2 −2 r Φ |D / S α(F )| ≤ Φ (r(u, u0 )) r2 |D / S α(F )|2 C(u;[u0 ,u]) −2

≤ cΦ

C(u;[u0 ,u])

−'

r |α(LS F )| + Φ 2

(r(u, u0 ))

2

≤ cΦ

¯ e4 ) + Q(LS LO F )(K,

(r(u, u0 ))

r |α(F )| 2

(r(u, u0 ))

C(u;[u0 ,u]) −(3+')

C(u;[u0 ,u])

C(u;[u0 ,u])

2

¯ e4 ) Q( LO F )(K, C(u;[u0 ,u])

(5.130) for any 5 > 0. Using the Maxwell equation D / 3 α − (∂r Φ + 40 We

Φ / σ = 0, )α − ∇ / ρ −∗ ∇ r

use the relation |D / S α(F )|

≤

|LS α(F )| + |DS||α(F )|

≤

|α(LS F )| + c|α(F )|(1 +

m | log Φ|) . r

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the second integral satisﬁes

2 r2 τ− |D / 3 α(F )|2 C(u;[u0 ,u])

2 2 2 ≤c τ− |α(F )|2 + r2 τ− |∇ / ρ(F )|2 + r2 τ− |∇ / σ(F )|2 C(u;[u0 ,u])

≤ c Φ−1 (r(u, u0 ))

¯ e4 ) Q( LO F )(K, C(u;[u0 ,u])

+ C(u;[u0 ,u])

2 2 r2 τ− |∇ / ρ(F )|2 + r2 τ− |∇ / σ(F )|2

(5.131)

As in the Minkowski case, see eq. (3.59) of [Ch-Kl1], the following inequality holds

2 τ− (|r∇ / ρ(F )|2 + |r∇ / σ(F )|2 ) C(u;[u0 ,u])

2 2 τ− |ρ( LO F )|2 + τ− |σ( LO F )|2 ≤c C(u;[u0 ,u])

¯ e4 ) ≤ cΦ−1 (r(u, u0 )) Q( LO F )(K, (5.132) C(u;[u0 ,u])

and from it

C(u;[u0 ,u])

2 r2 τ− |D / 3 α(F )|2 ≤ cΦ−1 (r(u, u0 ))

¯ e4 ) Q( LO F )(K, C(u;[u0 ,u])

(5.133) Finally

41

r4 |D / 4 α(F )|2 C(u;[u0 ,u]) −1

≤ cΦ

(r(u, u0 )) Φ−(2+') (r(u, u0 ))

−4

≤ cΦ

(r(u, u0 ))

¯ e4 )+ Q(LS LO F )(K, C(u;[u0 ,u])

¯ Q( LO F )(K, e4 )

C(u;[u0 ,u])

¯ e4 ) + Q(LS LO F )(K, C(u;[u0 ,u])

¯ e4 ) Q( LO F )(K, C(u;[u0 ,u])

(5.134) 41 Φ−(2+!)

could in fact be substituted by Φ−2 | log Φ|.

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935

The last integral of 5.127 is estimated in the same way, obtaining

r6 |∇ /D / 4 α(F )|2 ≤ cΦ−4 (r(u, u0 ))

C(u;[u0 ,u])

¯ e4 ) Q(LS LO F )(K, C(u;[u0 ,u])

¯ e4 ) + Q( LO F )(K,

+ C(u;[u0 ,u])

C(u;[u0 ,u])

¯ e4 ) Q( L2O F )(K,

(5.135)

Collecting all these estimates we have

|r 2 α(F )(u, u)| ≤ cΦ−2 (r(u, u0 )) 5

¯ e4 ) Q(LS LO F )(K, C(u;[u0 ,u])

12

¯ e4 ) + Q( LO F )(K,

+ C(u;[u0 ,u])

C(u;[u0 ,u])

¯ e4 ) Q( L2O F )(K,

1 ≤ cΦ−2 (r(u, u0 )) I0O (≥ t¯; u, u) + I0S (≥ t¯; u, u) 2 (5.136) Substituting in eq. 2.25 U with τ− α(F ) it follows that to control |α(F )(u, u)| we have to control the following integrals 42 :

−2 2 2 2 2 τ− |α(F )| , τ− r |∇ / α(F )|2 Φ (r(u, u)) C(u;[u0 ,u]) C(u;[u0 ,u])

4 2 2 4 τ− |D / 3 α(F )| , τ− r |∇ / 2 α(F )|2 C(u;[u0 ,u]) C(u;[u0 ,u])

4 2 τ− r |∇ /D / 3 α(F )|2 (5.137) C(u;[u0 ,u])

These integrals are estimated as before, with the obvious substitutions, and the ﬁnal result is

3 −1 − 2 −2 ¯ e3 ) |α(F )(u, u)| ≤ cr τ− Φ (r(u, u)) Q(LS LO F )(K, C(u;[u0 ,u])

12

¯ e3 ) + Q( LO F )(K,

+ C(u;[u0 ,u])

C(u;[u0 ,u])

¯ e3 ) Q( L2O F )(K,

12 S ¯ ¯ ≤ cΦ−2 (r(u, u)) I O 0 (≥ t; u, u) + I 0 (≥ t; u, u) 42 The

factor Φ−2 (r(u, u)) in front of the ﬁrst integral arises from the inequality

2 2 4 τ− |D / 3 τ− α(F )|2 ≤ Φ−2 (r(u, u)) τ− |α(F )|2 + τ− |D / 3 α(F )|2 .

C(u;[u0 ,u])

C(u;[u0 ,u])

C(u;[u0 ,u])

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It is important to remark that the choice of estimating |α| in terms of integrals along the backward cones C(u), instead that along the forward cones C(u) as was done for |α|, is obliged. In fact these integrals have to be estimated in terms of the norms built with the energy momentum tensor Q and, see eqs. 3.38, the integrals of Q along the C(u) cones do not contain α. The null components ρ and σ, viceversa, can be estimated using both types of integrals. 5.3.2 ρ and σ We discuss here only the bound for the scalar function ρ, as for σ the proof is exactly the same 43 . Diﬀerently from the one form α, ρ satisﬁes the following equation

|ρ(LΩij F )|2 = |LΩij ρ(F )|2 = | LO ρ(F )|2 = r2 |∇ / ρ|2 , i<j

i<j

therefore the integrals of Q( LO F ) are not suﬃcient to control ρ. On the other side from the Poincar´e inequality

r2 |ρ − ρ¯|2 ≤ c r4 |∇ / ρ|2 ≤ c r2 |ρ( LO F )|2 (5.138) S(u,u)

S(u,u)

S(u,u)

we expect, using the previous energy type norms, to be able to control (ρ − ρ¯). We divide the problem in two parts writing: |ρ| ≤ |ρ − ρ¯| + |¯ ρ|. 5.3.3 ρ − ρ¯ Substituting U with r(ρ − ρ¯) in eq. 2.22, using the Poincar´e inequality and the 1

equation D3 ρ¯ = D3 ρ we obtain that |r2 τ−2 (ρ − ρ¯)| is bounded by the sum of the following three integrals

2 r4 |∇ / ρ|2 , r6 |∇ / 2 ρ|2 , r4 τ− |∇ /D / 3 ρ|2 C(u,[u0 ,u])

C(u,[u0 ,u])

C(u,[u0 ,u])

From eq. 3.38 the ﬁrst two integrals are controlled by

¯ e3 ) and Φ−1 (r(u, u)) Q( LO F )(K, Φ−1 (r(u, u)) C(u,[u0 ,u])

¯ e3 ). Q( L2O F )(K,

C(u,[u0 ,u])

Using the Maxwell equation D3 ρ − 2 Φr ρ + div / α = 0, the third integral is controlled by the following ones

2 2 r2 τ− |∇ / ρ|2 ≤ r2 τ+ |∇ / ρ|2 , C(u,[u0 ,u]) C(u,[u0 ,u])

2 r4 τ− |∇ / 2 α|2 . C(u,[u0 ,u])

43 The two functions will be treated diﬀerently only when we discuss their initial conditions as they have a diﬀerent physical meaning.

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Asymptotics of the Electromagnetic Field in Schwarzschild Spacetime

These integrals are bounded by

−1 −1 ¯ Φ (r(u, u)) Q( LO F )(K, e3 ) , Φ (r(u, u)) C(u,[u0 ,u])

C(u,[u0 ,u])

937

¯ e3 ) Q( L2O F )(K,

so that ﬁnally 1 2

−1

|r τ− (ρ − ρ¯)(F )(u, u)| ≤ cΦ 2

+ C(u;[u0 ,u])

We have to control ρ¯(u, u) ≡ r(u, u) satisﬁes r(u, u) + 2m log( ∂ ∂u r

12 ≤ cΦ−1 (r(u, u)) I O 0 (u, u)

¯ e3 ) Q( L2O F )(K,

5.3.4 ρ¯

From

C(u;[u0 ,u])

12

1 |S(u,u)|

¯ e3 ) Q( LO F )(K,

(r(u, u))

S(u,u)

(5.139)

ρ, where |S(u, u)| = 4πr2 (u, u) and

r(u, u) 1 − 1) = (u − u) = r∗ (u, u). 2m 2

2

= − Φ2

∂ Φ2 |S(u, u)| = −|S(u, u)| ∂u r follows and ∂ ∂u

ρ= S(u,u)

−Φ2 r

ρ+ S(u,u)

S(u,u)

∂ ρ ∂u

(5.140)

so that ﬁnally ∂ ρ¯ ∂u

= =

∂ 1 −1 ∂ ρ+ ( |S(u, u)|)¯ |S(u, u)| ∂u |S(u, u)| ∂u

∂ ρ¯ 1 |S(u, u)| S(u,u) ∂u

As on the scalar functions

∂ ∂u

=

Φ 2 D3

ρ S(u,u)

(5.141)

using the Maxwell equations we obtain

Φ2 Φ ∂ Φ2 ρ¯ = ρ¯ − div ρ¯ / α= ∂u r 2 r so that

∂ 2 ¯= ∂u r ρ

0 and, ﬁnally, |¯ ρ(u, u)| ≤

1 r2 (u, u)

sup |r2 ρ¯|. Σt=0

(5.142)

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5.4 Proof of Lemma 4.9 Let us deﬁne ˜ µν ≡ Q(LS LΩij F )µν Q

= F˜µρ F˜νρ + ∗ F˜µρ ∗ F˜νρ 1 = 2F˜µρ F˜νρ − gµν F˜ρσ F˜ ρσ 2

(5.143)

where, for a generic couple (i, j) where i, j ∈ {1, 2, 3}, F˜µρ ≡ LS LΩij Fµρ . It is easy to prove that ˜ µν = 2(Dµ F˜µρ )F˜νρ Dµ Q

(5.144)

We are therefore reduced to studying Dµ F˜µρ = Dµ LS LO Fµρ where LO F denotes the generic LΩij F . As = (S λ Dλ LO F )µρ + Dµ S α ( LO F )αρ + Dρ S β ( LO F )µβ = S λ (Dλ LO F )µρ + Dµ S α ( LO F )αρ + Dρ S β ( LO F )µβ (5.145)

(LS ( LO F ))µρ

then (Dσ F˜ )µρ

≡ = + +

(Dσ LS LO F )µρ (Dσ S)λ (Dλ LO F )µρ + S λ (Dσ Dλ LO F )µρ (Dσ Dµ S)α ( LO F )αρ + (Dσ Dρ S)β ( LO F )µβ (Dµ S)α (Dσ LO F )αρ + (Dρ S)β (Dσ LO F )µβ

(5.146)

Moreover Dσ Dµ Sλ

= [Dσ , Dµ ]Sλ + Dµ Dσ Sλ =

Rλβσµ S β + Dµ (S) πσλ − Dµ Dλ Sσ

= Rλβσµ S β − Rσβµλ S β + Dµ (S) πσλ − Dλ (S) πµσ + Dλ Dσ Sµ = ( Rλβσµ − Rσβµλ + Rµβλσ )S β + Dµ (S) πσλ − Dλ (S) πµσ + Dσ (S) πλµ − Dσ Dµ Sλ

(5.147)

From it 1 1 Dσ Dµ Sλ = − Rβ(λσµ) S β + Rβσµλ S β + (Dµ (S) πσλ − Dλ (S) πµσ + Dσ (S) πλµ ) 2 2 (5.148) = Rλµσβ S β + (S) Γσµ λ where we used the Bianchi identity Rβ(λσµ) = 0 and denoted (S)

Γσµ λ ≡

1 Dµ ((S) πσλ ) − Dλ ((S) πµσ ) + Dσ ((S) πλµ ) . 2

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From it (Dσ LS ( LO F ))µρ

= (Dσ S λ )(Dλ LO F )µρ + S λ (Dσ Dλ LO F )µρ + (Dσ Dµ S α )( LO F )αρ + (Dσ Dρ S α )( LO F )µα + (Dµ S α )(Dσ LO F )αρ + (Dρ S α )(Dσ LO F )µα (5.149)

and (LS Dσ ( LO F ))µρ

= S λ (Dλ Dσ LO F )µρ + (Dσ S λ )(Dλ LO F )µρ + (Dµ S α )(Dσ LO F )λρ + (Dρ S λ )(Dσ LO F )µλ (5.150)

Subtracting these expressions we obtain [Dσ , LS ] LO Fµρ

= + = + =

S λ (Dσ Dλ − Dλ Dσ ) LO Fµρ (Dσ Dµ S α )( LO F )αρ + (Dσ Dρ S α )( LO F )µα S λ ( Rµβσλ LO Fρβ + Rρβσλ LO Fµβ ) (Dσ Dµ S α )( LO F )αρ + (Dσ Dρ S α )( LO F )µα S λ ( Rµβσλ LO Fρβ + Rρβσλ LO Fµβ )

+ ( Rλµσβ S β + + (Rλρσβ S β + =

(S)

(S) (S)

Γσµ λ ) LO F λρ

Γσρ λ ) LO F λµ

Γσµ λ LO F λρ +

(S)

Γσρ λ LO F λµ

(5.151)

As from the Maxwell equations Dµ LO Fµρ = 0 and as LS g µσ = −(S) π µσ , we obtain Dµ F˜µρ

= g µσ Dσ F˜µρ = g µσ Dσ LS LO Fµρ = g µσ LS Dσ LO Fµρ + g µσ ( (S) Γσµ λ LO F λρ + = =

(S) µσ

Γσρ λ LO F λµ )

Γσρ λ LO Fµλ ] (S) µσ π ˆ Dσ LO Fµρ + (S) Γλ LO Fρλ + (S) Γσρ λ LO F σλ (5.152) π

Dσ LO Fµρ + g µσ [ (S) Γσµ λ LO Fρλ +

(S)

where (S) Γλ ≡ g µσ (S) Γσµ λ and Therefore from eq. 5.144

(S) µσ

π ˆ

(S)

≡ (S) π µσ − (1/4)g µσ tr(S) π.

˜ µν = 2(Jρ(1) + Jρ(2) + Jρ(3) )F˜νρ Dµ Q 3

(l) (l) (l) (l) −J0 (e3 )eσ4 − J0 (e4 )eσ3 + 2J0 (eθ )eσθ + 2J0 (eφ )eσφ (LS LO F )νσ = l=1

(5.153)

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where ˆ µσ Dσ LO Fµρ Jρ(1) =(S) π Jρ(2) =

(S)

Jρ(3)

(S)

=

Γλ LO F λρ Γσρ λ LO F

(5.154) σλ

Recalling that 2 ν 2 ν ¯ ν = Φ (τ+ K e4 + τ− e3 ) 2 ν σ (LS LO F )νσ e3 e4 = ρ(LS LO F ) (LS LO F )νσ eνa eσ3 = α(LS LO F )(ea ) (LS LO F )νσ eνa eσ4 = α(LS LO F )(ea )

(5.155)

we conclude that ¯ν = (DivQ(LS LO F ))ν K

3

[−J (l) (e3 )eσ4 − J (l) (e4 )eσ3 l=1

+ =

2J

(l)

(eθ )eσθ

+ 2J

(l)

(eφ )eσφ ](LS

2 ν 2 ν LO F )νσ (Φ/2)(τ+ e4 + τ− e3 )

2 Φ τ+ [J (l) (e4 )ρ(LS LO F ) − I (l) · α(LS LO F )]

3

(5.156)

l=1

2 τ− [J (l) (e3 )ρ(LS LO F ) − I (l) · α(LS LO F )] where I (l) · α(LS LO F ) ≡ a J (l) (ea ) · α(LS LO F )(ea ) and the similar expression for I (l) · α(LS LO F ). The explicit expressions of (S) Γλ and (S) Γσµ λ are easily obtained by direct computation. −

5.5 The currents J (l) (e4 ), J (l) (e3 ), J (l) (eθ ), J (l) (eφ ) We start considering the currents J (1) ≡ J (1) ( LO F ) whose explicit expression is ˆ µσ Dσ LO Fµρ Jρ(1) =(S) π

(5.157)

From eqs. 2.11, 2.12 it follows that (S) µσ

π ˆ

Φ2 ) = sign(α)gαβ 1 + r∗ (Φ∂r Φ − r

(5.158)

and the only components in the null frame diﬀerent from zero are e3µ e4ν

≡

(S)

j = −2µ(S)

(S) µν µ ν π ˆ ea ea

≡

(S)

iab = −(δaθ δbθ + δaφ δbφ )µ(S)

(S) µν

π ˆ

Denoting Fˆ ≡ LO F we write Jρ(1)

≡

(S) µσ

π

Dσ Fˆµρ =(S) π ˆ µσ g µµ g σσ Dσ Fˆµρ

(5.159)

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Asymptotics of the Electromagnetic Field in Schwarzschild Spacetime

=

(S) µσ

π ˆ

941

[(−1/2)(e3µ e4µ + e4µ e3µ ) + eθµ eθµ + eφµ eφµ ]

[(−1/2)(e3σ e4σ + e4σ e3σ ) + eθσ eθσ + eφσ eφσ ]Dσ Fˆµρ 1 = − µ(S) (De4 Fˆµρ ) + (De3 Fˆµρ )e3µ 2 −µ(S) (De Fˆµρ )e µ + (De Fˆµρ )e µ θ

θ

φ

φ

(5.160)

Therefore 1 J (1) (e4 ) = − µ(S) (De3 Fˆµρ )eµ4 eρ4 + (De4 Fˆµρ )eµ3 eρ4 2 − µ(S) (Deθ Fˆµρ )eθµ eρ4 + (Deφ Fˆµρ )eφµ eρ4 1 J (1) (e3 ) = − µ(S) (De3 Fˆµρ )eµ4 eρ3 + (De4 Fˆµρ )eµ3 eρ3 2 − µ(S) (Deθ Fˆµρ )eθµ eρ3 + (Deφ Fˆµρ )eφµ eρ3 1 J (1) (eθ ) = − µ(S) (De3 Fˆµρ )eµ4 eρθ + (De4 Fˆµρ )eµ3 eρθ 2 − µ(S) (Deθ Fˆµρ )eθµ eρθ + (Deφ Fˆµρ )eφµ eρθ 1 J (1) (eφ ) = − µ(S) (De3 Fˆµρ )eµ4 eρφ + (De4 Fˆµρ )eµ3 eρφ 2 − µ(S) (Deφ Fˆµρ )eθµ eρφ + (Deφ Fˆµρ )eφµ eρφ .

(5.161)

As easily (De4 Fˆµρ )e3µ e4ρ (De Fˆµρ )e µ e ρ

= 2∂e4 ρ(Fˆ ) = ∂e (α(Fˆ )(eθ )) − (Φ/r)ρ(Fˆ )

(Deφ Fˆµρ )eφµ e4ρ

= ∂eφ (α(Fˆ )(eφ )) − (Φ/r)ρ(Fˆ ) + (1/r)(cot θ)(α(Fˆ )(eθ ))

(De3 Fˆµρ )e4µ eρ3 (De Fˆµρ )e µ eρ

= −2∂e3 ρ(Fˆ ) = ∂e (α(Fˆ )(eθ )) − (Φ/r)ρ(Fˆ )

(Deφ Fˆµρ )eφµ eρ3

= ∂eφ (α(Fˆ )(eφ )) − (Φ/r)ρ(Fˆ ) + (1/r)(cot θ)(α(Fˆ )(eθ ))

θ

θ

θ

θ

4

3

θ

θ

(De4 Fˆµρ )e3µ eθρ (De3 Fˆµρ )e µ e ρ

= −∂e4 (α(Fˆ )(eθ )) − ∂r Φ(α(Fˆ )(eθ )) = −∂e3 (α(Fˆ )(eθ )) + ∂r Φ(α(Fˆ )(eθ ))

(De4 Fˆµρ )e3µ eρφ

= −∂e4 (α(Fˆ )(eφ )) − ∂r Φ(α(Fˆ )(eφ ))

(De3 Fˆµρ )e3µ eρφ (Deφ Fˆµρ )eφµ eθρ (Deθ Fˆµρ )eθ µ eρφ

= −∂e3 (α(Fˆ )(eφ )) + ∂r Φ(α(Fˆ )(eφ ))

4

θ

= −∂eφ σ(Fˆ ) − (Φ/2r)[α(Fˆ )(eθ ) − α(Fˆ )(eθ )] = ∂eφ σ(Fˆ ) − (Φ/2r)[α(Fˆ )(eφ ) − α(Fˆ )(eφ )]

(5.162)

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substituting these expressions in the various components of the J (1) current and using the Maxwell equations we obtain Φ J (1) (e4 ) = µ(S) (r)[4 ρ − 2div / α] r Φ J (1) (e3 ) = µ(S) (r)[4 ρ − 2div / α] r Φ J (1) (eθ ) = µ(S) (r)[2∂eφ σ + (α(eθ ) − α(eθ ))] r Φ J (1) (eφ ) = µ(S) (r)[−2∂eθ σ + (α(eφ ) − α(eφ ))] r where with α, α, ρ, σ we indicate α( LO F ), α( LO F ), ρ( LO F ), σ( LO F ).

5.6 Proof of Lemma 4.12

¯ αβ We start estimating the term Σt ∩V (u,u) |(K)π Q( LaO F )αβ |. Using eq. 4.77

¯ αβ |(K)π Q( LaO F )αβ | Σt ∩V (u,u)

r t 1 + | log( 2m − 1)| ¯ e3 ) Q( LaO F )(K, 2r τ+ Σt ∩V (u,u)

r t 1 + | log( 2m − 1)| m ¯ e3 ) ≤c Q( LaO F )(K, 2r Φ(r(u, u0 (t))) Σt (r≥r(u,t)) τ+

m ≤c Φ(r(u, u0 (t)))

(5.163) where r(u, t) is the radius of the two dimensional surface S(u, t) = Σt ∩ C(u). To r t(1+| log( 2m −1)|) we observe that, for r ≤ 4m, control the factor τ2 r +

t ≤ 2|r∗ (δ0 )|, τ12 ≤ 1 and + 2 r t 1 + | log( 2m − 1)| 1 δ0 δ0 ≤ |r∗ (δ0 )| 1 + | log( )| ≤ c 1 + | log( )| r m 2m 2m (5.164) Using these estimates we have for t ≤ |r∗ (δ0 )|:

r t 1 + | log( 2m − 1)| m ¯ e3 ) Q( LaO F )(K, c (r(u, u0 (t))) 2r Φ τ+ Σt (r≥r(u,t)) 2

1 δ0 m ¯ e3 ) )| Q( LaO F )(K, ≤ c (r(u, u0 (t))) 1 + | log( Φ 2m 1 + t2 Σt (r≥4m) 2

m δ0 ¯ e3 ) + c (r(u, u0 (t))) 1 + | log( )| Q( LaO F )(K, Φ 2m Σt (r≤4m) 3 δ0 )| H(t) (5.165) ≤ c 1 + | log( 2m

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943

¯ αβ The same estimate holds for the term Σt ∩V (u,u) |(K)π Q(LS LO F )αβ |. ¯ α | is done in the same The estimate of the terms in Σt ∩V (u,u) |(DivQ(LS LO F ))α K way. The ﬁnal result is 2

¯ α | ≤ c 1 + | log( δ0 )| H(t) |(DivQ(LS LO F ))α K (5.166) 2m Σt ∩V (u,u) Acknowledgements. One of the authors, F.N., wants to thank K.Osterwalder for many useful discussions at the ﬁrst stages of this work and the Math. Department of the Swiss Federal Institute of Technology of Zurich, E.T.H., where he was invited for some months.

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J.M.Bardeen, W.H.Press “Radiation ﬁelds in the Schwarzschild background” J. Math. Phys., Vol.14, No.1, 1973, 7–19.

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D.Christodoulou, S.Klainerman “Asymptotic properties of linear ﬁeld equations in Minkowski space” Comm. Pure Appl. Math. XLIII, 1990, 137–199.

[Ch-Kl2]

D.Christodoulou, S.Klainerman “The global non linear stability of the Minkowski space” Princeton Mathematical series, 41, 1993.

[Ch-Kl-Ni] D.Christodoulou, S.Klainerman, F.Nicol` o “On a null hypersurface approach to the proof of stability of the Minkowski space” Under preparation. [Fr]

F.G.Friedlander “The Wave Equation on a Curved Space-time” Cambridge University Press, 1975.

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S.W.Hawking, G.F.R.Hellis “The Large Scale Structure of Spacetime” Cambridge Monographs on Mathematical Physics, 1973.

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S.Klainerman “Einstein Geometry and Hyperbolic Equations” Contemporary Mathematics Vol. 71, AMS 1988,125–156.

[Kl2]

S.Klainerman “Uniform decay estimates and the Lorentz invariance of the classical wave equation” Comm. Pure appl. Math. 38, 1985, 321–332.

[Kl-Ni]

S.Klainerman, F.Nicol` o “On local and global aspects of the Cauchy problem in General Relativity” Class. Q. Grav. To appear.

[McL]

R.G.McLenaghan “An explicit determination of the empty spacetimes on which the wave equation satisﬁes Huygens’ principle” Proc.Camb.Phil.Soc., Vol. 65, 1969, 139–155.

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Ann. Henri Poincar´ e

[M-Th-W]

C.W.Misner, K.S.Thorne, J.A.Wheeler “Gravitation” Freeman, 1973.

[Pe1]

R.Penrose “Conformal treatment of inﬁnity” Relativity, Groups, and Topology B. deWitt and C. deWitt, (eds), Gordon and Breach,, NY, 1963.

[Pe2]

R.Penrose “Zero rest mass ﬁelds including gravitation: asymptotic behavior” Proc. Roy. Soc. Lond. A284, 1962, 159–203.

[Pr1]

R.H.Price “Nonspherical Perturbations of Relativistic Gravitational Collapse. I.” Phys. Rev. D 5 10, 1972, 2419–2438.

[Pr2]

R.H.Price “Nonspherical Perturbations of Relativistic Gravitational Collapse. II. Integer-spin, Zero-Rest-Mass Fields” Phys. Rev. D5 10, 1972, 2439-2454.

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J.Stewart “Advanced General Relativity” Cambridge Monographs on Mathematical Physics, 1991.

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J. Porril, J.Stewart “Electromagnetic and gravitational ﬁelds in a Schwarzschild space-time” Proc.Roy.Soc.Lond. serie A, Vol. 376, 1981, 451–463.

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B.G.Schmidt, J.Stewart “The scalar wave equation in a Schwarzschild space-time” Proc.Roy.Soc.Lond. serie A, Vol.367, 1979, 503–525.

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W.-T. Shu ”Asymptotic Properties of the Solutions of Linear and Nonlinear Spin Field Equations in Minkowski Space” Comm. Math. Phys. 140 (1991) 449–480.

W. Inglese, F. Nicol` o Dipartimento di Matematica Universit` a degli Studi di Roma ”Tor Vergata” Via della Ricerca Scientiﬁca I-00133-Roma Italy Communicated by Sergiu Klainerman submitted 25/10/99, accepted 31/12/99

To access this journal online: http://www.birkhauser.ch

Ann. Henri Poincar´ e 1 (2000) 945 – 976 c Birkh¨ auser Verlag, Basel, 2000 1424-0637/00/050945-32 $ 1.50+0.20/0

Annales Henri Poincar´ e

On Static Stars in Newtonian Gravity and Lane-Emden Type Equations Urs M. Schaudt

Abstract. The equations governing static stellar models in Newtonian gravity are equivalent to a Lane-Emden type equation. For such equations existence, uniqueness, and regularity of global solutions is shown for a large class of right-hand sides, including a subclass of non-Lipschitz continuous equations of state which is relevant if e.g. phase transitions occur. Furthermore, it is shown that for a star of ﬁnite radius the polytropic index of the equation of state is not necessarily bounded near the star’s surface. R´esum´e Les ´ equations qui r´ egissent les mod`eles stellaires statiques en gravit´e newtonienne sont ´equivalentes ` a une ´ equation de type Lane-Emden. Pour de telles ´ equations, l’existence, l’unicit´e et la r´ egularit´e de solutions globales sont montr´ ees pour une large classe de membres de droite, incluant une sous-classe d’´equations d’´ etat continues et non lipschitziennes qui s’appliquent lorsque par exemple une transition de phase a lieu. De plus, il est montr´e que pour une ´ etoile de rayon ﬁni, l’indice polytropique de l’´equation d’´ etat n’est pas n´ ecessairement ﬁni pr`es de la surface.

1 Introduction The ﬁeld equations for equilibrium conﬁgurations of rotating stars, within general relativity or Newtonian gravity, can be written in suitable coordinates as a system of semilinear elliptic partial diﬀerential equations where the elliptic operators on the left-hand side are equivalent to Laplacians in ﬂat space (see [2], [19]). Due to Poisson’s integral formulas this system (with a free boundary!) is equivalent to a ﬁxed point problem u = T u in suitable function spaces. The principal idea, namely to start with “reasonable” functions u0 and iteratively apply the mapping T in order to get approximate solutions of the ﬁxed point problem, led to one of the most eﬃcient numerical solution techniques [2] for rotating stellar models and for more general conﬁgurations. Therefore, it is a natural task (which is also important for the reliability of such numerical solutions) to prove rigorously that these approximate solutions converge to the solution of the original problem. In this article the most simple case toward this goal is investigated, namely static (i.e. non-rotating) stars within the framework of Newton’s theory of gravitation. In this case the above mentioned ﬁxed point problem is equivalent to a singular ordinary diﬀerential equation of second order of so-called Lane-Emden type (see e.g. [8] and [7]). It turns out that, apart from “essentially” polytropic

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equations of state (see [8], [7], [11], [18] and [16]), there are still some interesting open questions for this classical problem. For example, I have found in the literature no general existence or uniqueness results for global solutions if the equation of state is not Lipschitz continous, as is the case if e.g. phase transitions occur (and for realistic models such situations cannot be ruled out). Furthermore, the question for which equations of state the stellar radius is ﬁnite is not yet completely solved. Below, it is shown that existence, uniqueness, and regularity of global solutions of Lane-Emden type equations can be established (along the line of the principal idea of the iterative scheme, and with some a priori error estimates) for a large class of right-hand sides, including equations of state with phase transitions. Moreover, it is shown that essentially polytropic behaviour with index strictly less than ﬁve of the equation of state near the star’s surface is not necessary for the star to be of ﬁnite size. The article is organized as follows: In Sect. 2 the physical problem is brought into an appropriate mathematical form. In Sect. 3 the existence, uniqueness, and regularity results for global solutions of Lane-Emden type equations are presented. In Sect. 4 the relation between the equation of state and the ﬁniteness of the stellar radius is investigated. Finally, in Sect. 5 the obtained results are summarized and some generalizations are pointed out.

2

Mathematical Formulation of the Problem

A static star of ideal ﬂuid has three physical degrees of freedom in Newtonian gravity: the gravitational potential U , the mass density , and the pressure p. These three quantities are scalar ﬁelds on 3-dimensional ﬂat space R3 . The three basic equations governing such an equilibrium conﬁguration are (i) Poisson’s equation, (ii) Euler’s equation, and (iii) an equation of state (EOS). Using the notations ∆ for the Laplacian and ∇ for the gradient in R3 , Poisson’s and Euler’s equation read1 : ∆U = 4π , (1) ∇p = − ∇U ,

(2)

respectively. For most astrophysically interesting objects suﬃciently close to equilibrium it is permissible to presuppose an EOS of the form (see e.g. [7], [20]) = (p) .

(3)

Basically, in this article it is assumed that the real-valued function p → (p), deﬁned on the interval [0, pmax ] ⊂ R+ 0 , obeys the following properties: 1. (p) > 0 for all p > 0, 1 Throughout this article “geometrized units” are used where the gravitational constant is set equal to one.

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On Static Stars in Newtonian Gravity and Lane-Emden Type Equations

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2. (0) = 0, 3. (p) is increasing, 4. (p) is piecewise continuous, and 5. the integral

p

F (p) := 0

1 d˜ p (˜ p)

(4)

exists for all p ∈ [0, pmax ]. Remark 1. Property 1. ensures that the mass density is positive. Property 2. is used for convenience. In the following limp↓0 (p) > 0 is permissible (corresponding to stiﬀ matter at the star’s surface)! Property 3. implies that the matter described by the EOS is “microscopically stable” (see e.g. [10]). The regularity property 4. will be slightly strengthened below (see Deﬁnition 2). Since (0) = 0, property 5. is essentially a condition on the behaviour of (p) as p → 0. Furthermore, it is presupposed that the pressure p, as a function on R3 , is at least continuous everywhere. Let I := {x ∈ R3 | p(x) > 0} be the interior of the star2 , S := ∂I (the boundary of I, i.e.) the star’s surface, and E := R3 \ (I ∪ S) the exterior of the star. Then Euler’s Eq. (2) can be integrated in I: To this end, let Γ ⊂ I be any C 1 -path from a point xS in S to a point x in I. According to the assumptions, p > 0 and = (p) > 0 in I, and p = 0 on S. Therefore −1 (5) F (p(x)) = ∇p, ds = − ∇U, ds = U (xS ) − U (x) Γ

Γ

if U is at least C on I ∪ S. This equation has immediate consequences: 1

1. U is constant on every connected component of S (since F , p, and U are continuous, and F (0) = 0). Carleman [6] (for incompressible matter) and Lichtenstein [14] (for the general case) proved that S consists of only one component. Thus, let US := U (xS ) be the gravitational potential on the star’s surface S. 2. U (x) < US for all x ∈ I, since F (p) > 0 for p > 0. 1 3. Since F (p) exists almost everywhere (a.e.), and F (p) = (p) > 0 a.e. (due + to the assumptions), the function [0, pmax ] p → F (p) ∈ R0 is invertible. Thus, for all x ∈ I ∪ S

p(x) = F −1 (US − U (x)) ,

(6)

i.e. in the interior of the star and on the star’s surface the pressure can be expressed in terms of the gravitational potential. 2 Note

that I = p−1 (R+ ) is an open subset of R3 , for p is C 0 .

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For the following it is convenient to use u(x) := US − U (x)

(7)

as the basic potential3 . Assuming at present that U (x) > US for all x ∈ E (this will be shown below, see Corollary 2) relation (6) can be extended to all x ∈ I ∪S ∪E = R3 : (8) p(x) = F −1 (u(x)+ ) where the abbreviation u+ := sup{u, 0} is used for the restriction of u to its positive part. Therefore, introducing the function [0, F (pmax ) =: umax ] u → µ(u) := (F −1 (u)) ∈ [0, (pmax ) =: max ] ,

(9)

the three basic Eqs. (1)–(3) can be condensed into the single equation ∆u = −4π µ(u+ )

(10)

on R3 . Again, Carleman [6] and Lichtenstein [14] proved that every bounded (i.e. physically relevant) solution of this equation is necessarily spherically symmetric 4 . Remembering that in Eq. (10) one integration constant is free (corresponding to US ) it is convenient to ﬁx this constant by demanding that u takes a given value 0 < uc ≤ umax at the center of symmetry. According to Eq. (8), this is equivalent to ﬁxing the pressure at the star’s center: pc = F −1 (uc ) ⇔ uc = F (pc ) . !

!

(11)

It is shown below (see Corollary 2) that pc = pmax . Convention: In the following, the center of symmetry is always chosen as the origin of the particular coordinate system. Due to its scaling property Eq. (10) can be transformed into a “standard form”. To this end let a > 0 and x = a ξ. Then in the ξ-coordinates Eq. (10) reads: u(ξ) µ(u ) c uc u(ξ) µ(uc ) + · = −4πa2 . (12) ∆ξ uc uc µ(uc ) uc )1/2 , u ˜ := u/uc , and µ ˜(˜ u) := µ(uc u ˜)/µ(uc ). Then the Therefore, let a := ( 4π µ(u c) problem takes the following form:

Given: An increasing function u ˜ → µ ˜(˜ u) with µ ˜(0) = 0 and µ ˜(1) = 1. 3 Note that only such a diﬀerence of U has a physical invariant meaning, for if U is a solution of Eqs. (1) and (2), so is U + c (c ∈ R). Usually, c is ﬁxed by demanding that lim|x|→∞ U (x) = 0. 4 If u ≤ 0, this is a consequence of Liouville’s theorem.

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Wanted: A bounded function u ˜ on R3 satisfying u ˜(0) = u ˜c = 1 and ∆˜ u = −˜ µ(˜ u+ ) .

(13)

Remark 2. From the mathematical point of view it is not necessary in the following that the function µ ˜(˜ u) corresponds to an EOS according to Eq. (9). Convention: For simplicity, the symbols “˜” on u and µ are omitted in the following. Remark 3. Considering that a solution for the above problem is necessarily spherically symmetric Eq.(13) reads in spherical coordinates: 1 d 2 2 du(r) r = u (r) + u (r) = −µ(u(r)+ ) . (14) 2 r dr dr r For a polytropic EOS, i.e. µ(u) = uν with ν > 0 (see Lemma 5 below), this is the so-called Lane-Emden equation 5 (see [8], [7]). With Poisson’s integral, Eq. (13) is equivalent (at least in the distributional sense; see e.g. [15], Theorem 6.21) to 1 µ(u(y)+ ) µ(u(y)+ ) 1 u(ξ) = 1 − (15) dy + dy , ∀ξ ∈ R3 . 4π R3 |y| 4π R3 |ξ − y| Using the spherical symmetry of u and the abbreviation u(r) instead of u(r ξ/|ξ|) for any ξ = 0 in R3 and r ≥ 0, a straightforward computation of the integrals in Eq. (15) yields the following (generally) nonlinear integral equation of Volterra type: r u(r) = 1 − g(r, s) µ(u(s)+ ) ds =: (T u)(r) , (16) 0

for all r := |ξ| ≥ 0, where

s g(r, s) := s 1 − r

(17)

(for r = 0 see Lemma 1 below). Note that 0 ≤ g(r, s) < s for all 0 < s ≤ r and g is not symmetric. In summary, it has been demonstrated that the global problem of a static star in Newtonian gravity is equivalent to the ﬁxed point problem (16) for a 1dimensional real-valued function u. The rest of this article is devoted to the investigation of this ﬁxed point problem and its solutions. This investigation will be similar to the treatment of Picard and Lindel¨ of of the initial value problem for a non-singular ordinary diﬀerential equation of ﬁrst order. 5 Usually, this equation is considered only in the interior and it is understood that the interior solution is matched to an exterior solution where the interior solution vanishes, i.e. the “+”subscript for u on the right hand side of the equation is omitted.

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Remark 4. A similar integral equation approach to static Newtonian stellar models is taken in the work of Bucerius [3]–[5]. In contrast to this article, where the pressure at the center of symmetry, pc , is ﬁxed, Bucerius ﬁxes the stellar radius. However, if the stellar radius is ﬁxed the solutions are not unique in general as the linear case µ(u) = u shows: In this case the solutions are given in the interior sin(r) I ∼ = [0, π) by u(r) = a r where a > 0 is arbitrary. Furthermore, only a few explicit EOS are treated to obtain approximate solutions in form of truncated series in terms of eigenfunctions of the corresponding linear problem.

3

Existence, Uniqueness, and Regularity Results

3.1 A Priori Properties Deﬁnition 1. For v ∈ L∞ (R+ 0 ) let (Qv)(r) :=

r

g(r, s)v(s) ds ,

∀r ≥ 0.

(18)

0

Lemma 1 (Properties of Q). For all v, w ∈ L∞ (R+ 0 ): 1. The mapping v → Qv is linear. 2. If v ≤ w then Qv ≤ Qw. 1 g(1, σ)v(rσ) dσ. 3. (Qv)(r) = r2 0

4. limr↓0 (Qv)(r) = 0. 5. r → (Qv)(r) is (at least) H¨ older continuous diﬀerentiable on every compact 1,α + , i.e. Qv ∈ C subset K ⊂ R+ 0 loc (R0 ) with α ∈ (0, 1), and (Qv) (r) =

r

s 2

0

r

1

σ2 v(rσ) dσ ,

v(s) ds = r

∀r ≥ 0 .

(19)

0

Especially (Qv) (0) = 0. 6. Let δ > 0. If v ∈ C 0 ([0, δ]) then the second derivative of Qv at r = 0 exists and (Qv) (0) = v(0)/3. Especially for all r ∈ [0, δ]: v(0) (Qv)(r) = + qv (r) r2 , 6 with qv ∈ C 0 ([0, δ]) and qv (0) = 0.

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Proof. 1. Distributive law and linearity of the integral. 2. Since g(r, s) ≥ 0 for all 0 < s ≤ r, the assertion follows from the analogous property of the integral. 3. Substitute σ = s/r in (18). 4. Let δ0 > 0, and c0 := sups∈[0,δ0 ] |v(s)|. By assumption 0 ≤ c0 < ∞. Then for all 0 < r ≤ δ0 r r s s r2 v(s) ds ≤ c0 ds = c0 . |(Qv)(r)| = s 1− s 1− r r 6 0

0

Thus limr↓0 (Qv)(r) = 0.

v(|y|) v(|y|) 1 1 5. Since by assumption (Qv)(r := |ξ|) = 4π dy − 4π dy for all R3 |y| R3 |ξ−y| ξ ∈ R3 and v ∈ L∞ (R) the regularity statement follows directly from the regularity properties of the Poisson integral (see e.g. [15], Theorem 10.2). Moreover in the previous equation, and thus in Eq. (18), diﬀerentiation commutes with the integral sign. Hence for all r > 0: (Qv) (r) = r(1 − r 1 r d s(1 − s/r)v(s) ds = 0 (s/r)2 v(s) ds = r 0 σ2 v(rσ) dσ where r/r)v(r) + 0 dr in the last step the substitution σ = s/r was used. An analogous argument as for limr↓0 Qv(r) = 0 shows that limr↓0 (Qv) (r) = 0. 6. Let r > 0. Then r (Qv) (r) − (Qv) (0) = r−0

1 0

σ2 v(rσ) dσ − 0 = r

1

σ2 v(rσ) dσ =: f (r) . 0

Note that f (0) = v(0)/3. Therefore, to prove the assertion it is suﬃcient to show that the function f is continuous on [0, δ]. To this end let ε > 0. For v is continuous on the compact interval [0, δ] (by assumption) it is even uniformly continuous, i.e. ∃ηε > 0 such that |v(r1 ) − v(r2 )| < ε for all r1 , r2 ∈ [0, δ] with |r1 − r2 | < ηε . Since r1 σ, r2 σ ∈ [0, δ] and |r1 σ − r2 σ| ≤ |r1 − r2 | < ηε 1 for all σ ∈ [0, 1], it follows that |f (r1 ) − f (r2 )| ≤ 0 σ2 |v(r1 σ) − v(r2 σ)|dσ < 1 ε 0 σ2 dσ = ε/3 < ε, i.e. f is continuous on [0, δ]. Corollary 1. Every solution u of the ﬁxed point problem (16) has the following properties: 1. limr↓0 u(r) = 1. 1,α (R) with α ∈ (0, 1) and 2. u ∈ Cloc r 2 s u (r) = − µ(u(s)+ ) ds , r 0

∀r ≥ 0 .

Especially u (0) = 0 and if µ ≡ 0 a.e. then u (r) < 0 for all r > 0 .

(20)

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U.M. Schaudt

3. Let µ1 := limu↑1 µ(u). Then u(r) = 1 −

µ1 2 6 r

Ann. Henri Poincar´ e

+ o(r2 ) as r ↓ 0.

Proof. Since u = T u = 1 − Qµ(u+ ), 0 ≤ µ ≤ 1, and the function u → µ(u) is increasing by assumption, the assertions are immediate consequences of Lemma 1. Note: If µ ≡ 0 a.e. then µ(u+ ) > 0 at least on a neighbourhood of r = 0 (since u is continuous and u(0) = 1.) Corollary 2. Let u be a solution of the ﬁxed point problem (16) and µ ≡ 0 a.e.. Then: 1. The functions u, p = F −1 (u+ ), and = µ(u+ ) take their maximal values only at r = 0, i.e. at the center of symmetry. 2. There is at the most one 0 < rS ≤ ∞ with u(rS ) = 0. If such an rS exists it is the stellar radius (otherwise let rS := ∞). If rS < ∞ then u (rS ) < 0. 3. If rS < ∞, the following holds for the gravitational potential U (corresponding to u):      I   <  S = . (21) U , for all x ∈ U (x)   S   E > 4. |u| is bounded. Proof. 1.–3. are immediate consequences of u (r) < 0 for all r > 0. To 4.: u ≤ 1 by 1. If rS = ∞ then u ≥ 0. If rS < ∞ then6 µ(u(r)+ ) ≤ 1[0,rS ] (r) by assumption. r r Hence u(r) ≥ 1 − 0 S s(1 − s/r) ds ≥ 1 − 0 S s ds = 1 − rS2 /2 > −∞ for all r > 0.

3.2 Existence, Uniqueness, and Regularity for Lipschitz Continuous µ Lemma 2. For all v, w ∈ R: |v+ − w+ | ≤ |v − w|. Proof. Since v+ = 12 (v + |v|), it follows that |v+ − w+ | = 12 |(v + |v|) − (w + |w|)| = 1 1 1 2 |(v − w) + (|v| − |w|)| ≤ 2 (|v − w| + ||v| − |w||) ≤ 2 (|v − w| + |v − w|)| = |v − w|. c rj : r < r0 + + Lemma 3. Let 0 < r0 ≤ ∞, c, j ∈ R0 , and R0 r → µ ˆ(r) := . 0 : r ≥ r0 Then  c  (j+2)(j+3) rj+2 : r ∈ [0, r0 ) (22) 0 ≤ (Qˆ µ)(r) =  ≤ c rj+2 + : r ∈ R . 0 j+2 0 Proof. Let r0 < ∞. Evaluating the integral in Eq. (18) yields  c j+2 : 0 ≤ r ≤ r0  (j+2)(j+3) r (Qˆ µ)(r) = j+2 c c  : r ≥ r0 . + j+3 r0j+3 r10 − 1r (j+2)(j+3) r0 6 Let

1X (x) := {1 if x ∈ X, 0 if x ∈ X} be the characteristic function of a set X.

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Since [r0 , ∞) r → j+2 c j+2 r0

c (j+2)(j+3)

c r0j+2 + j+3 r0j+3

1 r0

−

1 r

953

is increasing and tends to

as r → ∞ the assertion follows. Eq. (22) still holds if r0 = ∞.

Lemma 4 (Contraction). Assume that: 1. The function µ is Lipschitz continuous on [0, 1]: µ ∈ C 0,1 ([0, 1]), i.e. ∃& ∈ R+ 0 such that |µ(v) − µ(w)| ≤ &|v − w| for all v, w ∈ [0, 1]. 2. ∃X ⊂ BR := C 0 ([0, R)), .R := supr∈[0,R) |.| with R > 0 and T (X) ⊂ X. Then for all v, w ∈ X: T n (v) − T n (w)R ≤

√

2n & R =: C(&, R) (2n + 1)!

v − wR .

(23)

If in addition R = ∞ and | v(r) ≥ 0} v ∈ X <∞ r∗ := sup rv := sup{r ∈ R+ 0 then for all v, w ∈ X: T n (v) − T n (w)∞ ≤

C(&, r∗ )2n v − w∞ . (2n)!

(24)

Therefore, ∃n0 ∈ N such that T n0 is a contraction on X. Proof. If R < ∞, let r∗ := R. To prove estimates (23) and (24) it is suﬃcient to establish the inequality  n 2n : r ∈ [0, r∗ )   (2n+1)! v − w r n n |(T v)(r) − (T w)(r)| ≤ √ 2n (25)   r∗ v − w : r ∈ [0, R) , (2n)! where the index R of .R is omitted for simplicity. By induction: 1. n = 0: Inequality (25) is true. 2. Step n → n + 1: Assume that (25) is valid for an n, then |T n+1 (v) − T n+1 (w)| = |T (T n v) − T (T n w)| by (16), (18) = |Qµ((T n v)+ ) − Qµ((T n w)+ )| by Lemma 1(1.) = Q µ((T n v)+ ) − µ((T n w)+ ) by Lemma 1(2.) µ Lipschitz7 by Lemma 2 by (25) 7 Note

≤ ≤ ≤ ≤

Q |µ((T n v)+ ) − µ((T n w)+ )| Q (& |(T n v)+ − (T n w)+ |) Q (& |T n v − T n w|) Qˆ µ,

that (T n v)(r)+ , (T n w)(r)+ ∈ [0, 1], since T (X) ⊂ X by assumption.

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c rj : r < r0 n , with c = & (2n+1)! v − w, j = 2n, and 0 : r ≥ r0 r0 = r∗ . Hence, with (22):

where µ ˆ(r) =

|(T n+1 v)(r) − (T n+1 w)(r)| ≤  n (2n+1)!

v−w 2n+2  (n+1)  = (2(n+1)+1)! v − w r2(n+1)  (2n+2)(2n+3) r √ 2(n+1)  n  r∗  (2n+1)! v−w r2n+2 = ∗ 2n+2 (2(n+1))! v − w

: r ∈ [0, r∗ ) : r ∈ [0, R) ,

i.e. inequality (25) is valid for n + 1.

Remark 5. In Lemma 4, µ is assumed to be continuous in particular. Since µ(0) = 0 and µ(1) = 1 by assumption, limv↓0 µ(v) = 0 and limv↑0 µ(v) = 1. Note that it is not necessary that µ is increasing. Proposition 1 (Existence, uniqueness, regularity). Assume µ ∈ C 0,1 ([0, 1]). Then the ﬁxed point problem (16), u = T u, has a unique 8 solution u ∈ C 2,α (R+ 0 ) with α < 1. If in addition the trivially extended function + k,α k+2,α µ ∈ C ((−∞, 1]) with k ≥ 1 then u ∈ C (R0 ). Proof. 1. Let R > 0 and v ∈ XR := {w ∈ C 0 ([0, R)) | w ≤ 1}. Then v(r)+ ∈ [0, 1] for all r ∈ [0, R). Thus µ(v+ ) is deﬁned, and µ(v+ ) ≥ 0. Hence Qµ(v+ ) ≥ 0 ⇒ T v = 1 − Qµ(v+ ) ≤ 1, i.e. T v ∈ XR . Therefore by Lemma 4, there is an n0 such that T n0 is a contraction on XR which is a subset of the Banach space BR := (C 0 ([0, R)), .R := supr∈[0,R) |.|). By virtue of the Banach ﬁxed point theorem (see e.g. [13], Theorem 5.1-2 & Lemma 5.4-3) v = T v has a (unique) ﬁxed point uR ∈ XR and limn→∞ T n u0 = uR for every u0 ∈ XR . 2. Let 0 < R1 ≤ R2 and uR1 , uR2 be the corresponding solutions, which exist by 1. Then uR1 ≡ uR2 on [0, R1 )∩[0, R2 ) = [0, R1 ): By assumption, T n uR1 ≡ uR1 and uR2 ≡ T n uR2 for all n ∈ N. Thus, due to the estimate (23) it follows that uR1 − uR2 R1 = T n uR1 − T n uR2 R1 ≤ Cn uR1 − uR2 R1 with limn→∞ Cn = 0. Hence uR1 − uR2 R1 = 0 ⇔ uR1 ≡ uR2 on [0, R1 ). 3. Since R1 > 0 is arbitrary in 2., there is a unique ﬁxed point u ∈ C 0 (R+ 0 ). 0,1 4. By Corollary 1, u ∈ C 1 (R+ ([0, 1]) and µ(0) = 0 it 0 ) and u ≤ 1. Since µ ∈ C + + 0,1 0,α follows that µ(u+ ) ∈ C (R0 ) ⊂ C (R0 ) with α < 1. Therefore, due to the regularity properties of the Poisson integral u ∈ C 2,α (R+ 0 ) (cf. Corollary 1, and see e.g. [15], Theorem 10.3). If k ≥ 1, the assertion follows by induction using the above argument again. 8 I.e.

µ(v) = 0 for all v < 0.

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Remark 6. 1. u is bounded by Corollary 2, i.e. u∞ < ∞. 2. Due to the regularity properties, diﬀerentiation commutes with the integral sign for the solution u of the ﬁxed point problem (16). Therefore, u is a classical solution of Eq. (14). 3. If µ ≡ 0 a.e. then u ≡ 1 is the unique solution of the ﬁxed point problem (16). 4. In case that r∗ < ∞ (cf. Lemma 4), only the analogue of steps 1. and 4. is needed in the above proof. Furthermore, then the stellar radius rS is ﬁnite. For more details on the question whether the stellar radius is ﬁnite or not see Sect. 4 below. Corollary 3 (A priori error estimates). Assumptions as in Proposition 1. Then for every 1 ≥ u0 ∈ C 0 (R+ 0 ) the sequence ui := T ui−1 , ∀i ∈ N, converges uniformly to the unique ﬁxed point u on every intervall [0, R > 0) and u − un R ≤ cosh C(&, R) · with C(&, R) =

C(&, R)2n · u1 − u0 R (2n + 1)!

(26)

√ &R (where & is the Lipschitz constant of µ). If r∗ < ∞ then

u − un ∞ ≤ cosh C(&, r∗ ) ·

C(&, r∗ )2n · u1 − u0 ∞ . (2n)!

(27)

Proof. Using the triangle inequality it follows that for all n, m ∈ N: T n+m u0 − T n u0 R

≤

by (23) ≤

m−1 k=0 m−1 k=0

T n+k+1 u0 − T n+k u0 R C(&, R)2(n+k) u1 − u0 R (2(n + k) + 1)! ∞

≤

C(&, R)2k C(&, R)2n u1 − u0 R · . (2n + 1)! (2k)! k=0

C(,R)2k Since limm→∞ T n+m u0 = u by Banach’s ﬁxed point principle and ∞ = k=0 (2k)! cosh C(&, R) the ﬁrst a priori estimate follows. If r∗ < ∞, the argument is analogous. Remark 7. Note that the estimates (26) and (27) imply that the convergence is even faster than exponential.

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3.3 “Characteristic” Examples of EOS A physically important class of EOS are the polytropes. In the literature, nearly exclusively this class has been used for static stellar models in Newtonian gravity (see e.g. [8], [7], [11]). Lemma 5 (Polytropic EOS). Given a polytropic equation of state: p() = k γ ⇐⇒ (p) =

p γ1

,

(28)

µ ˜(˜ u) = u ˜ν ,

(29)

k

with k > 0 and γ > 1. Then µ(u) = κ uν , i.e. where ν :=

1 γ−1

> 0 (⇔ γ = 1 + ν1 ) is the index 9 of the polytropic EOS, and κ = (νγ k)−ν ⇐⇒ k =

Proof. By deﬁnition (4): p (˜ p)−1 d˜ p= F (p) =

k1/γ (1− γ1 )

0

The integral exists iﬀ 1 − given by

1 γ

1 . νγ κ(γ−1)

(30)

1

p1− γ = νγ k1/γ p1/(νγ) .

> 0 ⇔ γ > 1, and the inverse of the function F is

u = F (p) = νγ k1/γ p1/(νγ) ⇐⇒ p =

uνγ = F −1 (u) . (νγ)νγ kν

Since µ(u) := ( ◦ F −1 )(u) (by deﬁnition (9)) it follows that µ(u) =

F −1 (u) k

γ1 =

˜ν . and µ ˜(˜ u) = κuν /κuνc = (u/uc )ν = u

uν (νγ)ν

k

ν+1 γ

=

1 uν , (νγ k)ν

Therefore, Proposition 1 applies to a polytropic EOS only if the index ν ≥ 1 since for 0 ≤ ν < 1 the corresponding function [0, 1] u → µ(u) = uν is no more Lipschitz continuous at u = 0, i.e. at the star’s surface10 . To include this case ν < 1, and furthermore, to permit EOS with phase transitions, Proposition 1 must be generalized. This generalization will be developed in Sect. 3.4. It is helpful to have an idealized EOS modeling matter with N phases: 9 In the literature, the notation N or n is frequently used for the index of a polytropic EOS, instead of ν. 10 It is known, that this surface exists if ν ∈ [0, 5); see e.g. [7], [11].

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Lemma 6 (Step function EOS). Given an EOS being a step function: (p) = {i , if p ∈ (pi−1 , pi ), and i = 1, . . . , N } ,

(31)

with N ≥ 1, 0 < 1 ≤ . . . ≤ N , and 0 =: p0 < p1 < . . . < pN =: pmax . Then (32) µ(u) = {i , if u ∈ (ui−1 , ui ), and i = 1, . . . , N } , i pj −pj−1 with u0 = 0, and ui = j=1 for i = 1, . . . , N . Note that the values j of at pi , and µ at ui , are irrelevant for the ﬁxed point problem (16). Proof. Straightforward, following the proof of the preceding Lemma.

3.4 More General µ: Existence, Uniqueness, and Regularity Deﬁnition 2 (Admissible µ). In the following µ : [0, 1] → [0, 1] is called admissible iﬀ 1. µ(0) = 0 and µ(1) = 1. 2. µ is an increasing function. 3. Extend µ by µ(v) := 0 for all v < 0. Then, for every v ≤ 1 there are constants δv (µ) > 0, &v (µ) ≥ 0, αv (µ) ∈ (0, 1] such that for all w, z in the open interval (v − δv (µ), v) the inequality |µ(w) − µ(z)| ≤ &v (µ)|(v − w)αv (µ) − (v − z)αv (µ) |

(33)

holds. 4. D := {v ≤ 1 | µ is not continuous in v} ⊂ [0, 1] is a set of measure zero. Since µ is increasing the left- and right-hand limit exist for all v ≤ 1: 0 ≤ lim µ(w) =: µ− (v) ≤ µ(v) ≤ µ+ (v) := lim µ(w) ≤ 1 . w↑v

w↓v

Especially lim µ(v) =: µ0 ≥ 0 v↓0

and

lim µ(v) =: µ1 ≤ 1 . v↑1

Remark 8. 1. With the trivial extension of µ for negative values the index “+” (for the positive part of a function) can be omitted in the deﬁnition of T (cf. (16)). Furthermore, for all v ≤ 0 condition (33) is trivial. (This extension is only introduced to simplify some of the following proofs.) 2. The functions µ corresponding to a polytropic (especially for ν ∈ [0, 1)) or a step function11 EOS are admissible. 3. If µ corresponds to an EOS then µ(v) > 0 for all v ∈ (0, 1]. 11 Note

that if µ is a step function, then v (µ) = 0, for all v ≤ 1.

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Lemma 7. Assume that µ ≡ 0 a.e. is admissible. Then: 1. w ≤ v ≤ 1 ⇒ T v ≤ T w ≤ 1. 2. Let u0 :≡ 1 and ui := T ui−1 for all i ∈ N. Then: 1,α<1 (a) For all i ∈ N: ui ∈ Cloc (R+ 0 ) and ui (r) < 0 for all r > 0.

(b) For all i ∈ N0 := N ∪ {0}: u2i+1 ≤ u2i+3 ≤ u2i+2 ≤ u2i ≤ 1 .

(34)

Especially, the subsequence {u2i }i∈N0 is decreasing and the subsequence {u2i+1 }i∈N0 is increasing. Therefore, both subsequences converge (at least pointwise on R+ 0 ): lim u2i+1 =: u ≤ u ¯ := lim u2i ≤ 1 .

i→∞

i→∞

¯ and u ¯ = T u , i.e. u and u ¯ are ﬁxed points of T 2 . (c) u = T u 1,α<1 (R+ (d) If v is a ﬁxed point of T 2 then v ∈ Cloc 0 ) and v(r) < 0 for all 1,α<1 + r > 0; i.e. u , u ¯ ∈ Cloc (R0 ) and u (r), u ¯(r) < 0 for all r > 0.

Proof. 1. If w(r) ≤ v(r) ≤ 1 for all r ∈ R+ 0 then 0 ≤ µ(w(r)) ≤ µ(v(r)) ≤ 1 for all since 0 ≤ µ ≤ 1 is an increasing function. Thus by Lemma 1(2.), r ∈ R+ 0 1 ≥ T w = 1 − Qµ(w) ≥ 1 − Qµ(v) = T v. 2. (a) Since 0 ≤ µ ≤ 1 and µ ≡ 0 a.e., the assertion is an immediate consequence of Lemma 1(5.). (b) Since 0 ≤ u0 ≡ 1, it follows by 1. that u1 = T u0 ≤ T 0 = 1 = u0 . Then again by 1., u0 = 1 ≥ u2 = T u1 ≥ T u0 = u1 , u1 = T u0 ≤ T u2 = u3 , and u3 = T u2 ≤ T u1 = u2 . Therefore u1 ≤ u3 ≤ u2 ≤ u0 . By induction the assertion follows. (c) For all r > 0 the sequence {vi (.) := g(r, .)µ(u2i (.))}i∈N0 of functions on (0, r) is bounded: 0 ≤ vi ≤ r. Let D := {v ≤ 1 | µ is not continuous in v} ⊂ [0, 1]. By assumption D is a setof measure zero. Since −1 + ˜ := u2i < 0 on (0, r) for all i ∈ N, the set D i∈N u2i (D) ⊂ R0 has + ˜ measure zero and vi is continuous on R0 \ D for all i ∈ N. Therefore, limi→∞ vi (.) = limi→∞ g(r, .)µ(u2i (.)) = g(r, .)µ(¯ u(.)) a.e. on (0, r). By Lebesgue’s dominated convergence theorem (see e.g. [15], Theorem 1.8) limit and integration sign commute. Thus for all r > 0: u(r)

= =

lim u2i+1 (r) = lim (T u2i )(r) i→∞ r g(r, s)µ(u2i (s)) ds) lim (1 −

i→∞

i→∞

0

Vol. 1, 2000

On Static Stars in Newtonian Gravity and Lane-Emden Type Equations

= 1−

959

r

g(r, s)µ(¯ u(s)) ds 0

= (T u ¯)(r) , ¯ = T (T u) = T 2 u and and u ¯ = T u by analogy. Therefore, u = T u 2 u ¯=Tu=T u ¯. (d) Again an immediate consequence of Lemma 1(5.).

Remark 9. For all compact intervals [0, R > 0] the sequences of functions {u2i }i∈N , {u2i+1 }i∈N are subsets of C 1 ([0, R]). These sequences and its derivatives are uni1 formly bounded since 1− 16 R2 ≤ u1 ≤ ui ≤ 1 and |ui (r)| = r 0 σ2 µ(ui−1 (rσ)+ ) dσ 1 ≤ r 0 σ2 dσ = r3 ≤ R 3 for all i ∈ N. Hence these sequences are equicontinuous (see e.g. [12], Theorem 5.19). Thus by the Arzel` a-Ascoli theorem (see e.g. [12], Theorem 5.20) and by the monotonicity of both sequences it follows that u2i → u ¯ and u2i+1 → u uniformly on [0, R]. Lemma 8. Let 0 < R ≤ ∞. If T 2 has a unique ﬁxed point on [0, R), so does T and both ﬁxed points are equal on [0, R). Proof. Let u = T 2 u. Then T 2 (T u) = T (T 2 (u)) = T (u). Hence T u is a ﬁxed point of T 2 . Since the ﬁxed point of T 2 is unique T u = u, i.e. u is a ﬁxed point of T . Let u ˜ be another ﬁxed point of T . Then u ˜ is also a ﬁxed point of T 2 . However, this ﬁxed point is unique. Thus u ˜ = u. Remark 10. Note that the converse of Lemma 8 is not true in general. Lemma 9. Assume that µ ≡ 0 a.e. is admissible. Let X := T 2 ({˜ v : R+ ˜ ≤ 1}) ⊂ C 1 (R+ 0 →R|v 0), and for all r0 > 0, w ∈ X Xr0 (w) := {v ∈ X | v(r) = w(r) , ∀r ∈ [0, r0 ]} .12 Then there exist injective mappings q: X → C 0 (R+ 0 ), v → q(v) : Xr0 (w) → C 0 (R+ lrw0 (v) 0 ), v →

lrw0 such that: 1. ∀r ∈ R+ 0: v(r) v(r0 + r)

+ q(v)(r) r2 , ∀v ∈ X (35) 6 = w(r0 ) − |w (r0 )| + lrw0 (v)(r) r , ∀v ∈ Xr0 (w) . (36) = 1−

µ

1

12 Note that X (w) is the equivalence class containing w with respect to the equivalence r0 relation ∼r0 on X: v ∼r0 w :⇔ v = w on [0, r0 ].

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+ 2. There are constants η > 0, H, Lw r0 ∈ R , β ∈ (0, 2] such that ∀r ∈ [0, η]:

(a) (b)

|q(v)(r)| w |lr0 (v)(r)|

≤ H rβ , ≤ Lw r0 r ,

∀v ∈ X ∀v ∈ Xr0 (w) .

(37) (38)

z 3. There are constants δ0 , δrz0 > 0, C0 , Crz0 ∈ R+ 0 , α0 , αr0 ∈ (0, 2] such that

(a) ∀r ∈ [0, δ0 ] and ∀v, w ∈ X: |q(T v)(r) − q(T w)(r)| ≤ C0 0

r

1 |q(v)(s) − q(w)(s)| ds . s1−α0

(39)

(b) ∀r ∈ [0, δrz0 ], ∀z ∈ X, and ∀v, w ∈ Xr0 (z): |lrT0z (T v)(r) − lrT0z (T w)(r)| ≤ Crz0

0

r

1 z z z |lr (v)(s) − lr0 (w)(s)| ds . s1−αr0 0 (40)

Proof. 1. If v ∈ X then (by Lemma 1) v ∈ C 1 (R+ 0 ), v (0) = 0, v (r) < 0 for all r > 0, and v (0) = limw↑1 µ(w)/3 = µ1 /3 > 0. Therefore the mappings q and l exist. These mappings are injective since Eq. (35) resp. (36) can be uniquely solved for q resp. l.

2. (a) Let v ∈ X. By deﬁnition there is a vˆ ≤ 1 with v = T 2 vˆ. Let v˜ := T vˆ, then v = T v˜ and v˜ is bounded by 1 from above, and by r → 1 − µ61 r2 from below. Let η := 6δµ1 (µ) > 0, i.e. ∀r ∈ (0, η): v˜(r) ∈ (1 − δ1 (µ), 1). 1 Then, it follows that for all r ∈ (0, η): µ1 2 |q(v)(r)| = r−2 v(r) − 1 − r 6 v ))(r) − 1 − (Qµ1 )(r) = r−2 1 − (Qµ(˜ (by deﬁnition of T ) ≤ r−2 lim (Q |µ(˜ v ) − µ(1 − ε)|)(r) ε↓0

(using Lemma 1) ≤ r−2 lim &1 (µ) Q (1 − v˜)α1 (µ) − (1 − (1 − ε))α1 (µ) (r) ε↓0

(using inequality (33)) = r−2 &1 (µ) Q (1 − v˜)α1 (µ) (r)

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On Static Stars in Newtonian Gravity and Lane-Emden Type Equations

≤ r−2 &1 (µ)

r

g(r, s)

µ

1

6 (since 0 ≤ (1 − v˜(s)) ≤

s2

961

α1 (µ) ds

0

µ1 6

s2 )

α (µ)

=

&1 (µ)µ1 1 r2α1 (µ) 6α1 (µ) (2 + 2α1 (µ))(3 + 2α1 (µ))

≤

&1 (µ)µ1 1 61+α1 (µ)

α (µ)

r2α1 (µ) . α1 (µ)

1 ∈ R+ , and β := 2α1 (µ) ∈ (0, 2] inTherefore, with H := 16(µ)µ 1+α1 (µ) equality (37) follows. Note that H and β are independent of v.

(b) Let r0 > 0, w ∈ X, and v ∈ Xr0 (w). Then by deﬁnition, v(r) = w(r) for all r ∈ [0, r0 ], v (r0 ) = w (r0 ) < 0, and ∃w ˜ ≤ 1 such that w = T w. ˜ Thus, the function R+ 0 r → v(r + r0 ) is bounded by w(r0 ) from above. Since µ(w) ˜ ≤ µ1 ≤ 1 a.e., it is bounded from below by the strictly monotone function r → hw ˜ · 1[0,r0 ) + µ1 · 1[r0 ,∞) (r0 + r) r0 (r) := 1 − Q µ(w) µ1 2 r = w(r0 ) + w (r0 )r0 + 2 0 2 µ1 r03 µ1 r − (r0 + r)2 . (41) −w (r0 ) 0 − r0 + r 3 r0 + r 6 Therefore |lrw0 (v)(r)|

= r−1 w(r0 ) − v(r0 + r) − |w (r0 )| (by (36)) (r) + w (r ) ≤ r−1 w(r0 ) − hw 0 r0 (since hw r0 (r) ≤ v(r0 + r) ≤ w(r0 ), and w (r0 ) < 0) µ1 (3r0 + r) + 6w (r0 ) r =: Lw = r0 (r) r . 6(r0 + r)

w Hence, with 0 ≤ Lw r0 := supr∈[0,η] Lr0 (r) < ∞ for η > 0, inequality (38) w follows. Note that Lr0 is independent of v.

3. (a) Let v, w ∈ X. Then T v, T w ∈ X by deﬁnition. Furthermore, the functions v, w, T v, T w are bounded by 1 from above and from below by 6δ1 (µ) µ1 2 ˜ r → 1 − r . Let δ0 := > 0. Hence ∀r ∈ (0, δ˜0 ): v(r), w(r), 6

µ1

(T v)(r), and (T w)(r) ∈ (1 − δ1 (µ), 1).

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Therefore, due to the properties of µ it follows ∀v, w ∈ X and r ∈ (0, δ˜0 ): |q(T v)(r) − q(T w)(r)| r2 = = |(T v)(r) − (T w)(r)| (by (35)) r g(r, s) µ(v(s)) − µ(w(s)) ds = 0 r ≤ &1 (µ) s (1 − v(s))α1 (µ) − (1 − w(s))α1 (µ) ds 0

(g(r, s) ≤ s, and property (33)) r µ 1 + q(v)(s))α1 (µ) − s1+2α1 (µ) ( = &1 (µ) 6 0 µ1 −( + q(w)(s))α1 (µ) ds 6 (by (35)). Due to estimate (37) and µ61 > 0, for every 0 < λ < µ61 there is a δ0 = δ0 (λ) ∈ (0, δ˜0 ] (independent of v, w !) such that µ61 + q(v)(s) and µ1 µ1 µ1 α 6 + q(w)(s) are in ( 6 − λ, 6 + λ) for all s ∈ [0, δ0 ). Since |(c + x) − α α (c + y) | ≤ (c−d)1−α |x − y|, ∀α < 1, c > 0, and |x|, |y| ≤ d < c, it follows that for all r ∈ (0, δ0 ): |q(T v)(r) − q(T w)(r)| r2 ≤ r 1 (µ)α1 (µ) s1+2α1 (µ) |q(v)(s) − q(w)(s)| ds . ≤ ( µ1 −λ)1−α1 (µ) 6

Thus, with C0 :=

0

(

1 (µ)α1 (µ) µ1 1−α1 (µ) 6 −λ)

< ∞, α0 := 2α1 (µ) ∈ (0, 2], and r−2 ≤

s−2 estimate (39) follows.

(b) Let r0 > 0, z ∈ X, and v, w ∈ Xr0 (z). Then T v, T w ∈ Xr0 (T z). Furthermore as in 2.(b), the functions R+ 0 r → v(r + r0 ), w(r + r0 ) are bounded by z(r0 ) from above and from below by the strictly monotone function r → hzr0 (r) (see Eq. (41)). Let δ˜rz0 := (hzr0 )−1 (z(r0 )− δz(r0 ) (µ)) > 0, then v(r0 + r), w(r0 + r) ∈ (z(r0 ) − δz(r0 ) (µ), z(r0 )) for all r ∈ (0, δ˜rz0 ). Therefore, due to the properties of µ it follows for all v, w ∈ Xr0 (z) and r ∈ (0, δ˜rz0 ): |lrT0z (T v)(r) − lrT0z (T w)(r)| r = = |(T v)(r0 + r) − (T w)(r0 + r)| (by (36)) r0 +r = g(r0 + r, s) µ(v(s)) − µ(w(s)) ds r0

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963

(since v = w on [0, r0 ]) r |µ(v(r0 + s)) − µ(w(r0 + s))| ds ≤ (r0 + δ˜rz0 ) 0

(s → r0 + s, and g(r0 + r, r0 + s) ≤ (r0 + s) ≤ (r0 + δ˜rz0 )) r ≤ (r0 + δ˜rz0 )&z(r0 ) (µ) (z(r0 ) − v(r0 + s))αz(r0 ) (µ) − 0 − (z(r0 ) − w(r0 + s))αz(r0 ) (µ) ds (by (33))

= (r0 + δ˜rz0 )&z(r0 ) (µ)

r

0

sαz(r0 ) (µ) (|z (r0 )| + lrz0 (v)(s))αz(r0 ) (µ) − − (|z (r0 )| + lrz0 (w)(s))αz(r0 ) (µ) ds

(by (36)). In analogy to (a): Due to estimate (38) and |z (r0 )| > 0, for every λ ∈ (0, |z (r0 )|) there is a δrz0 = δrz0 (λ) ∈ (0, δ˜rz0 ] (independent of v, w !) such that |z (r0 )| + lrz0 (v)(s) and |z (r0 )| + lrz0 (w)(s) are in (|z (r0 )| − α λ, |z (r0 )|+λ) for all s ∈ [0, δrz0 ). Since |(c+x)α −(c+y)α | ≤ (c−d) 1−α |x− y|, ∀α < 1, c > 0, and |x|, |y| ≤ d < c, it follows that for all r ∈ (0, δrz0 ): |lrT0z (T v)(r) − lrT0z (T w)(r)| r ≤ (r0 +δ˜rz0 )z(r0 ) (µ)αz(r0 ) (µ) ≤ 1−αz(r ) (µ) (|z (r0 )|−λ)

Thus, with Crz0 := (0, 1] ⊂ (0, 2], and r

0

0

r

sαz(r0 ) (µ) lrz0 (v)(s) − lrz0 (w)(s) ds .

(r0 +δ˜rz0 )z(r0 ) (µ)αz(r0 ) (µ) 1−αz(r ) (µ) 0

(|z (r0 )|−λ) −1 −1

≤s

< ∞, αrz0 := αz(r0 ) (µ) ∈

estimate (40) follows.

Lemma 10 (Contraction). Let R > 0, X ⊂ C 0 ([0, R]), and R : X → X. If there are constants K ∈ R+ 0 , α > 0 such that for all x, y ∈ X and r ∈ [0, R] r 1 |x(s) − y(s)| ds , |(Rx)(r) − (Ry)(r)| ≤ K 1−α 0 s then for all n ∈ N0 and r ∈ [0, R]: K n |(R x)(r) − (R y)(r)| ≤ n

n

α

n!

KRα n r

nα

x − yR ≤

α

n!

x − yR .

(42)

Proof. By induction : 1. n = 0: Since x, y are continuous functions on the compact interval [0, R] the supremum norm x − yR := supr∈[0,R] |x(r) − y(r)| is ﬁnite and inequality (42) is true.

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2. Step n → n + 1: |(Rn+1 x)(r) − (Rn+1 y)(r)| = = |(R(Rn x))(r) − (R(Rn y))(r)| r 1 ≤ K |(Rn x)(s) − (Rn y)(s)| ds 1−α s 0 K n r nα s α ≤ K x − yR ds 1−α n! 0 s K n+1 =

α

(n + 1)!

r(n+1)α x − yR .

Proposition 2 (Existence, uniqueness, regularity). For every admissible µ the ﬁxed point problem (16), u = T u, has a unique solution 1,α<1 in Cloc (R+ 0 ). If µ ≡ 0 a.e. the solution u is a strictly decreasing function. Moreover, if for 0 ≤ r1 < r2 ≤ ∞ the function µ ∈ C k,α<1 (u((r1 , r2 ))) with k ≥ 0 then u ∈ C k+2,α ((r1 , r2 )). Proof. If µ ≡ 0 a.e., then due to the deﬁnition of T (see (16)) u ≡ 1 is the unique ﬁxed point. Therefore, let µ ≡ 0 a.e. in the following. 1. Existence and uniqueness: Due to Lemma 7 and Lemma 8 it is suﬃcient to show that T 2 has at the most one ﬁxed point on R+ ¯ =: u 0 since then u = u is the unique solution of u = T u. To this end, let v, w ∈ F2 := {z ∈ + + 2 C 0 (R+ 0 ) | z = T z on R0 } and A := {r ∈ R0 | v(s) = (T v)(s) = w(s) = (T w)(s) , ∀s ∈ [0, r]}. A = ∅ because 0 ∈ A by Lemma 1. Furthermore, A is closed since v, w and T v, T w are continuous. Moreover A is open in R+ 0 : Let r0 ∈ A. By deﬁnition of A, T n v, T n w ∈ Xr0 (v) = Xr0 (w) for all n ∈ N0 . Then due to Lemma 9 : (a) If r0 = 0: Let X := n∈N0 {q(T n v), q(T n w)} ⊂ q(X) ⊂ C 0 (R+ 0 ) and −1 R := q ◦ T ◦ q . Then R(X) ⊂ X and for all x, y ∈ X: r 1 |(Rx)(r) − (Ry)(r)| ≤ K |x(s) − y(s)| ds 1−α s 0 ∀r ∈ [0, R] by (39), with R := δ0 > 0, K := C0 ∈ R+ 0 , α := α0 > 0. v Tv w Tw (b) If r0 > 0: Then lr0 = lr0 = lr0 = lr0 since T v, T w ∈ Xr0 (T v) = Xr0 (v). Let X := n∈N0 {lrv0 (T n v), lrv0 (T n w)} ⊂ lrv0 (Xr0 (v)) ⊂ C 0 (R+ 0) and R := lrv0 ◦ T ◦ (lrv0 )−1 . Then R(X) ⊂ X and for all x, y ∈ X: r 1 |(Rx)(r) − (Ry)(r)| ≤ K |x(s) − y(s)| ds 1−α s 0 v ∀r ∈ [0, R] by (40), with R := δrv0 > 0, K := Crv0 ∈ R+ 0 , α := αr0 > 0.

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Therefore, in both cases Lemma 10 applies: Let x := q(v), y := q(w) if r0 = 0 and x := lrv0 (v), y := lrv0 (w) if r0 > 0. Then by estimate (42): x − yR = R2n x − R2n yR ≤ c2n x − yR where v = T 2 v, w = T 2 w ⇒ x = R2 x, y = R2 y was used. Since c2n → 0 as n → ∞ it follows that x − yR = 0, i.e. v = w on [0, r0 ] ∪ [r0 , r0 + R] = [0, r0 + R]. Then by Lemma 8, T v = v = w = T w on [0, r0 + R]. Thus, the set A is open because R > 0. In summary, A is open, closed, and not empty. + Hence, A = R+ 0 , i.e. if v, w ∈ F2 then v ≡ w on R0 . 2. The conditional higher regularity properties follow as in Propositon 1. Remark 11. 1. The proof shows that (a) the sequence ui = T ui−1 with u0 ≡ 1 converges to the unique ﬁxed point u and due to (34) the following a posteriori estimates hold for all i ∈ N: u2i+1 ≤ u ≤ u2i . (43) (b) there is (at least) a neighbourhood [0, δ0 > 0] of r0 = 0 such that the uniform convergence of ui on [0, δ0 ] is even faster than exponential. 2. If there is an rS < ∞ with u(rS ) = 0, i.e. the stellar radius rS is ﬁnite, then u ∈ C ∞ ((rS , ∞)). More precisely, then Eq. (16) implies u(r) = a + b/r for r r ≥ rS with a < 0 and b = M := 0 S µ(u(s))s2 ds > 0. 3. If µ is “merely” continuous, then it is straightforward to prove existence of solutions of Eq. (16) in analogy to the Peano existence theorem for nonsingular ordinary diﬀerential equations: The set X := {v ∈ C 0 ([0, R]) | v ≤ 1} is closed and convex13 . Furthermore, the mapping X u → T u ∈ X is continuous14 , T (X) ⊂ X, and T (X) ⊂ C 1,α<1 ([0, R]) ⊂ C 1 ([0, R]) is precompact by the Arzel` a-Ascoli theorem15 . Hence, Schauder’s ﬁxed point theorem (see e.g. [9], Corollary 11.2) guarantees that T has a ﬁxed point in X. However, such a ﬁxed point is not necessarily unique. Corollary 4. If µ is a step function with N “phases” 16 then the sequence u0 :≡ 1, ui≥1 := T ui−1 coincides(!) with the unique ﬁxed point u = T u for all i ≥ 2N + 2. for all v, w ∈ X, and λ ∈ [0, 1]: λv + (1 − λ)w ≤ λ + (1 − λ) = 1. that µ is uniformly continuous for it is continuous on the compact interval [0, 1] by assumption. 15 Because 1 − R2 ≤ v ≤ 1 and |v | ≤ R for all v ∈ T (X), it follows that T (X) is a set of 6 3 uniformly bounded and equicontinuous functions (see e.g. [12], Theorem 5.19). 16 Here, it is not necessary that µ corresponds to a step function EOS, i.e. lim v↓0 µ(v) = 0 is allowed. 13 Since 14 Note

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Proof. Due to Proposition 2, u = T u exists and is unique. If µ ≡ 0 a.e. then ui ≡ 1 = T 1 for all i ∈ N. Therefore let µ(v) ≡ 0 a.e. in the following, i.e. µ(v) = {k , if v ∈ (vk−1 , vk ), k = 1, . . . , N } with −∞ =: v−1 < 0 =: v0 < v1 < . . . < vN := 1 and 0 =: 0 ≤ 1 < . . . < N ≤ 1. By Lemma 7, the functions ui≥1 are strictly decreasing. Thus, for every k = 0, . . . , N and i ∈ N there is a unique 0 < rik ≤ ∞ such that either ui (rik ) = vk or ui > vk on [0, ∞ =: rik ). Especially riN = 1 for all k k k i ≥ 1. Since u2i+1 ≤ u2i+3 ≤ u2i+2 ≤ u2i , it follows that r2i+1 ≤ r2i+3 ≤ r2i+2 ≤ k r2i . Assumption : ∃k0 ∈ {1, . . . , N } and j0 = 2i0 + 1 ≥ 1 such that rjk00 < ∞ and uj = uj0 on [0, rjk00 ] for all j ≥ j0 (⇒ rjk0 = rjk00 , ∀j ≥ j0 ). Then uj≥j0 ∈ (vk0 −1 , vk0 ) k0 −1 k0 −1 k0 −1 (⇒ µ(uj ) = k0 ) on (rjk00 , rjk0 −1 ). Since rjk0 −1 ≤ rj+2 ≤ rj+3 ≤ rj+1 for all k0 −1 k0 −1 j ≥ j0 + 2n (with n ∈ N0 ), it follows that µ(uj≥j0 +1 ) = k0 on (rj0 , rj0 +2 ). −1 Hence, µ(uj≥j0 +1 ) = µ(uj0 +1 ) on [0, rjk00+2 ). This implies (since ui+1 = 1−Qµ(ui )) k0 −1 −1 = ∞, then uj0 +2 is already that for all j ≥ j0 +2: uj = uj0 +2 on [0, rj0 +2 ). If rjk00+2 k0 −1 the (unique) ﬁxed point u. If rj0 +2 < ∞, then (by continuity) uj≥j0 +2 = uj0 +2 on −1 [0, rjk00+2 ]. In summary, either uj0 +2 is the ﬁxed point u or the assumption holds for k0 − 1 ∈ {0, . . . , N − 1} and j0 + 2 ≥ 3. If the assumption is true for k0 = 0, then uj≥j0 +1 = uj0 +1 on R+ 0 (i.e. the ﬁxed point is reached) because µ(uj ) = 0 on (rj00 , ∞) for all j ≥ j0 . Since ui≥1 (0) = 1, the assumption is true for k0 = N and j0 = 1. Therefore (note that k0 ∈ {0, . . . , N }) it follows that at the most after 1 + N · 2 + 1 = 2N + 2 iterations the ﬁxed point of T is reached. Remark 12. 1. It can be shown by similar arguments that even for any u0 ≤ 1 the sequence ui = T ui−1 coincides with the ﬁxed point after ﬁnite steps of iterations. This “amazing” convergence property is closely related to the fact that in (33) the constants &v (µ) = 0 for all v ≤ 1. 2. If µ is a step function, the ﬁxed point u can be constructed explicitly (at least in principe): If u(r) ∈ (vk−1 , vk ) then u must have the form u(r) = ak +

k 2 bk − r r 6

(because u (r) + 2r u (r) = −k , cf. Eq. (14)). Since u ∈ C 1 (R+ 0 ), the constants ak , bk ∈ R are uniquely determined by the condition that u and u are N 2 continuous on R+ 0 . Starting with u(0) = 1 and u (0) = 0, i.e. u(r) = 1− 6 r as long as u(r) ≥ vN−1 , these conditions lead at the most to cubic equations.

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4 Relations between µ and Finiteness of rS For physical applications, one of the most important question is whether there is a ﬁnite rS with u(rS ) = 0 for a given admissible µ or not17 , i.e. in the context of static stars in Newtonian gravity whether for a given EOS the stellar radius rS is ﬁnite or not. This question will be investigated in this section. The following facts are known about this question in the literature: For a polytropic EOS the stellar radius is ﬁnite if the index ν ∈ [0, 5) and it is inﬁnite if ν ≥ 5 (see [7] and [11]). If µ behaves “essentially” polytropic near µ = 0, then the stellar radius is ﬁnite if the polytropic index ν ∈ [0, 3]. And if ν ∈ (3, 5) the radius can be ﬁnite or inﬁnite (see [18], [16], and [17], p. 20). Rendall and Schmidt [18] raised the question: Is it necessary for a ﬁnite stellar radius that the function µ behaves essentially polytropic with index strictly less than ﬁve near µ = 0 (i.e. at the star’s surface)? Below (see Remark 15 ), it is shown that the answer for Newtonian gravity is: No. Lemma 11 (A priori criteria). Assume that µ is admissible. By Proposition 2 the corresponding ﬁxed point problem (16) has a unique solution u. Then the following holds: 1. If limv↓0 µ(v) =: µ0 > 0 then rS is ﬁnite. 2. Let u0 :≡ 1 and ui≥1 := T ui−1 . If ∃ j = 2i with i ≥ 1 such that ∃ rj < ∞ with uj (rj ) = 0 then rS ≤ rj , i.e. rS is ﬁnite. Proof. 1. Assume, in contrary to the assertion that rS is not ﬁnite, i.e. u > 0 on R+ 0. Therefore, µ(u) ≥ µ0 > 0 on R+ by assumption. Hence u(r) = (T u)(r) = 0 1 − (Qµ(u))(r) ≤ 1 − (Q(µ0 · 1[0,∞) ))(r) = 1 − µ60 r2 =: u1 (r), for all r ∈ R+ 0, which is a contradiction since u1 (r) < 0 for r > 6/µ0 < ∞. 2. Since limi→∞ ui = u by the proof of Proposition 2 and u ≤ u2i≥0 due to the estimates (34) in Lemma 7 the assertion follows (because u2i is decreasing). Lemma 12 (General sub- and supersolution). For every admissible µ and corresponding ﬁxed point u the following holds: ˇ(r) := 1 − 1. Let u ˇ : R+ 0 → R, r → u

µ1 6

r2 . Then u ˇ ≤ u, i.e. u ˇ is a subsolution.

ˆ := {v ∈ (0, 1] | µ(v) > 0} 1, and 2. Let µ ≡ 0 a.e., D ˆ v → rˆ(v) := 6(1−v) D µ(v) µ(v) 2 R+ ˆv (r) := 1 − 6 r ≥ v : r ≤ rˆ(v) < ∞ 0 r → u v : r ≥ rˆ(v) R+ 0 r

→ u ˆ(r) := inf u ˆv (r) ≥ 0 . ˆ v∈D

Then u ≤ u+ ≤ u ˆ, i.e. u ˆ is a supersolution. 17 Note

that rS is unique if it exists (if not, let rS := ∞), since u is strictly monotone.

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Proof. 1. µ(u+ ) ≤ µ1 a.e. because u(0) = 1, u is decreasing (by Corollary 1), and µ : [0, 1] → [0, 1] is increasing with µ ≤ µ1 a.e. (by assumption). Hence by Lemma 1, u(r) = (T u)(r) = 1 − (Qµ(u+ ))(r) ≥ 1 − (Q(µ1 · 1[0,∞) ))(r) = 1 − µ61 r2 =: u ˇ(r) for all r ∈ R+ 0. ˆ with v0 ∈ u(R+ )+ . ˆ ⊂ u(R+ )+ . For assume this is false, i.e. ∃v0 ∈ D 2. D 0 0 + Then by deﬁnition, rˆ(v0 ) < ∞ and u > v0 on R0 (because u is decreasing). Therefore (since µ is increasing) 0 ≤ v0 < u(r) = (T u)(r) ≤ 1 − (Q(µ(v0 ) · 1[0,∞) ))(r) = 1 − µ(v6 0 ) r2 → −∞ as r → ∞, which is a contradiction. It ˆ To this end let v0 ∈ D. ˆ remains to show that u ˆv ≥ u+ for all v ∈ D. + Hence by the preceding v0 ∈ u(R+ ) , i.e. ∃r ∈ R with u(r ) = v > 0. 0 0 0 0 + 0 < Since u is strictly decreasing, r0 is unique and u(r) > v for all r r . Thus < 0 > 0 µ(u+ ) ≥ µ(v0 ) on [0, r0 ]. Then for all r ∈ [0, r0 ]: 0 < v0 ≤ u(r)+ = u(r) = ˆv0 (r) (⇒ r0 ≤ rˆ(v0 )). (T u)(r) ≤ 1 − (Q(µ(v0 ) · 1[0,r0 ] ))(r) = 1 − µ(v6 0 ) r2 = u Since u ˆv0 ≥ v0 > 0 on R+ and v ≥ u(r) ≥ u(r) for all r ≥ r0 , it follows 0 + 0 that u ˆv0 ≥ u+ ≥ u on [0, r0 ] ∪ [r0 , ∞) = R+ . 0 Corollary 5. If µ(v) > 0 for all v > 0 and limv↓0 µ(v) ≥ 0 for an admissible function µ, then lim u(r)+ = 0 r→∞

for the corresponding ﬁxed point u of T . ˆ Then by Proof. Let ε > 0 and vε := min{ε, 1} ∈ (0, 1]. By assumption vε ∈ D. ˆ(r) ≤ u ˆvε (r) = vε ≤ ε for all Lemma 12(2.), 0 < rˆ(vε ) < ∞ and u(r)+ ≤ u r ≥ rˆ(vε ), i.e. limr→∞ u+ (r) = 0. Remark 13. For instance, the Corollary applies to all EOS, especially to polytropic with index ν > 5. Lemma 13 (Necessary and suﬃcient condition). Let u be the unique ﬁxed point of T for an admissible µ. Equivalent are: 1. rS is ﬁnite. 2. ∃ r0 ∈ (0, ∞) such that 3. ∃ r0 ∈ (0, ∞) such that

u(r0 ) < |u (r0 )| = −u (r0 ) r0 r0 1< s µ(u(s)+ ) ds . 0

Proof. (1. ⇒ 2.): Assume rS is ﬁnite. Then r0 := rS > 0 since u(0) = 1 and u ∈ C 0 . Hence u(r0 ) = 0 and u (r0 ) < 0 by Corollary 1. Thus inequality 2. holds.

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(2. ⇒ 1.): Assume inequality 2. holds, i.e. ∃ r0 ∈ (0, ∞) such that u(r0 ) + r0 u (r0 ) < 0. Let 0 ≤ v := u+ · 1[0,r0 ] . Then v ≤ u+ and µ(v) ≤ µ(u+ ) (since µ is increasing by assumption). Therefore (by Lemma 1) T v = 1 − Qµ(v) ≥ 1 − Qµ(u+ ) = T u = u and ∀r ≥ r0 (using Eqs. (16) and (20)): r0 s (T v)(r) = 1 − s 1− µ(u(s)+ ) ds r 0 r0 2 r0 s µ(u(s)+ ) ds b =: a + = 1 − 0 s µ(u(s)+ ) ds + 0 r r r0 2 s b (T v) (r) = − µ(u(s)+ ) ds = − 2 . r r 0 Since T v = T u = u on [0, r0 ] and T v, u ∈ C 1 (R+ 0 ), it follows that u(r0 ) = (T v)(r0 ) = a + rb0 and u (r0 ) = (T v) (r0 ) = − rb2 . Thus 0

0 > u(r0 ) + r0 u (r0 ) = a . Therefore limr→∞ u ≤ limr→∞ T v = limr→∞ a + b/r = a < 0. Because u(0) = 1 and u is decreasing there is an rS ∈ (0, ∞) with u(rS ) = 0, i.e. 1. holds. r (2. ⇔ 3.): 0 > u(r0 ) + r0 u (r0 ) = a = 1 − 0 0 s µ(u(s)+ ) ds. Remark 14. 1. Note that limr0 ↓0

u(r0 ) r0

= ∞ and limr0 ↓0 u (r0 ) = 0.

2. If u(r0 ) > 0, then r0 < rS and the knowledge of the ﬁxed point u(r) for r > r0 is not needed in order to guarantee the ﬁniteness of the stellar radius rS ! 3. For the polytropic EOS with µ(u) = u5 ): u(r) = (1 + ∞index ν = 5 (i.e. ∞ 1 2 −1/2 (⇒ rS = ∞) and 0 s µ(u(s)+ ) ds = 0 s (1 + 13 s2 )−5/2 ds = 1. 3 r ) Corollary 6. Let u be the unique ﬁxed point of T for an admissible µ. Then lim u (r) = 0 .

r→∞

Proof. If rS < ∞, then Eq. (16) implies that u(r) = a + b/r for all r > rS with a, b ∈ rR. Hence the assertion follows. If rS = ∞ then u ≥ 0. Thus u+ = u and 0 s µ(u(s)) ds ≤ 1 for all r ∈ R+ 0 by Lemma 13(3.). Therefore |u (r)| = r 2 −1 r (s/r) µ(u(s)) ds ≤ r s µ(u(s)) ds ≤ 1/r. Hence the assertion follows. 0 0 Corollary 7 (Suﬃcient conditions). Assume µ is admissible and u is the corresponding ﬁxed point of T .

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1. If there is u ˜ ≤ u and r0 < ∞ with 1 < 2. If

µ1 3

<

1 0

r0 0

Ann. Henri Poincar´ e

s µ(˜ u(s)+ ) ds ⇒ rS is ﬁnite.

µ(v) dv ⇒ rS is ﬁnite.

3. If there is u ˜ ≥ u with

∞ 0

s µ(˜ u(s)+ ) ds ≤ 1 ⇒ rS is inﬁnite.

Proof. 1. Since µ is increasing u+ ) ≤ µ(u+ ) and therefore 1 < r0by assumption, µ(˜ r0 s µ(˜ u (s) ) ds ≤ s µ(u(s) ) ds. Hence by Lemma 13(3.) the assertion + + 0 0 follows. 2. If 0 < µ1 ≤ 1, then by Lemma 12(1.) u ˇ(s) = 1 − µ61 s2 ≤ u(s) and u ˇ ≥ 0 on [0, r0 := 6/µ1 < ∞]. Thus, using the substitution v = u ˇ(s) r0 0

s µ(ˇ u(s)+ ) ds =

3 µ1

1 0

µ(v) dv > 1 .

Hence by 1. the assertion follows. If µ1 = 0 then µ ≡ 0 a.e. by deﬁnition, i.e. the assumption does not hold. by assumption. Therefore, it follows 3. µ(˜ u+ ) ≥ µ(u+ ) since µ is increasing ∞ r ∞ that 1 ≥ 0 s µ(˜ u(s)+ ) ds ≥ 0 s µ(u(s)+ ) ds, i.e. 1 ≥ 0 0 s µ(u(s)+ ) ds for all r0 ∈ (0, ∞). Hence by Lemma 13(3.) the assertion follows. Remark 15. 1. Note that statement 2. in Corollary 7 is an a priori criterion. Furthermore, this criterion answers (within Newtonian gravity) the mentioned question raised by Rendall and Schmidt: Since there are admissible µ having polytropic behaviour with index ν ≥ 5 near µ = 0 and satisfying this criterion 2. (e.g. µ(u) = uν≥5 · 1[0,v< 23 ) + 1[v,1] ), the answer to their question is: The essentially polytropic behaviour with index strictly less than ﬁve near µ = 0 is not necessary for a ﬁnite stellar radius. Joseph and Lundgren remark (see [11], p. 243, Footnote :) that Lebovitz made a similar observation. 2. Criterion 2. in Corollary 7 reads for polytropic EOS with index ν ∈ R+ 0: 1 ν 1 1 v dv = > , which is valid for ν ∈ [0, 2). Since it is known that ν+1 3 0 for all polytropic EOS with ν < 5 the stellar radius rS is ﬁnite (see [7] and [11]), this shows that the criterion 2. in Corollary 7 is not necessary. Another suﬃcient a priori criterion for ﬁnite radius, which is in the psharp p˜ p polytropic case, was given by Simon [21], Eq. (14) : F (p) = 0 (dp) ≤ 6 (p) ˜ for all p ∈ (0, pc ). Since for polytropic EOS this (“pointwise”) condition is valid only if the index ν ≤ 5, the argument in 1. shows that this conditon is not necessary either.

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3. An illuminating example for the general case is the following: For the polytropic EOS µ(v) = v ν with index ν ≥ 5 the stellar radius is inﬁnite. However, for all µδ (v) = δ ν · 1[δ,1] (v) (note that µδ ≤ µ) with 0 < δ < 23 the stellar radius is ﬁnite! 4. The supersolution u ˆ in Lemma 12 is too weak in order to give (together with condition 3. in Corollary 7) a reasonable (suﬃcient) criterion for an (realistic) EOS so that the radius is inﬁnite. However, if µ is admissible and of the form µ = 0 on [0, v0 ] with 0 < v0 < 1 and 0 < µ0 := µ+ (v0 ) ≤ µ ∈ C 1 ((v0 , 1)) ∞ 1 1 u(s)+ ) ds = v0 rˆ(v) µ(v)| rˆ (v)| dv = 3 v0 (1 + (1 − on (v0 , 1], then 0 s µ(ˆ 1 (v) v) µµ(v) ) dv = 3(1 − v0 )(1 + | ln µ0 |) − 3 v0 | ln µ(v)| dv, where the substitution s = rˆ(v) = 6(1−v) ˆ(ˆ r(v)) = v was used. For example, if µ = µ0 · 1[v0 ,1] µ(v) and u ∞ then 0 s µ(ˆ u(s)+ ) ds = 3(1 − v0 ). Hence, if v0 ≥ 23 then 3(1 − v0 ) ≤ 1 and rS is inﬁnite by condition 3. in Corollary 7, which is the optimal result in this special case. 5. Since the subsolutions u2i+1 and the supersolutions u2i (cf. (43)) converge monotonically to the unique solution u as i → ∞, these sub- and supersolutions can be used (at least in principle) in condition 1. and 3. of Corollary 7 in order to provide sequences of (suﬃcient) conditions of increasing sharpness for a given admissible µ. Since criterion 3. in Lemma 13 is suﬃcient and necessary these sequences of conditions are optimal in the “limit” i → ∞ (e.g. for the polytropic EOS with “critical” index ν = 5 it was already men∞ tioned that 0 s u(s)5 ds = 1). Lemma 14 (Gronwall type). Let d ∈ C 0 ([0, R)) with r d(r) ≤ a r2 + b g(r, s)d(s) ds , ∀r ∈ [0, R) 0

and 0 < R ≤ ∞, a ∈ R, b ≥ 0. Then for all r ∈ [0, R): √ 6a sinh( b r) 1 √ − 1 = a r2 1 + 20 d(r) ≤ b r2 + O(b2 r4 ) . b br Proof. For every ε > 0 let fε (r) := (ε + integration shows that for all r ∈

R+ 0:

fε (r) = ε + a r2 + b

√ b r) 6a sinh( √ b ) br

r 0

−

(44)

6a b .

Then, an elementary

g(r, s)fε (s) ds .

(45)

Hence, in order to prove estimate (44) it is suﬃcient to show that d(r) < fε (r) for all r ∈ [0, R): Since d(0) ≤ 0 < ε = limr↓0 fε (r) this inequality is true for r = 0. Assume r0 := inf{r ∈ [0, R) | d(r) = fε (r)} > 0. Since d, fε ∈ C 0 , it follows that

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d(r0 ) = fε (r0 ) and d ≤ fε on [0, r0 ] (otherwise, the intermediate value theorem yields a contradiction). Therefore, r0 d(r0 ) ≤ a r02 + b g(r0 , s) d(s) ds 0

(by assumption) r0 g(r0 , s) fε (s) ds < ε + a r02 + b 0

(since ε > 0, b ≥ 0, and ∀s ∈ [0, r0 ]: g(r0 , s) ≥ 0, d(s) ≤ fε (s)) = fε (r0 ) (by (45)) ,

which is a contradiction.

Remark 16. Since g(r, s) ≤ s, Gronwall’s Lemma (see e.g. [1], p. 99) can be used to obtain a similar estimate. However, this estimate is weaker than (44). Corollary 8 (“Approximation”). Assume µ1 , µ2 are admissible functions and u1 , u2 are the corresponding unique ﬁxed points due to Proposition 2. Let δ := µ1 − µ2 1 := supv∈[0,1] |µ1 (v) − µ2 (v)| ∈ [0, 1]. If in addition µ1 is Lipschitz continuous on [0, 1], with Lipschitz constant & > 0, then for all r ∈ R+ 0: √ δ sinh( & r) 1 √ |u1 (r) − u2 (r)| ≤ & r2 + O(&2 r4 ) . (46) − 1 = δ r2 16 + 120 & &r Proof. Let d(r) := |u1 (r) − u2 (r)| (note that d ∈ C 0 (R+ 0 )). Then r g(r, s) µ2 (u2 (s)+ ) − µ1 (u1 (s)+ ) d(r) = 0

(by Eq. (16) r ≤ g(r, s)|µ2 (u2 (s)+ ) − µ1 (u2 (s)+ )| ds 0 r g(r, s)|µ1 (u2 (s)+ ) − µ1 (u1 (s)+ )| ds + 0r r ≤ δ g(r, s) ds + & g(r, s)d(s) ds 0 0 r δ 2 r +& g(r, s)d(s) ds . = 6 0 By Lemma 14 the assertion follows.

Remark 17. If, for example, µ2 is a step function with ﬁnite “stellar radius” (note that this can be explicitly decided, at least in principle), then estimate (46) implies that all Lipschitz continuous µ1 with µ1 − µ2 1 ≤ δ ( 1 have also ﬁnite stellar radius.

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4.1 Relation between EOS and Surface Potential In the following, the omitted symbols “˜” on µ and u are restored for accuracy ˜ := {˜ (see Eq. (12) and the following convention). Let A µ admissible | ∃ pc > 0 and −1 ˜ it follows that the ∃ EOS such that µ = ◦ F and r˜S (˜ µ) < ∞}. If µ ˜ ∈ A, support of the positive part of the corresponding solution u˜ = u ˜(˜ µ) and the mass density ˜ = µ ˜(˜ u+ ), viewed as spherically symmetric functions on R3 , is a ball with radius r˜S < ∞. Then, due to Poisson’s equation (1) the corresponding Newtonian ˜(y) dy vanishes at inﬁnity. Therefore, by gravitational potential U (x) ∝ − R3 |x−y| ˜ Eq. (7) it follows that limr→∞ u ˜(r) = US := U (rS ξ/|ξ|)/uc (for every 0 = ξ ∈ R3 ). Since the solution u ˜ is unique, the mapping ˜ −→ (−∞, 0) = R− , φ˜ : A ∞ ˜ ˜(˜ µ, r) = 1 − sµ ˜(˜ u(˜ µ, s)+ ) ds µ ˜ −→ φ(˜ µ) := lim u r→∞

(47)

0

(by Eq. (16) , ˜ the corresponding normalized Newtonian which assigns to every admissible µ ˜∈A ˜S , is well-deﬁned. surface potential U Remark 18. The following diagram holds : ˜

φ ˜S . (pc , ) −→ (uc , µ) −→ µ ˜ −→ U

Note that all symbols “→” represent well-deﬁned mappings. It is known that φ˜ restricted to polytropic EOS with index ν ∈ (0, 5) is injective. ˜ → R− is not injective, i.e. in general the value Lemma 15. The mapping φ˜ : A of the normalized surface potential of a solution does not uniquely determine the standard form of the EOS. Proof. Let µ ˜ := · 1[0,1] with ∈ (0, 1]. Then, it is straightforward to show that ˜ φ(˜ µ ) = −2 for all ∈ (0, 1]. Remark 19. Another more interesting example, which shows that φ˜ is not injective √ u) = u ˜, and µ ˜2 = 9−18 33 · 1[0, 12 ) + 1[ 12 ,1] (where in general, is the following: Let µ ˜1 (˜ √ 9− 33 ˜ µ1 ) = φ(˜ ˜ µ2 ) = −1. ≈ 0.18). Again, it is straightforward to show that φ(˜ 18

(Note that u ˜(˜ µ1 , r) =

sin(r) r

for r ∈ (0, π].)

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Conclusions

It was shown that existence, uniqueness, and regularity of global solutions for Lane-Emden type equations can be established using a simple iterative scheme for quite general right-hand sides, including equations of state with phase transitions. The iteration converges uniformly for Lipschitz continuous right-hand sides at a rate even faster than exponential (at least on every compact set). For a large subclass of the non-Lipschitz continuous right-hand sides, the same convergence behaviour could be established only near the center of symmetry. Whether this rate of convergence still holds outside a neighbourhood of the center of symmetry or not remains an open question (apart for step function, where the solution is reached after ﬁnite steps of iteration!). Furthermore, two equivalent criteria were given which are necessary and suﬃcient so that the stellar radius is ﬁnite. These criteria lead to a suﬃcient (however not necessary) a priori condition on the equation of state which shows that essentially polytropic behaviour with index strictly less than ﬁve of the equation of state near the star’s surface is not necessary in order to have a star of ﬁnite size. Still, the question for a “practicable” suﬃcient and necessary a priori criterion is open. Moreover, it was shown that the relation between the equation of state and the surface potential is not injective in general. Since the ﬁeld equations for equilibrium states of rotating stars in Newton’s as well as in Einstein’s theory of gravitation have essentially the same structure as the ﬁeld equations in the static Newtonian case (as was pointed out in the introduction), the numerical results obtained with the method [2] give hope that some of the main ideas in this article can be generalized to obtain existence results for realistic models of rotating stars within general relativity. Acknowledgements. I thank Herbert Pﬁster for encouragement, helpful discussions, and for reading the manuscript. And I am grateful to Eric Gourgoulhon for translating the abstract into French. This work was supported by Deutsche Forschungsgemeinschaft.

References [1] H. Amann, Gew¨ ohnliche Diﬀerentialgleichungen. Berlin, New York: de Gruyter 1983 [2] S. Bonazzola, E. Gourgoulhon, M. Salgado, J.A. Marck, Axisymmetric rotating relativistic bodies: a new numerical approach for “exact” solutions. Astron. Astrophys. 278 (1993) 421–443. [3] H. Bucerius, Integralgleichungstheorie des Sternaufbaus. I. Die polytrope Gaskugel. Astron. Nachr. 265 (1938) 145–158. [4] H. Bucerius, Integralgleichungstheorie des Sternaufbaus. II. Die numerische L¨osungsmethode. Astron. Nachr. 266 (1938) 49–62.

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[5] H. Bucerius, Integralgleichungstheorie des Sternaufbaus. III. Zusammengesetzte polytrope Gaskugeln. Astron. Nachr. 267 (1939) 253–272. ¨ [6] T. Carleman, Uber eine isoperimetrische Aufgabe und ihre physikalischen Anwendungen. Math. Zeitschr. 3 (1918) 1–7. [7] S. Chandrasekhar, An Introduction to the Study of Stellar Structure. New York: Dover Publications, Inc. 1967. [8] R. Emden, Gaskugeln. Leipzig: Teubner-Verlag 1907. [9] D. Gilbarg, N.S. Trudinger, Elliptic Partial Diﬀerential Equations of Second Order. Berlin, Heidelberg, New York: Springer-Verlag 1998. [10] B.K. Harrison, K. Thorne, M. Wakano, J.A. Wheeler, Gravitation Theory and Gravitational Collapse. Chicago: University of Chicago Press 1965 Wiley & Sons, Inc. 1983. [11] D.D Joseph, T.S. Lundgren, Quasilinear Dirichlet Problems Driven by Positive Sources. Arch. Rational Mech. Anal. 49 (1973) 241–269. [12] J. Jost, Postmodern Analysis. Berlin, Heidelberg, New York: Springer-Verlag 1998. [13] E. Kreyszig, Introductory functional analysis with applications. New York: John Wiley & Sons, Inc. 1978. ¨ [14] L. Lichtenstein, Uber eine Eigenschaft der Gleichgewichtsﬁguren rotierender Fl¨ ussigkeiten, deren Teilchen einander nach dem Newtonschen Gesetze anziehen. Math. Zeitschr. 28 (1928) 635–640. [15] E.H. Lieb, M. Loss, Analysis. Graduate Studies in Mathematics, Vol. 14. American Mathematical Society 1997. [16] T. Makino, On the existence of positive solutions at inﬁnity for ordinary diﬀerential equations of Emden type. Funkcialaj Ekvacioj. 27 (1984) 319–329. [17] G. Rein, A.D. Rendall, Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics. Math. Proc. Camb. Phil. Soc. 128 (2000) 363–380. [18] A.D. Rendall, B.G. Schmidt, Existence and properties of spherically symmetric static ﬂuid bodies with a given equation of state. Classical Quantum Grav. 8 (1991) 985–1000. [19] U.M. Schaudt, On the Dirichlet Problem for the Stationary and Axisymmetric Einstein Equations. Commun. Math. Phys. 190 (1998) 509–540.

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[20] S.L. Shapiro, S.A. Teukolsky, Black Holes, White Dwarfs, and Neutron Stars. The Physics of Compact Objects. New York: John Wiley & Sons, Inc. 1983. [21] W. Simon, On journeys to the Moon by balloon. Class. Quantum Grav. 10 (1993) 177–181. Urs M. Schaudt Institut f¨ ur Theoretische Physik Universit¨ at T¨ ubingen Auf der Morgenstelle 14 D–72076 T¨ ubingen, Germany E-mail : [email protected] Communicated by Sergiu Klainerman submitted 30/11/99, accepted 27/01/2000

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Annales Henri Poincar´ e

On Stationary Vacuum Solutions to the Einstein Equations Michael T. Anderson

0. Introduction A stationary space-time (M, g) is a 4-manifold M with a smooth Lorentzian metric g, of signature (−, +, +, +), which has a smooth 1-parameter group G ≈ R of isometries whose orbits are time-like curves in M . We assume throughout the paper that M is a chronological space-time, i.e. M admits no closed time-like curves, c.f. Section 1.1 for further discussion. Let S be the orbit space of the action G. Then S is a smooth 3-manifold and the projection π:M →S is a principle R-bundle, with ﬁber G. The chronology condition implies that S is Hausdorﬀ and paracompact, c.f. [Ha] for example. The inﬁnitesimal generator of G ≈ R is a time-like Killing vector ﬁeld X on M , so that LX g = 0. The metric g = gM restricted to the horizontal subspaces of T M , i.e. the orthogonal complement of < X > ⊂ T M then induces a Riemannian metric gS on S. Since X is non-vanishing on M , X may be viewed as a time-like coordinate vector ﬁeld, i.e. X = ∂/∂t, where t is a global time function on M . The time function t gives a global trivialization of the bundle π and so induces a diﬀeomorphism from M to R × S. The metric gM on M may be then written globally in the form gM = −u2 (dt + θ)2 + π ∗ gS ,

(0.1)

where θ is a connection 1-form for the R-bundle π and u2 = −X, X > 0.

(0.2)

The 1-form ξ dual to X is thus given by ξ = −u2 (dt + θ). The 1-form θ is uniquely determined by gM and the time function t, but of course changes by an exact 1-form if the trivialization of π is changed. We point out that (M, gM ) is geodesically complete as a Lorentzian manifold if and only if (S, gS ) is complete as a Riemannian manifold, c.f. Lemma 1.1.

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The vacuum Einstein ﬁeld equations on the space-time (M, g) are rM = 0,

(0.3)

where rM is the Ricci curvature of (M, gM ). Stationary vacuum space-times are usually considered as the possible ﬁnal, i.e. time-independent, states of evolution of a physical system, in particular isolated physical systems such as isolated stars or black holes, outside regions of matter. The most important non-trivial example is the Kerr metric, c.f. [W], modeling the time-independent gravitational ﬁeld outside a rotating star. It is easy to see from the ﬁeld equations, c.f.(1.4) below, that there are no nonﬂat stationary vacuum solutions of the ﬁeld equations (0.3) whose orbit space is a closed 3-manifold S. Hence, we will always assume that S is an open 3-manifold. Next, it is natural to consider the class of stationary vacuum space-times which are geodesically complete. In this respect, Lichnerowicz [L, §90] proved that any such solution (M, g) for which the 3-manifold (S, gS ) is complete and asymptotically ﬂat is necessarily ﬂat Minkowski space. The assumption that S is asymptotically ﬂat is very common in general relativity in that such space-times serve as natural models for isolated physical systems, e.g. stars or black holes. The reasoning here is that as one moves further and further away from an isolated gravitational source, the corresponding gravitational ﬁeld should decay as it does in Newtonian gravity, giving in the limit of inﬁnite distance the empty Minkowski space-time. However, mathematically the requirement that S is asymptotically ﬂat is a very strong assumption on both the topology and geometry of S outside large compact sets. Further, the reasoning above is not at all rigorous. It presupposes that a geodesically complete stationary solution of the vacuum equations, i.e. a stationary solution without sources, is necessarily empty, and so in particular ﬂat. Consider the fact that there are geodesically complete, non-stationary vacuum space-times consisting of gravitational waves, c.f. [MTW, §35.9] or [R, §8.8] for example. Again, physically, such space-times can be considered as idealized limiting conﬁgurations at inﬁnite distance from radiating sources. Similarly, if there does in fact exist a complete non-ﬂat stationary vacuum solution, say (M∞ , g∞ ), then there could well exist models (M, g) for isolated physical systems which are asymptotic to (M∞ , g∞ ) at space-like inﬁnity. For instance, it is not even clear apriori that the curvature of a stationary space-time, vacuum outside a compact source region, should decay anywhere at inﬁnity. The ﬁrst main result of this paper is that in fact there are no such nontrivial stationary space-times; this of course places the physical reasoning above on stronger footing. Theorem 0.1. Let (M, g) be a geodesically complete, chronological, stationary vacuum space-time. Then (M, g) is the ﬂat (i.e. empty) Minkowski space (R4 , η), or a quotient of Minkowski space by a discrete group Γ of isometries of R3 , commuting with G. In particular, M is diﬀeomorphic to S × R, dθ = 0 and u = const.

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This result, together with Lemma 1.1 below implies that if (M, g) is a nonﬂat stationary vacuum space-time, then the orbit space S must have a non-empty metric boundary. More precisely, since (S, gS ) is Riemannian, let S denote the metric, (equivalently the Cauchy), completion of S and let ∂S = S \ S. Hence Σ = ∂S = ∅,

(0.4)

if (M, g) is not ﬂat. In order to avoid trivial ambiguities, we will only consider maximal stationary quotients S. For example any domain Ω in R3 with the ﬂat metric, u a positive constant, and θ = 0 generates a stationary vacuum solution, (namely a domain in Minkowski space). In this case, the metric boundary ∂Ω is artiﬁcial, and has no intrinsic relation with the geometry of the solution. The solution obviously extends to a larger domain, i.e. all of Minkowski space. Thus, we only consider maximal solutions (S, gS , u, θ), in the sense that the data (S, gS , u, θ) does not extend to a larger domain (S , gS , u , θ ) ⊃ (S, gS , u, θ) with u > 0 on M . It follows that in any neighborhood of a point q ∈ Σ = ∂S, either the metric gS or the connection 1-form θ degenerates in some way, or u approaches 0 in some way, or both. Without any further restrictions, the behavior of the data near ∂S can be quite complicated; numerous concrete examples of this can be found among the axi-symmetric stationary, or even axi-symmetric static, i.e. Weyl, solutions; c.f. [A1] for further discussion. In particular, singularities, both of curvature type and of non-curvature type, may form at the boundary. The horizon H = {u = 0}, viewed as a subset of S, may or may not be well-deﬁned in this generality; of course it corresponds to the locus in M where the Killing vector X becomes null. Even when H is well-deﬁned and smooth, in general there may be other, possibly singular, parts to ∂S. Theorem 0.1 leads to the following apriori estimate on the norm of the curvature of a stationary vacuum solution away from the boundary of S, and on the rate of curvature blow-up on approach to the boundary. Theorem 0.2. There is a constant K < ∞ such that if (M, g) is any chronological stationary vacuum solution, (not geodesically complete), then |RM |[x] ≤ K/ρ2 [x],

(0.5)

where RM is the curvature tensor of (M, g), [x] is the Killing orbit through x ∈ M and ρ(x) = distgS ([x], ∂S). The constant K is independent of the data (M, g). Note that Theorem 0.2 implies Theorem 0.1 by letting ρ → ∞. On the other hand, Theorem 0.2 requires Theorem 0.1 for its proof. In particular, this result shows that the curvature of (M, g) decays at least quadratically w.r.t. the distance from ∂S. The contents of the paper are as follows. We discuss some background information and preliminary results in Section 1, needed for the work to follow. Theorem 0.1 is proved in Section 2 and Theorem 0.2 is proved in Section 3.

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I would like to thank Piotr Chrusciel and Jim Isenberg for useful discussions, the referee for pointing out some needed clariﬁcations and Grisha Perelman for pointing out an error in a previous version of the paper.

1. Background and Preliminary Results. 1.1. A stationary space-time (M, g) uniquely determines the orbit data (S, gS , u, Ω) described in Section 0, where Ω = dθ is the curvature 2-form of the bundle π on S. Conversely, given arbitrary orbit data (S, gS , u, Ω), u > 0, satisfying certain equations, (c.f. (1.3)–(1.6) below), there is a unique stationary space-time (M, g) in the sense of Section 0, i.e. a chronological space-time with a global isometric R-action with the given orbit data. Of course, if (M, g) is not chronological, then it will not be uniquely determined by the orbit data. One may for instance take a Z-quotient of (M, g), preserving the orbit data. More importantly, if (M, g) is not chronological, then the orbit space S may not be a manifold; even if S is a manifold, it may not be Hausdorﬀ, c.f. [Ha]. Since the arguments to follow are global on S, we require that S is globally well-behaved, which is ensured by the chronology condition. It is not known for instance if Theorem 0.1 is valid without this assumption. Recall that a space-time (M, g) is geodesically complete if all geodesics in (M, g), parametrized by an aﬃne parameter s, are deﬁned for all s ∈ R. The vertical subspace of T M is the subspace spanned by the Killing ﬁeld X and the horizontal distribution H is its orthogonal complement in T M , deﬁned by the metric gM . Lemma 1.1. A stationary space-time (M, gM ) is geodesically complete if and only if the orbit space (S, gS ) is geodesically complete. Proof. Suppose (M, gM ) is geodesically complete. Let γ be a geodesic in S. Since the projection π : M → S is a principle ﬁber bundle, with horizontal spaces H ⊂ T M , the geodesic γ may be lifted to a horizontal geodesic γ¯ in (M, gM ), with the same parametrization. Since (M, gM ) is complete, γ¯ is deﬁned for all values of the parameter, and hence so is γ. Conversely, suppose (S, gS ) is geodesically complete, and hence complete as a metric space. Let γ be a geodesic in M , with aﬃne parameter s and tangent vector T . Then the projection σ = π ◦ γ is a curve in S, whose acceleration is given by ∇V V =

1 2 −2 1 κ ∇u − κL(V ). 2 2

(1.1)

Here V = dσ/ds = π∗ T, ∇ is the covariant derivative in (S, gS ), κ =< X, T >= const and L is the linear map deﬁned by < L(A), B > X = [A, B]v where A, B are horizontal vector ﬁelds on M and v is the vertical projection, c.f. [T, Ch.18.3] for example. Conversely, any curve σ satisfying (1.1) lifts to a geodesic in (M, g).

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The equations (1.1) form a 2nd order system of ODE w.r.t. the parameter s; note that L(V ) is linear in V , while κ is a constant in s, depending linearly on V . By local existence and uniqueness, there exist locally deﬁned solutions σ for arbitrary initial data (x, V (x)) ∈ T S. Since S is complete, it follows that σ exists for all values of s ∈ R. Hence (M, g) is geodesically complete. Remark 1.2. It is easy to verify that if (M, g) is a stationary, (strongly) globally hyperbolic space-time, in the sense that (M, g) admits a geodesically complete Cauchy surface L, (w.r.t. the induced metric), then (M, g) is geodesically complete. The converse issue however, i.e. whether a chronological, stationary and geodesically complete space-time is necessarily globally hyperbolic, is not clear to the author, at least without further assumptions on u and θ. For brevity, we will often say that (M, gM ) or (S, gS ) is complete instead of geodesically complete. 1.2. Let ξ = −u2 (dt + θ) be the 1-form dual to the Killing vector X, as in Section 0. The twist potential ω is the 1-form on M deﬁned by 1 ω = ∗ (ξ ∧ dξ), (1.2) 2 It is easily veriﬁed that ω is G-invariant, and that it descends to a 1-form ω on the base space S. The form ω represents the obstruction to integrability of the horizontal distribution in T M , and so is related to the curvature 2-form Ω of the connection 1-form θ. In fact, one easily veriﬁes that 2ω = −u4 ∗ dθ = −u4 ∗ Ω, on (S, gS ). The vacuum Einstein equations (0.3) on (M, g) are G-invariant, and so also descend to equations on S. The vacuum equations are equivalent to the following equations on (S, gS ) : r = u1 D2 u + 2u−4 (ω ⊗ ω − |ω|2 · g), ∆u = −2u−3 |ω|2 , divω = 3dlogu, ω, dω = 0.

(1.3) (1.4) (1.5) (1.6)

Here r = rS is the the Ricci curvature of (S, gS ), D2 u is Hessian of u on (S, gS ), ∆u = trgS D2 u and log is the natural logarithm; we refer for instance to [Kr, Ch. 16] for a derivation of these equations, (but note that [Kr] does not use the factor 1 2 in (1.2)). The equation (1.3) comes from the pure space-like (or horizontal) part of rM , the equation (1.4) from the vertical part of rM , i.e. rM (X, X), while the equations (1.5)–(1.6) come from the mixed directions. The equation (1.6) implies that ω is locally exact, i.e. there exists φ, the twist potential, such that 2ω = dφ locally. On the universal cover S of S, (1.7) holds globally.

(1.7)

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Observe that these equations are invariant under the substitutions u → λu, ω → λ2 ω,

(1.8)

corresponding to ξ → λξ, and θ → λ−2 θ. 1.3. To prove Theorems 0.1 and 0.2, we will need to study sequences of stationary (vacuum) solutions, where all the data (S, gS , u, ω) are allowed to vary. Thus, in eﬀect, we need to understand aspects of the moduli space of stationary solutions. For this, we will frequently use the following two Lemmas, which will be proved together. Lemma 1.3. (Convergence). Let (Ωi , gi , ui , ωi ) represent data for a sequence of solutions to the stationary vacuum equations (0.1). Suppose on the domains (Ωi , gi ), |ri | ≤ Λ, diam Ωi ≤ D, vol Ωi ≥ νo ,

(1.9)

dist(xi , ∂Ωi ) ≥ δ,

(1.10)

and

for some xi ∈ Ωi and positive constants νo , Λ, D, δ. Then, for any ε = ε(δ) > 0 suﬃciently small, there are domains Ui ⊂ Ωi , with ε/2 ≤ dist(∂Ui , ∂Ωi ) ≤ ε, and xi ∈ Ui such that a subsequence of the Riemannian manifolds (Ui , gi , xi ) converges, in the C ∞ topology, modulo diﬀeomorphisms, to a limit manifold (U, g, x), with limit base point x = lim xi . Further, the potentials ui and 1-forms ωi may be renormalized by scalars λi , as in (1.8), so that they converge smoothly to limit potential u and 1-form ω. The limit (U, g, x, u, ω) represents a smooth solution to the stationary vacuum equations. Lemma 1.4. (Collapse). Let (Ωi , gi , ui , ωi ) represent data for a sequence of solutions to the stationary vacuum equations (0.1). Suppose on the domains (Ωi , gi ), |ri | ≤ Λ, diam Ωi ≤ D, vol Ωi → 0

(1.11)

dist(xi , ∂Ωi ) ≥ δ,

(1.12)

and

for some xi ∈ Ωi and constants Λ, D, δ. Then, for any ε = ε(δ) > 0 suﬃciently small, there are domains Ui ⊂ Ωi , with ε/2 ≤ dist(∂Ui , ∂Ωi ) ≤ ε with xi ∈ Ui , such that Ui is either a Seifert ﬁbered space or a torus bundle over an interval. In both cases, the gi -diameter of any ﬁber F , (necessarily a circle S 1 or torus T 2 ), goes to 0 as i → ∞, and π1 (F ) injects in π1 (Ui ). i of Ui does not collapse and hence has a Consequently, the universal cover U , g, x), with x = lim x , x a lift of subsequence converging smoothly to a limit (U i i i . In addition, the limit (U , g, x) admits a free isometric R-action. xi to U

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As above, the potentials ui and 1-forms ωi , after possible renormalization by , g, x, u, ω) is a smooth scalars, converge smoothly to limits u and ω. The limit (U solution of the stationary vacuum equations, and all data are invariant under a . free isometric R-action on U Proofs. The proofs of the ﬁrst parts of Lemmas 1.3 and 1.4 are essentially immediate consequences of the well-known Cheeger-Gromov theory on convergence and collapse of Riemannian manifolds with bounded curvature, c.f. [CG1,2], [Ka], [A3,§2] for example. We note that we are implicitly using the fact, special to dimension 3, that the full curvature is determined by the Ricci curvature. More precisely, under the bounds (1.9)–(1.10), one obtains convergence of a subsequence of {gi } to a C 1,α limit metric g on the domain U ; the convergence is in the C 1,α topology, for any α < α < 1. For a clear introduction to this theory, c.f. [P, Ch. 10]. In particular, the bounds (1.9) imply a lower bound on the injectivity radius of every point in Ui ; this is Cheeger’s lemma, c.f. [C], [P, 10.4.5] Under the bounds (1.11)–(1.12), the sequence of domains collapses with bounded curvature in the sense that the injectivity radius at every point in Ui tends to 0. This implies that the domains Ui admit an F-structure, [CG1,2]. In dimension 3, this means that Ui is topologically a graph manifold, i.e. a union of Seifert ﬁbered spaces (S 1 ﬁbrations over a surface) or torus bundles over an interval, glued together along toral boundary components of such, c.f. [Ro, §3]. A result of Fukaya, c.f. [F, Ch.11,12] and references therein, implies that on domains of bounded diameter, i.e. under (1.11)–(1.12), for i suﬃciently large, the F-structure may be chosen to be pure, so that Ui itself is either a Seifert ﬁbered space or a torus bundle over an interval. The collapse takes place by shrinking the ﬁbers, (circles or tori), to points. From the theory of Seifert ﬁbered spaces, c.f. [O] or [Ro, Thm. 4.3], the ﬁbers inject in π1 whenever Ui is not covered by S 3 . But this is necessarily the case here, since Ui is an open domain, (c.f. the remark following (0.3). Thus, one may unwrap the collapse by passing to covers, for instance the universal cover, that unwind the ﬁbers. This ability to unwrap collapse on domains of controlled diameter is special to dimension 3. It remains to show that the convergence is actually smooth (C ∞ ), and that the limit, in either case of Lemma 1.3 or 1.4, is a smooth solution to the stationary vacuum equations. This is done by showing that the equations (1.3)–(1.6) form essentially an elliptic system and using elliptic regularity. By taking the trace of (1.3) and using (1.4), one derives that s = −6u−4 · |ω|2 ,

(1.13)

where s is the scalar curvature of (S, gS ), so that (1.4) is equivalent to ∆u =

s u. 3

(1.14)

Since, by hypothesis, the Ricci curvature is uniformly bounded on (Ωi , gi ), so is the scalar curvature si . Now the potential functions ui may be unbounded, or

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converge to 0, in neighborhoods of the base points xi . Thus, we renormalize ui by setting u ¯i = ui /u(xi ),

(1.15)

so that u ¯i (xi ) = 1. The equation (1.14) is of course invariant under this renormalization. Moreover, since ui > 0 everywhere, and since the local geometry of (Ωi , gi ) is uniformly controlled in C 1,α away from ∂Ωi , i.e. within Ui , the Harnack inequality, (c.f. [GT, Thm. 8.20]), applied to the elliptic equation (1.14) implies that there is a constant κ > 0, independent of i, such that κ≤

sup¯ ui ≤ κ−1 ; inf u ¯i

(1.16)

here the sup and inf are taken over Ui , or more precisely over an ε/4 thickening of Ui . Of course the diameter bound in (1.9) or (1.11) is being used here. It then ¯i are follows from L2 elliptic theory, c.f. [GT, Thm. 9.11], that the functions u uniformly bounded in L2,p (Ui ), p < ∞. Next, as in (1.15), we renormalize the twist 1-forms ωi by ω ¯ i = ωi /(u(xi ))2 ,

(1.17)

c.f. (1.8). It then follows from (1.13), (1.15), (1.17) and the uniform L∞ bound on si that the forms ω ¯ i are uniformly bounded in L∞ on Ui . Next, to obtain higher regularity, consider the equations (1.5)–(1.6) ¯i , dφi , ∆φi = 3dlog u locally, i.e. in neighborhoods where the twist potential φ = φi is deﬁned; (we omit the overbar from the notation for φ). We may add a constant to φi and assume φi (xi ) = 0. By the bound on ω ¯ i above, |dφi | is uniformly bounded, as is ¯ i is |dlog u ¯i |, so by elliptic regularity, φi is bounded locally in L2,p , and hence ω uniformly bounded locally in L1,p everywhere in Ui . By (1.13) again, this implies si is bounded in L1,p , and so by elliptic regularity applied to (1.14), u ¯i is uniformly bounded locally in L3,p . Hence, the right side of (1.3) is bounded in L1,p , and so the Ricci curvature ri is uniformly controlled locally in L1,p everywhere in Ui . This implies that the metrics gi are uniformly controlled in L3,p in local harmonic coordinates, c.f. [A3. §3] for example. Hence, by the Sobolev embedding theorem, the sequence {gi } is uniformly bounded in C 2,α , α < 1. This process may now be iterated inductively to give uniform C k control on {gi }, for any k < ∞, away from the boundary, as well as uniform C k control on {¯ ui } and on {¯ ωi }. This proves that the convergence to the limit is in the C ∞ topology, as well as C ∞ convergence to limits u ¯ and ω ¯ . Since the metrics gi are stationary vacuum solutions, it is obvious that the limit (U, g, u ¯, ω ¯ ) is also. As an application of these results, we prove the following Lemma, which shows that a given complete stationary vacuum solution gives rise to another one with uniformly bounded curvature.

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Lemma 1.5. Let (S, g, u, ω), g = gS , represent data for a complete non-ﬂat stationary vacuum solution. Then there exists another complete non-ﬂat stationary vacuum solution given by data (S , g , u , ω ), g = gS , obtained as a geometric limit at inﬁnity of (S, g), which has uniformly bounded curvature, i.e. |rg | ≤ 1 and |rg |(y) > 0,

(1.18)

for some y ∈ S . Proof. We may assume that (S, g) itself has unbounded curvature, for otherwise there is nothing to prove since (1.18) can then be obtained by a ﬁxed rescaling of (S, g) if necessary. Let {xi } be a sequence in S such that |r|(xi ) → ∞, as i → ∞.

(1.19)

Let Bi = Bxi (1) and let di (x) = dist(xi , ∂Bi ). Consider the scale-invariant ratio (d2i · |r|)(x), for x ∈ Bi , and choose points yi ∈ Bi realizing the maximum value of (d2i · |r|)(x) on Bi . Since (d2i · |r|)(x) is 0 on ∂Bi , yi is in the interior of Bi . By (1.19), we have d2i (yi ) · |r|(yi ) → ∞, as i → ∞

(1.20)

and so in particular |r|(yi ) → ∞. Now consider the pointed rescaled sequence (Bi , gi , yi ), where gi = |r|(yi ) · g. By construction, |ri |(yi ) = 1, where ri is the Ricci curvature of gi . This, together with (1.20) and its scale-invariance, implies that δi (yi ) ≡ distgi (yi , ∂Bi ) → ∞. Further, by the maximality property of yi , |ri |(x) ≤ |ri |(yi ) ·

δi (x) δi (x) = . δi (yi ) δi (yi )

(1.21)

It follows from (1.20) that |ri |(x) ≤ 2, at all points x of uniformly bounded gi distance to yi , (for i suﬃciently large, depending on distgi (x, yi )). If the pointed sequence (Bi , gi , yi ), (or a subsequence), is not collapsing at yi , i.e. the volume of the unit gi -ball at yi is bounded below as i → ∞, then by Lemma 1.3, {(Bi , gi , yi )} has a subsequence converging, smoothly and uniformly on compact subsets, to a limit (U , g , y), y = lim yi . The limit is a complete stationary vacuum solution, (since δi (yi ) → ∞), and by the smooth convergence, |rg | ≤ 2 everywhere and |rg (y)| = 1, where y = limyi . A further bounded rescaling then gives (1.18). The limit potential u and twist form ω are obtained as in Lemma 1.3. On the other hand, suppose this sequence is collapsing at yi , so that the volume of the unit gi -ball at yi converges to 0, (in some subsequence). Then by

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Lemmas 1.3 and 1.4, it is collapsing everywhere within gi -bounded distance to yi , i.e. within (Byi (R), gi ), for any ﬁxed R < ∞. For any such R, if i is suﬃciently large, there are domains Ui (R) ⊂ Byi (R), with ∂Ui (R) near ∂Byi (R) w.r.t. gi , which are highly collapsed along an injective Seifert ﬁbered structure or torus i (R), gi ) is not collapsing. bundle structure on Ui (R). Hence the universal cover (U For any sequence Rj → ∞, there is then a suitable diagonal subsequence Uij such ij converge smoothly, as above, to a complete stationary vacuum that the covers U solution; again a bounded rescaling then gives (1.18).

2. Proof of Theorem 0.1. Let (M, gM ) be a complete stationary vacuum solution. As above in Section 1.2 and Section 1.3, we will work exclusively on the 3-manifold quotient S, with data u, ω and g satisfying the ﬁeld equations (1.3)–(1.6). By passing to the universal cover, we may and will assume for this section that S is simply connected. It is very useful to rewrite the metric gM in (0.1) in the form gM = −u2 (dt + θ)2 +

1 g¯S , u2

(2.1)

where g¯S is the conformally equivalent metric g¯S = u2 · gS

(2.2)

on S. Using standard formulas for behavior under conformal changes, c.f. [B, Ch. 1J], w.r.t this metric the ﬁeld equations (1.3)–(1.5) are equivalent to: r¯ = 2(dlogu)2 + 2u−4 (ω)2 ,

(2.3)

¯ ∆logu = −2u−4 |ω|2 ,

(2.4)

divω = 4dlogu, ω,

(2.5)

c.f. also [Kr, Ch. 16]. All metric quantities in (2.3)–(2.5) are w.r.t. the g¯ = g¯S metric. There are two reasons for preferring g¯ to g = gS . First, it is apparent from (2.3) that r¯ ≥ 0,

(2.6)

so that (S, g¯) has non-negative Ricci curvature. Second, the ﬁeld equations (2.3)– (2.5) are exactly the Euler-Lagrange equations for the functional 1 |du2 |2 + |dφ|2 Sef f = (s − ( ))dV. (2.7) 2 u4 S

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Here we are using the fact that S is simply connected, so that the relation (1.7) holds globally on S. This functional is the Einstein-Hilbert functional on Ginvariant metrics on M , dimensionally reduced to a functional on data (¯ g , u, φ) on S, when gM is expressed in the form (2.1). It corresponds to a coupling of 3-dimensional gravity to the energy (or σ-model) of the mapping E = (φ, u2 ) from S to the hyperbolic plane. The mapping E is called the Ernst potential and the Euler-Lagrange equations (2.3)–(2.5) imply that E : (S, g¯S ) → (H 2 (−1), g−1 )

(2.8)

is a harmonic map. Here H 2 (−1) is the hyperbolic plane, given as the upper halfplane (R2 )+ = {(x, y) : y > 0}, with metric g−1 =

dx2 + dy 2 . y2

(2.9)

We refer for instance to [H1] or [H2] for further details and discussion on Sef f . From the equation (2.3), we see that r¯ =

1 ∗ E (g−1 ). 2

(2.10)

In particular, the energy density of e(E) of E, given by e(E) =

1 |E∗ |2 2

satisﬁes s¯ = e(E) =

1 trg¯ E ∗ (g−1 ). 2

(2.11)

For clarity, we break the proof up at this stage into two steps. Step I. Assume the metric (S, g¯S ) is complete. The space (S, g¯S ) may or may not have uniformly bounded curvature, i.e. possibly after a bounded rescaling, |¯ r| ≤ 1,

(2.12)

everywhere on S, where the norm is taken w.r.t. g¯S . If (2.12) holds, then the arguments below are applied to (S, g¯S ). If instead the curvature of (S, g¯S ) is unbounded, (and hence (S, g¯S ) is not ﬂat), we apply Lemma 1.5 to obtain a new non-ﬂat stationary space-time (S , g¯S , u , ω ) satisfying (2.12). The arguments below are then applied to (S , g¯S ). With this understood, we drop the prime from the notation and assume that (S, g¯S ) satisﬁes (2.12).

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We now apply the well-known Bochner formula, c.f. [EL, (3.12)], to the harmonic Ernst map E, to obtain 2 ¯ ¯ ∆e(E) = |∇DE| + rM , E ∗ (g−1 ) −

3

(E ∗ R−1 )(ei , ej , ej , ei ).

(2.13)

i,j=1

Here the sign of the curvature tensor for the last term is such that R−1 (X, Y, Y, X) is the sectional curvature of g−1 for an orthonormal pair (X, Y ). We claim that the last two terms in (2.13) are given by ¯ r , E ∗ (g−1 ) = 2|¯ r|2 ,

(2.14)

s2 − |¯ r|2 ) ≥ 0. −(E ∗ Rg−1 )(ei , ej , ej , ei ) = 4(¯

(2.15)

The equation (2.14) follows immediately from (2.10). For (2.15), using the fact that g−1 is of constant sectional curvature −1, we have −(E ∗ R−1 )(ei , ej , ej , ei ) = g−1 (E∗ ei , E∗ ei ) · g−1 (E∗ ej , E∗ ej ) − g−1 (E∗ ei , E∗ ej )2 . Choosing {ei } to be an orthonormal basis in (S, g¯S ) diagonalizing the Ricci curvature r¯, and using (2.10), gives (2.15). In particular, the equations (2.13)–(2.15) show that the energy density e(E) is a subharmonic function on (S, g¯S ). Since (2.12) holds on (S, g¯S ), (2.11) implies that e(E) is uniformly bounded above on (S, g¯S ). Thus, let {xi } be a maximizing sequence for e(E), i.e. e(E)(xi ) → sup e(E) < ∞.

(2.16)

Since the curvature of (S, g¯S ) is bounded, and this space is complete, it follows from elementary properties of the Laplacian that ∆e(E)(xi ) ≤ εi , where εi → 0, as i → ∞. However, (2.13)–(2.15) then imply that |¯ r|2 (xi ) ≤ εi → 0. This of course forces e(E)(xi ) = s¯(xi ) → 0. Since xi is a maximizing sequence, this is only possible if e(E) ≡ 0, i.e. E is a constant map. This means that u = const > 0, φ = const, and hence (M, g) is ﬂat. Thus (M, g) is Minkowski space, (since S is simply connected). Observe that this argument now implies that the passage to the geometric limit (S , g¯S ) at the beginning of Step I was not in fact necessary. Step II. We now remove the assumption that g¯ is complete, by transfering the estimates above back to the complete manifold (S, gS ).

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Exactly as in the beginning of Step I however, since (S, gS ) is complete, if necessary we use Lemma 1.5 ﬁrst to pass to a non-ﬂat geometric limit (S , gS ) with uniformly bounded g -curvature, i.e. satisfying (1.18). As before, we drop the prime from the notation below. Since g¯S = u2 gS , we have the following relation between the Laplacians of gS and g¯S , c.f. [B, Ch. 1J] for example: ¯ = u−2 ∆f + u−3 du, df , ∆f for any function f , where metric quantities on the right are w.r.t. gS . Setting f = s¯ then gives ¯ s − dlogu, d¯ ∆¯ s = u2 ∆¯ s.

(2.17)

Now the function s¯ may well be an unbounded function on (S, gS ); (in fact the unboundedness may cause the incompleteness of g¯S ). However, in terms of the metric g, we have 1 s¯ = u−2 (2|dlogu|2 + u−4 |dφ|2 ) ≡ u−2 · h, (2.18) 2 where the last inequality deﬁnes h and the norms on the right are w.r.t. gS . This follows by taking the trace of (2.3). Since the curvature of gS is uniformly bounded, i.e. (1.18) holds, the same arguments as in the proof of Lemma 1.3-1.4 imply that |dlogu|2 + u−4 |dφ|2 ≤ C,

(2.19)

for some C < ∞. The estimate (2.19) can also be deduced directly from (1.13) and (1.3)–(1.7). Hence, h is uniformly bounded above on (S, gS ). Returning to (2.17), we then have ∆¯ s = ∆u−2 h = u−2 ∆h + h∆u−2 + 2du−2 , dh.

(2.20)

∆u−2 = −2u−3 ∆u + 6u−4 |du|2 = u−6 |dφ|2 + 6u−4 |du|2 ,

(2.21)

Now

where the last equality uses (1.4) and (1.7). Hence, combining (2.20)–(2.21), we obtain ∆h = u2 ∆¯ s − (u−4 |dφ|2 + 6u−2 |du|2 )h − 2u2 du−2 , dh. Substituting (2.17) gives ¯ s − (u−4 |dφ|2 + 6u−2 |du|2 )h − 2u2 du−2 , dh − u2 dlogu, d¯ ∆h = u4 ∆¯ s. (2.22) Since s¯ = u−2 · h, d¯ s = −2u−3 hdu + u−2 dh, and so (2.22) becomes ¯ s − (u−4 |dφ|2 + 6u−2 |du|2 )h + 4dlogu, dh + 2u−2 h|du|2 − dlogu, dh, ∆h = u4 ∆¯ i.e.

¯ s − (u−4 |dφ|2 + 4u−2 |du|2 )h + 3dlogu, dh. ∆h = u4 ∆¯

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By (2.18) again, the middle term on the right above equals −2h2 = −2u4 s¯2 . Hence, we have ¯ s − 2u4 s¯2 . ∆h − 3dlogu, dh = u4 ∆¯

(2.23)

On the other hand, from the Bochner formula (2.13) and (2.14)–(2.15), we have 2 ¯ s = |∇DE| ¯ + 2|¯ r|2 + 4(¯ s2 − |¯ r|2 ), ∆¯

where all quantities are w.r.t. the g¯ metric. Substituting this in (2.23) then gives 2 ¯ + 2u4 (¯ s2 − |¯ r|2 ) ≥ 0, ∆h − 3dlogu, dh = u4 |∇DE|

(2.24)

where the terms on the left are in the g metric while those on the right are in the g¯ metric. We now basically repeat the argument above in Step I to prove that h ≡ 0.

(2.25)

Thus, recalling from (2.19) that h is bounded on (S, gS ), let {xi } be a maximizing sequence for h. It follows as before that ∆h(xi ) ≤ =i , |dh|(xi ) ≤ =i while |dlogu|(xi ) remains uniformly bounded. To prove (2.25), it is most convienient to pass to the limit of the pointed sequence (S, gS , xi ) by use of Lemmas 1.3-1.4. Thus, a subsequence of {(S, gS , xi )} converges smoothly, (passing to covers if necessary in the case of collapse), to a complete stationary vacuum solution (S∞ , g∞ , x∞ ). Here the limit potentials u∞ and φ∞ are limits of the renormalized potentials ui = u/u(xi ), φi = φ/u(xi )2 . Observe that h and dlogu are invariant under such renormalizations, as is the right side of (2.24) under the changes u → ui , g¯S → g¯i = u2i · gS . It follows from these estimates and (2.24), together with the maximum principle, that the limit (S∞ , g∞ , x∞ , u∞ , φ∞ ) satisﬁes ¯ h ≡ h∞ = const, |∇DE| = 0, |¯ r|2 − s¯2 = 0,

(2.26)

where g¯∞ = u2∞ · g∞ and h∞ = supS h.

(2.27)

¯ r = 0, i.e. the Ricci curvature To see that h∞ = 0, (2.26) and (2.10) imply that ∇¯ r¯∞ of g¯∞ is parallel. By the Bianchi identity this implies that the scalar curvature s¯∞ of g¯∞ is constant. Since h = h∞ is constant, (2.18) shows that u∞ is also constant on (S∞ , g∞ ). Hence by (2.4) on (S∞ , g∞ ), it follows that dφ∞ = 0. By the deﬁnition of h in (2.18), this of course gives h∞ ≡ 0, which by (2.27) gives (2.25). The equation (2.25) means that u is a constant function and ω = 0, so that dθ = 0. It follows that (S, gS ) and (M, gM ) are both ﬂat, which proves the result.

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3. Proof of Theorem 0.2. The following result gives Theorem 0.2 essentially as an immediate corollary. The proof is a standard consequence of the global result in Theorem 0.1, together with the control on moduli of stationary vacuum solutions given in Lemmas 1.3 and 1.4. Theorem 3.1. Let (M, gM ) be a stationary vacuum solution, with orbit data (S, gS , u, θ), and U ⊂⊂ S a domain with smooth boundary, so that u > 0 on U . Then there is an (absolute) constant K < ∞, independent of (M, gM ) and U , such that for all x ∈ U, |rS |(x) ≤

K , ρ(x)2

(3.1)

where ρ(x) = distgS (x, ∂U ). Proof. The proof is by contradiction. Thus, assume that (3.1) does not hold. Then there are stationary vacuum solutions (Mi , gMi ), with orbit data (Si , gSi , ui , ωi ), smooth domains Ui ⊂⊂ Si on which ui > 0 and points xi ∈ Ui such that ρ2 (xi )|ri |(xi ) → ∞, as i → ∞.

(3.2)

Let ρi = ρ(xi ). Since it may not be possible to choose the points xi so that they maximize |ri | (over large domains), we shift the base points xi as follows; compare with the proof of Lemma 1.5. Choose ti ∈ [0, ρi ) such that t2i supBxi (ρi −ti ) |ri | = supt∈[0,ρi ) t2 · supBxi (ρi −t) |ri | → ∞, as i → ∞,

(3.3)

where the last estimate follows from (3.2), (set t = ρi ). Let yi ∈ Bxi (ρi − ti ) be points such that |ri |(yi ) = supBxi (ρi −ti ) |ri |.

(3.4)

Further, setting t = ti (1 − k1 ), k > 1, in (3.3), one obtains the estimate t2i |ri |(yi ) ≥ t2i (1 −

1 2 1 ) · supBx (ρi −ti (1− k1 )) |ri | ≥ t2i (1 − )2 · supByi (ti /k) |ri |, i k k (3.5)

so that supByi (ti /k) |ri | ≤ (1 −

1 −2 ) |ri |(yi ), k

(3.6)

Now rescale or blow-up the metric so that | ri |(yi ) = 1 by setting gi = |ri |(yi )· g, and consider the pointed sequence (Ui , gi , yi ). We have | ri |(yi ) = 1,

(3.7)

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and by (3.3) and scale invariance, distgi (yi , ∂Ui ) → ∞, as i → ∞.

(3.8)

Also, (compare with (1.21)), it follows from (3.6) that | ri |(x) ≤ C(distgi (x, yi )).

(3.9)

We also normalize u by setting u ˜i (x) =

u(x) , u(yi )

(3.10)

and note that u ˜i > 0 on Ui . We may now apply Lemmas 1.3 and 1.4, exactly as in the proof of Lemma 1.5 ˜i , ω ˜ i , yi ) converges to conclude that a subsequence of the pointed sequence (Ui , gi , u in the C ∞ topology on compact subsets, to a limit stationary vacuum solution (U∞ , g∞ , u ˜∞ , ω ˜ ∞ , y), which is complete and satisﬁes u ˜∞ > 0 everywhere. Here, one must pass to the universal cover in case of collapse, as in Lemma 1.4, and the potential u ˜i and 1-form ω ˜ i are normalized so that u ˜i (yi ) = 1 and |˜ ωi (yi )| is bounded. ˜∞ constant and d˜ ω∞ = 0. However, By Theorem 0.1, g∞ must be ﬂat, u the smooth convergence of the sequence (Ui , gi ) guarantees that the equality (3.7) passes to the limit, contradicting the fact that g∞ is ﬂat. As in the proof of Lemmas 1.3 and 1.4, it follows from (3.1) that |dlogu|(x) ≤

K , ρ(x)

(3.11)

u−2 |ω|(x) ≤

K . ρ(x)

(3.12)

and

Combining the estimates (3.1) and (3.11)–(3.12), one obtains the same bound on the full curvature tensor RM of (M, g). Note that since K is independent of the domain U , (3.1) holds for ρ the distance to the boundary Σ of S, even if Σ is singular. To see this, just apply Theorem 3.1 to a smooth exhaustion Uj of S, with ∂Uj converging to ∂S in the Hausdorﬀ metric on subsets of (S, gS ). In particular, these results together prove Theorem 0.2. We note that elliptic regularity further implies that, for any j ≥ 1, |∇j RM |(x) ≤

K(j) K(j) , |∇j logu|(x) ≤ j . ρ2+j (x) t (x)

(3.13)

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Theorem 0.2, when combined with Lemmas 1.3 and 1.4, shows that the moduli space of stationary vacuum solutions is apriori well-controlled away from the boundary Σ = ∂S. Thus, away from the boundary, sequences of such metrics either have a smoothly convergent subsequence, or they collapse, in which case the universal covers have a convergent subsequence. Theorems 0.1 and 0.2 give new proofs of similar results for static vacuum solutions in [An2, Thm. 3.2]. Similarly, in work to follow, we plan to consider generalizations of the results on the asymptotic structure of static vacuum space-times in [A1] to stationary space-times as well as consider the Riemannian analogues of these questions.

References [A1]

M. Anderson, On the structure of solutions to the static vacuum Einstein equations, (preprint, S.U.N.Y. Stony Brook, July 1998), http://www.math.sunysb.edu/∼anderson

[A2]

M. Anderson, Scalar curvature, metric degenerations and the static vacuum Einstein equations on 3-manifolds I, Geom. and Funct. Anal., vol 9:5, (1999), 855-967; http://www.math.sunysb.edu/ ∼ anderson

[A3]

M. Anderson, Extrema of curvature functionals on the space of metrics on 3-manifolds, Calc. Var. and P.D.E., vol. 5, (1997), 199-269.

[B]

A. Besse, Einstein Manifolds, Springer Verlag, New York, (1987).

[C]

J. Cheeger, Finiteness theorems for Riemannian manifolds, Amer. Jour. Math., vol. 92, (1970), 61-75.

[CG1]

J. Cheeger and M. Gromov, Collapsing Riemannian manifold while keeping their curvature bounded I, Jour. Diﬀ. Geom., vol. 23, (1986), 309-346.

[CG2]

J. Cheeger and M. Gromov, Collapsing Riemannian manifold while keeping their curvature bounded II, Jour. Diﬀ. Geom., vol. 32, (1990), 269298.

[EL]

J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc., vol. 10, (1978), 1-68.

[F]

K. Fukaya, Hausdorﬀ convergence of Riemannian manifolds and its applications, Recent. Topics in Diﬀ. and Analytic Geom. (T. Ochiai, ed.), Kinokuniya, Tokyo, (1990), 143-283.

[GT]

D. Gilbarg and N. Trudinger, Elliptic Partial Diﬀerential Equations of 2nd Order, 2nd Edition, Springer Verlag, New York, (1983).

[Ha]

S. Harris, Conformally stationary space-times, Class. Quantum Gravity, vol. 9, (1992), 1823-1827.

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[H1]

M. Heusler, Black Hole Uniqueness Theorems, Cambridge U. Press, Cambridge, (1996).

[H2]

M. Heusler, Stationary Black Holes: Uniqueness and Beyond, Living Reviews in Relativity, vol. 1, No. 6, (1998): http://www.livingreviews.org

[Ka]

A. Kasue, A convergence theorem for Riemannian manifolds and some applications, Nagoya Math. J., vol. 114, (1989), 21-51.

[Kr]

D. Kramers, H.Stephani, M. MacCallum, E. Herlt, Exact Solutions of Einstein’s Field Equations, Cambridge U. Press, Cambridge, (1980).

[L]

A. Lichnerowicz, Theories Relativistes de la Gravitation et de L’Electromagnetisme, Masson and Cie., Paris, (1955).

[MTW] C.W.Misner, K.S. Thorne and J.A. Wheeler, Gravitation, W.H. Freeman, New York, (1973). [O]

P. Orlik, Seifert Manifolds, Lecture Notes in Math., vol. 291, Springer Verlag, New York, (1972).

[P]

P. Petersen, Riemannian Geometry, Grad. Texts in Math., vol. 171, Springer Verlag, New York (1998).

[Ri]

W. Rindler, Essential Relativity, 2nd Edition, Springer Verlag, New York, (1977).

[Ro]

X. Rong, The limiting eta invariants of collapsed three-manifolds, Jour. Diﬀ. Geom., vol. 37, (1993), 535-568.

[T]

M. Taylor, Partial Diﬀerential Equations III, Applied Math. Sciences, vol. 117, Springer Verlag, New York, 1996.

[W]

R. Wald, General Relativity, Univ. of Chicago Press, Chicago (1984).

Michael Anderson Department of Mathematics S.U.N.Y. at Stony Brook Stony Brook, N.Y. 11794-3651 Partially supported by NSF Grant DMS 9802722 E-mail : [email protected] Communicated by Sergiu Klainerman submitted 21/10/99, accepted 31/05/00

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Annales Henri Poincar´ e

On the Structure of Solutions to the Static Vacuum Einstein Equations Michael T. Anderson

0 Introduction The static vacuum Einstein equations are the equations ur = D2 u,

(0.1)

∆u = 0, on a Riemannian 3-manifold (M, g), with u a positive function on M . Here r denotes the Ricci curvature, D2 the Hessian, and ∆ = trD2 the Laplacian on (M, g). Solutions of these equations deﬁne a Ricci-ﬂat 4-manifold N , of the form N = M ×u S 1 or N = M ×u R, with Riemannian or Lorentzian metric of the form gN = gM ± u2 dt2 .

(0.2)

These equations are the simplest equations for Ricci-ﬂat 4-manifolds. They have been extensively studied in the physics literature on classical relativity, where the solutions represent space-times outside regions of matter which are translation and reﬂection invariant in the time direction t. However, with the exception of some notable instances, (c.f. Theorem 0.1 below), many of the global properties of solutions have not been rigorously examined, either from mathematical or physical points of view, c.f. [Br] for example. This paper is also motivated by the fact that solutions of the static vacuum equations arise in the study of degenerations of Yamabe metrics (or metrics of constant scalar curvature) on 3-manifolds, c.f. [A1]. Because of this and other related applications of these equations to the geometry of 3-manifolds, we are interested in general mathematical aspects of the equations and their solutions which might not be physically relevant; for example, we allow solutions with negative mass. In this paper, we will be mostly concerned with the geometry of the 3manifold solutions (M, g, u) of (0.1), (i.e. the space-like hypersurfaces), and not with the 4-manifold metric. Thus, the choice of Riemannian or Lorentzian geometry on N in (0.2) will play no role. This considerably simpliﬁes the discussion of singularities and boundary structure, but still allows for a large variety of behaviors; c.f. [ES] for a survey on singularities of space-times. Obviously, there are no non-ﬂat solutions to (0.1) on closed manifolds, and ¯ so it will be assumed that M is an open, connected oriented 3-manifold. Let M

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be the metric (or Cauchy) completion of M and ∂M the metric boundary, so that ¯ = M ∪ ∂M is complete as a metric space. M In order to avoid trivial ambiguities, we will only consider maximal solutions of the equations (0.1). For example any domain Ω in R3 with the ﬂat metric, and u a positive constant, satisﬁes (0.1). In this case, the metric boundary ∂Ω is artiﬁcial, and has no intrinsic relation with the geometry of the solution. The solution obviously extends to a larger domain, i.e. R3 . Thus, we only consider maximal solutions (M, g, u), in the sense that (M, g, u) does not extend to a larger domain (M , g ) ⊃ (M, g) with u > 0 on M . It follows that at the metric boundary ∂M of M , either the metric or u degenerates in some way or u approaches 0 in some way, (or a combination of such). A classical result of Lichnerowicz [L1, p.137] implies that if the metric (M, g) satisfying (0.1) is complete, (i.e. ∂M = ∅), and u → 1 at inﬁnity, then u ≡ 1 and M is ﬂat, i.e. R3 or a quotient of R3 . More generally, it is proved in [A1, Thm.3.2] that if (M, g) is a complete solution to (0.1), (hence u > 0 everywhere), then (M, g) is ﬂat and u is constant, i.e. the assumption on the asymptotic behavior of u is not necessary, c.f. also Theorem 1.1 below. Thus, there are no complete non-trivial solutions to (0.1) and hence ∂M must be non-empty. The set formally given by ¯, Σ = {u = 0} ⊂ M is called the horizon. It is closely related to the notion of event horizon in general relativity. More precisely, Σ may be deﬁned as the set of limit points of Cauchy sequences on (M, g) on which u converges to 0. Although ∂M = ∅, it is possible that Σ = ∅. However, most solutions of physical interest do have Σ = ∅. In the framework of classical relativity, the non-triviality of a static vacuum solution, i.e. the non-vanishing of its curvature, is due to the presence of matter or ﬁeld sources at ∂M, or ’inside‘ the horizon Σ in case (M, g) extends as a vacuum solution past Σ. It is natural to consider the situation where (M, g) is not complete and for which the metric boundary ∂M of M coincides with the horizon Σ. More precisely, we will say that (M, g) is complete away from the horizon Σ if for any sequence ¯ = M ∪ ∂M one has u(pi ) → 0. pi → p ∈ ∂M in the metric topology on M Conversely if {pi } is a bounded sequence in M with u(pi ) → 0, then the deﬁnition of (M, g, u) implies that a subsequence of {pi } converges to a point p ∈ ∂M. Thus, ∂M = Σ is given by the Hausdorﬀ limit of the ε -levels Lε of u as ε → 0. While most solutions (M, g) of physical interest are complete away from Σ, there are many solutions for which this is not the case, c.f. §2 for further discussion. In such examples, the curvature typically blows up within a ﬁnite distance to Σ. Among the solutions which are complete away from Σ, most all are singular at Σ, again in the sense that the curvature of the metric g blows up on approach to Σ. This is closely related to the fact that the equations (0.1) are formally degenerate at

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Σ. Thus, in general, even when M is complete away from Σ, the metric completion M ∪ Σ need not be a smooth manifold with boundary. We will say that (M, g, u) extends smoothly to Σ, if (i): the set Σ is a smooth ¯ is a smooth manifold with surface and the partial completion M = M ∪ Σ ⊂ M boundary, (ii): the metric g extends smoothly to a smooth Riemannian metric on M , and (iii): the potential u extends smoothly to Σ with Σ = {u = 0}; smoothness here means at least C 2 . Note that it might be possible that Σ has inﬁnitely many components, or non-compact components of inﬁnite topological type. In any case, one immediate strong consequence of (0.1) is that if g extends smoothly to Σ, then at Σ one has D2 u|Σ = 0. (0.3) ¯ and |∇u| is a non-zero constant on each Thus, Σ is a totally geodesic surface in M component of Σ, c.f. Remark 1.5 below. Observe that if (M, g) is smooth up to Σ, and complete away from Σ, then the isometric double of M across Σ is a smooth complete Riemannian manifold. The harmonic function u extends smoothly across Σ as harmonic function, odd w.r.t. reﬂection in Σ. By far the most signiﬁcant solution of the static vacuum equations is the Schwarzschild metric, of mass m, given on the space-like hypersurface M by gS = (1 −

2m −1 2 2m 1/2 ) dr + r2 ds2S 2 , u = (1 − ) , r > 2m. r r

(0.4)

This metric models the vacuum exterior region of an isolated static star or black hole. It is a spherically symmetric metric on M = (2m, ∞) × S 2 and has Σ given by a (totally geodesic) symmetric S 2 , of radius 2m. The mass m is usually assumed to be positive, but we will not make this assumption here. Thus, we allow m ≤ 0. Of course if m = 0, then gS is just the ﬂat metric with u = 1. If m < 0, (0.4) is understood to be deﬁned for r > 0. The Schwarzschild metric is asymptotically ﬂat in the sense that there is a compact set K in M such that M \ K is diﬀeomorphic to R3 \ B(R), and in a suitable chart on M \ K, the metric approaches the Euclidean metric at a rate of 1/r, i.e. 2m gij = (1 + )δij + O(1/r2 ), (0.5) r with curvature decay of order 1/r3 , r = |x|, and with m ∈ R. The function u (up to −2 a multiplicative constant) has the asymptotic form u = 1 − m ) with |∇u| = r +O(r 2 O(1/r ). A triple (M, g, u) satisfying these conditions is called asymptotically ﬂat. We note the following remarkable characterization of the Schwarzschild metric. Theorem 0.1. (Black-hole uniqueness),[I1],[Ro],[BM] Let (M, g, u) be a solution of the static vacuum Einstein equations, which is smooth up to Σ and complete away from Σ. If Σ is a compact, (possibly disconnected) surface and (M, g, u) is asymptotically ﬂat, then (M, g, u) is the Schwarzschild metric, with m > 0.

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The hypothesis that the space-like hypersurface (M, g, u) is asymptotically ﬂat is very common in physics. Namely, in modeling a static space-time outside an isolated, i.e. compact, ﬁeld or matter source, it is natural to assume that in regions far away from the source the geometry of space approximates that of R3 , i.e. empty space. Nevertheless, mathematically the asymptotically ﬂat assumption is quite strong in that it severely restricts both the topology and geometry of (M, g) outside a large compact set. Further, the physical reasoning above presupposes that there are no complete non-ﬂat solutions to the vacuum equations (0.1), i.e. that a static gravitational ﬁeld is non-empty solely due the presence of matter somewhere. Remark 0.2. This latter issue in fact led Einstein to hypothesize that space M is compact, in order to avoid dealing with ’artiﬁcial‘ boundary value problems at inﬁnity, c.f. [E, p.98ﬀ.]. This issue, closely related to Mach’s Principle, is discussed in some detail in work of Lichnerowicz, (Propositions A and B of [L1, §31] and [L2, Ch.II]); c.f. also [MTW, §21.12,],[Ri, §9.12] for instance. As remarked above, there are in fact no complete non-trivial static vacuum solutions, so that the asymptotically ﬂat assumption in Theorem 0.1 may not be unreasonable, c.f. however [El], [G]. In fact, the main result of this paper is that this hypothesis is not necessary in most circumstances; it follows from much weaker assumptions. To explain this, we ﬁrst need to consider a weakening of the condition that ∂M is compact. Let t(x) = dist(x, ∂M ).

(0.6)

Deﬁne ∂M to be pseudo-compact if there is a tubular neighborhood U of ∂M whose boundary ∂U in M , ∂U ∩ M , is compact, i.e. {t(x) = so } is compact, for some so > 0, (and hence all 0 < so < ∞). As will be seen in §2, there are numerous examples of static vacuum solutions with ∂M pseudo-compact but not compact. ¯ \ U. The mass of E Let E be an end, i.e. an unbounded component, of M may be deﬁned by 1 mE = lims→∞ mE (s) = lims→∞ 4π

< ∇ log u, ∇t > dA,

(0.7)

SE (s)

where SE (s) = t−1 (s) ∩ E. Since u is harmonic, log u is superharmonic, so that the divergence theorem implies that mE (s) is monotone decreasing in s. Hence the limit (0.7) is well-deﬁned, (possibly −∞). Note that the static vacuum equations are invariant under multiplication of u by positive constants. We use the log u term in (0.7) in place of u so that the mass is independent of this rescaling in u. The following is the main result of this paper.

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Theorem 0.3. Let (M, g, u) be a solution of the static vacuum Einstein equations. (i). Suppose ∂M is pseudo-compact. ¯ \ U has only ﬁnitely many ends {Ei }, 0 ≤ i ≤ q < ∞. Supposing q > Then M 0 and (i) holds, let E ∈ {Ei } be any end of M satisfying (ii). u(xi ) ≥ uo , for some constant uo > 0 and some sequence xi ∈ E with t(xi ) → ∞. Then the end (E, g, u) is either asymptotically ﬂat, or small, in the sense that the area growth of geodesic spheres satisﬁes ∞ 1 ds = ∞. (0.8) areaSE (s) Further, if mE = 0 and supE u < ∞, then the end E is asymptotically ﬂat. We make several remarks on this result. First, if ∂M is not pseudo-compact, ¯ \ U has inﬁnitely then it is easy to construct static vacuum solutions for which M many ends, c.f. the end of §1 or §2(I). Further, there are examples of ends E with compact boundary, on which (ii) does not hold and which are neither asymptotically ﬂat nor small in the sense of (0.8), c.f. Example 2.11. Thus, both hypotheses ¯ compact, (i) and (ii) are necessary in Theorem 0.3. There are also examples with M ¯ c.f. §2(I), so that M \ U may have no ends, (q = 0). On the other hand, both alternatives in Theorem 0.3, namely asymptotically ﬂat or small ends, do occur. Asymptotically ﬂat ends satisfy areaSE (s) ∼ 4πs2 as s → ∞, while small ends have small area growth. For example, (0.8) implies, at least, that there is a sequence si → ∞ such that areaSE (si ) ≤ si · (log si )1+ε , for any ﬁxed ε > 0. All known examples of solutions satisfying (0.8) are topologically of the form (R2 \ B) × S 1 outside a compact set and the geodesic spheres have at most linear area growth. The main example of a static vacuum solution with a small end is the family of Kasner metrics, c.f. Example 2.11 below. To illustrate the sharpness of the last statement in Theorem 0.3, we construct in Remark 3.8 a (dipole-type) static vacuum solution (M, g, u) with a single end E on which mE = 0, supE < ∞, and which satisﬁes (0.8). Similarly, Example 2.11 provides static vacuum solutions with mE > 0, supE u = ∞ and satisfying (0.8). Thus, the last result is also sharp. Assuming ∂M is pseudo-compact, it is easy to see that if mE > 0 then (ii) holds, so that (ii) may be replaced by the assumption mE > 0. Thus, for the physically very reasonable class of solutions such that ∂M is pseudo-compact, mE > 0 for all ends E, and u is bounded, all ends E of M are asymptotically ﬂat. We also point out that if M is complete away from Σ, then (ii) holds at least on some end E. The proof of Theorem 0.3 gives some further information on the asymptotic structure of the small ends. For instance, the curvature in such ends decays at least quadratically, c.f. (1.3), and in the geodesic annuli AE ( 12 si , 2si ) in E, the metric

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approaches in a natural sense that of a Weyl solution, i.e. a static axisymmetric solution, as si → ∞. Thus, asymptotically, small ends have at least one non-trivial Killing ﬁeld. However, it is not known for example if the metric is asymptotic to a unique Weyl solution, or even if small ends are necessarily of ﬁnite topological type. It would also be of interest to prove that (M, g, u) has a unique end if ∂M is pseudo-compact. However, we have not been able to do this without further assumptions, c.f. Remark 3.10. The deﬁnition of asymptotically ﬂat rules out the possibility that M is smooth up to Σ and complete away from Σ, with Σ non-compact, since for example u = 0 on Σ while u → 1 on any asymptotically ﬂat end. Consider however the following (B1) solution, gB1 = (1 −

2m −1 2 2m ) dr + (1 − )dφ2 + r2 dθ2 , r r

(0.9)

where φ/4m ∈ [0, 2π], θ ∈ [0, π], r ≥ 2m > 0, with u = r · sin(θ). Note that the potential u is unbounded. The B1 metric is just a 3-dimensional slice of the 4-dimensional Schwarzschild metric (N, gS + u2 dθ2 ), (t changed to θ in (0.2)), obtained by dividing N by an S 1 ⊂ Isom(S 2 ) orthogonal to dθ; this is the slice ’orthogonal’ to the usual slice giving the Schwarzschild metric (0.4). This metric has Σ given by two disjoint, isometric copies of R2 , each of positive Gauss curvature and asymptotic to a ﬂat cylinder. It is smooth up to Σ and complete away from Σ. The metric is globally asymptotic to the ﬂat metric on R2 × S 1 , again with curvature decay of order O(1/r3 ) and with u of linear growth in distance to Σ. Such solutions will be called asymptotically cylindrical. In fact a large class of Weyl solutions have ’dual‘ solutions in this sense which are asymptotically cylindrical, c.f. Remark 2.9. This paper is organized as follows. Following discussion of some general topics on static space-times in §1, we analyze in some detail the class of Weyl vacuum solutions in §2. Several new results on the structure of these solutions are given; for instance Proposition 2.2 gives a new characterization of Weyl metrics. In addition, some eﬀorts have been made to give a reasonably clear and organized account of the breadth of possibilities and behavior of Weyl metrics, since their treatment in the literature is rather sketchy and since they serve as a large class of models on which to test Theorem 0.3. Theorem 0.3 is proved in §3, and the paper concludes with several remarks on generalizations, and some open questions. I would like to thank the referee for suggesting a number of improvements in the exposition of the paper.

1 Background Discussion Let (M, g, u) be an open, connected oriented Riemannian 3-manifold and N = M ×u R or N = M ×u S 1 , as in (0.2). Thus N represents a static space-time and

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M ⊂ N is totally geodesic. The Einstein ﬁeld equations on N are rN −

sN · gN = T, 2

(1.1)

where T is the (R or S 1 -invariant) stress-energy tensor. (We are ignoring physical constants here). These equations may be expressed on the space-like hypersurface M as the system 1 r − u−1 D2 u + (u−1 ∆u − s) · g = TH , 2

(1.2)

1 s = TV , 2 where TH is the horizontal or space-like part of T and TV is the vertical or time-like part of T . These are the equations on M for a Lorentzian space-time N ; in case N is Riemannian, the ﬁrst equation is the same while the second is - 12 s = TV . When T = 0, one obtains the vacuum equations (0.1), which are the same for Lorentzian or Riemannian signature. A common example, with T = 0 is a static perfect ﬂuid, c.f. [Wd, Ch.4], with T given by T = (µ + ρ)dt2 + ρg, where µ, ρ are time independent scalar ﬁelds representing the energy density and pressure respectively. The equations (1.2) imply that the full Riemann curvature RN of N is determined by r, u and T . The horizon Σ = {u = 0} corresponds formally to the ﬁxed point set of the S 1 action on N and requires special consideration. For example, even if M is smooth up to Σ the Riemannian 4-manifold (N, gN ) might not be smooth across Σ, even though the curvature RN is smooth. Namely, assuming the S 1 parameter t ∈ [0, 2π), if |∇u|Σ = 1, then N has cone singularities (with constant angle by (0.3)) along and normal to the totally geodesic submanifold Σ ⊂ N. (This issue does not arise for Lorentzian metrics). By multiplying the potential function u by a suitable constant, one can make the metric gN smooth across any given component of Σ; one cannot expect however in general that this can be done simultaneously for all components of Σ, if there are more than one. This issue will reappear in §2. The following result is proved in [A1, Cor.A.3]. It implies, (by letting t → ∞), that if (M, g, u) is a complete solution to the static vacuum equations with u > 0 everywhere, then M is ﬂat, and u is constant. Theorem 1.1. Let (M, g, u) be a solution to static vacuum equations (0.1). Let t(x) = dist(x, ∂M ) as in (0.6). Then there is a constant K < ∞, independent of (M, g, u), such that |r|(x) ≤

K K , |∇ log u|(x) ≤ . t(x)2 t(x)

(1.3)

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Remark 1.2. The same result has recently been proved for stationary vacuum solutions, i.e. space-times admitting a complete time-like Killing ﬁeld, c.f. [A2]. We will discuss elsewhere to what extent Theorem 0.3 generalizes to stationary vacuum solutions. Recall from §0 that the potential function u of a static vacuum solution may be freely renormalized by arbitrary positive constants; hence the appearance of log u in (1.3), as in (0.7). Theorem 1.1 implies that the curvature of (M, g) is controlled away from ∂M, and hence the local geometry of solutions is controlled away from ∂M by lower bounds on the local volume or injectivity radius. More precisely, we have the following results which are essentially a standard application of the CheegerGromov theory of convergence/collapse of Riemannian manifolds, c.f. [CG], [A3, §2], or also [P, Ch.12] for an introduction to these results. Further details of the proofs of these results are given in [A2], (for the more general class of stationary space-times), and also in [A1,App.]. Lemma 1.3. (Non-Collapse). Let (Mi , gi , ui ) be a sequence of solutions to the static vacuum equations (0.1). Suppose |ri | ≤ Λ, diamMi ≤ D, volMi ≥ νo , and dist(xi , ∂Mi ) ≥ δ, for some xi ∈ Mi and positive constants νo , Λ, D, δ. Assume also that ui is normalized so that ui (xi ) = 1. Then, for any / > 0 suﬃciently small, there are domains Ui ⊂ Mi , with //2 ≤ dist(∂Ui , ∂Mi ) ≤ /, and xi ∈ Ui such that a subsequence of the Riemannian manifolds (Ui , gi ) converges, in the C ∞ topology, modulo diﬀeomorphisms, to a limit manifold (U, g), with limit function u and base point x = lim xi . The triple (U, g, u) is a smooth solution of the static equations with u(x) = 1. Lemma 1.4. (Collapse). Let (Mi , gi , ui ) be a sequence of solutions to the static vacuum equations (0.1). Suppose |ri | ≤ Λ, diamMi ≤ D, volMi → 0 and dist(xi , ∂Mi ) ≥ δ, for some xi ∈ Mi and constants Λ, D, δ. Assume also that ui is normalized so that ui (xi ) = 1. Then, for any / > 0 suﬃciently small, there are domains Ui ⊂ Mi , with //2 ≤ dist(∂Ui , ∂Mi ) ≤ / with xi ∈ Ui , such that Ui is either a Seifert ﬁbered space or a torus bundle over an interval. In both cases, the gi -diameter of any

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ﬁber F, (necessarily a circle S 1 or torus T 2 ), goes to 0 as i → ∞, and π1 (F ) injects in π1 (Ui ). i of Ui , such that Consequently, there are inﬁnite Z or Z ⊕ Z covers U i , gi , xi } does not collapse and hence has a subsequence converging smoothly {U , g, x) of the static vacuum equations with x = lim x , x a lift of xi to to a limit (U i i , g, x) admits a free isometric R or R ⊕ R action, (c.f. §2), which Ui . The limit (U also leaves the potential function u invariant, and u(x) = 1. In studying static solutions, it is often very useful to consider the conformally equivalent metric g˜ = u2 · g on M . An easy calculation using the behavior of Ricci curvature under conformal deformations, c.f. [Bes, p.59], shows that the Ricci curvature r˜ of g˜, in the vacuum case (0.1), is given by r˜ = 2(d log u)2 ≥ 0.

(1.4)

˜ denotes the Laplacian of g˜, then Further, if ∆ ˜ log u = 0. ∆

(1.5)

The equations (1.4)-(1.5) are equivalent to the static vacuum equations (0.1). Since these equations are invariant under the substitution u → −u, it follows that if (M, g, u) is a static vacuum solution, then so is (M, g , u−1 ), with g given by g = u4 · g. Similarly, observe that if (N, gN ) is the associated Ricci-ﬂat 4-manifold (0.2), then ∆N log u = 0.

(1.6)

Here and below, log always denotes the natural logarithm. We discuss brieﬂy some of the simplest static vacuum solutions: Levi-Civita Solutions. There are 7 classes of so-called degenerate static vacuum solutions, where the eigenvalues λi of the Ricci curvature r satisfy λ1 = λ2 = −2λ3 , called A1-A3, B1-B3, C, c.f. [EK, §2-3.6]. The B metrics are dual to the A metrics, as mentioned in §0, c.f. §2 for details. The A1 metric is the Schwarzschild metric. It is of interest to examine the A2 metric, given in standard cylindrical coordinates on R3 by gA2 = z 2 (dr2 + (sinh2 r)dφ2 ) + (

2m − 1)−1 dz 2 , z

(1.7)

1/2 with u = ( 2m and z ∈ [0, 2m], m > 0. The horizon Σ = {u = 0} is given by z − 1) the set {z = 2m} and hence is the complete hyperbolic metric H 2 , with curvature −(2m)−2 . It is easily veriﬁed that the A2 metric is smooth up to Σ. However, Σ = ∂M ; the set {z = 0} is at ﬁnite distance to Σ, and so ∂M has another

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(singular) component obtained by crushing (compact subsets of) the hyperbolic metric to a point. Let Γ be any discrete group of hyperbolic isometries. Then Γ extends in an obvious way to a group of isometries of gA2 . The uniformization theorem for surfaces implies that any orientable surface except S 2 and T 2 , including surfaces of inﬁnite topological type and inﬁnitely many ends, admits a complete hyperbolic metric, i.e. is the quotient H 2 /Γ, for some Γ. Hence, any such surface and hyperbolic metric can be realized as the horizon Σ of a static vacuum solution. Topologically, for Σ = H 2 /Γ, we have M = Σ × I. Hence for example if Σ has inﬁnitely many ends, then M also has inﬁnitely many ends; in particular, this shows that the hypothesis that ∂M is pseudo-compact in Theorem 0.3 is necessary. The A3 metric is gA3 = z 2 (dr2 + r2 dφ2 ) + zdz 2 ,

(1.8)

with u = z −1/2 > 0, r > 0. Hence Σ is empty in this case - it occurs at inﬁnity in the metric. This metric may be realized as a pointed limit of the A2 metric as m → ∞ and also as a limit, in a certain sense, of the A1 metric, c.f. Example 2.11. Remark 1.5. The discussion above raises the natural question if any orientable connected Riemannian surface (Σ, g) can be realized as the horizon of a static vacuum solution, smooth up to Σ, which is deﬁned at least in a neighborhood of Σ. In general, this appears to be unknown. Observe that any complete constant curvature metric on an orientable surface can be realized in this way. The Schwarzschild metric gives the constant curvature metric on S 2 , the quotients of the A2 metric give all hyperbolic surfaces, and quotients of the ﬂat metric, with u a linear function, give all ﬂat metrics on a surface, (T 2 , S 1 × R, or R2 ). Geroch-Hartle show in [GH] that any rotation-invariant metric on S 2 or T 2 can be realized at the horizon. Except for the Schwarzschild metric, such solutions are not complete away from Σ. Observe that the full 1-jet of (M, g) at Σ (assumed connected) is determined solely by the surface metric (Σ, g), since Σ is totally geodesic and, renormalizing u if necessary, |∇u| ≡ 1 on Σ by (0.3). Observe also that one cannot have |∇u| ≡ 0 on Σ, since u is harmonic and the divergence theorem applied to a small neighborhood U of Σ would imply that u ≡ 0 on U , which is ruled out. On the other hand, the metric (M, g) is not uniquely determined by its boundary values (Σ, g). Namely, the ﬂat metric on T 2 is realized by the ﬂat vacuum solution M = T 2 × R+ , u = t = dist(Σ, ·) and also (locally) by a non-ﬂat metric, c.f. [T],[P]. Similar remarks hold for local perturbations of the Schwarzschild metric, c.f. [GH].

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2 Weyl Solutions A large and very interesting class of explicit solutions of the static vacuum equations are given by the Weyl solutions [W], c.f. also [EK,§2.3-9] or [Kr, Ch.1618]. In fact, it appears that essentially all known explicit solutions of the static vacuum equations are of this form. Since the literature on these solutions is not very organized or rigorous, especially regarding their global structure, we discuss these solutions in some detail. These metrics will also illustrate the necessity of the hypotheses in Theorem 0.3. Deﬁnition 2.1. A Weyl solution is a solution (M, g, u) of the static vacuum equations (0.1) which admits an isometric R-action, i.e. a non-zero homomorphism R → Isom(M ), leaving u invariant. A priori, the topology of a Weyl solution could be quite non-trivial; for example M could be any Seifert ﬁbered space. The ﬁrst result shows that only the simplest topology (and geometry) is possible. For the moment, we exclude any possible ﬁxed point set of the R-action from the discussion. Proposition 2.2. Let (M,g,u) be a Weyl solution with R-action without ﬁxed points, which does not admit a (local) free isometric R×R action. Then the universal cover , g) of (M, g) is a warped product of the form (M = V ×f R, g = gV + f 2 dφ2 , M

(2.1)

with (V, gV ) a Riemannian surface and f a positive function on V. The R-action is by translation on the second factor. on M Proof. This result, whose proof is purely local, is a strengthening in this situation of a well-known result in general relativity, Papapetrou’s theorem, c.f. [Wd,Thm.7.1.1], which requires certain global assumptions, (e.g. smoothness up to Σ). Let K denote the (complete) Killing ﬁeld generated by the R-action on M , and f = |K|. We may assume that u is not a constant function on M , since if u is constant, the metric is ﬂat, and so admits a local R × R action. Since (M, g, u) is real-analytic, u is not constant on any open set in M . We thus choose a neighborhood U of any point p where ∇u(p) = 0 on which |∇u| > 0. Deﬁne e1 by e1 = ∇u/|∇u|, and extend it to a local orthonormal frame e1 , e2 , e3 for which e3 = K/|K| = K/f . Note that this is possible since u is required to be invariant under the ﬂow of K, so that < ∇u, K >= 0. (2.2) In U , the metric g may be written as g = π ∗ gV + f 2 (dφ + θ)2 ,

(2.3)

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where π : U → V is a Riemannian submersion onto a local surface (V, gV ), θ is a connection 1-form, K = ∂/∂φ and f is a function on the orbit space V . If θ = 0, then the result follows. Thus, we assume |θ| > 0 in U and show this implies that g has a free isometric local R × R action. Consider the 1-parameter family of metrics gs = π ∗ gV + s2 f 2 (dφ + s−2 θ)2 ,

(2.4)

for s > 0, with g1 = g. Geometrically, this corresponds to rescaling the length of the ﬁbers of π and changing the horizontal distribution of π, (when θ = 0). Now it is a standard fact that the 1-parameter family of 4-metrics gs4 = gs ± u2 dt2

(2.5)

remains Ricci-ﬂat for all s. This can be seen from standard formulas for Riemannian line bundles, c.f. [Bes, 9.36, 9G], [Kr, 16.1-3] or [A2, §1.2]. Thus, the metrics gs all satisfy the static vacuum equations urs = Ds2 u from (0.1), with the same potential u. Equivalently, the conformal metrics g˜s = u2 gs satisfy r˜s = 2(d log u)2 , (2.6) for all s, c.f. (1.4) The right side of (2.6) is of course independent of s. We claim that the metrics gs on U are all locally isometric. While this could be proved by a lengthy direct computation, we argue more conceptually as follows. Let esi be a local orthonormal frame for gs , determined as above for g. We then have es1 = e1 , es3 = s−1 e3 while es2 varies in the plane < e2 , e3 > . The same relations hold w.r.t. g˜s . Recall also that the full curvature tensor is determined by the Ricci curvature in dimension 3. It then follows from these remarks and (2.6) that for each q ∈ U and s > 0, there is a sectional curvature preserving isomorphism Fs : Tq M → Tq M, i.e. ˜ s (Fs (P )) = K ˜ 1 (P ), K ˜ s is the sectional curvature w.r.t. g˜s . Clearly Fs where P is any 2-plane and K varies smoothly with q and s. Using the expression (2.6), a result of Kulkarni [Ku] then implies that the metrics g˜s are locally isometric and hence so are the metrics gs . Let Ω = dθ be the curvature form of the line bundle π. Then w.r.t. the metric g, |Ω| = |Ω(e1 , e2 )| = | < ∇e1 e2 , e3 > |. The same equalities hold w.r.t. Ωs = dθs = s−2 dθ and the gs metric. A short computation then gives |Ωs |gs → 0, as s → ∞.

(2.7)

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Hence consider the behavior of the metrics gs as s → ∞. We are then expanding or blowing up the metric in the ﬁber direction, at a given base point. Since the metrics gs are isometric, there are (local) diﬀeomorphisms ψs such that ψs∗ gs converges to a limit metric g∞ . At a given base point, the diﬀeomorphisms ψs expand or blow up smaller and smaller intervals of the parameter φ to unit size, giving rise to a limit parameter φ∞ . When θ = 0, this is the only change; the limit metric g∞ is the same as g with the parameter φ replaced by φ∞ . (This is completely analogous to passing from the ﬂat metric on R2 \ {0} to the ﬂat metric on its universal cover, i.e. unwrapping the circles to lines). However, when θ = 0, the e2 direction is also being expanded or blown-up in a similar way. The function u is left invariant under the family {ψs }. It follows from (2.7) that the limit metric g∞ is a static vacuum solution of the form 2 g∞ = π ∗ gV∞ + f∞ (dφ∞ )2 , (2.8) i.e. the 1-form θ∞ = 0 in the limit. Further, since the e2 direction has been blown up, the function f∞ varies only in the e1 direction, i.e. f∞ = f∞ (u). Metrics of the form (2.8) are analyzed in detail below. Referring to (2.10), let r = f∞ ·u = h(u). In a possibly smaller open subset of U , we may invert h and write u = u(r), where r is a local coordinate on V∞ . It is easy to see, (c.f. (2.12)-(2.13) below for example), that g∞ admits a non-vanishing Killing ﬁeld ∂/∂z, tangent to V∞ but orthogonal to ∂/∂r, and hence g∞ admits a free isometric local R × R action. The metrics gs are all locally isometric and so the metric g = g1 also has a free isometric local R × R action. Since the proof above is completely local, Proposition 2.2 holds locally, (in suitably modiﬁed form), even if (M, g) admits only a local or partial R -action. Static vacuum solutions admitting a free isometric local R × R action are completely classiﬁed; they are either ﬂat or belong to the family of Kasner metrics, c.f. Example 2.11 below or [EK, Thm.2-3.12]. Such solutions do have Killing ﬁelds K which are not hypersurface orthogonal, i.e. dθ = 0 in (2.3). For example, if ∂/∂ψ and ∂/∂z are standard generators of the (local) R × R action, then linear combinations such as K = ∂/∂ψ + ∂/∂z are non-hypersurface orthogonal Killing ﬁelds. Nevertheless, all such solutions do admit, of course, hypersurface orthogonal Killing ﬁelds and so may be written in the form (2.1). For the remainder of the paper, we thus assume that a Weyl solution has the form (2.1) locally. In addition, we will always work with the Z-quotient of the metric (2.1) and so consider Weyl solutions as warped products of the form V ×f S 1 ; it will not be assumed in general that V is simply connected. Duality. Observe that Weyl solutions naturally come in ’dual’ pairs. Namely the Ricci-ﬂat 4-manifold (N, gN ) has the form gN = gV + f 2 dφ2 + u2 dt2 ,

(2.9)

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and so both Mu = V ×f S 1 and Mf = V ×u S 1 are static vacuum solutions on the 3-manifolds, with potentials u, resp. f . Consider the product of the lengths of circles, or equivalently, the area of the torus ﬁber in N , r = f · u.

(2.10)

This is a globally deﬁned positive harmonic function on (V, gV ). To see this, on M = Mu , by (2.5), we have 0 = ∆u = ∆V u+ < ∇ log f, ∇u >, and the same formula, with u and f reversed, holds on Mf . Hence ∆V f u = f ∆V u + u∆V f + 2 < ∇u, ∇f >= 0. Charts. We now describe a collection of preferred charts in which to express the Weyl solution (M, g); this description is due to Weyl [Wl]. The surface V may be partitioned into a collection of maximal domains Vi on which the harmonic conjugate z of r is single-valued, so that F = r + iz is a well-deﬁned holomorphic function from Vi into the right half-plane C+ = {(r, z): r > 0, z ∈ R}. One might also pass to a suitable cover, for instance the universal cover, of V to obtain a globally deﬁned conjugate harmonic function, but it is preferable not to do so. Now each Vi may be further partitioned into a collection of domains Vij on which F is a conformal embedding into C+ , so that g|Vij = F ∗ (dr2 + dz 2 ). We will thus simply view Vij as a domain in C+ , with gV a metric pointwise conformal to the ﬂat metric dr2 + dz 2 . It follows that the corresponding domain Mij = Vij ×f S 1 is embedded as a domain Ω = Ωij in R3 endowed with cylindrical coordinates (r, z, φ), φ ∈ [0, 2π) with the background (unphysical) complete ﬂat metric dr2 + dz 2 + r2 dφ2 . We note that all the data above are canonically determined by the two Killing ﬁelds on N and thus the coordinates (r, z, φ) are called canonical cylindrical or Weyl coordinates for (M, g). Of course Ω is axially symmetric, i.e. symmetric w.r.t. rotation about the z-axis. To express the metric g|Ω in these coordinates, the ﬁeld equations (0.1) imply that the function ν = log u (2.11) is an axially symmetric (independent of φ) harmonic function on Ω ⊂ R3 ; this again follows in a straightforward way from computation of the Laplacian of u and f on Mf and Mu as above. A computation of the conformal factor for the metric gV , c.f. [Wd,Ch.7.1], then leads to the expression of g in these coordinates: g = u−2 (e2λ (dr2 + dz 2 ) + r2 dφ2 ),

(2.12)

where λ is determined by ν as a solution to the integrability equations λr = r(νr2 − νz2 ), λz = 2rνr νz .

(2.13)

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The equations (2.13) mean that the 1-form ω = r(νr2 − νz2 )dr + 2rνr νz dz is closed on Ω. Conversely, given any axially symmetric harmonic function ν on a connected open set Ω in R3 , if the closed 1-form ω is exact, (for example if π1 (Ω ∩ C+ ) = 0), the equations (2.13) determine λ up to a constant and the metric (2.12) gives a solution to the static vacuum equations with S 1 symmetry. (The addition of constants to ν or λ changes the metric at most by diﬀeomorphism or homothety). It is remarkable that solutions to the non-linear vacuum equations (0.1) can be generated in this way by solutions to the linear Laplace equation on R3 . Remark 2.3. (i) The Levi-Civita solutions in §1 are all Weyl solutions. However, the expressions for the A1-A3 metrics in (0.4),(1.7),(1.8) and the B1 metric in (0.9) are not in Weyl canonical coordinates. Note that the quotients of the A2 metric discussed above are no longer Weyl solutions, although they could be considered as local Weyl solutions; the R-action is only locally deﬁned on the quotients. (ii) There seems to be no known Weyl solutions which cannot be expressed globally in the form (2.12). Fixed Point Set. The behavior of solutions at the part of ∂M where either one of the two S 1 or R actions on N has ﬁxed points requires special considerations. ¯ ∩A This is of course the locus where u = 0 or f = 0, and hence includes the part Ω of the z-axis A = {r = 0} in any canonical coordinate chart. It is not necessarily the case however that this locus is contained in A, c.f. the end of Example 2.10. Note that given any Weyl solution (2.12), any covering of R3 \ A induces another solution of the form (2.12), but with φ parametrizing a circle of length 2πk. For the universal cover (k = ∞), the φ-circle is replaced by a line. In fact, (2.12) is well-deﬁned when φ runs over any parameter interval [0, 2πα). Observe however that any asymptotically ﬂat Weyl solution must have α = 1, since the metric must be smooth near inﬁnity. Thus, we will assume α = 1 in the following, unless stated otherwise. Now suppose there is an open interval J in A such that the functions u and λ, and hence the form (2.12) extend continuously to J. The form g then represents a continuous metric in a neighborhood of J if and only if the elementary ﬂatness condition λ = 0, (2.14) is satisﬁed on J. On intervals where (2.14) does not hold, the metric g has cone singularities, so that it is not locally Euclidean. From (2.13), it is clear that if λ has a C 1 extension to J, then λ is constant on J. However, such constants may ¯ ∩ A. This will be analyzed further in Remark vary over diﬀering components of Ω 2.8. For the remainder of §2, we assume that M = Ω,

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so that the Weyl solution is given globally in the form (2.12). Let I be the set where ν = −∞, i.e. the Gδ set in Ω ⊂ R3 given by ν −1 (−∞, n), (2.15) I= n

where n runs over negative integers. It is usually assumed in physics that I is nonempty, although this need not be the case; this will also be discussed further below. The set I corresponds to the horizon Σ of the Weyl solution (M, g), since u = 0 on I. This correspondence is formal however, since the geometry and topology of ¯ , g), c.f. most of the examples below. I ⊂ R3 is very diﬀerent than that of Σ ⊂ (M For the same reasons, although M = Ω topologically, the metric boundary ∂M of (M, g) is (most always) very diﬀerent than the Euclidean boundary of Ω ⊂ R3 . In the following, we discuss some of the most signiﬁcant possible behaviors for the potential function u, and the associated Weyl solution, in order to illustrate the breadth of these solutions. The discussion is by no means complete or exhaustive. ¯ compact. (I) Ω Let M = Ω be any bounded, C ∞ smooth axisymmetric domain (i.e. connected open set with smooth compact boundary) in R3 and let φ be any C k,α function on ∂Ω, k ≥ 0, α ∈ (0,1), which is axially symmetric about the z-axis. For simplicity, assume that Ω ∩ C+ is simply connected. Let ν be the solution to the Dirichlet problem ∆ν = 0, ν|∂Ω = φ. Then ν is also axi-symmetric about the z-axis, and hence ν generates a Weyl solution as in (2.12). Suppose that for a given k ≥ 1 α ∈ (0,1), φ as above is C k,α on ∂Ω, but is nowhere C k+1 on ∂Ω. Then ν extends to a C k,α function on the Euclidean closure ¯ and hence, from (2.13), the function λ in (2.12) is also uniformly bounded. This Ω means that the metric g is quasi-isometric to the ﬂat metric on Ω, and hence the metric boundary of Ω w.r.t. the Weyl metric g is the same as its Euclidean boundary. Since ν is not C k+1 anywhere on ∂Ω, this solution M = Ω is maximal, i.e. admits no larger static vacuum extension; C 2 smooth solutions of the static vacuum equations are analytic. Further ν is bounded, so that u = eν is bounded away from 0, and hence the solution (Ω, g) has no horizon. As noted in §0, the presence of the boundary ∂M is physically assumed due to the presence of matter or ﬁeld sources. Thus, at least when k ≥ 2, the vacuum solution (M, g) can be extended to a larger space-like domain (M , g ) ⊃ (M, g) with non-zero stress-energy T in M \ M. On the other hand, if k = 0 above, then the geometry of the metric boundary (∂M, g) will in general be very diﬀerent than the smooth geometry of ∂Ω in R3 . Further, one can of course consider non-smooth domains Ω ⊂ R3 in this situation. These remarks indicate that the structure of the metric boundary ∂M seemingly can be quite arbitrary.

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¯ is non-compact in R3 . For the remainder of this section, we assume that Ω The same remarks as above hold for non-compact domains with smooth (surface) boundary. Thus for example M = Ω might have inﬁnitely many ends if ∂Ω is non-compact, showing that the assumption (i) in Theorem 0.3 is necessary. For simplicity, we only consider the following situation from now on. (II) Suppose dimH ∂Ω ≤ 1,

(2.16)

where the boundary is in the topology of R3 and dimH is the Hausdorﬀ dimension. Thus ∂Ω is a closed set of capacity 0, c.f. [H, Thm.5.14], and so in particular is a ¯ = R3 . polar set. Clearly Ω (A) (Positive Case). Suppose that ν is locally bounded above, i.e. sup ν < ∞, ∀x ∈ ∂Ω.

(2.17)

Bx (1)

It follows, c.f. [H, Thm.5.18] that ν extends uniquely to a globally deﬁned subharmonic function on R3 . Hence, one may use the value distribution theory of subharmonic functions on R3 to analyze the geometry of Weyl solutions. The Riesz representation theorem c.f. [H,Thm.3.9], implies that any subharmonic function ν on R3 may be represented semi-globally, i.e. on B(R) = B0 (R) ⊂ R3 for any R < ∞, as 1 ν(x) = − dµξ + h(x), (2.18) |x − ξ| B(R) where dµξ is a positive Radon measure on B(R) called the Riesz measure of ν and h is a harmonic function on B(R); both dµ and h are axi-symmetric if ν is. (A Radon measure is a Borel measure which is ﬁnite on compact subsets). For the moment, we only consider the situation where there exists K < ∞, independent of R, s.t. 1 (2.19) dµξ ≤ K. B(R) |x − ξ| In this case, one obtains a global representation of ν as 1 dµξ + h(x), ν(x) = − R3 |x − ξ|

(2.20)

where dµξ is a positive measure and h a harmonic function on R3 . (In (D) below, we brieﬂy discuss the situation where (2.19) is not assumed). In particular, if ν is uniformly bounded above, say sup ν = 0, then the Liouville theorem for harmonic functions implies that h ≡ 0, and one has the expression 1 ν(x) = − dµξ . (2.21) |x − ξ| 3 R

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Note that since ν is harmonic on Ω, I¯ ⊂ suppdµ ⊆ ∂Ω,

(2.22)

but in many situations, as will be seen below, the ﬁrst inclusion is strict. (A)(i) Pure harmonic potentials. Suppose that ν is a smooth harmonic function deﬁned on all of R3 , so that ν = h in (2.20). It is clear that in this case Σ = ∅ in the sense that I = ∅ in R3 . Since ν is axisymmetric, ν may be viewed as an expansion in Legendre polynomials, i.e. z ν= ak Rk · Pk ( ), R k≥0

z z where R2 = r2 + z 2 . For instance, R · P1 ( R ) = z, R2 · P2 ( R ) = 3z 2 − R2 . 3 While these solutions are deﬁned on all of R , no such solution gives a complete Weyl metric g on M = R3 , by Theorem 1.1. For instance, for ν = z, the Weyl metric is

g = e−2z (e−r (dr2 + dz 2 ) + r2 dφ2 )), u = ez . 2

Any straight ray in the (r, z) half-plane has ﬁnite length in this metric, except a ray parallel to the negative z-axis. The horizon Σ occurs formally at {z = −∞}, of inﬁnite g-distance to any point in R3 . (A)(ii) Newtonian potentials. Suppose that h = 0 in (2.20), so that ν is the Newtonian potential of an axisymmetric positive mass distribution dµ as in (2.21). This situation corresponds exactly to the Newtonian theory of gravity, (or equivalently the electrostatics of a positively charged distribution). While there is a vast classical literature on this subject, we will only consider the most interesting situation where ¯ supp dµ = I,

(2.23)

so that ν approaches −∞ on a dense set in supp dµ. The following Lemma characterizes this situation. Lemma 2.4. Let dµ be an axi-symmetric positive Radon measure on R3 . Then supp dµ = I¯ ⇔ supp dµ ⊂ A.

(2.24)

Proof. Suppose ﬁrst that supp dµ is not contained in A. Since dµ is axially symmetric, part of supp dµ, namely the part not contained in A, is then given by a union of circles about the z-axis. Suppose ﬁrst that there is a circle C which is an isolated component of supp dµ, so that dµ|C is a multiple of Lebesgue measure on C. This case has been examined in [Wl],[BW], and we refer there for details. In particular in this case the potential ν is bounded below on and near C, and hence

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¯ If C is not isolated, then using (2.21), the same reasoning holds, supp dµ = I. since the measure dµ is then even less concentrated on the circles. On the other hand, if supp dµ ⊂ A, then dµ is a positive Radon measure on A. Standard measure theory implies that the upper density of dµ w.r.t. Lebesgue measure dA at a ∈ A, i.e. lim supr→0 µ(Bra (r)) , is positive, for Lebesgue almost all a ∈ supp dµ. From the expression (2.21), it is clear that for any such a, ν(x) → −∞ as x → a. This gives the converse. For the remainder of the discussion in (A), we assume (2.23) holds. From the theory of subharmonic functions on R3 , the set I given by (2.15) may be an arbitrary Gδ set in A ⊂ R3 , i.e. a polar set in A. Since countable unions of polar sets are polar, note that I is not necessarily closed in A ⊂ R3 . (For example, let {zi } ∈ A be an increasing sequence −i converging to a limit point z, with say dist(zi , zi+1 ) = i−2 , and let dµ = 2 δzi , where δzi is the Dirac measure based at zi . Then I = {zi } and supp dµ = {zi ∪ z}). Given any x ∈ R3 , let mx (r) be the mass of the measure dµ in the ball Bx (r), i.e. mx (r) = dµ. (2.25) Bx (r)

This is a non-negative increasing function on R+ , for any given x and the limit m = limr→∞ mx (r) > 0,

(2.26)

is the total mass of dµ. This agrees, up to a universal constant factor, with the (ADM or Komar) mass in general relativity, when the latter is deﬁned, and with (0.7) for solutions with pseudo-compact boundary. Note that one may have m = +∞. Lemma 2.4 and a standard result from potential theory, c.f. [H, Thm.3.20], characterize the possible Riesz measures satisfying (2.23). Lemma 2.5. A necessary and suﬃcient condition that a positive Radon measure dµ is the Riesz measure of an axi-symmetric subharmonic function ν on R3 with sup ν = 0 and supp dµ = I¯ is that supp dµ ⊂ A, and, for any given x ∈ A, ∞ mx (r) dr < ∞. (2.27) r2 1 It is easy to see that a Weyl solution (M, g) generated by a potential ν as in (2.21) for which supp dµ = I¯ is a compact subset of the axis A, is asymptotically ﬂat, in the sense preceding Theorem 0.1. Further, the simplest or most natural ¯ and intersecting surfaces enclosing any ﬁnite number of compact components of I, ¯ A outside I, are 2-spheres in M . Of course if supp dµ ⊂ A is non-compact, then the solution cannot be asymptotically ﬂat. A simple example is the solution generated by the measure 1 dµ = dAζ , 1 + |ζ|

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where ζ parametrizes A and dA is Lebesgue measure on A. Observe also that such solutions do not have pseudo-compact boundary. It is worthwhile to discuss some standard examples of Weyl solutions and their corresponding measures. Example 2.6. (i) (Curzon Solution). From the point of view of the Riesz measure, perhaps the simplest example is the measure dµ given by a multiple of the Dirac measure at some point on A, so that ν = −m/R, R(x) = |x|, is a multiple of the Green’s function on R3 . This gives rise to the Curzon (or monopole) solution, c.f. [Kr,(18.4)], 2 2 4 gC = e2m/R [e−m r /R (dr2 + dz 2 ) + r2 dφ2 ], (2.28) with u = e−m/R . Here Ω = R3 \ {0}, ∂Ω = {0}, and it is often stated that gC has a point-like singularity (monopole) at the origin. However the geometry of ∂M is very diﬀerent than that of a point. Namely the circles about the z-axis have length diverging to inﬁnity as R → 0. Thus, small spheres R = / about {0} become very long in the φ direction, and very short in the transverse θ direction, forming a very long, thin cigar. In particular, as a metric space, ∂M = R. This is the ﬁrst example where ∂M is non-compact but pseudo-compact. Of course ∂M = Σ, so that (M, g) is complete away from Σ. A more detailed analysis of the Curzon singularity is given in [SS]. Note that one could not have solutions with both directions expanding at ∂M, so that area ∂M = ∞, with ∂M pseudo-compact. This can be seen by use of minimal surface arguments, c.f. [G]. (ii) (Schwarzschild Solution). The Schwarzschild metric (0.4) is a Weyl metric, with measure dµ = 12 dA on [−m, m], where dA is the standard Lebesgue measure on A. The resulting potential ν in (2.21) is the Newtonian potential of a rod on the z-axis with mass density 12 , given by νS =

R+ + R− − 2m 1 2 log( ), where R± = r2 + (z ± m)2 . 2 R+ + R− + 2m

(2.29)

As mentioned before, the horizon Σ here is a smooth totally geodesic 2-sphere of radius 2m and ∂M = Σ. The Weyl solution generated by the potential a · νS , for νS as in (2.29) with a > 0 and a = 1, is not isometric or homothetic to the Schwarzschild metric. The associated Weyl metric is no longer smooth up to the horizon; in fact Σ is not even a 2-sphere unless a = 1. Remark 2.7. More generally, consider any Weyl solution generated by a Riesz measure dµ satisfying (2.23). Observe that f = ur , the length of the φ circles in the Weyl metric, stays bounded away from 0 and ∞ on approach to supp dµ, if and only if log r − C ≤ ν ≤ log r + C, (2.30)

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for some C < ∞, since ν = log u. From the expression (2.21), this occurs only for the Schwarzschild potential νS . Brieﬂy, the reason for this is as follows. The estimate (2.30) implies that the potential υ = ν − νS is bounded and given by convolution of dist−1 with a signed Radon measure dλ. However, as in the proof of Lemma 2.4, if υ is bounded then one sees that necessarily dλ << dA and further the Radon-Nikodym derivative dλ/dA is 0, a.e. (dA). In other words, any point of non-zero density for dλ w.r.t. dA gives rise to approximating points on which ν is unbounded. It follows that dλ = 0, and hence υ = 0. In all other cases, where f → 0 on approach to a region in supp dµ, the length of the φ circles goes to 0, and hence this portion of Σ is singular, of dimension ≤ 1, while where f → ∞, the length of the circles goes to ∞ and the corresponding portion of Σ is singular and non-compact, (as in the Curzon solution). Note that if supp dµ is compact, then in all cases, ∂M is pseudo-compact. Thus among the Weyl solutions given by a Newtonian potential, only the Schwarzschild metric is smooth up to Σ. This gives a very simple (local) conﬁrmation of Theorem 0.1 in this special case. Example 2.6.(iii) (Superposition/Multiple Holes). Subharmonic functions of the form (2.21) form a convex cone. In particular, one thus has a natural linear superposition principle for Weyl solutions. This feature is another remarkable property of Weyl solutions. For example, one may choose the measure dµ = 12 dA on two, or any number of disjoint intervals {Ij } on the axis A, provided (2.27) holds. These correspond to solutions with ’multiple black holes’, each interval Ij giving a component of Σ which is a 2-sphere of radius equal to the length of Ij . Although such solutions are essentially smooth up to Σ, they do not deﬁne smooth vacuum solutions on R3 \ ∪Bj . There are cone singularities, (called struts or rods in the physics literature), along geodesics (corresponding to A \ {Ij }) joining the 2-spheres of Σ, so that the metric g is not locally Euclidean along such curves. Thus, the elementary ﬂatness condition (2.14) is violated on A \ ∪Ij . Nevertheless, the curvature of such metrics is uniformly bounded everywhere. These cone singularities must be considered part of ∂M, so ∂M in this case is a union of 2-spheres joined by a collection of curves and thus connected. Of course, the black hole uniqueness theorem, Theorem 0.1, also implies that such solutions cannot be smooth everywhere, when the number of intervals is ﬁnite. However, the proof of this result strongly uses the asymptotically ﬂat assumption. It seems to be unknown whether there are any Schwarzschild type metrics with inﬁnitely many black holes, i.e. metrics complete away from Σ and smooth up to Σ with Σ consisting of inﬁnitely many 2-spheres, and which satisfy (2.17) or (2.21). It is natural to conjecture that such solutions do not exist, c.f. however the end of Remark 2.8 below. Remark 2.8. Whenever supp dµ ⊂ A is not connected but compact, there will exist such cone singularities on A\ supp dµ. When supp dµ, or a suﬃciently small smoothing of supp dµ, is interpreted to represent a matter source, then

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this statement corresponds exactly to the fact that there are no equilibrium (i.e. time-independent) many-body solutions in Newtonian gravity of this character, c.f. [Bn]. Even when supp dµ is non-compact and disconnected, this seems very likely to be true, c.f. the expressions for sums of Schwarzschild rods (2.29) in [IK, p.336-337], which generalize to inﬁnitely many rods. The components {Ci } of A\ supp dµ represent idealized matter sources (struts or rods) keeping the components of supp dµ apart in static equilibrium. The cone angle α = αi is constant on each C = Ci , and corresponds to a concen¯ given by a multiple of the Lebesgue measure tration of scalar curvature on C ⊂ M on C s = (1 − α)dAC , Thus the vacuum equations (0.1) are not satisﬁed across {Ci }. If a very small tubular neighborhood of radius r of such a rod is smoothly ﬁlled in with a perfect ﬂuid source of constant pressure ρ and energy density µ, then one has the relation limr→0 rρ = −limr→0 rµ > 0. The eﬀective mass of such rods is zero, i.e. they do not contribute to the gravitational potential ν, c.f. [I2] for a detailed discussion. By passing to covering spaces, it is always possible to create such cone singularities in Weyl solutions, even if none existed to begin with. For instance, for the Schwarzschild solution (0.4), with potential (2.29), take any covering, including the universal covering, of R3 \ A = (S 2 \ {a ∪ −a}) × R+ , where {a, −a} are two antipodal points on S 2 . This gives a solution whose metric completion has cone singularities along two (radial) geodesics starting at the antipodal points on S 2 = Σ and going to inﬁnity. Note that this discussion assumes that ν is given by a Newtonian potential (2.21). In fact, there are Weyl solutions (M, g, u) everywhere smooth up to the axis A, with Σ = ∂M disconnected, with no cone singularities or struts keeping the components of Σ apart. Namely, the B1 solution (0.9), dual to the Schwarzschild solution, has this property. Another, more remarkable, example is given in [KN]. These authors construct a Weyl solution of the form (2.12), which is complete away from Σ and smooth up to Σ, with Σ consisting of inﬁnitely many Schwarzschild-like 2-spheres. In fact, the solution is periodic in the z-direction. This metric is not of the form (2.21), but is a limit of a sequence of solutions of the form (2.20), c.f. (D) below. Remark 2.9. If (M, g, u) is a Weyl solution of the form (2.12) with ν = νu = log u, then the dual solution (M , g , f ), discussed in (2.9), is also a Weyl solution of the form (2.12), with potential νf = log f given by νf = log r − νu .

(2.31)

Hence if one potential is Newtonian, the dual one is not. Note that the sets Iu , If where νu and νf are −∞ are disjoint, with I¯u ∪ I¯f = A. Hence, if νu is a Newtonian potential with supp dµ compact, so that the associated Weyl solution is asymptotically ﬂat, then the dual Weyl solution is asymptotically cylindrical.

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Another example of a potential where both terms in (2.20) are non-trivial is the situation considered (locally) in [GH], where h is a smooth axi-symmetric harmonic function deﬁned on a neighborhood of supp dµ, c.f. Remark 1.5. As vacuum solutions, these metrics cannot be complete away from Σ, as in the discussion on pure harmonic potentials. This completes our discussion of the Positive Case. (B) (Negative Case). Under the assumption (2.16), suppose now instead that ν is locally bounded below in R3 , i.e. inf ν > −∞, ∀x ∈ ∂Ω.

Bx (1)

(2.32)

Then ν extends uniquely to a globally deﬁned superharmonic function on R3 . Exactly the same discussion as in (A) above holds here, under the substitution ν → −ν. (This corresponds to the transformation u → u−1 following (1.5)). In this case, the Riesz measure is a negative measure, so that one has solutions with negative mass. Note that here the potentials ν or u are unbounded above within supp dµ, i.e. u or ν go to +∞. (C) (Mixed Case). Next, one may superimpose Weyl solutions with positive and negative measures dµ, i.e. consider ν of the form (2.21), with dµ a signed Radon measure. For example, one may form dipole-type solutions with potential of the form ν = ν+ + ν− where ν+ and ν− are (for instance) Curzon or Schwarzschild solutions of positive and negative mass placed at diﬀerent regions on the axis. This gives for instance examples of asymptotically ﬂat solutions with mass m assuming any value in R. More generally, since positivity is no longer assumed, the measure dµ may be replaced by distributions, for example weak derivatives of measures. Example 2.10. (Multipole Solutions). As a typical example, one may take potentials corresponding to derivatives of the Dirac measure based at a point a ∈ A, i.e. the multipole potentials, z R−n−1 · Pn ( ), R where Pn is the nth Legendre polynomial, or arbitrary linear combinations of such; c.f. [MF, p.1276ﬀ]. Such potentials are limits of combinations of Newtonian potentials with positive and negative mass. Thus, it is reasonable to expect that there are (Newtonian) equilibrium solutions, i.e. solutions with no cone singularities on the axis. This is proved in [Sz], where explicit equilibrium conditions are given. Note that one may have inﬁnitely many multipole ’particles‘ in equilibrium. Another example is the potential of a dipole ring z(x) ν(x) = − dξ, |x − ξ|3 C

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where x = (r, z, φ) and dξ is the Lebesgue measure on the unit circle C = {r = 1} in the z = 0 plane. Here ν(x) → −∞, as x → C along the rays r = 1, z > 0. Hence, in this case, the set I = C is not contained in the axis A. (D) (Limits). Finally, one may consider potentials which are limits of potentials of the type (A)-(C) above, (besides those in Remark 2.9, Example 2.10). We consider just one important instance of this here. Example 2.11. (Kasner Metric). It is easily seen that the potential ν = log r generates the ﬂat metric g = dr2 + dz 2 + dφ2 , on (R3 )+ , with potential u = r. Observe that since the φ-lines have constant length, the function r is now an aﬃne (in fact linear) function on (R3 )+ . On the other hand, an equally simple computation shows that the potential ν = a · log r, for any a ∈ R, generates the metric 2

g = r2a

−2a 2

Equivalently, setting s = ra

(dr2 + dz 2 ) + r2−2a dφ2 , u = ra .

−a+1

,

g = ds2 + sα dz 2 + sβ dφ2 , u = sγ ,

(2.33)

where α = (2a−2)/(a−1+a−1 ), β = (2a−1 −2)/(a−1+a−1 ), γ = (a−1+a−1 )−1 . Here s ∈ R+ , z ∈ R and φ ∈ [0, 2π] or any other interval, including R, (by passing to covering and quotient spaces). These metrics are all non-homothetic, provided a ∈ [−1, 0) ∪ (0, 1]; a = 0 gives the ﬂat metric with u = 1 while a = −1 gives the A3 metric (1.8). The potential ν = a · log r can be considered as the limit ν = limm→∞ a[νS (m) − log 2m],

(2.34)

where νS (m) is the Schwarzschild potential (2.29) of mass m. Thus it is a limit of potentials of the form (2.18), where both terms are non-zero, (the harmonic term h is of course constant here). These metrics are dual, in the sense discussed in (2.9) to the Kasner (or Bianchi I) vacuum cosmological models, with homogeneous (ﬂat) but anisotropic space-like hypersurfaces, c.f. [Wd, Ch.7.2]. It is easy to see that the Kasner metrics are the only Weyl solutions (M, g) which have an isometric R × R action, even locally. (The axisymmetric potential ν on Ω ⊂ R3 must be invariant under an orthogonal R-action, hence giving a rotationally invariant harmonic function on R2 . Thus the potential must be a multiple of log r). Consider these metrics on the quotient M = R+ × S 1 × S 1 . In case a > 0, we have α < 0, β > 0, γ > 0 and so Σ = ∂M. As in the discussion with the Curzon metric, the z-circles have unbounded length as s → 0, so that ∂M = R and the levels t = /, (t(x) = distg (x, ∂M )), are long, thin cigars. Thus, ∂M is ¯ \ U is obviously small. non-compact, but pseudo-compact. The end of M

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If a < 0, then α > 0, β > 0, γ < 0 and so Σ = ∅, (it occurs at inﬁnity), ¯ \ U is not small; the area growth of with ∂M = {pt}. In this case, the end of M 1−γ geodesic spheres is O(r ). However, none of these solutions are asymptotically ﬂat even in a weak sense, except of course when a = 1. Namely, the curvature decays only quadratically in the g-distance to ∂M, i.e. |r| = O(t−2 ), and not any faster. Hence, the case a > 0 shows that the conclusion of Theorem 0.3, (asymptotically ﬂat or small ends), cannot be strengthened to only asymptotically ﬂat ends, while the case a < 0 shows that the assumption (ii) on u in Theorem 0.3 is necessary. Another metric of this (limit) type is that constructed in [KN], referred to in Example 2.8. This metric has the same asymptotics as the Kasner metric.

3 Characterization of Asymptotically Flat Solutions In this section, we prove Theorem 0.3. The proof of the ﬁrst statement on ﬁniteness of the number of ends is quite easy, so we begin with this. Throughout this section, let (M, g, u) be a static vacuum solution with ∂M pseudo-compact. We recall from §0 that M is connected and oriented. As in §0, let t(x) = distg (x, ∂M ), and suppose U = t−1 (0, so ), so that ∂U ∩ M is compact. For r, s ≥ so , let S(s) = t−1 (s), A(r, s) = t−1 (r, s) be the geodesic spheres and annuli about ∂U. It is important to note that neither S(s) nor A(r, s) are necessarily connected, even if M has only one end. (Of course if E is a given end of M , then SE (s) = S(s) ∩ E must be connected for some sequence s = sj → ∞). Let Sc (s) and Ac (r, s) denote any component of S(s) resp. A(r, s), so that S(s) = ∪Sc (s), A(r, s) = ∪Ac (r, s). Of ¯ \ U, so that these sets have compact course t is a proper exhaustion function on M closure in M . Let diami Ac (r, s) denote the intrinsic diameter of Ac (r, s), i.e. the diameter of the connected metric space (Ac (r, s), g). Lemma 3.1. There exists a constant do < ∞, independent of s, such that the number of components of A( 12 s, 2s) is at most do and 1 diami Ac ( s, 2s) ≤ do · s. 2

(3.1)

¯ \ U, g) has a ﬁnite number of ends {Ei }. In particular, the manifold (M Proof. Consider ﬁrst the 4-manifold (N, gN ), as in (0.2) which is smooth and Ricci¯ = π −1 (U ), where π : N → M is projection on the ﬁrst factor. Since ﬂat outside U

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¯ ∩ N is compact, it follows from results of [Lu] that Lemma 3.1 holds on N , so ∂U ¯ \ U also that in particular N has a ﬁnite number of ends. Since N = M × S 1 , M has a ﬁnite number of ends. The choice of the time parameter on N deﬁnes a totally geodesic embedding ¯ . A geodesic M ⊂ N and we have tN |M = t where tN (x) = distN (x, ∂M ), ∂M ⊂ N ball or annulus in M embeds in the geodesic ball or annulus of the same size in N . Hence (3.1) also holds for M . Lemma 3.1 of course proves the ﬁrst statement of Theorem 0.3. Observe that the estimate (3.1) is invariant under rescaling of the metric g. For the remainder of the proof, we (usually) work with a given end E from the ﬁnite collection {Ei }. The main statement of Theorem 0.3 is that if ∞ 1 ds < ∞, (3.2) areaSE (s) then the end E is asymptotically ﬂat. The proof of this result is rather long, so we outline here the overall strategy. The asymptotic behavior of (E, g, u) is studied in general by examining the structure of the possible tangent cones at inﬁnity, deﬁned below. Basically, tangent cones at inﬁnity fall into two classes, according to whether the asymptotic geometry near a given divergent sequence of base points is non-collapsing or collapsing, c.f. Lemmas 1.3-1.4. The main point is to prove that under the bound (3.2), all tangent cones at inﬁnity are ﬂat manifolds, and further that no collapse behavior is possible. Once this is established, the proof that (E, g, u) is asymptotically ﬂat is relatively straightforward. A priori, the end E may be very complicated topologically, for instance of inﬁnite topological type; consider for instance that E might be of the form S∞ ×S 1 , where S∞ is any non-compact surface of inﬁnite topological type and one end. A main idea is to use the behavior of the potential function u, in particular its value distribution theory, to control the topology and geometry of E in the large. We have already seen in §2 that the potential u controls quite strongly the geometry of Weyl solutions. Lemma 3.6 below is the key technical lemma which expresses this control for general static vacuum solutions (with pseudo-compact boundary). Further remarks on the strategy of proof precede the Lemmas below. The discussion to follow, until the end of Lemma 3.6, holds in general for ends E of static vacuum solutions with compact boundary. The estimate (3.2) will only be used after this. We now deﬁne the tangent cones at inﬁnity of a given end E. (While this is a commonly used terminology, such limit metric spaces are not necessarily metric cones in general). First, we recall by Theorem 1.1 that there is a constant K < ∞ such that, ∀x ∈ M, K K |r|(x) ≤ 2 , |d log u|(x) ≤ . (3.3) t (x) t(x)

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The scale-invariant estimates (3.3) give quite strong initial control on the asymptotic geometry of (E, g, u) which allows one to get started. Observe that an immediate consequence of (3.1) and (3.3), by integration along paths in Ac ( 13 s, 3s), is the following Harnack inequality: supu ≤ K1 , inf u

(3.4)

where the sup and inf are taken over any component Ac ( 12 s, 2s) and K1 is independent of Ac ( 12 s, 2s). Let xi be any divergent sequence of points in E, with ti = t(xi ) → ∞. Consider the connected geodesic annuli Ai = Ai (κ) = Ac (κ−1 ti , κti ), xi ∈ Ai , w.r.t. the rescaled or blow-down metric gi = t−2 i · g;

(3.5)

here κ is any ﬁxed positive constant > 1. By the curvature estimate (3.3), the metrics gi have uniformly bounded curvature on Ai - the curvature bound depends only on K and κ. Further, by (3.1), the diameter of Ai w.r.t. gi is also uniformly bounded. Hence, if the sequence is non-collapsing, i.e. if there is a lower volume bound volg Ai ≥ νo ·t3i , for some νo > 0, (equivalent to volgi Ai ≥ νo by scaling), then Lemma 1.3 implies that a subsequence of the pointed sequence {(Ai , gi , xi )}, converges smoothly, away from the boundary, to a limiting smooth metric (A∞ (κ), g∞ , x∞ ). The limit is a solution of the static vacuum equations; as noted in Lemma 1.3, the potential u is renormalized to ui = u/u(xi ), so that the limit potential u∞ satisﬁes u∞ (x∞ ) = 1, c.f. also (3.4). Choosing a sequence κj → ∞ and a suitable diagonal subsequence, gives the maximal static vacuum solution (A∞ , g∞ , u∞ , x∞ ). Observe here also that the estimate (3.1) implies that ∂A∞ = {pt}. On the other hand, if the sequence (Ai , gi ) is collapsing, in the sense that volg Ai << t3i , as ti → ∞, (equivalent to volgi Ai → 0), then Lemma 1.4 implies that Ai is a Seifert ﬁbered space or torus bundle over an interval. As disi (κ) of Ai (κ) (or more cussed there, one may then pass to Z or Z ⊕ Z covers A precisely smooth interior approximations to Ai (κ)) to obtain a non-collapsing sei (κ), gi , xi ), smoothly convergent to a limit (A ∞ (κ), g∞ , x∞ ); (gi here is quence (A lifted to the cover Ai , as is xi ). As above, one may then choose a sequence κj → ∞ ∞ , g∞ , u∞ ). This and pass to a diagonal subsequence to obtain a maximal limit (A limit static vacuum solution has an isometric R action, or R ⊕ R action in the case of a rank 2 collapse. Hence by Proposition 2.2 it is a Weyl solution. In the latter case, the solution is then a (possibly ﬂat) Kasner metric, c.f. Example 2.11. ∞ is a trivial R, (or R ⊕ R), bundle over a surface V , (or Topologically, the limit A interval), again by Proposition 2.2. ∞ , (or Z⊕Z quotient in As in §2, we will always work in a Z-quotient A¯∞ of A ¯ the case of rank 2 collapse), and ﬁnite covers Ai (κ) converging to A¯∞ (κ). For any

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ﬁxed κ > 0, the manifolds (A¯i (κ), gi , xi ) thus have uniformly bounded curvature and diameter, a uniform lower bound on their volume, and converge smoothly to the limit (A¯∞ (κ), g∞ ) ⊂ (A¯∞ , g∞ ). The limit has a free isometric S 1 or S 1 × S 1 action, and so in particular is an S 1 or T 2 bundle. Hence A¯i (κ) is also topologically an S 1 or T 2 bundle, (although not metrically). To be deﬁnite, the ﬁnite covers are chosen so that the length of the S 1 factor or factors at the base point xi ∈ A¯i (κ) converge to 1 in the limit. Recall by Lemma 1.4 that the inclusion map of the ﬁbers induces an injection on π1 . The coverings A¯i (κ) are obtained by taking large ﬁnite unwrappings of the S 1 or T 2 ﬁbers, (corresponding to taking subgroups of π1 (S 1 ) or π1 (T 2 ) of large but ﬁnite index). All ﬁnite covering spaces of S 1 or T 2 are still S 1 or T 2 , and hence we may, and do, choose the unwrappings so that, as smooth manifolds, A¯i (κ) = Ai (κ), for any κ > 0. In the limit, the unwrapping of the collapse thus just corresponds to expanding the length of the collapsing S 1 factor (or factors), preserving the holonomy, if any, of the S 1 bundle; compare with the proof of Proposition 2.2. The limit spaces (A∞ , g∞ , u∞ , x∞ ) or (A¯∞ , g∞ , u∞ , x∞ ) constructed above are called tangent cones at inﬁnity of (E, g, u). Note that such tangent cones are only attached to some subsequence of a given divergent sequence of base points {xi }. Hence, a priori, the tangent cones at inﬁnity could be highly non-unique as Riemannian manifolds. In general, there may be no relation between the geometry of diﬀerent tangent cones based on (subsequences of) distinct divergent sequences 2 {xi }; for example, tangent cones based on sequences with t(xi ) = 2i and t(xi ) = 3 2i . The tangent cones only detect behavior of the end E in gi -bounded distance to the base points xi . On the other hand, since tangent cones at inﬁnity attached to any divergent sequence always exist, for any s suﬃciently large, say s ≥ so , the geometry of (Ac ( 12 , 2), gs ) or (A¯c ( 12 , 2), gs ) is always close to that of some tangent cone at inﬁnity A∞ or A¯∞ . Further, by construction, the tangent cones are always connected and, since M is oriented, so is each tangent cone. The following lemma is a typical application of the use of tangent cones at inﬁnity. Lemma 3.2. Suppose the curvature r decays faster than quadratically in the end (E, g, u), i.e. ε(t) |r|(x) ≤ 2 , (3.6) t (x) where ε(t) → 0 as t → ∞. Then there is a compact set K ⊂ E such that E \ K is diﬀeomorphic either to R3 \ B or to (R2 \ B) × S 1 , where B is a 3-ball, (resp. a 2-ball), i.e. E is of standard topological type. Further, the annuli AE ( 12 s, 2s) are connected, for all s ≥ so , for some so < ∞.

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Proof. By the preceding discussion, the condition (3.6) is equivalent to the statement that all tangent cones at inﬁnity (A∞ , g∞ ) or (A¯∞ , g∞ ) of E are ﬂat, (as well as connected and oriented). The two possible conclusions of Lemma 3.2 correspond to the two possibilities of non-collapse and collapse in the formation of the tangent cones. Suppose ﬁrst that g is non-collapsing on E, i.e. there exists νo > 0 such that volg Ac ( 12 s, 2s) ≥ νo · s3 , for all s large, and all components Ac . Recall that ∂A∞ = {pt} in this situation. It then follows that for s suﬃciently large, each Ac ( 12 s, 2s) is diﬀeomorphic, and almost isometric to the standard ﬂat annulus A = r−1 ( 12 s, 2s) in R3 , r(x) = |x|, (away from the boundary). In fact, each tangent cone at inﬁnity A∞ is isometric to R3 \ {0} in this situation. Here we are implicitly using the fact that the only complete oriented ﬂat 3-manifold with an isolated singularity is R3 \ {0}, c.f. [AC] for example. Similarly, a smooth approximation to Sc (s) is diﬀeomorphic and almost isometric to S 2 (s) ⊂ R3 . By the isotopy extension theorem, these diﬀeomorphisms from Ac ( 12 s, 2s) to the standard annulus may then be assembled to a global diﬀeomorphism, and almost isometry, of E \ K into R3 \ B, for some compact set K ⊂ E. We refer to [AC, Thm.1.18] for the proof of these statements, (in a slightly diﬀerent but equivalent form), which are now quite standard. The main point is of course that since the family of annuli Ac ( 12 s, 2s) as s varies is topologically rigid, i.e. one has a unique topological type, there is no value of s at which the topology can change or bifurcate. Suppose on the other hand that g is collapsing on Ac ( 12 si , 2si ), for some sequence si → ∞, and some sequence of components Ac . As discussed above, one may then pass to suitable covers A¯i = A¯c ( 12 si , 2si ) so that, in a subsequence, (A¯i , gi ) is diﬀeomorphic and almost isometric to its limit A¯∞ ( 12 , 2) ⊂ A¯∞ . The maximal limit A¯∞ is a ﬂat manifold with either a free isometric S 1 or S 1 × S 1 action. Hence there are two possibilities for A¯∞ , namely either V × S 1 or R+ × S 1 × S 1 , where V is a ﬂat 2-manifold and the metric is a product metric on each S 1 factor. In the former case, the diameter estimate (3.1) implies, (as in the non-collapse case above), that V is a complete ﬂat cone, possibly with an isolated singularity at {0}. Hence, although these two possibilities for the limiting metric of A¯∞ ( 12 , 2) are distinct, both are the same topologically, i.e. A¯∞ ( 12 , 2) is topologically I × S 1 × S 1 . Now recall from the discussion on tangent cones that A¯i is diﬀeomorphic to 1 Ac ( 2 si , 2si ); metrically A¯i approximates one of the types of ﬂat manifolds above, with S 1 factors shrinking to very short circles. In both cases, (a smoothing of) S¯c (si ) is diﬀeomorphic and almost isometric to a ﬂat torus T 2 . In particular, the topological type of A¯i is distinct from that of the annuli Ai above in the non-collapse case, which are topologically always of the form I × S 2 ⊂ R3 \ {0}, (for any choice of base point and component). This implies ﬁrst that the family {Ac ( 12 s, 2s)} must be collapsing for all s, as s → ∞, and all components Ac . Second, the topological type of the annuli A¯s , and hence that of As = Ac ( 12 s, 2s) is unique, and given for all s large and all c by I × S 1 × S 1 . Use

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of the isotopy extension theorem in the same way as above then proves that the end E itself is diﬀeomorphic to (R2 \ B) × S 1 . Remark 3.3. (i) It is easily seen from the vacuum equations (0.1) that the condition (3.6) is equivalent to ε(t) |∇ log u|(x) ≤ , (3.7) t(x) as t → ∞, c.f. also the proof of Theorem 1.1. (ii). In the context of Lemma 3.2, the tangent cones at inﬁnity (A¯∞ , g∞ ) may not be unique up to isometry in the collapse case, and so may vary within the moduli space M1 of ﬂat product metrics of the form V × S 1 or within the moduli space M2 of ﬂat product metrics of the form R+ × S 1 × S 1 . Note that the moduli space Mo of ﬂat metrics on R3 or R3 \ {0} is just one point, (c.f. again [AC] for the latter statement for example). Similarly, by the normalization preceding Lemma 3.2 that the S 1 factors have length 1, the moduli space M2 ⊂ M2 normalized in this way is also just one point. The moduli space M1 ⊂ M1 where the S 1 factor has length 1 is naturally identiﬁed with R+ , parametrized by the cone angle at {0}. Observe however that these two moduli spaces M1 and M2 are disjoint; they cannot be connected (or even approximated) by a curve of ﬂat metrics. Now the geometry of the annuli (AE ( 12 s, 2s), gs ) varies continuously with s. By the remarks preceding Lemma 3.2, this induces a continuous variation of the possible tangent cones (A¯∞ , g∞ ) in M1 or M2 . Hence, on a given end E, one cannot obtain two diﬀerent tangent cones, one of the form V ×S 1 and another of the form R+ ×S 1 ×S 1 ; c.f. also the proof of Lemma 3.7 below. (iii). We will need a slight generalization of Lemma 3.2 for the next lemma below. Thus let γ(s) be any properly embedded curve in E with t(γ(s)) → ∞ as s → ∞, and suppose (3.6), (or (3.7)), holds in the balls Bγ(s) (δ · t(γ(s))), for some ﬁxed δ > 0. Then the conclusion of Lemma 3.2 also holds. To see this, consider the blow-downs gs = t−2 (γ(s)) · g, based at γ(s), and the associated tangent cones at inﬁnity. The scale invariant condition (3.6), when applied to Bγ(s) (δ · t(γ(s))), implies that all such tangent cones are ﬂat in (Bx∞ (δ), g∞ ), and thus ﬂat everywhere in their maximal domain A∞ , by the fact that smooth solutions of the vacuum equations are real analytic. The proof then proceeds exactly as in Lemma 3.2 To prove that the estimates (3.6) and (3.7) do in fact hold on E, we need to understand in more detail the value distribution of the potential u. The main result needed for this is given in Lemma 3.6, and then (3.6)-(3.7) follow rather easily in (3.17)-(3.18) below. However, some preliminary results are required for the proof of Lemma 3.6. The main diﬃculty is that u may not, in this generality, be a proper function onto its image, (c.f. the remark following the proof of Lemma 3.4). Lemma 3.4 below is a slightly weaker substitute for this property.

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Let U = t−1 (0, so ) be a neighborhood of ∂M as in the beginning of §3. Let γ(τ ) be a maximal ﬂow line of ∇u. We will say that γ is divergent if γ does not intersect U at two diﬀerent times, i.e. if γ(τ ) exits U at some time, then γ(τ ) never reenters (a possibly distinct component of) U , and if further γ(τ ) does not ¯ \U as τ → ±∞. It follows that if γ is divergent, terminate at a critical set of u in M then γ is complete in at least one direction, (τ → +∞, τ → −∞ or both), and in any such direction, γ(τ ) diverges to inﬁnity in M . Since the potential u has no local maxima or minima in M , the set of ﬂow lines terminating on a critical set of u in M \ U is a closed set of measure 0 in M . This follows for instance from the fact that the measure |du|dA, where dA is Lebesgue measure on the level sets of u, is preserved under the gradient ﬂow of u, and this measure tends to 0 on approach to critical points of u. Thus among the set of ﬂow lines not joining points of U , the divergent ﬂow lines are generic in terms of measure on M . Of course the ﬂow lines are curves of steepest ascent for u as τ increases, and of steepest descent for u as τ decreases. ¯ , with U ¯ ⊂ K, such that any Lemma 3.4. There exists a compact set K ⊂ M divergent ﬂow line γ(τ ) intersects K. Proof. Suppose that this were not the case, so that there exist, necessarily complete, ¯ . We may choose K ﬂow lines γ(τ ), τ ∈ R, which do not intersect a given K ⊃ U ¯ \ U, since suﬃciently large so that γ(τ ) is then contained in a ﬁxed end E ⊂ M there are only ﬁnitely many ends. Let Aγ (τ ) = Aγ ( 12 t(γ(τ )), 2t(γ(τ ))) be the component of the geodesic annulus containing the base point γ(τ ) and let Eγ be the part of E swept out by such annuli, Eγ = ∪τ Aγ (τ ) ⊂ E. As τ → ∞, the function u(γ(τ )) is monotone increasing. If u(γ(τ )) increases to +∞, then u → ∞ uniformly as τ → ∞ in Eγ+ = ∪τ >0 Aγ (τ ) ⊂ E, by the Harnack estimate (3.4). Since E is an end, there exists some sequence τj → ∞ such that, for tj = t(γ(τj )), the spheres Sγ (tj ) ⊂ Aγ (τj ) satisfy Sγ (tj ) = SE (tj ), i.e. the spheres SE (tj ) are connected, and hence Aγ (τj ) = AE (τj ). Thus, u becomes uniformly unbounded on AE (τj ), as j → ∞. However, as τ → −∞, the curve γ(τ ) also diverges to inﬁnity in E and u(γ(τ )) is decreasing, (and so in particular bounded), as τ → −∞. This contradiction implies that u(γ(τ )) → u+ < +∞, as τ → +∞. Of course u+ > 0. Suppose ﬁrst, (for simplicity), that limsupE u = u+ , i.e. limt→∞ m(t) = u+ , where m(t) = supSE (t) u. The maximum principle for the harmonic function u implies that for t suﬃciently large, the function m(t) is either monotone increasing or monotone decreasing in t and hence approaches the value u+ as t → ∞. Consider the annuli Aγ (τ ) in the scale gt = t−2 · g, t = t(γ(τ )), as in (3.5). Any sequence τi → ∞ has a subsequence such that the corresponding annuli (Aγ (τi ), gti ) converge to a limiting domain A∞ ( 12 , 2) in a tangent cone at inﬁnity (A∞ , g∞ ), or (A¯∞ , g∞ ), passing to covers as described above in the case of collapse. By construction, we then see that the potential function u∞ for this limit static vacuum solution achieves its maximal value u+ > 0 at an interior point. Since u∞ is harmonic, the maximum principle implies that u∞ ≡ u+ , and hence by the vacuum

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equations (0.1), the limit (A∞ , g∞ ) or (A¯∞ , g∞ ) is ﬂat. This argument holds for any subsequence, and since the convergence to the limit is smooth, we see that t2 · r|Aγ ( 12 t,2t) → 0, t = t(γ(τ )), as τ → +∞, by the scale-invariance of this expression. It follows from Lemma 3.2 and Remark 3.3(iii) that the end E is topologically standard, and the annuli AE ( 12 t, 2t), t = t(γ(τ )), are connected in E, for all τ suﬃciently large. From the prior argument, this implies in particular that u → u+ uniformly at inﬁnity in E. However, as before, as τ → −∞, u(γ(τ )) is monotone decreasing to a value u− ≥ 0. It follows that u+ = u− . This is of course impossible, and shows that γ(τ ) must have exited E at some (negative) time. Thus it remains to prove that L ≡ limsupE u = u+ . Since the annuli AE (τj ) above are connected, the Harnack inequality (3.4), together with the fact that u+ < ∞, implies that L < ∞. Now choose points xj ∈ Sγ (tj ) such that u(xj ) → L. As above, the functions u|AE (τj ) have a subsequence converging to a limit harmonic function u∞ on a tangent cone at inﬁnity based at x∞ = limxj . Then as before u has an interior maximum at x∞ and hence u∞ ≡ L; this gives L = u+ . It follows from Lemma 3.4 and the discussion preceding it that any maximal ﬂow line of ∇u intersects an a priori given large compact set K ⊂ M , except those exceptional ﬂow lines which start or end at a critical point of u far out in M . In particular, a set of full measure in any given level set L of u may be connected to points in K by ﬂow lines of ∇u. In this sense, u is ’almost proper‘, in that it behaves almost like a proper function in terms of the gradient ﬂow. Observe that this does not necessarily imply that the level sets of u are compact, i.e. that u is proper. For instance, the Weyl solution generated by the dipole potential ν = ν+ + ν− considered in §2(IIC) satisﬁes (3.6), (it is even asymptotically ﬂat), but the 0-level of ν is non-compact if ν+ and ν− are chosen so that the mass is 0. In this example, the only divergent ﬂow lines of ∇u are the two ends of the z-axis. Next, as in §2, let ν = log u. The following result is quite standard. Lemma 3.5. On (N, gN ) as in (0.2), with Riemannian metric, we have ∆N |∇ν| ≥ 0. Proof. This standard estimate is a simple consequence of the Bochner-Lichnerowicz formula 1 ∆|∇ν|2 = |D2 ν|2 + < ∇∆ν, ∇ν > +r(∇ν, ∇ν), 2

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on (N, gN ), where we have dropped the subscript N from the notation. Since ∆ν = 0, by (1.6), and since (N, gN ) is Ricci-ﬂat, this gives 1 ∆|∇ν|2 ≥ |D2 ν|2 . 2 One computes ∆|∇ν| =

1 1 |∇ν|−1 ∆|∇ν|2 − |∇ν|−3 |∇|∇ν|2 |2 , 2 4

and, by the Cauchy-Schwarz inequality |∇|∇ν|2 |2 ≤ 4|D2 ν|2 |∇ν|2 , so the result follows. Lemmas 3.4 and 3.5 lead to the following key result relating the behavior of |∇u| to the area growth of geodesic spheres. This result is a straightforward consequence of the divergence theorem for proper harmonic functions u on manifolds of non-negative Ricci curvature. Lemma 3.4 allows one to remove the assumption that u is proper. Lemma 3.6. There is a constant C < ∞ such that for any component Sc (s) of S(s) ⊂ M , s ≥ 1, supSc (s) |∇u| ≤ C · areaSc (s)−1 . (3.8) Proof. We work on the Riemannian 4-manifold (N, gN ) until the end of the proof. Let Aˆc = Aˆc (s) = π −1 (Ac ( 12 s, 2s)) and Sˆc = Sˆc (s) = π −1 (Sc (s)), where π : N → M is projection on the ﬁrst factor, with S 1 ﬁbers. From the coarea formula, we have |∇ν|2 = |∇ν|dσv dv, (3.9) ˆc A

v

ˆc Lv ∩A

where Lv is the v-level set of ν in N and the outer integral in (3.9) is over the range of values in R of ν in Aˆc . Now as remarked following Lemma 3.4, up to a set Zˆv of measure 0 in Lv ∩ Aˆc , all points in the set (Lv ∩ Aˆc ) \ Zˆv may be joined by ˆ independent of ﬂow lines of ∇ν to points in a ﬁxed bounded hypersurface Tˆ in K, ˆ ˆ v, s; here K is the compact set from Lemma 3.4, and T = ∂ K for instance. Hence, by the divergence theorem applied to the harmonic function ν on N , |∇ν| ≤ |∇ν| ≤ c1 , (3.10) ˆc Lv ∩A

Tˆ

for some c1 < ∞, independent of s and Aˆc . Now |∇ν| is subharmonic on (N, gN ) by Lemma 3.5, and diamiN Aˆc ≤ c · s, by (the proof of) Lemma 3.1. A standard sub-mean value inequality for manifolds of non-negative Ricci curvature, c.f. [SY,Thm.II.6.2], then gives c2 supSˆc (s) |∇ν|2 ≤ |∇ν|2 . (3.11) volAˆc Aˆc

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supSˆc (s) |∇ν|2 ≤ c3 · oscAˆc ν · (volAˆc )−1 .

(3.12)

Hence the estimates (3.9)-(3.11) imply

To estimate the right hand side of (3.12), again by (the proof of) Lemma 3.1, osc ν ≤ c4 · sup|∇ν|s on Aˆc . Further, we claim that supAˆc (s) |∇ν| ≤ csupSˆc (s) |∇ν|, for some constant c independent of s and Aˆc . To see this, by scale-invariance, it suﬃces to prove that supAˆc (1) |∇ν| ≤ csupSˆc (1) |∇ν| w.r.t. the rescaled metrics gs = s−2 g. By the curvature and diameter bounds (3.3) and (3.4) and Lemmas 1.3 and 1.4, the metrics (Aˆc (1), gs ) form a compact family of metrics in the C ∞ topology, unwrapping in the case of collapse. Thus, one has uniform control on the metrics gs on Aˆc (1). Similarly, when normalized if necessary by additive and multiplicative constants so that supSˆc (1) ν = supSˆc (1) |∇ν| = 1, the positive harmonic functions ν on (Aˆc (1), gs ) also form a compact family of functions in the C ∞ topology, i.e. a normal family. This follows by the Harnack estimate (3.4) and the Harnack principle (elliptic regularity) for harmonic functions, c.f. [GT, Thm 2.11, Ch. 8]. This compactness of the metrics and functions from elliptic theory proves the claim above. Similarly, the metric compactness above also implies there is a constant c < ∞ such that c−1 · areaSc (1) ≤ volAc (1) ≤ c · areaSc (1), w.r.t. the metrics gs . (This estimate can also be derived directly from the BishopGromov volume comparison theorem). Rescaling back to the metric g then gives c−1 5 · s · areaSc (s) ≤ volAc ≤ c5 · s · areaSc (s), for some constant c5 < ∞. Note that by deﬁnition, areaSˆc (s) = udA,

(3.13)

(3.14)

Sc (s)

and the same for volAˆc . Hence by (3.4), the estimate (3.13) holds also for Sˆc (s) and Aˆc in place of Sc (s) and Ac . Thus, by combining these estimates above, (3.12) gives supSˆc (s) |∇ν| ≤ c6 · areaSˆc (s)−1 . Using (3.14) and (3.4) again, this estimate implies (3.8).

We are now in a position to begin the proof of Theorem 0.3 itself. Observe that the previous results in §3 have not used the assumption (3.2), nor the assumption (ii) in Theorem 0.3 that u does not approach 0 everywhere at inﬁnity in E. Only

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the assumption that ∂M is pseudo-compact has been used. Hence, at this stage, we do not even know that E has ﬁnite topological type. The main point initially is to prove that the estimates (3.6)-(3.7) above do hold on E under these assumptions. Recall that we have SE (s) = ∪Sc (s), for Sc (s) ⊂ E. Each geodesic ray σ(s) in E, i.e. an integral curve of ∇t, with σ(s) ∈ SE (s), determines a component Sσ (s) = Sc (s) s.t. σ(s) ∈ Sc (s); the union of such components sweep out a part Eσ of the end E. Of course E is the union of Eσ among all (non-homotopic) rays σ. From Lemma 3.1 and from the obvious areaSE (s) = areaSc (s), we have ∞ ∞ areaSE (s)−1 ds < ∞ ⇔ areaSσ (s)−1 ds < ∞, for some geodesic ray σ ⊂ E. (Here the integrals start at some ﬁxed value s ≥ so > 0). Hence, under the assumption (3.2), we have ∞ areaSσ (s)−1 ds ≤ K < ∞, (3.15) for some ray σ ⊂ E and constant K. By integrating along the curve σ, (3.15), Lemma 3.6 and the Harnack estimate (3.4) imply that u is uniformly bounded in Eσ . In fact, we claim that u∞ = limt(x)→∞ u(x) < ∞

(3.16)

exists, where the limit is taken in Eσ . To see this, let γ(s) be any ”quasi-geodesic” in Eσ , i.e. γ is a smooth curve with γ(s) ∈ Sσ (s) and |dγ/ds| ≤ C1 , for some ∞ C1 < ∞. By (3.8) and (3.15), we then have |du(γ(s))|ds ≤ C · C1 · K < ∞, and so u(γ(s1 )) − u(γ(s2 )) → 0 whenever s1 , s2 → ∞. Hence the limit u∞ (γ) is well-deﬁned. The diameter estimate (3.1) implies that all points in Eσ lie on quasi-geodesics, (with a ﬁxed C1 ), in Eσ , starting on Sσ (so ). Further, the limit u∞ (γ) is clearly independent of γ, since for instance (3.8) and (3.15) imply that oscAσ ( 12 sj ,2sj ) u → 0, on some sequence sj → ∞. Hence (3.16) follows. Next, since E is an end, there exists some sequence tj → ∞ such that the geodesic spheres SE (tj ) are connected, and hence Sσ (tj ) = SE (tj ). By (3.16), u|SE (tj ) → u∞ as tj → ∞. The maximum principle applied to the harmonic function u thus implies that u|AE (tj ,tk ) → u∞ , whenever tj , tk → ∞. Thus, we see that (3.16) holds where the limit is taken in the full end E, and not just in Eσ . Now we use the assumption (ii) of Theorem 0.3, which says that u(xj ) ≥ uo > 0, for some constant uo > 0 and some divergent sequence xj ∈ E. It follows from this and the existence of the limit (3.16) in E that u∞ > 0.

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Hence, we may, and will, renormalize the potential function u of the static vacuum solution (M, g, u) so that, on E, limt(x)→∞ u(x) = 1.

(3.17)

The estimate (3.17) essentially immediately implies the scale-invariant estimates (3.18) supSE (s) |r| << s−2 , supSE (s) |∇u| << s−1 , as s → ∞, strengthening the bounds (3.3). For as discussed following (3.3), (3.18) is equivalent to the statement that all tangent cones at inﬁnity of E are ﬂat, (c.f. also the proof of Lemma 3.2). But, as noted above, all tangent cones at inﬁnity are static vacuum solutions, and (3.17) implies that the limit potential u∞ satisﬁes u∞ ≡ 1. Hence, the static vacuum equations (0.1) of course imply the limit metrics are ﬂat. Lemma 3.2 now determines the topology of the end E, as one of two (standard) alternatives, according to non-collapse or collapse behavior at inﬁnity. Before proceeding to the analysis of these cases, note that (3.18) implies that the metrics g and g˜ = u2 · g from (1.4) are quasi-isometric on E, and almost isometric near inﬁnity. Since g˜ has non-negative Ricci curvature on E, standard volume comparison theory implies that the area and volume ratios ˜ ˜ areag˜ (S(s)) volg˜ (B(s)) , 2 3 s s

(3.19)

are monotone non-increasing in s. Hence, their limits at s = ∞ exist, and by the equality of g and g˜ at inﬁnity, the limits at s = ∞ of the ratios in (3.19) w.r.t. the g metric and g-geodesic spheres and balls also exist, and equal the g˜ limits. To proceed further, we now separate the discussion into non-collapse and collapse cases. Case A (Non-Collapse). Suppose that E is non-collapsing at inﬁnity, i.e. by the remarks above, limsups→∞

v(s) v(s) = lims→∞ 3 > 0, 3 s s

(3.20)

where v(s) denotes volg (BE (s)). This implies, via (3.13), that areaS(s)−1 ≤ c·s−2 , and hence, by Lemma 3.6, supS(s) |∇u| ≤ cs−2 .

(3.21)

The volume condition (3.20) implies that all tangent cones at inﬁnity A∞ of (E, g) exist, without passing to covering spaces, and, as discussed above, are ﬂat solutions of the static vacuum equations with potential u∞ ≡ 1. As discussed in the proof of Lemma 3.2, the tangent cone at inﬁnity is unique, (up to isometry), and given by R3 \ {0}, and further E \ K is diﬀeomorphic to R3 \ B, for some compact

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set K ⊂ E. The blow-down metrics gs = s−2 · g on all annuli AE ( 12 s, 2s) converge smoothly to the ﬂat metric on A(1, 2) ⊂ R3 , uniformly as s → ∞, and hence there are local (harmonic) coordinates on AE ( 12 s, 2s) in which g has the expansion gij = δij + γij , where |γij (x)| → 0 uniformly as s → ∞. Again as discussed in the proof of Lemma 3.2, these local coordinates, e.g. on {AE (2i−1 , 2i+1 )}, i > 0, may be assembled into a global chart, mapping E \ K onto R3 \ B, c.f. [AC] for further details if desired. With respect to such a chart, the metric gij has the form gij = δij + γij ,

(3.22)

on all of E \ K, with |γij | → 0 uniformly at inﬁnity in E. In other words, g is C o asymptotic to the ﬂat metric at inﬁnity. To prove that the metric on E is (strongly) asymptotically ﬂat, as deﬁned preceding Theorem 0.1, consider again the metric g˜ = u2 ·g. From (1.4) and (3.21), the curvature of g˜ decays as |˜ r| ≤ Ct−4 , as t → ∞. (Note also that t and t˜ are approximately equal for t large). It follows that the expansion (3.22) may be improved, for g˜, to g˜ij = δij + O(t−2 ),

(3.23)

in a suitable (harmonic) coordinate chart. We refer to [BKN] or [BM] for instance for further details here. Brieﬂy, elliptic regularity theory applied to the equations (1.4)-(1.5), together with the curvature decay above, implies that the 2nd derivatives of the metric g˜ in the coordinate chart decay as t−4 , so that the metric g˜ decays to the ﬂat metric at a rate of t−2 . Hence gij = u−2 g˜ij = (1 + 2υ)δij + O(t−2 ),

(3.24)

where υ = 1 − u. Here we are using that fact that since |∇u| = O(t−2 ), u has an expansion of the form u = 1 +O(t−1 ). Further, since log u is harmonic w.r.t. g˜, the decay (3.21) and (3.23) implies that ∆f log u = O(t−4 ) for t large, where ∆f is −2 the ﬂat Laplacian on R3 . This means that u has an expansion u = 1 − m ), t + O(t where m is the mass of E deﬁned in (0.7). In particular, these estimates show that the end E is asymptotically ﬂat in the sense preceding Theorem 0.1. Note that, to ﬁrst order in t−1 , the function υ = 1 − u corresponds to the Green’s function in R3 , i.e. the fundamental solution of the Laplacian, weighted by the mass m. It is of course possible to have m = 0, as for instance for the dipole-type Weyl solutions in §2(IIC), or also m < 0. Further, since u has been normalized so that u → 1 at inﬁnity in E, the expression (0.7) for the mass is equivalent to the usual deﬁnition 1 mE = < ∇u, ∇t > dA, (3.25) 4π SE (s)

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where s is suﬃciently large so that SE (s) ∩ ∂E = ∅. This is because the expression (3.25) is independent of s, since u is harmonic, and the fact that it is asymptotic to the expression (0.7) as s → ∞. This completes the analysis of Case A. Case B (Collapse). Under the standing assumption (3.15), suppose that the end E is collapsing at inﬁnity, i.e. v(s) v(s) limsups→∞ 3 = lims→∞ 3 = 0. (3.26) s s We will prove that this situation is impossible. The results preceding Case A remain valid, so that (3.17)-(3.18) hold, all tangent cones at inﬁnity A¯∞ of E are ﬂat products of the form V × S 1 or R+ × S 1 × S 1 , where V is a ﬂat 2-dimensional cone. Further E \K is diﬀeomorphic to (R2 \B)×S 1 , for some compact set K ⊂ E. By (3.15) and Lemma 3.6, we have ∞ supS(s) |∇u|(s)ds ≤ K1 < ∞. (3.27) The main point is now to show that (E, g) itself, (and not just its tangent cones), is asymptotic to a ﬂat quotient of R3 , and hence has at most quadratic volume growth of geodesic balls or linear area growth of geodesic spheres. This is done in the following result, which is a strengthening of Lemma 3.2. Lemma 3.7. Under the assumptions (3.26) and (3.27) above, there is a compact set K ⊂ E such that (E \ K, g) is quasi-isometric to a ﬂat product (R2 \ B) × S 1 or R+ × S 1 × S 1 . Proof. As in Case A, it is useful to work with the metric g˜ = u2 g; again, this makes no signiﬁcant diﬀerence, since (3.17) holds. All the metric quantities below are thus in the g˜ metric. For notational simplicity however, we drop the tilde from the notation. Let t∞ (x) = lims→∞ (dist(x, SE (s))−s). As in the construction of Busemann functions, the limit here exists, c.f. [Wu] for a discussion of such functions. By construction, t∞ is a Lipschitz distance function, i.e. t∞ realizes everywhere the distance between its level sets. Observe that on R3 , t∞ is just the distance function to {0} ∈ R3 , on V × S 1 , t∞ is the distance function to {0} ∈ V pulled back to V × S 1 , for any cone V with vertex {0}, while on R+ × S 1 × S 1 , t∞ is the distance function on R+ pulled back to the total space. By renormalization, (as with the potential u), t∞ induces a distance function t¯∞ on each tangent cone A¯∞ by deﬁning t¯∞ (x) = lim(t∞ (x)/t∞ (xi )), where t(xi ) → ∞ and xi are the base points converging to the base point x∞ ∈ A¯∞ . Thus t¯∞ is the function above on V × S 1 or R+ × S 1 × S 1 . The map t∞ : (E, g˜) → R+ is distance non-increasing, and preserves distance along the integral curves of t∞ ; thus where smooth, it is a Riemannian submersion.

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We will show that t∞ gives rise to a Lipschitz quasi-isometry by examining the asymptotics of the second fundamental form of its level sets. Thus, let σ(s) be any geodesic ray in E which is an integral curve of ∇t∞ , and let B = Bσ (s) denote the second fundamental form of the level surface t∞−1 (s) at σ(s). The form B is well-deﬁned and smooth along any such ray σ. Recall that B satisﬁes the Riccati equation B + B 2 + RT = 0,

(3.28)

where T is the unit tangent vector along σ. Consider the behavior of s · B(s) as s → ∞. This quantity is scale-invariant, and thus converges smoothly, (in s) on any tangent cone at inﬁnity. subsequences), to the limiting expression s¯ · B∞ (¯ Since the parameters s and t∞ |σ are the same up to additive constants, s¯ = t¯∞ . Similarly, by the deﬁnition of t¯∞ , B∞ is the second fundamental form of the levels t¯∞ . Hence, either s) = (dθ/|dθ|)2 , when the tangent cone is of the form V × S 1 , and θ is (i): s¯ · B∞ (¯ the angle variable about {0} ∈ V, or s) = 0, when the tangent cone is of the form R+ × S 1 × S 1 . (ii): s¯ · B∞ (¯ s) is either of rank 1, with eigenvalue 1, or identically 0. Note Thus s¯ · B∞ (¯ that the expression in case (i) is independent of the cone V , i.e. the cone angle at {0}. As noted preceding Lemma 3.2 and in Remark 3.3(ii), the geometry of ¯ 1 s, 2s) smoothly approximates that of a limit tangent cone A¯∞ , for s large, A( 2 and varies continuously in s. Since the two alternatives (i) and (ii) above for the structure of B∞ are rigid, it follows that all tangent cones A¯∞ are of the same type, i.e. they are all of the form V × S 1 , or all of the form R+ × S 1 × S 1 . The main task now is to show that the deviation of s · B(s) from its limit s) has bounded integral. To do this, we use the Riccati equation, and s¯ · B∞ (¯ estimate the decay of the curvature term RT , using basically standard methods in comparison geometry, c.f. [P, Ch.6.2] for instance. Thus, from (1.4), the sectional curvature K of (M, g˜) satisﬁes KXZ = |∇ν|2 ≥ 0, KXY = −|∇ν|2 ≤ 0, where Z = ∇u/|∇u|, X, Y are vectors orthogonal to Z, and ν = log u. Hence |RT | ≤ |∇ν|2 . Substituting this in (3.28) gives |B + B 2 | ≤ |∇ν|2 .

(3.29)

Let λ be any eigenvalue of B, with unit eigenvector e; (note that B is symmetric). Observe then that s · λ(s) converges either to 1 or to 0, as s → ∞. The estimate (3.29) when applied to (e, e) gives ±(sλ + sλ2 ) ≤ s|∇ν|2 .

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Integrate this by parts along any ﬁnite interval I to obtain |sλ|∂I + (sλ2 − λ)ds| ≤ s|∇ν|2 ds. I

I

If sλ → 0 as s → ∞, choose I to be any interval on which sλ = 0 at ∂I and sλ = 0 on I, so that sλ has a deﬁnite sign on I. If there are no such boundary points, choose I to be an inﬁnite half-line. Similarly, if sλ → 1, choose I to be intervals such that sλ = 1 at ∂I with sλ − 1 = 0 on I. Then summing up the estimate above over all such intervals gives 2 |sλ − λ|ds ≤ s|∇ν|2 ds + Co , σ

σ

for some constant Co < ∞. Now the estimate (3.18), together with (3.27) gives s|∇ν|2 ds ≤ C1 , for some constant C1 < ∞. Suppose ﬁrst s · λ(s) → 1 as s → ∞. We then obtain 1 |sλ(λ − )|ds ≤ C2 < ∞, s and hence

1 |λ − |ds ≤ C3 < ∞. s

Similarly, if s · λ(s) → 0 as s → ∞, one obtains |λ|ds ≤ C3 < ∞.

(3.30)

(3.31)

Now the second fundamental form B gives the logarithmic derivative of the norm of Jacobi ﬁelds formed by the family of t∞ -rays in E starting at some level t−1 ∞ (so ). Thus, if J is any Jacobi ﬁeld formed from the t∞ -congruence, and v = J/|J|, we have along any t∞ -ray, B(v, v) =

d (log |J|(s)). ds

Suppose ﬁrst that the end E has tangent cones at inﬁnity of the form R+ ×S 1 ×S 1 . Then (3.31) implies the uniform bound C4−1 ≤ |J|(s) ≤ C4 ,

(3.32)

with C4 = eC3 . This means that the geometry of the level surfaces of t∞ is uniformly bounded as s → ∞, i.e. the diameter and area of the level surfaces is

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uniformly bounded away from 0 and ∞. It is then clear that there is a Lipschitz quasi-isometry of (E \ K, g˜) to R+ × S 1 × S 1 induced by t∞ . Since g˜ and g are also quasi-isometric, by (3.17), the lemma follows in this case. If E has tangent cones at inﬁnity of the form V × S 1 , then there is a basis of Jacobi ﬁelds whose elements satisfy either (3.32), or, from (3.30), C4−1 s ≤ |J|(s) ≤ C4 s.

(3.33)

As before, this implies that t∞ gives rise to a quasi-isometry of (E \ K, g) to (R2 \ B) × S 1 . Of course Lemma 3.7, in both cases, immediately implies that areaSE (s) ≤ c · s,

(3.34)

for some constant c < ∞. However, (3.34) violates the standing assumption (3.15), (or (3.2)). It follows that no end (E, g) can satisfy the assumptions of Case B. Together with Case A, this completes the proof of the second statement in Theorem 0.3. We now turn to the proof of the last statement in Theorem 0.3. We will assume that the end E is small, i.e. 1 ds = ∞, (3.35) areaS E (s) E and derive a contradiction from the assumptions supu < ∞ and mE = 0. The proof is based on a result of Varopoulos [V] which states that ends of Riemannian manifolds satisfying (3.35) are parabolic, i.e. admit no non-constant positive superharmonic functions v which tend uniformly toward their inﬁmum at inﬁnity. (Actually, the result in [V] is a condition on the volume growth of geodesic balls, but this is equivalent to the bound (3.35) under the estimate (3.13)). We will prove that the potential u is such a non-constant function, giving the required contradiction. Thus, suppose ﬁrst that supE u < ∞, (3.36) Arguing as in (3.9), but now on E ⊂ M in place of N , we have |∇u|2 = |∇u|dAv dv, E

and as in (3.10),

v

Lv ∩E

Thus, these estimates imply that

(3.37)

Lv ∩E

|∇u| ≤

|∇u| ≤ C. T

|∇u|2 < ∞, E

(3.38)

1036

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M.T. Anderson

Ann. Henri Poincar´ e

|∇u|2 → 0 as s → ∞.

(3.39)

AE (s,∞)

On the other hand, again referring to the proof of Lemma 3.6, since almost all (in terms of measure) points in Lv ∩ E may be joined by ﬂow lines of ∇u to a ﬁxed bounded surface T in K, by the divergence theorem there is a subsurface T ⊂ T such that |∇u| ≥ |∇u| ≥ c, (3.40) Lv ∩E

T

for some constant c > 0. It follows from (3.37)-(3.40) that oscAE (s,∞) u → 0 as s → ∞.

(3.41)

By assumption (ii) in Theorem 0.3, we may thus assume w.l.o.g. that limt→∞ u = 1,

(3.42)

in E. Thus, as noted in (3.18), |r| ≤ ε(t)/t2 , where ε(t) → 0 as t → ∞, and so Lemma 3.2 holds on E. Now choose a smooth approximation S to a large geodesic sphere SE (s) with SE (s) ∩ ∂E = ∅, so that S separates E into two components, one being the outside containing the end E. We claim that if, in addition to (3.36), mE = 0,

(3.43)

then there is a set of ﬂow lines γ of ∇u or −∇u, starting on S, of positive measure on S, and pointing out of S, which never intersects S again at later times. Hence such γ diverge to inﬁnity in E, since up to a set of measure 0, γ does not terminate in a critical point of u. (Compare with the earlier discussion regarding divergent ﬂow lines and Weyl dipole solutions concerning Lemma 3.4). To see this, suppose instead that all ﬂow lines say of ∇u which initially point out of S eventually intersect S again, with the exception of those terminating in critical points. Consider the measure dµ =< ∇u, ν > dA,

(3.44)

on S, where dA is the Lebesgue measure and ν is the unit outward normal on S; dµ is absolutely continuous w.r.t. dA. Since u is harmonic, the divergence theorem implies that the gradient ﬂow of u preserves the measure dµ in the following sense. Let D be a domain in S and let Ω be the domain in E formed by a collection of ﬂow lines outside S, whose endpoints form a smooth surface D . If ν denotes the unit outward normal to Ω at D , then the ﬂow from D to D carries the measure dµ to the measure dµ =< ∇u, ν > dA , where dA is Lebesgue measure on D . In particular, the ﬂow preserves the masses of the measures.

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Thus, under the assumptions above, the gradient ﬂow (with varying ﬂowtimes), induces a homeomorphism of S \ Z into itself, where Z is a set of Lebesgue measure 0, corresponding to ﬂow lines terminating in critical points. However, this homeomorphism inverts the direction of ∇u w.r.t. the ﬁxed normal ν, and hence maps domains D+ on which the measure dµ is positive onto domains D− on which dµ is negative, in such a way that that mµ (D+ ) = |mµ (D− )|.

(3.45)

This of course implies that the total mass of dµ on S is 0. However using (3.44) and the remarks concerning (3.25), the mass mE of E equals the total mass of dµ. This contradiction proves the claim. We may now complete the proof as follows. Assuming E is an end satisfying (3.36) and (3.43), there exists an open set O of ﬂow lines γ = γ(τ ) of either ∇u or −∇u which start on a set of positive measure on S and diverge to inﬁnity in E. Consider the former case, which corresponds to mE > 0. A generic ﬂow line of ∇u in E tends to the maximal value of u in E and hence a generic ﬂow line in O also tends this maximal value. By (3.42), it then follows that supE\K u = 1, for some compact set K ⊂ E. The function v = −u is thus a bounded (non-constant) harmonic function on E, which tends uniformly to its inﬁmum at inﬁnity. Hence E cannot be parabolic. This contradiction shows that (3.35) cannot hold for E, and thus, by the proofs in Cases A and B above, the end E is asymptotically ﬂat. The proof in case mE < 0 is the same. This completes the proof of Theorem 0.3. Remark 3.8 (i). There are (non-ﬂat) static vacuum solutions with a small end, namely the Kasner metrics (2.33), with a > 0. These solutions have volB(s) ∼ s2−δ , volS(s) ∼ s1−δ , where δ = (a + a−1 − 1)−1 ∈ (0, 1), and hence the end is small. Note that u ∼ sδ is unbounded. There are other Weyl solutions which are complete away from Σ with ∂M pseudo-compact and with one small end, (take for instance the potential given by the Green’s function on R2 ×S 1 , see (ii) below), but all known examples with small ends are either asymptotic to the Kasner metric at inﬁnity or have faster than quadratic curvature decay, i.e. satisfy (3.18). It is an open problem to understand in more detail the structure of small ends of static vacuum solutions. It follows from the results above in §3 that all tangent cones at inﬁnity of E are collapsing, and hence they are all Weyl solutions. But the metric uniqueness of tangent cones at inﬁnity is unknown, as is the question of whether small ends have ﬁnite topological type. (ii). We construct an example which illustrates the sharpness of the last statement of Theorem 0.3. Let G1 and G2 be the Green’s functions for the Laplacian on the ﬂat product R2 × S 1 , with poles at (0, pi ), for p1 ,p2 distinct points in S 1 . Here we consider S 1 as the z-axis in R3 quotiented out by an isometric Z-action. As in (2.34), Gi (x) = G(x, pi ), viewed as a function on the universal cover R3 ,

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may be written as Gi = limn→∞

n 1 − cn , r j=−n j

Ann. Henri Poincar´ e

(3.46)

where rj is the Euclidean distance to the collection of lifts of pi in R3 and cn is a suitable normalizing constant with cn → ∞ as n → ∞, chosen so that G(x, pi ) is ﬁnite. Thus Gi is an axisymmetric and z-periodic harmonic function on (R2 )+ ×S 1 , where the S 1 now means rotations about the z-axis in R3 . Hence the potential ν = G1 − G2 generates a Weyl solution as in (2.12), with u = eν and which has an isometric Z-action along the z-axis. Let M denote the Z quotient of this solution. Then the metric boundary of M is pseudo-compact. Since Gi ∼ log r as r → ∞, u → 1, at inﬁnity in M . The end E = R+ × T 2 is small and has mass 0 in the sense of (0.7). These solutions of course resemble the dipole-type solutions discussed in §2(IIC), but with a collapsing end. Remark 3.9 (i). Although we will not detail it here, an examination of the proof shows that Theorem 0.3 holds for non-vacuum static solutions of the Einstein ﬁeld equations (1.2), provided suitable decay conditions are imposed on the stressenergy tensor T . This is the case for example, if T is any tensor with compact support, or more generally if T satisﬁes an estimate of the form |T |(x) ≤ c · t−3 (x), for some c > 0 and all x with t(x) ≥ so , together with u1 ∆u ∈ L1 (M \ U ). This latter condition is needed to obtain the bound (3.10). By (1.2), note that u1 ∆u = 12 trT. The starting estimate (3.3) may be obtained by applying (a suitable version of) [An1,Thm.3.3]. (ii). Also, Theorem 0.3 can be given a ﬁnite or quantitative formulation, i.e. one can relax the assumption of completeness, in basically the same way as the local estimates (3.3) follow from the non-existence of global static vacuum solutions with ∂M = ∅, c.f. [An1,App.]. Thus, if (3.15) holds and (M, g) is ’suﬃciently large‘, depending on K, then suﬃciently far out in (M, g), the metric is close to a ﬂat metric. We leave a precise formulation and proof, (based on Theorem 0.3), to the reader. Remark 3.10 As noted in §0, it would be of interest to prove that (M, g, u) has a unique end. Under the hypotheses of Theorem 0.3, we conjecture this is the case at least when M is complete away from Σ. If M is in addition smooth up to Σ, this has been proved by Galloway [G]. Following essentially the same arguments as in [G], it is not diﬃcult to show that if M is complete away from Σ and if the Riemannian 4-manifold N admits a compact smoothing of a neighborhood of Σ having non-negative Ricci curvature, then M has a unique end.

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Remark 3.11 We point out that Theorem 0.3 and Theorem 0.1 are false in higher dimensions, due to the existence of Einstein metrics on compact manifolds, which are not of constant curvature, in dimensions ≥ 3. (The equations (0.1) on any n-dimensional manifold M n generate Ricci-ﬂat manifolds on N n+1 ). Thus, let (Σ, g) be any compact (n − 2) dimensional Einstein manifold, with Ricg = (n − 3) · g and deﬁne the warped product metric g¯ on R2 × Σ by g¯ = dt2 +

4(f (t))2 2 dφ + f 2 (t) · g, (n − 2)2

where f is the unique function on [0, ∞) such that f (0) = 1, f > 0 and (f )2 = 1 −f 1−n . A simple computation shows that (R2 × Σ, g¯) is complete and Ricci-ﬂat, c.f. [Bes, p.271] and the space-like hypersurface R+ × Σ, with metric dt2 + f 2 (t) · g, is a solution to the static vacuum equations, with potential u = f , (up to a multiplicative constant). Thus the horizon is Σ and the solution is smooth up to Σ and complete away from Σ. This metric is asymptotically conical, i.e. asymptotic to the complete (Euclidean) cone on (Σ, g), but is asymptotically ﬂat only in the case that (Σ, g) = S n−2 (1), corresponding to the n-dimensional Schwarzschild metric.

References [A1]

M. Anderson, Scalar curvature, metric degenerations and the static vacuum Einstein equations on 3-manifolds I, Geom. & Funct. Analysis, vol. 9:5, (1999), 855-967; Los Alamos archive, math.DG/9912162.

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M. Anderson, On stationary solutions to the vacuum Einstein equations, to appear in Annales Henri Poincare; Los Alamos archive, math.grqc/0001091.

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M. Anderson, Extrema of curvature functionals on the space of metrics on 3-manifolds, Calc. Var. & P.D.E., vol. 5, (1997), 199-269.

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M. Anderson and J. Cheeger, Diﬀeomorphism ﬁniteness for manifolds with Ricci curvature and Ln/2 -norm of curvature bounded, Geom. & Funct. Analysis, vol.1:3, (1991), 231-252.

[BW] R. Bach and H. Weyl, Neue L¨ osungen der Einsteinschen Gravitationsgleichungen, Math. Zeitschrift, vol. 13, (1922), 134-145. [BKN] S. Bando, A. Kasue and H. Nakajima, On a construction of coordinates at inﬁnity on manifolds with fast curvature decay and maximal volume growth, Inventiones Math., vol. 97, (1989), 313-349. [Bes]

A. Besse, Einstein Manifolds, Springer Verlag, New York, (1987).

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H. Bondi, Negative mass in general relativity, Rev. Mod. Phys., vol. 29:3, (1957), 423-428.

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W.B. Bonnor, Physical interpretation of vacuum solutions of Einstein’s equations I, Gen. Rel. and Grav., vol. 24:5, (1992), 551-574.

[BM] G. Bunting and A. Massoud-ul-Alam, Non-existence of multiple black holes in asymptotically Euclidean static vacuum space-times, Gen. Rel. and Grav., vol 19, (1987), 147-154. [CG]

J. Cheeger and M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded I, II, Jour. Diﬀ. Geom., vol. 23, (1986), 309-346, and vol. 32, (1990), 269-298.

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J. Ehlers and W. Kundt, Exact solutions of the gravitational ﬁeld equations, in Gravitation: An Introduction to Current Research, Ed. L. Witten, Wiley, N.Y. (1962), 49-101.

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A. Einstein, The Meaning of Relativity, 5th Edition, Princeton U. Press, Princeton, (1956).

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G.F.R. Ellis, Exact and inexact solutions of the Einstein ﬁeld equations, in The Renaissance of General Relativity and Cosmology, Ed. G.Ellis et al, Cambridge U. Press, Cambridge, (1993), 20-39.

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G.F.R. Ellis and B.G. Schmidt, Singular space-times, Gen. Rel. and Grav., vol.8:11, (1977), 915-953.

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G. Galloway, On the topology of black holes, Comm. Math. Phys., vol 151, (1993), 153-166.

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R. Geroch and J.B. Hartle, Distorted black holes, Jour. Math. Phys, vol. 23:4, (1982), 680-692.

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D. Gilbarg and N. Trudinger, Elliptic Partial Diﬀerential Equations of Second Order, 2nd edition, Springer-Verlag, New York, (1983).

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W. Hayman, Subharmonic Functions, Academic Press, London, (1976).

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W. Israel, Event horizons in static vacuum space-times, Phys. Review, vol. 164:5, (1967), 1776-1779.

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W. Israel, Line sources in general relativity, Phys. Review D, vol. 15:4, (1977), 935-941.

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W. Israel and K.A. Khan, Collinear particles and Bondi dipoles in General Relativity, Il Nuovo Cimento, vol. 33:2, (1964), 331-344.

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[KN]

D. Korotkin and H. Nicolai, A periodic analogue of the Schwarzschild solution, Los Alamos Archive, gr-qc/9403029, March 1994.

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D. Kramers, H.Stephani, M. MacCallum, E. Herlt, Exact Solutions of Einstein’s Field Equations, Cambridge U. Press, Cambridge, (1980).

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R. S. Kulkarni, Curvature and metric, Annals of Math., vol. 91, (1970), 311-331.

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A. Lichnerowicz, Theories Relativistes de la Gravitation et de L’Electromagnetisme, Masson, Paris, (1955).

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A. Lichnerowicz, Problemes Globaux en Mechanique Relativiste, Hermann Co., Paris, (1939).

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X.-d. Liu, Ball covering on manifolds with non-negative Ricci curvature near inﬁnity, Proc. Symp. Pure Math., Amer. Math. Soc., vol. 54:3, (1993), 459-464.

[MTW] C.W.Misner, K.S. Thorne and J.A. Wheeler, Gravitation, W.H. Freeman, New York, (1973). [MF] P.M. Morse and H. Feshbach, Methods of Theoretical Physics, McGrawHill, New York, (1953). [P]

P.C. Peters, Toroidal black holes? Jour. Math. Phys, vol. 20:7, (1979), 14811485.

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W. Rindler, Essential Relativity, 2nd Edition, Springer Verlag, New York, (1977).

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D. Robinson, A simple proof of the generalization of Israel’s theorem, Gen. Rel. and Grav., vol 8:8, (1977), 695-698.

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R. Schoen and S.T. Yau, Lectures on Diﬀerential Geometry, International Press, Cambridge, MA (1994).

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S.M. Scott and P. Szekeres, The Curzon singularity, I: Spatial sections, Gen. Rel. and Grav., vol. 18:6, (1986), 557-570.

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K.S. Thorne, A toroidal solution of the vacuum Einstein ﬁeld equations, Jour. Math. Phys, vol 16:9, (1975), 1860-1865.

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N. Varopoulos, Potential theory and diﬀusion on Riemannian manifolds, Conf. on Harmonic Analysis in Honor of A. Zygmund, Wadsworth, Belmont, Ca. (1983), 821-837.

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Michael T. Anderson Department of Mathematics S.U.N.Y. at Stony Brook Stony Brook, N.Y. 11794-3651 e-mail : [email protected] Partially supported by NSF Grants DMS 9505744 and 9802722 Communicated by Sergiu Klainerman submitted 16/11/99, accepted 21/09/00

To access this journal online: http://www.birkhauser.ch

Ann. Henri Poincar´ e 1 (2000) 1043 – 1095 c Birkh¨ auser Verlag, Basel, 2000 1424-0637/00/0601043-53 $ 1.50+0.20/0

Annales Henri Poincar´ e

Creation of Fermions at the Charged Black-Hole Horizon Alain Bachelot Abstract. We investigate the quantum state of the Dirac ﬁeld at the horizon of a charged black-hole formed by a spherical gravitational collapse. We prove this state satisﬁes a KMS condition with the Hawking temperature and the chemical potential associated with the mass and the charge of the black-hole. Moreover, the fermions with charge of same sign to that of the black-hole are emitted more readily than those of opposite charge. It is a spontaneous loss of charge of the black-hole due to the quantum vacuum polarization.

I Introduction The purpose of this paper is to investigate the quantum state of charged spinor ﬁelds at the horizon of a black-hole created by the collapse of a spherical charged star. Our main result expresses that the ground state, that is given by the Boulware vacuum in the past, is of Unruh type at the future horizon. It is the famous Hawking eﬀect : a static observer at inﬁnity, interprets this state as a thermal radiation of particles and antiparticles outgoing from the black-hole. Moreover the black-hole emits more readily fermions whose the charge is of same sign as its own charge. A similar phenomenon for the scalar ﬁelds has been discussed by G. W. Gibbons in [21]. The space time outside the collapsing star is given by a four-dimensional, globally hyperbolic manifold (M,g), of Kruskal-Reissner-Nordstrøm type, endowed with a 1-form Aµ dxµ (the electromagnetic potential due to the star). The boundary ∂M of M has two pieces : a time-like part S (the moving surface of the star), a characteristic part H + (the future black-hole horizon). We ﬁrst solve the mixed hyperbolic problem for the Dirac system for the particles of mass m and charge q, iγ µ (∇µ + iqAµ )Ψ − mΨ = 0, in M,

(I.1)

with initial data on a space-like hypersurface Σ0 . To take the interaction between the matter of the star and the ﬁeld, into account, we add some boundary condition, nµ γ µ Ψ = BΨ, on S,

(I.2)

where B belongs to a large class of operators on S, including in particular the MIT condition [13] : 5

B = ieiαγ .

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Then we construct the local algebra of observables A(M), and given the Fock vacuum on Σ0 , corresponding to the Boulware vacuum in the past, we prove that the quantum state on H + satisﬁes a KMS condition involving the mass and the charge of the star (Unruh state). Moreover the temperature and the chemical potential are independent of the history of the collapse, and the boundary condition. We also investigate the role of the cosmological constant in the case of the charged black-holes in an expanding universe (De Sitter-Reissner-Nordstrøm metric). From a mathematical point of view, we adopt the framework of our previous studies on the Hawking eﬀect for the Klein-Gordon ﬁelds [2], [3], [4]. It is convenient to choose a frame for which H + is pushed away to the null inﬁnity (Schwarzschild-like coordinates). Then the surface of the star is an asymptotically characteristic moving boundary, and an asymptotically inﬁnite Doppler eﬀect appears (blue shift). Hence the problem is reduced to a very sharp analysis of the asymptotic behaviour of the Dirac propagator. The key point consists in proving that the approximation by the geometrical optics is valid. This paper is organized as follows : we precise the geometric assumptions in part two; we present the key asymptotic estimate for the classical Dirac equation in the third part; we construct the quantum ﬁeld and we state the main result on the Hawking eﬀect in part 4; this result is discussed in part 5, especially the interpretation in terms of particles, and we study the role of the cosmological constant; part 6 is devoted to the mathematical proofs of the results of parts 3 and 4; taking advantage of the spherical invariance, we reduce the problem to studying a system in one space dimension, which we investigate. After the conclusion, we give in the appendix, by sake of completeness, the main tools of the quantization of the spin ﬁelds on the stationary space-times. We end by giving some bibliographic information. Obviously, the list of references is very incomplete. We cite only the works that we have used. Among a huge literature on the Hawking eﬀect, we can mention [3], [4], [10], [18], [19], [21], [22], [31], [36], [37], [40], [41], [42]. More generally, the quantum ﬁeld theory on curved spacetime is investigated in [1], [8], [16], [17], [20], [24], [30]. The Dirac equation on a black-hole background has been studied in [5], [12], [27], [32], [33], [34], [35] [37], [40].

II Geometrical Framework The space-time outside a static spherical black-hole is a four dimensional, globally hyperbolic manifold (MBH , g) 2 MBH = Rt ×]r0 , r+ [r ×Sθ,ϕ , 0 < r0 < r+ ≤ ∞,

gµν dxµ dxν = F dt2 − F −1 dr2 − r2 (dθ2 + sin2 θdϕ2 ).

(II.1)

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Here F is a C 1 function of r > 0 satisfying F (r0 ) = 0, F (r0 ) > 0, r0 < r < r+ ⇒ F (r) > 0.

(II.2)

r0 is the radius of the black-hole horizon, and we introduce κ0 :=

1 F (r0 ), 2

(II.3)

the surface gravity at the black-hole horizon. As regards r+ , we assume, either that r+ < ∞, F (r+ ) = 0, F (r+ ) < 0,

(II.4)

then MBH is asymptotically of DeSitter type, and r+ is the radius of the cosmological horizon, or r+ = ∞,

lim F (r) = F (∞) > 0,

r→∞

lim F (r) = 0,

r→∞

(II.5)

in which case MBH is asymptotically ﬂat in a weak sense. The fundamental example is the Reissner-Nordstrøm metric, which is the unique spherically symmetric solution of the Einstein-Maxwell equations in the vacuum : F (r) = 1 −

2M Q2 + 2 , 0 ≤| Q |< M. r r

(II.6)

0 < M and Q are respectively the mass and the electric charge of the black-hole, the radius and the surface gravity of which are : M 2 − Q2 r0 = M + M 2 − Q2 , κ0 = (II.7) 2 (and r+ = ∞). M + M 2 − Q2 More generally we could consider spherical charged black-holes in an expanding universe, described by the DeSitter-Reissner-Nordstrøm metric F (r) = 1 −

2M Λ Q2 + 2 − r2 , r r 3

(II.8)

where Λ > 0 is the cosmological constant. It is well-know that the black-hole horizon is a ﬁctitious singularity of the metric, that can be removed by a suitable change of variables. It is convenient to introduce a tortoise radial coordinate x ∈ R satisfying : dx = F −1 , dr by choosing for r ∈]r0 , r+ [ r 1 1 2κ0 x= ln | r − r0 | − dr + x0 . − 2κ0 F (r) r0 r − r0

(II.9)

(II.10)

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We note that the map : x ∈ R → r ∈]r0 , r+ [ is one-to-one with the asymptotic behaviour : | r − r0 |= O(e2κ0 x ), x → −∞.

(II.11)

We can extend x(r) for r ∈]r− , r0 [ by the formula (II.10) for r− such that 0 < r− < r0 , r− < r < r0 ⇒ F (r) < 0. We deﬁne the Kruskal-Szekeres coordinates (T, X, θ, ϕ) : T =

1 r − r0 1 κ0 x ηκ0 t e − ηe−ηκ0 t , X = eκ0 x eηκ0 t + ηe−ηκ0 t , η = e . 2 2 | r − r0 | (II.12)

The Schwarzschild type coordinates (t, r, θ, ϕ) give two local maps with domains 2 2 M = Rt ×]r0 , r+ [r ×Sθ,ϕ and Rt ×]r− , r0 [r ×Sθ,ϕ , but fail to represent the blackhole horizon {r = r0 }. Kruskal-Szekeres coordinates deﬁne an atlas with a single map with domain a neighborhood of MBH = {(T, X, ω); X ≥| T |, ω ∈ S 2 }, and the black-hole horizon appears as the characteristic submanifold {X =| T |} × Sω2 . In fact we are concerned with the realistic black-holes created by the gravitational collapse of a spherical star. So we consider a star, stationary in the past, contracting to a black-hole in the future. In Kruskal-Szekeres coordinates, its boundary is given by {(T, X = Z(T ), ω), T ∈ R, ω ∈ S 2 } where Z is a C 2 function of T ∈ R. Since the star boundary is necessarily time like, we have : −1 < Z (T ) ≤ 0,

(II.13)

and the creation of the black-hole is expressed by : ∃T0 > 0; Z(T0 ) = T0 .

(II.14)

Hence we deal with the manifold M = (T, X, ω) ∈ R × R × S 2 ; T ≤ T0 ⇒ X ≥ Z(T ), T0 ≤ T ⇒ X ≥ T , (II.15) and its boundary ∂M consists of the world lines of the star boundary : S = {(T, X = Z(T )); T ≤ T0 } × S 2 ,

(II.16)

and the future black-hole horizon : H + = {(T, X = T ); T ≥ T0 } × S 2 .

(II.17)

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To study the structure of the quantum state on H + we should have to solve a Characteristic Cauchy problem with data on H + , for a hyperbolic system with T -dependent coeﬃcients, on the manifold M that is singular at S ∩ H + = {(T0 , X = T0 )} × S 2 . Therefore we prefer to adopt the Schwarzschild-like coordinates (t, x, θ, ϕ) for wich the coeﬃcients of the Dirac system are t-independent and the singularity is pushed away to inﬁnity. Instead of solving a Goursat problem, we develop a time dependent Scattering type theory on : M = (t, x, ω); t ∈ R, x ≥ z(t), ω ∈ S 2 , (II.18) where the boundary is described by the function z(t) deﬁned by X = Z(T ) ⇐⇒ x = z(t). We can easily prove (see [2]) that this function satisﬁes :  z ∈ C 2 (R),     ∀t ∈ R, −1 < z(t) ˙ ≤ 0,  z(t) = −t − Ae−2κ0 t + ζ(t), A > 0,    ˙ |= O(e−4κ0 t ), t −→ +∞, | ζ(t) | + | ζ(t)

(II.19)

(II.20)

and for commodity, we choose x0 in (II.10) such that ∀t ≤ 0, z(t) = z(0) < 0.

(II.21)

Here A depends only on κ0 , and we remark that the physics of the collapse is hidden in the rest ζ(t), when the leading term −t − Ae−2κ0 t involves only the surface gravity. This fact leads to the No Hair property of the Hawking eﬀect. With this choice of frame, the star boundary S seems to be asymptotically characteristic : S = (t, x = z(t), ω); t ∈ R, ω ∈ S 2 , (II.22) and a point of the future horizon H + is reached at the inﬁnity of a null ray (t, x = −t + t0 , ω)t∈R as t → +∞, in short : 1 ln(2T1 ), ω = (T1 , X1 = T1 , ω) ∈ H + . (II.23) lim t, x = −t + t→+∞ κ0 Finally the geometrical framework of a generic spherical gravitational collapse is given by (II.1), (II.2), (II.4), (II.5), (II.18), (II.20), (II.21).

III The Classical Dirac Equation We consider the Dirac equation for particles with charge q ∈ R and mass m ≥ 0 outside a collapsing charged spherical star : iγ µ (∇µ + iqAµ )Ψ − mΨ = 0.

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Here the 1-form Aµ dxµ deﬁnes the electromagnetic ﬁeld created by the star. The geometrical framework is given in Part II. Then the Dirac equation has the form in (t, r, θ, ϕ) coordinates (see e.g. [33], [34], [27]) : 0=

1 ∂ ∂ 1 F − 12 0 1 2 iF γ + iqAt + iF γ + + + iqAr ∂t ∂r r 4F 1 ∂ i ∂ i + + iqAθ + γ3 + iqAϕ − m Ψ, + γ2 r ∂θ 2 tan θ r sin θ ∂ϕ

where the Dirac matrices are : 0 σ0 0 0 a , γ =i γ =i −σ 0 0 σa

σa 0

(III.1)

, a = 1, 2, 3,

with the Pauli matrices : 1 0 1 0 0 1 0 −i 0 1 2 3 , σ = , σ = , σ = . σ = 0 1 0 −1 1 0 i 0 We have to add a boundary condition on the star surface. We write this condition as nµ γ µ Ψ = BΨ

(III.2)

where nµ the unit outgoing normal and B is some operator on the surface of the star. It is natural to assume B is local in time, rotationally invariant and such that the L2 -norm of the spinor is conserved. Such a boundary condition, which is local in space-time, is the generalized MIT boundary condition [13], [26] (see also [5]) : 5

BM IT Ψ = ieiαγ Ψ,

(III.3)

where α ∈ R is the Chiral Angle and γ 5 := −iγ 0 γ 1 γ 2 γ 3 =

σ0 0

0 −σ 0

.

(III.4)

When the spinor ﬁeld is massless (m = 0), the system is chiral invariant : we can choose any real α. When m is non zero, (III.3) deﬁnes a family of non equivalent boundary conditions. In this case, and if the space-time is asymptotically ﬂat, we must restrict the range of the chiral angle : m = 0, r+ = ∞ =⇒ α = (2k + 1)π, k ∈ Z.

(III.5)

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We are interested by proving that the Hawking eﬀect does not depend of the interaction between the ﬁeld and the matter of the star. Hence we consider a very large class of boundary conditions given by the family of zero order pseudodiﬀerential operators 5 ieiα,n γ Π,n , (III.6) B= ,n

where the sequence (α,n ) satisﬁes (III.5), and Π,n is the orthogonal projector on the (*, n)-space of the spinoidal spherical harmonics expansion (VI.4). We assume that the electromagnetic potential satisﬁes At = A(r) ∈ C 1 ([r0 , r+ ]), (III.7) Ar = Aθ = Aϕ = 0. These hypotheses are fullﬁlled in the important case of the Reissner-Nordstrøm Black-Hole since (see e.g. [11]) : A=

Q . r

(III.8)

In fact the Dirac equation is obviously well deﬁned on the whole space-time. In Kruskal coordinates (T, X, ω) we introduce the spinor ﬁeld 1

ΦK (T, X, ω) = rF 4 (r)eitqA(r0 ) M Ψ(t, r, ω), with



1 0  0 F − 12 M :=   0 0 0 0

0 0 F−2 0 1

(III.9)

 0 0  , 0  1

and the Dirac system becomes : 0=

∂

−1 ∂ q + γ0γ1 + i (A − A(r0 )) X1 + T γ 0 γ 1 ∂T ∂X κ0

−1 1 1 1 1 + (X + T )−1 F F 2 1 + γ 0 γ 1 + F 2 X1 + T γ 0 γ 1 4κ0 κ0 1 ∂ 1 1 0 2 3 ∂ γ + + γ + im M −1 ΦK Mγ r ∂θ 2 tan θ r sin θ ∂ϕ

(III.10)

We can easily check that all the coeﬃcients, which are (T, X, ω) dependent, are regular on the horizon H + . We could use the theory of the mixed hyperbolic systems to solve it, but since we want to get some precise information on the ﬁelds near the horizon, it is more convenient to use the tortoise coordinate x ∈ R given

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by (II.10) instead of r ∈]r0 , r+ [, and to change the representation of the spinor again, by introducing 1

Φ(t, x, θ, ϕ) = rF 4 Ψ(t, x, θ, ϕ).

(III.11)

Given a spinor ﬁeld Φs deﬁned on the Cauchy hypersurface Σs =]z(s), ∞[x ×Sω2 ,

(III.12)

the mixed problem becomes : 1 ∂ ∂ Φ + γ 0 γ 1 Φ + iqAΦ + F 2 γ 0 ∂t ∂x 1 2 ∂ ∂ 1 1 γ + + γ3 + im Φ = 0, x > z(t), r ∂θ 2 tan θ r sin θ ∂ϕ

x = z(t) =⇒ √

0

1 zγ ˙ − γ 1 Φ = iBΦ, 1 − z˙ 2

Φ(t = s, .) = Φs (.).

(III.13)

(III.14)

(III.15)

To construct the functional framework, we introduce the Hilbert spaces : 4 4 L2t := L2 (Σt , dxdω) , L2∞ := L2 (Rx × Sω2 , dxdω) .

(III.16)

For s < t ≤ ∞, L2s is naturally embedded in L2t ; this amounts to extending the function by zero for x ≤ z(s). We denote . the norm in L2t . We deﬁne on L2t the operator Ht : 1 1 2 ∂ ∂ ∂ 1 1 Ht Φ = iγ 0 γ 1 Φ − qAΦ + iF 2 γ 0 γ + + γ3 + im Φ, ∂x r ∂θ 2 tan θ r sin θ ∂ϕ (III.17)

t ∈ R, D(Ht ) =

0

1 zγ ˙ − γ 1 Φ = iBΦ , Φ ∈ L2t ; Ht Φ ∈ L2t , x = z(t) ⇒ √ 1 − z˙ 2 (III.18) D(H∞ ) = Φ ∈ L2∞ ; H∞ Φ ∈ L2∞ .

(III.19)

Lemma III.1. The operator iHt is maximal accretive for any t, and skew-adjoint for t ≤ 0 and t = ∞. Moreover the point spectrum of H∞ is empty.

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Since H∞ has no eigenvalue, there exists no time-periodic Dirac ﬁelds with ﬁnite energy on the whole Reissner-Nordstrøm space-time. We let open the problem of the existence of such solutions outside a stationary star, and we solve the Dirac equation outside the collapsing star. The mixed problem has the form : ∂ Φ = iHt Φ, ∂t

(III.20)

Φ(t) = U (t, s)Φs .

(III.21)

and it is solved by a propagator

More precisely we apply a Trotter-Kato method to get the following : Proposition III.2. For Φs ∈ D(Hs ), there exists a unique solution Φ ∈ C 1 (Rt ; L2∞ ) of (III.13), (III.14), (III.15), (III.6), satisfying for any real t : Φ(t) ∈ D(Ht ).

(III.22)

Φ(t) = Φs ,

(III.23)

Moreover we have :

and U (t, s) can be extended in an isometric strongly continuous propagator from L2s onto L2t satisfying

∀Φs ∈ D(Hs ), (t → U (t, s)Φs ) ∈ C 1 Rt , L2∞ ,

d U (t, s)Φs = iHt U (t, s)Φs , dt (III.24)

∀Φs0 ∈ [C0∞ (Σs0 )]4 , ∃h > 0, (III.25)

d U (t, s)Φs0 = −iU (t, s)Hs Φs0 , (s → U (t, s)Φs0 ) ∈ C 1 ]s0 − h, s0 + h[, L2t , ds

x > R ⇒ Φs (x, ω) = 0 ⇒ x > R+ | t − s |⇒ [U (t, s)Φs ](x, ω) = 0 .

(III.26)

Now we state that, given ΨBH a ﬁeld falling into the future black-hole, there exists a unique Dirac ﬁeld Ψ which is equal to ΨBH at the horizon, and Ψ = 0 at the null inﬁnity. Since we have chosen the Schwarzschild coordinates (t, x, ω), this characteristic problem becomes a scattering problem : the characteristic data becomes an asymptotic data. The link between Φ and ΦK makes clear the suitable set of asymptotic data near the horizon. Since ΦK is well deﬁned on H + , the relation Φ = e−itqA(r0 ) M −1 ΦK implies that Φ2 (t, x = −t + s, ω) and Φ3 (t, x =

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−t + s, ω) tends to zero as t → ∞. Therefore we introduce the subspaces of ﬁelds Φ, falling into the black-hole as t tends to inﬁnity : L2BH = Φ ∈ L2∞ ; Φ2 = Φ3 = 0, x < 0 ⇒ Φ(x, ω) = 0 , (III.27) or going out to inﬁnity : L2out = Φ ∈ L2∞ ; Φ1 = Φ4 = 0, x < 0 ⇒ Φ(x, ω) = 0 .

(III.28)

At the black-hole horizon the electromagnetic potential equals to A(r0 ) and F is zero, hence we compare the dynamics (III.20) with : ∂ Φ = iHBH Φ, ∂t HBH Φ = iγ 0 γ 1

(III.29)

∂ Φ − qA(r0 )Φ, D(HBH ) = Φ ∈ L2∞ ; HBH Φ ∈ L2∞ . ∂x (III.30)

HBH is selfadjoint and the Cauchy problem for (III.29) is solved by the unitary group on L2∞ : UBH (t) := eitHBH .

(III.31)

Since A(r) → A(r0 ) and F (r) → 0 exponentially, the Cook method allows to construct the wave operator that gives the solution of the asymptotic problem : Proposition III.3. Assume Φ ∈ L2BH . Then the strong limit : ΩBH Φ = lim U (0, T )UBH (T )Φ in L20 T →+∞

(III.32)

exists and deﬁnes an isometry from L2BH to L20 . We can now state the main result of asymptotic behaviour of the propagator near the horizon. To express this estimate we shift the Cauchy data toward the black-hole horizon by the following way. For Φ ∈ L2∞ and T > 0 we put : ΦT (x, ω) = Φ(x + T, ω).

(III.33)

Theorem III.4 (Key Estimate). Given Φout ∈ L2out , ΦBH ∈ L2BH , we have for J = [0, ∞[ or ]0, ∞[ :

lim 1J (H0 )U (0, T ) ΦTout + ΦTBH 2 T →∞ −1 2π 2π (III.34) =< Φout , ζe κ0 HBH 1 + ζe κ0 HBH Φout >L2∞ + 1J (H0 ) (ΩBH ΦBH ) 2 . with 2π

ζ = e κ0 qA(r0 ) .

(III.35)

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We remark that the term involving Φout in the right member of (III.34) does not depend on the boundary condition of type (III.2), (III.6), and on the mass of the ﬁeld. It is also independent of the history of the collapse deﬁned by function z(t). We brieﬂy describe the main ideas of the proof. The key phenomenon is the asymptotically inﬁnite Doppler eﬀect due to the collapse to a black-hole : in Schwarzschild coordinates, the contracting surface of the star is asymptotically characteristic. Hence it appears a blue shift and we establish that the approximation of the geometrical optics is valid : for large T , the main part of the energy of t → U (t, T )ΦTout , 0 ≤ t ≤ T , propagates near the null hypersurface {(t, x = −t, ω), 0 ≤ t ≤ T, ω ∈ S 2 }. Furthermore we can evaluate the leading term of U (0, T )ΦTout : U (0, T )ΦTout ∼ Φ∗T + o(1), T → ∞, with Φ∗T (x, ω) := eiqA(r0 )T

1

y := 2T + We have :

1

| κ0 x | 2

(−Φout,3 (y, ω), 0, 0, Φout,2 (y, ω)) ,

1 1 ln(−x) − ln(A). κ0 κ0

1J (H0 ) U (0, T )ΦTout ∼ 1J (HBH ) (Φ∗T ) ,

moreover, an explicit calculation by Fourier transform gives the fundamental identity for any T > 0 : −1 2π 2π 1J (HBH ) (Φ∗T ) 2 =< Φout , ζe κ0 HBH 1 + ζe κ0 HBH Φout >L2∞ . Finally, we remark that U (0, T )ΦTout and U (0, T )ΦTBH are asymptotically orthogonal as T → ∞, since U (0, T )ΦTout weakly converges to zero because of the Doppler eﬀect, and U (0, T )ΦTBH strongly converges to ΩBH ΦBH .

IV The Quantum Dirac Fields We interpret the crucial result (Theorem III.4) in the framework of the Quantum Field Theory. Since we deal with a curved space-time with moving boundary, the concept of particles is not appropriate, hence we adopt the approach of the algebras of local observables in the spirit of [16], [19]. According to J. Dimock [17], we construct the algebra of quantum spin ﬁelds on a curved space-time as following. We consider a globally hyperbolic manifold U with a foliation by a family of Cauchy hypersurfaces Σt , i.e. U = ∪t∈R Σt .

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We choose a representation of the CAR (= canonical anticommutation relations) on Σ0 . It consists of a Hilbert space H and some antilinear continuous function Ψ0 between the space of spinors on Σ0 , that we represent as L2 (Σ0 , C4 ), and L(H), satisfying in particular Ψ∗0 (F1 )Ψ0 (F2 ) + Ψ0 (F2 )Ψ∗0 (F1 ) =< F1 , F2 > 1. Now a classical spin ﬁeld structure is deﬁned by a propagator B(t, s) that is an isometry from L2 (Σs , C4 ) onto L2 (Σt , C4 ). We introduce the operator ∞ B(0, t)Φ(t)dt ∈ L2 (Σ0 , C4 ). (IV.1) S : Φ ∈ C0∞ (U, C4 ) −→ S(F ) := −∞

Then the quantum spin ﬁeld is the operator valued distribution Ψ : Φ ∈ C0∞ (U, C4 ) −→ Ψ(Φ) := Ψ0 (SΦ) ∈ L(H).

(IV.2)

For any open set O ⊂ U, the algebra of observables is A(O) := C∗ algebra generated by Ψ∗ (Φ1 )Ψ(Φ2 ), supp Φj ⊂ O. A beautiful result due to Dimock assures that the collection A(O) is independent (up to a net isomorphism) of the representation of the CAR, the Cauchy hypersurface, and the choice of spin structure. Now we consider a state ωΣ0 on the C∗ algebra generated by Ψ∗0 (F1 )Ψ0 (F2 ), Fj ∈ L2 (Σ0 , C4 ). Then we deﬁne a ground state ωU on A(U) by putting for Φj ∈ C0∞ (U, C4 ) : ωU (Ψ∗ (Φ1 )Ψ(Φ2 )) := ωΣ0 (Ψ∗0 (SΦ1 )Ψ0 (SΦ2 )).

(IV.3)

In the case of the stationary space-times, we have Σt = {t}×Σ0 , the generator of the propagator is a densely deﬁned self adjoint operator H on L2 (Σ0 , C4 ) : B(t, s) = ei(t−s)H , and the Fock vacuum state on Σ0 is deﬁned by 0 (Ψ∗0 (F1 )Ψ0 (F2 )) :=< 1]0,∞[ (H)F1 , F2 > . ωΣ 0

(IV.4)

β,µ satisﬁes the (β, µ)-KMS condition, 0 < β, µ ∈ R, More generally, a state ωΣ 0

if

−1 β,µ (Ψ∗0 (F1 )Ψ0 (F2 )) :=< zeβH 1 + zeβH F1 , F2 >, z = eβµ . ωΣ 0

(IV.5)

We immediately associate a state ωUβ,µ on U by (IV.1), (IV.3), (IV.5). In fact describes a double Gibbs equilibrium state : on the one hand, an ideal Fermi particle gas with temperature 0 < T = β −1 and chemical potential µ, and on

ωUβ,µ

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the other hand an ideal Fermi antiparticle gas with the same temperature T but an opposite chemical potential −µ. If q is the charge of the particles, the charge density of the gaz is (see Lemma A.2) : 9=

1 qµ. π

(IV.6)

For sake of completeness, we present, in the Appendix, the details of the second quantization of the Dirac ﬁeld with a time-independent Hamiltonian. We apply these procedures to our problem. First the quantization on M, the space-time outside the collapsing star, is deﬁned by choosing the foliation Σt :=]z(t), ∞[x ×S 2 , the Fock quantization and the Fock vacuum on Σ0 given by (IV.4) with H = H0 . In the past, this state is the so called Boulware vacuum that corresponds to the familiar concept of an empty state for a static observer. Then the quantum ground state on M is characterized by the two-point function for Φj ∈ C0∞ (M, C4 ) : ∞ ∞ ∗ U (0, t)Φ1 (t)dt, U (0, t)Φ2 (t)dt . ωM (Ψ (Φ1 )Ψ(Φ2 )) := 1]0,∞[ (H0 ) −∞

−∞

L20

(IV.7) To describe the ﬁelds near the future Black-Hole horizon, we have introduced the self-adjoint operator HBH on the stationary space-time MBH = Rt × Rx × S 2 . The quantum ﬁelds ΨBH (Φ) for Φ ∈ C0∞ (MBH , C4 ) are constructed as before, by taking the Fock quantization on Rx × S 2 and S in (IV.2) equals to : ∞ UBH (−t)Φ(t)dt. (IV.8) SBH Φ := −∞

According to the previous deﬁnitions, the two point function given for Φj ∈ C0∞ (MBH , C4 ) by : " !

−1 β,µ ωBH (Ψ∗ BH (Φ1 )ΨBH (Φ2 )) := zeβHBH 1 + zeβHBH SBH Φ1 , SBH Φ2 2 , L∞

βµ

z=e

,

(IV.9)

deﬁnes a thermal state on A(MBH ). In fact it will be useful to split the ﬁelds into a part outgoing to inﬁnity, and a part falling into the black-hole, as t → +∞, by putting for Ψ ∈ C4 : P out Ψ := (0, Ψ2 , Ψ3 , 0), P in Ψ := (Ψ1 , 0, 0, Ψ4 ), and we have for F ∈ L2∞ : itHBH out e P F (x, ω) = e−iqA(r0 )t P out F (x − t, ω), itHBH in e P F (x, ω) = e−iqA(r0 )t P in F (x + t, ω).

(IV.10)

(IV.11)

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We are mainly concerned with the subalgebra of outgoing local observables. Given an open set O ⊂ MBH Aout (O) = C∗ algebra generated by Ψ∗BH (P out Φ1 )ΨBH (P out Φ2 ), Φj ∈ C0∞ (O, C4 ).

(IV.12)

We are interested in formulating the Hawking eﬀect at the Black-Hole horizon, in terms of KMS state on a local algebra. According to (II.23), the points of the future horizon are reached at the inﬁnity of the incoming radial null geodesics {(t, x = −t + x0 , ω), t ∈ R}. We introduce Min := (t, x = −t + x0 , ω), t ∈ R, 0 < x0 , ω ∈ S 2 (IV.13) Then, given Φj0 ∈ C0∞ (Min , C4 ), the two point function of the ground state at the horizon is characterized by

lim ωM Ψ∗ (Φ1T )Ψ(Φ2T ) T →∞

where we put for T > 0 : ΦjT (t, x, θ, ϕ) = Φj0 (t − T, x + T, θ, ϕ).

(IV.14)

Theorem IV.1 (Main Result). Given Φj0 ∈ C0∞ (Min , C4 ) we have

β,µ ∗ ΨBH (P out Φ10 )ΨBH (P out Φ20 ) lim ωM Ψ∗ (Φ1T )Ψ(Φ2T ) = ωBH

T →∞

∗

0 Ψ0 (ΩBH SBH P in Φ10 )Ψ0 (ΩBH SBH P in Φ20 ) , +ωΣ 0

(IV.15)

with β=

2π , µ = qA(r0 ). κ0

(IV.16)

We remark that, near the horizon, the ground state ωM is asymptotically equals, on Aout (Min ), to a thermal state. Therefore this theorem expresses that the ground state, that is the Boulware vacuum in the past, has exactly the structure of the Unruh state near the future horizon (see e.g. [3], [18], [40]). According to

β,µ ∗ Lemma A.2, ωBH ΨBH (P out Φ10 )ΨBH (P out Φ20 ) corresponds to a ﬂux of particles leaving the vicinity of the black hole, and streaming outwards. These outgoing modes are thermally distributed with the Hawking temperature : TBH =

κ0 , 2π

(IV.17)

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and the charge density of the ﬂux is equal to : 9BH =

1 2 q A(r0 ). π

(IV.18)

Moreover, the state of outgoing modes is independent of the nature of the collapse, and of the boundary condition : one need not worry about the exact history of the collapse, or about interactions between the quantum ﬁeld and the matter of the star subsumed in the large class of boundary conditions (III.2), (III.6) (No Hair result). In the case of the Reissner-Nordstrøm Black-Hole created by a star of mass 0 < M and charge Q, | Q |< M , we have : M 2 − Q2 (IV.19) TBH = 2 , 2π M + M 2 − Q2

9BH =

q2 Q . π M + M 2 − Q2

(IV.20)

An important fact is that Q and 9BH have the same sign, hence the black-hole preferentially emits fermions whose charge is of same sign as its own charge, rather than fermions of opposite charge.

V Discussion In fact, the previous interpretation of the main result in terms of particles, is relevant only for a static observer at inﬁnity. In the framework of the curved space-times, we have to be very carefull to describe a ﬁeld with some ”particles”. Such a description of the ﬁeld crucially depends on how the ”particles” are deﬁned and detected, i.e. the description of some state, as a vacuum state, a thermal state, etc., speciﬁcally depends on the choice of the observer. We ﬁrst consider a radially freely falling observer released from rest at inﬁnity 1 in the distant past. Its radial is V r = −(1 − F ) 2 and he carries a nat −1 velocity ural orthonormal frame F ∂t + V r ∂r , F −1 V r ∂t + ∂r , r−1 ∂θ , (r sin θ)−1 ∂ϕ . On the hand, the nature of the outgoing particles in the vicinity of the future horizon is ill deﬁned for such an observer, because (IV.16) and (A.57) show that the average wavelength of the emitted quanta is comparable with the size of the hole. In some sense, this observer is inside these particles. On the other hand, a particle detector (e.g. an Unruh box) will react to states which have positive frequency with respect its propertime. Hence a geodesic detector freely falling across the future ∂ horizon, will respond to the presence of ∂U -positive frequency with U = T − X, in i.e. of the in-modes P Φ. We conclude that the response of this detector is deter ∗ 0 in 1 Ψ mined by ωΣ (Ω S P Φ )Ψ (Ω SBH P in Φ20 ) . (In fact, an Unruh box BH BH 0 BH 0 0 0

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is a ”ﬂuctuometer” and contains information both about the ﬂuctuations of the ﬁeld and its own motion; see e.g. [10] and a lucid contribution by Unruh in [1]). Finally an observer falling through the horizon will see no particles pouring out of the collapsing star. In opposite, a static observer at inﬁnity deﬁnes the particles outgoing from the ∂ black hole, as the positive frequency modes for ∂V with V = T + X, which exactly β,µ ∗ out ΨBH (P out Φ10 ) are the out-modes P Φ. Therefore this observer interprets ωBH

out 2 ΨBH (P Φ0 ) as a thermal radiation of particle and anti-particles leaving the black hole. We mention an alternative analysis based on the equivalence principle [37] : an observer in free fall deﬁnes locally a ﬁeld theory and a local vacuum state that always look the same, just like ﬂat space time ﬁeld theory in an empty local neighborhood; this quantum ﬁeld theory and this vacuum state are viewed by a static external observer as constantly redeﬁned; the set of these local vacua determines the global ground state, which is interpreted by the static observer as an outgoing stream of particles. In summary, the two terms in the right member of (IV.15), correspond to two diﬀerent kinds of particles cannot be detected by the same observer. The in that

0 modes of the state ωΣ Ψ∗0 (ΩBH SBH P in Φ10 )Ψ0 (ΩBH SBH P in Φ20 ) are detected 0 by the freely falling observer across the horizon, who cannot see the outgoing modes. Nevertheless, the gravitational disturbance produced by the collapsing star, actually induces the creation of an outgoing thermal charged ﬂux of radiation, as seen by a static observer at inﬁnity. Our result (IV.15) completely agrees with the analysis by Hawking [22] (see so [40]). We shall investigate, in a future work, a more precise estimate of the response of the detectors, involving the computation of the renormalized stress energy momentum tensor. We now investigate the rather subtle role of the cosmological constant in the case of the DeSitter-Reissner-Nordstrøm Black-Hole. The spherical black-hole with mass M > 0, electric charge Q ∈ R, in an asymptotically ﬂat universe (cosmological constant Λ = 0), or expanding universe (Λ > 0), is described by the DeSitter-Reissner-Nordstrøm metric −1 Q2 Q2 2M Λ 2 2M Λ 2 2 2 ds = 1 − + 2 − r dt − 1 − + 2 − r r r 3 r r 3 dr2 − r2 (dθ2 + sin2 θdϕ2 ), which, by the uniqueness theorem of Birkhoﬀ, is the unique spherically symmetric solution of the Einstein-Maxwell equations (with cosmological constant Λ ≥ 0) (see e.g. [11]). Given M > 0 the mass of the black-hole, we deal with the radius of the black-hole horizon r0 (Q, Λ), the Surface Gravity at the black-hole horizon κ0 (Q, Λ), the Temperature of the quantum state at the horizon TBH (Q, Λ) and the Charge Density of the gaz of particles and antiparticles outgoing from the black-hole to inﬁnity 9BH (Q, Λ). We could give the terrifying expressions of these quantities

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computed using some program of formal calculus (e.g. MAPLE is convenient). It is more interesting to investigate their behaviours with respect to the charge of the black-hole Q and the cosmological constant Λ. We deduce these results from Theorem IV.1, (IV.17) and (IV.18), by elementary but tedious calculations of expansions with respect to the small parameter.

V.1 Charged black-hole in an asymptotically ﬂat universe (ReissnerNordstrøm Black-Hole) It is the case where Λ = 0 and 0 ≤| Q |< M . r0 (Q, 0) = M + M 2 − Q2 ,

(V.1)

M 2 − Q2 κ0 (Q, 0) = 2 , M + M 2 − Q2

(V.2)

M 2 − Q2 TBH (Q, 0) = 2 , 2π M + M 2 − Q2

(V.3)

9BH (Q, 0) =

q2 Q . π M + M 2 − Q2

(V.4)

We note that the radius of the black-hole and the temperature are decreasing functions of | Q |∈ [0, M [, and we have : TBH (Q, 0) = TBH (0, 0) −

1 Q4 + O(Q6 ). 128πM 5

(V.5)

V.2 Neutral black-hole in an expanding universe (DeSitter-Schwarzschild Black-Hole) It is the case where 0 < 9ΛM 2 < 1 and Q = 0 (introduced by Kottler in 1918). 1 2 5π 1 − arccos(3M Λ 2 ) , r0 (0, Λ) = 1 cos (V.6) 3 3 Λ2 κ0 (0, Λ) =

TBH (0, Λ) =

1 (3M − r0 (0, Λ)), r02 (0, Λ)

(V.7)

1 (3M − r0 (0, Λ)), 2πr02 (0, Λ)

(V.8)

9BH (0, Λ) = 0.

(V.9)

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We check that : r0 (0, Λ) = with

Ann. Henri Poincar´ e

3M 1 + y 2 √ , y 3+y

1 arccos(3M Λ 2 ) √ ∈] 3, ∞[. y=# 1 1 − cos2 13 arccos(3M Λ 2 )

cos

1 3

We deduce that the radius of the black-hole (resp. the temperature) is an increasing 1 (resp. decreasing) function of the cosmological constant Λ ∈]0, 9M 2 [, and we have : 2M < r0 (0, Λ) < 3M.

(V.10)

V.3 Weakly charged black-hole in an expanding universe It is the case where Λ > 0 and | Q |→ 0. 1 Q2 + O(Q4 ), 2(3M − r0 (0, Λ))

(V.11)

r0 (0, Λ) − 2M 2 3 Q + O(Q4 ), 2r03 (0, Λ) 3M − r0 (0, Λ)

(V.12)

r0 (Q, Λ) = r0 (0, Λ) −

κ0 (Q, Λ) = κ0 (0, Λ) +

TBH (Q, Λ) = TBH (0, Λ) +

9BH (Q, Λ) =

r0 (0, Λ) − 2M 2 3 Q + O(Q4 ), 4πr03 (0, Λ) 3M − r0 (0, Λ)

q 2 Q3 q2 Q + + O(Q5 ). πr0 (0, Λ) 2πr02 (0, Λ)(3M − r0 (0, Λ))

(V.13)

(V.14)

By (V.10) we see that a small charge of the black-hole decreases its radius as in the Reissner-Nordstrøm case. But in opposite with (V.5), the temperature is an increasing function of the charge near zero.

V.4 Black-hole in a weakly expanding universe It is the case where 0 ≤| Q |< M and Λ → 0+ . We obtain : 4 M + M 2 − Q2 r0 (Q, Λ) = r0 (Q, 0) + Λ + O(Λ2 ), 6 M 2 − Q2

(V.15)

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Λ κ0 (Q, Λ) = κ0 (Q, 0) + −4M 2 − 4M M 2 − Q2 + 5Q2 + O(Λ2 ), 6 M 2 − Q2 (V.16) TBH (Q, Λ) = TBH (Q, 0) +

Λ −4M 2 − 4M M 2 − Q2 + 5Q2 + O(Λ2 ), 12π M 2 − Q2 (V.17) 

3  2 − Q2 M M +   Λ + O(Λ2 ). 9BH (Q, Λ) = 9BH (Q, 0) 1 − 2 2 6 M −Q

(V.18)

We constat that the radius of the black-hole is an increasing function of the cosmological constant near zero, and the absolute value of the charge density is decreasing. As regards the temperature, it appears that the ratio of the charge with respect to the mass plaies a rather subtle role : if the charge of the black-hole is not too large, more precisely # 24 0 ≤| Q |< M, (V.19) 25 the temperature is a decreasing function of Λ. But for a strongly charged blackhole, i.e. # 24 M <| Q |< M, (V.20) 25 the temperature is an increasing function of Λ. For the critical value # 24 | Q |= M, 25 we evaluate that $ κ0 $ TBH

#

Q=± #

Q=±

% 24 5 324 3 2 M, Λ = − M Λ + O(Λ3 ), 25 36M 125

% 5 162 3 2 24 M, Λ = − M Λ + O(Λ3 ), 25 72πM 125π

(V.21)

(V.22)

(V.23)

hence the temperature is decreasing with respect to Λ again. At our knowledge, this phenomenon had not been previously noted.

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VI Proof of the asymptotic estimates This part is devoted to the proofs of the results of Parts III and IV. Taking advantage of the spherical invariance we reduce our 3D+1 problem to a 1D+1 problem. Therefore the strategy consists in using the expansion in spinoidal spherical harmonics. It is well known (see e.g. [33] , [34]) that there exists two Hilbert bases T 1l ,n (ϕ, θ), T−l 1 ,n (ϕ, θ) of L2 (S 2 ), 2

2

1 1 l ∈ N + , n ∈ Z, l− | n |∈ N, 2 2 such that :

∂ 1 + ∂θ 2 tan θ

T± 1 ,n = ± 2

(VI.1)

n 1 T± 1 ,n − i l + T∓ 1 ,n , 2 sin θ 2 2

(VI.2)

∂ T 1 = −inT± 1 ,n . 2 ∂ϕ ± 2 ,n

(VI.3)

Then we have 2 2 4 & 2 L,n , L2,n := CT− 1 ,n ⊕ CT+ 1 ,n ⊕ CT− 1 ,n ⊕ CT+ 1 ,n . (VI.4) L (S ) = 2

2

2

2

,n

4 If we denote Πl,n the projector from L2 (S 2 ) onto L2,n , a spin ﬁeld Φ(t, x, ϕ, θ) is solution of the Dirac equation (III.13) satisfying the boundary condition (III.14), (III.6), iﬀ u(t, x) given by u(t, x) := eitqA(r0 ) ei

α,n 2

γ5

Π,n [Φ(t, x, .)] ,

(VI.5)

is solution of ∂u ∂u +L + iV,n u = 0, t ∈ R, x > z(t), ∂t ∂x ' 1 + z(t) ˙ ∀t ∈ R, u2 (t, x = z(t)) = u4 (t, x = z(t)), 1 − z(t) ˙

(VI.6)

' u3 (t, x = z(t)) = − Here L is the matrix



−1  0 L=  0 0

1 + z(t) ˙ u1 (t, x = z(t)), 1 − z(t) ˙

0 1 0 0

 0 0 0 0  , 1 0  0 −1

(VI.7)

(VI.8)

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Creation of Fermions at the Charged Black-Hole Horizon

and the potential is given by :



0 0

1  V,n (x) = q(A − A(r0 )) + mF 2   −ie−iα,n 0

 0 1 1 F2   1 + l+ 2 r  0 0

0 0 0

−ie−iα,n

ieiα,n 0 0 0

1063

0



ieiα,n    0 0

 1 0 0 0 0 0  . 0 0 −1  0 −1 0

(VI.9)

Therefore we have transformed our 3D+1 problem into a family of 1D+1 simple problems (VI.6), where the potential V,n has nice properties of asymptotic behaviours as x → ±∞. To get now the key estimate (III.34), it will be suﬃcient to prove a similar asymptotic result for these 1D+1 problems (Theorem VI.5).

VI.1

One dimensional problem

We consider the mixed hyperbolic problem with unknown u =t (u1 , u2 , u3 , u4 ) : ∂u ∂u +L + iV u = 0, t ∈ R, x > z(t), ∂t ∂x

(VI.10)

∀t ∈ R, u2 (t, x = z(t))=λ(t)u4 (t, x = z(t)), u3 (t, x = z(t)) = −λ(t)u1 (t, x = z(t)), (VI.11) ∀x > z(s), u(s, x) = f (x).

(VI.12)

Here V is a matrix valued map of x satisfying :

V ∈ C 1 Rx ; C4×4 ,

(VI.13)

∀x ∈ R, V ∗ (x) = V (x).

(VI.14)

Moreover we assume the following asymptotic behaviours : there exists ε > 0, C > 0, µ ≥ 0, 9 ∈ R and two hermitian matrices Γ, V∞ such that : ∞ sup {| V (x) | + | V (x) |}dt < ∞, (VI.15) 0

x<−t

lim V (x) = V∞ ,

x→+∞

lim

x→+∞

d V (x) = 0, dx

V∞ = µΓ − 9Id, ΓL + LΓ = 0, Γ2 = Id,

(VI.16) (VI.17)

and if µ > 0, there exists η ∈]0, 1] such that : ∀f ∈ C4 , f1 + f3 = f2 − f4 = 0 ⇒ i < ΓLf, f >C4 ≥ (η 2 − 1) | f |2 .

(VI.18)

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The boundary condition is given by functions λ(t) and z(t) satisfying :   z ∈ C 2 (R),       ∀t ≤ 0, z(t) = z(0) < 0, ∀t ∈ R, −1 < z(t) ˙ ≤ 0,    z(t) = −t − Ae−2κt + ζ(t), A > 0, κ > 0,     | ζ(t) | + | ζ(t) ˙ |= O(e−4κt ), t −→ +∞,

(VI.19)

' 1 + z(t) ˙ . 1 − z(t) ˙

λ(t) :=

(VI.20)

We introduce the function spaces : 0 ≤ t, Lpt := [Lp (]z(t), ∞[x , dx)]4 , 1 ≤ p ≤ ∞,

(VI.21)

Lp∞ := [Lp (Rx , dx)]4 .

(VI.22)

For s < t, Lps is naturally embedded in Lpt ; this amounts to extending the function by zero inside [z(t), z(s)]. For 0 ≤ t ≤ ∞ we denote . the L2t norm and | . |∞ the norm in L∞ t . We consider some spaces of more regular data : dk 0 ≤ t ≤ ∞, k ∈ N∗ , Htk := f ∈ L2t ; k f ∈ L2t (VI.23) dx and we denote . k the norm of Htk deﬁned by : $ f k =

∞

z(t)

% 12 dk 2 | f (x) | + | k f (x) | dx . dx 2

(VI.24)

Thanks to the Sobolev embedding 4 Ht1 ⊂ C 0 ([z(t), ∞[) ∩ L∞ t ,

(VI.25)

we can introduce the family of densely deﬁned operators on L2t : HV,t := iL

d − V, dx

(VI.26)

with domain 1 1 Wt1 := f ∈ Ht1 ; f2 (z(t)) = λ(t)f4 (z(t)), f3 (z(t)) = −λ(t)f1 (z(t)) , W∞ := H∞ . (VI.27)

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Lemma VI.1. The operator iHV,t is maximal accretive for any t, and skew-adjoint for t ≤ 0 and t = ∞. Moreover the point spectrum of HV,∞ is empty. Proof of Lemma VI.1. Let f be in Wt1 . Since the Sobolev inequality assures that f ∈ Wt1 ⇒ lim | f (x) |= 0, x→∞

(VI.28)

an integration by part gives :

˙ | f (z(t) |2 ≤ 0, 2! < iHV,t f, f >L2t = λ2 (t) − 1 | f1 (z(t)) |2 + | f4 (z(t)) |2 = z(t) (VI.29) hence iHV,t is accretive. Now we easily check that its adjoint (iHV,t )∗ is deﬁned in the sense of distributions by (iHV,t )∗ f = −iHV,t f, with domain :

∗

D (iHV,t )

= f ∈ Ht1 ; f2 (z(t)) =

1 1 f4 (z(t)), f3 (z(t)) = − f1 (z(t)) . λ(t) λ(t)

Hence we have again : ∗

˙ | f (z(t) |2 ≤ 0. 2! < (iHV,t ) f, f >L2t = z(t) Since both iHV,t and its adjoint are accretive, we conclude that iHV,t is maximal accretive and, if λ(t) = 1, skew-adjoint. In the same manner, HV,∞ is selfadjoint on L2∞ . If u ∈ L2∞ is an eigenvector of HV,∞ for the eigenvalue λ ∈ R, then 1 v(x) = e−iλLx u(x) is a solution in H∞ to : v (x) + iLeiλLx V e−iλLx v = 0. Since v(x) → 0 as x → −∞, and Gronwall lemma that v = 0.

(0 −∞

| V (x) | dx < ∞, we conclude by the

The solution of (VI.10), (VI.11), (VI.12), is formally expressed via a propagator UV (t, s) : u(t) = UV (t, s)f.

(VI.30)

More precisely the mixed problem is solved by the following : Proposition VI.2. For f ∈ Ws1 , there exists a unique solution u ∈ C 1 (Rt ; L2∞ ) of (VI.10), (VI.11), (VI.12) satisfying for any real t : u(t) ∈ Wt1 .

(VI.31)

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Ann. Henri Poincar´ e

Moreover we have : u(t) = f ,

(VI.32)

and UV (t, s) can be extended in an isometric strongly continuous propagator from L2s onto L2t satisfying

∀f ∈ Ws1 , (t → UV (t, s)f ) ∈ C 1 Rt , L2∞ ,

d UV (t, s)f = iHV,t UV (t, s)f, dt (VI.33)

∀f ∈ [C0∞ (]z(s0 ), ∞[)] , 4

d ∃h>0, (s → UV (t, s)f )∈C 1 ]s0 − h, s0 + h[, L2t , UV (t, s)f = −iUV (t, s)HV,s f, ds (VI.34)

x > R ⇒ f (x) = 0 ⇒ x > R+ | t − s |⇒ [UV (t, s)f ](x) = 0 .

(VI.35)

Proof of Proposition VI.2. The uniqueness follows from the conservation of the norm (VI.32) which is established by evaluating d ∞ | u(t, x) |2 dx = 2! < iHV,t u(t), u(t) >L2t −z(t) ˙ | u(t, z(t)) |2 = 0. dt z(t) To prove the existence we introduce the operators :  cos θ(t) 0 sin θ(t) 0  0 cos θ(t) 0 sin θ(t)  R(t) :=  − sin θ(t) 0 cos θ(t) 0 0 − sin θ(t) 0 cos θ(t)

   , θ(t) := arctan λ(t), 

T (t) : f ∈ L20 → T (t)f ∈ L2t , [T (t)f ](x) = f (x − z(t) + z(0)). We remark that

0 ˙ ˙ = −iθ(t)γ , [R(t)]−1 R(t) 4 4 , T˙ (t) = −z(t)T ˙ (t)∂x . T ∈ C 1 Rt , L C01 (Rx ) , C00 (Rx )

Then u is a solution to the problem iﬀ w(t) = [R(t)]−1 [T (t)]−1 u(t) is solution to ∂t w + A(t)w = 0, x > z(0), w2 (t, z(0)) = w3 (t, z(0)) = 0,

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Creation of Fermions at the Charged Black-Hole Horizon

1067

where A(t) :=

0 ˙ ˙ ∂x − iθ(t)γ + i[R(t)]−1 [T (t)]−1 V R(t)T (t). [R(t)]−1 [T (t)]−1 LR(t)T (t) − z(t) We can easily show that the operator A(t) with dense domain (independent of t) D(A(t)) = f ∈ H01 ; f2 (z(0)) = f3 (z(0)) = 0 is skew-adjoint on L20 . Moreover, since z is C 2 and V is uniformly continuous on R, the map t → A(t) is norm continuous from R to L(D(A(0)), L20 ). Then the Theorems of T. Kato [28] assure that there exists a unique strongly continuous propagator S(t, s) on L20 such that : w(t) = S(t, s)w(s), and for f ∈ D(A(0)), S(t, s)f ∈ D(A(0)) is a strongly diﬀerentiable map from Rt × Rs to L20 satisfying : d S(t, s)f = −A(t)S(t, s)f, dt

d S(t, s)f = S(t, s)A(s)f. ds

Then the propagator deﬁned by : UV (t, s) = R(t)T (t)S(t, s)[R(s)]−1 [T (s)]−1 , satisﬁes (VI.33) and (VI.34). To establish (VI.35) we check that : d ∞ | u(t, x) |2 dx = dt R+|t−s| −(| u2 |2 + | u3 |2 )(t, R+t−s)1[0,∞[ (t−s)+(| u1 |2 +| u4 |2 )(t, R+s−t)1[0,∞[ (s−t). It will be useful to have the explicit form of the free propagator U0 (s, t) : Lemma VI.3. For t ≤ s, given f ∈ L2s , u(t) = U0 (t, s)f is given by x > z(t) ⇒ u2 (t, x) = f2 (x − t + s), u3 (t, x) = f3 (x − t + s),

(VI.36)

x > s + z(s) − t ⇒ u1 (t, x) = f1 (x + t − s), u4 (t, x) = f4 (x + t − s),

(VI.37)

' z(t)< x < s + z(s) − t⇒u1 (t, x)= −

1 − z(τ ˙ (x + t)) f3 (x + t + s − 2(τ (x + t))), 1 + z(τ ˙ (x + t)) (VI.38)

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' z(t) < x < s + z(s) − t⇒u4 (t, x) =

Ann. Henri Poincar´ e

1 − z(τ ˙ (x + t)) f2 (x + t + s − 2(τ (x + t))), 1 + z(τ ˙ (x + t)) (VI.39)

where function τ is the unique solution of z(0) ≤ y < 0, z(τ (y)) + τ (y) = y,

(VI.40)

and satisﬁes τ (y) = −

1 1 ln(−y) + ln(A) + O(y), y → 0− , 2κ 2κ

1 + z(τ ˙ (y)) = −2κy + O(y 2 ), y → 0− .

(VI.41)

(VI.42)

Proof of Lemma VI.3. (VI.36) and (VI.37) are consequences of (VI.10), (VI.12) with V = 0. We have also by (VI.11) : u1(4) (t, x) = u1(4) (τ (x+t), z(τ (x+t)))= −(+)

1 u3(2) (τ (x+t), z(τ (x+t))). λ(τ (x + t))

Hence (VI.39) and (VI.38) follow from (VI.36). The properties of function τ are established in [2], Proposition I.2. We denote U0 (t) the unitary group on L2∞ solving the Cauchy problem associated with the hyperbolic system on R2 : ∂u ∂u +L = 0, t ∈ R, x ∈ R, ∂t ∂x

(VI.43)

with inﬁnitesimal generator 1 H0,∞ := iL∂x , D(H0,∞ ) = H∞ .

We introduce the subspaces of the left and right propagating ﬁelds : L2in = f ∈ L2∞ ; f2 = f3 = 0, x < 0 ⇒ f (x) = 0, , L2out = f ∈ L2∞ ; f1 = f4 = 0, x < 0 ⇒ f (x) = 0, .

(VI.44)

(VI.45) (VI.46)

Proposition VI.4. Assume f ∈ L2in . Then the strong limit : 2 Ωin V f = lim UV (0, T )U0 (T )f in L0 T →+∞

exists and deﬁnes an isometry from L2in to L20 .

(VI.47)

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Creation of Fermions at the Charged Black-Hole Horizon

1069

Proof of Proposition VI.4. Since UV (0, T ) and U0 (T ) are isometric it is suﬃcient to consider f ∈ L2in ∩ [C0∞ (]0, R[)]4 . For such a data we have U0 (T )f = U0 (T, 0)f = f (T + .), hence applying (VI.33), (VI.34) we obtain : d (UV (0, T )U0 (T )f ) = UV (0, T )V U0 (T )f, dT and by (VI.15) we conclude that ∞ ∞ d UV (0, T )U0 (T )f dT ≤ sup{| V (x) |; x < −T }dT < ∞. dT 0 −R

The result follows from the Cook’s method. We now state the main result of this part. Given f ∈ L2∞ we denote f T (x) = f (x + T ).

(VI.48)

Theorem VI.5. For f ∈ L2out we have : −1

2π 2π f >L2∞ . lim 1[+,∞[ (HV,0 ) UV (0, T )f T 2 =< f, e κ H0,∞ 1 + e κ H0,∞

T →∞

(VI.49)

The proof is rather long and technical, so we begin by sketching the main steps of the method. The idea consists in comparing UV (0, T )f T with fT∗ deﬁned by :   1 − | κx |− 2 f3 (2T + κ1 ln(−x) − κ1 ln(A))   0 , fT∗ (x) =  (VI.50)   0 1 | κx |− 2 f2 (2T + κ1 ln(−x) − κ1 ln(A)) i.e. we rigorously justify the approximation of the geometrical optics. We prove in Lemma VI.8 that U0 (0, T )f T ∼ fT∗ . We explicitly calculate 1[0,∞[ (H0,0 ) fT∗ by Fourier transform in Lemma VI.6, and we show in Lemma VI.7 that 1[+,∞[ (HV∞ ,0 ) fT∗ tends to the same limit. To replace 1[+,∞[ (HV∞ ,0 ) by 1[+,∞[ (HV,0 ) we establish in Lemma VI.10 that the diﬀerence between these both operators is compact. At last, we make the link with UV (0, T )f T and U0 (0, T )f T by using the fast decay of V as x → −∞.

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Lemma VI.6. For f ∈ L2out we have : fT∗ D 0 in L20 − weak∗, T → ∞,

(VI.51)

moreover, for any T > 0 we have : fT∗ = f , −1 2π 2π 1[0,∞[ (H0,0 ) fT∗ 2 =< f, e κ H0,∞ 1 + e κ H0,∞ f >L2∞ .

(VI.52)

(VI.53)

Proof of Lemma VI.6. We may assume that f is smooth and compactly supported. By the change of variables y = 2T + κ1 ln(−x) − κ1 ln(A) we directly obtain (VI.52) and √ κ | fT∗ (x) | dx = e−κT Aκ e 2 y | f (y) | dy → 0, T → ∞, which implies (VI.51). To prove the key identity (VI.53) we introduce a map P from L20 into L2∞ by putting for g = P f : x ≤ z(0) ⇒ g(x) = f (x),

(VI.54)

 g1 (x) = −f3 (2z(0) − x),     g (x) = f (2z(0) − x), 2 4 x ≤ z(0) ⇒  g (x) = −f 3 1 (2z(0) − x),    g4 (x) = f2 (2z(0) − x).

(VI.55)

For f ∈ W01 we have : L

d d P f = P L f. dx dx

(VI.56)

Hence, using the Fourier transform F(ϕ) = ϕ, ˆ we see that H0,0 is unitarily equivalent to the operator : ξ − √ L, 4π densely deﬁned on the Hilbert space : 4 L2 := fˆ ∈ L2 (Rξ ) ; fˆ1 (ξ) + fˆ3 (−ξ) = fˆ2 (ξ) − fˆ4 (−ξ) = 0 . ∗

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Creation of Fermions at the Charged Black-Hole Horizon

We have

1071



 0 0 0 1[0,∞[ (ξ)   0 1]−∞,0] (ξ) 0 0  FP. 1[0,∞[ (H0,0 ) = P −1 F −1    0 0 1]−∞,0] (ξ) 0 0 0 0 1[0,∞[ (ξ) (VI.57)

Hence : 2π

1[0,∞[ (H0,0 ) fT∗

= lim+ Aκ ε→0

0

|

∞

| F(fT∗ )(ξ) |2 dξ

0 ∞

ei(A+iε)ζe

κJy

−∞

κ

e 2 y f (y)dy |2 dζ

1 κ f (y1 ).f¯(y2 )dy1 dy2 ε→0 R×R ε cosh 2 (y1 − y2 ) − iA sinh 2 (y1 − y2 ) (VI.58) ∞ 1 Aκ (−ξ)dξ. = lim+ | fˆ(ξ) |2 F ε cosh( κ2 x) − iA sinh( κ2 x) ε→0 4π −∞

= lim+

Aκ 2

∞

= 2

κ

Now given ε = 0, ξ < 0 and N > 0, M > 0, we evaluate ) e−ixξ h(x)dx, h(x) := , κ ε cosh( 2 x) − iA sinh( κ2 x) along the path {−N ≤ !x ≤ N, %x = 0, M } ∪ {0 ≤ %x ≤ M, !x = ±N }. First we have :   ∞   ±N+iM κ   h(x)dx ≤ Ce− 2 N exξ dx → 0, N → ∞,    ±N 0    N+iM  ∞ κ   h(x)dx ≤ CeM ξ e− 2 |x| dx → 0, M → ∞.   −N+iM  −∞ We deduce that

∞

h(x)dx = 2iπ −∞

∞

ρn (ε)

n=1

where ρn (ε) are the residues of h(x) at the poles zn (ε) ∈ {z ∈ C; %z > 0}. We easily check that : 2i ε zn (ε) = nπ − arctan( ) , κ A 2nπ 2i sup | ρn (ε) − (−1)n e κ ξ |≤ Cε, Aκ 1≤n

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Ann. Henri Poincar´ e

hence we get that for ξ < 0 we have :  −1    2π 4π 2π 1 ξ ξ  ≤ Cε. F (ξ) − e κ 1+e κ κ κ   ε cosh( 2 x) − iA sinh( 2 x) Aκ In the same manner, for ξ > 0, we choose M < 0 and considering the poles zn (ε) ∈ {z ∈ C; %z < 0} we obtain :

∞

h(x)dx = 2iπ −∞

−∞

ρn (ε),

n=0

2nπ 2i (−1)n e− κ ξ |≤ Cε, Aκ n≤0  −1    1 4π − 2π ξ F  ≤ Cε. (ξ) − 1+e κ κ κ   ε cosh( 2 x) − iA sinh( 2 x) Aκ

sup | ρn (ε) −

Finally we conclude that : −1 2π 1 4π 2π ξ κ κ ξ F (ξ) = e 1 + e , 0 − iA sinh( κ2 x) Aκ

(VI.59)

and 1[0,∞[ (H0,0 ) fT∗ 2 = ∞ −1 −1 2π 2π 2π 2π 1 e− κ ξ 1 + e− κ ξ | fˆ(ξ) |2 dξ =< f, e κ H0,∞ 1 + e κ H0,∞ f >L2∞ . 2π −∞ Lemma VI.7. For f ∈ L2out , we have : + + + 1[+,∞[ (HV∞ ,0 ) fT∗ − 1[0,∞[ (H0,0 ) fT∗ + → 0, T → ∞.

(VI.60)

Proof of Lemma VI.7. We have : 1[+,∞[ (HV∞ ,0 ) = 1[0,∞[ (HµΓ,0 ) .

(VI.61)

We consider the case µ > 0. We introduce the self-adjoint operators on L2∞ : 1 HµΓ := iL∂x − µΓ, D(HµΓ ) = H∞ ,

− KµΓ := H− µΓ,0 ⊕ HµΓ,0 , D(KµΓ,0 ) = D(HµΓ,0 ) ⊕ D(HµΓ,0 ),

(VI.62)

(VI.63)

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Creation of Fermions at the Charged Black-Hole Horizon

1073

2 4 where H− deﬁned by : µΓ,0 is the self-adjoint operator on L (−∞, z(0)) H− µΓ,0 := iL∂x − µΓ,

(VI.64)

f1 (z(0)) = f3 (z(0)), f2 (z(0)) = −f4 (z(0))} .

(VI.65)

0 ⊕ 1[0,∞[ (HµΓ,0 ) (fT∗ ) = 1[0,∞[ (KµΓ ) (0 ⊕ fT∗ ) .

(VI.66)

4 4 ) = f ∈ L2 (−∞, z(0)) ; f ∈ L2 (−∞, z(0)) , D(H− µΓ,0

We have :

For f ∈ D(HµΓ ) we evaluate : HµΓ f 2 = f 2 +µ2 f 2 , and for f ∈ D(HµΓ,0 ) we get : HµΓ,0 f 2 = f 2 +µ2 f 2 +iµ < ΓLf (z(0)), f (z(0)) >C4 . We deduce from (VI.18) that : ∀f ∈ D(KµΓ ), KµΓ f ≥ µη f . Hence choosing χ ∈ C ∞ (R), such that : t ≤ 0 ⇒ χ(t) = 0, µη ≤ t ⇒ χ(t) = 1, we have : 1[0,∞[ (HµΓ ) = χ(HµΓ ), 1[0,∞[ (KµΓ ) = χ(KµΓ ). We remark that

(HµΓ + i)−1 − (KµΓ + i)−1

is of ﬁnite rank, thus compact on L2∞ . Hence by the Weyl criterion (see e.g. [15], Theorem B.1.1) the operator 1[0,∞[ (HµΓ ) − 1[0,∞[ (KµΓ ) = χ(HµΓ ) − χ(KµΓ ) is compact.

(VI.67)

Then we deduce from (VI.51), (VI.61), (VI.66) that : 0 ⊕ 1[+,∞[ (HV∞ ,0 ) (fT∗ ) − 1[0,∞[ (HµΓ ) (0 ⊕ fT∗ ) → 0, T → ∞.

(VI.68)

We calculate this last projector using the Fourier transform : % $ 1 1 −1 − (ξL + µΓ) F, 1[0,∞[ (HµΓ ) = F 2 2 ξ 2 + µ2 F (fT∗ ) (ξ) =

√

κAe−κT ϕ Ae−2κT ξ , ϕ(ζ) :=

∞

−∞

κy

κ

eiζe e 2 y f (y)dy.

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We obtain :

1[0,∞[ (H0,∞ ) − 1[0,∞[ (HµΓ ) (0 ⊕ fT∗ ) 2   +2 ++ + ζ ζ 1 + +  | ϕ(ζ) |2 dζ ≤ C + − + + 4κT 2 + ζ 2 + A2 e−4κT µ2 | ζ |+ e ζ + a2 µ2

(VI.69)

−→ 0, T → ∞. Since (fT∗ )2 = (fT∗ )3 = 0 we have : 1[0,∞[ (H0,∞ ) (0 ⊕ fT∗ ) 2 =

1 2π

∞

| F (fT∗ ) (ξ) |2 dξ = 1[0,∞[ (H0,0 ) fT∗ 2 ,

0

(VI.70)

hence (VI.60) follows from (VI.68), (VI.69) and (VI.70). Lemma VI.8. For f ∈ L2out we have : U0 (0, T )f T − fT∗ → 0, T → ∞,

(VI.71)

U0 (0, T )f T D 0 in L20 − weak∗, T → ∞.

(VI.72)

Proof of Lemma VI.8. Given ε > 0 we choose g ∈ L2out , continuous, compactly supported in [0, R] such that f − g ≤ ε. Then for any T > 0 we have

fT∗ − gT∗ ≤ ε,

hence we need to prove that : U0 (0, T )g T − gT∗ → 0, T → ∞. Thanks to Lemma VI.3 we have for T > (R − z(0))/2 : U0 (0, T )g T 2 (x) = U0 (0, T )g T 3 (x) = 0, U0 (0, T )g T 1(4) (x) = ' 2 + 2κx + O(x2 ) 1 1 g ln(−x) − ln(A) + O(x) . x + 2T + −(+) 3(2) −2κx + O(x2 ) κ κ We deduce that : U0 (0, T )g − T

gT∗

=κ 2

∞

∞

| g y + O eκy−2κT − g(y) |2 dy → 0, T → ∞,

and (VI.72) follows from (VI.51) and (VI.71).

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Creation of Fermions at the Charged Black-Hole Horizon

1075

We prove now a result of H 1 -regularity of the solution. We essentially show that the polarized wave front set propagates according to the Hamilton ﬂow and the usual law of reﬂection of singularities on the moving boundary. If the data of our problem, V (x) and z(t) were C ∞ , we could invoke the deep general theorems of Ivri˘ı [25]. Since V and z are less regular, we prefer to give an elementary proof for the solution of our simple system. We introduce the unitary group on L2∞ associated with HV,∞ : UV (t) := eitHV,∞ .

(VI.73)

Lemma VI.9. For f ∈ L2∞ satisfying f2 = f3 = 0, x < a ⇒ f (x) = 0, we have for any T > 0 : 1{x
(VI.74)

Moreover, if z(T ) < a, we have : 1{x
(VI.75)

Proof of Lemma VI.9. We denote u(t, x) :=[UV (t)f ] (x), and for g =t (g1 , g2 , g3 , g4 ), we put : [g]14 =t (g1 , 0, 0, g4 ), [g]23 =t (0, g2 , g3 , 0). We have : (∂t − ∂x )[u]14 + i[V u]14 = 0, x < a ⇒ [u]14 (0, x) = 0 (∂t + ∂x )[u]23 + i[V u]23 = 0, [u]23 (0, .) = 0.

(VI.76) (VI.77)

We get for all x : [∂x u]23 (−T, x) = 0 i [V u]23 (s, x + s + T ) + [V [∂x u]14 ]23 (s, x + s + T ) + [V [∂x u]23 ]23 (s, x + s + T )ds. −T

We deduce from (VI.76) that : [∂x u]14 (s, x + s + T ) =

1 d i [u(s, x + s + T )]14 + [V u]14 (s, x + s + T ), 2 ds 2

1076

A. Bachelot

Ann. Henri Poincar´ e

hence integration by parts gives : [∂x u]23 (−T, x) i i = [V (x + T )[f (x + T )]14 ]23 − [V (x)[u(−T, x)]14 ]23 2 2 0 i 1 +i [V u]23 + [V [V u]14 ]23 − [V [u]14 ]23 + [V [∂x u]23 ]23 (s, x + s + T )ds. 2 2 −T Since UV (t) is unitary and V and V are bounded, we get : 0 [∂x u]23 (s) ds, [∂x u]23 (−T ) ≤ C(1 + T ) f +C −T

and we conclude by the Gronwall Lemma that : [∂x u]23 (−T ) ≤ CT f .

(VI.78)

On the other hand we deduce from (VI.76) that for x < a + T we have : [∂x u]14 (−T, x) 0 =i ([V u]14 + [V [∂x u]23 ]14 + [V [∂x u]14 ]14 ) (s, s − s − T )ds. −T

Then we deduce from (VI.78) that 1{x
0

−T

1{x
hence using one more the Gronwall Lemma we obtain : 1{x
(VI.79)

and (VI.74) is established. To prove (VI.75) we choose α, β ∈ R, χ ∈ C ∞ (R) satisfying : z(T ) + T < α < β < a + T, x ≤ α ⇒ χ(x) = 0, x ≥ β ⇒ χ(x) = 1, and we deﬁne : v(t, x) = [UV (t, T )f ] (x) − χ(x + t) [UV (t − T )f ] (x). v is a solution to v(T, x) = 0, x > z(t) ⇒ ∂t v(t, x) = iHV,t v(t, x) + h(t), v2 (t, z(t)) = λ(t)v4 (t, z(t)), v3 (t, z(t)) = −λ(t)v1 (t, z(t)),

Vol. 1, 2000

Creation of Fermions at the Charged Black-Hole Horizon

with

1077

h(t, x) := χ (x + t)L [UV (t − T )f ] (x).

(VI.74) implies that : 1 ≤ CT f . sup h(t) H∞

h(t) ∈ D(HV,t ),

0≤t≤T

Whence

t

UV (t, s)h(s)ds ∈ Ht1 ,

v(t) = T

sup v(t) Ht1 ≤ CT f ,

0≤t≤T

and we get (VI.75) using (VI.74) again. Lemma VI.10. The operator 1[+,∞[ (HV,0 ) − 1[+,∞[ (HV∞ ,0 )

(VI.80)

is compact on L20 . Proof of Lemma VI.10. We have : 1[+,∞[ (HV,0 ) − 1[+,∞[ (HV∞ ,0 ) = 1[0,∞[ (H1 ) − 1[0,∞[ (H2 ), H1 := HV ++Id,0 , H2 := HµΓ,0 . The identities : a ∈ R∗ ,

a = |a|

0 ≤ t, e−a

2

t

∞

2 a √ e−a t dt, πt 0 ) e−z 1 dz, =− 2iπ γ a2 t − z

where γ is the path in the complex plane given by : 3π iθ π <θ< ∪ {z = α ± i; 0 ≤ α} , γ= z=e ; 2 2 yield : 1[0,∞[ (H1 ) − 1[0,∞[ (H2 ) =

i 3

4π 2

0

∞

,

−1

−1 dt dz √ . e−z H1 tH21 − z − H2 tH22 − z t γ

)

(VI.81)

We have : H2 = H1 + V1 , V1 := V + 9Id − µΓ, H22 = H21 + W, W := µ2 Id − (V + 9)2 + iLV + i[L(V + 9) + (V + 9)L]∂x ,

1078

A. Bachelot

Ann. Henri Poincar´ e

and by (VI.16), (VI.17) : | V1 (x) | + | µ2 Id − (V (x) + 9)2 + iLV (x) | + | L(V (x) + 9) + (V (x) + 9)L |→ 0, x → ∞.

(VI.82)

For t > 0, z ∈ γ we consider :

−1

−1 A(t, z) := H1 tH21 − z − H2 tH22 − z

−1 2

−1

−1 W tH2 − z − V1 tH22 − z = tH1 tH21 − z .

−1 Since tH22 − z is bounded from L20 to H02 , the Sobolev embedding and the decay

−1 2 2 estimates (VI.82) imply that W is compact from H0 to L0 and W tH22 − z

−1

−1 is bounded on are compact on L20 . Since tH1 tH21 − z and V1 tH22 − z L20 we conclude that A(t, z) is compact on L20 . Now to prove the Lemma, it suﬃces to establish that | e−z | √ A(t, z) L(L20 ) ∈ L1 (R+ t × γz ). t

(VI.83)

On the one hand :

−1 1 1 L(L20 ) ≤ sup | λ(tλ2 − z)−1 |= Hj tH2j − z ≤ √ , 2 t 2t(| z | −!z) λ∈R (VI.84) hence 0 < t, z ∈ γ, A(t, z) L(L20 ) ≤

1 . t

(VI.85)

On the other hand we write :

−1 2

−1

−1 W tH2 − z − V1 tH22 − z . A(t, z) = t H1 tH21 − z We have :

−1 L(L20 ) ≤ 1, 0 < t, z ∈ γ, tH22 − z

and we deduce from (VI.84) that :

−1 1 tH22 − z L(L20 ,H01 ) ≤ 1 + √ . t Since W ∈ L(H01 , L20 ) we get : 0 < t, z ∈ γ, A(t, z) L(L20 ) ≤ C(1 +

√ t),

(VI.86)

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Creation of Fermions at the Charged Black-Hole Horizon

1079

where C > 0 does not depend on t > 0, z ∈ γ. Therefore (VI.85), (VI.86) yield sup A(t, z) L(L20 ) ≤ z∈γ

2C , 1+t

and (VI.83) is established Proof of Theorem VI.5. For 0 < tε < T we write UV (0, T )f T = OT (ε) + UV (0, tε )gε,T ,

(VI.87)

with OT (ε) := UV (0, tε ) (UV (tε , T ) − U0 (tε , T )) f T , gε,T := U0 (tε , T )f T . If f is supported in [0, R], then UV (s, T )f T is supported in [z(s), −s + R], hence

T

OT (ε) ≤

V UV (s, T )f T ds ≤ f tε

∞

sup{| V (x) |; x ≤ −s + R}ds.

tε

Then, given ε > 0, we choose tε such that : sup OT (ε) ≤ ε.

T ≤tε

(VI.88)

Now for r ≥ 0 we deﬁne θr,T as the unique solution to z(θr,T ) − θr,T = r − 2T. By Lemma VI.3 we have : θr,T = T −

r + O(e−2κT ), 2

(VI.89)

and for 0 ≤ tε ≤ θR,T , gε,T is supported in [2(θ0,T − T ) − tε , 2(θR,T − T ) + R − tε ] ⊂ [−tε − O(e−2κT ), −tε ] and [gε,T ]23 = 0. We introduce the function Yε given by x < z(tε ) + tε ⇒ Yε (x) = 0, z(tε ) + tε ≤ x ⇒ Yε (x) = 1. The compact embedding H 1 ⊂ L2 with (VI.72) and Lemma VI.9, assure that : (1 − Yε )UV (0, tε )gε,T → 0, T → ∞,

(VI.90)

(1 − Yε )UV (−tε )gε,T → 0, T → ∞.

(VI.91)

We remark that : Yε UV (0, tε )gε,T = Yε UV (−tε )gε,T .

1080

A. Bachelot

Ann. Henri Poincar´ e

Therefore, thanks to (VI.88), (VI.90), it is suﬃcient to prove that : 2π

lim 1[+,∞[ (HV,0 ) (Yε UV (−tε )gε,T ) 2 =< f, e κ

H0,∞

T →∞

−1 2π 1 + e κ H0,∞ f >L2∞ . (VI.92)

In the sequel, we write u ∼ v for u, v ∈ C 0 (RT , L2∞ ) iﬀ u(T ) − v(T ) → 0 when T → ∞. Since (VI.72) implies that Yε UV (−tε )gε,T tends weakly to 0 in L20 , we deduce from Lemma VI.10 and (VI.61) that : 1[+,∞[ (HV,0 ) (Yε UV (−tε )gε,T ) ∼ 1[0,∞[ (HµΓ,0 ) (Yε UV (−tε )gε,T ) .

(VI.93)

We have also : 0 ⊕ 1[0,∞[ (HµΓ,0 ) (Yε UV (−tε )gε,T ) = 1[0,∞[ (KµΓ ) (Yε UV (−tε )gε,T ) .

(VI.94)

We deduce from (VI.67) that : 1[0,∞[ (KµΓ ) (Yε UV (−tε )gε,T ) ∼ 1[0,∞[ (HµΓ ) (Yε UV (−tε )gε,T ) ,

(VI.95)

and with (VI.91) : 1[0,∞[ (HµΓ ) (Yε UV (−tε )gε,T ) ∼ 1[0,∞[ (HµΓ ) (UV (−tε )gε,T ) .

(VI.96)

We introduce a potential Vε on R given by : x ≥ −2tε ⇒ Vε (x) = V (x), x ≤ −2tε ⇒ Vε (x) = V (2tε − x). Then the ﬁnite speed of propagation implies that : UVε (−tε )gε,T = UV (−tε )gε,T . Now putting H1 := HVε ++Id , H2 := HµΓ in the proof of Lemma VI.10, we get that 1[0,∞[ (HµΓ ) − 1[+,∞[ (HVε ) is compact on L2∞ . Then we obtain : 1[0,∞[ (HµΓ ) (UV (−tε )gε,T ) ∼ 1[+,∞[ (HVε )UVε (−tε )gε,T .

(VI.97)

We remark that : 1[+,∞[ (HVε )UVε (−tε )gε,T = 1[+,∞[ (HVε )gε,T ,

(VI.98)

and by the previous argument, we have again : 1[+,∞[ (HVε )gε,T ∼ 1[0,∞[ (HµΓ )gε,T . Now (VI.71) implies :

gε,T ∼ U0 (tε )fT∗

(VI.99)

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Creation of Fermions at the Charged Black-Hole Horizon

1081

Since U0 (tε )fT∗ (x) = fT∗ (x + tε ) and HµΓ commutes with ∂x , we have 1[0,∞[ (HµΓ )U0 (tε )fT∗ = 1[0,∞[ (HµΓ )fT∗ and we get with (VI.69), (VI.70) and Lemma VI.53 that : −1 2π 2π f >L2∞ , T → ∞. (VI.100) 1[0,∞[ (HµΓ )gε,T →< f, e κ H0,∞ 1 + e κ H0,∞ We conclude that (VI.92) is a consequence of (VI.93) to (VI.100).

Corollary VI.11. Given fout ∈ L2out , fin ∈ L2in , we have for J = [9, ∞[ or J = ]9, ∞[ :

2 T T lim 1J (HV,0 )UV (0, T ) fout + fin T →∞ −1 2π 2π =< fout , e κ H0,∞ 1 + e κ H0,∞ fout >L2∞ in 2 + 1J (HV,0 ) ΩV fin . (VI.101) Proof of Corollary VI.11. Since (VI.72) implies that gε,T tends weakly to 0 in L2tε as T → ∞, (VI.87) and (VI.88) assure that T D 0 in L20 − weak∗, T → ∞, UV (0, T )fout

(VI.102)

and because 1{0} (HV,0 ) is ﬁnite rank, we have : T = 0. lim 1{0} (HV,0 )UV (0, T )fout

T →∞

Then, using : T = UV (0, T )U0 (T )fin , UV (0, T )fin

the result follows from (VI.102), Proposition VI.4 and Theorem VI.5.

VI.2

Proofs of the asymptotic estimates

At present we are able to investigate the 3D+1 problem, so we return to the proofs of the results of Part III and IV. Proof of Lemma III.1. For 0 ≤ t ≤ ∞, we expand Φ ∈ L2t in the following way :   u1,n (x)T− 1 ,n (ϕ, θ) 2   α  u2,n (x)T+ 1 ,n (ϕ, θ)  −i ,n γ5   2 2 (VI.103) Φ= e  u3,n (x)T 1 (ϕ, θ)  , ,n −   ,n 2 u4,n (x)T+ 1 ,n (ϕ, θ) 2

1082

A. Bachelot

and we introduce :



R,n

Ann. Henri Poincar´ e



u1,n  u2,n :  u3,n u4,n

 u1,n (x)  u2,n (x)  2  Il,n : Φ ∈ L2t −→   u3,n (x)  ∈ Lt , u4,n (x)   u1,n (x)T− 1 ,n (ϕ, θ) 2  α,n 5  u  2,n (x)T+ 1 ,n (ϕ, θ) 2 −i γ  ∈ Lt −→ e  2 2  u3,n (x)T 1 (ϕ, θ)  − 2 ,n  u4,n (x)T+ 1 ,n (ϕ, θ)

(VI.104)     ∈ L2t .  

2

We make the link with the study of the one dimensional problem in the previous section by putting : & &

R,n L2t , Ht = R,n HV,n ,t − qA(r0 )Id I,n , (VI.105) L2t = ,n

,n

where HV,n ,t is given by (VI.26) and (VI.9). We note that the hypotheses (VI.13) to (VI.18) are satisﬁed for the choices : (VI.106) κ = κ0 , µ = m F (r+ ), 9 = qA(r0 ), η = inf(1, 1 + cos α,n ),   Γ= 

0 0

−iα,n

−ie

0 0 0

−ie−iα,n

0

ieiα,n 0 0 0

0



ieiα,n  .  0 0

(VI.107)

Hence the result follows from Lemma VI.1, in particular we have ˙ Φ(z(t), .) 2L2 (S 2 ) . 2! (iHt Φ, Φ) = z(t)

(VI.108)

Proof of Proposition III.2. The conservation law (III.23) is a consequence of (VI.108), and the existence of the solution is obtained by taking : & R,n UV,n (t, s)I,n . (VI.109) U (t, s) = ei(s−t)qA(r0 ) ,n

Then the result follows from Proposition VI.2. Proof of Proposition III.3. We remark that : & & L2BH = R,n L2in , L2out = R,n L2out , ,n

,n

(VI.110)

Vol. 1, 2000

HBH Φ =

Creation of Fermions at the Charged Black-Hole Horizon

&

1083

&

R,n H0,∞ − qA(r0 )Id I,n , UBH (t) = e−iqA(r0 )t R,n U0 (t)I,n .

,n

,n

(VI.111) Therefore, since U (0, T )UBH (T ) =

&

R,n UVl,n (0, T )U0 (T )I,n ,

,n

the existence of the wave operator follows from Proposition VI.4 by putting : & R,n Ωin (VI.112) ΩBH = V,n I,n . ,n

Proof of Theorem III.4. We apply Corollary VI.11 and the dominated convergence theorem to get :

1J (H0 )U (0, T ) ΦTout + ΦTBH 2

= 1J (HV,0 )UV,n (0, T ) I,n ΦTout + I,n ΦTBH 2 ,n

−→

T →∞

−1 2π 2π < I,n Φout , e κ0 H0,∞ 1 + e κ0 H0,∞ I,n Φout >L2∞

,n

2 + 1J (HV,n ,0 ) Ωin I Φ ,n BH V,n −1 2π 2π =< Φout , ζe κ0 HBH 1 + ζe κ0 HBH Φout >L2∞ + 1J (H0 ) (ΩBH ΦBH ) 2 . Proof of Theorem IV.1. By the identity of polarization it is suﬃcient to consider Φ10 = Φ20 = Φ0 and we assume that : supp Φ0 ⊂ (t, x, ω) ∈ [−R, +R] × [0, R] × S 2 ; 0 < x + t . For T > 0 we introduce the map TT : F ∈ L2∞ −→ (TT F ) (x, ω) = F (x + T, ω). Lemma VI.12. R U (T + R, T + t)TT Φ0 (t)dt −R −TT +R e+iqA(r0 )R UBH (2R)SBH P out Φ0 +e−iqA(r0 )R SBH P in Φ0 L2R+T −→ 0, T → ∞.

(VI.113)

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A. Bachelot

Ann. Henri Poincar´ e

Proof of Lemma VI.12. We write by using (VI.109) :

R

−R

U (T + R, T + t)TT Φ0 (t)dt =

R,n

,n

R

−R

eiqA(r0 )(t−R) UV,n (T + R, T + t)I,n TT Φ0 (t)dt,

and we get by (VI.111) and (IV.11) : TT +R e+iqA(r0 )R UBH (2R)SBH P out Φ0 + e−iqA(r0 )R SBH P in Φ0 = R R,n eiqA(r0 )(t−R) U0 (R − t)I,n TT Φ0 (t)dt. ,n

−R

The hypothesis on the support of Φ implies that : UV,n (T + R, T + t)I,n TT Φ0 (t) = UV,n (R − t)I,n TT Φ0 (t), and the Duhamel formula with (II.11) assures that : UV,n (R − t)I,n TT Φ0 (t) − U0 (R − t)I,n TT Φ0 (t) L2∞ ≤ C(R, Φ)e−κ0 T .

Hence we easily deduce (VI.113).

To prove the Theorem, we get by (IV.7), Theorem III.4, Lemma VI.12, (IV.9) and (IV.4) : ωM (Ψ∗T (ΦT )ΨT (ΦT ))

∞

U (0, t)ΦT (t)dt 2 ∞ = 1]0,∞[ (H0 )U (0, T + R) U (T + R, t + T )TT Φ0 (t)dt 2

= 1]0,∞[ (H0 )

−∞

−∞

−→ < SBH P Φ0 , e e κ0 HBH T →∞ −1 2π 2π 1 + e κ0 qA(r0 ) e κ0 HBH SBH P out Φ0 >L2∞

+ 1]0,∞[ (H0 ) ΩBH P in SBH Φ0 2 2π ∗

κ0 ,qA(r0 ) ΨBH (P out Φ0 )ΨBH (P out Φ0 ) = ωBH ∗

0 Ψ0 (ΩBH P in SBH Φ0 )Ψ0 (ΩBH P in SBH Φ0 ) . + ωΣ 0 out

2π κ0

qA(r0 )

2π

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Creation of Fermions at the Charged Black-Hole Horizon

1085

VII Conclusion We have considered a charged Dirac ﬁeld outside a spherical charged star, stationary in the past and collapsing to a black hole in the future. The interaction between the ﬁeld and the matter of the star is subsumed in a boundary condition belonging to a large class. We have rigorously established the famous result on the thermalization of the vacuum by the collapse : if the ground quantum state in the past is the Boulware vacuum, then, this state becomes of Unruh type near the future black-hole horizon. Moreover the temperature and the chemical potential are independent of the history of the collapse and of the boundary condition (in the class that we introduced). A static observer at inﬁnity interprets this state as a stream of particules and antiparticles outgoing from the black hole to inﬁnity. Furthermore, the black-hole preferentially emits fermions whose charge is of same sign as its own charge, rather than fermions of opposite charge. We have investigated the rather subtle role of the cosmological constant in the case of the DeSitter-Reissner-Nordstrøm Black-Hole : in the case of a weakly charged black hole in an expanding universe, the temperature is an increasing function of the charge, unlike the asymptotically ﬂat case; in the case of a strongly charge, 24M 2 < 25Q2 < 25M 2 , the temperature is an increasing function of the cosmological constant. We have only studied the two-point function which carries the information on the vacuum ﬂuctuations. A subsequent work will be devoted to the investigation of the stress energy momentum tensor. Another interesting problem consists in treating the matter of the star (ﬂuid or dust) instead of considering a boundary condition. It goes without saying that we leave open the huge problem of the back reaction of this vacuum polarization on the metric, nevertheless we make some comments on the subject. The previous remarks suggest that the black-hole loses mass and charge [8]. Since the propagator of the Dirac system is unitary, there is no supperadiance of fermion ﬁelds [12], despite the existence of the generalized ergosphere for classical particles [14]. Therefore we may expect that the rate of the spontaneous loss of charge of the black-hole through charged fermion ﬁelds, is weak in the semiclassical regime, unlike the scalar case for wich supperradiant modes appear [21]. All these conjectures require the solution of monstrously non linear problems.

A Second Quantization of the Dirac Fields To be able to construct the Boulware vacuum in the past, and the thermal state at the horizon, we describe here the essential features of the quantization of the Dirac ﬁeld, convenient for the stationary space-times (for more details in the case of the ﬂat space, see e.g. [6],[7],[9],[23],[38],[39]). We ﬁrst consider the case of one kind of non interacting fermions. Let h be a complex Hilbert space (the one-fermion space), and we denote <, > its scalar prod-

1086

A. Bachelot

Ann. Henri Poincar´ e

uct, linear with respect to the ﬁrst argument. We deﬁne the space of n-fermions as the antisymmetric n-tensor product of h : n .

F(0) (h) := C, 1 ≤ n ⇒ F(n) (h) :=

h,

(A.1)

ν=1

and the F ermi − F ock space : F(h) :=

∞ &

F(n) (h).

(A.2)

n=0

For f ∈ h we construct the fermion annihilation operator ah (f ), and the fermion creation operator a∗h (f ) by putting : ah (f ) : F(0) (h) → {0}, 1 ≤ n, ah (f ) : F(n) (h) → F(n−1) (h),

(A.3)

√ n ah (f ) (f1 ∧ ... ∧ fn ) = ε(σ) < fσ(1) , f > fσ(2) ∧ ... ∧ fσ(n) , n! σ

(A.4)

0 ≤ n, a∗h (f ) : F(n) (h) → F(n+1) (h), √ a∗h (f ) (f1

∧ ... ∧ fn ) =

n+1 ε(σ)f ∧ fσ(1) ∧ ... ∧ fσ(n) , n! σ

(A.5)

(A.6)

where the sum is taken over all the permutations σ of {1, 2, ...n} and ε(σ) is one if σ is even and minus one if σ is odd. We have for f (n) ∈ F(n) (h) : ah (f )(f (n) ) 2 + a∗h (f )(f (n) ) 2 = f 2 f (n) 2 ,

(A.7)

hence ah (f ) and a∗h (f ) have bounded extensions on F(h). Moreover these operators satisfy : ah (f ) = a∗h (f ) = f ,

(A.8)

a∗h (f ) = (ah (f ))∗ ,

(A.9)

and the canonical anti-commutation relations (CAR’s) : ah (f )ah (g) + ah (g)ah (f ) = 0,

(A.10)

a∗h (f )a∗h (g) + a∗h (g)a∗h (f ) = 0,

(A.11)

a∗h (f )ah (g) + ah (g)a∗h (f ) =< f, g > 1.

(A.12)

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The CAR Algebra on h is the C∗ -algebra A(h) generated by the identity 1 and the ah (f ), f ∈ h. There exist interesting operators that do not belong to A(h). For instance, given a closed separable subspace F of h, the number operator NF (n) deﬁned on ∪∞ by n=0 F NF :=

∞

a∗h (fj )ah (fj )

(A.13)

j=0

where (fj )j∈N is an orthonormal basis of F (obviously NF does not depend of the choice of the basis). When the classical ﬁelds obey the Schr¨ odinger type equation dψ = iHψ, dt where H is a selfadjoint operator on h, a gauge-invariant quasi-free state ω on A(h) satisﬁes the (β, µ)-KMS condition, 0 < β, µ ∈ R, if it is characterized by the two-point function ! "

−1 ω(a∗h (f )ah (g)) = zeβH 1 + zeβH f, g (A.14) where z is the activity given by : z = eβµ .

(A.15)

(Note that we have written the Schr¨ odinger/Dirac equation as ∂t ψ = iHψ, instead of the traditional form i∂t ψ = Hψ adopted in [9] or [39]. Hence we must change H into −H to ﬁnd the conventions of these authors.) This state is a model for the ideal Fermi gas with temperature 0 < T = β −1 and chemical potential µ. In statistical mechanics, such a state is called Gibbs grand canonical equilibrium state. In the case of charged spinor ﬁelds, the situation is more intricate since we have to consider both kinds of fermions, the particles and the antiparticles. We consider a complex Hilbert space H (the space of the classical charged spin ﬁelds), and an antiunitary operator C on H (the charge conjugation). We assume H is split into two orthogonal spaces H = H+ ⊕ H − .

(A.16)

h + = H+ ,

(A.17)

h− = CH− .

(A.18)

We deﬁne the one particle space

and the one antiparticle space

Then the space of n particles and m antiparticles is given by the tensor product of the previous spaces : F(n,m) := F(n) (h+ ) ⊗ F(m) (h− ),

(A.19)

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and to be able to treat arbitrary numbers of particles and antiparticles simultaneously, we introduce the Dirac − F ermi − F ock space : F(H) :=

∞ &

F(n,m) .

(A.20)

n,m=0

If we denote the elements ψ of F by a sequence : ψ = (ψ(n,m) )n,m∈N , ψ(n,m) ∈ F(n,m) , the vacuum vector is the vector Ωvac deﬁned by : (n,m) = 0. Ω(0,0) vac = 1, (n, m) = (0, 0) ⇒ Ωvac

(A.21)

Now for ϕ+/− ∈ h+/− we deﬁne the particle annihilation operator, a(ϕ+ ), the particle creation operator, a∗ (ϕ+ ), the antiparticle annihilation operator, b(ϕ− ), (n) (m) the antiparticle creation operator, b∗ (ϕ− ), by putting for ψ+ ⊗ ψ− ∈ F(n,m) : (n) (m) (n) (m) = ah+ (ϕ+ ) ψ+ ⊗ ψ− ∈ Fn−1,m , (A.22) a(ϕ+ ) ψ+ ⊗ ψ− (n) (m) (n) (m) = a∗h+ (ϕ+ ) ψ+ ⊗ ψ− ∈ Fn+1,m , a∗ (ϕ+ ) ψ+ ⊗ ψ−

(A.23)

(n) (m) (n) (m) b(ϕ− ) ψ+ ⊗ ψ− = ψ+ ⊗ bh− (ϕ− ) ψ− ∈ Fn,m−1 ,

(A.24)

(n) (m) (n) (m) = ψ+ ⊗ b∗h− (ϕ− ) ψ− ∈ Fn,m+1 . b∗ (ϕ− ) ψ+ ⊗ ψ−

(A.25)

All these operators have bounded extensions on F(H) and satisfy the CAR’s. The main object of the theory is the quantized Dirac ﬁeld operator Ψ : f ∈ H −→ Ψ(f ) := a(P+ f ) + b∗ (CP− f ) ∈ L(F(H)),

(A.26)

where we have denoted by P+/− the orthogonal projector from H onto H+/− . The mapping f ∈ H −→ Ψ(f ) is antilinear and bounded : Ψ(f ) = f .

(A.27)

∗

Its adjoint denoted by Ψ (f ) is given by Ψ∗ (f ) = a∗ (P+ f ) + b(CP− f ),

(A.28)

and the CAR’s are satisﬁed : Ψ(f )Ψ(g) + Ψ(g)Ψ(f ) = 0,

(A.29)

Ψ∗ (f )Ψ∗ (g) + Ψ∗ (g)Ψ∗ (f ) = 0,

(A.30)

Ψ∗ (f )Ψ(g) + Ψ(g)Ψ∗ (f ) =< f, g > 1.

(A.31)

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The Field Algebra is the C∗ -algebra A(H) generated by 1 and the Ψ(f ), f ∈ H. If we take f only in H+ (−) we get a subalgebra isometric to A(h+(−) ). The vacuum state ωvac on A(H) is deﬁned by A ∈ A(H), ωvac (A) := (AΩvac , Ωvac ) ,

(A.32)

or by the two point function : ωvac (Ψ∗ (f )Ψ(g)) =< P− f, P− g > .

(A.33)

Now we assume the classical ﬁelds to satisfy a Dirac type equation dΨ = iHΨ dt

(A.34)

where H is selfadjoint on H and leaves H+ and H− invariant. Then H+ := H|h+ , H− := −CH|H− C −1 ,

(A.35)

are respectively selfadjoint on h+ and h− , and the classical ﬁelds of one particle, ϕ+ , and of one antiparticle, ϕ− , are solutions to a Schr¨ odinger type equation on h+(−) : dϕ+(−) = iH+(−) ϕ+(−) . dt

(A.36)

A usual splitting of H (with the remark following A.15) is the choice H+ = 1]−∞,0[ (H) H− = 1]0,∞[ (H).

(A.37)

We say that a state ωβ,µ on A(H) satisﬁes the (β, µ)-KMS condition, 0 < β, µ ∈ R, if it is characterized by the two-point function ! "

−1 ωβ,µ (Ψ∗ (f )Ψ(g)) = zeβH 1 + zeβH f, g , z = eβµ . (A.38) The link with the Gibbs equilibrium states for particles and antiparticles is given explicitly in the following : Lemma A.1. Given ϕj+(−) ∈ h+(−) , we have : + + −1 zeβH 1 + zeβH ϕ1+ , ϕ2+ ,

(A.39)

− − −1 z −1 eβH 1 + z −1 eβH ϕ1− , ϕ2− .

(A.40)

ωβ,µ (a∗ (ϕ1+ )a(ϕ2+ )) =

ωβ,µ (b∗ (ϕ1− )b(ϕ2− )) =

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Therefore the restrictions of ωβ,µ to A(h+ ) and to A(h− ), describe a double Gibbs equilibrium state : on the one hand, an ideal Fermi particle gas with temperature 0 < T = β −1 and chemical potential µ, and on the other hand an ideal Fermi antiparticle gas with the same temperature T but an opposite chemical potential −µ. Proof of Lemma A.1. Taking f = ϕ1+ , g = ϕ2+ in (A.38), we obtain (A.39). Choosing f = C −1 ϕ2− , g = C −1 ϕ1− , we get : ωβ,µ (b(ϕ2− )b∗ (ϕ1− ))

! "

−1 −1 2 = zeβH 1 + zeβH C ϕ− , C −1 ϕ1− !

−1 −1 2 " = ϕ1− , zCeβH 1 + zeβH C ϕ− − −1 = ϕ1− , 1 + z −1 eβH ϕ2− .

Then we deduce (A.40) using the normality of the state, ωβ,µ (1) = 1, and the CAR : b∗ (ϕ2− )b(ϕ1− ) + b(ϕ1− )b∗ (ϕ2− ) =< ϕ2− , ϕ1− > 1. We apply these procedures to deﬁne the Boulware state in the past, and the thermal state at the horizon. First the quantization at time t = 0 is deﬁned by choosing H = L20 , H = H0 ,

(A.41)

CΦ = t(Φ4 , Φ3 , −Φ2 , −Φ1 ).

(A.42)

If we stress the charge of the spin ﬁeld q by denoting the Hamiltonian (III.17) by H0 = H0 (q), we remark that H− = H0 (−q), and CΦ satisﬁes the boundary condition (III.14) for Φ ∈ D(H0 ). Hence C is actually a charge conjugation. As regards the deﬁnition of particles and antiparticles, appears a slight ambiguity due to the fact that 0 is a possible eigenvalue of H0 , unlike the case of the whole Reissner-Nordstrøm manifold for which H∞ has no eigenvalue (Lemma III.1). We leave open the problem of the point spectrum of H0 and we choose :

(P+ , P− ) = 1]−∞,0] (H0 ), 1]0,∞[ (H0 ) or 1]−∞,0[ (H0 ), 1[0,∞[ (H0 ) , (A.43) and we denote Ψ0 the quantum ﬁeld at time t = 0 constructed in the previous way. We deﬁne the Boulware quantum state ω0 on the ﬁeld algebra A(L20 ) as the vacuum state : 0

/ Φj0 ∈ L20 , ω0 Ψ∗0 (Φ10 )Ψ0 (Φ20 ) = P− Φ10 , P− Φ20 . (A.44)

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To quantize at the black-hole horizon, we choose HBH := L2∞ , H = HBH ,

(A.45)

P+ = 1]−∞,0[ (HBH ), P− = 1]0,∞[ (HBH ),

(A.46)

C is given by (A.42) again, and ΨBH (Φ) denotes the quantum ﬁeld deﬁned by (A.28). We can easily express these operators using the partial Fourier transform with respect to x, fˆ(ξ, ω) of f (x, ω) ∈ L2 (Rx × Sω2 , dxdω) : ˆ 1 (ξ, ω) = Φ ˆ 4 (ξ, ω) = 0, P+ HBH = Φ ∈ L2∞ ; ξ ≥ qA(r0 ) ⇒ Φ ˆ 2 (ξ, ω) = Φ ˆ 3 (ξ, ω) = 0 , ξ ≤ −qA(r0 ) ⇒ Φ (A.47) ˆ 1 (ξ, ω) = Φ ˆ 4 (ξ, ω) = 0, P− HBH = Φ ∈ L2∞ ; ξ ≤ qA(r0 ) ⇒ Φ ˆ 2 (ξ, ω) = Φ ˆ 3 (ξ, ω) = 0 . ξ ≥ −qA(r0 ) ⇒ Φ

(A.48)

In fact it will be useful to split the ﬁelds into a part outgoing to inﬁnity, and a part falling into the black-hole, as t → +∞, by putting : Hout := Φ ∈ L2∞ ; Φ1 = Φ4 = 0 , (A.49) Hin := Φ ∈ L2∞ ; Φ2 = Φ3 = 0 .

(A.50)

We denote P out and P in the orthogonal projectors from HBH onto Hout and Hin . We are mainly concerned with the outgoing (anti)particles. Let ω out be a state on out A(H ). Given a Lebesgue measurable subset Λ of R×S 2 , of measure | Λ |∈]0, ∞[, we introduce out Hout ; (x, ω) ∈ / Λ ⇒ Φ(x, ω) = 0 . (A.51) Λ := Φ ∈ H We choose an orthonormal basis (Φj )j∈N of Hout Λ , we deﬁne the following numbers : ωout (a∗ (P+ Φj )a(P+ Φj )), (A.52) NΛ+ (ωout ) :=| Λ |−1 j

NΛ− (ωout ) :=| Λ |−1

ωout (b∗ (CP− Φj )b(CP− Φj )),

(A.53)

j

and if these numbers are ﬁnite

9Λ (ωout ) := q NΛ+ (ωout ) − NΛ+ (ωout ) .

(A.54)

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We could understand these quantities as, respectively, the density of particles, the density of antiparticles, the charge density (in Λ). But this interpretation is somewhat misleading since according to the Paley-Wiener theorem P+ Hout Λ ∩ out out = P H ∩ H = {0}, hence no particle and no antiparticle is localized in Hout − Λ Λ Λ Λ. Nevertheless, in the case of the Gibbs states, these quantities do not depend on Λ, hence these concepts are meaningful: Lemma A.2. NΛ± (ωout ) is independent of the choice of the basis (Φj )j∈N . Moreover, out on A(Hout ), we have given a (β, µ) − KM S state ωβ,µ out NΛ+ (ωβ,µ )=

1 1 1 out out )= ) = qµ. ln 1 + eβµ , NΛ− (ωβ,µ ln 1 + e−βµ , 9Λ (ωβ,µ πβ πβ π (A.55)

Proof of Lemma A.2. If (Ψn )n∈N is another orthonormal basis of Hout Λ we have :    NΛ+ (ωout ) =| Λ |−1 < Φj , Ψn > < Φj , Ψm > ωout (a∗ (P+ Ψn )a(P+ Ψm )) n,m −1

=| Λ |

j

ω

out

(a∗ (P+ Ψn )a(P+ Ψn )).

n

By using the Fourier transform and (A.39), (A.47) and (A.49), we calculate out NΛ+ (ωβ,µ )= (A.56) ∞ β(µ−qA(r0 )) −βξ e e 1 ˆ j (ξ) |2 + | Φ ˆ j (ξ) |2 dξ.

|Φ 2 3 2π | Λ | −qA(r0 ) 1 + eβ(µ−qA(r0 )) e−βξ j

Now given fh ∈ L2 (Λ) we have : f2 2 + f3 2 =

|< f2 , Φj2 > + < f3 , Φj3 >|2 ,

j

hence by choosing fk (x) = e−ixξ 1Λ (x), fl = 0 we deduce that : j ˆ (ξ) |2 + | Φ ˆ j (ξ) |2 = 2 | Λ | |Φ 2 3 j out out and we easily get the value of NΛ+ (ωβ,µ ). Finally we obtain NΛ− (ωβ,µ ) thanks to Lemma A.1 by changing µ into −µ.

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References [1] J. Audretsch, V. de Sabbata, editors. Quantum Mechanics in Curved Space-Time, volume 230 of NATO ASI Series B. Plenum Press, 1989. [2] A. Bachelot. Scattering of Scalar Fields by Spherical Gravitational Collapse. J. Math. Pures Appl., 76, 155–210, 1997. [3] A. Bachelot. Quantum Vacuum Polarization at the Black-Hole Horizon. Ann. Inst. Henri Poincar´e - Physique th´eorique, 67 (2), 181–222, 1997. [4] A. Bachelot. The Hawking Eﬀect. Ann. Inst. Henri Poincar´e - Physique th´eorique, 70 (1), 41–99, 1999. [5] A. Bachelot-Motet. Nonlinear Dirac ﬁelds on the Schwarzschild metric. Class. Quantum Grav., 15 (7), 1815–1825, 1998. [6] J. Bellissard. Quantized ﬁelds in interaction with external ﬁelds. I. Exact solutions and perturbative expansion. Comm. Math. Phys., 41, 235–266, 1975. [7] J. Bellissard. Quantized ﬁelds in external ﬁelds. II. Existence theorems. Comm. Math. Phys., 46, 53–74, 1976. [8] N. D. Birrell, P.C.W. Davies. Quantum ﬁelds in curved space. Cambridge University Press, 1986. [9] O. Bratteli, D. W. Robinson. Operator Algebras and Quantum Statistical Mechanics II. Springer Verlag, 1981. [10] P. Candelas. Vacuum polarization in Schwarzschild spacetime. Phys. Rev. D, 21 (8), 2185–2202, 1980. [11] B. Carter. Black Hole Equilibrium States. In Black Holes, Les Astres Occlus, Les Houches 1972, B.S. de Witt, C. de Witt Edts., Gordon and Breach, 1973. [12] S. Chandrasekhar. Mathematical Theory of Black Holes. Oxford University Press, 1983. [13] A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorne, V.F. Weisskopf. New extended model of hadrons. Phys. Rev. D(3), 9 (12), 3471–3495, 1974. [14] G. Denardo, R. Ruffini. On the Energetics of Reissner Nordstrøm Geometries. Phys. Lett., 45B (3), 259–262, 1973. ´ ski, C. G´ [15] J. Derezin erard. Scattering Theory of Classical and Quantum N -particle Systems. Springer Verlag, 1997. [16] J. Dimock. Algebras of Local Observables on a Manifold. Commun. Math. Phys., 77, 219–228, 1980. [17] J. Dimock. Dirac Quantum Fields on a Manifold. Transac. Amer. Math. Soc.,269, 133–147, 1982.

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[18] J. Dimock, B. S. Kay. Classical and Quantum Scattering Theory for Linear Scalar Fields on the Schwarzschild Metric. Ann. Phys. (NY), 175, 366–426, 1987. [19] K. Fredenhagen, R. Haag. On the Derivation of Hawking Radiation Associated with the Formation of a Black Hole. Comm. Math. Phys., 127, 273–284, 1990. [20] S. A. Fulling. Aspects of Quantum Field Theory in Curved Space-Time. Cambridge University Press, 1989. [21] G. W. Gibbons. Vacuum Polarization and the Spontaneous Loss of Charge by Black Holes. Comm. Math. Phys., 44, 245–264, 1975. [22] S. Hawking. Particle Creation by Black Holes. Comm. Math. Phys., 43, 199–220, 1975. [23] B. Helffer, H. Siedentop. Form Perturbations of the Second Quantized Dirac Field. Math. Phys. Electron. J., 4, paper 4, 16 pages. 1998. [24] C. J. Isham. Quantum ﬁeld theory in Curved Space-Times, a general mathematical framework. In Diﬀerential Geometric Methods in Mathematical Physics II, volume 676 of Lecture Notes in Math., pages 459–512. Springer Verlag, 1977. [25] V. Ja. Ivri˘ı. Wave fronts of solutions of boundary value problems for symmetric hyperbolic systems. II. Systems with characteristics of constant multiplicity. (Russian). Sibirsk. Mat. Zh., 20 (5), 1022–1038, 1979. [26] R.L. Jaffe, A. Manohar. Bound states of the Dirac equation outside a hard sphere. Ann. Physics, 192 (2), 321–330, 1989. [27] W. M Jin. Scattering of massive Dirac ﬁelds on the Schwarzschild black hole spacetime. Class. Quantum Grav., 15 (10), 3163–3175, 1998. [28] T. Kato. Linear evolution equations of ”hyperbolic” type. J. Fac. Sc. Univ. Tokyo, 17, 241–258, 1970. [29] T. Kato. Perturbation Theory for Linear Operators. Springer Verlag, second edition, 1980. [30] B. S. Kay. Quantum Mechanics in Curved Space-Times and Scattering Theory. In Diﬀerential Geometric Methods in Mathematical Physics, volume 905 of Lecture Notes in Math., pages 272–295. Springer Verlag, 1980. [31] C. M. Massacand, C. Schmid. Particle Production by Tidal Forces and the Trace Anomaly. Ann. Phys (NY), 231, 363–415, 1994. [32] F. Melnyk. Wave operators for the massive charged linear Dirac ﬁelds on the Reissner-Nordstrøm metric. to appear in Class. Quantum Grav., 2000. [33] J-P. Nicolas. Scattering of linear Dirac ﬁelds by a spherically symmetric Black-Hole. Ann. Inst. Henri Poincar´e - Physique th´eorique, 62 (2), 145–179, 1995.

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[34] J-P. Nicolas. Global Exterior Cauchy Problem for Spin 3/2 Zero Rest-Mass Fields in the Schwarzschild Space-Time. Comm. P. D. E., 22 (3-4), 465–502, 1997. [35] J-P. Nicolas. Dirac ﬁelds on asymptotically ﬂat space-times. Preprint 99-22, Centre de Math´ematiques, Ecole Polytechnique, Paris, 1999. [36] I.D. Novikov, V. P. Frolov. Physics of Black Holes. Kluwer Academic Publishers, 1989. [37] B. Punsly. Black-Hole evaporation and the equivalence principle. Phys. Rev. D, 46 (4), 1288–1311, 1992. [38] S.N.M. Ruijsenaars. Charged particles in external ﬁelds. II. The quantized Dirac and Klein-Gordon theories. Comm. Math. Phys., 52, 267–294, 1977. [39] B. Thaller. The Dirac Equation. Springer-Verlag, 1992.

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[40] W.G. Unruh. Notes on Black-Hole Evaporation. Phys. Rev. D, 14 (4), 870–892, 1976. [41] W.G. Unruh. Origin of Particles in Black-Hole Evaporation. Phys. Rev. D, 15 (2), 365–369, 1977. [42] R. Wald. Quantum ﬁeld theory in curved space-time and black-hole thermodynamics. University of Chicago Press, 1994.

A. Bachelot Math´ematiques Appliqu´ees de Bordeaux CNRS Universit´e Bordeaux-1 F-33405 Talence Cedex e-mail : [email protected] Communicated by Detlev Buchholz submitted 21/09/99, accepted 05/06/00

To access this journal online: http://www.birkhauser.ch

Ann. Henri Poincar´ e 1 (2000) 1097 – 1122 c Birkh¨ auser Verlag, Basel, 2000 1424-0637/00/0601097-26 $ 1.50+0.20/0

Annales Henri Poincar´ e

Quantum Group Actions on the Cuntz Algebra A. L. Carey, A. Paolucci, R.B. Zhang Abstract. The Cuntz algebra carries in a natural way the structure of a module algebra over the quantized universal enveloping algebra Uq (g), and the structure of a co-module algebra over the quantum group Gq associated with Uq (g). We determine the ﬁxed and co-ﬁxed point algebras in the case of the classical quantum algebras and groups. We also show that the Hopf algebraic structure of the classical quantum groups can be recovered from a knowledge of the co-ﬁxed point algebras.

1 Introduction This paper is motivated by earlier work of one of us [17] on the co-action of the (Woronowicz’ [20]) quantum group SUq (d) on the Cuntz algebra Od [3] and by studies by Doplicher and Roberts [5] of group actions on Cuntz algebras. The general theory of Hopf algebras [16] suggests that we should seek actions of the Drinfeld-Jimbo [6, 13] deformations of universal enveloping algebras of semi-simple Lie algebras dual to the co-actions (found in [17]) of the Woronowicz quantum groups. We describe these actions and co-actions. In the case of the classical quantum groups we determine the ﬁxed and co-ﬁxed point algebras in terms of the braid group generators in Od together with certain speciﬁc additional elements (the q− antisymmetric tensor and the q−trace) which serve to distinguish the diﬀerent classical groups. By analogy with [5] this paper provides interesting parallels with the work on braid group statistics in two dimensional quantum ﬁeld theory [8]. To make the paper accessible we include some expository material in Section 2 where we introduce notation and basic facts. We denote by Uq (g) the quantized universal enveloping algebra of a simple Lie algebra g. When 0 < q < 1 the al(π)

gebraic dual of Uq (g) contains the Woronowicz quantum group Gq . (The bar (π) indicates closure in Woronowicz’ C ∗ -norm of a dense Hopf subalgebra Gq introduced in subsection 2.2). In Section 3 we describe realizations of the braid group in the Cuntz algebra for generic q (that is, not a root of unity). (π)

When 0 < q < 1 a co-action of Gq on the Cuntz algebra Od on d generators (where d is the dimension of the representation π of Uq (g)) was discovered in [17] in the case of the quantum group SUq (d). See Remarks in subsection 2.3 for a (π)

discussion of the dependence of Gq on π. It was also shown in this case that there is a homomorphism α of the braid group Bn on n generators into Od such (π)

that the co-ﬁxed points under the Gq

co-action were generated by α(Bn ). In

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section 3 we review this embedding of the braid groups in Od as preparation for our main results in Sections 4 and 5. In section 2 we construct for generic q an action of Uq (g) on the Cuntz algebra by non-unital non-star densely deﬁned endomorphisms with the following properties: (i) The domain of the endomorphisms is the dense subalgebra Od0 of Od given by polynomials in the generators. (π) (ii) There is a co-action of Gq on Od0 dual to this Uq (g) action. (In the case of SUq (d), this co-action coincides for 0 < q < 1 with that discovered in [17].) In sections 4 and 5 we show that the braid group elements in Od0 are both ﬁxed and co-ﬁxed. For the quantum groups associated with the classical series of Lie algebras, this result can be further sharpened. We introduce in section 5 two additional elements of the Cuntz algebra Od , the rank d q−antisymmetric tensor Sq and the q−trace τ . Theorem 1. Let g be one of the simple Lie algebras Am (d = m+1), Bm (d = 2m+1), Cm (d = 2m) or Dm (d = 2m), and π be the natural representation of Uq (g). Assume that 0 < q < 1. The ﬁxed point algebra of Od0 under the Uq (g) action is generated, algebraically, by (a) the braid group elements together with the rank d q-antisymmetric tensor in the case Am ; (b) the braid group elements, the rank d q-antisymmetric tensor, and the q-trace in Bm and Dm cases; (c) the braid group elements together with the q-trace in the Cm case. A natural question to ask is what happens if one dualizes the co-action and section 6 is devoted to showing that we recover the given action of Uq (g). There is (π) a slight technical point in that the Hopf dual of Gq is not obviously Uq (g). There is a possibly non-trivial ideal which one needs to factor out. However in the Cuntz realization this ideal acts trivially so that we have duality working fully. Finally in Section 7 we turn to the question of whether a knowledge of the co-ﬁxed point algebra in Od0 under the co-action of a ∗-Hopf algebra determines the algebraic structure of this Hopf algebra. Theorem 2. Let A be a ∗-Hopf algebra deﬁned in the following way. The generators are ¯ij , i, j = 1, 2, . . . , d, uij , u ¯ij . The relations in A and the co-algebraic structure are deterwhere ∗(uij ) = u mined by the conditions (i) the dense subalgebra Od0 of Od forms a co-module algebra over A with the co-action ω : Od0 → Od0 ⊗ A,

si →

d j=1

sj ⊗ uji ,

s∗i →

d j=1

s∗j ⊗ u ¯ji .

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(ii) the co-ﬁxed point algebra (Od0 )A is given by either Theorem 1 (a), (b) or (c). Then A is respectively either (a) SUq (d), (b) SOq (d) or (c) SPq (d), that is, one of the classical quantum groups as listed in [7, 11]. In order to understand the observations of this paper at a more fundamental level we conclude our discussion in the ﬁnal section by developing the action and co-action within the framework of braided tensor categories (cf [17]). The viewpoint in this paper is algebraic in that we have not attempted to analyze the role played by the C ∗ -algebra topology on the Cuntz algebra from the viewpoint of the Uq (g) action.

2 Quantized universal enveloping algebras and dual quantum groups 2.1 Quantized universal enveloping algebra Uq (g) Let g be any ﬁnite dimensional simple Lie algebra over the complex ﬁeld C. Denote by Φ+ the set of the positive roots g relative to a base Π = {α1 , . . . , αr }, where of r r is the rank of g. Deﬁne E = i=1 Rαi . Let ( , ) : E × E → R be an inner product of E such that the Cartan matrix A of g is given by r

A = (aij )ij=1 , aij =

2(αi , αj ) . (αi , αi )

The Jimbo version [13] of the quantized universal enveloping algebra Uq (g) is deﬁned to be the unital associative algebra over C, generated by {ki±1 , ei , fi | i = 1, . . . , r} with the following relations ki kj = kj ki , ki ki−1 = 1, −aij /2

ki ej ki−1 = qi ij ej , ki fj ki−1 = qi a /2

[ei , fj ] = δij

1−aij t

(−1)

t=0 1−aij

t=0

t

(−1)

1 − aij t 1 − aij t

fj ,

ki2 − ki−2 , qi − qi−1

(ei )t ej (ei )1−aij −t = 0,

i = j,

(fi )t fj (fi )1−aij −t = 0,

i = j,

qi

(1)

qi

where q is a complex which is assumed to be non-zero, and is not a parameter, s is the Gauss polynomial, and qi = q (αi , αi )/2 . root of unity. Also, t q

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The algebra Uq (g) has in addition the structure of a Hopf algebra. We take the following co-multiplication ∆(ki±1 ) = ki±1 ⊗ ki±1 , ∆(ei ) = ei ⊗ ki + ki−1 ⊗ ei , ∆(fi ) = fi ⊗ ki + ki−1 ⊗ fi . The co-unit " : Uq (g) → C and antipode γ : Uq (g) → Uq (g) are respectively given by "(ei ) = "(fi ) = 0, "(ki±1 ) = "(1) = 1, γ(ei ) = −qi ei , γ(fi ) = −qi−1 fi , γ(ki±1 ) = ki∓1 . Later in the paper we will need the notion of a ‘universal R-matrix’ as a technical device. This R-matrix does not live naturally in the Jimbo algebra Uq (g) but in the Drinfeld version [4]. If we set q = exp(ζ), and regard ζ as a formal indeterminate, then the Drinfeld version [6] of the quantized universal enveloping algebra is an associative algebra over C[[ζ]] completed with respect to the ζ-adic topology for C[[ζ]]. It is generated by {ei , fi , hi , i = 1, 2, . . . , r} with ki±1 = q ±hi /2 , subject to the same relations (1). We use the notation Uζ (g) to denote this algebra which in the terminology of Drinfeld is a quasi-triangular Hopf algebra. This means it admits an invertible ˆ ζ (g) ( ⊗ ˆ represents tensor product completed with respect to the R ∈ Uζ (g)⊗U ζ - adic topology ), called the universal R-matrix, which satisﬁes the following deﬁning relations R∆(a) = ∆ (a)R, (∆ ⊗ id)R = R13 R23 ,

∀a ∈ Uζ (g), (id ⊗∆)R = R13 R12 .

Further general properties of R are (γ ⊗ id)R = (id ⊗γ)R = R−1 (" ⊗ id)R = (id ⊗")R = 1, R12 R13 R23 = R23 R13 R12 , where ∆ is the opposite co-multiplication deﬁned by ∆ (x) = T (∆(x)) with T ˆ ζ (g). The key relation is the last one: the quantum being the ﬂip map on Uζ (g)⊗U Yang-Baxter equation, which is a direct consequence of the deﬁning relations of R. The universal R-matrix is of the form −1 R = q i,j (B )ij hi ⊗hj [1 ⊗ 1 + (qα − qα−1 )Xα ⊗ X−α + higher order terms], α∈Φ+

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where B = ((αi , αj )) and qα = q (α,α)/2 . Here the Xα are quantum analogs of the root vectors with kj Xα kj−1 = q (α,αj )/2 Xα , α ∈ Φ+ ∪ (−Φ+ ). In particular, Xαi = ki ei , X−αi = fi ki although the exact form of the “composite root vectors” is not important for us here. The higher order terms are all of the form C(s) E (s) ⊗F (s) where C(s) is a rational function in qα and qα−1 and α ∈ Φ+ , E (s) is a monomial in Xα , α ∈ Φ+ , of order ≥ 2 and F (s) is a monomial in X−α , α ∈ Φ+ of order ≥ 2. Let π be a nontrivial irreducible representation of Uq (g) with highest weight µ. That is, if v+ is the highest weight vector of the associated module W , we require the actions of ki±1 on v+ to be ki±1 v+ = q ±(µ,αi )/2 v+ ,

∀i.

We also assume that the basis for W corresponding to π is chosen in such a way that it reduces to the basis of a g−module in the q → 1 limit (this is always possible [12][18]). Then π deﬁnes a unique Uζ (g) representation by simply interpreting q as exp(ζ) and we will use π to denote this as well. Deﬁne RT = T (R) and let T (+) = (π ⊗ id)RT ,

T (−) = (π ⊗ id)R−1 .

(2)

By examining the structure of the universal R-matrix, we can easily see that (1) Both T (±) can be expressed solely in terms of ei , fi , ki±1 , and kπ±1 = q ±hµ , where hµ is a linear combination of the hi such that [hµ , ei ] = (µ, αi )ei . For some Lie algebras it can happen that µ is not in the root lattice of g. In that case, kπ±1 can not be expressed as products of integer powers of the ki±1 . (2) The diagonal elements of T (±) are products of kπ±1 ki±1 ; the nearest oﬀ-diagonal elements are ei , fi multiplied by products of kπ±1 , kj±1 . (3) The kπ±1 , ki±1 , ei , fi , for all i can be recovered explicitly from the diagonal and nearest oﬀ diagonal elements of T (±) . Let us now specialize to the situation where q is a complex parameter. Deﬁnition 2.1. The matrix elements of T (±) generate, algebraically, an associative ˜q (g) over the complex ﬁeld C. algebra U If µ belongs to the root lattice of g, this algebra coincides with the Jimbo quantized universal enveloping algebra , otherwise, it contains Uq (g) as a subal˜q (g) has additional generators kπ±1 . The action of the Jimbo algebra to gebra as U ˜q (g) but not uniquely in the latter be deﬁned in section 4 extends to the algebra U case. The extensions diﬀer however only by the fact that one needs to choose a particular (complex) root of q, and are related to one another by an action of the appropriate group of roots of unity as automorphisms. At generic q, this diﬀerence ˜q (g) is thus not at all important. between the two algebras Uq (g) and U

2.2 Uq (g) as a ∗-Hopf algebra Let us now further assume that q is real and satisﬁes 0 < q < 1. In this case, Uq (g) admits a variety of ∗-algebra structures. Of particular importance for the analysis

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of this paper is that deﬁned by the following conjugate linear anti-involution ∗(ei ) = fi , ∗(fi ) = ei , ∗(ki±1 ) = ki±1 .

(3)

Now ∆ and " are ∗ - homomorphisms. These properties together with the deﬁning relations of the antipode imply that γ ∗ γ∗ = id. Therefore Uq (g) acquires the structure of a ∗-Hopf algebra. For later use we introduce the quantum Cartan involution θ = ∗γ. A Uq (g)-module W will be called unitary if it admits a nondegenerate positive deﬁnite sesquilinear form ( , ) satisfying (xv, w) = (v, x∗ w), ∀v, w ∈ W, x ∈ Uq (g). It is an easy exercise to directly prove that the natural representations of the Uq (g) associated with the A, B, C and D series of simple Lie algebras are unitary. Let W be a unitary ﬁnite dimensional irreducible Uq (g)-module. Then we can introduce an orthonormal basis {wi }| i = 1, 2, . . . , d} (d = dimW ) such that (wi , wj ) = δij . We denote the irreducible Uq (g)-representation associated with this orthonormal basis by π. The orthonormal basis of W induces a basis {w ¯i } for W ∗ deﬁned by w ¯i (v) = (wi , v),

∀v ∈ W.

W ∗ carries a natural Uq (g)-module structure with the module action deﬁned by (xw)(v) ¯

= w(γ(x)v), ¯

∀x ∈ Uq (g), w ¯ ∈ W ∗ , v ∈ W.

Denote by π † the Uq (g)-representation on W ∗ relative to the basis {w ¯i }, and call π † the dual representation of π.

2.3

(π)

The quantum group Gq

We now move on to set up the notation and properties of the dual to Uq (g), the Woronowicz quantum group. The ﬁnite dual (Uq (g))0 of Uq (g) has a natural Hopf algebra structure, with the multiplication m0 , co-multiplication ∆0 , unit 10 , co-unit "0 , and antipode γ0 deﬁned in the standard fashion [16]. We consider (π) a subalgebra Gq of (Uq (g))0 deﬁned in the following way. Let π be a nontrivial ﬁnite dimensional irreducible Uq (g) representation deﬁned with respect to an orthonormal basis as discussed above. Consider the matrix d

U = (uij )i,j=1 , uij ∈ (Uq (g))0 , deﬁned by uij , a = π(a)ij ,

∀a ∈ Uq (g).

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(π)

Deﬁnition 2.2. We deﬁne Gq as the associative subalgebra of (Uq (g))0 generated by the matrix elements of U , with the multiplication deﬁned by πij (a(1) )πkl (a(2) ), ∀a ∈ Uq (g). uij ukl , a = (a)

In the case when g is a classical Lie algebra and for 0 < q < 1 in [20] this algebra is completed in an appropriate C ∗ -algebra norm. (A diﬀerent approach to the C ∗ -completion can be found in [10]) For the most part we will not need this topology here and will work with the uncompleted algebra. When necessary we (π) denote the C ∗ completion by a bar. Clearly the unit of Gq coincides with the co-unit " of Uq (g). Set R(π) = (π ⊗ π)R, where R is the universal R-matrix of Uζ (g). Then R(π) involves qi ’s only and hence one may specialize q to a complex parameter. The following is well known and straightforward to check: Lemma 2. U satisﬁes the quadratic relation (π)

(π)

R12 U1 U2 = U2 U1 R12 . (π)

It is also easy to see that Gq multiplication is given by

(4)

has the structure of a bi-algebra. The co-

∆0 (uij ) =

uik ⊗ ukj ,

(5)

k

which follows from the equation ∆0 (uij ), a ⊗ b = πij (ab),

∀a, b ∈ Uq (g),

while the co-unit is 1Uq (g) uij (1Uq (g) ) = δij . (π)

Furthermore, Gq is equivalent to

admits an antipode γ0 . The deﬁning relations of the antipode γ0 (U ) = U −1 ,

(6)

where U −1 is a d ⊗ d matrix in (Uq (g))0 satisfying (U −1 )ij (a) = π † (a)ji ,

∀a ∈ Uq (g).

Here π † is the irreducible representation of Uq (g) dual to π. For γ0 to deﬁne an (π) antipode of Gq , we need to show that

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Ann. Henri Poincar´e

Lemma 3. The elements of U −1 belong to Gq . (π)

Proof. The Lemma is equivalent to the statement that some repeated tensor product of π (with respect to the co-multiplication ∆) contains the dual representation π † as an irreducible component. For this to be true, it suﬃces to show that a one dimensional representation can arise from nontrivial tensor products of π. We claim that there exists a nonvanishing Λ ∈ W ⊗d which generates a one dimensional representation of Uq (g). As the representation theory of Uq (g) at generic q is the same as that of U (g), let us ﬁrst examine the classical situation. In the q → 1 limit, the Uq (g) action on W yields a U (g) action. The totally antisymmetric rank d tensor of W is one dimensional, and we denote its basis element by Λ(0) . It is clearly true that x · Λ(0) = 0,

∀x ∈ g.

By calling upon the Lusztig-Rosso theorem, [12, 18] we conclude that there exists a nonvanishing Λ ∈ W ⊗d , which reduces to Λ(0) in the q → 1 limit, such that a ◦ Λ = "(a),

∀a ∈ Uq (g).

This completes the proof of the Lemma. Deﬁnition 2.3. We will call Λ the rank d(= dim π) q-antisymmetric tensor of W . The Cartan involution θ of Uq (g) induces a natural Hopf ∗-algebra structure (π) (π) for Gq with the ∗-operation deﬁned by ∗(a), x = a, θ(x), ∀a ∈ Gq , x ∈ ¯ Uq (g). Let U = (¯ uij ) with u ¯ij = γ0 (uji ). Then it can be easily shown that ∗(uij ) = u ¯ij . Note that the u ¯ij also satisﬁes π † ij (x)

= u ¯ij (x),

∀x ∈ Uq (g).

Remarks. We need to comment on the representation π and the dependence on it (π) of the quantum group Gq . If π can generate all the ﬁnite dimensional represen(π) (π ) tations of Uq by repeated tensor products, then Gq will contain all Gq as Hopf subalgebras, where π is any representation of Uq (g). The natural representations (i.e., the deformations of the deﬁning representations of the corresponding classical Lie algebras) of Uq (sl(n)) and Uq (sp(2n)) all have this property, as does the spinor representation of Uq (so(n)). The classical quantum groups SUq (n), SPq (2n) and SOq (n) are associated with the natural representations. In the so(n) case, the quantum group generated by the matrix elements of the spinor representation(s) of Uq (so(n)) is the ‘quantum spin group’.

2.4 Fixed and co-ﬁxed points We begin with some generalities on co-actions for Hopf algebras. Let A be a Hopf algebra with co-multiplication ∆, co-unit " and antipode γ. Let V be a left A-module, which is assumed to be locally ﬁnite, i.e., it satisﬁes the following

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properties: corresponding to each v ∈ V , we can ﬁnd a ﬁnite set of elements vi ∈ V , i = 1, 2, . . . , N , such that v=

N

ci ∈ C;

ci vi ,

a ◦ vi =

i=1

N

ψji (a)vj ,

∀a ∈ A,

j=1

where ψji (a) ∈ C. Let A0 be the ﬁnite dual Hopf algebra with multiplication m0 , unit ", co-unit 1A , co-multiplication ∆0 and antipode γ0 . The left A-module V automatically carries a right A0 co-module structure ω : V → V ⊗ A0 deﬁned in the following way. For any element v ∈ V , if we write v(1) ⊗ v(2) v(2) ∈ A0 , ω(v) = (v)

then for all a ∈ A, ω(v)(a) =

v(1) v(2) (a) = a ◦ v

(v)

Introduce the notation (V )A for the ﬁxed point set of V under the action of A, and (V )A0 for the co-ﬁxed point set of V under the co-action of A0 . Thus (V )A = {v ∈ V | a ◦ v = "(a)v, ∀a ∈ A};

(V )A0 = {u ∈ V | w(u) = u ⊗ 1A0 }.

Consider v ∈ V . If v is co-ﬁxed by A , i.e., ω(v) = v ⊗ ", then 0

a ◦ v = ω(v)(a) = "(a)v, hence v is also ﬁxed by A. On the other hand, if v ∈ V is ﬁxed by A, then v(1) v(2) (a), ∀a ∈ A, a ◦ v = "(a)v = (v)

that is, ω(v) = v ⊗ ", so that v is also co-ﬁxed. Therefore, (V )A = (V )A0 .

2.5 The q-determinant Returning now to the discussion begun in subsection 2.2 we will need the notion (π) of q-determinant of the matrix U used to deﬁne Gq . By the discussion of the (π) preceding two subsections we may consider the co-action of Gq , denoted by ω, on W . Now this co-action when applied to the q-antisymmetric tensor Λ yields (π) ω(Λ) = Λ ⊗ detq U where detq U is some element of Gq . Deﬁnition 2.4. We call detq U the q-determinant of U . However, invoking the argument of the preceding subsection, Λ must be a (π) co-ﬁxed point of Gq as Λ generates a trivial module of Uq (g). Hence detq U = ", i.e., U has q-determinant 1Gq (= "). Note that in [20] the quantum determinant is set equal to the identity as an additional relation, while this relation is built into (π) the deﬁnition of Gq used here.

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(π)

2.6 The dual Hopf algebra of Gq (π)

In this subsection we regard Gq as a Hopf algebra in its own right. We want (π) to investigate the ﬁnite dual of Gq . To this end consider the set of elements (±) (π) ∗ (±) {lij ∈ (Gq ) |i, j = 1, . . . , d}, deﬁned by writing L(±) = eij ⊗ lij . Then 1, U ⊗r = I ⊗r and (+)

(+)

. . . Rr(+) ,

(−)

(−)

. . . Rr(−) ,

L(+) , U1 U2 . . . Ur = R1 R2

L(−) , U1 U2 . . . Ur = R1 R2

(7)

where R(−) = (R(π) )−1 ,

R(+) = P R(π) P,

P is the ﬂip map on W ⊗ W and our notation is largely the same as that of [7]. (π) By considering the co-multiplication of Gq , one can show [7] that L(±) satisfy the following relations (+)

(±)

(±)

= L2 L1 R12 ,

(+)

(+)

(−)

= L2 L1 R12 .

R12 L1 L2 R12 L1 L2

(±)

(±)

(+)

(−)

(+)

(+)

(8)

Let us denote by Uq the algebra generated by the matrix elements of L(±) . A co-multiplication for Uq is given by (±) (±) (±) ∆(L(±) ) = L(±) ⊗ L(±) (i.e. ∆(lij ) = lik ⊗ lkj ), k

and the corresponding co-unit and antipode are respectively given by "(L(±) ) = I,

γ(L(±) ) = (L(±) )−1 .

Deﬁne a linear map L(+) → T (+) ,

L(−) → T (−) .

This map extends in a unique way to a Hopf algebra homomorphism φ : Uq → ˜q (g), which is clearly surjective. The kernel of φ is a Hopf ideal of Uq , and U ˜q (g) as Hopf algebras. Uq /Kerφ ∼ =U

3 Cuntz algebra realizations of the braid group 3.1 Representations of the braid group Consider R(π) , where π is a d-dimensional representation of Uq (g) and let P : W ⊗ W → W ⊗ W be the ﬂip: P (a ⊗ b) = b ⊗ a. Deﬁne σ = P R(π) .

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It follows from the deﬁning relations of the universal R matrix that [σ, (π ⊗ π)∆(a)] = 0, ∀ a ∈ Uq (g), (σ ⊗ 1)(1 ⊗ σ)(σ ⊗ 1) = (1 ⊗ σ)(σ ⊗ 1)(1 ⊗ σ) Note that σ acts on W ⊗ W , while the above equation holds as endomorphisms of W ⊗ W ⊗ W. Deﬁne bi = 1 ⊗ · · · ⊗ 1 ⊗σ ⊗ 1 ⊗ · · · ⊗ 1 , i−1

i = 1, 2, . . . , n

n−i

Then we have the braid relations bi bi+1 bi = bi+1 bi bi+1 bi bj = bj bi , |i − j| ≥ 2 Also,

bi , π ⊗(n+1) ∆(n) (a) = 0,

(9)

∀ a ∈ Uq (g)

3.2 Cuntz algebra realization of the braid group Let Od be the Cuntz algebra [3] on d generators. This is the universal C∗ -algebra ∗ generated by {sj |j = 1, 2, . . . , d} satisfying si sj = δij and j sj s∗j = 1 where 1 denotes the identity of Od . Let {eij |i, j = 1, 2, . . . , d} be the matrix units for End W , which obey eij ekl = δjk eil . There exists a well known algebra homomorphism η : EndW → Od , deﬁned by η(eij ) = si s∗j . This extends to an algebra homomorphism η : EndW ⊗m → Od for each m, deﬁned by η (ei1 j1 ⊗ ei2 j2 ⊗ · · · ⊗ eim jm ) = si1 si2 . . . sim−1 sim s∗jm s∗jm−1 . . . s∗j2 s∗j1 . Express the braid σ ∈ EndW as σ=

d

σij,kl eij ⊗ ekl .

i,j,k,l=1

Then its image under η is given by θ = η(σ) =

σij,kl si sk s∗l s∗j .

More generally, we put θi = η(bi ) then sl1 sl2 . . . sli−1 θs∗li−1 . . . s∗l2 s∗l1 , θi = {l}

which satisfy the braid group relations in Od .

i = 1, 2, . . . , n.

(10)

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Ann. Henri Poincar´e

4 The Cuntz algebra as a module algebra over Uq (g) Introduce the notation: H=

d

H r = H ⊗r ,

Csi ,

H=

∞

Hr,

H∗ =

r=0

i=1

d

Cs∗i , ,

1

where H is the algebraic direct sum, H 0 = C and H ∗ is the dual vector space of H with pairing H ∗ ⊗ H → C given by s∗i , sj = δij . Set H ∗r = H ∗⊗r ,

H∗ =

∞

H ∗r .

r=0

The pairing between H ∗r , H t is given by s∗jr s∗jr−1 . . . s∗j1 , si1 si2 . . . sit = δrt δi1 j1 δi2 j2 . . . δir jr . This deﬁnition is compatible with Cuntz multiplication when r and t are equal. We now build up our action of Uq (g) on the dense subalgebra Od0 of Od consisting of polynomials in the generators in three steps outlined in the following subsections.

4.1 Uq (g) action on H and H∗ We begin by deﬁning the action of Uq (g) on C by a ◦ c = "(a)c.

(11)

That is, we regard C as a trivial module over Uq (g). Now let π be the d-dimensional nontrivial irreducible representation of Uq (g) introduced earlier. A Uq (g)-module action ◦

Uq (g) ⊗ H −→H can be deﬁned by setting a ◦ si =

d

π(a)ji sj .

(12)

j=1

Since π is irreducible, H contains a unique ( up to scalar multiples) highest weight vector sπ . Now each H r furnishes a Uq (g)-module with the module action deﬁned by the co-multiplication: a ◦ (si1 si2 . . . sir ) = π(a(1) )j1 i1 π(a(2) )j2 i2 . . . π(a(r) )jr ir × sj1 sj2 . . . sjr . {j}

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This introduces to H a Uq (g)-module structure, provided only ﬁnite linear combinations of vectors in the direct sum H are considered. The dual vector spaces H ∗r have a natural Uq (g)-module structure. On H∗ , the Uq (g) action is deﬁned by requiring a ◦ s∗i , sj = s∗i , γ(a) ◦ sj = π(γ(a))ij . Then each H ∗r becomes a tensor product module, and H∗ is the module obtained as the algebraic direct sum of the H ∗r ’s. Explicitly the action of Uq (g) on H ∗r is given by a ◦ (s∗jr s∗jr−1 . . . s∗j1 ), si1 si2 . . . sir = (πj1 i1 ⊗ · · · ⊗ πjr ir )∆(r−1) (γ(a)). Note that if we write ∆(k−1) (a) = then ∆(k−1) (γ(a)) =

a(1) ⊗ · · · ⊗ a(k) ,

γ(a(k) ) ⊗ γ(a(k−1) ) ⊗ · · · ⊗ γ(a(1) )

(a)

Hence

(a(1) ◦ s∗jr )(a(2) ◦ s∗jr−1 ) . . . (a(r) ◦ s∗j1 ) a ◦ s∗jr s∗jr−1 . . . s∗j1 = (a)

Uq (g) actions on H ⊗ H∗ and H∗ ⊗ H

4.2

The actions are deﬁned by the co-multiplication in the obvious way, namely, for u ∈ H, v ∗ ∈ H∗ , a(1) ◦ u ⊗ a(2) ◦ v ∗ , a ◦ (u ⊗ v ∗ ) = a ◦ (v ∗ ⊗ u) = a(1) ◦ v ∗ ⊗ a(2) ◦ u. They have the following useful properties a ◦ (usi ⊗ s∗i v ∗ ) = (a(1) ◦ u)sj ⊗ s∗j (a(2) ◦ v ∗ ), i

a◦

(v ∗ s∗i

j

(a)

kl

(a)

⊗ sj u) = (a(1) ◦ v ∗ )π(γ(a(2) ))ik s∗k ⊗ π(a(3) )lj sl (a(4) ◦ u).

We wish to examine the properties of the module actions under Cuntz multiplication. Consider

a ◦ s∗jr s∗jr−1 . . . s∗j1 ⊗ si1 si2 . . . sit (a(1) ◦ s∗jr )(a(2) ◦ s∗jr−1 ) . . . (a(r) ◦ s∗j1 ) ⊗ (a(r+1) ◦ si1 ) . . . (a(r+t) ◦ sit ). = (a)

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Direct calculations can establish that

a ◦ s∗jr s∗jr−1 . . . s∗j1 si1 si2 . . . sit (a(1) ◦ s∗jr )(a(2) ◦ s∗jr−1 ) . . . (a(r) ◦ s∗j1 )(a(r+1) ◦ si1 ) . . . (a(r+t) ◦ sit ), = (a)

where both sides of the above equation are regarded as elements of the Cuntz algebra. It is also clearly true that

a ◦ si1 si2 . . . sit s∗jr s∗jr−1 . . . s∗j1 (a(1) ◦ si1 )(a(2) ◦ si2 ) . . . (a(t) ◦ sit )(a(t+1) ◦ s∗jr ) = (a)

(a(t+2) ◦ s∗jr−1 ) . . . (a(r+t) ◦ s∗j1 ). Therefore, the Uq (g) action preserves the Cuntz multiplication.

4.3 Od0 as a module algebra over Uq (g) The results of the last subsection suggest that a Uq (g)-module action on Od0 : ◦

Uq (g) ⊗ Od0 −→Od0 , can be introduced directly in which each element of Uq (g) acts by a non-unital endomorphism (by which we mean it preserves the multiplication but not the *-operation or identity) of Od0 . This is achieved by deﬁning a ◦ 1 = "(a) a ◦ si = π(a)ji sj , j

a◦

s∗i

=

π(γ(a))ij s∗j ,

j

(a(1) ◦ x)(a(2) ◦ y), a ◦ (xy) =

x, y ∈ Od0 .

(a)

The deﬁning relations of the Cuntz algebra are clearly preserved, (a(1) ◦ s∗i )(a(2) ◦ sj ) = δij a ◦ 1, a ◦ (s∗i sj ) = (a) d

a◦(

1

si s∗i ) =

d (a(1) ◦ si )(a(2) ◦ s∗i ) = a ◦ 1. 1

(a)

Therefore, Od0 deﬁnes a module algebra over Uq (g) under the action (13).

(13)

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This module algebra structure of Od0 over Uq (g) can be straightforwardly ˜q (g), deﬁned in Deﬁnition 2.1, by specifying the action extended to the algebra U of the additional generators kπ±1 on the highest weight vector sπ of H, kπ .sπ = q (µ,

µ) π

s .

(14)

Note that when the highest weight µ of π does not belong to the root lattice, (µ, µ) is in general a rational number, and in that case we make a choice of the complex value of q (µ, µ) .

4.4 Braids as ﬁxed points The ﬁxed point set of Od0 under the Uq (g) action (13) is {u ∈ Od0

| a ◦ u = "(a)u,

∀a ∈ Uq (g)}.

deﬁnes a subalgebra of Od0 . Since Od0 is a Uq (g) module algebra, the ﬁxed point set Consider the Uq (g) action on the braid generator θ = σij,kl si sk s∗l s∗j (π ⊗ π)∆(a(1) ) · σ · (π ⊗ π)∆(γ(a(2) )) ij,kl si sk s∗l s∗j a◦θ = (π ⊗ π)∆(a(1) γ(a(2) )) · σ ij,kl si sk s∗l s∗j = = "(a)θ, where we have used the fact that the braid generator σ commutes with (π⊗π)∆(a), ∀a ∈ Uq (g). More generally, a ◦ θr+1 = a(1) ◦ (si1 . . . sir ) · [a(2) ◦ η(σ)]a(3) ◦ (s∗ir . . . s∗i1 ) = "(a)θr+1 . Thus we have shown that the braids θr ∈ Od0 , ∀r = 0, 1, 2, . . . , are ﬁxed points of the Uq (g)-action. (π)

5 Od0 as a co-module algebra over Gq

(π)

5.1 The algebra Od0 as a co-module algebra over Gq

Following the general method discussed in Section 2 we can deﬁne a co-module (π) action of Gq on Od0 by ω(1) = 1 ⊗ ",

ω(si ) =

d

sj ⊗ uji ,

j=1

ω(s∗i ) =

d j=1

s∗j ⊗ u ¯ji ,

ω(xy) = ω(x)ω(y),

∀ x, y ∈ Od0 ,

(15)

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where the multiplication on the right hand side of the last equation is the natural (π) (π) one for the algebra Od0 ⊗Gq induced by the multiplication of Od0 and that of Gq . The consistency of this deﬁnition is conﬁrmed by the simple calculations below: ω(x)ω(y)(a) = x(1) y(1) x(2) y(2) , a (x),(y)

=

x(1) x(2) , a(1) y(1) y(2) , a(2)

(qa) (x),(y)

=

(a(1) ◦ x) (a(2) ◦ y) = a ◦ (xy) (a)

= ω(xy)(a),

∀a ∈ Uq (g).

(π) Gq .

Hence Od0 is a co-module algebra over (π) An important fact to be observed is that the Gq co-module structure of Od0 is compatible with the ∗-structure in the following sense: ω∗

= (∗ ⊗ ∗)ω,

(16) (π) Gq .

where both sides are regarded as conjugate linear maps Od0 → Od0 ⊗ This (π) follows directly from the deﬁnition (15) of the Gq co-action on Od0 . An immediate (π) consequence is that if φ ∈ Od0 is co-ﬁxed by Gq , i.e., ω(φ) = φ ⊗ ", ∗

then its conjugate φ is also co-ﬁxed, ω(φ∗ ) = φ∗ ⊗ ".

5.2 The braids as co-ﬁxed points We have already shown that the braids are ﬁxed by the Uq (g) action, thus they (π) must be co-ﬁxed by Gq . Nevertheless, we look at an example to illustrate the general result. Let θ= σij,kl si sk s∗l s∗j be a braid embedded in Od0 . The coeﬃcient matrix σ commutes with (π ⊗ π)∆(a) (π) ∀a ∈ Uq (g). This translates, for the dual Hopf algebra Gq , to the following relations σ12 U1 U2 = U1 U2 σ12 . Now ω(θ) =

si sk s∗l s∗j ⊗ (U1 U2 σ12 γ0 (U1 U2 ))ij,kl

= θ ⊗ ".

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5.3 Proof of Theorem 1 In this subsection we assume that the Lie algebra g belongs to the four classical series Am , Bm , Cm and Dm . Let π be the natural representation of Uq (g), namely, the deformation of the deﬁning representation of g. Denote by CBN−1 the algebra generated by the braids bi , i = 1, 2, . . . , N − 1, deﬁned in subsection 3.1. Note that when g is Am , CBN−1 furnishes a representation of the Hecke algebra, while for g = Bm , Cm or Dm , CBN−1 gives rise to a representation of the Birman-WenzlMurakami algebra. Recall that the BWM algebra has some extra generators apart from the braids. The (bi −q)(bi −q −1 ) in CBN−1 account for these extra generators. Theorem 10.2.5 in [2] states that Proposition. CBN−1 is the commutant of π ⊗N (Uq (g)). This result was proven by Jimbo [14], and Kirillov and Reshetikhin [15]. It will be of crucial importance for proving Theorem 2. Let (Od0 )kl be the subspace of Od0 spanned by the monomials si1 si2 . . . sik ∗ ∗ sj1 sj2 . . . s∗jl . Then (Od0 )kl ⊃ (Od0 )k−1,l−1 ⊃ (Od0 )k−2,l−2 ⊃ . . . . We call an element f ∈ Od0 homogeneous of bi-degree (k, l) if f ∈ (Od0 )kl with both k and l being minimal. For each element φ of the ﬁxed point algebra (Od0 )Uq (g) , there exists a ﬁnite set Θ ⊂ Z+ × Z+ with the property that if (m, n) ∈ Θ, then (m + z, n + z) ∈ Θ, 0 = z ∈ Z, such that φ =

φk,l ,

(k,l)∈Θ

where φk,l ∈ Od0 is a homogeneous element with a bi-degree (k, l). Note that the Uq (g) action on Od0 does not alter the bi-degree of a homogeneous element, thus, every φk,l also belongs to (Od0 )Uq (g) . Furthermore, every homogeneous φk,l ∈ (Od0 )Uq (g) of bi-degree (k, l) may be interpreted as a Uq (g)-module homomorphism φk,l : H l → H k . We denote composition of module homomorphisms by . It is an immediate consequence of the Proposition that Lemma 4. Every homogeneous element of (Od0 )Uq (g) with bi-degree (k, k) belongs to the subalgebra η(CB∞ ) of Od0 generated by the braids θi , i = 1, 2, . . . .. To analyze the ﬁxed points of bi-degree (k, l) with k = l, we need to consider the four series of Lie algebras separately. Let us ﬁrst examine Am . This case has been thoroughly understood in [17]. Our analysis here will not yield any new results, but will direct the way for studying the other cases. Set d = m + 1. It follows from the results of subsection 2.3 that there exists a nonvanishing rank d q-antisymmetric tensor Sq ∈ H d , which generates a trivial Uq (g) module, a ◦ Sq = "(a)Sq .

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Similarly, its conjugate Sq∗ ∈ (H ∗ )d is also ﬁxed by Uq (g). We will normalize Sq and Sq∗ in such a way that Sq∗ Sq = 1. Observe the following rather obvious but important fact: if there exits any nontrivial Uq (g)-module homomorphism H l → H k , then by comparing the Uq (g)weights of the vectors on both sides, we easily see that |k−l| must be a nonnegative multiple of d. Consider a homogeneous φ ∈ (Od0 )Uq (g) of bi-degree (k + id, k). It leads to a nontrivial Uq (g)-module homomorphism φ : H k → H k+id , i > 0. Then φ (Sq∗ )i : H k+id → H k+id deﬁnes a new Uq (g)-module homomorphism. It follows from Lemma 4 that φ (Sq∗ )i ∈ η(CB∞ ). Now φ = {φ (Sq∗ )i } (Sq )i , that is, the ﬁxed point φ can be expressed in terms of braids and the rank d q-antisymmetric tensors Sq and Sq∗ . Similarly, a homogeneous element ψ ∈ (Od0 )Uq (g) of bi-degree (k, k + id) leads to a Uq (g)-module homomorphism ψ : H k+id → H k . From ψ we can construct another Uq (g)-module homomorphism (Sq )i ψ : H k+id → H k+id . Again (Sq )i ψ can be expressed solely in terms of the braids, and ψ = (Sq∗ )i {(Sq )i ψ}. This completes the analysis of the ﬁxed point algebra (Od0 )Uq (g) for the case when g is Am . When g is either Cm or Dm , there exists a unique (up to scalar multiples) nontrivial Uq (g)-module homomorphism τ , called the q-trace, and its conjugate, with τ ∗ : H 2 → C, τ : C → H 2, as well as the analogs of Sq and Sq∗ . Appropriate normalizations will render τ and τ ∗ satisfying τ ∗ τ = 1. By examining the weights of the repeated tensor products of H, we easily see that nontrivial Uq (g)-module homomorphisms H l → H k exist only when k − l is divisible by 2. By repeating the same arguments as in the Am case, but with τ and τ ∗ replacing Sq and Sq∗ respectively, we again arrive at the conclusion that any ﬁxed point can be expressed in terms of the braids θi , τ and τ ∗ . Finally we consider Bm . Analogs of Sq , Sq∗ , τ and τ ∗ also exist in this case. However, for any value of k − l, there can exist nontrivial Uq (g)-module homomorphisms H k → H l . We ﬁrst consider the case when k and l diﬀer by an even integer. Given homogeneous elements φ, ψ ∈ (Od0 )Uq (g) respectively of bi-degrees (k + 2i, k) and (k, k + 2i), we can construct Uq (g)-module homomorphisms φ (τ ∗ )i : H k+2i → H k+2i , (τ )i ψ : H k+2i → H k+2i .

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By Lemma 4, φ (τ ∗ )i and (τ )i ψ belong to η(CB∞ ). Now φ = {φ (τ ∗ )i } (τ )i , ψ = (τ ∗ )i {(τ )i ψ}. Now we consider the homogeneous elements Φ, Ψ ∈ (Od0 )Uq (g) of bi-degrees (k + 2i + 1, k) and (k, k + 2i + 1) respectively. We deﬁne a map ω : Z → Z+ by z, z > 0, z → We also decree that (τ )0 = (τ ∗ )0 = 1. Then (τ )ω(m−i) Φ 0, z ≤ 0. Sq∗ (τ ∗ )ω(i−m) and (τ )ω(i−m) Ψ Sq∗ (τ ∗ )ω(m−i) belong to the subalgebra η(CB∞ ) of Od0 generated by the braids. Now Φ = (τ ∗ )ω(m−i) {(τ )ω(m−i) Φ Sq∗ (τ ∗ )ω(i−m) } (τ )ω(i−m) Sq, Ψ = (τ ∗ )ω(i−m) {(τ )ω(i−m) Ψ Sq∗ (τ ∗ )ω(m−i) } (τ )ω(m−i) Sq . Therefore, in the case g = Bm , the ﬁxed point algebra (Od0 )Uq (g) is generated by the braids, the q-traces τ and τ ∗ , and the rank 2m + 1 q-antisymmetric tensors Sq and Sq∗ . Remark. The ﬁxed and co-ﬁxed points of Od0 co-incide so it follows that (Od0 )Uq (g) (π) generates the co-ﬁxed subalgebra (Od ) (π) for the C ∗ -closure of Gq . This is beGq

cause the co-action does not change the bi-degree so every co-ﬁxed element of Od must be a linear combination of co-ﬁxed elements of given bi-degree and the above argument extends.

6

(π)

The Gq

co-module Od0 as a Uq module

In the above calculations we started with Od0 as a Uq (g) module and showed how (π) to obtain the dual co-action of Gq . It is of some interest to understand whether, (π) when the reverse procedure is adopted, namely regarding Od0 as a Gq co-module as in [17] and constructing the dual action, one recovers the given Uq (g) action. This is indeed the case but it requires us to understand Uq (g) in a diﬀerent way namely in terms of Uq . In the notation of Section 2, A is a Hopf algebra with the ﬁnite dual A0 , which is also a Hopf algebra. Let V be a co-module of A, ω

V −→V ⊗ A. 0

Then V has a natural A0 -module structure A0 ⊗ V −→V deﬁned, for any v ∈ V by, a ◦ v = (idV ⊗a)w(v),

a ∈ A0 .

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Now we consider Od0 as a module over Uq ⊂ (Gq )∗ . The co-action of Gq on Od0 is deﬁned by (15). It follows from the discussion above that this co-action dualizes an action of Uq on the Cuntz algebra: (π)

a ◦ 1 = "(a), a ◦ si = sj a, uji ,

a ◦ s∗i =

j

a ◦ (xy) =

a(w(xy)) =

(π)

s∗j a, γ0 (uij ),

j

x(1) y(1) a, x(2) y(2) =

(a(1) ◦ x)(a(2) ◦ y). (a)

By recalling the deﬁning relations (7) of Uq , we can see that Kerφ annihilates Od0 . ˜q (g) on O0 coincide, Hence the actions of Uq and U d a ◦ x = φ(a) ◦ x, ∀a ∈ Uq , x ∈ Od0 , ˜q (g) on Od appearing on the right hand side is that deﬁned by where the action of U ˜q (g) action and hence a fortiori (13) and (14). Therefore, we have recovered the U (π) the Uq (g) action on the Cuntz algebra from the Gq co-action.

7 Proof of theorem 2 Up to now we have assumed that a quantized universal enveloping algebra and the dual quantum group are given, and we sought to understand their action and co-action on the Cuntz algebra, and, in particular, the ﬁxed point and co-ﬁxed point algebras. In this section, we will go in the reverse direction. We have the following Lemma 5. Let A be a ∗-Hopf algebra with generators uij , u ¯ij ,

i, j = 1, 2, . . . , d,

where ∗(uij ) = u ¯ij . Assume that the dense subalgebra Od0 of Od forms a co-module algebra over A where the co-action is ω : Od0 si

→ Od0 ⊗ A, →

d

sj ⊗ uji ,

j=1

s∗i

→

d j=1

s∗j ⊗ u ¯ji .

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Then the co-multiplication ∆0 , co-unit "0 and antipode γ0 of A are given by ∆0 (uij ) =

d

uik ⊗ ukj ,

k=1

uij ) = ∆0 (¯

d

u ¯ik ⊗ u ¯kj ,

k=1

uij ) = δij , "0 (uij ) = "0 (¯ ¯ji . γ0 (uij ) = u

(17)

Furthermore, the co-action respects the ∗-operations of Od and A in the sense that ω∗ = (∗ ⊗ ∗)ω. Proof. Everything is obvious except for the antipode, which requires some explana ∗ tion. Note that the deﬁning relations of Od , i.e., s∗i sj = δij , s s i i i = 1, require that d

u ¯ki ukj

k=1

=

d

uik u ¯jk = δij ,

(18)

k=1

which implies the deﬁning relations, k γ0 (uik )ukj = k uik γ0 (ukj ) = δij , of the antipode γ0 . However, note that due to the order of u and u ¯ in equation (18), we cannot express γ0 (¯ uij ) so simply in terms of ukl . This of course is not surprising, as si and s∗j are treated diﬀerently in the Cuntz algebra, and hence the asymmetry between u and u ¯. As we will see later, in the examples to be considered, the u ¯ij are related to ukl . Let us now turn to the co-ﬁxed point algebra of the co-action of A on Od . We have the following Lemma 6. If we have a co-ﬁxed point of the form φ= ci1 ,...,ik ;j1 ,...,jl si1 . . . sik s∗j1 . . . s∗jl i1 ,...,ik ;j1 ,...,jl

then the following relations hold cr1 ,...,rk ;j1 ,...,jl ui1 r1 . . . uik rk r1 ,...,rk

=

ci1 ,...,ik ;s1 ,...,sl usl jl usl−1 jl−1 . . . us1 j1 .

(19)

s1 ,...,sl

Proof. As φ is co-ﬁxed we have φ⊗1 = i1 ,ii ,...,ik ,ik ;j1 ,j1 ,...,jl ,jl

ci1 ,...,ik ;j1 ,...,jl si1 . . . sik s∗j1 . . . s∗jl ⊗ ui1 i1 . . . uik ik u ¯j1 j1 . . . u ¯jl jl ,

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which immediately leads to ci1 ,...,ik ;j1 ,...,jl ui1 i1 . . . uik ik u ¯j1 j1 . . . u ¯jl jl = ci1 ,...,ik ;j1 ,...,jl . i1 ,...,ik ;j1 ,...,jl

By using (18) we can rewrite this relation into the desired form. Lemmas 5 and 6 justify our formulation of Theorem 2: the co-algebra structure and relations among the generators of A are forced on us by requiring the particular co-action on the Cuntz algebra. For the compact quantum groups associated with the four classical series of Lie algebras, the deﬁning relations are all of the form (19), and can be shown to arise from co-ﬁxed points. Consider, for example, a ﬁxed point algebra generated by Sq = q |w| Ew(1)...w(d) sw(1) . . . sw(d) , w∈W

θ

=

σij,kl si sk s∗l s∗j ,

i,j,k,l

and their conjugates, where W is the symmetric group, |w| is the length of the element w, and q is real and satisﬁes 0 < q < 1. Ei1 ...id is totally antisymmetric in the d indices with E1...d = 1, and σ = P R, P =

d

eab ⊗ eba ,

R=q

a

eaa ⊗eaa

{1 ⊗ 1 + (q − q −1 )

a,b=1

ebc ⊗ ecb }.

b,c:b
The θ and Sq respectively lead, through equation (19), to

R12 U1 U2 Eq (i1 , . . . , id )uj1 i1 . . . ujd id

= U2 U1 R12 , = Eq (j1 , . . . , jd ),

(20)

i1 ,...,id

where Eq (w(1), . . . , w(d)) = q |w| Ew(1)...w(d) . It is clear but important to note that the relation in A arising from any co-ﬁxed point through Lemma 6 can be expressed in terms of the relations (20). Equation (20) is the deﬁning relations of the quantum group SUq (d), where the quantum determinant condition (the second equation) also allows us to express u ¯ij in terms of ukl . To see that the hypotheses of the theorem do not imply additional relations between the generators in A we observe that there is a well deﬁned co-action of SUq (d) on Od . Hence if the hypotheses implied additional relations in A then this would force relations in SUq (d) which could not be satisﬁed. This proves Theorem 2 in the case of SUq (d). We will not present the detailed analysis of the B, C and D series of quantum groups here, as the method of study is very much the same as in the A case. It suﬃces to say that appropriate choices of a braid θ, a q−trace τ , and a rank d

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q−antisymmetric tensor as the generators of the co-ﬁxed point algebra will make A coincide with SOq (2m+1) (d = 2m+1), SPq (2m) (d = 2m), or SOq (2m) (d = 2m), where the q-trace τ in particular requires the transpose of U = (uij ) to be its inverse up to a conjugation by a real matrix. A further remark is that the rank d q-antisymmetric tensor does not lead to new relations in the case of SPq (2m). Details of these quantum groups can be found in [11].

8 Categorical interpretation There is a way to interpret our earlier constructions more abstractly. This may be useful in order to relate this paper to other examples. Let A be a Hopf ∗-algebra generated by the elements of u = (uij )di,j=1 . We assume that u is unitary, that is, d < ∞, which furnishes a uu∗ = I = uu∗ . Let H be a Hilbert space of dimension co-representation of A, ω : H → H ⊗ A, ω(ei ) = j ej ⊗ uji . Tensor products of H with itself yield co-representations of A in a natural way. We have not speciﬁed the relations satisﬁed by the matrix elements of u at this stage. In fact, it is our purpose here to show that the deﬁning relations can be recovered from the co-algebraic structure and the co-ﬁxed points of the A co-action on the braided tensor category generated by H. Consider the tensor category Fd , whose objects are H ⊗r , and arrows are the linear mappings T : H ⊗r → H ⊗s . As in Doplicher [4], we associate to Fd the Cuntz algebra Od . Then there is a natural A coaction on Od deﬁned by Γ : Od → Od ⊗ A Γ(si ) =

d

sj ⊗ uji .

j=1

This co-action, when restricted to the UHF-algebra Md∞ of Od through the embedding η(ei1 j1 ⊗ · · · ⊗ eik jk ) = si1 . . . sik s∗jk . . . s∗j1 , gives ea1 b1 ⊗ · · · ⊗ eak bk ⊗ ua1 i1 . . . uak ik u∗bk jk . . . u∗b1 j1 ΓUHF (ei1 j1 ⊗ · · · ⊗ eik jk ) = a1 ...ak b1 ...bk

for all positive integers k. Set uk = u⊗· · ·⊗u. Then for any x ∈ Mdk , Φ(x) = ΓUHF (x) = uk (x⊗1)(uk )∗ . The ﬁxed points of Φ are those x such that uk (x⊗ 1)(uk )∗ = (x⊗ 1) ⇒ uk (x⊗ 1) = (x ⊗ 1)uk . Let R be the solution of quantum Yang-Baxter equation such that R : H ⊗ H → H ⊗ H. Let P be theﬂip. Then σ = PR satisﬁes the braid relations. As elements of Od , θ = η(σ) = σijkl η(eij ⊗ ekl ) = σijkl si sk s∗l s∗j and θi = η(bi ) = sl1 sl2 . . . sli−1 ⊗ σ ⊗ s∗li−1 . . . s∗l2 s∗l1 {l}

satisfy the braid group relations.

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Form then a braided tensor category (Fd , R) by replacing the commutativity of Fd with the braiding bi . Denote by (Od )Γ the ﬁxed point algebra of (Fd , R) under the coaction Γ: (Od )Γ = {x : Γ(x) = x ⊗ 1}. Now we require that the ﬁxed point algebra (Od )Γ is generated by the braids bi . Then it follows from u2 (σ ⊗ 1) = (σ ⊗ 1)u2 that R(u ⊗ 1)(1 ⊗ u) = (1 ⊗ u)(u ⊗ 1)R, which is precisely the deﬁning relations of a quantum group in the formulation of [7]. Therefore, the relations and co-algebra structure of A are implied by the co-action and the co-ﬁxed point algebra (Od )Γ associated to the braided category (Fd , R). Conversely, starting with our Hilbert space H of ﬁnite dimension d as an irreducible U module, where U is a Hopf ∗-algebra, we can consider the tensor category whose objects are H ⊗n and the arrows are linear mappings H ⊗n → H ⊗n . Assume there exists a nondegenerate pairing between A and U which relates ∆, γ, " of A with the multiplication, unit of U , and relates the algebra structure of A with the comultiplication and co-unit of U . As for the antipode, we have γ(x), a = x, γ(a), x ∈ A, a ∈ U. As before Od is associated to the tensor category Fd . Then there exists a ∗ natural U -module action on O given by a◦1 = "(a)1, a◦s = i j π(a)ji sj , a◦si = d ∗ j π(γ(a))ij sj , a ◦ (xy) = (l) (a(1) x) ◦ (a(2) y) ∀x, y ∈ Od . This module action gives rise to a co-module action of A on Od in the standard fashion. We can continue in this vein, reinterpreting the earlier explicit constructions from the more abstract viewpoint. However we desist after making one ﬁnal comment. If we also assume that there exists a braid operator σ acting on H ⊗ H, such that [σ, (π ⊗ π)∆(a)] = 0, ∀a ∈ U then θ = η(σ) ∈ Od can be easily seen to be co-ﬁxed by A. Furthermore, Ru1 u2 = u2 u1 R, where R = P σ and u = (uij )di,j=1 .

References [1] A. L. Carey and D. E. Evans, On an automorphism action of U (n, 1) on On , J. Funct. Analys., 70 (1987) 90. [2] V. Chari and A. Pressley, A guide to quantum groups, Cambridge University Press, Cambridge (1994). [3] J. Cuntz, Simple C ∗ -algebras generated by isometries, Commun. Math. Phys., 57 (1977) 173.

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[4] S. Doplicher, Abstract compact group duals, operator algebras and quantum ﬁeld theory, Proc. Internl. Cong. Math. Kyoto (1990). [5] S. Doplicher, J. E. Roberts, Duals of compact Lie groups realized in the Cuntz algebras, J. Funct. Analys., 74 (1987) 90. [6] V. G. Drinfeld, Quantum groups, Proc. Internl. Cong. Math., Berkeley, 1 (1986) 789. [7] L. D. Faddeev, N. Yu. Reshetikhin and L. A. Takhtajan, Quantization of Lie groups and Lie algebras, Leningrad Math. J., 1 (1990) 193. [8] K. Fredenhagen, K. H. Rehren and B. Schroer, Superselection sectors with braid group statistics and exchange algebras I, Commun. Math. Phys., 125 (1989) 201. [9] M. Gerstenhaber, On the deformation of rings and algebras, Ann. Math. (2), 79 (1964) 59. [10] R. Gover and R. Zhang, Geometry of quantum homogeneous vector bundles and the representation theory of quantum groups I. Rev. Math. Phys., 11 (1999) 533–552. [11] T. Hayashi, Deformation of classical groups, Publ. RIMS, Kyoto Univ., 28 (1992) 57–82. [12] G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras, Adv. Math., 70 (1988) 237–249. [13] M. Jimbo, A q-diﬀerence analog of U (g) and the Yang-Baxter equation, Lett. math. Phys., 10 (1985) 63. [14] M. Jimbo, Quantum R-matrix related to the generalized Toda system: an algebraic approach. In “Field Theory, Quantum Gravity and Strings”, H.J. de Vega and N. Sanchez (eds), Lecture Notes in Physics 246 pp 335–361, Springer, Berlin (1986) [15] A. N. Kirillov and N. Yu. Reshetikhin Representations of the algebra Uq (sl(2)), q−orthogonal polynomials and invariants of links, in “Inﬁnite dimensional Lie algebras and groups” V.G. Kac (ed) pp 285–339, World Scientiﬁc Singapore (1989). [16] S. Montgomery, Hopf algebras and their actions on rings, Regional Conference Series in Math., 82 (1993). [17] A. Paolucci, Co-actions of Hopf algebras on the Cuntz algebras and their ﬁxed point algebras, Proc. American Math. Soc., 125 (1997) 1033–1042.

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A. L. Carey, A. Paolucci, R. B. Zhang

Ann. Henri Poincar´e

[18] M. Rosso, Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra, Commu. Math. Phys. 117 (1988) 581–593. [19] M. Sweedler, Hopf algebras, Benjamin, New York (1969). [20] S. L. Woronowicz, Compact matrix pseudogroups, Commun. Math. Phys. 111 (1987) 613; Tannaha-Krein duality for compact matrix pseudogroups, Inv. Math., 93 (1988) 35, Compact Quantum Groups, preprint Warsaw, 1993 A. L. Carey and R. B. Zhang Department of Pure Mathematics University of Adelaide Adelaide, Australia e-mail : [email protected] A. Paolucci Department of Mathematics University of Leeds Leeds, LS2 9JT, UK Communicated by Jean Bellissard submitted 16/09/98, accepted 18/05/00

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Ann. Henri Poincar´ e 1 (2000) 1123 – 1157 c Birkh¨ auser Verlag, Basel, 2000 1424-0637/00/0501123-35 $ 1.50+0.20/0

Annales Henri Poincar´ e

Solutions of the Dirac-Fock Equations without Projector Eric Paturel Abstract. In this paper we prove the existence of inﬁnitely many solutions of the Dirac-Fock equations with N electrons turning around a nucleus of atomic charge Z, 1 is the fundamental satisfying N < Z + 1 and α max(Z, N ) < 2 2 π , where α ≈ 137 π

+2

constant of the electromagnetic interaction. This work is an improvement of an article of Esteban-S´ er´ e, where the same result was proved under more restrictive assumptions on N .

1 Introduction This paper deals with the Dirac-Fock (DF) model in relativistic quantum mechanics. Many atomic or molecular models have been studied from a mathematical point of view, during this last ﬁfty years. The Hartree (H) model, and its more realistic variant, called Hartree-Fock (HF), are among the most famous ones. Many results are known about them, for example the existence of a ground state [12], and inﬁnitely many stationary states [13]. The link between these atomic models and other approximations, like Thomas-Fermi (TF) and Thomas-Fermi-von Weizs¨acker (TFW) has been widely studied (cf. the Selecta of E.H. Lieb [14]). The HF model is very useful in computational chemistry, where it yields results on the energy levels of atoms, that agree with experimental data, as long as the atomic charge is not too large. These models are nevertheless based on a nonrelativistic approximation: the kinetic energy is represented by the stationary Schr¨ odinger operator −∆. In [16], Swirles introduced the Dirac-Fock energy functional: this functional approximates the relativistic energy of a system of N electrons moving around a static nucleus of large positive charge Z. The approximations are still important: the nucleus is supposed to be ﬁxed (Born-Oppenheimer approximation), the spin interactions are neglected, the wave function of N electrons is represented by a Slater determinant, and the only interaction considered (i.e. the electrostatic interaction) is not Lorentz invariant. However, the numerical results obtained with the DF model are in very good agreement with experimental data (see [11], [10], [5]), especially for the innershell electrons of heavy atoms. The model also includes the case of molecules, with a distribution of positive charge Zµ, where µ is a probability measure, and Z the total positive charge. Note that it is also possible to take into account non purely electrostatic interaction as perturbations of the electrostatic case. We refer to [9] for a study of a multi-determinant Dirac-Fock model.

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In [6], M.J. Esteban and E. S´er´e proved the existence of inﬁnitely many stationary states for the DF model, provided N and Z satisfy the following conditions: N < Z + 1 , α max(Z, 3N − 1) <

2 π

2 +

π 2

.

1 stands for the fundamental constant of the electromagnetic interacHere α ≈ 137 tion. Their method is variational and, after changes in the functional (smoothing the potential and penalizing the constraint), they reduce it thanks to a concavity property: the major advantage is that the new functional is no more strongly indeﬁnite. Once this reduction is made, they build min-max sequences by a linking argument. This result is the ﬁrst existence result for this model. Unfortunately, as big atoms have many electrons, Esteban-S´er´e’s result does not cover them: the theorem is not valid for neutral atoms with atomic charge greater than 41. The 2 reason is that the concavity argument does not work for α(3N − 1) > 2 + π . π

2

Our aim is to improve this result, so that it becomes true for every N and Z satisfying 2 N < Z + 1 , α max(Z, N ) < 2 π . π + 2 Note that the same assumptions are needed for the existence of a ground state in the Brown-Ravenhall model (see [6]). To set out our result, we ﬁrst recall the necessary deﬁnitions for the DF model. In the whole paper, we use the system of units such that = c = 1, and the mass me of the electron is also normalized to 1. The state of a free electron is represented by a wave function Ψ(t, x), with Ψ(t, .) ∈ L2 (R3 , C4 ) for any t, which satisﬁes the free Dirac equation: i∂t Ψ = H0 Ψ, with H0 = −i

3

αk ∂k + β .

(1.1)

k=1

We denote (X, X ) the standard Hermitian product in C4 , and (φ, ψ)L2 the usual Hermitian product in L2 (R3 , C4 ). The Dirac equation contains the 4 × 4 complex matrices α1 , α2 , α3 and β, whose standard form is 0 σk I 0 β= , αk = , 0 −I σk 0 with the Pauli matrices 0 1 0 −i 1 0 , σ2 = , σ3 = . σ1 = 1 0 i 0 0 −1 These matrices satisfy some algebraic conditions which ensure that H0 is a symmetric operator such that H02 = −∆ + 1 .

(1.2)

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The following lemma contains some properties of H0 and the Coulombic 1 potential V (x) = |x| , whose proof can be found in [2], [19], [18]. This lemma was already stated in [6], we repeat it for the reader’s convenience. Lemma 1.1. H0 is a self-adjoint operator on L2 (R3 , C4 ), with domain H 1 (R3 , C4 ). Its spectrum is (−∞, −1] ∪ [1, +∞). There are two orthogonal projectors on L2 (R3 , C4 ), Λ+ and Λ− = 1L2 − Λ+ , both with inﬁnite rank, and such that √ √ 1 − ∆Λ+ = Λ+ 1 − H0 Λ+ = Λ+ H0 = √ √∆ (1.3) H0 Λ− = Λ− H0 = − 1 − ∆Λ− = −Λ− 1 − ∆ The Coulombic potential V (x) = (ϕ, V ϕ)L2

1 |x|

satisﬁes the following Hardy-type inequalities: 1 π 2 ≤ (1.4) + (ϕ, |H0 |ϕ)L2 , 2 2 π

for all ϕ ∈ Λ+ (H 2 ) ∪ Λ− (H 2 ). Moreover, 1 π (ϕ, |H0 |ϕ)L2 , ∀ϕ ∈ H 2 , (ϕ, V ϕ)L2 ≤ 2 ||V ϕ||L2 ≤ 2||∇ϕ||L2 , ∀ϕ ∈ H 1 . 1

1

(1.5) (1.6)

We now introduce the DF model. Although our result can be stated for molecules (see Remark 1.4 below), we restrict ourselves, for the sake of simplicity, to the case of electrons around one nucleus of atomic charge Z. We introduce the functional space of the variational problem associated to the DF model. Let E = H 2 (R3 , C4 ) , E + = Λ+ E , E − = Λ− E . 1

E is a Hilbert space with Hermitian product √ (ϕ, ψ)E = ϕ, 1 − ∆ψ 2 = ϕ+ , ψ+ E + ϕ− , ψ− E .

(1.7)

L

Now, the DF energy functional, denoted by E, is deﬁned on the space E N , with E(Φ)=

N N (ϕl , H0 ϕl )L2 − αZ (ϕl , V ϕl )L2 l=1

α + 2

l=1

R3 ×R3

V (x − y) [ρ(x)ρ(y) − tr (R(x, y)R(y, x))] d3 xd3 y , (1.8)

where α is positive, with physical value α ≈ complex matrix, given by ρ(x) =

N l=1

1 137 ,

(ϕl (x), ϕl (x)) , R(x, y) =

ρ is a scalar and R is a 4 × 4

N

ϕl (x) ⊗ ϕ∗l (y) .

l=1

Physically, ρ is the electronic density, R is the exchange matrix.

(1.9)

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We look for critical points of E under the normalization constraint Φ ∈ Σ = {Φ ∈ E N | GramL2 Φ = 1l} .

(1.10)

The manifold Σ and the DF functional are invariant under the action of the group U(N ) of N × N self-adjoint complex matrices with determinant of modulus 1. We denote

u1l ϕl , ..., uNl ϕl , u ∈ U(N ) , Φ ∈ E N . (1.11) u.Φ = l

l

Thanks to this invariance, we may diagonalize the Lagrange multiplier matrix and write the Euler-Lagrange equations in the following form, called Dirac-Fock equations H Φ ϕk = εk ϕk , where

(1.12)

H Φ ψ = H0 ψ − αZV ψ + α(ρ ∗ V )ψ − α

R3

R(x, y)ψ(y)V (x − y)dy .

(1.13)

Physically, H Φ represents the Hamiltonian of an electron in the mean ﬁeld due to the nuclei and the electrons. The eigenvalues ε1 , ..., εN are the energies of each electron in this mean ﬁeld. With the unit convention, the physically interesting states correspond to 0 < εk < 1: a positive energy less than the rest mass of the electron. We recall that the result [6] makes use of a concavity argument, more precisely 2 the fact that, under the assumption that α(3N − 1) < 2 + π , the DF-functional π

2

is concave in the E − directions. To get rid of this, we have to handle directly a strongly indeﬁnite functional, which moreover does not satisfy the Palais-Smale property. This is not very comfortable, and we have to do some preliminary work before the usual variational tools can be applied. Our result, which is an extension of [6, Theorem 1.2], is the following:

2 Theorem 1.2. Assume that α max(Z, N ) < 2 + π , N < Z + 1, with Z > 0 the π 2 nuclear charge. Then there exists a sequence (Φn ) of critical points for the DF functional E on Σ. The functions ϕn1 , ..., ϕnN satisfy the normalization constraints GramL2 Φn = 1l. They are smooth outside 0, decay exponentially as well as their 3 derivatives as |x| → +∞, and are strong solutions, in W 1, 2 (R3 , C4 ), of the DiracFock equations

H Φn ϕnk = εnk ϕnk , 1 ≤ k ≤ N ,

(1.14)

∀k , 0 < εnk < 1 .

(1.15)

with

Moreover, as n goes to inﬁnity, the energies εnk go to 1.

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Remark 1.3. The corresponding physical bound for N and Z is 124, with N ≤ Z. Remark 1.4. As in [6], it is easy to change the formulation of the problem to consider others distributions of positive charge than a point-like nucleus. Theorem 1.2 extends easily to the case of a nuclear charge density Zµ, where Z > 0 is the total nuclear charge and µ is a probability measure deﬁned on R3 . In this general point-like situation, Φn is smooth outside Supp(µ). In the particular case of m m nuclei, each one having atomic number Z at a ﬁxed location x , Zµ = i i i=1 Zi δxi , and Z = m Z . i i=1 The paper is organized as follows: the next section is devoted to a sketch of the proof of Theorem 1.2. Section 3 deals with the reductions of the initial problem to a more comfortable one. We expose in Section 4 a linking procedure. Finally, Section 5 deals with the compactness properties and the limits of the min-max sequences.

2 Sketch of the proof In order to prove this result, it is hard to use the variational method directly, for many reasons: the corresponding functional is strongly indeﬁnite, there is no compactness property available for this functional, and the Coulombic potential is hard to deal with. To avoid successively these obstacles, we are led to consider diﬀerent stages of approximate problems. Indeed, we use here many approximation tools so that the ﬁnal variational problem is simpler to handle with. The strategy can be described like this: (a) In order to get rid of the roughness of the Coulombic potential V , we replace it with an approximate one, actually a convolution of V with a normalized Gaussian depending on a parameter ν. This Gaussian function tends to the Dirac distribution as ν goes to zero. Moreover, as the initial problem is constrained, and the physically interesting states are governed by the Lagrange multipliers given by the constraint, we deﬁne instead a problem with penalization of the constraint. The choice of the penalization gives directly a lower bound for the energy of the electrons. This step is exactly the same as in [6]. (b) The use of variational tools needs at least some compactness properties. To get this, we change the problem into a periodic one, that is, we consider a functional space of periodic functions. We have to change one more time the shape of the potential. The corresponding functional satisﬁes for example the Palais-Smale compactness property, which is enough to take the limit of min-max sequences. (c) The last (not the least!) problem is that the functional is still strongly indeﬁnite. To deal with this, we operate a Lyapunov-Schmidt reduction to a ﬁnite dimensional problem . This reduction is made possible because of the

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compactness properties of the functional, once the steps (a) and (b) are fulﬁlled. Such a tool was used in a famous paper by Conley and Zehnder [3] on Arnold’s conjecture. (d) We develop a linking method in the ﬁnite dimensional case. This method is based on the construction of a pseudogradient vector ﬁeld. The LyapunovSchmidt reduction gives then critical points for the periodic problem (b). We have then to translate the estimates on the (Conley-Zehnder) Morse index of the critical points for the ﬁnite dimensional problem onto an estimate on the eigenvalues εk . (e) We pass to the limit as the length of the period goes to inﬁnity. There remains to take the limit of the solutions as the penalization parameter p and the smoothness constant ν go respectively to inﬁnity and 0: we will use again the same arguments as in [6].

2.1 First reduction To begin this program, we will write the variational problems corresponding to its successive steps. We will explain the reasons why we consider these problems and their advantages relatively to the preceding formulations. Recall that the ﬁrst variational formulation of the problem of ﬁnding solutions to the DF equations is Problem 2.1. Find critical points for the functional E(Φ) =

N N (ϕl , H0 ϕl )L2 − αZ (ϕl , V ϕl )L2 l=1

α + 2

l=1

R3 ×R3

V (x − y) [ρ(x)ρ(y) − tr (R(x, y)R(y, x))] d3 xd3 y ,

under the constraint

Φ ∈ Σ = Φ ∈ E N |GramL2 Φ = 1l , with the following condition on the matrix of Lagrange multipliers Λ, ∃u ∈ U(N ) , uΛu∗ = Diag(ε1 , ..., εN ) with ∀k, 0 < εk < 1 .

(2.1)

The ﬁrst transformation of Problem 2.1 into a more practical one was introduced in [6]. We follow the authors and brieﬂy explain the reasons of this transformation to Problem 2.2. We refer to [6] for more details. As in [6], we have to deal with the fact that the Coulombic potential V is not a compact perturbation of H0 . That is why we replace V by the regularized potential, for ν > 0: Vν (x) = (gν ∗ V )(x) ,

(2.2)

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with gν (x) =

1129

2 1 − |x| 2ν . 3 e (2πν) 2

This replacement is made for the attractive potential of the nucleus, as well as for the electronic repulsion and exchange terms. We denote by Eν the corresponding ν functional, and the associated one-particle Hamiltonian is denoted by H Φ . As announced in (a), we also replace the constraint “Φ ∈ Σ” by a penalization term πp depending on an integer p. We put (2.3) πp (Φ) = tr (GramL2 Φ)p (1l − GramL2 Φ)−1 , and we deﬁne the penalized functional Fν,p = Eν − πp ,

(2.4)

A = Φ ∈ E N | 0 < GramL2 Φ < 1l .

(2.5)

in the domain

Note that Fν,p is invariant under the U(N ) action. Any U(N ) orbit in A contains a point Φ such that GramL2 Φ is diagonal, with eigenvalues in nondecreasing order: GramL2 Φ = Diag(σ1 , ..., σN ), 0 < σ1 ≤ ... ≤ σN < 1 .

(2.6)

We call O the set of points of A satisfying (2.6). If Φ ∈ O, then ∂Fν,p ν (Φ) = H Φ ϕk − εk ϕk , ∂ϕk

(2.7)

with εk = ep (σk ) , ep (x) =

d dx

xp 1−x

=

pxp−1 − (p − 1)xp . (1 − x)2

(2.8)

The advantage of this penalization is that ep is a positive increasing function on (0, 1), so that 0 < ε1 ≤ ... ≤ εN . Hence, the critical points of Fν,p in O are solutions of a nonlinear eigenvalue problem with positive eigenvalues. So we may deﬁne the following problem Problem 2.2. Find critical points Φ of the functional Fν,p on the domain A, satisfying moreover: ∃u ∈ U(N ) | GramL2 (u.Φ) = Diag(σ1 , ...σN ) , (2.9) with, for all k , εk = ep (σk ) < 1.

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As we see later, we need a control, not only on the critical points for Fν,p , but also on some of its Palais-Smale sequences. Even if Fν,p does not satisfy the PalaisSmale property, we may use a compactness result: [6, Lemma 2.1], which indicates the behaviour of some Palais-Smale sequences, and ensures that the solutions of Problem 2.2 we will ﬁnd actually converge to solutions of Problem 2.1. Lemma 2.3 ([6]-Lemma 2.1). Assume that α max(Z, N ) <

2 π 2 2 +π

.

(a) Let (νn ) be a sequence of real numbers in (0, 1), (pn ) a sequence of positive integers, and (Φn ) a sequence in O, i.e. such that GramL2 Φn = Diag(σ1,n , ..., σN,n ), 0 < σ1,n ≤ ... ≤ σN,n < 1 . We denote εk,n = epn (σk,n ). We assume that Fν n ,pn (Φn ) −−−→ 0 , n→∞

(2.10)

N ∗ 1 for the strong topology of H − 2 (R3 , C4 ) = E N . We also assume that lim inf σ1,n > 0 .

(2.11)

lim inf σ1,n ≥ h0 ,

(2.12)

n→∞

Then, n→∞

where h0 ∈ (0, 1) is a constant which depends only on αZ, αN . (b) If moreover, lim sup σN,n < 1 ,

(2.13)

n→∞

then, after extraction of a subsequence, the functions ϕk,n converge to N 1 functions ϕk ∈ E ∩ 1≤q≤ 3 W 1,q (R3 , C4 ), for the strong H 2 topology. 2

(b.1) In the case νn → ν ∈ (0, 1), and pn = p for n large, Φ = (ϕ1 , ..., ϕN ) is a critical point of Fν,p in O. Moreover, Fν,p = limn→∞ Fνn ,pn (Φn ). (b.2) In the case νn → 0, and pn → +∞, ϕ1 , ..., ϕN satisfy the orthonormality constraints (ϕl , ϕk )L2 = δkl . They are strong solutions, in Σ ∩ 1,q (R3 , C4 ), of the DF-equations 1≤q≤ 3 W 2

H Φ ϕk = εk ϕk , εk = lim εk,n ∈ [h0 , 1) , n→∞

(2.14)

and the DF-energy is E(Φ) = lim Fνn ,pn (Φn ) . n→∞

(2.15)

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2.2 Reduction to the periodic setting From now on, we consider ﬁxed ν ∈ (0, 1) and p > 3. It is possible to proceed to another transformation of the variational problem, as announced in (b). Let L ∈ R, L > 0 and ω = 2π L , 1 L L 2 ((− , )3 , C4 ) , Eω = Hper 2 2 deﬁned as functions with the following Fourier series, ψ(x) = aωp eiωp.x

p∈Z3

where the Fourier coeﬃcients aωp ∈ C4 satisfy 1 1 + |ωp|2 2 |aωp |2 < ∞ . p∈Z3

We may deﬁne the periodic free Dirac operator acting on L2per ((− L2 , L2 )3 , C4 ) with domain Eω with the same formula (1.1). Since H02 = −∆ + 1, the spectrum of this operator is contained in (−∞, −1] ∪ [1, +∞). This allows to deﬁne two orthogonal projectors on L2per ((− L2 , L2 )3 , C4 ), still denoted by Λ+ and Λ− , having the same properties (1.3) as their equivalent in the non-periodic case. In order to deﬁne a DF-functional on Eω , we have to deﬁne a periodic potential corresponding to the Coulombic potential. Let Gν,ω be the periodic potential deﬁned by its Fourier series gν (p)eiωp.x ω Gν,ω (x) = . 4π |p|2 3 p∈Z ,p =0

Remark that the coeﬃcients are, for p = 0, cωp =

ωgν (p) = Vν (p) . 4π|p|2

Unlike the Coulombic potential, this periodic potential is no more positive everywhere, but we get the following property: there exists a positive constant C0 independent of ω such that Gν,ω ≥ −ωC0 .

(2.16)

This causes a little change in the computations, relatively to the non-periodic case, but, as we will see, the essential properties are preserved, as ω is close enough to 0. We may see Gν,ω as a function looking like Vν in a neighborhood of 0 and the other points of the form Lp, with p ∈ Z3 . Note that x 1 Gν,ω = G1 (x) , ω ω where G1 satisﬁes −∆G1 = λ∈2πZ3 Vν (. − λ) − 1.

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Thanks to the Fourier series computation, we are able to translate in the periodic case the results of Lemma 1.1: we express these inequalities using the Fourier transform and pass to the limit from a regular transform to a sum of Dirac distributions, corresponding to a Fourier series. The details of the computations are left to the reader. Lemma 2.4. The periodic potential Gν,ω satisﬁes the following Hardy-type inequalities, uniformly in ν ∈ (0, 1): 2 1 π (φ, Gν,ω φ)L2 ≤ + (φ, |H0 |φ)L2per , (2.17) per 2 2 π for all φ ∈ Λ+ (Eω ) ∪ Λ− (Eω ). Moreover, π (φ, |H0 |φ)L2per , ∀φ ∈ Eω , ≤ (φ, Gν,ω φ)L2 per 2 1 . ||Gν,ω φ||L2per ≤ 2||∇φ||L2per , ∀φ ∈ Hper

(2.18) (2.19)

We may deﬁne the periodic DF-functional, depending on the parameters ν, p (ﬁxed) and ω: Pω (Φ) = Eν,ω (Φ) − πp,ω (Φ) ,

(2.20)

that is, Pω (Φ) =

N

(ϕk , H0 ϕk )L2per − αZ

k=1

+

α 2

N

(ϕk , Gν,ω ϕk )L2per

k=1

L 3 L L 3 (− L 2 , 2 ) ×(− 2 , 2 )

Gν,ω (x − y) [ρ(x)ρ(y) − tr(R(x, y)R(y, x))] dxdy

−tr (GramL2per Φ)p (1l − GramL2per Φ)−1 . This functional is deﬁned in the space Aω = Φ ∈ EωN | 0 < GramL2per Φ < 1l .

(2.21)

(2.22)

We write the Euler-Lagrange equations related to the Problem 2.7: ω

ΠΦ ϕk =

∂πp,ω , ∂ϕk

(2.23)

ω

where ΠΦ is the one-particle Hamiltonian associated to the periodic problem, i.e. ω

ΠΦ ψ

= H0 ψ − αZGν,ω ψ + α(ρ ∗ Gν,ω )ψ −α R(x, y)ψ(y)Gν,ω (x − y)dy .

(2.24)

L 3 (− L 2,2)

The most interesting properties of the new functional are the two following lemmas.

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Lemma 2.5. Assume that α(N − 1) <

1133

, and let ν ∈ (0, 1), p ≥ 3 be ﬁxed, ¯ ν) > 0 and and M > 0. Then, there exists ε0 > 0, and two constants K(M, σ ¯ (M, ν, p) < 1 such that, given any sequence (Φn )n≥0 ∈ Aω verifying, for n large enough |Pω (Φn )| ≤M, ||Pω (Φn )|| − 12 ≤ ε0 , 2

π 2 2 +π

Hper

we get the following estimates, for n large enough 2

2 ¯ ||ϕk,n ||Eω ≤ K(M, ν) ,

(2.25)

k

and ¯ (M, ν, p) . σN,n ≤ σ

(2.26)

In other words, Palais-Smale sequences for Pω stay in a closed ball of EωN . This allows to prove the Palais-Smale property for Pω . Lemma 2.6. Given δ ∈ (0, 12 ), the functional Pω satisﬁes the Palais-Smale condition in Bδ,ω = {Φ ∈ EωN | δ1l ≤ GramL2per Φ < 1l}, i.e. any Palais-Smale sequence in Bδ,ω is bounded and, up to a subsequence, converges to a critical point for Pω . Note that, as in the non-periodic case, this functional is invariant by the action of U(N ). We may deﬁne the periodic variational problem: Problem 2.7. Find critical points Ψ of the functional Pω in the space Aω , with the following property ∃u ∈ U(N ) | GramL2per (u.Ψ) = Diag(σ1 , ...σN ) , (2.27) ∃b0 , ∀k , εk = ep (σk ) ≤ b0 < 1. The resolution of this variational framework will give solutions of Problem 2.2 provided we get a convergence lemma, which achieves the point (b) of the program: Lemma 2.8. Let (ωn ) be a sequence of positive real numbers such that limn→∞ ωn = 0 and (Φn ) a sequence of critical points of Pωn in Aωn satisfying: ∃un ∈ U(N ) | GramL2per (un .Φn ) = Diag(σ1,n , ..., σN,n ) ,

(2.28)

∃b0 ∈ (0, 1) | lim max ep (σk,n ) ≤ b0 .

(2.29)

and n→∞

k

Then, after extraction of a subsequence, the sequence (Φn ) converges, in E, to a critical point Φ for Fν,p , with h0 1l ≤ GramL2per Φ < 1l , where h0 is the constant of Lemma 2.3.

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2.3 Lyapunov-Schmidt reduction So we are led to ﬁnd critical points for the functional Pω corresponding to the periodic problem. Unfortunately, this functional is still strongly indeﬁnite. One way to deal with this problem is to reduce the functional space to a ﬁnite dimensional space, thanks to a Lyapunov-Schmidt method. This reduction is based on the fact that we may split any wave function Φ into two components: a “high frequency” component ΦU and a “low frequency” one ΦD , where frequency is meant to be the eigenvalue for the free Dirac operator H0 . Thanks to a contraction property of H0−1 , we solve, using a ﬁxed point theorem, the part of the DF-equations related to the high frequency part: this solution depends of course of the low frequency component. ˜ > 1. We consider the projection ΛU (resp. ΛD ) on the eigenspaces for Let λ ˜ (resp. |λ| ≤ λ). ˜ We get of course H0 corresponding to eigenvalues λ with |λ| > λ H0 ΛU = ΛU H0 , and H0 ΛD = ΛD H0 . We put ΦU = ΛU Φ, and ΦD = ΛD Φ. Using the projectors ΛU and ΛD on (2.23), together with the commutation properties, we get the following system:   (ϕk )D = H −1 ΛD Mρ ϕk + NR ϕk + ∂πp,ω 0 ∂ϕk (2.30)  (ϕk )U = H −1 ΛU Mρ ϕk + NR ϕk + ∂πp,ω , 0 ∂ϕk where Mρ = αZGν,ω − α(ρ ∗ Gν,ω ) and NR is the operator R(x, y)ψ(y)Gν,ω (x − y)dy . NR (ψ) = L 3 (− L 2,2)

¯ = (ϕ¯k ) ∈ ΛD Eω for every k, a solution to the We observe that, given Φ second equation of (2.30) may be written as Ξ = (ξk )1≤k≤N , with ξk ∈ ΛU Eω for all k, and Ξ is a ﬁxed point of the operator IΦ¯ mapping (ΛU Eω )N into itself, given by ∂πp,ω ¯ IΦ ¯ H0−1 ΛU Mρ(Φ+Ξ) ξk + NR(Φ+Ξ) ξk + (Φ + Ξ) . Ξ −→ ¯ ¯ ∂ϕk 1≤k≤N The next lemma shows that we may apply the Banach ﬁxed point theorem ¯ associated to Φ. ¯ The following estimate will depend on ν, to ﬁnd an unique Ξ ¯ +Ξ ¯ is not too close to the boundary and we have to check that the considered Φ {Φ | GramL2per Φ = 1l}. Lemma 2.9. Let µ ∈ (0, 1). Recall that we have ﬁxed ν ∈ (0, 1) and p > 3. ˜ > 0 such that, given any λ ≥ λ ˜ and Φ ¯ ∈ (ΛD Eω )N satisfyThere exists λ ing 0 < GramL2per Φ < (1 − µ)1l, (where the projections ΛD and ΛU are deﬁned with λ), the application IΦ¯ deﬁned above, restricted to a ball of radius µ2

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1135

˜ < 1 such that for every Ξ1 , Ξ2 in around 0, is contracting , i.e. there exists k(λ) µ N (ΛU (Eω )) ∩ B(L2per )N (0, 2 ), ˜ ||I(Ξ1 ) − I(Ξ2 )||L2per ≤ k(λ)||Ξ 1 − Ξ2 ||L2per . ˜ large enough, the ball (ΛU (Eω ))N ∩ B(L2 )N (0, µ ) is globally Moreover, taking λ 2 per invariant relatively to the action of IΦ¯ . Then the application of the Banach ﬁxed point theorem gives a ﬁxed point for IΦ¯ in the ball (ΛU (Eω ))N ∩ B(L2per )N (0, µ2 ). Thus we may deﬁne the map hλ˜ ¯ the unique from (ΛD Eω )N to (ΛU Eω )N ∩ B(L2per )N (0, µ2 ), associating to every Φ ¯ ¯ ﬁxed point of IΦ¯ in the ball. We put Ξ = hλ˜ (Φ). Thanks to this property, we can reduce our problem to a ﬁnite dimensional one: if we put, for Φ ∈ (ΛD Eω )N Qω (Φ) = Pω Φ + hλ˜ (Φ) , (2.31) then Problem 2.7 is solved when the following problem is solved: Problem 2.10. Find critical points Φ ∈ (ΛD Eω )N of the functional Qω , such that Ψ = Φ + hλ˜ (Φ) satisﬁes condition (2.27).

2.4 End of the proof Once this reduction is done, we may use variational tools on the function Qω acting on the ﬁnite dimensional space (ΛD Eω )N . Brieﬂy speaking, we build a min-max using a linking property. In particular, we have to avoid critical points corresponding to solutions with N electrons, for N < N , i.e. critical points Φ with a null determinant Gram matrix. For more details, see Section 4. The result of this construction is the following lemma. 2 Lemma 2.11. Let N and Z satisfy α max(N, Z) < π + 2 , and recall that ν and p 2 π are ﬁxed. Given ω > 0 small, there exists J(ω) ∈ N going to inﬁnity as ω goes to 0, and a sequence (Ψjω )0≤j≤J(ω) , of critical points of Pω . These critical points satisfy 0 < GramL2per Ψjω < 1l ,

and there exists a positive sequence cj such that, for any j ≥ 0, we have 0 < cj < N , limj→∞ cj = N , with N > Pω (Ψjω ) ≥ cj .

(2.32)

Moreover, each Ψjω is of the form Ψjω = Φjω + hλˆ (Φjω ) ,

(2.33)

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ˆ ˜ The function Φj is a critical point of Qω , i.e. the reduced for some λ(ω, ν, p) ≥ λ. ω ˆ and its Morse index iω,j satisﬁes functional associated to ω and λ, ˆ + N 2 + 2N j , iω,j (Φjω ) ≤ N m(ω, λ)

(2.34)

ˆ is the (real) dimension of (ΛD Eω )N . where 2N m(ω, λ) We remark that, for the construction of linkings of higher order, the functional space needs to have a suﬃciently large dimension. Then, the Lyapunov-Schmidt j reduction gives a critical point Ψjω = (ψk,ω )1≤k≤N for Pω if a critical point for Qω is found. We take advantage of the Morse-type information (2.34), as we see in the next lemma, to ﬁnd an upper bound for the eigenvalues εk strictly lower than 1. Lemma 2.12. Let ν, p ﬁxed. With the preceding notations, there exists two nondecreasing sequences a(j) and b(j) ∈ (0, 1), independent of ω small, such that ω

j j ΠΨ ψk,ω = εjk,ω ψk,ω ,

(2.35)

0 < a(j) ≤ εj1 ≤ ... ≤ εjN ≤ b(j) < 1 .

(2.36)

with

Now, it is possible, thanks to Lemma 2.8, to pass to the limit as the pulsation ˜ j of critical points for the functional Fν,p , ω goes to 0: we obtain then a sequence Ψ j with eigenvalues ε˜k . Finally, the use of Lemma 2.3 allows to pass to the limit as the parameters ν and p go respectively to 0 and ∞. Remark 2.13. One may probably get rid of the last reduction (Lyapunov-Schmidt reduction) by the use of the work of Abbondandolo [1], where a new Conley-Zehnder index is presented, allowing to handle directly strongly indeﬁnite functionals. The end of this paper is organized as follows: Section 3 is devoted to the proof of Lemmas 2.5, 2.6, 2.8, 2.9. Section 4 is devoted to the linking procedure in the ﬁnite dimensional setting, i.e. the proof of Lemma 2.11. Section 5 deals with the estimates on the eigenvalues related to the Morse index, i.e. the proof of 2.12.

3 Convergence of approximate problems 3.1 Preliminary results We are going to prove some results that will be used in the following proofs. ω They deal with the “periodic one-electron Hamiltonian” ΠΦ . These results are the analogues of [6, Lemma 3.1] in the periodic case. The main diﬀerence is the appearance of a term, which vanishes as ω goes to 0.

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Lemma 3.1. Assume α max(Z, N ) <

2 π π+2

1137

.

(i) There exists two constants h0 > 0 and ω0 > 0, such that, for any ν ∈ [0, 1], 0 < ω < ω0 and Φ ∈ EωN such that GramL2per Φ ≤ 1l, and ψ ∈ Eω , h0 ||ψ||

ω

1

2 Hper

≤ ||ΠΦ ψ||

−1

Hper2

.

(3.1)

ω

In other words, for ω small enough, ΠΦ is a self-adjoint isomorphism between − 12

1 2

Hper and its dual Hper , whose inverse is bounded independently of Φ, ν. (ii) Fix ν ∈ (0, 1] and p ≥ 3. Let ωn > 0 and Φn ∈ EωNn , with GramL2per Φn ≤ 1l. We assume that ||ϕk,n ||Eωn is a bounded sequence. Let ψn ∈ Eωn be ωn such that the sequence ||ΠΦn ψn ||L2per is bounded. then ψn is precompact in 1

2 Hloc (R3 , C4 ).

Proof. (i) We only point out the diﬀerences with [6, Lemma 3.1]. Let ψ ± = Λ± ψ. From inequalities (2.17) and (2.16), we have ( π + π2 )αZ ω ω (ψ+ , ΠΦ ψ+ )Eω ×Eω∗ − (ψ− , ΠΦ ψ− )Eω ×Eω∗ ≥ 1 − 2 − ωN C1 ||ψ+ ||2Eω 2 π ( 2 + π2 )αN − ωZC2 ||ψ− ||2Eω , + 1− 2 where C1 ,C2 are constants coming from (2.16). With the assumption on N and Z, there exists ω0 > 0 such that, given any ω ∈ (0, ω0 ), the two factors are positive. ( π + 2 )α max(N,Z) − ω0 max(C1 N, C2 Z). We get, for ω ≤ ω0 , Let us choose h0 = 1 − 2 π 2 ω

||ψ||Eω ||ΠΦ ψ||Eω∗

ω

≥ Re(ψ+ − ψ− , ΠΦ ψ)Eω ×Eω∗ = ≥

ω (ψ , ΠΦ ψ+ )Eω ×Eω∗ h0 ||ψ||2Eω , +

− (ψ

−

ω , ΠΦ ψ− )Eω ×Eω∗

(3.2) (3.3) (3.4)

and (3.1) is proved. ωn

(ii) From (i), a bound on ||ΠΦn ψn ||L2per implies a bound on ||ψn ||Eωn , hence a bound of the norm of ψn in Lr ((− ωπn , ωπn )3 , C4 ), for any 2 ≤ r < 3. Moreover, the potential Gν,ωn is in the Marcinkiewicz space M3 ((− ωπn , ωπn )3 , C4 ), and its norm in this space is bounded independently of ωn. So Gν,ωn ψn is bounded in any Lqloc (R3 , C4 ), with 1 ≤ q < 32 , and (ρΦn∗Gν,ωn )ψn − Gν,ωn (x−y)RΦn (x, y)ψn (y)dy

is in any Lqloc (R3 , C4 ), with 1 ≤ q < 3. As a consequence, using (i), −1 ψn = H0 αZGν,ωn ψn − α(ρΦn ∗ Gν,ωn )ψn + ω α Gν,ωn (x − y)RΦn (x, y)ψn (y)dy + ΠΦn ψn 1

2 (R3 , C4 ). is precompact in Hloc

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3.2 Proof of Lemmas 2.5 and 2.6 Consider a sequence (Φn ) ∈ Aω satisfying

|Pω (Φn )| ≤ M , ||Pω (Φn )|| ≤ ε0 ,

(3.5)

with GramL2per Φ = Diag(σ1,n , ..., σN,n ), 0 < σ1,n ≤ ... ≤ σN,n < 1 . We have to prove that such a sequence is bounded in EωN , if ε0 is close enough to 0. For ν > 0 ﬁxed, there is a constant K1 (ν) > 0 such that, for any Φ ∈ Aω , Eν,ω (Φ).Φ ≤ 2Eν,ω (Φ) + K1 (ν) .

From (3.5), with the notations θp (t) = Eν,ω (Φn ) ≤ M +

tp 1−t ,

(3.6)

θp (t) = ep (t), we infer

θp (σk,n ) ,

(3.7)

 12  ep (σk,n )σk,n − ε0  ||ϕk,n ||2Eω  .

(3.8)

k

and (Φn ).Φn ≥ Eν,ω

k

k

Combining (3.5), (3.6), (3.7) and (3.8), we get

ep (σk,n )σk,n − 2θp (σk,n ) ≤ Eν,ω (Φn ).Φn − 2Eν,ω (Φn ) + 2M

k

+ε0

12 ||ϕk,n ||2Eω

k

≤ K2 (ν, M ) + ε0

, 12 ||ϕk,n ||2Eω

.

k

Since

ep (t)t θp (t)

=

ntn n≥p n n≥p t

k

≥ p, the assumption p ≥ 3 leads to

ep (σk,n )σk,n ≤ K3 (ν, M ) + ε0

k

12 ||ϕk,n ||2Eω

.

(3.9)

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1139

− On the other side, consider the product of Pω (Φn ) with Φ+ n − Φn . We recall ± ± ± the notation Φ = (Λ ϕ1 , ..., Λ ϕN ). We get − + − + − Pω (Φn ).(Φ+ n − Φn ) = Eν,ω (Φn ).(Φn − Φn ) − πp,ω (Φn ).(Φn − Φn ) 12 ( π2 + π2 )α max(Z, N ) 2 ≥ 1− ||ϕk,n ||Eω 2 k − ep (σk,n )σk,n + 2πp,ω (Φn )Φ− (3.10) n k

Since πp,ω (Φ)ψ = 2

ep (σk )Re(Φk , ψk ) ,

(3.11)

k

the last term is nonnegative and we get, with c1 = 1 −

12 ||ϕk,n ||2Eω

≤

c−1 1

k

2 (π 2 + π )α max(Z,N ) 2

,

ep (σk,n )σk,n +

Pω (Φn ).(Φ+ n

−

Φ− n)

,

(3.12)

k

Using (3.5), we estimate the last term, and get, for every ε0 < c1 ,

12 ||ϕk,n ||2Eω

≤ (c1 − ε0 )−1

k

ep (σk,n )σk,n .

(3.13)

k

1 This and (3.9) imply that [ k ||ϕk,n ||Eω ] 2 and k ep (σk,n )σk,n are both bounded. ¯ We call the bound K(M, ν). Since ep (t) → +∞ as t → 1− , this implies the estimate

σN,n ≤ σ ¯ (M, ν, p) < 1 .

(3.14)

and the proof of Lemma 2.5 is over.

To prove Lemma 2.6, it is suﬃcient to apply Lemma 2.5 to a Palais-Smale sequence for Pω in Bδ,ω . Then, using the usual Sobolev inequalities, implying compact embeddings in the periodic case, the precompactness claimed is proved.

3.3 Proof of Lemma 2.8 After extraction of a subsequence, we may impose n→∞

εk,n = ep (σk,n ) −−−→ εk ≤ b0 . Consider a sequence of critical points (Φn ) satisfying (2.28) and (2.29). Using the Euler-Lagrange equation (2.23), their corresponding critical values are

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bounded by N b0 , and Lemma 3.1 gives a bound of the norm of (Φn ) in EωNn . We 1 may cut these functions and consider them as functions in H 2 ((− L2n , L2n )3 , C4 ) ⊂ 2π L2 (R3 , C4 ), with Ln = ωn . We apply Lemma 3.1 (ii) and obtain that the sequence 1

2 Φn is precompact in Hloc . Then, using arguments similar to [6, Proof of Lemma 2.1 (b)], we ﬁnd that the cut functions actually converge in E to critical points for Fν,p , with the bound b0 on each energy level εk .

3.4 Proof of Lemma 2.9 In order to prove this lemma, we take advantage of the smoothing of the potential Gν,ω . We will obtain a contraction coeﬃcient depending on ν. Indeed, we may write, calling λ the cut-oﬀ frequency, " ||I(Ξ1 ) − I(Ξ2 )||L2per ≤ H0−1 ΛU Mρ(Φ+Ξ ξ − Mρ(Φ+Ξ ξ ¯ ¯ 1 ) 1,k 2 ) 2,k + NR(Φ+Ξ ξ − NR(Φ+Ξ ξ ¯ ¯ 1 ) 1,k 2 ) 2,k ∂πp,ω ¯ ∂πp,ω ¯ + (Φ + Ξ1 ) − (Φ + Ξ2 ) ] ∂ϕk ∂ϕk 1 K3 (ν, p, ω, µ)||Ξ1 − Ξ2 ||L2per , ≤ λ where K3 is obtained with the use of Lemma 2.4 (the tedious but straightforward ˜ > 0 such that details of this computation are left to the reader). Then, deﬁning λ ˜ K−1 , we obtain the desired contraction property. λ 3 The proof of the stability property is based on the same kind of estimate: for ˜ large enough, we have for example IΦ¯ (0) ∈ BL2 (0, µ ). λ 10 per

4 The linking construction This section is devoted to the proof of Lemma 2.11. Recall that we now consider a functional Qω deﬁned on a subset of a ﬁnite dimensional space (ΛD Eω )N , and ν, ˜ are ﬁxed. The splitting Eω = ΛU Eω ⊕ ΛD Eω is made at a frequency λ ˆ ≥ λ, ˜ p, ω, λ ˆ be the real dimension of ΛD Eω . We which will be precised later. Let 2m(ω, λ) recall that, due to the Lyapunov-Schmidt reduction, Qω is deﬁned on Aµ , where Aµ denotes (4.1) Aµ = Φ ∈ (ΛD Eω )N | GramL2per Φ < (1 − µ)1l . In this deﬁnition, µ is very small and will be chosen later. Note that GramL2per Φ may be degenerate, if Φ ∈ Aµ . According to Lemma 2.5, the limitation GramL2per Φ < (1 − µ)1l is not restrictive, since we know that there is no critical point outside Aµ , for µ suﬃciently small. On the contrary, this limitation parametrized with µ is a protection against a diverging πp,ω -term or a diverging ||Φ||EωN .

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Solutions of the Dirac-Fock Equations without Projector

We deﬁne the degenerate set D in Aµ , D = Φ ∈ Aµ | det GramL2per Φ = 0 .

1141

(4.2)

We will deﬁne pairs of submanifolds (Vj , Σ+ j )1≤j≤J(ω) which link together, following [6], but we ﬁrst explain the idea of our linking procedure with the pair corresponding to j = 0. At the end of this section, we will check that the pairs deﬁned for the other linkings satisfy the same properties. Deﬁne, for δ ∈ (0, 1) such that δ > µ, the following U(N )-invariant manifold without boundary + Σ+ (4.3) δ = Φ ∈ Λ Aµ | GramL2per Φ = (1 − δ)1l , and, if V denotes a linear subspace of Λ+ Eω , with (complex) dimension N , V = Aµ ∩ (Λ− Eω ⊕ V )N .

(4.4)

+ Under these conditions, the submanifolds Σ+ δ and ∂V link, i.e. Σδ ∩ ∂V = ∅

and #

+ Σδ ∩ V /U(N ) = 1 .

(4.5)

Indeed Λ− Eω ⊕ V and Λ+ Eω intersect transversally (as linear subspaces of Eω ). Their intersection is a complex subspace of dimension N (namely V ), and the intersection above is exactly one orbit by the U(N ) action. The key point of the proof of Lemma 2.11 is the construction of a pseudogra dient vector ﬁeld X in Aµ = Φ ∈ (ΛD Eω )N | 0 ≤ GramL2per Φ < (1 − µ)1l having good properties with respect to the degenerate set D. Following a linking idea, the image of V by the ﬂow of this pseudogradient will always intersect Σ+ δ . Let us deﬁne the vector ﬁeld X . Let θε : R → [0, 1] be a smooth real function satisfying   θε (s) = 0 for s ≤ 2ε , θε (s) = 1 for s ≥ ε , (4.6)  θε (s) > 0 for s ∈ ( 2ε , ε) . + Let Φ ∈ Aµ . We consider Φ+ = Λ+ Φ = (ϕ+ 1 , ..., ϕN ), and + O+ = Φ ∈ Aµ | GramL2per Φ+ = Diag(σ1+ , ..., σN ) .

(4.7)

+ ). With Ψ = (ψk )1≤k≤N = Let Φ ∈ O+ , with GramL2per Φ+ = Diag(σ1+ , ..., σN Φ + hλ˜ (Φ), we deﬁne the function 

 12 ||ψk ||2Eω − Kb (V ) , τ (Φ) = θ1  k

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¯ ¯ is deﬁned in where Kb (V ) ≥ K(N, ν) will be deﬁned more precisely later, and K (2.25). The vector ﬁeld X is given by ∂Qω + Xk (Φ) := θ1 (Qω (Φ) + 2) (θε (σk ) + τ (Φ)) − (Φ) + ∂ϕk # " (4.8) (1 − τ (Φ))(1 − θε (σk+ )) Λ− ϕk − Λ+ ϕk . Remark that we are only interested in positive levels for Qω , so we deﬁne a nonzero X at points Φ where Qω > −2. The function τ gives the following indication: τ (Φ) > 0 and Qω (Φ) ≤ N implies that ||Qω (Φ)||(EωN )∗ ≥ ε0 , according to Lemma 2.5. The main feature of this vector ﬁeld is that it is a positive linear − + ω combination of the gradient − ∂Q ∂ϕk of Qω and the gradient Λ ϕk − Λ ϕk of the periodic free Hamiltonian. As every Φ ∈ Aµ is obtained by the action of U(N ) on O+ , we may then extend X to an equivariant vector ﬁeld, for u ∈ U(N ), X (u.Φ) := u(X (Φ)) .

(4.9)

By construction, this vector ﬁeld is locally Lipschitz. Lemma 4.1. The vector ﬁeld −X is a pseudogradient for Qω , i.e. it is locally Lipschitz and there exists κ > 0 such that for every Φ ∈ Aµ , ∂Qω ∂Qω (Φ).Xk ≤ −κ|| (Φ)||Eω∗ ||Xk ||Eω . ∂ϕk ∂ϕk

(4.10)

Moreover, there exists θ1 , θ2 > 0 such that, for Qω (Φ) ≥ −1, θ1 ||

∂Qω ∂Qω ||Eω∗ ≤ ||Xk ||Eω ≤ θ2 || ||E ∗ . ∂ϕk ∂ϕk ω

(4.11)

Proof. We just have to prove (4.10) for Φ ∈ O+ since Qω is invariant by the U(N ) + action. Let Φ ∈ O+ satisfy GramL2 Φ+ = Diag(σ1+ , ..., σN ). there is nothing to + prove if every σk is greater than ε: in this case, X is the gradient of Qω . Suppose now that 0 ≤ σk+ ≤ ε. Using the chain rule, we get ∂Qω (Φ).ξk ∂ϕk

= =

∂Pω (Φ + hλˆ (Φ)).(ξk + Dhλˆ (ξk )) , ∂ϕk ∂Pω (Φ + hλˆ (Φ)).ξk , ∂ϕk

(4.12)

by construction of the Lyapunov-Schmidt reduction. Recall that hλˆ has been deﬁned after Lemma 2.9 and is bounded, by construction ||hλˆ (Φ)||EωN ≤

C3 ||Φ||EωN . ˆ λ

(4.13)

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1143

Using this inequality and Lemma 3.1, one ﬁnds ∂Eν,ω (Φ + hλˆ (Φ)). Λ+ ϕk − Λ− ϕk ≥ ∂ϕk ( π + π2 )α max(N, Z) C5 (1 − − ω max(N C1 , ZC2 ) ||ϕk ||2Eω (.4.14) ) 1− 2 ˆ 2 λ There remains to compute the terms involving πp,ω . To do this, we need to extend some deﬁnitions. We deﬁne, for Q a N × N hermitian matrix such that 0 ≤ Q < 1l : # " Sp (Q) = tr Qp (1l − Q)−1 = tr(Qn ) , (4.15) n≥p

# " ep (Q) = pQp−1 − (p − 1)Qp [1l − Q]−2 = nQn−1 .

(4.16)

n≥p

Then we may write Sp (Q).h = tr(ep (Q).h) and πp,ω = Sp ◦ GramL2per . This allows to write the remaining terms like ∂πp,ω (Φ + hλˆ (Φ)).ξk = (4.17) ∂ϕk |ξk | tr ep GramL2per (Φ + hλˆ (Φ)) Mk (Φ + hλˆ (Φ), ξk ) + O( ), ˆ λ where the last term comes from the Lyapunov Schmidt reduction, and does not ˆ is large enough. Moreover Mk (ζ, ξk ) is the following N × N change the result, if λ complex matrix, with αj = (ζj , ξk ), :   O α1 O   ..   .  ∗  ∗   Mk (ξ, ζ) =  α1 . . . 2Re(αk ) . . . αN  .   ..   . O O αN For the sake of simplicity, we put Ψ = Φ + hλˆ (Φ), and explain the computations in the case k = 1, the other values of k implying the same calculations. With the notation eij = (ep (GramL2per Ψ))ij , the coeﬃcients of the matrix, we have ∂πp,ω (Ψ).ξ1 ∂ϕ1

=

j

ej1 (ξ1 , ψj ) + 

= 2Re 

j

j

e1j (ψj , ξ1 ) 

e1j (ψj , Λ+ ϕ1 − Λ− ϕ1 ) ,

(4.18)

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E. Paturel

replacing ξ1 = Λ+ ϕ1 − Λ− ϕ1 . We may write ∂πp,ω (Ψ).(Λ+ ϕ1 − Λ− ϕ1 ) ∂ϕ1

=

2Re(e11 σ1+ )

− 2Re

= 4Re(e11 σ1+ ) − 2Re

N

Ann. Henri Poincar´ e

−

e1i (Λ ϕi , Λ ϕ1 )

i=1 N

−

e1i (ϕi , ϕ1 )

,

(4.19)

i=1

because (ϕi , ϕ1 ) = (Λ+ ϕi , Λ+ ϕ1 ) + (Λ− ϕi , Λ− ϕ1 ) = δi1 σ1+ + (Λ− ϕi , Λ− ϕ1 ), and this is due to the diagonalization of the Λ+ -terms. But N

e1i (ϕi , ϕ1 ) = ep (GramL2per Ψ)GramL2per Φ

11

i=1

≥ 0,

(4.20)

and the sum is nonnegative because it is a diagonal term of a positive hermitian matrix. Hence, ∂πp,ω (Ψ).(Λ+ ϕ1 − Λ− ϕ1 ) ≤ 4Re(e11 σ1+ ) , ∂ϕ1

(4.21)

Now, we claim that e11 (and, for k = 1, ekk ) is bounded above with a constant depending only on µ, on the set where X = 0 and θε (σ1+ ) ≥ 0 (resp. θε (σk+ ) ≥ 0). Indeed, for µ close enough to 1, for any Ψ such that GramL2per Ψ ≥ (1 − µ)1l}, we 1 ¯ get either Pω (Ψ) < −2 or ( k ||ψk ||2Eω ) 2 > K(N, ν). The last alternative is given by a simple comparison between the Eν,ω -term and the πp,ω -term in the functional. In both cases, the coeﬃcient of Λ+ ϕk − Λ− ϕk in the vector ﬁeld X is zero, hence the only set where Λ+ ϕk −Λ− ϕk occurs in X is Ψ ∈ {Φ ∈ Aµ | GramL2per Φ ≤ (1 − µ)1l}, where we get an uniform estimate for e11 : if GramL2per Φ ≤ (1 − µ)1l, then e11 ≤ C (µ) .

(4.22)

Re(e11 σ1+ )

Then we prove, using this result, that is bounded √ above with a function of ε going to 0 as ε goes to 0. Indeed, either ||ϕ1 ||2L2 ≥ ε, and we get per the estimate 2 Re(e11 σ1+ ) ≤ |e11 |||ϕ+ 1 ||L2per √ ≤ εC (µ)||ϕ1 ||2L2per ,

√ or ||ϕ1 ||2L2 ≤ ε. In this case, we may use a property of the function ep , given per in the following proposition, whose proof is left to the reader: Proposition 4.2. Let µ ∈ (0, 1). There exists a nondecreasing continuous function ¯µ (s) = 0 and, given any self-adjoint N × N χ ¯µ : R+ → R+ , such that lims→0 χ matrix A = (aij ) satisfying 0 ≤ A ≤ (1 − µ)1l, we get ¯µ (a11 ) . [ep (A)]11 ≤ χ

(4.23)

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Then we may write ¯µ ( e11 ≤ χ

1145

C3 √ ε) , ˆ λ

and Re(e11 σ1+ ) ≤ χ ¯µ (

C3 √ ε)||ϕ1 ||2L2per . ˆ λ

(4.24)

In any case, we get the following bound ∂πp,ω (Ψ).(Λ+ ϕ1 − Λ− ϕ1 ) ≤ χµ (ε)||ϕ1 ||2L2per , (4.25) ∂ϕ1 √ √ with χµ (ε) = max( εC (µ), χ ¯µ ( Cλˆ3 ε)), and limε→0 χµ (ε) = 0. Now, combining (4.14) and (4.25), we get ( π2 + π2 )α max(N, Z) ∂Qω C5 + − ||ϕ1 ||2Eω (Ψ).(Λ ϕ1 − Λ ϕ1 ) ≥ (1 − ) 1− ˆ ∂ϕ1 2 λ C5 −(1 − )ω max(N C1 , ZC2 )||ϕ1 ||2Eω ˆ λ −χµ (ε)||ϕ1 ||2Eω .

(4.26)

The same computation gives the same result for k = 1. Taking ε small enough, we obtain a positive lower bound, and (4.10) comes from the estimates ||

∂Eν,ω || ∗ N ≤ C10 ||ϕi ||Eω , ∂ϕi (Eω )

(4.27)

||

∂πp,ω ||Eω∗ ≤ C11 (µ)||ϕi ||Eω . ∂ϕi

(4.28)

and

Now we prove (4.11). The ﬁrst inequality comes directly from (4.26). It is suﬃcient to check the second property with the free Hamiltonian gradient, and ||Λ+ ϕk − Λ− ϕk ||2Eω

≤ ||Λ+ ϕk |||2Eω + ||Λ− ϕk |||2Eω ≤ ||ϕk |||2Eω ∂Qω ≤ C12 .(Λ+ ϕk − Λ− ϕk ) ∂ϕk ∂Qω ||E ∗ ||Λ+ ϕk − Λ− ϕk ||Eω , ≤ C12 || ∂ϕk ω

and the proof is over.

We call η s the ﬂow of the pseudogradient X . From now on, “ﬂow lines” are meant to be the ﬂow lines of the pseudogradient −X . We deﬁne now the constant Kb (V ).

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¯ Lemma 4.3. Given the linear subspace V , there exists Kb (V ) ≥ K(max V Qω , ν) s and µ ∈ (0, 1) such that the ﬂow η of the pseudogradient X (deﬁned with this Kb (V )) has the following property: For any Φ ∈ V such that Qω (Φ) ≤ M , for any s ≥ 0 such that Qω (η s (Φ)) ≥ 0, we get 0 ≤ GramL2per (η s (Φ)) ≤ (1 − µ)1l .

(4.29)

¯ Proof. Let K > K(max V Qω , ν). Suppose that there exists 0 ≤ s1 ≤ s2 such that

N

12 ||(η s1 (Φ))k ||2Eω

k=1 N

¯ = K(max Qω , ν) V

12 = K

||(η s2 (Φ))k ||2Eω

k=1

and, for any s ∈ [s1 , s2 ],

N

12 ||(η

s2

¯ ∈ [K(max Qω , ν), K ] .

(Φ))k ||2Eω

V

k=1

Since −X is a pseudogradient for Qω , we know that Qω (η s (Φ)) ≤ Qω (Φ) ≤ max Qω . V

This implies, following Lemma 2.5, that for any s ≥ 0 such that

N

12 ||(η

s

(Φ))k ||2Eω

¯ > K(max Qω , ν) , V

k=1

we have ||

∂Qω s (η (Φ))||(EωN )∗ ≥ ε0 . ∂Φ

Then,

s2

∂Qω s (η (Φ)).X (η s (Φ))ds ∂Φ s1 s2 ≤ −κε0 ||X (η s (Φ))||ds

Qω (η s2 (Φ)) − Qω (η s1 (Φ)) =

s1

¯ Qω , ν)) . ≤ −κε0 (K + K(max V

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1147

Deﬁning 2 + maxV Qω ¯ Kb (V ) = K(max Qω , ν) + , (4.30) V κε0 ¯ we get that any ﬂow line joining a point of EωN -norm K(max V Qω , ν) to a point N of Eω -norm Kb (V ) attains this point with Qω (η s2 (Φ)) ≤ −2 . ¯ Hence there exists a K ∈ [K(max V Qω , ν), Kb (V )] such that for every Φ ∈ V and for every s ≥ 0 verifying Qω (η s (Φ)) ≥ 0, we get 12

N ||(η s (Φ))k ||2Eω ≤ K , k=1

which implies, as in Lemma 2.5, a bound σ ¯ (uniform in s and Φ) depending on maxV Qω , ν and p, for the eigenvalues of GramL2per (η s (Φ)). Denoting 1 − µ this bound, (4.29) is proved. The pseudogradient X has the following interesting properties: Lemma 4.4. If Φ ∈ O+ satisﬁes σ1+ < 2ε , then for any s ≥ 0, the matrix GramL2per ([η s (Φ)]+ ) has an eigenvalue lower than e−s σ1+ . As a consequence, if η s (Φ) tends to a critical point Φc for Qω as s goes to −∞, if moreover 0 is an eigenvalue of GramL2per Φc , then, for any s ∈ R, 0 is an eigenvalue of GramL2per ([η s (Φ)]+ ). This is very important for our linking procedure: according to this lemma, there is no ﬂow line going from a point in a ε-neighborhood of D to a point in Σ+ δ . In particular, the ﬂow lines joining V to Σ+ stay away from D, at a distance of at δ least ε. Proof of Lemma 4.4. It is suﬃcient to prove that, calling σ + (s) the smallest eigenvalue of GramL2per ([η s (Φ)]+ ), we get, lim

σ+ (s + h) − σ + (s) ≤ −σ + (s) . h h>0,h→0 sup

We can take Φ ∈ O+ after action of an element in U(N ). We put then + (s) . GramL2per ([η s (Φ)]+ ) = Diag σ+ (s), σ2+ (s), ..., σN Then σ+ (s + h) =

inf

x∈CN ,||x||=1

≤ ||Λ η

+ s+h

||

(4.31)

xi Λ+ η s+h (Φ)i ||2L2per

i (Φ)i ||2L2per

≤ ||Λ+ η s (Φ)i ||2L2per + 2h Λ+ η s (Φ)i , −Λ+ η s (Φ)i L2

per

+ o(h) ,

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and we conclude lim

σ+ (s + h) − σ + (s) ≤ −σ + (s) < 0 , h h>0,h→0 sup

(4.32)

which implies the lemma.

For the linking procedure, we follow now ideas inspired by [15]. We will suppose that Qω is a Morse function; if it is not true, we will indicate later the means to overcome the problem. It is possible, since we work on a ﬁnite dimensional setting, to perturb slightly V into V¯ such that it is transverse to every stable manifold of any critical point for Qω . ¯ for s ≥ 0. From the linking property, given any Consider the image η s (V) s ≥ 0, we have

# η s (V) ∩ Σ+ δ /U(N ) = 1. ¯ in the sequence As V is compact, there is a limit point Φ

−s s ¯ ∩ Σ+ /U(N ) , η (V) η δ as s goes to inﬁnity. We now consider the following point: ¯ , ¯ ∗ = lim η s (Φ) Φ s→+∞

(4.33)

and claim that ¯ ∗ is a critical point for Qω , and Lemma 4.5. Φ ¯ ∗ ) ≥ c = min (Qω (Φ)) > 0 . Qω (Φ + Φ∈Σδ

¯ ∗ ∈ D and its Morse index is bounded above by the real dimension of Moreover, Φ ˆ + N 2. V/U(N ), i.e. N m(ω, λ) ¯ ∗ is a critical point for Qω is a consequence of the linking Proof. The fact that Φ ¯ ∗) ≥ c > 0 property, and the pseudogradient properties of X . The inequality Qω (Φ also follows from this linking property. ¯ ∗ ∈ D: actually, this is due to the construction of We have to prove that Φ X . Looking at the ﬂow lines issued from η −s η s (V) ∩ Σ+ δ /U(N ) , they are all connecting a point in V/U(N ) to a point in Σ+ δ /U(N ). Since ﬂow lines arriving at ¯ to belong to D. Σ+ /U(N ) cannot come close to D, it is impossible for Φ δ ¯ ∗ , and, Finally, V/U(N ) intersects the stable manifold of the critical point Φ by assumption, the intersection is transverse, but this is only possible if ¯ ∗ ) ≤ dim (V/U(N )) codimW s (Φ

(4.34)

and, since Qω is a Morse function, the codimension above is exactly the Morse ¯ ∗ for this function. index of Φ

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1149

If Qω is not a Morse function, it is possible , adding a small U(N )-invariant perturbation τn acting on Aµ , to change Qω , when acting on Aµ /U(N ), into a Morse function Qω + τn . From the Morse Lemma, we may even impose τn to be as small as we want, for example we take lim ||τn ||C 1 = 0 .

n→+∞

We also replace X by the vector ﬁeld Xn obtained from Qω +τn with formula (4.8). ¯ n , exactly as above. There remains We obtain a nondegenerate critical point Φ then to pass to the limit as n goes to inﬁnity. This is an easy task: since we work ¯ n is precompact, and a in a ﬁnite dimensional setting, the bounded sequence Φ ¯ subsequence converges, say, to Φ∗ . Now, ¯ n ) = −τn (Φ ¯ n ) −n→+∞ Qω (Φ −−−−→ 0 , ¯ ∗ is a critical point for Qω . Since the distance between D and Φ ¯ n is unihence Φ ¯ ∗ ∈ D. Finally, since we consider only strict Morse index formly bounded below, Φ (i.e. the number of strictly decreasing directions, or negative eigenvalues of the Hessian matrix) of a ﬁnite dimensional function, the equation (4.34) still holds true. We may now deﬁne precisely the submanifolds (Vj , Σ+ j ), and check that the announced properties are veriﬁed. To this aim, we use a periodic version of [6, Lemma 5.2]: 2 Lemma 4.6. Assume that αZ < 2 + π . Then there exists a nondecreasing sequence π 2 {λj , j ≥ 0} in (0, 1), with limj→+∞ λj = 1 and a sequence {Gj,ω , 0 ≤ j ≤ J(ω)} of complex linear subspaces of ΛD Eω+ , with

dimC (ΛD Eω+ /Gj,ω ) = j ,

(4.35)

and

ϕ+ , H0 − αZGν,ω ϕ+

L2per

≥ λj ||ϕ+ ||L2per , ∀ϕ+ ∈ Gj,ω .

Proof. We refer to [6] or, for a similar but nonperiodic case [17]. We deﬁne the following submanifold without boundary & j +1 + N + 2 Σ+ 1 l . = Φ ∈ (G ) | Gram Φ = j,ω Lper j j +2

(4.36)

(4.37)

Now, if Vj is a linear subspace of Λ+ Eω , with complex dimension j + N , deﬁned as the orthogonal space of Gj+N,ω in ΛD Eω+ , we may deﬁne Vj = Aµ ∩ (Λ− Eω ⊕ Vj )N .

(4.38)

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Now Vj and Σ+ j link and we have the same formula (4.5). The properties of the pseudogradient stay valid, particularly the fact that no ﬂow line can join D to ˆ ˆ Σ+ j . Given ω and λ, we may deﬁne a J(ω) = m(ω, λ) and a constant Kb uniformly for every linking (i.e. for 1 ≤ j ≤ J(ω)) replacing Kb (V ): Kb =

max (Kb (Vj )) .

1≤j≤J(ω)

The critical points are obtained exactly in the same way, and the Morse index estimate reads now ˆ + 2N (N + j) − N 2 , iM (Φjω ) ≤ dimR Vj /U(N ) = N m(ω, λ)

(4.39)

and we get (2.34). Finally, we get (2.32) using Lemma 4.6: we get of course Qω (Φjω ) ≥

min Qω (Φ)

Φ∈Σ+ j

≥ N λj

j +1 , j +2

(4.40) (4.41)

as soon as the j-th linking is possible. We deﬁne cj = N λj j+1 j+2 and this gives ˆ (2.32). Now, using µ deﬁned in Lemma 4.3 and λ large enough, we may deﬁne Ψjω = Φjω + hλˆ (Φjω ) , which satisﬁes the conditions of Lemma 2.11.

(4.42)

5 Morse-type estimates In this section, devoted to the proof of Lemma 2.12, we look at the Morse index of the critical point and min-max sequences developed in the preceding section, in order to get some a priori estimates on the eigenvalues εk . Obtaining estimates from Morse-type information is now a usual technique (see [7], or [8] for a complete exposition). Thanks to the U(N ) invariance, we consider Φjω one of the critical points found in the last section for Qω , such that GramL2per Φjω = Diag(σ1 , ..., σN ) , with 0 < σ1 ≤ ... ≤ σN < 1 .

(5.1)

We denote its Morse index by k0 . To get the upper bound for the corresponding eigenvalues εk of the inﬁnite dimensional problem, we will use the deﬁnition of the Morse index: if V is a linear subspace of (ΛD Eω )N such that dim V ≥ k0 + 1, then Qω (Φjω ) cannot be negative on the whole space V . Therefore we are going to construct a subspace V of (ΛD Eω )N , whose dimension will be greater than k0 + 1 and where we have an estimate like ∃0 < b(k0 ) < 1, ∀Ψ ∈ V, Qω (Φjω ).Ψ2 < (b(k0 ) − εN )||Ψ||2 .

(5.2)

As Qω cannot be negative on V , there exists a Ψ = 0 in V such that Qω (Φjω ).Ψ2 ≥ 0 and thus εN ≤ b(k0 ) < 1.

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1151

In order to construct such a subspace, we need the following preliminary result: Proposition 5.1. Recall that ν and p are ﬁxed. There exists m(ν) ∈ N such that, given any critical point Ψjω of Pω that we found, there exists a subspace WΦjω of EωN , with dimension m, such that ⊥

(EL− )N = WΨjω ⊕ FΨ−j , ω

and 1 (5.3) ∀χ ∈ FΨ−j , Pω (Ψjω ).χ2 ≤ − ||χ||2 . ω 2 Proof. Using the Cauchy-Schwarz inequality, one can easily prove the estimate, for any χ ∈ (Λ− Eω )N , Pω (Ψjω ).χ2

≤ −

N

√ (χk , 1 − ∆χk ) + 3

ρΨjω (x)ρχ (y)Gν,ω (x − y)dxdy

k=1 N √ χk , − 1 − ∆ + 3(ρΨjω ∗ Gν,ω ) χk . ≤

(5.4)

k=1

Deﬁne WΨjω =

Ker

√ λ≤ 12 ,λ∈σ( 1−∆−3(ρ

√

1 − ∆ − 3(ρΨjω ∗ Gν,ω ) − λId .

j ∗Gν,ω )) Ψω

j Since ν ∈ (0, 1) is ﬁxed, we may see Uν,ω = 3(ρΨjω ∗ Gν,ω ) as a compact √ perturbation of 1 − ∆. Hence, by a classical argument of spectral theory, WΨjω is ﬁnite dimensional. But we can say more: its dimension may be bounded above with a constant m depending on ν. This comes from the fact that, for each q ≥ 1, there is a constant Cν,q such that

1 j ||(Uν,ω − )+ ||Lqper ≤ Cν,q , 2

(5.5)

where (U )+ denotes the positive part of U . Then, [4, Section 2] gives the estimate (to use this result, it is in fact suﬃcient to get a L3 bound). Then, the orthogonal space of WΨjω in (Λ− Eω )N satisﬁes (5.3). Note that by construction, the vectors in WΨjω are regular functions: indeed, for ν ﬁxed, one can easily prove, by a bootstrap method, that for each q ≥ 1, there is a constant Cν,q such that, for all χ ∈ WΦjω , 1,q ≤ C ||χ||Wper ν,q ||χ||L2per .

Thanks to this estimate, we get ||ΛU |W

j Ψω

1 ||L(Eω ,Eω ) = O( ) , ˆ λ

(5.6)

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˜ j , and consider F j , or simply F for so we can replace WΦjω by ΛD WΦjω = W Φω Φω ˆ great enough, an ˜ j in ΛD Λ− Eω . Given λ convenience, the orthogonal space of W Φω estimate like (5.3) is conserved: 1 (5.7) ∀χ ∈ FΦ j , Qω (Ψjω ).χ2 ≤ − ||χ||2 . ω 4 Let Ud be a subspace with dimension d of radial smooth functions with exponential decay deﬁned in R3 and values in R. We may associate, for u ∈ N, u > 0,     f ( u. )        0    , f ∈ U Wd,u = χ =  . d 0        0 Since we work in a periodic setting, we have to “periodize” these functions: we consider, for each ψ ∈ Wd,u , a corresponding periodic function ψ˜ ∈ Eω , such ˜ ˜ d,u,ω the vector = ψ(x). We denote W that, for every x ∈ (− ωπ , ωπ )3 , we get ψ(x) ˜ ˜ d,u,ω by the space spanned by the ψ, for ψ ∈ Wd,u . We denote W the image of W + projector ΛD Λ . Deﬁne now the following subspace of (ΛD Eω )N , for ω small enough V = {[{0} × ... × {0} × W ] ⊕ F } ∩ {ϕj1,ω , ..., ϕjN,ω }⊥ . We have to check that V satisﬁes the estimate (5.2). The following result will be helpful. Lemma 5.2. For any Ψ ∈ Aω and ϕ ∈ Eω , of the form ϕ(x) = f (|x|) for every x ∈ (− ωπ , ωπ )3 , taking Φ(x) = (0, ..., 0, ϕ(x)), we have 1 E (Ψ) [Φ, Φ] ≤ (ϕ, H0 ϕ)L2per + α(1 + v(ω))(N − Z − 1)(ϕ, Gν,ω ϕ)L2per , 2 ν,ω with v(ω) going to 0 as ω goes to 0. Proof. The proof is the same as in [6, Lemma 4.4], taking into account the fact that we work in a periodic setting. More precisely, we may write Eν,ω (φ1 ...φN ) = Eν,ω (φ1 ...φN−1 ) + (φN , H0 φN )L2per − αZ(φN , Gν,ω φN )L2per N−1 +α Gν,ω (x − y) |φk (x)|2 |φN (x)|2 k=1

−α

Gν,ω (x − y)

N−1

(φk (y), φN (y)) (φN (x), φk (x)) .(5.8)

k=1

So Eν,ω is a quadratic form in φN , when φ1 , ..., φN−1 are ﬁxed. Note that Gν,ω (x − y) (φk (y), ψ(y)) (ψ(x), φk (x))

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1153

is nonnegative, since the Fourier series associated to Gν,ω has only positive coeﬃcients. Hence, 1 E (Ψ) [Φ, Φ] ≤ (ϕ, H0 ϕ)L2per − αZ(ϕ, Gν,ω ϕ)L2per 2 ν,ω N−1 +α Gν,ω (x − y) |ψk (x)|2 |ϕ(x)|2 , π π 3 π π 3 (− ω , ω ) ×(− ω ,ω)

(5.9)

k=1

for any Φ = (0, ..., 0, ϕ(x)), with ϕ ∈ Eω . Now, if ρ ∈ L1 is radial and non-negative, then an easy computation shows that ρ(x)Vν (x − x0 )dx ≤ ρ(x)Vν (x)dx, ∀x0 ∈ R3 . R3

R3

We deduce the following property for periodic functions: |ϕ(x)|2 Gν,ω (x − x0 )dx ≤ (1 + v(ω)) |ϕ(x)|2 Gν,ω (x)dx , (5.10)

π π 3 (− ω ,ω)

π π 3 (− ω ,ω)

with v(ω) ≥ 0 going to 0 as ω goes to 0. As a consequence, dy L 3 (− L 2,2)

N−1

|φk (y)|2

k=1

≤ (1 + v(ω))

L 3 (− L 2,2)

dy L 3 (− L 2,2)

N−1

|ψ(x)|2 Gν,ω (x − y)dx ≤ |φk (y)|2

k=1

L 3 (− L 2,2)

|ψ(x)|2 Gν,ω (x)dx

≤ (1 + v(ω))(N − 1)(ψ, Gν,ω ψ)L2per ,

and the lemma follows.

˜ d,u,ω by the projector ΛD Λ . For u large The space W is the image of W enough, ΛD Λ+ |W is close to this identity. Indeed, there exists c∗ > 0, depend˜ d,u,ω ˜ d,u,ω , ing on d, such that, for every ψ ∈ W +

||Λ− ψ||2L2per

≤

c∗ ||ψ||2L2per . u2

(5.11)

Moreover, scaling arguments imply that there exists a constant 0 < c∗ < c∗ depending on d such that, for any ψ ∈ W and u large, (H0 ψ, ψ)

= ||ψ||2L2per ,

||∇ψ||2L2per

≤

(ψ, Gν,ω ψ)

≥

(5.12)

∗

c ||ψ||2L2per , u2 c∗ ||ψ||2L2per , −C0 ω||ψ||2L2per . u

(5.13) (5.14)

We obtain the estimate (5.2) as follows. In the following, we take the notation (χ)k for the k-th coordinate of the vector χ ∈ EωN . Note that the decomposition

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χu + χ− follows the splitting in Proposition 5.1. We get: 1 j P (Ψ ).χ2 2 ω ω

=

1 1 E (Φi )(χ, χ) − πp,ω (Ψjω )(χ, χ) 2 ν,ω 2

=

1 1 Eν,ω (Ψjω )(χu + χ− , χu + χ− ) − 2ep (σk )||(χ)k ||2L2per , 2 2 N

k=1

this follows form the fact that every (Ψ)k is orthogonal to every ϕik . Now, 1 j 1 1 Pω (Ψω ).χ2 = Eν,ω (Ψjω )(χu , χu ) + Eν,ω (Ψjω )(χ− , χ− ) 2 2 2 N " (Ψjω )(χu , χ− ] − ep (σk )||(χ)k ||2L2per + Re Eν,ω k=1

≤ (χu , H0 χu ) + α(1 + v(ω))(N − Z − 1)(χu , Gν,ω χu ) −

N k=1

1 ep (σk )||(χ)k ||2L2per− ||χ− ||2L2per+ c1 ||∇χu ||L2per ||Λ− χ− ||L2per , 2

the last inequality is coming from Lemma 5.2, the fact that χ− ∈ F and Proposition 5.1. The real part term (involving c1 ) is obtained with Hardy type inequality. We follow using the estimates (5.12), (5.13), (5.14) and (5.11): 1 j P (Ψ ).χ2 ≤ 2 ω ω 1 α(Z + 1 − N ) − ||χ− ||2L2per + C0 ω||χu ||2L2per ≤ ||χu ||2L2per 1 − u 2 +

+

N c∗ c1 ep (σk )||(χu )k + (χ− )k ||2L2per ||χu ||L2per ||χ− ||L2per − u k=1 α(Z + 1 − N ) 1 ≤ ||χu ||2L2per 1 + C0 ω − − ep (σN ) − ||χ− ||2L2per u 2 N c∗ c1 ||χu ||L2per ||χ− ||L2per − ep (σk )||(χ− )k ||2L2per − 2ep (σN )Re(χu , (χ− )N ) u k+1 α(Z + 1 − N ) 1 ≤ ||χu ||2L2per 1 + C0 ω − − ep (σN ) − ||χ− ||2L2per u 2 c∗ c1 + ||χu ||L2per ||χ− ||L2per + 2ep (σN )||Λ− χu ||L2per ||χ− ||L2per u α(Z + 1 − N ) 1 ≤ ||χu ||2L2per 1 + C0 ω − − ep (σN ) − ||χ− ||2L2per u 2 c + ||χu ||L2per ||χ− ||L2per u

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1155

α(Z + 1 − N ) (c )2 − ep (σN ) + ≤ ||χu ||2L2per 1 + C0 ω − u 2u2 α(Z + 1 − N ) (c )2 − ep (σN ) + . ≤ ||χ||2L2per 1 + C0 ω − u 2u2 From this computation, we obtain the following estimate on Qω : 1 j Q (Φ ).χ2 2 ω ω

(5.15) α(Z + 1 − N ) (c ) 1 − ep (σN ) + + O( ) , 1 + C0 ω − 2 ˆ u 2u λ

≤ ||χ||2L2per

2

where the last term is due to the Lyapunov-Schmidt reduction. ˆ large enough and ω small enough, there exists u such that Given λ 1 α(Z + 1 − N ) (c )2 +O − ep (σN ) + < 0. C0 ω − ˆ u 2u2 λ Then we put b = 1 + C0 ω −

α(Z + 1 − N ) (c )2 + δλˆ ∈ (0, 1) , + u 2u2

and the estimate follows. Now, for d large enough (d = m + 2N + 1, for example), dim V is greater than k0 + 1. According to our argument, we proved that every εk is bounded above by b. Although the bound b on the eigenvalues εk obtained above is not uniform in ν, we still have, for every k, εk ≤ 1. So, using Lemma 3.1, we ﬁnd a bound for the Eω -norm of (Ψjω )k which is independent of ν: GramL2per Ψjω < 1l implies '(Ψjω )k 'L2per ≤ 1, then 'εk (Ψjω )k ' − 21 ≤ 1 and this implies, by (3.1), H

ω −1 1 (εk (Ψjω )k 'Eω ≤ . '(Ψjω )k 'Eω = ' ΠΨjω h0

(5.16)

j = 3(ρΨjω ∗ Gν,ω ) in any Lq ,for q ≥ 1, independently This allows to estimate Uν,ω of ν. In other terms, the constant Cν,q in 5.5 may be chosen independent of ν. We get also that Cν,q in 5.6 may be chosen independent of ν. This implies that the estimate b is in fact independent of ν. On the other hand, using the equation, we get N εjk ≥ Pω (Ψjω ) ≥ cj , k=1

the real cj being the constant of Lemma 2.11. So εjk ≥ cj − N + 1 = a(j), with <

<

a(j) −→ 1, since cj −→ N , as j goes to inﬁnity, and Lemma 2.12 is proved. Acknowledgement. The author wishes to thank E. S´er´e for his constant attention and his precious remarks all along this work.

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References [1] A. Abbondandolo. A new cohomology for the Morse theory of strongly indeﬁnite functionals on Hilbert spaces. Topol. Methods Nonlinear Anal., 9(2):325– 382, 1997. [2] V. I. Burenkov and W. D. Evans. On the evaluation of the norm of an integral operator associated with the stability of one-electron atoms. Proc. Roy. Soc. Edinburgh Sect. A, 128(5):993–1005, 1998. [3] C. Conley and E. Zehnder. Morse-type index theory for ﬂows and periodic solutions for Hamiltonian equations. Comm. Pure Appl. Math., 37(2):207– 253, 1984. [4] I. Daubechies. An uncertainty principle for fermions with generalized kinetic energy. Comm. Math. Phys., 90(4):511–520, 1983. [5] J. Desclaux. Relativistic Dirac-Fock expectation values for atoms with Z = 1 to Z = 120. Atomic Data and Nuclear Data Table, 12:311–406, 1973. [6] M. J. Esteban and E. S´er´e. Solutions of the Dirac-Fock equations for atoms and molecules. Comm. Math. Phys., 203(3):499–530, 1999. [7] G. Fang and N. Ghoussoub. Morse-type information on Palais-Smale sequences obtained by min-max principles. Comm. Pure Appl. Math., 47(12):1595–1653, 1994. [8] N. Ghoussoub. Duality and perturbation methods in critical point theory. Cambridge University Press, Cambridge, 1993. With appendices by David Robinson. [9] O. Gorceix, P. Indelicato, and J.P. Desclaux. Multiconﬁguration Dirac-Fock studies of two-electron ions: I. electron-electron interaction. J. Phys. B: At. Mol. Phys., 20:639–649, 1987. [10] I.P. Grant. Relativistic Calculation of Atomic Structures. Adv. Phys., 19:747– 811, 1970. [11] Y.K. Kim. Relativistic self-consistent ﬁeld theory for closed-shell atoms. Phys. Rev., 154:17–39, 1967. [12] E.H. Lieb and B. Simon. The Hartree-Fock theory for Coulomb systems. Comm. Math. Phys., 53(3):185–194, 1977. [13] P.-L. Lions. Solutions of Hartree-Fock equations for Coulomb systems. Comm. Math. Phys., 109(1):33–97, 1987.

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[14] Selecta of E.H. Lieb. The stability of matter: From atoms to stars. SpringerVerlag, Berlin-Heidelberg-New York, second edition, 1997. Edited by W. Thirring. [15] D. Salamon. Morse theory, the Conley index and Floer homology. Bull. London Math. Soc., 22(2):113–140, 1990. [16] B. Swirles. The relativistic self-consistent ﬁeld. Proc. Roy. Soc., A 152:625– 649, 1935. [17] B. Thaller. The Dirac equation. Springer-Verlag, Berlin, 1992. [18] C. Tix. Lower bound for the ground state energy of the no-pair Hamiltonian. Phys. Lett. B, 405(3-4):293–296, 1997. [19] C. Tix. Strict positivity of a relativistic Hamiltonian due to Brown and Ravenhall. Bull. London Math. Soc., 30(3):283–290, 1998. Eric Paturel CEREMADE URA CNRS 749 Universit´e Paris IX-Dauphine place du Mar´echal de Lattre de Tassigny F-75775 Paris cedex 16, France e-mail: [email protected]

Communicated by Rafael D. Benguria submitted 30/12/99, accepted 18/04/00

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Ann. Henri Poincar´ e 1 (2000) 1159 – 1202 c Birkh¨ auser Verlag, Basel, 2000 1424-0637/00/0601159-44 $ 1.50+0.20/0

Annales Henri Poincar´ e

Pressure in Classical Statistical Mechanics and Interacting Brownian Particles in Multi-dimensions Kˆ ohei Uchiyama Abstract. − Let P (u) denote the pressure at the density u deﬁned in the Gibbs statistical mechanics determined by a 2-body potential U (qi − qj ). The function U (x) is supposed rotationally invariant and of ﬁnite range but may be unbounded about the origin. We establish a representation of P (u) by means of the law of large numbers for the virial i,j qi · ∇U (qi − qj ), whether or not there occur phase transitions. This result on P (u) is motivated by a study of the hydrodynamic behavior of a system of a large number of interacting Brownian particles moving on a ddimensional torus (d = 1, 2, ...) in which the interaction is given by binary potential forces of potential U . Employing our representation of P (u), we also show that in the hydrodynamic limit of such a system there arises a non-linear evolution equation of the form ut = 12 ∆P (u) under a certain hypothetical postulate concerning concentration of particles.

0. Introduction A system of Brownian particles interacting through 2-body-potential drift terms is one of the simplest natural models that are thought to exhibit a mathematical structure of how a macroscopic evolution equation comes out from microscopic dynamics of a statistically large system regulated by conservation laws. While the model is interesting as a physical system of small particles suspended in a ﬂuid (see [10: Part II] for physical interpretation as well as recent developments of the subject), apart from such interests in the model its hydrodynamic scaling limit has long been studied from Mathematical point of view, through which it has been well understood that under the hypothesis of the local equilibrium a nonlinear diﬀusion equation for the limiting density of a suitably scaled distribution of particles must be derived and its diﬀusion coeﬃcient be determined as a function of the density which reﬂects the microscopic structure of the interactions (cf., eg., [2][7][9]). The argument for the derivation is convincing but had been lacking in any mathematical vindication until Varadhan [16] gave a rigorous derivation in one-dimensional case of smooth repulsive potential. In this paper we primarily consider multidimensional models in an approach similar to that of [16]. The nonlinear diﬀusion equation envisaged to appear in the hydrodynamic limit is of the form ut = 12 ∆P (u), where the function P (u) is the pressure at the density u deﬁned in the Gibbs statistical mechanics on Rd determined by the pair potential function, U (x) say. For its derivation, it is crucial to represent the pressure P (u) by means of average of the virial − i,j qi · ∇U (qi − qj ) = − 12 i,j (qi − qj ) · ∇U (qi − qj ). In this paper we establish such a representation of

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the pressure. In multi-dimensions there may occur some kind of phase transitions: we do not know of uniqueness of grand-canonical Gibbs measures nor of the validity of equivalence of ensembles. This however does not cause essential diﬃculty for the veriﬁcation of the representation. In a similar situation of lattice Ginzburg-Landau models, which exhibit phase transitions, Rezakhanlou [6] obtains the corresponding result and accordingly the hydrodynamic limit for the model. We suppose the pair potential to be superstable and of ﬁnite range, but do not suppose it to be entirely smooth as in [16] (our potential may be unbounded about the origin). If the space dimension is one, the method of [16] with the help of the representation of P (u) by means of the virial mentioned above (and an extension of two-block estimate to unbounded potentials given in [15]) establishes the validity of the hydrodynamic limit. In multi-dimensions, however, the method does not (at least directly) apply: there arises a serious diﬃculty (in proving the so-called two-block estimate; see (5.2) and a comment following it). Roughly speaking we can modify the method under a certain uniform bound of the space-time average of the p-th moment of a T scaled empirical density, 0 dt E[ρN (θ, t)]p dθ, for some p > 3, or something like it, but such a bound, though convincingly plausible, is diﬃcult to verify. In this paper we suppose such a bound to be true from the outset and deduce from it the expected scaling limit. We do not know for the moment of any deﬁnite argument that approves this supposition for any initial measure of interest; it or something similar to it seems indispensable for proving the validity of the hydrodynamic limit at least in the present approach. Our model, being essentially a gradient system, can be handled by the relative entropy method devised by Yau [18], but it forces the initial densities to take values from the region of no phase transitions, so that the result thus obtained would not cover the intrinsic generality of the issue.

1 Statements of the main results Let U (x) be a radial function on Rd , namely it is given in the form U (x) = V (|x|) (|x| denotes the Euclidean length of x ∈ Rd ). We shall suppose that V is a twice continuously diﬀerentiable function of r > 0 which is non-increasing in a neighborhood of 0 and satisﬁes the following conditions (i) for some constant co > 0, V (r) = 0 for r > co ; 1 (ii) either (a) V ≥ 0 or (b) 0 V (r)rd−1 dr = ∞; (iii) if V is bounded, then it can be extended to a twice continuously differentiable function of r ≥ 0 and V (0) := limr→0 V (r) > 0; if V is unbounded, then lim sup r2 |V (r)|2 + r2 |V (r)| e−2V (r) < ∞. (1.1) r→0

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1.1. In this subsection will be given a deﬁnition of pressure (denoted by P (ρ)) determined by the potential function U , and stated a theorem on its virial representation. For a d-tuple = (1 , ..., d ) with positive entries i > 0 let Λ() denote a hyper-interval (d-dimensional interval) [−1 , 1 ] × · · · × [−d , d ]. The canonical partition function for n particles in Λ() with the empty-boundary condition is deﬁned by dq 0 0 Z ,0 = 1 and Z ,n = for n ≥ 1 exp − U (qi − qj ) n! [Λ( )]n i,j(=)

where q = (q1 , ..., qn ) ∈ [Λ()]n is an n-particle conﬁguration (ordered n-tuple) and dq = dq1 · · · dqn (a d × n-dimensional volume element). Put ∗ := min{1 , ..., d } and let ∗ → ∞ and n → ∞ in such a way that n/|Λ()| → ρ (ρ ≥ 0). (|A| denotes the d-dimensional volume of a measurable set A.) Then there exists a limit Φ(ρ) := lim

∗ →∞

−1 0 log Z ,n ; |Λ()|

Φ(ρ), called Helmholtz’ free energy, is a continuous convex function of ρ ≥ 0 such that Φ(0) = 0 and Φ(ρ) ≥ ρ log ρ − Cρ for a constant C (cf. [1], [8]). The pressure (sometimes called Gibbs’ free energy) as a function of chemical potential λ ∈ R is given by F (λ) = sup[λρ − Φ(ρ)]. ρ≥0

The function Φ is diﬀerentiable for ρ > 0; the derivative Φ (ρ), ρ > 0 is necessarily non-decreasing and continuous; hence F (λ) may be regarded as a continuous function of ρ ≥ 0, which deﬁnes our pressure P (ρ), or, what amounts to the same thing, P (ρ) = Φ (ρ)ρ − Φ(ρ),

ρ>0

(1.2)

and P (0) = 0; in particular P (ρ) is non-decreasing and continuous (for ρ ≥ 0). It may be worth noticing that we do not know whether P is strictly increasing or not. For the derivation of the hydrodynamic scaling limit discussed in the next subsection it is crucial that P (ρ) − ρ can be represented as a limit of averages of −(qi −qj )·∇U (qi −qj )/d over conﬁgurations q = (qi ) in the box Λ(). Let µω ,n (dq) denote the canonical Gibbs measure on Λ() of n particles with a boundary conﬁguration ω. Put for a conﬁguration q ∈ [Rd ]n , Ψ αβ (q) =

1 |Λ()|

i,j(=):qi ,qj ∈Λ( )

ψαβ (qi − qj )

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where ψαβ (z) = −zβ ∇α U (z) (z = (z1 , ..., zd ) ∈ Rd ). Actually we shall prove the following theorem (see Sections 2 and 8 for more details). Theorem 1.1. For each r > 0 and η > 1 and each pair of indices 1 ≤ α, β ≤ d , sup

sup

ω:ω≤exp( ∗ ) n≤r|Λ( )|

[Λ(η )]n

2 Ψαβ (q) − [P (ρη ) − ρη ]δαβ µω η ,n (dq) −→ 0,

as ∗ → ∞, where η = (η1 , ..., ηd ), ρη = n/|Λ(η)|, and the supω is taken over all conﬁgurations ω = (ω1 , ..., ωm ) in Λ( + co ) \ Λ() ( + co = (1 + co , ..., d + co ) ) such that %ω := m ≤ e ∗ . If d = 1 and V is bounded, the assertion of Theorem 1.1 is proved in [16] based on the uniqueness for grand canonical Gibbs measures. As a byproduct of the proof of Theorem 1.1 we shall show, assuming a hyperstability condition on the function U (rather than the conditions (i) through (iii)), that P (ρ) is Lipschitz-continuous. This provides a proof diﬀerent from those of Dobrushin and Minlos [1] or Ruelle [8]. (See Theorem 8.2 in Section 8.) 1.2. The virial representation of P (ρ) given in Theorem 1.1 is motivated by a study of hydrodynamic behavior of interacting Brownian particles. Let Td be the d-dimensional unit torus represented by the hyper-cube [0, 1)d , and (x1 (t), ..., xN (t)) a system of interacting Brownian particles evolving on Td according to the following system of stochastic diﬀerential equations:

1 xi (t) − xj (t) dt + dBi (t), i = 1, 2, ..., N. ∇U dxi (t) = − ' ' j=i

Here ' is a small positive parameter (representing the size of the particles in a macroscopic scale), B1 , B2 , ... are independent standard Brownian motions moving on Td deﬁned on some probability space (Ω, F, P); U (x) is the function introduced at the beginning of this section. d N whose The process xN t := (x1 (t), ..., xN (t)) is a diﬀusion process on (T ) inﬁnitesimal generator is given by 1 (N) 1 ∆ − ∇U 2 ' i=1 N

LN =

j=i

xi − xj '

∂ , ∂xi

where ∆(N) denotes the Laplace operator on (Td )N and ∂/∂xi the gradient operator with respect to xi ∈ Td . The process xN t is ergodic. The invariant probability law is given by xi − xj 1 νN (dx) = dx1 · · · dxN , exp − U ZN ' i,j(=)

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relative to which LN is symmetric. Here (and also in what follows) the summation indicated by i,j(=) extends over all ordered pairs (i, j) such that 1 ≤ i, j ≤ N and i = j; ZN is the normalization constant. Let ξtN be the empirical distribution of the particles x1 (t), ..., xN (t) with weight 'd per particle, namely ξtN is the weighted counting measure on Td deﬁned by N J(x)ξtN (dx) = 'd J(xi (t)), J ∈ C ∞ (Td ). Td

i=1

The problem is to determine the limit of ξtN as N → ∞ and ' ↓ 0 in such a way that the average density N 'd remains (asymptotically) constant. The limit measure is expected to have a density which solves the non-linear diﬀusion equation ∂ 1 u(θ, t) = ∆P (u(θ, t)), ∂t 2

(θ, t) ∈ Td × (0, ∞),

(1.3)

where ∆ is the Laplace operator on Td and the function P (ρ), ρ ≥ 0, is the pressure deﬁned in the preceding subsection. We suppose that the diﬀusion process xN t starts from an initial law which has a density, denoted by f0N , relative to νN . The evolution of the process may be analytically characterized by the forward equation ∂ftN = LN ftN , ∂t where ftN is the density relative to νN of the law of xN t . On the family of initial densities {f0N } we impose the following bound of their entropies f0N log f0N dνN = o(N 1+2/d ) as N → ∞. (1.4) The particle size ' is supposed to be given as a function of N such that 'd N converges to a positive constant; hence it does not appear in the notations. We will regard ξtN as a stochastic process taking values in the space of all ﬁnite measures M(Td ), which is viewed as a metric space whose topology agrees with that of weak convergence of ﬁnite measures. For any T > 0 and a non-random element u0 ∈ M(Td ) we shall concern weak solutions of (1.3) on the time interval (0, T ) that satisfy the initial condition u(θ, t)dθ −→ u0 (dθ)

as t → 0

(1.5)

as well as the integrability condition

T

P (u(θ, t))dθ < ∞.

dt 0

Td

(1.6)

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It is known [14] that such a solution if any is unique if d = 1; in the case d ≥ 2 it is unique at least if u0 is absolutely continuous and its density is square integrable. We put ψ(r) = −rV (r) r > 0. In one-dimension the method of [16] may be adapted to the case of unbounded V to deduce from Theorem 1.1 the next theorem. Theorem 1.2. Let d = 1. Suppose, in addition to (i) through (iii), that either 1 1 ψ(r) ≥ 0 for all r > 0; or 0 V (r)dr = ∞ and 0 [ψ (r) ∨ 0]dr < ∞. (Here a ∨ b = max{a, b}.) Further suppose that (1.4) is satisﬁed and ξ0N converges in probability to a non-random element u0 ∈ M(T1 ). Then the random trajectory ξtN (dθ), t ≥ 0, converges in probability to a single trajectory u(θ, t)dθ, t ≥ 0 in the topology of locally uniform convergence of continuous trajectories in M(T1 ) and the limit function u(θ, t) is a (unique) solution of the non-linear diﬀusion equation (1.3) satisfying (1.5) and (1.6). We derive a corresponding result in multi-dimensions under a hypothetical d postulate. Let h be a smooth non-negative function on Rd having a compact support such that h(q)dq = 1 and h(0) > 0; put for θ ∈ T , xi − θ ρo (θ) = ρo (θ, x) = h ' i and o

o

S (θ) = S (θ, x) =

i

j=i

|ψ|

|xi − xj | '

xi − θ h '

where |ψ|(r) = |ψ(r)|. Our third result, Theorem∗ 1.3, is to reduce the problem to the following condition: for any δ > 0 T

3/2 o N S o (θ, xN dt ) I(S (θ, x ) > r)dθ > δ =0 (Ha) lim sup P t t r→∞ N

and

Td

0

lim sup P

M →∞ N

T

dt Td

0

3 [ρo (θ, xN )] dθ > M = 0. t

(Hb)

−1 If ψ(0+) < ∞, then ρo (θ) ≥ C S o (Aθ) for some positive constants A and C. o o If ψ(0+) > 0, then ρ (θ) ≤ C S (θ) + 1 , so that (Hb) follows from (Ha). The condition (H) does not depend on particular choice of the function h. Theorem* 1.3. Suppose that (H) holds, (1.4) is satisﬁed, ξ0N converges in probability to a non-random element u0 ∈ M(Td ) and the sequence of initial conﬁgurations xN 0 satisﬁes

o N 2 lim sup P [ρ (θ, x0 )] dθ > M = 0. (1.7) M →∞ N

Td

Then the same conclusion as in Theorem 1.2 holds with obvious modiﬁcation.

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The hypothesis (H) is obviously implied by sup N

T

dt 0

Td

p o N 3 E [S o (θ, xN t )] + [ρ (θ, xt )] dθ < ∞ for some p > 3/2.

(H )

The conditions (H) and (H ) measure the degree of non-concentration of particles in average: they would be violated if excessively many particles accumulate in a small region. They should be veriﬁed for non-trivial initial conditions, but the present author do not know how to prove it whether V is bounded or not. For this reason we did and will mark the theorems or lemmas that are proved by using (H) with ∗. If the process starts with the invariant measure νN , it holds that for all p>0 sup E N

T

dt

0

Td

and in the case d ≥ 2 lim sup E sup M →∞ N

0≤t≤T

Td

p o N 2p [S o (θ, xN dθ < ∞ )] + [ρ (θ, x )] t t

(1.8)

p o N 2p [S o (θ, xN dθ > M = 0, )] + [ρ (θ, x )] t t

which with p > 3/2 of course imply (H ) and (H), respectively. (1.8) is also valid for independent Brownian motions starting from initial distributions subject to a certain mild condition if ψ is replaced, in the deﬁnition S o (θ, x), by any function of p ϕ having a compact support and satisfying that |ϕ| dx < ∞. The validity of both (H) and (H ) is plausible for a wide class of initial distributions since the evolution law governed by LN does not seem to develop accumulation of particles, our potential being essentially repulsive so that it must exercise a dispersing eﬀect on the particle conﬁgurations, although we do not know of any eﬀective argument that approves such plausibility. 1 the bound (1.8) with an initial density f0N NFor p = N is implied by the condition f0 log f0 dνN = O(N ), provided that |ψ| ≤ C(1+V ) (cf. [16]). The rest of the paper is organized as follows. In Section 2 we shall prove Theorem 1.1 in the case (a) of the condition (ii). In Section 3 we shall prove the existence of the diﬀusion process xN t , which is not entirely obvious in the case when V is unbounded, as well as the compactness of the family of laws of the measure valued processes {(ξtN ; 0 ≤ t ≤ T )}∞ N=1 . In Section 4 we shall give some results on the local equilibrium state which are consequences of a result of [16] and Theorem 1.1. In Section 5 the convergence of ξtN in a strong topology (the two-block estimate) will be proved. The hypothesis (H) will be used for this (and in a sense only for this purpose). The proofs of Theorems 1.3 and 1.2 will be given in Sections 6 and 7, respectively. In the ﬁnal section (Appendix) we shall provide a proof of Theorem 1.1 in the case (b) of the condition (ii); and also some results on properties of the functions Φ(ρ) and P (ρ).

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2 The representation of the pressure P (ρ) by means of the virial In this section we prove Theorem 1.1 assuming (ii.a), namely V ≥ 0. Its proof in the other case (ii.b) is somewhat involved and relegated to Appendix (Section 8). We introduce some notations. Let co be the smallest positive constant such that V (r) = 0 if r ≥ co . Given = (1 , ..., d ), i > co , we take a conﬁguration, ω = (ωk ) say, on the outer shell Λ( + co ) \ Λ(), where co = (co , ..., co ) and + co = (1 + co , ..., d + co ), and for a conﬁguration q = (q1 , ..., qn ), qi ∈ Λ(), such that qi = qj if i = j, we put U(q) = U (qi − qj ), (2.1) i,j(=)

Hω (q) = U(q) + 2

U (qi − ωj ).

(2.2)

i,j n The canonical Gibbs measure µω ,n is then a measure on [Λ()] = Λ() × · · · × Λ() (the n-fold Cartesian product of Λ()) given by

µω ,n (dq) = where

1 dq exp{−Hω (q)} , Zn (, ω) n!

exp{−Hω (q)}

Zn (, ω) = [Λ( )]n

dq n!

if n ≥ 1,

and Z0 (, ω) = 0.

The canonical Gibbs measures on any translation of Λ() by y ∈ Rd is deﬁned as a τ (−y)ω by y, where τ (−y)ω denotes the translation of the boundary translation of µ ,n conﬁguration ω by −y. The rest of this section will be divided into three subsections. In the ﬁrst two subsections we shall prove preliminary lemmas on the canonical Gibbs measures. Although some of them are reformulation of well-known facts, we shall include their proofs for completeness or for convenience of their citation in the later arguments. In the third we shall prove Theorem 1.1 (in a stronger form) under (ii.a). Throughout all these subsections we shall write ρ = n/|Λ()|. 2.1. For the following two lemmas we do not need the assumption (ii) nor (iii) (the condition (i) may even be replaced by a super-stability condition (cf. Appendix 8.2) with possible minor changes in the statements of results). Lemma 2.1. Let δ(ρ) = 2−1 (2vd ρ)−1/d and C(ρ) = 2d+1 ρ exp 2d+2 ρ |x|>δ(ρ)

U+ (x)dx ,

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where U+ (x) = [U (x)]+ (a+ = max{a, 0}) and vd stands for the volume of ddimensional unit ball. Then for n > 1 and ∗ > 2co , Zn−1 (, ω) ≤ C(ρ ). Zn (, ω)

(2.3)

Proof. Write q = (q , qn ) for q ∈ [Λ()]n with q ∈ [Λ()](n−1) , qn ∈ Λ(). For a given q , deﬁne D = {qn ∈ Λ( − co ) : |qi − qn | > δ for all i = n}. Put δ = 2−1 (2ρ vd )−1/d , so that if ∗ > 2co |D| ≥ |Λ( − co )| − nvd δ d ≥ |Λ()|(2−d − ρ vd δ d ) = |Λ()|2−(d+1) (|D| denotes the volume of D). Then, by Jensen’s inequality, 2 exp − 2 U (qn − qi ) dqn ≥ |D| exp − U (qn − qi )dqn |D| D D i=n i=n 2n U+ (x)dx . ≥ |D| exp − |D| |x|>δ Hence

exp{−H (q , qn )}dqn ≥ exp ω

− H (q ) − 2 ω

d+2

ρ

Λ( )

|Λ()| U+ (x)dx (d+1) . 2 |x|>δ

Integrating both sides on [Λ()]n−1 yields the inequality (2.3).

Lemma 2.2. Let c be a positive constant such that i ≥ c + co , i = 1, ..., d, and deﬁne mi = (2i − co )/(2c + co ) and m = m1 · · · md . Then, Zn (, ω) ≥ (2c)dm for n < m; and m 0 0 , Zc,n/m +1 } Zn (, ω) ≥ min{Zc,n/m

for

n ≥ m.

(2.4)

Here c = (c, ..., c) and a denotes the integral part of a. Proof. We can divide Λ() into m hyper-cubes of side length 2c and corridors (a union of hyper-slabs) of width equal to or larger than co so that each hyper-cube is separated from the other ones as well as from Rd \ Λ() by a distance more than or equal to co . Obviously ∗ 1 exp{−Hω }dq, Zn (, ω) ≥ n ! · · · n ! 1 m (n ,..,n ) 1 m n +···+n =n 1

∗

m

where (n1 ,..,nm ) indicates that the integration extends over such q that the ith hyper-cube contains just ni particles from q. Put k = n/m and suppose

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k > 0. Then n = mk + q, 0 ≤ q < m, so that among m-tuples (n1 , ..., nm ) with n1 + · · · + nm = n there is one whose entries are k or k + 1. Neglecting the other ones we obtain (2.2). The case n < m is easy to dispose of since then Hω = 0 if ni ≤ 1 for all i. 2.2. Throughout this subsection (and the next one as well) we suppose that V ≥0

(ii.a)

(in addition to (i) and (iii)). For a set K ⊂ Rd and a conﬁguration q = (qi ) on Rd write q ∩ K and q \ K for the conﬁgurations (qi : qi ∈ K) and (qi : qi ∈ / K), respectively. Let NK = NK (q) denote the number of points qi contained in K: NK (q) = %(q ∩ K) = %{i : qi ∈ K}. ω ω = HK (q) for Hω∪(q\K) (q ∩ K), or, to be precise, Also write HK   ω  (q) = U (qi − qj ) + 2 U (qi − qj ) + 2 U (qi − ωk ) . HK i:qi ∈K

j=i:qj ∈K

j:qj ∈K /

k

ω (q). It obviously follows that Hω (q) = Hω (q \ K) + HK

Lemma 2.3. There exist positive constants A and B (depending only on U ) such that if a Borel set K is covered by m hyper-cubes of side length co and ∗ > 2co , then for 0 ≤ γ ≤ 1 and for every positive integer k, (C(ρ )|K|e(1−γ)A )k ω exp{−(1 − γ)m−1 Bk2 }, exp{γHK (q)}µω ,n (dq) ≤ k! {NK =k} where C(ρ) is the same as in Lemma 2.1 and |K| denotes the volume of K. Proof. The assumptions (i) and (ii) imposed on U imply the hyper-stability 1 U(q ∩ K) = U (qi − qj ) ≥ Bk2 − Ak if k = NK m qi ,qj ∈K,i=j

for some positive constants A, B, provided that K is covered by m hyper-cubes of side length co . (Cf. [8]; see also Lemma 8.1 in Section 8.) Ignoring the interactions across the boundary of K, which reduces the interaction energy by virtue of (ii.a), ω we obtain Hω (q) − γHK (q) ≥ Hω (q \ K) + (1 − γ)U(q ∩ K). Hence {NK =k}

ω exp{γHK (q)}µω ,n (dq)

≤

|K|k n exp{−(1 − γ)(m−1 Bk2 − Ak)} exp{−Hω (q)}dq n! k Zn (, ω) n−k (Λ( )\K)

≤

Zn−k (, ω) |K|k exp{−(1 − γ)(m−1 Bk2 − Ak)} · . k! Zn (, ω)

The inequality (2.3) then concludes the estimate of the lemma.

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Lemma 2.4. If a Borel set K is covered by m hyper-cubes of side length co and ∗ > 2co , then µω ,n (NK = k) ≤

(C(ρ )|K|eA )k exp{−m−1 Bk2 }. k!

Proof. This is a special case of the previous lemma.

Lemma 2.5. For p ≥ 1 there exists a continuous function Mp (ρ) of ρ ≥ 0 depending only on U and p such that if a non-negative function χ(q), q ∈ Rd \ {0}, satisﬁes χ(q) ≤ AI(|q| < co )e2U (q)/p

(2.5)

for some positive constant A, then for any hyper-interval K with all its sides ≥ co p p p χ(qi − qj ) µω ,n (dq) ≤ |K| A Mp (ρ ). [Λ( )]n

i:qi ∈K j=i

ω Proof. Since, by virtue of (iia), HK (q) ≥ 2U (qi − qj ) if qi ∈ K and j = i, the assumption on χ of the lemma yields p p ω χ(qi − qj ) exp{−HK (q)} ≤ AI(|qi − qj | < co ) i:qi ∈K j=i

i:qi ∈K j=i

≤ Ap (NK )2p , where K is the co neighborhood of K. Hence the p-th moment to be estimated is at most ω ω Ap (NK )2p exp{HK (q)}µ ,n (dq) [Λ( )]n

The inequality of the lemma then follows from Lemma 2.3 (with γ = 1 and K in place of K) if the diameter of K is less than a ﬁxed constant, a say. In the general case we have only to represent K as a disjoint sum subintervals K(i) of congruent ∅ ∅ having a diameter less than a, so that HK,χ ≤ i HK(i),χ (where the notation would be evident), and to apply H¨ older’s inequality. Lemma 2.6. Let M1 (ρ) be as in Lemma 2.5. If χ(q) ≤ AI(|q| < co )e2U (q) , then n χ(qi − ωk )µω (2.6) ,n (dq) ≤ AM1 (ρ ) (k = 1, 2, ...). [Λ( )]n i=1

Proof. One can proceed as in the previous proof with trivial modiﬁcation.

Lemma 2.7. If n, ∗ → ∞ so that n/|Λ()| → ρ¯, then −

1 log Zn (, ω) −→ Φ(¯ ρ) |Λ()|

uniformly with respect to ω and ρ¯ ≤ r, where r may be an arbitrary positive constant.

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0 0 Proof. Immediate from the trivial inequalities Z −c ≤ Zn (, ω) ≤ Z ,n and the o ,n 0 fact that −|Λ()|−1 log Z ,n converges to Φ(¯ ρ) as n/|Λ()| → ρ¯. That the convergence is uniform in ρ¯ ≤ r is redundant from the very way of convergence just proved.

For the proof of the next lemma we use the fact that Φ is continuously diﬀerentiable. In the next sub-section (in the case when V ≥ 0) and the Appendix (in a general case) we shall give a proof of this fact, in which this lemma will not be used. (It, including its corollary, lemma 2.9, is used only in the proof of Theorem 2.1 given at the end of the next subsection.) Lemma 2.8. For each triplet of numbers r > 0, 0 < δ < 1 and α > 0 there exist positive constants η and L such that if ρ ≤ r, ∗ > L and if K is a hyper-interval included in Λ() with all its sides > 2co and |K| ≥ δ|Λ()|, then for ∗ large enough −η|Λ( )| P (ρ ) − P (ρ ) , µω K >α <e ,n where ρK = |K|−1 NK . Proof. Write Λ = Λ() and ρ = ρ . We suppose |K| < (1 − δ )|Λ| for a (small) constant δ > 0. [This gives rise to no loss of generality because the inequality in the opposite direction implies on the one hand that |ρ−ρK | ≤ NΛ\K |Λ|−1 +[δ /(1−δ )]ρ and on the other hand, in view of Lemma 2.4, that for any δ > 0 there exists a positive constant C such that −1 ≥ δ ) ≤ exp{−C|Λ|/(δ ∨ −1 µω ,n (NΛ\K |Λ| ∗ )}

provided δ > 0 is small enough.] Let = (1 , ..., d ) be such that K is a shift of Λ( ). Then µω ,n (P (ρK ) − P (ρ) > α) 1 = Zn (, ω) ≤

1 Zn (, ω)

1 k!(n − k)! k:P (k/|K|)>P (ρ)+α 0 Z ,k k:P (k/|K|)>P (ρ)+α

exp{−Hω }dq2

dq1 [Λ\K]n−k

[Λ\K]n−k

Kk

exp{−U(q1 )}

dq1 . (n − k)!

By decomposing Λ() \ K (for lower bound) and Λ() (for upper bound) into appropriate subregions and corridors the method used in the proof of Lemma 2.2 readily deduces that if (n − k)/|Λ \ K| → ρ1 , then 1 log |Λ \ K|

exp{−U(q1 )} [Λ\K]n−k

dq1 −→ −Φ(ρ1 ) (n − k)!

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uniformly for k ≤ n as long as ρ ≤ r (entailing n/|Λ \ K| ≤ r/δ ). Hence, if ρ ≤ r and η > 0, then for ∗ suﬃciently large, dq 1 1 log Z 0 ,k + log exp{−U(q1 )} |Λ| (n − k)! [Λ\K]n−k

|K| |Λ \ K| ≤− Φ(|K|−1 k) + Φ(|Λ \ K|−1 (n − k)) + η |Λ| |Λ| for all k ≤ n and, by Lemma 2.7, −

1 log Zn (, ω) ≤ Φ(ρ) + η. |Λ|

Now we apply the continuity of the derivative Φ as well as the convexity of Φ. Although we do not know if it is strictly convex, the deﬁnition of P (ρ) = F (λ(ρ)) given by means of the Legendre transform of Φ entails that we can choose η = η(r, δ, α) > 0 so that if P (k/|K|) > P (ρ) + α and ρ ≤ r, then

|K| |Λ \ K| Φ(ρ) − Φ(|K|−1 k) + Φ(|Λ \ K|−1 (n − k)) ≤ −3η. |Λ| |Λ| (It is here that we need the assumption |K| > δ|Λ|. The existence of η which may depends on ρ would be clear; for choosing η independently of ρ ≤ r we make use of the continuity of P (ρ).) Combining what are obtained above we see that lim sup |Λ()|−1 log µω ,n (P (ρK ) − P (ρ) > α) ≤ −η. In the same way we obtain a similar estimate for µω ,n (P (ρK ) − P (ρ) < −α), which completes the proof of Lemma 2.8. Lemma 2.9. If f (t) is a continuous and increasing function of t ≥ 0 and δ is a positive constant, then lim sup sup sup sup f (P (ρK ))dµω ,n ≤ f (P (r)), ∗ →∞

ω,i n≤r|Λ( )| |K|≥δ|Λ( )|

[Λ( )]n

where the last supremum is taken over all hyper-intervals K included in Λ() with all sides larger than 2co such that |K| ≥ δ|Λ()|. Proof. This is an immediate corollary of Lemma 2.8 since ρK ≤ r/δ so that the bounded convergence theorem can be applied. 2.3. We continue to assume the case (a) of the condition (ii). Deﬁne zα zβ ψαβ (z) = −zβ ∇α U (z) = ψ(|z|), ψ(r) = −rV (r), |z|2 1 ψαβ (qi − qj ). Ψ αβ = Ψ αβ (q) = |Λ()| i,j(=):qi ,qj ∈Λ( )

Here we shall prove the following theorem, which obviously implies Theorem 1.1.

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Theorem 2.1. For each r > 0 and each pair of indices 1 ≤ α, β ≤ d Ψαβ − [P (ρ ) − ρ ]δαβ 2 µω lim sup sup ,n (dq) = 0, ∗ →∞ ω n≤r|Λ( )|

[Λ( )]n

where ρ = n/|Λ()| and the ﬁrst sup is taken over all conﬁgurations ω in the shell Λ( + co ) \ Λ(). Let δ > 0 and ω and be as above. For 1 ≤ s ≤ 1 + δ put ω s = (ωks : k = 1, ..., m) ∈ Rd×m ,

ωks = (sωk,1 , ωk,2 , ..., ωk,d ) ∈ Rd ,

s = (s1 , 2 , ..., d ) in analogy of the proof given in [16] to the corresponding result in one-dimension. Then 1+δ d 1+δ 1+δ log Zn (s , ωs )ds. (2.7) log Zn ( , ω ) − log Zn (, ω) = ds 1 Setting qis = (sqi,1 , qi,2 , ..., qi,d ) and accordingly changing the variables of integration we have Zn (s , ωs ) = sn Zn, ,ω (s),

where

s

exp{−Hω (qs )}

Zn, ,ω (s) = [Λ( )]n

dq . n!

Hence (s) n Zn, ,ω d log Zn (s , ωs ) = + . ds s Zn, ,ω (s)

(2.8)

Here Zn, ,ω (s) denotes the derivative relative to s, and is given by s 1 dq Zn, ,ω ψ11 (qis − qjs ) + 2 (s) = ψ11 (qis − ωks ) exp{−Hω (qs )} . s n! n [Λ( )]

(Recall ψ11 (z) = −z1 ∇1 U (z) = −(z12 /|z|)V (|z|), z ∈ Rd .) [At this point it may ψ11 (qi − be worth making a remark. We are going to identify the limit of |Λ|−1 qj )dµω ¯) with P (¯ ρ) − ρ¯. Changing the variable back in the integral ,n (as ρ → ρ above we get s Zn, ,ω (s) 1 = ψ11 (qi − qj ) + 2 ψ11 (qi − ωks ) µω s ,n (dq). Zn, ,ω (s) s [Λ( s )]n The identiﬁcation is easily deduced from this relation together with (2.5,6) if d = 1 since then we have the uniqueness for grand canonical Gibbs measures (accordingly

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the nice ergodicity), while, with such uniqueness result unavailable, we need grope some suitable device if d > 1.] One calculates the derivative of s−1 ψ11 (z s ) relative to s to see that if

2 2 z 4 z1 z1 1 − |z|V (|z|) + χ11 (z) = 1 − |z|2 V (|z|), |z| |z| |z|

1 d 1 s ψ11 (z ) = − 2 χ11 (z s ). ds s s Making an elementary calculation upon the (ﬁrst) expression of Zn, ,ω (s) given above and then changing the variable back show then

(s) d Zn, ,ω ds Zn, ,ω (s) 1 ψ11 (qi − qj ) + 2 ψ11 (qi − ωks ) = 2 the variance of s

(2.7)

i,k

i,j(=)

s µω s ,n (dq)

with respect to s 1 χ11 (qi − qj ) + 2 χ11 (qi − ωks ) µω − 2 s ,n (dq). s [Λ( s )]n i,k

i,j(=)

We are to exploit this relation to consider log Zn, ,ω (s) nearly convex. By the hypothesis (iii) on U we can choose a positive number A so that |ψ11 (q)|, |χ11 (q)| ≤ AI(|q| < co )e2U (q) for every q ∈ Rd . Applying Lemmas 2.5 and 2.6 we therefore obtain s s χ11 (qi − qj ) + 2 χ11 (qi − ωk ) µω (2.8a) s ,n (dq) [Λ( s )]n

i,k

i,j(=)

≤ AM1 (ρ ) |Λ()| + 2(%ω) as well as

ψ11 (qi −

[Λ( )]n

ωk )µω ,n (dq)

≤ AM1 (ρ )%ω,

(2.8b)

where %ω stands for the number of ωk ’s (presupposed to be contained in Λ( + co ) \ Λ()). Putting f (s) := Zn, ,ω (s)/Zn, ,ω (s) and recalling (2.6) we may rewrite (2.5) as log Zn (1+δ , ω1+δ ) − log Zn (, ω) = 1

1+δ

n s

+ f (s) ds

= n log(1 + δ) + δf (1) +

1+δ

s

ds 1

1

f (t)dt.

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Finally, by (2.7) and (2.8), we obtain that for all δ small enough, log Zn (1+δ , ω1+δ ) − log Zn (, ω) ≥ n log(1 + δ) + δ ψ11 (qi − qj )µω ,n (dq)

(2.9)

[Λ( )]n i,j(=)

− δAM1 (ρ )(2%ω) −

δ2 AM1 (ρ ) |Λ()| + 4(%ω) . 2

In the same way we get log Zn (1−δ , ω1−δ ) − log Zn (, ω) ψ11 (qi − qj )µω ≥ n log(1 − δ) − δ ,n (dq)

(2.10)

[Λ( )]n i,j(=)

δ 2 (1 + O(δ)) AM1 (ρ ) |Λ()| + 2(%ω) . 2 Theorem 2.2. Let ρ¯ be a non-negative constant. Then Ψ 11 (q)µω −→ P (¯ ρ) − ρ¯ ,n (dq) − δAM1 (ρ )(2%ω) −

[Λ( )]n

as n, ∗ → ∞ in such a way that n/|Λ()| → ρ¯, provided that ω, conﬁgurations on Λ( + co ) \ Λ(), are subject to the condition that %ω/|Λ()| → 0. For each constant r > 0 and each positive function γ() approaching 0 as ∗ → ∞, the convergence is uniform with respect to (¯ ρ, ω) such that ρ¯ ≤ r and %ω < γ()|Λ()|. Proof. We pass to the limit as ∗ → ∞ in (2.9). Noticing that |Λ(1+δ )| = (1 + δ)|Λ()|, we then obtain 1 − (1 + δ)Φ(¯ ρ/(1 + δ)) − Φ(¯ ρ) δ Ψ 11 (q)µω ≥ ρ¯ + lim sup ,n (dq) + O(δ). [Λ( )]n

We deduce from (2.10) a similar inequality (but in the opposite direction) with δ replaced by −δ and lim sup by lim inf . By introducing the variable η determined by (1 + δ)(1 − η) = 1 these two inequalities may be written as 1 Φ((1 − η)¯ ρ) − Φ(¯ ρ) + o(1) −η ≥ Φ(¯ ρ) + ρ¯ + lim sup Ψ 11 (q)µω ,n (dq) n [Λ( )] Ψ 11 (q)µω ≥ Φ(¯ ρ) + ρ¯ + lim inf ,n (dq) [Λ( )]n

1 Φ((1 + η)¯ ρ) − Φ(¯ ρ) + o(1), ≥ η

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where o(1) → 0 as η ↓ 0 ( we have made use of the fact that Φ is continuous). Noticing that the derivative of Φ from the right is larger than or equal to that from the left since Φ(ρ) is convex, we conclude that Φ is continuously diﬀerentiable and ρ¯Φ (¯ ρ) = Φ(¯ ρ) + ρ¯ + lim Ψ 11 (q)µω ,n (dq). ∗ →∞

[Λ( )]n

The asserted uniformity is automatical from the very fact that there exists a limit which is independent of ω. The relation of Theorem 2.2 then follows from P (ρ) = ρΦ (ρ) − Φ(ρ). Theorem 2.3. In the same sense of convergence as in Theorem 2.2, Ψ αβ (q)µω −→ (P (¯ ρ) − ρ¯)δαβ . ,n (dq)

(2.11)

[Λ( )]n

Proof. Let Tαβ be a limit point of the left-hand side of (2.11). Let a = (a1 , ..., ad ) be a unit vector in Rd . We have α,β aα ψαβ aβ = (z · a)2 |z|−2 ψ(|z|). Taking a small positive number δ, we cover Λ() by identical and disjoint hyper-cubes of side length ∗ δ whose edges are parallel or perpendicular to the vector a. Theorem 2.2 may be applied to Ψ’s corresponding to these hyper-cubes. The error that arises from the interaction between neighboring hyper-cubes is estimated from above by AM1 (ρ ) ×

the total volume of corridors ≤ const M1 (ρ )[δ∗ ]−1 , |Λ()|

where the corridors consist of the hyper-slabs of width co tiling the inner walls of the hyper-cubes; and the other contribution to the error, which comes from those sitting on the border of Λ(), is at most AM1 (ρ ) ×

the total volume of hyper-cubes on the border ≤ const M1 (ρ )δ |Λ()|

as is deduced from Lemma 2.5 and the hypothesis (iii) on V . These bounds for errors vanish in the limit as → ∞ and δ → 0 in this order. We can therefore conclude that aα Tαβ aβ = P (¯ ρ) − ρ¯, α,β

proving that T is a constant times the identity matrix. The proof of Theorem 2.3 is complete. Proof of Theorem 2.1. The proof is carried out only in the case α = β = 1 since the other case can be similarly dealt with. For a large positive integer m we partition Λ( − co ) into md hyper-intervals which are shifts of Λ(( − co )/m). We write |Λ()|Ψ 11 = |Λ( )|

i

τy(i) Ψ 11 + R ,m ,

=

− co m

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where y(i) denotes the center of the i-th hyper-interval, τy the translation operator and R ,m the remainder term, which consists of the contribution to Ψ 11 of the interaction among the hyper-intervals across their borders and the contribution of the conﬁgurations on Λ() \ Λ( − co ). We are going to take limit as ∗ → ∞ and m → ∞ in this order. From Lemma 2.5 together with the assumption (iii) it follows that 2 |R ,m |2 dµω ,n ≤ CM2 (r) (m|∂Λ()|) , where |∂Λ| denotes the surface area of Λ and C some constant, so that the contribution of R ,m vanishes in the limit. We have to prove 2 |Λ( )| lim lim sup sup sup τy(i) Ψ11 − [P (ρ ) − ρ ] dµω ,n = 0. m→∞ ∗ →∞ ω n≤r|Λ( )| |Λ()| i According to Lemma 2.8 we may replace P (ρ ) − ρ by

1 1 N [P (ρ ) − ρ ] ρ = K(i) K(i) K(i) K(i) , md i |K(i)| where K(i) denotes the i-th hyper-interval (namely the box Λ( ) shifted by y(i)). By Lemma 2.5 and (iii) again 2 sup sup sup sup τy(i) Ψ 11 dµω ,n < ∞; m

,i

ω

n≤r|Λ( )|

and by Lemma 2.9 sup lim sup sup m

∗ →∞

sup

ω,i n≤r|Λ( )|

[P (ρK(i) ) − ρK(i) ]2 dµω ,n < ∞.

Hence it suﬃces to show that for each m, τy(i) Ψ 11 − [P (ρK(i) ) − ρK(i) ] τy(j) Ψ 11 − [P (ρK(j) ) − ρK(j) ] dµω ,n → 0 as ∗ → ∞ uniformly in ω and n ≤ r|Λ()| whenever K(i) and K(j) are separated by a distance more than co from each other. By the DLR equation and the Schwarz inequality this follows if we show that, uniformly with respect to ω and n ≤ r|Λ()|, [Λ( )]k

τ (−y(i))ω

Ψ 11 (q)µ ,k

(dq)

2 − [P (ρK(i) ) − ρK(i) ] dµω ,n → 0 k=NK(i)

(ω is a random conﬁguration in the shifted shell τy(i) [Λ( + co ) \ Λ( )]), which in turn is deduced from Theorem 2.2 by employing Lemma 2.4 to control the eﬀect of %ω . The proof of Theorem 2.1 is complete.

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3 Existence of the processes and tightness for ξtN This section consists of two subsections. In the ﬁrst one we provide a proof for the existence of the diﬀusion process xN t , which is not entirely obvious in the case when V is unbounded. In the second we prove that the set of laws of the processes {ξtN , 0 ≤ t ≤ T } is relatively compact. 3.1. In this subsection N > 1 is ﬁxed and suppressed from xN t and νN and we put ' = 1 for simplifying notations. We suppose co < 1/2, which will cause no loss of generality. The diﬀusion process is well deﬁned at least up to the collision time T ∗ := inf{t ≥ 0 : xi (t) = xj (t)

for some pair i, j(=)}.

To be precise T ∗ is deﬁned as the monotone limit of the stopping times τ (δ) := inf{t ≥ 0 : |xi (t) − xj (t)| ≤ δ for some i, j(=)}

(3.1)

as δ ↓ 0. The probability law of the process xt , 0 ≤ t < T ∗ starting at x will be denoted by Px and the law initialized by the invariant measure ν by Pν . In the multidimensional case the problem is reduced to showing Theorem 3.1. If d ≥ 2, then Px [T ∗ = ∞] = 1 for any starting point x such that xi = xj if i = j. The following lemma is due to [4; Lemma 1.4]. Lemma 3.1. Let F (x) be a smooth function on {x ∈ (Td )N : xi = xj if i = j} such that F (x) → ∞ as mini,j(=) |xi − xj | → 0 and put for a positive number M τM = inf{t ≥ 0 : F (xt ) ≥ M }. Suppose that ∇F = 0 on the boundary {x : F (x) = M }. Then for t > 0, e ν F 2 + 12 t|∇F |2 . Pν [τM ≤ t] ≤ M (Here ∇ denotes the gradient operator on (Td )N .) Proof. Put D = {x ∈ (Td )N : F (x) < M }, so that τ = τM is the ﬁrst leaving time from D. The boundary of D is smooth owing to the assumption on ∇F. Hence the Laplace transform ϕλ (x) = Ex [e−λτ ] (λ > 0) solves the Dirichlet problem Lϕλ = λϕλ in D with the boundary condition ϕλ (x) = 1 on Dc . We can then follow [4] for the rest of the proof. Proof of Theorem 3.1 Let g(r) be a smooth non-negative function of r > 0 such that g(r) = 0 for r ≥ 1 and g(r) = log log r−1

for 0 < r < e−2

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and deﬁne a function F of conﬁguration x = (x1 , ..., xN ) by F (x) =

g(|xi − xj |).

(3.2)

i,j(=)

Then lim inf max ∇{g(| · |)}(xi − xj ) = ∞, M →∞ x:F (x)=M i

(3.3)

j=i

implying that ∇F = 0 on {x : F (x) = M } for all suﬃciently large M ; hence Lemma 3.1 (for M large enough) is applicable with F given by (3.2). [For veriﬁcation of (3.3) consider a limit set {y1 , ..., yk }, as m → ∞, of a m sequence of the sets each of which consists of exactly N points xm 1 , ..., xN from d m m T such that F (x1 , ..., xN ) = m (m = 1, 2, ...). Clearly k < N . For convenience m of exposition assume that k = 1 and the sequence ({xm 1 , ..., xN }; m = 1, 2, ...) is m convergent (namely, all the points xi converge to one and the same point), and choose for each m one of the ‘most salient’ extremal points, xm i(m) say, of the convex m m m hull of the set {x1 , ..., xN }. Then one sees that | j=i(m) ∇{g(|·|)}(xm i(m) −xj )| → ∞.] 1/2 Now one observes that ν F 2 + |∇F |2 ≤ const(1 + 0 [r log r]−2 rd−1 dr) < ∞, provided that d ≥ 2. Hence, according to Lemma 3.1, Pν [T ∗ = ∞] = 1. Let τM be the stopping time deﬁned in Lemma 3.1. The point-wise result then follows from the relation Px [T ∗ = ∞] = lim Ex [Pxt [T ∗ = ∞]; τM > t], t→0

which is valid for all suﬃciently large M , combined with the fact that the measure Px [xt ∈ · ; τM > t] is absolutely continuous relative to ν. In one dimension collisions take place with positive probability if, eg., V (r) = −a log r for small r with a < 1/2. Even in such a case, however, we can continue the process. In fact it is not hard to construct the diﬀusion process up to the time T ∗∗ when a multiple collision occurs for the ﬁrst time: T ∗∗ = inf{t ≥ 0 : xi (t) = xj (t) = xk (t)

for some triplet i, j, k(=)}.

since the origin is always an entrance boundary for the one dimensional diﬀusion process on [0, ∞] governed by the generator Lf = 12 f + [V (r) + b]f for every constant b. Thus the problem is solved by showing the following lemma. Lemma 3.2. Let d = 1. Suppose U (r) ≤ 0 for all suﬃciently small r > 0. If the initial conﬁguration, x say, involves no multiple collision, then so does the process for ever, namely Px [T ∗∗ = ∞] = 1.

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Proof. We deﬁne F (x) =

1179

2 2 2 g (xi − xj ) + (xj − xk ) + (xk − xi ) ,

i,j,k(=)

where g is the same as in the proof of Lemma 3.1. By the same reasoning as before it suﬃces to prove that ν F 2 + |∇F |2 < ∞, which is easily reduced to 1/2 [r(log r)2 ]−1 dr < ∞. 0 3.2. In this subsection we notice that compactness of the laws of our empirical processes ξtN , which is easy to prove if (H) is assumed, follows without assuming (H) from the condition as N → ∞, (3.4) f0N log f0N dνN = O(N ) which, though stronger than (1.4), is quite sound as a condition to be imposed on the initial density f0N . Theorem 3.2. Suppose that the initial densities {f0N } satisfy (3.4). Then for each time T the set of random processes {(ξtN , 0 ≤ t ≤ T )} is tight relative to the topology of uniform convergence of continuous trajectories in M(Td ). Proof. The proof is similar to one given by Varadhan [17]. Here we indicate key steps of the proof for reader’s convenience. (The details are also found in [3; Section 7.6] .) t Let T ≥ 1. For J ∈ C ∞ (Td ) consider the process XtN = 0 LN {ξ(J)}(xN s )ds (0 ≤ t ≤ T ). Then ξtN (J) − ξ0N (J) = XtN + mt with E[sup0≤t≤T m2t ] = O(N −1 ). We have only to show that P sup|t−s|<δ |XtN − XsN | > α vanishes as N → ∞ and δ ↓ 0 in this order. Put ΩN,δ (t − s) = sup δ/|t − s|N νN (LN {ξ(J)}f ) − IN (f ) , N 1

νN (f )=1,f ≥0

where IN (f ) = 8 i=1 νN ((∇i f )2 /f ). With the help of the inequality 8β|XtN − XsN | β exp exp √ sup |XtN − XsN | ≤ 4e4 δ −2 dtds δ |t−s|<δ |t − s| [0,T ]2 √ (β > 0), which is obtained by taking Ψ(u) = e8u − 1 and p(u) = u/β in the Garsia-Rodemich-Rumsey theorem given in [13; p.47] (with simple manipulations), a standard argument using the entropy inequality and Kac’s formula reduces the problem to showing 1 lim lim sup log exp{|t − s|ΩN,δ (t − s)}dtds = 0. (3.5) δ→0 N→∞ N [0,T ]2 √ Integrating by parts we obtain N νN (LN {ξ(J)}f ) ≤ ∇J∞ 2dN IN (f ), so that ΩN,δ (t − s) ≤ ∇J2∞ (d/2)N δ /|t − s|; hence (3.5).

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4 Local Gibbs States In this section we state a result on local equilibrium (Theorem 4.1 below), which is essentially the same as that obtained in [16] for the one-dimensional model, and then give a consequence of the combination of it with Theorem 1.1. The entropy bound (1.4) is supposed to hold in what follows. Let h be a function on Rd of the form h(z1 , ..., zd ) = χ(z1 ) · · · χ(zd ), where χ is a smooth non-negative function on R such that χ(t) = 0 if |t| > 1, χ(t)dt = 1 and tχ (t) < 0 if 0 < |t| < 1. For λ > 0 and a conﬁguration x = (x1 , ..., xN ) ∈ (Td )N , put ρλ (θ) = ρλ (θ; x) =

N

hλ

i=1

xi − θ '

(hλ (θ) = λ−d h(λ−1 θ)).

(Although we have used the notations ρ with a d-tuple = (1 , ..., d ) and ρK (q) with a set K in Section 2, there will be no fear of confusing them with ρλ introduced above; in particular the symbol ρ is in use in Sections 2 and 8 only.) Let xθ be the conﬁguration x viewed from θ ∈ Td : xθ = (x1 − θ, ..., xN − θ) and deﬁne

1 f¯N (x) = T

T

ftN (xθ )dθ.

dt Td

0

Theorem 4.1. For each λ > 0, any limit point, as N → ∞, of the law of the point process {'−1 xi : '−1 xi ∈ Λ(λ)} (λ = (λ, ..., λ)) induced from f¯N (x)νN (dx) is a convex combination of canonical Gibbs measures µω λ,n over varying particle numbers n and boundary conﬁgurations ω, and has average particle density not greater than lim 'd N . Proof. The proof given to Lemma 7.5 of [16] in one dimension may be followed word for word. Put for λ ≥ 1 S

(λ)

(θ) = S

(λ)

(θ, x) = ρ2λ (θ) +

|ψ|

i,j(=)

and (λ) Ψαβ (θ)

=

(λ) Ψαβ (θ, x)

=

i,j(=)

ψαβ

|xi − xj | '

xi − xj '

h2λ

hλ

xi − θ '

xi − θ '

.

Theorem 4.2. For each r > 0 and 1 ≤ α, β ≤ d ¯N (λ) lim lim sup E f Ψαβ (0) − [P (ρλ (0)) − ρλ (0)]δαβ ; S (λ) (0) ≤ r = 0. λ→∞ N→∞

(4.1)

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Proof. For microscopic variables q = (q1 , ..., qn ), n = 1, 2, ... put ρ∗λ (q) = ρλ (0, 'q) ∗(λ) (λ) and Ψαβ (q) = Ψαβ (0, 'q), namely ρ∗λ (q) =

n

∗(λ)

hλ (qi ) and Ψαβ (q) =

i=1

ψαβ (qi − qj ) hλ (qi ) .

i,j(=)

Since the integrand under the expectation is uniformly bounded due to the truncation by means of S (λ) , it suﬃces, in view of Theorem 4.1, to prove that for each r > 0 and for some η, 0 < η < 1, ∗(λ) lim sup sup Ψαβ (q) − [P (ρ∗λ (q)) − ρ∗λ (q)]δαβ µω ηλ,n (dq) = 0, λ→∞ ω<eλ n
[Λ(ηλ)]n

(4.2) where λ = (λ, ..., λ) and %ω denotes the number of particles constituting ω. For very large λ the function hλ being locally almost constant in the sense that sup |hλ (qi − q) − hλ (qi )| ≤

|q|≤c

c Ch · λ λd

for c > 0, the relation (4.2) is easily deduced from Theorem 1.1 with the help of Lemmas 2.4 and 2.8 (or Lemmas 8.2 and 8.4 in the case (iib)) by partitioning the hyper-cube Λ(λ) into relatively small ones K(j) as in the proof of Theorem (λ) 2.1. [Indeed, if one deﬁnes aj = |Kj |hλ (yj ) with yj ∈ K(j) suitably chosen so (λ) (λ) ∗(λ) aj ρK(j) that aj = 1, then ρ∗λ (q) and Ψαβ (q) are well approximated by (λ) −1 and aj |K(j)| i,k(=):qi ,qk ∈K(j) ψαβ (qi − qk ), respectively; and, by virtue of (λ) ∗ Lemma 2.8, P (ρλ (q)) by aj P (ρK(j) ). The details are easy to carry out.]

5 Strong convergence of P (ρλ (θ, xN t )) Let hλ and ρλ (θ) = ρλ (θ, x) be the same as in the previous section. We wish to compare the microscopic density ρλ (θ) with the macroscopic one ρλ ∗ hη (θ) where λ is taken large and η small. With the possibility of phase transitions and without knowledge of growth rate of P (ρ) for large ρ we consider f (P (ρλ )) − f (P (ρλ ∗ hη )) for each bounded continuous f instead of ρλ − ρλ ∗ hη . This section is devoted to proving the following theorem. Theorem 5.1. Suppose that the hypothesis (H) holds in addition to the entropy bound (1.4) and the condition (1.7) on initial conﬁgurations. For any subsequence of N = 1, 2, ... there exists its subsequence {N } such that the law of {ξtN : 0 ≤ t ≤ T } is convergent and that for each bounded continuous function f on [0, ∞), T N lim lim sup lim sup E dt ))) − f (P (ρ ∗ h (θ, x ))) dθ = 0, f (P (ρλ (θ, xN λ η t t η↓0

λ→∞

N →∞

0

Td

where ρλ ∗ hη (θ, , xN t ) denotes the convolution

ρλ (θ − θ , xN t )hη (θ )dθ .

1182

Kˆ ohei Uchiyama

Ann. Henri Poincar´e

Let G = G(θ) be a smooth function on Td . Put F (x) = ρλ (θ)(G ∗ ρλ )(θ)dθ,

(5.1)

Td

where G ∗ ρλ (θ) = G ∗ ρλ (θ, x) = Td G(θ − θ )ρλ (θ , x)dθ . Then, after carrying out integration by parts, we obtain

∂ xi − θ ∇G ∗ ρλ (θ)dθ F (x) = 2 hλ ∂xi ' Td and

LN F =

ρλ (θ)∆G ∗ ρλ (θ)dθ

xi − θ xi − θ + ∆G(θ − θ)hλ dθdθ hλ ' ' i=1

xi − θ 1 xi − xj − 2 ρλ (θ )dθ hλ ∇U dθ. ∇G(θ − θ) · ' ' '

N

i,j(=)

Here the domain Td is omitted from the integration sign. Since ∇U (−x) = −∇U (x), in the last integral 2hλ ((xi − θ)/') may be replaced by

hλ

xi − θ '

− hλ

xj − θ '

1

∇(hλ )

= 0

xi − s(xi − xj ) − θ '

ds ·

xi − xj . '

Substituting the expression on the right-hand side and performing integration by parts once more, we arrive at LN F = ρλ (θ)∆G ∗ ρλ (θ)dθ + Y λ (5.2) +

d d

ρλ (θ )dθ

˜ (λ) (θ)dθ. ∇α ∇β G(θ − θ) · Ψ αβ

α=1 β=1

where

˜ (λ) (θ) = Ψ ˜ (λ) (θ, x) = Ψ αβ αβ

i,j(=)

1

hλ

0

xi − s(xi − xj ) − θ '

ds ψαβ

xi − xj '

and Y

λ

=Y

λ,N

(x) =

N i=1

hλ

xi − θ '

∆G(θ − θ)hλ

xi − θ '

dθdθ .

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The diﬃculty we encounter in the multidimensions is caused by the cross terms (i.e., the terms with α = β) on the right-hand side of (5.2). We are to deﬁne G by (5.3) below, as in [12], to substitute xN t for x and to integrate both sides of (5.2) with respect to t ∈ [0, T ]. The term Y λ is then easy to deal with: see (5.15). The cross terms must vanish in the limit because of the local equilibrium (Lemma 5.1 below) once a relevant uniform integrability is established, and for the latter purpose we cannot help employing some bound of S (λ) (θ) like (H) along with Lemma 5.2 below. Lemma 5.1. For each r > 0 and 1 ≤ α, β ≤ d ¯N ˜ (λ) (λ) lim lim sup E f Ψ (0) − [P (ρ (0)) − ρ (0)]δ ; S (0) ≤ r = 0. λ λ αβ αβ λ→∞ N→∞

˜ (λ) (0, x)−Ψ(λ) (0, x)| Proof. The lemma follows from Theorem 4.2 if we show that |Ψ αβ αβ ≤ const λ−1 S (λ) (0, x). But this inequality is clear from the inequality

hλ xi − s(xi − xj ) − θ − hλ xi − θ ≤ Ch hλ xi − θ ' ' λ 2' which is valid whenever ψαβ ((xi − xj )/') = 0 and λ ≥ 2co .

Let pt (y − x), x, y ∈ T , t > 0, be the fundamental solution for the heat equation ∂t u = 12 ∆u on Td . Its Fourier expansion is given by pt (y − x) = exp{−2π 2 |k|2 t}ek (x)ek (y), (5.3) d

k∈Zd d 2 2 2 where the sum 1 , ..., kd ) ∈ Z , |k| = k1 + · · · + kd and √ extends over all k = (k 2 ek (x) = exp{ −1 (2π)k · x}. Let b > ' . In (5.1) we take as G(x) b pt (x)dt (5.4) Gb (x) = 22

from now on. Let Fb denote the corresponding F : Fb (x) = ρλ (θ)(Gb ∗ ρλ )(θ)dθ. From

Gb (x)dx = b − '2 and Gb ≥ 0 the Schwarz inequality deduces that Fb (x) ≤ bρλ (·, x)22 ,

where f 2 = that

Td

|f |2 dx

∇Gb ∗ f 22 =

1/2

(5.5)

. From (5.3) and the Parseval identity it also follows

2 d

|kα | −2(π|k|)2 22 −2(π|k|)2 b ˆ e | f(k)| − e π|k|2 α=1 k=0

≤ f 22 , where f is any square integrable function on Td .

(5.6)

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Kˆ ohei Uchiyama

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A similar bound for ∇α ∇β Gb ∗ f may be obtained in the same way but we need an estimate in Lp with p = 3/2, which is provided by the next lemma. Lemma 5.2. For each p > 1 there exists a constant Ap independent of b (≤ 1) and ' such that for any Lp -function f on Td and for 1 ≤ α, β ≤ d, ∇α ∇β Gb ∗ f p ≤ Ap f p where f p =

Td

|f |p dx

1/p

(p ≥ 1).

Proof. For any C 2 -function f on Td ∇α ∇β f p ≤ Ap ∆f p , which is a periodized version of a similar formula in the space Lp (Rd ) and follows from the identity ∇α ∇β f = −Rα Rβ ∆f , where Rα , α = 1, ..., d, are the periodic Riesz transforms, whose multipliers are −ixα /|x| and which satisfy the bound Rα f p ≤ cp f p (cf. [11, pages 243, 263 and 264]). Noticing ∆Gb = 2(pb − p22 ), we accordingly obtain that ∇α ∇β Gb ∗ f p ≤ 2Ap (pb − p22 ) ∗ f p ≤ 4Ap f p .

The proof is complete.

Lemma 5.3. There exists a constant Ch depending only on the function h such that 'd N dθ p22 (θ − θ )|ρλ (θ) − ρλ (θ )|dθ ≤ √ Ch (λ > 2). λ Td Td

xi − θ xi − θ − hλ we see Proof. Applying the mean value theorem to hλ ' ' that √ |ρλ (θ) − ρλ (θ )| ≤ Cλ−1/2 ρ2λ (θ) if |θ − θ | < ' λ, where C depends on h only. The double integral in the inequality of the lemma is then bounded by √ C'd N λ−1/2 + 2 ρλ (θ )dθ I(|θ − θ | > ' λ)p22 (θ − θ)dθ. An easy calculus shows that the second term is O('d N λd/2−1 e−λ/2 ). Lemma 5.4. Let Mt be deﬁned by the equation t N ) = F (x ) + LN Fb (xN Fb (xN b t 0 s )ds + Mt . 0

Then Mt ≤ C

'd N λd

t

[ρλ (θ, s)]2 dθ.

ds 0

Td

(5.7)

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Proof. Mt is a martingale and its quadratic variational process is written as

t

Mt =

(∇Fb · ∇Fb )(xs )ds 0

t

=

4

0

hλ

i

xi (s) − θ '

By applying Schwarz inequality, the relation the right-hand side is bounded by 'd N 4 d λ

2

h (q)dq Rd

2 ∇Gb ∗ ρλ (θ, s)dθ

h2λ (q)dq = λ−d

2

ds 0

h2 (q)dq and (5.6),

t

ds.

[ρλ (θ, s)] dθ , Td

which proves the lemma.

Let g be a continuous non-increasing function on [0, ∞) such that g(t) = 1 if t < 1 and g(t) = 0 if t > 2. Put λ,r ρλ (θ)∆Gb ∗ ρλ (θ)(1 − g(S (λ) (θ)/r))dθ, R0 (x) = Td

R1λ,r (x) =

d d

ρλ (θ )dθ

Td

α=1 β=1

˜ (θ)(1 − g(S (λ) (θ)/r))dθ ∇α ∇β Gb (θ − θ) · Ψ αβ (λ)

Td

and λ,r Rα,β (x) =

ρλ (θ )dθ

Td

×

∇α ∇β Gb (θ − θ) × Td ˜ (λ) (θ) − [P (ρλ (θ)) − ρλ (θ)]δαβ }g(S (λ) (θ)/r)dθ. {Ψ αβ

Then, by (5.2) LN Fb = Y λ + R0λ,r + R1λ,r (x) +

d d α=1 β=1

λ,r Rα,β

P (ρλ (θ))g(S (λ) (θ)/r)dθ

+ Td

∆Gb (θ − θ) · ρλ (θ )dθ .

Td

We substitute the relation 1 ∆Gb = pb − p22 2

into the last integral. In the (p22 )-part, i.e., in the integral p22 (θ − θ)ρλ (θ )dθ , we wish to replace ρλ (θ ) by ρλ (θ). If we denote by R2λ,r the error arising in this

1186

Kˆ ohei Uchiyama

Ann. Henri Poincar´e

replacement, then LN Fb = Y λ + R0λ,r + R1λ,r + R2λ,r +

d d

λ,r Rα,β

(5.8)

α=1 β=1

+

P (ρλ (θ))g(S

(λ)

Td

−

(θ)/r)dθ

pb (θ − θ) · ρλ (θ )dθ

Td

P (ρλ (θ))ρλ (θ)g(S (λ) (θ)/r)dθ Td

and R2λ,r =

P (ρλ (θ))[ρλ (θ) − ρλ ∗ p22 (θ)]g(S (λ) (θ)/r)dθ. Td

We obtain, by Lemma 5.3 together with the inequality ρλ (θ) ≤ 2S (λ) (θ), the uniform bound |R2λ,r (x)| ≤ ['d N Ch P (4r)]λ−1/2 ,

(5.9)

and, by Lemma 5.2, the inequalities |R1λ,r (x)| ≤ Cρλ 3

˜ λαβ (1 − g(S (λ) /r))3/2 , Ψ

(5.10)

α,β

and (λ)

λ,r (λ) ˜ |Rα,β (x)| ≤ Cρλ 3 {Ψ /r)3/2 . αβ − [P (ρλ ) − ρλ ]δαβ }g(S

(5.11)

Here the norm · p is taken with respect to the variable θ, so that the right-hand side of (5.10) and (5.11) are still functions of x (eg. ρλ p = ( [ρλ (θ, x)]p dθ)1/p ). The hypothesis (Ha) will be applied through the following lemma. The function h involved in (H) may be the same as used above, so that ρo (θ) = ρ(1/2) (θ) and S (1/2) (θ) = S o (θ) + ρo (θ). Lemma* 5.5. For each δ > 0 T dt lim lim sup lim sup P r→∞ λ→∞

N→∞

0

Td

3/2 (λ) N [S (λ) (θ, xN )] I(S (θ, x ) ≥ r)dθ > δ = 0. t t

Proof. By using Markov’s inequality one derives the inequality −3/2 ) [S (λ) (θ)]3/2 I(S (λ) (θ) ≥ r)dθ (1 − 2 Td [S (λ) (θ)]3/2 − r3/2 I(S (λ) (θ) ≥ r)dθ + r3/2 I(r < S (λ) (θ) ≤ 2r)dθ. ≤ Td

Td

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An application of Jensen’s inequality with the convex function f (x) = (x3/2 − r3/2 ) · I(x > r) with the help of the relations S (λ) (θ) ≤ CS (λ) ∗ h2 (θ) = CS (1/2) ∗ h2λ (θ) = C · (S o + ρo ) ∗ h2λ (θ) (for some constant C) shows that the ﬁrst integral on the right-hand side above is dominated by a constant multiple of o [S (θ)]3/2 I(S o (θ) ≥ r/2C) + [ρo (θ)]3/2 I(ρo (θ) ≥ r/2C) dθ Td

(for λ ≥ 1). (Here the Jensen is applied both for the averaging by convolution and to the hypothesis (H) it therefore suﬃces to for the average (S o + ρo )/2.) Owing prove that if ζλ,k (x) = (2k )3/2 I(2k < S (λ) (θ, x) ≤ 2k+1 )dθ, then for each δ > 0 lim lim sup lim sup P

k→∞ λ→∞

N→∞

T

ζλ,k (xN )dt ≥ δ = 0. t

0

But this can be easily veriﬁed by using the bound ∞

ζλ,k ≤

[S (λ) (θ)]3/2 dθ ≤ C 3/2

(λ ≥ 1)

[S (1/2) (θ)]3/2 dθ

k=1

and the fact that the laws of (also owing to (H)).

T

3/2 dt [S (1/2) (θ, xN dθ constitute a tight family t )]

0

By using Young’s inequality we deduce from Lemma∗ 5.5 and (5.10) that for every δ > 0 T

lim lim sup lim sup P |R1λ,r (xN )|dt > δ = 0. (5.12) t r→∞ λ→∞

N→∞

0

Similarly lim lim sup lim sup P

r→∞ λ→∞

N→∞

T

|R0λ,r (xN )|dt > δ = 0. t

(5.13)

0

Lemma 5.1 yields that lim lim sup E

λ→∞ N→∞

T

0

λ,r |Rα,β (xN )|dt = 0. t

(5.14)

It is easy to see that |Y λ |2 ≤ 4N C 2

i

h2λ

xi − θ '

2 dθ

=4

N 'd C λd

2 h42 .

(5.15)

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Kˆ ohei Uchiyama

Ann. Henri Poincar´e

With the help of these estimates we can follow Varadhan’s argument of using the Young measure with some modiﬁcations. Put ρN,λ (θ, t) := ρλ (θ, xN t ) and as in [12] or [15] deﬁne a random measure π N,λ on [0, T ] × Td × [0, ∞) by N,λ (du) = dtdθδρN,λ (θ,t) (du). π N,λ (dtdθdu) = dtdθπt,θ

Then {π N,λ = π N,λ (dtdθdu), N = 1, 2, ..., λ = 1, 2, ...} constitutes a tight family of measure-valued random variables. The set of random quantities ξ N,λ = (ξtN,λ , 0 ≤ t ≤ T ), where ξtN,λ (dθ) = ρN,λ (θ, t)dθ, is also tight as a family of measure-valued continuous processes (see Section 3.2). Let QN,λ be the probability law induced by (π N,λ , ξ N,λ ) and Q any limit point of {QN,λ } as N → ∞ and λ → ∞ in this order. Under the law Q the ﬁrst component, π say, is of the form πt,θ (dθ) as directly inferred from the deﬁnition of π N,λ ; also for the second component ξ = (ξt (dθ)), we have ξt (dθ) = ρ(θ, t)dθ with ρ(θ, t) := uπθ,t (du) a.s. (dtdθdQ) by virtue of (H). We substitute xN t for x in (5.8) and integrate both sides with respect to t over [0, T ]; then rewrite principal terms in the resulting relation by means of (π N,λ , ξ N,λ ) and pass to the limit as N → ∞, λ → ∞ and r → ∞ in this order. All the contributions of the error terms on the right-hand side of (5.8) have been shown to vanish in the limit in the estimates (5.9) and (5.12-15). Moreover, in T N N view of Lemma 5.4, 0 LN Fb (xN s )ds may be replaced by Fb (xT ) − Fb (x0 ). Thus if we deﬁne T M λ,N,r λ,N W1,M = dt dθg(S (λ) (θ, xN )/r) uP (u)πt,θ (du). t Td

0

λ,N,r W2,b =

T

0

ρN,λ (θ , t)dθ

dt

Td ×[0,∞)

Td

0

λ,N pb (θ − θ) · P (u)g(u/r)πt,θ (du)dθ,

then it follows that for every δ > 0 and M > 0 λ,N,r λ,N,r = 1, < δ + Fb (xN lim lim inf lim inf P Fb (xN T ) + W1,M 0 ) + W2,b r→∞ λ→∞

N→∞

(5.16)

where we have also applied the fact that P (u) ≥ 0. Since the mapping T ξt ∗ pb (θ)P (u)g(u/r)π(dtdθdu) (π, ξ) → 0

Td ×[0,∞)

T

dt

= 0

pb (θ − θ )ξt (dθ )

dθ Td

Td

P (u)g(u/r)πt,θ (du), [0,∞)

is continuous and bounded for each b > 0 and r > 0, since Fb ≥ 0, and since T →0 dt [1 − g(S (λ) (θ, xN lim lim sup lim sup P t )/r)]dθ > δ r→∞ λ→∞

N→∞

0

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for every δ > 0, it follows from (5.16), (1.7) (assumption on xN 0 ) and (5.5) that T M dt dθ uP (u)πt,θ (du) (5.17) 0

≤

0

ρ(·, 0)22 b

+

T

dt

pb (θ − θ )ρ(θ , t)dθ

dθ

0

∞

P (u)πt,θ (du) 0

for every M > 0 with Q-probability one; hence, letting b ↓ 0 and M → ∞, we have T dt dθ uP (u)πt,θ (du) (5.18) 0

T

≤

dt

dθ

u πt,θ (du )

P (u)πt,θ (du)

a.s.(Q).

0

Since P (u) is non-decreasing, the inequality in the opposite direction automatically holds. In order to infer a consequence from the inequality (5.18) we wish to have ! " ∞ T Q dt dθ uP (u)πt,θ (du) < ∞ = 1. (5.19) Td

0

0

P (u)π (du)dθ < ∞ = 1, which t,θ 0 T in turn follows from Theorem 4.2 since the law of 0 dt P (u)πt,θ (du)dθ under Q is a limit point of those of T N (λ) dt P (ρλ (θ, xN (θ, xN t ))I(ρλ (θ, xt ) < M )g(S t )/r)dθ

In view of (5.17) this follows from Q

T

dt

0

T as N → ∞, λ, r → ∞, and M → ∞ and the laws of 0 dt Ψλ11 (θ, xN t )dθ form a tight family owing to Lemma∗ 5.5. Under (5.19) the inequality (5.18) is possible only if for every bounded continuous function f (u) on [0, ∞) ∞ f (P (u))πt,θ (du) = f (P (ρ(θ, t))) a.s. (dQdtdθ). (5.20) 0

Now it is easy to complete the proof of Theorem∗ 5.1. We choose a subse quence N of a given subsequence of {N } so that {QN ,λ } converges for every λ = 1, 2, ... The left-hand side of the formula of Theorem∗ 5.1 may be rewritten by means of QN,λ as T QN ,λ E |f (P (u)) − f (P (ρ ∗ hη (θ, t)))|πt,θ (du)dθdt, lim sup lim sup lim η→0

λ→∞ N →∞

0

which, in view of (5.20), is not greater than T lim sup sup E Q dt |f (P (ρ(θ, t))) − f (P (ρ ∗ hη (θ, t)))|dθ, η→0

Q

0

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where supQ is taken over all limit points Q of {QN ,λ } as N → ∞, λ → ∞ in this order. Since the laws of ρ(θ, t) under Q are identical, we may remove the supremum over Q; hence the last limit supremum vanishes. The proof of Theorem∗ 5.1 is complete.

6 Proof of Theorem∗ 1.3 The proof of Theorem∗ 1.3 is based on the results of the preceding section and the uniqueness result for weak solutions of (1.3) as stated in Section 1. Suppose that ξ0N converges to u0 ∈ M(Td ). Let J be a smooth function on the torus Td . Let Q be a limit point of the probability laws of the measure valued processes ρN,λ (θ, t)dθ (= ρλ (θ, xN t )dθ), 0 ≤ t ≤ T, as N → ∞. The limit does not depend on λ at all in view of the trivial bound N N ≤ d∇J∞ λ' , ξt (J) − J(θ)ρ (θ, x )dθ (6.1) λ t Td

where ∇J∞ = supq,α |∇α J(q)| and ξtN (J) = Td J(θ)ξtN (dθ). Recall that under Q the measure valued process possesses a density, ρ(θ, t) say, for almost all t with probability one. According to the uniqueness result on the equation (1.3) stated in Section 1 it suﬃces to prove that the (potentially random) function ρ(θ, t) is a weak solution of (1.3) (with this ρ(θ, t) in place of u(θ, t)) satisfying the initial condition (1.5) a.s.(Q) since the integrability condition (1.6) is valid for ρ(θ, t) by virtue of (5.19) and (5.20). As in the previous section (see (5.2)) we obtain N J(θ)ρλ (θ, xt )dθ − J(θ)ρλ (θ, xN (6.2) 0 )dθ Td

=

1 2

t

ds 0

d

d

Td

N ˜ (λ) (θ, xN ∇α ∇β J(θ) Ψ s ) + ρλ (θ, xs )δαβ dθ + mt αβ

α=1 β=1

where mt =

t 0

i

dθhλ

Td

xi (s) − θ '

∇J(θ) · dBi (s) .

A simple computation yields E[|mt |2 ] ≤ d∇J2∞ tN '2d = O('d ). We decompose the integral on the right-hand side of (6.2) by dividing the domain of integration according as ρλ (θ, xN s )≤M

or ρλ (θ, xN s ) > M.

(6.3)

The contribution of the second part can be easily shown to be negligible by using N ˜ (λ) (θ, xN (H). For the ﬁrst part in (6.3) we may replace Ψ s ) by [P (ρλ (θ, xs )) − αβ

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ρλ (θ, xN s )]δαβ since the error produced by the replacement converges to zero in probability according to Lemma 5.1. Therefore we can write 1 t N,λ,M ξtN (J) − ξ0N (J) = ds ∆J(θ) PM (ρλ (θ, xN , (6.4) s ))dθ + R 2 0 d T where PM (ρ) = P (ρ)g(P (ρ)/M ) and the error term RN,λ,M converges to zero in the sense that for every δ > 0 lim lim sup lim sup P |RN,λ,M | > δ = 0. M →∞ λ→∞

N→∞

∗

Now we apply Theorem 5.1 (together with (5.19)) to deduce from (6.4) the relation 1 t J(θ)ρ(θ, t)dθ − J(θ)u0 (dθ) = ds ∆J(θ) P (ρ(θ, s))dθ a.s. (Q). 2 0 Td Td Td Thus Theorem∗ 1.3 has been proved.

7 Proof of Theorem 1.2 Let d = 1. Then, in the same way as in [15], we obtain the bound E

N T

0

ψ(N |xi (t) − xj (t)|)h(N (xi (t) − xk (t)))dt < CN.

(7.1)

i=1 j=i k=i

E 0

N T i=1 j=i

ψ(N |xi (t) − xj (t)|) dt < CN. N |xi (t) − xj (t)|

(7.2)

where h is the function introduced in Section 1. In the case when ψ attains negative values we make use of the following lemma (use Lemma 8.1 of the next section for the proof). 1 1 Lemma 7.1. Suppose that 0 ψ(r)dr = ∞ and 0 [ψ (r)]+ dr < ∞. Let ϕ(x) be a positive, integrable continuous function on R such that ϕ(y)/ϕ(x) is bounded on {|x| < |y|}. Then for each δ > 0 there exists a constant C (independent of N ) such that for every conﬁguration x1 , ..., xN (=) on R

2 N N ϕ(N (xi − xk )) ≤ CN + δ ψ(N |xi − xj |)h(N (xi − xk )). i=1 k=i

i=1 j=i k=i

The bounds (7.1) and (7.2) together with Lemma 7.1 imply that the function H(i, x) := j=i ψ(N |xi − xj |) is uniformly integrable relative to the measure N −1 j δ0 (· − j) × fN (x)νN (dx). On using this uniform integrability as well as Theorem 4.2 the proof of Theorem 1.2 may proceed as in [16].

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8 Appendix This appendix consists of two subsections. In the ﬁrst one we shall prove Theorem 1.1 in the case (ii.b). Some asymptotic or continuity properties of Φ and P will be obtained in the second subsection. We shall use the same notations as in Section 2. 8.1. In this subsection we assume 1 V (r)rd−1 dr = ∞

(ii.b)

0

instead of the positivity V ≥ 0 and prove Theorem 1.1 under (ii.b). Lemma 8.1. For any non-empty cube K and any positive constant M there exists a constant C such that for any k ∈ N and any conﬁguration q1 , ..., qk on K, U (qi − qj ) ≥ M k2 − C. i,j(=) (k)

Proof. Let (qi )1≤i≤k , k = 1, 2, ... be an arbitrarily given sequence of conﬁgurations on K. We have to and only have to deduce from (ii.b) that 1 (k) (k) U (qi − qj ) = ∞. k→∞ k 2 lim

(8.1)

i,j(=)

Let µ = µ(dx) be any limit point of the sequence of counting measures µk (dx) := k−1 i δq(k) (dx). Since i

k−2

(k)

U (qi

(k)

− qj ) ≥

i,j(=)

U (x − y) ∧ K×K

an application of Fatou’s lemma shows that 1 (k) (k) U (qi − qj ) ≥ lim inf ∗ 2 k→∞ k i,j(=)

√ 1 kµk (dx)µk (dy) − √ , k

U (x − y)µ(dx)µ(dy) K×K

where the limit inﬁmum is taken along a subsequence which leads to µ. Put −d I(|x − y| < r)µ(dx)µ(dy) f (r) = r K×K

co

and substitute U (x − y) = 0 −V (r)I(|x − y| < r)dr. Remembering that V is supposed non-increasing near the origin, we obtain co U (x − y)µ(dx)µ(dy) = (−V (r))rd f (r)dr. K×K

0

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By using Fatou’s lemma it is not hard to see that vd K ϕ2 dx if µ(dx) = ϕ(x)dx lim inf f (r) ≥ r↓0 ∞ if µ has a singular part (use, if you wish, the multidimensional extension of De la Vall´ee Poussin’s decomposition theorem [9; Theorem 15.7 of Chapter IV], which says that µ is absolutely continuous on the complement of the set of (singular) points at which the derivac tives of µ are inﬁnite); in particular lim inf r↓0 f (r) > 0. Hence 0 o (−V (r))rd f (r) dr = ∞ under (ii.b), showing (8.1) as required. In the rest of this subsection we suppose, for simplifying the exposition, that 1 = · · · = d = co p for some positive integer p. Let Qt , t ∈ T denote a family of mutually disjoint cubes of side length co which together cover Λ(+co )\Λ(). Similarly let Rs , s ∈ S denote a family of those which cover Λ(). Let the indices t and s stand for the centers of the cubes Qt and Rs , respectively. The Euclidean distance of a point x ∈ R from a set K ⊂ R is denoted by d(x, K). For a conﬁguration q = (q1 , ..., qn ) and a set D we denote by q ∩ D a conﬁguration q restricted on D as in Section 2. Lemma 2.3, not generally valid under (ii.b), must be suitably modiﬁed and the following version is natural in the case of potentials of ﬁnite range. Lemma 8.2. For any positive constants a, b, and B and any constant 0 ≤ γ < 1 there exists a continuous function A(ρ), ρ ≥ 0 (independent of n and ) such that if K is a cube of side length ≤ co and mt = mt (ω) := %(ω ∩ Qt ) ≤ aeb d(t,K) then

for

t ∈ T,

{NK =k}

k 2 exp{γ U(q ∩ K)}µω ,n (dq) ≤ A(ρ )|K| exp{−Bk },

where the integration extends over q ∈ [Λ()]n such that NK (q) = k. Proof. The following proof is adapted from Dobrushin and Minlos [1], in which a similar result with empty boundary condition is established in the case of LennardJones type potentials. We suppose for simplicity that so ∈ K for some so ∈ S and ˜ s = Rs if s = so and R ˜ so = Rso \ K. Let 1 ≤ k ≤ n be given K ⊂ Rso , and put R arbitrarily and ω as in the lemma. Deﬁne a function λ on S by λs = aeb|s−so | . Given a subset S ⊂ S and k = (ks )s∈S , an integer valued function on S , such that ks ≥ λs and |k | := ks ≤ n, (8.2) s∈S

1194

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we denote by D(S , k ) the set of conﬁgurations q ∈ [Λ()]n such that if s ∈ S , ˜ s ∩ q) = = ks %(R < λs if s ∈ S \ S and put Jk (S , k ) = so that

{NK

=k}∩D(S ,k )

exp{γ U(q ∩ K)}µω ,n (dq),

{NK =k}

exp{γ U(q ∩ K)}µω ,n (dq) =

S ⊂S

Jk (S , k ),

(8.3)

k

where the second sum extends over k = (ks )s∈S satisfying (8.2). Let # ˜s . D = s∈S\S R Then, by Lemma 8.1, for any positive constant M there exists a constant C such ˜ s ∩ q) for s ∈ S \ S , then that if q ∈ {NK = k} ∩ D(S , k ) and ks = ks (q) := %(R − γ U(q ∩ K) + Hω (q)

≥ Hω (q ∩ D) + (1 − γ)(M k2 − C) + − vo kkso +

s∈S:s∼so

kks +

t∈T :t∼so

+

(M ks2 − C)

s ∈S

s ∈S

s∈S:s∼s

kmt +

ks ks

mt ks ,

t∈T s ∈S :s ∼t

√ where vo = − inf r>0 V (r)(≥ 0) and s ∼ s means that 0 < |s − s | ≤ dco . Recalling the deﬁnition of λs and the hypothesis on mt we estimate the ﬁrst three terms in the square bracket on the right-hand side above as follows

√ 1 2 1 d b dco kkso + k + ae kks + kmt ≤ 3 k + ks2 . 2 2 s∈S:s∼so

s ∈S :s ∼so or =so

t∈T :t∼so

√ b dco

√ b dco

Noticing that λs /λs ≤ e if s ∼ s and mt /λs ≤ e estimate the other two terms by √ ks2 . 3d eb dco

if s ∼ t, we similarly

s ∈S √

Choose M so that B ≤ (1 − γ)M − vo 3d and M := M − vo [3d eb dco + 12 ] > 0. We then see that −γ U(q ∩ K) + Hω (q) ≥ Hω (q ∩ D) + Bk2 + (M ks2 − C) − C , s∈S

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where C = (1−γ)C + 12 vo 3d (aeb that deﬁnes Jk we obtain Jk (S , k ) ≤

√

1195

dco 2

) . Substituting this bound into the expression

1 (M ks2 − C) + C × exp − Bk2 − Zn (, ω) s∈S d|k | ω |K|k co $ × e−H (q) dq . k! [ s∈S ks !](n − |k |)! Dn−|k |

By using Lemma 2.1 (which is valid for the present V ) ω 1 e−H (q) dq ≤ Zn−|k | (, ω) (n − |k |)! Dn−|k |

≤ Zn (, ω)[C(ρ )]|k | . Hence Jk (S , k ) ≤

2 |K|k e−Bk [C(ρ )cdo ]|k | eC exp{− s∈S (M ks2 − C)} $ · . k! s∈S ks !

The sum over k : ks ≥ λs , s ∈ S of the second ratio on the right-hand side is at C 2 d ˜ )]} where C(ρ) ˜ most e exp{− C(ρ s∈S $ [M λs − = C + C(ρ)co . Finally apply the relation S ⊂S s∈S βs = s (1+βs ) ≤ exp{ s βs }, which is valid whenever βs ≥ 0, and you will deduce the inequality of the lemma from (8.3) with ˜ ) < ∞. A(ρ ) = eC exp exp − a2 M e2bco |x| + C(ρ x∈Zd

Remark. Lemma 8.2 can be extended to general compact sets K even when K depends on and is unbounded as gets large if the statement of Lemma 8.2 is suitably modiﬁed. In fact, the same proof shows the following result (not applied in this paper): For any positive constants a, b, and B and a constant 0 ≤ γ < 1, there exists a continuous function A(ρ) such that if S1 is a subset of S and mt := %(ω ∩ Qt ) ≤ aeb d(t,Rs )

for t ∈ T, s ∈ S1 ,

then for ks ∈ N given for s ∈ S1 , exp γ U(q ∩ Rs ) µω ks2 . ,n (dq) ≤ A(ρ ) exp − B {NRs =ks ,s∈S1 }

s∈S1

s∈S1

Lemma 8.3. There exists a continuous function A(ρ) of ρ ≥ 0 such that o Zn (, ω) ≤ Z ,n exp vo m2t + A(ρ )|∂Λ()| , t∈T

where vo = − inf r>0 V (r) and |∂Λ| is the (d − 1)-dimensional area of the surface ∂Λ.

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Proof. The proof is quite similar to (and rather simpler than) the previous one. Deﬁne the function λs , s ∈ S by λs = c−1 o d(s, ∂Λ()). Then with S and k = (ks ; s ∈ S ) given and D(S , k ), D, and ks = ks (q) (s ∈ / S) deﬁned as in the previous proof we see that for q ∈ D(S , k ), Hω (q) ≥ U(q ∩ D) + (M ks2 − C) − vo

s∈S

ks ks +

s ∈S s∈S:s∼s

≥ U(q ∩ D) +

(M ks2

t∈T s∈S:s∼t

− C) − vo

s∈S

mt ks

m2t ,

t∈T

+ (Notice that ks = 0 if s ∈ / S , d(s, ∂Λ()) = co /2 where M = M − vo (3 and q ∈ D(S , k ).) Proceeding as before arrive at the inequality of the one can ˜ ) ≤ A(ρ )|∂Λ|. lemma by choosing A(ρ) so that C + s∈S exp − M λ2s + C(ρ d+1

1 2 ).

Having Lemmas 8.2 and 8.3 on hand we can easily obtain analogues of Lemmas 2.5, 2.6 and 2.7 (accordingly modiﬁed). We also have that of Lemma 2.8 as given in the following form. Lemma 8.4. For each triplet of numbers r > 0, 0 < δ < 1 and α > 0 there exist positive constants a and L such that if ρ ≤ r, ∗ > L and K is a hyper-interval included in Λ() and if δ ≤ |K|/|Λ()| ≤ (1 − δ), then 2 µω ,n P (ρK ) − P (ρ ) > α < exp − a|Λ()| + vo t mt , where ρK = |K|−1 NK and ρ = n/|Λ()|. Proof. The proof of Lemma 2.8 can be adapted for the present purpose if we show that 1 1 dq1 exp{−Hω (q1 , q2 )}dq2 log |Λ| k!(n − k)! [Λ\K]n−k Kk

|K| |Λ \ K| vo 2 m , Φ(|K|−1 k) + Φ(|Λ \ K|−1 (n − k)) + η + ≤− |Λ| |Λ| |Λ| t t where Λ = Λ(). Let T˜ be the set of all s ∈ S such that Rs intersects the co neighborhood of K, and put m ˜ s (q1 ) = %(q1 ∩Rs ) for s ∈ T˜ and q1 , a conﬁguration on Λ \ K. By virtue of Lemma 8.3 it therefore suﬃces to show that dq 1 exp − Hω (q1 ) + m ˜ 2t (q1 ) log (n − k)! [Λ\K]n−k t∈T˜ m2t + η|Λ|. ≤ −|Λ \ K|Φ(|Λ \ K|−1 (n − k)) + vo t∈T

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To this end we can proceed as in the proof of Lemma 8.2 if we deﬁne the function λs by λs = c−1 o min{d(s, ∂Λ()), d(s, K)}.

The details are omitted.

Proof of Theorem 1.1 under (ii.b). We consider only the case α = β = 1. The general case can be dealt with as in Section 2. Notice that on account of Lemma 8.2 3 ω Ψ11 µη ,n (dq) < ∞. sup sup sup

ω≤exp ∗ n≤r|Λ( )|

[Λ(η )]n

(Here r > 0 and η > 1 are ﬁxed constants.) Then, proceeding as in the proof given under V ≥ 0 at the end of Section 2 (with the help of Lemma 8.4 in place of Lemma 2.8), we observe that our task is reduced to proving that for any δ > 0, τ (−y(i))ω ω µη ,n Ψ11 (q)µ ,k (dq) − [P (ρK(i) ) − ρK(i) ] > δ → 0 k=NK(i) k [Λ( )]

uniformly with respect to i, ω satisfying %ω ≤ e ∗ and n ≤ r|Λ(η)|. This is veriﬁed by showing an analogue of Theorem 2.1 suitably modiﬁed. By Lemmas 8.2 and 8.3 we have, in place of (2.9), that 0 log Z 01+δ ,n − log Z ,n ≥ n log(1 + δ) + δ

[Λ( )]n

with

1 δ|Λ()|

[Λ(η )]n

ψ11 (qi − qj ) µω ,n (dq) + R(, δ, ω )

|R(, δ, ω (q))|µω η ,n (dq) → 0

as ∗ → ∞ and δ ↓ 0 in this order, where ω (q) = q ∩ [Λ( + co ) \ Λ()]; and a similar relation in place of (2.10). By taking this remark into account it is easy to adapt the proof of Theorem 2.1 to accomplish the veriﬁcation mentioned above. 8.2. In this subsection V is not required to satisfy any of conditions (i), (ii) or (iii). Instead of them V is assumed to be of the form V = Vs + Vp , 2 where Vp is a non-negative C function of r > 0 with Vp (0+) > 0 and Vs satisﬁes the stability condition that 1≤i<j≤n Vs (|qi − qj |) ≥ −Cn for some constant C. Further suppose that ∞ sup |V (r)| td−1 dt < ∞. 1

r≥t

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Ann. Henri Poincar´e

By “ → ∞(F )” we mean the procedure of passing to the limit as ∗ → ∞ in such a way that lim sup max{1 , ..., d }/ min{1 , ..., d } < ∞. It holds under the conditions on V imposed above that for every ρ ≥ 0, there exists 0 |Λ()|−1 log Zn, , Φ(ρ) = lim →∞(F ), ρ →ρ

where (and in what follows) we write ρ = n/|Λ()| as in Section 2. The next theorem provides an upper bound of P (ρ) for ρ large as well as a crude lower bound. The result is not used in this paper. Lemma 8.5. It holds that Φ(ρ) ≥ Bρ2 − Cρ + ρ log ρ for some constants B > 0 and C; moreover lim supρ→∞ Φ(ρ)/ρ2 < ∞ if 0+ V (r)rd−1 dr < ∞; and limρ→∞ Φ(ρ)/ρ2 = ∞ if 0+ V (r)rd−1 dr = ∞. If for some constant p > 0, V (r) = O(r−p ) as r ↓ 0, then as ρ → ∞ O(ρ2 log ρ) if p = d, Φ(ρ) = (8.4) O(ρ1+p/d ) if p > d. 0 Proof. Obviously, log Z ,n ≤ − inf q∈[Λ( )]n U(q) + log(|Λ()|n /n!). By the hyperstability (valid for the present V and applied in the proof of Lemma 2.3) we see that 0 −|Λ()|−1 log Z ,n ≥ Bρ2 − Cρ + ρ log ρ (B > 0). If 0+ V (r)rd−1 dr = ∞, then, according to Lemma 8.1, B may be arbitrarily large, so that limρ→∞ Φ(ρ)/ρ2 = ∞. The asserted lower bounds of Φ(ρ) have been shown. For the upper bounds we need to apply an analogue of Lemma 2.2, in which the conclusion is modiﬁed as follows: for some constant A 0 0 0 + An2 /m for n > m, − log Z ,n ≤ −m min log Zc,n/m , log Zc,n/m +1

and 0 ≤ −m log(2c)d + An for n ≤ m. − log Z ,n

(8.5)

(For the proof, that of Lemma 2.2 may be readily adapted.) In view of the ﬁrst estimate above, proving the bound (8.4) is reduced to proving that as k → ∞ if p = d, O(k2 log k) 0 − log Z1,k = (8.6) 1+p/d O(k ) if p > d, where 1 = (1, ..., 1). But by a little reﬂection one ﬁnds that 0 − log Z1,k

≤ const k

1/d k

j=1

j d−1 V (jk−1/d ),

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from which (8.6) is immediate. This bound also shows that lim supρ→∞ Φ(ρ)/ρ2 < ∞ if 0+ V (r)rd−1 dr < ∞. Let µ0 ,n denote the canonical Gibbs measure with no boundary particle. Lemma 8.6. For m = 1, 2, ..., 0 [U(q) + Bn + m − 1]m µ0 ,n (dq) ≤ − log Z ,n − n log ρ + (B + 1)n + m

m

,

[Λ( )]n

where B may be any constant such that U(q) ≥ −B%q. It follows in particular that as → ∞(F ) so that n/|Λ()| → ρ¯, 1 lim sup ρ) − ρ¯ log ρ¯ + ρ¯. (8.7) U(q)µ0 ,n (dq) ≤ Φ(¯ |Λ()| m

Proof. We use the identity [U + Bn + m − 1] = F (exp{U + Bn + m − 1}), where F (Y ) = (log Y )m . Since F (Y ) is concave for Y ≥ em−1 and log(|Λ()|n /n!) ≤ −n log ρ + n − 1, an application of Jensen’s inequality then immediately leads to the inequality of the lemma. It is incidentally veriﬁed in the proof of Theorem 2.2 that Φ is continuously diﬀerentiable. The method employed there may be extended to show Lipschitz continuity of P for a broad class of functions V . While the result has been established by Ruelle [8] and we need to require an additional condition ((8.8) below; see also Remark after the proof of the next theorem), our proof is simpler than that of [8]. Put χ(r) = [−rV (r)]+ ∨ [r2 V (r)]+ . Theorem 8.1. Let V satisfy the conditions stated at the beginning of this subsection. Further suppose that χ(r) ≤ ϕ(r)

for

r>1

and

χ(r) ≤ M V (r) + ϕ(r)

for

r ≤ 1,

(8.8)

where M is a constant and ϕ(r) is a positive continuous function of r ≥ 0 which is decreasing and integrable. Then Φ(ρ) is continuously diﬀerentiable; for any u > 0 there exists a constant L such that |P (ρ) − P (¯ ρ)| ≤ L|ρ − ρ¯| lim

sup

→∞(F ) n≤r|Λ( )|

[Λ( )]n

if

0 ≤ ρ, ρ¯ < u;

1 ψ11 (qi − qj )µ0 ,n (dq) − [P (ρ ) − ρ ] = 0 |Λ()| i,j(=)

(8.9) for each r > 0; and lim sup P (ρ)/Φ(ρ) < ∞. ρ→∞

(8.10)

1200

Kˆ ohei Uchiyama

Proof. Let us prove that A(ρ) :=

1 →∞(F ), ρ →ρ |Λ()| lim sup

[Λ( )]n i,j(=)

Ann. Henri Poincar´e

χ(|qi − qj |)µ0 ,n (dq)

(8.11)

is locally bounded on [0, ∞). To this end it suﬃces, according to (8.7), to show that for any conﬁguration q of n particles χ(|qi − qj |) ≤ C1 U(q) + C2 n (8.12) Vp (|qi − qj |) etc. We for some constants C1 and C2 . Let us write U (Vp ) (q) = observe that there exist constants A and B such that U (ϕ) ∨ U (V− ) ≤ BU (Vp ) + An and hence by (8.8) χ(|qi − qj |) ≤ M U (V+ ) + C(U (Vp ) + n) with another constant C. Thus (8.12) follows from the stability condition for Vs = V −Vp . Now recalling the arguments made in the proof of Theorem 2.2 and noticing χ11 ≤ χ we infer (8.9) and the continuity of the derivative Φ . The relation (8.10) is easily deduced from (8.7), (8.9), (8.12) and Lemma 8.5. Letting → ∞(F ) so that n/|Λ()| → ρ in (2.9) and (2.10) (with the factor of δ 2 /2 replaced by the integral on the left-hand side of (2.8)) after dividing each terms of them by |Λ()| we obtain 1 − (1 + δ)Φ(ρ/(1 + δ)) − Φ(ρ) − ρ log(1 + δ) + A(ρ)δ 2 + o(δ 2 ) 2 1 ≥ (1 − δ)Φ(ρ/(1 − δ)) − Φ(ρ) + ρ log(1 − δ) − A(ρ)δ 2 + o(δ 2 ) 2 with o(δ 2 ) locally uniform. Introducing ' = δρ we observe that as ' → 0 (with ρ ﬁxed), (1 + δ)Φ(ρ/(1 + δ)) + (1 − δ)Φ(ρ/(1 − δ)) = Φ(ρ + ') + Φ(ρ − ') + o('2 ); then that the inequality above may be rewritten in the form 1 1 [ρ + A(ρ) + o(1)] ≥ 2 [Φ(ρ + ') + Φ(ρ − ') − 2Φ(ρ)]. 2 ρ '

(8.13)

This bound implies, as is proved at the end of this proof, that 1 1 lim sup [Φ (ρ + h) − Φ (ρ)] ≤ 2 [ρ + A(ρ)] h ρ h→0 or, what amounts to the same thing, that 1 1 lim sup [P (ρ + h) − P (ρ)] ≤ 1 + A(ρ). h ρ h→0

(8.14)

Vol. 1, 2000

Pressure and Interacting Brownian Particles in Multi-Dimensions

1201

To complete the proof we must show that A(ρ) = O(ρ) as ρ ↓ 0. We apply (8.5), where we determine c by (2c + 2co )d = 1/ρ so that n ≤ m if ∗ is large and δ small, to have Φ(ρ) ≤ ρ log ρ + Cρ for ρ < 1. Combining this with (8.12) and (8.7) we conclude A(ρ) ≤ C ρ as required. It remains to deduce (8.14) from (8.13). Let f2 (ρ) denote the right-hand side ρ+2 ρ of (8.13). Put F2+ (ρ) = ρ [Φ (u) − Φ (ρ)]du and F2− (ρ) = ρ−2 [Φ (ρ) − Φ (u)]du. Then for any (large) positive integer n it holds that n−1 ' [Φ (ρ + n') − Φ (ρ)] = F2+ (ρ) + '2 f2 (ρ + k') + F2− (ρ + n'). k=1

We also have F2+ (ρ) + F2− (ρ + ') = '2 f2 (ρ) and F2± ≥ 0. For h > 0, take ' = h/n in the above, and you see h−1 [Φ (ρ + h) − Φ (ρ)] ≤ n−1 nk=0 f2 (ρ + kh/n). The same argument shows that the last inequality holds for negative h if h is small enough. Now it is clear that (8.13) implies (8.14) since A(ρ) is upper semi-continuous. Remark. While the second condition of (8.8) is not satisﬁed if V (r) diverges as r ↓ 0 to inﬁnity faster than any polynomial order, it may be replaced by the condition (1.1) (as well as (ii.b)) for the local boundedness of A(ρ) deﬁned by (8.11) (to this end employ a bound as given in Lemma 8.2; see [1] for V not of ﬁnite range), which implies (8.9) as well as the local Lipshitz continuity of P (ρ), but does not the bound (8.10) (in fact (8.10) is not in general valid without (8.8)).

References [1] R.L. Dobrushin and R.A. Minlos, Existence and continuity of pressure in classic statistical physics, Teorija Verojatn. i ee Prim. 12, 596–618 (1967). [2] M.Z. Guo, G.C. Papanicolaou, S.R.S. Varadhan, Non-linear diﬀusion limit for a system of nearest neighbor interactions, Comm. Math. Phys. 118, 31–59 (1988). [3] C. Kipnis, C. Landim, Scaling limits of interacting particle systems. GMW 320, Springer-Verlag, Berlin-Heidelberg-New York (1999). [4] C. Kipnis, S.R.S. Varadhan, Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions, Comm. Math. Phys. 104, 1–19 (1986). [5] E. Presutti, A mechanical deﬁnition of the thermodynamic pressure, J. Stat. Phys. 13, 301–314 (1975). [6] F. Rezakhanlou, F., Hydrodynamic limit for a system with ﬁnite range interactions, Comm. Math. Phys. 129, 448–480 (1990). [7] H. Rost, Diﬀusion de sph`eres dures dans la droite r´eelle : comportement macroscopique et ´equilibre local, L.N.Math. 1059, 127–143 (1984).

1202

Kˆ ohei Uchiyama

Ann. Henri Poincar´e

[8] D. Ruelle, Superstable interactions in classical statistical mechanics, Comm. Math. Phys. 18, 127–159 (1970). [9] S. Saks, Theory of the integral. 2nd revised ed. Dover, New York. (1964) (reprint of 2nd revised ed., 1937 [10] Spohn, Large scale dynamics of interacting particles. Texts and Monographs in Physics, Springer-Verlag, Berlin-Heidelberg-New York (1991). [11] E.M. Stein, G. Weis, G., Fourier analysis on Euclidean spaces. Princeton Univ. Press, Princeton (1971). [12] Y. Suzuki, K. Uchiyama, Hydrodynamic limit for a spin system on a multidimensional lattice Probab. Theory Relat. Fields, 95, 47–74 (1993). [13] D.W. Stroock, S.R.S. Varadhan, Multidimensional diﬀusion processes. Springer Verlag, New York (1979). [14] K. Uchiyama, Scaling limits of interacting diﬀusions with arbitrary initial distributions Probab. Theory Relat. Fields, 99, 97–110 (1994). [15] K. Uchiyama, Scaling limit for a mechanical system of interacting particles Comm. Math. Phys. 177, 103–128 (1996). [16] S.R.S Varadhan, Scaling limit for interacting diﬀusions Comm. Math. Phys., 135, 313–353 (1991). [17] S.R.S. Varadhan, Non-linear diﬀusion limit for a system with nearest neighbor interactions-II Asymptotic problems in probability theory: stochastic models and diﬀusions on fractals (eds. Elworthy and Ikeda), Longman (1993) 75– 128. [18] H.T. Yau, Relative entropy and the hydrodynamics of Ginzburg-Landau models Lett. Math. Phys. 22, 63–80 (1991). K. Uchiyama Department of Mathematics Tokyo Institute of Technology Oh-okayama Meguro-ku, Tokyo 152-8551 Research partially supported by Japan Society for the Promotion of Science e-mail : [email protected] Communicated by Michael Aizenman submitted 21/06/99, accepted 25/07/00

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