)naon iviatnemb i ice.`
2 68
Spectral Asymptotics in the Semi-Classical Limit M. Dimassi & J. Sjostrand
X
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London Mathematical Society Lecture Note Series. 268
Spectral Asymptotics in the Semi-Classical Limit
Mouez Dimassi Universite de Paris-Nord
Johannes Sjostrand Ecole Polytechnique
;.R.,.,. CAMBRIDGE UNIVERSITY PRESS
CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo
Cambridge University Press The Edinburgh Building, Cambridge C132 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521665445
© Cambridge University Press 1999
This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999
A catalogue record for this publication is available from the British Library ISBN-13 978-0-521-66544-5 paperback ISBN-10 0-521-66544-2 paperback Transferred to digital printing 2007
Contents 0. Introduction 1. Local symplectic geometry
vii 1
2. The WKB-method
11
3. The WKB-method for a potential minimum
17
4. Self-adjoint operators
27
5. The method of stationary phase
43
6. Tunnel effect and interaction matrix
49
7. h-pseudodifferential operators
75
8. Functional calculus for pseudodifferential operators
93
9. Trace class operators and applications of the functional calculus
111
10. More precise spectral asymptotics for non-critical Hamiltonians
119
11. Improvement when the periodic trajectories form a set of measure 0
125
12. A more general study of the trace
139
13. Spectral theory for perturbed periodic problems
155
14. Normal forms for some scalar pseudodifferential operators
189
15. Spectrum of operators with periodic bicharacteristics
201
References
209
Index
221
Index of notation
226
0. Introduction A new branch of mathematical analysis, so-called microlocal analysis, started to be more systematically developed about 30 years ago by Kohn-Nirenberg, Hormander, Maslov and Sato, soon followed by many others. Originally the motivations came from problems in partial differential equations, but it soon became increasingly clear that many aspects of microlocal analysis are reminiscent of quantum mechanics, and for instance the Heisenberg uncertainty principle plays a fundamental role in both theories. Mathematically, a version of this principle says that if u E L2(Rn) and we define the Fourier transform by
u() = fe_iu(x)dx, so that Parseval's relation IIUI12
=
(27r1
)n Ilull2,
holds, where the norms are those of L2, then if we take n = 1 for simplicity, and let x0, o E R: Ilu112 < 211(. - xo)ull
27r
II( - o)ull, u E S(R).
(0.2)
Here S(R) is the Schwartz space of smooth functions on R which decay rapidly at infinity together with all their derivatives. A rough interpretation of this is that if most of the L2-norm ('energy') of u is concentrated to an interval of length a and most of the energy of ii is concentrated to an interval of length b, then: ab > 27r.
(0.3)
The reason for putting this precise numerical constant comes from well-known asymptotic formulas for the counting of eigenvalues (of Weyl type) which can be interpreted by saying that each eigenfunction occupies a volume (27r) in phase space.
Another similarity between the two theories is the interplay between classical and quantum objects. In microlocal analysis, the quantum objects are given
by pseudodifferential and Fourier integral operators etc. and the classical ones by those of symplectic geometry: canonical transformations, Poisson brackets etc. In quantum mechanics the same duality appears in the semiclassical limit. If we consider for instance the stationary Schrodinger operator -h 2A + V (X),
(0.4)
when h becomes very small, then the quantum objects are wave functions, eigenvalues etc., while the classical ones are given by the classical trajectories
viii
Spectral Asymptotics in the Semi-Classical Limit
of the associated classical Hamiltonian p := S2+V(x), i.e. the integral curves of the corresponding Hamilton field Hp = 21; (8/8x) - V'(x) (a/5 ). Thanks to microlocal analysis it has been possible to get refined results about the distributions of eigenvalues for differential operators (mostly elliptic ones) on compact manifolds and in bounded domains (Hormander, DuistermaatGuillemin, Ivrii and others), and while the Weyl asymptotics gives the leading terms in such results and is simply a phase space volume, the further terms or remainder estimates depend on dynamical properties of the Hamilton flow. These notes are about the analogous developments for the semiclassical limit.
The motivation among specialists (such as Chazarain, Helffer-Robert and later many others) was that microlocal analysis should provide a tool for a more rigorous understanding of many spectral problems also in this field. To some extent the early work consisted of carrying over the above mentioned spectral results to the study of, say, (0.4), but the area turned out to be much richer and new problems and results appeared, and the microlocal analysis itself has received new impulses from these efforts.
The contents of these notes are: 1. Local symplectic geometry. Here we develop some of the standard theory, following closely one of the chapters in [GrSj].
2. The WKB-method. We discuss the construction of local asymptotic solutions of (P - E)u = 0, where P is the operator (0.4), and get an example of the interplay between classical and quantum objects. 3. The WKB-method for a potential minimum. Here we follow some work by Helffer and one of the authors, and show how to construct asymptotic eigenvalues and eigenfunctions near a non-degenerate minimum of the potential.
.¢. Self-adjoint operators. This is mostly a compilation of abstract spectral theory, and at the end of the chapter, we determine the low-lying eigenvalues for potentials with a non-degenerate minimum. This also justifies the more complete asymptotics of eigenvalues obtained in Chapter 3. 5. The method of stationary phase. We followed closely the presentation of [GrSj], based on the classical work of Hormander [Hol]. A small variation leads to some refined remainder estimates, which may be new. This method is one of the fundamental ingredients of microlocal analysis, even though in the present notes we choose not to appeal explicitly to this method when presenting the theory of pseudodifferential operators. ([GrSj] shows how to get everything from stationary phase.)
0. Introduction
ix
6. Tunnel effect and interaction matrix. This chapter is devoted to exponentially small corrections to eigenvalues of (0.4), due to the interaction of potential wells through the classically forbidden region. An essential tool is the use of exponentially weighted L2-estimates, developed for second order operators by Lithner and Agmon. We have followed some work of Helffer and one of the authors.
7. h-pseudodifferential operators. In this chapter the basic theory of pseudodifferential operators is developed, without trying to reach maximal generality or refinement. These operators are of the form P(x, hD; h), where P(x, f ; h) belongs to some suitable space of symbols. The most standard case is when P(x, ; h) is uniformly bounded together with all its derivatives, uniformly
with respect to h. In the case n = 1 (for simplicity) the symbol P(x, h) varies only a little in rectangles of the form II x JJ if II and JJ are intervals of length eo and eo/h respectively, for some small but fixed constant eo > 0.
The area of I. x JJ is eo/h, and the uncertainty principle is satisfied with a good margin, when h is small enough. The symbolic calculus is developed and in particular it is established that h-pseudodifferential operators form an algebra, and the symbols of the composition of two operators is the product of the symbols plus an error which is roughly of the order h smaller. 8. Functional calculus for pseudodifferential operators. We base this calculus on a functional formula using almost analytic extensions, and a semi-classical
version of an important lemma of Beals which permits us to characterize pseudodifferential operators. One of the main results (which is due to Helffer
and Robert in the semi-classical case) says that if P is a self-adjoint hpseudodifferential operator (from now on sometimes called pseudor for short) and f E C000, then f (P) is again a h-pseudor with leading symbol f (p(x, l;)),
where p(x,1;) is the leading symbol of P. We follow some joint papers of Helffer and one of the authors. This approach to the calculus is also extended to the case of several commuting self-adjoint operators.
9. Trace class operators and applications of the functional calculus. Here we derive asymptotic expansions for the trace and get as a corollary the leading (Weyl-)asymptotics for the number of eigenvalues in an interval. 10. More precise spectral asymptotics for non-critical Hamiltonians. Here we
study the unitary evolution group as a Fourier integral operator and the singularity of its trace near the time 0. This leads to an estimate of the remainder in the spectral asymptotic formula, which in general is optimal. 11. Improvement when the periodic trajectories form a set of measure 0. Here we estimate the trace of the evolution group also for large times. The methods and the results of this chapter as well as the preceding one are fairly standard,
x
Spectral Asymptotics in the Semi-Classical Limit
first due to Hormander, Guillemin-Duistermaat in the non-semiclassical case, then extended to the semi-classical case (and improved) by Ivrii, Petkov and Robert.
12. A more general study of the trace. Here we extend the results of Chapter 10 to the case of microhyperbolic systems. The presentation is inspired by works of Ivrii, which avoid explicit constructions (which might be impossible anyway), but we have used a stationary approach, which in later work by one of the authors has been extended to situations with an implicit dependence of the spectral parameter. Such implicit spectral problems appear frequently when making so-called Grushin reductions of a spectral problem. 13. Spectral theory for perturbed periodic problems. For slowly varying pertur-
bations of periodic Schrodinger operators, one can make a reduction to the study of an h-pseudor, a so called effective Hamiltonian, and it then becomes possible to obtain asymptotic results about the eigenvalues of the perturbed operator in a gap of the spectrum of an unperturbed one. We have followed work by Gerard-Martinez-Sjostrand and Dimassi, related to earlier works by Buslaev, Guillot-Ralston-Trubowitz and Helffer-Sjostrand. The reduction used is an example of a so-called Grushin reduction, a technique which has turned out to be extremely useful in many situations, in particular when combined with functional formulas of the type given in Chapters 8, 9 and 12.
14. Normal forms for some scalar pseudodifferential operators. Here we return to non-degenerate potential wells, studied in Chapters 3, 4 and 6, and establish a quantum Birkhoff normal form, which permits (under a non-resonance condition) to obtain complete asymptotic expansions of all eigenvalues in an interval [0, h6 J, where S > 0 is arbitrary. This chapter is based on a work of Sjostrand, in a cirle of ideas developed by Lazutkin, Colin de Verdiere, Graffi-Paul, Bellissard-Vittot, Iantchenko and many others. 15. Spectrum of operators with periodic bicharacteristics. When the Hamilton flow is periodic there is a phenomenon of clustering of eigenvalues, that we study, following works by Colin de Verdiere, Weinstein, Helffer-Robert and others.
We hope that these notes may serve as an introduction to a still very active
subject, and they correspond largely to a course given by the authors at the universities of Rennes (,where the first impulse to write these notes was received), Paris Sud, Paris Nord, as well as the Ecole Polytechnique. They cover more recent material than the now classical book by Robert [Rol], but remain hopefully at an introductory level. A fairly large portion can be covered in a one semester course. For further and deeper study, we can recommend the recent book by Ivrii [I11. See also the book of
0. Introduction
xi
Safarov-Vassiliev [SaVa] which deals with asymptotics of large eigenvalues for boundary value problems.
We would like to thank N. Lerner and G. Metivier for giving one of the authors the original impulse to write these notes. We have also profited more or less directly from a long collaboration and many stimulating discussions with B. Helffer, who we thank particularly. We also thank A. Grigis for the permission to use two chapters from [GrSj] with only minor changes, and one of the referees who indicated some important references.
1. Local symplectic geometry We assume that the reader is familiar with some basic objects of differential geometry, such as manifolds, tangent and cotangent vectors, differential forms and vector bundles. We shall, however, review briefly some of these notions.
For a smooth manifold X of dimension n we shall denote by Ck(X) the space of k times continuously differentiable complex valued functions on X if k E N, and we set C°°(X) = nkENCk(X). Tangent and cotangent vectors. Let X be a smooth manifold of dimension n. Let xo E X. If -y, =y :] - 1,1[- X are two C1 curves with y(0) = ry(O) = xo, we say that y, ry are equivalent if 11y(t) - ry(t)II = o(t), t -+ 0. (Here we choose some local coordinates xl, ... , xn, near xo, so that I h y(t) - 3 (011 is well defined, and we notice that the choice of local coordinates and of the corresponding norm does not influence the definition.) The equivalence class of -y will be denoted by y'(0) or dty(0), and will be called a tangent vector at xo. The set of all tangent vectors at a point xo is denoted by T,,0X and is called the tangent space of X at xo. It is easy to see (by working in a system of local coordinates) that TT0X is a real vector space of dimension n.
If f, f : X -> R are two C' functions, we say that f, f are equivalent if (f (x) - f (xo)) - (f (x) - f(xo)) = o(IIx - xoII), x --> xo. We let df (xo)
(called the differential of f at xo) denote the equivalence class of f. It is (by definition) a differential 1 form at xo, also called a cotangent vector at xo. The set T*. X of cotangent vectors at a point xo is a real vector space of dimension n. It is called the cotangent space at xo of X. There is a natural duality between TT0X and TT0X, given by (df (xo), -y '(t)) _
\dt) t_of (-Y(t))
If x1,. .. , x,,, are local coordinates defined in a neighborhood of xo, then dx1(xo),... , dx,,,(xo) (or dxl,... , dxn for short) form a basis of TT0X. A corresponding dual basis in Txo X is given by aal , ... , aan , where aaj is the tangent vector induced by the curve t H xo + tee. Here we work in the local
coordinates above, and ej denotes the jth unit vector in R. It is easy to
check that df = Ei a dxj, y'(0) = Eni d aa at the point xo.
The sets TX = UxoExTx0X and T*X = Ux0ExTT*0X are vector bundles and
in particular C°° manifolds. If xl,... , x,,, are local coordinates on X, then we get the corresponding local coordinates (x, t) =(x1, ... , xn, tl, ... , tn) on TX and (x, e) = (x1, ... , xn, i, . . . , fin) on T*X by representing v E TX and p E T*X by their base point x (given by the coordinates ( X1 ,--- , xn)) and the corresponding tangent vector E tj aaj and cotangent vector E j dxj. If
2
Spectral Asymptotics in the Semi-Classical Limit
yl, ... , yn is a second system of local coordinates, then in the intersection
of the two open sets in X parameterized by the two systems of local coordinates, we have the point-wise relations t = ay s, rl =t (ay )t for the corresponding local coordinates (x, t), (y, s) on TX and (x, ), (y, 77) on T*X. ate; Here aay = (ayk )1<j,k
If p E T*X, we let 7r
E X be the associated base point so that
7r(p)
: T*X -> X is the natural projection map. A section in T*X is a right
inverse of ir. The same definitions can be given for TX and more generally for any vector bundle. Sections in T*X are called differential 1 forms, and sections in TX are called vector fields. Most of the time we will only consider sections of class C°° and we will also most of the time only consider locally defined sections (i.e. sections of T*U and TU where U is some small open subset of X).
A vector field can be written in local coordinates as v = E tj(x) aaj and a differential 1 form as w =
j (x)dxj.
If Y is a second manifold and f : Y - X a map of class C', yo E Y, xo = f(yo) E X, then we have a natural map f* = df : TyoY ---> TT0X, which The in local coordinates is given by the ordinary Jacobian matrix ay adjoint is f * : T* X -> TyoY and we note that if u is a C1 function on X and y : I -+ Y is a C' curve, with y(0) = yo, 0 in the interior Int I of the interval
I, then (f o y)'(0) = f. (-y'(0)), d(u o f)(yo) = f*(du(xo)). More generally, if Z is a third manifold, g : Z -+ Y is of class C' and zo E Z, g(zo) = yo, then (f o g)* = f* o g*, (f o g)* = g* o f *. When passing to sections, we see that if w is a 1 form on X, then f *w is a well defined 1-form on Y (called the pull-back of w by means of f). The corresponding push-forward f*v of a vector field v can be defined when f is a diffeomorphism, but not in general. If -y : ]a, b[--4 X is a C' curve and to E]a, b[, we recover the tangent vector y'(to) of -y at to as -y. (-2- (to)).
The elementary theory of ordinary differential equations gives the following
fact: if v is a C°° vector field on X, then for every xo E X, we can find T+(xo), T_ (xo) in ] 0, +oo], such that we have a smooth (i.e. C°°) curve: ] - T_ (xo), T+ (xo) [D t'-i y(t) =: exp(tv) (xo) E X with -y(O) = xo, ry'(t) = v(y(t)). If we choose T± maximal, we get lower semi-continuous functions X E) x
T±(x) and a smooth map {(t, x) E R x X; -T_(x) < t < T+(x)} D (t, x) --> 4)(t, x) = exp(tv)(x) with 4(0,x) = x, at4D (t, x) = v(4) (t, x)). When t is fixed, we can in general
not define exp tv on all of X but only on the set of x E X for which
1. Local symplectic geometry
3
-T_ (x) < t < T+ (x). We have exp tv(exp sv(x)) = exp(t + s)v(x), for t, s, x such that the left member is defined.
The canonical 1 and 2 forms. Let 7r : T*X --j X be the natural projection (which is simply (x, t;) H x in canonical coordinates). For p E T*X, consider 7r* : T,*(P)X -* Tp (T*X). Since p E T,T(P)X, we can define the canonical 1 form wP E TP (T*X), by wP = it*(p). Varying p, we get a smooth 1 form w on on T*X, which in canonical coordinates has the expression: w = E j dxj .
We next recall some facts about forms of higher degree. If L is a finite dimensional real vector space, and L* the dual space, then we have a natural duality between the k fold exterior product spaces AkL and AkL*, given by
(ul A... Auk,vl A... Avk) = det((uj,vk)), ui E L, vk E L*. Without repeating the definition of exterior products and exterior product spaces, it may be useful to recall that exterior products ul A ... A uk are linear in each of their factors and change sign if we permute two neighboring factors. Moreover, if e 1, . . . , e.,,, form a basis for L, then a basis for AkL is formed by the ell A ... A elk, for all 1 < j1 < j2 < ... < jk < n.
If M is a C°° manifold of dimension m, then a differential k form v is a section of the vector bundle AkT*M. In local coordinates x 1 , . . v=
vi(x)dxl,
.
, x,,,,:
(1.1)
III=k
where in general, I = (ii, ... , ik) E {1, 2, ... , m}k, III = k, dxI = dxil A ... A dxik. The representation above becomes unique if we restrict the sum to the set of I with it < i2 < ... < ik. If v is a k form of class C' locally given by (1.1), we define the k + 1 form dv = E dvj A dx1.
(1.2)
III=k
dv is called the exterior differential of v and it can be shown that its definition
does not depend on the choice of local coordinates or on the choice of the representation (1.1). We have the following facts: d2 = 0,
(1.3)
4
Spectral Asymptotics in the Semi- Classical Limit
If w is a k + 1 form of class C°°, which is closed in the sense that dw = 0, then in every open set in M which is diffeomorphic to a ball, we can find a smooth k form v, such that dv = w. (1.4) If f : Y --> X is a smooth map between two smooth manifolds, then there is a unique way of extending the pull-back f * from 1 forms to k forms by multilinearity. If v is a smooth k form on X, then d(f*v) = f * (dv). (1.5) We now return to the canonical 1 form w on T*X, and define the canonical 2 form a on T*X as a = dw. In canonical coordinates: n
a=
<3 A dxj.
For p E T * X, ap E A2Tp (T * X) can be viewed as a linear form on A2TP (T * X)
or equivalently as an antisymmetric bilinear form on Tp(T*X) x Tp(T*X). Mixing the two points of view, we write: ap(t, s) = (ap, t AS), t, s E Tp(T*X). In canonical coordinates, and with the notation t = (tm,tg) (so that t =
> (tx; aa; + tg,
we get
s = (sx,
aP(t, s) = (h, sx) - (s£, ty) _
(hi sxj - k txj).
From this it is clear that ap is a non-degenerate bilinear form and consequently there is a bijection H : TP (T*X) -* Tp(T*X) determined by: o, (s, Hu) = (s, u) s E Tp(T*X), u E TP (T* X). we In canonical coordinates, if u = uxdx + E(uxjdxj + get Hu = ug aL - ux aL . If f (x, ) is of class C' on X (or on some open
subset of X), we define the Hamilton field of f by Hf = H(df). In canonical coordinates, n
Hf _ 1
of a
of a
Oa j Ox; - ax,
a manifold, p E M, t E TPM, then we define the contraction : AkTT M - Ak_1TP M as the adjoint of the left exterior multiplication to : Ak-1TPM --> AkTPM. Then with M = T*X, the Hamilton field is tj
(equivalently) defined by the pointwise relation,
Hfja = -df.
(1.7)
1. Local symplectic geometry
5
If f, g are two C' functions defined on some open set in T*X, we define their Poisson bracket as the continuous function
If, g} = Hf(g) = (Hf, dg) = o,(Hf, H9), where, in the second expression, we view Hf as a first order differential operator. In canonical coordinates,
Of 8g
a f ag
Notice that { f, g} = -{g, f }, and in particular that { f, f } = 0.
Lie derivatives. Let v be a smooth vector field on a manifold M and let w be a smooth k form on M. Then the Lie derivative of w along v is defined pointwise by Gvw =
(d) ((expty)*w). dt t-o
If u is a another smooth vector field on M, we also define Gvu =
((exp -tv)*u). (d) dt t=o
Here we need of course to observe that the push-forward of a vector field by means of a local diffeomorphism can be defined locally. We have the following facts:
(1) When w is a 0 form, i.e. a function, then Gvw = v(w).
(2) Gvu = [v, u] = vu - uv, where u, v are viewed as first order differential operators in the last two expressions. (3) Gv(dw) = d(Gvw), (4) Gv (wi A W2) = (Gvwl) A W2 + wl A (Gvw2),
(5) Gv(ujw) = (Gvu)jw + uj (Gvw),
(6) Gvw = vjdw+d(vjw), (7)
Gvl+v2 = Gvl + Gv2
Lemma 1.1. If f is a smooth function on some open subset in T*X, then GHfO=0.
6
Spectral Asymptotics in the Semi-Classical Limit
Proof. It suffices to make the calculation,
LHfQ = HfJdo + d(HfJo) = HfJd2w - d2 f = 0.
Locally, we can define the maps Dt = exptHf, when tj is sufficiently small and we have 4)t *o, = o-. In fact, we have pointwise:
dtDt =
(ds)S=o4)t1DsQ = 4)trHfa = 0.
Lemma 1.2. If f, g are two smooth functions defined on some open subset ofT*X, then [Hf,H9] =H{f,g} Proof. We have to show that [Hf, Hg] Ja = -d{ f, g}. This follows from the computation:
-d{f,g} = -d(GHfg) = -(LHfdg) =GHf(Hgja) = [Hf,Hg]Jo,+Hgj(GHfa) = [Hf,Hg]Ja.
Using the preceding lemma, it is easy to prove the Jacobi identity for three smooth functions,
If, {g,h}}+{g,{h, f}}+{h,{f,g}} = 0. Lagrangian manifolds. A submanifold A C T*X is called a Lagrangian manifold if dim A = dim X and a1 A = 0. In general, we define the restriction
of a differential k form to a submanifold as the the pull-back of this form by means of the natural inclusion map, and there is a corresponding natural way of viewing the tangent space of a submanifold at some given point as a subspace of the tangent space of the ambient manifold at the same point. If A is a submanifold of T*X and p E A, then we define TPA' C TP(T*X) as the orthogonal space with respect to a of TPA C TP(T*X). The sum of the dimensions of TA and TPA' add up to the dimension of TP(T*X), but there is no reason for TPA and TPA' to have zero intersection. As a matter of fact, it is clear that a submanifold A C T*X is Lagrangian if and only if TPA = TPA'
for every p E A. That there are plenty of Lagrangian submanifolds follows from the following result.
1. Local symplectic geometry
7
Theorem 1.3. Let A C T*X be a submanifold with dim A = dim X and such that 7rIA : A - X is a local diffeomorphism (in the sense that every point p in A has a neighborhood in A which is mapped diffeomorphically by it onto a neighborhood of 7r(p)). Then A is Lagrangian if and only if
for each point p E A, we can find a (real) C°° function q(x) defined near 7r(p), such that A coincides near p with the manifold {(x, do(x)); x E some neighborhood of ir(p)}.
Proof. If w is the canonical 1 form, we notice that d(wIA) = alA Therefore the following three statements are equivalent: (1) A is Lagrangian. (2)
WIA is closed (i.e. d(WIA) = 0).
(3) Locally on A, we can find a smooth function (P with WIA =
dq5.
If x1,.. . , x,-,, are local coordinates on X, we can also view them (or rather their compositions with 7r) as local coordinates on A, and represent A by equations t; = t; (x) in the corresponding canonical coordinates. Then (3) is equivalent to e (x) = i.e. Et j(x)dxj = dO. # Hamilton-Jacobi equations. These equations are of the form p(x, (P') = 0, where p is a real-valued C°° function defined on some open subset of T*X. Here we shall also assume that dp(x, t;) 54 0, when p(x, ) = 0. The basic idea in treating a Hamilton-Jacobi equation is to consider the Lagrangian manifold A = A, associated with 0 as in the preceding theorem, and try to find such a manifold inside the hypersurface H defined by p(x, ) = 0. If p E A, we shall then have TPA C TpH (considering these tangent spaces as subspaces of TpT*X), and hence TP H C TPA, since TPAL = TpA. It is easy to see that TPHL = RHp, so we must have Hp E TPA at every point p E A, or in other words Hp should be tangent to A at every point of A.
Proposition 1.4. Let A' C H be an isotropic submanifold (in the sense that QI A' = 0) of dimension n - 1 passing through some given point po E H
and such that HP(po) 0 Tp(,A'. Then in a neighborhood of po we can find a Lagrangian manifold A such that A' C A C H (in that neighborhood).
Proof. According to the observation above it is natural to try A = {exp(tHP)(p); Itl < e, p E A', Ip - pol < e} for some sufficiently small e > 0. (Here l p - po 1 is well-defined, if we choose
some local coordinates.) Then A' C A (near po) and since Hp is tangent to
Spectral Asymptotics in the Semi-Classical Limit
8
H (by the relation Hpp = 0) we also have A C H. From the assumption Hp(po) ¢ Tp0A' and the implicit function theorem, it also follows that A is a smooth manifold of dimension n. In order to verify that A is Lagrangian, we first take p E A' (with I p-poI < e) and consider TPA = TPA' ®RHp. Then UPIT.A.TPA = 0 since ° ITA'.TA' 0, Qp(Hp, Hp) = 0, ap(t, Hp) = (t, dp) = 0 for all t E TPA' C TPH.
More generally, at the point pt = exp(tHp)(p), p E A', we have TPt (A) = exp(tHp)* (TPA)
and for u, v E TA we get, using the fact that exp(tHp)*op, = Qp: up, (exp(tHp).u, exp(tHp)*v) = ap(u, v) = 0.
We have then verified that aIA = 0, which suffices since A has the right
#
dimension.
In the following we write x = (x', xn) E Rn, x' = (x1,. .. , xn_1) E Rn-1. Theorem 1.5. Let p(x, t;) be a real valued C°° function, defined in a neighborhood of some point (0, eeo) E T*Rn, such that p(0, o) = 0, - (0, t o) # 0. a real valued C°° function defined near 0 in Rn-1 such that Then there exists a real valued smooth function O(x), defined in a (0) a neighborhood of 0 E R', such that in that neighborhood: p(x, O',(x)) = 0, O(x', 0) = O(x'), O'X(0) = o.
(1.8)
If fi(x) is a second function with the same properties, then O(x) = cb(x) in some neighborhood of 0.
Proof. In a suitable neighborhood of (0, eo) E Rn-1 x Rn we have p(x', 0, ) = 0 if and only if n = A(x', c'), where .\ is a real valued C°° function, with A(0, o) = (bo)n. Let
A' _ {(x, ); xn = 0, ' = ax' W), Sn = A(x', '), x' E neigh (0)}, where neigh (0) indicates some sufficiently small neighborhood of 0.
Then A' C p-1 (0) is isotropic of dimension n - 1 and Hp is nowhere tangent . aan with a 0. Let A C p-1(0) be a to A' since Hp has a component Lagrangian manifold as in Proposition 1.4. The differential of 'rI A : A --> Rn
is bijective at (0, to), so if we restrict the attention to a sufficiently small
1. Local symplectic geometry
9
neighborhood of that point, we can apply Theorem 1.3 and see that A is of the form = 0'(x), x E neigh (0). We have then p(x, 0'(x)) = 0, 0'(0) = o. Since A' C A, we get a (x') = a ; (x', 0), so modifying 0 by a constant, we get O(x', 0) = z/0(x'). We leave the verification of the uniqueness statement as
#
an exercise.
We can view A as a union of integral curves of Hp, passing through A'. The
projection of such an integral curve is an integral curve of the field v = E' 2f- (x, 0' (x))-I , which can be identified with Hp1n via the projection 'CIA
If q(x,
we have the trivial identity
a (x, n
ap
a-
i
a
(x, 0x(x)) axe
= q(x, 0x(x)) i
If x = x(t) is an integral curve of v with xn(0) = 0, then we get Ox(t)) = zb(x'(0)) + fo
where t;(s) _ ¢'(x(s)), so that
s F---> (x(s), c(s)) is the integral curve of Hp with xn(0) = 0, t;'(0) = a (x'(0)),
n(0) _ A(x'(0), '(0))
If 0 _
«
depends smoothly on some parameters a E Rc, then
_
4(x, a) will be a smooth function of (x, a), and differentiating the equation p(x, 0' (x)) = 0, we see that as is constant along the bicharacteristics curves, i.e. the x-space projections of the Hp integral curves in A0. In order to recall the roots of symplectic geometry in classical mechanics, let us consider the case when p = 2 n + V (x), where V (x) is some smooth real potential and m > 0 is a constant. The equations for the Hp integral v, e'(t) = -V'(x(t)), and if we eliminate fi(t), we curves are x'(t) = vntt . We can view this as get the differential equation for x(t): x"(t) the motion of a classical partical of mass m. is the momentum, so that is a constant e(t) = mv(t). v(t) is the velocity. The total energy of the motion (i.e. constant on every integral curve). Finally - "m is the acceleration induced by the force -V'(x).
Another motivation for working on the cotangent bundle comes from the general theory of partial differential equations. Consider a differential operator with smooth coefficients, P : C°° (X) -* C°° (X ), where X is a manifold. For every choice of smooth local coordinates, P takes the form P = E1ai<,,, aa(x)D', where m E N = 10,1,2 ....} is the order of the operator and we use standard multi-index notation: a = (al, ... , an) E Nn, D = Dx1 ... DXn lal = al + ... + an, Dx = (Dx...... In the corresponding canonical coordinates (x, ), we define the principal
Spectral Asymptotics in the Semi-Classical Limit
10
symbol,
E
where
JaI='m
Then one can check that is a well-defined function in C°°(T*X). A quick way to see this is to notice that if a, 0 E C°°(X) and c6 is real valued, then P(a(x)ei'Wx) ) = eia¢(x)(p(x,0'(x))Ama(x)
+O(Am-1)), when A
, oo.
Notes The presentation here is close to that of [GrSj] and has been inspired by that
of Duistermaat [Du]. The reader who wants to study the subject in depth could turn to the books of Sternberg [St] and Hofer-Zehnder [HoZe].
2. The WKB-method This is a very general method for constructing rapidly oscillating approximate solutions to differential equations. We restrict our attention to the problem of constructing approximate solutions to the stationary Schrodinger equation (-h2A + V (X) - E)u = 0
(2.1)
in the semi-classical limit: h --> 0. Here V is a smooth real valued function, defined on some open set X C R', and 0 = Ei axe is the Laplace operator. With -h 2A + V (X) we associate the classical Hamiltonian
p(x, e) = 2 + V W'
(2.2)
and in some sense we may say that (2.1) is a quantum mechanical problem and that the corresponding problem of classical mechanics is to understand the nature of the trajectories of
Hp = 2 . 0 - V(x) .
(2.3)
in the energy surface p(x, ) = E. The general problem of so-called semiclassical analysis (or semi-classical approximation) is to relate the two
problems in the limit when h \ 0. Both problems are difficult when considered globally, and easier when considered locally.
Let A be a Lagrangian submanifold of p-1(E), and assume, possibly after restricting our attention to some part of A, that A is of the form A0 = {(x, /'(x)); x E S2}, 52 open subset of X,
(2.4)
where 0 E C°° (Q; R), so that 0 solves the characteristic equation:
V (x) - E = 0.
(2.5)
One way of producing such a manifold is the following. Let r be a hypersurface in R' and let 0 E C°° (F; R) satisfy (2.6)
where 0' denotes the gradient of 0 when F is viewed as a Riemannian manifold with the induced metric, and (05)2 is the corresponding square of the norm. In fact, choose local coordinates yi, ... , y'n on F and corresponding local coordinates y1, ... , yn_ 1, yn on Rn, such that yi1 r = Jj, Yn I r = 0, and such that ay- is orthogonal to TF at every point. In the corresponding canonical coordinates (y, 77), the restriction of S2 to I, is equal to q(y', r7') +
Spectral Asymptotics in the Semi-Classical Limit
12
an,,n(y')rln, where q is a positive definite quadratic form in rl' and a,,,,,,, > 0, and the condition (2.6) becomes q(y', *'(y')) +V(y') - E < 0 on P. Theorem
1.5 can now be applied in a neighborhood of any fixed point xo E P, and gives two solutions ¢ = 0± of the Hamilton-Jacobi problem
(0X)2 +V(x) - E = 0, OIr = Moreover irx (HP IA,,) is transversal to F.
Now return to the more general situation with A = A,5 of the form (2.4). We try to construct an approximate solution of (2.1) of the form u(x; h) = e'O(x)/ha(x; h).
(2.8)
We get
e-2O/h(-h20 +V(x) - E)(eiOlha(x; h)) _ (E(hDx, + ax; 0)2 + V (X) - E)a(x; h)
_ ((41)2+V(x)-E)a+>h(Dxj o5x;0+,9x;0oDx3)a-h20a =
2h
(4)'(x) ax +
20O)a - h2Aa,
(2.9)
where the last equality follows from the eikonal equation (2.5). We now look for a(x; h) of the form a(x;h) - ao(x) + a,(x)h + a2(x)h2 +..., aj E C°°(S2),
(2.10)
in the sense that N
(ate (a - Eai (x)hj)I
CK,a,NhN+1 x E K,
(2.11)
0
for allK CC 1,aENn,NEN. We remark here that in general if aj E C°°(SZ), 0 < h < 1, then we can find a E C°°(Q) which satisfies (2.10). In fact, this follows from the classical Borel construction of a smooth function with a given formal Taylor series at some given point: Let X E CO '(R) be equal to 1 near 0 and put 00
a(x;h) = ) 'aj (x) W X
h)
(*)
for a suitable sequence )j -> oo. By CO '(Q) we denote the space of all u E C°° (S2) with compact support. Let Kk CC S2 be an increasing sequence
2. The WKB-method
13
of compact sets tending to Q. Choosing Aj sufficiently large, we can arrange
that
IIh'X(Ajh)aj(')IIck(xk) < h'-12-1
j > k, j = 1,2,...
Then (*) converges in the Ck topology for every fixed k. Moreover, N
N
a-
00
aj (x)(X(Aj h) - 1)h' + E aj X(Aj h) hi,
aj (x) hi _
N+1
0
0
and we check that the Ck (Kk) norm of each of the sums to the right is 0(h N+1). In this discussion, we can also replace hi by hki where kj / 00. Substituting (2.10) into the last expression in (2.9) and requiring that each term in the resulting asymptotic expansion (of the same type as (2.10)) vanish, we get the sequence of transport equations: W (x) ' ax + 1 AO)ao(x) = 0,
(To)
(0'(x) (9. + I AO)a1(x) = 2 Oao,
(Ti)
W (x) ' ax + 1 Ao)a2(x) = 2Aa1,
(T2)
ax + 200)an,(x) = 2Oa,,-1,
(T.)
Remark 2.1. In the one-dimensional case it is sometimes convenient to look for u of the form
b xh (E -
V;(x))1/4eiO/h, where now (01)2 + V(x) - E = 0,
(2.12)
E - V (x). The reason for this is that (To) becomes (0'(x)8 + 24"(x))ao(x) = 0 and this ordinary differential equation
so that for instance 0' = has the general solution
ao(x) = Cexp(-
12
Cexp(-12logo') _
C
(2.13)
so if b(x; h) - bo(x)+hb1(x)+... in (2.12), then we have bo = const. One can also look for solutions of the form r-1/2 exp f x 0(y)dy, with 0 > 0, leading to a non-linear 2nd order (Ricati) equation for 0, which can be formally
Spectral Asymptotics in the Semi-Classical Limit
14
solved by an asymptotic series: 0(x; h) - Oo(x) + h2zb2(x) + h4*4(x) + ...,
with Oo(x) = E _-V (x). Assuming that we have solved successively all the transport equations in fl and that a(x; h) is an asymptotic sum in the sense of (2.10), (2.11), we get
(-h2A + V(x) - E)(e'O(x)iha(x; h)) = e'O(x)/hr(x; h),
(2.14)
with r - 0:
axr(x;h)I
CK,x,Nhn'+1 x E K cc fl, a E Nn, N E N.
(2.15)
We can solve the sequence of transport equations in the following situation. Assume that ¢'(x) 0, and choose local coordinates (t, y) in a neighborhood of some point, so that q'(x) ax becomes at and so that the neighborhood
becomes: ft = {(t, y) E R x Rn-1; Itl < e, Iyj < e} for some small e > 0. Then we get a unique solution of the transport equations, if we prescribe the
restrictions to t = 0 of ao, al, .... If we want to construct WKB-solutions globally (that is in some large given region) many difficulties may appear:
- 0 may become singular in some region (caustics), in particular the construction above with a real valued phase 0 will always be restricted to the classically allowed region: V (x) - E < 0. - ¢ may be multivalued. - The integral curves of 0' ax may behave in such a way that we get problems with the transport equations.
The consideration of these difficulties (which are of a great mathematical and physical interest) by Keller [Ke] and Maslov [Mal] are at the origin of various versions of the theory of Fourier integral operators, in particular the one by Hormander [Hol].
In the classically forbidden region: V (x) - E > 0, we can construct local solutions of the form e-O(x)/ha(x; h) with 0 real valued. Formally, we may write -0 = i(io) so the earlier constructions work, replacing everywhere 0 by i4. Hence we obtain the eikonal equation
-(0')2+V-E=0, and the sequence of transport equations ((P/ .
ax + AO)ao = 0, 2
(T' )
2. The WKB-method
15
(0' - a. + 2AO)ai = 20ao,
(Ti)
etc. When choosing an asymptotic sum of the resulting formal symbol a, we get the obvious analogue of (2.14), (2.15). The classical Hamiltonian is now
q(x, ) + E :_ -(p(x,
E) = 2 - V (x) + E.
Notes WKB-solutions are of great interest in spectral theory and can be used in proving existence results. Under suitable conditions exact eigenfunctions for some problems can be well approximated by the WKB-solutions. The reader is referred to Chapter 6 for further information in this context. There have been many generalizations and extensions of the WKB-method which cannot be described here, such as the behavior of WKB-solutions near caustics and the WKB-method in the analytic case. We refer the reader to [Mall, [La], [Ke], [Gr], [Vol and [CaNoPh].
3. The WKB-method for a potential minimum We shall construct approximate eigenvalues and eigenfunctions for the Schrodinger operator in a multi-dimensional case. Consider
P=-1h20+V(x) in a neighborhood of 0 E Rn and assume that V is smooth, real valued with a non-degenerate (local) minimum at 0:
V(0) = 0, V'(0) = 0, V"(0) > 0.
(3.2)
Here the last inequality is to be understood in the sense of symmetric 2 t2 - V (x). matrices. Put p(x, ) = 2 t2 + V (x), -p(x,
After a Euclidean change of variables x and the corresponding dual change of variables e, and still denoting by q the function q expressed in the new variables, we may assume that: q(x,
21 2
=
-
n n1
x? + O((x,)3),
(3.3)
where aj > 0 are the eigenvalues of the Hessian of V at 0. Introduce the new
variables yj = yj/ bj and the corresponding dual variables qj = j
bj.
Then the function q becomes: u2
1
q(y, rl) =
3
bjaj
2
3
2
Choose bj so that e, = bjaj, i.e. bj = 1/ aj. Then we get
' (pj - yj) + 0((y, r1)3), Aj =
q(y, rl) =
aj.
(3.5)
The Hamilton field becomes: n
Hq =
Aj(r7jayl
+yjanj)+0((y,ri)2).
(3.6)
This field vanishes at (0, 0) and, consequently, we cannot apply the general
result about existence of solutions of the Hamilton-Jacobi equations in Chapter 1.
18
Spectral Asymptotics in the Semi-Classical Limit
In general, if v(x, 8x) = Ei aj (x)8xj is a smooth real vector field on R" 1 which vanishes at 0, then we consider the corresponding linearized vector field:
vo(x, 8X) = E
8xk aj (0)xk8Xj _ (Ax, 8x),
(3.7)
where A = (8Xka1(0)) is called the matrix of the linearization. Using the fact that v vanishes at 0, we see that vo is an invariantly defined linear vector field on TOR' and that A is invariantly defined as a map: TOR' , TOR'. In fact, A maps v(0) to µ(0) if v, p are real vector fields, related by y = C,v = [v, v].
The eigenvalues and eigenvectors of the linearization of a vector field are of importance for the properties of the integral curves near a stationary point (i.e. a point where the vector field vanishes). For instance, we have the socalled stable manifold theorem (see for instance Abraham-Marsden [AbMa], Abraham-Robbin [AbRo]) which exists in various versions. Here is one of them:
Theorem 3.1. Let v be a C°° vector field defined near 0 in R' which vanishes at 0. Let d be the number of eigenvalues with real part > 0, counted
with their algebraic multiplicity, of the linearization 8v(0)/8x of v at 0. Then in a suitable neighborhood U of 0, there is a unique closed smooth connected d-dimensional submanifold A C U containing 0 and with the following properties:
(1) v is tangent to A at every point of A: v. E TPA, Vp E A, (2) The complexification C ® TOA is the sum of the generalized eigenspaces corresponding to the eigenvalues with real part > 0.
Moreover, 3 C > 0 such that for all x E A, we have I1exp (-tv) (x) 11 _< is the standard Euclidean norm. Replacing Ce-t/Cjlxii, t > 0. Here 11 11
this norm by another suitable one, we may even arrange so that the prefactor `C' to the right, in the last estimate, can be replaced by 1.
Returning to v = H9 in (3.6), we get the linearization at (0, 0): 0
Al
I Al
0 0
F=
A2
A2 0
3. The WKB-method for a potential minimum
19
which has the structure of a block-matrix, when the coordinates are enumerated as yl, 771, y2, 'q2,-.., and where the non-diagonal blocks are 0. The linearization of a Hamilton field is sometimes called a fundamental matrix. In general, if F is a fundamental matrix, then exp (tF) is a symplectic matrix: a(exp (tF)v, exp (tF) p) = a(v, µ) and differentiating this and putting t = 0, we see that F is anti-symmetric with respect to the symplectic form: a(Fv, µ) + a(v, Fµ) = 0.
(3.9)
Let
C2n _ ® Eµ µE-(F)
be the Jordan decomposition of C2, into generalized eigenspaces. Then we see that EN, and EA are orthogonal with respect to a if p + A # 0. In fact,
assume that µ + A: 0 and let x E Ej,,, y E E. Then (a - F)mIEµ = 0, for some m > 0, and similarly with p replaced by A, so
(µ+F)mE,
=((p+A)I+(F-A))mIE,
: EA -+Ea
is bijective. We can therefore write y = (p + F)"nz, z E EA and
a(x, y) = a(x, (µ +
F)m'z)
= a((µ - F)''x, z) = 0,
so Eµ and Ea are orthogonal when µ + A 0 0. Then clearly Eµ and E_µ cannot be orthogonal, and a even gives a natural way of identifying E_N, with the dual space of E. If we regroup the distinct eigenvalues into {0}, {µl, -All,- -,{µd, -11d}, then we get the decomposition into d + 1 spaces -
which are orthogonal for a: C2n = Eo ® (Eµ1 (D E-µi) ® ... ® (Eµd ®E-µd)
Notice also that the restriction of the symplectic form to each of these subspaces is non-degenerate, that the dimension of Eo is even, and that the dimension of Eµ is equal to the dimension of E_µ. Using also the fact that F is real, we see that the non-zero eigenvalues split into groups: A3, -Ai, for
A3 > 0, iaj, -iaj for aj > 0 and (j, (j, -(j,
where Re (j > 0, Im c > 0.
Returning to F in (3.8), we get the real eigenvalues ±A3, j = 1, ... , n. The corresponding eigenvectors are immediate to compute and the sum of the eigenspaces corresponding to positive eigenvalues is given by q = y, y E R'. Applying the stable manifold theorem, we see that in a suitable neighborhood
of (0, 0), we can find a closed n-dimensional submanifold A+ such that
20
Spectral Asymptotics in the Semi-Classical Limit
(0, 0) E A+, T(o,o) A+ _ {(y, y)} and such that Hq is tangent to A+ at every point.
Reversing the sign of q we also obtain a closed n-dimensional submanifold A_ with (0,0) E A_, T(o,o)A_ = {(y, -y)} such that Hq is tangent to A_. We also have Ilexp (tHq)(p)II < Ce-Itl/°IIpII, for p E Af,
t > 0.
(We say that Hq is expansive on A+ and contractive on A_.) We call A+ and A_ respectively the stable outgoing and the stable incoming manifolds. (A more standard terminology is to call A+ the unstable manifold and to call A_ the stable manifold, however we use the slightly different terminology, thinking about the stability of the manifolds themselves under the Hq-flow.)
Lemma 3.2. qI A+ = 0. Proof. Since q(p) = q(exp (tHq)(p)), we get q(p) = limt.
q(exp (tHq) (p)) = q(0) = 0,
#
for p E A±.
Lemma 3.3. A± are Lagrangian manifolds. Proof. Using X1, ... , xn as local coordinates on A+, we have v := Hq IA _ + E vj(x)a,,j, and av/ax(0) has the eigenvalues A1, ... , A. Recall that if x(t) is an integral curve of v then by the stable manifold theorem, or by a direct argument, we have IIx(t)II
Ce-ItI/CIIx(0)II, t -> -oo,
(3.10)
where we may even replace the first factor C on the right by 1. We look for the evolution of tangent vectors. Let x(t, s) be an integral curve, depending smoothly on the additional parameter s. Differentiating the equation atx = v(x), (viewing v as a vector depending on x) we get atasx
= ax
(x(t, s))asx.
Using (3.10) and the fact that av(0)/ax has only eigenvalues with strictly positive real part, we get IIasx(t,s)II i Ce-ItIlc Iasx(0,s)II, t < 0.
(3.11)
3. The WKB-method for a potential minimum
21
In other words, if p(t) = exp (tHq)(p(0)), t < 0, S(t) = (exp (tHq))*6(0) E Tp(t) A+, then
116(t)JI < Ce-Itl/CIIb(0)JI, t < 0.
It is now easy to see that A+ is Lagrangian. First we notice that the dimension is the right one, secondly that for v, a E TPA+: QP(V, 1t) = QexptHq(p)((exptHq)*v, (exptHq)*l,t) - 0, t -> -oo, so UP (V, µ) = 0.
In the original coordinates we get a smooth real function O(x) defined in a neighborhood of 0, such that A+ is given by = q'(x) (and A_ is given by = -0'(x)), P(0) = 0, 0'(0) = 0, 0"(0) > 0. (To get the last fact, just notice that in the y-coordinates, we have 0(y) = 2 +O(y3).) We also have of course, the eikonal equation: q(x, 0, or more explicitly: (01)2 - V(x) = 0. We can now start the WKB-construction. We want to find a(x; h) - ao(x) +
hai(x) +... and E - J:o Ej hj such that (P - hE)(ae-Olh) = re-m(x)lh, r = O(hN), VN > 0.
(3.12)
Here we notice in general that P(ae-0l h) = be-,Pl h, with b = hL(a)-h2Oa/2, where 1 = VO V. V + 0O/2. Thus, if we try a with the asymptotic expansion above, it is enough to solve (in some fixed neighborhood of 0) the sequence of transport equations: (L - E0)ao = 0, (To) 1
(G - Eo)ai = Elao + 2Oao,
(Ti)
(G - Eo)a2 = E2ao + E1a1 + 2Oa1.
(T2)
etc. Since the gradient of 0 vanishes at 0 it is not obvious how to solve these equations and we have to study G more closely. Recall that q = 2 t2 V (X), 2 x + O(x3). Denote by Vo the leading quadratic part of the V(x)
-
last expression, and write similarly, O(x) = Oo(x) + O(x3). Then from q(x, 0' ) = 0, we get (01)2 = Vo(x), and knowing also that 00 is positive 2 x , and consequently, definite, we get the unique solution Oo (x) _ D00 =
Aj. We conclude that
G = 1:(Ajxj + O(x2))8X, +
2' + O(x).
Spectral Asymptotics in the Semi-Classical Limit
22
Put Go=
Ajxj8;+i
2.
We notice that if Prom is the space of polynomials in n variables which are homogeneous of degree m E N, then Co(Phom) C Ph°m, and the monomials, xa, Ial = m, constitute a basis of eigenvectors of the restriction of Go to Phom. The corresponding eigenvalues are j )j (c + ), so Co - Eo is a bijection in m
if E0
Aj (% +2);Iam}.
2
Consider a homogeneous transport equation (like (To)):
(G - Eo)f = 0,
(3.13)
where f E C°° does not vanish to infinite order at 0. Let f = fm (x) + O(xm+1) 0 7' fm E P. Then we get (,Co - Eo) fm = 0. We conclude that Eo has to be an eigenvalue of Lo: Eo = E(aj + )Aj, for some a E Nn of length m. We now assume for a given Eo:
2
There is precisely one a E N' such that Eo =
Aj
(1 + aj).
(H)
Let ao denote the unique a in the above assumption, and put mo = laoL,
fo = xa0. We now work with formal power series at 0. Then we can construct a solution f = E°°mofk, fk E Phom, to (3.13) in the following way. Put fmo = x°0. Then (in the sense of formal power series at 0): (G - Eo)fmo = E°0°+1 gk, where gk is homogeneous of degree k. Let fmo+i solve (,Co - Eo) fmo+l = -gmo+1 Then (G - Eo)(fmo + f--+l) = Emo+2 hk, etc.
Let fmo be the (formal power series) solution of (3.13) that we have just constructed. More generally, inhomogeneous equations can be treated the same way, and we obtain
Proposition 3.4. For every formal power series g at x = 0 there is a unique scalar )(g) such that (G - Eo)f = g - A(g)fmo has a solution. The solution is unique up to a multiple of fmo. It is now clear how to construct formal power series solutions ao, al, a2, etc. to the sequence of transport equations, as well as a corresponding sequence
Eo, El, .... We leave as an exercise to the reader to show that the Ej are uniquely determined, once Eo has been chosen (satisfying (H)). We also
notice that if 2j < mo, then aj vanishes to the order mo - 2j at x = 0. The final step in our construction is to pass from formal power series at x = 0 to actual C°°-functions defined in some fixed j-independent neighborhood of
3. The WKB-method for a potential minimum
23
x = 0. If M is the open set where V and 0 are defined, we let fZ C M be an open neighborhood of 0 and we say that S2 is star-shaped if the following statements hold: (1) If x E Q, then exp (tV O.Ox) (x) is well-defined and belongs to 1 fort < 0. Moreover exp (tVO. 8x)(x) converges to 0 when t -+ -oo.
(2) For every compact set K C S2, the set K = {0} U {exp (tVq Ox) (x); x E K, t < 0} is a compact subset of Q.
In view of (3.10) and the subsequent remark, we see that B(0, ro) = {x E Rn; lixII < ro} is star-shaped if ro > 0 is sufficiently small. We now let S2 be star-shaped, and consider the equation
(L - Eo)u=v in the space of C°° functions on fl which vanish to infinite order at 0. With v = Vc Ox we only retain that this equation is of the form
(v + k(x))u = v,
(3.14)
where k E C' (Q). Let ] - 00,0] D t '-4 -y(t) be the v-integral curve with -y(0) = x for some given x E Q. Along ly the differential equation takes the form: dt +k(Y(t)))u(Y(t)) = v(Y(t))
Here v(-y(t)) = O(e-CItI) for every C > 0 (since y(t) approaches 0 exponentially fast when t -> -oo), and we have a unique solution with the same properties, given by u('Y(t)) = J
e- ft k(7(,))d,v(-y(s))ds.
t 00
Since y(O) = x, we see that the only possible solution of (3.14) which vanishes to infinite order at 0 is given by: o
U (x) =
LO
e-
k(exp ov(x))da
v(exp sv(x))ds,
(3.16)
and it only remains to show that this expression defines a function in the space we want. For x in a compact in fl, there is a constant C > 0 such that ilexpsv(x)ll < Ce-1s11cllxll, Ildxexpsv(x)II < Ce-1S1/c
and for the higher differentials we have estimates of the same type. Taking repeatedly higher and higher differentials of the expression (3.16) we verify the required properties, and we obtain:
24
Spectral Asymptotics in the Semi-Classical Limit
Proposition 3.5. Let S2 be star-shaped and let v E C°° (S2) vanish to infinite
order at 0. Then the equation (3.14) has a unique solution with the same properties.
We now consider the sequence of transport equations (Tj) in some starshaped domain Q. We recall that we already know how to solve these equations in the space of formal power series as x = 0, and we let ao, al, a2i .. .
be smooth functions on St which represent a formal power series solution. Then we look for aj = aj + b;, where bj E C°° (S2) vanishes to infinite order at 0. Using the last proposition it is easy to see that there are such uniquely determined functions bj. Now let a(x; h) - E aj (x) hi. We have proved:
Theorem 3.6. Let E0 satisfy (H) and let ao and mo be defined as after (H), and let Il be star-shaped. Then we can find aj(x) E C°° (1) with ao(x) = x0'0 + O(Ixlmo+1) aj(x) = O(jxjm0-23), 2j < mo, and uniquely determined real numbers El, E2 such that if E(h) - Eo + Elh +..., then
(P -
hE(h))(e-01ha)
= re-01h,
where J8"r(x; h) I < CK,w,ah`v, x E K for every K CC Q, a E N'2, N E N. This theorem can be generalized to the case of arbitrary values E0 of the form Aj (aj + ), not necessarily satisfying (H), but the argument above becomes a complicated, and moreover one will in general get half-powers a little more of h in the expansion of E(h).
We end this chapter by recalling how the values E0 can be viewed as eigenvalues of a certain harmonic oscillator. We start with the case of the standard harmonic oscillator on R: 2
2P
= ( dx2 + x).
One eigenvalue is Ao = 1 and the corresponding normalized eigenfunction is uo = 7C- 4
e-y2/2
To generate the other eigenvalues, introduce the annihilation operator Z = d + x. Then if we dx + x and its adjoint, the creation operator Z*
use the standard notation [A, B] = AB - BA for the commutator of the operators A, B, we get [Z, Z*] = 2, P = ZZ* - 1 = Z*Z + 1. Assuming we already found a function uj with Pub _ Ajuj, we try uj+l = Z*uj. Then PZ*ui = (Z*Z + 1)Z*uj = Z*(ZZ* + 1)uj = Z*(P+2)uj = (Aj +2)Z*uj. Hence uj+1 is an eigenfunction with eigenvalue Aj+1 = A3 + 2. Thus we get where the sequence of eigenfunctions ej = Cj(Z*)i(e-x2/2) = pj(x)e-272/2,
3. The WKB-method for a potential minimum
25
C; > 0 is a normalization constant, determined by the requirement that the L2-norm of ej should be equal to 1. Here the polynomials pj are called Hermite polynomials. It is clear that pj is of degree j exactly, so an arbitrary polynomial is a linear combination of the pj. Since the space of functions of p(x)e_x2/2, the form with p polynomial, is dense in L2(R), it follows that the orthonormal family eo, el,... is an orthonormal basis. Anticipating a little on the general spectral theory for selfadjoint operators, that will be reviewed in chapter 4, we see that the spectrum of our one-dimensional harmonic oscillator is given by the eigenvalues )j = 1 + 2j for j = 0, 1, 2.... and that each of these eigenvalues is simple. In R, we consider the generalized harmonic oscillator
P=-1A+Vo(x) where Vo is a positive definite quadratic form. Then as we have seen in the beginning of this chapter, we can make a linear change of variables and reduce
P to the operator P
where D = D. We then get the eigenvalues tta = A, (aj + 2 ), for a E Nn, and a corresponding orthonormal basis of eigenfunctions:
u.(y) =
C«(-8y, +yi)«' ....(-8yn
+yn)«"(e
Y2/2)
=p«(y)e-y2/2.
If we consider the semi-classical harmonic oscillator
Ph=- 2h2A+Vo(x), reduces Ph to hP1, so we get the then the change of variables x = eigenvalues h E \j (aj + ), and the corresponding eigenfunctions expressed i In other words, the WKBin the y coordinates: h-n/4pa(h-1/2y)e-y2/(2h).
constructions earlier produce exact eigenvalues and eigenfunctions in the case of a (generalized) harmonic oscillator.
Notes In this chapter we have followed Helffer-Sjostrand [HeSj2].
4. Self-adjoint operators In this chapter we review some of the standard theory, and apply it to a semi-
classical Schrodinger operator with a potential well. Let 7-l be a complex separable Hilbert space. The corresponding norm and inner product are denoted by II II, (.1.). By definition, an unbounded operator S : 7-l --> 7-l is given by a subspace D(S) c 7-l, called the domain of S, and a linear operator S : D(S) -> H. (It might be better to speak about not necessarily bounded operators, since bounded operators are not excluded from the class of unbounded operators.) The graph of S is defined by: -
graph (S) = {(x, Sx); x E D(S)}.
(4.1)
This is a subspace of 7-l x 71 that we equip with the norm of 7-l x 7-l: II (u, v) II2 = IIull2 + IIvlI2. We say that S is closed if graph (S) is closed.
Proposition 4.1. Let S : 7-l - 7-1 be an unbounded operator with a dense domain D(S). Then there exists an unbounded operator S* : 7{ - 7-l given by:
D(S*) = {v E 7{; 3C(v) > 0 such that I (Sul v) I < C(v) II ull , u E D(S)},
(SuI v) = (uI S*v), Vu E D(S), by E D(S*).
(4.2) (4.3)
Proof. We define D(S*) by (4.2). Then for v E D(S*), the linear form D(S) D u H (Sulv) has a unique continuous extension to 7-1, hence there exists a unique w E 71 such that (Suly) = (ulw), `du E D(S). The map D(S*) D v'--f w is linear and we can define S*v to be equal to w.
#
Notice that if we drop the assumption that D(S) is dense, then we can still define D(S*) by (4.2), and at least one element S*v E 71 by (4.3). However, the vector S*v is no longer uniquely defined, since it can be changed by addition of an arbitrary element of D(S)' := {u E 7{; (ulv) = 0, Vv E D(S)}. On the other hand, if we assume that S is bounded (so that D(S) = 7-1 and IISull < CIIull, and IISII denotes the smallest possible constant in the previous estimate) then S* is bounded and I I S* I I = IISII.
Let J : 7-l x 7i be given by J((u, v)) _ (-v, u). Then J2 = -I, where I denotes the identity operator. Viewing 71 x 7{ as a Hilbert space with the scalar product ((ui, u2) I (vl, v2)) = (ui I vi) + (u2Iv2), we see that J* _ -J, and that J is unitary. (We recall that a bounded operator U : 7-I -> 7-l is unitary if it is isometric: IIUull = lull and surjective. Equivalently, a unitary operator U is a bounded operator which satisfies: U*U = UU* = I. One can
28
Spectral Asymptotics in the Semi-Classical Limit
also define unitary operators between two different Hilbert spaces.) We have the following relation:
graph (S*) = (J(graph (S)))1 = J((graph (S))1). In particular, S* is always a closed operator. Notice also that graph (S) = (graph (S))11 = J((graph (S)-L-L) = (J((Jgraph (S))1))-L. Hence if V(S*) is dense, then S** is the closure of S ; S** = S in the sense that graph (S) graph (S) = graph (S**).
In general it is not obvious that the closure of the graph of S is the graph of an operator. If this is the case, we say that S is closable. The above discussion shows that if the domains of S and S* are dense, then S is closable. Conversely, it is easy to see that if D(S) is dense and S is closable, then D(S*) is dense and the closure of S is equal to S**.
In general when S is densely defined we have the useful identity:
Im (S)1 = Ker (S*).
Here we put Ker (A) = {u E D(A); Au = 0} and Im (A) _ {Au; u E D(A)}. Definition 4.2. If A, B : 7-1 -> 7-l are unbounded operators, we say that A C B, if D(A) C D(B) and Bu = Au for all u E D(A). Equivalently, A is contained in B if and only if we have the corresponding inclusion for the graphs.
If D(A) is dense and A C B, then B* C A*. Definition 4.3. Let A : 7-1 -p 7-l be a densely defined unbounded operator. We say that A is symmetric if A C A* and selfadjoint if A = A*.
Self-adjoint operators have very nice properties, so if A is a given symmetric operator we are interested in finding a selfadjoint operator B which contains
A. We then say that B is a selfadjoint extension of A. Notice that every selfadjoint operator is closed, so if B is a selfadjoint extension of a symmetric operator A then, necessarily, q C B. (Here we notice that every symmetric operator is closable.)
Definition 4.4. A symmetric operator A is called essentially selfadjoint if q is selfadjoint.
If A is essentially selfadjoint, then the closure of A is the only selfadjoint extension of A.
4. Self-adjoint operators
29
Theorem 4.5. Let A : N -; N be symmetric. Then (1), (2), (3), are equivalent, where:
(1) A is selfadjoint,
(2) A is closed and Ker (A* ± i) = {0} for the two signs,
(3) Im (A ± i) =H for the two signs. Proof. In order to show that (1) (2), it suffices to show that if S : N - N is symmetric, then Ker (S ± i) = {0}. If u E Ker (S + i), then 0 = ((S + i)uIu) = (Sulu) + illu112.
(4.4)
Since (Sulu) = (ulSu) = (Sulu), we see that (Sulu) is real, and hence (4.4) implies that u = 0. Also notice that for u in the domain of S we have the inequality: Jlull2 < Im ((S + i)uIu) < JI(S + i)ullllull, and after dividing by hull we get the first half of Hull <- 11 (S + i)ull, (lull <-
(4.5)
II (S - i)ull.
The second half is obtained in the same way. From (4.5) we deduce that if S is closed and symmetric then Im (S ± i) are closed subspaces of H. We now show that (2) = (3), so we assume that (2) holds. Since Im (A + i) _
(Ker (A* f i))1 = N, and since A is closed and symmetric, we have Im (A
i) = Im (A
i), and (3) follows.
We finally show that (3) = (1). Assume (3) and let x E D(A*). Using (3) we know that there exists y E D(A) such that (A* -i)x = (A-i)y. Since A C A*,
we have (A* - i)(x - y) = 0. Since Ker (A* - i) _ (Im (A +i))' _ N' = {0}, we have Ker (A* - i) = {0}, hence x - y = 0 and hence D(A*) = D(A). # From the preceding result, it is easy to get the following characterization of essentially selfadjoint operators.
Corollary 4.6. Let A : N -* N be symmetric. Then (1)
(2)
(3), where:
(1) A is essentially selfadjoint, (2) Ker (A* ± i) = {0} for both the signs,
(3) Im (A ± i) is dense in N for both the signs.
In both the results above, we can replace ±i by two arbitrary points z+ and z_ with ±Im z± > 0.
30
Spectral Asymptotics in the Semi-Classical Limit
Let A : H - N be densely defined. We say that zo E C belongs to the resolvent set, p(A), if (zo - A) : D(A) -> 1-t is bijective and the inverse (zo - A)-1 belongs to £(H, H), the space of bounded operators: N - N. We then put R(zo) = (zo - A)-1. For z E C, we have (z - A)R(zo) = I + (z zo)R(zo), and since the norm of the last term is 1z - zo I IIR(zo) II, we see that
for Iz-zoI < 1/IIR(zo)II, (z-A) has aright inverse, R(zo)(I+(z-zo)R(zo))-1 Similarly we have the left inverse, (I + (z - zo)R(zo))-1R(zo). We conclude that if zo E p(A), Iz-zol < 1/IIR(zo)MJ, then z E p(A). This implies that p(A) is an open set. By definition its complement in C (which is then a closed set) is called the spectrum of A. It is usually denoted by a(A) or by sp (A).
In general, if z, w E p(A), then the corresponding resolvents commute: R(z)R(w) = R(w)R(z), and we have the important resolvent identity: R(z) - R(w) = (w - z)R(z)R(w). If A is selfadjoint, then a(A) C R, and essentially from the proof of (4.5) it follows that IIR(z)II < 1/IImzI. Later we will see that we even have IIR(z) II = (dist (z, in the selfadjoint case. (Indeed, this will be immediate from the spectral theorem.) a(A)))-1
Defect indices. The following is a general theorem on selfadjoint extensions of symmetric operators, that we state without proof.
Theorem 4.7. Let A : N --> N be a closed symmetric operator. Then: (1) dim Ker (z - A*) is constant for z in the open upper half-plane. The same holds for z in the open lower half-plane. We put n± = dim Ker (±i - A*). (2) One of the following holds: a(A) is the closed upper half-plane, the closed lower half-plane, the whole complex plane, or a subset of the real line.
(3) A is selfadjoint iff a(A) C R.
(4) A is selfadjoint if n+ = n_ = 0. (5) A has a selfadjoint extension if n+ = n-.
(6) If n+ = n_, then there is a bijection between the set of selfadjoint extensions of A and the set of unitary operators U : x+ - N_, where Hf = Ker (±i - A*). R) and let S = -A + V (x), where A = a=3 is Remark. Let V E Li, the Laplace operator, equipped with the domain. D(S) = Co (Rn). Then S is symmetric and the antilinear operator r of complex conjugation of functions.
4. Self-adjoint operators
31
Fu = u commutes with S and with S* = A*, where A denotes the closure of S. Then r(7-I±) = 7-l:F, so we conclude that there exists at least one selfadjoint extension of S.
Friedrichs extensions. Let S be a symmetric operator such that S > I in the sense that (4.6) (Sulu) > Ilull2, Vu E D(S). We can then associate with S the quadratic form:
q(u,u) = (Sulu) = Illulll2,
(4.7)
where the last equality defines a norm on D(S). Let D(q) be the completion of D(S) for this norm. (This completion is the abstract Hilbert space obtained as the set of equivalence classes of Cauchy sequences on D(q) for the norm I 11, where the Cauchy sequences (uj) i° and (vj) i° are said to be equivalent if l uj - vj I I - 0.) Since every Cauchy sequence for the norm I is also we have a natural bounded linear map a Cauchy sequence for the norm I I
I
I
I
I
I
I
I
I
I
(of norm < 1) j : D(q) -* R.
Lemma 4.8. j is injective. Proof. Let u E D(q), j(u) = 0. If un E D(S), Illun-ulll -* 0, then Ilunll -* 0. We have
IIIulll2 = lim lim q(un,u,,,,) = lim lim (SunI um) = lim 0 = 0. n-oo m-.oo
n-oo m-.oo
n-.oo
We can then view D(q) as a subspace of 7-l, and extend q to a quadratic form on D(q) x D(q). In general, we say that a quadratic form q(u, u) > IIull2 defined on the subspace D(q) x D(q) is closed, if D(q) is complete for the norm I ' I = q('>) We have just verified that the quadratic form q in (4.7) extended to D(q) x D(q) is closed. Moreover, it is densely defined in the sense that D(q) is dense. I
I
I
I
Theorem 4.9. Let q(u,u) > Ilull2 be a closed and densely defined quadratic form with domain D(q). Then there exists a unique selfadjoint operator Q : 7-l -* 7-l with domain D(Q) contained in D(q) and with (Qulu) = q(u, u) for every u E D(Q).
Theorem 4.10. Let S : 7-l , 7-l be a symmetric operator which is semibounded from below in the sense that (Sulu) > -Mllull2, u E D(S).
32
Spectral Asymptotics in the Semi-Classical Limit
Then there exists a unique selfadjoint extension A of S with the property that D(A) C D(q), where q is the closed quadratic form associated with S + (M + 1)1, discussed prior to Theorem 4.9. The operator A in the last result is called the Friedrichs extension of S. Example. Let S2 C R"` be open and let 0 < V E Coo (Q). Then the symmetric operator S = -A + V with domain CO '(Q) is semi-bounded from below, so we can define a corresponding Friedrichs extension. If we assume that St is bounded with a smooth (C°°) boundary, and V is bounded and continuous on the closure of S2, then the domain of the corresponding quadratic form is Ho (a), the closure of CO '(Q) for the norm IIDauIIL2
IIuIIH1 = 1a1<1
The domain of the corresponding Friedrichs extension then turns out to be Ho (S2)f1H2(1l), and since all functions in this domain vanish on the boundary (in a sense defined by means of the Sobolev spaces), we may call the Friedrichs
extension (in this case) the Dirichlet realization of -A + V on Q. Stronger results exist in the literature:
Theorem 4.11 (Kato [Kal]). If 0 < V E Ll
S = -A + V, with
domain D(S) = Co (R'), then S is essentially selfadjoint. Theorem 4.12 (Farris-Lavine [FaLa]). Let V = V1 + V2 with V1 E Ll °(R"`) V1 > _CJ X12 - D for some constants C, D, and with V2 E LP(R'), where we
assume that p > 2 when n < 3, p> 2 when n = 4 and p > n/2 whenn > 5. Then -A + V with domain Co (Rn) is essentially selfadjoint.
It follows from the last result that when n > 3 and V E Li °, V > -C/IxI -DIxI2-E, then -A+V with domain D(S) is essentially selfadjoint. We next recall various forms of the spectral theorem for selfadjoint operators.
Theorem 4.13. Let A be a selfadjoint operator on a separable Hilbert space R. Then there exist a measure space (M, M, µ), where µ is a finite measure, a measurable function a : M -+ R and a unitary operator U : 'H -* L2(dp) such that: (1) A vector zl> E H belongs to D(A) iff a UO E L2(dp).
(2) If
E D(A), then UAW = alto.
4. Self-adjoint operators
33
One may even arrange so that L2(dµ) is the direct countable orthogonal sum of L2 (R; d i ) , for j = 1, 2, ..., where j is a finite Borel measure on R and a(x) = x. More generally, in the situation described in the theorem, we have a(A) = Imess(a) = {A E R; p(a-1([A - e, A + e))) > 0, `de > 0}.
Functional calculus. If h : R - C is a bounded Borel function, and U and a are as in the preceding theorem, we define h(A) = U 1(h o a) U. We then obtain the existence part of the following theorem: Theorem 4.14. Let 7{ be a separable Hilbert space, and let A : f -+ 7-l be a selfadjoint operator. Let Bb(R) = {h : R -* C; h is a bounded Borel function }. Then there exists a unique map: 13b(R) E) h
h(A) E G(7-1,N),
with the properties (1)-(4):
(1) h(A)* = T(A), h1(A)+h2(A) _ (h1+h2)(A), h1(A)h2(A) _ (h1h2)(A), (2) IIh(A)IIc(,H,x) <- supR Jhj (It is even enough to take the supremum over the spectrum of A.),
(3) If hn E Bb, hn(x) --> x, n -> oo, Ihn(x)I <- IxI, then hn(A)V) -> AV) for every V) E D(A),
(4) If hn -> h pointwise and sup I hn I _< C, then hn (A) (This means that hn(A)cb -* h(A)q5 for every 0 E 7-1.)
h(A) strongly,
Moreover, we have
(5) If 0 E D(A), A E It, AV) = A b, then h(A)zv = h(A)O,
(6) If h > 0, then h(A) > 0 (in the sense that (h(A)0I0) > 0 for all 0 E D(A)).
Spectral measures. For every 0 E 7-l there is a unique Borel measure µO of finite mass, such that for every bounded Borel function g: (g(A)4I4) = f g(A)j (dA). By polarization, we can also construct a measure 110,,, such that (g(A)0IV%) = f g(A)p0,,,(dA). These measures are called spectral measures. In the multiplicative representation of Theorem 4.13, we have µo =
34
Spectral Asymptotics in the Semi-Classical Limit
a.(j012µ) (i.e. direct image of the measure 10I2µ under a). In fact, if g is a bounded Borel function, then f g(A)(a.(10I2µ)(dA) = f g(a(m))I0(m)I2µ(dm)
= (g(A)*I*) =
fg(A)(dA).
From measure theory, we now recall that every Borel measure on R has a unique decomposition: µ = µpp + µac + /µsc, where the three measures to the right are mutually singular, that is carried by disjoint sets, and where: 11PP is a pure point measure: µpp = E ajS,,; , where the sum is countable or finite and 6 denotes the Dirac measure at the point xj,
µac is an absolutely continuous measure, i.e. there exist a locally integrable function f with respect to the Lebesgue measure dx, such that µ,,c = f dx, µsc is singular continuous, that is µsc has no point masses (µsc({x}) = 0 for every x E R) and is carried by a set of Lebesgue measure 0.
Using this decomposition one can construct an orthogonal decomposition: x = H pp ®xac ® xsc such that each of the closed subspaces to the right is invariant under (A+i)-1, and such that if 0 E lpp, E 'Hac or E xsc, then µ,5 is pp, ac or sc respectively. There is an obvious way of defining the (selfadjoint)
restriction of A to these subspaces, and AInpp has an orthonormal (O.N.) basis of eigenvectors. Moreover, every eigenvector of A belongs to 7-lpp. It is clear that the decomposition of 7-l above is unique.
Let app(A), aac(A), asc(A) be the spectra of the restrictions of A to the corresponding three subspaces. Then app (A) is the closure of the the set app(A) of eigenvalues of A. (We recall that if 0 E D(A), A (=- R, then 0 is called an eigenvector of A and A is the corresponding eigenvalue.) It is easy to see that the spectrum of A is equal to app (A) U aac(A) U asc(A).
Spectral projectors. If Sl is a Borel set in R, we put P- = 10 (A), where 1ci denotes the characteristic function of SZ (equal to 1 on Sl and equal to 0 on R \ S2). Then the PP form a spectral family: (a) PQ is an orthogonal projection, (b) PO = 0,
(c) If S2 j = 1, 2.... is a sequence of Borel sets with S2j / 1, when j ---> oo, then PP, -+ Po strongly,
4. Self-adjoint operators
35
(d) Pn1Pn2 = Pn1nn2.
Put Pa =
If 0 E 7-l, then (OIPac) is a bounded increasing function of A and the corresponding Stieltjes measure is the spectral measure µ,5: d(OI PAO) = /to.
If g is a complex valued Borel function on R, we put
D(g(A)) = {o E N;
J
I9(A)I2d(OIPPO) < oo}.
This space is dense in 1-l and we can define g(A) : D(g(A)) -p H by
(9(A)0I0) = f 9(A)d(Pa0Io), 0,V) E D(g(A)). If g is real-valued, then g(A) is selfadjoint. Formally, we write:
g(A) = fg(A)dPA.
In the situation of Theorem 4.13, if g : R --> C is a Borel function, and 0 E N ^-, L2 (µ), then
f I9(t)I2zm(dt) = f I9(t)I2a*(IOI2µ) = f I9(a(m))O(m)I21t(dm).
Hence, 0 E D(g(A)) if (g o a)-O E L2 and g(A) can be identified with the operator of multiplication by g o a. In the special case when g(A) _ A, we get g(A) = A, and we arrive at a second version of the spectral theorem:
A= JAdPA. We also get:
Theorem 4.15. If f, g : R --+ C are Borel functions, 0 E D(g(A)) and g(A)o E D(f (A)), then ¢ E D(fg(A)) and (fg)(A) = f (A)g(A)O. We also notice that for A E C \ Q(A), we have (A - A)-1 = f (t - A)-1dPt.
Stone's formula. For e > 0 and -oo < a < b < oo consider b
BE,a,b =
(27ri)-1 f
a
((A - A -
ie)-1
- (A - A +
ie)-1)dA,
Spectral Asymptotics in the Semi-Classical Limit
36
where the integral can be defined as an operator valued Riemann integral. If [a, b) E) A F--> ga E C(R) n L°° (R) is continuous, it is easy to see that:
f9A(A)dA =
(f
(.)dA)(A)a
and hence BE,a,b = ff(A), where
fE(t)
27ri
f((t - A - ie)-1 - (t - A + ie)-1)dA =
27r
f b (t - A)2 + E2
dA.
We see that 0 < ff(t) < 1 and that fE _* (1[a,b] + 1[a,b[), when e tends to 0. 2 We then obtain Stone's formula: 2 (P[a,b] + Pa,b[) = strong
with BE,a,b defined above.
Essential spectrum. Let A : 7-l -> 7-l be selfadjoint. Then A E R belongs to o(A) if 0 for every e > 0. Definition 4.16. We say that A E R belongs to the essential spectrum aess(A) if Pea-E,a+E[ is of infinite rank for every E > 0.
It is easy to see that the essential spectrum is a closed set, contained in the spectrum. We also define the discrete spectrum Qdis°(A) = Q(A) \ Qes(A). The discrete spectrum is then the union of all eigenvalues of A of finite multiplicity which are isolated from the rest of the spectrum. We have the following
Weyl criterion. Let A E C. Then
(1) A belongs to the spectrum of A if 2 a normalized sequence On E D(A), n = 1, 2, ... such that (A - A)On 1 0,
(2) A belongs to the essential spectrum of A if there exists an infinite orthonormal sequence On E D(A) such that (A - A)Qin -> 0.
The following two results (which extend a classical theorem of H.Weyl) say roughly that the essential spectrum is unchanged under compact perturbations.
Theorem 4.17. Let A, B : 7-l -> 7-l be selfadjoint operators such that (A+i)-1 - (B+i)-1 is compact. Then Qess(A) = Qess(B)
4. Self-adjoint operators
37
Definition .4.18. Let A : N -* 7-l be selfadjoint, C : N -> N, with D(A) C D(C) and C(A + i)-1 compact. Then we say that C is relatively compact with respect to A, or simply that C is A-compact. Notice that if C is compact, then C is A-compact for any selfadjoint operator acting in the same Hilbert space.
Theorem 4.19. Let A be selfadjoint and let C be symmetric and A-compact.
Define B = A + C with domain D(B) = D(A). Then B is selfadjoint and Qess(A) = Qess(B)
Example. Let V E L°O(RT;R) and assume that V(x) -* 0, when I xI -> oo.
Let A = -A with D(A) = H2(Rn). Then V(-A + i)-1 is compact, so aess(A + V) = aess(A) = [0, oo[. This means that o, (A + V) is the union of [0, oo[ and a finite or countable subset of ] - oo, 0[ with 0 as the only possible accumulation point, consisting of eigenvalues of finite multiplicity. Example. More generally, let V E Li °(Rn; R) be bounded from below and
put c = lim
Then inf Qess(-A + V) > c. In fact, assume for instance that c < +oo, put A = -A + max(V, c), B = V - max(V, c) so that B(x) is bounded and tends to 0 when jxj -> oo. Since D(A) C H1(Rn),
(A + i) is bounded: L2 -* H1 and consequently B(A + i)-1 is compact. Hence, oess(A + B) = vess(A) C a(A) C [c, oo[. When limV = +oo the same arguments works if we replace c by any real number. In this case -A + V has a purely discrete spectrum, given by a sequence of eigenvalues (of finite multiplicity) which tends to +oo. See also Persson [Pe].
The mini-max principle is a an important tool in the applications, and when applied to selfadjoint operators, bounded from below, it takes the form of a maxi-min principle. Let A : N -+ N be an unbounded selfadjoint operator which is semi-bounded from below in the sense that there exists a constant
C E R such that A > -CI. Assume that N is infinite dimensional. Let Al < µ2 < ... be an increasing enumeration of all the eigenvalues of A with µj < Qess(A), repeated according to their multiplicity. If the number No of such eigenvalues is finite, we define I No+1 = INo+2 = ... to be inf oess(A)
Let 1 < N < oo, let u1, ... , UN-1 E N be linearly independent and let E = (u1, ... , UN-1)1, where (u1, ... , UN-1) is the linear space spanned by U1i...,uN_1.
Lemma 4.20. D(A) fl E is dense in E. Proof. Let v1, ... , vN_1 E D(A) be close to u1, ... , UN-1 in norm. Then the space F = (v1, ... , VN_1) is of dimension N - 1 and transversal to E,
38
Spectral Asymptotics in the Semi-Classical Limit
and there is a unique bounded projection II 7-L -> E with Im (II) = E, Ker (II) = F. Then II maps D(A) into D(A), and we also see that II(D(A)) is dense in E. # :
Lemma 4.21. We have inf
(Aulu) < AN(A).
uED(A)nE, IIuII=1
Proof. W e first assume that AN(A) is among the first N eigenvalues (repeated according to their multiplicities) strictly below Qess(A). Let el, ... , eN be a corresponding orthonormal family of eigenfunctions. Then (el,... , eN) n E {0} and we let u = EN Ajej be a normalized vector in the intersection,
so that E I'; I2 = 1. Clearly u E D(A) n E, and N
N
(Aulu) =
A;IA;I2 <_ ANIA.7I2 =AN
In the case when AN = inf aess(A), the space 1l_-,µN+El(A)(7-l) C D(A) is infinite dimensional for every e > 0 and has a non-trivial intersection with E. If u is a normalized vector in the intersection, we have (Aul u) <_ AN + e, by the spectral theorem, and since e > 0 can be taken arbitrarily small, we obtain the lemma in this case also. #
Theorem 4.22. Under the assumptions above, we have the maxi-min formula:
AN(A) =
sup
u1,... ,' N-1E7{,
uE(ul,.
linearly independent
inf
uN-1)1'nD(A),
(Aulu).
1+11=1
Proof. We have already seen that the RHS is < [IN (A). When AN (A) is the Nth eigenvalue < inf Qess(A), we can take uj = ej, 1 < j < N - 1 with ej as in the proof of the last lemma, and see that inf
uE (el ,...,e N- l )1 nD(A),
(Aulu) = AN (A).
I1u11=1
The second case AN (A) = inf oess (A) can be treated similarly.
We end this chapter with a rough determination of the low-lying eigenvalues of semi-classical Schrodinger operators of the type considered in Chapter 3. Further and deeper results in this direction will be given in Chapter 14. Let
4. Self-adjoint operators
39
M be either R' or a closed bounded subset of R' with smooth boundary and such that 0 belongs to the interior: 0 E int (M). The arguments below will also work with only minor modifications in the case when M is a smooth compact Riemannian manifold, possibly with a boundary. Let V E C°°(M; [0, oo[)
with V(x) = 0 precisely for x = 0 and assume that liminfV(x) > 0, in the case when M = R'. Let P = -h 2A + V,
(4.8)
and denote by P also the corresponding Friedrichs extension, when starting from the symmetric operator (4.8) with domain Co (int (M)). We know that P > 0 has purely discrete spectrum in [0, c] for some c > 0, and we shall now determine the first approximation of the eigenvalues of P in any interval of the form [0, Coh] in the limit h -j 0. (V" (0)x, x) be the leading term in the Taylor expansion of V 2 at 0, and introduce the harmonic oscillator,
Let Vo (x) =
Po = -h2 p + Vo (x), x E Rn.
(4.9)
This operator (realized through the Friedrichs extension) has a purely discrete spectrum, which we essentially computed in Chapter 3, and we saw
there that the eigenvalues of Po = Po(h) are of the form 0 < Eoh < Elh < E2h < ..., where Ej are the eigenvalues of Po(1). A corresponding O.N. basis of eigenfunctions is given by ej (x; h) = h-n/4ej (h-1/2x). Here ej(x) E S(Rn) are Hermite functions up to a linear change of variables.
Let 0 < C o ¢ {Eo, E1,. ..} and let No be the number of Ejs in [0, Co], so that EN,,-1 < Co < EN,. Theorem 4.23. Under the assumptions above, there exists ho > 0, such that for 0 < h < ho, P has precisely No eigenvalues, 0 < A0(h) < ... < )Na_1(h) in [0, Coh] Moreover, )j(h) = Ejh + O(h3/2).
Proof. Let X E Co (int (M)) be equal to 1 near 0. Since the L2-norm of (V(x) - V2(x))X(x)ej (x; h) is 0(h3/2), we get
(P - Ej h)(X(x)ej (x; h)) = rj (x; h), II rj M = 0(h3/2),
(4.10)
and from this we conclude that for each j < No -1, there exists an eigenvalue pj of P with µj = Ejh + O(h3/2), but to show that we get No eigenvalues of P (counted with multiplicity) in this way, requires a little more work except when the Ej are all distinct.
Let R >> 1 and choose a quadratic partition of unity, X0
(X)2
+ X1(x)2 = 1,
(4.11)
40
Spectral Asymptotics in the Semi-Classical Limit
with Xo, X1 E C°°(M; [0, 1]), Xo E Co (B(0, 2Rh1/2)), Xo = 1 on B(0., Rh1/2) Oa((Rh1/2)-kkI). Here B(0, r) denotes the open ball in Rn of center BaXj = 0 and radius r. Notice that
X0 [A, Xo] +X1 [A, X11 = Xo0(Xo) + X1A(X1) = O(R2h),
(4.12)
which gives the so-called IMS localization formula,
(-Lulu) = (-LXouIXou) +(-LX1uIXlu) + ((Xoz(Xo) + Xiz(Xi))uIu), u E D(P). (4.13) Combining (4.12), (4.13), we get for u E D(P): (PuIu) = (Pxoulxou) + (PX1ulXlu) + O(R2 )IIuII2.
(4.14)
(PX1ul Xiu) > (VX1ulXiu) >- ENohIIX1uII2,
(4.15)
Here
if we choose first R large enough and then h small enough, so that inf
Ixl>Rhl/2
V > EN,,h.
On the other hand, (Pxoulxou) = (PoXoulXou) + O(R3h3/2)IIXoull2,
(4.16)
when h is small enough, depending on R.
Assume that Xou 1 ej
h), 0 < j < No - 1, so that (PoXoulXou) >
(ENo h) IIXouI12. Then from (4.14)-(4.16), we get
(PuIu) > (ENo -O(R2 +R3h1I2))hIIuII2
(4.17)
From the maxi-min principle, it follows that the (No + 1) st eigenvalue of P is > (ENo - o(1))h, when h -* 0.
It is easy to find a simple closed loop -y in {z E C; Re z < Coh} such that
dist (z, a(P) U a(Po(h))) > eoh, z E ry and such that hEj, j < No - 1 are in the interior of 'y. Here Eo > 0 is some fixed number independent of h. Returning to (4.10), let E C L2(M) be the No-dimensional space spanned by Xej ( ; h), j = 0, ... , No - 1, and observe that the functions Xej form an almost O.N. basis in E in the sense that (XejIXek) = 6j,k +0(h°°).
(4.18)
4. Self-adjoint operators
41
Rewrite (4.10) as
(z - P)(Xej) = (z - Ejh)Xej - rj> and apply (z - P)-1(z - Ejh)-1 for z E
(z -
P)-1Xej
'y:
= (z - Ejh)-1Xej + (z - P)-1(z - Ejh)-1rj.
(4.19)
Let lr
21 I (z - P)-1dz 7ri Y
be the spectral projection associated with P, and the intersection of R and the interior of -y and let F = 7r(L2(M)). From the conclusion after (4.17), we know that dim F < No. From (4.19) we get
lr(Xej) = Xej + kj, IIkj II = 0(h112),
(4.20)
and it follows (when h > 0 is small enough) that the dimension of F is at least equal to No, so we have finally dimF = No. Moreover, fj = lr(Xej),
0 < j < No - 1 form a basis in F, and if we use (4.10) again, we get Pfd = hE3 fj +0(h3/2). Let f = (fo,..., fNo_1) be the corresponding row vector and introduce the orthonormalized basis g = f ((fj fk))-112. The ... < ANp_1 denote matrix of PI F is then diag(hEj) + 0(h3/2). If Ao the eigenvalues of PIF, it follows that A3 = hEj + 0(h3/2). This completes the proof of the Theorem.
#
Let Ej be one of the eigenvalues of the harmonic oscillator Po(1) with j < No - 1, and assume that Ej is a simple eigenvalue. It follows from Theorem 3.6 that the corresponding eigenvalue A3 (h) has an asymptotic
expansion - h(ao + a1h + a2h2 + ...), where ao = Ej. If we drop the assumption that Ej is simple then we still have an asymptotic expansion for As(h)/h with leading term Ej, provided that we allow for half powers of h. See Simon [Sil], Helffer-Sjostrand [HeSjl].
Notes Most of this chapter is a compilation of general and well-known facts for spectral theory. We have used [NaRi], [ReSi] and [CFKS]. The asymptotic behavior of the lowest eigenvalues of the Schrodinger operator in the semiclassical regime has been studied by many authors. See [Sill, [HeSjll. In the one-dimensional case precise asymptotic expansions were computed by Combes, Duclos and Seiler [CDS]. The case of the asymptotic behavior when the minimum is degenerate has been studied by Martinez and Rouleux [MR].
5. The method of stationary phase Let X C Rn be an open set, 0 E C°°(X; R) (i.e. a real valued smooth function) such that do 54 0 everywhere. If u c CO '(X), then the integral
I(A) _ f eiaq(x)u(x)dx
(5.1)
+oo. This can be seen by repeated integra-
is rapidly decreasing when A
tions by parts, using for instance the operator tL = iAIq E a" -L Oxj which satisfies
= ex\O. More precisely, we obtain:
tL(ei'\O)
For every compact K C X and every N E N, there is
a constant C = CK,O,N, such that II(A)I < C( sup I-I
for A > 1, u E Co (X), supp u C K.
(5.2)
This means that if 0 E C°° (X; R), u E Co (X), the asymptotic behaviour of I(A) when A -> +oo is determined by 0, u in a neighborhood of the set of critical points of 0. (Recall that a critical point of a function is a point where the gradient of the function vanishes.) The most important (and most easy) case is the one of non-degenerate critical points. We say that the critical point xo E X of 0 is non-degenerate if det q"(xo) 0, where "(xo) = (a a ck )1<j,k
differential of the map X 3 x H 0'(x) E R" at xo, it follows that xo is an isolated critical point, if it is a non-degenerate one. The Morse lemma gives local coordinates near a non-degenerate critical point xo of 0 for which O(x) - q(xo) becomes a quadratic form.
Lemma 5.1. Let ¢ E C°°(X; R) and let xo E X be a non-degenerate critical point. Then there are neighborhoods U of 0 E Rn and V of xo, and a C°° difeomorphism lc : V -> U, such that 0 0 K-1(x) - O(xo) +
1
21
+ ... + x2 _ x2+l - ... - xn).
Here r and n - r are the numbers of positive and negative eigenvalues, respectively, of /"(xo). Proof. After a translation and a linear change of coordinates, we may assume
that xo = 0 and that O(x) =
1
(x1 + ... + xr - xr+1
x -> 0.
44
Spectral Asymptotics in the Semi-Classical Limit
By Taylor's formula, 92
fi(x) = f(i - t)((tx))dt =
qj,k(x)xjxk =
where Q(x) = (qj,k(x)), qj,k(x) = 2 fo (1 - t)ea , k(tx)dt, so that Q(O) _ ¢"(0) is the diagonal matrix with diagonal elements equal to 1 at the first r
places and with the remaining n - r diagonal elements equal to -1. We look for is of the form ic(x) = A(x)x, where the matrix A(x) depends smoothly on x and satisfies A(O) = I. Then A(x) should satisfy (x, Q(x)x) _ (A(x)x, Q(O)A(x)x), so it suffices to have Q(x) ='A(x)Q(O)A(x).
Let Sym (n, R) be the space of real symmetric n x n matrices and consider the map .F : Mat (n, R) E) A-->tAQ(0)A E Sym (n, R),
where Mat (n, R) denotes the space of all real n x n matrices. The differential
at the point A = I is dF : Mat (n, R) D SA Ht(SA)Q(0) + Q(0)(6A) E Sym (n, R). dF is surjective with a right inverse given by: 613 ,-+
2Q(0)-1SB.
By the implicit function theorem, F has a local smooth right inverse mapping a neighborhood of zero in Sym (n, R) into a neighborhood of 0 in Mat (n, R), with F o 9 = id. We get A(x) with the required properties by taking A(x) _ c(Q(x)). The map ic(x) = A(x)x is a diffeomorphism from a neighborhood of 0 onto a neighborhood of 0, since dre(0) = A(0) = I. #
For u E L1(R), we define the Fourier transform
f
and we extend the definition to the case u E S'(R'S) in the usual way. If Q is a non-degenerate symmetric complex n x n matrix with Im Q > 0 (in the sense of Hermitian matrices), then we know that ei(x,Qx)/2
i
Here we choose the continuous branch of the square root which is positive when !Q is real and positive. In particular, when Q is real, we get ei(x,Qx)/2 I
Q
eigenvalues of Q.
.
(27f)n/lei a sg° Q I det
= r - (n - r), with r being the number of positive
5. The method of stationary phase
45
For u E C0 (R'), we get from Parseval's formula:
f ia(x,Qx)/2 e u(x)dx =
(27rA)_
J
x"sgnQ a/2 e I det QI1/2
fe
u( )d
1
.
(5.4)
We now replace A by 1/t and consider I (t, U) =
e-i4sg'nQ
I det Q1 -21
n
(27rt) 2
f
ei(x,QS)l(2t)u(x)dx
For U E CO' this is a smooth function of t E [0, +oo[, and
a I(t, u) = I (t, Pu), I(0, u) = u(0), where P = - 2 (Dx, Q-1Dx). Taylor's formula gives
= N- 1 1(0' I (t, u)
1: k=O
k
kU)
= tk +
N!
(P uk (0)tk
RN (u, t)
+ N! RN (u t)
k=O
where 1
RN(u,t) =
Nf (1 - s)N-1I(st,PNu)ds 0
so that II7PNUII
I RN (U, t) I < (21
LI < C-
aaPNUII L1. lal
This gives eia(Qx,x)/2u(x)dx N-1
27r z ei a sgn Q
ak+2 (2a k=O k! I det QI 2
Q-1Dx))ku(0)
(D
1
ISN(U,A)I <
N!IdetQI2(27r)nII.F(1
+ a-N- 211 SN(u, a), (5.5)
(Dx,Q-1Dx))NUIILI IIaa(2i(Dx,Q-1Dx))NUIIL1.
N!IdetQl2 aI_ +1
(5.6)
Spectral Asymptotics in the Semi-Classical Limit
46
Here the last sum can be replaced by II ((D, Q-1Dx))NUII HZ+E for any e > 0, provided that we allow Cn to depend on E.
The following variant gives a sharper estimate on SN and avoids the use of the Fourier transform. We start with the immediate estimate:
II(t,u)I <-C.IdetQl2IIUIIL" With f (t) = I (t, u), we Taylor expand at t = 1:
f () 1) (t -
f(t) _
k + (M
11)! f 1(1- s)M-1 f(M)(1 - s + st)ds.
0
Here for 0 < t < 1:
1 101 (1(1
s + t)n/2
ds
OM,n(1)A(M - 2 , t),
where A(m, t) is 1 when m > 0, 1 + loge when m = 0, and t' when - 2 < m < 0. If we Taylor expand 1(t, u), we therefore get M-1
II(t,u)I <- Cn,MIdetQI2(
IIPkUIIL1 +
IIPMUIIL1A(M-
2,t)).
Taking M = [2], the largest integer < 2, we get [2]-1
II(t,u)I <- CnIdetQI2( Y IIPkuIILI + IIP[21uIIL=Bn(t)),
where Bn(t) is equal to 1 + log i when n is even and t-1/2 when n is odd. Using this in the formula for RN, with u replaced by PNU, we get: [2]-1
IRN(u,t)I
It follows that [2]-1
ISN(u,A)I <- CT,N(E IIPN+kuIILI + 0
IIP[2]+NUIIL1Bn(_)).
(5.7)
5. The method of stationary phase
47
The following special case is one of the main ingredients in the theory of pseudodifferential operators. We replace n by 2n and x by (x, y) E Rn, and (0I
we put Q = Q-1 =
2
0 ) . Then
((x, y), Q(x, y)) = -x y, 2i (D(x,y), Q 1 D(x,y)) = i
,
Oy3
Then (5.5), (5.6) become: N-1 ( 27
y)dxdy =
)n if
laI
E k ((2 E ax1 ayi)ku)(0, 0) + SN (u, A) k= 0
1IaI
lI
IaI
n
I£I
(5.5')
(aaay u) (0, 0) + SN (u, A),
where ISN(u,A)I
N!A
E
IIaxe, aa(Ox ay)NUIILI,
(5.6')
I a+131 <2n+1
or where we could use the estimates resulting from (5.7).
We now return to the general situation. Combining the Morse lemma and (5.5), (5.6), we get Proposition 5.2. Let 0 E C°O(X; R) have the non-degenerate critical point xo E X and assume that c'(x) 0 for x 0 xo. Then there are differential
of order < 2v, such that for every compact K C X and every N E N, there is a constant C = CK,N such that for every operators
u E COO (X) n V(K) eZam(x)u(x)dx
N-1
- (1:
(A2,.(Dx)u)(xo)A-v-2)e"\O(x0)
0
<
CA-N-
sup I aau(x) 1, A > 1. Ial<2N+n+1
Here AO =
(2,) '1 e' M4 saw 0"(=0)
Idetk"(xo)I
Recall that a differential operator of order < m on an open set X C RI is an operator of the form P(x, Dx) = EaC-Nn, a,<. aa(x)Dx. Proof. Let V be a neighborhood of x0 as in Lemma 5.1, and let x E Co (V) be equal to 1 near xo. Then f ei),(kxudx can be reduced to an integral of the
48
Spectral Asymptotics in the Semi-Classical Limit
form (5.4) (with a new u), by means of the Morse lemma. The value of the Jacobian of the change of variables at the point 0 can be determined from the relation /"(xo) =tdn(0) o Q(0) o dn(0) (in the notations of Lemma 5.1 and its proof). The integral f eia'k(x)(1 - x(x))u(x)dx can be treated by means of repeated integrations by parts. #
There are many variants of the situation discussed above. The phase and amplitude may depend smoothly on additional parameters, in which case, one
has to check that the arguments above go through, with smooth parameter dependence everywhere. The phase may also be critical on a submanifold instead of at isolated points, and using Fubini's theorem this case can be reduced to the parameter dependent case just mentioned.
Notes The main reference for this chapter is [Hol], See also [Ho4]. We have followed closely [GrSj], while (5.7) may be new.
6. Tunnel effect and interaction matrix In the study of semi-classical approximation one is often interested in the Schrodinger equation, either in its stationary version Pu = Eu, or in its time dependent version, as u = Pu. Here P = -h 2A + V (x) and the goal is to compare the (semi-classical) limit h -> 0 for these equations, with properties of the corresponding classical dynamical system generated e2 + V(x). We then consider the energy by the Hamiltonian surface p(x, ) = E and the corresponding classical particles are the Hamilton trajectories in p-1(E): d
dt
(x(t), fi(t)) = HP(x(t), fi(t))
where Hp is the Hamilton vector field:
-
ap a
ap
Hp_
I
Oajax;
and we are working over some open set in R". For the stationary Schrodinger
equation we expect the solutions to be concentrated to the energy surface p-1(E); and for the time dependent Schrodinger equation we expect the solutions to decompose into wave packets that move along the classical trajectories. In this chapter, we shall study phenomena where the expected correspondence between classical and quantum mechanics is not so clear, specifically we shall study what happens in the classically forbidden region, {x; V (x) > E}, where no classical particles of energy E can exist. It turns out that the corresponding eigenfunctions have to be exponentially small in this region. How-
ever, they are not identically zero, and this is at the origin of the interesting tunnel effect which we shall discuss. a. Lithner-Agmon estimates.
Proposition 6.1. Let Il C R' be bounded with C2-boundary. Let V E C(S2; R), 4 E Lip (ft R) (the space of real valued Lipschitz continuous functions). Then the gradient V4) is well defined in L°°(Sl) as the almost x(E) E everywhere limit of V (XE*4D) = xE*V4D, when e -+ 0, where Co (B(0, e)) is a standard mollifier. For every u E C2(S2) satisfying ulaq = 0, we have IIV(e41/hu)II2dx+J
h2J
=Re
J
(V(x)-IV I2)e241/hlul2dx
e241/hPu(x)u(x)dx.
(6.1)
Spectral Asymptotics in the Semi-Classical Limit
50
Gformula:
Proof. Let first -D E C2(12). Then by
f
h2 f IIV(eu)II2dx = -h2A(e4/hu) e4lhudx
f
2
f
04) h au axj axj
-udx
- f h20(e1Dlh)ue11)lhudx = I + II + III. Here
f
Re (II) _ -
8(b h
a
axj axj
f
Iul2dx =
h a (e2,b/h &D )Iul2dx
axj
axj
= f e24)/h (2 11 V4)II2 + hob) I ul2dx. For III, we use: -h20(e4'lh) =
-e1,/h(IIVDII2
+ hz I ). Then
I + II + III = -Re f e2,D/hh21 (u)udx + fe2IIVII2IuI2dx, and (6.1) follows in this case. Now let (D E Lip (S2), and denote by the same letter some Lipschitz extension of this function to R. Put 'I = XE * 4). Then VIE is bounded in L°° and V (D E L°° (Q), where VD is defined in the sense of distributions. For the last
statement, we let ¢ E Co (R'; Rn): I
f
V
OJ . (pdxl = I Elim
//
V
. Odxl
/r
V IIOIIJ.17
and recall that L°° is the dual of L1. By integration theory, we have V 1 = limE-(,. V a.e. since V = XE * V4. To get (6.1) in general, we first write (6.1) with c replaced by -E, let E -+ 0 and use the dominated
#
convergence theorem.
We indicate a shorter proof of (6.1) for smooth 4, using the natural L2norms for differential 1 forms ; II F_uj (x)dxj II2 = E Iluj II2 and letting
(dV')u(x) = u(x)d4)(x), d4)l (E uj (x)dxj) = E uj (x) a k (x), du(x) _
E ft dxj, d* (> uj (x) dxj) _ -Eau (x). This proof is easy to extend to the case of Riemannian manifolds. Notice that if v = e1b/hu, then for u E C2(S2) vanishing on aQ: Re
= Re
(e24)l h(-h2A)ul u) (e4)lh(-h2A)e-1,lhvl v)
= Re (e-4>lh(-h2A)e1,/hvlv).
6. Tunnel effect and interaction matrix
51
Here h(hd)*eIDl h)(e-4'/h(hd)e4ll e-'/h(-h20)e4>l h = (e-Dl h) = ((hd)* - d4)l)(hd + dV^) = ((hd)*(hd) ((hd)*(dp^) - (dbl)(hd))
The last term in the last member is anti-symmetric, while the first term in the last member is symmetric, and we get Re
(e"/h(-h20)e-1,l
hvl v) = Ilhdvll2 - (Ild-D lI2vlv),
which gives (6.1).
Proposition 6.2. Under the assumptions of Proposition 6.1, let F+, F_ E L°° (S2), FF > 0 satisfy
V(x) - (V4 (x))2 = F+(x)2 - F_(x)2 a.e. Then
h2llV(e4/hu)II2 +
I
IIF+eID/hull2
2
< II
1
F+ + F_
e4l/hPull2 + 3IIF-e4'/hujI2. 2 (6.2)
Proof. From (6.1) we get h2lI V(e"lhu)II2 + IIF+ II
II
F++F_ eD/hpull2 + 14 Il eD/hF+u + eD/hF_ull2 + IIF_ul/hII2 1
< II
F++F_ e4/hPull2 +
which implies (6.2).
1
1Ile'D/hF+hII2 + 2 2
3lle4)/hF
hII2,
#
The preceding results have simple extensions to the case when we replace Rn by an n-dimensional complete Riemannian manifold M. In that case we let 0 be the Laplace-Beltrami operator, and we choose the natural norms and scalar products for gradients according to the recipes of Riemannian geometry and adapt the alternative proof of (6.1) indicated above.
In Chapter 4 we obtained a lower bound for the essential spectrum of a Schrodinger operator on Rn with a lower semi-bounded potential. We
Spectral Asymptotics in the Semi-Classical Limit
52
shall now reexamine such operators using Lithner-Agmon estimates. Let V E C°(R'; R) be bounded from below: infR. V > E° for some E° E R. Let P denote the Friedrichs extension of -A+V (x) for Co (Rn) ; P = -O+V, so that the domain of P is contained in the closure of Co (Rn) for the quadratic form
q(u) = f (IIVu112 + (V - E°)lul2)dx.
Let X E Co (Rn) be equal to 1 near 0 and let 4) E Lip (Rn) be constant for large Ixl. If u E D(P), then (6.1) is valid if we replace u by uR(x) = X(R)u(x).
Moreover, when R -> oo, we have with convergence in L2: uR
u,
IV - 1,74,1211/2 UR -,_, I V - I VY I2I1/2u, PUR -> Pu, so, passing to the limit, we
obtain (6.1) also for u E D(P). Here we get the last convergence and the fact that uR E D(P) by considering the commutator [P, X(Hence we also get (6.2) for u E D(P), when 4) is constant outside a compact set. R)].
Now assume that V (x) > 2a > 0 for all x E R'. Then inf a(P) > 2a. Let v E L2 have compact support in B(0, Ro). Let u E D(P) be the solution of Pu = v. Put 4)R(X) = \1Ro
R(x)(R - R°).
Then (V R)2 < a a.e., so V - (V R)2 > a a.e. We choose F_ = 0, F+(x) =
V(x) - (V4R(x))2 > IIV(e'Ru)II2 +
and get from (6.2)
2
Ile'DRull2 < I Ile4)Rvll2.
Combining this with
IIe'RVuII <- IIV(e'Ru)II+v IIeIRull, we get 2IIeIRVII.
Put
Proposition'6.3. Assume V > 2a > 0 and let v E L2 have support in lxl < Ro. If u E D(P) is the unique solution of Pu = v, then
v' Ile'Vull +alle"ull C 21le'vll, where 1(x) = \1{Ixl>RO(x)(lxI - Ro).
Using this result, it is easy to verify more directly that
-
inf o ess (P) > lim inf V (x),
Ixl--
when V is continuous and bounded from below.
6. Tunnel effect and interaction matrix
53
b. The Lithner-Agmon metric and decay of eigenfunctions. Let M denote either a compact connected Riemannian manifold of dimension
n, or Rn. Let V E C°°(M; R) and assume in the second case that Eo :=
liminfV(x) > -oo. Let E E R, with E < Eo in the second case, and introduce the Lithner-Agmon (LA) metric: (V (x) - E)+dx2,
(6.3)
where a+ = max(a, 0) and dx2 denotes the the Riemannian metric on M. For a piecewise Cl curve -y, we can define its length IyI in the LA-metric, and if x, y E M, we define the LA distance d(x, y) between x and y as inf Iyj over
all piecewise C' curves -y joining y to x. This distance may be degenerate in the sense that we may have d(x, y) = 0 when x y. We have, however, standard properties such as: d(x, y) = d(y, x), d(x, z) < d(x, y) + d(y, z),
(6.4)
I d(x, z) - d(x, y) I < d(y, z).
(6.5)
Moreover, y' -> d(x, y) is a locally Lipschitz function and I d(x, z) - d(x, y)I < ((V (y) - E)+ + o(1))1/2IIz - yII y,
(6.6)
when z --> y and where II . IIy denotes the natural norm induced from the natural norm on the tangent space TyM via the standard identification of a neighborhood of y in M and a neighborhood of 0 in T.M. It follows that for every x, (V (y) - E)+ 2, for a.a. y, (6.7) IIV yd(x, y) II and for all y: II V.d(x, y) II < (V (x) - E)+2, for a.a. x.
(6.8)
If U C M, we put d(x, U)
ynf d(x, y).
Again, we have I d(x, U) - d(y, U) I < d(x, y), so that II V,,d(x, U) II < (V (x) -
E)+ 2 a.e. on M. Let
E
liminfl11
,
= 0 for simplicity, and assume consequently that V(x) > 0, in the case when M = R. Let U = {x E M;V(x) <
0}. Then U is compact. Let u = u(x; h) be a normalized eigenfunction of the Schrodinger operator P = -h20 + V (x), with eigenvalue A(h) which tends to zero, when h tends to zero. Here, we assume that h varies in some subset of 10, hol, ho > 0 which has 0 as an accumulation point.
Spectral Asymptotics in the Semi-Classical Limit
54
Proposition 6.4. If fi(x) = d(U, x), then for every E > 0, we have for h > 0 small enough depending on E:
CeElh, IIV(e41/hu)II
whenM=R",
+ IIe4)/hull < CeE/h when M is compact.
Proof. We give the proof in the case when M is a compact manifold, and then we simply mention the additional argument that is needed in order to treat the RTh case. Apply Proposition 6.2 with 1(x) = (1 and with V replaced by V - A(h). Then
V- (h)- (VD)2=V- -(1-E)211V4112>V-A-(1-E)2V In the complement of U, we get
V - A(h) - (VD)2 > (1 - (1 - E)2)V - A(h) = (2e - E2)V - A. Let U. = {x E M; V (x) < E}. Then outside UE:
V - A - (V4))2 > 2E2 - E3 - A(h) > E2 - A(h) >
2 E2,
assuming that e < 1 and that h is sufficiently small depending on E. Now take Ft such that F+ = V - A(h) - (V4))2 and F_ = 0 outside UE. Then (6.2) gives CEIIe(1-E)i/huIlu,
which gives with a new constant CE: IIe(1_E)i/hhVuII2 + IIe(1-E)i;/huII2 < CEII e(1-E) '/hull2
Let K denote the maximum of 6(E)->0when E--k0. Then
on M and let S(e) = supu, ), so that
IIe"'/hhVull2 + IIeI/huII2 < Ce2EK/h+26(E)/hlluli2 < Ce2(EK+6(E))1h.
This implies the desired estimate with a new C and with a new e, which can be chosen arbitrarily small. When M = R' we take 4)R(x) = (1-E)XR(1(x)), where 0 < R is a constant and XR(t) = t1[o,Rl(t)+R1{t>R}(t). Since 1R is a
6. Tunnel effect and interaction matrix
55
constant for large x, we can apply Proposition 6.2. Using the same argument as above we get: h2IJV(e4'R1hu)II2
+
((2e - e2)V(x) -
J (2E-E2) V
For A < a(e) (a(E) -> 0 when E -> 0), the right hand side can be estimated by C(E)eEt h. Letting R -> oo, and using the fact that (2E - E2)V (x) - A > C(E) > 0 when fi(x) > E, we get the desired estimate. # From this one can obtain pointwise estimates on the eigenfunctions, using classical a priori estimates for the Laplace operator. We will return to this question and the problem of finding more refined estimates later on.
Remark. Proposition 6.4 remains valid if M is a compact Riemannian C2 manifold with boundary, and P the corresponding Dirichlet realization of
-h2A+V. c. The interaction matrix.
We give the discussion in the case when M is a compact Riemannian manifold. The case M = Rn can be treated in the same way with only minor changes under the assumption that lim inf
V (x) > 0.
Assume that
{xEM;V(x) <0}=U1U...UUN, where Uj are compact disjoint sets (the `potential wells'). Let d be the LA distance at energy zero. Assume that diamd(Uj) = 0, b'j,
(6.9)
where diam d denotes the diameter with respect to d. For a small value of i > 0, we introduce B(Uj, rt) = {x E M; d(x, Uj) < 711. We shall assume without loss of generality that the boundary of B(U;, ij) is smooth, since otherwise it is easy to make small changes in the arguments below. Consider the operator PM,, defined to be the Dirichlet realization of P on Mi = M \ Uk#j B(Uk,77).
(6.10)
(Equivalently PM; is the Friedrichs extension of P from Co (int (Mj)). It is easy to show that the domain of PMT is equal to H2 (Mj) fl Ho (Mj), where
56
Spectral Asymptotics in the Semi-Classical Limit
Hk (Mj) denotes the classical Sobolev space of order k and Ho is the closure
of Co (int(Mj)) in H1(Mj). It is also well-known that PM, has a purely discrete spectrum. Let NP,,,, (A) denote the number of eigenvalues < A. If V > -Co, then the mini-max principle shows that NpMj (A) < N_h2o_co(A) = #{ eigenvalues of - h2OM,, < A + Co}
= #{eigenvalues of - OMj, < h-2(A+Co)}. Here OM; denotes the Dirichlet realization of the Laplace operator on Mi. Now it is a classical result that N_/M, (A) < C(1 + A)n/2, so we get NpMi (A) < C(A + Co)n/2h-n.
We shall use only that the number of eigenvalues of PMT in a fixed interval grows at most as a polynomial in 1/h.
Let I(h) = [a(h),/3(h)] be an interval and let a(h) > 0 be a function defined for h E J C]0,1] with 0 E J. We assume I(h) -> {0}, h -> 0,
e_E/h
a(h) >
for every e > 0,
(6.11)
(6.12)
CE
Vj, PMT has no eigenvalues in ]a(h) - 2a(h), a(h)[U]/3(h), 13(h) + 2a(h)[. (6.13)
The purpose is to study the spectrum of P near I (h) in terms of the spectral information that we may have about the PM3 . Let us first consider the resolvent of P.
Definition. Let A = Ah, h E J be a family of operators L2(M) -* H1(M) and let f E C°(M x M; R). We say that the kernel A(x, y) of A (using the same letter for an operator and its distribution kernel) is (5(e- f(x,y)/h) if for all xo, yo E M and e > 0, there exist neighborhoods V, U C M of xo and yo and a constant C, such that IIAuIIH1(v) <
Ce-(f(xoyo)-,)IhIInhIL2(u),
for all u E L2 (M), with support in U. By (5(e-fl 1h +. .. + e-fk/h), we mean 6(e- min(f;)/h) when fl, ... , fk are finitely many functions with the same properties as f. Here we make two observations.
6. Tunnel effect and interaction matrix
(1) If A(x, y) _ 6(e-fl'), B(x, y) =
57
O(e-s/h), then Ao B(x, y) =
(5(e-kph),
where k(x,y) = minZEM (f (x, z) + g(z, y).
(2) Let A(x,y) = (5(e-f (x,y)/h) and let 0,0 E C°(M) satisfy: ¢(x) > -f (x, y) + 0(y), for all x, y E M. Then IIe-0fhAull < CeE/hIle-1,/hujj, for all u E L2(M).
As an example, let V > 0 be a positive potential on M. Then (P - z)-1 exists for z E K(h) when h > 0 is small enough, if K(h) is some bounded subset of C which tends to {O} when h -* 0, and we have uniformly w.r.t. z
(P -
z)-1(x, y) = (7(e-d(x,y)Ih).
This still holds when M is a compact manifold with boundary.
To see this, let v have its support in a small neighborhood of yo. It then suffices to estimate the solution of u of the equation (P - z)u = v by means of LA estimates.
Next we consider the case of one potential well, U C M, which is assumed to have diameter 0 for the LA distance (at energy 0). Let K(h) C C tend to {0}, when h -> 0 and assume that for every e > 0,
dist (K(h), a(P)) > CE e-'1h
Proposition 6.5. Under the above assumptions, we have
(P - z)-1(x, y) = (7(e-d(-,y)/h), uniformly, w.r.t. z E K(h).
Proof. Let M° = M \ B(U, r7), and let 9 E Co (B (U, 277)) be equal to 1 near B(U, 77). Let 9 E Co (B (U, 377)) be equal to 1 near B(U, 277). Using the resolvent identity, we get the following representation:
(PM - z)-1 = (1 - 9)(PM0
- z)-1(1 - 9) + (P -
z)-'
+(P - z)-19[P, 9](PMo - z)-1(1 - 9).
(6.14)
From the decay estimates and the fact that II(P-z)-1IIc(L2,L2) < CEeEth for every e > 0, we see that
((P - z)
(x, y) _
(e
1(-d(x,U)+3y)).
58
Spectral Asymptotics in the Semi-Classical Limit
then (6.14) implies that O(e"(-d(x,y)+6n)).
(P - z)-1(x, y) =
Here we can choose 77 > 0 arbitrarily small, and the proposition follows.
We now return to the general situation with N potential wells of diameter 0 for the LA distance. We assume that K(h) --* {0} when h --i 0 and that for arbitrarily small r) > 0 and for all e > 0, there are constants CE = CE,,, such
that dist (K(h), o,(PM; )) >
e-E/ CE
when h is small enough depending on rl.
Proposition 6.6. Under the above assumptions and for h small enough, we
have K(h) fl o (P) = 0 and (P - z)-1(x, y) = z E K(h).
O(e-d(x,y)/h), uniformly for
Proof. Let Bj E Co (B(Uj, 2r7)), Bj = 1 near B(Uj, r7) and put Xj = 1 - Ek j Ok. Also choose Xj E Co (int (Mj)), = 1 near B(Uj, Co), Co >> r7 independent of r7, s.t. T Xj = 1. In order to construct the resolvent of P, we put N
Ro(z)
Xj(PMj - z) 1zj
O(e-d(x,y)/h) and that Xj is equal to 1 near the Notice that Ro(z)(x,y) = support of j (when 77 is small enough). We have
(P - z)Ro = I - K, K=
E[P,ek](PM; -z)-1Xj.
k¢j
Since (PM, - z)-1 =
O(e-d(x,y)/h)
IIKIIG(L2,L2) <
it is clear that CEe-(Co-E)/h
He > 0,
where C o = minj#k d(supp Xj, B(Uk, 277)) > 0. Hence (1 - K)
exists for
h sufficiently small, and then (P - z)-1 exists and is given by (P - z)-1 = Ro(z)(I - K)-1. Let D be the diameter of M for the LA-distance. Then for No sufficiently large, we have No
(I - K)-1 = E Kj + O(e-Dlh), in C(L2, L2). 0
6. Tunnel effect and interaction matrix
59
Denoting the remainder in the last equation by L, it is clear that (Ro o L)(x,y) =
(5(e-d(x,y)/h).
If we relax the definition of (0 by replacing Hl by L2, then K(x, y) _ (5(e-d(x°y)/h). Hence Kj (x, y) = (5(e-d(x,y)/h), and consequently Ro o Kj (x, y) = (5(e-d(x,y)/h)
where we again use the original definition of 0. This gives the Proposition.
Let µj,1, ... , µj,,,,,j be the eigenvalues of Pmj in I(h) and let cbj,,,,,j E L2(Mj) be a corresponding O.N. system. It will be convenient to
write a = (j, k), 1 < k < mj, and put j(a) = j. (These quantities also depend on 77, but the discussion below will also imply that this dependence is very weak, so we shall not insist too heavily on this dependence.)
Let 9j E Co (B(Uj, 2rl)) be equal to 1 near B(Uj,7)) and put Xj = 1 F-kOj Ok,
Y)a = Xj(«)0
Then for every e > 0:
=
uniformly on Mj,
and similarly for all derivatives of 0a,
(Actually, we only need
O(eE/h) for all e > 0, and similarly for 0«,
Then
(P - µa)'« = r«, where suppra C Uk$j(a)B(Uk,27J), ra =
0(e(E-d(U,(a),x))/h) for all
c > 0,
and similarly for all derivatives.
Let F C L2(M) be the space associated with o,(P) (1 (I(h) + [- 2 and let E C L2(M) be the space spanned by the
a2h)1) 1
Proposition 6.7. dim E = dim F. Before the proof we make some general considerations. If El, E2 are closed subspaces of a Hilbert space 7-l, we introduce the non-symmetric distance d (E1,E2) =
sup xEE1, Ilxll=l
d(x,E2) = II(1 -7rE2)IEIII
= 1I7rE1 - 71Ez1rE1 11 =
II7fE1- 7tE17rE2I1,
Spectral Asymptotics in the Semi-Classical Limit
60
where d denotes the natural distance in 7-l and 7rE, is the orthogonal projection onto Ej. Notice that we have the `oriented' triangle inequality: d (El, Es) < d (El, E2) + d (E2, E3).
Lemma 6.8. If d (El, E2) < 1, then (a) IrE2
E,
: E1 --4E2 is injective,
I
and (b) _7rE1 I E2
:
E2 --* El is surjective.
In particular dim E1 < dim E2.
Proof. Let us first prove (a). If 7rE21E, is not injective, then there exists x1 E E1 with lix1ll = 1, such that _7rE2x1 = 0. Then d(xl, E2) = lixl -
I
7rE2x1 11 = 1lxi 11 = 1, so d (El, E2) = 1, contrary to the assumption in the lemma.
To show (b) it is enough to show the surjectivity of 7rE1 IrE2 I E1
l E1 - (1E1 - 7rE1 IrE2) I E,
= I - K.
But here the assumptions of the lemma imply that I1K1I < 1.
Put Ai,k = IrE; I Ek
Ek -' Ej, for j
k E {1, 2}. Notice that A2,1 and A1,2
are adjoints to each other.
Lemma 6.9. If d (El, E2) and d (E2i El) are both < 1, then they are equal.
Proof. We have d (El, E2)2 = sup
1 - 11A2,1x1II2,
II=111=1
so
inf 11111=1
11A2,1x1
112
= 1 - d (El, E2) 2,
which implies that A2,1 is injective with a bounded left inverse. Similarly, A1,2 is injective with a bounded left inverse. Since these two operators are
6. Tunnel effect and interaction matrix
61
adjoints to each other, it follows that they are bijective and that their inverses have the same norm. The last identity now shows that
IIA2,iII = (1 - d (El,
E2)2)-1/2,
and since the norm to the right is equal to the norm of the inverse of A1,2, which has an analogous expression, we conclude that d (El, E2) _
#
d (E2, E1)
Proof of Proposition 6.7. If j(a)
j(/3), then
(O«I`)'Q)I < Ce(E-d(Uj (.)+Uj (p)))lh
for every e > 0. If j (a) = j(3), then +0E(e*(E-2
(W«IV5,0) = 6«,(3 O(e-
= 6« Q +
77
mink#i(.) d(U1(.)+Uk)+4r7))
mink0j(.) d(U3(a)+Uk)).
Here we recall that most of our quantities depend on the small parameter OE,,)(e(+E(7l)-f)/h), where E(q) - 0 when rl > 0, and write (O(e-flh) for rl -> 0. Let D' be the N x N matrix with diagonal elements 0 and with
the off-diagonal element equal to resumed by:
e-d(Ui,vk)/h. The above inequalities can be
= I +O(D' +D12(()«I00))
where the estimates for matrices are to be understood elementwise, uniformly
with respect to the row and column indices, so that O((D' +
6«,0 _
D2)f(.),j(,3)).
Let So = minf,6k d(Uj, Uk). Then IF := (( «I1GQ)) = I + (5(e-S.1 h).
Here we get the corresponding estimate in the ordinary matrix norm, since the size of the matrix is Q(h-n). It follows that p-1/2 = I + (O(e-S°/h). Let i/' denote the row vector of all the i«. Then we get an O.N. basis in E:
We introduce v, = 1rF'w«, where 7rF =
1
27ri
f(z - P)-ldz,
Spectral Asymptotics in the Semi-Classical Limit
62
and ry is the simple positively oriented loop around 1(h) given by the set of complex points in C at a distance a(h) from 1(h). Since (P-µj(a))Oa = ra, we have
- ra,
(z - P)2ba = (z - I-tj(a))Wa
so that
(z - P)-1Oa = (z - µj(a))-''b. + (z - P)-1(z - µj(a))-1ra" and
va = 00+
1
27fi 7
(z -
P)-1(z
-
pj(a))-1radz.
From the estimateson ra and on the resolvent, we get va - Wa = O(e-6j(.)(x)/h) = (5(e-SO /h) in the L2 and in the H1 sense, where 6j (x) = kn d(Uj, Uk) + d(Uk, x).
Using also that dim E = O(h-'), we deduce that d (E, F) = (5(e-S0/h). It remains to estimate d (F, E). Let u E D(P) be a normalized eigenfunction: (P -.)u = 0, A E I(h) + [-2, 2]. We return to the formula for Ro(z) in the proof of Proposition 6.6. We have (PM, - z)(Xju) = (A - z)Xju + (7(e-Co/h),
and it follows as above that 1
27ri
(z - PM; )-1dzXju = Xju + O(e-c01 h). y
Hence if we put 7ro = - 2 i f Ro (z)dz, which has its range in E, we get 7rou = u +
O(e-Co/h).
Again, using that dim F = O(h-'), we deduce that d (F, E) = O(e-Colh) Hence,
d(E,F)= d(F,E)<1, which concludes the proof of Proposition 6.7, in view of the lemmas 6.8, 6.9.
6. Tunnel effect and interaction matrix
63
The next problem is to determine the matrix of PIF for some suitable basis. We shall orthonormalize the basis va. Recall that
va - a =
(6.15)
(5(e-ds(.)(x)/h).
Since va - 0a and va - zip are orthogonal to F, we have (vaIva) - (0. 100) = (vaI vo - a) + (va - 4)aI ")Q)
-(va - V)al vo - 0a)
Using (6.15), we get (7(e-7-°`,o1h)'
(vaIv/j) - (4akb ) = where
Ta a =
min
d(Uj(a), Uj) + d(U;, Uk) + d(Uk, Uj(p)).
J549(a),k549(,Q)
This can also be written in matrix form: ((v.Ivo)) - ((Y'.100)) = (5 (D'(1 + D') D') = O(D'2 + Di3).
Let us introduce T = (ta,p), where ta,,3 = 0 when j(a) = j(f3), and ta,Q =
when j(a)
j(3). Then
((0al V)O)) = I + T + (O(Di2 + D13), so
((va I v,3)) = I+ T + (5(D,2 + V'3).
6.16)
We do the same calculations for (PvaIvo). Since we control the exponential decrease of va - 4ba in H1, we obtain similarly ((PV)a100))
((PvaI va)) =
+
(5 (D12
+ D13).
(6.17)
Here (PbaIV)Q) =
=2
(P4baI @) +
2(Y'aIP
o)
(l-ta + µa) (0a I00) + 2 ( ( r .1 0 , 3 ) + (0a l ro))
Using the fact that ((V)aIV),3)) = I + T + (0(D12)' we can write this in matrix form,
((PO. 10Q)) = diag (µa) + T diag (µa) + 1 diag (µa) T 1
+2((ral0a) + (Y'aIro)) + (5(Di2 +V'3).
(6.18)
Spectral Asymptotics in the Semi-Classical Limit
64
Here
(0a1 ro) =
-h2J(2Vxj() -VOQ+V)a(OXj(O))O,Q)dx,
and
- h2 I V5«(Oxj(a))OAdx l
= h2
J
Xj(«)VO. VXj(O)OOdx + h2
J
Xj(a)O«VXj(Q) . V0pdx + (5(D 12),
so
(4'aIrR) = h2
f
Xj(a)(OOVXj(/3) . D0a -OaVXj(,Q)
.VOO)dx+O(Di2).
(6.19)
(Here we made the computations as if M were equal to R' at least locally,
but it is easy to see that (6.19) holds in the more general Riemannian case.) If j(a) = j(3), this quantity is O(V'2). Let wa,$ be 0 when a = 3 and wa,Q = h2 f Xj(a)(0p0Oa - OaVO,3) Vxj(p)dx otherwise. We also introduce the symmetrized quantity, wa,/i = (w,,,,3 + wo,a) and let W be the corresponding matrix. Then by (6.17-19): a (Pval va) = diag (µa) + 2Tdiag (µa) + 2diag (lca)T
+(wa,a) +O(Di2 +V'3).
(6.20)
Put V = ((valv,3)). Then if v denotes the row vector (va), VV-1/2 is an O.N. basis in F. The matrix of PIF with respect to this basis is V-1/2
((Pva Iv a))V
-1/2
(6.21)
From (6.16) we get
V-1/2 = I -
T + (5(V'2 + V13),
(6.22)
and combining this with (6.20), we see that the matrix (6.21) becomes
(I - 2T + (5(E)12 + D13)) x (diag (µa) + 2Tdiag (µa) + 2diag (/!.,)T + W + (5(D,2 + v'3)) 21
x (I - T + O(D12 + Di3)).
6. Tunnel effect and interaction matrix
65
Now W, T = O(D'), so the matrix (6.21) is equal to (5(D12
diag (µa) + f V-
+ D13).
We have proved:
Theorem 6.10. Under the assumptions above, the matrix of PIF w.r.t. the O.N. basis VV-1/2 is of the form diag (µa) + f V_
O(Di2 + D13).
We now discuss the problem of `computing' the interaction coefficients appearing in the matrix W. Assume that i(a) j(3) and that there is a E]0, d(Uj(a), UF(O)) - 271[ such that µa - µ6 = O(e-(°-E)/h) for every fixed e > 0. By integration by parts, we have
wa,d = (µa -,U,3)
J
Wa 4)Odx + w,0,a,
so
wa,6 - wO,a =
O(e-(d(Uj(.),Uj(a))+°-e)/h)
for every e > 0. Let E be the `ellipsoid' defined by d(Uj(a), x) + d(Uj(O), x) < d(Uj(a), UF(O)) + a.
Assume that E does not intersect any of the B(Uk, 2i) with k Let St C M be open with smooth boundary such that SZ D B(U,(a), 277),
M\fD
{j (a), j()3)}.
B(U,(O), 27l).
Let r = 5 fl E, and let n denote the unit normal on r oriented into Q. Using Green's formula, one can then prove that
wa"° _ h2
f
43 - 4 On
O
O4'_)dS + O(eOn
U1(R))+a
(6.23)
In view of the weighted estimates we have for 0a, 00, we expect the main contribution to the integral in (6.23) to come from the intersection between r and the union of minimal LA-geodesics joining Uj(a) and UF(O). We refer to Remark 2.13 in Helffer-Sjostrand [HeSj2] for further details. Under additional assumptions it is possible to determine the order of magnitude of the interaction coefficients wa,6 and even to give asymptotic expansions. Let us describe one such situation. Assume that all the wells are
Spectral Asymptotics in the Semi-Classical Limit
66
reduced to single points: U, = xj, and that V has a non-degenerate minimum in each of these points. Let us further assume that the localized harmonic oscillators for each of these points have the same lowest eigenvalue E0, and choose the interval I(h) to be of the form [h(Eo - co), h(Eo + co)] for some fixed sufficiently small co. Then we have precisely one eigenvalue µa = µj(a) for each j, and 1 a = h(Eo + 0(h)), according to the discussion at the end of Chapter 4. In the appendix of the present chapter, we show (also for some excited eigenvalues) that the eigenfunction 0a is well approximated by the WKB-eigenfunction constructed in Chapter 3, provided that we restrict the attention to a star-shaped subdomain of Ma, (where we write Ma = Mj(a)). Since we assumed Eo to be the ground state of the local harmonic oscillator for each well, we also know that the amplitude for each eigenfunction is an elliptic symbol (i.e. with a non-vanishing leading term in its asymptotic expansion) of order n/4. For given a, ,3 as in (6.23), let us assume that there is no other Ury (so that y {a, i3}) with d(Ua, Up) = d(Ua, U-1) + d(Uy, Up). If we choose ri (in the definition of the Mj) sufficiently small, then the minimal geodesics from Ua to Up are contained in the intersection of Ma and MO, except in B(Ua, rt) U B(UU, 77). In particular, the intersection of r with the union of minimal geodesics from Ua to Up is contained in the region where both c a and q'p are well-approximated by the corresponding WKB-functions constructed in Chapter 3. In the appendix, we established this approxima-
tion only in L2 and H' norms, but further easy arguments permit us to extend this approximation to the uniform norms and for all derivatives. Alternatively, we can reexamine how (6.23) is obtained by means of the Green formula and use only the approximation result of the appendix. Both methods permit us to replace (pa, 0,3 by their WKB-approximations in (6.23). After those substitutions, we see that the integrand in (6.23) is of the order h-1-n/2 e-(d(v,,x)+d(x,U,))/h and we get an estimate, of magnitude hl/2e-d(U..,p)/h <
-wa,p <
Chl-n/2e-d(U..,p)/h.
(6.24)
If we do not want to assume the exponential smallness of iGa - µp, leading
to (6.23), we can still obtain (6.24) directly from the definition of wa,p after (6.19). Now add the assumption that there are only finitely many LAgeodesics y joining Ua to Up and that they are all non-degenerate in the following sense. Let H be a hypersurface intersecting y transversally at some point xo, strictly between Ua and Up, then for x E H close to xo we have d(Ua, x) + d(UU, x) - d(Ua, Up) - IX - xo 12.
(6.25)
It follows from general facts in Riemannian geometry that this definition is independent of the choice of the hypersurface H. Under this additional assumption, we can apply the method of stationary phase (see Chapter 5) to the integral in (6.23) (after replacing 4a, qp by their WKB-approximations), and get -wa,,o =
h'/2a(h)e-d(U.,UP)/h
a(h) ,., ao + hal + ... , ao > 0.
(6.26)
6. Tunnel effect and interaction matrix
67
The leading coefficient ao can be computed, and we refer to Wilkinson [Wi], Dobrokotov-Kolokol'tsov-Maslov [DoKoMa] for further details. In the following, we shall discuss some examples where some symmetry group implies that the eigenvalues associated with the different wells are all equal. We will assume that we are in the situation where we only have to consider one eigenvalue for each well, and that the interaction coefficients have the expected order of magnitude, as in the special situation discussed above.
Example 1. We consider the classical case of a symmetric double well potential. Let N = 2 and assume that there is an isometry t : M -+ M such that t(Ul) = U2, t(U2) = U1, V o t = V. Then PM, and PM2 have the same eigenvalues. Let p be an isolated eigenvalue of PM, such that ],a - 2a(h), p + 2a(h) [ contains no other eigenvalues, where a(h) > cE e-E/h, VE > 0. If So = d(U1, U2), then the matrix of PI F in a suitable O.N. basis is given by
l (A ww µ
J
+
2So/h).
As indicated above, we may assume that 1 e(So-E)/h < Iwj <
/
CEe(-So+E)/h'
for every c > 0. In order to handle the remainder appearing in the interaction matrices, we shall use the following simple consequence of the mini-max principle:
Lemma 6.11. Let A, B be two Hermitian N x N matrices with eigenvalues
Al < A2 < ... < AN and µl < µ2 < ... < µN respectively. Then IA - µ;I < IIA - BII, for every j. Applying this, we conclude that the eigenvalues of P in ],a - a, it + a[ are of the form Af = Ft ± w + CO(e-2So1h)
In particular, we observe a splitting: A+ - A_ = 2w + 6(e-2Solh)
Example 2. Let V be a N -periodic potential on S' = M with N consecutive potential wells: U1,. .. , UN. If t : M --+ M denotes rotation by N , then t(Uj) = Uj+1 (and we put UN+1 = U1, t*(V) = V).
More generally, we assume that we have N wells and an isometry t M - M leaving M invariant and permuting U1,. .. , UN cyclically. In the
:
68
Spectral Asymptotics in the Semi-Classical Limit
general case, we also assume that So = d(Ul, U2) is strictly less than Sl = minj_k0_1,o,1 d(U Uk), where we consider that j, k E Z/NZ. Since Si < 2So, we get the matrix for PI F:
+2
0
w
0
...
0
w
0
... W0 + 6(e-Sllh).
w
0
...
w
0
Let us compute the eigenvalues of 0
1
0
...
1
1
1
0
1
...
0
2
0
1
0
...
0
1
0
1
0
.
.
.
which can be viewed as 2(r_1+r+1) on 22(Z/NZ), where r±lu(k) = Try an eigenvector of the form u(k) = ea8k. The obvious periodicity condition
implies that ON = 21rj, so j E Z, 0 = 27r!, for j = 0,1,...,N-1. We get the N linearly independent eigenvectors uj(k) = ei27rjk/N with the corresponding eigenvalues, cos(27rj/N), j = 0,1, ... , N - 1.
The conclusion is that P has N eigenvalues in [µ - a, µ + a], and they are of the form: µ + w cos(2.7rj/N) + (5(e-S'/h) for j = 0,1, ... , N - 1. We see that the value µ + w + O(e-Sl/h) (for j = 0) and when N is even, the value µ - w + O(e-Sl/h) (for j = N/2), are simple. The other eigenvalues seem to be double and we may ask whether this is really the case. Usually, multiple eigenvalues are due to invariance of the operator under group actions and the question ending the last example can be answered by finding such an invariance. Let us consider such invariances more generally, and let us restrict our attention to the actions of finite groups. Let G be a finite group of isometries M --+ M leaving V invariant. If g E G, u E L2(M), we put g*u = u(g-lx). Then g* is unitary and commutes with P. If F is the spectral subspace defined before, then g* : F -+ F, and [g*, 7rF] = 0 for all
gEG. Lemma 6.12. We can choose the 0j above such that g*0j = Ba9(j), where o is the permutation o f {1, ... , N} given by g(U,) = Uo9(j).
The functions 0; are constructed by a procedure of averaging over the elements of the group, and we leave the simple details to the reader. Now
6. Tunnel effect and interaction matrix
69
take Off, Xj, Oj = Xj0j, vi = 7rFZb as before. Then g*oj = ± 9(j), and we assume for simplicity that only the plus sign appears here. (This will be the case if we work with the lowest eigenvalue for each PM,.) Then g*vj = vQ9(j).
Recall the construction of the O.N. basis e1,. .. , eN in F: We look for a positive selfadjoint matrix (sj,k), such that vk = si,kej, with e1, ... , eN orthonormal. Then (v7I vk) ' (T, S.7,Uev I
Sk,µeµ),
v
so ((v3Ivk)) = S o
eµ)) o S. Thus
S2 = ((vjlvk)),
(6.27)
and S is the unique positive selfadjoint matrix satisfying (6.27). (Uniqueness follows if we consider the spectral decomposition of S.) To sum up, there is a unique positive selfadjoint matrix S (now viewed as an operator F -* F) such that vk = Sek, with e1,.. . , eN orthonormal. Then Se-9(k) = v°e(k) = g*vk = g*Sek = (g*Sg* 1)g*ek,
and the uniqueness of S implies that S = g*Sg* 1 and ev9(k) = 9*ek.
In other words, the action of G on F is induced by the action of the corresponding well permutation group on the basis vectors e1, ... , eN. In practice, we combine this with the fact that every eigenspace of PI F is invariant under the G action.
Back to Example 1. Let the full matrix of PIF with respect to e1, e2 be
M=
(ab
b
I
, and use the fact that this matrix must commute with the
matrix induced by the permutation of e1 and e2;
I
0
1
1
0
I
.
It follows that
a = c and the eigenvalues of M are then a ± b.
Back to Example 2. Let M be the matrix of PIF with respect to the basis e1, ... , eN. We know that M commutes with the matrix of cyclic permutations:
7r=
0 0 0 0
\1
1
0
0
1
0 0
0
0
1
0
0
0
... ... ... ...
0
0
0
... 0/
0 0 0 0
70
Spectral Asymptotics in the Semi-Classical Limit
The eigenvalues of 7r are simple and of the form A = e27rij/N, j = 0,1, ... 1, and the corresponding eigenvectors are of the form
, N-
1
I
A
I
(6.28)
Because of the commutativity, the corresponding one-dimensional eigenspaces are also eigenspaces of M, or in other words, the eigenvectors of 7r are also eigenvectors of M, so the earlier computation of the eigenvectors remains
valid when the remainder terms are taken into account. The eigenvectors (6.28) of M are non-proportional to real vectors, except for j = 0 and when N is even for j = N/2, and since M is a real matrix, this implies that the eigenvalues of M associated with the cases j 54 0, N/2 are at least double. The earlier computation then also shows that they are exactly double when h is small enough. Example 3. Assume that we have N potential wells and a symmetry group G with the property that for all (i, j), (i, j) in {1, ... N}2 with i 54 j, i
there exists a g E G such that (og(i),Qg(j)) is equal to (i, j) or (j,i). Let M = (mij) be the real symmetric matrix of PIF. Then mij = (eilMej) = (g*eijMg*ej) = ... = Also assume_ that for every (i, i) E 11,...,N 12, there exists a g E G such that Qg(i) = Z. Then we obtain mi,i = mZ i. We conclude that
M=pI+
O
w w w ...
W
w
0
w1
ww w
w
w
w w ... 0
w
w
0
... ...
W
W
w w w ... 0!
The eigenvalues of the rank 1 matrix with the entry 1 everywhere are: N with multiplicity 1 and 0 with multiplicity N - 1. It follows that the eigenvalues of M (assuming that w 0) are p - w with multiplicity N -1 and p+ (N -1)w with multiplicity 1. As a special case of this example, we may think of the case when n = 3 and we have four wells at the summits of a regular tetrahedron. In more general situations, it is useful to use representation theory for groups, and in particular Shur's lemma. We refer to Helffer-Sjostrand [HeSjl-2] and Serre [Se] for further details.
6. Tunnel effect and interaction matrix
71
Schrodinger operators with potential wells have turned out to be useful in topology, and questions concerning the multiplicity of eigenvalues for the Laplacian in Riemannian geometry, and there is now a rather considerable amount of articles on these subjects. See Bismut [Bis], Helffer-Sjostrand [HeSj3], Witten [W], Kappeler [Kap], Colin de Verdiere [CdV1-2]. See also [Si2].
Appendix In this appendix, we shall show that the exact eigenfunctions for the one well problem are well approximated in suitable star-shaped regions by the WKB-
expressions in Chapter 3. Let M be a bounded closed connected subset of Rn with smooth boundary. Everything works with only minor modifications in the case when M is a smooth compact Riemannian manifold, possibly with boundary, or when M = R'. Let xo E int M and let 0 < V E C°°(M) vanish at xo precisely and have a non-degenerate minimum there, so that V"(xo) > 0. We shall study P = -h2A + V defined as before by means of the Friedrichs extension.
Let O(x) be the phase function constructed in Chapter 3 in some open neighborhood of xo, and let d(x) = d(x, xo) denote the LA-distance from xo to x. Recall that O(x) is a solution of the eikonal equation q(x, O') = 0, where q(x, f) = 2 - V (X).
Proposition A.1. In a sufficiently small neighborhood of x0, we have q5(x) = d(x).
Proof. Let y : [0, 1] -> M be a a Cl curve. Then
I
f 1 V0('y(t)) .7 (t)dtI < 0
f
i V('Y(t))I'Y'(t)Idt = I'yI>
0
(A.1)
where by definition the last expression is the length of -y for the LA-metric. Taking the infimum over all Cl curves from x to y, we get, 10(x) - 0(y) I
d(x, y),
and in particular for y = xo: 0 < O(x) < d(x).
If y is an integral curve of 170 up to reparametrization, then we have equality in (A.1): 0(7(0))I = I'yI ? d(x,y).
Spectral Asymptotics in the Semi-Classical Limit
72
Hence 10(x) - ¢(y) I < d(x, y), with equality if there is 170-integral curve from
x to y. But for any x near 0, there is a unique 170 integral curve from x to 0, and, consequently, O(x) = d(x), asserted. #
Let xl E int M and let -y : ] - oo, 01 --+ int M be a minimal geodesic for the LA-metric from xo to xi. If we write d(y, xo) = d(y, B(xo, e)) + c, when B(xo, e), and where we let B(xo, r) denote the open ball of radius xo
y
and center r with respect to the LA-metric, and notice that B(xo, e) has smooth boundary, it follows from general facts about Riemannian manifolds, see Milnor [Mil, that there is a neighborhood 1 of xo U -y(] - oo, 0[), such that (1) O(x) = d(x, xo) E Coo (Q),
(2) fl is star-shaped in the sense of Chapter 3. (The backward integral curve of 170 through a given point x E SZ then gives (up to reparametrization) the unique LA geodesic joining x to xo.)
Let u E C°° (S2) be an approximate eigenfunction of P constructed as in
Chapter 3, with the associated eigenvalue hE(h), where E(h) - Eo + hEl + ..., and where Eo is a simple eigenvalue of the localized harmonic oscillator -A + 2 (V"(xo)x, x). As we saw in the end of Chapter 4, there is a corresponding exact eigenvalue of the form hE(h) + 0(h°°), and this is the only eigenvalue in some interval [hE(h) - eoh, hE(h) + eoh], for some eo > 0. Without loss of generality we may assume that the exact eigenvalue is precisely hE(h). Let v E C°°(M) be the corresponding exact normalized eigenfunction. If X E Co (f1) is equal to 1 near xo, we know that IIXu - vii = O(h°°).
Here we write g(h) = 0(h°°) if Ig(h)I < CNhN for every N > 0. We shall first establish a sharpened LA-estimate for v, where we shall not yet use the assumption that Eo is a simple eigenvalue of the localized harmonic oscillator.
Proposition A.2. There exist constants C, C, such that h2II(1 + d)-°edlhV 112 + hII(1 +
)-°ed/hvII2 < Ch,
where d = d(x, xo).
Proof. Put ,D (x) = d(x) - Ch log max(d(x) , C).
Then for d(x) < Ch, we have V - V, 2 = 0, and for d(x) > Ch: 174D(X) = (1 -
d(x)
)Vd(x), a.e.,
6. Tunnel effect and interaction matrix
73
in view of the following general observation. Let f E C' and let 0 be a Lipschitz function defined in suitable domains. Let 0E = 0 * XE, where XE is a
standard regularizer. Then the Lipschitz function f o 0 is the limit of f o 0E
in the sense of distributions when e - 0, and V(f o 0E) = (f' o (f' o (P) V O a.e. Consequently V (f o -0) = (f' o 0) V O a.e.
It follows that for d(x) > Ch: 2
V(x)-V(D2>(1-(1- dh)2)V(x)=V(x)(2dh-( dh-Chd(x) Now Glo -,
< Co, for some constant Co > 1, so
V(x) - (V(D)2 > Ch a.e. in the region d(x) > Ch. Here we can assume that C is as big as we like by choosing C sufficiently large. In the following we shall assume that we have chosen C > Eo. Choose F+, F_ as in Proposition 6.2 with V replaced by V (x) - hE(h),
F_(x) = 0 and F+(x) =
V - hE(h) -
for d(x) > Ch,
F+ + F_ - vrh-, for d(x) < Ch. Then hell
V(e1D/hv)ll2
+
Chll ell hull
C
B(O,Ch)
< Ch.
Here we notice that e-D/h
(1 + d)-Ced/h.
Moreover, IlhV(eI,/hv)ll
and ID4)I < V1/2 < Ch1/2
S Ile4'/hhVvIl +
V1/2
ll(V4)e1,1hvIl,
< Ch1/2(h)1/2, so we get Proposition A.2. #
An immediate consequence of Proposition A.2 is that there is a number No (depending on Eo) such that Ilea/ zVvll + Iled/hvlI =
We now come to the main result of this appendix, where we recall that u is the asymptotic eigenfunction, constructed in Chapter 3, and that Sl is a star-shaped neighborhood of a minimal LA-geodesic.
Spectral Asymptotics in the Semi-Classical Limit
74
Theorem A.3. For every compact set K C f and every N E N, we have IiedihV(u - v)II L2(K) + II ed'h(u - v)II L2(K) = 0(hN).
Proof. For a given K, let K be the union of all minimal LA-geodesics from K to x0. Choose X E CO '(Q), equal to 1 in a neighborhood of K. For some sufficiently small e > 0, put ,DE (x) = min (-D (x), FE (x)),
where
FE(x) _ (1 - e)yEsinfvX(d(O,y) +d(y,x)) Then:
(1) There exists h0> 0 such that for 0 < h < ho, 'E(x) is equal to 4)(x) in a neighborhood of k, and equal to (1 - e)d(x) on supp VX. (2) We have V - IVFE(x)I2 > (1- (1- e)2)V(x) > eV(x), so V(x) - hE(h) (V4),)2 is > co for d(x) > Ch and = 0(h) for d(x) < Ch. (3) Put w = u - v. Applying the weighted L2-estimates to Xw, we get: h2IIV
II L2(K) + hII
L2 (K)
IIe4` h(X(P - hE)w + [P,
C(h = 0(h°°).
The proof is complete.
X]w)112 + hII e"'`/hwII L2({d(x)
#
Notes Proposition 6.2 is a semi-classical version of results of Lithner [Li] and Agmon [Ag]. In the one dimensional case, sharp estimates can be obtained by using techniques of ordinary differential equations, see Gerard-Grigis [GeGr], Harrel [Hal], Kirsch-Simon [KiSil] and Nakamura [Na]. In higher dimensions
we mention in addition to the work of [HeSj1,2] on which this chapter is based, the papers of Simon [Sil,2], Kirsch-Simon [KiSi2], Martinez [M1,2]. The case of Schrodinger operators with magnetic field has been studied by B.Helffer-Sjostrand [HeSj8].
Results concerning the tunnel effect in periodic semi-classical situations are due to Harrel [Ha2], Simon [Si3], A.Outassourt [Ou], and the more general case of infinitely many potential wells has been treated by Carlsson [Car]. A broad survey of the tunnel effect and interaction matrices can be found in the book of Helffer [He].
7. h-pseudodifferential operators
Let V - R' be an n-dimensional vector space and let V' be its dual. The pseudo-differential operators (from now on pseudors for short) that we consider will be operators from S(V) to S'(V) (and most of the time with
better continuity properties), associated with a symbol a(x, ) defined on V x V' by means of a rule of quantization, depending on a fixed parameter t E [0, 1] and on a "Planck's constant" h E]0, 1]. Formally the operator associated with a is Au(x) = Oph,t(a)u(x) _ (2 h)n
ff
ei(x-y)-',qlha(tx
+ (1 - t)y, i7)u(y)dydr7. (7.1)
Here the case t = 1/2 corresponds to the Weyl quantization, which has the most pleasant general properties, and the the case t = 1 corresponds to the standard quantization, which is traditionally used in the theory of partial differential equations. In the formula (7.1) dy denotes the Lebesgue measure associated with some basis in V, and di7 denotes the Lebesgue measure associated with the corresponding dual basis. Then dydr7 is invariantly defined and is indeed the symplectic volume element associated with the invariantly defined symplectic 2 form a(x, ; y, 77) _ y - x r7 on (V X V')2. Here we write l; y = y = y) = E iyi
Proposition 7.1. If a E S(V x V'), then Oph,t(a) is continuous S' - S. Proof. If we work with the h-Fourier transform, .Fhu(l;) =
f
then the distribution kernel of A = Oph,t(a) is KA (X, y) = a(tx + (1 - t)y, x - y),
(7.2)
where a(x, y) = (.FW1),j_y(a(x, i7)) denotes the partial inverse Fourier trans-
form in the second variable. Hence a E S, so KA E S and the proposition
#
follows.
Notice that if we define the adjoint A* u, v E S, (uI v) = f uvdx, then
:
S' --* S by (Auly) = (uIA*v),
A* = Oph,1-t(a) if A = Oph,t(a).
(7.3)
Proposition 7.2. If a E S'(V X V'), then A = Oph,t(a) is well-defined and continuous: S --> S' in the sense that A maps a convergent sequence in S to a weakly convergent sequence in S.
Spectral Asymptotics in the Semi-Classical Limit
76
Proof. In this case we can define the distribution kernel KA E S' by (7.2) and the corresponding operator has then the required continuity property.# It is also easy to see that if aj is a (weakly) convergent sequence in S', then Oph,t(aj) is convergent in the space of continuous operators S -> S'. Using this remark, many of the arguments below can be justified, by approximation of a given symbol in S' by a sequence of symbols in S. As a converse to the last proposition, one can show that if A : S(V) - S(V) is continuous, then the corresponding distribution kernel KA belongs to S'(V x V), and for given h, t as above we have a unique a satisfying (7.2), given by
a(x, y) = KA (X + (1 - t)y, x - ty),
(7.4)
so A = Oph,t(a) for a uniquely determined a E S'(V x V'), given by hKA(x + (1 - t)y, x - ty)dy.
a(x, rt) = J
Notice also that Oph,t(a) =
(7.5)
In the case of the Weyl
quantization, we shall use the simplified notation Oph (a) = Oph(a) _ aw(x, hDx). Similarly for the standard quantization (t =2 1) we sometimes write Oph 1(a) = a(x, hDx). In order to give some motivation for the Weyl quantization, we consider the E V. Then Oph,t(L) real linear form L(x, t;) = x* x + * , x* E V', hDx. This operator is independent of t and equals L(x, hD) = x* x +
is symmetric, when equipped with the domain S(V). We claim that L is essentially selfadjoint. In fact, let u E D(L*), and let v = L*u E L2. Then we
first see that Lu = v in the sense of distributions. Let X E S(V), x(0) = 1, and put uE = X(ex)X(fD)u E S(V), for e > 0. Then, if II ' II denotes the norm in L2, we get IIu -
IIX(ex)(X(ED) - 1)uII + II(X(ex) - 1)uII - 0,
IILu - vuI < II [L, x((:x)1x(eD)uII + IIx(ex)[L, x(eD)]uiI + IIx(ex)x(cD)v - vii -> 0,
when e -+ 0, using Parseval's formula to see that II [L, x(eD)]uii --p 0. It follows that L* is the closure of L, so L is essentially selfadjoint, as claimed.
Consider now the problem of constructing the unitary group: Ut = e-itL(x,hD)lh
7. h-pseudodifferential operators
77
In other words, for u E S we want to find v(t,x), C' in t with values in S such that hDtv(t, x) + L(x, hD)v = 0, v(0, x) = u(x) where u is an arbitrary element of S. Indeed, by the spectral theorem, if u E D(L), then v(t) = Utu is the unique solution in C°(R; D(L))f1C1(R; L2) of the initial value problem above. We try v(t, x) = =: Utu. Then v(0, x) = u(x), and Oph(e-itL(x,g)/h)u
if we write at = exp(-itL(x, l;)/h), we have
hDtUtu = Oph(hDt(at))u, hDt(at) =
-L(x,e)e-itL(x,£)/h.
(7.6)
Moreover, we may apply L(x, hD) inside the sign of integration and obtain L(x, hD)Utu(x)
_ (2 1h)- ff
n + hD)(at(x 2 y, r7))u(y)dydi7.
Here
L(x,r1+hD)(at(x y,rt)) = L(x 2x1*( 2 2
y>rl)at(x
2
Y,77)+
x - y)at(x
2
h
y>rl) +
2
axat)(x 2 y,rl)
The second term (contributes to the integral by
h)n
hD,l)(e2(x-y)'nlh)at(x
2(x*
JJ (2 _ (2 h)- If e2(x-y)',nlh(-2(x*
.
2
-,rl)u(y)dydi
hD,,)at)(x
2
y,rl)u(y)dydrt.
More generally we have proved that for any a E S':
L(x, hD)Oph(a) = Oph(b), where b(x, l;) = L(x, )a(x, e) + 2a {L, a}. (7.7)
For the special symbol at (and more generally for any symbol of the form f (L (x, ))), we have {L, at} = 0, so L(x, hD)OPh(at) = OPh(Lat).
(7.8)
Combining this with (7.6), we get (hDt + L(x, hD))v = 0, v(0, x) = u. It will follow from Lemma 7.8 below that v is a C°° function of t with values in S C D(L), so we obtain Utu =
OPh(e-ithlh)u,
u E S.
(7.9)
e-itL(x,hD)/h = e-itx* .x/h (multiplication When L(x, ) = x* x, we have e-itL(x,hD)/h = operator) and when L(x, ) = * , we get is the operator of translation by the vector to*. where rt£.u(x) = u(x -
Spectral Asymptotics in the Semi-Classical Limit
78
From (7.9) we get the formula: e-iL(x,hD)/h = e-ix* x/2h
and it follows that if M(x, t) = y* x + if
e- ix* x/2h
(7.10)
is a second real linear form,
then e-iL(x,hD)/he-iM(x,hD)/h = ei{L,M}/2he-i(L+M)(x,hD)/h
(7.11)
e-iL(x,hD)/h with the mapping (x, t) H (x, t`) + -x*) is the Hamilton vector of L. Notice also that
Heuristically, we associate
HL, where HL = L(x,e) = Using (7.9),
we shall decompose a general pseudor. Assume that
a(x, t) E S(V x V') and start by writing Fourier's inversion formula:
a(x, t) =
(2nh)2n
ffei**)ma(x*,*)dx*d*, t
(7.12)
where
a(x*,t:*) _ ff is the h-Fourier transform. Taking the Weyl quantization, we get
a' (x hD)
2n
(27rh)
I ] a(x* * )e n (x*
hD)
dx*dt* S
(f)e'e(x,hD)df, 1 2n (21rh)
(7.13)
with uniform convergence in the space 1(L2, L2) since the operators eie(x'hD)lh are unitary.
Composition of symbols. Let a, b E S(V x V'). Using the preceding representation, we get aW
o
bw
a(Q)b(m)e' e(x'hD) o enm(x,hD)dfdm
= (29fh)4n f f
a(t)b(m)e 2n {e,rrc} e-n (e+m)(x,hD) dPdm
1
_ (21rh)2n
f c(r)e*r(x'hD)dr = cw
where
c(r)
(27rh)2n
L
7. h-pseudodifferential operators
79
We shall show that c = c, where
c(x) = (e2°(Dx;Dy)a(x)b(y))ly=x = (e°(hD,;hDY)a(x)b(y))ly=x,
and where we shorten the notation by letting x, y denote points in V x V. More explicitly,
e(x) _
1
(27Th)4-
ffeh(Y2(Y))a(x)b(Y)dxdY, *
so
c(r) _ 1 (
1
eh(x*'x+y*'x-r.x)dx)e I a(x*;Y*)a(x*)b(y*)dx*dy*
(27rh )2n
27Th )2n
(
Here the parenthesis is equal to 5(x* +y* -r), and Q(x*; y*) = {x*, y*} when x*, y* are identified with linear forms. We then see that c = c, so that a= c. Going back to the original notation, we have shown:
Theorem 7.3. If a, b E S(V X V'), then aw(x, hD)bw(x, hD) = c- (x, hD), where c(x,t;) = e 2 °(Dx,De;DY
(7.14)
This result can also be proved by a more direct method (exercise) but we have chosen a method that emphasizes the advantages of the Weyl quantization.
By the same more direct method one can also show that for the classical quantization (t = 1), we have a(x, hD)b(x, hD) = c(x, hD), where
c(x, ) =
eihD D, a(x,
(7.15)
e)b(y, 71) I
Notice that the formula (7.14), contrary to the formula (7.15), is naturally invariant under composition of the symbols by an affine canonical transformation.
We next derive a formula which connects the symbols for the different quantizations. If f = x* x + * is a real linear form, we have
0 Ph,c
ek
x*.xT-e*ez -t x*
x=
e'(sh t)
x*OPh,s(eie/h).
In particular, OPh,t(eie(x,l)/h) = eh
(2-c)£*-x*eie(x,hD)/h
Spectral Asymptotics in the Semi-Classical Limit
80
Using (7.13), we get
a' (x, hD) 1
h)2n
ff a(X* ' *)e*(t
OPh,t(at),
where at = e h, (t- z )hDs hDe a = eih(t- 2 )Dx.DE a.
From this we easily get the more general relation: ei(t-s)hDx D4a.(x,S),
(7.16)
when Oph,t(at) = OPh,s(as). It is now easy to obtain the composition formula (7.15) from (7.16) and Theorem 7.3. Denote the Weyl symbols by a,,,, b,,,, c,,, (still of class S) and let a, b, c be the classical symbols of the same operators: a',,, (x, hD) = a(x, hD), bw(x, hD) = b(x, hl)), cw(x, hD) _ c(x, hD), cw (x, hD) = aw (x, hD) o bw (x, hD), c(x, hD) = a(x, hD) o b(x, hD). Then e'2 Dx DE(e'2 v(Dx,DE Dv
c = e`2 D: (e`2
)b(J, rI)) 1 Y=x,,=
=
In order to prepare the symbolic calculus of pseudors, we consider in general eihA(D)u, for u E S(Rn), where (Qt;, l;`) is a real quadratic form on 2 R. By Fourier's inversion formula, we have IeihA(D)u(x)
u(x)I < hNC(s, N) - E (ihA(D))' II D«A(D)NUII r,2, j! j
if s > n/2 is an integer. (Here we also use the fact that leit - Eo -1 LL for t E R). We also want to estimate eihA(D)u away from the support of u. Assume that A is non-degenerate and let A-1(x) = (Q-lx, x) be the dual quadratic form
on R. Then
a
z e a s n Q (e-iA-1(x)/h * u) = Kh * U, \27rh/n IdetQI2 g
eihA(D)u = r
-
1
7. h-pseudodifferential operators
where f * g(x) = fR. f (x
81
- y)g(y)dy.
Let O(x) = -A-1(x), so that 10'(x)l - jxj. Let X E Co (Rn) be equal to 1 near 0, and make repeated integrations by parts, using (7.18)
tL = (0'(x))-20'(x) hDx. Then we get I((1 - X)Kh) * u(x)I
CNhN 2
ii(x - y)-NDau(y)II Ll
(7.19)
[-]
Here we write (x) =
1 + x2.
Definition 7.4. We say that m : Rn -> [0, oo[ is an order function if there are constants Co > 0, No > 0, such that m(x) < Co(x - y)Nom(y).
Definition 7.5. Let m be an order function on Rn. We let S(Rn, M) = S(m) be the set of a E C°°(Rn) such that for every a E Nn, there exists Cc. > 0, such that iaaa(x)i < Cam(x).
Notice that the product m1m2 of two order functions m1i m2 is an order function, that S(m) is a Frechet space and that the map S(ml) X S(m2) 3 (al, a2) '--> ala2 E S(mlm2) is continuous. Also S(Rn) is dense in S(m) for the topology of S(m(x)') for every e > 0. In fact, if a E S(m), consider the sequence aj(x) = X E S(Rn) and X(0) = 1.
If a = a(x; h) depends on h E]0,1], we say that a E S(m) if a(.; h) is uniformly bounded in S(m) when h varies in ]0, 11. For k E R, we put Sk(m) = h-kS(m) (in the sense that the elements of Sk(m) are functions of the form h-ka(x; h) for a E S(m)). If 6 E [0, 1], we let Sb (m) be the space of functions a(x; h) on Rn x]0,1] which belong to S(m) for every fixed h and satisfy
i9'a(x)i < C«m(x)h-61"l-k.
If aj E Sb' (m), kj \ -oo, we say that a
(7.20)
1o aj, if a - j:N o aj
E
SaA}1(m) for every N E N. For a given sequence aj as above, we can always find such an asymptotic sum a (by the Borel argument, explained in Chapter
2) and a is unique up to an element in S-°°(m) := lkS,,(m).
Proposition 7.6. Let A be a non-degenerate quadratic form on (Rn)*. Let 0 < 6 < 2 and let m be an order function. Then eihA(D) : S' -> S'
Spectral Asymptotics in the Semi-Classical Limit
82
is continuous: S,5 (m) -+ S6 (m). Moreover, if 6 < 2, then eihA(D)u
E0 ah
)k U k!
in Sb (m), for every u E Sb (m).
Proof. The proof is fairly straightforward in the case 6 < 2, using (7.17) (stationary phase) and (7.19) (integration by parts). We here only treat the limiting case 6 = 2, and we then have to review the integration by parts argument. As already noticed, eihA(D)u(x) = C,,,h- z f eiO(y)/hu(x - y)dy.
Here, we split the integral into two parts by using the cutoff functions X(*) and 1 - X(Oh-), where X E Co (R) is equal to 1 near 0. The first integral is easy to estimate, and in the second integral, we integrate by parts, using tL in (7.18) with x replaced by y. The second integral becomes
O(1)h 2
f I(hDy
O'(y)2
I (y)
J
)N((1
- X(y))u(x - y))I dy
Putting y = v"h-y, we get
0(1) / I(Dy J
01(0)N((1 101(y)12
-
and here the integrand is
(y)(y)
0(1)m(x -N y) < 0(1)m(x)(
Ny)N.
which is integrable with integral 0(m(x)). The derivatives can be estimated
similarly, since 8x'eihA(D)u = eihA(D)O u
#
Replacing n by 4n, we obtain:
Proposition 7.7. The map S x SE) (a1,a2) -* a1Hha2 := (e 2 v(Dx,DE (7.21)
has a bilinear continuous extension: Sa°(ml) X Sb(m2) --* S6(mlm2) for all and all order functions ml, m2. When 6 < 2, we have 6 E [0, a] 00 H
a1Nha2 -
1
ih
E k ((2 (Dx, D£; Dy, Dn))kal(x, )a2(y, rl))I k=O
Y= X,
'i
--f
7. h-pseudodifferential operators
83
in S6 (mIm2) for all aj E Sb (mj ), j = 1, 2.
One can also show that the extension in Proposition 7.7 is unique for a suitable topology.
Lemma 7.8. Let h = 1 and let m be an order function. For a E S(m), Op (a) is continuous S -+ S, and S' -4 S', and is a continuous function of a E S(m) with values in the space of continuous operators S -> S and in the space of continous operators S' -* S'. Proof. We only give an outline of the proof. It is enough to consider the continuity in S since the one in S', will follow by duality. We first consider the case when a E S. Using repeated integrations by parts with the help of the operators (x - y)-2(1 + (x - y) D,7) and (q)-'(1 - i Dy), we see that Op (a) is uniformly continuous S -> L°°, when a varies in a bounded set in S(m). By a density argument, it follows that Op (a) can be defined as a continuous operator S -p L°°, when a E S(m). This result, which holds not only for the Weyl-quantization, but also for the other t-quantizations with 0 < t < 1, can then easily be extended to operators of the form x'DQOp (a),
#
and the lemma follows.
Theorem 7.9. Let ml, m2 be order functions and let 0 < 6 < 1/2. For aj E Sb(mj), we have OPh(al)OPh(a2) = OPh(a10ha2)
Proof. Here it is enough to treat the case of a fixed value of h, say 1, and it is then clear how to combine Proposition 7.7, Lemma 7.8 and a simple density argument. # Exercise. Let m be an order function, let 6 E [0, 11 and 0 < t, s < 1. Show
that if as E S6 (m) and at is given by (7.16), then at E Sb (m). Use this to extend the composition result in the preceding theorem to the case of operators of the form Oph t (a).
We next discuss L2-continuity. For that we shall use the Cotlar-Stein
Lemma 7.10. Let A 1, A2, ... E L(E, F), where E, F are Hilbert spaces, and assume that for some M > 0, we have: 00
sup E II A, Ak 112 < M 7
k=1 00
SUP E I I Aj A* Ii 2 < M. 7
k=1
Spectral Asymptotics in the Semi-Classical Limit
84
Then A = ETA. converges strongly and A is a bounded operator with IIAII < M.
Proof. We first assume that only finitely many of the Aj are different from zero: Aj = 0 for j > N + 1. We have IIAII2 = IIA*AII, and more generally from the spectral theorem for bounded selfadjoint operators, we II(A*A)-II. Here,
get
(A*A)m =
Aj2m.
The norm of
is bounded by IIA;1Aj2II
IIA72, -1Aj2mhi and also by IIAj*1II IIAj2A;3II
(7.22)
'
IIA73Aj4II
IIAj2,,,II Taking the geometric
mean of the two bounds, we get:
...
A,2m-1Aj2m II <- M ' IIA;,Aj2II Ii Aj2
IIAii 32 ...
II 2
since II Aj II < M. Then by (7.22):
< IIAIl2-
N
N
M
IIAj1A3211
IIA.72Aj3II2
.. IIAj2m_1Aj2m III.
(7.23)
Here we first estimate the sum over j211. and obtain M2
IIAj*,Aj2IIIIIA;2Aj3II2 ...IIAj2m-2Aj2m-,II2
j1,--j2m-1
We then estimate the j2-._1-sum and so on, and we obtain: <M2m-1EM=N.M2m.
IIAII2-.
j, Hence IIAII < N2 M, and letting m tend to infinity, we get IIAII
M.
We then consider the general case. If u = A*v for some k, then 00
00
IlAjuil =
IIAjAkvhI j=1
1
00
00
IIAjA%II2MIIvII = (E IIAkA; II2)MIIvII < 00, j=1 j=1
so EAju converges when u is in the sum E of the ranges of the A*. Since all partial sums EN Aj are uniformly bounded, we conclude that Ei° Aju converges, when u E E.
7. h-pseudodifferential operators
85
If U E El, then u is in the intersection of the kernels of all the Aj, so we
#
trivially have convergence in that case too. We can now prove:
Theorem 7.11. Let a E SO (l), 0 < 6 < 2. Then Oph(a) is bounded: and there is a constant C independent of h such that
T--i
IIOph(aJ1 f(L2(R^),L2(R^)) < C for 0 < h < 1.
Proof. We first reduce the proof to the case when h = 1 by observing that the change of variables x = f x can be used to define a unitary equivalence between a'(x, hD) and Op(ah), where ah(x, h) = a(fx, fl;; h) satisfies: 18 O, ah (x, ) I < CC,p, with constants Ca,p which are independent of h.
It remains then to show that if a E S(1) is independent of h, then Op(a) is bounded as an operator in L2 and has a norm which can be bounded by some semi-norm of a in S(1). Let X E Co (R2, ) have the property that EaEZ2n Xa (x, l;) = 1, where Xa (x, t) = X((x, t;) - a). Put as = Xaa E S, A. = Op(aa). Then, as we saw in the beginning of this chapter, IIAa11 < C11IIaal1L1 < oo, where as denotes the standard Fourier transform of a,,,, and the constant C,,, depends on the dimension only.
Put ba,Q = as ap. From the estimates (7.17), (7.19), it is easy to see that for every N E N and every -y E N2,:
a+,3)-N,
ba,,3(x,b)I
lary
where pN,-y is a semi-norm on S(1). It follows that pN(a)2(a
IIAaAQ11 <_ C'n.llba,011L1
where PN is a semi-norm on S(1). From this we immediately obtain that
supEIIA*AQ112
a
where p is some semi-norm on S(1). It then suffices to apply the Cotlar-Stein Lemma. #
Theorem 7.12. (Semi-classical sharp Carding inequality). Let P(x, £) E S°(R2n) satisfy:
P(x,
0, V(x, ) E R2ri.
(7.24)
Spectral Asymptotics in the Semi-Classical Limit
86
Then there exist C, ho > 0 such that:
(Pw(x, hDx)uIu)L2(Rn) > -Ch IIuIIL2(Rn),
(7.25)
for every u E L2(Rn) and all h E ]0, ho].
Proof. Let Y E R2n. By Taylor's formula we have ;
0 < P((x,t;) +Y) = P(x, ) +
e), Y)
+
f(i - s)(OP((x,) + sY)Y,Y)ds.
(7.26)
We take Y = with E > 0 sufficiently small, using (7.26) and the fact that ax £P is uniformly bounded, and get: 0 < P(x, e) - EI VX,ePI2 + so for Cc < 121 IVx,fPI2 <-
EP(x,O.
Consequently, we get the standard estimate for non-negative smooth functions:
V Ial = 1.
Hence for0
I aa,£P(x, ) I (P(x, ) + A)-1 < C.
(7.27)
For a, 0 E Nn, we have: 0.0'013
(P(x, ) + A)-1 = (P(x, ) + A)-1 k tt \\ Cg1,a1...,k,,Ok fl(P(x, S) + /1)-1 j=1
x k>1,
/
axc7
Q,P
,
x).(7.28)
11+ +IPkL
Using (7.27), (7.28) and the fact that (P(x, ) + A)-11 ax 8QP(x, ) I <_ Co,p
A-1
for lal + 101 > 2, we get:
ax aQ(P(x, ) + A)
< Ca,Q(P + A) -1A
2
a1
(7.29)
7. h-pseudodifferential operators
87
for all a,,3 E N. Let C > 1 be a fixed constant and put Q(x, e; h) = (P(x, ) + Ch)-1,
(7.29) shows that hQ(x, l;; h) E S° (c) uniformly for C > 1. By Taylor's formula we have
(P + Ch)OhQ(x, t;; h) - 1 +
z2
{P + Ch, Q}
1
t)e 2`a(D=,DecD,,D )
(2 o, (Dx, D£; D, Dn))2
0
(P(x, ) + Ch)Q(y, 77;
Using the fact that for every Ic < 2, h28,' E S° (C) uniformly for C > 1, 2 as well as the fact that {P + Ch, Q} = {P, Q} = 0, we get from Proposition 7.7:
(P + Ch)OhQ(x, t;; h) - 1 E S° (C ),
uniformly for C > 1. Now from Theorem 7.11 we get:
(Pw(x, hDx) + Ch) o Q' (x, hDx; h) = I + R' (x, hDx; h),
with IIRW(x,hDx;h)IIc(L2,L2) = C(C). Consequently for C large enough o o (I + R'")-1 = I, then v(Pw (x, hD,)) c (P'w(x, hD,) [-Ch, +oo[. From this we get (7.25). #
Appendix: Metaplectic operators This theory is an easy model theory to the theory of Fourier integral operators (sometimes fouriors from now on) and contains all the essential ideas. Let 4)(x, 0) be a real quadratic form on Rn x RN and assume
dae, , ... , da9N are linearly independent.
(A.1)
{(x, 0); 10(x, 0) = 0} is an n-dimensional space, and it is easy to see that the linear map Then CD
C,t E) (x, 0) H (x, VX(x, 0)) E Rn x Rh = T*Rn
(A.2)
is injective and that the image A4) is a (linear) Lagrangian space. Indeed, the injectivity is a direct consequence of (A.1), and if we identify Ad) with C'D by means of (A.2), we get dxIA.
dxlc. = (,D' dx + B d0)lc. = d(I c.),
Spectral Asymptotics in the Semi-Classical Limit
88 so 1; dxI A
is closed and hence CIA. =
d(1;
dx) A. = 0.
It also follows from (A.1), that &D (x, 0) # 0 for x = 0, 6 integrations by parts by means of the operators
0, and making
hDe)
in a region jxi < 0(1)101, we see that
(ID, 0) = h- 2
-a ffe*'(x)dxd6,
(A.3)
is well defined for 0 E S(R') (for instance as the limit when e -+ 0 of the corresponding expression with the additional convergence factor X(e(x, 0)), where X E S(Rn+N), X(0, 0) = 1), and defines a distribution Is E S'(Rn). Formally,
I(x) = h2
fe*0)d6.
(A.4)
Lemma A.1. For almost all * real quadratic forms q(x) on Rn, 1(x, 0)-q(x) is a non-degenerate quadratic form on Rn+N Proof. D(x, 0) - q(x) is degenerate precisely when det(4"- I
0
11
qXX
I) = 0,
which is an algebraic condition on q, so it is enough to find one quadratic form q which does not fulfill this condition, and for that, it is enough to find a complex q. Choose q with Im q < 0 (negative definite). If t = (tm, to) E Cn+N is in the kernel of the Hessian of 4 - q, we have
0 = Im ((V" - { q00
00
))t) = -Im (q"tx, tx-) - G 11 tX
II2
so tx = 0. Then we get Vx'oto = 0, V'00 to = 0, which implies that to = 0, by
#
(A.1).
Fix q as in the lemma. Then for every q E Rn, the function (x, 0) F--> D(x, 0) - q(x) - x
71
(A.5)
has a unique critical point (x(r7), 0(77)). Equivalently, for every q E Rn, the
(affine) Lagrangian space A,, := {(x, l;); l; = q + q'(x)} intersects A., at a unique point (x(77),1;(77)) (the image of (x(r7), 0(77)) E C4), under (A.2)).
* The precise sense of this is given in the proof
7. h-pseudodifferential operators
89
Moreover, the intersection is transversal. Let H(i7) be the critical value of (A.5). Then,
_ -x(0,
H'07) = \art ('D(x, 0) - q(x) - x - rl))
(A.6)
and since H(77) is a quadratic form, we see that H(77) only depends on A = A4, and on q, but not on the choice of -D for which A = Ap.
Using the method of stationary phase (in its exact quadratic version) we see
that .Fh(e-zq(x)1hI)(ii) =
C4,,ghn/4eiH(n)/h,
(A.7)
where C4,,q # 0 is independent of h. We conclude that if fi is a second phase
satisfying (A.1), such that A = A4,, then there exists a constant C 0 0, independent of h, such that 14, = CIi. In a more fancy way, we can say that with A4,, we have associated a one-dimensional space of functions of h with values in S': {h hi a4; a E C}, which depends on A4,, but not on 4). To complete the picture, let us verify that if A is a linear Lagrangian space, then A = A4, for some 1 satisfying (A.1). If q(x) is a quadratic form, then the non-transversal intersection between A and Aq = {(x, ); = q'(x)} can be expressed as an algebraic condition on q. On the other hand, if we take q complex with Im q > 0, then (a, t A t) > 0 for all t E Aq (now viewed as a complex Lagrangian space), while it is easy
to see that
(v, t A t) = 0 for all t in the complexification of A. Hence the complexifications of A and Aq intersect transversally. As in the proof of Lemma A. 1, we conclude that Aq and A intersect transversally for most real quadratic forms q. We fix such a form q. Define A,7 as before (parallel to Aq). Then A and A,, intersect at a unique point (x(77), (r7)), and we can parameterize A by R' E) q --' (x(77), 6(11)) E A.
(A.8)
Using the fact that Z; (77) =,q + q'(x(r7)), we get
d(-x(r7) d77) _
_
dr7j A dxj (77)
d(ee(rl)) Ad(xj(rl)) - Ed(a.;q)(x(rl)) Ad(xj(i7))
aln - d(j:(a.,q)(x(rl))d(xj(rl))) = 0 - d2(q(x(i7))) = 0. Let H(r7) be the real quadratic form with dH(77) = -x(77) d77, and put -,b(x,,q) =
Clearly T satisfies (A.1), and the corresponding
90
Spectral Asymptotics in the Semi-Classical Limit
critical space C4, is given by x = -H'(i) = x(71), and A4, is the space of all (x(71), rl + q'(x(rl))) = (x (77), (,q)), so A4,
A.
Let L(x, ) = E x xj+E l;j*t=j = x * l; be a linear form which vanishes on the linear Lagrangian space A = A4,. Then L(x, hD)I. = h- 2 - 4 f ei4,(x,O)/h(x* x + l;*
8x 4P (x, 9))d9.
Here the parenthesis in the integral is a linear form which vanishes on C4, and is therefore of the form Ei ajae for some constants aj. Hence, N
L(x, hD)I4, = E ash-2 1
a9 dB
j
0,
where the last identity follows by integrations by parts. (All the integrals are here given a sense as for the one in (A.3).)
As a special case, we consider operators. Replace n by 2n and Rn by R2n = Rn, X R. Let 1' = 4) (x, y; 0) satisfy (A.1). If
A') = { (x, t=; y, -71); (x, e; y, 71) E A4, }
is the graph of a linear map and hence of a linear canonical transformation ic, and we let J4, : S -> S' be the operator with distribution kernel I4,, then we say that JD is associated with it. It is easy to see that in this case J4, : S -> S. Conversely, if is : T*Rn -> T*Rn is a linear canonical transformation, then (graph ic)' is a linear Lagrangian space. Thus to every linear canonical transformation we have have a corresponding one-dimensional linear space of mappings h --> aJ4,, a E C, where J4, : S -> S. We have:
- is = id corresponds the space of multiples of the identity operator.
- If J4, corresponds to ic, then the complex adjoint J, corresponds to is-1.
- If J4,; corresponds to rcj, j = 1, 2, then J4,1 o JD2 = JD corresponds to icl o rc2. In fact, we get'P(x, y; z, Oi, B2) = (D1(x, z, 91) + 42(z, y, 92), and we check that this function satisfies (A.1) with 0 = (z, 01i 02) etc.
Combining these facts, we see that J,J4 = J4J, = C4I, where C4, > 0. Hence J4, extends to an operator which is bounded in L2, and if 0 54 a E C satisfies lal2C4, = 1, aJ4, is unitary. We say that aJ4, is a metaplectic operator.
From the earlier discussion, we see that if L(x, ), M(x, t;) are linear forms
with L o is = M, and JD is associated with it, then LJ4, = J4M, for if
7. h-pseudodifferential operators
91
ID (x, y) is the kernel of J4, then the kernel of LJD - JAM is (L(x, hDx) M(y, -hDy))4, and L(x, l;) - M(y, -97) vanishes on CD = (graph ic)'. Under the same assumptions, it follows that eitL J = J4,eitM, and since Weyl quantizations of symbols are superpositions of operators of the form eitL(x,hD) and the Weyl symbol of such an operator is we get
Theorem A.2. Let a E S(m), where m is an order function on T*R' and let is : T*Rfl -+ T*R' be a linear canonical transformation. Let Jb be an associated operator as above. Then Oph(a)J = JOph(b), where b = aolcES(mon). Notes Among the books devoted to the theory of pseudodifferential operators and related topics, we can mention: Hormander [H64], Treves [Tn], Taylor [Ta], Alinhac and Gerard [A1Ge], Robert [Rol], Ivrii [I1], Grigis-Sjostrand [GrSj]. Many symbol classes have been introduced since the work by KohnNirenberg [Ko-Ni]: Hormander [H65], Beals-Fefferman [BeFe], [Be], [Sj2]. For
the Carding inequality and its various extensions, see [GA], [H66], [LaNi], [CoFe], [Ta], [FePh], [Sjl]. The particular approach to Weyl-quantization of this chapter is inspired by [BGH], though the ideas are classical and many other references doubtless could be added. The same ideas appear in the study of operators with magnetic fields, see [HeSj4] and further references given there.
8. Functional calculus for pseudodifferential operators If f (A) is a bounded continuous function on R and H is a selfadjoint operator, then by the spectral theorem (see Chapter 4) f (H) is a well defined bounded
operator. It is useful to obtain a more precise description of f (H). Under appropriate conditions Strichartz [Str] showed that a function of an elliptic pseudor on a compact manifold is also a pseudor (see also Taylor [Ta]). The purpose of this chapter is to show that a smooth function of an h-pseudor H is also an h-pseudor. This result was obtained by Helffer-Robert via Mellin
transformation. Our method is based on a standard Cauchy formula (see Theorem 8.1 below). One of its main advantages is that it allows us to pass easily from resolvent estimates to estimates of other functions of H. To know the properties of the resolvent as a pseudodifferential operator we shall use a characterization of pseudodifferential operators due to Beals [Be], and adapted to the h-pseudo differential setting by Helffer-Sjostrand [HeSj5],
see also Robert [Rol], Bony-Chemin [BonCh]. Our method works in the case of functions of several variables and we discuss functional calculus of several commuting selfadjoint operators. For further results in this direction, see Charbonnel [Charl], Colin de Verdiere [CdV3] and the recent paper of Andersson [An].
If f E Co (R) we can find an almost analytic extension f E Co (C) with the properties (8.1) Iafl < CNIImzIN, b'N > 0,
AR = f .
(8.2)
Here a=az=2(a +Zay This idea was introduced by Hormander [H62] and has subsequently been used by many people: Nirenberg [Ni], Melin-Sjostrand [MeSj], Maslov [Ma2], Kucherenko [Ku], Dyn'kin [Dy], [Dy2], etc. The original approach
by Hormander was to adapt the Borel construction and put f (x + iy) = (k) x1(iy)kx(Aky) with X E Co (R) equal to 1 near 0 and with .\k tendf ki ing to +oo sufficiently fast when k - oo. A second construction was intro-
duced by Mather [Mat] and more recently by Jensen and Nakamura [JeNa]. If V)(x) E C0 is equal to 1 in a neighborhood of supp (f), and if X is a standard cutoff function as above, then we can put
Ax + iy) _ ow f ei(x+iW 2
where f is the Fourier transform off . We check that the last formula produces an almost analytic extension (i.e. a function which verifies (8.1), (8.2) and
Spectral Asymptotics in the Semi-Classical Limit
94
which can be further truncated in y). First, we see that (8.2) follows from the Fourier inversion formula. Moreover,
of = 2
-
- Jejx'(yI
yN2 2xr) 2 2
2 02(x)
f
0'(x)
(x)dxd
27r
=I+ II, where XN(t) = t-NX'(t) E Co (R). Here III < CNIyINIIfN+1f(e)IIL1. To estimate II, we use the fact that x - x 0, on the support of "'(x) f (x), and get by integrations by parts:
II =
4
i
(x)
4ir
x
ff
2 4'(x) yN
=
'(x) yN
2
ff
ei(x-x+iv)f X
X( Y)) y f
x - x + iy f
ei(xffi -
47r
(x)dxdt
i)2 (x DX)2(-Dx)N(ei(x
V) I(x)yN Jf ei(x- +iy)
(t;+i)
(x)dxd
a
x + y)(S + i)2 f (x)di<
+iy)C)
Nay+)
(i + D-)2DY (
f(x) ( + i)2 dxd
f (x) )dxdt; x-x+iy
= O(IYIN).
In both constructions we notice that we can take f with support in an arbitrarily small neighborhood of the support of f. We also notice that if g is the difference of two almost analytic extensions of the same function, so that 91R = 0, 89 = ON(IImxIN), VN, then g = ON(IImxIN), VN.
Theorem 8.1. Let P be a selfadjoint operator on a Hilbe_rt_space R. Let (C) be an extension off with 8f = O(IIm zl). f E Co (R) and let f E Col(C) Then
.f (P) _
_2
af(
fz)(z -P)-1L(dz).
(8.3)
Here L(dz) = dxdy is the Lebesgue measure on C, and we notice that the integral in (8.3) converges as a Riemann integral for functions with values in
G(l,7-l). Notice that if we replace P by a complex number, then (8.3) simply reflects that 7rz is a fundamental solution of D.
8. Functional calculus for pseudodifferential operators
95
Proof. (Dimassi [Dill) Let Q E L(7-1,7-1) be the RHS of (8.3). Let u,v E 7-t and write ((z - P)-1ulv) = (z - t)-1(dEtul v),
J
where Et = 11_,, t] (P) is the family of spectral projections associated with P. (Here, we use the fact that (z - P)-1 = f (z - t)-1dEt.) Consequently,
(QuIv) _ -
J
8 f (z)
f(z - t)(dEtuI v)L(dz),
and we can apply Fubini's theorem to get (QuIv) =
f(-I ff(z)(z - t)-1L(dz))(dEtIv)
The inner integral is equal to f f (z)az (-1 (z - t)-1)L(dz) = f (t), since 7rZ- is a fundamental solution of 8 (which is proved by means of Green-Riemann's formula). Consequently,
(QuIv) = ff(t)(dEtulv) and hence Q = f (P).
An even shorter proof can be given, using the representation of P as a multiplication operator (Theorem 4.13). This result can be generalized to classes of non selfadjoint operators acting in Banach spaces. Then we may use (8.3) as a definition. In such a case, we should write f (P) to the left, and require that az f vanishes to a sufficiently high order on the spectrum of P (which now is some closed subset of the complex plane). See Dyn'kin [Dy], Droste [Dr], Davies [Dal,2], Jensen-Nakamura [JeNa]. We now develop R. Beals' characterization of pseudodifferential operators in its semi-classical variant. Assume that a E S'(R2n) and that a(x, D) (in the
standard (t = 1) quantization) is L2-bounded. Let 0, z/) E S be fixed and consider C1101111011 >_ K(a(x, D)V)I 0)I
= I (2zr)n
a(x,
lie-ie'g (corresponding to new Here we can replace 0 by Oe-ix'x* and zli by functions 0, 0 with the same L2 norms), and we get aezx
C CII01I11'bll.
Spectral Asymptotics in the Semi-Classical Limit
96
If X(x, ) E Co (R2n), choose 0, 0 with 0 in a neighborhood of supp X so that x = 4?(x, for some 1 E CO. It follows that Cxllallc(L2 j,2).
(8.4)
Here the constant C. only depends on the support of X and on >lal<2n+1 Ilaax!IL-. We get the same constant C. if we replace X by any translate of X, or equivalently if we replace a by any translate of a. The reason for this is that if (y, 77) E R2n, then a(y + x, y + Dx) is obtained from a(x, Dx) by conjugation by the unitary operator Tyeix".
If (ate a) (x, D) is L2-bounded for jai < 2n + 1, we get from (8.4) and the subsequent remark that lal<2n+1
where C. is invariant under translations of X. In particular,
IIaIIL°° < C E II
a) (x, D)II G(L2 L2).
lal<2n+1
Let a,, be the Weyl symbol of a(x,D) so that a(x,D) = aw (x, D) and aw = eiDs.D4/2 a. We shall now prove that with a new `translation invariant' constant Cx, we have for every X E Co (R2n): IF(Xaw)(x*,S*)I C
Cx((x*,S*))-(2n+1) E
II
L2).
IaI<2n+1
(8.6)
Here (U £a)(x, D) = (a,,eaw)'°(x, D), so the sums in (8.5) and (8.6) are equal.
Up to a constant factor, which we shall neglect, we have aw = ei4' * a, where is a real non-degenerate quadratic form on R2n. Let 1 = EjEZ2n Xj be a partition of unity on R2n with Xj = TjXo, Xo E CO. Let Xo satisfy XoXo = Xo and put Xk = TkXo. Temporarily we simplify the notation by writing x instead of (x, e) and t instead of (x*, t;*). Then for j, k E Z2n, we have: .1:7(Xj(ei.D
*
fff e
Xj,k(x, y)Xka(r7)dydxdr7,
8. Functional calculus for pseudodifferential operators
97
where F is real-valued and where Xi,k (x, y) = Xj(x)Xk
Here we also used the Taylor sum formula:
4'(x-y) = Notice that the modulus of any derivative of X3,k(x, y) can be bounded by a constant which is independent of x, y, j, k.
We make 2N integrations by parts using the operators
1-( a.-D(j - k))2
1+(77 (77 - a.-D(j
- k))
After estimating the resulting x, y integrals in a straightforward way, we get: -F(Xj
(ei4)
= ON(1)
* Xka))(e)
f ( - 0.4)(j -
k))-N(,q - 0.4,(j
-
k))-NIXka(,R)IdR.
(8.7)
Since D is non-degenerate, ( - a,4) (j - k)) is of the same order of magnitude as (45"-1 - j + k), and similarly for (71 - 5 4'(j - k)). With N > 2n, we get:
( - ax4'(j - k))-N(l1 -
a..D (j
_,q)-N. - k)) -N < CN(
kEZ2'n
Summing over k in (8.7), we get IF(Xi(e24) * a))(
)I <- CN f (- n)-N Sup IXka(q)Id?7
Here we use (8.5) with C., independent of k and conclude that (8.6) holds with X = X; and C 3 independent of j. To get (8.6) for arbitrary X, we simply represent X as a linear combination with smooth coefficients of a finite number of the X;. As a special case, we have proved:
Proposition 8.2. Let a E S'(R2') and assume that Op (a9,a) E £(L2, L2) for J al < 2n + 1. Then a E L°° and IIaMILOO < Cn E IIOp (a.,£a) II c(L2,L2). dal <2n.+1
Spectral Asymptotics in the Semi-Classical Limit
98
The proof above gives the same result for any quantization Op 1,t with
0
Remark. Let B C D'(R2n) be a translation invariant Banach space with a translation invariant norm, and assume that the inclusion B --> D' is sequentially continuous. Then it can be proved, using the Banach-Steinhaus theorem, that for every x E Co (R2"), there are constants Cx, Nx > 0, such
that
and it is easy to see that C. and N. can be chosen invariant under translations of X. Bony [Bon] made essentially this observation and applied it with B equal to the space of Weyl symbols of L2 bounded operators, which
in our case leads to Proposition 8.2 with 2n + 1 replaced by some finite unspecified number N > 0.
If A = Op (a), then the commutators [xj, A] and [Dx,, A) have the symbols {xj, a} = i8 3 a and a} = axe a respectively, so the assumption in the proposition can be reformulated as: ade1(x,D) ... adek(x,D)Op (a) E £(L2, L2) for all k < 2n + 1 and all linear forms Pj (x, (Here we use
the standard notation: adAB = [A, B], and notice that by the Jacobi identity for commutators: [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0, we have ad[A,B1 = [adA, adB]). We next turn to the h-dependent case:
Proposition 8.3. Let A = Ah : S(Rn) , S'(Rn), 0 < h < 1. The following two statements are equivalent:
(1) A = Oph(a), for some a = a(x, l;; h) E S°(1). (2) For every N E N and for every sequence 21(x, t;),... ,2N (x, l;) of linear forms on R2n, the operator ade, (x,hD) o...oadeN (x,hD) Ah belongs to £(L2, L2)
and is of norm O(hN) in that space. Proof. That (1)x(2) follows from the calculus in Chapter 7. In the opposite
direction, we assume (2), and notice that we have the general identity Uhbw(x, hD)Uh 1 = bw(v"h-x, VKD), if Uh is the unitary operator given by: Uhu(x) = hn/4u(v/h-x). Then (2) can be reformulated as: ade1(h1/2x,h1/2D) 0 ... o adeN(h1i2x,h1i2D)ah (x, D) = O(hN) in C(L2, L2),
with ah(x, ) = a(h1/2x, h1/21;; h), or rather as: ade1(x,D) o... o adeN(x,D)ah (x, D) = O(h 2) in C(L2, L2).
8. Functional calculus for pseudodifferential operators
99
Applying Proposition 8.2 to ah and to its derivatives, we get Cahl«1/2
#
which implies that a E S°(1).
We will also need the following generalization of the implication (2) .(1) in the preceding proposition. Assume that a = a(x, t:, z; h) depends on the additional parameters z and that for some function 6 = 6(z) with values in 10, 1] we have
adel(x,hD) o ... o adjN(x,hD)Oph(a) = O(S-NhN) in £(L2, L2)
for all N > 0 and all linear forms 21i ... , EN on
(*)
R2n.
Proposition 8.4. Under the assumption (*), we have Ie x a a ( x,
) 2n+16-Ia1-I$1 .
< Ca,o max (1 , , z; h) I _
(8 . 8)
b
Proof. As in the proof of Proposition 8.3, we introduce ah (x, t) _ a(vlh-x,
z; h) and reformulate (*) as: adel(x,D) o ... o adtN(x,D)OP (ah) = 0(6-NhN/2).
(8.9)
Proposition 8.2 then gives Ilaxa'3ahIILoo
Cn I
<2n+1
< Cn max(6-1a+131h1a+01/2 6-la+0l-(2n+1)h(Ia+0I+2n+1)/2)
= Cn max(1, v '° )2n+16-Ia+Qihla+/3I/2 S
and since 99BQah = hla+Q1/2(888,3 a)(Vh-x,Vh-t:,z;h), we get (8.8).
#
Let p(x, ; h) E S°(1) be elliptic in the sense that there is a constant C > 0
such that IpI > 1/C. Then q = 1/p belongs to S°(1) and p'(x, hD) o
q (x, hD) = 1 - hrw(x, hD), where r E S°(1), so for h < h° > 0 small enough, llhr'(x,hD)IIc(L2,L2) < 1/2 and (I - hrw(x,hD))-1 exists in G(L2, L2) and has norm < 2. Put Ql = 4' (x, hD) (I - hrw(x, hD))-1, so that IIQI IIc(L2,L2) < C and pw o Q1 = I (where we drop `(x, hD)' in order to
shorten the notation). Similarly, we can construct Q2 E £(L2, L2) of norm < C such that Q2 o pW = I. Then, Q2 = Q2(P' Q1) = (Q2PW )Q1 = Q1
Q.
Spectral Asymptotics in the Semi-Classical Limit
100
We claim that Q = qW (x, hD), q E S°(1).
(8.10)
Proof. Let 21 i 22, ... be linear forms on R2n and put Lj = fj (x, hD). If L is one of the Lj, we first recall that eitL o p' o e-itL = pt where pt = p o exp(tH,). It follows that this operator is a C1 function of t with values in the space £(L2, L2). If A : S -' S' is any continuous operator, we (Dt)t=o(eitLAe-itL) = adL(A). It follows that notice that adL(Q) = =
(Dt)t_o(eitLpwe-itL)-1
-Q((Dt)t-o(eitLpwe-itL))Q
= -QadL(p')Q
By iteration we then see that adL,k o ... o adL(Q belongs to £(L2, L2) and is equal to a finite linear combination of terms of the form: Q((adL)J1 P)Q((adL)J2P)Q ... Q((adL)J`P)Q'
where J1,... Je is a partition of {1, 2, ... , k} and where we write (adL) J = This notation is justified by the fact that the adL3 commute, which can be seen from the Jacobi identity for commutators, and the fact fl3EJadL;.
that ad[L,,Lk] = adconst, = 0. Then adL,k o ... o adL(Q = O(hc) in C(L2, L2) and we can use Proposition 8.3 to conclude. #
We can also obtain an asymptotic expansion for the symbol of the inverse operator. Write PQ = I - hR, where P = pw (x, hD), Q = (111)711(x, hD), R = rw(x, hD). Put QN = Q(I + hR + h2R2 + ... + hNRN). Then
PQN = I - hN+1RN+1 SO QN = QPQN = Q - QhN+1RN+1 = Q mod (S-(N+1)(1)). If Q = Oph(q), we obtain
q- p +h( I Ohr) + h2(1 Ohrghr) + ...
(8.11)
More generally, let m be an order function, and let p E S°(m) be elliptic in the sense that l p(x, t:; h) I > C m(x, ) for some C > 0. Define q = P E S°(1) and observe as above that pw(x, hD)q (x, hD) = I - hrw(x, hD), r E S°(1). Again I - hr21 is invertible as an L2 bounded operator when h < ho for some ho > 0 small enough, and we now apply Beals' lemma
directly to I - hrw(x, hD), to see that (I - hrw)-1 E Oph(S°(1)). Then
q o (I -
E Oph(S°(m-1)) is a right inverse. Similarly we get a left inverse, and as before we get Q E Oph(S°(m-1)) with PQ = QP = I, writing P = pw(x, hD). hrv')-1
We now apply the preceding results to the functional calculus of pseudodifferential operators. Let m > 1 be an order function on R27z and let
8. Functional calculus for pseudodifferential operators
101
P = p' (x, hD; h), where p E S° (m) is real-valued. When m is unbounded, we also assume that p + i is elliptic, and in that case, the discussion below will be valid only for 0 < h < h° for some sufficiently small ho > 0. These assumptions will be valid throughout the discussion of the functional calculus for one operator below. We will also frequently write p' for pw (x, hD; h)
We know that (p' ± i)-1 exists and belongs to Oph(,1-n) (provided that h is sufficiently small in the case when m is not bounded). It is easy to see that (pw±i)-1(L2(Rn)) =: Vp is a space independent of the choice of the sign in front of the `i' and that u E Vp can be equipped with any of the equivalent norms 11(p- + i)u!I or II (pw - i)uII. (Of course, Dp = L2 in the case when m is bounded.) We may view P = pw as a symmetric operator with domain S(Rn).
Proposition 8.5. P is essentially selfadjoint and the unique selfadjoint extension is given by pw equipped with the domain Dp.
Proof. Let P denote the closure of P. Then Pu = v means that uj -> u, vj
-+ v in L2, where Puj = vj, with uj, vj E S. Then pwu = v,
(pw+i)u = v+iu, so u = (pw +i)-1 (v +iu) E Dp. Conversely, if u E Dp, and
v = pwu, we let fj E S with fj -. (pw+i)u in L2 and put uj = (pw+i)-lfj, vj = pw(pw + i)-1 fj, so that uj --+ u, vj --+ v in L2. We have then shown that P is given by pw (x, hD; h) with domain Dp.
Let u E D(P*), and P*u = v. It follows that (pw + i)u = v + iu, so u = (pw + i)-1(v + iu) E Dp. Hence D(P*) = Dp and P* = P.
#
From now on, we let P denote also the selfadjoint extension. We shall consider
the resolvent (z - P)-1 for Im z # 0.
Proposition 8.6. For IzI < const., Imz # 0, we have (z - P)-1 = rw(x, hD, z; h), where I ax aQr(x, , z; h) I < c«,P max(1, IIhl )2n+1 IIm zI-(IaI+IRI)-1.
(8.12)
The same result holds for (P + i)(z - P)-1. Proof. We treat (P + i) (z - P)-1. Let 21, f2, ... be linear forms on R2n and put Lj = 2j (x, hD). Then adL,k o ... o adL1 (P + i) (z - P) is a finite linear combination of terms of the form,
(adL°(P +i))(z
- P)-1(adi'P)(z -
P)-1
... (adL`P)(z
- P)-1,
102
Spectral Asymptotics in the Semi-Classical Limit
where J 0 , . .. , J e is a partition of { 1, ... , k}, Jj # 0 for j 54 0 and we allow
Jo to be empty and use the convention that adO (P + i) = P + i. This is clear, since we know that (z - P)1 = Oph(gh,z), with qh,z E S(m-1) for Im z 0 0 and for every fixed h. We also see that (adLP) (z - P)-1 is in L(L2, L2) and we obtain that adLk o ... o adL, (P + W Z -
zI
P)-1 = °(IIm zlk+1)
in L(L2, L2). The estimates (8.12) for the symbol of (P + i)(z - P)-1 then
#
follow from Proposition 8.4.
Theorem 8.7. Let f E C0 0"(R). Then f (P) E Oph(S°(m-k)), for every
kEN.
Proof. W e apply (8.3) with f satisfying (8.1). Let r(x, t , z; h) be the symbol in (8.12). Then f (P) = Oph(a), where a(x, t:; h) = --1 f of (z)r(x, t=, z; h)L(dz),
(8.13)
and from (8.1) and (8.12) we easily obtain that a E S°(1) so f (P) E Oph(S°(1)). Writing f (t) = (t + i)-k fk(t), we see that f (P) = (P + # i)-kfk(P) = OPh(S°(m-k)) We finally show how to get an asymptotic expansion in powers in h of the symbol of f (P), when P = Oph(p), with p - po(x, t)+hp1(x, )+h2p2(x, )+ ... in S°(m) and p + i is elliptic in the case when m is not bounded. We notice that if 6 > 0 and if we restrict the integral in (8.13) to the domain IImzj < h6 then we get an element in the symbol class S-°°(1) = lkSk(1).
On the other hand, if 6 < 2, and we restrict our attention to the domain IIm zI > h6, then by Proposition 8.6, we have r E Sb (1) and we want to find
the asymptotic expansion of r in this space. Clearly we can find a formal asymptotic expansion: 1
z - po(x,)
+h
q, (x, S, z) (z -
+ h2
q2 (X, S, z)
+
.. .
(z - po(x,e))
(8.14)
with qj (x, t=, z) a polynomial in z with smooth coefficients, so that in the sense of formal asymptotic expansions in powers of h, we have (z - p)Ohr = 1, rOh(z - p) = 1,
8. Functional calculus for pseudodifferential operators
103
where we put
aghb - E
l (( 20' (Dx, Dg; Dy, D,7 ))k(a(x, )b(y, ,q)))I
y,C=Tl
(When pi = 0, we also have qi = 0.) Letting jzj tend to infinity, we see that qj is a polynomial of degree < 2j. If we let z take different values of the order
of magnitude m(x, ), we see that 2i
qi(x, , z) = L'gi,k(x, C)zk, qi,k E
S(m2k).
k=0
If we restrict our attention to jzj < const., jIm zj > h6, we see that we can give a meaning to (8.14) in S66 (m) and that (z 1- k, k E S-°° (1). Then (by the Beals lemma) (1 - kw)-1 = (1 - kw), k E S-°°(1) and consequently
r = i h(1 - k) belongs to Sb(1-) and also has the asymptotic expansion (8.14).
It follows that f(P) = Oph(a), a E S°(m), a - a° + hal + h2a2 + ..., ai E S(m) , 1 it
IIm zI>hb
8f (z)
qi (x, z) L(dz). (z - 170(x, S))2i+1 ,,
(8.15)
Modulo S-'(-!-), we can replace ai by
a_-1 7r
_ (2.)!
a
f
L(dz) _ _ qi f (z - po)2j+i (2j)! 1
1
8fgi(-8z)2'j z - 1PO L(dz)
ipo)L(dz) (8.16)
_ (2 .), acj (qj (x, , t)f
In particular, ao = f (po(x, )), ai = pl (x, )f'(po(x, )). We end this chapter by discussing functional calculus for several commuting (formally) selfadjoint operators. In order to avoid some abstract difficulties, we consider right away the case of pseudodifferential operators. Let M(x, ) >
1 be an order function on R2' and let pi,... , p,,,, E S°(M) be real-valued. Put Pi = Oph(pi) and define
Q:= mI +
P? E Oph(S°(M2)).
(8.17)
Spectral Asymptotics in the Semi-Classical Limit
104
When M is unbounded, we assume that Q is elliptic and restrict our attention
to a region h > 0 small enough. If f E Co (Rm) or more generally if f E C°O(Rm) is constant near infinity, we want to define and study f (P1, ... , P.). We assume that Pj commute: [Pj, Pk] = 0 for all j, k. To start with we need some formalism from the theory of several complex variables. If u(z) is a distribution defined on some open set in Cm, we define the (1, 0) and (0,1) forms m
m
azu = E(a;u)dzj, azu = j:(az;u)dz;, 1
1
where azi = a (a.; + ay; ), az; = 2 (a.,; - ay; ), when writing zj = xj + iyj. The formal complex adjoint of a,, is given by
az =E-az;dz; =taz, when dzl,... , dz,,,,, is considered to be an orthonormal basis for the (1,0) forms at each point. More explicitly:
azv =
-az,vj, when v =
vj (z) dzj.
Then az az = - a A, where A is the standard Laplacian on R2m N Cm. Let Eo(z) be the standard fundamental solution of A, so that Eo(z) _ Iz12(1-m) when m > 2. Here CI log(Izj2) when m = 1, and Eo(z) = -C Cm > 0. Put Fo(z) = -4azEo so that azFo = So. Notice that
Iz1Iz1
4C,n (2,
F,o -
m) z dz _
-C,,,
2
n,
when m >_ 2
zdz =-C1lzl2, whenm=1. Fo=- 4C1zdz Iz12
As in the case of one complex variable, if f E Co (Rm), it is easy to construct f, '5f= O(IImzIN), for every N E N. We now f E CO- (Cm) with put,
f (P1, ... Pm) _
P)* dz f (af, ((z,(z- -P,)(z, - P,)) m
m)L(dz)
_-CmEf azif(z)(z.7-P.7)*(E(zv-Pv)*(zv-Pte))-mL(dz) j=1
_ -Cm
1
f ((z - P)* af)(E(zv 1
Pte)*(zv
-
Pte))-mL(dz).
(8.18)
8. Functional calculus for pseudodifferential operators
105
Notice that if we replace P by (P1, p2, ... , p.,,,,) E R', then (8.18) holds. For the understanding of (8.18) we make two comments:
(1) Assuming as before that h > 0 is sufficiently small when M is not bounded, we see that Ii -PI2 = E(i - Pj)*(i - 13) is essentially selfadjoint and invertible, and that the inverse is a pseudor. We next look at Iz - PI2 =
Q(z)=1: (z. -Pj)*(z. -Pj). ForuES, we have (Iz - PI2uIu) _
II(z; - Pj)uII2 >- IImzI2lIuIl2.
Also
II (zj-Pj)uIl2 = (I z-PI2uIu) < II Iz-PI2uII IIuII
2II Iz-PI2uII2+2 IIuhI2,
for every E > 0, so we get
(E I1(zj
- P;)ull2)
< Elllz - PI2u11 + CEIIull-
We can then compare two different operators Iz - PI2 and Iw - PI2. Since
(Iz-PI2-Iw-PI2) = E(Izj-wjI2+(zj - wj)(wj-Pi)+(z'-wj)(wj-P,)'), we get for every u E S, when z, w belong to some bounded set: II(Iz - PI2 - Iw - PI2)uhh <- Elliz - PI2uII + C'EIIuII
From this estimate it is clear that the closures of Iz - P12 and Iw - PI2 have the same domain, namely the closure of S for the norm Il Ii - PI2uII. Let Q denote the unique selfadjoint extension of Ii - P12 and let Q be the closure of lz - PI2 with the same domain. Then for 0 < R < +oo:
(Ri + Q)(Ri + Q)-1 = I + (Q - Q)(Ri + Q)-1, and
II (Q - Q)(Ri + Q)-lull <- EIIQ(Ri + Q)-lull + C'(E) II (Ri + Q)-lull < (C+ C(6) )llull.
Here E + R< 1 if we first choose E = a and then take R > 0 sufficiently large. It follows that Ri + Q is surjective if R is sufficiently large. Similarly -Ri + Q is surjective. We conclude that the operators Q(z) with the same
domain as the unique selfadjoint extension of Q(i) are selfadjoint. From
Spectral Asymptotics in the Semi-Classical Limit
106
now on we let Iz - P12 denote these selfadjoint operators, and we also let 1z - P1 denote the corresponding non-negative square-roots. It also follows that Iz - P12 is invertible for Im z # 0 and that the norm of the inverse is < We conclude that the integrals in (8.18) are well-defined and give rise to a bounded operator. (2) For IIm zI # 0, we notice that m
(z - Pv)*(zv -
taz((z -
Pv))-
dz) = 0,
or equivalently that Pv)*(zv
az; ((zj -
j
- Pv))-m) = 0,
1
or still:
m
M
dz(T J(z)(zj - Pj)*(E(zv - Pv)*(zv j=1
A dz))
1
m
m
az;f(z) (zj j=1
- Pj)*(T(zv -
p)*(zv
- Pv))-mdz A dz,
1
where dz A dz = dzl A dz2 ... A dzm n dzI A ...Adz,,,,
and (-1)m+j-ldzl h ... A dz,», n dzl n ... dzj ... A dzm,
dz (dz n dz) _
the hat indicating an absent factor. Writing the integrals as limits of the corresponding integrals over jImzj > e, when e -> 0, we then obtain from Stokes' formula:
f(PI,..., Pm) f
m
E f (z) (zj - Pj)*((z - P)*(z = C,,,, lim J e-.0 Imzj=ej=1
Adz).
We conclude that if f = 0, then since jf j < CN I Im z j 1, VN E N, we have f (P1i . . P,,,) = 0. In other words, for arbitrary f E Co (R'), our .
,
definition of f (P1, ... , Pm) does not depend on the choice of the almost analytic extension f .
8. Functional calculus for pseudodifferential operators
107
We next show that f (P1, ... , P,,,,) E Oph(S°(1)) with the help of Beals' lemma. Let Lj = xj, 1 < j < n, Lj = hDxj_ , n+1 <j :52n. Let adL = adL1
... adL2 ,
a E N2n . Then
adL((zj - Pj)*Iz - PI-2m) is a finite linear combination of terms of the form
(adL (zj -
Pj)*)Iz
- Pl-Zk1adL1(lz - Pl2)Iz -
Pl-2k2
...
Iz - pl-2keadLe(Iz - PI2)Iz - P1-2ke}1,
(8.19)
with kl + ... + kt+l = m + f, k,,... , ke+i 0, a = a0 + a1 + ... + at, all ... , at 0 0. Here a° may vanish and we then put ado ((zj - Pj)*) _ (zj - Pj)*. We notice that ad' (lz - P12) is a linear combination of terms of the form (ado' with a' _ ,Q4ry'. Recall that lz-PI denotes the positive square root of Iz - PJ2 and let Iz - PI-1 be the inverse of Iz - P1, and hence the square root of lz - Pl-2, the inverse of Iz - P12. If R E Oph(S°(M)), we let Aj = Rhi - Pl-2(i - P)*i E Oph(S°(1)) so that
RAj(i-Pj).Then, RAj(zj -Pj)+B, with B=EAj(i-zj)E OPh(S°(1)), so IIRull <- C(E II (zz - P;)uII + (lull) <- C(Illz - Plull + IIull)
- PIuI12 = Ell (zj - Pj)ull2. Since l lull < II 1 lJlz - Pull, we IImzIIIIz - PIull, so get (with z in some compact set in C): IlRuhh since
11 Iz
IIRiz - PI-1 11
IIa MZJ'
Using this we can now estimate the norm of the expression (8.19) by
As in Proposition 8.4, we then get
(zj - Pj)l z - PI-2m = Oph(aj), with lad aoajI
Ca,p(max(1,
))2-+1IIm
II
(
We have then proved the case N = 0 in:
zl-IaI-IQI-(2m,-I).
(8.20)
108
Spectral Asymptotics in the Semi-Classical Limit
Theorem 8.8. Let f E Co (R'). Then f (PI, ... , Pm) E Oph(S°(M-N)) for every N E N.
Proof. For larger N, we write
f(Al,...,Am) = (m2+Ai+...+A2m)-kfk(Al,...,Am), where we also notice that m2 + A2 + ... + am = Em Ii - Aj 12 when Aj are real. We will clearly get the theorem for N = 2k, if we manage to prove: Ii-Pj-2kfk(Pl,...,Pm),
f(Pl,...,Pm) =
or equivalently (when acting on functions in S):
Ii - pl2kf(pl, ... , Pm) = fk(1 1, ... , Pm). It then suffices to show that
Pjf(F1,...,Pm) = gi(P1,...,Pm), with gj (A) = Aj f (A). Here,
g (1') - Pi f (P)
-Cm J((z - P)* . az.f (z))(zz - pj)I z -
Pl-2"'.L(dz),
and
(zj - Pj)Iz - Pl
-2m = 1
177L8zj I Z
- p12-2m
so we get a constant times
f
((z - P)* - 8,z f (z))8z; (1Z
- p12-2m)L(dz)
_ - f(&J(z) + (z - P)* 8z8; f (z))I z - pj2-2mL(dz)
_ - f(&Iz - p12-2mL(dz) +
f 8f(z)0 ,z o (z - P)* Iz -
p12-2mL(dz).
Here
8z o (z - P)*Iz - p12-2m =rIz-p12-2m+(z-P)*
so we getgj (P) - Pj f (P)=0.
(z-P)(1-m)Iz-Pl-2m =
Iz-PI2-2m
#
8. Functional calculus for pseudodifferential operators
109
When pj - pj,o + hpj,l +... in S° (M), we also get, as in the case of functions of one operator, that
f(Pi,...,Pm) = OPh(a), with
a - ao + hal + h2a2 + ... in S°(M-N), VN, where
ao(x, ) = f (pi,o,... ,Pm,o).
Assume that f (P) has only a discrete spectrum in some open interval I and let p E I be an eigenvalue. Let Fµ C L2 be the corresponding finite dimensional eigenspace. Since Pj and f (P) commute, we see that P3 : Fµ -* F. It is then easy to see that FN, has a basis e1,.. . , eN such that Peek = )j,kek. From this we get f (Pi, ... , Pm)ek = pek = f (A1,k, ..., Am,k)ek
(8.21)
In fact, let us prove the last equality in the slightly more general case when e E L2, Pje = Ale for all j. Then e is in the domain of any power of Iz - P1, and
f(P)e = -Cm f 8 f (z - P)* I z - Pl-2meL(dz)
_ -Cm
ff(z) (z - A)1z - Al-2mL(dz)e
= f(A)e.
Notes Functional calculus occurs frequently in spectral theory and will be useful in the next chapters. There are several different approaches to stating a functional calculus for a pseudodifferential operator. In the case of pseudodifferential operators with discrete spectrum one can use the method of Strichartz [Str], which was adapted by Robert [Ro2] to the case of h-pseudors with compact resolvents. Functions of elliptic selfadjoint operators are studied by Taylor [Ta], using the formula: f (A) = (27r) f '. eitA f(t)dt, and considering eitA as a Fourier integral operator. The Mellin transform was used by Helffer-Robert [HeRol] to prove Theorem 8.7. Using this method Charbonnel [Char2] developed a functional calculus for functions of several commuting h-pseudors.
A functional calculus for bounded operators on a Banach space, based on the Cauchy formula (8.3), was constructed by Dyn'kin [Dyl] and in the case of
110
Spectral Asymptotics in the Semi-Classical Limit
several commuting operators by Droste [Dr]. Any formula which represents a function as a superposition of simpler functions can (at least in principle) be taken as a starting point for a functional calculus (by replacing the argument by an operator). The considerable usefulness of the Cauchy formula (8.3) in practical applications was exploited in [HeSj6] (there combined with so-called Grushin reductions) and in many subsequent works. See for instance [HeSj4], [Da2,3], [JeNa], [Ge], [Sk], [Sj4], [SjZw] and [Dil].
9. Trace class operators and applications of the functional calculus We start by reviewing some standard facts about Hilbert-Schmidt and trace class operators, and refer for instance to [GoKr] for more details. Let E, F
be separable Hilbert spaces and let A E £(E, F) be compact. Then A*A and AA* are compact selfadjoint operators > 0, and they have the same non-vanishing eigenvalues, sl (A)2, S2(A) .. .... with sj (A) \ 0, j --* oo in the case when there are infinitely many such values. The sj (A) are called the characteristic or singular values of A. Definition 9.1. A is Hilbert-Schmidt if IIAIIHS
sj(A)2)2 < oo, and A
is of trace class if IIAIItr = E0° sj(A) < 00.
The space of Hilbert-Schmidt (HS) and trace class operators are complete normed (i.e. Banach) spaces, and a bounded operator A is HS respective of trace class if A* is HS respective of trace class. If (ej), (fj) are O.N. bases in E and F respectively, and (aj,k) is the corresponding matrix of A, then
IIAIIHSla;,k 12IIAe;II2=T
IIA*fjII2.
If B E G(F, H) and A is HS, then BA is HS, and IIBAIIHS < IIBIIIIAIIHS. Similarly, if C E G(D, E) and A is HS, then AC is HS and IIACIIHS IIAIIHSIIc'II
If E = L2(Y, v), F = L2(X, µ), where (Y, v) and (X,,a) are some measure spaces, then A : E -+ F is Hilbert-Schmidt if A is an integral operator of the form: Au(x) = f K(x, y)u(y)v(dy) with a kernel Kin L2 (X X Y;,a x v). Moreover, we have IIKIIL2 = IIAIIHS, when A is such an operator.
For trace class operators, we have
IIAIItr= sup EI(AejI.fi)I, (ei),(f3 )
where (ej) and (fj) are O.N. bases in E, F. (We only consider the non-trivial case, when both E and F are of infinite (countable) dimension.)
If B is a bounded operator and A is of trace class, then BA is of trace class and IIBAIItr <_ IIBIIIIAIItr, and we have the similar result for compositions of
the type AB, still with B bounded and A of trace class. If A : E -> E is of trace class, then tr A F_ (Aej I ej) is independent of the choice of the O.N. basis (ej) and ItrAI IIAIItr. If A : E --> F is of trace class and B : F --> E is bounded, then tr AB = tr BA.
112
Spectral Asymptotics in the Semi-Classical Limit
If A : E -> F and B : F
G are HS, then BA is of trace class and IIBAIItr < IIBIIHSIIAIIHS In the case when G = E, we also have tr BA = tr AB.
The HS operators E - F form a Hilbert space with scalar product: (AI B) = trB*A.
If A = Opt(a), then the distribution kernel of A is K(x, y) = a(tx + (1 t)y, x - y), where a denotes the inverse Fourier transform with respect to the
last variable. The change of variables, x = tx + (1 - t)y,
x - y has a
Jacobian of absolute value 1, so we get:
Proposition 9.2. Let a E S'(R2,) Then Opt(a) is HS iff a E L2(R2"), we .
also have IIOpt(a)IIHs = (2,r)n Jf Ia(x,9)I2dxd6.
The study of pseudors of trace class is slightly more delicate, and we cannot give a complete characterization as in the case of HS operators. We shall only give a sufficient condition for a pseudor to be of trace class. We start
by looking at a class of integral operators. Let Au(x) = f K(x, y)u(y)dy be an integral operator L2(Rn) -4L 2 (Rn), say with K E S(R2n), to start with. We shall estimate the trace class norm of A. Let 1 = E(j k)EZ2^ Xj,k be a partition of unity with Xj,k = T(j,k)X0,0, Xo,o E Co (R2n). Then K(x,y) _ E Kj,k(x, y), where Kj,k(x, y) = Xj,k(x, y)K(x, y). We estimate the trace norm of the operator corresponding to Kj,k, and without loss of generality, we may assume that (j, k) = (0, 0). Choose 0 E Co (Rn) such that O(x)V)(y) = 1 near supp (Ko,o). Then K0,o(x,y) = (27r)1 2n
ff
represents K0,0 as a superposition of rank 1 kernels. In general, if B : u '--k (ul e) f is a rank 1 operator with f, e in some Hilbert space 7-l, then IIBIItr = IIeIIIIfII. Thus if Ao,o is the operator corresponding to K0,0, we get IIAo,olltr <-
(2 )2n
Jf
II2.
We get the same estimates for Aj,k given by Kj,k and we conclude that IIAIItr < C
IIXj,kKIIL', (j,k)EZ2^
(9.1)
9. Trace class operators and the functional calculus
113
where C depends on the partition of unity. From this we get
IIAlltr < C E Ilaf,yKIIL"
(9.2)
IaI<<2n+1
By density, we conclude that if K E S'(R2n ) and if the right hand side of (9.1) is finite, then A is of trace class and we have (9.1) and (9.2) respectively.
Also notice that when the RHS of (9.1) (or (9.2)) is finite, then K is a bounded continuous function so we can define K(x, x) which is integrable. When K E S(R2n), we see (for instance by approximating K by finite rank kernels) that
tr (A) =
J
K(x, x)dx,
(9.3)
and this extends to general Ks for which the RHS of (9.1) is finite.
We next turn to pseudors, and we start with the case of the classical quantization (t = 1). If A = Op, (a), we can write
Au(x) =
(21r)n
f e"£a(x, )Fu(e)de,
and since F is a bounded operator, it is enough to estimate the trace class norm of the operator with kernel K(x, ) = e"£a(x, ). Writing k(x
a (x* - k, :
S))(x* b*) =
Let Xo,o E Co be equal to 1 near the support of Xo,o and put xj,k = Tj,kxo,o Then the expression above becomes e-29
k, S* -j) = e-'j k (27r)1
where Y'j,k = Tj,k?o,o,
0,0 = ezx'£Xo,o Since j,k and,bo,o have the same L1
2n
(4'j,k * X2,ka)(x* - k, e* -j),
norm, we get
a)IILI < CIIXj,kah" We conclude that A is of trace class if IlxkaIILl < 00,
(9.4)
(j,k)EZ2n
or if we make the stronger assumption II
IaI <2n+1
,oalIL= < 00.
(9.5)
Spectral Asymptotics in the Semi-Classical Limit
114
By a density argument, we also see that
ffa(x,)dxd,
tr Op, (a) = (2ir)-
when (9.4) holds. The corresponding formula for the h-quantization is: trO Pl,h(a)
=
( 27Th 1 )7'
ff a(x, )dxdt;.
We next look at the Weyl quantization.
Lemma 9.3. Let 4P(x) be a non-degenerate quadratic form on R' and let 1 = EjEZm Xj, Xj = TjXo, Xo E Co . Then there is a constant C > 0 such that
; II T (xj
(ei
u))IIL1
foralluES. Proof. As in (8.7) we have _77(Xj(e"t *
0(1)
ik) - b)N
f
88-b(j - k))N I xku(1)Id'q.
Consequently, if we fix N < m + 1: II-17(Xj(e" * (Xku)))IIL1 = 0(1) f
1
(n - a.,D (j - k))N
l xu(n)I d'q
and if we sum over j and use the fact that 4) is non-degenerate, we get IIF(Xj(ei'
* (xku)))IIL1 = 0(1)IIXkuIILI.
3
Summing over k, we get (9.8).
If Op (am) = Op, (a), then a =
* a,,,, where 4D is a non-degenerate quadratic form, so if (9.4) holds for a,,,, then it also holds for a, and Cei'D
Ilxj,kaIILI <- C E II Xj,kawll Ll j,k
We then get
j,k
9. Trace class operators and the functional calculus
115
Theorem 9.4. Let a E S'(R2n) satisfy (9.4) (which is a weaker assumption than (9.5)). Then Op (a) is of trace class and IIOp (a)Iltr C Cn E IIXj,kaI L1,
(9.9)
j,kf
tr Op (a) = (2,,)n
JJ
a(x,
(9.10)
Let m > 1 be an order function and let p E S° (m) be real-valued. For simplicity, we assume p to be independent of h. If m is unbounded, we also assume that p+i is elliptic. Let I CC R be an open interval and assume that (p(x, ), I) > 0. We can assume, without loss of generality,
that limp(x, ) - sup I > 0. Let p(x, l;) E S(m) be real-valued coinciding with p for large (x, ), and have the property that p > sup I everywhere. For 0 < h < h° > 0 sufficiently small, p' = pw(x, hD) and jW = p (x, hD) are essentially selfadjoint, with the same domain, and (z-p)-1 is well defined and holomorphic for z in some fixed complex neighborhood of I. If f E Co (I) and if we choose the almost analytic extension f with support sufficiently close to I, we see that f (p`°) = 0, either by abstract theory, since pw has no spectrum in I, or by an integration by parts in the formula analogous to (8.3). Continuing to follow a trick of Dimassi [Dil], we write the resolvent identity for Im z 54 0:
(z-p°)-1=(z-p )-1+(z-p"')-1(pw-p )(z-p )-1' substitute into (8.3) and integrate. We get:
.f(pW)_-IfJ(z)(z - p'(p° -p )(z-p')-1L(dz).
(9.11)
Here pw - pw is of trace class, with trace class norm =O(h-n), so we conclude that f (pw) is of trace class and of trace class norm = 0(h-n). It follows that
tr f (p') = > f (Aj) = 0(h-n). In this argument, we may replace I by a slightly larger interval, and choose f > 0 of class Co with support in the slightly larger interval and equal to 1 on I. It follows that the number of eigenvalues of p' in I is 0(h-n). In order to get further, we observe the following simple result:
Proposition 9.5. Let aj E S°(mj), j = 1, 2 and assume that supp (al) C K1 CC R2n, where K1 is independent of h. Then for every N E N and
C>0:
10a(albha2)I < CN(h"dist ((x, ), K1)-N),
Spectral Asymptotics in the Semi-Classical Limit
116
for dist ((x, ), Kl) > 1 This can be proved either directly by integrations by parts, or by using the pseudor calculus of Chapter 7, with suitable new choices of order functions.
Let X E Co (R2n) be equal to 1 in a neighborhood of supp (p - pw) and consider
f (Pw)(1 - Xw) = -1
f of (z)(z -
pw)-1(pw - p
)(z -
Xw)L(dz).
Using the preceding proposition, we see that all derivatives of the symbol of
(p-p)w(z-p )-1(1-Xw) are O(h'((x, ))-N) for every N E N. The trace class norm of this expression is therefore 0(h°°), and consequently Ilf(Pw)(1 - XW)Iltr = 0(h°°),
in particular, tr f (Pw) = tr (f (PL)Xw) + 0(h°°).
(9.12)
Now recall that
f V) = Oph(a), where a E S°(1), a - a° + hal + h2a2 + ... with a° = f (p(x, )) and with a1 = 0 (assuming that f is independent of h). Moreover, aj (x, l;) vanishes outside supp (p - p). Consequently, we obtain
Theorem 9.6. We have 00
tr f (Pw)
(27rh7
j=0
h' f aj(x,
Here aj have the properties recalled above. In particular a° = f (p) and a1 = 0.
Corollary 9.7. Let [a,/3] CC I be an h-independent sub-interval, and let h) denote the number of eigenvalues of pw in [a,)3]. Then when h--*0: 1
(2 h)n (V ([a, Q]) + a(1)) <- N([a, l3]; h) <where
V ([a, (3]) = lim E\0
f Jf
V ([a, a]) = lim f
J
(27rh)n
(V ([a, a]) + o(1)),
dxd, dxde.
9. Trace class operators and the functional calculus
117
Proof. Assume for simplicity that a < /3. For every small e > 0, choose f, f E Co (I; [0,1]) with 1[a+E,Y-E] C f<
f C 1[«-E,/j+E]
It then suffices to observe that
tr f (p') < N([a, /3]; h) < tr 7(p'), and to apply the preceding theorem.
Notes Theorem 9.4 and its proof are close to the arguments of [Sj1,2]. See also Boulkhemair [Bo]. Theorem 9.6 is due to Helffer-Robert [HeRo] using a functional calculus based on the Mellin transformation. Here we used the Cauchy formula (8.3) together with an idea from [Dil].
10. More precise spectral asymptotics for non-critical Hamiltonians Under an additional assumption, we shall improve Corollary 9.7. The idea
is to study the unitary group e-itP/h and more precisely, we shall study the trace of e-itP/h f (P), where f E Co (R) is independent of h. We assume that P, f, I satisfy the assumptions given after Theorem 9.4. As we could have done already in the preceding chapter, we allow the symbol p of P to depend
on h and assume that it has an asymptotic expansion p " po + hp1 + ... in S°(m), where m > 1 is an order function. As noticed there, we assume without loss of generality that supI > 0. Then the full symbol is bounded from below, and it is easy to prove that for h > 0 sufficiently small, the operator P is bounded from below: P > -Co for some constant Co > 0. Increasing Co if necessary, we may also assume that I C] - Co + 1, Co - 1[. We can find a asmooth real-valued function g(A) such that g(A) = A in a neighborhood of I, g is increasing on [-Co, oo[ and g is equal to a constant plus a Co function. Then g(P) and P will have the same eigenvalues in I. Moreover {p E R2n;po(p) E I} _ {p E R2n;g(po(p)) E I}
and po = 9(po) on this set. Also, g(P) = a9 w, where a9 E S°(1) has an asymptotic expansion in powers of h, and this asymptotic expansion coincides with that of p in a neighborhood of the (compact) set above. This means that as long as we are only interested in the eigenvalues of P in I, we may replace P by g(P), and m by 1. So from now on we may assume that m = 1.
Let f (P) = of (x, hDx; h) and let X E CO' (R2n) be equal to 1 near po 1(I) From the results of Chapter 9, it is easy to show that
f (P)u(x) = (2h)n
ei(=-y)-BIhX(x 9)af (x
y, 0; h)X(y, 0)u(y)dyd9
2
+K(h)u,
(10.1)
where IIK(h)Iltr = 0(h°°).
We want to approximate Vf(t) := e-itP/h f(P) for Itl small by a Fourier integral operator of the form
U1(t)u(x) =
1
(27rh)n
ei(,k(t,x,e)-y.e)lhb(t x, y, 0; h)u(y)dyd0,
(10.2)
where b E C°°(] - c, [; S°(1)) should have uniformly compact support
c should be real, smooth and well-defined near the in (x, y, 0), and 0(t, x, 0) support of b. Trying to solve the equation (hDt + P)Uf = 0, we are led to the problem:
1 e-i¢/h(hDt +P)(eicblhb) E C°°(] - 1, C [;S-(10.3)
120
Spectral Asymptotics in the Semi-Classical Limit eiO/hb!t=o
=
eix OlhX(x,
8)af(x +
2
y, 8; h)X(y, 8),
(10.4)
where P acts in the x variables. We choose 0 to solve
atcb + po(x, e-0) = 0, 01t=0 = x 6,
(10.5)
for Itl small enough and (x, 8) in some sufficiently large but fixed compact set. (Actually, the problem can be solved globally in (x, 8), with 0(t, x, 8) - x 8 E C°°(] - ', [; S(1)).) By integrations by parts and the method of stationary c that if b E C°° (] - c , c [; S° (1)) has uniformly compact support phase, we see
in (x, y, 8), and b - bo + hbl + ..., then
p(ei(Plhb) = ei(P/hd
S° (1)) has an asymptotic expansion
where d E C°°
d - do(t,x,y,8) +dl(t,x,y,8)h+..., with do (t, x, y, 8) = po (x, a.0)bo (t, x, y, 8),
di(t,x,y,8)
a9po(x,aaq) O co+go(x,8)co plus terms depending on pl and cl.
Here qo is a smooth function which can be expressed in terms of po, pi, 0. Moreover, cj vanish for (x, y, 6) outside some fixed compact set, and c and its derivatives are O(hn'((x, y, 8))-N) away from that set. Here we have assumed that 0 has been suitably defined far away, either by solving the eikonal equation globally or by making 0 equal to a constant far away.
Then (hDt + P)(eiO/hb) = e'Olhhe, c E C°°(] - I, I [; S°(1)), and c co + hcl +..., where co =
i atbo + (i 8 (po) (x, 0.0) -0. + qo(t, x, 8))bo.
(10.6)
By solving the usual transport equations, we get b E C°° (] - C , I [; S'(1)) with uniformly compact support in (x, y, 8), such that (hDt + P)(eiolhb) _ eidlhhe, with c E C°°(] - C, c [; S-N(((x, y, 8))-N)) for every N E N. Then
(hDt + P)Uf(t) = K(t; h), Uf(0) = f (P) - K(h),
(10.7)
10. More precise spectral asymptotics
121
with k, K = 0(h°°) in trace norm.
We want to compare Uf(t) with Vf(t) := e-itP/h f(P). Put Wf(t) _ Uf(t) - Vf(t). Then (hDt + P)Wf(t) = 0(h°°), Wf(0) = 0(h°°) in trace norm. Consider eitPlhW f (t):
hDt(eitP/hWf(t)) = eitP/h(hDt +P)Wf(t) = 0(h°°) in trace norm.
Hence, eitP/hWf(t) = 0(h°°) and consequently Wf(t) = 0(h°°) in trace norm, so we obtain:
e-itP/h f(P) = Uf(t) + K(h), where K(h) = 0(h°°) in trace norm. (10.8) We assume, (possibly after shrinking I) that
po(x,e) E I = dpo(x,.) # 0,
(10.9)
J et -P)/hb(t)f(P) dt = tr (Fh 1(VG)(A - P)) f (P),
(10.10)
and we want to study
tr
27rh
when 0 E Coo`(] - c, [), A E I. In view of (10.8) we see that the expression c plus (10.10) is equal to 0(h°°) 1
ei(At+O(t,.,O)-x-O)1hV)(t)b(t,
1
(27rh) (2-7rh)-
1ff
x, x, 0; h)dxd0dt.
At + 0(t, x, 0) - x 0 = t(A - po(x, 0) + 0(t))
(10.11)
(10.12)
with t small. Putting the t derivative equal to 0, we see that po(x, 0) = A + Ca(t), and since dpo # 0, when po = A, the phase is critical for Itl small precisely when t = 0, po (x, 0) = A. Near any such critical point we choose local coordinates t, PO, wl, ... , w2n-1 and consider the Hessian of (10.12) with
respect to t, po at the critical point:
(_1
01)
This is a non-degenerate matrix of determinant -1 and of signature 0. After introducing some partition of unity on the energy surface po(x, 0) = A and applying the method of stationary phase with respect to the t, po-integration,
Spectral Asymptotics in the Semi-Classical Limit
122
we see that the expression (10.11) has an asymptotic expansion in powers of h: (A)hl-n +... (10.13) + tl Here we use the fact that X = 1, b = f (po) + 0(h) in the interesting region, to(,\)h-n
,.,
and get to(A) = 0(0)f (A) (21
JPo_
Idet
wd(x, d(po,
0)w2n-1) I dwl ... dw2n-l, (10.14)
where in order to simplify the notation we neglected to write down the intermediate step with the partition of unity on the energy surface. Recall that the Liouville form on the hypersurface po = A is the 2n - 1 form LA = g,\ (w)dwl A... Adw2n_1 such that LAAdpo = dxAdB, and we shall view La as a density. The definition of the Liouville form does not depend on the choice of local coordinates, since dx A dO is the symplectic volume element and since we have (M A dpo) I p"=;k = 0 at the points where po = A, if M is any 2n - 1 form whose restriction to po = A is zero. (10.14) becomes to(A)
=
0)f(A) fo_a
L,(dw).
(10.15)
The right hand side of (10.10) can be written as
(Fh 1') Pf (A),
(10.16)
" where .Th 1O(A) = hW(h) _ h(A), b = .I'1 lw, µf = Ef(Aj)E(A - Aj), g, with and Aj denote the eigenvalues of P in I. If 0 is of the form g E Co (] - 2c, 2c [) real, g(t) = g(-t), then zb = IgI2 > 0. We may also arrange that Vi(0) = 1 or, equivalently, that f dA = 1. /'
A consequence of (10.16,13,14) is that if N(J; h) denotes the number of eigenvalues of P in a subinterval J of I and if the length IJI of J is h, then
N(J; h) = 0(hl'n). Put 2(A) = fpo-a LA(dw), so that (Fh
10
* µf)(A) = (2 h)n (f (A)f(A) + 0(h)), A E I.
(10.17)
This relation extends to all A E R, and we can there replace 0(h) by 0((-). We integrate this from -oo to A E I:
J
(J 00
h(A' - y)dA')µf (dy)
- (2lrh)n (f[ 1
P.(x,)
f (po(x, £))dxde + 0(h)),
(10.18)
10. More precise spectral asymptotics
123
and here
-00 Oh (A' - y) dA' = f CO
(' h y + t)dt =
(y)
+O((A h y)
According to the observation prior to (10.17), we have
f(A
00
h
y)--,af(dy) = 0(1)(
(k)--)hl-n = O(hl-n),
k=0 so (10.18) gives
f
00
fo(x,£)
1 pf (dy) = (2 irh)n
(10.19)
p(f
We now drop the assumption that all values in I are non-critical for po.
Theorem 10.1. Let [a, b] C I, a < b and assume that a, b are not critical values of po. Then the number of eigenvalues of P in [a, b] is N([a, b]' h)
dxd + O(h)). (2 h)n (fJpo(-,£)E[a b]
Proof. Choose fj E Co (R) such that f2 has its support in ]a, b[, such that fl and f3 have their supports in small neighborhoods of a and b respectively and such that fl + f2 + f3 = 1 on [a, b]. Then we can use (10.19) and its obvious analogue to study fl(Af),
f3 (A1) and A3
Aj>a
while > f2(A1) has already been studied in Chapter 9.
Notes Theorem 10.1 was proved in the case of Schrodinger operators with compact resolvent by Chazarain [Ch], who constructed parametrices for small times
for the associated evolution equation. In the general case the theorem is due to Helffer-Robert [HeRo2], Ivrii [11]. The leading term in the expansion,
`the Weyl term', can be obtained by many other methods. In the case of a selfadjoint elliptic operator of arbitrary positive order on a compact manifold, a Weyl asymptotic with small remainder is due to Hormander [H63].
11. Improvement when the periodic trajectories form a set of measure 0 In this chapter we shall estimate the large time behaviour of the unitary group and see that, under an additional assumption, we can get a two term asymptotic result for the number of eigenvalues in an interval.
As a preparation, we discuss (a special case of) the Egorov theorem for conjugation of a pseudor with a Fourier integral operator (from now on fourior for short). For simplicity we work in trace classes, and only prove what we shall need later.
Let p - po+hp, +... E S°(1) be real-valued. Let q - qo+hql +... E S°(m), where the order function m is bounded and integrable. We shall study Qt = eitP/hQe-itP/h modulo 0(h°°) in trace norm, where P = p" Q = qw To do this we construct an approximation for Qt. Notice that
hDt(Qt) = adpQt,
(11.1)
atgt = h(p hqt - gtOhp), qlt-° = q.
(11.2)
or if we write Qt =qt"':
We look for an approximate solution g - go (t, x, )+hgl(t, x, 6) +... E S° (m) of (11.2) with an error in S-°°(m). We first get atgo = Hpogo, golt=o = qo,
(11.3)
which has the unique solution E S(m),
and the higher order symbols are obtained by solving transport equations of similar type. We get supp (gj (t, )) C exp(-tHpo)(K), if K is the union of the supports of the qj. Choosing a suitable asymptotic sum for g, we obtain a smooth family of operators Qt with (hDt - adp)(Qt) = Rt with Rt E Oph (S-°° (m)) uniformly for t in any compact interval, and with Qo = Q. In particular, IIRtIItr = 0(h°°) uniformly for t in any bounded interval. We write
hDt(e-itP/hQteitP/h) = e-itP/hRteitP/h and integrate in t, and obtain: lie-itP/hj teitP/h
- Qlltr = 0(h°°)
126
Spectral Asymptotics in the Semi-Classical Limit
uniformly on any bounded interval. Consequently,
eitPlhQe-itP/h = Qt + O(h°°) in trace norm,
(11.4)
uniformly for tin any bounded interval. (The more precise and general forms of Egorov's theorem describe eitP/hQe-itP/h more completely as a pseudor, even with e-itP/h replaced by a more general fourior. We give such a result in the appendix to this chapter.)
Let P = p' satisfy essentially the same assumptions as in Chapter 10: p - po+hpl+h2p2+... in S°(m), m > 1, p real, p+i elliptic and h sufficiently small when m is unbounded. We also let I cc R be an open interval and assume that (po(x, t ), I) > 0, 0 < f E CO' (I; R). As before, we may assume that limpo-sup I > 0, and replacing P by g(P) for a suitable we function g, we may also assume that m = 1. Assume that every value in I is non-critical for po (an assumption that will be relaxed in the final result), and take 0 < f E Co (I; R).
We shall study the trace norm of (Th' )(A - P) f (P)Xw for A E I, E Co (R) also when supp (O) is large. We start, X E Co (R2n; [0, 1]), however, with the case when z/i E Co (] - c, [) for C > 0 large enough so c can find E Co (] - c , c [) that the analysis of Chapter 10 can be applied. We of the form g * g with 0 < g E Co (] - ac , 2c [) such that > 0 on the support
of 0, and write '=kO for kECo (] - c,1[).Then
1k*.PhlO
and II(-Tfh1V))(A - P)f(P)X"'IItr
<
J
I1 k(A')IIITh 1O(A - A'- P).f (P)XwII trdA'.
(11.5)
Here the integral over IA'I > Vh- is O(h°°), and for IA'I < v"h-, we write A instead of A - A' and estimate: II(T 1O)(X - P)f (P)Xwlltr = II 27h
feitm(t)f(P)xdtIltr.
Let X E Co (R2n; R) be equal to 1 in a neighborhood of {exp(tH 0)(p); p E supp X, ItI < }. Then by Egorov's theorem above:
c
(1 - j(w)eit(A-P)1h f(P)Xw = 0(h°°) in trace norm, when ItI
and consequently,
P)f(P)Xwlltr = IIXw(Th 1O)(A - P)f(P)Xwlltr +O(h°°) = IIXw(Fh
1O)(A-P)f(P)XWXwlltr+0(h°°)
< IIXW II IIXw(117h 10) (A
P) .f (P)Xwlltr + O(h°°).
11. When the periodic trajectories form a set of measure 0
127
By construction F, 10 > 0. The trace norm and the trace of a positive selfadjoint operator coincide, moreover, using the fact that IXI < 1 we see by the semi-classical sharp Garding inequality (Theorem 7.12) that IIX' 11 < 1 + 0(h), so we get ll(1:h
10)(A- P)f(P)Xwlltr
< (1 + 0(h))tr
P) f (P) X ') + C` (h°°)
Here the trace can be evaluated as in Chapter 10, and we get II(J7 1O)(A - P)f(P)Xwlltr < C(O)h-'L(fp _ IXI2La(dw) +o(1)). po-a
Notice that the integral to the right changes only by o(1) if we replace A by A + o(1). Returning to (11.5), we get ll(Yh 1V)(A - P)f (P)Xwlltr <-
C(O)h-n(f
IXI2La(dw) + o(1)).
(11.6)
po-)'
For to E R, we have Fh 1(Ttp4/')(A) = eit0A/h(.Ph lV))(A) and since is unitary, we see that the LHS of (11.6) does not change, if we replace '% by eito(A-P)
Tto 0.
If 0 E Co (R) is arbitrary, we can write V) as a finite sum of translates of functions in Co (] - c, [) and applying (11.6) to each of these functions, we see that (11.6) is valid cfor general Eli E CO '(J) (J C R is an open bounded interval), with X E Co (R2n) equal to 1 near supp X.
Looking now at a slightly different situation, we keep the assumptions on and X, and assume in addition that b vanishes near 0 and that exp(tHp) (supp x) fl supp x = 0, for t E supp V.
(11.7)
Then fort E supp Vi, if we let Xt E Co be equal to 1 near exp(tHp(,) (supp X), we have by Egorov's theorem: 11(1 - Xt
)e-itP/h f(P)Xwlltr
= 0(h°°).
Hence
tr (e-itP/hf (P)X') = tr (Xt e-itP/hf (P)X-) + 0(h°°) = tr (e-itP/hf (P)XwXt) + C(h-) We can choose Xt with support disjoint from that of X, and conclude that
tr (e-itP/h f(P)Xf) = 0(h°°), for tin a neighborhood of supp 0,
Spectral Asymptotics in the Semi-Classical Limit
128
and consequently, under the assumption (11.7):
tr (.Fh' b)(A - P) f (P) X- = 0(h°°).
(11.8)
Notice that we get the same conclusion if we replace the assumption (11.7) by the assumption that supp X is disjoint from the energy surface po = A (in view of the results of Chapter 9). We now fix an energy AO E I and assume The union of periodic Hpo trajectories in po = AO is of Liouville measure 0. (11.9)
Fix R > 0. Then {p;po(p) = Ao, exptHpo(p) = p, for some 0 < Itl < R} is a closed set of measure 0, and for every e > 0, we can find Xi = XR,E E Co (R2n; [0, 1]) equal to 1 near that set, and a corresponding X E Co (R2n), which is equal to 1 near supp X, such that fpo-ao LXI2Lao (dw) < e. We can then find X2,
Co such that
, XN E
N
Xj is equal to 1 near po 1(supp f ),
X
and for j > 2, we have either exp(tHpo) (supp Xj) n supp Xj = 0 for 21 < Itl < R, .or
supp Xi n {po = Ao } = 0.
Let 0 E Co (] - R, R[) and let o E Co (] - C, [) be equal to
near
c for JA - Aol < e(h), [-2c> 2c] Then for A - AO = o(1) (or more precisely where e(h) > 0 is some fixed function tending to 0, when h -> 0): tr ((I: 1(0 - io))(A - P) f (P)) = tr (( 1( - Vo))(A - P)f (P)Xw) + O(h°°) N
_
tr ((.F
Vo))(A - P) f (P) Xw) + O(hoo)
= tr ((.F
-V)o))(A - P) f (P) xw) + 0(h°°)
= O0(1)h-'(e + o(1)), h -* 0.
Here tr (.F, 1(0o)(A - P) f (P)) has been studied in Chapter 10, and we get for general 0 E Co (R): tr.F'h 1O(A - P)f (P) =
h-n,(,O(O)f
(Ao)
Po=Ao
Lao (dw) + o(1)),
(11.11)
11. When the periodic trajectories form a set of measure 0
129
when A - Ao = o(1), h -> 0.
Let b E Co - C,
() be even with
= 1 near 0, with % :_ Pi 1(41) > 0,
and put h = hh)c = Fh 14p. One of the results of the preceding chapter is then that:
Oh * pf (A) = to(A)h-n + ti (A)h' -n + O(h2-n),
(11.12)
where,uf is defined in Chapter 10 and where to(A) is given in (10.15), t1(A)
has similar properties and depends on the subprincipal symbol of P. See Remark 11.2 for a computation. As with (10.17), this relation extends to all real A and we can replace 0(h2-n) by O(h2-n(A)-n') for every N. Since .7h
4Yeh, we can also apply (11.11) and get
Oeh * Af (A) = (to(A) + o(1))h-n, h --> 0, A - Ao = 0(1)
(11.13)
for every fixed E > 0. This implies that if we restrict our attention to some interval around A0 which shrinks to {Ao} when h -* 0, then the number of eigenvalues in every subinterval of length eh is O(1)Ehl-n, provided that h is small enough depending on E. Let H = 1[o,+,,[ be the Heaviside function and define Oh by
hcbh.
Using the fact that f z/1dy = 1, we get Oh (A) = n h where 0 is an odd function with a jump discontinuity at 0 and which is of class S on each half A axis. We have O(A) = f-.4/b(y)dy when A < 0. We want to study f (Aj),
(H * ILf)(A) = a;
and we know from (11.12) that
H * h * 1f (A) =
To(A)h-n
+T1(A)hl-n
(11.14)
+ O(h2-n),
where Tj = H * t3. We are therefore interested in
H*ch*af -H*µf =hcbh*µf The idea is to approximate hOh by an odd superposition of translations of the function Eh and to use (11.13) to get a cancellation. Let X E Co (R; [0, 1]) We first look be equal to 1 near 0 and even. Put OR = X(*)0, OR = hOR(h).
at h(cbh -
OR)
* µf (A) =
f
(1- X(
RhtN(A
h
t)µf(dt)
(k)-Nh'-n =
= 0(1) IkI>R
0(R)hl-n.
(11.16)
Spectral Asymptotics in the Semi-Classical Limit
130
We next want to replace OR by OR * Eh = (OR * E)h. Here ,kR
* IE(x) - OR (X) = f(R(x - Ey) -
Assume for instance that x > 0. Then the last integral is ON(1)
_
Jy<x/E
_ ON,M(1)
IOR(x - Ey) - 0R (x)I (y)-Ndy + ON(1)
f
x/E oo
f
y>x/E
(y)-Ndy
Max ((x - tfy)-M)EI yl (y)-Ndy + ON(1)(E )1-N.
Here C(x - tcy) (tcy) > (x), so
(x - tEy)-M < C(x) M(tEy)M < C(x)-M(y)M and choosing M, N suitably we get for any (new) given N E N: I0R *
E(x) - OR(x)I = ON(1)(E(x)-N + (E)-N).
We then look at E)-N * µf(Ao)
= (11-R>RJ(Eh)(hE) _N) * uf(A0) + ((1 In order to estimate the first term to the right, we notice that the support of the function 1 [_R R) (hE) can be covered by 0(R) intervals of the length Eh, and by arguments already used before and the observation after (11.13), we
see that the first term to the right in the last equation is 0(Rch1-n). As for the second term, we observe that _Eh
Rh
T._5
and the corresponding convolution is then U` (R-1hl-"''). We conclude that for h > 0 sufficiently small depending on E > 0, we have O(6(E)h1-n,)
Eh )-N * ['f (Ao) =
for some function b(c) which tends to 0 when c tends to 0. We get the same estimate for E(h)-N * pf(Ao), so
h(Oh * Zh - Oh) * I'f (Ao) = 0(6(R,
e)hl-"''),
6 (E, R) --> 0, E -> 0. (11.18)
11. When the periodic trajectories form a set of measure 0
131
Now look at hOR * Oh * µf (Ao) = h f
OR(Y)h * µf) (A0 - y)dy
On the support of OR (y) we have y = OR(h), so we can use (11.13) and get OR,E(1)h1-n = OR,E(1)h1-",
huh * Weh * /tf (Ao) = h f OR(y)dy to(Ao)h-' +
(11.19)
using also the fact that OR is odd. From (11.16,18,19), we get hOh * µf (Ao) =
o(1)(h1-T''),
(11.20)
and combining this with (11.15) and (11.14), we get +T1(Ao)hl-" + o(hl-"`).
H * µf (Ao) = To(Ao)h-n
(11.21)
Theorem 11.1. Let m > 1 be an order function, p - po+hp,+h2p2+... E S° (m) real valued with p + i elliptic when m is unbounded. Let [a, b] C R be
an interval such that lim,_,,,.dist (po(x, l;), [a, b]) > 0 and assume that a, b are not critical values, and that the unions of periodic Hp,, trajectories in the energy surfaces po = a and po = b are of measure 0. Then for h > 0 small enough, the number N([a, b]; h) of eigenvalues of p' in [a, b] satisfies: N([a, b]; h) = (2
where
a=
h)n (f f o(x'£)E[a'b] dxd + ah + o(h)),
f
f
{po=a}
po=b}
(11.22)
pl(x,e)Lb(d(x, ))
Proof. Let a > 0 be small enough so that po has no critical values in
[a-a,a+a]U[b-a,b+a]. Let fl E Co (]a-a,a+a[;[0,1]), f2 E
Co (]a+2,b- 2[; [0,1]), f3 E Co (]b-a,b+a[; [0,1]) satisfy fl+f2+f3 = 1 on [a - 2,b+ 21. Let yo (h) < µ1(h) < ... < µN(h)(h) be the eigenvalues of p' (x, hDx; h) (counted with their multiplicities) in [a - a, b + a]. We have
N([a,b];h) _ E (fi +f2+f3)(µj (h)) a<µj(h)
a
E
a(i (h)
µj(h)
µj(h)
f3(A (h)).(11.23)
Spectral Asymptotics in the Semi-Classical Limit
132
According to Theorem 9.6 we have tr f2 (pw (x, hDx; h) )
(2h)- (f f2 (po (x, e))dxd 0(h2)).
+h f
(11.24)
From (11.21) we have: T1(b)hl-n + o(h1-n), (11.25)
To(b)h-n +
E f3(µ.Y (h)) = H * µf3 (b) = µj (h)
f1(uj(h)) = tr fi(pw) a<j (h)
_ (f f,(po(x, )) (2 +
fl(1i (h)) µj(h)
)-
To (a))
h-n
(f p, (x, )fi (po (x, )) (2)n- - Tl (a)) hl-n + o(hl-n).
(11.26)
Combining (11.23-26), we get (11.22), where it remains to compute a. Let cp E C0 00(]b- a , b+ [), using (11.14) and the fact that 0(h 2-n) is uniform
onAE]b-2,b+2[,wea get
ff =f (
W(T +
(H * µf3)(A)dA =
J(A)H * h * Ap, (A) dA
P(A)To(A)dA h-n + f(A)Ti(A)dA h1-n + O(h2-n).
0 (which follows from the fact that As f T and f 41 (T)dr = 1, it follows that
f
lp(T +
f(hr +
(11.27)
= 1 near zero)
W(A) + 0(h2).
(11.28)
Using (11.27), (11.28) and the fact that H * µf3 (A) = 0(h-n), we get:
f(A)(H * pf3)(A)dA = f W(A)To(A)dA h-n +
f
+ 0(h2-"`). (11.29)
11. When the periodic trajectories form a set of measure 0
133
If ¢(\) = f , W(r)dr, then the left member in the last equation is equal to tr (f3o) (P). From the functional calculus of Chapter 8 (formula (8.16)) we have:
tr (f3O) = (2 h)- If 0(po(X, ))f3(PO(x, ))dxd + (27fh)n if
(11.30)
Using the fact that
f
f(A)f3(A)(f0=A
P1 (X,
we deduce from (11.29), (11.30) that p1(x,)f(po(x,))dxd - f() f T1(A) = f p1(x,)LA(dw). po=a} po{x,£)
(11.31)
Now (11.23-26), (11.31) and the fact that f1 + f2 + f3 = 1 on [a, b] give a = f
fpo=b}
po=a}
Remark 11.2. By definition T1(A) = H * ti (A) = f>i t1(y)dy. From (11.31) we deduce that t1(A) =
aas
T1(A) =
-f3(A)a(f as
(11.32)
po=a
By the same method we can compute all the coefficients, tj (A), of the expression (10.13), see [Roll.
Appendix: Egorov's theorem We proceed in the spirit of the general theory of fouriors, followed in the appendix about metaplectic operators in Chapter 7, but for simplicity we do not develop the global theory. A smooth real valued function, defined in a neighborhood SZ of (xo, 00) E R'ti x RN, with 0'' (xo, 00) = 0 is called a non-degenerate phase function if due, 0, ... , d 90N 0 are linearly independent at every point of Cp,
(A. 1)
134
Spectral Asymptotics in the Semi-Classical Limit
where CO := {(x, 8) E SZ; 0'0(x, 8) = 0}.
(A.2)
Then Ck is a smooth n-dimensional manifold and the map CO E) (x, 8) H (x, 0' (x, 8)) E T*Rn
(A.3)
has injective differential at every point. Possibly after shrinking Il around (x0, 80), we can assume that (A.3) is injective and that the image A4, C T*Rn
is an n-dimensional manifold. As in the appendix to Chapter 7, we can identify dxIAO with d(OlC.), so f dxI Alp is an exact form and al AO _ d( dxIA,,) = 0. Consequently A4, is a Lagrangian submanifold of T*Rn. Let t 0 = 0' (xo, 8o), so that (xo, o) is the image of (xo, 8o) under (A.3).
As in Lemma A.1 of Chapter 7, we see that for almost all real quadratic forms q(x): det Hess ((O(x, 8) - q(x)).,o,oo
(A.4)
0.
Let q be such a quadratic form. Then for rt in a small neighborhood of o, the function (x,8) (A.5) has a unique critical point (x(q), 0(rl)) in some neighborhood of (xo, 00), and (x(0), 0(0)) = (x0, 80). Moreover, this critical point is non-degenerate and
(as in the appendix to Chapter 7) the critical value H(q) depends only on A4,, q and q(xo, 80) (i.e. a normalization constant in 0).
Let a(x, 0; h) E S°(1) have its support in a small neighborhood of (xo, 80), and put 4 f e2O(x,e)/ha(x, 8; h) d8. (A.6) I (a, 0; h) (x) = h- 2 -
J
For q in a neighborhood of o, the method of stationary phase (Chapter 5) shows that fh(e-Zq(x-xo)/hI(a, 0; h)) (77; h) = e2H(77)/hA(r,; h),
(A.7)
where A E S-'/4 satisfies hnl4e2 4
a(x(rl), 8(rl), h) mod S- A(rl; h) = (2ir) 2 +° det(O" - q")(x(77), 8(rl)) 1
n/4
(A.8)
Here t((p, q) is the signature of the matrix in (A.4), i.e. the number of positive eigenvalues minus the number of negative eigenvalues. In (A.8) we have given only the first term in an asymptotic expansion, this term as well as the others all vanish when (x(rl), 8(r])) is outside the support of a and A is of class S-°° in that region.
11. When the periodic trajectories form a set of measure 0
135
For rl outside a neighborhood of to, we can estimate the Fourier transform in (A.7) by integration by parts, and we get in this region: a,a,
Fh(e-29(.-xo)/h j(a,
0; h)) I C
CN,ahN(1l)-N
(A.9)
for all a, N. Let *(x, w) be a second non-degenerate phase function, defined in a neigh-
borhood of (xo, wo) E Rn x RM with (xo, wo) E C,p and assume that ''(xo, wo) = o, V (xo, wo) = O(xo, 9o) and that A0 = A near (x0, 0) Then we have
Proposition A.1. For every sufficiently small neighborhood S2 of (x0,00), there exists a neighborhood 52 of (x0, wo) such that for every a E S°(Rn X R N, 1) with support in Il, there exists b E S°(Rn x RM, 1) with support in St, such that
I(a,0) = I
and conversely for every b
< CN,ahN(x)-N, N > 0, a E Nn
....
(A.11)
there exists a ...
Proof. Observe that Fh(e-i9(x-xo)lhI(b,
0)) (,q; h) = eiH(v)/hB(77; h),
(A.12)
for 77 in a neighborhood of t:o, where B E S-'/4 is given by a formula analogous to (A.8). For a given a with support close enough to (xo, Bo), we can then construct b so that B coincides with A (in (A.7)) modulo S-°° in a neighborhood of eo. Outside such a neighborhood, we have (A.9) for I(a, 0) and similarly for I(b,,b). Then (A.10) follows from Fourier's inversion # formula. Note that the proposition remains valid for classical symbols a E S°\(Rn X RN, 1), b E S° (Rn x RM), where in general we define SkI (Rn, m) to be the space of symbols a(x; h) in S" (Rn, m) (with m some order function
and k E R) with an asymptotic expansion in the latter space: 00
a(x; h) - h-k E h'aj (x). j=o
To complete the picture, we notice (by adapting the argument in the appendix
of Chapter 7) that if A C T*Rn is a Lagrangian manifold, then in a
136
Spectral Asymptotics in the Semi-Classical Limit
neighborhood of any given point (xo, eo) E A, we have A = AO for some non-degenerate phase function.
We now discuss Fourier integral operators, and replace Rx by R2 y = Rn x R. Let (P(x,y,0) be a non-degenerate phase function defined near (xo, yo 9o) E R2n+N with 0'' (xo, yo, Bo) = 0 and assume that A
{(x, ; y, -i7); (x, ; y, 77) E AO}
is the graph of a diffeomorphism ,(p mapping a neighborhood of (yo, i 0) onto a neighborhood of (xo, o), where t=o = 0' (xo, yo, Oo), -77o = (x0, yo, 00) Then rc is a canonical transformation, i.e. a local diffeomorphism with the
property that the pull-back of the symplectic 2 form under t is again the symplectic 2 form. Let J(a, 0) be the operator with distribution kernel I(a, 0), where a is a symbol of class S° with support close to (xo, yo, Bo). We say that the fourior J(a, 0) is associated with rc = ic,5 and of order 0. Notice that the complex adjoint J(a, 0)* is associated with is-1, and that h-pseudors can be viewed as fouriors associated with the identity transformation rc = id.
Here and below, the following observation is useful (and standard in the
theory of pseudors). Let m be an order function on Ran and let b E S°(R3n,m). Then the proof of Lemma 7.8 shows that
Au(x) :_
(2h)n ff
ei(x-y).Olhb(x,
y, 9; h)u(y)dyd9
is a well-defined and continuous operator S --f S and S' -* S' which depends continuously on b.
Applying A under the sign of integration in Fourier's inversion formula for u E S, we see that A = Oph,1(a), where
a(x, ; h) = e-ix E/hA(ei(
Proposition 7.6, applied with the parameter dependent order function m. (y, 0) = m(x, y, 0), shows that
a(x, ; h) _
(2h)n ff e h
y, 0; h)dydO
belongs to S°(R2n, m,), with m(x,) = m(x, x, e), and has the asymptotic expansion in that space:
a(x, ; h) ^'
°°hk 1: (a9 Dy)'(b(x, y, B; h))I 1
0
h1"I
aEN^
a!
b(x, y, ))
l
y-x
-
11. When the periodic trajectories form a set of measure 0
137
Proposition A.2. Let J(a, 0) be as above and let J(b, V)) be a second fourior of order 0 associated with a second canonical transformation ic, mapping a neighborhood of (x0, eo) onto a neighborhood of (z0, (o). Then J(b, 0) o J(a, 0) is a fourior of order 0, associated with is o K.
This result is an immediate consequence of the fact that 0(z, x, w)+O(x, y, 0) is a non-degenerate phase function with (x, w, 0) as fibre variables.
Let J(a, 0) be as above, and assume in addition that a is elliptic near (xo, yo, 00) and (mainly to fix the ideas) that O(x0i yo, 00) = 0. Then J(a, 0)* J(a, 0) and J(a, O)J(a, 0)* are h-pseudors, elliptic near (yo, rjo) and (xo, o) respectively, and we can compose them to the left and the right respectively to get h-pseudors with symbols in S° (1) which are equal to 1 mod S°° in some neighborhoods of (yo, 770) and (x0i l o) respectively. If '(x, y, 0) = -¢(y, x, 0)
is a phase associated with J(a, 0)* (generating is-1), it follows that there exists b E S° with support near (yo, xo, 00), such that
J(b,O)J(a, O) = OPh(jl), J(a, O)J(b,1G) = Oph(j2),
(A.13)
where 31,32 E S°(R2,,1), and jl-1, j2-1 are of class S-°° in a neighborhood of (yo, r7o) and (x0, to) respectively. We say that J(b, z/)) is a microlocal inverse
of J(a, 0). A semi-classical version of Egorov's theorem says that if J(a, 0), J(b, b) are as above, and if P = Oph(p) with p E S°(1), then Q := J(b,,O)PJ(a, 0) is of the form Oph(q) with q E S°(1) and moreover, q = p o it mod S-1(1) in a neighborhood of (yo, r7o).
(A.14)
We leave the proof as an exercise or refer for instance to [GrSj].
Notes Theorem 11.1 is due to Petkov-Robert [PeRo] and Ivrii [I1]. For the proof of (11.22) we have followed [12,3]. For the computation of the coefficient, a, of the expression (11.22) we have used the idea of [PeRo]. When the measure of periodic trajectories is non-zero clustering of eigenvalues may appear, see Petkov-Popov [PePo] for a very general result in this direction, corresponding to earlier results of Guriev-Safarov and Safarov. In the case of a compact manifold without boundary, Duistermaat-Guillemin [DuGu] proved two term asymptotics supposing that the measure of periodic
bicharacteristics equals zero. Under a similar condition on the periodic trajectories, this result was obtained by Ivrii [14] for second order differential operators in compact manifold with boundary. See also Zelditch [Zel].
12. A more general study of the trace In many situations, it is impossible to construct parametrices for small times for the Schrodinger equation associated with an h-pseudor. This happens for instance when we are interested in systems or in boundary value problems. For such problems, Ivrii (see [I1]) made the beautiful observation that one can nevertheless get full asymptotic expansions for the trace. By suitable scaling arguments and hyperbolic energy estimates, he showed that the trace in the
region hS < It! < 1/C is 0(h°°), for 0 < S < 1. Then in the region Iti < hb, he observed that very rough parametrix constructions suffice to get the full asymptotics of the trace. In this chapter, we shall not follow the original approach by Ivrii, but rather develop a corresponding stationary version. Many ideas of Ivrii are still present in this version. A new amusing feature is the very systematic use of almost analytic extensions, and a corresponding absence of propagation estimates. An advantage of this presentation is that
we can also treat situations where the spectral parameter is implicit, and where there is no really natural associated evolution equation, and we get a flexible tool which can be combined with Grushin reductions and effective Hamiltonians (see Chapter 13). For the transparency of the presentation, however, we shall mainly be concerned with the resolvent. We now formulate the assumptions and the results.
Let a < 3 be two real numbers, and let P(h) be an h-pseudor with symbol >o hipj (x, ) in S°(R2n, 1; G(Cm, Cm)) and denote the scalar products on R2n and C'n by (, ) and (, ) respectively. We assume:
p(x, ; h) - >
(H1) P(h) is symmetric for every h E]0, ho]. In particular pj(x, e) is a Hermitian matrix for every j and every (x, 6) E R2n.
(H2) Let A, (x, ) < ... < A,,,,, (x, ) be the eigenvalues of po(x, ) arranged in increasing order. Then lim inf d(Aj (x, ), [a,,3]) > 0
for all j, where d denotes the standard distance on R.
Definition 12.1. We say that po(x, ) is microhyperbolic at (x, e) in the direction T E R2n, if there are constants Co, C1, C2 > 0 such that ((dpo(x, ),T)W,w) ?
1
IIWII2
-
for all (x, ) E R2n with II (x, ) - (x, ) II < 1 and for all w E C'n. (In this definition, we may replace po by any other smooth function with values
Spectral Asymptotics in the Semi-Classical Limit
140
in the Hermitian matrices.) If for some constants Co, C1 > 0 the above estimate holds for all (x, t;) E R2n, w E C'n, we say that po is uniformly microhyperbolic on R2n in the direction T.
For r E R, we put ET
{(x, e) E R2n; det(po - r) = 0}. We introduce the
following assumption:
(H3) (po - r) is microhyperbolic (in some direction) at every point of E, when r E {a, /3}. The assumptions (H1), (H2) imply that if q > 0 is small enough, then (for h > 0 small enough) the spectrum of P(h) in [a - r7, ,3 + 77] is discrete. We choose 77 small, so that (H3) still holds, when T varies in [a -q, a + 77] U [/3 - 77,0 + 77].
Let (Aj(h))o<j
Let 0 E Co (R) have its support in a small neighborhood of the origin and be equal to one near the origin. Let f E Co (]r-r7,T+r7[), T E {a,/3}. Under the assumptions (H1-3), we have,
Theorem 12.2. For every integer N > 1, we have
tr
10)(A
- P)f(P)) = (2-7rh) -n
(f (A) N-1
j-o
hi-yj (A) + 0(hN)),
uniformly for A E [T - 77,,r + 771. Here, ryj are smooth functions of A, independent of 0, f, and for j = 0, we have
'Yo(A)=-2 i
f
where X E C°(R2) is equal to 1 in a neighborhood of
From Theorem 12.2 and the arguments of Chapter 10, we will deduce
Theorem 12.3. We have Nh(a, )3) = (2,h)n 1
ff
ii«,(jl)
dxde + 0(h1-n'), h -, 0.
12. A more general study of the trace
141
The most essential step in the proof of Theorem 12.2 will be the following result:
Proposition 12.4. There exists Co > 0, such that if 9 E Co (] 2,1[), then for every b E]0, 1], we have
tr (f (P) r, 19E(A - P)) = O(h°°),
(12.1)
uniformly for A E]r - 77, T + i[, e E [h1-6, 1 ], where we have written 9(E).
Proof. Let f be an almost analytic extension of f, such that: f E Co (C),
(12.2)
f(z)= f (z) for all z E R,
(12.3)
af(z) = O(IImzIN) for all N E N.
(12.4)
Let 0(t) E C°°(R; [0, 1]) be equal to 1 for t < 1 and equal to 0 for t > 2. Let M > 0 be a sufficiently large constant, to be fixed later, and put Mh ]og
elm z
h
(z) _( Mh log i ). h
We then get
"(f W Mh log h ( O(IImzIN), if Imz < 0 Sl O(%Mh
]Og
r (z)JImzIN +
Mh --T g 7; 1[1,2]
(MhImg )), Imz > 0. (12.5)
The starting point will be the Cauchy formula:
f w)(Th
lee)(X-P) =
-1 f(J
. log
h)(z)(Th'OE)(A-z)(z-P)-1L(dz).
Let X E Co (R 2,; [0, 1]) be equal to 1 in a small neighborhood of and put P(h) = P(h)+ix'(x, hDx)I. By definition of X, (P-z) is elliptic for z in a complex neighborhood K of [r -,q,,r +,q]. Let X1, X2 be two functions in Co (R2n) which are equal to 1 in a neighborhood of supp X. From now on we shall sometimes use the same symbol for an h-pseudor and for its
Weyl-symbol. Replacing (z - P)-1 in the three last terms of the identity:
(z - P)-1 = X1 (Z - P)-1X2 + (1 - X1)(z - P)-1X2 +X1(z - P)-1(1 - X2) + (1 - X1)(z - P)-1(1 - X2)
Spectral Asymptotics in the Semi-Classical Limit
142 by
(z - P)-1 = (z -
P)-1
- (z - P)-1(P - P)(z - P)-1,
and using the fact that (z -
P)-I is holomorphic in K, as well as the fact that supp (P - P) fl supp (1- Xj) = 0 for j = 1, 2, we get, using the cyclicity of the trace: tr (f (P).F/ 18E ((A - P)) _
-tr 1 5(f*- log h)(Z)'rh 1BE(A - z)XI(z - P)-1x2L(dz) + J ... L(dz) - tr 1 -tr ... JIm z>O L(dz) + O(h°°). m z<0
f
(12.6)
7r
Here we also used the (Paley-Wiener) estimate,
f x_10, A -z_ h ( )
O(f)e 2h forlmz<0.
(12.7)
Remark 12.5. In the domain Imz > 0, we have V)Mh log
IImzIN < (2Mh log 1)N
(12.8)
h
for all N > 0, and using also (12.7), we will be able to neglect this contribution from (12.5) to (12.6).
(12.4) and (12.7) imply that the first term of the third member of (12.6) is O(h°°). In view of (12.6) and Remark 12.5, (12.1) will follow from
-tr 17r
f
Djf b
h log
)(z).Fn 18E(A - z)X1(z -
P)-1X2L(dz)
m z> Mh log h
_ 0(h°°).
(12.9)
Xi(x, ) where each Xi has its support in If we choose X1, X2 of the form a small neighborhood of some point in we see that it suffices to show:
Lemma 12.6. For every (i, j) with supp Xi fl supp Xj 54 0, and for every N E N, there exists M(N) > 0, such that for M > M(N): 1
7r
Im z> Mh log
= O(hN).
EF
a(f'pMh log 1)(z).Fh
19E(A
- z)tr (Xi(z - P)-1Xj)L(dz) (12.10)
12. A more general study of the trace
143
In fact, by the cyclicity of the trace, it is easy to see that the corresponding integrals with supp Xl fl supp Xk = 0, are 0(h°°). In the following, we fix (i, j) as in the lemma. To prove Lemma 12.6, we need
Lemma 12.7. Let P = p(x, hD,,) + P(h) - po(x, hD,:) be selfadjoint, where p E S(R2n,1; C(Cm, C"°)) and p(x, ) = po(x, ) in a small neighborhood of supp Xi U supp Xj. Then for every N E N, there exists M(N) > 0, such that
for M > M(N):
tr
1
J Im z> Mh log' h
tr 1 7n
D (,/o M log h)('z)Tjl leE(^ -'z)(Xi(z - P)-1Xj)L(dz)-
JI'.z>Mh log a(f
logh )(z).F
19E(X
- z)(Xi(z - P)-1Xj)L(dz)
= O(hN).
(12.11)
Proof of Lemma 12.7. Let X E C'°(R2) be equal to one in a small neighborhood of supp Xj U supp Xj and have its support contained in the interior of an h-independent set where the symbols of P(h) and P(h) coincide. The identity
(z - P)X(z -
P)-IXj
_
Xj+ X(P - P)(z - P)-IXj- [P, ](z - P)-IXj+ 0(ht),
(12.12)
and the fact that XjX = Xl +0(h°°) in trace norm, imply
Xi(Z - P)-IXi - Xi(Z - P)-IXj = Xi(Z -
+0(II
P)-IX(P
- P)(z -
P)-1X1 - Xi(Z - P)-1[P, X- (z
- P)-IX.;
oo
(12.13)
Here the first term of the second member is 0(h°°IIm z1-2) in trace norm. For the second term, we shall use the fact that modulo 0(h°°) in trace norm [P, x]
has a symbol with support in a compact set k such that supp Xi fl k = 0. Let Go E Co (R2n) be real-valued and such that
Go = 1 near supp Xi, Go = 0 near K. Put G = aGo, a > 0. We notice that the symbol a = eG 11g * is of class Sb (1) for every 6 > 0. By eG log 7' we also denote the corresponding h-pseudor,
Spectral Asymptotics in the Semi-Classical Limit
144
which is elliptic and has an inverse operator same classes. It is clear that for some k E N:
(eGlogh)-1 with symbol in the
eGlog*(z-P)(eGlog*)-1 =z-P+O(ah log-)IIVCoIIck,
(12.14)
in operator norm, for h < h(a), where h(a) > 0 is some continuous function.
Using the fact that G = a near supp Xi and that G = 0 near K, we get in trace norm: ec> log*xi(z -.P)-'[P, X-1
G log h i (z - P) = Xie
1
(e
00
G log I
fi)
1
[P, X-] +
'l
=Xi(z-e G log 77 P(e G log h ) - )[P, a = min(
Imz Ch log
1
zl
)
h-
(12.15)
X-1
O(1)),
where 0(1) is some arbitrarily large and fixed constant and C is sufficiently large. Then the expression (12.15) is 0((h' IImzl) in trace norm, and we get with a new constant Cl > 0:
Xi(z - P)-1 [P,
= O(h-nlIm
zl-1e-O' log h )
=O(h-'IImzl-lmax(h°(1),e I)). < Imz < 2 Mh log h, we get, by using also (12.7):
For Mh log h
(Th 1M(A - z)Xi(z - P)
1P, X1 _ I O(eh-'-1IIm zl-1 max(h°(1), e- T1 IM z(*
I
(12.16)
where we recall that e < Co. We choose Co > Cl. Then the LHS of (12.11) is
O(hm) +
f
e
O(1)( Mh
log ti1
`
)2Eh--1 max(h°(1) e-'L
-))L(dz)
log *:51m.:52AL& log jRe z I
= 0(h°°) + O(1)
e 1
eh-"-1 max(h°(1),
e-cOM(log h)(*-*)).
Mh log h
Choosing first M sufficiently large and then the exponent in h°(1) sufficiently large, we see that this is O(hN) for any given N. #
Proof of Lemma 12.6. Combining Lemma 12.7 with the result of the appendix to this chapter, we see that we may assume (after changing the symbol
12. A more general study of the trace
145
away from supp Xi U supp Xj), that the leading symbol p has the property that for r E {a, i3}, (,r - p) is globally and uniformly microhyperbolic with respect to HO, where 0 is a linear function which may depend on r. After a linear canonical transformation (see the appendix to Chapter 7), we may assume that 0 = x1.
For t E R, we put Xk,t = e-ate/hXkeit(k/h, k = Pt = e-itO1hpeit(k/h.
ij,
Notice that the symbol of Pt is obtained from that of P by translating by -t in the b1-direction. Introducing almost analytic extensions of the symbols Xi, Xj, P, we can define Xj,t, Xk,t, Pt for t E C, such that J=Q = O(JImtl°°) in S°,
where Qt denotes one of Xi,t, Xj,t, Pt, and such that Xj,t have uniformly compact support in (x, t;) whenever t is restricted to a compact set. We put
Qt =
iRetO/h e--
Qilmte
iRet4'/h
Lemma 12.8. We have in S°: atQ
O(IImtl°°),
(12.17)
Ot Qt = 0(1), k e N,
(12.18)
Qt - Qt = O(IImtlO°).
(12.19)
Proof. Qt satisfies: (*)
hDtQt + [0, Qt] = 0 for t real at Qt = Vt Qt = 0(1) Vk E N.
and this implies that hDtQt + [0, Qt] = O(hIIm t100). Also if Qt is a CO° function of t with values in S° satisfying (*) and Qo = Qo, then
Qt - Qt = O(JImtl°°) in So.
Spectral Asymptotics in the Semi-Classical Limit
146
Let us verify that Qt is such a solution: We have:
a Qt = -ie-iRe tc/h [0, QiIm t}eiRe t¢/h a a (QiIm haimtQt = e -iRe to/h halmt h
aRe t
t)eiRe tcL/h
From the construction of Qt, we have
[0,QiImt _
a h(aQ)ilmt+hO(jImtj°°) h
i aIm t
(QiIm t) + hO(IIm tl°°),
and it follows that at Qt = O(IImtl°°), so Qt is a solution of (*). Then (12.19)
#
follows.
We put k(t, z)
= tr (Xi,t(z - Pt)-lk7,t)
k is well defined in the zone jImtj < ?1F011mzI and has the following properties:
k(t, z) = 0(h-' Jim zj -1), k(t, z) = O(h-' Jim tj°° Jim zI-2). Ft Since k(t,z) only depends on Imt, it follows that ao
k(iImt, z) = k(0, z) + O(himtjz12 We can then replace tr (Xi (z - P) - 1 Xj) in the integral in the first member of (12.10) by k(iC EE log h, z), where Cl does not depend on the other constants.
By construction of Pt we see that the leading symbol of z - Pilm t is of the form
z - piImt(x,
+O(1Imt12),
writing pilm t instead of po,ilm t for short. Put i = Im t - EE log(-!) and let - 2 < Im z < 0. Using the global and uniform microhyperbolicity (Definition 12.1) with T = (0; (-1, 0, 0, ... , 0)), we obtain for IRe z - rI < rt: Im (z - piµ(x, t;)) > (2it + Im z)I - O(1)µ(Re z - p(x, e))2
12. A more general study of the trace
147
in the sense of Hermitian matrices, where we assumed for simplicity that the constant Co in Definition 12.1 is equal to 2. Then, if h is small enough, there exists a constant C2 > 0 such that, IM (z (A + Im z)I, C2µ(2 (12.20)
where the star indicates that we take the usual complex adjoint. Now the semi-classical version of the sharp Garding inequality (see Theorem 7.12) implies,
IM (GPh (z - pi,)uI u) + CWIIOPh(z - piµ)uII2 (GPh '(Im (-pz,7)
+C2p(z -ptµ)*(z -pjµ))uIu)
>- (µ + Im Z) IIUI12 - O(1) (h + ph) II UI12 >
It
- O(1)hiIIUII2
IIuII2,
(12.21)
for all u E L2(Rn; Ctm), for h small enough. (12.21) combined with the inequality ab < s a2 + 2µb2 gives u IIuI12 <_ IIOPh(z - piµ)uII I uII + C1ulloph(z - p2, )UII2
-+C1µ)Iloph(z-Piµ)uII2.
6IIUII2+(
Then there exists C2 > 0 such that, (12.22)
C2 IIuII <_ IIoPh (z - pzµ)uII,
for all u E L2(Rn; Cm). We conclude that (z - P log 1)-1 extends as a holomorphic function of z to the zone Im z > - I log h and consequently the LHS of (12.10) is equal to 7r
Imz<0
a(f P- log
=C
I)(z).Fh 1BE(. - z)k(i
log
Mh
log h, z)L(dz)
+O(hN), where we define Oo,p = i(- Q z)(I «z ). Using Remark 12.5, which extends to the zone Im z < 0, and the fact that Mh k(iC log h,z) = O(1)(h-nelMhlog h)
in the zone -
log 1
*71
lmz<0
< Im z < 0, we get
h -
f8(Wn?h log 1,
log*).b9 E(A- z)k(i
h-n
I
= 0(1) (Mh log h)2
log
1
E2h-n-2
- 2-CMg log
1
-O(1)Mloghe
I
=
-n.-2+ M 2c? 0(j),2 E2h 1
(
log h,z)L(dz)
e 217L(dz)
R. =El' -7,'r+17[,
log Qm z<-
Mh
Mlogh
Spectral Asymptotics in the Semi-Classical Limit
148
To finish the proof, it suffices to choose M sufficiently large. This also ends the proof of Proposition 12.4. ## Remark 12.9. Proposition 12.4 remains valid if supp 6 C] - 1, -1 2[.
Proof of Theorem 12.2. In the following we let C be a large fixed constant and 6 a constant in ] 0, 1 [. According to the preceding remark and Proposition 12.1, we have:
tr (F18(\ - P) f (P)) =
-tr 17r
J
a(0h6f)(z)Th 16E(A - z)(z - P)-1XL(dz) +O(h°°). (12.23)
Here X, 6 are the functions given in Theorem 12.2, e = h1-6 and zvh6 = 1/i(I) where 1li E Co (] - 2,2[) is equal to 1 on [-1 1]. In fact, we can represent (9 - 8E) as a sum of functions of the type appearing in Proposition
2.4, Remark 12.9, which leads to (12.23) without X = Xw(x, hDx). The insertion of X can then be justified in the same way as the insertion of X1, X2 in (12.6). Since a(?,bh6 f) = i,1' oaf + f a(Jh6 ), we can again use Remark 12.5 to reduce the domain of integration in (12.22) to K6 = {z E C; IImzj > h6}.
Here we also use the fact that Fh 196(A - z) = O(he%IImzI)/isby PayleyWiener. According to Proposition 8.6, the symbol of (z - P) in K6 by:
Eo(x, ) + hEl (x, t:, z) +... + hNEN (x, , z) +
O(hN(1-26) ).
given for z (12.24)
Here Ej (x, , z) is a finite sum of terms of the form (z -
po)
with k < 2j + 1 and bk E SO(R272,1; L(Cm, Ctm)) uniformly in z. We then get:
tr ((Ph 10) (X - P) f (P)) _
(2irh)n
hj a + O(h N(1-26)-n ),
(12.25)
j=o
where
aj = - 1
J
tr
z)Ej (x, , z)X(x,
Lemma 12.10. The functions ej (z) = if tr Ej (x, t;, z)X(x,
(12.26)
defined
on ]T - rl, T +7l[±i]0, a[ for some constant a > 0, extend to C°°-functions on IT - 7717* + 77[±2[O, a[.
12. A more general study of the trace
149
Proof. For the simplicity of the notation, we only treat the case j = 0. Decomposing X by a partition of unity, we may also assume that X(x, ) has its support in some small neighborhood of some point in 1"[--,7,-+,71- Let U be a constant direction of microhyperbolicity for po - A in a neighborhood of supp X.
Let po be an almost analytic extension of po. As with the formula (12.22),
we show that (z -
itU))-1
exists and is of norm O(t) for
Re z E]T - rt, T + r7[, Im z > 0, t > 0, Im z + t > 0. Stokes' formula gives for Imz > 0:
G(z) = ff(z - po(x, ))-1X(x, ) dx<
ff2' -itoU (z - po)-1X(x, )dxd +
fff r*(((z - po)-lapo(z - po)-1 + (z - po)
1
)A
o
where F : (t, x, ) '--f (x, ) - itU and X is an almost analytic extension of X. Here the first term of the last member extends to a C°° function on ]r - 77, 7- + 77[+i]0, a]. The same is true for the last term, since the integrand is 0(t°°) as well as all its derivatives with respect to z. We have then proved
that eo extends to a smooth function on IT - 97,,r + r7[+i[0, a[. Using the Hermitian property we also get the extension to IT -q, T + i1 [-i [0, CV[.
#
From (12.26), Lemma 12.10 and Stokes' formula, we get
aj = - 121rifJ(A/)F_1o( A - A')(ej (A' + io) - ej (A' - i0))dA'.
(12.27)
Here Th 10, = Th7EB = F-1(o(h')) = So + O(h°°) E
in the sense of distributions, so (12.27) gives aj
2 - f(A)(e1(A + i0) - ej(A - i0)) + O(h°°),
(12.28)
7ri
and we get Theorem 12.2 with 2I (ej (A + i0) - ej(A - iO)).
ryj(A) = -
7ri
(12.29)
Spectral Asymptotics in the Semi-Classical Limit
150
Notice that for f E Co (]rr -q, T' +,q[), (x, e) E supp x: 1
f (A) (tr (A + i0 - po(x, )) 1 - tr (A - i0 - po(x, )) 1)dA
2'7ri
_
f(Aj(x, j=1
Consequently,
f
f (A) Yo(A)dA _ m
f f f (,\j (x, ))x(x> )dxdt _
j=1
ff f (,Nj (x, ))dxd . j=1 (12.30)
Proof of Theorem 12.3. Let f1 E CO '(]a - rt, a + r)[; [0, 1]), f2 E Co (]a + 2, 0 - 2 [ ; [0,1]) f 3 E
[ 0,
satisfy fl + f2 + f3 = 1 on
[a - 2, 3 + 2]. Let Ao(h) < A, (h) < ... < AN(h) be the eigenvalues of P(h) (counted with their multiplicity) in [a - 77, 0 + TI]. We have Nh(a,13) _
(f1 + f2 + f3)(Aj(h))
a
= E f1(Aj(h)) + E f2(.j(h)) + E f3(Aj(h)) a
= E fl(.j(h)) +tr f2(P(h)) + Aj(h)
f3(Aj(h)) (12.31) Aj(h)5,3
According to Theorem 9.6, we have:
tr f2(P(h)) =
(27th)-n
If f2(Aj (x, ))dxdt; + O(hl-n). E j=1
(12.32)
Using the argument of Chapter 10 (Theorem 10.1), we have: fl (Aj (h)) = (27Th)-' a
Ja
+ f1(A)-yo(A)dA + O(hl-n),
(12.33)
a
f3(Aj(h)) = (27rh) -n f Aj(h)
A pproaching
+ O(h1-?').
(12.34)
0o
fl, f f_00 B] f3 from above and from below by Co -functions
and using the fact that -yo(A) is continuous, we get: Q
m
f f3(A)'yo(A)dA_ E f 00
j=1
f
(12.35)
12. A more general study of the trace
f
"0 .fl(A)yo(A)dA=
j=1
ff
151
(12.36)
and combining this with (12.31), (12.32), we get Theorem 12.3.
Appendix: Extension of a microhyperbolic function Let p(x) be smooth in a neighborhood of 0 E Rn with values in the Hermitian
m x m-matrices, and assume that p(x)is microhyperbolic in the constant direction t E R', so that t(ar)p
-
(t,axp(x))
-C1(p(x))2,
(A.1)
for some Co > 0 in the sense of Hermitian matrices. After a conjugation with
a constant unitary transformation, we may assume that p takes the block matrix form p(x) =
p11(x)
p21(x)
p12(x)
P22(X))
(A.2)
with p11(0) = 0, p21(0) = 0, p12(0) = 0 and with p22(0) bijective. Put po(x) _
((dPll(O)x)
(0
0)
)
pz
0
Lemma A.1. po(x) is uniformly microhyperbolic on Rn in the direction t.
Proof. Let
7o- (0 0) Then for every 6 E]0, 1], 0
t(ax)po = (t, dp(0)) + (0(1) 0(1) )
> 1 - Cl(p(0))2 -
0(1)(67r2 +
(I - lro)2).
Choosing 6 > 0 small enough, we get t(ax)po ? 2I
uniformly with respect to x.
-
0(1)(po(x))2,
(A.3)
#
152
Spectral Asymptotics in the Semi-Classical Limit
Let X E Co (R"; [0, 1]) be equal to 1 near 0 and put with a (new) sufficiently small 6 > 0:
ps(x) = X(s)(p(x) - po(x)) + po(x).
(A.4)
If jxj = 0(6), we have p6 (x) = (0
f (x)) + 0(6),
where f (x), f (x)-1 = 0(1), uniformly with respect to x. Hence, p6(x)2 = (0(62) 0(6)
f (x)2
+ 0(6))
>(
\
0(62) 0
Using the fact that p6(x) = po(x) for jxj > 0(6), we see that (A.5) extends uniformly to all x E RTh.
For 6 small enough, we have OM P6
((tPii(0)) + 0(6) 0(1) (0(1)
0(1)
and the argument that gave (A.3) now shows that
t(846 ? k'- (0 0(1)) Thanks to (A.5), we get for 6 small enough,
t(Mp6 > 4I I - 0(1)p6(x)2.
(A.6)
We have then proved
Lemma A.2. If 6 > 0 is small enough, then p6 is uniformly microhyperbolic in the direction t.
In the following, we let 6 be a small fixed constant. We need to further modify p6 away from x = 0 in order to become of class S(1). Notice that all the derivatives of p6 are uniformly bounded on R'2. If f E Co (R), we know from Chapter 8, that
f(p6(x)) = --
fJ(z)(z -p6(x))-'L(dz)
(A.7)
12. A more general study of the trace
153
is in S(1), and if we let f vary in a class of Co (R) functions with uniform bounds on the diameter of the supports and on the supremum of every derivative, then f (p6) varies in a bounded set in S(1). If f E S(R, 1), then we can find such a class such that for every xo E R', we
can find fl, ... , f, in the class such that .f (P6 (X)) = f1(p6(x)) + ... + frn.(po(x))
for x in some neighborhood of xo. It follows that f (p6) E S(R',1). Now, choose f E S(R, 1) real-valued, such that f (t) = t for Itl _< 1, f (t) > 1 for t > 1, f (t) < -1 for t < -1, and put p(x) = f (p6 (x)). Without any loss of generality, we may assume that the norm of p6 (x) is smaller than 1 for x in some neighborhood of 0.
Proposition A.3. (i) p(x) = p(x) in a small neighborhood of 0, (ii) p E S(R", 1; L (Cm Cm)) (iii) p is globally and uniformly microhyperbolic in the direction t.
Proof. (i) follows from the construction and we have verified (ii).
2], such that ±[s, s +X111] are For every xo E Rn, we can find s E disjoint from the spectrum of p6(x) for x in some neighborhood W,0 of xo. [(1),
(Here the 0(1) is uniform with respect to xo.) Then 7r = ir(x) is C°° and p = lrp6lr + (1 - 7r)a(1 - 7r), [a, 7r] = 0,
(p6(x)) (A.8)
where 0 < a, 8xaa, a-1, 8c,it are 0,,(1) for x E Woo, uniformly with respect to xo. Let A, B be Hermitian matrices. Then for every a > 0, we have:
AB + BA > -aA2 - 1 B2.
a
(A.9)
From (A.8), we get, writing t(ax) = t, t(p) = t(lr)p6lr + 7rp6t(7r) + irt(p6)ir-
t(ir)a(1 -7r) - (1 - 7r)at(ir) - (1 - ir)t(a)(1 - of .10) Using (A.6), as well as the fact that 7r2 = ir, we get with a new constants Co, C1 > 0, 7rt(p6)ir >
Co
- C1irp67r.
(A.11)
154
Spectral Asymptotics in the Semi-Classical Limit
From (A.8) and the properties of a, we obtain 7rt(pb)7r >-
I-
O(1)p
(A.12)
The argument used above, combined with (A.9), gives for arbitrary a > 0, 7rp,5 t(7r) + t(ir)pblr > -a - 0 (1)p
a
-t(7r)a(1 - 7r) - (1 - ir)at(7r) > -a -
0(1) p a
(1 - 7r)t(a)(1 - 7r) > -O(1)p . Choosing a sufficiently small, we obtain
t(ar)p ? 21 1 -
0(1)p
.
(A.13)
Notes Trace formulae have been studied and used by many authors, see for instance
[CdV2], [DuGu]. In the semi-classical regime, a trace formula has been studied in detail by Chazarain for the Schrodinger operator and by HelfferRobert [HeRol] and Ivrii [I1] for a general class of h-pseudors. We mention also the papers of Brummelhuis-Uribe [BrUr] and Petkov-Popov [PePo]. A trace formula for several commuting operators was established by Colin de Verdiere [CdV3], Uribe and Zelditch [UrZe]. See also the recent paper of Charbonnel-Popov [CharPo]. The presentation of this chapter follows a paper of [DiSj]. The present work is generalized to the case where the dependence on spectral parameter is non-linear in [Di3]. Applications for the periodic Schrodinger operator with slowly and strong varying perturbations are treated in [Di3].
13. Spectral theory for perturbed periodic problems Let F = ®' 1 Zej be the lattice generated by some basis e1 , .. , en in R. .1
Consider the Schrodinger operator n
(Dyj + Aj(hy))2 +V(y) + cp(hy), (h > 0, h -> 0),
PA,w =
where v is F-periodic: v (x + y) = v (x), dry E F, and cp is bounded with all its derivatives. A(x) = (Al (x), ... , A, (x)) is a magnetic potential such that all derivatives of non-vanishing order are bounded. In solid state physics, the Hamiltonian PA,,, describes the motion of an electron in a periodic crystal with external electric and magnetic fields. Such problems arise naturally in the investigation of impurity levels in the one-electron model of solids, and
in particular in the theory of the colour of crystals. We refer the reader to [DHJ. Let F* = {y* E Rn; y* y E 27rZ, Vy E F} be the dual lattice
so that F* _ EB i Zei, where ei is the dual basis, ei ek = 6ik27r. For A2O < ... be the eigenvalues of the E Rn/F*, let a fixed operator (Dy + )2 + V(y) L2(Rn/F) - L2(R' /F). It is well-known :
(see [ReSi]) that the spectrum of the non-perturbed periodic Schrodinger operator, Po = -A + V (y), consists of the closed intervals J1 = Ai (Rn/F* ), J2 = A2(Rn/F*),...There are many papers dealing with different aspects of the spectral theory of PA,, (see [ADH], [ReSi] and [Bil,2]). To study the spectrum of PA,w we use the method of the effective Hamiltonian. This method was introduced in solid state physics and has subsequently been used by many people: Buslaev [Bull, Guillot-Ralston-Trubowitz [GRT], Nenciu [Nel,2], Helffer-Sjostrand [HeSj6], Gerard-Martinez-Sjostrand [GMS] etc. The effective Hamiltonian approximation is to replace, for h small, PA,w by the collection of h-pseudors:
A(x)) + p(x) for j E N,
(13.0)
are the Bloch eigenvalues described above. In the case of Schrowhere )j dinger operators with constant magnetic fields and no external electric field (i.e. when Aj are linear and (p = 0), rigorous reductions from PA = PA,O to (13.0) have been given by Nenciu [Nel], and Helffer-Sjostrand [HeSj6]. To construct asymptotic solutions, u, of PA,wu = .ou near some energy
level Ao, Buslaev (also Guillot-Ralston-Trubowitz) uses the following idea: if u(x, y) E D'(R,,, x Ry) is a solution of
P(x, y, hD,: + Dy + A(x))u n
_ (E(hD.j + Dyj + Aj(x))2 + V(y) +
156
Spectral Asymptotics in the Semi-Classical Limit
which is r-periodic in y, then u = u(hy, y) satisfies PA,wu = Aou. They construct u by considering P(x, y, hDx+Dy+A(x)) as an h- (pseudo) differential
operator in x with operator valued symbol. This in turn is related to the study of operators of the form (13.0). Gerard-Martinez-Sjostrand [GMS] use this idea to relate the spectrum of PA,w near an energy level )° to that of P = P(x, y, They obtain an N x N system of h-pseudors of order 0, E'+ (x, hD,,, + A(x), Ao; h) (which is the effective Hamiltonian) such that the symbol E_+ (x, t;, AO; h) is F*-periodic in and has an asymptotic expansion in powers of h in S°(R2n, G(CN, CN)). Here and in the following, the order function will be 1 when nothing else is indicated, and we then write S°(R2n) = S°(R2n, 1). The spectra of PA,w and P are related by: A E Q(PA,w)
0 E u(Ew+(x, hDx + A(x), A; h)),
for A in a neighborhood of A0, where this last operator is considered as a bounded operator: UN - UN and V° is the Hilbert space {u = EryEr cyS(xh-y); (c7)-,Er E 12(P)}. One can use the effective Hamiltonian to prove that a gap in the spectrum of P° is stable under small perturbations of the magnetic field (see Proposition 13.27), a result due to Avron-Simon [AvSi] and Nenciu [Ne2]. This reduction from (PA,w - A) to E_'+(x, hDx + A(x), AO; h) is used
by Dimassi [Dil] to study the discrete spectrum of PA,w. He obtains an asymptotic expansion in powers of h of tr f (PA,w) for f E Co (I), where I is an interval disjoint from the essential spectrum. In this chapter, we will develop the method of the effective Hamiltonian to the study of the spectrum
of P = P(hy, y, Dy + A(hy)) on Rn (h > 0, h -> 0), where P(x, y, rl)
is
elliptic, periodic in y and has smooth bounded coefficients in (x, y). A(x) is a magnetic potential with bounded derivatives. We follow essentially the papers of [GMS] and [Dil]. Let P(x, y, 77) E C°° (R3n) be real valued and have the following properties:
(H1) P(x,y,77) _ Ela,<maa(x,y)77a,
(H2) as(x,y) = as(x,y+ry),Vlal < m,V'y E I', where r is a lattice ®i 1 Zei for a basis (el, e2, ... , en) of Rn. (H3) I 6r.1By as (x, y) l <_ Ca,Q,7 da, ,S,1' E Nn, la l < m.
(H4) pm (x, y, 77)
Ial=m a. (x, y)rla
co
I771
' for some Co > 0.
Let A(x) = (Al (x), A2 (x), ... , An (x)) E C°° (R"; Rn ). We assume that:
13. Spectral theory for perturbed periodic problems
157
(H5) Va E Nn \ {0} there exists Ca such that Ie,,A(x)I < Ca. For m E N, we put H,9 = {u E L2(Rn); (hDx + A(x))au E L2(Rn), b IaH < m}, £m = {u E L2(R2n); (Dy + hDx + A(x))au E L2(R2n) d Ian < m},
which are Hilbert spaces with the natural norm.
Notice that the commutator 1 8Ak _ 5Aj [Dx, + A Dxk + Ak] = i (ax7 axk )'
is a C°° function which is bounded with all its derivatives. Hence, if we change the order of the factors (hDxj + A; (x)) in the definition of HT, the space HA (as a vector space) does not change and only its norm changes into an equivalent norm. Let X E Co (B(0, 2)) with x(x) = 1 for IxI < 1, where B(x,r) = {y E Rn; Iy - xI < r}. Put Xj (x) = X(x/j). For u E HA and IaH < m we have: hIQI C,a,a(hDaxj)(hDx +
[(hD1 + A(x))a, x,lu =
A(x))a-Ou,
000,I/3I
where Ca,p are constants. As supp (DaXj) C {x E Rn; (j < IxI < 2j)} and I(D'3X;)I < CCj-I' I for all 03 E Nn \ {0} with C,3 independent of j, the right hand side of the last equality tends to zero in L2(Rn). Hence, every u in HA can be approximated by elements with compact support, and each u in HA with compact support can be approximated by Co -functions by means of a standard regularization. Then we have proved:
Proposition 13.1. HA is a Hilbert space in which Co is dense. Suppose that P is independent of y, and put:
Op' (P) = Pv'(x, hDx + A(x)).
Proposition 13.2. If we consider Opw (P) as a symmetric operator on L2(Rn) with domain Co (Rn), then Op'(P) is essentially selfadjoint and the domain of the selfadjoint extension is HA (Rn).
Proof. Here it is enough to treat the case of a fixed value of h, say 1. Using the composition formula (7.14) for Weyl quantization and the fact that A(x)
158
Spectral Asymptotics in the Semi-Classical Limit
has bounded derivatives as well as the fact that for any a E N', a E N2'", 1,31 > 1, we have
A(x))') _
aa,a,7(x)( + A(x))', I71
where the as p,7(x) and all their derivatives are bounded functions on R', we see that for any a E Nm and any function a(x) bounded with all its derivatives,
a(x)(Dx + A(x))' = [a(x)( + A(x))a]w +
A(x))13]w+ 1/31
(13.1)
where the ba,p and all their derivatives are bounded. From (13.1) we deduce
[a(x)( + A(x))a]w = a(x)(Dx + A(x))a + E ca,p(x)(Dx + A(x))13, IQI
(13.2)
where the ca,p have the same property as the ba,p. Then OpA (P) can be written:
ba(x)(Dx+A(x))a,
Opwq(P)=
(13.3)
IaI<m
where ba and all their derivatives are bounded on R'. If we fix xo E Rn and put o = A(xo), we have ba(x)(Dx + A(x) - A(xo))ao
Opw-,o(P) _ lal<m
(13.4)
which is a bounded operator: HA o -' If Sl is an open set in R' we define: II (Dx + A(x))aull L2(n)
IIUIIHA(o) =
Ial
The operator (13.4) can be written
D' ca(x,xo)Dx, Ia1<m
(13.5)
13. Spectral theory for perturbed periodic problems
159
with ca(x,xo) = aa(x) for IaI = m, I9 ca(x,xo)I < Cp for all 3 E Nn and all x E B(xo, 2) with C,3 independent of xo, as follows from assumption (H5). The norms II
IIHk_£o(B(x(,,2))
and II IIHk(B(xo,2)) are equivalent uniformly with
respect to xo. Standard a priori estimates for elliptic operators give for any u E HA+Ic(B(xo, 2)), Ck[IIOpa(P)uIIHk(B(x(,,2)) +
II1IIHk+m(B(xo,l)) c
IIUIIL2(B(xo,2))],
(13.6)
where ck is independent of xo. Considering a covering of R' by a family of balls B(xo, 2) with xo in a lattice, we obtain from (13.6):
Lemma 13.3. If u
E
L2(R"`) and OpA(P)u E HA(Rn), then u
E HA+k(Rn) and we have: IIUIIHA+m < Ck(IIOpA(P)uII Hk + IIuIIL2),
(13.7)
where Ck is independent of u .
Using this lemma with k = 0, we get Proposition 13.2.
Remark 13.4. If we replace (D,, + A(x)) which appears in the definitions of
HA", HA and Op'(P) by (Dx + A(x + y)) with y parameter in R', then (13.7) remains true with C independent of y.
We now drop the assumption that P is independent of y and consider the operator P = P' (x, y, hDx + Dy + A(x)), which is the Weyl quantization of P(x, y, h +,q + A(x)). The change of variable (x,y) - (x, y) = (x - hy,y),
(13.8)
transforms P into:
P=Pw(x+hy,y,Dy+A(x+hy)). Lemma 13.3 and Remark 13.4 show that P is essentially selfadjoint with domain
£, = {u E L2(R2n); (Dy + A(x +
E L2(R2n), V IaI < m}.
Now the change of variables (13.8) (which can be realized as a unitary transformation) shows that P is essentially selfadjoint with domain £,,,,.
Spectral Asymptotics in the Semi-Classical Limit
160
To study P we will use the Floquet-Bloch reduction in the y variable. Let F* be the dual lattice of r, given by r* = {y* E Rn; ry* -y E 27rZ, V -y E F}. ,
(13.9)
ForuES(R2n) and9ERn, we put e'-Y'au(x,
Uu(x, y; 0) =
y - ry).
(13.10)
-YEr
We notice that Uu(x, y, 0) only depends on 0 modulo the dual lattice r*, so we may view Uu(x, y, 0) as a smooth function on R2n x (Rn/r*). For 0 E Rn/r*, we put DB = {u E D'(R2n); u(x, y +'y) = eiry*eu(x, y) } and
xe = {Li
(R2n)
f
n Dei
u(x, y)I2dxdy < +oo},
Rx x E
where E is a fundamental domain of F. When 0 = 0 we write simply L2(Rn x Ry/F). From (13.10) we see that Uu(., 0) E He,
(13.11)
and if we view NB as a bundle over R/r*, we can view Uu as a smooth section of this bundle (thinking of He as a subspace of Li (R2n)). We write Uu E C°O(Rn/F*;No). Every v E L2(Rn/F*) can be expanded in a Fourier series: v(9) =
v(`y)eilo, 'yEF
where v(-y) = vol (E*)-1 fE= e-i-yev(9)d0. We have
vol (E*)-1
JE
*
Iv(0)I2d0 = E Iv(ry)I2.
(13.12)
'yEr
For every x, y we can view (13.10) as the Fourier series expansion of Uu(x, y; .), so (13.12) gives: vol
(E*)-1
E I u(x, y -'y)I2. JE* IUu(x, y; 0)I2d0 ='yEr
(13.13)
13. Spectral theory for perturbed periodic problems
161
Integrating this over R' x E, we obtain
(vol (E*)-1 fff R xExE (
RX x Ry
IUu(x, y; 9)I2dxdyd9)112 (13.14)
Iu(x,y)I2dxdy)1/2.
This means that U can be extended to an isometry: L2(R2n) L2(Rn/F*; fle), if we equip the latter Hilbert space (which by definition is the space of v E L,,,(R3n) such that v(x, y + y; 6) = eiryev(x, y; 0) and v(x, y; 9 + ry*) = v(x, y; 0) for all ry E IF and all ry* E F*) with the norm given
by the left hand side of (13.14). Let v(x, y; 0) E Co" (Rn/F*; 7-lo), and write the Fourier series expansion with respect to 9: v(x,y;0) _
(13.15)
vry(x,y)eirye,
^yEr
vry(x, y) = vol (E*)-1 fE* e
°v(x, y; B)d9.
(13.16)
Using the property v(x, y - -y; 6) = e-iry'ev(x, y; 9), we see that vry (x,
y) = vo (x, y - -Y)
(13.17)
Hence when v is smooth we have v = UWv, where: Wv(x, y) = vol (E*)-1 fE* v(x, y; 9)d9 = vo(x, y).
(13.18)
(13.18) combined with (13.15) and (13.17) gives
IIWVIIL2(Rnz xR,-)
=> f fRsxE Ivo(x,y-'Y)I2dxdy ryEr
ryEr
xE
I3 (x, y)I2dxdy =
2
xe)
(13.19)
(13.19) shows that W is an isometry: L2(Rn/I'*;he) -+ L2(Rn), so W is a bounded right inverse of U. Since U is also an isometry we conclude that U
is unitary and that W = U-1. Notice that U commutes formally with (hDx + Dy + A(x))", so for every
k E N, U is unitary from Eye into L2(Rn/F*;7-le), where 7-le = {u E (hDx + Dy + A(x))"u E 7-Hei blal < k}. Let PO denote the selfadjoint operator Pw(x, y, hDx + Dy + A(x)) acting in the sense of distributions on ?1
162
Spectral Asymptotics in the Semi-Classical Limit
the space ?-le with domain ?-LB . Thanks to (13.3) and using the fact that (hD., + Dy + A(x))a commutes with U, we see that
UPU-1 =
f®
J
Ped6,
(13.20)
where by definition the last expression is the selfadjoint operator Q on L2(R"/F*;?-Lg) with domain L2(Rf/F*;?-le) given by Qv(x, y; 0) = (Pev(., ; 0))(x, y)
For 0 E Rf/F*, PO is unitarily equivalent to the operator pe = ei(xlh-y)
acting on Ko = {u E Li JR2,); u(x, y + _Y) = u(x, y),
Iu(x, y)12dxdy < +oo} ff Rx xR, /r
with domain K,,,, = {u E Ko; (hDx+Dy+A(x))au E Ko,V jal < m}. Thanks to the definition of Pe and using the fact that P has the form (13.3) as well as the fact that (hDx + Dy + A(x))"' commutes with ei(x/h-y)e we see that Pe is the same differential operator P acting on Ko with domain K,-,,,. Hence the spectrum of Pe is 0-independent. Using (13.20) we get:
Proposition 13.5. The spectrum of P acting on L2(R2n) with domain E,, is the same as the spectrum of P acting on Ko with domain IC,,,. Remark 13.6. From (13.1), (13.2) we have Km = {u c Ko; [(he + rl + A(x))a]v'u E Ko, VIal < m}.
Now we consider in more detail the operator P. We will give a reduction of the study of o,(P) by considering P as a h-pseudor in the x variables with an operator-valued symbol P(x, + A(x)) = P- (x, y, Dy + 6 + A(x)) as an KO), and by introducing a suitable auxiliary (soelement of called Grushin) problem, which will permit a reduction to an h-pseudor ; an effective Hamiltonian. Here
Ko = L2(Rn/F),
13. Spectral theory for perturbed periodic problems
163
K.m, = {u E Ko; (Dy + )au E Ko, Vial < m}.
Notice that only the norm of Km, depends on l;, not the space itself, and under the hypothesis (H5) we have: II UII Km,£+A(x)
C((S - ) + (x - z))mIl UII Km,S+,a(=)
Vu E Km,o (x, z,
E R4"`,
(13.21)
Ila.'ag P(x, + A(x))IIG(K-,e+A(=),Ko) < Ca,o.
(13.22)
We need some basic results about operator valued pseudors. Our main reference is here the unpublished work of Balazard-Konlein [BaKo]. We shall consider a family of Hilbert spaces Ax, X = (x, l;) E R2n satisfying:
Ax = AY as vector spaces for all X, Y E R2n,
(13.23)
there exist No > 0 and C > 0 such that IlullAx < C(X - Y)NOIIuIIAY for all u E Ao,X,Y E R2n.
(13.24)
Let tax, X E R2n be a second family with the same properties. We say that p E C°°(R2n; £(Ao, Bo)) belongs to S°(R2n; £(Ax, Bx)) if for every a E N2n, there is a constant Ca such that Ik9 PII.c(Ax;Bx) <_ Ca, for all X E R2n.
(13.25)
We can then associate with p the operator pw(x, hDx), and similarly to Proposition 7.7, Lemma 7.8 and Theorem 7.11 we have:
Proposition 13.7. Let p E S°(R2n; L(Ax, Bx)), where Ax, t3x satisfy (13.23),(13.24). Then Oph (p) = p"°(x, hD,,) is uniformly continuous S(Rn; .Ao) --> S(R'; Bo).
Proposition 13.8. Assume Ax = Ao, 13x = Bo, `dX E R2n. If p E S°(R2n; G(Ao, 5o)) (i.e. II'XpII c(Ao;13o) < Ca, for all X E R2n) then Oph (p) is uniformly bounded
L2(Rn; Ao) - L2(Rn; t3o). Let Cx be a third family of Hilbert spaces which also satisfies (13.23), (13.24).
Proposition 13.9. Let p E S°(R 2n;G(Bx,Cx)),q E So(R2n;.C(Ax,Bx)) Then Oph (p) o Oph (q) = Oph (r), where r E S°(R2n; £(Ax,Cx)) is given by
ih
r = exp (2
Dg; Dy, Dv))(p(x,
)
q(yl))I
>
Spectral Asymptotics in the Semi-Classical Limit
164
where a is the usual symplectic 2form. We have the asymptotic formula: °O
r
k=0
1
k!
ih
(2 a(D., De; Dy, D,,))kp(x, e)q(y, 77)Ix=y,e=v
in the sense of Chapter 7. Denote by To g the space {u E LI0,(Ry); u(y+-y) = for E Rn/r*, and by the space {u E Dyu E lal < m}. We will fix some energy level Ao E R+.
Proposition 13.10. There exist N E N, a complex neighborhood 19 of n C°° (Rnx X R£ x Ry) for 1 < j < N, such that for each (x, ) E R2n and each A E 79 the operator: A0, and functions cod (x, , y) E C°° (RX £;
P(x, e, A) (
uu ) = I (Pw (x, y, Dy R)(x, )u + R_ (x, )u
(13.26)
is invertible from K,,,,,g X CN into Ko X CN with an inverse So (x, l;, A) _ Eo 0 o uniformly bounded with respect to (x, , A) together with all
E- E-+ / ( Eo derivatives in L(Ko X CN, K n, X CN) for (x, l;) E Rn, A E (R+ (x, )u)j = (u, coy (x, , ))p0.0
and R_ (x, )u = .N u3 .
Cpj (x,
19. ,
Here
')
Moreover, the functions coj satisfy the estimates: C C«,3, a; ,3 E Nn, X, S E Rn d'Y* E r*, cj (x, + ^y*, y) = e-zy7* p5 (x, , y) Ilax 5 c o
lIi< ,,
(13.27)
Proof. By the appendix to this chapter, for a given Al E R, there exist N Oj : (Rn*/P*)£ -+ linearly independent for every analytic and C o > 0 such that for all u E Vect (01i ... , ON)':
(((-o)"`i2
- Al)u,
co llull'o,£.
(13.28)
£-
(13.29)
Using hypothesis (H3-4), we get (Pw(x, y, Dy)u, u)Fo,E
((-Q)m/2u, u)To
Cl
uniformly for x E Rn, E Rn/r*, U E .Fm,g.
13. Spectral theory for perturbed periodic problems
165
Combining (13.28), (13.29), we obtain:
(C C,1 +Ci (13.30)
and taking Al large enough we get Re ((P' (x, y, Dy) - A)u,
c2 IluI12
with C2 > 0, uniformly for x E Rn, A E 19, Vect (1'1 i ... , ON) I
(13.31)
E Rn/P* and u E Fm, fl
-
We define
by:
R_ () ui , u2 , ... , un,) _ FN 1 u3
and (u, 1j e). As in the appendix, (13.31) im-
plies that the Grushin problem
(P' (x, y, Dy) - A)u + R-(l;)u- = v
1 R+()u = v+ is bijective from x CN into .' o, x CN with an inverse uniformly bounded for x E R', A E 19, l; E Rn/P*. With cpj (y, ) = e-i11'10j (y, t ) we define P(x, l;, A) as in the proposition, noticing also that u -> is into K,,,,,C. Then P(x,1;, A) is invertible with a uniformly unitary from bounded inverse So in C(Ko X CN, Km,e X CN) . From P o Eo = I and
EooP=I we get
for jal=1.As 9xCP=0(1)in
,C(K,,,,,£ X CN, KO X CN), then a9 So = 0(1) in G(Ko X CN, Km, X CN).
Continuing to take higher and higher order derivatives of P o Eo and using the fact that axo' £P is bounded in C(K,,,,,g X CN, KO X CN) for all a, we see # that ax £Eo = 0(1) in C(Ko X CN, K"',g X CN).
Proposition 13.11. The operator So (x, hDx + A(x), A) is continuous from
S(R'; Ko X CN) into S(Rn; K,,,,,o X CN), from S'(Rn; Ko X CN) into S'(Rn; K,,,,,o X CN) and uniformly bounded from 1Co x L2(Rn; CN) into 1C,n x L2(Rn; CN) for A E 19. Moreover, we have: PW (x, hDx + A(x), A) o Eo (x, hDx + A(x), A) = 1 + hW°(x, hDx + A(x), A; h),
(13.32)
where R(x, , A; h) ^' E000 Rj (x, , A)hj in So(R2n; G(Ko X CN)) and R, Rj depend holomorphically on A.
Proof. The continuity of So (x, hDx + A(x), A) in S and S' follows from the propetties of Eo (Proposition 13.10) and Proposition 13.7. Using Remark
Spectral Asymptotics in the Semi-Classical Limit
166
13.6, to show the L2-boundedness, it suffices to show that for all Icxl < m C ((hD. + Dy + A(x))`x)w 0
1
o £w is 0(1) in °
L(Ko x L2(Rn; CN); Co x L2(Rn; CN)).
The fact that
(13.33)
can be viewed as an h-pseudor with operator Ko and Proposition 13.9 show
valued symbol
that C(
+Dy+A(x))a 01 -So' 1 0
E
S°(R2n; £(Ko x CN, KO X CN)),
combining this with Proposition 13.8 we get (13.33). The formula (13.32) follows from Proposition 13.9.
#
Proposition 13.12. Assume (HI) to (H5). For h sufficiently small and A E 19, Pw(x, hDx + A(x), A) has a uniformly bounded inverse of the form £w (x, hDx + A(x), A; h), where £(x>, A, h) E So(R2n; ,C(Ko X CN, Km, X CN))
has an asymptotic expansion
£'(x, , A, h) - 1: £j (x, , A)h', Co = P(x, , A)-i j=0
as above. This inverse has the same continuity properties as So(x, A(x), A).
Proof. The Calderon-Vaillancourt result (Proposition 13.8) and Proposition 13.11 show that for h small enough II hR°(x, hD,,, + A(x), A; h)IIc(L2(RnxRn/r).L2(Rn;C^')) < 1/2.
Then for h sufficiently small (1 + hR')-1 exists in £(L2(Rn x Rn/P)x L2(Rn; CN)). Consequently Pw(x, hDx + A(x)) has a right inverse £w (x, hDx + A(x), A; h) = So(x, hD,, + A(x), A) o (1 + hRw)-1. We know
from Chapter 8 that if p(x, ; h) is an elliptic scalar valued symbol in S°(R2n) then there exists q(x, , h) E S°(R2n) such that pw(x, hDx; h) o qw (x, hD.; h) = qw (x, hDx; h) o pw (x, hDx; h) = I. We also know that q(x, ; h)
when p(x,l;;h) - Ej>opj(x,l;)hj in S°(R2n). Using the
13. Spectral theory for perturbed periodic problems
167
fact that this remains true in the case of operators with operator valued symbol, we see that for h small enough
(1 + hRw)-1 = 1 + hRw,
where R' is an h-pseudor with the same properties as R'. Consequently £(x, F,, A; h) has an asymptotic expansion £w(x, , A; h) - Ej'° £j (x, l , A) hi in S°(R2n; L (K° X CN, Km, X CN)). For A E 19 f1 R, P"' (x, hDx + A(x)) is selfadjoint and then £w(x, hDx+A(x), A; h) is also a left inverse for A E 19f1R, and also for A E 19 by analytic continuation. #
Remark 13.13.
(1) Proposition 13.10 remains true if we replace A° by a compact interval.
(2) Let P(A) _ R A 13.10 and E (A))
/
E (A) J
be the operator constructed in Proposition the matrix of the inverse £(A) = E"' (x, hDx+
A(x), A; h). (E_+(A) is called the effective Hamiltonian associated with (P - A) and is selfadjoint when A is real.) We have the following properties: (i) £(A) is analytic for A E V.
(P - A)-1 = E(A) E-+(A)-1
- E+(A)(E-+(A))-1E-(A),
= -R+(A)(P - A)-1R-(A).
(13.34) (13.35)
(13.34), (13.35) follow from the identities: P(A) o£(A) = I, £(A) o P(A) = I and can be used to show that many properties of (P - A) are the same as for E-+(A) (iii) Using the fact that R+, R_ are independent of A we get
aaE-+(A) = E-(A)E+(A).
(13.36)
Proposition 13.14. Under assumptions (Hl) to (H5), for A E 19, h small enough, one has the following equivalence: A E Q(P)
0 E o (E"+ (x, hDx + A(x), A; h)).
Spectral Asymptotics in the Semi-Classical Limit
168
Proof. This follows from Remark 13.13 (formulas (13.34), (13.35)) and the continuity property of S obtained in Proposition 13.12. # Remark 13.15. Because of (13.27), we have: R- (x, R+ (x,
+'y*) = e-''y'Y* R- (x, +'Y*) = R+ (x, )ezyry*
On the operator level we get: hDx + A(x))eixy*/h = e-iy7*R'(x, hDx + A(x)) f e-ixry*IhR-"'(x, e-'xy*/hR+(x,
hDx + A(x))eix7*Ih = R+(x hDx + A(x))eiy7*.
Combining this with the fact that C
Pw (x, y, S + 7* + DO = e-2y7* PW (x, y, S + Dy)eiyry+,
we get
[Pw(x, hDx + A(x)), (
ei(x/h-y)'Y* 0
0
eixy*/h) i = 0.
(13.37)
Let us consider P0 = Pw (hy, y, Dy + A(hy)) with domain H,,,.,A = {u E L2(R'); (Dy +A(hy))au E L2(R'), V Ic l < m}. From Proposition 13.2 Po is essentially selfadjoint on Co (Rn) with domain H,,,.,A. To reduce the study of Po, we apply the same method as for P. We get
Proposition 13.16. The spectrum of Po = P'(hy, y, Dy, +A(hy)) acting on L2(R') with domain Hm,A is the same as the spectrum of P = Pw(x, y, hDx+Dy+A(x)) acting on Lo := {EryEr v(x)6(x-by+hry); h-"l2v E L2(R')} with domain L,,,, :_ {u E Lo; (hDx + Dy + A(x))au E Lo, jctil 5 m}. Here Lo, L,,, are equipped with the natural norms. Remark 13.17.
(1) The nature of the spectrum of the two operators P and Po may conceivably differ.
(2) Using (13.1) and the fact that ((he + y + A(x))a)w can be viewed as an h-pseudor with operator valued symbol (t + A(x) + Dy )a : K,,,,,,t+A(x) - Ko, we see that L,,, = {u E Lo; ((he + r/ + A(x))c)wu E Lo, Jc < m}. It follows from the last result that we can reduce the study of the spectrum of Po to that of P acting on Lo with domain Lm. Let S(R' x Ry/P) = {co E
13. Spectral theory for perturbed periodic problems
169
C°°(R2 ); (x)Nt9 yip E L2(Rn x Ry/F),VN,a}. For W E S(Rn X (Ry/F)), we have p(x,y) _
c7*(x)e'7 y,
(13.38)
7'Er*
with c7= (x) = vol (E)
e-2-'*'ycp(x, y)dy.
JE
(13.39)
Integrating by parts in (13.39) and using the Schwartz inequality, we get:
Ic7'(x)I
(13.40)
I2dy)1i2(1+Iy*I)-N,
lal
with C independent of x in Rn. Combining (13.38) and (13.40) we get (with a new constant C) for every fixed N > n,
f ,I
(hy, y)I2dy
Chff IaI
a(x,
zxE
2dydx.
(13.41)
A distribution u = Er v(x)S(x - h(y - -y)) in Lo can be written u = Er v(h(y - y))S(x - h(y -'y)). If p E S(RX X (Ry/F)), then
(U" P) := ff
u (x, y)cp(x, y)dxdy
x,y)ERx xE
_
f v(h(y -'Y)),P(h(y -'Y),y)dy =
7Er E
f
n
v(hy),P(hy,y)dy.(13.42)
R
(13.41) and (13.42) show that u can be viewed as an element of S'(Rx' x (Ry/F)). Hence we can hope to adapt Proposition 13.12 to Lo. We will characterize more precisely the space Lo and L,,,,. Let us denote by Vo the subspace of S'(Rx) consisting of distributions of the form:
f7S(x - hy)
w(x) = 7Er
with (f7)7Er E 12(r). Vo is equipped with its natural Hilbert space structure,
and we first study this space in more detail. Let r7. be the operator of multiplication by eix7*lh.
Lemma 13.18. Let X E Co (R'°) satisfy E7.Er. x( +'Y*) = 1. Then for u, v E Vo, the quantity (xw(hDx)u,V)L2(Rn.) is well defined and independent of the choice of X. Moreover; 1
(xw(hDx)u, v)L2(Rn) = hnvol (Rn/F)
(u,
v)va.
Spectral Asymptotics in the Semi-Classical Limit
170
Proof. If u = EryEr fu(-Y)6(' - hry) E Vo, then.Fhu(b) = E-Er fu(--y)e'-" E
If V = EryEr f'(Y)S( -
L2(Rn/r*)) (where rhu(e) = fRn hy) E Vo, we have
(.Fhu,.Fhv)L2(R,,/r*) = vol (Rn/I'*)(u, v)vo.
(Xw(hDx)u,v) can be defined by the Parseval formula: (Xw(hDx)u,v)L2(Rn) = (27h)-n(.Fh(Xf(hD.)u), Fhv)L2(Rn)
=
(27th)-n fRn
f
(27th)-n
E*
ry*Er*
Since
+ry*Er*
for all ry* in r*, we get
t 1 and .Fhu(e+'y*) = .Fhu(6), -hv(S+'I*) =.Fhv(6)
(Xu'(hDx)u,
v)L2(R.,.) = (27rh)-n(.Fhu, Fhv)L2(R-/r*)
_
1
hnvol (Rn/r)
(13.43)
(u, v)vo,
where we also used the fact that vol (Rn/I'*) x vol (Rn/I') = (27r)n.
Proposition 13.19. For every u E Vo there exists uo E L2(Rn) such that with u.y* = Try*uo:
u = E u.y* with convergence in S'(R').
(13.44)
ry* Er*
Moreover, 3 C > 0 independent of u for which uo can be chosen such that for every bounded set 13 C S°(R2n) and every N E N, there exists a constant CN > 0 such that VA E B with dist (supp A, R1x x {0}) > C, we have II Aw(x, hDx)uoll L2(Rn) <- CNh- 2 (dist (supp A, Rn x
{0}))-N
(13.45)
uniformly for h > 0 small enough. The constants CN can be taken 0(IIuIIv.) uniformly with respect to u.
Conversely, if uo E L2(Rn) satisfies (13.45) and if u,y. = Try*uo, then the sum Ery*Er* ury* converges in S'(Rn) towards an element u of Vo, with IIuIIuo bounded by a constant times the sum of h2 IIuoIIL2 and a finite number of the
CN in 0 3.4 5) .
13. Spectral theory for perturbed periodic problems
171
Proof. If u E Vo, we take (2irh)-n
uo = X(hD.)u = (.'Fh 1X) * u =
fu('Y)X('Y -YEr
with X E Co (Rn, [0, 1]), Ey'EF and satisfies
1. Then uo E HS for every s E R
II (hDx)at01I L2 < C«h 2 IIuIlvo
(13.46)
By integrating by parts in the oscillatory integral which gives Aw(x, hDx)uo, we obtain for every k E N,
Au'(x, hDx)uo =
h2k-lal b-(x, hDx)(h101 aQ(uo(x))), I7+a1=2k
with the Ok,y(1) factor having bounded derivatives to all orders. Using Theorem 7.11 we get:
IIAw(x,hDx)uoll
{0}))-2k
lR(hDx)auoll, 101<2k
(13.47)
with Ok(1) independent of h and uo. Now (13.45) follows from (13.46) and (13.47).
To prove (13.44) we only remark that for u E Vo we have
u and
u.y. = X(hD. - -Y*)u.
Let us prove the converse statement. Let A E Co (Rn) be real valued and 1 in a sufficiently large bounded neighborhood of 0. If uo such that satisfies (13.45) then for every N E N, we have; II(1 - Aw(hDx))(hDx)NUOII L2(R-) = CNh-n/2,
where (x) = (1 + (x12)1/2. By Theorem 7.11 we have II Aw(hDx)(hDx)NUOII L2(R') = ON(1)IIUOII
Consequently, (21rh)-n/2
rhU0IIL2 = II(hDx) ruoIILZ = ON(1)(IIUOII +CNh-nl2). (13.48)
Put U = Ery' Er' Try' u0. From (13.48) we deduce that
E L2(Rn/r*)
.Fh(U)(S) = 'Y' E r'
172
Spectral Asymptotics in the Semi-Classical Limit
and IIFh(u)(0IIL2(Rn/r*) = result.
CN(1)(hn/2IIuoII
+ CN). Now (13.43) gives the
#
Proposition 13.20. Let B(x, ) E S°(R2n) with B(x, l; + y*) = B(x, l;) for any y* E r*. Then Bw(x, hDx) is bounded on V0i uniformly with respect to h small enough.
Proof. Let u E V0 and let us write u = Ey. uy as in Proposition 13.19. Then BW(x, hDx)u = Ey.
where
vy. = B' (x, hDx )uy. = ry B' (x, hDx + y* )uo.
Since B(x, + y*) = B (x, e), we get vy. = ry. v0, where vo = B' (x, hDx )uo.
Using the same argument as in (13,47), we see that v0 satisfies (13.45) with constants CN estimated by similar constants for u0. Then Proposition 13.19 shows that v E V0 and IIvilvo <_ Co(h 2 II Bw(x, hDx)u°II L2 + IIuIIuo) < C(h2 IIuoIIL2 + IIuIIuo). Thanks to (13.46) we obtain the result. #
Proposition 13.21. Let Q1(x,.), Q2 (x, l;), B(x, t;) E S° (R2,). We assume that Q1, Q2 and B are r*-periodic in t; and lIx supp Q1 = {x E R'; (x, ) E supp Q1 for some E R' } is compact . We also assume that dist (llx supp Q1, IIx supp Q2) > e0 > 0 for some fixed eo > 0. Then for every
N E N there exists a constant CN such that IIQi (x, hDx) o B"'(x, hDx) o Q2 (x, hDx)II L(v0) < CNhN.
Proof. Let u E V0 and let us write u = Ey. uy. as in Proposition 13.19. vy., where Then Q' (x, hDx)B'(x, hDx)Q2 (x, hDx)u vy* = Q1 (x, hDx) o B' (x, hDx) o Q2 (x, hDx)uy..
As in the proof of the preceding proposition, vy = ry. voi with v0 = Q1 (x, hDx)B' (x, hDx)Q2 (x, hDx) uo.
From the assumption on the support of Q1, Q2, we see by the Weyl calculus (Chapter 7) that for every N E N IIVOIIL2(R') = /N(hN)IIUOII
Let A E S°(R2') be as in Proposition 13.19. By the argument used in (13.47), we see that for every integer N E N, II Aw(x, hDx)voII L2(R2n) < CNhN(1 + disc (supp A, R" x {0}))-N, (13.49)
13. Spectral theory for perturbed periodic problems
173
with CN = O (I I u I I vo) Now by Proposition 13.19 we can conclude.
We will generalize the above results to the space Lo. If u = E v(x)S(x - by + hy) E Lo, 7Er
we have
IIuIILo = h-nIIvlii2 =
fn/r Ilv(hy+h-) IIi (r) dy.
Let X be as in Proposition 13.18 and consider u as an element of ThyVO with y
as a parameter. Let w = ElEr a(x)S(x - h(y - -y)) E Lo. Similarly to (13.43) we have (Xv'(hDx)uy, wy)L2(Rx-)
_
1
v(h(y + a))a(h(y + 0))
«,QEF
f
e i(«-a)-e X()<.
'
Since Ey,Er X( + y*) = 1, then x(y) = fR
0 for all y E 1
and x(0) = vol (Rn/I'*). Consequently (Xv'(hDx)uy, wy)L2(R^) _
(2irh)n
u(h(y + a))a(h(y + a)). «Er
Integrating this over R' ,/]F, we get 1
(XW (hDx)u, w)L2(R.nxR./r*) = vol (R1/I')hn h-n(v, a)L2(R.).
Then the analogue of Proposition 13.18 holds for Lo. Specifically: 1
(X'(hDx)u,u)L2(RnxRy/r*) = vol(R' /r)h' IullL.. We will now prove the analogue of Proposition 13.19 for Lo. Let T.y be the ei(x/h-y)7` and notice that u for u E Lo operator of multiplication by and y* E r*.
Proposition 13.22. For any u E Lo there exists uo E L2(Rn x Ry/P) such that with u.y = T7-uo: U = E,y.Er* u.y* with convergence in S'(Rx x Ry/P). Moreover, 3 C > 0 independent of no such that uo can be chosen such that for every bounded set B C S°(R2n) and every N E N, there exists a constant
174
Spectral Asymptotics in the Semi-Classical Limit
CN > 0 such that VA E B (independent of (y, ii)) with dist (supp A, R' x {0}) > C, we have
CNh-2(dist(suppA,Rn x
{0}))-N
(*)
uniformly for h > 0 small enough. The constants CN are (9(IIuJIL(,). Conversely, if uo E L2 (Rn x Ry /I') satisfies (*) and if u,y. = Ty. uo, then the series u.y. converges in S'(Rn x Ry/I') towards a distribution u in Lo, with IIuIILo bounded by a constant times the sum of hi IIUOIIL2(RrXRy/r) and a finite number of the CN in (*) .
The proof is similar to that of Proposition 13.19 and we omit it.
Theorem 13.23. Assume (Hl) to (H5). Then for h > 0 small enough and A E 29, P' (x, hD,, + A(x), A) is uniformly bounded from L,,, X (V0)N to Lox (V0)N and has the uniformly bounded two sided inverse Sw(x, hDx + A(x), A, h) from Lox (V0)N to L,,,, x (Vo)1` Proof. We know that P"' (x, y, hDx + Dy + A(x)) is uniformly bounded from
L, to Lo. It then remains to show in view of Proposition 13.12 that the following operators are uniformly bounded: R+ (x, hDx + A(x)) : Lo -> V N,
(13.50)
R' (x, hD,, + A(x)) : VN .
(13.51)
ET (x, hD,,, + A(x), A; h) : VN
L,,,,, L,,,,,
(13.52)
Ew (x, hDx + A(x), A; h) : Lo -+ V N,
(13.53)
E"' (x, hDx + A(x), A; h) : Lo
(13.54)
Lm,
Ew+(x, hDx + A(x), A; h) : vN -* UN.
(13.55)
Here (13.55) is a consequence of Proposition 13.20 and the same proof gives (13.50) and (13.53). Let us show (13.52) ((13.51) and (13.54) can be proved similarly). From Remark 13.17 it suffices to show that for all a E Nn with j al < m, ((hl; +rl+A(x))a)' o f+ (x, hDx + A(x), A; h) is uniformly bounded from VN to Lo. This composed operator is of the form Cw(x, hDx+A(x), A; h) with C(x, e, A; h) in So (R 2n;,C(CN ;L 2(R"/r))) and T,y. C" (x, hDx + A(x), A; h) = C" (x, hDx + A(x), A;
for all ry* E I * ((13.56) follows from (13.37) with P replaced by E).
(13.56)
13. Spectral theory for perturbed periodic problems
175
Let U E and decompose u = Ey, u.y. as in Proposition 13.21. Then we have Cw(x, hDx + A(x), A; h)u = E v* with vy. = Cw (x, hDx + A(x), A; h)ry. uo = Try. Cw (x, hDx + A(x), A; h)uo,
and thus we are in the situation where Proposition 13.21 applies and gives the result. # Using Theorem 13.23 and Remark 13.13 we get:
Corollary 13.24. For A E V and h > 0 sufficiently small, we have: A E o,(Pw(hy, y, Dy + A(hy))
(where the operator is equipped with the domain HA) if and only if 0 E a(E'+(x, hDx + A(x), A; h)), where the last operator is considered as a bounded operator: UN -> UN
It is of some interest to see what kind of Grushin problem we obtain for the original operator Po = Pw(hy, y, Dy + A(hy)) if we compose the Grushin problem of Theorem 13.23 with the earlier identifications. We recall that we
have the unitary map L2(Rn) D u H f = r_ryEr v(x)6(x - h(y - y)) E Lo, defined by v(hy) = u(y) (see Proposition 13.16). We shall compute the j th component of R' (x, hDx + A(x)) f for some fixed j, 1 < j < N in terms of u. We recall that cpj (x, l;, y) is r-periodic in y, and cps (x, t + ry*, y) e-iY Y
cps (x, t=, y) for all ry* in r*. We also recall that
(R+(x, l;)u)j = f u(y)cpj(x, , y)dy,
(13.57)
E
where E is a fundamental domain of r (cpj and R+ (x, t:) are given by Proposition 13.10). For simplicity we will drop the index j: (27rh)'R+(x, hDx + A(x)) f (x)
R+(x2
_ E
YEr
fRIRfEy
P(x 8(x -
+A(x2x))f(x,.)dxd x 2
,
x
+ A(x
), y)f (x, y)dydxd
2
fry -
P(x2x,,+A(x2x),y)dydxdt ryEr
f
£
y))
E W(x+h(2y+ry)
(13.58)
Spectral Asymptotics in the Semi-Classical Limit
176
Using the I'-periodicity in y of cp, we get
R' (x, hDx + A(x)) f (x) _ (27Th)
hu(y)V (X
JR{ JRy
2
by, + A(x
2 by),
(13.59)
recalling that v(hy) = u(y). Introducing the function '(x, e, y) _ y), we obtain (R' (x, hDx + A(x)) f) (x)
_
(27Th)-"'.
f eiA( 2
-u(y)
ei
(x
fR1
Ry
_: (R+u)(x).
(13.60)
Rn
So if we make the change of variables ij = 1; + A( 2hy ), we get
(R' (x, hDx + A(x)) f)(x)
_
eia( +hy )(y-xlh)u(y) f
(27Th)-T
e2(xnlh)'%(x
JRy
2
(13.61)
In general, if f (77) is a F*-periodic function, then
a(y - h),
(27Th)-n f ef ()d= '
(13.62)
7Er
with a.y = (vol (E*))-1 fE. ei7r/ f (77)dr7. Since zb(x,77, y) is r-periodic in q and O(x, rl, y + ry) = ei701,0(x,77, y) for all 'y E r, we get from (13.61) and (13.62):
(R+u) (x) +hy
eiA(
7Er 7Er
_E(
)(y- /h)u(y) f eZ7n%(x2hy,77,y)vol(E*)dy)6(x-hry)
E
Ry
J
eiA("
K)(y-7)u(y) [ (h`y 2
hy,7l,y-'y)vol(E*)dy)6(x-hry)
V
u(y)WA(hy
'YE7
2
y,y -'y)dy)6(x - h-y),
(13.63)
with WA (x, y) _ vol (E,) e-2y A(x) fE, Vi(x,77, y)dr7. Taking into account the
index j, we get: (R+u)j(x) = E7Er(R+u)j(y)6(x - h-y), where (R+u)j E 12(r) is given by: (R+uW'Y) = f u(y)WA,i (h'Y R^
2
Y, y - -y) dy,
13. Spectral theory for perturbed periodic problems
177
with WA ,j (x, y) = v°1 E, e-iy a(x) fE. Oj (x, y, y)dy, for all 1 < j < N. Since the various identifications in our computation are unitary and since
R--(x, hD,, + A(x)) = R+(x, hD,, + A(x))*, it is clear that this operator is naturally identified with R_ = R. Summing up, we have proved: Corollary 13.25. For A in a neighborhood of A0, the operator C Pw (hy,
y, Dy + A(hy)) - A R_
Hm,A x l2 (r; CN) , L2 x l2 (r; CN)
0f
R+
is bijective with bounded inverse I
EE_
EE+
I
-+
. The matrix of E_+ is equal
to the matrix of Ew+(x, hD,, + A(x), A, h) acting on VN, if we identify the latter space with l2(r; CN) in the natural way. We end this chapter by discussing Schrodinger operators with slowly varying perturbations. Let V (y), cp(x) E C°° (Rn, R), where V (y) is r-periodic and
cp is bounded with all its derivatives and tends to zero at infinity. We are interested in the operator n
(Dy, + Aj (hy) )2 + V (y) + p(hy) = Pw (hy, y, Dy + A(hy)),
PA,,v = j=1
where P(x, y, 71) = rl2 + V (Y) + O(x).
Let I CC R be an open interval and put I = I - cp(Rn). In the appendix we construct 01(L;, y), '1'2 (C, y), ... , ON (t:, y) smooth in all variables, r*-periodic (t;, y), such that the problem: in t and with Oj y + y) = P(t;,
_
\
z k-
A
0() 1
:
'F2, x CN
F£ x CN
is bijective for e E Rn and /z in a neighborhood of\I, with
(R+(C)u)j = (u, z/)j). Let I E ( z,,) E +(( z,) I be the inverse of P(C, -z). Taking z = z - V(x) we get an inverse E' (x, C, z) E°- (x, C, z)
E+' (x, C, z) ) E°--+(x, C, z) /
for the operator P(C,z - (p(x))
R+( )eZy£, P(x, , z) =
.
(E(C, z - P(x))
E- (C, z - P(x))
_E+(C, z - po(x))
E-+(C, z - W(x)) /
Putting R_(C) =
((D+eT(Y)_z R_0(e)) Jl
R+(e) = we know from
Proposition 13.11 that when h is small enough, Pt0(x, hDx + A(x), z) : K2 x L2(Rn; CN) - L2(Rn x Rn/r) x L2(Rn; CN)
Spectral Asymptotics in the Semi-Classical Limit
178
is bijective and has the uniformly bounded inverse Ew(x, hD,,; +A(x), z; h). If
E_+(x, + A(x), z; h) is the N x N matrix which appears in the lower right corner of E(x, +A(x), z; h) then E_+(x, e, z; h) E So(R2n; L(CN, CN) ) has a complete asymptotic expansion in powers of h and the leading term is E° +(x, , z) = E_+ is 1 *-periodic with respect
to , and if z E I, then z E a(PA,,) (as an operator acting on L2(Rn)) if 0 E a(Ew+(x, hD,, + A(x), z, h)), where Ew+ now acts on the space VN. In the following we denote by E_+ the matrix of E"' (x, hDx +A(x), A, h) acting on VN, if we identify the latter space with 12(1'; CN) in the natural way. Remark 13.26.
(1) From Remark 13.13 and the construction above we have z))-1IIL(CN,CN)
= O(JImzI-1), det E°+(x, , z) = 0 iff z E a(P£,W), dim ker E° + (x, , z) = dim ker (PC,, - z), where PC,, =
(13.64)
is considered as a non bounded-operator
in Ko = L2(Rn/F)). From simple general results on elliptic operators on compact manifolds, we know that PP = (Dy + )2 + V (y) has a discrete spectrum with eigenvalues counted with multiplicity: A, (6) < A2 with )j() j that for fixed and A3 () is even an analytic function of near every point o E Rn/F* where )j (co) is a simple eigenvalue of PC,,. The )j are called the Floquet eigenvalues. The sets Jk = Ak(Rf/r*)
are closed intervals, and the spectrum a(Po) of Po = -A + V (y) (as a non bounded operator in L2(Rn)) is given by a(Po) = aess(Po) = U' 1Jk. Then we deduce:
det E° +(x, , z) = 0 if there exists k > 1 such that z = Ak () + O(x). (13.65)
(2) Let zo E R, d = dim Ker E2 +(x, , z) for a fixed (x, e). By ordinary perturbation theory (see Kato [Ka2]) we can reorder the eigenvalues (A3(z))1<j CjIImzj,
so A (zo) # 0 for all 1 < j multiplicity d.
N. Hence, z --f det E° +(x, , z) has a root zo of
Proposition 13.27. Suppose that zo 0 a(Po). Then there exist e > 0, ho > 0 such that z 0 a(PA) when Iz - zoI < e, h E]0, ho[. Here PA = E=i (Dy3 + Aj (hy)) 2 +V(y)
13. Spectral theory for perturbed periodic problems
179
Proof. W e have E° +(x, + A(x), z) = E_+( + A(x), z), where
z) is The assumption
the effective Hamiltonian associated with z° ¢ Q(P°) and (13.65) imply:
1det E° +(x, , z°) I > C° with CO is independent of (x, ). Theorem 8.3 shows that (E"'+ (x, hD., + A(x), z; h))
exists for Iz - z°l < e,
0 < h < h° small enough, and is equal to Oph r(x, + A(x), z; h), with r E S°(R2n, L (CN, CN) and r(x, , z; h) = r(x, + y*, z; h). From Proposition 13.20 we conclude that (E"+)-1 is bounded on VN for h small enough, and by Corollary 13.24 we see that z v(PA) when Iz - z°I < e. # Now we assume that
o(-0+ V)nI=0.
(H)
Assumption (H) and Proposition 13.27 imply that
Q(PA)nI=0, for h small enough. Using the Weyl criterion (Chapter 4), and the fact that cp(x) tends to zero at infinity, we see that for h small enough the spectrum of PA,,, in I is discrete. Let f E C0 (I). We have:
Theorem 13.28.
tr f (P) -
(2rrh)-n E ajhj,
(h \,0),
(13.66)
dxdt;.
(13.67)
j>0 with
a° = E* J JRx k>1 f (,p (x) + Ak
Proof. Let cp(x) E C°° (R') be real valued, coincide with cp for large x and satisfy:
(v(-A+V) + P(x), X E Rx}) n I = 0. Let E°+(x, e, z) be the effective Hamiltonian associated with P(x, y, e2 + V (y) + ;p(x) and put
E-+(x, , z, h) = E° +(x, , z) + E-+(x, , z, h) - E° +(x, , z). From the properties of ip we have
E-+(x, e, z, h) = E-+ (x, e, z, h) for large x
180
Spectral Asymptotics in the Semi-Classical Limit
and there exists C > 0 such that for h small enough Idet E_ +I > C uniformly in x,1;, z, h.
From Corollary 13.25 and Remark 13.13 we have:
(E-+)-1 = R+(z - PA,w)-1Rand
(z -
PA,,)-1
= -E +
E+(E_+)-1E_.
(13.68)
As R_ and R+ are bounded, IIE-+IIc(0(r;cN)) = O(IImzI-1).
(13.69)
Let f E CO '(C) be an almost analytic extension of f with support close to that of f such that for all N E N
&f(z) = O(IImzIN),
(13.70)
(see Chapter 8 for such a construction). By Theorem 8.1 we have
f (PA,w) =7r - f and the identity
z
(z) (z - PA,v)L(dz),
_-1 _ -
E+ = E-+
-
E+(E-+ - -Z--+)E-+i
with E_+ defined as in Corollary 13.25, combined with (13.68) and the fact
that E_+, E are holomorphic in z near I, give
f
(z)(E+E-+(E_+ - E-+)E-+E_L(dz).
(13.71)
Lemma 13.29. Let Q(x, Z;) E S°(R") with Q(x, + ry*) = Q(x, t;) for any ry* E r*. We assume that K = lIx supp Q is compact. Then Q' (x, hDx) is of trace class on Vo and tr (QW (x, hD.,)) =
1
(2 h)- fE- fR.'
Q(x, e) dxd + O(h°°).
(13.72)
13. Spectral theory for perturbed periodic problems
181
Proof. We denote by A(«,,y) the coefficient of the matrix Q when we identify Vo with 12(r). Using the P*-periodicity of Q on we obtain, using (13.62): A(«,y) = fE* e
ha + h-y i(«-y) Q( ha ,)
d6
vol (E*)
By integrating by parts we see that A(«,y) = ON((
1
K )N(
1+dist(a,h)
1
1+dist(ry,hK)
)N) for every N E N.
Consequently Qw(x, hDx) is of trace class and < i«,p IA«,1I = O(h-'). To prove (13.72) it suffices to use the following lemma: IIQWIltr
Lemma 13.30. If f E S(R'), then
JR
f (x)dx = h"'vol (E)
f (hry) + O(h°°), h --> 0.
(13.73)
yEr
Proof. Define Fh(x) = h"vol (E)
f (h (-y + x)).
yEr
Fh is C°°, F-periodic and for all a E N', ax Fs(x) = O«(hH«l),
(13.74)
uniformly on x in R. Moreover, for every x E R"
Fh(x) _ E c.y
iy*x
(13.75)
y*Er*
with
cy. = (vol (E))-1
JE
e-Zy*xFh(x)dx.
(13.76)
Integrating by parts in (13.76) and using (13.74) we get
cy. = ON(1)hN(1 + I Y*I)-N,
(13.77)
for all 'y* # 0 and all N E N. (13.75) and (13.77) imply
Fh(x) = co + O(h°°) =
Jn
R
f (x)dx + 0(h'),
and (13.73) follows as the special case x = 0.
#
Spectral Asymptotics in the Semi-Classical Limit
182
Lemma 13.29 follows.
By Lemma 13.27 (E_+ - E_+) is of trace class and we can take the trace and permute integration and the operator `tr' in (13.71). The identity azE_+ = E_ E+ shows that for Im z 54 0, _ -1 _ _ _ -1 _
tr (E+E+(E-+ - E-+)E-+E-) = tr (E-+(E-+ - E-+)E-+azE-+)
(13.78)
Let X E Co (RT) be equal to 1 in a neigborhood of II,, (supp (E° +(x, Z;, Z) E° +(x, Z;, z))), and denote by X the matrix associated with the operator of multiplication by x(x) on O''. Since Ex (supp (E°+(x, t, z) -E°+ (x, , z))) fl supp (1 - x) = 0, (13.69) and Proposition 13.21 show that:
I-+(E-+ - E-+)E-+azE-+(1 - x)Iltr = O(h°°IImz 1) IE
f
so tr f (PA,w) =
(z)E-+(E-+-E-+)E-+azE-+XL(dz)+O(h°°). a Splitting the integral into two terms and using the fact that E_+azE_+ is holomorphic in z, we get
tr
tr f (PA, v)
f
J
z (z)E-+azE-+XL(dz) + 0(h°°).
(13.79)
Lemma 13.31. There exists r(x, t;; h) E S°(R2n, L(CN, CN)) such that r(x, e; h) - Ej>o hire (x, 6) and Oph (r (x, t; + A(x); h))
_
of (z)(E'+(x, hDx + A(x), z; h))-1azE"'+L(dz).
1
7r jImzl>hb az
(13.80)
Moreover, rj is 1'* -periodic in t; for all j > 0 and ro (x, l;) is independent of A with: ro(x,
-7r
f
(13.81)
Proof. Let us recall that the results of Chapter 8 remain true in the case of operators with operator valued symbol. Let l1 i 12, ... be linear forms on R2, and put Lj = lj (x, hD,,). From the identity E'+ o (E"'+)-1 = I we have adL3 (E-+)-1 = _(E, w+)-1 o adLj
o
(E'+)-1,
13. Spectral theory for perturbed periodic problems
183
where adLj A denotes the commutator [Lj, A]. As adL, (A o B) = (adLj A) o
B+AoadL3B, ((Ew+)-1azE'+)
adLj
_ -(Ew+) -1 o adLj Ew+ o (Ew+)-1 o azEw'+ + (Ew+)
o adLj azEw+. (13.82)
Using (13.82), the fact that II(Ew+)-1II = O(IImzI-1) as in (13.69) and the fact that Ew+,azEw+ are h-pseudors with symbol in S°(R2n, G(CN, CN)), we see that IIadL; (Ew+)-1 o azEw+II = O(II zI2 ).
An easy induction (just indicated in the proof above) then shows that:
IIadL; o...oadLN((Ew+) 1 o azEw+)II = O(
hN
IImzIN+1)
(13.83)
Now by the Beals characterization of h-pseudors we can apply the same proof
as for Theorem 8.7. The periodicity of rj follows from that of Ej +(x, , z).
If we restrict the integral in the right hand side of (13.79) to the domain IIm zI < h6 then we get a term O(h°°) in trace norm. If we restrict our attention to the domain IIm zI > h6 then by Lemma 13.29 and Lemma 13.31 we get (13.66). To finish let us compute ao. We have
ao = ff
t r [ro(x, + A(x))]dxd = ,x E*
f
R, x E'
= JJR.xE (- 1Ir J of az (z)tr (E° +(x, , z))-1azE°+(x, , (13.84)
Thanks to Liouville's formula (i.e. tr (aA(z)A-1(z)) = -Titin the sense of matrices), we get Oz
ao
- fRxxE* (
0
L(dz))dxd.
(z)
E°_+(xx z) To prove (13.67) we use Remark 13.64 and the following Lemma: 7r
a-z-
Lemma 13.32. Let g be an analytic function. Let (zk)k>1 be the roots (counted with their multiplicity) of g in supp J). We have:
7
1
g,(z) L(dz) = &-Z g W
f (zk). k>1
184
Spectral Asymptotics in the Semi-Classical Limit
Proof. This follows from the formula -ID( Z-Zp1 ) = S( - zo) and the fact that 9 (_) = Ek>1 Z1=k + k(z), where k is holomorphic for z in a small IT
neigborhood of supp f .
Appendix: Grushin problem We will construct a suitable auxiliary (so-called Grushin) problem associated with the operator (-A+V(y) - Ao), for some fixed energy level A0. The same
proof applies to the operator P,,,, = ((-A)2 - Ao).
Theorem A.1. There exist N analytic functions cpj
Rn*/I'* 1 < j < N, such that for every l; E Rn*/F* the Grushin problem, (PP - Ao)u + R_
v,
:
v+,
Fo,g,
--.S
(A.1)
has a unique solution (u, u-) E .F2,£ X CN for every (v, v+) E Fo,£ x CN. Here we have put
PP = D2 +V(y),
u (j)coj(
(u, 1<j
In the next proposition we prove that if pl (c), ... , c,0N (l;) are linearly independent, then the well posedness of (A.1) only depends on the vector spaces Te generated by W1 (t;), ... , cpN(l;).
Proposition A.2. Fix
cpN(1;) are linearly and assume that independent. Then the problem (A.1) is well posed if and only if (1- 7r) (PP -
Ao) as an operator from T£ fl .F2, to El is bijective. Here it denotes the orthogonal projection onto T.
Proof. In the case v = O,v+ = 0 the problem A.1 takes the form 0, u E T.l fl .F2,g. Then (1 - 7r) (PP - Ao)u = 0, (PP - Ao)u + R_ so the injectivity of the restriction of (1 - 7r)(PP - Ao) to T£ fl .F2,g and are linearly independent imply that we have the fact that cpl (c), ... , coN uniqueness for the problem (A.1). If the restriction of (1 - ir)(PP - Ao) to
T, fl .F2,6 is not injective, then there is u 0 0 E T fl F2,£ such that (PP - Ao)u = ir(PP - Ao)u E Range of R_(i;), and we see that there is u- such that (A.1) holds with v = 0, v+ = 0. We have then established the equivalence between uniqueness for (A.1) and injectivity of the restriction of (1 - 7r) (PC - Ao) to T fl T2,g. Since cpl(t=), ... , cpN(l;) are independent, R+ (l;) is surjective and the solvability of (A. 1) is equivalent to the solvability
13. Spectral theory for perturbed periodic problems
185
of the first equation of (A.1), with u E TL n .F2,g. Since the image of R_ (e) is TT, we see that solvability of (A. 1) is equivalent to the surjectivity of the restriction of (1 - 7r) (PP - Ao) to T
Proposition A.3. Let P be a second order elliptic selfadjoint operator on a compact manifold M with smooth coefficients. Let 01, ... , cON E L2 (M) be linearly independent functions and assume that that there is a constant Co > 0 such that (Pu, u) > Co' I I u l I 2 , u E H'(M) n [ V 1 ,- .. , cPNI',
(A.2)
where [c1i...... PN] denotes the linear span o f the functions V 1 ,. .. , coN, and H9(M) (for s E R) is the classical Sobolev space on M of orders. Then if P is another second order selfadjoint operator and ;51, ... , coN E L2(M), with co1 II, IIP - PII c(H1,H-1) and , II1N - wN II small enough, there exists
a constant C1 > 0 such that
N
(Pu, U) > Ci I IIuII I
-
C1
I (u, ;) l2,
(A.3)
1
for all u E H1(M).
Proof. Without loss of generality, we may assume that 01, ... , cpN is an orthonormal system. Choose 01, ... , ON E H2 with (0j, cPk) = Sj,k. For u E H1, we put u = u - E1 (u, coj)oj E H1(M) n [cPl,... , cPN]1, so we can apply (A.2) to u: (Pu, u) > C6-1 llull2 Now, N
N
(Pi,ii)<(Pu,u)+CIIuli
I(u,cv,)I+C
I(u,W;)I2, 1
1
N
IIuII2 < 2llull2 + C
I (u, co) I2,
so with a new constant we get N
(Pu, U) > C-1llull2 - C
I (U' W,)I2
Combining (A.4) with the Garding inequality (Pu, U) > C-1llulli - C'llull2,
(A.4)
Spectral Asymptotics in the Semi-Classical Limit
186
we obtain (C2 + 1)(Pu, u) > C-1IIUII2
-0
N
I(u,
O,)I2
From this we get (A.2) if 11F- PII r(H1,H-1), III -'P111, small enough.
(A.5)
II PN - 1PN II are
#
Proof of Theorem A.1. For a fixed eo E Rn*/F* we can find cpi eo) and a constant CO > 0 such that ((Pa - Ao)u,u) ? Co 'IIuII2 for all u E
n (A.6)
Let E C Rn be a fundamental domain of I'. Modifying Wq by terms with small norm (which will not destroy (A.6)), we may assume that supp (W°) n8E = 0 , so that Wj (x, o) = E °(x 1, with Ojo E L2(E) n &'(int (E)). Put Wj (x, ) = >7 V)j' (x - -y)eiO. Proposition A.3 shows that for close to t o we have with a new constant Co > 0 Y
((P£ - Ao)u, u) >- C
'IIuII2
for all u E Pl,eo n [1Pl (', f ), ... , PN(', )]1 (A.7)
Clearly, if we add more functions to our system W1, ... , cON then (A-7) remains true for t; in the same neighborhood of o and with the same constant Co. Varying the point to, and using the compactness of Rn*/F*, we obtain with a new N a system Wj(x, t ), such that (A.7) holds for all E Rn*/F* with a new constant Co > 0 which is independent of e. Without changing (A.7) we may eliminate successively all the Ojo is which are linear combinations of the others (and make the corresponding elimination of W3). We then obtain cPN (A.7) with cPl 1'*-periodic and analytic in t and linearly
independent for every e E Rn*/r*. It is easy to show from (A.7) that (1 -7r£)(PP - A0) : n [Pl(',S),...,cON(',S)]1 is bijective, and this completes the proof of Theorem A.I.
#
Notes In this chapter we have only discussed one (semi-classical) aspect of a very wide subject. In addition to the papers, [GMS], [Dil], on which the chapter is based, we can mention [HeSj4], which gives precise information on the density of states of the periodic Schrodinger operator with magnetic field. The time-dependent periodic Schrodinger operator is discussed in the papers of Gerard [Ge] and Ralston [Ra]. In the one-dimensional case many results were obtained by Buslaev [Bul,2,3] and Buslaev-Dimitrieva [BuDi]. The results of the present chapter can be applied to study the eigenvalues in gaps of the essential spectrum for certain perturbations with large coupling
13. Spectral theory for perturbed periodic problems
187
constant. Let A be a selfadjoint operator and consider the quantity N(E, A)
(where A > 0 and E is a regular point for A) defined as the number of eigenvalues of At := A+tW crossing E as t increases from 0 to A. Here W is a perturbation decaying at infinity. The behaviour of N(E, A) has been studied
in detail for the periodic Schrodinger operator. Since the work of [ADH] we know that the behaviour of N(E, A) is dramatically different for nonnegative and non-positive perturbations. Specifically, the leading term of the asymptotics for non-positive potentials is given by the classical Weyl formula and does not contain any information on the periodic background. See [ADH], [Bil,2]. On the contrary, for non-negative perturbations the answer contains the density of states associated with the unperturbed operator and depends
only on the asymptotics of the perturbation at infinity. An asymptotic expansion of tr f (At) for f E Co (I), where I is an open interval disjoint from the essential spectrum, was obtained by [Di2]. In the one dimensional case more precise results were obtained by Sobolev [So]. The situation is more difficult if the perturbation is alternating. At present only partial results are known to this effect, see [ADH], [Heml,2] and [Le]. The case when A is the Schrodinger operator with magnetic field was studied by Birman-Raikov [BiRa]. See also [GeSi].
14. Normal forms for some scalar pseudodifferential operators In this chapter, we shall give local normal forms for classical pseudors valid
near a non-degenerate minimum of the symbol. For a formal selfadjoint operator P = pw(x, hDx) whose symbol admits a non-degenerate minimum at (0, 0), we show that there exists a unitary fourior U such that the symbol of U*PU in a neighborhood of (0,0) is - E' o pj(x, )hj, with Hro,2pj = O((x, )°°), where po,2 = F_j ( + x ). Here )j are the eigenvalues of the 2 if MA :_ {a E ZTh; .\ a = 0} _ {0} then matrix (ate p)(0, 0). In particular, fj(T1,...,T,,,) is a real-valued smooth function defined in a neighborhood of (0,,. , 0). As indicated at the end of Chapter 4, we apply this result to study more .
precisely the asymptotic behaviour of the lowest eigenvalues of an h-pseudor: pw(x, hDx) when p(x, ) has a non-degenerate minimum at (0, 0), p(x, ) > 0
for (x, ) 0 (0, 0) and lim
p(x, ) > 0. For any fixed N > 0, we get asymptotic formulas for the eigenvalues up to 0(hN). We now formulate the assumptions. Let 1 C R2n be a neighborhood of the origin. We let S, ',(Q) be the space of formal asymptotic sums a(x, ; h) - E o aj (x, )hj, with aj E C°° (S2) (in the sense of Chapter 2 formula (2.10)). With such a symbol, we associate a formal h-Weyl quantization: aw(x, hDx; h)u(x) =
(2xh)-n
ff
y
;
(14.1)
2
and if b is a second symbol of the same kind we have (formally)
a' (x, hDx; h) o b' (x, hDx; h) = c' (x, hDx; h),
with c(x, ; h) - E' o cj (x, ) hi given by Proposition 7.7. We shall mostly (0, 0), so if aj, bj are defined only consider formal power series at (x, simply as formal power series at (0, 0) then the composition formula still defines c(x, ; h) - E' o cj (x, ) hi with cj as formal power series at (0, 0). Denote the corresponding symbol spaces by S. Put S,''(fl) = h-mSol St =
h--S°. Let p(x, ; h)
o pj (x,
)W E S.1 be a real valued symbol, such that
po(x, e) has a non-degenerate minimum at (0, 0). We shall use the following
well-known fact (see for instance [HoZe] for a proof). If q is a positive definite quadratic form on R2,, then there exists a real linear canonical transformation ic and Ar i ... , A,,, > 0, which can be invariantly defined in terms of the (linear) flow Hq, such that q o t = E Aj (x + ). Applying a
Spectral Asymptotics in the Semi-Classical Limit
190
this to the quadratic part of the Taylor expansion of p at (0, 0), and using Theorem A.2 of Chapter 7, we may assume that: po(x,
(14.2)
)3)
j=1
) _ =1 2Aj (xj + j) the quadratic
In the following we denote by po,2 (x,
Let Ma = {a E Zn; Ej=1 ajAj =
part of po(x, ), and by p the point (x, 0}.
Definition 14.1. Put 12-yj = t j + ixj, vf2-77j = -i(t j - ixj). A resonant function is a smooth function with a Taylor series of the form
f
aa,Qy'r)a.
a,Q;(c -/3)ENIa
Notice that dr7jdyj =
so (y,
are complex symplectic coordinates.
Remark 14.2.
(1) In the
variables we have n PO =
and
jyjrlj
n
(14.3)
n
2Ajyj?lj, Hp.,,, =
po,2 =
j=1
a Z)`j (yj
j=1
ayj
- 77j
a
).
ayj
Let f be a smooth function with Taylor series of the form f E-,OEN fa,Qy"'rla. Then
2(A, a - 0)fa,OyaijR
Hpo.2f c,QEN
Hence, f is resonant if Hp,,,, f = 0((x, )°°). (2) If Q(x, t;) is a quadratic form and a " > ajhj E S° (S2), then we have [Qw(x, hDx), a'(x, hD,,)] _ -ihOph {Q, a} _ -ihOph (HQ (a)). This follows from the Weyl calculus (see Chapter 7). With po,2 as above we have: the symbol of [po 2(x, hDx), a' (x, hDx; h)] is equal to zero in St 1, if each aj is resonant.
14. Normal forms for some scalar pseudodifferential operators (3) If A1, A 2 ,- .. , A,,,
191
are Z-independent (i.e. MA = {0}), then the resonant xi ), ... , xn)) + O((x, )°°) with
functions are of the form g j(1
2
f E COO.
In the appendix to Chapter 11, we reviewed some theory of h-fouriors, and in particular how to associate such an operator with a canonical transformation r, between two neighborhoods (0, 0) in R2n, with i(0, 0) = (0, 0). Let U be such an operator of order 0 with a compactly supported symbol of class S°1.
Then U* is associated with /C-1, and if we normalize the phase in U by adding a suitable constant, then U*U = Oph(jl), UU* = Oph(j2), with jj, j2 in S°1, of compact support modulo S-°°(((x, ))-N) for every N. Choosing the symbol of U suitably, we can arrange so that j1-1, j2 -1 are of class S-°° near 0, and we then say that U is microlocally unitary near (0, 0). In most of this chapter, we only consider the symbols near (0, 0). If Sl is a sufficiently small neighborhood of that point and p - E' O pj hi E S°,(1) is real-valued, then U*p'U is a well-defined formal h-pseudor with a real-valued symbol Ax, ; h) - E' o pj (x, ) hi E S° (S2), where S2 = -1(S2) po = po n.
If d/c(0, 0) is close to the identity, then K is given by a smooth generating function cp defined near (0, 0), so that: K : a(x, 77), r7) H (x, a In fact, this can be seen as in the proof of Theorem 1.3, using the fact that the symplectic form E d j A dx; + E dy; A dr7j vanishes on the graph of ic, and letting (x, 17) play the role of the x-variables in Theorem 1.3. We can choose U of the form
Uu(x) =
(27th)-n
Jf ei(v(x,'))-y-"?)1 ha(x, 77; h)u(y)dyd77,
(14.4)
with a E S1 0A.
Remark 14.3.
(1) Let a(x, ; h) ^' E°° o aj(x,
p(x, ; h) ^' > °o pj(x, )hi E So(R2n)
be two real valued symbols. Following the procedure of Chapters 10-11, e-ita"'/hpw is a fourior, for small t, with associated canonical transformation Ot = exp tHao, where (Pt is the flow generated by the Hamiltonian field Hao . e-i(t+s)a-/h = e-ita'/he-isa"'/h as well as the fact (2) Using the fact that that the composition of two fourior is again a fourior (see the appendix of Chapter 11) we see that the above remark remains true for all t E R.
We have the so-called Birkhoff normal form for the principal symbol po (x, l;) (recalling that after linear canonical transformation po satisfies (14.3) in the (y, r7) variables):
Spectral Asymptotics in the Semi-Classical Limit
192
Proposition 14.4. There exists a real smooth canonical transformation K from a neighborhood of (0,0) onto a neighborhood of (0,0) E R2n, with K(0, 0) = (0, 0), dre(0, 0) = id such that po o r, is resonant.
For the proof, we shall need two lemmas.
Lemma 14.5. Let b be a real-valued function defined in a neighborhood of (0, 0) such that b(p) = 0(p2) near that point. If q is a real-valued smooth function defined near (0, 0) which vanishes to the order m > 2 there, then b(expHq(p)) - b(p) = Hq(b)(p)
+0(p2("'-1))
Proof. From the mean value theorem and the fact that Hq(p) = O(pm-1), we have
exptHq(p) - p = U' (tpm-1), for all t e [0, 11.
(14.5)
Taylor's formula and (14.5) show that Hq(b)(exptHq(p)) - Hq(b)(p) = U'
(tp2(m-1))
for all t E [0,1].
(14.6)
Now the lemma follows from (14.6) and the equality
b(expHq(p)) - b(p) =
J
1
8t(b(exptHq(p)))dt =
0
J
1
Hq(b)(exptHq(p))dt.
o
Lemma 14.6. If g is a smooth real valued function defined near (0, 0) E R2n, vanishing to the order m > 0 there, then there is a smooth real valued function f vanishing to the order m at (0, 0) such that
H0(f)=g+r,
(14.7)
where r is a resonant function.
Proof. Let f be solution of (14.7). Then we have
HH0(Re f) =g+Rer.
(14.8)
Since a function r is resonant if Hp(,,, (r) - 0, it follows that Re r is resonant. Hence if f is a solution of (14.7) then Ref has the properties required in the lemma. Then we can take f real valued. It remains to show the existence of a complex valued solution of (14.7). Let us introduce the (y, ,q) variables. We are looking for a function f defined near (0, 0) with Taylor-Maclaurin series
14. Normal forms for some scalar pseudodifferential operators f = E«,8EN'+ f«,13y«770. Put g = El«1+I/31>m po satisfies (14.3), we get
193
g«,ay«,qR Using the fact that
Hp. (f) - g = E(if«,Oj A, a - 0) +
ga,e)y` 771,
(14.9)
where F«,a(f«,p) is a finite linear combination of terms f«,,Q, with la'I+113'I < jal + 1,31. For jc + 1,31 < m, g«,p = 0, and we take f« Q = 0. For jal + 101 = m, the coefficient of y«r70 becomes i f«,/3 (A, a - 3) - g«,(3.
If a - 3 E MA, we can choose f«,p arbitrarily and g«,py«y/3 is a resonant term. If a - /3 MA, we take f«,p = -ig«,p/(A, a - 0). For jal + I,13I > m, we are in the same situation as above with g«,p replaced by a known number
g«,p - F«,p(f«,p) Given f«,p for all a,,3 E N', we can construct by Borel's theorem (see Chapter 2) a C°° function such that f f«,Qy«r73 in a neighborhood of (0, 0) and f has the required properties. #
Proof of Proposition 14.4. Write po(P) = Po,2(P) + P3 (P), so that p3 = 0(p3). Lemma 14.6 applied to P3 gives a C°° function q3 = 0(p3) such that Hpo(g3) = p3 - r3,
(14.10)
where r3 is a resonant function. From Lemma 14.5 we have
po(expHg3(P)) - po(p) = -Hp.(g3) +P4,
(14.11)
with p4 = 0(p4). (14.10) combined with (14.11) gives: po(exp Hq3 (P)) = Po,2 + r3 + p4.
Consider the sequence m1 = 3, m2 = 4, ... , mj+l = 2(mj -1),.... Assume by induction that we have constructed smooth real functions q,,,,,, q,2.... ) 4r,,k, with q,,, Ca(p'r`i) such that poexp Hq,,,1 o...oexp Hgmk = P0,2+r3,k+pmk+1 with r3,k = 0(p3) resonant and Pm.k+1 = 0(pmk+l). Using again the two preceding lemmas, we find qlk+l = Q(pmk+l) with p o exp Hq,,,, o ... o exp Hgmk+l = Po,2 + r3,k+1 +Pmk}2.
From q, ,
= 0(pm3) and (14.5) we have expHq
(p) - p = 0(pm3
(14.12)
If icy = exp Hg-1 o ... o exp Hq,,,k+1(p), then (14.12) implies that Kj+k - Ki =
0 (pm3 } 1-1) for k > 0, and if we consider generating functions for our
194
Spectral Asymptotics in the Semi-Classical Limit
canonical transformations, we see that there exists a smooth canonical transformation i with r,(0) = 0, dre(0) = I, and fC - Kj = O(pmi+1-1) for
#
all j > 1. It follows that ic has the required properties.
Theorem 14.7. Let p(x, l;, h) - E o pj (x, )hj be a real valued symbol defined in a neighborhood of (0, 0) and assume that po(x, t;) has the form (14.3) with Aj > 0. Then we can find a real canonical transformation t from a neighborhood of (0, 0) onto a neighborhood of (0, 0) with ,c(0, 0) = (0, 0), dtc(0, 0) = I and an associated unitary fourior U (defined microlocally near (0, 0)) such that every term in the asymptotic expansion of the (formal) h-pseudor [E 2' )2 + x ), U*pwU] vanishes to infinite order at (x, 0 = (0, 0).
0
Proof. Let i be the real valued canonical transformation given in Proposition
14.4. Let Uo be a unitary fourior given by (14.4) associated with /£ (we choose a(x, rl, h) in (14.4) such that Uo becomes unitary). Then UopwUois a formal h-pseudor with a real valued symbol p = E S° (12) for some neighborhood S2 of (0, 0) such that po = po o r,, is resonant (i.e.
HPo,2po = 0(p°°)). We recall that po,2 is the quadratic part of po and p = (x, Since k(0, 0) = (0, 0) and dic(0, 0) = id, po has the same quadratic part po,2 = po,2. From now on, we drop the tilde in connection with this new operator, and simply denote it by p.
Let A - E' o ajhj be a real valued symbol in S°1. Then, formally, B' _ exp iA' is a unitary elliptic h-pseudor and Bw*p'1,Bw = p'1'
+ Bw*
[p'1', Bw]
=: V.
(14.13)
Using the pseudor-calculus of Chapter 7 (more precisely the fact that the leading term of Bw*[pw, Bw] equals hHP(ao)), we see that j5 - Ej>o pjhj with po = po, pl = pi + HPo (ao). Using Lemma 14.6 we can then construct a real valued symbol ao, such that p1 is resonant. Assume inductively that we have found ao, al, . . , aN_1 such that p - E 'O pj hi, where pj is resonant .
for j < N. We then look for Bw = exp(ihNAw) with AN E S° (S2), and aN as its principal symbol. Thanks again to (14.13), the leading term of Bw*[pw, Bw] is hN+1HPo(aN), so Bw*pwBw = p '
'
with p 'v E'' j jO,
where pj = pj for j < N and with pN+1 = PN+1 + HP0(aN). We choose aN such that pN+1 becomes resonant. We finally obtain a microlocally unitary fourior as an infinite (asymptotically convergent) product U = Uo o exp(iao) o exp(ihaw) o exp(ih2a2) o ... such that if the symbol of U*pwU is - E' o pjhj, then HPo,2pj = O(p°°) for every j E N, where # P0,2 = Fj-1 a Aj (x + ) From Remark 14.2 we obtain the result. Using Remark 14.3 and Theorem 14.7 we get:
14. Normal forms for some scalar pseudodifferential operators
195
Corollary 14.8. Let p(x, ; h) - > pj (x, Z;)hj E S°1 with po(x, ) as above and assume that MA = {0}. Then there is a real valued smooth symbol f (r1, ... , Tn; h) - Ej>0 f j (Tl...... n)hj defined for (T1, ... , Tn) in a neighborhood o f (0, ... , 0), with fo(ri, ... ,Tn) = E AjTj + O(ITI2), and a formal fourior U, which is unitary, and whose associated canonical transformation is defined in a neighborhood of (0, 0), and maps this point onto itself, such that U*PU = P = p (x, hDx; h) microlocally near (0, 0), where p(x, ; h) -
Ej>Opj(x)hj E 5'°1(S2), with
2(Sn +xn)) +
fj(2( 1 +xl),.
O((x, e)°°). Here S2 is a neighborhood of (0, 0).
As mentioned at the end of Chapter 4, we will study more precisely the asymptotic behaviour of the lowest eigenvalues of a Schrodinger op-
erator when the potential has a single non-degenerate minimum. Let p(x, l;; h) E S° (R2'') be real valued with asymptotic expansion p(x, T;; h) Ej>o pj (x, hi. We assume that p0 (x, t;) > 0 with equality only at (0, 0), and lim infI(x,e)I.,, po(x, ) > 0. After a linear canonical transformation (implemented as in the appendix of Chapter 7), we may assume that po(x,
) _ E 1Aj(
(14.14)
j=1
We know (by the semi-classical sharp Garding inequality, Theorem 7.11) that pw(x, hDx; h) is semi-bounded from below by -Ch for some C > 0. Combining this with results of Chapter 9, we see that the spectrum of the operator P = p'°(x, hDx; h) is discrete and of total multiplicity 0(h-') in po(x) ). an interval ] - oo, rl] with rl < lim
Theorem 14.9. Assume that MA = {a E Zn; a A = 0} = {(0,. .. , 0)}. Then there exists a real valued smooth function F(7-1i...,Tn;h)
-F'o(T,,...,Tn)+Fl(T1,...,7-n)h+...,
(T1...... n) E Rn
with Fj = const. for T1 + ... + r, > 1, Fo(T) = > Ajr + O(ITI2), F0 > 0 when Tj > 0, r # 0, such that for every fixed 6 > 0, the eigenvalues of P = p- (x, hDx; h) in ] - oo, h6] are of the form F((k1 + 2 )h, ... , (kn + )h; h) + ((h°°), k E N. 2
(14.15)
More precisely, if we let a1 < a2 < ... be the increasing sequence of eigenvalues of P and if we let ,Q1 < ,(32 < ... be the increasing sequence of values of the form F((k1 + )h, ... , (kn + )h; h) with k E Nn, then 2 2 aj -,6j = O(h°°) uniformly, as long as aj (or /3j) is < V.
Spectral Asymptotics in the Semi-Classical Limit
196
As a preparation we need the following lemma:
Lemma 14.10. Fix e E [0,
Let X E Co (R2n; [0, 1]) be equal to 1 in a neighborhood of (0, 0). We assume that the support of X is large enough such that po (x, ) + X(h-E (x, 6)) > (1 + )h2E, for some C > 0. Let X < h2E and 1[.
u E L2(Rn) such that IuII = 1 andc(P - A)u = 0. Then
u = X"'u + O(h°°), in L2,
(14.16)
uniformly with respect to A and u. Here XW'E = X'(h-Ex, h-EhDx).
Proof. Let X E Co (R2n; [0, 1]) be equal to 1 near (0, 0) and have its support contained in the interior of the set where X = 1. We choose the support of X large so that (1 +
p(x, ) + X(h-Ex,
2C
)h2E.
Introducing x = h-Ex, we can apply the semi-classical sharp Garding inequality with
h'-2E
as the new parameter `h', and get:
PE := P + Xw'E > (1 + 3C )h2E, for h small enough, in the sense of selfadjoint operators. Hence for A < h2E
(PE - X)-1 =
O(h-2E),
in C(L2, L2).
(14.17)
Let Xo, X1, , XN E Co (R2n; [0,1]) with Xj = 1 near supp Xj_1 for j = 1,, .. , N and with Xo = X, XN = X. Then [Xi 'E, Xk'E] = 0(h°°) in
£(L2, L2) when k
j, and X 'E(1 - Xw'E) = 0(h°°), [Xw'E, PE]X = O(h°°) E
in C(L2, L2) when k > j. From (P - A)u = 0 we get u = (PE -
A)-1Xw'Eu
Then (1 - Xw'E)U = [(1 - Xw'E), (PE - A)-1]Xw'Eu + (PE - A)-1(1 - Xw'E)Xw'EU. (PE
-
A)-1 [Xw'E,
PE](PE
-
A)-1Xw'Eu
+ R1u,
where IIRI II = O(h°°). Using the fact that X,w\,'E 1 ... xw,E = XW,E + 0(h°°) as well as the fact that [Xk'E, PE]XW,E1 = O(h°°) in £(L2, L2), we get (1 - Xw'E)u = (PE - A) [X1
[X°'E, PE] (PE - X)
[XN,E
1' PE] (PE -
A)-1
.. .
PE](PE - A)-1Xw'Eu + RNU,
where IIRNII = 0(h°°). As [Xw'E, PE] = [Xk'E, P] + O(h°°), then \ ... (1 - Xw'E)u = (PE P](PE - X) [XN'E P](PE - A) X)1 [Xw'E, P](PE - X)-1Zw,Eu + RNU, IIRNII = O(h°°). (14.18)
[X,'E
14. Normal forms for some scalar pseudodifferential operators
197
Using the fact that the principal symbol of P vanishes to the second order at (0, 0) we see by the symbolic calculus of Chapter 7 (more precisely Proposition 7.7) that the symbols of [X, P], [Xj, P] are of the form hr(x, e; h)
with r(x, ; h) E S°(1), and Proposition 7.11 shows that [X, P], [Xj, P] 0(h) as bounded operators on L2. Now (14.17), (14.18) imply 11(1 - Xu',e)ull =
(hN(l-2e)-2e).
Here we may choose N as large as we like and this gives (14.16). Remark 14.11. Fix e = 0 and let 6 be a positive constant. Following the proof of the lemma, when A < 6 we can choose the support of X as small as we like provided that we take 6 small enough. In the following proposition we will reduce the spectral study of P in ] - oo, 77[
for r7 small enough to that of P = U*PU, defined in Theorem 14.7. Recall that P = U*PU = p (x, hDx; h) is only defined near (0, 0) and that Ax, ; h) ^' E pj (x, t=) hi E S°(1Z), j>0
(14.19)
with {p0,2, pj } = 0((x, e)°°). Here p0,2 is the common quadratic part of the Taylor expansions of po (x, l;) and of p0 (x, t ). We may assume that ; (i) The symbol (14.19) is globally defined and of class S°(R2n). (ii) For every (x, l;) E R2n and every h E]0, h0] (0 < h0 is a small constant p(x, t:; h) E R and p0 (x, t;) > 0 with equality only at (0, 0).
(iii) lim infl £l-,, po(x, t:) > 0. Then P = p (x, hD,; h) is a well defined selfadjoint operator with the same general properties as P. In particular it is semi-bounded from below by -Ch and has discrete spectrum in ] - 001 77] for some q > 0, independent of h.
Proposition 14.12. Fix 77 small enough. Let al < a2 < ... and 131 < Q2 < ... be the eigenvalues of P and P respectively in the interval ] - oo, r7]. Then 711, where 771 is any aj -,3j = 0(h°°) uniformly for all j such that fixed number in ]0, 77[.
Proof. If a E o,(P)fl ] - oo, 77], Pu = au and (lull = 1 then by Proposition 14.10 u = Xw(x, hD,,)u + 0(h°°) in L2, where X E Co (R2n) is equal to 1 in a neighborhood of (0, 0). If 77 is small enough, then by Remark 14.11 we
198
Spectral Asymptotics in the Semi-Classical Limit
can take x with supp x contained in the region where U and U* are defined as unitary operators and U*u is then well defined in L2 mod O(h°°), and IIU*ull = 1. We have PU*u = aU*u + O(h°°), and hence
dist (a, o(P)) = 0(h°°).
(14.20)
As P has the same general properties as P, then by the same argument we show that if a E o, (P) n ] - oo, rt], then
dist (a, o(P)) = 0(h°°).
(14.21)
In particular, if [f -hN1, f +hNh]no(P) = 0 or [f -hN1, f +hNhJno(P) = 0 for some fixed Nl and for f < rl, then
[f - 12 hN1, f + 12 hN1J n (o(P) U o(P)) = 0,
(14.22)
when h is sufficiently small. Let -Ch < fl < f2 < rl be two values satisfying (14.22) and let E, F be the spectral subspaces associated with [fl, f2J nor (P) and [fl, f2] n a(P) respectively. Let e1, ... , eN be an orthonormal basis of
eigenvectors in E: Pej = ajej. Then with ej = U*ej, we have (ej, ek) _ bj,k + 0(h°°), and
Pej = ajej +7'j, with rj = O(h°°).
(14.23)
Let k be the N-dimensional space spanned by the ej . From (14.23) we have
(z - P)-lej = (z - aj)-lej + (z - aj)-1(z - P)-lrj, integrating this with respect to z over the rectangular contour I' with the corners fj ± ih, we conclude that 7rFe3 = ej + 0(h°°).
(14.24)
We recall that irF = ZZ fr(z - P)-ldz. Thanks to the non-symmetric distance, d , introduced in Chapter 6, we deduce from (14.23) that d (E, F) _ 0(h°°). Hence from Lemma 6.10 we get dim F > dim k = dim E, when h is sufficiently small. Similarly, dim F < dim E, so dim E = dim F. Combining this with (14.20), (14.21) we get the result.
Proof of Theorem 14.9. The proof is now reduced to the study of o(P). By Corollary 14.8, we have:
pj(X,0 = fj(xl 2
1 ...
2
tt 2
Xn 2 Sn) +0((x,t)°°),
14. Normal forms for some scalar pseudodifferential operators
199
with fo(Ti,... , Tn) = EA,jrj + O(ITI2). Following the construction of Ax, 1;; h), we can assume that fj = const. in the region T1 +... +Tn > 1 and fo > 0 with equality only when T = 0.
By the functional calculus for functions of several commuting h-pseudors discussed at the end of Chapter 8, we can construct a real valued function F(T1i ... , Tn; h) in S°(R2n), with asymptotic expansion
F(T1i...,Tn;h) `v F'o(T1...... n)+Fi(T1...... n)h+...,
(14.25)
where Fj have the same general properties as the fj above and where fo = Fo,
such that if R := Rw (x, hDx; h) = F(2 ((hD.i )2 + x1) , .. ., 2 ((hDxn )2 + x,22); h), (14.26)
then R(x, l;; h) - Eo r,j (x, ) hi E S°(R2n), and p.j (x, Z) = rj (x, ) + O((x,)°°), near (0, 0).
(14.27)
(Notice that pseudors of the form (14.26) must have resonant symbols.) Fix 6 E]0, 2 [. Let A < h6 and u E L2(Rn) such that Ijull = 1 and (P-A)u = 0. Then
(R - A)u = (R - P)u = (R -
P)Xw, 2 u + (R
- P)(1 - X°, 2 )u,
where X E C'°(R2) and Xw 2 = X"' (h- 2 x, h- 2 hD,,). From Lemma 14.10
we have (R - P)(1 - X', 2)u = 0(h°°). Using the fact that the symbol of (R- P) is - E kj (x, t;) hi with kj (x, ) = O((x, )°O), as well as the fact that (x, e) _= O(h2) on supp X(h-2 (x, C)), we get, using the symbolic calculus:
(R - P)Xw4u = O(h°°) so
(R - P)u = 0(h°°). This result holds after permutation of P, R. Using the same arguments as in the proof of Proposition 14.12 we get:
Lemma 14.13. Fix rl > 0 small enough. Let a1 < a2 < ... and,31 < ,(32 < ... be the eigenvalues of P and R respectively in the interval ] - oo, rl]. Then a. - ,Q3 = 0(h°°) uniformly for all j such that a,j, )3.j < h6. Now Theorem 14.9 follows from Proposition 14.13, Lemma 14.14 and the
fact that in ] - oo, rl[ the spectrum of R is given by the values F(h(k1 + 2),...,h(kn+ 2);h), k E Nn.
200
Spectral Asymptotics in the Semi-Classical Limit
Remark 14.15. We have seen in Theorem 4.23 that all the eigenvalues in an interval [0, Coh] of the Schrodinger operator, -h2A+V, described at the end of Chapter 4, have asymptotics of the form
where Eo < Co runs through the eigenvalues of the associated harmonic oscillator. As indicated at the end of Chapter 4, if Eo is a simple eigenvalue
then (*) has an asymptotic expansion - h(Ec + Elh + E2h2 + ...). This follows from Theorem 14.9 when MA = {0}.
Notes Proposition 14.4 and further references can be found in [GDFGS]. Birkhoff normal forms for small perturbations of harmonic oscillators were given by Bellissard-Vittot [BeVi]. The relationship between the classical Birkhoff
theorem and corresponding quantum perturbation theory is discussed in the paper of Graffi-Paul [GrPa]. More about Birkhoff normal forms can be found in Gutzwiller [Gut], Francoise-Guillemin, [FrGu], Zelditch [Ze2] and Guillemin [Gu].
In the one-dimensional case Theorem 14.9 is due to [HeRo3,4], [HeSj7], [CdV4]. A similar result in the case when the symbol has a saddle point was given by [HeSj7]. In multiple dimensions the theorem is due to Sjostrand [Sj5], which we have followed in this chapter. More recent results in the same direction can be found in Kaidi-Kerdelhue [KaKer], Iantchenko [Ia], Popov [Po] and Bambusi-Graffi-Paul [BaGrPa].
15. Spectrum of operators with periodic bicharacteristics In this chapter we treat scalar h-pseudors with periodic Hamiltonian flow. Recall the eigenvalue asymptotics of Theorem 10.1 (under suitable assumptions, and in particular that E1 and E2 are not critical values): N([E1, E2]; h) =
(27rh)-n
ff
d xd +
0(h1).
(15.1)
As we have seen, the analysis of the remainder term 0(h-n+1) depends on the set of periodic trajectories in po 1({E1, E2}). If the Liouville measure
of this set is equal to zero then we can replace 0(h-n+1) in (15.1) by (cl(El, E2)h-n+1 + (See Theorem 11.1.) There are, however, o(h-n+l)).
many natural and interesting situations when all the Hp,, solution curves are periodic. In such cases, after a functional reduction to the case when all trajectories have the same period, we prove the existence of two constants 6, a
such that the spectrum of P(h) = Oph (p0) is concentrated near the points (TE 1 + a)h + S, 1 = 1, 2, 3, ... (a result due to Colin de Verdiere [CdV4], Duistermaat-Guillemin [DuGul, Helffer-Robert [HeRo3]). We will give an asymptotic expansion in powers of h of the counting function N(Ij; h) when
Il is a subinterval of length 0(h2) centered at T (T = (T 1 + a)h + 6 E ]El, E2 [). In the one dimensional case we get a more precise Bohr-Sommerfeld quantization condition (Theorem 15.10).
Let E1 < E2 be two real numbers, and let p = p(x, ) E S°(R2n) be independent of h for simplicity. We assume that p is real valued and that for e > 0 small enough, we have (H1) lim
d(p(x, ), [El - E, E2 + E]) > 0,
(H2) dp 34 0, for all (x, ) E p-1([E1 - E, E2 + E]), (H3) p-1(E) is connected for every E E [E1 - E, E2 + E],
(H4) there exists a smooth function T =
0 defined in p-1([E1 -
E, E2 + e]) such that exp(T(x, ) Hp) (x, 6) = (x, 6), V(x, 6) E p 1([E1 - e, E2 + E]).
Remark 15.1. Assumption (H1) implies that if h > 0 is small enough, then the spectrum of P(h) = Oph (p) is discrete in a neighborhood of [El - e, E2 + E].
Let y(x, ): [0, T (x, )] E) t F-, exp tHp(x, ), so that y(x, ) is a closed curve.
Spectral Asymptotics in the Semi-Classical Limit
202
Lemma 15.2. T(x, .) and J(x, ) := ff(S ) dx only depend on p(x,
writing T(p(x, )), J(p(x, l;)), (x, e) E p-1([El - E, E2 + e]), we have J'(p) _ T(p). Proof. Let [0, 1] D s --> (x(s),1;'(s)) E p-1([Ej - e, E2 + E]) be a C1 curve
and put cp(t, s) = exp(tHp) (x(s), (s)), 0 < s < 1, 0 < t < T(x(s), (s)) so that cp(T(x(s), (s)), s) = V(0, s). Put `ys = {0(t, s); 0 < t < T(x(s), (s))}. Let a be the symplectic 2 form on R2n and let W* (a) be the corresponding pull-back. Then cp* (a) = a(t, s)dt A ds, where (with co* = dip), a(t,s)
(a,HpAW*(a
=
J(p(co(t,s)))
= -(dp,,p*(as)) a, Stokes' formula gives
Since
r
J
dx -
JI.
dx =
'Y
Tc.T(s'),f(s'))
JIS J 0
a
as'
o
(p(cp(t s')))dtds'.
(15.2)
Here
p(cp(t, s)) =: p(s)
(15.3)
is independent of t, so (15.2) reduces to
f dx - f edx = If rys
J°
T(x(s'),(s'))(s').
(15.4)
all belong to the same energy surface p(x, ) = E, we see that
J(x(s),e(s)) is independent of s. Hence (by abuse of notation) Ax' O _ J(p(x,t;)). Then (15.4) shows that aPJ(p) =T(p) with T(p(x,)) =T(x, Choose J E S° (R) (with order function 1 when nothing else is indicated),
real valued, such that J(E) = J(E) for E E 1 , J(E) > J(E2 + E) for
E > E2 + E and J(E) < J(E, - e) for E < El - E IE = [El - e, E2 + e], I = Io. Put Q(h) =
-LJ(P(h)).
Here we write From Chapter 8,
Q(h) = Oph (q(h)) with S° E) q- >j>o g3hi, qj independent of h and of class S°, qo (x, ) = 2L J(p(x, )), ql (x, ) = 0. Since Hqo = 2L J'(p)Hp we have:
Lemma 15.3. exp(tHgo)(x, t;) is 27r-periodic for (x, l;) in a neighborhood of qo 1([J(E,), J(E2)])
15. Spectrum of operators with periodic bicharacteristics
203
As J is a CO° diffeomorphism from a neighborhood of I onto a neighborhood
of J(I), the properties of the discrete spectrum of P(h) in I can be deduced from those of Q(h) in J(I). Without any loss of generality we can assume:
(H5) TE=2lrforallEEI,. Put
6=
2 f (E) tdx - E.
(15.5)
It follows from Lemma 15.2 and assumption (H5) that 6 is a constant independent of E in IE. As long as we are only interested in the eigenvalues of P(h) in I, we may replace P(h) by P(h)b(P(h)) with zb E Co (]E1-e, E2+e[) equal to 1 in a neighborhood of [E1i E2]. Let U.0 (t) = e-itP/ho (P(h)). Here we are interested in Up (27r). From results of Chapter 10 and for small t, UP (t) is a fourior with associated canonical transformation exp (tHpo) (x, ). In our situation exp (27rHp) (x, ) = (x, ) for all (x, ) in supp (?P(p)). Then we have
the well-known result (which to a fairly large extent can be deduced from the appendix in Chapter 11, see Duistermaat [Du], Asada-Fujiwara [AsFu], Chazarain [Ch]):
Theorem 15.4. Assume (H1) to (H5). Then Ui,(2ir) = Oph (q(h)), with Ej>o qjhi, qj independent of h and supp RD C p-1(]El S°(R2") c, E2 + e[). Moreover,
4o (x, 0 _ (p(x,
))e-2zriv(h)
(15.6)
where o,(h) = 4 + h, 6 is defined by (15.5) and a E Z is the Maslov index of ) the trajectory {exp(tHp,,)(x, t;), t E [0, 2ir]} in p-1(]El - e, E2 + co. A consequence of Theorem 15.4 is that: e(-27ri/h)(P(h)-hv(h))O(P(h)) = V)(P(h)) + hR(h),
(15.7)
where the symbol of R(h) is in S.0,,(R 2- Let b1 E CO -(]E, - e, E2 + e[) be such that bbl = 01. In view of (15.7) we have: e(-2ari/h)(P(h)-ho(h))V)1 (P(h)) = VJ1(P(h))(I + hR(h)).
(15.8)
Then we can use Chapter 8, to see that for h > 0 small enough:
(I + hR(h))-1 = I + hR(h), where R(h) is an h-pseudor with the same properties as R(h). For h small enough we put:
W(h) = (1/2irih)log (I + h7 (h)).
204
Spectral Asymptotics in the Semi- Classical Limit
Using Chapter 8 again, we see that W (h) is of class S,,,. It is clear that -W (h) commutes with P(h). Summing up we have proved:
Proposition 15.5. Under the assumptions of Theorem 15.4, there exists W(h) = Oph (w(h)), with S°1(R2'") E) w(h) with P(h) such that:
wjhj, which commutes
e(-27ri/h)(P(h)-ho(h)-h2W(h)),bi(P(h)) = 01(P(h)).
We now decrease r: > 0 so that [E1-e, E2+e] becomes contained in the region where 01 = 1. Let (Aj(h))o<j
can choose Uj,h to be eigenvectors of P(h) := P(h) - ha(h) - h2W(h) also, and we let vj(h) be the corresponding eigenvalues. Proposition 15.5 implies that vj(h) E hZ for h small enough. In view of the definition of P(h) we can find Co > 0 (independent of h) such that (A1(h) - vj(h) - h 4 - 61 < Coh2.
(15.9)
Using (15.9) and the fact that a(P(h)) n [E1i E2] C v(Q(h)) as well as the fact that vj(h) E hZ, we get:
Theorem 15.6. Under the assumptions above, there exist Co > 0, ho > 0 such that (o(P(h)) n [El, E2]) C UIEZI!(h),
for all h E]0, ho] with I,(h) = [(l + )h + 6 - Coh2, (1 + )h + 6 + Coh2]. 4 4 Remark 15.7. If ho is small enough, then It(h) n II, (h) = 0 for all h E]0, ho] and all 1# 1'. Now we assume
(H6) exp(tHH)(x,i;)
(x,e), for all (x,e) E p-1([E1 - e, E2 +E]) and all
t E]o, 2ir[.
Let 1/i E Co (] El - E, E2 + e[. We define for 1 E Z: V)(A)
Nl,,p (h) = AEv(P(h))ni, (h)
15. Spectrum of operators with periodic bicharacteristics
205
When is equal to 1 in a neighborhood of [El, E2] we get the number NA(h) of eigenvalues in 11(h).
Theorem 15.8. Under the assumptions (H1-6), we have N1,, (h) - E rj (h (1 + 4) + 6)h'-n, h -' 0, j>1
where I,j E Co (]E1 - e, E2 + e[), and for j = 1 we have LA(dw), rl(A) = V)(A)(27r)-" f p(W)=A
where LA is the Liouville measure of the hypersurface p = A (cf. (10.15)).
Proof. Recall that o,(P(h)) n [El - e, E2 + e] C hZ, for all h E]0, ho]. Then
(W(P(h))e-ith-1P(h))
tr
(15.10)
_ IEZ AEa(P(h))nIj
so Nl, , (h) are the Fourier coefficients of the preceding quantity, 27r
Nl,,p (h) =
eitltr(4'(P(h))e-ith-1P(h))dt.
27r JO
11 [) be such that -+mEZ X(t - 2irm) = 1. Thanks to the expression of o(h) we get
Let X E Co (] -3 ,
NI,,G(h)
2
f
eatT/hX(t)tr(,O(P(h))eithw(h)e-itP(h)/h)dt
21 R
where ,r = 6+(1+2)h E]E1- e, E2 +e[. Expressions of the form (15.11) have 4 been studied in Chapters 10-12. Assumption (H6) implies that t = 0 is the unique period in suppx of the Hp solution curves in p-1(]Ei - e, E2 + e[), and the arguments of Chapter 11 show that 0(h°°) for t E suppx \ S2, if Q is any neighborhood of 0. Then the analysis tr(zf)(P(h))eathw(h)e-itP(h)/h) is
of Chapter 10 can be applied and Theorem 15.8 follows from (15.11),(10.13) and (10.15). #
Now we restrict our attention to pseudors in one dimension. We assume that
p satisfies (H1-6). We further notice in this case that p-1(E) is a closed
Spectral Asymptotics in the Semi-Classical Limit
206
smooth curve, y(E), with Maslov index equal to 2 (see Maslov [Mall ) for all E E]E1 - e, E2 + e[. In particular we have
dx = f
J(E) =
const.,
(E )
and
X(E) =
JP(W)=E
LE(dw).
Combining this with Theorem 15.8, we get
Corollary 15.9. N1(h) = 1 for all h E]0, ho] such that ((l + 2)h + 6) E [E1, E2]
Let (1, h) E Z x ]O, ho] be such that (1 + 1)h + 6 E [E1, E2] and let A, (h) be the unique eigenvalue of P(h) in I,(h) (given by Corollary 15.9). We have:
Theorem 15.10. For every integer N > 2 we have N
hkfk'(Ac(h)) = ((l + 2)h + 6) +
A, (h) + k=2
uniformly for l E Z. Here fk E C°°(]E1 - e, E2 + e[; R). Proof. Let z/) E Co (]E1-e, E2+e[) be such that ',b(A) = of [E1, E2]. Applying Theorem 15.8 to
on a neighborhood
AEo(P(h))nli(h)
we get
al (h) - (h(l + 2) + 6) + Er (h(l + 2) + 6)hj-1, h -> 0. j>2
Coming back to the formula (15.11), using the fact that the subprincipal symbol of p(x, t ; h) equals zero (we recall that here p(x, ; h) = p(x, ) is independent of h) we get, using formula (11.32), that r2(r) = 0. Consequently, Al(h)
rj(h(l+
(h(l + 2)+6)+
+6)hj 1, h-> 0.
(15.12)
i>3
Now Theorem 15.10 follows from (15.12).
#
15. Spectrum of operators with periodic bicharacteristics
207
Notes Results on clustering of eigenvalues for compact manifolds are due to Colin de Verdiere [CdV2], Wenstein [We] and J.J.Duistermaat-Guillemin [DuGu]. They showed that most of the eigenvalues are concentrated near the lattice points T k+,Q, k = 1, 2, ..., where T is the common period of the Hamiltonian flow and ,d is a constant. Theorem 15.6 is due to Chazarain [Ch] in the case of a Schrodinger operator, -h20 + V, in the general case the theorem is due to Helffer-Robert [HeRo3]. The proofs of Theorem 15.6 and Theorem 15.8 are taken from [HeRo3]. If the assumption (H4) is only satisfied at a fixed energy level E, then Theorem 15.6 remains true provided that we restrict the interval [El, E2] to [E - 0(h), E + 0(h)]. That result is due to Brummelhuis-Uribe [BruUri] in the case of Schrodinger operators, -h 20+V, when 1/h E N, and to Dozias [Doz] in the general case. We refer to Petkov-Popov [PePo] for the more general case when the union of closed trajectories is of non-vanishing measure.
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Index affine canonical transformation affine Lagrangian space
94
almost analytic extension almost O.N.
100
46
asymptotic expansion
19
asymptotic solutions
130
Beals' characterization Beals' lemma
101
109
bicharacteristic curve
15
Birkhoff normal form
197
Bloch eigenvalue
161
Bohr-Sommerfeld quantization Borel constructions bundle
85
207
18
7
Calderon and Vaillancourt's theorem canonical cordinates canonical 1 form
9
canonical 2 form
9
9
canonical transformation caustics
96
20
characteristic equation classical Hamiltonian
17 17
89
Spectral Asymptotics in the Semi-Classical Limit
222
classically forbidden region
complex Lagrangian space
20 95
complex symplectic coordinates
cotangent space
7
cotangent vector
7
Cotlar-Stein lemma critical point
89
49
defect indices
36
differential form
7
discrete spectrum
dual lattice
196
40
166
effective Hamiltonian
162
effective Hamiltonian approximation Egorov theorem
139
eikonal equation
18
elliptic
106
energy surface
17
essential spectrum
40
essentially self-adjoint operator
Farris-Lavine theorem
38
Floquet-Bloch reduction Floquet eigenvalues
184
166
34
161
Index
Fourier integral operator (fourior) Friedrichs extension
37
fundamental matrix
24
generating function
197
Grushin problem
168
Grushin reduction Hamilton field
168
10
Hamilton-Jacobi equations harmonic oscillator
30
Hilbert-Schmidt class
117
IMS localization formula
LA-metric
61
72 59
Lagrangian manifold
lattice
46
15
interaction matrix LA-geodesics
13
30
Hermite polynomial
integral curve
93
12
161
Lie derivative
11
linear canonical transformation linearized vector field Liouville form
128
24
96
223
Spectral Asymptotics in the Semi-Classical Limit
224
Liouville's formula
189
Lithner-Agmon estimate Lithner-Agmon metric maxi-min principle
59
44
metaplectic operators microhyperbolic
93
145
mini-max principle Morse lemma
55
43
49
periodic bicharacteristics
periodic trajectories
134
phase function
141
Poisson bracket
11
pseudor pull-back
195
3 8
push-forward
8
relatively compact
43
resonant function
196
Schrodinger equation semi-classical analysis
17 17
semi-classical approximation
17
semi-classical Garding inequality
91
semi-classical harmonic oscillator
31
signature
140
spectral measure
39 40
spectral projector
spectral theorem
38
stable incoming manifold
26
stable manifold theorem
24
stable outgoing manifold
26
star-shaped
29
standard quantization Stone's formula
81
41
symplectic matrix
tangent space
25
7
tangent vector
7
trace class operator trace formula
117
160
transport equations
19
unbounded operator
33
Vector bundle
7
Weyl criterion
42
Weyl quantization Weyl term
81
129
WKB-approximations WKB expansions
19
72
Notation-index A
34
adAB
104
a = (al, ..., a,,) 39
13b (R)
Co (St) d
18
65
DaX = Da1...D«n X1 Xn D'
166
D(S)
33 163
Em
9
f*l f* 7-1
33 38
Ho (S2)
Hm
15
163
Hf
10
He
163
/C,,,,
168
K.,£
169 176 11
Lip(S2; R)
Af
26
A,
16
55
15
Index lit.-,
II
II
117
IIHS
9 (O(e-f/h)
78
0(h°°)
28
horn
PQ
62
40
S(R)
1
S(Rn X Ry /I')
S(V), S'(V) sj (A)
175 81
117
Soi (Rn, M)
Sb (m), S-00(m) S*
87
33
a=d Adx cress, ad
42
app, asc, aac
to
11
tj
11
10
40
T*X, TX, TAX, TxX Vo
175
(V (x) - E)+dx2 '`Y = "1 l ..."nn
59 16
7
227
