Commun. Math. Phys. 308, 1–21 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1316-8
Communications in
Mathematical Physics
On the Correlation Function of the Characteristic Polynomials of the Hermitian Wigner Ensemble Tatyana Shcherbina Institute for Low Temperature Physics Ukr.Ac.Sci., 47 Lenin Ave., 61103 Kharkov, Ukraine. E-mail:
[email protected] Received: 7 July 2010 / Accepted: 19 March 2011 Published online: 27 August 2011 – © Springer-Verlag 2011
Abstract: We consider the asymptotic of the correlation functions of the characteristic polynomials of the hermitian Wigner matrices Hn = n −1/2 Wn . We show that for the correlation function of any even order the asymptotic coincides with this for the Gaussian Unitary Ensemble up to a factor, depending only on the fourth moment of the common probability law of entries W jk , W jk , i.e. that the higher moments do not contribute to the above limit. 1. Introduction Characteristic polynomials of random matrices have been actively studied in recent years. The interest was initially stimulated by the similarity between the asymptotic behavior of the moments of characteristic polynomials of a random matrix from the Circular Unitary Ensemble and the moments of the Riemann ζ -function along its critical line (see [8]). But with the emerging connections to the quantum chaos, integrable systems, combinatorics, representation theory and others, it has become apparent that the characteristic polynomials of random matrices are also of independent interest. This motivates the asymptotic studies of the moments of characteristic polynomials for other random matrix ensembles (see e.g. [3,10]). In this paper we consider the hermitian Wigner ensemble with symmetric entries distribution, i.e. hermitian n × n random matrices, Hn = n −1/2 Wn
(1)
with independent (modulo symmetry) and identically distributed entries W jk and W jk such that E{W jk } = E{(W jk )2 } = 0, E{2l+1 W jk } = E{2l+1 W jk } = 0,
E{|W jk |2 } = 1, j, k = 1, .., n, l, n ∈ N.
(2)
2
T. Shcherbina (n)
(n)
Denote by λ1 , . . . , λn the eigenvalues of Hn and define their Normalized Counting Measure (NCM) as Nn () = {λ(n) j ∈ , j = 1, .., n}/n,
Nn (R) = 1,
(3)
where is an arbitrary interval of the real axis. The global regime of the random matrix theory, centered around the weak convergence of the NCM of eigenvalues, is well-studied for many ensembles. It is shown that Nn converges weakly to a non-random limiting measure N known as the Integrated Density of States (IDS). The IDS is normalized to unity and is absolutely continuous in many cases N (R) = 1, N () = ρ(λ)d λ. (4)
The non-negative function ρ in (4) is called the limiting density of states of the ensemble. In the case of the hermitian Wigner ensemble ρ is given by the well-known semicircle law (see, e.g.,[11]): 1 ρsc (λ) = 4 − λ2 . (5) 2π The mixed moments (or the correlation functions) of characteristic polynomials are F2m () =
2m
Hn j=1
det(λ j − H )Pn (d Hn ),
(6)
where Hn is the space of hermitian n × n matrices, d Hn =
n
d Hjj
j=1
H j,k H j,k
(7)
1≤ j
is the standard Lebesgues measure on Hn , Pn (d Hn ) is a probability law of the n ×n random matrix Hn , and = {λ j }2m j=1 are real or complex parameters that may depend on n. We are interested in the asymptotic behavior of (6) for matrices (1) as n → ∞ and λ j = λ0 +
ξj , nρsc (λ0 )
j = 1, .., 2m,
where λ0 ∈ (−2, 2), ρsc is defined in (5) and ξ = {ξ j }2m j=1 are real parameters varying in a compact set K ⊂ R. In the case of the hermitian matrix model, i.e. the matrices with Pn (d Hn ) = Z n−1 e−n tr V (Hn ) d Hn , where V is bounded from below and grows sufficiently fast at infinity, the asymptotic behavior of (6) is known. Using the method of orthogonal polynomials, it was shown (see [2,13]) that 1 (nρ(λ0
2 ))m
F2m 0 + ξ /(nρ(λ0 ))
emnV (λ0 )+αV (λ0 ) = Cn
2m
j=1 ξ j
det
sin(π(ξi −ξm+ j )) m π(ξi −ξm+ j ) i, j=1
(ξ1 , .., ξm )(ξm+1 , .., ξ2m )
(1 + o(1)),
(8)
Characteristic Polynomials of the Wigner Ensemble
3
as n → ∞, where 0 = (λ0 , . . . , λ0 ) ∈ R2m , αV (λ) =
V (λ) , 2ρ(λ)
(9)
ρ is a density of (4), λ0 is such that ρ(λ0 ) > 0 and (x1 , . . . , xm ) is the Vandermonde determinants of x1 , . . . , xm . Unfortunately, the method of orthogonal polynomials can not be applied to the general case of the hermitian Wigner Ensembles (1)–(2). Thus, to find the asymptotic behavior of (6) other methods should be used. In [7] Gotze and Kosters use the exponential generating function to study the second moment, i.e. the case m = 1 in (6). It was shown that 1 F2 (λ0 + ξ1 /(nρsc (λ0 )), λ0 + ξ2 /(nρsc (λ0 ))) nρ(λ0 ) sin(π(ξ1 − ξ2 )) = 2π e−n exp{nλ20 /2 + α(λ0 )(ξ1 + ξ2 ) + κ4 } (1 + o(1)), π(ξ1 − ξ2 ) where α(λ) =
λ , κ4 = μ4 − 3/4, 2ρsc (λ)
(10)
and μ4 is the fourth moment of the common probability law of W jk , W jk . In this paper we consider the general case m ≥ 1 of (6) for the random matrices (1). Following [7], define 1 ξ ξ F2 λ0 + , λ0 + D (n) (ξ ) = nρ(λ0 ) nρsc (λ0 ) nρsc (λ0 )
2 nλ0 + 2α(λ0 )ξ + κ4 (1 + o(1)). = 2π e−n exp (11) 2 The main result of the paper is Theorem 1. Let the entries W jk , W jk of matrices (1) have a symmetric probability distribution with finite first 4m moments. Then we have for m ≥ 1, lim
n→∞
=
(nρsc (λ0 ))
m2
1 2m l=1
D (n) (ξl )
F2m 0 + ξ /(nρsc (λ0 ))
exp{m(m − 1)κ4 (λ20 − 2)2 /2} sin(π(ξi − ξm+ j )) m det , (ξ1 , . . ., ξm ) (ξm+1 , . . ., ξ2m ) π(ξi − ξm+ j ) i, j=1
(12)
where F2m and ρsc (λ) are defined in (6) and (5), 0 = (λ0 , . . . , λ0 ) ∈ R2m , λ0 ∈ (−2, 2), ξ = {ξ j }2m j=1 , and α(λ) and κ4 are defined in (10). The theorem shows that the above limits for the mixed moments of characteristic polynomials for random matrices (1) coincide with those for the Gaussian Unitary Ensemble corresponding to V = λ2 /2 and ρ = ρsc in (8) up to a factor, depending only on the fourth moment of the common probability law of entries W jk , W jk , i.e. that the higher moments of the law do not contribute to the above limit. This is a manifestation of the universality, that can be compared with the universality of the local bulk regime for Wigner matrices (see [5] and references therein).
4
T. Shcherbina
The paper is organized as follows. In Sect. 2 we obtain a convenient integral representation for F2m in the case of symmetric probability distribution of entries with 4m finite moments by using the integration over the Grassmann variables and Harish Chandra/Itzykson-Zuber formula for integrals over the unitary group. In Sect. 3 we prove Theorem 1 by applying the steepest descent method to the integral representation. We denote by C, C1 , etc. and c, c1 , etc. various n-independent quantities below, which can be different in different formulas. Integrals without limits denote the integrals over the whole real axis. 2. The Integral Representation In this section we obtain the integral representation for the mixed moments F2m (6) of the characteristic polynomials. To this end we use the integration over the Grassmann variables. The integration was introduced by Berezin and widely used in the physics literature (see [1] and [4]). For the reader’s convenience we give a brief outline of this technique here. 2.1. Grassmann integration. Let us consider the two sets of formal variables {ψ j }nj=1 , {ψ j }nj=1 , which satisfy the anticommutation conditions ψ j ψk + ψk ψ j = ψ j ψk + ψk ψ j = ψ j ψ k + ψ k ψ j = 0,
j, k = 1, .., n.
These two sets of variables {ψ j }nj=1 and {ψ j }nj=1 generate the Grassmann algebra A. Taking into account that ψ 2j = 0, we have that all elements of A are polynomials of {ψ j }nj=1 and {ψ j }nj=1 . We can also define functions of the Grassmann variables. Let χ be an element of A, which absolute term is a. For any analytical function f we mean by f (χ ) the element of A obtained by substituting χ − a in the Taylor series of f at the point a. Since χ is a polynomial of {ψ j }nj=1 , {ψ j }nj=1 , which absolute term is a, there exists such l that (χ − a)l = 0, and hence the series terminates after a finite number of terms and so f (χ ) ∈ A. Following Berezin [1], we define the operation of integration with respect to the anticommuting variables in a formal way: d ψ j = d ψ j = 0, ψ j d ψ j = ψ j d ψ j = 1. (13) This definition can be extended on the general element of A by the linearity. A multiple integral is defined to be a repeated integral. The “differentials” d ψ j and d ψ k anticommute with each other and with the variables ψ j and ψ k . Thus, if f (χ1 , . . . , χm ) = a0 +
m j1 =1
a j1 χ j1 +
a j1 j2 χ j1 χ j2 + · · · + a1,2,...,m χ1 . . . χm ,
j1 < j2
where {χ j }mj=1 are independent Grassmann variables, then f (χ1 , . . . , χm )d χm . . . d χ1 = a1,2,...,m . Let A be an ordinary hermitian matrix. The following Gaussian integral is wellknown:
Characteristic Polynomials of the Wigner Ensemble
5
n n
d z j d z j 1 = . exp − A j,k z j z k π det A j,k=1
(14)
j=1
One of the important formulas of the Grassmann variables theory is the analog of (14) for the Grassmann algebra (see [1]): n n
exp A j,k ψ j ψk d ψ j d ψ j = det A. (15) j,k=1
j=1
Besides, we have q n n
ψ l p ψs p exp A j,k ψ j ψk d ψ j d ψ j = det Al1 ,..,lq ;s1 ,..,sq , p=1
j,k=1
(16)
j=1
where Al1 ,..,lq ;s1 ,..,sq is a (n − q) × (n − q) minor of the matrix A without rows l1 , .., lq and columns s1 , .., sq .
2.2. Asymptotic integral representation for F2 . In this subsection we obtain the asymptotic integral representation of (6) for m = 1 by using the Grassmann integrals. This formula was obtained in [7] by using another method. We give here a detailed proof to show the basic ingredients of our technique that will be elaborated in the next subsection to obtain the asymptotic integral representation of (6) for m > 1. Denote 2 D2 = D (n) (ξl ), (17) l=1
where D (n) (ξ ) is defined in (11). In follows from (11) and the steepest descent method that 2 n 4 − λ 0 α(λ )ξ +κ /2 4 D (n) (ξ ) = e 0 2 n 2 (18) × (t − iλ0 /2)n e− 2 (t+iλ0 /2) d t (1 + o(1)) as n → ∞. Using (15), we obtain from (6), D2−1 F2 () = D2−1 E
=
D2−1 E
e
2
s=1 λs
e n
2 l=1
n
j,k=1 (λl −H ) j,k ψ jl ψkl
p=1 ψ ps ψ ps
d ψ qr d ψqr
r =1 q=1
exp
n 2
−
2 w j,k √ (ψ jl ψkl + ψ kl ψ jl ) n j
n 2 2 n iw j,k wjj − √ (ψ jl ψkl − ψ kl ψ jl ) − ψ jl ψ jl d ψ qr d ψqr , (19) √ n n j=1
l=1
r =1 q=1
6
T. Shcherbina
2 where {ψ jl }nj,l=1 are the Grassmann variables (n variables for each determinant in (6)). Denote
χ +j,k = χ +j, j =
2 2 (ψ jl ψkl + ψ kl ψ jl ), χ − = (ψ jl ψkl − ψ kl ψ jl ), j = k, j,k l=1 2
l=1
ψ jl ψ jl ,
j, k = 1, .., n.
(20)
l=1 2 s Using that (χ ± j,k ) = 0 for s > 4, j, k = 1, .., n (since ψ js = ψ js = 0 for any j = 1, .., n, s = 1, 2), we expand the second exponent under the integral in (19) into the series and integrate with respect to the measure (2). We get then n (χ +j,k )2 2 μ4 + 4 1+ + (χ ) D2−1 F2 () = D2−1 e s=1 λs p=1 ψ ps ψ ps 4n 4!n 2 j,k j
j
r =1 q=1
j=1
where μ4 is 4th moment of the common probability law of W jk , W jk of (2). Writing 1 ± 2 μ4 1 ± 2 κ4 ± 4 ± 4 1± (χ ) + (χ ) = exp ± (χ j,k ) + (χ ) , j = k, 4n j,k 4!n 2 j,k 4n 4!n 2 j,k 1 + 2 1 + 2 (χ j, j ) = exp (χ ) , j, k = 1, .., n, 1+ 2n 2n j, j where κ4 is defined in (10), and using (21), we get D2−1 F2 () = D2−1
2
e
s=1 λs
n
κ4 1 p=1 ψ ps ψ ps − 2n σ1 + n 2 σ2
2 n
d ψ qr d ψqr ,
(22)
ψ ks ψkl ,
(23)
r =1 q=1
where 1 + 2 2 (χ j,k ) − (χ − − ) (χ +j, j )2 j,k 2 n
σ1 = −
j
=
2 n l=1
j=1
j=1
ψ jl ψ jl
2
+2
n
1≤l<s≤2
j=1
ψ jl ψ js ·
n k=1
2 1 + 4 4 (χ j,k ) + (χ − = ψ j1 ψ j2 ψ j1 ψ j2 . j,k ) 4! n
σ2 =
j
j=1
Now we use the formulas π 2 exp{ab } = exp{−ax 2 − 2abx}d x, a π exp{abc} = exp{−auu − abu − acu}d ud u, a
(24)
Characteristic Polynomials of the Wigner Ensemble
7
where b, c are complex numbers or even Grassmann variables (i.e. sums of the products of even number of Grassmann variables), and a is a positive number. For the case of even Grassmann variables this formula can be obtained by expanding the exponent into the series and integrating each term by using (13). We obtain 1 n2 exp − σ1 = exp − 2n 2π 2 H2 n 2 exp iτ p ψ j p ψ j p + j=1
p=1
n 2 τq + 2 2 2
q=1
u ab u ab
1≤a
iu cd ψ jc ψ jd + iu cd ψ jd ψ jc d Q,
(25)
1≤c
where
Q=
τ1 u 12
u 12 τ2
,
(26)
H2 is the space of 2 × 2 hermitian matrices and d Q is given in (7) for n = 2. Write the formula (cf. (24)) n κ
|κ | 2 p ε(κ4 ) 2 4 4 exp 2 σ2 = exp ψ j1 ψ j2 ψ j1 ψ j2 d p (27) e−|κ4 | p n π n j=1
with σ2 of (23) and ε(x) =
x, x > 0, −i x, x < 0.
(28)
Substituting (25)–(27) in (22) and using (15)–(16) for integration over the Grassmann variables, we obtain D2−1 F2 () = Z2 dp
H2
n
d Qe− 2 tr Q
2 −|κ
4| p
2
2 p ε(κ4 ) n det(Q − i) + , n
(29)
where Q is defined in (26) and = diag {λ1 , λ2 },
Z2 =
(−1)n n 2 D2−1 . 2π 2 π |κ4 |−1
(30)
Recall that we are interested in = 0 + ξ /nρsc (λ0 ), where 0 = diag{λ0 , λ0 } and ξ = diag{ξ1 , ξ2 }. Change variables to τ j − iλ0 /2 − iξ j /nρsc (λ0 ) → τ j , j = 1, 2 and note that after the change we can move the integration with respect to τ j from line z = λ0 /2 + ξ j /nρsc (λ0 ) back to the real axis. Indeed, consider the contour C j R , which is the rectangle with vertices at (−R, 0), (−R, λ0 /2 + ξ j /n), (R, λ0 /2 + ξ j /n) and (R, 0). Since the integrand in (29) is analytic in {τ j }2j=1 , the integral with respect to τ j of this function over C j R is equal to 0. Besides, the integral over the segments of lines z = ±R tends to 0 as R → ∞, since the integrand in (29) is a polynomial of τ j multiplied by exp{−nτ 2j /2}. Thus, letting R → ∞, we obtain that the integral with
8
T. Shcherbina
respect to τ j over the line z = λ0 /2 + ξ j /n is equal to the integral over the real axis. We obtain i 2 i − n tr Q+ 2 0 − ρsc i(λ ) tr Q+ 2 0 ξ 0 e 2 D2−1 F2 () = Z 2 H2
i0 2 p ε(κ4 ) n det(Q − )+ dp d Q 2 n i0 i tr Q + ξ d Q exp − = Z 2 dp ρsc (λ0 ) 2 H2 n ξ2
1 2 p ε(κ4 ) −|κ4 | p 2 − tr μ (Q) 1 + , n 2n ρsc (λ0 )2 ndet(Q − i0 /2) ×e
1 −|κ4 | p 2 − 2n tr
ξ2 ρsc (λ0 )2
(31)
where Q is again a hermitian matrix (see (26)) and n
μn (Q) = det n (Q − i0 /2)e− 2 tr (Q+i0 /2) . 2
(32)
Let q1 , q2 be the eigenvalues of Q. Set n = {(Q, p) : a ≤ |ql − iλ0 /2| ≤ A, l = 1, 2, | p| ≤ log n}, nQ = {Q ∈ H2 : a ≤ |ql − iλ0 /2| ≤ A},
(33)
for sufficiently small a and sufficiently big A (note that if |λ0 | ≥ δ, then |ql − iλ0 /2| ≥ δ 2 /4 and we can omit the first inequality in (33)). The integral in (31) over the 2 domain maxl=1,2 |ql | ≥ A is O(e−n A /4 ), A → ∞ and the integral over the domain −1 minl=1,2 |ql | ≤ a is O(e−n log a ), a → 0. If a ≤ |ql − iλ0 /2| ≤ A and | p| ≥ log n, then according to (17), (18), and (26), the corresponding integral is bounded by 2 2 Z2 |μn (Q)|d Q (1 + C p/n)n e−|κ4 | p dp = O(e−C log n ), (34) nQ
| p|≥log n
and we can write D2−1 F2 () = Z 2 ×e
n
μn (Q)e
1 −|κ4 | p 2 − 2n tr
−i tr(Q+
i0 ξ 2 ) ρsc (λ0 )
(1 + f n (det(Q − i0 /2), p))
i ξ2 +2 p ε(κ4 ) det−1 (Q− 2 0 ) ρsc (λ0 )2
d p d Q + O(e−c log n ), (35) 2
where f n (det(Q − i0 /2), p) =e
−2 p ε(κ4 ) det−1 (Q−
i0 2 )
1+
2 p ε(κ4 ) n det(Q −
i0 2 )
n − 1.
(36)
If we write the term in the brackets above as (1 + T )n = exp{n log(1 + T )} and expand the exponent with respect to T , we obtain on n , | f n (det(Q − i0 /2), p)| ≤
logk n , n
(37)
Characteristic Polynomials of the Wigner Ensemble
9
where k is independent of n. Note also that f n is an analytic function in p and entries of Q. Besides, it is easy to check that i0 2 −1 e−|κ4 | p +2 p ε(κ4 ) det (Q− 2 ) dp I := | p|≤log n
=
π κ4 det−2 (Q−i0 /2) 2 e + O(e−c log n ), |κ4 |
n (see (33)). Thus, (35) yields hence |I | > C3 > 0 on n 2 D2−1 D2−1 F2 () = μn (Q) exp −itr(Q + i0 /2) ξ /ρsc (λ0 ) n 2 (−1) 2π nQ
+ κ4 det −2 (Q − i0 /2) 1 + f n(1) (det(Q − i0 /2)) d Q +O(e−c log n ), 2
(38)
where f n(1) (det (Q
×
− i0 /2)) = e
| p|≤log n
e−|κ4
| p 2 +2 p ε(κ
ξ2 ρsc (λ0 )2
1 − 2n tr
4
) det−1 (Q−i
−1+ I 0 /2)
1 −1 − 2n tr
e
ξ2 ρsc (λ0 )2
f n (det(Q − i0 /2), p)dp.
(39)
nQ and It follows from (37), that f n(1) (det(Q − i0 /2)) is analytic in elements of Q on | f n(1) (det(Q − i0 /2))| ≤ logk n/n,
(40)
where k is independent of n. Let us change variables to Q = U ∗ T U , where U is a unitary matrix and T = diag{t1 , t2 }. Then d Q of (7) for n = 2 becomes π (t1 − t2 )2 d t1 d t2 d μ(U ), 2 where μ(U ) is the normalized to unity Haar measure on the unitary group U (2) (see e.g. [9], Sect. 3.3). Since functions det(Q − i0 /2) and tr (Q + i0 /2)2 are unitary invariant, (38) implies D2−1 F2 () =
n 2 (−1)n 4π D2 ×e
− n2
2
U (2)
dμ(U )
iλ0 2 s=1 (ts + 2 ) −
tr
L aA ×L aA
d t1 d t2
2
(tl − iλ0 /2)n
l=1
i iλ ξ U ∗ (T + 2 0 )U ρsci(λ +κ4 r2=1 (tr − 20 )−2 0)
i0 2 )) + O(e−c log n ), ×(t1 − t2 )2 1 + f n(1) (det(T − 2
(41)
where L aA = {t ∈ R : a ≤ |t − iλ0 /2| ≤ A}.
(42)
The integral over the unitary group U (2) can be computed using the well-known Harish Chandra/Itsykson-Zuber formula (see e.g. [9], App. 5)
10
T. Shcherbina
n Proposition 1. Let A be the normal n × n matrix with distinct eigenvalues {ai }i=1 and B = diag{b1 , . . . , bn }. Then
1 exp − tr(A − U ∗ BU )2 2 (B) f (B)d U d B 2 U (n) n
(B) 1 f (b1 , . . . , bn )d B, j! exp − tr(a j − b j )2 (43) = 2 (A) j=1 where f (B) is any symmetric function of {b j }nj=1 , d B = nj=1 d b j and (A), (B) n , {b }n are the Vandermonde determinants for the eigenvalues {ai }i=1 i i=1 of A and B. We obtain finally from (41) 2 iλ0 2 n 2 iρsc (λ0 )n 2 D2−1 F2 () = (tl − iλ0 /2)n e− 2 l=1 (tl + 2 ) n 2π(−1) D2 L aA ×L aA −
l=1
2
iξl iλ0 −2 −2 l=1 ρsc (λ0 ) (tl + 2 )+κ4 (t1 −iλ0 /2) (t2 −iλ0 /2)
×e t 1 − t2 2 1 + f n(2) (T ) d t1 d t2 + O(e−c log n ), × ξ1 − ξ2
(44)
where L aA is defined in (42) and f n(2) (T ) = f n(1) (det(T −i0 /2)) is analytic and bounded by logk n/n if tl ∈ L aA , l = 1, 2. This asymptotic integral representation is used in Sect. 3 for the proof of Theorem 1 for m = 1. 2.3. Asymptotic integral representation for F2m . Denote 2m D2m = D (n) (ξl ),
(45)
l=1
where D (n) (ξ ) is defined in (11). Using (15), we obtain from (6) (cf. (19)) −1 −1 D2m F2m () = D2m E
−1 = D2m E
×e
−
2m n l=1
e
j,k=1 (λl −H ) j,k ψ jl ψkl
n 2m
d ψ qr d ψqr
r =1 q=1
2m
e
j
s=1 λs
w j,k 2m √ p=1 ψ ps ψ ps − j
n
n
iw j,k 2m w j j 2m n 2m √ √ l=1 (ψ jl ψkl −ψ kl ψ jl )− j=1 l=1 ψ jl ψ jl n n
n
d ψ qr d ψqr ,
r =1 q=1
(46) 2m {ψ jl }n, j,l=1
where are the Grassmann variables (n variables for each determinant). As in (20) we denote χ +j,k = χ +j, j
=
2m 2m (ψ jl ψkl + ψ kl ψ jl ), χ − = (ψ jl ψkl − ψ kl ψ jl ), j,k l=1 2m l=1
l=1
ψ jl ψ jl ,
j, k = 1, . . . , n.
j = k, (47)
Characteristic Polynomials of the Wigner Ensemble
11
2 s Using that (χ ± j,k ) = 0 for s > 4m, j, k = 1, .., 2m (since ψ jl = ψ jl = 0 for any j = 1, .., n, l = 1, .., 2m), we expand the exponent under the integral in (46) into the series and integrate with respect to the measure (2). We get then similarly to (22), n 2m n 2m κ2 p 2m 1 −1 −1 D2m F2m () = D2m e s=1 λs k=1 ψ ks ψks − 2n σ1 + p=2 n p σ p d ψ qr d ψqr , 2
r =1 q=1
(48) where κ2 p are cumulants of the entries w jk , w jk of (2), i.e. the coefficients in the expansion s κq (it)q + o(t s ), t → 0. l(t) := log E{eitw jk } = q! q=0
The function σ1 in (48) is the same as in (23) (but with χ ± j,k of (47) and the sums from 1 to 2m instead of from 1 to 2), n 1 + 4 2 + 4 4 σ2 = ((χ j,k ) + (χ − ) ) + (χ j, j ) j,k 4! 4! j
=2
+
j=1
n
ψ jl1 ψ jl2 ψ js1 ψ js2 ·
l1
1 4
n
ψkl1 ψkl2 ψks1 ψks2
k=1 n
ψ jl1 ψ js1 ψ jl2 ψ js2 ·
l1 =s1 ,l2 =s2 j=1
ψkl1 ψks1 ψ kl2 ψ ks2 ,
(49)
k=1
and we have for p ≥ 3, 1 + 2p 2 + 2p 2p σp = ((χ j,k ) + (−1) p (χ − (χ j, j ) j,k ) ) + (2 p)! (2 p)! j
2m
=
j
[ 2p ]
( p)
cs,l
n
l1 ,..,l2 p =1 s=0
×
n
ψ jl1 ..ψ jl p+2s ψ jl p+2s+1 ..ψ jl2 p
j=1
ψkl1 ..ψkl p+2s ψ kl p+2s+1 ..ψ kl2 p ,
k=1 ( p)
where cs,l are n-independent positive coefficients and l = (l1 , . . ., l2 p ). Using (24), we obtain −|κ | 2 w w + v v −2 4 l l s s l l s s l l s s l l s s l<s l1 =s1 ,l2 =s2 1 2 1 2 1 2 1 2 12 1 2 12 1 2 en κ4 σ2 = C2 e ×
n
e
ε(κ4 ) 2n
a1 =b1 ,a2 =b2
va1 a2 b1 b2 ψ ja1 ψ jb1 ψ ja2 ψ jb2 +v a1 a2 b1 b2 ψ ja1 ψ jb1 ψ ja2 ψ jb2
j=1
×
n
e
2ε(κ4 ) c
wc1 c2 d1 d2 ψ jc1 ψ jc2 ψ jd1 ψ jd2+w c1 c2 d1 d2 ψ jc1 ψ jc2 ψ jd1 ψ jd2
d W d V,
j=1
(50)
12
T. Shcherbina
where c < d and l < s mean that c1 < c2 < d1 < d2 , l1 < l2 < s1 < s2 and dW =
l<s
dV =
d wl1 l2 s1 s2 d wl1 l2 s1 s2 ,
d vl1 l2 s1 s2 d vl1 l2 s1 s2 ,
l1 =s1 ,l2 =s2
C2
=
π 2|κ4 |
−(2m ) 4
π |κ4 |
−(2m)2 (2m−1)2
(51)
,
and for p ≥ 3,
exp
×
κ
2p σp np
n
=
C p
exp
− |κ2 p |
2m
p
[2]
r l,s rl,s
l1 ,..,l2 p =1 s=0
exp
j=1
ε(κ2 p ) n p/2
2m
[ 2p ]
( p) (cl,q )1/2 (rl,q ψ jl1 ..ψ jl p+2q ψ jl p+2q+1 ..ψ jl2 p
l1 ,..,l2 p =1 q=0
+r l,q ψ jl1 ..ψ jl p+2q ψ jl p+2q+1 ..ψ jl2 p ) d R
(52)
with l = (l1 , . . . , l2 p ) and dR =
2m
p
[2]
C p
d rl,s d rl,s ,
=
l1 ,..,l2 p =1 s=0
π |κ2 p |
−[ p ](2m)2 p 2
,
p ≥ 3.
(53)
Substituting (50)–(52) and (25) (with sums from 1 to 2m) to (48), using (15)–(16), and integrating over the Grassmann variables in (48), we get −1 D2m F2m () = Zm
H2m
dQ
n
d V d R d W e− 2 tr Q νn (v, w, r )n (i Q + , v, w, r ), 2
(54)
where H2m is the space of 2m × 2m hermitian matrices, v w rp r
= {va1 a2 b1 b2 |a1 = b1 , a2 = b2 , a1 , a2 , b1 , b2 = 1, .., 2m}, = {wa1 a2 b1 b2 |a1 < a2 < b1 < b2 , a1 , a2 , b1 , b2 = 1, .., 2m}, = {rl,s |l1 , .., l2 p = 1, .., 2m, s = 0, .., [ p/2]}, = (r3 , . . . , r2m ),
(55)
and 2m
|κ2 p |r p r p . νn (v, w, r ) = exp − |κ4 |vv − 2|κ4 |ww − p=3
(56)
Characteristic Polynomials of the Wigner Ensemble
13
d Q, d V, d R, and d W are defined in (7) for n = 2m, (51), and (53), and ⎛
τ1 u 12 u 13 ..
u 12 τ2 u 23 ..
u 13 u 23 τ3 ..
⎜ ⎜ ⎜ Q=⎜ ⎜ ⎝u 1,2m−1 u 2,2m−1 u 3,2m−1 u 2m,1 u 2m,2 u 2m,3
⎞ .. u 1,2m−1 u 1,2m .. u 2,2m−1 u 2,2m ⎟ ⎟ .. u 3,2m−1 u 3,2m ⎟ ⎟ .. .. .. ⎟ .. τ2m−1 u 2m,2m−1 ⎠ .. u 2m−1,2m τ2m
(57)
is obviously hermitian. We denote also = diag{λ1 , λ2 , . . . , λ2m },
Zm =
2m 2m 2 −1 n D2m C p . 2 2m π 2m p=2
(58)
According to (15)–(16), (i Q+, v, w, r ) in (54) is a polynomial of the entries of i Q+ and of {vl1 l2 s1 s2 /n}, {wl1 l2 s1 s2 /n}, {rl1 ,..,l2 p ,s /n p/2 } with n-independent coefficients and degree at most 2m. Besides, possesses the properties: 1. The degree of each variable in (i Q + , v, w, r ) is at most one. 2. (i Q + , v, w, r ) does not contain terms C(i Q + )wl1 l2 s1 s2 /n or C(i Q +)wl1 l2 s1 s2 /n, since the terms ψ jl1 ψ jl2 ψ js1 ψ js2 or ψ jl1 ψ jl2 ψ js1 ψ js2 cannot 2m be completed to l=1 ψ jl ψ jl only by terms ψ jl ψ js . 3. (i Q + , v, w, r ) can be written as (i Q +, v, w, r ) = det(i Q + ) −
2ε(κ4 ) σ1 + f n (i Q + , v/n, w/n, r p /n p/2 ), n
(59)
where f n (i Q + , v/n, w/n, r p /n p/2 ) contains all the terms of (i Q + , v, w, r ) which are O(n −3/2 ) as n → ∞ but Q, v, w, r are fixed, and σ1 contains linear with respect to v terms. In view of (16),
σ1 =
(vl1 l2 s1 s2 ql1 ,s1 ,l2 ,s2 + vl1 l2 s1 s2 ql2 ,s2 ,l1 ,s1 ),
(60)
l1 =s1 ,l2 =s2
where qs,l, p,r is (2m − 2) × (2m − 2) minor of the matrix i Q + without rows with numbers s and l and columns with numbers p and r . ξ /nρsc (λ0 ), where 0 = diag{λ0 , .., λ0 } and Recall that we are interested in = 0 + ξ = diag{ξ1 , . . . , ξ2m }. Shift now τ j − iλ0 /2 − iξ j /nρsc (λ0 ) → τ j , j = 1, .., 2m to obtain in new variables (cf. (31)) −1 D2m F2m () = Z m
×e
H2m
dQ
− n2 tr (Q+
νn (v, w, r )n (i Q + 0 /2, v, w, r )
i0 2 2 ) −i
ξ2 sc (λ0 )2
tr (Q+ i20 )ξ /ρsc (λ0 )− 2n1 tr ρ
d V d R d W,
(61)
14
T. Shcherbina
where Q is the hermitian matrix of (57) and d Q, d V, d R and d W are defined in (7) for n = 2m, (51), and (53). We have in view of property (1), v 0 llss , v, w, r )| ≤ 1+C 12 1 2 (1 + C|(i Q + 0 /2)qs |) |(i Q + 2 n q,s l =s
×
a1
2m w aabb 1+C 1 2 1 2 n
2m
[ 2p ]
r l,s 1 + C p/2 n
(62)
p=3 l1 ,..,l2 p =1 s=0
with n-independent C and where l = s means l1 = s1 , l2 = s2 , and (59) yields v llss 1 + C(Q) 1 2 1 2 |(i Q + 0 , v, w, r )| ≤ |det(i Q + 0 /2)| n l1 =s1 ,l2 =s2
×
l1
w 2m llss 1 + C(Q) 1 2 1 2 n
2m
p
[2] r l,s 1 + C(Q) p/2 . n
(63)
p=3 l1 ,..,l2 p =1 s=0
2m are the Here C(Q) is bounded if a ≤ |ql − iλ0 /2| ≤ A, l = 1, .., 2m, and {ql }l=1 2 eigenvalues of Q. Note that if |λ0 | > δ > 0, then |ql − iλ0 /2| ≥ δ everywhere. Denote
n = {(Q, v, w, r ) : a ≤ |qs − iλ0 /2| ≤ A, |vl1 l2 s1 s2 | ≤ log n, |wl1 l2 s1 s2 | ≤ log n, |rl,s | ≤ log n}, nQ
(64)
= {Q ∈ H2m : a ≤ |qs − iλ0 /2| ≤ A, s = 1, .., 2m}.
According to (62), the integral over the domain maxl=1,..,2m |ql − iλ0 /2| ≥ A in (61) 2 is O(e−n A /4 ), A → ∞ and the integral over the domain minl=1,..,2m |ql − iλ0 /2| ≤ a −1 is O(e−n log a ), a → 0. Moreover, the bound (63) implies that this integral over the domain, where the absolute value of at least one of {vl1 l2 s1 s2 }, {wl1 l2 s1 s2 } or {rl,s } is 2 greater than log n but a ≤ |qs − iλ0 /2| ≤ A, s = 1, .., 2m, can be bounded by e−c log n (similarly to (34)). Hence, using (18), (45), and (58) for the estimate of the integral with |μn (Q)|, we can write 2 − 1 tr ξ −2ε(κ4 )det−1 (i Q+0 /2)σ1
−1 D2m F2m () = Z m νn (v, w, r )e 2n ρsc (λ0 )2 ×μn (Q)e
−tr
n i0 ξ (Q+ 2 ) ρsci(λ 0)
(1 + f n (Q, v, w, r )) d D + O(e−c log n ), 2
(65)
where d D = d Q d V d R d W, μn , σ1 and νn (v, w, r ) are defined in (32), (60), and (56) respectively, and n f n (i Q + 20 , v, w, r ) − 2ε(κ4 )σ1 /n 2ε(κ4 )det−1 (i Q+ 20 )σ1
f n (Q, v, w, r ) = e − 1. 1+ det(i Q + 0 /2) (66) If we write the term in the brackets above as (1 + T )n = exp{n log(1 + T )}, expand the exponent with respect to T and use property (3) and formula (63), we obtain on n , | f n (Q, v, w, r )| ≤ n −1/2 logk n,
(67)
Characteristic Polynomials of the Wigner Ensemble
15
where k is independent of n. Note also that f n is analytic in entries of Q. It is easy to check that I := νn (Q, v, w, r )d V d R d W n
=
2m
(C p )−1 eκ4 σ (i Q+0 /2)det
−2 (i Q+
0 /2)
+ O(e−c log n ), 2
(68)
p=2
where νn (Q, v, w, r ) = exp{−2ε(κ4 )det−1 (i Q + 0 /2)σ1 } νn (v, w, r ), ql1 ,s1 ,l2 ,s2 ql2 ,s2 ,l1 ,s1 σ (i Q + 0 /2) =
(69)
l1 =s1 ,l2 =s2
with ql1 ,s1 ,l2 ,s2 defined in (60) (but for the matrix i Q + 0 /2 instead of i Q + 0 ). Moreover, according to the Cauchy-Binet formula (see [6]), σ (i Q + 0 /2) is the sum S2m−2 (A) of the principal minors of order (2m − 2) × (2m − 2) for the matrix A = (i Q ∗ + 0 /2)(i Q + 0 /2) = U ∗ (i T0 + 0 /2)2 U, where U is a unitary 2m × 2m matrix diagonalizing Q and T0 = diag{q1 , .., q2m }, i.e. Q = U ∗ T0 U . Since S2m−2 (A) is the coefficient in front of λ2 in the characteristic polynomial det(A − λI ), S2m−2 (A) is unitary invariant, and therefore σ (i Q + 0 /2) is also unitary invariant. Thus, we have on n of (64), 0 1 0 ≤ C, )det −2 (i Q + ) = σ (i Q + λ0 2 λ0 2 2 2 (iq + ) (iq + ) s l 2 2 1≤s C > 0.
(70)
This, (65), and (68) yield −1 n 2m D2m 2
−1 D2m F2m () =
2m π
2m 2
e
Q
n
−tr (Q+
i0 0 0 2 i ξ 2 ) ρsc (λ0 ) +κ4 σ (i Q+ 2 )/det(i Q+ 2 )
2 ×μn (Q) 1 + f n(1) (Q) d Q + O(e−c log n ),
(71)
where μn is defined in (32) and f n(1) (Q) = e
1 − 2n tr
+I −1 e
ξ2 ρsc (λ0 )2 1 − 2n tr
−1
ξ2 ρsc (λ0 )2
n
νn (Q, v, w, r ) f n (Q, v, w, r )d V d R d W (72)
with I of (68) and νn of (69). According to (66) and (70), we have | f n(1) (Q)| ≤ logk n/n 1/2 . (1)
Besides, we will use below that f n (Q) is analytic in elements of Q.
(73)
16
T. Shcherbina
Let us change variables to Q = U ∗ T U , where U is a unitary 2m × 2m matrix and T = diag{t1 , . . . , t2m }. The differential d Q in (71) is π m(2m−1)
2m
j!
−1
2 (T )d T d μ(U )
j=1
2m d tl , (T ) is the Vandermonde determinant of in the new variables, where d T = l=1 2m {tl }l=1 , and μ(U ) is the normalized to unity Haar measure on the unitary group U (2m) (see e.g. [9], Sect. 3.3). Functions det(i Q + 20 ), tr (Q + i2 0 )2 , and σ (i Q + 20 ) are unitary invariant. Hence, (71) implies 2 2m (−1)mn n 2m iλ0 n −1 F2m () = dμ(U ) dT (tl − D2m ) 2m l=1 A 2m 2 D2m (2π )m j=1 j! U (2m) (L a ) 2m
tr U ∗ (T + i20 )U ρsci(λξ 0 ) +κ4 1≤l<s≤2m (tl − iλ20 )−2 (ts − iλ20 )−2 2 ×2 (T ) 1 + f n(1) (U ∗ T U ) + O(e−c log n ), (74) ×e
iλ0 2 l=1 (tl + 2 ) −
− n2
where L aA is defined in (42). Using Proposition 1, we have 2 iλ0 2 n 2m (−1)mn n 2m −1 D2m F2m () = e− 2 l=1 (tl + 2 ) m D2m (2π ) (L aA )2m ×e
κ4
× where
iλ0 −2 iλ0 −2 1≤i< j≤2m (ti − 2 ) (t j − 2 )
−i πme
2m
l=1 (tl +
(T ) 2
2m
(tl −
l=1 ξl iλ0 2 ) ρsc (λ0 )
(T ) (−i ξ /ρsc (λ0 ))
iλ0 n ) 2
2 + f n(2) (T ) d T + O(e−c log n ),
(75)
ξ (1) ∗ i0 )U f (U T U )d μ(U ). exp − itr U ∗ (T + 2 ρsc (λ0 ) n According to (73), we get that f n(2) (T ) =
| f n(2) (T )| ≤ n −1/2 logk n, tl ∈ L aA , l = 1, .., 2m.
(76)
We obtain finally (cf. (44)) −1 F2m () = D2m
(−1)mn n 2m D2m 2m π m ×e
− n2
2m
2
(L aA )2m
iλ0 2 l=1 (tl + 2 ) −i
2m
(tl −
l=1 2m
iλ0 n ) (iρsc (λ0 ))m(2m−1) 2
ξl iλ0 l=1 (tl + 2 ) ρsc (λ0 ) +κ4
iλ0 −2 iλ0 −2 l1
(T ) 2 1 + f n(2) (T ) d t j + O(e−c log n ), (ξ ) j=1 2m
× where
f n(2) (T ) = (T ) (−i ξ /ρsc (λ0 ))ei (2)
2m
l=1 tl ξl /ρsc (λ0 )
f n(2) (T ).
f n (T ) is an analytic function bounded by n −1/2 logk n if tl ∈ L aA , l = 1, .., 2m.
(77)
Characteristic Polynomials of the Wigner Ensemble
17
3. Asymptotic Analysis In this section we prove Theorem 1, passing to the limit n → ∞ in (77) for λ j = λ0 + ξ j /nρsc (λ0 ), where ρsc is defined in (5), λ0 ∈ (−2, 2) and ξ j ∈ [−M, M] ⊂ R, j = 1, .., 2m. To this end consider the function 4 − λ20 t 2 iλ0 + t − log(t − iλ0 /2) − . 2 2 8
V (t, λ0 ) =
(78)
Then (77) yields
−1 D2m
(nρsc (λ0
2 ))m
F2m () = Z m,n
Wn (t1 , . . . , t2m )d T + O(e−c log n ), 2
(L aA )2m
(79)
where D2m is defined in (45), 2m
2m
(T ) ( ξ) iλ0 −2 iλ0 −2 ×eκ4 1≤l<s≤2m (tl − 2 ) (ts − 2 ) 1 + f n(2) (T ) ,
Wn (t1 , . . . , t2m ) = e
−n
l=1
V (tl ,λ0 )−i
ξl l=1 ρsc (λ0 ) tl
(80)
and (−1)mn n m ρsc (λ0 )m(m−1) e−mκ4 = . (−i)m(2m−1) 22m π 2m 2
Z m,n
(81)
Now we need Lemma 1. The function V (t, λ0 ) for t ∈ R attains its minimum at 4 − λ20 t = x± := ± . 2
(82)
Moreover, if t ∈ Un (x± ) := (x± − n −1/2 log n, x± + n −1/2 log n), then we have for sufficiently big n, C log2 n . n
(83)
12 t − (4 − λ20 )/4 − log(t 2 + λ20 /4) , 2
(84)
V (t, λ) ≥ Proof. Note that for t ∈ R, V (t, λ0 ) = thus
d t V (t, λ0 ) = t − 2 , dt t + λ20 /4 d2 1 2t 2 V (t, λ ) = 1 − , + 0 d t2 t 2 + λ20 /4 (t 2 + λ20 /4)2
(85)
18
T. Shcherbina
and t = x± of (82) are the minimum points of V (t, λ0 ). Writing iλ0 4 − λ20 V+ := V (x+ , λ0 ) = − i arcsin(−λ0 /2), 4 iλ0 4 − λ20 V− := V (x− , λ0 ) = − + i arcsin(−λ0 /2) − iπ, 4 we conclude that
(86)
V (x± , λ0 ) = 0. Expanding V (t, λ0 ) into the Taylor series and using (85)–(86), we obtain for t ∈ Un (x± ): 4 − λ20 (t − x± )2 + O(n −3/2 log3 n), 4 where x± is defined in (82). This implies for t ∈ Un (x± ), V (t, λ0 ) =
V (t, λ) ≥
(87)
C log2 n . n
The lemma is proved. Taking into account that |t j − iλ0 /2| > a for t j ∈ L aA , j = 1, .., 2m, we have
−2 −2 exp κ4 (88) (tl − iλ0 /2) (ts − iλ0 /2) ≤ C. 1≤l<s≤2m
Since ξ1 , . . . , ξ2m are distinct, the inequality |(T )/( ξ )| ≤ C1 for |t j | ≤ A, j = 1, .., 2m, (83), and (88) yield 2 2 Z m,n .. Wn (t1 , . . . , t2m )d T ≤ C1 n m e−C2 log n , L aA \(U+ ∪U− )
L aA
L aA
where L aA , Wn and Z m,n are defined in (42), (80), and (81) respectively, and U± = {t ∈ R : |t − x± | ≤ n −1/2 log n}
(89)
with x± of (82). Note that we have for t ∈ U± in view of (78) and (86), (t − x± )2 1 + f ± (t − x± ), n → ∞, (90) V (t, λ0 ) = V± + 1 + (x± − iλ0 /2)2 2 where f ± (t − x± ) = O((t − x± )3 ). Shifting t j − x± → t j for t j ∈ U± and using (86) we obtain for the r.h.s. of (79), 2m 2m ncα j iξ t − t 2 − j j −n f (t ) Z m,n d tj e 2 j ρsc (λ0 ) α j j (Un )2m j=1 α j=1 κ4 1≤l<s≤2m (tl + pαl )−2 (ts + pαs )−2 − 2m j=1 (nVα j +i x α j ξ j /ρsc (λ0 ))
×e (t1 + xα1 , . . . , t2m + xα2m ) 2 (1 + f n(2) (T )) + O(e−c log n ), × (ξ1 , . . . , ξ2m )
(91)
Characteristic Polynomials of the Wigner Ensemble
19
where the sum is over the collection α = {α j }2m j=1 , α j = ±, j = 1, .., 2m and −2 , c± = 1 + p±
p± = x± − iλ0 /2, Un = (−2n −1/2 log n, 2n −1/2 log n). (92)
Define I :=
(Un )2m
2m 2m 2m
ncα j 2 iξ j t j exp − n f α j (t j ) tj − − 2 ρsc (λ0 ) j=1
j=1
×(t1 + xα1 , . . . , t2m + xα2m )
2m
j=1
d tj
j=1
t j + xα j −
k−1 iξ j nρsc (λ0 )cα j Un, j
2m ncα j 2 iξ j t − n fα j t j − × exp − , dt j 2 j nρsc (λ0 )cα j j,k=1
= det
(93)
where U j,n =
− 2n −1/2 log n +
iξ j iξ j . , 2n −1/2 log n + nρsc (λ0 )cα j nρsc (λ0 )cα j
Since f ± (t) = O(t 3 ), changing variables to
√
√ nt j → t j , expanding exp{−n f α j (t j / n−
iξ j /nρsc (λ0 )cα j )} in (93), and keeping the terms up to the order n −4m , we obtain 2
I =
(+)
k−1 iξ j 2π det xα j − ncα j nρsc (λ0 )cα j j=1 2m 1 (α ) + Pk,mj (ξ j /n) (1 + o(1)), n → ∞, n j,k=1 2m
(94)
(−)
where Pk,m and Pk,m are polynomials with n- and j-independent (but k-dependent) coefficients of degree at most 4m 2 . Introducing D(ξ/n, λ) = det
xα j −
k−1 iξ j (α ) + λPk,mj (ξ j /n) nρsc (λ0 )cα j
2m ,
(95)
j,k=1
taking into account that D(ξ/n, λ) is a polynomial in {ξ j /nρsc (λ0 )}2m j=1 and λ, and putting α1 = . . . = αs = +, αs+1 = . . . = α2m = −, we find that if ξ j = ξl for j, l = 1, .., s or j, l = s + 1, .., 2m, then D(ξ/n, λ) = 0. This implies D(ξ/n, λ) = (ξ1 /n, . . . , ξs /n) (ξs+1 /n, . . . , ξ2m /n)(C0 + λF(ξ/n, λ)), (96)
20
T. Shcherbina
where F(ξ/n, λ) is a polynomial with bounded coefficients. Substituting λ = 0 in (95) and computing the Vandermonde determinant, we obtain s(s−1) (2m−s)(2m−s−1) 2 2 −i −i ρsc (λ0 )c+ ρsc (λ0 )c− s 2m iξ j iξk x+ − x− − + nρsc (λ0 )c+ nρsc (λ0 )c−
C0 =
j=1 k=s+1
=
−i ρsc (λ0 )c+
s(s−1) 2
−i ρsc (λ0 )c−
(2m−s)(2m−s−1) 2
(x+ − x− )s(2m−s) (1 + o(1)).
Hence, for α1 = . . . = αs = +, αs+1 = . . . = α2m = −, we get from (96) 2
2
nm I 2m π m (−i/ρsc (λ0 ))m(m−1)+(m−s) = 2 2 ( ξ) (c+ )s /2 (c− )(2m−s) /2 2 n −(m−s) ( 4 − λ20 )s(2m−s) (1 + o(1)) × , n → ∞. s 2m j=1 l=s+1 (ξ j − ξl )
(97)
This expression is of order O(1), and for s = m it is of order o(1). Hence, only terms of (91), where exactly m of {α j }2m j=1 are pluses contribute in the limit (12). Consider one of such terms in (91), e.g. the term with α1 = .. = αm = 1, αm+1 = .. = α2m = −1. Using (81), (92), and (97) with s = m we can rewrite this term as i m(m+1) eiπ(ξm+1 +..+ξ2m −ξ1 −..−ξm ) m(m−1)κ4 (λ2 −2)2 /2 0 e . (2iπ )m i,m j=1 (ξi − ξm+ j ) In view of the identity
sin(π(ξ j −ξm+k )) m det π(ξ j −ξm+k )
j,k=1
det
iπ(ξ j −ξm+k )
iπ(ξm+k −ξ j )
−e ξ j −ξm+k
m j,k=1
, (2iπ )m (ξ1 , .., ξm ) (ξm+1 , .., ξ2m ) the determinant in the l.h.s. is the linear combination of exp{iπ 2m j=1 α j ξ j } over the 2m collection {α j } j=1 , in which exactly m elements are pluses. According to the identity (see [12], Prob. 7.3) # $m m(m−1) 1 k
=
e
(98)
the coefficient of exp{iπ(ξm+1 + .. + ξ2m − ξ1 − .. − ξm )} is
m m(m−1) det ξm+k1−ξ j (−1) 2 j,k=1 . = 2 (2iπ )m (ξ1 , .., ξm ) (ξm+1 , .., ξ2m ) (−1)m (2iπ )m i,m j=1 (ξi − ξm+ j ) Other coefficients can be computed analogously. Thus, restricting the sum in (91) to that over the collection {α j }2m j=1 , in which m elements are pluses, and m are minuses, and using (98), we obtain Theorem 1 after certain algebra.
Characteristic Polynomials of the Wigner Ensemble
21
Acknowledgements. The author is grateful to Prof.L.Pastur for many interesting discussions of the problem. This work was partially supported by joint project 17-01-10 of National Academy of Sciences of Ukraine and Russian Foundation for Basic Research.
References 1. Berezin, F.A.: Introduction to the algebra and analysis of anticommuting variables. Moscow: Moscow State University Publ., 1983 (Russian) 2. Brezin, E., Hikami, S.: Characteristic polynomials of random matrices. Commun. Math. Phys. 214, 111– 135 (2000) 3. Brezin, E., Hikami, S.: Characteristic polynomials of real symmetric random matrices. Commun. Math. Phys. 223, 363–382 (2001) 4. Efetov, K.: Supersymmetry in disorder and chaos. New York: Cambridge University Press, 1997 5. Erdos, L.: Universality of Wigner random matrices: a survey of recent results. Uspekhi Mat. Nauk. 66, 67–198 (2011) 6. Gantmacher, F.R.: The Theory of Matrices. New York: Chelsea, 1959 7. Gotze, F., Kosters, H.: On the second-ordered correlation function of the characteristic polynomial of a hermitian Wigner matrix. Commun. Math. Phys. 285, 1183–1205 (2008) 8. Keating, J.P., Snaith, N.C.: Random matrix theory and ζ (1/2 + it). Commun. Math. Phys. 214, 57–89 (2000) 9. Mehta, M.L.: Random Matrices. New York: Academic Press, 1991 10. Mehta M.L. Normand J.-M.: Moments of the characteristic polynomial in the three ensembles of random matrices. J. Phys. A: Math. Gen. 34, 4627–4639 (2001) 11. Pastur, L.: The spectrum of random matrices. Teoret. Mat. Fiz. 10, 102–112 (1972) (Russian) 12. Polya, G., Szego, G.: Problems and theorems in analysis, Vol. II. Die Grundlehren der math. Wissenschaften. Berlin-Heidelberg-New York: Springer-Verlag, 1976 13. Strahov, E., Fyodorov, Y.V.: Universal Results for Correlations of Characteristic Polynomials: RiemannHilbert Approach. Commun. Math. Phys. 241, 343–382 (2003) Communicated by P. Forrester
Commun. Math. Phys. 308, 23–47 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1324-8
Communications in
Mathematical Physics
Existence of Axially Symmetric Static Solutions of the Einstein-Vlasov System Håkan Andréasson1 , Markus Kunze2 , Gerhard Rein3 1 Mathematical Sciences, Chalmers University of Technology, Göteborg University, 41296 Göteborg,
Sweden. E-mail: [email protected]
2 Fakultät für Mathematik, Universität Duisburg-Essen, 45117 Essen, Germany.
E-mail: [email protected]
3 Fakultät für Mathematik, Physik und Informatik, Universität Bayreuth, 95440 Bayreuth, Germany.
E-mail: [email protected] Received: 24 August 2010 / Accepted: 1 April 2011 Published online: 4 September 2011 – © Springer-Verlag 2011
Abstract: We prove the existence of static, asymptotically flat non-vacuum spacetimes with axial symmetry where the matter is modeled as a collisionless gas. The axially symmetric solutions of the resulting Einstein-Vlasov system are obtained via the implicit function theorem by perturbing off a suitable spherically symmetric steady state of the Vlasov-Poisson system. 1. Introduction The aim of the present investigation is to prove the existence of static, asymptotically flat, and axially symmetric solutions of the Einstein-Vlasov system. This system describes, in the context of general relativity, the evolution of an ensemble of particles which interact only via gravity. Examples from astrophysics of such ensembles include galaxies or globular clusters where the stars play the role of the particles and where collisions among these particles are usually sufficiently rare to be neglected. The particle distribution is given by a density function f on the tangent bundle T M of the spacetime manifold M. We assume that all particles have the same rest mass which is normalized to unity. Hence the particle distribution function is supported on the mass shell P M = {gαβ p α p β = −c2 and p α is future pointing} ⊂ T M. Here gαβ denotes the Lorentz metric on the spacetime M and if x α are coordinates on M, then p α denote the corresponding canonical momentum coordinates; Greek indices always run from 0 to 3, and we have a specific reason for making the dependence on the speed of light c explicit. We assume that the coordinates are chosen such that ds 2 = c2 g00 dt 2 + gab d x a d x b , where Latin indices run from 1 to 3 and t = x 0 should be thought of as a timelike coordinate. On the mass shell p 0 can be expressed by the remaining coordinates, p 0 = −g 00 1 + c−2 gab pa p b ,
24
H. Andréasson, M. Kunze, G. Rein
and f = f (t, x a , p b ) ≥ 0. The Einstein-Vlasov system now consists of the Einstein field equations G αβ = 8π c−4 Tαβ
(1.1)
a p 0 ∂t f + pa ∂x a f − βγ p β p γ ∂ pa f = 0
(1.2)
coupled to the Vlasov equation
via the following definition of the energy momentum tensor: dp 1 dp 2 dp 3 1/2 . Tαβ = c|g| pα pβ f − p0
(1.3)
α are the ChrisHere |g| denotes the modulus of the determinant of the metric, and βγ toffel symbols induced by the metric. We note that the characteristic system of the Vlasov equation (1.2) are the geodesic equations written as a first order system on the mass shell P M which is invariant under the geodesic flow. For more background on the Einstein-Vlasov equation we refer to [3]. In [8,16,18,19] the existence of a broad variety of static, asymptotically flat solutions of this system has been established, all of which share the restriction that they are spherically symmetric. The purpose of the present investigation is to remove this restriction and prove the existence of static, asymptotically flat solutions to the EinsteinVlasov system which are axially symmetric but not spherically symmetric. From the applications point of view this symmetry assumption is more “realistic” than spherical symmetry, and from the mathematics point of view the complexity of the Einstein field equations increases drastically if one gives up spherical symmetry. We use usual axial coordinates t ∈ R, ρ ∈ [0, ∞[, z ∈ R, ϕ ∈ [0, 2π ] and write the metric in the form
ds 2 = −c2 e2ν/c dt 2 + e2μ dρ 2 + e2μ dz 2 + ρ 2 B 2 e−2ν/c dϕ 2 2
2
(1.4)
for functions ν, B, μ depending on ρ and z. The Killing vector fields ∂t and ∂φ correspond to the stationarity and axial symmetry of the spacetime. The additional assumption that the spacetime is static implies that the total angular momentum of the spacetime is zero, cf. [21, Ch. 11, Prob. 6]. Since the generalization to stationary spacetimes with non-zero angular momentum induces qualitatively new, additional difficulties it is postponed to a later investigation. The reason for writing ν/c2 instead of ν is so that below ν converges to the Newtonian potential U N in the limit c → ∞. The metric is to be asymptotically flat in the sense that the boundary values lim
|(ρ,z)|→∞
ν(ρ, z) =
lim
|(ρ,z)|→∞
μ(ρ, z) = 0,
lim
|(ρ,z)|→∞
B(ρ, z) = 1
(1.5)
are attained at spatial infinity with certain rates which are specified later. In addition we need to require the condition that the metric is locally flat at the axis of symmetry, i.e., ν(0, z)/c2 + μ(0, z) = ln B(0, z), z ∈ R.
(1.6)
We refer to [4] for more information on axially symmetric spacetimes and state our main result.
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
25
Theorem 1.1. There exist static solutions of the Einstein-Vlasov system (1.1), (1.2), (1.3) with c = 1 such that the metric is of the form (1.4) and satisfies the boundary conditions (1.5), (1.6), and the spacetime is axially symmetric, but not spherically symmetric. The strategy of the proof of this result is as follows. Due to the symmetries of the metric the following quantities are constant along geodesics: E := −g(∂/∂t, p α ) = c2 e2ν/c p 0 2 2 = c2 eν/c 1 + c−2 e2μ ( p 1 )2 + e2μ ( p 2 )2 + ρ 2 B 2 e−2ν/c ( p 3 )2 , 2
L := g(∂/∂ϕ, p α ) = ρ 2 B 2 e
−2ν/c2
p3 ;
(1.7) (1.8)
E can be thought of as a local or particle energy and L is the angular momentum of a particle with respect to the axis of symmetry. Since up to regularity issues a distribution function f satisfies the Vlasov equation if and only if it is constant along geodesics, any distribution function f which depends only on E and L satisfies the Vlasov equation with a metric of the above form. Hence we make the ansatz f (x a , p b ) = φ(E, L),
(1.9)
and the Vlasov equation (1.2) holds. Upon insertion of this ansatz into the definition (1.3) of the energy momentum tensor, the latter becomes a functional Tαβ = Tαβ (ν, B, μ) of the yet unknown metric functions ν, B, μ, and we are left with the problem of solving the field Einstein equations (1.1) with this right-hand side. We obtain solutions by perturbing off spherically symmetric steady states of the Vlasov-Poisson system via the implicit function theorem; the latter system arises as the Newtonian limit of the Einstein-Vlasov system. Our main result specifies conditions on the ansatz function φ above such that a two parameter family of axially symmetric solutions of the Einstein-Vlasov system passes through the corresponding spherically symmetric, Newtonian steady state. The parameter γ = 1/c2 turns on general relativity and the second parameter λ turns on the dependence on L and hence axial symmetry; notice that L is not invariant under arbitrary rotations about the origin, so if f actually depends on L the solution is not spherically symmetric. The scaling symmetry of the Einstein-Vlasov system can then be used to obtain the desired solutions for the physically correct value of c. The detailed formulation of our result is stated in the next section together with the basic set up of its proof. The remaining sections of the paper are then devoted to establishing the various features of the basic set up which are needed to apply the implicit function theorem, and to prove various properties of the solutions we obtain. We conclude this introduction with some further references to the literature. The idea of using the implicit function theorem to obtain equilibrium configurations of self-gravitating matter distributions from already known solutions can be traced back to L. Lichtenstein who argued the existence of axially symmetric, stationary, selfgravitating fluid balls in this way [12,13]. His arguments were put into a rigorous and modern framework in [9]. The analogous approach was used in [17] to obtain axially symmetric steady states of the Vlasov-Poisson system, see also [20]. The approach has also been used to construct axially symmetric stationary solutions of the Einstein equations coupled to a matter model: In [10] matter was described as an ideal fluid whereas in [1,2] it was described as a static or a rotating elastic body respectively. Besides the different matter model our investigation differs from the latter two in that we employ the rather explicit form of the metric stated above and a reduced version of the Einstein field equations which closely follows [4].
26
H. Andréasson, M. Kunze, G. Rein
2. Set Up of the Proof In what follows we also use the Cartesian coordinates (x 1 , x 2 , x 3 ) = (ρ cos ϕ, ρ sin ϕ, z) ∈ R3 , which correspond to the axial coordinates ρ ∈ [0, ∞[, z ∈ R, ϕ ∈ [0, 2π ]; it should be noted that tensor indices always refer to the spacetime coordinates t, ρ, z, ϕ. By abuse of notation we write ν(ρ, z) = ν(x), etc. In Sect. 3 we collect the relevant information on the relation between regularity properties of axially symmetric functions expressed in the variables x ∈ R3 or ρ ∈ [0, ∞[, z ∈ R, respectively. We introduce two (small) parameters γ = 1/c2 ∈ [0, ∞[ and λ ∈ R. In order to obtain the correct Newtonian limit below we adjust the ansatz for f as follows. Let v 1 = eμ p 1 , v 2 = eμ p 2 , v 3 = ρ Be−γ ν p 3 , so that p =e 0
−γ ν
1 + γ |v|2 .
For the particle distribution function we make the ansatz f (x, v) = φ (E − 1/γ ) ψ(λL). The important point here is that eγ ν(x) 1 + γ |v|2 − 1 1 E − 1/γ = → |v|2 + ν(x) as γ → 0, γ 2
(2.1)
(2.2)
i.e., the limit is the non-relativistic energy of a particle with phase space coordinates (x, v) in case ν = U N is the Newtonian gravitational potential. For γ = 0 this limit is to replace the argument of φ in (2.1). We now specify the conditions on the functions φ and ψ. Conditions on φ and ψ. (φ1) φ ∈ C 2 (R) and there exists E 0 > 0 such that φ(η) = 0 for η ≥ E 0 and φ(η) > 0 for η < E 0 . (φ2) The ansatz f (x, v) = φ 21 |v|2 + U (x) leads to a compactly supported steady state of the Vlasov-Poisson system, i.e., there exists a solution U = U N ∈ C 2 (R3 ) of the semilinear Poisson equation 1 2 |v| + U dv, U (0) = 0, U = 4πρ N = 4π φ 2 U N (x) = U N (|x|) is spherically symmetric, and the support of ρ N ∈ C 2 (R3 ) is the closed ball B R N (0), where U N (R N ) = E 0 and U N (r ) < E 0 for 0 ≤ r < R N , U N (r ) > E 0 for r > R N . (φ3) 6 + 4πr 2 a N (r ) > 0, r ∈ [0, ∞[, where
1 2 |v| + U N (r ) dv. a N (r ) := φ 2 R3 (ψ) ψ ∈ C ∞ (R) is even with ψ(L) = 1 iff L = 0, and ψ ≥ 0.
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
27
For such a steady state, lim U N (x) = U N (∞) > E 0 .
|x|→∞
The normalization condition U N (0) = 0 instead of U N (∞) = 0 is unconventional from the physics point of view, but it has technical advantages below. Examples for ansatz functions φ which satisfy (φ1) and (φ2) are found in [5,19], the most well-known ones being the polytropes φ(E) := (E 0 − E)k+
(2.3)
for 2 < k < 7/2; here E 0 > 0 and (·)+ denotes the positive part. In Sect. 7 we show that for this class of ansatz functions also (φ3) holds. Numerical checks indicate that (φ3) holds for general isotropic steady states of the Vlasov-Poisson system. We can now give a more detailed formulation of our result. Theorem 2.1. There exists δ > 0 and a two parameter family (νγ ,λ , Bγ ,λ , μγ ,λ )(γ ,λ)∈[0,δ[×]−δ,δ[ ⊂ C 2 (R3 )3 with the following properties: (i) (ν0,0 , B0,0 , μ0,0 ) = (U N , 1, 0), where U N is the potential of the Newtonian steady state specified in (φ2). (ii) If for γ > 0 a distribution function is defined by Eq. (2.1) and a Lorentz metric √ by (1.4) with c = 1/ γ then this defines a solution of the Einstein-Vlasov system (1.1), (1.2), (1.3) which satisfies the boundary condition (1.6) and is asymptotically flat. For λ = 0 this solution is not spherically symmetric. (iii) If for γ = 0 a distribution function is defined by Eq. (2.1), observing (2.2), this yields a steady state of the Vlasov-Poisson system with gravitational potential ν0,λ which is not spherically symmetric for λ = 0. (iv) In all cases the matter distribution is compactly supported both in phase space and in space. Remark. (a) The smallness restriction to γ = 1/c2 is undesired because c is, in a given set of units, a definite number. However, if ( f, ν, B, μ) is a static solution for some choice of c ∈ ]0, ∞[ then the rescaling f˜(ρ, z, p 1 , p 2 , p 3 ) = ν˜ (ρ, z) = ˜ B(ρ, z) = μ(ρ, ˜ z) =
c−3 f (cρ, cz, cp 1 , cp 2 , p 3 ), c−2 ν(cρ, cz), B(cρ, cz), μ(cρ, cz)
yields a solution of the Einstein-Vlasov system with c = 1. The factor c2 in the metric (1.4) is removed by a rescaling of time. (b) The smallness restriction to λ means that the solutions obtained are close to being spherically symmetric.
28
H. Andréasson, M. Kunze, G. Rein
(c) The metric does not satisfy the boundary conditions (1.5), but lim
|(ρ,z)|→∞
ν(ρ, z) = ν∞ ,
lim
|(ρ,z)|→∞
μ(ρ, z) = −ν∞ /c2 ,
lim
|(ρ,z)|→∞
B(ρ, z) = 1. (2.4)
However, if we by abuse of notation redefine ν = ν − ν∞ and μ = μ + ν∞ /c2 , then the original condition (1.5) is restored and the metric (1.4) takes the form
2 2 ds 2 = −c2 e2ν/c c12 dt 2 + c22 e2μ dρ 2 + e2μ dz 2 + ρ 2 B 2 e−2ν/c dϕ 2 (2.5) with constants c1 , c2 > 0, which simply amounts to a choice of different units of time and space. By general covariance of the Einstein-Vlasov system (1.1), (1.2), (1.3) the equations still hold. (d) In view of [17], part (iii) of the theorem does not give new information on steady states of the Vlasov-Poisson system and is stated mainly in order to understand the obtained two parameter family of states as a whole. However, we note that for the Newtonian set-up in [17] axially symmetric steady states were obtained as deformations of a spherically symmetric one. The present approach differs considerably from this and in principle is more direct. (e) In the course of the proof of the theorem additional regularity properties and specific rates at which the boundary values at infinity are approached will emerge. (f) An alternative approach to the one followed here would be to start with static, spherical symmetric solutions of the Einstein-Vlasov system and use the implicit function theorem to perturb off from these. The technical reason why we adopted the present approach is that we have much better information on the linearized equations in the Newtonian case than in the relativistic one, which is essential in the application of the implicit function theorem. The alternative of perturbing off spherically symmetric Einstein-Vlasov solutions is currently under investigation. In the rest of this section we transform the problem of finding the desired solutions into the problem of finding zeros of a suitably defined operator. The Newtonian steady state specified in (φ2) will be a zero of this operator for γ = λ = 0, and the implicit function theorem will yield our result. In order that the overall course of the argument becomes clear we will go through its various steps, postponing the corresponding detailed proofs to later sections. The Einstein field equations are overdetermined, and we need to identify a suitable subset of (combinations of) these equations which, on the one hand, suffice to determine ν, B, μ, and which are such that at the end of the day all the field equations hold once this reduced system is solved. We introduce the auxiliary metric function ξ = γ ν + μ. Let and ∇ denote the Cartesian Laplace and gradient operator respectively. Taking suitable combinations of the field equations one finds that ∇B 1 ν + (2.6) · ∇ν = 4π γ γ e(2ξ −4γ ν) T00 + T11 + T22 + 2 2 e2ξ T33 , B ρ B ∇ρ · ∇ B = 8π γ 2 B (T11 + T22 ), B + (2.7) ρ
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
∂ρ B ∂z B 1+ρ ∂ρ ξ − ρ ∂z ξ B B
1 ρ = ∂ρ (ρ 2 ∂ρ B) − ∂zz B + γ 2 ρ (∂ρ ν)2 − (∂z ν)2 , 2ρ B 2B ∂ρ B ∂ρ (ρ∂z B) ∂z B 1+ρ ∂z ξ + ρ ∂ρ ξ = + 2γ 2 ρ ∂ρ ν∂z ν. B B B
29
(2.8) (2.9)
The last two equations arise from ρ (G 11 − G 22 ) = 0 and ρ G 12 = 0 respectively; note that due to (2.1), T11 = T22 and T12 = 0. Because of the asymptotic behavior of B and the structure of the left-hand side of (2.7) we write B = 1 + h/ρ. Next, we observe that by taking suitable combinations of (2.8) and (2.9) we obtain equations which contain only ∂ρ ξ or ∂z ξ respectively, and we chose the former. In the above equations the components of the energy momentum tensor are functions of the unknown quantities ν, h, ξ = γ ν + μ for which we therefore have obtained the following reduced system of equations: 1 ∇(h/ρ) · ∇ν, B ∂ρρ h + ∂zz h = 8π γ 2 ρ B11 (ν, B, ξ, ρ; γ , λ),
(1 + ∂ρ h)2 + (∂z h)2 ∂ρ ξ = ∂z h ∂zρ h + 2γ 2 (ρ + h)∂ρ ν∂z ν
1 2 2 2 (∂ρρ h − ∂zz h) + γ (ρ + h) (∂ρ ν) − (∂z ν) , + (1 + ∂ρ h) 2 ν = 4π (00 + γ 11 + γ 33 ) (ν, B, ξ, ρ; γ , λ) −
(2.10) (2.11)
(2.12)
where αβ are defined below in (2.14)–(2.16). We supplement this with the boundary condition (1.6) which in terms of the new unknowns and since necessarily h(0, z) = 0, reads ξ(0, z) = ln 1 + ∂ρ h(0, z) .
(2.13)
It remains to determine precisely the dependence of the functions αβ on the unknown quantities ν, h, ξ . Since the ansatz (2.1) is even in the momentum variables p 1 , p 2 , p 3 — the fact that ψ is even is needed here, all the off-diagonal elements of the energymomentum tensor vanish. The computation of its non-trivial components uses the new integration variables eγ ν 1 + γ |v|2 − 1 , s = v3, η= γ the abbreviation
m(η, B, ν, γ ) = Be−γ ν
e−2γ ν (1 + γ η)2 − 1 , γ
30
H. Andréasson, M. Kunze, G. Rein
and yields 00 (ν, B, ξ, ρ; γ , λ) = γ 2 e(2ξ −4γ ν) T00 m(η,B,ν,γ ) 4π (2ξ −4γ ν) ∞ e = φ(η)(1 + γ η)2 ψ(λρs) ds dη, (2.14) B (eγ ν −1)/γ 0 11 (ν, B, ξ, ρ; γ , λ) = T11 + T22 ∞ m(η,B,ν,γ ) 4π = 3 e2ξ φ(η) ψ(λρs)(m 2 (η, B, ν, γ ) − s 2 ) ds dη, (2.15) B (eγ ν −1)/γ 0 e2ξ T33 ρ2 B2 m(η,B,ν,γ ) φ(η) ψ(λρs) s 2 ds dη;
33 (ν, B, ξ, ρ; γ , λ) = 4π = 3 e2ξ B
∞
(eγ ν −1)/γ
(2.16)
0
we recall that T11 = T22 . The reason for keeping B as argument on the right-hand sides above is that the matter terms are differentiable in this variable, but taking a derivative with respect to h would yield an irritating factor 1/ρ. For elements of the function space chosen below h/ρ extends smoothly to the axis of symmetry ρ = 0. We now define the function spaces in which we will obtain the solutions of the system (2.10), (2.11), (2.12). As noted above we write, by abuse of notation, axially symmetric functions as functions of x ∈ R3 or of ρ ≥ 0, z ∈ R; regularity properties of axially symmetric functions are considered in Sect. 3. We fix 0 < α < 1/2 and 0 < β < 1, and consider the Banach spaces X1 := ν ∈ C 3,α (R3 ) | ν(x) = ν(ρ, z) = ν(ρ, −z) and ν X1 < ∞ , X2 := h ∈ C 4,α (R2 ) | h(ρ, z) = −h(−ρ, z) = h(ρ, −z) and h X2 < ∞ , X3 := ξ ∈ C 2,α (Z R ) | ξ(x) = ξ(ρ, z) = ξ(ρ, −z) and ξ X3 < ∞ , where Z R := {x ∈ R3 | ρ < R} is the cylinder of radius R > 0, the latter being defined in (2.17) below. The norms are defined by ν X1 := ν C 3,α (R3 ) + (1 + |x|)1+β ∇ν ∞ , h X2 := h C 4,α (R2 ) + (1 + |(ρ, z)|)2 ∇(h/ρ) ∞ , ξ X3 := ξ C 2,α (Z R ) , and X := X1 × X2 × X3 ,
(ν, h, ξ ) X := ν X1 + h X2 + ξ X3 .
L ∞ -norm, functions in C k,α (Rn ) have by definition continuous
Here · ∞ denotes the derivatives up to order k and all the highest order derivatives are Hölder continuous with exponent α, |D σ g(x) − D σ g(y)| g C k,α (Rn ) := D σ g ∞ + sup , |x − y|α x,y∈Rn ,x = y |σ |≤k
|σ |=k
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
31
and D σ denotes the derivative corresponding to a multi-index σ ∈ Nn0 . We note that if h ∈ X2 , then B = 1+h/ρ ∈ C 3 (R3 ), cf. Lemma 3.2. Moreover, it will be straightforward to extend ξ to R3 once a solution is obtained in the above space. Now we recall the properties of the Newtonian steady state specified in (φ2). That condition implies that there exists R > R N > 0 such that U N (r ) > (E 0 + U N (∞))/2, r > R.
(2.17)
If ||ν − U N ||∞ < |E 0 − U N (∞)|/4 and 0 ≤ γ < γ0 , with γ0 > 0 sufficiently small, depending on E 0 and U N , then eγ ν(x) − 1 > E 0 for all |x| > R. γ This implies that there exists some δ > 0 such that for all (ν, h, ξ ; γ , λ) ∈ U, the matter terms resulting from (2.14)–(2.16) are compactly supported in B R (0), where U := {(ν, h, ξ ; γ , λ) ∈ X × [0, δ[×] − δ, δ[ | (ν, h, ξ ) − (U N , 0, 0) X < δ}. In addition we require that δ > 0 is sufficiently small so that for all elements in U it holds that B = 1 + h/ρ > 1/2, and the factor in front of ∂ρ ξ in (2.12) is larger than 1/2; since h vanishes on the axis of symmetry, h/ρ is controlled by ∇h. Now let an element (ν, h, ξ ; γ , λ) ∈ U be given and substitute it into the matter terms defined in (2.14)–(2.16). With the right hand sides obtained in this way the Eq. (2.10)–(2.12) can then be solved, observing the boundary condition (2.13) and the fact that we require h to vanish on the axis of symmetry. We define the corresponding solution operators by 1 1 G 1 (ν, h, ξ ; γ , λ)(x) := − − M1 (y) dy |y| R3 |x − y| 1 ∇(h/ρ)(y) · ∇ν(y) dy + , 4π R3 B(y) |x − y| G 2 (ν, h, ξ ; γ , λ)(x) := 4 ln |(ρ − ρ, ˜ z − z˜ )| ρ˜ M2 (ρ, ˜ z˜ ) d ρ˜ d z˜ , R2 ρ G 3 (ν, h, ξ ; γ , λ)(x) := ln 1 + ∂ρ h(0, z) + g(s, z) ds, 0 ≤ ρ < R. 0
Here M1 (x) := (00 + γ 11 + γ 33 )(ν(x), B(x), ξ(x), ρ, ; γ , λ), M2 (ρ, z) := γ 2 B(x) 11 (ν(x), B(x), ξ(x), ρ; γ , λ), M2 (ρ, z) = M2 (−ρ, z) for ρ < 0 and z ∈ R, and
−1
g := (1 + ∂ρ h)2 + (∂z h)2 ∂z h ∂zρ h + 2γ 2 (ρ + h)∂ρ ν∂z ν
1 2 2 2 (∂ρρ h − ∂zz h) + γ (ρ + h) (∂ρ ν) − (∂z ν) . + (1 + ∂ρ h) 2
(2.18)
32
H. Andréasson, M. Kunze, G. Rein
Finally we define the mapping to which we are going to apply the implicit function theorem as F : U → X , (ν, h, ξ ; γ , λ) → (ν, h, ξ ) − (G 1 , G 2 , G 3 )(ν, h, ξ ; γ , λ). The proof of Theorem 2.1 now proceeds in a number of steps. Step 1. As a first step we need to check that the mapping F is well defined, in particular it preserves the various regularity and decay assumptions. This is done in Sect. 4. Step 2. The next step is to see that F(U N , 0, 0; 0, 0) = 0. This is due to the fact that for γ = λ = 0 the choice h = ξ = 0 trivially satisfies (2.11), (2.12), while (2.10) reduces to ν = 4π 00 (ν, 1, 0; 0, 0) with
00 (ν, 1, 0; 0, 0) = 4π
∞ ν
φ(η) 2(η − ν) dη =
R3
φ
1 2 |v| + ν 2
dv;
notice that h = 0 implies that B = 1. By (φ2), ν = U N is a solution of this equation, and the fact that U N ∈ X1 is part of what was shown in the previous step. Step 3. Next we show that F is continuous, and continuously Fréchet differentiable with respect to (ν, h, ξ ). The fairly technical but straightforward details are covered in Sect. 5. Step 4. The crucial step is to see that the Fréchet derivative L := DF(U N , 0, 0; 0, 0) : X → X is one-to-one and onto. Indeed, L(δν, δh, δξ ) = (δν − L 1 (δν) − L 2 (δh, δξ ), δh, δξ − L 3 (δh)) , where
L 1 (δν)(x) := −
R3
1 1 − |x − y| |y|
a N (y)δν(y) dy,
1 dy ∇(δh/ρ)(y) · ∇U N (y) 4π R3 |x − y| 1 1 − δξ(y)ρ N (y) dy, +2 |y| R3 |x − y| ρ 1 (∂ρρ δh − ∂zz δh)(s, z) ds, 0 ≤ ρ < R, L 3 (δh)(x) := ∂ρ δh(0, z) + 2 0
L 2 (δh, δξ )(x) :=
with a N as defined in (φ3). To see that L is one-to-one let L(δν, δh, δξ ) = 0. Then the second component of this identity implies that δh = 0, and hence also δξ = 0 by the third component. It therefore remains to show that δν = 0 is the only solution of the equation δν = L 1 (δν), i.e., of the equation δν = 4πa N δν, δν(0) = 0
(2.19)
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
33
in the space X1 . Under the assumption on a N stated in (φ3) this is correct and shown in Sect. 6. It is at this point that our unconventional normalization condition in (φ2) together with the shift in the solution operator G 1 become important; notice that L 1 (δν)(0) = 0. To see that L is onto let (g1 , g2 , g3 ) ∈ X be given. We need to show that there exists (δν, δh, δξ ) ∈ X such that L(δν, δh, δξ ) = (g1 , g2 , g3 ). The second component of this equation simply says that δh = g2 . Now δh ∈ X2 implies that L 3 (δh) ∈ X3 , cf. Lemma 3.1 (b). Hence we set δξ = g3 + L 3 (δh) to satisfy the third component of the onto equation, and it remains to show that the equation δν − L 1 (δν) = g1 + L 2 (δh, δξ )
(2.20)
has a solution δν ∈ X1 . Firstly, L 2 (δh, δξ ) ∈ X1 . The assertion therefore follows from the fact that L 1 : X1 → X1 is compact, as is shown in Lemma 6.2. We are now ready to apply the implicit function theorem, cf. [7, Thm. 15.1], to the mapping F : U → X ; strictly speaking we should suitably extend F to γ < 0, but this is not essential. We obtain the following result. Theorem 2.2. There exists δ1 , δ2 ∈]0, δ[ and a unique, continuous solution map S : [0, δ1 [×] − δ1 , δ1 [→ Bδ2 (U N , 0, 0) ⊂ X such that S(0, 0) = (U N , 0, 0) and F(S(γ , λ); γ , λ) = 0 for all(γ , λ) ∈ [0, δ1 [×] − δ1 , δ1 [. The definition of F implies that for any (γ , λ) the functions (ν, h, ξ ) = S(γ , λ) are a solution of Eqs. (2.10)–(2.12), and if f is defined by (2.1) then the equations (2.6), (2.7), (2.12) hold with the induced energy momentum tensor. We can extend ξ to the whole space using the solution operator G 3 for all x ∈ R3 . Also, the boundary condition (1.6) on the axis of symmetry is satisfied: ξ(0, z) = G 3 (ν, h, ξ )(0, z) = ln(1 + ∂ρ h(0, z)) = ln B(0, z); recall that ξ = γ ν + μ. For γ = 0 we conclude first that h = 0, cf. (2.11) or the G 2 -part of the solution operator respectively, then the G 3 -part implies that ξ = 0 so that the solution reduces to (ν, 0, 0), where ν solves ν = 4π 00 (ν, 1, 0, ρ; 0, λ). Since 00 (ν, 1, 0, ρ; 0, λ) = 4π
ν
∞
φ(η)
√
2(η−ν)
ψ(λρs) ds dη
0
coincides with the spatial density induced by the ansatz (2.1) for the Newtonian case, cf. [17, Lemma 2.1], part (iii) of Theorem 2.1 is established. If λ = 0 then condition (ψ) implies that f really depends on the angular momentum variable L which is not invariant under all rotations about the origin, but only invariant under rotations about the axis ρ = 0. Moreover, if the metric were spherically symmetric then the explicit dependence of the quantities j j on ρ would imply that the induced energy momentum tensor would not be spherically symmetric which is a contradiction. Hence the obtained solutions are not spherically symmetric if λ = 0. To complete the proof of Theorem 2.1 we must show that indeed all the field equations are satisfied by the obtained metric (1.4).
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H. Andréasson, M. Kunze, G. Rein
The corresponding argument relies on the Bianchi identity ∇α G αβ = 0 which holds for the Einstein tensor induced by any (sufficiently regular) metric, and on the identity ∇α T αβ = 0 which is a direct consequence of the Vlasov equation (1.2); ∇α denotes the covariant derivative corresponding to the metric (1.4). The details are carried out in Sect. 8. Finally we collect the additional information on the solution which we obtain in the course of the proof. Proposition 2.3. Let (ν, h, ξ ) = S(γ , λ) be any of the solutions obtained in Theorem 2.2 and define μ := ξ − ν/c2 and B = 1 + h/ρ. Then the limit ν∞ := lim|x|→∞ ν(x) exists, and for any σ ∈ N30 with |σ | ≤ 1 and x ∈ R3 the following estimates hold: |D σ (ν(x) − ν∞ )| ≤ C(1 + |x|)−(1+|σ |) , |D σ (B − 1)(x)| ≤ C(1 + |x|)−(2+|σ |) , |D σ ξ(x)| ≤ C(1 + |x|)−(2+|σ |) . In particular, the spacetime equipped with the metric (1.4) is asymptotically flat in the sense that (2.4) and, after a trivial change of coordinates, also (1.5) holds. 1 (y) Proof. By definition of G 1 , lim|x|→∞ ν(x) = M|y| dy. The first two estimates are established in Lemma 4.2. As to the third one we observe that by the boundary condition (2.13) and Lemma 4.2, |ξ(0, z)| ≤ C|∂ρ h(0, z)| ≤
C . (1 + |z|)2
By (2.12) and the known asymptotic behavior of the coefficients in that equation which are given in terms of ν and h and their derivatives, C , + |z|3
|∂ρ ξ(ρ, z)| ≤ cf. Lemma 4.2. Hence
ρ
|ξ(ρ, z)| ≤ |ξ(0, z)| + 0
≤
C +C 1 + |z|2
1 + ρ3
|∂ρ ξ(s, z)| ds ∞
0
ds C ≤ , 1 + s 3 + |z|3 1 + |z|2
which is the desired estimate for ξ(ρ, z), provided ρ < |z|. Since we already know that the metric under consideration satisfies the full set of the Einstein equations, we can now use (2.8) and (2.9) to see that also ∂z ξ is given in terms of ν and h and their derivatives and satisfies the same decay estimate as ∂ρ ξ . Starting from ρ |ξ(ρ, z)| ≤ |ξ(ρ, ρ)| + |∂z ξ(ρ, s)| ds, z
we can use the decay of ∂z ξ to obtain the decay estimate for ξ(ρ, z) for ρ ≥ z ≥ 0 (or ρ ≥ −z ≥ 0), and the proof is complete. Remark. The decay results obtained in Proposition 2.3 coincide with the ones derived in [4] for stationary, axisymmetric spacetimes.
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
35
3. Regularity of Axially Symmetric Functions We call a function f : R3 → R axially symmetric if there exists a function f˜ : [0, ∞[×R → R such that f (x) = f˜(ρ, z), where ρ = x11 + x22 and z = x3 for x ∈ R3 . In this section we collect some results on the relation between the regularity properties of f and those of f˜. Lemma 3.1. Let f : R3 → R be axially symmetric and f (x) = f˜(ρ, z), where f˜ : [0, ∞[×R → R. Let k ∈ {1, 2, 3} and α ∈ ]0, 1[. (a) f ∈ C k (R3 ) iff f˜ ∈ C k ([0, ∞[×R) and all derivatives of f˜ of order up to k which are of odd order in ρ vanish for ρ = 0. (b) f is Hölder continuous with exponent α ∈ ]0, 1[ iff f˜ is. Proof. As to part (a) let f ∈ C k (R3 ) be axially symmetric. Then f is even in x1 and x2 and f˜(ρ, z) = f (ρ, 0, z). This proves the “only-if” part. For the “if” part one checks that the corresponding derivatives of f , which exist for ρ = 0, extend continuously to ρ = 0. As to part (b) one only needs to observe that x → ρ(x) = since |∇ρ(x)| = 1.
x12 + x22 is Lipschitz,
At several places in our analysis it is convenient to extend functions of (ρ, z) to negative values of ρ. Lemma 3.2. Let h = h(ρ, z) ∈ C 4 (R2 ) be odd in ρ and define h(ρ, z)/ρ, ρ = 0, b(ρ, z) := ∂ρ h(0, z), ρ = 0. Then b ∈ C 3 (R2 ) and all derivatives of b up to order 3 which are of odd order in ρ vanish for ρ = 0. By abuse of notation, b ∈ C 3 (R3 ). Proof. The regularity of b only needs to be checked at ρ = 0. Since h is odd in ρ it follows that h(0, z) = ∂ρρ h(0, z) = 0 for z ∈ R. Hence as ρ → 0, b(ρ, z) =
1 (h(ρ, z) − h(0, z)) → ∂ρ h(0, z), ρ
and by Taylor expansion, 1 1 ∂ρ h(ρ, z) − 2 h(ρ, z) ρ ρ 1 = ∂ρ h(0, z) + ∂ρρ h(τ, z)ρ ρ 1 1 2 − 2 h(0, z) + ∂ρ h(0, z)ρ + ∂ρρ h(σ, z)ρ ρ 2 1 = ∂ρρ h(τ, z) − ∂ρρ h(σ, z) 2 1 → ∂ρρ h(0, z) = 0, 2
∂ρ b(ρ, z) =
(3.1)
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H. Andréasson, M. Kunze, G. Rein
where σ, τ are between 0 and ρ. All other derivatives can be treated in a similar fashion, where one should observe that ∂z h(0, z) = 0. The regularity with respect to x then follows by Lemma 3.1. 4. F is Well Defined As a first step we investigate the regularity properties of the functions j j , j = 0, . . . , 3, and of the induced matter terms M1 , M2 . Lemma 4.1. Let φ and ψ satisfy the conditions (φ1) and (ψ) respectively. (a) The functions 00 and 33 have derivatives with respect to ν, ξ, ρ ∈ R and B ∈ ]1/2, 3/2[ up to order three and these are continuous in ν, ξ, B, ρ, γ , λ. The same is true for 11 for derivatives up to order four. (b) For (ν, ξ, h; γ , λ) ∈ U, M1 ∈ C 2 (R3 ) and M2 ∈ C 2,α (R2 ) are both compactly supported. Proof. As to part (a) we note that differentiability with respect to ξ and ρ is straightforward. Concerning differentiability with respect to ν and B we observe that for j = 0, . . . , 3 the expression j j is differentiable once with respect to the indicated variables, provided φ ∈ L ∞ loc , cf. the proof of [17, Lemma 2.1]. Under Assumption (φ1) we can first differentiate twice before the change to the integration variables η and s and obtain expressions which are essentially of the same form as j j , but with φ or φ instead of φ so that the resulting expression can be differentiated once more. The reason why 11 is one order more differentiable is that when differentiating this expression with respect to ν or B the integral with respect to s is preserved, its integrand is differentiated, and the resulting expression is qualitatively of the same type as 00 and can be differentiated three more times. Part (b) follows since the functions ν, B, ξ which are now substituted into j j are all at least in C 2,α (R3 ); the fact that ξ is defined only on the cylinder Z R does not matter here because the integrals in the definitions of j j yield functions with support in Z R . We now show that F is well defined, more precisely: Lemma 4.2. Let (ν, ξ, h; γ , λ) ∈ U. Then the following holds : (a) G 1 = G 1 (ν, ξ, h; γ , λ) ∈ C 3,α (R3 ) is axially symmetric, even z = x3 , and M1in (y) 2 (1 + |x|)(G 1 − a) ∞ , (1 + |x|) ∇G 1 ∞ < ∞, where a = |y| dy. (b) G 2 = G 2 (ν, ξ, h; γ , λ) ∈ C 4,α (R2 ) is odd in ρ, even in z, and (1 + |(ρ, z)|)G 2 ∞ , (1 + |(ρ, z)|)2 D 1 G 2 ∞ , (1 + |(ρ, z)|)3 D 2 G 2 ∞ < ∞. Moreover, (1 + |(ρ, z)|)2 (G 2 /ρ) ∞ , (1 + |(ρ, z)|)3 D 1 (G 2 /ρ) ∞ < ∞. Here D j stands for any derivative of order j with respect to (ρ, z) ∈ R2 . (c) G 3 = G 3 (ν, ξ, h; γ , λ) ∈ C 2,α (Z R ) is axially symmetric, even in z = x3 , and G 3 C 2,α (Z R ) < ∞. (d) F(ν, ξ, h; γ , λ) ∈ X .
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
37
Proof. As to part (a) the potential induced by the matter term M1 , which is in C 1,α (R3 ) by Lemma 4.1 (b), has the desired regularity and decay properties due to standard regularity results in Hölder spaces, cf. [14, Thms. 10.2, 10.3], and the decay of 1/|x − y| and its derivatives together with the compact support of M1 . As to the source term g = ∇(h/ρ) · ∇ν of the second term in G 1 we notice that ν ∈ X1 and h ∈ X2 implies that g ∈ C 1,α (R3 ) with |g(x)| ≤ C(1 + |x|)−3−β , in particular g ∈ L 1 ∩ L ∞ (R3 ). This implies the regularity of the potential induced by g and also its decay:
|g(y)| dy ≤ |x − y|
|x−y|≤|x|/2
... +
|x−y|>|x|/2
2 dy + (1 + |y|)−3−β |g(y)| dy |x − y| |x| |x−y|≤|x|/2 C C dy + ≤ ≤ C(1 + |x|/2)−3−β |x − y| |x| |x| |x−y|≤|x|/2
≤C
for large |x| as desired; for the gradient of the potential induced by g we argue completely analogously. As to part (b) we first recall that M2 = M2 (ρ, z) is even in ρ, and the actual source term ρ M2 is odd, compactly supported, and by Lemma 4.1 (b) and Lemma 3.1 (b), M2 ∈ C 2,α (R2 ). Hence G 2 ∈ C 4,α (R2 ) is odd in ρ ∈ R. As to the decay of G 2 let supp M2 ⊂ B R (0) ⊂ R2 . Then for |(ρ, z)| ≥ 2R and (ρ, ˜ z˜ ) ∈ supp M2 the estimate |ln |(ρ − ρ, ˜ z − z˜ )| − ln |(ρ, z)|| ≤ holds, and since
2R |(ρ, z)|
ρ˜ M2 = 0 this implies that
C ; |G 2 (ρ, z)| = G 2 (ρ, z) − 4 ln |(ρ, z)| ρ˜ M2 (ρ, ˜ z˜ ) d z˜ d ρ˜ ≤ |(ρ, z)| the estimates for the derivatives of G 2 follow along the same lines. Finally, ∂ρ (G 2 /ρ) = −G 2 /ρ 2 + ∂ρ G 2 /ρ which implies that |∂ρ (G 2 /ρ)(ρ, z)| ≤
C C . + |(ρ, z)|ρ 2 |(ρ, z)|2 |ρ|
This yields the asserted decay when |ρ| becomes large. But we can also use (3.1) to see that |∂ρ (G 2 /ρ)(ρ, z)| ≤ C/|z|3 . Both estimates together yield the asserted decay for ∂ρ (G 2 /ρ), and the decay for G 2 /ρ and ∂z (G 2 /ρ) can be dealt with similarly. In order to prove part (c) we observe that (2.18) and the regularity of ν and h imply that g and hence G 3 ∈ C 2,α (Z R ). By construction, ∂ρ G 3 = g. Since h is odd in ρ we find that h(0, z) = ∂z h(0, z) = ∂zz h(0, z) = ∂ρρ h(0, z) = 0, which implies that g(0, z) = 0. Thus by Lemma 3.1, G 3 ∈ C 2,α (Z R ), and the proof is complete.
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H. Andréasson, M. Kunze, G. Rein
5. F is Continuous and Continuously Differentiable with Respect to ν, h, ξ In this section we give some details of the proof of the following result: Lemma 5.1. The mappings G i : U → Xi , i = 1, 2, 3 are continuous and continuously Fréchet differentiable with respect to ν, h, and ξ . Proof. We only show the differentiability assertion and focus on G 1 . Defining = 00 + γ 11 + γ 33 we consider the differentiability only with respect to ν, and neglecting the dependence on the remaining variables we look at the prototype mapping (ν(y)) G : V → X1 , G(ν)(x) := dy, R3 |x − y| where V ⊂ X1 is open, ∈ C 3 (R) and ◦ ν has support in a fixed ball for all ν ∈ V. Our first claim is that G has the Fréchet derivative (ν(y))δν(y) dy, ν ∈ V, δν ∈ X1 . [DG(ν)δν](x) = |x − y| R3 In order to prove this claim we need to show that for ν ∈ V there exists > 0 such that for δν ∈ B (0) ⊂ X1 , ||G(ν + δν) − G(ν) − DG(ν)δν||X1 = o(||δν||X1 ). The support property and the standard elliptic estimate imply that ||G(ν + δν) − G(ν) − DG(ν)δν||X1 ≤ C ||G(ν + δν) − G(ν) − DG(ν)δν||C 3,α (R3 ) ≤ C ||(ν + δν) − (ν) − (ν)δν||C 1,α (R3 ) ≤ C ||(ν + δν) − (ν) − (ν)δν||C 2 (R3 ) . b
Clearly, ||(ν + δν) − (ν) − (ν)δν||∞ = o(||δν||∞ ) ≤ o(||δν||X1 ). We need to establish analogous estimates for expressions where we take derivatives with respect to x up to second order of the left hand side. Let i, j ∈ {1, 2, 3}. Then ∂xi (ν + δν) − (ν) − (ν)δν = (ν + δν) − (ν) ∂xi δν + (ν + δν) − (ν) − (ν)δν ∂xi ν, where both terms on the right are o(||δν||X1 ). Similarly, ∂xi x j (ν + δν) − (ν) − (ν)δν = (ν + δν) − (ν) − (ν)δν ∂xi ν ∂x j ν + (ν + δν) − (ν) ∂xi ν ∂x j δν + ∂x j ν ∂xi δν + (ν + δν) − (ν) − (ν)δν ∂xi x j ν + (ν + δν) − (ν) ∂xi x j δν + (ν + δν) ∂xi δν ∂x j δν,
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
39
and all the terms appearing on the right are o(||δν||X1 ). This proves the differentiability assertion for G. As to the continuity of this derivative, ( (ν) − (˜ν )) δν (y) dy ||DG(ν) − DG(˜ν )|| L(X1 ,X1 ) = sup 3 | · −y| R ||δν||X ≤1 1
≤C
sup
||δν||X1 ≤1
||( (ν) − (˜ν )) δν||C 1,α (R3 )
X1
≤ C || (ν) − (˜ν )||C 2 (R3 ) → 0 as ν˜ → ν in X1 . b
These arguments prove the continuous Fréchet differentiability of the first part of G 1 with respect to ν. The derivatives with respect to h or ξ can be dealt with in exactly the same manner. The source term in the potential which represents the second part of G 1 can be expanded explicitly in powers of δh and δν which together with the standard elliptic estimate proves the assertion for that term; note that both B and B + δh/ρ are bounded away from 0. The mapping G 2 is treated in the same way as our prototype G above, except that we have to estimate the source term including its third order derivatives, observing that 11 has derivatives up to order four with respect to ν, B, ξ . The mapping G 3 is easier since the term g defined in (2.18) can be expanded explicitly in powers of δν and δh, where again we observe that the denominator in that expression is bounded away from 0. 6. DF (U N , 0, 0; 0, 0) is One-to-One and Onto We recall from Sect. 2 and Eq. (2.19) that in order to prove that the map L is one-to-one it remains to show that g = 0 is the only solution of g = 4πa N g, g(0) = 0,
(6.1)
in the space X1 . Inspired by the method in [17] we expand g into spherical harmonics Ylm , l ∈ N0 , m = −l, . . . , l, where we use the notation of [11]; for a more mathematical reference on spherical harmonics see [15]. Denote by (r, θ, ϕ) and (s, τ, ψ) the spherical coordinates of a point x ∈ R3 and y ∈ R3 respectively. For l ∈ N0 and m = −l, . . . , l we define glm (r ) :=
1 r2
|x|=r
∗ Ylm (θ, ϕ) g(x) d Sx .
(6.2)
The symmetry assumptions in the function space X1 imply that g1−1 = g10 = g11 = 0, since up to multiplicative constants the spherical harmonics with l = 1 are given by sin θ e±iϕ and cos θ . To proceed, we use the following expansion, cf. [11], ∞ l l 4π r< 1 = Y ∗ (τ, ψ) Ylm (θ, ϕ), l+1 lm |x − y| 2l + 1 r> l=0 m=−l
40
H. Andréasson, M. Kunze, G. Rein
where r< := min (r, s) and r> := max (r, s). In view of (6.1), 1 1 Y ∗ (θ, ϕ) d Sx a N (s) g(y)dy glm (r ) = − 2 r R3 |x|=r |x − y| lm ∞ l r< 4π a N (s) l+1 Y ∗ (τ, ψ) g(y) d S y ds =− 2l + 1 0 r> |y|=s lm ∞ l 4π r< =− a N (s) l+1 s 2 glm (s)ds 2l + 1 0 r> r ∞ 4π s l+2 rl =− a N (s) l+1 glm (s) ds + a N (s) l−1 glm (s) ds . 2l + 1 0 r s r By a straightforward computation we find that glm satisfies the equation
r 2 glm = l(l + 1) + 4π r 2 a N (r ) glm ,
(6.3)
where prime denotes a derivative with respect to r . (s)| so We use this to show that g00 = 0 as follows. We define w(r ) := sup0≤s≤r |g00 that |g00 (r )| ≤ r w(r ); at this point it becomes essential that g(0) = g00 (0) = 0. Now (6.3) can be integrated to yield the Gronwall estimate r s|a N (s)| w(s) ds, r ≥ 0, w(r ) ≤ 4π 0
so that w = 0, and hence g00 = 0 as desired. It therefore remains to consider glm with l ≥ 2. For these we prove the following auxiliary result. Lemma 6.1. Let a ∈ Cc ([0, ∞[) and λ > 0 be such that λ + 4πr 2 a(r ) > 0 for r ∈ [0, ∞[. Let u ∈ C 2 ([0, ∞[) be a bounded solution to (r 2 u ) = (λ + 4πr 2 a(r ))u.
(6.4)
Then u = 0. Proof. We fix ra > 0 such that a(r ) = 0 for r ≥ ra . Multiplying (6.4) with u and integrating by parts we obtain for r > 0, r r 2 2 (λ + 4π s a(s)) u (s) ds = (s 2 u (s)) u(s) ds 0 0 r = r 2 u (r ) u(r ) − s 2 (u (s))2 ds. (6.5) 0
Now if there exists r0 > 0 so that u(r0 ) = 0 or u (r0 ) = 0 then (6.5) implies that u(r ) = u (r ) = 0 for r ∈ [0, r0 ]. The unique solvability of (6.4) for r ≥ r0 then shows that u = 0 as claimed. So we assume now that u(r ) = 0 and u (r ) = 0 for r > 0. Since (6.4) is invariant under u → −u, we may suppose that u(r ) > 0 and u (r ) > 0 for all r ∈]0, ∞[; note that (6.5) enforces uu > 0 on ]0, ∞[. For r ≥ ra > 0 (6.4) simplifies to (r 2 u ) = λu, which has the solution (l + 1) u(ra ) + ra u (ra ) r l l u(ra ) − ra u (ra ) ra l+1 u(r ) = + . (2l + 1) ra (2l + 1) r Therefore u is unbounded which is a contradiction.
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
41
Since g ∈ X1 , Eq. (6.2) implies that glm is bounded. Due to (φ3) we can apply Lemma 6.1 to conclude that glm = 0 for all l ≥ 2, and thus g = 0 as desired. We now prove the compactness result which was needed to show that L is onto. Lemma 6.2. The mapping K : X1 → X1 , a N (y) w(y) dy (K w)(x) = |x − y| R3 is compact. We remark that the operator L 1 has the form L 1 (δν)(x) = −K (δν)(x) + K (δν)(0) and is compact if K is, since the mapping ν → ν(0) is continuous on X1 . Proof. First we observe that the mapping √ ∞ √ 1 2 u → |v| + u dv = 2 2π φ φ (E) E − u d E 2 R3 u is in C 2 (R), and since U N ∈ C 2 (R3 ) the function a N is in Cc2 (R3 ). Hence a N w ∈ C 1,1/2 (R3 ) for any w ∈ X1 , and since α < 1/2 the mapping K is well defined. We fix a function χ ∈ Cc∞ (R3 ) such that 0 ≤ χ ≤ 1, χ (x) = 1 for |x| ≤ 1, and χ (x) = 0 for |x| ≥ 2. Let χ R (x) = χ (x/R) for R > 0 and define (K R w)(x) = χ R (x)(K w)(x). We show that K R → K in the operator norm as R → ∞. To this end, let ζ R = 1 − χ R so that for w ∈ X1 and x ∈ R3 , (K w − K R w)(x) = ζ R (x)(K w)(x),
(6.6)
and the latter vanishes for |x| ≤ R. Now let w X1 ≤ 1. For σ ∈ N30 with |σ | ≤ 3, it follows that σ D (K w − K R w)(x) ≤ ζ R (x) D σ (K w)(x) cτ D τ ζ R (x)D σ −τ (K w)(x) + 0<τ ≤σ
≤ 1{|x|≥R}
1 C D σ −τ (K w)(x) ≤ C ; +C |x| R R 0<τ ≤σ
constants denoted by C do not depend on x or R. In order to estimate the Hölder norm of D σ (K w − K R w) for |σ | = 3 we take x, x˜ ∈ R3 with |x| ˜ ≥ |x| and again apply the product rule to the expression (6.6). Adding and subtracting terms we have to estimate expressions like C τ σ −τ (D ζ R (x) − D τ ζ R (x))D ˜ ˜ (K w)(x) ≤ |x − x| R and terms like the following: τ σ −τ −1 D ζ R (x) |y| |(a N w)(x − y) − (a N w)(x˜ − y)| dy; D R3
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H. Andréasson, M. Kunze, G. Rein
if |σ − τ | = 3 we throw one derivative onto a N w. The latter quantity together with its first order derivatives is Hölder continuous. The factor in front of the integral vanishes for |x| ≤ R, so we need only consider |x| ˜ ≥ |x| ≥ R. Since the domain of integration extends only over y with |y − x| ≤ R N or |y − x| ˜ ≤ R N we can on the domain of integration estimate |y| ≥ |x| − |x − y| ≥ R − R N ≥ R/2or analogously with x˜ instead of x, where we assume that R > 2R N . Since D σ −τ |y|−1 ≤ |y|− j with j ≥ 1 the term under consideration can be estimated by C R −1 |x − x| ˜ α and altogether we conclude that K w − K R w C 3,α (R3 ) ≤ C/R. Recalling the definition of the norm · X1 we see that the following chain of estimates finally shows that K R → K in the corresponding operator norm as desired: |a N (y)w(y)| dy |∇(K w)(x) − ∇(K R w)(x)| ≤ ζ R (x) 3 |x − y|2 R |a N (y)w(y)| dy + R −1 |∇χ (x/R)| 3 |x − y| R C C ≤ 1{|x|≥R} 2 + R −1 1{R≤|x|≤2R} |x| |x| ≤ C(1 + |x|)−β−1 R −(1−β) . To complete the proof we have to show that K R is compact for any R > 0 on the space X1 . First the fact that a N ∈ Cc2 (R3 ) implies that K R : C 3,α (R3 ) → C 3,1/2 (R3 ) is continuous, and the same is true for K R : C 3,α (R3 ) → C 3,1/2 (B 3R (0)), where we note that all the functions K R w with w ∈ C 3,α (R3 ) are supported in B3R (0). Since α < 1/2 the embedding C 3,1/2 (B 3R (0)) → C 3,α (B 3R (0)) is compact, and because of the support property we conclude that K R : X1 → X1 is compact; on ∇ K R w the weight (1 + |x|)1+β only amounts to multiplication with a bounded function. 7. Discussion of Condition (φ3) In this section we investigate Condition (φ3) for the case of the polytropic steady states (2.3). We first allow for the general range k ∈]−1/2, 7/2[ of polytropic exponent. Using the elementary integration formula, 1 2 k (k + 1) k+ 32 s − |v| dv = (2π )3/2 (7.1) s+ , s ∈ R, 2 (k + 25 ) R3 +
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
43
the Poisson equation in (φ2) is found to be 1 2 k+ 23 3/2 (k + 1) (r U ) = 4π(2π ) − U ) (E 0 N + N r2 (k + 25 ) for U N = U N (r ). According to [19] there exists a solution U N such that U N (0) < E 0 , U N (0) = 0, U N (R N ) = E 0 , U N (r ) > E 0 for r > R N , and U N (r ) > 0 for r ∈]0, R N [. For z := E 0 − U N this means that −
1 2 (k + 1) 3 (r z ) = 4π cn z +n , where n := k + ∈ ]1, 5[, cn := (2π )3/2 , r2 2 (k + 25 )
and furthermore z(0) > 0, z (0) = 0, z(R N ) = 0, and z (r ) < 0 for r ∈ ]0, R N [. In terms of z the function a N from (φ3) reads a N (r ) = −(2π )3/2
k(k) (k +
3 2)
k+ 21
z(r )+
= −n cn z(r )n−1 + ,
where once more (7.1) was used. Thus condition (φ3) is equivalent to 4π n cn r 2 z(r )n−1 < 6. +
(7.2)
Now consider the function ζ (s) := z(αs) for α := (4π cn )−1/2 . It is found to satisfy the Emden-Fowler equation −
1 2 (s ζ ) = ζ+n s2
(7.3)
and ζ (0) > 0, ζ (0) = 0, ζ (s0 ) = 0 for s0 := R N /α, as well as ζ (s) < 0 for s ∈ ]0, s0 [. In terms of s = α −1r condition (7.2) becomes s 2 ζ (s)n−1 < +
6 . n
(7.4)
The left-hand side can be conveniently expressed by means of the dynamical systems representation of (7.3). For, let U (t) := −
sζ (s)n sζ (s) ≥ 0, V (t) := − ≥ 0, t := ln s, ζ (s) ζ (s)
where we consider t ∈] − ∞, ln s0 [. Then U˙ = U (3 − U − nV ), V˙ = V (U + V − 1),
(7.5)
and U (t)V (t) = s 2 ζ (s)n−1 + , which provides the relation to (7.4). Thus we have to verify that U (t)V (t) < 6/n. In the terminology of [6, p. 501], where m = 0, ζ is an E-solution to (7.3). Thus [6, Prop. 5.5] implies that (U (t), V (t)) lies in the unstable manifold of the fixed point P3 = (3, 0) of (7.5). In particular, we have limt→−∞ (U (t), V (t)) = (3, 0). Also note that P3 is of saddle type with eigenvalues −3 and 2; the corresponding eigenvectors are (1, 0) and (−3n/5, 1). Since the line V = n1 (3 − U ) separates the regions U˙ > 0 (below the line) and U˙ < 0 (above the line), a phase plane analysis reveals that
44
H. Andréasson, M. Kunze, G. Rein
we must always have U (t) ≤ 3, so that W (t) := U (t)V (t) ≤ 3V (t). In addition, it is calculated that V and W are solutions to the system V˙ = V (V − 1) + W, W˙ = W (2 − (n − 1)V ),
(7.6)
such that limt→−∞ (V (t), W (t)) = (0, 0). The origin is a fixed point of saddle type for (7.6), the eigenvalues are −1 and 2 with corresponding eigenvectors (1, 0) and (1, 3). 2 2 Note that W˙ > 0 for V < n−1 , W˙ < 0 for V > n−1 , V˙ > 0 above the curve V → V (1 − V ), and V˙ < 0 below this curve. Since the curve has unity slope at V = 0, it follows that (V (t), W (t)), lying in the unstable manifold of the origin, will be above the curve for t very negative. Then a phase plane analysis shows that this property persists for all times. In particular, we always have V˙ > 0, and W is increasing until 2 it reaches its maximal value for t0 such that V (t0 ) = n−1 . Thus our original problem of proving (φ3) is equivalent to showing that W (t0 ) = max W < 6/n. Thanks to the preceding observations, the parametrized curve t → (V (t), W (t)) for t ∈] − ∞, t0 ] can be rewritten as a curve W = W (V ) in the (V, W )-plane which solves dW W (2 − (n − 1)V ) = , dV V (V − 1) + W
(7.7)
2 and which is such that W (0) = 0 and W ( n−1 ) = max W .
Lemma 7.1. If k < 7/2 is sufficiently close to 7/2, then (φ3) holds for φ given by (2.3). 2 Proof. If W (V ) < 1 < 6/n for all V ∈]0, n−1 ], then we are done. Hence we assume 2 that W (V0 ) = 1 for some V0 ∈]0, n−1 ]. Then 1 = W (V0 ) ≤ 3V0 yields V0 ≥ 1/3. Since W (V ) ≥ 1 for V ≥ V0 , it follows that V (V − 1) + W = (V − 1)2 + V + W − 1 ≥ V , so that by (7.7),
ln(max W ) =
max W W =1
dW ≤ W
2 n−1
V0
(2 − (n − 1)V˜ ) ˜ dV V˜
(2 − (n − 1)V˜ ) ˜ dV V˜ 1/3 6 1 = 2 ln − 2 + (n − 1). n−1 3 ≤
2 n−1
Therefore 36 1 (n − 1) − 2 . max W ≤ exp (n − 1)2 3
(7.8)
At n = 5 the relation 9 −2/3 6 < e 4 5 holds. Hence it follows from (7.8) that max W < 6/n is verified for n sufficiently close to n = 5.
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
45
The method of proof for the preceding lemma can be refined as follows. Fix A < 6/n. 2 Then W (V ) < A for V ∈ [0, n−1 ] would be acceptable. Hence we can assume that 2 W (V0 ) = A for some V0 ∈]0, n−1 ]. Then A = W (V0 ) ≤ 3V0 shows that V0 ≥ A/3. From W (V ) ≥ A for V ≥ V0 we obtain ln(max W ) − ln A 2 max W ˜) n−1 (2 − (n − 1) V dW ≤ d V˜ = 2 W V˜ − V˜ + A V0 W (V0 ) 2 ˜) n−1 (2 − (n − 1) V d V˜ ≤ 2 V˜ − V˜ + A A/3 5−n 5−n 3 − 2A =√ arctan √ + arctan √ 4A − 1 4 A − 1(n − 1) 3 4A − 1 2 9[An − 2(A + 1)n + 6 + A] n−1 ln . − 2 A(A + 6)(n − 1)2 Therefore max W ≤ A (n), where n−1 2 A(A + 6)(n − 1)2 A (n) := A 9[An 2 − 2(A + 1)n + 6 + A] 5−n 3 − 2A 5−n arctan √ . + arctan √ × exp √ 4 A − 1(n − 1) 4A − 1 3 4A − 1
For different A it can be checked (e.g. using Maple) for which values n ∈ ]1, min{6/A, 5}[ the relation A (n) < 6/n is verified. Taking A = 1, we get at least n ∈ [2.6, 5[, for A = 6/5 we get at least n ∈ [2.35, 4.85], and for A = 2, we get at least n ∈ [2.1, 2.5]. In summary, the desired relation max W < 6/n can be obtained for at least n ∈ [2.1, 5[, which corresponds to at least k ∈ [0.6, 3.5[ in (2.3). Notice however that the regularity assumption on φ requires k > 2. 8. The Field Equations Hold For a metric of the form (1.4) the components 00, 11, 12, 22, and 33 of the field equations are nontrivial. We have so far obtained a solution ν, B, ξ of the reduced system (2.6), (2.7), (2.12), where the appearing components of the energy momentum tensor are induced by a phase space density f which satisfies the Vlasov equation (1.2). We define E αβ := G αβ − 8π c−4 Tαβ so that the Einstein field equations become E αβ = 0. By (2.7), E 11 + E 22 = 0.
(8.1)
Using this information (2.6) says that 2
ρ 2 B 2 E 00 + c2 e4ν/c E 33 = 0 or 2
c2 e4ν/c E 00 + ρ 2 B 2 E 33 = 0.
(8.2)
46
H. Andréasson, M. Kunze, G. Rein
The Vlasov equation implies that ∇α T αβ = 0, and ∇α G αβ = 0 due to the contracted Bianchi identity, where ∇α denotes the covariant derivative corresponding to the metric (1.4). We want to use these relations to show that the remaining components of E αβ vanish also, but there is a technical catch: The metric, more specifically ξ , is only C 2 . To overcome this complication we approximate ξ by C 3 functions ξn . The induced Einstein αβ tensor G n again satisfies the Bianchi identity. Taking β = 1 and letting n → ∞ we obtain the equation ∂z B E 12 − ρ Be−2ξ (B + ρ∂ρ B) E 33 = 0, ∂z E 12 + 4∂z μ + B
(8.3)
where (8.2) has been used to eliminate E 00 and we recall that ξ = ν/c2 + μ. Here ∂z E 12 is at first a distributional derivative, but since all other terms in the equation are continuous this derivative indeed exists in the classical sense. The same approximation maneuver can be performed for β = 2 to obtain the equation ∂ρ E
12
1 ∂ρ B E 12 − ρ 2 Be−2ξ ∂z B E 33 = 0, + 4∂ρ μ + + ρ B
(8.4)
which holds for ρ > 0. However, if we multiply this equation with ρ we obtain an equation which holds for ρ ≥ 0. This is because E 12 (0, z) = 0 which is nothing but the boundary condition (2.13) on the axis of symmetry which we have incorporated into our integration of (2.12). We eliminate E 33 from (8.3), (8.4) and write the resulting equation for E 12 in terms of X := ρe4μ B E 12 . The result is the equation ∂ρ X −
ρ∂z B ∂z X = 0, B + ρ∂ρ B
which again holds for ρ ≥ 0. Since X (0, z) = 0 and since any characteristic curve of this equation intersects the axis of symmetry ρ = 0, we conclude that X vanishes identically. By (8.3) the same is true for E 33 so that E 12 = E 33 = E 00 = 0. Finally we observe that by (2.12), ∂ρ B ∂z B 1+ρ E 12 = 0. (E 11 − E 22 ) + ρ B B Since E 12 = 0 this means that E 11 = E 22 , and with (8.1) we conclude that E 11 = E 22 = 0, and all the non-trivial field equations are satisfied. Acknowledgement. The authors would like to thank Marcus Ansorg for useful discussions.
Existence of Axially Symmetric Static Solutions of Einstein-Vlasov System
47
References 1. Andersson, L., Beig, R., Schmidt, B.: Static self-gravitating elastic bodies in Einstein gravity. Commun. Pure Appl. Math. 61, 988–1023 (2008) 2. Andersson, L., Beig, R., Schmidt, B.: Rotating elastic bodies in Einstein gravity. Commun. Pure Appl. Math. 63, 559–589 (2009) 3. Andréasson, H.: The Einstein-Vlasov System/Kinetic Theory. Living Rev. Relativity 8 (2005), available at http://relativity.livingreviews.org/Articles/lrr-2005-z, 2005 4. Bardeen, J.: Rapidly rotating stars, disks, and black holes. In: Black Holes / Les Astres Occlus, ed. by C. DeWitt, B. S. DeWitt, Les Houches, 1972, London-NewYork-Paris: Gordon and Breach, 1973 5. Batt, J., Faltenbacher, W., Horst, E.: Stationary spherically symmetric models in stellar dynamics. Arch. Rat. Mech. Anal. 93, 159–183 (1986) 6. Batt, J., Pfaffelmoser, K.: On the radius continuity of the models of polytropic gas spheres which correspond to positive solutions of the generalized Emden-Fowler equations. Math. Meth. Appl. Sci. 10, 499–516 (1988) 7. Deimling, K.: Nonlinear Functional Analysis, Berlin-New York: Springer, 1985 8. Fjällborg, M., Heinzle, M., Uggla, C.: Self-gravitating stationary spherically symmetric systems in relativistic galactic dynamics. Math. Proc. Cambridge Philos. Soc. 143, 731–752 (2007) 9. Heilig, U.: On Lichtenstein’s analysis of rotating Newtonian stars. Ann. de l’Inst. H. Poincaré, Physique Théorique 60, 457–487 (1994) 10. Heilig, U.: On the existence of rotating stars in general relativity. Commun. Math. Phys. 166, 457–493 (1995) 11. Jackson, D.: Classical Electrodynamics, New York: Wiley, 1975 12. Lichtenstein, L.: Untersuchung über die Gleichgewichtsfiguren rotierender Flüssigkeiten, deren Teilchen einander nach dem Newtonschen Gesetze anziehen. Erste Abhandlung. Homogene Flüssigkeiten. Allgemeine Existenzsätze. Math. Z. 1, 229–284 (1918) 13. Lichtenstein, L.: Gleichgewichtsfiguren rotierender Flüssigkeiten, Berlin: Springer, 1933 14. Lieb, E., Loss, M.: Analysis, Providence, RI: Amer. Math. Soc., 1997 15. Müller, C.: Spherical Harmonics. Lecture Notes in Mathematics 17, Berlin: Springer, 1966 16. Rein, G.: Static solutions of the spherically symmetric Vlasov-Einstein system. Math. Proc. Camb. Phil. Soc. 115, 559–570 (1994) 17. Rein, G.: Stationary and static stellar dynamic models with axial symmetry. Nonlinear Analysis; Theory, Methods & Applications 41, 313–344 (2000) 18. Rein, G., Rendall, A.: Smooth static solutions of the spherically symmetric Vlasov-Einstein system. Ann. de l’Inst. H. Poincaré, Physique Théorique 59, 383–397 (1993) 19. Rein, G., Rendall, A.: Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics. Math. Proc. Camb. Phil. Soc. 128, 363–380 (2000) 20. Schulze, A.: Existence of axially symmetric solutions to the Vlasov-Poisson system depending on Jacobi’s integral. Commun. Math. Sci. 6, 711–727 (2008) 21. Wald, R.: General Relativity, Chicago, IL: Chicago University Press, 1984 Communicated by P.T. Chru´sciel
Commun. Math. Phys. 308, 49–80 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1343-5
Communications in
Mathematical Physics
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles Krzysztof Gaw¸edzki1 , David P. Herzog2,3 , Jan Wehr2 1 Laboratoire de Physique, C.N.R.S., ENS-Lyon, Université de Lyon, 46 Allée d’Italie, 69364 Lyon, France 2 Department of Mathematics, The University of Arizona, 617 N. Santa Rita Ave., P. O. Box 210089, Tucson,
AZ 85721-0089, USA. E-mail: [email protected]
3 Department of Mathematics, Duke University, Box 90320, Durham, NC 27708-0320, USA
Received: 3 September 2010 / Accepted: 1 July 2011 Published online: 30 September 2011 – © Springer-Verlag 2011
Abstract: We study a simple stochastic differential equation that models the dispersion of close heavy particles moving in a turbulent flow. In one and two dimensions, the model is closely related to the one-dimensional stationary Schrödinger equation in a random δ-correlated potential. The ergodic properties of the dispersion process are investigated by proving that its generator is hypoelliptic and using control theory. 1. Introduction Transport by turbulent flows belongs to phenomena whose understanding is both important for practical applications and abounds in intellectual challenges. Unlike the reputedly difficult problem of turbulence per se, turbulent transport allows simple modeling that accounts, at least qualitatively, for many of its observable features. The simplest of such models study transport properties of synthetic random velocity fields with presupposed distributions that only vaguely render the statistics of realistic turbulent velocities. The advection by velocity fields of quantities like temperature or tracer density may be derived from the dynamics of the Lagrangian trajectories of fluid elements. In synthetic velocity ensembles, such dynamics is described by a random dynamical system. One of the best studied schemes of this type is the so called Kraichnan model based on a Gaussian ensemble of velocities decorrelated in time but with long-range spatial correlations [8,18]. In this case, the random dynamical system that describes the Lagrangian flow is given by stochastic differential equations (SDE’s). It was successfully studied with the standard tools of the theory of random dynamical systems, but it also led to non-trivial extensions of that theory [12,19,20]. The problem of turbulent transport of matter composed of small but heavy particles (like water droplets in turbulent atmosphere) may be also studied by modeling turbulent velocities by a random synthetic ensemble, but it requires a modification of the previous approach. The reason is that heavy particles do not follow Lagrangian trajectories due to their inertia. On the other hand, the assumptions of the time decorrelation of random
50
K. Gaw¸edzki, D. P. Herzog, J. Wehr
velocities may be more realistic for inertial particles on scales where the typical relaxation time of particle trajectories (called the Stokes time) is much longer than the typical correlation time of fluid velocities. There have been a number of papers that pursued the study of dynamics of inertial particles with various simplifying assumptions, see e.g. [1–7,9,17,23,24,30,34]. The primary focus of those studies, combining analytical and numerical approaches, was the phenomenon of intermittent clustering of inertial particles transported by turbulent flow. A good understanding of that phenomenon is of crucial importance for practical applications. The aim of the present article is to show that the simplest among the models of inertial particles dynamics are amenable to rigorous mathematical analysis. More concretely, we study the SDE’s that describe the pair dispersion of close inertial particles in shortly correlated moderately turbulent homogeneous and isotropic d-dimensional velocity fields (not necessarily compressible). Such models were discussed in some detail in [2,17,24,25,30,34]. In particular, it was noted in [34] that the d = 1 version of the model is closely related to the one-dimensional stationary Schrödinger equation with δ-correlated potential studied already in the sixties of the last century [14] as a model for Anderson localization. As was stressed in [17], the d = 2 model for the inertial particle dispersion is also related to the one-dimensional stationary Schrödinger equation, but this time with δ-correlated complex potential. The models for dispersion were used to extract information about the (top) Lyapunov exponent for the inertial particles which is a rough measure of the tendency of particles to separate or to cluster [2,34]. The numerical calculations of the Lyapunov exponents in two or more dimensional models of particle dispersion presumed certain ergodic properties that seemed consistent with the results of simulations but were not obvious. From the point of view of the first order SDE for the particle dispersion and its time derivative, the ergodic properties of the dispersion process are a cumulative effect of the noisy advection in the subspace of the time-derivatives of dispersion and of the deterministic drift acting in the transverse directions of the phase space. We shall establish such properties rigorously by showing the hypoellipticity of the generator of the Markov process solving the corresponding SDE and by proving the irreducibility of the process with the help of control theory. For a quick introduction to such, by now standard, methods, we refer the reader to [11,31]. More information about the ergodic theory of Markov processes may be found in the treatise [26]. The main trouble in our analysis comes from configurations where two inertial particles (almost) coincide in space but have different velocities, leading to phase-space caustics [35] and enhanced spatial concentrations of particles (the “sling effect” of [7]). The reason is that these are the configurations where the diffusive part of the generator of the dispersion process vanishes. We show that in more than one space dimension two close inertial particles avoid such situations since, almost surely, the pair dispersion does not vanish. Some quantitative measure of such avoidance is provided. The result does not preclude the presence of caustic-like configurations in the evolution of three or more close inertial particles [35]. The paper is organized as follows. In Sect. 2, we present the SDE modeling the inertial particle dispersion. In Sect. 3, we recall its relation to models of one-dimensional Anderson localization. Section 4 establishes the hypoelliptic properties of the generator of the dispersion process. In Sect. 6, we introduce the (real-)projective version of the dispersion process whose compact space of states may be identified with the (2d − 1)dimensional sphere S 2d−1 . Section 5 is devoted to proving that the dispersion process is controllable. Together with the hypoelliptic properties of the generator, this implies that the projectivized version of the process has a unique invariant probability measure
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
51
with a smooth strictly positive density. The analytic expression for such a measure may be written down explicitly in d = 1 but not in higher dimensions. The smoothness and strict positivity of its density provide, however, in conjunction with the isotropy assumption, valuable information about the equal-time statistics of the projectivized dispersion. The isotropy permits to project further the projectivized dispersion to the quotient space S 2d−1 /S O(d). For d = 2, this space may be identified with with the complex projective space PC1 = C ∪ ∞ and the projected process with the complex-projectivized dispersion. In fact, the point at infinity, corresponding to phase-space caustics, may be dropped from PC1 since the complex-projectivized dispersion process stays in C with probability one. For d ≥ 3, the quotient space S 2d−1 /S O(d) is not smooth but has an open dense subset that may be identified with the complex upper-half-plane that the projected process never leaves. These non-explosive behaviors are established in Sect. 8 by constructing a Lyapunov function with appropriate properties. Physically they mean that the relative motion of a pair of close inertial particles avoids the caustic configurations, and for d ≥ 3, also the direction along the line that joins the particles. Results about behavior of the invariant density of the dispersion process projected to S 2d−1 /S O(d), established in Sect. 7, provide some qualitative information about such avoidance. Finally, in Sect. 9 and Appendix B, we demonstrate how the ergodic properties of the projectivized dispersion process proven in the paper lead to the formulae for the top Lyapunov exponent for inertial particles that were used in the physical literature. Appendix A derives a formula, used in the main text, expressing the S O(2d)-invariant measure on S 2d−1 in terms of S O(d) invariants. 2. Basic Equations The motion in a turbulent flow of a small body of large density, called below an inertial particle, is well described by the equation [1,2,23,24,34] 1 r¨ = − τ r˙ − u(t, r) , (2.1) where r(t) is the position of the particle at time t and u(t, r) is the fluid velocity field. Relation (2.1) is the Newton equation with the particle acceleration determined by a viscous friction force proportional to the relative velocity of the particle with respect to the fluid. Constant τ is the Stokes time. Much of the characteristic features of the distribution of non-interacting inertial particles moving in the flow according to Eqs. (2.1) is determined by the dynamics of the separation δr(t) ≡ ρ(t), called particle dispersion, of very close trajectories. In a moderately turbulent flow, the particle dispersion evolves according to the linearized equation: 1 ρ¨ = − τ ρ˙ − (ρ · ∇)u(t, r(t)) (2.2) or, in the first-order form: 1
ρ˙ = τ χ ,
1
χ˙ = − τ χ + (ρ · ∇)u(t, r(t)).
(2.3)
For sufficiently heavy particles, the correlation time of (∇u)(t, r(t)) is short with respect to the Stokes time τ and one may set in good approximation [2] ∇ j u i (t, r(t)) dt = d S ij (t),
(2.4)
52
K. Gaw¸edzki, D. P. Herzog, J. Wehr
where d S(t) is a matrix-valued white noise with the isotropic covariance i ik i k i k D ik d S j (t) d Slk (t ) = D ik jl δ(t − t ) dt dt , jl = A δ δ jl + B(δ j δl + δl δ j ). (2.5) Positivity of the covariance requires that A ≥ |B|,
A + (d + 1)B ≥ 0.
(2.6)
Incompressibility implies that A + (d + 1)B = 0, but we shall not impose it, in general. We shall only assume that A + 2B > 0 for d = 1 and that A > 0 for d ≥ 2. After the substitution of (2.4), Eq. (2.2) becomes the linear SDE 1
ρ¨ = − τ ρ˙ +
1 d S(t) ρ τ dt
(2.7)
that may be written in the first order form in a more standard notation employing differentials as 1 0 ρ ρ τ dt d = . (2.8) χ χ d S(t) − τ1 dt We shall interpret the latter SDE using the Itô convention, but the Stratonovich convention would lead to the same process. The solution of Eq. (2.8) exists with probability 1 for all times and has the form
t 0 1 ρ(t) ρ(0) τ ds = T exp , (2.9) χ (t) χ (0) d S(s) − τ1 ds 0 where the time ordered exponential may be defined as the sum of its Wiener chaos decomposition
t 0 T exp d S(s) 0 ·
0 d S(sn )
0 0
1 τ ds − τ1 ds
(sn −sn−1 )
e
0 0
=
∞
0 0
e
1 τ − τ1
0<s1 <···<sn
n=0
1 τ − τ1
(t−sn )
···
0 d S(s1 )
0 0
s1
e
0 0
1 τ − τ1
(2.10)
that converges in the L 2 -norm for functionals of the white noise d S(t). The resulting stochastic process (ρ(t), χ (t)) ≡ p(t) is Markov and has generator 1 j l ik 1 ρ ρ D jl ∇χ i ∇χ k . (2.11) L = τ χ · ∇ρ − χ · ∇χ + 2 i, j,k,l
In other words, for smooth functions f , d f ( p(t)) = (L f )( p(t)) (2.12) dt for · · · denoting the expectation. For the process p(t) given by Eq. (2.9), p(t) = 0 for all t ≥ 0 if p(0) = 0. On the other hand, if p(0) = 0 then p(t) = 0 with probability 1 for all t ≥ 0 so that we may restrict the space of states of the Markov process p(t) to R2d \{0} ≡ R2d =0 .
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
53
3. Relation to One-Dimensional Localization In the lowest dimensions, there is a simple relation between the stochastic process p(t) = (ρ(t), χ (t)) constructed above and simple models of Anderson localization in one space dimension. Let us set t
ψ(t) = e 2τ ρ(t)
(3.1)
exponentially blowing up the long-time values of ρ(t). Equation (2.7) implies that − ψ¨ +
1 d S(t) τ dt
ψ =−
1 4τ 2
ψ,
(3.2)
or, in the first order form, 0 ψ d = ξ d S(t) +
1 4τ dt
ψ . ξ 0
1 τ dt
(3.3)
Similarly as before, the above SDE defines a Markov process. Clearly, t 1 (ψ(t), ξ (t)) = e 2τ ρ(t), χ (t) + 2 ρ(t) .
(3.4)
Viewing t as the one-dimensional spatial coordinate, Eq. (3.2) takes the form of the vector-like stationary Schrödinger equation −
d2 ψ dt 2
+ V (t) ψ = E ψ,
(3.5)
where V (t) = τ1 d S(t) dt plays the role of the random matrix-valued white-noise potential and E = − 4τ1 2 of the (negative) energy. In particular, in d = 1, ψ(t) is a real scalar function and so is the δ-correlated potential V (t) =
1√ dβ(t) , A + 2B τ dt
(3.6)
where β(t) is the Brownian motion. The scalar version of Eq. (3.5) was studied in [14] as a model of one-dimensional Anderson localization, see also [21]. In d = 2, interpreting ψ as a complex number ψ 1 + iψ 2 , one may replace the matrix valued δ-correlated potential V (t) in the SDE (3.5) with the complex valued one V (t) =
1 τ
√
A + 2B
dβ 1 (t) √ dβ 2 (t) , +i A dt dt
(3.7)
where β 1 (t), β 2 (t) are two independent Brownian motions (the two realizations of V (t) lead to the Markov processes with the same generator and, consequently, with the same law). Consequently, as stressed in [17], Eq. (3.5) in d = 2 may be viewed as a model of localization for a one-dimensional non-hermitian random Schrödinger operator of the type not studied before.
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K. Gaw¸edzki, D. P. Herzog, J. Wehr
4. Hypoelliptic Properties of the Generator The generator (2.11) of the particle dispersion process p(t) = (ρ(t), χ (t)) has certain non-degeneracy properties which imply smoothness of the transition probabilities. Let us start with the following fact about the covariance (2.5) of the matrix-valued white noise d S(t): Lemma 4.1. We have D ik jl =
d
Eδ ij δnm + Fδ im δ jn + Gδni δ mj Eδlk δnm + Fδ km δln + Gδnk δlm
(4.1)
m,n=1
for √ E = d −1 − A + B + A + (d + 1)B , √ 1 √ F = 2 ( A + B + A − B), √ 1 √ G = 2 ( A + B − A − B).
(4.2)
Proof. The right hand side of Eq. (4.1) is: (E 2 d + 2E F + 2E G)δ ij δlk + (F 2 + G 2 )δ ik δ jl + 2F Gδli δ kj .
(4.3)
Hence, in order to satisfy Eq. (4.1), we must have A = F 2 + G2,
B = E 2 d + 2E(F + G) = 2F G.
(4.4)
The assumed values of E, F, G solve these equations.
Define the vector fields 1 X 0 = τ χ · ∇ρ − χ · ∇χ ,
ρ j Eδ ij δnm + Fδ im δ jn + Gδni δ mj ∇χ i , X nm = i, j
Ynm
=−
ρ j Eδ ij δnm + Fδ im δ jn + Gδni δ mj ∇ρ i
i, j
+ (χ j + ρ j ) Eδ ij δnm + Fδ im δ jn + Gδni δ mj ∇χ i = τ [X 0 , X nm ], i, j
Z nm
=−
(2χ j + ρ j ) Eδ ij δnm + Fδ im δ jn + Gδni δ mj ∇ρ i
i, j
+ (χ j + ρ j ) Eδ ij δnm + Fδ im δ jn + Gδni δ mj ∇χ i = τ [X 0 , Ynm ].
(4.5)
i, j
Using Eq. (4.1), one infers that Eq. (2.11) giving the generator L may be rewritten in the form:
1 (X nm )2 . (4.6) L = X0 + 2 m,n
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
55
Remark 4.2. The process p(t) may be equivalently obtained from the SDE d p = X 0 ( p) dt +
d
X nm ( p) dβmn (t),
(4.7)
m,n=1
where βmn (t) are independent Brownian motions. Here, we adopt the Stratonovich convention and hence the generator corresponding to that equation has the form (4.6) so that the latter SDE leads to a process with the same law as p(t). The convention of the stochastic integral we choose, however, is insignificant as the process ρ(t) is of bounded variation. In order to establish hypoelliptic properties of L, we shall use the following nondegeneracy relation satisfied by the vector fields X nm , Ynm and Z nm : Proposition 4.3. Suppose that p = (ρ, χ ) = 0. Then the vectors X nm ( p), Ynm ( p), Z nm ( p)
with m, n = 1, . . . , d
(4.8)
Proof. First suppose that ρ = 0 so that χ = 0. We have
χ j Eδ ij δnm + Fδ im δ jn + Gδni δ mj ∇χ i . Ynm (0, χ ) =
(4.9)
span the 2d-dimensional space.
i, j
Let φ ∈ Rd . Then
(α χ n χ m +β χ n φ m ) Ynm (0, χ ) = [E(αχ 2 +βχ ·φ)+ Fα χ 2 +G(αχ 2 +βχ ·φ)]χ ·∇χ m,n
+ Fβ χ 2 φ · ∇χ .
(4.10)
Setting α=−
χ ·φ E +G , (E + F + G)F (χ 2 )2
β=
1 F χ2
(4.11)
(note that F > 0 and E + F + G > 0), we obtain
(α χ n χ m + β χ n φ m ) Ynm (0, χ ) = φ · ∇χ .
(4.12)
m,n
Hence the vector φ · ∇χ is in the span of (4.8) for arbitrary φ. We have still to show that an arbitrary vector σ · ∇ρ is in that span. To this aim note that
Z nm (0, χ ) = −2 χ j Eδ ij δnm + Fδ im δ jn + Gδni δ mj ∇ρ i + (...)i ∇χ i . (4.13) i, j
i
Proceeding as before, we show that an appropriate combination of Z nm (0, χ ) gives the vector
σ · ∇ρ + (...)i ∇χ i (4.14) i
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K. Gaw¸edzki, D. P. Herzog, J. Wehr
from which the term (...)i ∇χ i may be removed by subtracting an appropriate combination of Ynm (0, χ ). That ends the proof of the claim of Proposition 4.3 for ρ = 0. Suppose now that ρ = 0. Proceeding as before, we see that arbitrary vector φ · ∇χ may be obtained by taking an appropriate combination of the vectors X nm (ρ, χ ). Similarly, an arbitrary vector σ · ∇ρ may be obtained as an appropriate combination of the vectors Ynm (ρ, χ ) and X nm (ρ, χ ). This completes the proof of Proposition 4.3.
The representation (4.6) and Proposition 4.3 imply, by virtue of Hörmander’s theory [11,16,28,29], the following result: Corollary 4.4. The operators L , L † , ∂t − L , ∂t − L † and 2∂t − L ⊗ 1 − 1 ⊗ L † are 2d 2d 2d hypoelliptic1 on R2d =0 , R+ × R=0 and R+ × R=0 × R=0 , respectively. In particular, the hypoellipticity of 2∂t − L ⊗ 1 − 1 ⊗ L † implies that the transition probabilities of the dispersion process p(t), Pt ( p0 , d p) = Pt ( p0 , p) d p,
(4.15)
have densities (annihilated by 2∂t − L ⊗ 1 − 1 ⊗ L † ) that are smooth functions of (t, p0 , p) for t > 0 and away from the origin in R2d . 5. Control Theory and Irreducibility The additional important property of the process p(t) restricted to R2d =0 is its irreducibility assured by the strict positivity of the smooth transition probability densities Pt ( p0 , p) for all t > 0 and p0 = 0 = p. The latter property results, according to Stroock-Varadhan’s Support Theorem [33], see also [31], from the controllability of the process p(t) on R2d =0 that is established in the following Proposition 5.1. For every T > 0 and p0 = 0 = p1 there exists a piecewise smooth 2 curve [0, T ] t → (u nm (t)) ∈ Rd such that the solution of the ODE
u nm (t) X nm ( p) (5.1) p˙ = X 0 ( p) + m,n
with the initial condition p(0) = p0 satisfies p(T ) = p1 . Proof. First suppose that ρ0 = 0 = ρ1 . Let [0, T ] t → ρ(t) be any curve such that ρ(0) = ρ0 , ρ(T ) = ρ1 ,
˙ τ ρ(0) = χ0 , ˙ ) = χ1 , τ ρ(T
(5.2)
˙ and such that ρ(t) = 0 for all t ∈ [0, T ]. Set χ (t) = τ ρ(t). Let φ(t) = τ χ˙ (t) + χ (t). Then the formula u nm =
1 α ρn ρm τ
+ β ρn φm ,
(5.3)
(5.4)
1 A differential operator D on a domain is hypoelliptic if for all distributions f, g such that D f = g, smoothness of g on an open subset U ⊂ implies smoothness of f on U .
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
57
where now α=−
ρ·φ E+G , (E + F + G)F (ρ 2 )2
β=
1 , F ρ2
(5.5)
defines smooth control functions [0, T ] t → (u nm (t)) such that Eq. (5.1) holds. Now suppose that ρ0 = 0 = ρ1 . Choose 0 < < 21 T and for 0 ≤ t ≤ , set t ρ(t) = 1 − e− τ χ0
(5.6)
˙ and χ (t) = τ ρ(t). Then p(t) = (ρ(t), χ (t)) satisfies Eq. (5.1) with u nm (t) ≡ 0 for 0 ≤ t ≤ , with the correct initial condition at t = 0. Note that (5.7) (ρ( ), χ ( )) = (1 − e− τ )χ0 , e− τ χ0 . Since, by the assumptions, χ0 = 0, we infer that ρ( ) = 0 and the solution of Eq. (5.1) for ≤ t ≤ T may be constructed as in the previous point but taking (5.7) as the initial conditions at t = . Similarly, if ρ0 = 0 = ρ1 then set for T − ≤ t ≤ T, T −t ρ(t) = 1 − e τ χ1 , (5.8) ˙ and χ (t) = τ ρ(t). Then p(t) = (ρ(t), χ (t)) satisfies Eq. (5.1) with u nm (t) ≡ 0 for T − ≤ t ≤ T , with the correct final condition at t = T . One has (5.9) (ρ(T − ), χ (T − )) = (1 − e τ )χ1 , e τ χ1 . Since, by the assumptions, χ1 = 0 now, we infer that ρ(T − ) = 0 and the solution of Eq. (5.1) for 0 ≤ t ≤ T − with ρ(t) = 0 may be constructed as in the first point but taking (5.9) as the final condition at t = T − . Finally, if ρ0 = 0 = ρ1 , we combine the above solutions for 0 ≤ t ≤ and T − ≤ t ≤ T with vanishing u nm with the solution with ρ(t) = 0 and appropriate u nm (t) for ≤ t ≤ T − ∈.
Remark 5.2. Note that the solution p(t) of the ODE (5.1) satisfying p(0) = p0 = 0 and p(T ) = p1 = 0 is everywhere nonzero. 6. Projection of the Dispersion to S2 d−1 The generator L of the process commutes with the multiplicative action of R+ on R2d given by
σ
p −→ σ p
(6.1)
for σ > 0. It follows that if p(0) = 0 then the projection [ p(t)] ≡ π (t)
(6.2)
R2d =0 /R+
is also a Markov process whose genof the process p(t) on the quotient space erator may be identified with L acting on functions on R2d =0 that are homogeneous of 2d degree zero. The quotient space R=0 /R+ may be naturally identified with the sphere S 2d−1 = { (ρ, χ ) | ρ 2 + χ 2 = R 2 }
(6.3)
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K. Gaw¸edzki, D. P. Herzog, J. Wehr
for a fixed R and we shall often use this identification below. The transition probabilities Pt (π0 ; dπ ) of the process π (t) are obtained by projecting the original transition probm m m abilities from R2d =0 to the quotient space. Note that the vector fields X 0 , X n , Yn , Z n 2d also commute with the action R+ so may be identified with vector fields on R=0 /R+ and Eq. (4.6) still holds. Viewed as vector fields on S 2d−1 , X nm , Ynm and Z nm still span at each point the tangent space to S 2d−1 . It follows that the operators L , L † , ∂t − L , ∂t − L † , 2∂t − L ⊗ 1 − 1 ⊗ L † (with the adjoints defined now with respect to an arbitrary measure with smooth positive density on S 2d−1 , e.g. the normalized standard S O(2d)-invariant one μ0 (dπ )) are still hypoelliptic and the transition probabilities of the projected process have smooth densities Pt (π0 ; π ) with respect to μ0 (dπ ) for t > 0. Consequently, the process π (t) is strongly Feller: for bounded measurable functions f on S 2d−1 , the functions Pt (π0 ; dπ ) f (π ) = Pt (π0 ; π ) f (π ) μ0 (dπ ) (6.4) (Tt f )(π0 ) = S 2d−1
S 2d−1
are continuous (and even smooth) for t > 0. Besides, the projected process is still irreducible since Pt (π0 ; π ) > 0 for all t > 0 and π0 , π ∈ S 2d−1 . The latter property follows from the relation between Pt (π0 ; π ) and Pt ( p0 ; p) and from the strict positivity of the latter away from the origin of R2d . The gain from projecting the process p(t) to the compact space S 2d−1 is that the projected process π (t) has necessarily invariant probability measures μ(dπ ). In particular, each weak-topology accumulation point for T → ∞ of the Cesaro means T
−1
T
Pt (π0 ; dπ ) dt
(6.5)
0
provides such a measure.2 Since the (a priori distributional) density n(π ) of an invariant measure is annihilated by L † , the hypoellipticity of the latter operator assures that n(π ) is a smooth function. The invariance relation Pt (π0 , π ) n(π0 ) μ0 (dπ0 ) = n(π ) (6.6) S 2d−1
together with the strict positivity of Pt (π0 , π ) implies then the strict positivity of the density n(π ) of the invariant measure and, in turn, the uniqueness of the latter (different ergodic invariant measures have to have disjoint supports, so that there may be only one such measure), see e.g. [31] for more details. One obtains this way Theorem 6.1. The projected process π (t) has a unique invariant probability measure μ(dπ ) with a smooth strictly positive density n(π ). The smoothness of the densities Pt (π0 ; π ) implies by the Arzelà-Ascoli Theorem that the operators of the semigroup Tt on the space C(S 2d−1 ) of continuous function on S 2d−1 with the sup-norm, defined by Eq. (6.4), are compact for t > 0. The uniqueness of the invariant measure implies then that the spectrum of Tt is strictly inside the unit disk except for the geometrically simple eigenvalue 1 corresponding to the constant eigenfunctions, see [31]. It follows that the process π (t) is exponentially mixing: 2 Probability measures on a compact space form a compact set in weak topology.
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
59
Theorem 6.2.
f 1 (π (t1 )) f 2 (π (t2 ))
−→
f 1 (π ) μ(dπ )
t1 →∞ t2 −t1 →∞
f 2 (π ) μ(dπ ),
(6.7)
exponentially fast for continuous functions f 1 , f 2 .
7. Properties of the Invariant Measure Due to the isotropy of the covariance (2.5), the generator L of the process π (t) commutes with the action of the rotation group S O(d) induced on S 2d−1 by the mappings
O
(ρ, χ ) −→ (Oρ, Oχ )
(7.1)
for O ∈ S O(d). As a consequence, the process π (t) stays Markov when projected to the quotient space Pd = S 2d−1 /S O(d). The unique invariant measure μ(dπ ) of the process π (t) has to be also invariant under S O(d) and its projection to Pd provides the unique invariant probability measure of the projected process.3 The projected invariant measure may be expressed in terms of invariants of the S O(d)-action. Such invariants will be chosen as the following dimensionless combinations: • for d = 1 where P1 = S 1 , x=
χ , ρ
(7.2)
• for d = 2 where P2 = PC1 , x=
ρ·χ ρ2
and
y=
ρ1χ 2 − ρ2χ 1 ρ2
(7.3)
with z = x + i y providing the inhomogeneous complex coordinate of PC1 , • for d ≥ 3, ρ·χ x= 2 ρ
and
ρ 2 χ 2 − (ρ · χ )2 y= . ρ2
(7.4)
Note that |x| → ∞ for d = 1 and |(x, y)| → ∞ in d ≥ 2 correspond to phase-space caustics where the dispersion ρ tends to zero with χ = τ ρ˙ staying finite. The right-hand side of the d ≥ 3 expression for y would give in d = 2 the absolute value of y. The configurations with y = 0 correspond to collinear ρ and χ . The quotient spaces Pd are not smooth for d ≥ 3. 3 To see the uniqueness, note that averaging over the action of S O(d) maps C(S 2d−1 ) to C(P ) and that d dual map sends invariant measures for the projected process to invariant measures of π (t).
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K. Gaw¸edzki, D. P. Herzog, J. Wehr
7.1. d = 1 case. In one dimension, Eq. (2.8) implies that 1 d x = − τ x + x 2 dt + d S(t).
(7.5)
The invariant probability measure on S 1 is easily found [14,34] to have the form dμ = η(x)d x with 2 x 2 1 1 x 3 +x 2 −1 − τ (A+2B) 3 x 3 +x 2 3 τ (A+2B) η(x) = Z d x d x, e (7.6) e −∞
where Z is the normalization constant. Since the normalized rotationally invariant meadx sure on S 1 = {(ρ, χ ) | ρ 2 + χ 2 = R 2 } has the form dμ0 = π(1+x 2 ) , it follows from our
general result that the density n(x) = π(1 + x 2 ) η(x) of the invariant measure relative to dμ0 must be smooth and positive at x = ∞, i.e. at the origin when expressed in the variable x −1 . In particular, η(x) = O(|x|−2 )
|x| → ∞,
for
(7.7)
which may also be easily checked directly. In one dimension, the generator L given by Eq. (2.11) acts on a function f (x) according to the formula: 1
1
(L f )(x) = − τ (x 2 + x) ∂x f (x) + 2 (A + 2B) ∂x2 f (x).
(7.8)
It coincides with the generator of the process satisfying the SDE (7.5). The trajectories of the latter process with probability one explode to −∞ in finite time but, in the version of the process that describes the projectivized dispersion of the one-dimensional inertial particle, they re-enter immediately from +∞. 7.2. d = 2 case. In two dimensions, the invariant measure on S 3 has to have the form dμ =
1 2π
η(z, z¯ ) d 2 z d arg(ρ).
(7.9)
On the other hand, the S O(4)-invariant normalized measure on S 3 is dμ0 =
1 2π
η0 (z, z¯ ) d 2 z d arg(ρ)
(7.10)
1 . π(1 + |z|2 )2
(7.11)
with η0 (z, z¯ ) =
It follows from the general result obtained above that the density of dμ relative to dμ0 , n(z, z¯ ) =
η(z, z¯ ) , η0 (z, z¯ )
(7.12)
has to extend to a smooth positive function on PC1 , i.e. to be smooth and positive at zero when expressed in the variables (z −1 , z¯ −1 ). In particular, η(z, z¯ ) = O(|z|−4 )
for
|z| → ∞.
(7.13)
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
61
The unique invariant probability measure of the Markov process obtained by projecting 1 π (t) from S 3 to S 3 /S O(2) = PC1 has the form (7.9) with 2π d arg(ρ) on the right hand side dropped. Note that the relation (7.13) implies that ∞ η(x, y) dy = O(|x|−3 ) for |x| → ∞, (7.14) −∞
√ by changing variables y → 1 + x 2 y in the integral. Such behavior was heuristically argued for and numerically checked in [2]. In two dimensions, the generator L of Eq. (2.11) acts on S O(d) invariant functions f (x, y) according to the formula: 1
(L f )(x, y) = − τ (x 2 − y 2 + x) ∂x f (x, y) 1
− τ (2x y + y) ∂ y f (x, y) 1
1
+ 2 (A + 2B) ∂x2 f (x, y) + 2 A ∂ y2 f (x, y).
(7.15)
It coincides with the generator of the process z(t) = (x + i y)(t) in the complex plane given by the SDE [30] √ √ 1 (7.16) dz = − τ z + z 2 dt + A + 2B dβ 1 (t) + i A dβ 2 (t), where β 1 (t) and β 2 (t) are two independent Brownian motions. 7.3. d ≥ 3 case. Finally, in three or more dimensions, the invariant measure on S 2d−1 has to have the form dμ = η(x, y) d x d y d[O],
(7.17)
where O ∈ S O(d) is the rotation matrix such that O −1 ρ is along the first positive half-axis in Rd and O −1 χ lies in the half-plane spanned by the first axis and the second positive half-axis. Note that, generically, O is determined modulo rotations in (d − 2) remaining directions. d[O] stands for the normalized S O(d)-invariant measure on S O(d)/S O(d − 2). In the same notation, the S O(2d)-invariant normalized measure on S 2d−1 takes the form dμ0 = η0 (x, y) d x d y d[O].
(7.18)
for η0 (x, y) =
(d − 1)2d−1 y d−2 , π(1 + x 2 + y 2 )d
(7.19)
as is shown in Appendix A. As before, it follows from the general analysis that the function n(x, y) =
η(x, y) η0 (x, y)
(7.20)
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K. Gaw¸edzki, D. P. Herzog, J. Wehr
is smooth and positive on the sphere S 2d−1 = {(ρ, χ ) | ρ 2 + χ 2 = R 2 }. In particular, this implies that η(x, y) = O(y d−2 )
for
y 0,
(7.21)
i.e. for ρ and χ becoming parallel or χ 2 becoming small and η(x, y) = O(|x|−2d )
for
|x| → ∞
(7.22)
when ρ 2 → 0 but the angle between ρ and χ stays away from a multiple of π2 . The smoothness and positivity of n(x, y) on S 2d−1 imply (again by changing variables √ y → 1 + x 2 y in the integral) that now ∞ η(x, y) dy = O(|x|−d−1 ) for |x| → ∞. (7.23) 0
A straightforward calculation shows that, in three or more dimensions, the action on L on S O(d)-invariant functions f (x, y) is given by a generalization of Eq. (7.15): 1
(L f )(x, y) = − τ (x 2 − y 2 + x) ∂x f (x, y) 1
− τ (2x y + y −
τ A(d−2) ) ∂y 2y
1
1
f (x, y)
+ 2 (A + 2B) ∂x2 f (x, y) + 2 A ∂ y2 f (x, y).
(7.24)
It coincides with the generator of the process z(t) = (x + i y)(t) in the complex plane given by the SDE [2,17] √ √ 1 τ A(d−2) dz = − τ z + z 2 − i dt + A + 2B dβ 1 (t) + i A dβ 2 (t), (7.25) 2 Im(z) which upon setting d = 2 reduces to the SDE (7.16). 8. Absence of Explosion in the Complex (Half-)Plane Let us set
Qd =
R2 H+
if if
d = 2, d ≥ 3,
(8.1)
where H+ = {(x, y) | y > 0} is the upper-half plane. Note that Q d may be identified with an open dense subset of the quotient space Pd = S 2d−1 /S O(d) using the S O(d)-invariants (7.3) or (7.4) on S 2d−1 . We shall often use the complex combination x + i y as a coordinate on Q d . In the present section, we shall show that for d ≥ 2 the unique solution of the SDE (7.25) starting from z ∈ Q d remains in Q d for all times t ≥ 0 with probability one. This will also have to be the property of the projection of the process π (t) to the quotient space Pd = S 2d−1 /S O(d) when described in the complex coordinate z = x + i y. Indeed, the
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
63
coincidence of the generators of the two processes will assure that they have the same law. Let us start by generalizing and simplifying (7.25). Let w(t) = z(t) + 1/2, where z(t) solves (7.25) with z(0) = x + i y ∈ Q d . Clearly, w(t) satisfies an SDE of the form dw =
2κ1 2κ2 τ b(d−2) 1 −w 2 + α + i dt + dβ (t) + i dβ 2 (t), τ τ Im(w)
1 τ
(8.2)
where α = a1 + ia2 ∈ C, b > 0, κ1 ≥ 0, κ2 > 0, and β 1 (t) and β 2 (t) are two independent Brownian motions. When d = 2, the term proportional to b(d − 2) is absent from (8.2). When d ≥ 3, we suppose that τ b(d − 2) ≥ κ2 . Clearly in (7.25), all of these assumptions are met under the given substitution. Since w(t) is a horizontal shift of z(t), w(t) stays in Q d with probability one for all times if and only if z(t) does. Employing methods of refs. [15,22,27,31], we shall estimate the time at which the process w(t) leaves Q d . To this end, it is easy to see that there exists a sequence of precompact open subsets {On | n ∈ N} of Q d such that On ↑ Q d as n → ∞. Thus we may define stopping times: τn = inf{s > 0 | w(s) ∈ Onc },
(8.3)
for n ∈ N. Let τ∞ be the finite or infinite limit of τn as n → ∞. Definition 8.1. We say that the solution w(t) is non-explosive if P [τ∞ = ∞] = 1.
(8.4)
Naturally, in order to show that w(t) remains in Q d for t ≥ 0 with probability one, it is enough to prove that w(t) is non-explosive. Let M be the generator of the process w(t) = x(t) + i y(t). We see that for f ∈ C ∞ (Q d ): (Md f )(x, y) = − τ (x 2 − y 2 − a1 )∂x f (x, y)− τ (2x y − a2 −τ b(d − 2)y −1 )∂ y f (x, y) 1
+
κ1 2 ∂ τ x
1
f (x, y) +
κ2 2 ∂ τ y
f (x, y),
(8.5)
where the term τ b(d − 2)y −1 is absent for d = 2. Let us define ∂ Qd =
∞ {(x, y) ∈ R2 | y = 0} ∪ ∞
if d = 2, if d ≥ 3,
(8.6)
with ∞ denoting the point compactifying R2 . To ensure condition (8.4), it suffices to construct a (Lyapunov) function d ∈ C∞ (Q d ) that satisfies:
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K. Gaw¸edzki, D. P. Herzog, J. Wehr
(I) d (x, y) ≥ 0 for all (x, y) ∈ Q d , (II) d (x, y) → ∞ as (x, y) → ∂ Q d , (x, y) ∈ Q d , (III) Md d (x, y) ≤ Cd (x, y) for all (x, y) ∈ Q d , where C > 0 is a positive constant. See, for example, [27]. We will show: Theorem 8.2. If κ1 ≥ 0 and κ2 > 0 for d = 2 or τ b(d − 2) ≥ κ2 > 0 for d ≥ 3 then there exists d ∈ C ∞ (Q d ) that satisfies (I), (II), and Md d (x, y) → −∞ as (x, y) → ∂ Q d , (x, y) ∈ Q d .
(IV)
Given such d , clearly d + 1 will satisfy (I), (II) and (III). We will then have: Theorem 8.3. Under the assumptions of Theorem 8.2, the solution w(t) of the SDE (8.2) stays in Q d for all times t > 0 with probability one if w(0) = x + i y ∈ Q d . Corollary 8.4. This implies the same result about the solution z(t) of the SDE (7.25) with A > 0 and A + 2B ≥ 0. The existence of the Lyapunov function with the properties asserted in Theorem 8.2 has another consequence. It allows to show that T 1 lim lim inf T Pt (w, Onc ) dt = 0 (8.7) n→∞ T →∞
0
for the SDE (8.2) and On ↑ Q d as before, implying the existence of an invariant measure on Q d , see Theorems 4.1 and 5.1 in Chap. III of [15]. If the generator of the process is elliptic, then the same tools that we used for the projectivized dispersion (i.e. hypoellipticity and control theory [31]) show that the invariant measure must have a smooth strictly positive density and be unique. This gives: Theorem 8.5. Under the assumptions of Theorem 8.2, the system (8.2) on Q d has an invariant measure which is unique and has a smooth strictly positive density if κ1 > 0. Remark 8.6. Theorem 8.5 allows to reaffirm and strengthen what has already been proven earlier since it implies the existence of an invariant measure for the system (7.25) if A > 0 and A + 2B ≥ 0 and its uniqueness if A + 2B > 0. Given the non-explosivity result of Corollary 8.4, the approach taken earlier implied the existence and the uniqueness of an invariant measure for the system (7.25) under more stringent conditions: A > 0, A ≥ |B| and A + (d + 1)B ≥ 0. The construction of the Lyapunov function d is split up into two cases: d = 2 and d ≥ 3. The existence of d for d ≥ 3 will be easy, given 2 . Thus we shall first construct 2 . 8.1. d = 2 case. It is not easy to write down a globally defined function 2 that satisfies (I), (II), and (IV) in all of Q 2 = R2 . This is because the signs of the coefficients of the vector fields in M2 vary over different regions in R2 . We shall thus construct functions that satisfy these properties in different regions, the union of which is R2 . We shall then glue together these functions to form one single globally defined 2 . One should note that this idea is similar in spirit to that of M. Scheutzow in [32]. Let r = x 2 + y 2 . For the rest of Subsection 8.1, we will drop the use of the subscript 2 in M2 and 2 . We first need the following:
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65
Definition 8.7. Let X ⊂ R2 be unbounded. We say that a function f (x, y) → ±∞ as r → ∞ in X if f (x, y) → ±∞ as (x, y) → ∞, (x, y) ∈ X . Definition 8.8. Let X ⊂ R2 be unbounded and let ϕ ∈ C ∞ (X ) satisfy (i) ϕ ≥ 0 for all (x, y) ∈ X , (ii) ϕ → ∞ as r → ∞ in X , (iii) Mϕ → −∞ as r → ∞ in X . We call ϕ a Lyapunov function in X corresponding to M and denote N (α, κ1 , κ2 , X ) = { Lyapunov functions in X corresponding to M } . We shall abbreviate “Lyapunov function” by LF. Definition 8.9. Let X ⊂ R2 be unbounded and f, g : X → R. We shall say that f is asymptotically equivalent to g in X and write f X g if lim
r →∞
f (x, y) = 1, g(x, y)
where the limit is taken only over points (x, y) ∈ X . It is clearly sufficient to construct LFs in regions that cover R2 , except, possibly, a large ball about the origin. The constructions will be done in a series of propositions. The possibly daunting multitude of parameters is designed to make the gluing possible. There is a total of five LFs in five different regions and the details that follow are not difficult to verify. The crucial LF is the fifth one, ϕ5 , defined in a region where explosion occurs in a nonrandom equation, i.e., when α = κ1 = κ2 = 0 in (8.2). Proposition 8.10. Let X 1 = {x ≥ 1} ⊂ R2 , C1 > 0, and δ ∈ (0, 1/2). Define ϕ1 (x, y) = C1 (x 2 + y 2 )δ/4 .
(LF1)
We claim that ϕ1 ∈ N (α, κ1 , κ2 , X 1 ) for all α ∈ C, κ1 ≥ 0, κ2 > 0. Proof. ϕ1 is nonnegative everywhere in R2 , hence everywhere in X 1 . ϕ → ∞ as r → ∞ in all of R2 , hence in all of X 1 . It is easy to check that ∂x x ϕ1 and ∂ yy ϕ1 both go to zero as r → ∞. Thus dropping second order terms in the expression for Mϕ1 , we have C1 δ C 1 δ a1 x + a2 y x(x 2 + y 2 )δ/4 + 2 2 (x 2 + y 2 )1−δ/4 C1 δ x(x 2 + y 2 )δ/4 → −∞ − 2
τ Mϕ1 X 1 − X1
(8.8)
as r → ∞ in X 1 , since x ≥ 1 in X 1 .
We need a remark before we move onto the next region. Let R ⊂ R2 be the real axis. Remark 8.11. Let f (x, y) = u(x, |y|) be a twice differentiable function in X \R. Then (τ M f )(x, y) = κ1 u x x (x, |y|) + κ2 u |y||y| (x, |y|) + (y 2 − x 2 + a1 ) u x (x, |y|) + (−2x|y| + sgn(y)a2 ) u |y| (x, |y|). Proof. Apply the chain rule to the operator ∂|y| .
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In the following arguments, often the function will be of the form f (x, y) = u(x, |y|). The above remark will allow for simplifications in the argument for property (iii) in Definition 8.8. Proposition 8.12. Let C2 > 0, δ ∈ (0, 1/2) and ϕ2 = C2 (−x + |y|δ/2 ).
(LF2)
Then ϕ2 ∈ N (α, κ1 , κ2 , X 2 ) for all α ∈ C and all κ1 ≥ 0, κ2 > 0, where X 2 = {−2 ≤ x ≤ 2} ∩ {|y| ≥ 22/δ }. Proof. ϕ2 is indeed smooth in X 2 since X 2 is bounded away from R. Note that the region was chosen so that ϕ2 ≥ 0 in X 2 . Moreover, since x is bounded in this region, r → ∞ in X 2 if and only if |y| → ∞. Hence, ϕ2 → ∞ in X 2 . By Remark 8.11 and noting that ∂x x ϕ2 = 0 and that ∂|y||y| ϕ2 → 0 as |y| → ∞, we have δ δ τ Mϕ2 (x, y) X 2 C2 (x 2 − y 2 − a1 ) + C2 (−2x|y| + sgn(y)a2 ) |y| 2 −1 2 X 2 − C2 y 2 → −∞
as r → ∞ in X 2 .
Proposition 8.13. Let C3 > 0 and δ ∈ (0, 1/2). Define 2 δ x + y2 ϕ3 = C3 |y|3/2
(LF3)
on X 3 = {x ≤ −1} ∩ {|y| ≥ 1}. Then ϕ3 ∈ N (α, κ1 , κ2 , X 3 ) for all α ∈ C and all κ1 ≥ 0, κ2 > 0. Proof. Smoothness of ϕ3 is not a problem in this region as we are bounded away from R in X 3 . Clearly, ϕ3 ≥ 0 and note that ϕ3 → ∞ as r → ∞ in X 3 . After dropping the δ(δ − 1)-terms which are negative, we obtain: δ−1
2 15x 2 x3 2 x + y2 1 τ Mϕ3 (x, y) ≤ C3 δ + κ + κ − 1 2 |y|3/2 |y|3/2 4|y|7/2 4|y|3/2 |y|3/2 2 3x 2a1 x 1 + x|y|1/2 + 3/2 + sgn(y)a2 − + 5/2 |y| 2|y|1/2 2|y| 2 δ−1 x3 x + y2 X 3 C3 δ + x|y|1/2 |y|3/2 |y|3/2 (8.9) = δ x ϕ3 → −∞ as r → ∞ in X 3 since x ≤ −1 in X 3 .
Proposition 8.14. Let C4 > 0, η > 1 and δ ∈ (0, 1/2). Define ϕ4 (x, y) = C4 on X 4 = {x ≤ −1} ∩
|x|2δ + |y|2δ
(LF4)
3
|y| 2 δ
1 3 3 η κ2 2 ( 2 δ + 1) √
|x|
≤ |y| ≤ 2 .
Then ϕ4 ∈ N (α, κ1 , κ2 , X 4 ) for all α ∈ C and all κ1 ≥ 0, κ2 > 0.
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67
Proof. Note that ϕ4 is smooth in X 4 since this region excludes both x and y axes. Moreover, ϕ4 → ∞ as r → ∞ in X 4 since then x must approach ∞ as r → ∞, and y is bounded above. Dropping insignificant terms in the expression for Mϕ4 , we see that in X 4: |x|2δ |x|2δ+1 3 3 + κ2 2 δ 2 δ + 1 + δ|x||y|δ/2 τ Mϕ4 (x, y) ≤ C4 − δ 3 3 δ δ+2 |y| 2 |y| 2 δ |x|2δ−1 |x|2δ 1 3 −1 2 − a1 2δ + sgn(y)a δ|y| − sgn(y)a δ 22 22 3 3 |y| 2 δ |y| 2 δ+1 |x|2δ |x|2δ+1 3 3 X 4 C4 −δ + κ2 2 δ 2 δ + 1 3δ 3 |y| 2 δ+2 |y| 2 1 3 3 |x|2δ+1 −1 + κ2 δ+1 = C4 δ 3 2 2 |x||y|2 |y| 2 δ |x|2δ+1 → −∞ ≤ − C4 δ(1 − 1/η2 ) 3 |y| 2 δ in X 4 as r → ∞.
Proposition 8.15. Let C5 , β > 0 and E > 0 such that 2κ2 > Eβ, let ξ > 1, and let ϕ5 (x, y) = C5 (E|x|β − y 2 |x|β+1 )
(LF5)
be defined on 1 E X 5 = {x ≤ −1} ∩ |y| ≤ ξ |x| . Then ϕ5 ∈ N (α, κ1 , κ2 , X 5 ) for all α ∈ C and all κ1 ≥ 0. Proof. The fact that ϕ5 is smooth in X 5 is clear as x ≤ −1 in X 5 . Again by the choice of X 5 , ϕ5 ≥ 0 and ϕ5 → ∞ as r → ∞ in X 5 . Dropping irrelevant terms in the expression for Mϕ5 , we see that in X 5 : τ Mϕ5 (x, y) ≤ C5 κ1 Eβ(β − 1)|x|β−2 − 2κ2 |x|β+1 + Eβ|x|β+1 + (β + 1)y 4 |x|β + C5 a1 (−Eβ|x|β−1 + (β + 1)y 2 |x|β ) − 2a2 y|x|β+1 X 5 C5 (Eβ − 2κ2 ) |x|β+1 → −∞ as r → ∞ in X 5 , as |x| must approach ∞ when r → ∞ in X 5 .
We now have our desired LFs. It is not obvious, however, that the regions X 1 , X 2 , . . . , X 5 cover R2 except, possibly, a bounded region about the origin. To assure that one has to show that X 4 and X 5 overlap. In order to make this more tangible, we will choose some of the parameters given in the previous propositions. With the choices that follow, however, we first need a lemma that says that varying the diffusion coefficients (κ1 , κ2 ) is permitted. This lemma will also be of crucial use later when we glue the LFs to form a globally defined .
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Lemma 8.16. Fix κ2 > 0 and suppose that ∈ N (α, κ1 , κ2 , R2 ) for all α ∈ C and all κ1 ≥ 0. Then for every ι2 > 0, we can find ∈ N (α, ι1 , ι2 , R2 ) for all α ∈ C and all ι1 ≥ 0. α For the proof of Lemma 8.16, we temporarily use the notation M(κ for the gen1 ,κ2 ) erator M given by (8.5).
Proof. Let η > 0 be such that η3 ι2 = κ2 . Define (x, y) = (ηx, ηy). For the function , smoothness and properties (I) and (II) are immediate. Let s = ηx and t = ηy. Then by the chain rule,
as r =
τ M(ια1 ,ι2 ) (x, y) =
τ η
M(ι
=
τ η
M(ι
αη2 3 3 (s, t) 1 η ,ι2 η ) αη2 (s, t) 3 1 η ,κ2 )
→ −∞,
x 2 + y 2 → ∞.
By Lemma 8.16, it is enough to find a function ∈ N (α, κ1 , κ2 , R2 ) for some fixed κ2 > 0 for all α ∈ C and all κ1 ≥ 0. All of the ϕi satisfy these criteria. In fact, ϕi for i = 1, 2, 3, 4 work more generally. The reason that ϕ5 only works for 2κ2 > Eβ is due to the fact that when α = κ1 = κ2 = 0, the solution to (8.2) has an explosive trajectory along the negative real axis. √ Now we choose some parameters. Let E = 5 and ξ = 5/2, so as to make X 5 = {x ≤ −1 ∩ {|y||x|1/2 ≤ 2}. 1 55 1 Let β = 11 4 δ, δ = 7 κ2 , and κ2 ∈ (0, 1). Then Eβ = 28 κ2 < 2κ2 and δ ∈ (0, 2 ). Hence for all i = 1, 2, . . . , 5, ϕi is a LF in the region X i . Decrease κ2 > 0 so that
X 4 ⊃ {x ≤ −1} ∩ {1 ≤ |y||x|1/2 ≤ 2}. One can easily check that X 4 and X 5 overlap in such a way that we have covered all of R2 with X 1 , X 2 , . . . , X 5 except a bounded region about the origin. We fix κ2 > 0 sufficiently small (this will be made precise later). We will construct a function ∈ N (α, κ1 , κ2 , R2 ) for all α ∈ C and all κ1 ≥ 0. The idea is as follows. Note that ϕ1 and ϕ2 are LFs in the region X 1 ∩ X 2 . We shall define a nonnegative smooth auxiliary function ζ (x) ∈ C ∞ (R) such that ζ (x) = 0 for x ≥ 2 and ζ (x) = 1 for x ≤ 1, satisfying some additional properties. We will then show that the combination (1 − ζ )ϕ1 + ζ ϕ2 is a LF in the larger region X 1 ∪ X 2 . Proceeding inductively this way, we shall construct a LF in all of R2 . Let us first define some auxiliary functions needed to construct such a . Let ζ : R → R+ be a C ∞ function such that 1 for x ≤ 1, ζ (x) = 0 for x ≥ 2, and ζ (x) < 0 for all x ∈ (1, 2). We define the smooth function μ : R → R+ as the horizontal shift of ζ , three units to the left, i.e., μ(x) = ζ (x + 3) for x ∈ R.
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
Let
⎧ ⎪ ⎨ ζ (|y|) ν(x, y) = 0 ⎪ ⎩0
69
for x ≤ −2, for |y| ≥ 2, for x > −1,
and assume that ν is C ∞ outside of the ball B4 . Let q : (−∞, −1] × R → R be defined by ⎧ ⎪ if |x|1/2 |y| ≥ 2, ⎨1 q(x, y) = |x|1/2 |y| − 1 if 1 < |x|1/2 |y| < 2, ⎪ ⎩0 if |x|1/2 |y| ≤ 1, and
r (t) =
1 exp − 1−(2t−1) 2
if 0 < t < 1,
0
otherwise.
Let s(x) =
1 N
x
−∞
r (t) dt,
where N = R r (t)dt. Now define a function on R2 by ⎧ ⎪ ⎨ s(q(x, y)) if x ≤ −1, ρ(x, y) = 1 if |y| ≥ 3, ⎪ ⎩1 if x ≥ −1/2. 2 ) outside of B . Clearly, ρ is C ∞ (R 4 √ ˙ ϕ for x 2 + y 2 ≥ r 2 (δ) by Let r (δ) = max 4, 24/δ + 4 . Define
⎧ ϕ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ζ ⎪ ⎪ ϕ2 + (1 − ζ )ϕ1 ⎪ ⎪ ϕ2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ μϕ3 + (1 − μ)ϕ2 ϕ(x, y) = ϕ3 ⎪ ⎪ ⎪ νϕ4 + (1 − ν)ϕ3 ⎪ ⎪ ⎪ ⎪ ϕ4 ⎪ ⎪ ⎪ ⎪ ⎪ ρϕ4 + (1 − ρ)ϕ5 ⎪ ⎪ ⎩ ϕ5 and now
(x, y) =
if if if if if if if if if
x ≥ 2, 1 < x < 2, − 1 ≤ x ≤ 1, − 2 < x < −1, x ≤ −2, |y| ≥ 2 x ≤ −2, 1 < |y| < 2, x ≤ −2, |y| ≤ 1, |x|1/2 |y| ≥ 2, x ≤ −2, 1 < |x|1/2 |y| < 2, x ≤ −2, |x|1/2 |y| ≤ 1,
ϕ(x, y) arbitrary positive and smooth
if x 2 + y 2 ≥ Br (δ) , if x 2 + y 2 < Br (δ) .
It is easy to see can be chosen to be nonnegative and C ∞ (R2 ). With the aid of Lemma 8.16, the following lemma implies Theorem 8.2 in the d = 2 case.
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Lemma 8.17. For κ2 sufficiently small, ∈ N (α, κ1 , κ2 , R2 ) for all α ∈ C and all κ1 ≥ 0. Proof. Clearly, is smooth and satisfies Properties I and II. Since Mϕi → −∞ as r → ∞ in X i for each i, all we must verify is that M → −∞ as r → ∞ in the overlapping regions. Let us recall the choices that have already been made: E = 5, ξ =
√
5 , 2
β=
11 δ, 4
δ=
κ2 , 7
and note that κ2 ∈ (0, 1) was chosen such that X 4 ⊃ {x ≤ −1} ∩ {1 ≤ |y||x|1/2 ≤ 2}. Pick C5 > C4 = C3 > C2 > C1 . Consider first ψ1 := ζ ϕ2 + (1 − ζ )ϕ1 defined in the region Y1 = {1 < x < 2} ∩ Brc(δ) . We have τ Mψ1 = ζ τ Mϕ2 + (1 − ζ )τ Mϕ1 + (y 2 − x 2 + a1 )ζ (ϕ2 − ϕ1 ) + κ1 (ζ (ϕ2 − ϕ1 ) + 2ζ (∂x ϕ2 − ∂x ϕ1 )) Y1 − ζ C2 y 2 − (1 − ζ )
C1 δ x(x 2 2
+ y 2 )δ/4 + (y 2 − x 2 + a1 )ζ (C2 − C1 )|y|δ/2
+ κ1 (ζ (C2 − C1 )|y|δ/2 − 2ζ C2 ).
(8.10)
Note that if x is bounded away from 1 and 2 in (1, 2), the dominant term above is y 2 ζ (x)(C2 − C1 )|y|δ/2 → −∞. Note also that as x → 1 or x → 2, ζ , ζ → 0. But, the first two terms in the expression above decay to −∞ at least as fast as −C|y|δ/2 , where C > 0 is a constant independent of ζ and x. Thus we may choose > 0 so that whenever x ∈ (1, 2)\(1 + , 2 − ), Mψ1 decays at least as fast as −D|y|δ/2 , where D is some positive constant. We now consider ψ2 := μϕ3 + (1 − μ)ϕ2 in the region Y2 = {−2 < x < −1} ∩ Brc(δ) . We have τ Mψ2 = μ τ Mϕ3 + (1 − μ)τ Mϕ2 + (y 2 − x 2 + a1 )μ (ϕ3 − ϕ2 ) + κ1 (μ (ϕ3 − ϕ2 ) + 2μ (∂x ϕ3 − ∂x ϕ2 ) Y2 μ δxϕ3 − (1 − μ)C2 y 2 + (y 2 − x 2 + a1 )μ (C3 − C2 )|y|δ/2 Y2
+ κ1 (μ (C3 − C2 )|y|δ/2 + 2μ C2 ) μ C3 x|y|δ/2 − (1 − μ)C2 y 2 + (y 2 − x 2 + a1 )μ (C3 − C2 )|y|δ/2 + κ1 (μ (C3 − C2 )|y|δ/2 + 2μ C2 ).
Note that, for the very same reasons as in the case of ψ1 , Mψ2 → −∞ as r → ∞ in Y2 .
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71
Let ψ3 := νϕ4 + (1 − ν)ϕ3 in the region Y3 = {x ≤ −2} ∩ {1 < |y| < 2} ∩ Brc(δ) . Thus τ Mψ3 = ν τ Mϕ4 + (1 − ν)τ Mϕ3 + (−2x|y| + sgn(y)a2 )(∂|y| ν)(ϕ4 − ϕ3 ) Y3
2 + κ2 (∂|y| ν)(ϕ4 − ϕ3 ) + 2κ2 (∂|y| ν)∂|y| (ϕ4 − ϕ3 ) ν τ Mϕ4 + (1 − ν)τ Mϕ3 → −∞.
This is true since we chose C3 = C4 . Hence the first order term (−2x|y| + sgn(y)a2 )(∂|y| ν)(ϕ4 − ϕ3 ) approaches infinity at worst as fast as C|x|, where C is a positive constant. The second 2 ν)(ϕ − ϕ ) is a bounded function in this region. Moreover the term order term κ2 (∂|y| 4 3 2κ2 (∂|y| ν)∂|y| (ϕ4 − ϕ3 ) at worst approaches infinity as fast as D|x|2δ , where D is some positive constant. But, both Mϕ3 and Mϕ4 approach negative infinity at least as fast as −D|x|2δ+1 , where D is another positive constant. This gives the desired result. Let ψ4 := ρϕ4 + (1 − ρ)ϕ5 , in the region Y4 = {1 < |x|1/2 |y| < 2} ∩ {x ≤ −2} ∩ Brc(δ) . Note that τ Mψ4 = s(q)τ Mϕ4 + (1 − s(q))τ Mϕ5 + ∂x (s(q))(ϕ4 − ϕ5 )(y 2 − x 2 + a1 ) + ∂|y| (s(q))(ϕ4 − ϕ5 )(−2x|y| + sgn(y)a2 ) + κ1 ∂x2 (s(q))(ϕ4 − ϕ5 ) 2 (s(q))(ϕ4 − ϕ5 ) + 2κ1 ∂x (s(q))∂x (ϕ4 − ϕ5 ) + κ2 ∂|y| + 2κ2 ∂|y| (s(q))∂|y| (ϕ4 − ϕ5 ).
Note that in the expression above, we may drop the κ1 ∂x2 (s(q))(ϕ4 − ϕ5 ) and 2κ1 ∂x (s(q))∂x (ϕ4 −ϕ5 ) terms, as they are asymptotically less than other terms. Dropping other obviously insignificant terms, we obtain: τ Mψ4 Y4 s(q)τ Mϕ4 + (1 − s(q))τ Mϕ5 − ∂x (s(q))(ϕ4 − ϕ5 )x 2 2 − ∂|y| (s(q))(ϕ4 − ϕ5 )2x|y| + κ2 ∂|y| (s(q))(ϕ4 − ϕ5 ) + 2κ2 ∂|y| (s(q))∂|y| (ϕ4 − ϕ5 ) ≤ F(x, y),
where F is a smooth function satisfying: C5 δ 11 δ+1 |x| 4 4 − ∂x (s(q))(ϕ4 − ϕ5 )x 2 − ∂|y| (s(q))(ϕ4 − ϕ5 )2x|y|
F Y4 − s(q)C4 (1 − 1/η2 )
δ
23δ/2
11
|x| 4 δ+1 − (1 − s(q))
2 (s(q))(ϕ4 − ϕ5 ) + 2κ2 ∂|y| (s(q))∂|y| (ϕ4 − ϕ5 ). + κ2 ∂|y|
(8.11)
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K. Gaw¸edzki, D. P. Herzog, J. Wehr
By the choice of C5 > C4 , it is easy to see that for large |x| in this region, there are constants C, D > 0 such that 11 |x| 4 δ+1 2 κ2 ∂|y| (s(q))(ϕ4 − ϕ5 ) + 2κ2 ∂|y| (s(q))∂|y| (ϕ4 − ϕ5 ) ≤ Cκ2 r (q) g(x, y)2
and 11
−∂x (s(q))(ϕ4 − ϕ5 )x 2 − ∂|y| (s(q))(ϕ4 − ϕ5 )2x|y| ≤ −Dr (q)|x| 4 δ+1 , where the constants C, D and the function r (q) are independent of κ2 . r (q) is a function that goes to zero faster than any power of the function g(x, y) := |(|x|1/2 |y| − 1)(|x|1/2 |y| − 2)| as |x|1/2 |y| → 1 or 2. But note that, for all > 0, we may choose κ2 so small so that D>
Cκ2 g(x, y)2
for 1 + ≤ |x|1/2 |y| ≤ 2 − . From the first two terms in (8.11), we obtain at least 11 −C δ x 4 δ+1 decay for some C > 0 independent of κ2 for all 1 ≤ x 1/2 |y| ≤ 2. But since every other term in the expression goes to zero faster than every power of g as |x|1/2 |y| → 1 or 2, we can choose an > 0 so small as above so that Mψ4 → −∞ for all 1 < |x|1/2 |y| < 2. This completes the proof.
8.2. d ≥ 3 case. Here we shall complete the proof of Theorem 8.2 for d ≥ 3. Recall that for d = 2 and > 0 sufficiently small, we defined 2 (x, y) := (x, y) ∈ C ∞ (Q 2 ) that satisfies (I), (II), and (IV) for all α ∈ C, κ1 ≥ 0, and κ2 ∈ (0, ). Lemma 8.16 implied then that for κ2 > 0 arbitrary, the function 2,η (x, y) := 2 (ηx, ηy) ∈ C ∞ (Q 2 ) satisfied (I), (II), and (IV) for all α, provided that η−3 = 2κ2 . Let us fix κ2 > 0. For d ≥ 3, we shall define d,η := 2,η + log(1 + log2 (ηy/2)).
(8.12)
Lemma 8.18. For fixed κ2 > 0, d,η is a smooth function on Q d that satisfies (I), (II), and (IV). Proof. By definition, d,η is smooth and nonnegative in Q d = H+ . Clearly, d,η → ∞ as (x, y) → ∂ Q d = {(x, y) ∈ R2 : y = 0} ∪ {∞}, (x, y) ∈ Q d . Thus we must verify property (IV). To this end, note that: τ Md d,η = τ Md 2,η + τ Md (log(1 + log2 (ηy/2))) b(d − 2) ∂ y 2,η = τ M2 2,η + τ y 2 log(ηy/2) + −2x y 2 + a2 y + τ b(d − 2) − κ2 2 y (1 + log2 (ηy/2)) 2κ2 (1 − log2 (ηy/2)) . + 2 y (1 + log2 (ηy/2))2
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
73
Case 1. Suppose first that y ≥ 2η−1 . It is easy to check that there exist positive constants K 1 , K 2 > 0 such that b(d − 2) ∂ y 2,η ≤ K 1 , y 2 log(ηy/2) 2κ2 (1 − log2 (ηy/2)) ≤ K2. + (a2 y + τ b(d − 2) − κ2 ) 2 y (1 + log2 (ηy/2)) y 2 (1 + log2 (ηy/2))2 τ
If also x > −2η−1 then − 2x
log(ηy/2) ≤ K3 1 + log2 (ηy/2)
(8.13)
for a positive constant K 3 , whereas for x ≤ −2η−1 , − 2x
log(ηy/2) ≤ −K 4 x. 1 + log2 (ηy/2)
(8.14)
Since M2 2,η → −∞ as (x, y) → ∂ Q d and y ≥ 2η−1 , and, besides, if x ≤ −2η−1 and y ≥ 2η−1 then 2,η (x, y) is equal to the rescaled function ϕ3 so that, by (8.9), M2 2,η ≤ K 5 x(x 2 + y 2 )δ/4 for some K 5 > 0, we infer that Md d,η → −∞ as (x, y) → ∂ Q d and y ≥ 2η−1 . Case 2. Suppose now that 0 < y < 2η−1 . If |x| < 2η−1 , then (x, y) → ∂ Q d if and only if y ↓ 0. Since 2,η is smooth on R2 , there exists a constant K 6 > 0 such that τ M2 2,η +
1 τ b(d − 2) ∂2,η ≤ K6 y ∂y y
for (x, y) ∈ (−2η−1 , 2η−1 ) × (0, 2η−1 ). Hence, recalling the assumption τ b(d − 2) ≥ κ2 , we have on this rectangle: 1 2 log(ηy/2) + −2x y 2 + a2 y + τ b(d − 2) − κ2 2 y y (1 + log2 (ηy/2)) 2 2κ2 (1 − log (ηy/2)) + 2 y (1 + log2 (ηy/2))2 1 2κ2 (1 − log2 (ηy/2)) ≤ K7 + 2 → −∞ y y (1 + log2 (ηy/2))2
τ Md d,η ≤ K 6
as y ↓ 0. If x ≥ 2η−1 , then 2,η is equal to the rescaled function ϕ1 . We see that by (8.8) there exist constants K 8 , K 9 , K 10 , K 11 > 0 such that τ Md d,η ≤ −K 8 x(x 2 + y 2 )δ/4 + K 9 (x 2 + y 2 )δ/4−1 2 log(ηy/2) + −2x y 2 + a2 y + τ b(d − 2) − κ2 2 y (1 + log2 (ηy/2))
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K. Gaw¸edzki, D. P. Herzog, J. Wehr
+
2κ2 (1 − log2 (ηy/2)) y 2 (1 + log2 (ηy/2))2
≤ −K 8 x(x 2 + y 2 )δ/4 + K 10 x + K 11
1 2κ2 (1 − log2 (ηy/2)) . + y y 2 (1 + log2 (ηy/2))2
Note that as (x, y) → ∂ Q d in this region then x → ∞ or y ↓ 0. It is thus easy to see that τ Md d,η → −∞. If x < −2η−1 , then it is easy to check that ∂ y 2,η is bounded above by the choice of C3 = C4 and C5 > C4 . Then, for some K 12 > 0, τ Md d,η ≤ τ M2 2,η + K 12
1 2κ2 (1 − log2 (ηy/2)) + y y 2 (1 + log2 (ηy/2))2
so that τ Md d,η → −∞ as (x, y) → ∂ Q d in this region.
9. Top Lyapunov Exponent The Lyapunov exponent λ for the dispersion process p(t) = (ρ(t), χ (t)) is the asymptotic rate of growth in time of the logarithm of the length ρ 2 + χ 2 . Suppose that the process starts at t = 0 from p0 = (ρ0 , χ0 ) = 0. Anticipating the existence of the limit below, we shall define: 1 T →∞ T
λ = lim
1 T →∞ T
T
= lim =
1 lim T →∞ T
=
1 lim T →∞ T
ln
0
0
T
T
ρ 2 (T ) + χ 2 (T ) − ln
ρ02 + χ02 )
d ln ρ 2 (t) + χ 2 (t) dt dt L ln ρ 2 (t) + χ 2 (t) dt
(L ln
ρ 2 + χ 2 ) Pt ( p0 , d p) dt,
(9.1)
0
whereL is the generator of the process p(t) given by Eq. (2.11). Note that the function L ln ρ 2 + χ 2 is smooth on R2d \{0} and homogeneous of degree zero. It may be viewed as a function f 0 (π ) on S 2d−1 that, besides, is S O(d)-invariant. We may then rewrite the definition of λ as T 1 f 0 (π ) Pt (π0 ; dπ ) dt. (9.2) λ = lim T T →∞
0
Now, the existence of the limit follows from the fact that the Cesaro means the transition probabilities Pt (π0 ; dπ ) tend in weak topology to the unique invariant probability measure μ(dπ ). Hence (9.3) λ= f 0 (π ) μ(dπ ) and is independent of p0 . The crucial input that allows to make the latter formula more explicit is the formula 1 ρ·χ x (9.4) L ln ρ 2 = = . 2 τ ρ τ
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
75
It implies that f 0 (π ) = L ln
1 χ2 x ρ 2 + χ 2 = + L ln 1 + 2 τ 2 ρ x 1 ln (1 + x 2 ) for d = 1 = + L 2 2 ln (1 + x + y ) for d ≥ 2 τ 2
(9.5)
in terms of the S O(d) invariants with L given by explicit formulae (7.8), (7.15) or (7.24). If the functions ln(1 + x 2 ) in d = 1 and ln (1 + x 2 + y 2 ) in d = 2 that are homogeneous of degree zero on R2d \{0} were smooth, then their contributions to the expectation with respect to the invariant measure on the right-hand side of (9.2) would drop out by the integration by parts. The problem is, however, the lack of smoothness of those functions at ρ = 0 and a more subtle argument is required. 9.1. d = 1 case. In one dimension, Eq. (9.3) reduces to the identity λ=
∞
x τ
−∞
+
1 L ln (1 + x 2 ) η(x) d x 2
(9.6)
with η(x) given by Eq. (7.6). Since the latter integral represents the integration of a smooth function against a smooth measure on S 1 , it converges absolutely. Consequently, the formula for λ may be rewritten in the form: λ = lim
n
n→∞ −n
x τ
+
1 L ln (1 + x 2 ) η(x) d x. 2
(9.7)
† Now the integration by parts and the formula L † η = 0 for the formal adjoint √ L defined with respect to the Lebesgue measure d x show that the term with L ln 1 + x 2 drops out (for the cancellation of the boundary terms it is crucial that the integral is over a symmetric finite interval [−n, n]). We obtain this way the identity
1 lim λ= τ n→∞
n
1 p.v. x η(x) d x ≡ τ −n
∞ −∞
x η(x) d x,
(9.8)
where “p.v.” stands for “principal value”. The result may be expressed [21] by the Airy functions [13]: λ=−
1 d 1 + ln Ai2 (c) + Bi2 (c) 1 2τ 4τ c 2 dc
for c =
1 2
(4τ (A + 2B)) 3
.
(9.9)
1 is the Lyapunov exponent for the one-dimensional Anderson problem The number λ+ 2τ (3.5), recall relation (3.1). It is always positive reflecting the permanent localization in one dimension. On the other hand, λ itself changes sign as a function of τ and A + 2B signaling a phase transition in the one-dimensional advection of inertial particles [34].
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9.2. d ≥ 2 case. In two or more dimensions, Eq. (9.3) becomes x 1 λ= + L ln (1 + x 2 + y 2 ) η(x, y) d x d y, τ 2
(9.10)
where η(x, y) is the density of the invariant measure from Eqs. (7.9) or (7.17). The asymptotic behavior of η(x, y) was established in Sect. 7.2 and Sect. 7.3. We show in Appendix B that it guarantees that the term with L ln (1 + x 2 + y 2 ) may, indeed, be dropped from the expectation on the right hand side of Eq. (9.2) so that 1 λ= x η(x, y) d x d y, (9.11) τ where the integral converges absolutely as follows from the estimates (7.14) and (7.23). In general, there is no closed analytic expressions for the right hand side, unlike in the one-dimensional case. The results of numerical simulations for λ, indicating its qualitative dependence on the parameters of the model, together with analytic arguments about its behavior when Aτ → ∞ with A/B = const. or when τ → 0 with A/τ = const. and B/τ = const. may be found in [2,4,17,24,25]. 10. Conclusions We have studied rigorously a simple stochastic differential equation (SDE) used to model the pair dispersion of close inertial particles moving in a moderately turbulent flow [2,17,24,25,30,34]. We have established the smoothness of the transition probabilities and the irreducibility of the dispersion process using Hörmander criteria for hypoellipticity and control theory. For the projectivized version of the dispersion process, these results implied the existence of the unique invariant probability measure with smooth positive density as well as exponential mixing. The latter properties permitted to substantiate the formulae for the top Lyapunov exponents for the inertial particles used in the physical literature. In two space dimensions, we also showed that the complex-projectivized version of the dispersion process is non-explosive when described in the inhomogeneous variable of the complex projective space, unlike the real-projective version of the dispersion in one space dimension. This shows that in d = 2 the particle dispersion, if non-zero initially, cannot vanish in finite time keeping a non-vanishing time-derivative, and hence avoids the (strict) caustic configurations [35]. A similar result was established in d ≥ 3 for the complex-valued process built from the S O(d) invariants of the projectivized dispersion that was shown to stay for all times in the upper halfplane, avoiding also the configurations when the particle dispersion becomes collinear with its time derivative. These non-explosive behaviors are the reason why the numerical simulation of the processes in the complex (half-)plane could lead to reliable numerical results [2]. There are other questions about the models studied here that may be amenable to rigorous analysis. Let us list some of them: What about dispersion processes for more than two particles and the expressions for the other 2d − 1 Lyapunov exponents (the 2d exponents have to sum to −d/τ [10])? Such multi-particle dispersion processes provide, among other things, more information about caustic-like configurations [35]. Can one establish the existence of the large deviations regime for the finite-time Lyapunov exponents (the corresponding rate function for the top exponent was numerically studied in the d = 2 model in [2,4]; it gives access to more subtle information about the clustering of inertial particles than the top Lyapunov exponent itself)? Is the SDE
Ergodic Properties of a Model for Turbulent Dispersion of Inertial Particles
77
modeling the inertial particle dispersion in fully developed turbulence, that was introduced and studied numerically in [3,4], amenable to rigorous analysis? All those open problems are left for a future study. Acknowledgements. We would like to thank J. Zabczyk for a reference to [32]. J.W. acknowledges the support from C.N.R.S. for an extended visit at ENS-Lyon during which this work was started. D.P.H. and J.W. acknowledge partial support under NSF Grant DMS 0623941.
Appendix A We establish here the expressions (7.18) and (7.19) for the S O(2d)-invariant normalized volume measure on S 2d−1 for d ≥ 3 (the same proof works also in d = 2, although there, the corresponding formulae are straightforward and well known). Note that for S 2d−1 identified with the set of (ρ, χ ) ∈ R2d such that ρ 2 + χ 2 = R 2 for fixed R, we may write dμ0 = const. δ(R − ρ 2 + χ 2 ) dρ dχ ,
(A.1)
provided that we identify functions on S 2d−1 with homogeneous function of degree zero on R2d \{0}. Let us parametrize: ρ = O(ρ, 0, . . . , 0),
χ = O(ρx, ρy, 0, . . . , 0),
(A.2)
where ρ = |ρ|, χ = |χ |, x and y are the S O(d)-invariants of Eq. (7.4), and O ∈ S O(d). Note that O −1 is the rotation that aligns ρ with first positive half-axis of Rd and brings χ into the half-plane spanned by the first axis and the second positive half-axis, as required in Sect. 7.3. O and O O , for O rotating in the subspace orthogonal to the first two axes, give the same (ρ, χ ). Let be a d × d antisymmetric matrix, i j = − ji . Setting O = e and differentiating Eqs. (A.2) at = 0, we obtain: (A.3) dρ = (dρ, ρ d21 , . . . , ρ dd1 ), dχ = (−ρy d21 , ρx d21 , ρx d31 + ρy d32 , . . . , ρx dd1 + ρy dd2 ) +(x dρ + ρ d x, y dρ + ρ dy, 0, . . . , 0). (A.4) Hence for the volume element, dρ dχ = ρ 2d−1 y d−2 dρ d x d y d21 · · · dd1 d32 · · · dd2 .
(A.5)
The product of di j gives, modulo normalization, the S O(d)-invariant volume element d[O] of the homogeneous space S O(d)/S O(d − 2) at point [1]. Using the S O(d)invariance, we infer that dρ dχ = const. ρ 2d−1 y d−2 dρ d x d y d[O].
(A.6)
Substituting the last expression to Eq. (A.1) and performing the integral δ(R − ρ 2 + χ 2 ) ρ 2d−1 dρ = δ(R − ρ 1 + x 2 + y 2 ) ρ 2d−1 dρ R 2d−1 (A.7) (1 + x 2 + y 2 )d that collects the entire ρ-dependence in the integration against homogeneous function of zero degree, we obtain Eq. (7.19), modulo a constant factor that is fixed by normalizing of the resulting measure. =
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Appendix B We show here that
(Lg)(x, y) η(x, y) d x d y = 0
(B.1)
in two or more dimensions, where g(x, y) = ln(1 + x 2 + y 2 ) and η(x, y) is the density of the invariant measure as defined by Eqs. (7.9) and (7.17). As mentioned in Sect. 9, the identity (B.1) does not follow immediately by integration by parts since the function g(x, y), is not smooth on S 2d−1 . We shall then replace g(x, y) by the functions g (x, y) = ln 1 +
ρ2 + χ 2 x 2 + y2 = ln 1 + (x 2 + y 2 ) ρ 2 + χ 2
(B.2)
that are smooth on S 2d−1 for > 0. The identity (B.1) will follow if we show that (Lg)(x, y) η(x, y) d x d y = lim (B.3) (Lg )(x, y) η(x, y) d x d y. 0
Note that 2x , + y 2 ))(1 + (x 2 + y 2 )) 2y , ∂ y g (x, y) = (1 + (1 + )(x 2 + y 2 ))(1 + (x 2 + y 2 )) 1 + (1 + 2 )(y 2 − x 2 ) + (1 + )(y 2 − 3x 2 ) ∂x2 g (x, y) = 2 , (1 + (1 + )(x 2 + y 2 ))2 (1 + (x 2 + y 2 ))2 1 + (1 + 2 )(x 2 − y 2 ) + (1 + )(x 2 − 3y 2 ) ∂ y2 g (x, y) = 2 (1 + (1 + )(x 2 + y 2 ))2 (1 + (x 2 + y 2 ))2 ∂x g (x, y) =
(1 + (1 + )(x 2
(B.4)
so that |∂x g (x, y)| ≤ |∂x2 g (x,
2|x| , 1 + x 2 + y2
y)| ≤ 10,
|∂ y g (x, y)| ≤ |∂ y2 g (x,
2|y| , 1 + x 2 + y2
(B.5)
y)| ≤ 10.
Using the explicit forms (7.15) and (7.24) of the generator L, we infer that |(Lg )(x, y)| ≤ C(1 + |x|) with an -independent constant C. Since the integral (1 + |x|) η(x, y) d x d y
(B.6)
(B.7)
converges due to the estimates (7.14) and (7.23), and point-wise lim (Lg )(x, y) = (Lg)(x, y),
0
relation (B.3) follows from the Dominated Convergence Theorem.
(B.8)
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References 1. Bec, J.: Multifractal concentrations of inertial particles in smooth random flows. J. Fluid Mech. 528, 255–277 (2005) 2. Bec, J., Cencini, M., Hillerbrand, R.: Heavy particles in incompressible flows: the large Stokes number asymptotics. Physica D 226, 11–22 (2007) 3. Bec, J., Cencini, M., Hillerbrand, R.: Clustering of heavy particles in random self-similar flow. Phys. Rev. E 75, 025301 (2007) 4. Bec, J., Cencini, M., Hillerbrand, R., Turitsyn, K.: Stochastic suspensions of heavy particles. Physica D 237, 2037–2050 (2008) 5. Duncan, K., Mehlig, B., Ostlund, S., Wilkinson, M.: Clustering in mixing flows. Phys. Rev. Lett. 95, 240602 (2005) 6. Elperin, T., Kleeorin, N., Rogachevskii, I.: Self-Excitation of fluctuations of inertial particle concentration in turbulent fluid flow. Phys. Rev. Lett. 77, 5373–5376 (1996) 7. Falkovich, G., Fouxon, A., Stepanov, M.G.: Acceleration of rain initiation by cloud turbulence. Nature 419, 151–154 (2002) 8. Falkovich, G., Gaw¸edzki, K., Vergassola, M.: Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913–975 (2001) 9. Fouxon, I., Horvai, P.: Separation of heavy particles in turbulence. Phys. Rev. Lett. 100, 040601 (2008) 10. Fouxon, I., Horvai, P.: Fluctuation relation and pairing rule for Lyapunov exponents of inertial particles in turbulence. J. Stat. Mech.: Theor. & Experim. 08, L08002 (2007) 11. Friz, P.K.: An Introduction to Malliavin Calculus, lecture notes, http://www.math.nyu.edu/phd-students/ frizpete/malliavin/mall.pdf, 2002 12. Gaw¸edzki, K.: Soluble models of turbulent transport. In: Non-Equilibrium Statistical Mechanics and Turbulence, eds. S. Nazarenko, O. Zaboronski, Cambridge: Cambridge University Press 2008, pp. 44–107 13. Gradstein, I.S., Rhyzik, I.M.: Table of Integrals, Series, and Products, Vth edition. New York: Academic Press 1994 14. Halperin, B.I.: Green’s functions for a particle in a one-dimensional random potential. Phys. Rev. 139, A104–A117 (1965) 15. Has’minskii, R.Z.: Stochastic Stability of Differential Equations. alphen aanden Rija, Netherlands: Sijthoff and Noordhoff, 1980 16. Hörmander, L.: The Analysis of Linear Partial Differential Operators. Vol. III, Berlin: Springer, 1985 17. Horvai, P.: Lyapunov exponent for inertial particles in the 2D Kraichnan model as a problem of Anderson localization with complex valued potential, http://arxiv.org/abs/nlin/0511023v1 [nlin.co], 2005 18. Kraichnan, R.H.: Small scale structure of a scalar field convected by turbulence. Phys. Fluids 11, 945–953 (1968) 19. Le Jan, Y., Raimond, O.: Integration of Brownian vector fields. Ann. Probab. 30, 826–873 (2002) 20. Le Jan, Y., Raimond, O.: Flows, coalescence and noise. Ann. Probab. 32, 1247–1315 (2004) 21. Lifshitz, I.M., Gredeskul, S., Pastur, L.: Introduction to the Theory of Disordered Systems. New York: Wiley, 1988 22. Mattingly, J.C., Stuart, A.M., Higham, D.J.: Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise, Stochastic Process. Appl. 101, 185–232 (2002). doi:10.1016/S03044149(02)00150-3 http://dx.doi.org/10.1016/S0304-4149(02)00150-3 23. Maxey, M.R., Riley, J.J.: Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26, 883–889 (1983) 24. Mehlig, B., Wilkinson, M.: Coagulation by random velocity fields as a Kramers problem. Phys. Rev. Lett 92, 250602 (2004) 25. Mehlig, B., Wilkinson, M., Duncan, K., Weber, T., Ljunggren, M.: On the aggregation of inertial particles in random flows. Phys. Rev. E 72, 051104 (2005) 26. Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Communication and Control Engineering Series, London: Springer-Verlag, 1993 27. Meyn, S.P., Tweedie, R.L.: Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes. Adv. Appl. Prob. 25, 518–548 (1993) 28. Norriss, J.: Simplified Malliavin calculus. In: Séminaire de probabilité XX, Lectures Note in Math. 1204, Berlin: Springer, 1986, pp. 101–130 29. Nualart, D.: Malliavin Calculus and Related Topics, 2nd edition, Berlin-Heidelberg: Springer, 2006 30. Piterbarg, L.: The top Lyapunov exponent for a stochastic flow modeling the upper ocean turbulence. SIAM J. Appl. Math. 62, 777–800 (2001) 31. Rey-Bellet, L.: Ergodic properties of Markov processes. In: Open Quantum systems II. The Markovian approach, Lecture notes in Mathematics 1881, Berlin: Springer, 2006, pp. 1–78
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32. Scheutzow, M.: Stabilization and destabilization by noise in the plane. Stoch. Anal. Appl. 11, 97–113 (1993) 33. Stroock, D.W., Varadhan, S.R.S.: On the support of diffusion processes with applications to the strong maximum principle. In: Proc. 6-th Berkeley Symp. Math. Stat. Prob., Vol. III, Berkeley: Univ. California Press, 1972, pp. 361368 34. Wilkinson, M., Mehlig, B.: The path-coalescence transition and its applications. Phys. Rev. E 68, 040101 (2003) 35. Wilkinson, M., Mehlig, B.: Caustics in turbulent aerosols. Europhys. Lett. 71, 186–192 (2005) Communicated by M. Aizenman
Commun. Math. Phys. 308, 81–113 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1325-7
Communications in
Mathematical Physics
On Quantization of Complex Symplectic Manifolds Andrea D’Agnolo1, , Masaki Kashiwara2 1 Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Trieste 63, 35121 Padova, Italy.
E-mail: [email protected]
2 Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan.
E-mail: [email protected] Received: 7 September 2010 / Accepted: 18 March 2011 Published online: 28 August 2011 – © Springer-Verlag 2011
Abstract: Let X be a complex symplectic manifold. By showing that any Lagrangian subvariety has a unique lift to a contactification, we associate to X a triangulated category of regular holonomic microdifferential modules. If X is compact, this is a Calabi-Yau category of complex dimension dim X + 1. We further show that regular holonomic microdifferential modules can be realized as modules over a quantization algebroid canonically associated to X . Contents 0. Introduction . . . . . . . . . . . . . . . . . . . 1. Gerbes and Algebroid Stacks . . . . . . . . . . 2. Contactification of Symplectic Manifolds . . . . 3. Holonomic Modules on Symplectic Manifolds . 4. Quantization Algebroid . . . . . . . . . . . . . 5. Quantization Modules . . . . . . . . . . . . . . Appendix A. Remarks on Deformation-Quantization References . . . . . . . . . . . . . . . . . . . . . . .
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0. Introduction Let X be a complex symplectic manifold. As shown in [16] (see also [13]), X is endowed with a canonical deformation quantization algebroid W X . Recall that an algebroid is to an algebra as a gerbe is to a group. The local model of W X is an algebra similar to the The first named author (A.D’A.) expresses his gratitude to the Research Institute for Mathematical Sciences of Kyoto University for hospitality during the preparation of this paper and acknowledges partial support from the Fondazione Cariparo through the project “Differential methods in Arithmetic, Geometry and Algebra”.
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one of microdifferential operators, with a central deformation parameter . The center of W X is a subfield k of formal Laurent series C[−1 , ]]. Deformation quantization modules have now been studied quite extensively (see [3,11,12] and also [14,19] for related results), and they turned out to be useful in other contexts as well (see e.g. [9]). Of particular interest are modules supported by Lagrangian subvarieties. It is conjectured in [11] that, if X is compact, the triangulated category of regular holonomic deformation-quantization modules is Calabi-Yau of dimension dim X over k. There are some cases (representation theory, homological mirror symmetry, quantization in the sense of [5]) where one would like to deal with categories whose center is C instead of k. In the first part of this paper, we associate to X a C-linear triangulated category of regular holonomic microdifferential modules. If X is compact, this category is Calabi-Yau of dimension dim X + 1 over C. Our construction goes as follows. For a possibly singular Lagrangian subvariety ⊂ X , we prove that there is a unique contactification ρ : Y − → X of a neighborhood of and a Lagrangian subvariety ⊂ Y such that ρ induces a homeomorphism between and . As shown in [6], the contact manifold Y is endowed with a canonical microdifferential algebroid EY . We define the triangulated category of regular holonomic microdifferential modules along as the bounded derived category of regular holonomic EY -modules along . We then take the direct limit over the inductive family of Lagrangian subvarieties ⊂ X . In the second part of this paper, we show that regular holonomic microdifferential modules can be realized as modules over a quantization algebroid E X canonically associated to X . More precisely, if ⊂ Y is a lift of ⊂ X as above, we prove that the category of coherent EY -modules supported on is fully faithfully embedded in the category of coherent E X -modules supported on . Our construction of E X is similar to the construction of W X in [16], which was in turn similar to the construction of EY in [6]. Here, we somewhat simplify matters by presenting an abstract way of obtaining an algebroid from the data of a gerbe endowed with an algebra valued functor. Let us briefly recall the constructions of EY , W X and present the construction of EX . Denote by P ∗ M the projective cotangent bundle to a complex manifold M and by E M the ring of microdifferential operators on P ∗ M as in [17]. Recall that, in a local system of coordinates, E M is endowed with the anti-involution given by the formal adjoint of total symbols. Let Y be a complex contact manifold. By Darboux theorem, the local model of Y is an open subset of P ∗ M. By definition, a microdifferential algebra E on an open subset V ⊂ Y is a C-algebra locally isomorphic to E M . Assume that E is endowed with an anti-involution ∗. Any two such pairs (E , ∗ ) and (E, ∗) are locally isomorphic. Such isomorphisms are not unique, and in general it is not possible to patch the algebras E together in order to get a globally defined microdifferential algebra on Y . However, the automorphisms of (E, ∗) are all inner and are in bijection with a subgroup of invertible elements of E. This is enough to prove the existence of a microdifferential algebroid EY , i.e. an algebroid locally represented by microdifferential algebras. Denote by T ∗ M the cotangent bundle to a complex manifold M, by (t; τ ) the symplectic coordinates on T ∗ C, and consider the projection ρ
→ T ∗ M, (x, t; ξ, τ ) → (x, ξ/τ ) P ∗ (M × C) −
On Quantization of Complex Symplectic Manifolds
83
defined for τ = 0. This is a principal C-bundle, with action given by translation in the t variable. Note that, for λ ∈ C, the outer isomorphism Ad(eλ∂t ) of ρ∗ E M×C acts by translation t → t + λ at the level of total symbols. Let X be a complex symplectic manifold. By Darboux theorem, the local model of X is an open subset of T ∗ M. Let ρ : V − → U be a contactification of an open subset U ⊂ X . By definition, this is a principal C-bundle whose local model is the projection {τ = 0} − → T ∗ M above. Consider a quadruple (ρ, E, ∗, ) of a contactification ρ: V − → U , a microdifferential algebra E on V , an anti-involution ∗ and an operator ∈ E locally corresponding to ∂t−1 . One could try to mimic the above construction of the microdifferential algebroid EY in order to get an algebroid from the algebras ρ∗ E. −1 This fails because the automorphisms of (ρ, E, ∗, ) given by Ad(eλ ) for λ ∈ C are not inner. There are two natural ways out. The first possibility, utilized in [16], is to replace the algebra ρ∗ E by its subalgebra W = C0 ρ∗ E of operators commuting with . Locally, this corresponds to the operators −1
of ρ∗ E M×C whose total symbol does not depend on t. Then the action of Ad(eλ ) is trivial on W, and these algebras patch together to give the deformation-quantization algebroid W X . −1 The second possibility, which we exploit here, is to make Ad(eλ ) an inner automorphism. This is obtained by replacing the algebra ρ∗ E by the algebra E =
−1 C∞ ρ∗ E eλ , λ∈C
where C∞ ρ∗ E = {a ∈ ρ∗ E; ad() N (a) = 0, ∃N ≥ 0} locally corresponds to operators in ρ∗ E M×C whose total symbol is polynomial in t. By patching these algebras we get the quantization algebroid E X . The deformation parameter is not central in E X . We −1 show that the centralizer of in E X is equivalent to the twist of W X ⊗C ( λ∈C Ceλ ) by the gerbe parameterizing the primitives of the symplectic 2-form. In an appendix at the end of the paper, we give an alternative construction of the deformation-quantization algebroid W X . Instead of using contactifications, we consider as objects deformation-quantization algebras endowed with compatible anti-involution and C-linear derivation. We thus show that W X itself is endowed with a canonical C-linear derivation. One could then easily prove along the lines of [15] that W X is the unique k-linear deformation-quantization algebroid which is trivial graded and is endowed with compatible anti-involution and C-linear derivation. Finally, we compare regular holonomic quantization modules with regular holonomic deformation-quantization modules. This paper is organized as follows. In Sect. 1, after recalling the definitions of gerbe and of algebroid on a topological space, we explain how to obtain an algebroid from the data of a gerbe endowed with an algebra valued functor. In Sect. 2, we review some notions from contact and symplectic geometry, discussing in particular the gerbe parameterizing the primitives of the symplectic 2-form. We further show how a Lagrangian subvariety lifts to a contactification. In Sect. 3, we first recall the construction of the microdifferential algebroid of [6] in terms of algebroid data. Then we show how to associate to a complex symplectic manifold a triangulated category of regular holonomic microdifferential modules.
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In Sect. 4, we start by giving a construction of the deformation-quantization algebroid of [16] in terms of algebroid data. Then, with the same algebroid data, we construct the algebroid EX . In Sect. 5, we prove coherency of quantization algebras and show how to realize regular holonomic microdifferential modules as modules over EX . In Appendix A, we give an alternative description of the deformation quantization algebroid using deformation-quantization algebras endowed with compatible anti-involution and C-linear derivation. We also compare regular holonomic deformation-quantization modules with regular holonomic quantization modules. The results of this paper were announced in [1], to which we refer. 1. Gerbes and Algebroid Stacks We review here some notions from the theory of stacks, in the sense of sheaves of categories, recalling in particular the definitions of gerbe and of algebroid (refer to [2,4,10,13]). We then explain how to obtain an algebroid from the data of a gerbe endowed with an algebra valued functor. 1.1. Review on stacks. Let X be a topological space. A prestack C on X is a lax analogue of a presheaf of categories, in the sense that for a chain of open subsets W ⊂ V ⊂ U the restriction functor C(U ) − → C(W ) coincides with the composition C(U ) − → C(V ) − → C(W ) only up to an invertible transformation (satisfying a natural cocycle condition for chains of four open subsets). The prestack C is called separated if for any U ⊂ X and any p, p ∈ C(U ) the presheaf U ⊃ V → Hom C(V ) ( p|V , p |V ) is a sheaf. We denote it by Hom C ( p, p ). A stack is a separated prestack satisfying a natural descent condition (see e.g. [10, Chap. 19]). If ρ : Y − → X is a continuous map, we denote by ρ −1 C the pull back on Y of a stack C on X . A groupoid is a category whose morphisms are all invertible. A gerbe on X is a stack of groupoids which is locally non empty and locally connected, i.e. any two objects are locally isomorphic. Let G be a sheaf of commutative groups. A G-gerbe is a gerbe P endowed with a group homomorphism G − → Aut(idP ). A G-gerbe P is called invertible if G|U − → AutP ( p) is an isomorphism of groups for any U ⊂ X and any p ∈ P(U ). We denote by P ×G Q the contracted product of two G-gerbes. This is the stack associated to the prestack whose objects are pairs ( p, q) of an object of P and an object of Q, with morphisms Hom
G
P×Q
G ( p, q), ( p , q ) = Hom P ( p, p ) × Hom Q (q, q ).
Let R be a commutative sheaf of rings. For an R-algebra A denote by Mod(A) the stack of left A-modules. An R-linear stack is a stack A such that for any U ⊂ X and any p, p ∈ A(U ) the sheaves Hom A ( p , p) have an R|U -module structure compatible with composition and restriction. The stack of left A-modules Mod(A) = FctR (A, Mod(R)) has R-linear functors as objects and transformations of functors as morphisms. Let L be a commutative R-algebra and A an R-linear stack. An action of L on A is the data of R|U -algebra morphisms L|U − → EndA ( p) for any U ⊂ X and any p ∈ A(U ), compatible with restriction. Then L acts as a Lie algebra on Hom A ( p , p) by [l, f ] = l p f − f l p , where l p denotes the image of l ∈ L(U ) in EndA ( p). This gives a filtration of A by the centralizer series
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0 CL Hom A ( p , p) = { f ; [l, f ] = 0, ∀l ∈ L},
i−1 i Hom A ( p , p) = { f ; [l, f ] ∈ CL , ∀l ∈ L} for any i > 0. CL 0 A and C ∞ A the substacks of A with the same objects as A and morphisms Denote by CL i L 0 0 A is an L-linear stack and C ∞ A Hom A , respectively. Note that CL CL Hom A and i CL L is an R-linear stack. An R-algebroid A is an R-linear stack which is locally non empty and locally connected by isomorphisms. Thus, an algebroid is to a sheaf of algebras as a gerbe is to a sheaf of groups. For p ∈ A(U ), set A p = EndA ( p). Then A|U is equivaop lent to the full substack of Mod(A p ) whose objects are locally free modules of rank op one. (Here A p denotes the opposite ring of A p .) Moreover, there is an equivalence Mod(A|U ) Mod(A p ). One says that A is represented by an R-algebra A if A A p for some p ∈ A(X ). The R-algebroid A is called invertible if A p R|U for any U ⊂ X and any p ∈ A(U ). The pull-back and tensor product of algebroids are still algebroids. The following lemma is obvious. 0 A is locally Lemma 1.1.1. Let A be an R-algebroid endowed with an action of L. If CL 0 ∞ connected by isomorphisms, then CL A and CL A are algebroids.
1.2. Algebroid data. Let R-Alg be the stack on X with R-algebras as objects and R-algebra homomorphisms as morphisms. Definition 1.2.1. An R-algebroid data is a triple (P, , ) with P a gerbe, : P − → R-Alg a functor of stacks and a collection of liftings of group homomorphisms
( p)× 7 o o p ooo Ad ooo o o oo
/ AutR-Alg ( ( p)) EndP ( p)
∀U ⊂ X, ∀ p ∈ P(U ),
(1.2.1)
compatible with restrictions and such that for any g ∈ Hom P ( p , p) and any φ ∈ EndP ( p ) one has p (gφ g −1 ) = (g)( p (φ )).
(1.2.2)
Remark 1.2.2. Denote by Grp the stack on X with sheaves of groups as objects and group homomorphisms as morphisms. The R-algebroid data (P, , ) induces three natural functors E, A, F : P − → Grp defined by E( p) = EndP ( p), A( p) = AutR-Alg ( ( p)), F( p) = ( p)× for p ∈ P and by E(g) = Ad(g), A(g) = Ad(g), F(g) = (g) for g : p − → p. Then condition (1.2.2) states that : E − → F is a transformation of functors and the commutative diagram (1.2.1) corresponds to a commutative diagram of transformations of functors ;F vv
vvv
E
v vv vv
Ad
/ A.
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Remark 1.2.3. There is a natural interpretation of R-algebroid data in terms of 2categories (refer to [18, §9], where 2-categories are called bicategories). Denote by R-Alg the 2-prestack on X obtained by enriching R-Alg with a set of 2-arrows f ⇒ f given by {b ∈ A; b f (a ) = f (a )b, ∀a ∈ A }, → A. In particular, f f if and only if for two R-algebra morphisms f, f : A − f = Ad(b) f for some b ∈ A× . The R-algebroid data (P, , ) is equivalent to the data of the lax functor of 2-prestacks : P − → R-Alg, where P has trivial 2-arrows and is obtained by enriching at the level of 2-arrows −1 → g in P( p). by (id g − →g ) = p (g g ) for a morphism g − We will prove in the next proposition that the following description associates an R-prestack A0 to the data (P, , ). (i) For an open subset U ⊂ X , objects of A0 (U ) are the same as those of P(U ). (ii) For p, p ∈ A0 (U ), the sheaf of morphisms is defined by Hom A0 ( p , p) = ( p)
E ndP ( p)
×
Hom P ( p , p).
→ p in A0 are equivalence classes [a, g] of pairs This means that morphisms p − (a, g) with a ∈ ( p) and g : p − → p in P, for the relation (a, φg) ∼ (a p (φ), g),
∀φ ∈ EndP ( p).
→ p and [a , g ] : p − → p is given by (iii) Composition of [a, g] : p − [a, g] ◦ [a , g ] = [ag(a ), gg ]. Here we set for short g(a ) = (g)(a ). (iv) For two morphisms [a, g], [a , g ] : p − → p and r ∈ R, the R-linear structure of A0 is given by r [a, g] = [ra, g], [a, g] + [a , g ] = [a + a p (g g −1 ), g]. (v) The restriction functors are the natural ones. Proposition 1.2.4. Let (P, , ) be an R-algebroid data. The description (i)–(v) above defines a separated R-prestack A0 on X . The associated stack A is an R-algebroid endowed with a functor J : P − → A such that EndA (J ( p)) ( p) for any p ∈ P. Proof. (a) Let us show that the composition is well defined. Consider two composable → p and [a , g ] : p − → p . At the level of representatives, morphisms [a, g] : p − set (a, g) ◦ (a , g ) = (ag(a ), gg ).
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(a-i) Let us show that for φ ∈ EndP ( p) we have (a, φg) ◦ (a , g ) ∼ (a p (φ), g) ◦ (a , g ). For this, we have to check that (aφ(g(a )), φgg ) ∼ (a p (φ)g(a ), gg ). This follows from p (φ)g(a ) = φ(g(a )) p (φ), which is a consequence of the commutativity of (1.2.1). (a-ii) Similarly, for φ ∈ EndP ( p ) we have to prove that (a, g) ◦ (a , φ g ) ∼ (a, g) ◦ (a p (φ ), g ). In other words, we have to check that (ag(a ), gφ g ) ∼ (ag(a p (φ )), gg ). This follows from gφ g = (gφ g −1 )gg and g(a p (φ )) = g(a )g( p (φ )) = g(a ) p (gφ g −1 ), where the last equality is due to (1.2.2). (a-iii) Associativity is easily checked. (b) The R-linear structure is well defined by an argument similar to that in part (a) above. (c) The functor J : P − → A is induced by the functor J0 : P − → A0 defined by p → p on objects and g → [1, g] on morphisms. The morphism ( p) − → EndA (J ( p)), a → [a, id] has an inverse given by [a, g] → a p (g). Note that the functor J : P − → A is neither faithful nor full, in general. Remark 1.2.5. For an R-algebroid A, denote by A× the gerbe with the same objects as A and isomorphisms as morphisms. Then A is the R-algebroid associated with the data (A× , A , ), where A ( p) = EndA ( p) and p is the identity. Example 1.2.6. Let X be a complex manifold and O X its structure sheaf. To an invertible O X -module L one associates an invertible Z/2Z-gerbe PL⊗1/2 defined as follows: ∼
→ (i) Objects on U are pairs (F, f ), where F is an invertible OU -module and f : F ⊗2 − L is an OU -linear isomorphism. (ii) If (F , f ) is another object, a morphism (F , f ) − → (F, f ) is an OU -linear ∼ ⊗2 → F, such that f = f ϕ . isomorphism ϕ : F − Note that any ψ ∈ EndPL⊗1/2 (F, f ) is a locally constant Z/2Z-valued function. Denote by CL⊗1/2 the invertible C-algebroid associated with the data (PL⊗1/2 , , ), where (F, f ) = CU , (ϕ) = id, (F , f ) (ψ) = ψ.
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2. Contactification of Symplectic Manifolds We first review here some notions from contact and symplectic geometry. In particular, we discuss the gerbe parameterizing the primitives of the symplectic 2-form. Then, we show how any Lagrangian subvariety of a complex symplectic manifold can be uniquely lifted to a local contactification. 2.1. The gerbe of primitives. Let X be a complex manifold and O X its structure sheaf. Denote by T X and T ∗ X the tangent and cotangent bundle, respectively, and by X and 1X their sheaves of sections. For k ∈ Z, denote by kX the sheaf of holomorphic k-forms k → k−1 the and by d : kX − → k+1 X X the differential. For v ∈ X denote by i v : X − k k inner derivative and by L v : X − → X the Lie derivative. Let ω ∈ Γ (X ; 2X ) be a 2-form which is closed, i.e. dω = 0. Definition 2.1.1. The gerbe Cω on X is the stack associated with the separated prestack defined as follows. (1) Objects on U ⊂ X are primitives of ω|U , i.e. 1-forms θ ∈ Γ (U ; 1X ) such that dθ = ω|U . (2) If θ is another object, a morphism θ − → θ is a function ϕ ∈ Γ (U ; O X ) such that dϕ = θ − θ . Composition with ϕ : θ − → θ is given by ϕ ◦ ϕ = ϕ + ϕ . The following result is clear. Lemma 2.1.2. (i) The stack Cω is an invertible C-gerbe. 2 (ii) If ω ∈ X (X ) is another closed 2-form, there is an equivalence C
∼
→ Cω+ω . Cω × Cω − For a principal C-bundle ρ : Y − → X , denote by →Y Tλ : Y − the action of λ ∈ C and by va =
d dλ Tλ λ=0
∈ Y
the infinitesimal generator of the C-action. Definition 2.1.3. The gerbe Cω on X is defined as follows: ρ
→ U, α) of a principal C-bundle ρ and a (1) Objects on U ⊂ X are pairs ρ = (V − 1-form α ∈ Γ (V ; 1V ) such that i va α = 1 and ρ ∗ ω = dα. In particular, L va α = 0. ρ
→ U, α ), morphisms χ : ρ − → ρ are morphisms of (2) For another object ρ = (V − principal C-bundles such that χ ∗ α = α . Denote by p1 : X ×C − → X the trivial principal C-bundle given by the first projection. Let t be the coordinate of C. For a primitive θ of ω, an object of Cω is given by ( p1 , p1∗ θ + dt). By the next lemma, any object ρ of Cω is locally of this form and any automorphism of ρ is locally of the form Tλ , for λ ∈ C. (See [16, Remark 9.3] for similar observations.)
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Lemma 2.1.4. There is a natural equivalence Cω − → Cω . In particular, Cω is an invertible C-gerbe. Proof. As above, denote by p1 : U × C − → U the first projection and by t the coordinate of C. Consider the functor B : Cω − → Cω given by θ → ( p1 , p1∗ θ + dt) on objects and ϕ → (x, t) → (x, t + ϕ(x)) on morphisms. As B is clearly faithful, we are left to prove that it is locally full and locally essentially ρ surjective. For the latter, let ρ = (V − → U, α) be an object of Cω (U ). Up to shrinking U , we may assume that the bundle ρ is trivial. Choose an isomorphism of principal C-bundles ξ : U × C − → V . As i va (ξ ∗ α − dt) = L va (ξ ∗ α − dt) = 0, there exists a 1 unique 1-form θ ∈ X (U ) such that ξ ∗ α − dt = p1∗ θ . Then ω|U = dθ and ρ B(θ ). It remains to show that any morphism χ : ρ − → ρ of Cω (U ) is in the image of B. Up to shrinking U , we may assume that ρ = ( p1 , p1∗ θ + dt) and ρ = ( p1 , p1∗ θ + dt). Then χ : U × C − → U × C is given by (x, t) → (x, t + ϕ(x)) for some ϕ ∈ O X (U ). Since χ ∗ ( p1∗ θ + dt) = p1∗ θ + dt, it follows that dϕ = θ − θ . Hence χ = B(ϕ). Definition 2.1.5. Let R be a commutative ring endowed with a group homomorphism : C − → R × . The stack Rω is the invertible R-algebroid associated with the data (Cω , R , ), where
R (ρ) = RU , R (χ ) = id RU , ρ (Tλ ) = (λ), ρ
for ρ = (V − → U, α), χ : ρ − → ρ and λ ∈ C. Note that by Lemma 2.1.2 there is an R-linear equivalence ∼
→ Rω+ω . Rω ⊗R X Rω − Remark 2.1.6. Equivalence classes of invertible C-gerbes and of invertible R-algebroids are classified by H 2 (X ; C) and H 2 (X ; R × ), respectively. The class of Cω coincides with the de Rham class [ω] of the closed 2-form ω, and the class of Rω is the image of [ω] by : H 2 (X ; C) − → H 2 (X ; R × ). 2.2. Symplectic manifolds. A complex symplectic manifold X = (X, ω) is a complex manifold X of even dimension endowed with a holomorphic closed 2-form ω ∈ Γ (X ; 2X ) which is non-degenerate, i.e. the n-fold exterior product ω ∧ · · · ∧ ω never vanishes for n = 21 dim X . ∼
→ X be the Hamiltonian isomorphism induced by the symplectic Let H : 1X − form ω. The Lie bracket of ϕ, ϕ ∈ O X is given by {ϕ, ϕ } = Hϕ (ϕ ), where Hϕ = H (dϕ) is the Hamiltonian vector field of ϕ. Example 2.2.1. Let M be a complex manifold. Its cotangent bundle T ∗ M has a natural symplectic structure (T ∗ M, dθ ), where θ denotes the canonical 1-form. Let (x) = (x1 , . . . , xn ) be a system of local coordinates on M. The associated system (x; u) of local symplectic coordinates on T ∗ M is given by p = i u i ( p)d xi . Then the canonical 1-form is written θ = i u i d xi and the Hamiltonian vector field of ϕ ∈ O M is written Hϕ = i ϕu i ∂xi − ϕxi ∂u i .
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An analytic subset ⊂ X is called involutive if for any f, g ∈ O X with f | = g| = 0 one has { f, g}| = 0. The analytic subset is called Lagrangian if it is involutive and dim X = 2 dim . Let X = (X , ω ) be another symplectic manifold. A symplectic transformation ψ : X − → X is a holomorphic isomorphism such that ψ ∗ ω = ω . By Darboux theorem, for any complex symplectic manifold X there locally exist symplectic transformations ψ
X ⊃U − → U M ⊂ T ∗ M, for a complex manifold M with dim M =
1 2
(2.2.1)
dim X .
2.3. Contact manifolds. Let γ : Z − → Y be a principal C× -bundle over a complex manifold Y . Denote by vm the infinitesimal generator of the C× -action on Z . For k ∈ Z, let O Z (k) be the sheaf of k-homogeneous functions, i.e. solutions ϕ ∈ O Z of vm ϕ = kϕ. Let OY (k) = γ∗ O Z (k) be the corresponding invertible OY -module, so that OY (−1) is × the sheaf of sections of the line bundle C ×C Z . γ → Y, θ ) is a complex manifold Y endowed with A complex contact manifold Y = (Z − a principal C× -bundle γ and a holomorphic 1-form θ ∈ Γ (Z ; 1Z ) such that (Z , dθ ) is a complex symplectic manifold, i vm θ = 0 and L vm θ = θ , i.e. θ is 1-homogeneous. Example 2.3.1. Let M be a complex manifold and θ the canonical 1-form on T ∗ M as in Example 2.2.1. The projective cotangent bundle P ∗ M has a natural contact structure (γ , θ ) with γ : T ∗ M\M − → P ∗ M the projection. Here T ∗ M \ M denotes the cotangent bundle with the zero-section removed. Note that the 1-form θ on Z may be considered as a global section of 1Y ⊗O OY (1). In particular, there is an embedding ι : OY (−1) − → 1Y , ϕ → ϕθ.
(2.3.1)
Note also that the symplectic manifold Z is homogeneous with respect to the C× -action, i.e. θ = i vm (dθ ). Moreover, there exists a unique C× -equivariant embedding Z → T ∗ Y such that θ is the pull-back of the canonical 1-form on T ∗ Y . Since dθ is 1-homogeneous, the Hamiltonian vector field Hϕ of ϕ ∈ O Z (k) is (k −1)homogeneous, i.e. [vm , Hϕ ] = (k − 1) Hϕ . An analytic subset of Y is called involutive (resp. Lagrangian) if γ −1 is involutive (resp. Lagrangian) in Z . γ
→ Y , θ ) be another contact manifold. A contact transformation Let Y = (Z − χ: Y − → Y is an isomorphism of principal C× -bundles Z
χ
γ
Y such that χ ∗ θ = θ .
χ
/Z /Y
γ
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By the Darboux theorem, for any complex contact manifold Y there locally exist contact transformations χ
Y ⊃V − → VM ⊂ P ∗ M,
(2.3.2)
for a complex manifold M with dim M = 21 (dim Y + 1). 2.4. Contactifications. Let X = (X, ω) be a complex symplectic manifold. A contactification of X is a global object of the stack Cω described in Definition 2.1.3. Morphisms of contactifications are morphisms in Cω . ρ For a contactification ρ = (Y − → X, α) of X , the total space Y of ρ has a natural q1 complex contact structure given by (Y × C× − → Y, τ q1∗ α), where q1 is the first pro× jection and τ is the coordinate of C ⊂ C. Note that, in terms of contact structures, a morphism ρ − → ρ of contactifications is a contact transformation χ : Y − → Y over X . Example 2.4.1. Let M be a complex manifold and denote by (t; τ ) the symplectic coordinates of T ∗ C. Consider the principal C-bundle, ρ
P ∗ (M × C) ⊃ {τ = 0} − → T ∗ M, (x, t; ξ, τ ) → (x; ξ/τ ), with the C-action given by translation in the t variable. Note that the bundle ρ is trivialized by ∼
→ (T ∗ M) × C, (x, t; ξ, τ ) → ((x; ξ/τ ), t). χ : {τ = 0} − Consider the projection p1 : (T ∗ M) × C − → T ∗ M. As in Example 2.2.1, denote by θ the canonical 1-form of T ∗ M. Then a contactification of (T ∗ M, dθ ) is given by (ρ, α), with ρ as above and α = χ ∗ ( p1∗ θ +dt). In a system (x; u) of local symplectic coordinates on T ∗ M, one has θ = u d x and α = (ξ/τ )d x +dt. As the canonical 1-form of T ∗ (M × C) is τ α = ξ d x + τ dt, the map (2.3.1) is given by ι : O P ∗ (M×C) (−1)|{τ =0} − → 1P ∗ (M×C) |{τ =0} , ϕ → ϕ τ α. 2.5. Contactification of Lagrangian subvarieties. In this section we show how any Lagrangian subvariety of a complex symplectic manifold lifts to a contactification (see e.g. [3, Lemma 8.4] for the case of Lagrangian submanifolds). Let us begin with a preliminary lemma. Lemma 2.5.1. Let M be a complex manifold, S ⊂ M a closed analytic subset and θ ∈ 1M a 1-form such that dθ | Sreg = 0. Then there locally exists a continuous function f on S such that f is holomorphic on the non-singular locus Sreg , and d f | Sreg = θ | Sreg . → S be a resolution of singularities and let p : S − → M be the composite Proof. Let S − S − → S → M. Thus S is a complex manifold, p is proper and p −1 (Sreg ) − → Sreg is an isomorphism. Consider the global section θ = p ∗ θ of 1S . As dθ | Sreg = 0 and p −1 (Sreg ) is dense in S , we have dθ = 0. Fix a point s0 ∈ S and set S0 = p −1 (s0 ). Since θ |(S0 )reg = 0, there exists a unique holomorphic function f defined on a neighborhood of S0 such that d f = θ and
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f | S0 = 0. As p is proper, replacing M by a neighborhood of s0 we may assume that f is globally defined on S . Set S = S × S S and S0 = S0 × S S0 . We may assume that S0 intersects each connected component of S . Consider the diagram Sreg
q
/ S
p1 p2
/
/ S
p
/ M,
where p1 and p2 are the projections S × S S − → S . To conclude, it is enough to prove that g = p1∗ f − p2∗ f vanishes, for then we can set f (w) = f (w ) with p(w ) = w. Since pp1 = pp2 , one has dq ∗ g = d( pp1 q)∗ θ − d( pp2 q)∗ θ = 0 so that g is locally . Hence g is locally constant by Sublemma 2.5.2 below with T = S constant on Sreg and U = Sreg . Since g vanishes on S0 , it vanishes everywhere. Sublemma 2.5.2. Let T be a Hausdorff topological space and U ⊂ T a dense open subset. Assume there exists a basis B of open subsets of T such that any B ∈ B is connected and B ∩ U has finitely many connected components. If a continuous function on T is locally constant on U , then it is locally constant on T . Let now X = (X, ω) be a complex symplectic manifold. Proposition 2.5.3. Let be a Lagrangian subvariety of X . Then there exist a neighborhood U of in X and a pair (ρ, ) with ρ : V − → U a contactification and a Lagrangian subvariety of V such that ρ| is a homeomorphism over and a holomorphic isomorphism over reg . Proof. Let {Ui }i∈I be an open cover of in X such that for each i ∈ I there is a primitive θi ∈ 1X (Ui ) of ω|Ui . Set i = ∩ Ui . Using Lemma 2.5.1, up to shrinking the cover we may assume that there is a continuous function f i on i such that f i |i,reg is a primitive of θi |i,reg . Set Ui j = Ui ∩ U j and similarly for i j . Up to further shrinking the cover we may assume that i j intersects each connected component of Ui j and that there is a function ϕi j ∈ O X (Ui j ) such that dϕi j = θi − θ j |Ui j and ϕi j |i j,reg = f i − f j |i j,reg . Set Ui jk = Ui ∩ U j ∩ Uk and similarly for i jk . Note that d(ϕi j + ϕ jk + ϕki ) = 0, so that ϕi j + ϕ jk + ϕki is locally constant on Ui jk . Since it vanishes on i jk , it vanishes everywhere. p1
Set ρi = (Vi −→ Ui , αi ), where Vi = Ui × C and αi = p1∗ θi + dt. Let (ρi , i ) be the pair with i = {(x, t) ∈ Vi ; x ∈ i , t + f i (x) = 0}. Then the pair (ρ, ) is obtained by patching the (ρi , i )’s via the maps (x, t) → (x, t + ϕi j (x)). Let us give an example that shows how, in general, and are not isomorphic as complex spaces. Example 2.5.4. Let X = (T ∗ C, dθ ) with symplectic coordinates (x; u), and Y = (X × C, α) with extra coordinate t. Then θ = u d x and α = u d x + dt. Take as ⊂ X a parametric curve = {(x(s), u(s)); s ∈ C}, with x(0) = u(0) = 0. Then = {(x, u, t); x = x(s), u = u(s), t + f (s) = 0},
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where f satisfies the equations f (s) = u(s)x (s) and f (0) = 0. For x(s) = s 3 , u(s) = s 7 + s 8 ,
f (s) =
3 10 10 s
+
3 11 11 s ,
we have an example where f cannot be written as an analytic function of (x, u). In fact, 11 ∈ s 11 = 11x(s)u(s) − 110 / C[[s 3 , s 7 + s 8 ]]. 3 f (s) and s 3. Holonomic Modules on Symplectic Manifolds We start by giving here a construction of the microdifferential algebroid of [6] in terms of algebroid data and by recalling some results on regular holonomic microdifferential modules. Then, using the results from the previous section, we show how it is possible to associate to a complex symplectic manifold a natural C-linear category of holonomic modules.
3.1. Microdifferential algebras. Let us review some notions from the theory of microdifferential operators (refer to [7,17]). Let M be a complex manifold. Denote by E M the sheaf on P ∗ M of microdifferential operators, and by Fk E M its subsheaf of operators of order at most k ∈ Z. Then E M is a sheaf of C-algebras on P ∗ M, filtered over Z by the Fk E M ’s. Take a local symplectic coordinate system (x; ξ ) on T ∗ M. For an open subset U ⊂ T ∗ M, a section a ∈ Γ (U ; Fk E M ) is represented by its total symbol, which is a formal series
a(x, ξ ) = a j (x, ξ ), a j ∈ Γ (U ; O P ∗ M ( j)) j≤k
satisfying suitable growth conditions. In terms of total symbols, the product in E M is given by Leibniz rule. More precisely, for a ∈ E M with total symbol a (x, ξ ), the product aa has total symbol
1 ∂ξJ a(x, ξ )∂xJ a (x, ξ ). J ! n
J ∈N
For a ∈ Fk E M , the top degree component ak ∈ O P ∗ M (k) of its total symbol does not depend on the choice of coordinates. The map σk : Fk E M − → O P ∗ M (k), a → ak induced by the isomorphism Fk E M /Fk−1 E M O P ∗ M (k) is called the symbol map. Recall that an operator a ∈ Fk E M \Fk−1 E M is invertible at p ∈ P ∗ M if and only if σk (a)( p) = 0. For a ∈ Fk E M and a ∈ Fk E M , one has {σk (a), σk (a )} = σk+k −1 ([a, a ]). op
An anti-involution of E M is an isomorphism of C-algebras ∗ : E M − → E M such that ∗∗ = id.
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Remark 3.1.1. In a local system of symplectic coordinates, an example of anti-involution ∗ of E M is given by the formal adjoint. This is described at the level of total symbols by
1 ∂ξJ ∂xJ a(x, −ξ ) . a ∗ (x, ξ ) = J ! n J ∈N
The formal adjoint depends on the choice of the top-degree form d x1 ∧ · · · ∧ d xn . Consider a contact transformation χ
P∗ M ⊃ V − → V ⊂ P ∗ M, where M, M are complex manifolds with the same dimension. It is a fundamental result of [17] that contact transformations can be locally quantized. Theorem 3.1.2. With the above notations: ∼ (i) Any C-algebra isomorphism f : χ∗ E M |V − → E M |V is a filtered isomorphism, and σk ( f (a )) = χ∗ σk (a ) for any a ∈ Fk E M . (ii) For any p ∈ V there exists a neighborhood U of p in V and a C-algebra isomor∼ phism f : χ∗ E M |U − → E M |U . (iii) Let ∗ and ∗ be anti-involutions of E M |V and E M |V , respectively. For any p ∈ V there exists a neighborhood U of p in V and a C-algebra isomorphism f as in (ii) such that f ∗ = ∗ f . An isomorphism f as in (ii) is called a quantized contact transformation over χ . Quantized contact transformations over χ are not unique. It was noticed in [6] that one can reduce the ambiguity to an inner automorphism by considering anti-involutions as in (iii) (see Lemma 3.2.4 below). The C-algebra E M is left and right Noetherian. It is another fundamental result of [17] that the support of a coherent E M -module is a closed involutive subvariety of P ∗ M. A coherent E M -module supported by a Lagrangian subvariety is called holonomic. We refer e.g. to [7] for the notion of regular holonomic E M -module. 3.2. Microdifferential algebroid. Let Y be a complex contact manifold. Definition 3.2.1. A microdifferential algebra E on Y is a sheaf of C-algebras such that, locally on Y , there is a C-algebra isomorphism E|V χ −1 E M in a Darboux chart (2.3.2). By Theorem 3.1.2, any C-algebra automorphism of E M is filtered and symbol preserving. It follows that a microdifferential algebra E on Y is filtered and has symbol maps σk : Fk E − → OY (k). Example 3.2.2. Let Y = P ∗ M be the projective cotangent bundle to a complex M the invertible O -module of top-degree manifold M and denote by M = dim M M forms. Consider the algebra of twisted microdifferential operators ⊗1/2
E⊗1/2 = M M
⊗−1/2
⊗O M E M ⊗O M M
.
Then E⊗1/2 is a microdifferential algebra on P ∗ M, and the formal adjoint ∗ of M Remark 3.1.1 gives a canonical anti-involution of E⊗1/2 . M
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Definition 3.2.3. The gerbe PY on Y is defined as follows: (1) For an open subset V ⊂ Y , objects of PY (V ) are pairs p = (E, ∗) of a microdifferential algebra E on V and an anti-involution ∗ of E. (2) If p = (E , ∗ ) is another object, Hom PY ( p , p) = { f ∈ Isom C-Alg (E , E); f ∗ = ∗ f }. (The fact that the stack of groupoids PY is a gerbe follows from Theorem 3.1.2.) Lemma 3.2.4 ([6, Lemma 1]). For any p = (E, ∗) ∈ PY there is an isomorphism of sheaves of groups ∼
ψ : {b ∈ E × ; b∗ b = 1, σ0 (b) = 1} − → EndPY ( p), b → Ad(b). By this lemma, we have a natural C-algebroid data on Y , and hence a C-algebroid, defined as follows. Definition 3.2.5. The microdifferential algebroid on Y is the C-algebroid EY associated to (PY , E , ), where
E ( p) = E, E ( f ) = f, p (g) = b, → p and g = ψ(b). for p = (E, ∗), f : p − By the construction in § 1.2, this means that objects of EY are microdifferential algebras (E, ∗) endowed with an anti-involution. Morphisms (E , ∗ ) − → (E, ∗) in EY are ∼ → E such that f ∗ = ∗ f . equivalence classes of pairs (a, f ) with a ∈ E and f : E − The equivalence relation is given by (a, Ad(b) f ) ∼ (ab, f ) for b ∈ E × with b∗ b = 1 and σ0 (b) = 1. Note that EY is locally represented by microdifferential algebras. In fact, the sheaf of endomorphisms of (E, ∗) in EY is isomorphic to E. Remark 3.2.6. Let Y = P ∗ M be the projective cotangent bundle to a complex manifold M. With notations as in Example 3.2.2, a global object of E P ∗ M is given by (E⊗1/2 , ∗). This implies that the algebroid E P ∗ M is represented by the microdifferential M algebra E⊗1/2 . M
γ
3.3. Holonomic modules on contact manifolds. Let Y = (Z − → Y, θ ) be a complex contact manifold. Consider the stack Mod(EY ) of modules over the microdifferential algebroid EY . For a subset S ⊂ Y , denote by Mod S (EY ) the full substack of Mod(EY ) of objects supported on S. By construction, EY is locally represented by microdifferential algebras. As the notions of coherent and regular holonomic microdifferential modules are local and invariant by quantized contact transformations, they make sense also for objects of Mod(EY ). Denote by Modcoh (EY ) and Modrh (EY ) the full substacks of Mod(EY ) whose objects are coherent and regular holonomic, respectively. Let R be an invertible C-algebroid. Then Mod(R) is locally equivalent to Mod(CY ). Hence the notion of local system makes sense for objects of Mod(R). Denote by LocSys(R) the full substack of Mod(R) whose objects are local systems. Consider the invertible C-algebroid C⊗1/2 on Y as in Example 1.2.6. Y By [6, Prop. 4] (see also [3, Cor. 6.4]), one has
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Proposition 3.3.1. For a smooth Lagrangian submanifold ⊂ Y there is an equivalence Mod,rh (EY ) p∗ LocSys( p −1 C⊗1/2 ),
where p :
γ −1
− → is the restriction of γ : Z − → Y.
Recall that a C-linear triangulated category T is called Calabi-Yau of dimension d if for each M, N ∈ T the vector spaces Hom T (M, N ) are finite-dimensional and there are isomorphisms Hom T (M, N )∨ Hom T (N , M[d]), functorial in M and N . Here H ∨ denotes the dual of a vector space H . Denote by Dbrh (EY ) the full triangulated subcategory of the bounded derived category of EY -modules whose objects have regular holonomic cohomologies. The following theorem is obtained in [11]1 as a corollary of results from [8]. Theorem 3.3.2. If Y is compact, then Dbrh (EY ) is a C-linear Calabi-Yau triangulated category of the same dimension as Y . 3.4. Holonomic modules on symplectic manifolds. Let X = (X, ω) be a complex symplectic manifold and ⊂ X a closed Lagrangian subvariety. By Proposition 2.5.3 there exists a neighborhood U ⊃ , a contactification ρ : V − → U and a closed Lagrangian subvariety ⊂ V such that ρ induces an isomorphism − → . Let us still denote by ρ the composition V − →U − → X . We set RH X, = ρ∗ Mod,rh (EV ), DRH (X ) = Db,rh (EV ). By unicity of the pair (ρ, ), the stack RH X, and the triangulated category DRH (X ) only depend on . For ⊂ , there are natural fully faithful, exact functors → RH X, , DRH (X ) − → DRH (X ). RH X, − The family of closed Lagrangian subvarieties of X , ordered by inclusion, is filtrant. Definition 3.4.1. (i) The stack of regular holonomic microdifferential modules on X is the C-linear abelian stack defined by RH X = lim RH X, . − →
(ii) The triangulated category of complexes of regular holonomic microdifferential modules on X is the C-linear triangulated category defined by DRH(X ) = lim DRH (X ). − →
1 The statement in [11, Theorem 9.2 (ii)] is not correct. It should be read as Theorem 3.3.2 in the present paper.
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As a corollary of Proposition 3.3.1, we get Theorem 3.4.2. For a closed smooth Lagrangian submanifold ⊂ X , there is an equivalence RH X, p1 ∗ LocSys( p1−1 C⊗1/2 ),
where p1 :
× C×
− → is the projection.
Remark 3.4.3. When X is reduced to a point, the category of regular holonomic microdifferential modules on X is equivalent to the category of local systems on C× . As a corollary of Theorem 3.3.2, we get Theorem 3.4.4. If X is compact, then DRH(X ) is a C-linear Calabi-Yau triangulated category of dimension dim X + 1. 4. Quantization Algebroid In this section, we first recall the construction of the deformation-quantization algebroid of [16] in terms of algebroid data. Then, with the same data, we construct a new C-algebroid where the deformation parameter is no longer central. Its centralizer is related to the deformation-quantization algebroid through a twist by the gerbe parameterizing the primitives of the symplectic 2-form. ρ
4.1. Quantization data. Let X be a complex symplectic manifold. Let ρ = (Y − → X, α) be a contactification of X and E a microdifferential algebra on Y . γ Recall that the contact structure on Y is given by Y = (Z − → Y, θ ), where Z = Y × C× , γ = q1 is the first projection and θ = τ q1∗ α for τ the coordinate of C× ⊂ C. Definition 4.1.1. A deformation parameter is an invertible section ∈ F−1 E such that ι(σ−1 ()) = α, under the embedding (2.3.1). Example 4.1.2. Let (t; τ ) be the symplectic coordinates on T ∗ C. Recall from Example 2.4.1 the contactification of the conormal bundle T ∗ M to a complex manifold M given by ρ
P ∗ (M × C) ⊃ {τ = 0} − → T ∗ M. In this case the condition ι(σ−1 ()) = α reads σ−1 () = τ −1 . Denote by ∂t ∈ F1 EC the operator with total symbol τ . It induces a deformation parameter = ∂t−1 in E M×C |{τ =0} . → Y (for λ ∈ C) denotes the C-action on Y and va denotes its Recall that Tλ : Y − infinitesimal generator. Note that d −1 Ad(eλ )|λ=0 dλ is a C-linear derivation of E inducing va on symbols. This derivation is integrable, and induces the isomorphism ad(−1 ) =
−1 )
eλ Ad(
−1
∼
= Ad(eλ ) : (T−λ )∗ E − → E.
This is a quantized contact transformation over T−λ .
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Definition 4.1.3. The gerbe P X on X is defined as follows: (1) Objects on U ⊂ X are quadruples q = (ρ, E, ∗, ) of a contactification ρ = ρ → U, α), a microdifferential algebra E on V , an anti-involution ∗ of E and a (V − deformation parameter ∈ F−1 E such that ∗ = −. (2) If q = (ρ , E , ∗ , ) is another object, Hom P X (q , q) = {(χ , f ); χ ∈ Hom Cω (ρ , ρ), f ∈ Isom C-Alg (χ∗ E , E),
f ∗ = ∗ f, f ( ) = },
with composition given by (χ , f ) ◦ (χ , f ) = χ χ , f (χ∗ f ) . −1
Note that Ad(eλ ) commutes with ∗ for λ ∈ C, since ∗ = −. Remark 4.1.4. Let M be a complex manifold. With notations as in Example 4.1.2, the operator ∂t ∈ F1 EC induces a deformation parameter = ∂t−1 in the algebra E⊗1/2 of M×C
twisted microdifferential operators. Hence PT ∗ M has a global object given by (ρ, E⊗1/2 {τ =0} , ∗, ∂t−1 ), M×C
with ∗ the anti-involution given by the formal adjoint. Lemma 4.1.5 ([16, Lemma 5.4]). For any q = (ρ, E, ∗, ) ∈ P X (U ), there is an isomorphism of sheaves of groups ∼
ψ : CU × {b ∈ ρ∗ F0 E × ; [, b] = 0, b∗ b = 1, σ0 (b) = 1} − → EndP X (q) −1 given by ψ(μ, b) = Tμ , Ad(beμ ) . One could now try to mimic the construction of the microdifferential algebroid EY in order to get an algebroid from the algebras ρ∗ E. This fails because the automorphisms −1 of (ρ, E, ∗, ) are not all inner, an outer automorphism being given by Ad(eλ ) for λ ∈ C. −1 There are two natural ways out: consider subalgebras where Ad(eλ ) acts as the −1 identity, or consider bigger algebras where Ad(eλ ) becomes inner. The first solution, utilized in [16] to construct the deformation-quantization algebroid, is recalled in Sect. 4.2. The second solution is presented in Sect. 4.3, and will allow us to construct the quantization algebroid. 4.2. Deformation-quantization algebroid. Let X be a complex symplectic manifold. We can now describe the deformation-quantization algebroid of [16] in terms of algebroid data. ρ Let ρ = (Y − → X, α) be a contactification of X . Let E be a microdifferential algebra on Y and ∈ F−1 E a deformation parameter. The deformation-quantization algebra associated with the data (ρ, E, ) is the algebra W = C0 ρ∗ E. This is the subalgebra of ρ∗ E of operators commuting with . Denote by k ⊂ C[−1 , ]] the center of W, a subfield of the field of formal Laurent series.
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Example 4.2.1. As in Example 4.1.2, consider the contactification of the conormal bundle T ∗ M to a complex manifold M given by ρ
P ∗ (M × C) ⊃ {τ = 0} − → T ∗ M. Then = ∂t−1 is a deformation parameter in E M×C . Set W M = C∂0t ρ∗ E M×C |{τ =0} . Take a local symplectic coordinate system (x; ξ ) on T ∗ M. Since an element a ∈ Fk W M commutes with ∂t , its total symbol is a formal series independent of t,
a j (x, ξ, τ ), a j ∈ O P ∗ (M×C) ( j),
j≤k
a− j (x, u, 1) and recalling that satisfying suitable growth conditions. Setting a j (x, u) = = ∂t−1 , the total symbol of a can be written as a(x, u, ) =
a j (x, u) j , a j ∈ OT ∗ M .
j≥−k
To make the link with usual deformation-quantization, consider two operators a, a ∈ F0 W M of degree zero with total symbol a(x, u), a (x, u) ∈ OT ∗ M . Then the product aa has a total symbol given by the Leibniz star-product a(x, u) a (x, u) =
|J | ∂uJ a0 (x, u)∂xJ a0 (x, u). J ! n
J ∈N
Note that there is a natural identification k = W{pt} . Recall the gerbe P X from Definition 4.1.3 and the isomorphism ψ of Lemma 4.1.5. Definition 4.2.2. The deformation-quantization algebroid on X is the k-algebroid W X associated to the data (P X , W , ), where
W (q) = W, W (χ , f ) = ρ∗ f, q (ψ(μ, b)) = b, for q = (ρ, E, ∗, ), W = C0 ρ∗ E, (χ , f ) : q − → q, and for (μ, b) as in Lemma 4.1.5. −1
Note that is indeed a lifting of since the action of Ad(eμ ) is trivial on W. Recall that objects of W X are objects q = (ρ, E, ∗, ) of P X . Morphisms q − → q in W X are equivalence classes [a, (χ , f )] with a ∈ W and (χ , f ) : q − → q a morphism in P X . M be a complex manifold and X = T ∗ M. With notations as the algebroid WT ∗ M is represented by the algebra W⊗1/2 = M . {τ =0}
Remark 4.2.3. Let in Remark 4.1.4, C 0 ρ∗ E ⊗1/2
M×C
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4.3. Quantization algebras. Let ρ = (Y − → X, α) be a contactification of the complex symplectic manifold X = (X, ω). Let E be a microdifferential algebra on Y and ∈ F−1 E a deformation parameter. Let us set E[ρ] = C∞ ρ∗ E, where C∞ E = {a ∈ E; ad() N (a) = 0, locally for some N > 0}. In local coordinates (x, t; ξ, τ ), sections of C∞ E are sections of E whose total symbol is polynomial in t. Definition 4.3.1. The quantization algebra associated with (ρ, E, ) is the C-algebra −1 E = E[ρ] eλ , λ∈C
whose product is given by −1
−1
eλ eλ
−1
−1
−1
−1
= e(λ+λ ) , eλ a = Ad(eλ )(a) eλ ,
for λ, λ ∈ C and a ∈ E[ρ] . Denote by R the group ring of the additive group C with coefficients in C, so that −1 R C eλ . λ∈C
Then one has an algebra isomorphism C0 E W ⊗C R, where W = ρ∗ C0 E is the deformation-quantization algebra associated with (ρ, E, ). In particular, C 0 E is a k ⊗ R-algebra.
C
4.4. Quantization algebroid. Let X = (X, ω) be a complex symplectic manifold. Recall the gerbe P X on X from Definition 4.1.3 and the isomorphism ψ of Lemma 4.1.5. Definition 4.4.1. The quantization algebroid on X is the C-algebroid E X associated to the data (P X , , ), where E μ−1
, E (q) = E, E (χ , f ) = ρ∗ f, q (ψ(μ, b)) = be
→ q, and for (μ, b) as in Lemma 4.1.5. for q = (ρ, E, ∗, ), (χ , f ) : q − Recall that objects of E X are objects q = (ρ, E, ∗, ) of P X . Morphisms q − →q in E X are equivalence classes [a, ˜ (χ , f )] with a˜ ∈ E and (χ , f ) : q − → q a morphism in EX . Note that there is a natural action of C[] on E X . With the notations of §1.1, we set for short 0 C0 E X = CC [] E X .
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Remark 4.4.2. Let M be a complex manifold and X = T ∗ M. With notations as in Remark 4.1.4, the algebroid ET ∗ M is represented by the algebra E⊗1/2 {τ =0} . M×C
Recall the notation R by Definition 2.1.5 for
λ∈C C e
λ−1 .
Let Rω be the invertible R-algebroid given −1
: C − → R × , λ → eλ . Note that Rω R X if X admits a contactification. The following proposition can be compared with [16, Remark 9.3]. Proposition 4.4.3. There is an equivalence of k ⊗C R-algebroids W X ⊗C X Rω C0 EX . → W X ⊗C X Rω defined by Proof. Consider the functor ψ : C0 EX − −1 −1 (ρ, E, ∗, ) → (ρ, E, ∗, ), ρ , [aeλ , (χ , f )] → [a, (χ , f )] ⊗ [eλ , χ ] −1
on objects and morphisms, respectively. Since a ∈ C0 E commutes with eλ , ψ is indeed compatible with composition of morphisms. The fact that ψ is an equivalence is a local problem, and thus follows from the isomorphism of the representative algebras C0 E W ⊗C R. In particular, W X is equivalent to the homogeneous component of degree zero in −1 C0 C eλ . E X ⊗R X R−ω W X ⊗C λ∈C
5. Quantization Modules Here, after establishing some algebraic properties of quantization algebras, we show how the category RH X of regular holonomic microdifferential modules can be embedded in the category of quantization modules.
5.1. A coherence criterion. Let us state a non-commutative version of Hilbert’s basis theorem. For a sheaf of rings A on a topological space, consider the sheaf of rings AS A ⊗Z Z[S] of polynomials in a variable S which is not central but satisfies the rule Sa = ϕ(a)S + ψ(a), ∀a ∈ A, where ϕ is an automorphism of A and ψ is a ϕ-twisted derivation, i.e. a linear map such that ψ(ab) = ψ(a)b + ϕ(a)ψ(b). The following result can be proved along the same lines as [7, Theorem A.26]. Theorem 5.1.1. If A is Noetherian, then AS is Noetherian.
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5.2. Algebraic properties of quantization algebras. As the results in the rest of this section are of a local nature, we will consider the geometrical situation of Example 2.4.1. In particular, for (t; τ ) the symplectic coordinates of T ∗ C, we consider the projection ρ
→ T ∗ M = X. P ∗ (M × C) ⊃ Y = {τ = 0} − For = ∂t−1 , we set E = E M×C |τ =0 , E[ρ] = C∞ ρ∗ E, W = C0 ρ∗ E, E =
−1
E[ρ] eλ .
λ∈C
Theorem 5.2.1. The ring E[ρ] is Noetherian. ∼
Proof. Note that there is an isomorphism WS − → E[ρ] given by S → t. Using the results of [7, Appendix], one proves that W is Noetherian. Then E[ρ] is also Noetherian by Theorem 5.1.1. Theorem 5.2.2. The sheaves of rings E and C0 E are coherent. as the arguments for C 0 E are similar. Proof. We shall only consider E, −1 For a finitely generated Z-submodule of C, set E = λ∈ E[ρ] eλ . By induction on the minimal number of generators of one proves that E is Noetherian. In fact, ∼ → E let = 0 + Zλ and assume that E0 is Noetherian. If 0 ⊕ Zλ, then E0 S − −1 by S → eλ . Hence E0 is Noetherian by Theorem 5.1.1. Otherwise, let N be the −1 smallest integer such that nλ ∈ 0 . Then E E0 S/S − enλ is again Noetherian. → E are As E is Noetherian, it is in particular coherent. Since the morphisms E − flat for ⊂ , coherence is preserved at the limit E lim E . − →
For M ∈ Mod(E[ρ] ), let us set for short ρE∗ M = E ⊗ρ −1 E[ρ] ρ −1 M, Supp(M) = supp(ρE∗ M) ⊂ Y. Let us denote by Modρ-f,coh (E[ρ] ) the full abelian substack of Modcoh (E[ρ] ) whose objects M are such that ρ is finite on Supp(M). Let us denote by Modρ-f,coh (E) the full abelian substack of Modcoh (E) whose objects N are such that ρ is finite on supp(N ). Proposition 5.2.3. (i) The ring E is flat over ρ −1 E[ρ] . (ii) There is an equivalence of categories Modρ-f,coh (E[ρ] ) o
ρE∗ ρ∗
/ ρ Mod ∗ ρ-f,coh (E),
meaning that the functors ρE∗ and ρ∗ are quasi-inverse to each other. Let us set for short Ak = ρ −1 Fk E[ρ] , Note that A−k =
k A
0
= A0
k ,
B−k =
k B0
Bk = Fk E. = B0
k
(5.2.1)
and
A0 /A−1 ρ −1 O X [t], B0 /B−1 OY . The above proposition is a non commutative analogue of the following classical result
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Proposition 5.2.4. (i) The ring OY is flat over ρ −1 O X [t]. (ii) There is an equivalence of categories ρ∗
Modρ-f,coh (O X [t]) o
ρ∗
/ ρ Mod ∗ ρ-f,coh (OY ).
Proof of Proposition 5.2.3 (i). With notations (5.2.1), it is enough to show that B0 is flat over A0 . Thus, for a coherent A0 -module M, we have to prove that L H −1 (B0 ⊗A M) = 0.
(5.2.2)
0
One says that u ∈ M is an element of -torsion if N u = 0 for some N ≥ 0, i.e. if A−N u = 0. Denote by Mtor ⊂ M the coherent submodule of -torsion elements. One says that M is an -torsion module if Mtor = M and that M has no -torsion if Mtor = 0. Considering the exact sequence 0− → Mtor − →M− → M/Mtor − → 0, it is enough to prove (5.2.2) in the case where M is either an -torsion module or has no -torsion. (a) Assume that M has no -torsion. Then the multiplication map A−1 ⊗A0 M − →M is injective. Setting M−1 = A−1 M = M, this implies the isomorphism (A0 /A−1 ) ⊗A0 M M/M−1 . By Proposition 5.2.4 (i), we have L L H −1 ((B0 /B−1 ) ⊗BL B0 ⊗A M) H −1 ((B0 /B−1 ) ⊗A (M/M−1 )) = 0. 0 0 0 /A−1
From the exact sequence 0 − → B−1 − → B0 − → B0 /B−1 − → 0 we thus obtain the exact sequence L L B−1 ⊗B0 H −1 (B0 ⊗A M) − → H −1 (B0 ⊗A M) − → 0. 0
0
L M) = 0. By Nakayama’s lemma, we get H −1 (B0 ⊗A 0 (b) Let M be an -torsion module. As M is coherent, there locally exists N > 0 such that N M = 0. Considering the exact sequence
0− → M−1 − →M− → M/M−1 − → 0, by induction on N one reduces to the case N = 1. Then M = M/M−1 has a structure of A0 /A−1 -module. Hence L L L L B0 ⊗A M B0 ⊗A A0 /A−1 ⊗A M B0 /B−1 ⊗A M, 0 0 0 /A−1 0 /A−1
and (5.2.2) follows from Proposition 5.2.4 (i).
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We shall consider an operator a ∈ F0 E[ρ] monic in the t variable, i.e. an operator of the form a = tm +
m−1
bi t i ,
m ∈ N>0 , bi ∈ F0 W.
(5.2.3)
i=0
Lemma 5.2.5. Let a be of the form (5.2.3). Then there are isomorphisms ρE∗ (E[ρ] /E[ρ] a) E/Ea, ρ∗ (E/Ea) E[ρ] /E[ρ] a. Proof. The first isomorphism is clear. For the second, note that ρ∗ (E/Ea) ρ∗ E/ρ∗ Ea since ρ is finite on supp(E/Ea). Note also that, by division, any c ∈ ρ∗ E can be written as ∼ → E[ρ] /E[ρ] a c = da + b with d ∈ ρ∗ E and b ∈ E[ρ] . Then the isomorphism ρ∗ E/ρ∗ Ea − is given by c → b. Proof of Proposition 5.2.3 (ii). (a) Let N0 be a coherent F0 E-module such that ρ is finite on supp N0 . We will show that N0 is F0 W-coherent. As this is a local problem on Y , we can assume that (x0 , t; ξ0 , 1) ∈ supp N0 only for t = 0. Thus supp N0 ⊂ {t p + ϕ(x, t, ξ/τ ) = 0} with ϕ ∈ O X [t] vanishing for t = 0 and of degree less than p in the t variable. Choose a system u 1 , . . . , u N of generators for N0 . By division, for each i there exists ai of the form (5.2.3) such that ai u i = 0. One thus gets an exact sequence 0− → N0 − →
N
F0 E/F0 Eai − → N0 − → 0.
i=1
As F0 E/F0 Eai is F0 W-coherent, N0 is a finitely generated F0 W-module. Since also N0 is finitely generated over F0 W, it follows that N0 is F0 W-coherent. In particular, this shows that any N ∈ ρ∗ Modρ-f,coh (E) is a coherent E[ρ] -module. (b) Let N ∈ ρ∗ Modρ-f,coh (E) and choose a system u 1 , . . . , u N ∈ N of generators. By (a), ρ∗ F0 Eu i is F0 W-coherent. Hence, {t j F0 Wu i } j>0 is stationary in ρ∗ F0 Eu i , so that there exist m i > 0 and bi j ∈ F0 W such that t m i u i = j<m i bi j t j u i . In other words, for each i there exists ai = t m i − j bi j t j of the form (5.2.3) such that ai u i = 0. One thus gets an exact sequence
0− →N − →
N
E/Eai − →N − → 0.
i=1
Applying the same argument to N one gets a presentation
N
E/Eai − →
i=1
N
E/Eai − →N − → 0.
i=1
Since ρ∗ = ρ! is exact on this sequence, by Lemma 5.2.5 the module ρ∗ N has the presentation
N i=1
E[ρ] /E[ρ] ai − →
N
E[ρ] /E[ρ] ai − → ρ∗ N − → 0.
i=1 ∼
Applying the exact functor ρE∗ and using again Lemma 5.2.5, we get that ρE∗ ρ∗ N − → N.
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(c) For M ∈ Modρ-f,coh (E[ρ] ), let us show that the map M − → ρ∗ ρE∗ M is injective. Let M0 be a lattice of M, that is a coherent sub-F0 E[ρ] -module such that E[ρ] M0 = M. Since ρ F∗0 E M0 is a lattice for ρE∗ M, it is enough to prove the injectivity of the map → ρ∗ ρ F∗0 E M0 . Assume that u ∈ M0 is sent to 0. By Proposition 5.2.4 there are M0 − isomorphisms ∼
M0 /F−1 EM0 − → ρ∗ ρ ∗ (M0 /F−1 EM0 ) ρ∗ ρ F∗0 E M0 /F−1 Eρ∗ ρ F∗0 E M0 . It follows that u ∈ F−1 EM0 . By induction we then get u ∈ k>0 F−k EM0 , so that u = 0. ∼ (d) We finally have to prove the isomorphism M − → ρ∗ ρE∗ M. Let u 1 , . . . , u N be a system of generators of M. By the same arguments as in (b), for each i there exists ai of the form (5.2.3) such that ai u i = 0 in ρE∗ M. By (c) this implies ai u i = 0 in M. As in (b) we thus get a resolution
N
E[ρ] /E[ρ] ai − →
i=1
N
E[ρ] /E[ρ] ai − →M− → 0,
i=1 ∼
→ ρ∗ ρE∗ M by Lemma 5.2.5. giving the isomorphism M −
For S ⊂ Y , let us denote by Mod S,coh (E[ρ] ) the full abelian substack of Modcoh (E[ρ] ) whose objects M are such that Supp(M) ⊂ S. For T ⊂ X , let us denote by ModT,coh (E) the full abelian substack of Modcoh (E) whose objects M are such that supp(M) ⊂ T . We set for short = E ⊗ M. EM E[ρ] Proposition 5.2.6. (i) The ring E is faithfully flat over E[ρ] . (ii) Let S ⊂ Y be an analytic subset such that ρ| S is proper and injective. Then the functor : Mod S,coh (E[ρ] ) − E(·) → Modρ(S),coh (EX ) is fully faithful. Proof. (i) is straightforward. (ii) For a coherent E[ρ] -module M, there is an isomorphism of E[ρ] -modules EM
−1
eλ M.
λ∈C −1
Here, the E[ρ] -module structure of eλ M is given by −1
a(eλ
−1
· b) = eλ
−1
· Ad(e−λ )(a)b, −1
for a ∈ E[ρ] and b ∈ M. Note that Supp(eλ M) = Tλ Supp(M).
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For M, M ∈ Mod S,coh (E[ρ] ), one has , EM) Hom E[ρ] (M , Hom E(EM
λ∈C
λ∈C
−1
eλ M)
λ∈C −1
Hom E[ρ] (M , eλ M) −1
Hom E (ρE∗ M , ρE∗ (eλ M))
Hom E (ρE∗ M , ρE∗ M) Hom E[ρ] (M , M), where the second to last isomorphism is due to the fact that Supp(M ) ∩ −1 Supp(eλ M) = ∅ for λ = 0. 5.3. Induced modules. Assume that the symplectic manifold X admits a contactificaρ tion ρ = (Y − → X, α). In this section we show how the constructions from the previous section can be globalized. Definition 5.3.1. For a contactification ρ of X , the gerbe Pρ on X is defined as follows: (1) Objects on U ⊂ X are triples p = (E, ∗, ) of a microdifferential algebra E on ρ −1 (U ), an anti-involution ∗ of E and a deformation parameter such that ∗ = −. (2) If p = (E , ∗ , ) is another object, Hom Pρ ( p , p) = { f ∈ Isom R-Alg (E , E); f ∗ = ∗ f, f ( ) = }. As a corollary of Lemma 3.2.4, one has Lemma 5.3.2. For any p = (E, ∗, ) ∈ Pρ there is an isomorphism of sheaves of groups ∼
→ EndPρ ( p) ψρ : {b ∈ E × ; [, b] = 0, b∗ b = 1, σ0 (b) = 1} − given by ψρ (b) = Ad(b). Definition 5.3.3. For a contactification ρ of X , the stack E[ρ] is the C-algebroid on X associated to the data (Pρ , E[ρ] , ) where
E[ρ] ( p) = E[ρ] , E[ρ] ( f ) = ρ∗ f, p (g) = b, for p = (E, ∗, ), f : p − → p and g = ψρ (b). Note that Proposition 4.4.3 implies W X C0 E[ρ] . As in the local case, for M ∈ Mod(E[ρ] ) we set for short Supp(M) = supp(ρE∗ M) ⊂ Y.
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Consider the faithful C-linear functors → EY , (E, ∗, ) → (E, ∗), ρ −1 E[ρ] − [a, f ] → [a, f ], E[ρ] − → E X , (E, ∗, ) → (ρ, E, ∗, ), −1 [a, f ] → [ae0 , idρ , f ],
on objects, on morphisms, on objects, on morphisms.
For S ⊂ Y they induce the functors → ρ∗ Modρ-f,coh (EY ), ρE∗ : Modρ-f,coh (E[ρ] ) − E(·) : Mod S,coh (E[ρ] ) − → Modρ(S),coh ( E X ). By Propositions 5.2.3 and 5.2.6 we have Proposition 5.3.4. (i) The functor ρE∗ is an equivalence. (ii) Let S ⊂ Y be an analytic subset such that ρ| S is proper and injective. Then E(·) is fully faithful. We can thus embed regular holonomic microdifferential modules in the stack of coherent E X -modules. Thus, with notations as in Definition 3.4.1, we have Corollary 5.3.5. There is a fully faithful embedding E X ). RH X ⊂ Modcoh ( Remark 5.3.6. We do not know if the above result extends to give an embedding DRH(X ) ⊂ Dbcoh ( E X ) at the level of derived categories. Appendix A. Remarks on Deformation-Quantization We give in this appendix an alternative description of the deformation quantization algebroid using triples (W, ∗, v) of a deformation-quantization algebra W endowed with an anti-involution ∗ and an order preserving C-linear derivation v. We also compare regular holonomic deformation-quantization modules with regular holonomic quantization modules. A.1. Deformation-quantization and derivations. Let X = (X, ω) be a complex symplectic manifold and W a deformation-quantization algebra on X . Lemma A.1.1. Let w be an order preserving k-linear derivation of W. Then w is locally of the form ad(−1 d) for some d ∈ F0 W. Proof. Let (x; u) be a local system of quantized symplectic coordinates (see [9, §2.2.3]). For i = 1, . . . , n, set ei = w(xi ) ∈ F−1 W. From w([xi , x j ]) = 0 we get [ei , x j ] = [e j , xi ] for any i, j = 1, . . . , n. Hence there locally exists e ∈ F0 W with ei = [xi , e]. Replacing w by w − ad(−1 e) we may assume w(xi ) = 0. Set di = w(u i ) ∈ F−1 W. From w([xi , u j ]) = 0 we get [xi , d j ] = 0, so that di = di (x) does not depend on u. From w([u i , u j ]) = 0 we get [di , u j ] = [d j , u i ]. Hence there locally exists d = d(x) ∈ F0 W with di = [u i , d]. Replacing w by w − ad(−1 d) we have w(xi ) = w(u j ) = 0, and hence w = 0.
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Definition A.1.2. Let PX be the stack on X associated with the separated prestack PX,0 defined as follows. (1) Objects on U ⊂ X are triples q = (W, ∗, v) of a deformation quantization algebra W on U , an anti-involution ∗ and an order preserving C-linear derivation v of W such that v() = and v∗ = ∗v. (2) If q = (W , ∗ , v ) is another object, Hom PX,0 (q , q) = {(g, d); g ∈ Isom R-Alg (W , W), d ∈ F0 W, g∗ = ∗g, d = d ∗ , v − gv g −1 = ad(−1 d)}, with composition given by (g, d) ◦ (g , d ) = (gg , d + g(d )). Using Lemma A.1.1 one gets Lemma A.1.3. The stack PX is a gerbe. Remark A.1.4. Let M be a complex manifold and X = T ∗ M. With notations as in Remark 4.1.4, where = ∂t−1 , a global object of PX is given by (W⊗1/2 , ∗, ad(t∂t )). M
Lemma A.1.5. For any q = (W, ∗, v) ∈
PX (U )
there is a group isomorphism ∼
ψω : CU × {b ∈ F0 W × ; b∗ b = 1, σ0 (b) = 1} − → EndPX (q) given by ψω (μ, b) = (Ad(b), μ + v(b)b−1 ). (i) Let us prove injectivity. Assume that Ad(b) = id and μ + v(b)b−1 = 0. Then b ∈ k(0), μ = 0 and v(b) = 0. As v(b) = ∂∂ b, we get b ∈ C. Since σ0 (b) = 1, this finally gives b = 1. (ii) Let us prove surjectivity. Take (g, d) ∈ EndPX (q). Since any k-algebra automorphisms of W is inner, we can locally write g = Ad(b) for some b ∈ F0 W × . As g commutes with the anti-involutions, we have Ad(b)(a ∗ ) = (Ad(b)(a))∗ = Ad(b∗−1 )(a ∗ ) for any a ∈ W. This implies Ad(b∗ b) = id, so that b∗ b ∈ k(0). Take k ∈ k(0) with k ∗ k = b∗ b. Up to replacing b with bk −1 we may thus assume that b∗ b = 1. This implies σ0 (b) = ±1 and we may further assume that σ (b) = 1. Replacing (g, d) by (g, d) · ψω (b−1 , 0) we may thus assume g = id. Since ad(−1 d) = 0, we have d ∈ k(0). As d ∗ = d and ∗ = −, the coefficients of the odd powers of in d vanish, and we may write d = μ + 2 d for μ ∈ C and d ∈ k(0). Take d ∈ k(0) such that ∂∂ d = d , and set b = exp(d ). Since v(b)b−1 = d , we have d = μ + v(b)b−1 . Hence ψω (μ, b) = (id, d).
Proof.
Definition A.1.6. The algebroid WX is the k-algebroid on X associated to the data (PX , W , ), where
W (q) = W, W (g, d) = g, q (h, e) = b, for q = (W, ∗, v), (g, d) : q − → q and (h, e) = ψω (μ, b). Proposition A.1.7. There is a k-linear equivalence WX W X .
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This follows from the following proposition. Proposition A.1.8. There is an equivalence of gerbes PX P X . Proof. Let us consider the gerbe PX whose objects on U ⊂ X are quintuples q = (ρ, E, ∗, , t) such that π(q) = (ρ, E, ∗, ) is an object of P X and t ∈ F0 E is an operator with [−1 , t] = 1. (The local model in a Darboux chart is obtained by Example 4.1.2 with −1 = ∂t and t = t.) We set Hom PX (q , q) = Hom P X (π(q ), π(q)). There is a natural equivalence ∼
→ P X , q → π(q). PX − Consider the functor ψ : PX − → PX given by q → (C0 ρ∗ E, ∗, ad(t−1 )), (χ , f ) → ρ∗ f, t − f (t ) ,
for q = (ρ, E, ∗, , t), for (χ , f ) : q − → q.
This is well defined since ad(t−1 ) − f ad(t −1 ) f −1 = ad((t − f (t ))−1 ). It follows from Lemmas A.1.5 and 4.1.5 that ψ is fully faithful. As PX and PX are gerbes, ψ is an equivalence. Recall that if q = (W, ∗, v) is an object of PX on an open subset U ⊂ X , then W X |U is represented by W. As shown in [15], the filtration and the anti-involution of W extend to W X . As we will now explain, also the derivation of W extends to W X . Let ε be a formal variable with ε2 = 0. Consider the natural morphisms i
π
W− → W[ε] − → W. Let us extend the anti-involution ∗ to W[ε] by setting ε∗ = −ε. Lemma A.1.9. Let ϕ : W − → W[ε] be an order preserving C-algebra morphism such that π ϕ = idW , ϕ() = + ε2 and ϕ∗ = ∗ϕ. Then ϕ = i + εv for an order preserving C-linear derivation v of W such that v∗ = ∗v. Remark A.1.10. There is an isomorphism of W ⊗C W op -modules (W[ε])ϕ C1 ρ∗ E such that the multiplication by ε corresponds to ad(−1 ). In local coordinates where −1 = ∂t and v = ad(t∂t ), this isomorphism is given by a + εb → at + b. The above lemma motivates the following definition. Definition A.1.11. A derivation of a C-linear stack A is the data of a pair ϕ = (C, ϕ), where C is an invertible C[ε]-algebroid such that C/ε is represented by C X and ϕ : A − → A ⊗C C is a C-linear functor such that π ϕ idA . Here π : A ⊗C C − → A is the functor induced by C − → C/ε.
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Consider the following algebroid. Definition A.1.12. The algebroid WεX is the k[ε]-algebroid associated to the data (PX , εW , ), where
εW (q) = W[ε], εW (g, d) = (1 + ε ad(d))g, q (h, e) = (1 + εμ)b, for q = (W, ∗, v), (g, d) : q − → q and (h, e) = ψω (μ, b). There is a natural morphism ϕ : WX − → WεX satisfying ϕ() = + ε2 and ϕ∗ = ∗ϕ. Similarly to Proposition 4.4.3, one proves that there is an equivalence of k[ε]-algebroids WεX W X ⊗C C[ε]ω , where C[ε]ω is the invertible C[ε]-algebroid given by Definition 2.1.5 for : C − → C[ε]× , λ → (1 + ελ). Thus W X is endowed with the derivation ϕ = (C[ε]ω , ϕ). Summarizing, W X is a filtered k-stack endowed with an anti-involution ∗ and with a C-linear derivation ϕ such that F0 W X /F−1 W X is represented by O X , ϕ() = and ϕ∗ = ∗ϕ. One can prove along the lines of [15] that W X is unique among the stacks which satisfy these properties and which are locally represented by deformation quantization algebras. A.2. Comparison of regular holonomic modules. We shall compare here regular holonomic quantization-modules with regular holonomic deformation-quantization modules. Let us start by recalling the definition of regular holonomic deformation-quantization modules from [11]. Let X be a complex symplectic manifold and a closed Lagrangian subvariety of X . Let W be a deformation-quantization algebra on X . Definition A.2.1. (i) One says that a coherent F0 W-module M0 is regular holonomic along if supp(M0 ) ⊂ and M0 /M0 is a coherent O -module. (ii) One says that a coherent W-module M is regular holonomic along if supp(M) ⊂ and there exists locally a coherent F0 W-submodule M0 of M such that M0 generates M over W and M0 is regular holonomic along . Recall that W X denotes the deformation-quantization algebroid on X . As the above definition is local, there is a natural notion of regular holonomic W X -module along . Let us denote by Mod,rh (W X ) the full substack of Modcoh (W X ) whose objects are regular holonomic along . Up to shrinking X , we may assume that there exist a contactification ρ : Y − → X and a Lagrangian subvariety of Y such that ρ induces an isomorphism − → . By definition, regular holonomic E X -modules along are equivalent to regular holonomic EY -modules along . In order to compare quantization and deformation-quantization modules, let us thus consider the forgetful functor for : ρ∗ Mod,rh (EY ) − → Mod,rh (W X ) induced by the equivalence W X C0 E[ρ] and the functor ρ −1 E[ρ] − → EY from §5.3.
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Proposition A.2.2. (i) The functor for is faithful but not locally full in general. (ii) If is a smooth submanifold, the functor for is locally essentially surjective but not essentially surjective in general. (iii) The functor for is not locally essentially surjective in general. Proof. (i) holds more generally for the forgetful functor ρ∗ Mod(EY ) − → Mod(W X ). (ii) Let be a smooth submanifold. Consider the commutative diagram / Mod,rh (W X ) O
for
ρ∗ Mod,rh (EY ) O ∼
ρ∗ p1 ∗ LocSys( p1−1 C⊗1/2 )
∼
/ LocSys(k⊗1/2 ),
where p1 : × C× − → is the projection. The vertical equivalences are due to Proposition 3.3.1 and [3, Cor. 9.2], respectively. The bottom arrow is given by L → k ⊗C L|s=1 , where s is the coordinate of C× . This shows that the forgetful functor is locally essentially surjective. To prove that it is not surjective in general, take X = T ∗ (C× ) and the zero section of T ∗ (C× ). Then the local system with monodromy 1 + around the origin is not in the essential image of the forgetful functor. (iii) follows from Proposition A.2.3 below. Before stating Proposition A.2.3 let us introduce some notations. Let M = C. Denote by (x, t; ξ, τ ) the symplectic coordinates of P ∗ (M × C) and by (x; u) those of T ∗ M. Let W = W M , and recall that = ∂t−1 . We will identify elements a ∈ W with their total symbol a(x, u, τ ), and write for example ax for the operator with total symbol ∂∂x a(x, u, τ ). Denote by OM = W/W∂x the canonical regular holonomic module along the zero section 1 = {(x, u); u = 0}. The quotient map W − → OM , b → [b] induces an isomorphism of vector spaces ∼ − C x0 W with the subring of operators whose total symbol does not depend on ∂x . OM ← For m ∈ Z>0 , consider the Lagrangian subvariety = 1 ∪ 2 , with 2 = {(x, u); u = x m }. For a ∈ C x0 W, let Ma be the regular holonomic module along with generators v1 , v2 and relations ∂x v1 = 0, (∂x − x m ∂t )v2 = av1 . Note that Ma C x0 W v1 ⊕ C x0 W v2 . ∼
→ Ma . Let a ∈ C x0 W be another operator. If [a − a ] ∈ (∂x − x m ∂t )OM , then Ma − ∼ 0 m → Ma is In fact, if e ∈ C x W satisfies a − a = ex − x e∂t , an isomorphism Ma −
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given by v1 → v1 , v2 → v2 + ev1 . Since OM /(∂x − x m ∂t )OM thus assume that
m−1 i=0
kx i , we may
a = a0 + a1 x + · · · + am−1 x m−1 with ai ∈ k. The following counterexample was developed by the second author (M.K.) while working with Pierre Schapira at [11]. Proposition A.2.3. If Ma for(N ) for some EY -module N , then a is homogeneous, i.e. a = ai0 x i0 for some i 0 ∈ {0, . . . , m − 1}. Proof. The existence of such an N is equivalent to the existence of an endomorphism t of Ma such that [t, x] = [t, ∂x ] = 0 and [t, ∂t ] = −1. (i) Let tv1 = bv1 + cv2 for b, c ∈ C x0 W. Then 0 = t∂x v1 = ∂x tv1 = ∂x (bv1 + cv2 ) = bx v1 + cx v2 + c(x m ∂t v2 + av1 ). Hence bx + ac = 0, x m c∂t + cx = 0. It follows from the second equation that c = 0. Thus the first equation implies that b ∈ k. Up to replacing t by t − b, we may assume that tv1 = 0. (ii) Let tv2 = bv1 + cv2 for b, c ∈ C x0 W. Then 0 = t (∂x − x m ∂t )v2 − av1 = (∂x − x m ∂t )tv2 + x m v2 − [t, a]v1 = (∂x − x m ∂t )(bv1 + cv2 ) + x m v2 − [t, a]v1 = bx v1 + cx v2 + c(x m ∂t v2 + av1 ) − x m b∂t v1 − x m c∂t v2 + x m v2 − [t, a]v1 . Hence ac + bx − x m b∂t − [t, a] = 0, cx + x m = 0.
(A.2.1)
m+1
The second equation gives c = − xm+1 + d for d ∈ k. Then, the first equation in (A.2.1) can be rewritten (ad(∂x ) − x m ∂t )(xa + (m + 1)b∂t ) − (xax − ea + (m + 1)∂t [t, a]) = 0, for e = (m + 1)d∂t − 1 ∈ k. Considering the degree in x, we have xa + (m + m−1 i 1)b∂t = xax − ea + (m + 1)∂t [t, a] = 0. Since a = i=0 ai x , it implies m−1 that i=0 ((e − i)ai − (m + 1)∂t [t, ai ])x i = 0. Hence we have (e − i)ai − (m + t [t,ai ] +i. Since 1)∂t [t, ai ] = 0 for every i. Thus we have either ai = 0 or e = (m+1)∂ ai (m+1)∂t [t,ai ] ai
∈ (m + 1)Z + F−1 k, this implies a = ai0 x i0 for some 0 ≤ i 0 ≤ m − 1.
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References 1. D’Agnolo, A., Kashiwara, M.: A note on quantization of complex symplectic manifolds. http://arxiv.org/ abs/1006.0306v2 [math.AG], 2010 2. D’Agnolo, A., Polesello, P.: Deformation quantization of complex involutive submanifolds. In: Noncommutative geometry and physics (Yokohama, 2004), Rivers Edge, NJ: World Scientific, 2005, pp. 127–137 3. D’Agnolo, A., Schapira, P.: Quantization of complex Lagrangian submanifolds. Adv. Math. 2131, 358– 379 (2007) 4. Giraud, J.: Cohomologie non abelienne. Grundlehren der Math. Wiss. 179, Berlin: Springer, 1971 5. Gukov, S., Witten, E.: Branes and quantization. Adv. Theor. Math. Phys. 13(5), 1445–1518 (2009) 6. Kashiwara, M.: Quantization of contact manifolds. Publ. Res. Inst. Math. Sci. 32(1), 1–7 (1996) 7. Kashiwara, M.: D-modules and Microlocal Calculus. Translations of Mathematical Monographs 217, Providence, RI: Amer. Math. Soc. 2003 8. Kashiwara, M., Kawai, T.: On holonomic systems of microdifferential equations III. Publ. RIMS Kyoto Univ. 17, 813–979 (1981) 9. Kashiwara, M., Rouquier, R.: Microlocalization of rational Cherednik algebras. Duke Math. J. 144(3), 525–573 (2008) 10. Kashiwara, M., Schapira, P.: Categories and sheaves. Grundlehren der Math. Wiss. 332, Berlin-Heidelberg-New York: Springer, 2006 11. Kashiwara, M., Schapira, P.: Constructibility and duality for simple holonomic modules on complex symplectic manifolds. Amer. J. Math. 130(1), 207–237 (2008) 12. Kashiwara, M., Schapira, P.: Deformation quantization modules, http://arxiv.org/abs/1003.3304v2. [math.AG], 2010, to appear in Astérisque 13. Kontsevich, M.: Deformation quantization of algebraic varieties. In: EuroConférence Moshé Flato, Part III (Dijon, 2000), Lett. Math. Phys. 56(3), 271–294 (2001) 14. Nest, R., Tsygan, B.: Remarks on modules over deformation quantization algebras. Mosc. Math. J. 4(4), 911–940 (2004) 15. Polesello, P.: Classification of deformation quantization algebroids on complex symplectic manifolds. Publ. Res. Inst. Math. Sci. 44(3), 725–748 (2008) 16. Polesello, P., Schapira, P.: Stacks of quantization-deformation modules on complex symplectic manifolds. Int. Math. Res. Notices 2004:49, 2637–2664 (2004) 17. Sato, M., Kawai, T., Kashiwara, M.: Microfunctions and pseudo-differential equations. In: Hyperfunctions and pseudo-differential equations (Katata 1971), Lecture Notes in Math. 287, Berlin-HeidelbergNew York: Springer, 1973, pp. 265–529 18. Street, R.: Categorical structures. In: Handbook of algebra, Vol. 1, Amsterdam: North-Holland, 1996, pp. 529–577 19. Tsygan, B.: Oscillatory modules. Lett. Math. Phys. 88(1–3), 343–369 (2009) Communicated by N.A. Nekrasov
Commun. Math. Phys. 308, 115–132 (2011) Digital Object Identifier (DOI) 10.1007/s00220-011-1323-9
Communications in
Mathematical Physics
Exactness of the Fock Space Representation of the q -Commutation Relations Matthew Kennedy, Alexandru Nica Pure Mathematics Department, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada. E-mail: [email protected]; [email protected] Received: 29 September 2010 / Accepted: 21 March 2011 Published online: 9 September 2011 – © Springer-Verlag 2011
Abstract: We show that for all q in the interval (−1, 1), the Fock representation of the q-commutation relations can be unitarily embedded into the Fock representation of the extended Cuntz algebra. In particular, this implies that the C ∗ -algebra generated by the Fock representation of the q-commutation relations is exact. An immediate consequence is that the q-Gaussian von Neumann algebra is weakly exact for all q in the interval (−1, 1). 1. Introduction The q-commutation relations provide a q-analogue of the bosonic (q = 1) and the fermionic (q = −1) commutation relations from quantum mechanics. These relations have a natural representation on a deformed Fock space which was introduced by Bozejko and Speicher in [1], and was subsequently studied by a number of authors (see e.g. [2,5–7,9,10]). For the entirety of this paper, we fix an integer d ≥ 2. Consider the usual full Fock space F over Cd , F = ⊕∞ n=0 Fn
(orthogonal direct sum),
(1.1)
where F0 = C and Fn = (Cd )⊗n for n ≥ 1. Corresponding to the vectors in the standard orthonormal basis of Cd , one has left creation operators L 1 , . . . , L d ∈ B(F). Define the C∗ -algebra C by C := C ∗ (L 1 , . . . , L d ) ⊆ B(F).
(1.2)
It is well known that C is isomorphic to the extended Cuntz algebra. (Although it is customary to denote the extended Cuntz algebra by E, we use C here to emphasize that we are working with a concrete C ∗ -algebra of operators.) Research supported by a CGS Scholarship from NSERC Canada (M. Kennedy) and by a Discovery Grant from NSERC Canada (A. Nica).
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Now let q ∈ (−1, 1) be a deformation parameter. We consider the q-deformation F (q) of F as defined in [1]. Thus (q)
F (q) = ⊕∞ n=0 Fn
(orthogonal direct sum),
(1.3)
(q)
where every Fn is obtained by placing a certain deformed inner product on (Cd )⊗n . (The precise definition will be reviewed in Subsect. 2.1 below.) For q = 0, one obtains the usual non-deformed Fock space F from above. (q) (q) In this deformed setting, one also has natural left creation operators L 1 , . . . , L d ∈ (q) B(F ), which satisfy the q-commutation relations (q)
(q)
(q)
(q)
(L i )∗ L j = δi j I + q L j (L i )∗ , 1 ≤ i, j ≤ d. Define the C∗ -algebra C (q) by (q)
(q)
C (q) := C ∗ (L 1 , . . . , L d ) ⊆ B(F (q) ).
(1.4)
For q = 0, this construction yields the extended Cuntz algebra C from above. It is widely believed that the algebra C and the deformed algebra C (q) are actually unitarily equivalent. In fact, this is known for sufficiently small q. In [5], a unitary U : F (q) → F was constructed which embeds C into C (q) for all q ∈ (−1, 1), i.e. C ⊆ U C (q) U ∗ , and it was shown that for |q| < 0.44 this embedding is actually surjective, i.e. C = U C (q) U ∗ . The main purpose of the present paper is to show that it is possible to unitarily embed C (q) into C for all q ∈ (−1, 1). Specifically, we construct a unitary operator ∗ ⊆ C. The unitary U Uopp : F (q) → F such that Uopp C (q) Uopp opp is closely related to the unitary U from [5], as we will now see. Definition 1.1. Let J : F → F be the unitary conjugation operator which reverses the order of the components in a tensor in (Cd )⊗n , i.e. J (η1 ⊗ · · · ⊗ ηn ) = ηn ⊗ · · · ⊗ η1 , ∀η1 , . . . , ηn ∈ Cd .
(1.5)
Note that for n = 0, Eq. (1.5) says that J () = . Let J (q) : F (q) → F (q) be the operator which acts as in Eq. (1.5), where the tensor (q) is now viewed as an element of the space Fn . It is known that J (q) is also unitary operator (see the review in Subsect. 2.1). Definition 1.2. Let q ∈ (−1, 1) be a deformation parameter and let U : F (q) → F be the unitary defined in [5]. Define a new unitary Uopp : F (q) → F by Uopp = J U J (q) . The following theorem is the main result of this paper. Theorem 1.3. For every q ∈ (−1, 1) the unitary Uopp from Definition 1.2 satisfies ∗ ⊆ C. Uopp C (q) Uopp
The following corollary follows immediately from Theorem 1.3. Corollary 1.4. For every q ∈ (−1, 1) the C ∗ -algebra C (q) is exact.
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To prove Theorem 1.3, we first consider the more general question of how to verify that an operator T ∈ B(F) belongs to the algebra C. It is well known that a necessary condition for T to be in C is that it commutes modulo the compact operators with the C ∗ -algebra generated by right creation operators on F. Unfortunately, this condition isn’t sufficient (and wouldn’t be sufficient even if we were to set d equal to 1, cf. [4]). Nonetheless, by restricting our attention to a ∗-subalgebra of “band-limited operators” on F and considering commutators modulo a suitable ideal of compact operators in this algebra, we do obtain a sufficient condition for T to belong to C. This bicommutant-type result is strong enough to help in the proof of Theorem 1.3. In addition to this Introduction, the paper has four other sections. In Sect. 2, we provide a brief review of the requisite background material. In Sect. 3, we prove the above-mentioned bicommutant-type result, Theorem 3.8. In Sect. 4, we establish the main results, Theorem 1.3 and Corollary 1.4. In Sect. 5, we apply these results to the family of q-Gaussian von Neumann algebras, showing in Theorem 5.1 that these algebras are weakly exact for every q ∈ (−1, 1). 2. Review of Background 2.1. Basic facts about the q-deformed Fock space. As explained in the Introduction, there is a fairly large body of research devoted to the q-deformed Fock framework and its generalizations. Here we provide only a brief review of the terminology and facts which will be needed in Sect. 4. 2.1.1. The q-deformed inner product As mentioned above, the integer d ≥ 2 will remain fixed throughout this paper. Also fixed throughout this paper will be an orthonormal basis ξ1 , . . . , ξd for Cd . For every n ≥ 1 this gives us a preferred basis for (Cd )⊗n , namely {ξi1 ⊗ · · · ⊗ ξin | 1 ≤ i 1 , . . . , i n ≤ d}.
(2.1) (Cd )⊗n
(obtained This basis is orthonormal with respect to the usual inner product on by tensoring n copies of the standard inner product on Cd ). As in the Introduction, we will use Fn to denote the Hilbert space (Cd )⊗n endowed with this inner product. The full Fock space over Cd is then the Hilbert space F from Eq. (1.1), with the convention that F0 = C for a distinguished unit vector , referred to as the “vacuum vector”. Now let q ∈ (−1, 1) be a deformation parameter. It was shown in [1] that there exists a positive definite inner product ·, · q on (Cd )⊗n , uniquely determined by the requirement that for vectors in the natural basis (2.1), one has the formula ξi1 ⊗ · · · ⊗ ξin , ξ j1 ⊗ · · · ⊗ ξ jn q = q inv(σ ) δi1 ,σ ( j1 ) · · · δin ,σ ( jn ) . (2.2) σ
The sum on the right-hand side of Eq. (2.2) is taken over all permutations σ of {1, . . . , n}, and inv(σ ) denotes the number of inversions of σ , i.e. inv(σ ) := |{(i, j) | 1 ≤ i < j ≤ n, σ (i) > σ ( j)}| . Note that under this new inner product, the natural basis (2.1) will no longer be orthogonal if q = 0. (q) We will use Fn to denote the Hilbert space (Cd )⊗n endowed with this deformed (q) inner product. In addition, we will use the convention that F0 is the same as F0 , i.e. it is spanned by the same vacuum vector . The q-deformed Fock space over Cd is then the Hilbert space F (q) from Eq. (1.3). For q = 0, the construction of F (q) yields the usual non-deformed Fock space F from Eq. (1.1).
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2.1.2. The deformed creation and annihilation operators For every 1 ≤ j ≤ d, one has (q) deformed left creation operators L j ∈ B(F (q) ) and deformed right creation operators (q)
Rj
(q)
(q)
(q)
∈ B(F (q) ), which act on the natural basis of Fn by L j () = R j () = ξ j and
(q)
L j (ξi1 ⊗ · · · ⊗ ξin ) = ξ j ⊗ ξi1 ⊗ · · · ⊗ ξin ,
(2.3)
(q)
R j (ξi1 ⊗ · · · ⊗ ξin ) = ξi1 ⊗ · · · ⊗ ξin ⊗ ξ j . (q)
Their adjoints are the deformed left annihilation operators (L j )∗ and the deformed (q)
(q)
right annihilation operators (R j )∗ , which act on the natural basis of Fn
by
⎧ (q) ⎪ (L j )∗ (ξi1 ⊗ · · · ⊗ ξin ) ⎪ ⎪ ⎪ ⎪ n ⎪ ⎪ ⎪ ⎪ = q m−1 δ j,im ξi1 ⊗ · · · ⊗ ξ ⎪ i m ⊗ · · · ⊗ ξi n , ⎪ ⎨ m=1
(2.4)
(q) ⎪ ⎪ (R j )∗ (ξi1 ⊗ · · · ⊗ ξin ) ⎪ ⎪ ⎪ ⎪ n ⎪ ⎪ ⎪ ⎪ = q n−m δim , j ξi1 ⊗ · · · ⊗ ξ ⎪ i m ⊗ · · · ⊗ ξi n , ⎩ m=1
where the “hat” symbol over the component ξim means that it is deleted from the tensor (e.g. ξi1 ⊗ ξ i2 ⊗ ξi3 = ξi1 ⊗ ξi3 ). It’s clear from these formulas that the left creation (left annihilation) operators commute with the right creation (right annihilation) operators. For the commutator of a left annihilation operator and a right creation operator, a direct calculation (see also Lemma 3.1 from [10]) gives the formula (q)
(q)
[(L i )∗ , R j ] | F (q) = δi j q n IF (q) , ∀n ≥ 1. n n
(2.5)
Taking adjoints gives the formula for the commutator of a left creation operator and a right annihilation operator. When we are working on the non-deformed Fock space F corresponding to the case when q = 0, it will be convenient to suppress the superscripts and write L j and R j for the left and right creation operators respectively. Note that in this case, Eq. (2.3) and Eq. (2.4) imply that d j=1
L j L ∗j =
d
R j R ∗j = 1 − P0 ,
(2.6)
j=1
where P0 is the orthogonal projection onto F0 . (q)
(q)
(q)
2.1.3. The unitary conjugation operator. For every n ≥ 1, let Jn : Fn → Fn be (q) the operator which reverses the order of the components in a tensor in (Cd )⊗n , i.e, Jn acts by the formula in Eq. (1.5) of the Introduction. A consequence of Eq. (2.2), which (q) (q) defines the inner product ·, · q , is that Jn is a unitary operator in B(Fn ). Indeed,
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this is easily seen to follow from Eq. (2.2) and the following basic fact about inversions of permutations: if θ denotes the special permutation which reverses the order on {1, . . . , n}, then one has inv(θ τ θ ) = inv(τ ) for every permutation τ of {1, . . . , n}. Therefore, we can speak of the unitary operator J (q) ∈ B(F (q) ) from Definition 1.1, (q) (q) is an involution, i.e. (J (q) )2 = which is obtained as J (q) := ⊕∞ n=0 Jn . Note that J IF (q) , and that it intertwines the left and right creation operators, i.e. (q)
Rj
(q)
= J (q) L j J (q) , 1 ≤ j ≤ d.
(2.7)
2.2. The unitary U. In this subsection, we review the construction of the unitary U : F (q) → F from [5], which appears in Definition 1.2. An important role in the construction of this unitary is played by the positive operator M (q) :=
d
(q)
(q)
L j (L j )∗ ∈ B(F (q) ).
j=1 (q)
(q)
Clearly M (q) can be written as a direct sum M (q) = ⊕∞ n=0 Mn , where Mn is a positive (q) (q) operator on Fn , for every n ≥ 0. Using Eq. (2.3) and Eq. (2.4), one can show that Mn (q) acts on the natural basis of Fn by (q)
Mn (ξi1 ⊗ · · · ⊗ ξin ) =
n
q m−1 ξim ⊗ ξi1 ⊗ · · · ⊗ ξ i m ⊗ · · · ⊗ ξi n .
(2.8)
m=1
(Recall that the “hat” symbol over the component ξim means that it is deleted from the tensor.) (q) (q) With the exception of M0 (which is zero), the operators Mn are invertible. This is implied by Lemma 4.1 of [5], which also gives the estimate (q)
(Mn )−1 ≤ (1 − |q|)
∞
1 + |q|k < ∞, ∀n ≥ 1. 1 − |q|k
(2.9)
k=1
An important thing to note about Eq. (2.9) is that the upper bound on the right-hand side is independent of n. The unitary operator U is defined as a direct sum, U := ⊕∞ n=0 Un , where the unitaries (q) Un : Fn → Fn are defined recursively as follows: we first define U0 by U0 () = , and for every n ≥ 1 we define Un by (q)
Un := (I ⊗ Un−1 )(Mn )1/2 .
(2.10)
In Proposition 3.2 of [5] it was shown that Un as defined in Eq. (2.10) is actually a unitary operator, and hence that U is a unitary operator. Moreover, in Sect. 4 of [5] it was shown that C ⊆ U C (q) U ∗ for every q ∈ (−1, 1).
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2.3. Summable band-limited operators. Throughout this subsection, we fix a Hilbert space H, and in addition we fix an orthogonal direct sum decomposition of H as H = ⊕∞ n=0 Hn .
(2.11)
We will study certain properties an operator T ∈ B(H) can have with respect to this decomposition of H. We would like to emphasize that the concepts considered here depend not only on H, but also on the orthogonal decomposition for H in Equation (2.11). Definition 2.1. Let T be an operator in B(H). If there exists a non-negative integer b such that T (Hn ) ⊆ Hm , ∀n ≥ 0, (2.12) m≥0 |m−n|≤b
then we will say that T is band-limited. A number b as in Eq. (2.12) will be called a band limit for T . The set of all band-limited operators in B(H) will be denoted by B. Definition 2.2. Let T be an operator in B. We will say that T is summable when it has the property that ∞
T | Hn < ∞,
n=0
where we have used T | Hn ∈ B(Hn , H) to denote the restriction of T to Hn . The set of all summable band-limited operators in B(H) will be denoted by S. Proposition 2.3. With respect to the preceding definitions, (1) B is a unital ∗-subalgebra of B(H) and, (2) S is a two-sided ideal of B which is closed under taking adjoints. Proof. The proof of (1) is left as an easy exercise for the reader. To verify (2), we first show that S is closed under taking adjoints. Suppose T ∈ S, and let b be a band limit for T . By examining the matrix representations of T and of T ∗ with respect to the orthogonal decomposition (2.11), it is easily verified that T ∗ |Hn ≤ T |Hm , ∀n ≥ 0. m≥0 |m−n|≤b
This implies that ∞
T ∗ |Hn ≤ (2b + 1)
n=0
∞
T |Hm < ∞,
m=0
which gives T ∗ ∈ S. Next, we show that S is a two-sided ideal of B. Since S was proved to be self-adjoint, it will suffice to show that it is a left ideal. It is clear that S is closed under linear combinations. The fact that S is a left ideal now follows from the simple observation that for T ∈ B and S ∈ S we have ∞ ∞ T S | Hn ≤ T S | Hn < ∞, n=0
which implies T S ∈ S.
n=0
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In the following definition, we identify some special types of band-limited operators. Definition 2.4. Let T be an operator in B. (1) If T satisfies T (Hn ) ⊆ Hn for all n ≥ 0, then we will say that T is block-diagonal. (2) If there is k ≥ 0 such that T satisfies T (Hn ) ⊆ Hn+k for n ≥ 0, then we will say that T is k-raising. (3) If there is k ≥ 0 such that T satisfies T (Hn ) ⊆ Hn−k for n ≥ k and T (Hn ) = {0} for n < k, then we will say that T is k-lowering. Note that a block-diagonal operator is both 0-raising and 0-lowering. The following proposition gives a Fourier-type decomposition for band-limited operators. As pointed out to us by the referee, it may be useful for intuition to think of this decomposition in terms of the group of block-diagonal unitaries {Dz | z ∈ T} on H, where for every z ∈ T (i.e. z ∈ C with |z| = 1) the unitary Dz sends ξ → z n ξ for all n ≥ 0 and ξ ∈ Hn . It is easily seen that for a decomposition like the one discussed in Eq. (2.13) below, one then has Dz∗ T Dz =
b
z −k X k +
k=0
b
z k Yk , ∀ z ∈ T.
k=1
This the components X k , Yk to be retrieved via integrals on T, e.g. X k = k allows ∗ T D dz. (This observation could have been used for an alternative proof of z D z z T Proposition 2.5. But it is also very easy, as shown below, to prove this proposition directly from the definitions.) Proposition 2.5. Let T be an operator in B with a band-limit b ≥ 0, as in Definition 2.1. Then we can decompose T as T =
b k=0
Xk +
b
Yk ,
(2.13)
k=1
where each X k is a k-raising operator for 0 ≤ k ≤ b, and each Yk is a k-lowering operator for 1 ≤ k ≤ b. This decomposition is unique. Moreover, if T is summable in the sense of Definition 2.2, then each of the X k and Yk are summable. Proof. First, fix an integer k satisfying 0 ≤ k ≤ b. For each n ≥ 0, consider the linear operator Pn+k T |Hn ∈ B(Hn , Hn+k ) which results from composing the orthogonal projection Pn+k onto Hn+k with the restriction T |Hn . Clearly Pn+k T |Hn ≤ T . This allows us to define an operator X k ∈ B(H) which acts on Hn by X k ξ = Pn+k T ξ, ∀ξ ∈ Hn .
(2.14)
It follows from this definition that X k is a k-raising operator. Similarly, for an integer k satisfying 1 ≤ k ≤ b, we can define a k-lowering operator Yk ∈ B(H) which acts on ξ ∈Hn by Pn−k T ξ if k ≤ n, Yk ξ = (2.15) 0 if k > n. It’s clear that Eq. (2.13) holds with each X k and Yk defined as above. Conversely, if Eq. (2.13) holds, then it’s clear that each X k and Yk is completely determined
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as in Eq. (2.14) and Eq. (2.15) respectively. This implies the uniqueness of this decomposition. Finally, suppose T is summable. The fact that each X k and Yk is summable then follows from the observation that Eq. (2.14) and Eq. (2.15) imply X k |Hn ≤ T |Hn and Yk |Hn ≤ T |Hn for every n ≥ 0. The following result about commutators will be needed in Sect. 4. Proposition 2.6. Let T ∈ B be a positive block-diagonal operator, and let V ∈ B be a 1-raising operator. Suppose that the commutator [T, V ] satisfies ∞
[T, V ] | Hn 1/2 < ∞.
(2.16)
n=0
Then the commutator [T 1/2 , V ] is a summable 1-raising operator. Proof. For every n ≥ 0, let Tn = T |Hn ∈ B(Hn ) and let Vn = V |Hn ∈ B(Hn , Hn+1 ). Since T is block-diagonal and V is 1-raising, it’s clear that [T, V ] and [T 1/2 , V ] are 1-raising operators which satisfy [T, V ] | Hn = Tn+1 Vn − Vn Tn , ∀n ≥ 0, and 1/2
1/2
[T 1/2 , V ] | Hn = Tn+1 Vn − Vn Tn , ∀n ≥ 0. It follows that the hypothesis (2.16) can be rewritten as ∞
Tn+1 Vn − Vn Tn 1/2 < ∞,
n=0
while the required conclusion that [T 1/2 , V ] ∈ S is equivalent to ∞
1/2
1/2
Tn+1 Vn − Vn Tn < ∞.
n=0
We will prove that this holds by showing that for every n ≥ 0, 1/2
1/2
Tn+1 Vn − Vn Tn ≤
5 V 1/2 Tn+1 Vn − Vn Tn 1/2 . 4
(2.17)
For the rest of the proof, fix n ≥ 0. Consider the operators A, B ∈ B(Hn ⊕ Hn+1 ) which, written as 2 × 2 matrices, are given by
0 Vn∗ 0 Tn . , B := A := 0 Tn+1 Vn 0 Since T is positive, it follows that A is positive, with 1/2 Tn 0 1/2 A = 1/2 . 0 Tn+1
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123
A well-known commutator inequality (see e.g. [8]) gives [A1/2 , B] ≤
5 B1/2 [A, B]1/2 . 4
(2.18)
From the definitions of A and B, we compute
0 (Tn+1 Vn − Vn Tn )∗ , [A, B] = Tn+1 Vn − Vn Tn 0 1/2
and this implies [A, B] = Tn+1 Vn − Vn Tn . Similarly, [A1/2 , B] = Tn+1 Vn − 1/2 Vn Tn , and it’s clear that B = Vn . By substituting these equalities into (2.18) we obtain 1/2
1/2
Tn+1 Vn − Vn Tn ≤
5 Vn 1/2 Tn+1 Vn − Vn Tn 1/2 . 4
Since Vn ≤ V , this clearly implies that (2.17) holds.
3. An Inclusion Criterion In this section, we work exclusively in the framework of the (non-deformed) extended Cuntz algebra C. We will use the terminology of Subsect. 2.3 with respect to the natural decomposition F = ⊕∞ n=0 Fn . In particular, we will refer to the unital ∗-subalgebra B ⊆ B(F) which consists of band-limited operators as in Definition 2.1, and to the ideal S of B which consists of summable band-limited operators as in Definition 2.2. The main result of this section is Theorem 3.8. This is an analogue in the C ∗ -framework of the bicommutant theorem from von Neumann algebra theory, where we restrict our attention to the ∗-algebra B and consider commutators modulo the ideal S. In this framework, the role of “commutant” is played by the C ∗ -algebra generated by right creation operators on F. For clarity, we will first consider the special case of a block-diagonal operator. Definition 3.1. Let T ∈ B be a block-diagonal operator. The sequence of C-approximants for T is the sequence (An )∞ n=0 of block-diagonal elements of C defined recursively as follows: we first define A0 by A0 = T (), IF , and for every n ≥ 0 we define An+1 by ∗ ci1 ,...,in+1 ; j1 ,..., jn+1 L i1 · · · L in+1 L j1 · · · L jn+1 , (3.1) An+1 := An + 1≤i 1 ,...,i n+1 ≤d 1≤ j1 ,..., jn+1 ≤d
where the coefficients ci1 ,...,in+1 ; j1 ,..., jn+1 are defined by ci1 ,...,in+1 ; j1 ,..., jn+1 := T (ξ j1 ⊗ · · · ⊗ ξ jn+1 ), ξi1 ⊗ · · · ⊗ ξin+1 −δin+1 , jn+1 · T (ξ j1 ⊗ · · · ⊗ ξ jn ), ξi1 ⊗ · · · ⊗ ξin .
(3.2)
The main property of the approximant An is that it agrees with the operator T on each subspace Fm for m ≤ n. More precisely, we have the following lemma.
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Lemma 3.2. Let T ∈ B be a block-diagonal operator, and let (An )∞ n=0 be the sequence of C-approximants for T , as in Definition 3.1. Then for every m ≥ 0, if m ≤ n, T | Fm A n | Fm = (3.3) (T | Fn ) ⊗ Im−n if m > n. Proof. We will show that for every fixed n ≥ 0, Eq. (3.3) holds for all m ≥ 0. The proof of this statment will proceed by induction on n. The base case n = 0 is left as an easy exercise for the reader. The remainder of the proof is devoted to the induction step. Fix n ≥ 0 and assume that Eq. (3.3) holds for this n and for all m ≥ 0. We will prove the analogous statement for n + 1. From Eq. (3.1), it is immediate that An+1 |Fm = An |Fm = T |Fm , ∀m ≤ n. Thus it remains to fix m ≥ n + 1 and verify that An+1 | Fm = (T | Fn+1 ) ⊗ Im−n−1 ∈ B(Fm ). In light of how (T | Fn+1 ) ⊗ Im−n−1 acts on the canonical basis of Fm , this amounts to showing that for every 1 ≤ k1 , . . . , km , 1 , . . . , m ≤ d, one has An+1 (ξ 1 ⊗ · · · ⊗ ξ m ), ξk1 ⊗ · · · ⊗ ξkm = δkn+2 , n+2 · · · δkm , m T (ξ 1 ⊗ · · · ⊗ ξ n+1 ), ξk1 ⊗ · · · ⊗ ξkn+1 .
(3.4)
On the left-hand side of Eq. (3.4) we substitute for An+1 using the recursive definition given by Eq. (3.1). This gives An+1 (ξ 1 ⊗ · · · ⊗ ξ m ), ξk1 ⊗ · · · ⊗ ξkm = An (ξ 1 ⊗ · · · ⊗ ξ m ), ξk1 ⊗ · · · ⊗ ξkm + ci1 ,...,in+1 ; j1 ,..., jn+1 α(i 1 , . . . , i n+1 ; j1 , . . . , jn+1 ),
(3.5)
i 1 ,...,i n+1 j1 ,..., jn+1
where for every 1 ≤ i 1 , . . . i n+1 , j1 , . . . , jn+1 ≤ d, we have written α(i 1 , . . . , i n+1 ; j1 , . . . , jn+1 ) ∗ = L i1 · · · L in+1 L j1 · · · L jn+1 (ξ 1 ⊗ · · · ⊗ ξ m ), (ξk1 ⊗ · · · ⊗ ξkm ) . It is clear that an inner product like the one just written simplifies as follows: ∗ L i1 · · · L in+1 L j1 · · · L jn+1 (ξ 1 ⊗ · · · ⊗ ξ m ), (ξk1 ⊗ · · · ⊗ ξkm ) ∗ = L j1 · · · L jn+1 (ξ 1 ⊗ · · · ⊗ ξ m ), (L i1 · · · L in+1 )∗ (ξk1 ⊗ · · · ⊗ ξkm ) = δi1 ,k1 · · · δin+1 ,kn+1 δ j1 , 1 · · · δ jn+1 , n+1 ξ n+2 ⊗ · · · ⊗ ξ m , ξkn+2 ⊗ · · · ⊗ ξkm = δi1 ,k1 · · · δin+1 ,kn+1 δ j1 , 1 · · · δ jn+1 , n+1 δ n+2 ,kn+2 · · · δ