Spectral Asymptotics in the Semi-Classical Limit (London Mathematical Society Lecture Note Series)

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)naon iviatnemb i ice.`

2 68

Spectral Asymptotics in the Semi-Classical Limit M. Dimassi & J. Sjostrand

X

CAMBRIDGE UNIVERSITY PRESS

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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


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


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


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 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=

(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)


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


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.


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


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


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 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') +


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


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


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


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


(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)


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).


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


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


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,


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


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.


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 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.


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


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 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.


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) x, n -> oo, Ihn(x)I 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


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,


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,


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)


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


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 0 and has a non-trivial intersection with E. If u is a normalized vector in the intersection, we have (Aul u) 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


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


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)


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 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 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


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 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)


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).


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


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 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 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.


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 which has 0 as an accumulation point.


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


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, 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


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)
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,


60

where d denotes the natural distance in 7-l and 7rE, is the orthogonal projection onto Ej. Notice that we have the òriented' 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


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,


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) 0.

(6.26)


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
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.,


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

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.


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) 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


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 -


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


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


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 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


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)

[-] [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'


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


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


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 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 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


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


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


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


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
lal 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).


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

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.


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


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 ì' 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


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,


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)


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)


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 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


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 .


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 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 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


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 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)) 0, choose f, f E Co (I; [0,1]) with 1[a+E,Y-E] C f
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,) 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


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

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 R

0(R)hl-n.

(11.16)


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 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.


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)


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, 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)


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,


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, t > 0, Im z + t > 0. Stokes' formula gives for Imz > 0:

G(z) = ff(z - po(x, ))-1X(x, ) dx
)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 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


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)


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


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) + 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 0.

Let A(x) = (Al (x), A2 (x), ... , An (x)) E C°° (R"; Rn ). We assume that:


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 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) 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) .


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 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)


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,


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)


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)


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


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)


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.

#


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,


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


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 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) C-1llull2 - C

I (U' W,)I2

Combining (A.4) with the Garding inequality (Pu, U) > C-1llulli - C'llull2,

(A.4)


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


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


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):


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'rì) 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


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:


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.


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


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


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)°°),


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


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.


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)])


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 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)


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


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).

#


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


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


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

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