Spectral theory of non-commutative harmonic oscillators

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1992 Alberto Parme...

Author: Alberto Parmeggiani

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1992

Alberto Parmeggiani

Spectral Theory of Non-Commutative Harmonic Oscillators: An Introduction

123

Alberto Parmeggiani Department of Mathematics University of Bologna Piazza di Porta San Donato, 5 40126 Bologna Italy [email protected]

ISBN: 978-3-642-11921-7 e-ISBN: 978-3-642-11922-4 DOI: 10.1007/978-3-642-11922-4 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2010924182 Mathematics Subject Classification (2000): 35P20, 35J45, 35P15, 35S05, 47A10, 47G30, 47N50, 58J52, 81Q20 c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper springer.com

To Serena & Lorenzo, Luisa & Pier Luigi

Preface

This book grew out of a series of lectures given at the Mathematics Department of Kyushu University in the Fall 2006, within the support of the 21st Century COE Program (2003–2007) “Development of Dynamical Mathematics with High Functionality” (Program Leader: prof. Mitsuhiro Nakao). It was initially published as the Kyushu University COE Lecture Note number 8 (COE Lecture Note, 8. Kyushu University, The 21st Century COE Program “DMHF”, Fukuoka, 2008. vi+234 pp.), and in the present form is an extended version of it (in particular, I have added a section dedicated to the Maslov index). The book is intended as a rapid (though not so straightforward) pseudodifferential introduction to the spectral theory of certain systems, mainly of the form a2 + a0 where the entries of a2 are homogeneous polynomials of degree 2 in the (x, ξ )-variables, (x, ξ ) ∈ Rn × Rn , and a0 is a constant matrix, the so-called noncommutative harmonic oscillators, with particular emphasis on a class of systems introduced by M. Wakayama and myself about ten years ago. The class of noncommutative harmonic oscillators is very rich, and many problems are still open, and worth of being pursued. I wish to thank Masato Wakayama, dearest friend and collaborator, and my friends and colleagues Nicola Arcozzi, Sandro Coriasco, Sandro Graffi, Frédéric Hérau, Takashi Ichinose, Miyuki Kuze, Lidia Maniccia, Luca Migliorini, Cesare Parenti and Cosimo Senni for their invaluable help in giving these notes a (hopefully) decent shape. Bologna December 2009

Alberto Parmeggiani

vii

Contents

1

Introduction .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .

1

2

The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 7 2.1 From the Hamiltonian to the Operator Acting on L2 . . . . . .. . . . . . . . . . . 7 2.2 The Spectrum of the Harmonic Oscillator . . . . . . . . . . . . . . . . .. . . . . . . . . . . 10

3

The Weyl–Hörmander Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.1 Review of the Weyl–Hörmander Calculus . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.2 Global Metrics and Global Pseudodifferential Operators .. . . . . . . . . . . 3.3 Spectral Properties of Globally Elliptic Pseudodifferential Operators.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.4 Notes . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .

15 15 25 41 53

The Spectral Counting Function N(λ ) and the Behavior of the Eigenvalues: Part 1 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 4.1 The Minimax Principle.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 4.2 The Spectral Counting Function .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 4.3 The Spectral Counting Function of the (Scalar) Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 4.4 Consequences on the Spectral Counting Function of an Elliptic Global ψ do . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .

65

5

The Heat-Semigroup, Functional Calculus and Kernels . . . . . .. . . . . . . . . . . 5.1 Elementary Properties of the Heat-Semigroup.. . . . . . . . . . . .. . . . . . . . . . . 5.2 Direct Definition of Tr e−tA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 5.3 Abstract Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 5.4 Kernels . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 5.5 f (A) as a Pseudodifferential Operator.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 5.6 Notes . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .

67 67 69 71 73 77 77

6

The Spectral Counting Function N(λ ) and the Behavior of the Eigenvalues: Part 2 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 79 6.1 A Parametrix Approximation of e−tA . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 79 6.2 The Karamata Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 83

4

55 55 58 61

ix

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Contents

6.3 6.4 6.5

Use of the Parametrix Approximation of e−tA for Obtaining the Weyl Asymptotics of N(λ ) . . . . . . . . . . . . .. . . . . . . . . . . 86 Remarks on the Heuristics on N(λ ) and ζA (s) . . . . . . . . . . . .. . . . . . . . . . . 91 Notes . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 92

7

The Spectral Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 93 7.1 Robert’s Construction of ζA by Complex Powers . . . . . . . . .. . . . . . . . . . . 93 7.2 The Meromorphic Continuation of ζA via the Parametrix Approximation of e−tA . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 96 7.3 The Ichinose–Wakayama Theorem .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .106 7.4 Notes . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .110

8

Some Properties of the Eigenvalues of Qw (α ,β ) (x, D) . . . . . . . . . . . .. . . . . . . . . . .111 8.1 The Ichinose and Wakayama Bounds . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .112 8.2 A Better Upper-Bound for the Lowest Eigenvalue . . . . . . . .. . . . . . . . . . .115 8.3 Notes . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .120

9

Some Tools from the Semiclassical Calculus . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .121 9.1 The Semiclassical Calculus .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .121 9.2 Decoupling a System .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .129 9.3 Some Estimates for Semiclassical Operators . . . . . . . . . . . . . .. . . . . . . . . . .140 9.4 Some Spectral Properties of Semiclassical GPDOs . . . . . . .. . . . . . . . . . .143

10 On Operators Induced by General Finite-Rank Orthogonal Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .149 10.1 Reductions by a Finite-Rank Orthogonal Projection . . . . . .. . . . . . . . . . .149 10.2 Semiclassical Reduction by a Finite-Rank Orthogonal Projection.. .155 11 Energy-Levels, Dynamics, and the Maslov Index . . . . . . . . . . . . . .. . . . . . . . . . .161 11.1 Introducing the Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .161 11.2 The Maslov Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .173 11.3 Notes . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .189 12 Localization and Multiplicity of a Self-Adjoint Elliptic 2×2 Positive NCHO in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .191 12.1 The Set Q2 and Its Semiclassical Deformation .. . . . . . . . . . .. . . . . . . . . . .192 12.2 Localization and Multiplicity of the Spectrum in the Scalar Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .206 12.3 Localization and Multiplicity of Spec(Aw 2 (x, D)), with A2 ∈Q2s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .227 12.4 Localization and Multiplicity of Spec(Qw (α ,β ) (x, D)) . . . . . .. . . . . . . . . . .234 12.5 Localization and Multiplicity of Spec(Aw 2 (x, D)), with A2 ∈ Q2 \ Q2s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .237 12.6 Notes . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .238

Contents

xi

Appendix . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .239 A.1 Almost-Analytic Extension and the Dyn’kin–Helffer– Sjöstrand Formula .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .239 Main Notation . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .245 B.1 General Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .245 B.2 Symbol, Function and Operator Spaces .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . .246 B.3 The Spectral Counting Function and the Spectral ζ -Function . . . . . . .247 B.4 Dynamical Quantities and Assumptions . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .247 B.5 Classes of Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .247 References .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .249 Index . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .253

Chapter 1

Introduction

A non-commutative harmonic oscillator (NCHO for short) is the Weyl-quantization aw (x, D) of an N × N system of the form a(x, ξ ) = a2 (x, ξ ) + a0 , (x, ξ ) ∈ Rn × Rn = T ∗ Rn , where a2 (x, ξ ) is an N × N matrix whose entries are homogeneous polynomials of degree 2 in the (x, ξ ) variables, and a0 is a constant N × N matrix. In other words, a NCHO is the Weyl-quantization of a matrix-valued quadratic form in (x, ξ ), plus a constant matrix term. The name NCHO (which was given by M. Wakayama and myself) originates on the one hand from the fact that a scalar harmonic oscillator is a single quadratic form in (x, ξ ), and on the other from the two levels of non-commutativity that one has to deal with when studying these systems: the level due to the matrix-valued nature of the symbol of the system, and the level due to the Weyl-quantization rule xk ξ j ↔ (xk Dx j + Dx j xk )/2 (where D = −i∂ , as usual), reflected through symplectic geometry by the Poisson-bracket relations {ξ j , xk } = δ jk , 1 ≤ j, k ≤ n. I shall say that a NCHO aw (x, D) is elliptic when det a2 (x, ξ ) behaves exactly like (|x|2 + |ξ |2 )N when |(x, ξ )| is large. When a2 and a0 are Hermitian matrices, the operator aw (x, D) is “formally self-adjoint” (that is, symmetric on S (Rn ; CN )), and when in addition it is positive elliptic (that is, the matrix a2 (x, ξ ) is positive definite for (x, ξ ) = (0, 0)), then it is self-adjoint as an unbounded operator in L2 (Rn ; CN ) with a discrete real spectrum. It is particularly striking to realize that, while (almost) everything is known for scalar harmonic oscillators, very little is known about the spectral properties of selfadjoint elliptic systems, even in the basic (and seemingly simple) case of NCHOs. This book is mainly intended as a pseudodifferential introduction to the spectral theory of elliptic NCHOs, although it contains also results for more general elliptic differential systems with polynomial coefficients, and is a first account of the properties of such systems. A particularly important example of NCHO, is given by the system ⎡ ⎤ ∂x2 x2 1 + − x − α ∂ + x ⎢ 2 2 2 ⎥ ⎢ ⎥, Qw (α ,β ) (x, D) = ⎣ 2 ∂x x2 ⎦ 1 x∂x + β − + 2 2 2

x ∈ R, α , β ∈ C,

A. Parmeggiani, Spectral Theory of Non-Commutative Harmonic Oscillators: An Introduction, Lecture Notes in Mathematics 1992, c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-11922-4 1,

1

2

1 Introduction

which is the Weyl-quantization of the matrix ⎡ 2 ⎤ ξ + x2 −ixξ ⎢α ⎥ 2 ⎥ Q(α ,β ) (x, ξ ) = ⎢ ξ 2 + x2 ⎦ , ⎣ ixξ β 2

(x, ξ ) ∈ R × R,

that was introduced by M. Wakayama and myself in a series of papers (see Parmeggiani-Wakayama [58, 59]). When α , β > 0 with αβ > 1, the system is positive elliptic, self-adjoint, and so it has a discrete spectrum in L2 (R; C2 ), and a very rich and remarkable structure. It is worth remarking that in [59] (see also [58]) the eigenvalues are described in terms of a scalar three-term recurrence, and hence in terms of a continued fraction (however, it is very difficult to have a direct and explicit expression of them). It is also worth mentioning that when α = β > 1, one has (see Parmeggiani-Wakayama [58, 59]) that Qw (α ,α ) (x, D) is unitarily equivalent to a scalar harmonic oscillator times the identity 2 × 2 matrix, so that its spectral properties are governed by the tensor product of the oscillator representation and the 2-dimensional trivial representation of sl2 (R) (see e.g. Howe-Tan [25]), i.e. one has matrix-valued creation/annihilation operators that can be used to “construct” the spectrum (see Parmeggiani-Wakayama [60] for further examples of NCHOs whose spectrum can be studied through the tensor product of the oscillator representation and the finite dimensional representation of sl2 (R) and its perturbations). Hence, when α , β > 0 and αβ > 1, Qw (α ,β ) (x, D) can be thought of as a matrix-valued deformation of the scalar harmonic oscillator (it is not yet known whether Qw (α ,β ) (x, D) when α = β admits creation/annihilation operators or not). System Qw (α ,β ) (x, D) for α , β > 0 and αβ > 1, will be the main example to bear in mind in this exploration of the basic spectral properties of elliptic NCHOs. Of course, it is a natural and important problem to understand the spectral behavior of Qw (α ,β ) (x, D) as α and β vary in C \ {αβ = 0}. I won’t be dealing with this aspect here, and concentrate mainly only on the theory of self-adjoint elliptic and positive NCHOs. My motivation for considering systems like Qw (α ,β ) (x, D) comes from PDEs, namely from the study of a-priori lower bound estimates, such as Melin’s or Hörmander’s or Fefferman-Phong’s , in the case of pseudodifferential systems (see Parenti-Parmeggiani [49], and also Parmeggiani [53, 54], and references therein). A basic tool for studying such inequalities is the use of the localized operator, introduced by L. Boutet de Monvel (and further developed by L. Boutet de Monvel, A. Grigis and B. Helffer in their study of the hypoellipticity with loss of derivatives; see [49] and the references therein). The localized operator is, loosely speaking, the Weyl quantization of the “meaningful” part of the Taylor-expansion of the symbol at characteristic points, whose spectral properties become relevant to the existence of the lower-bound estimate. In the case of systems, the localized operator is a differential system with polynomial coefficients, an example of which is given by Qw (α ,β ) (x, D). Another motivation comes from the search of vector-valued

1 Introduction

3

deformations of the scalar, fundamental, harmonic oscillator. In some sense, as the reader will see, the class of polynomial differential systems studied here is by all means a remarkable vector-valued deformation of the harmonic oscillator which inherits many beautiful spectral properties. From the point of view of Mathematical Physics, it is interesting to mention here the recent results of Hirokawa [22] on the Dicke-type crossings among the eigenvalues (of basic importance in many physical problems) for certain NCHOs. It is also conceivable that NCHOs (and more general differental systems with polynomial coefficients) appear naturally in questions related to the BornOppenheimer approximation or to Solid State Physics. For instance,1 when dealing with the Flux 1/2 problem relative to the spectral analysis of the Harper model, B. Helffer and J. Sjöstrand [21] are led to work with a semiclassical pseudodcos ξ cos x ifferental system whose symbol is , and one is concerned with cos x − cos ξ the analysis for energies near 0 and phase-space points (x, ξ ) near (π /2, π /2). After a phase-space translation, one is therefore forced to consider system

the sin ξ sin x ξ x , whose harmonic approximation is the system . The sin x − sin ξ x −ξ

2 D + x2 [D, x] square of the latter is the NCHO . −[D, x] D2 + x2 It is interesting to remark that the three-term recurrence equations I have mentioned above, appearing in the study of the spectrum of Qw (α ,β ) (x, D), are similar to certain ones appearing in Lay and Slavyanov’s book [39] about special functions. It is thus plausible that some physical applications of NCHOs may also be found by considering, in analogy, those given in [39]. The main results contained in this book are: • the meromorphic continuation of the spectral zeta function ζA of a general N × N

elliptic NCHO A in Rn , and the very elegant (and precise) result by Ichinose and Wakayama on the spectral zeta function of an elliptic Qw (α ,β ) (x, D); • some precise upper and lower bounds on the lowest eigenvalue of Qw (x, D) (α ,β ) (due to Ichinose and Wakayama, and to myself); • the “clustering” and “multiplicity” phenomena of eigenvalues in dependence on dynamical quantities, such as period and action (and Maslov index), related to the closed integral trajectories of the Hamilton vector-fields associated with the eigenvalues of the symbol of the system for a class of n-dimensional 2 × 2 positive elliptic NCHOs of order 2, that generalizes the class of positive elliptic Qw (α ,β ) (x, D) (these results are due to myself). In the course of achieving the afore-mentioned results, I have also developed some general tools, used here, that I hope will also prove useful to attack further spectral problems, beyond those treated in these notes.

1

I am indebted with B. Helffer for explaining this to me.

4

1 Introduction

Of some basic material I have not given a thoroughly self-contained presentation, at times preferring instead a kind of overall survey with the aim of giving a “userguide” of it, my choice being due to the existence of beautiful (and important) books and papers to which I address the reader. Here is now an account on the content of this book. In Chapter 2, I introduce the Weyl-quantization of the Hamiltonian of the harmonic oscillator and describe its spectral properties. This is the basic scalar model which will serve as a guide throughout this study. In Chapter 3, I give a rapid introduction to the general Weyl-Hörmander calculus, to be then specialized to the case of the “global metric”. This is the setting where I shall develop the spectral theory of NCHOs. In Chapter 4, I introduce the function that “counts” the eigenvalues of an operator with discrete spectrum (the spectral counting function), and give some of its elementary properties. I also describe the spectral counting function associated with a general harmonic oscillator, which is then used to derive a first crude information about the behavior of the spectral counting function associated with a general NCHO (information that, however, is already sufficient to obtain a first convergence region of the associated spectral zeta function). In Chapter 5, I give an account of the functional calculus of the operators I am interested in, and introduce the notion of “trace” of an operator. In Chapter 6, I decribe the construction of an approximation to e−tA , based on the parametrix construction for d/dt + A, where A is a positive elliptic global polynomial differential system. This is then used to compute the leading coefficient of the asymptotics of the spectral counting function for large eigenvalues, in terms of the symbol of the system. To achieve this, I use (as is classical) the Karamata Tauberian theorem, of which I give a complete proof. In Chapter 7, I recall Robert’s theorem on the meromorphic continuation of the spectral zeta function, and then give a proof of it for a positive elliptic N × N NCHO in Rn by using the approximation to e−tA constructed in Chapter 6, and finally recall (and sketch the proof of) the Ichinose-Wakayama theorem about the spectral zeta function associated with a positive elliptic Qw (α ,β ) (x, D). (The Ichinose-Wakayama result is a consequence of the more general result I give for an elliptic N × N NCHO, except for their very precise description of the “trivial zeros”. That is why I have decided to sketch their proof.) In Chapter 8, I recall Ichinose-Wakayama’s lower and upper bounds to the lowest eigenvalue of a positive elliptic Qw (α ,β ) (x, D), and then show a refinement (which is new) of the upper bound, whose proof uses only some elementary symplectic linear algebra. In Chapter 9, I review and construct some tools from the calculus of Semiclassical Pseudodifferential Operators (i.e. the Weyl-quantization of symbols of the kind p(x, hξ ; h), where h ∈ (0, 1] is the semiclassical parameter), proving in particular the decoupling construction (modulo operators that have a norm which is O(hN ) for every N ∈ Z+ ) of a given system whose principal symbol can be blockwise diagonalized.

1 Introduction

5

Then in Chapter 10, I introduce a useful approach to study large eigenvalues of self-adjoint (unbounded) operators (in the usual and in the semiclassical setting), consisting of a “reduction” of the operator by “cutting off low energies”. In Chapter 11, I recall some basic facts about the dynamics of the integral trajectories of the Hamilton vector-field associated with the symbol of a pseudodifferential operator (the bicharacteristics), with special interest in the periodic trajectories. I will then introduce the Maslov index of such trajectories, and give an effective way of computing it, following the approach and results of RobbinSalamon [62] (see also McDuff-Salamon [43]). In the notes to this chapter I rapidly recall some motivations for the introduction of the Maslov index, and give the appropriate references. In the final Chapter 12, I describe in the first place the clustering and multiplicity properties of the large eigenvalues of a scalar semiclassical pseudodifferential operator, which are described through the dynamical quantities introduced in Chapter 11. This is afterwards used to study, through the decoupling seen in Chapter 9 and the methods developed in Chapter 10, the large eigenvalues of an elliptic 2 × 2 NCHO in Rn whose symbol has distinct eigenvalues. Since the eigenvalues of a semiclassiw cal NCHO aw 2 (x, hD) (i.e. with a0 = 0) are h times the eigenvalues of a2 (x, D), this gives us precise clustering and multiplicity properties of the large eigenvalues of a NCHO, with no lower-order part a0 , in terms of the dynamical quantities associated with the eigenvalues of the symbol a2 (x, ξ ). This part is new, and generalizes the results obtained in Parmeggiani [55] for an elliptic Qw (x, D) to n-dimensional (α ,β ) 2 × 2 NCHOs. In the Appendix, I show a few properties of the almost-analytic extension to the complex domain of compactly supported functions in R, and (following DimassiSjöstrand [7]) recall a proof of the Dyn’kin-Helffer-Sjöstrand formula. Finally, a list of the main notation used throughout the book is given right before the Bibliography. I end this Introduction by saying that in this book there was no space for including other issues related to an elliptic Qw (α ,β ) (x, D), such as: • the representation-theoretic approach, due to H. Ochiai, which establishes the

• •

• •

deep relation of the spectral theory of these systems to the existence of particular holomorphic solutions of Heun differential operators in the complex domain (see Ochiai [46, 47]); the numerical study of the spectrum started by K. Nagatou, M.T. Nakao and M. Wakayama (see Nagatou-Nakao-Wakayama [45]); the number-theoretic investigation of the special values of the spectral zeta function associated with the spectrum initiated by T. Ichinose and M. Wakayama (see Ichinose-Wakayama [32, 33], Kimoto-Wakayama [36, 37], Kimoto-Yamasaki [38], and Ochiai [48]); the Poisson-like formulas on the spectral density of eigenvalues (see Parmeggiani [52]); the properties of the lowest eigenvalue and some perturbation theory in the parameters (α , β ) (see Ichinose-Wakayama [33], Parmeggiani [51, 52]).

They will be part of a wider book, that M. Wakayama and I are planning to write.

Chapter 2

The Harmonic Oscillator

In this chapter we shall review the spectral properties of the harmonic oscillator, i.e. of the operator given by the Weyl-quantization of the Hamiltonian (|ξ |2 + |x|2 )/2.

2.1 From the Hamiltonian to the Operator Acting on L2 Consider the phase-space Rn × Rn = T ∗ Rn = R2n (the cotangent bundle of Rn ), that we shall always consider endowed with the canonical symplectic form

σ=

n

∑ d ξ j ∧ dx j ,

(2.1)

j=1

where (x, ξ ) are symplectic coordinates in Rnx × Rnξ . Equivalently, for any given

δx δx , v = ∈ Rn × Rn one has tangent vector v = δξ δξ

σ (v, v ) = δ ξ , δ x − δ ξ , δ x ,

(2.2)

where ·, · denotes the usual inner-product in Rn . Remark that σ = d(∑nj=1 ξ j dx j ), where ∑nj=1 ξ j dx j is called the canonical 1-form. Next, consider in R2n the Hamiltonian 1 p0,n (x, ξ ) = (|ξ |2 + |x|2 ) = 2

n

∑ p0(x j , ξ j ),

j=1

where 1 p0 (x j , ξ j ) = p0,1 (x j , ξ j ) = (ξ j2 + x2j ), 2


7

8

2 The Harmonic Oscillator

is the one-dimensional harmonic oscillator in the variables (x j , ξ j ) ∈ R2 . Its Weyl-quantization is the operator (we shall throughout use Dx j or D j for −i∂x j or −i∂ j , respectively) 1 1 2 pw 0,n (x, D) = (−Δ x + |x| ) = 2 2

n

n

j=1

j=1

∑ (−∂x2j + x2j ) = ∑ pw0 (x j , Dx j ).

It is readily seen that pw 0,n (x, D) is continuous when thought of as an operaw n n tor p0,n (x, D) : S (R ) −→ S (Rn ) and as an operator pw 0,n (x, D) : S (R ) −→ n n n S (R ), where S (R ) and S (R ) are, respectively, the space of Schwartz functions and temperate distributions on Rn . The operator pw 0,n (x, D) is obtained from p0,n (x, ξ ) by replacing ξ j by D j , x j by x j and x j ξk by (x j Dk + xk D j )/2, that is to say, it is obtained as

−n pw 0,n (x, D)u(x) = (2π )

ei x−y,ξ p0,n (

x+y , ξ )u(y)dyd ξ , u ∈ S (Rn ), (2.3) 2

where it is understood that the integration with respect to y is effected first and the one with respect to ξ last. Exercise 2.1.1. Show the formula (2.3) indeed holds.

When studying the spectral properties of pw 0,n (x, D) it is also convenient to realize it as an unbounded operator in L2 (Rn ), that we will denote by H regardless the dimension n, H : D(H) ⊂ L2 (Rn ) −→ L2 (Rn ), with domain 2 n 2 n D(H) = {u ∈ L2 (Rn ); Hu = pw 0,n (x, D)u ∈ L (R )} =: B (R )

(2.4)

(the so-called maximal operator; we shall come back to this later on), where pw 0,n (x, D)u is understood in the sense of distributions. Note that one clearly has S (Rn ) ⊂ D(H), whence H is densely defined. It is also a closed operator. To see this, we have to show that ⎫ L2 ⎪ D(H) u j −→ u, as j → +∞ ⎪ ⎬ =⇒ u ∈ D(H) and Hu = v. ⎪ ⎪ L2 ⎭ Hu −→ v, as j → +∞ j

Since L2

S

u j −→ u =⇒ u j −→ u, we have on the one hand S

w pw 0,n (x, D)u j −→ p0,n (x, D)u, as j → +∞,

2.1 From the Hamiltonian to the Operator Acting on L2

9

and on the other, L2

pw 0,n (x, D)u j = Hu j −→ v, as j → +∞, whence 2 n pw 0,n (x, D)u = v ∈ L (R ),

i.e. u ∈ D(H) and Hu = v, which is what we wanted to prove. Let us restrict our attention, for the time being, to the 1-dimensional case n = 1. We have x2 + ξ 2 x + iξ x − iξ √ , (x, ξ ) ∈ R × R. p0 (x, ξ ) = = √ 2 2 2 Set hence x ∓ iξ ψ± (x, ξ ) := √ . 2 Note that |ψ± (x, ξ )| ≈ p0 (x, ξ )1/2 , where, for positive A and B, A ≈ B means that there are universal constants C1 ,C2 > 0 such that C1 A ≤ B ≤ C2 A. Notation. Let A, B > 0. We shall always write A B (or B A) when there is a universal constant C > 0 such that A ≤ CB. We hence have A ≈ B whenever A B and B A. Define next the Poisson bracket of two C1 functions f and g by { f , g} =

∂ f ∂g ∂ f ∂g . − ∂xj ∂ξj j=1 ∂ ξ j ∂ x j n

∑

Then, on the one hand p0 (x, ξ ) = ∏ ψ± (x, ξ )ψ∓ (x, ξ ), and {ψ+ , ψ− } = −i, ±

and on the other hand, upon setting x ∓ ∂x Ψ± := ψ±w (x, D) = √ , 2 we have, by a direct computation, 1 Ψ+Ψ− = H − , 2

1 Ψ−Ψ+ = H + , 2

[Ψ+ , Ψ− ] = −1.

(2.5)

The operators Ψ± are the celebrated creation (Ψ+ ) and annihilation (Ψ− ) operators. We now define, for a, b : R2 −→ C affine functions in x, ξ , the operation ab by i ab = ab − {a, b}, 2

10


so that

i 1 ψ+ ψ− = ψ+ ψ− − (−i) = p0 − , 2 2

and

i 1 ψ− ψ+ = ψ− ψ+ − i = p0 + . 2 2 We have therefore discovered the correspondence

Ψ±Ψ∓ = (ψ± ψ∓ )w (x, D). Notice, moreover, that on S (R) and S (R)

Ψ±∗ = Ψ∓ , Ψ±∗ = (ψ± )w (x, D), and 1 [Ψ+ , Ψ− ] = (ψ+ ψ− )w (x, D) − (ψ− ψ+ )w (x, D) = {ψ+ , ψ− }w (x, D). i

(2.6)

Hence the fact that the operators Ψ+ and Ψ− do not commute may be explained by the fact that {ψ+ , ψ− } = 0, which is hence a first level of non-commutativity when dealing with operators, due to the non-vanishing of the Poisson brackets {ξ j , x j } = 1. Moreover, the fact that a commutator is a “lower order” operator will no longer hold true in general for matrix-valued symbols (see Remark 3.1.9 below). This is the second level at which non-commutativity manifests itself when considering systems of operators.

2.2 The Spectrum of the Harmonic Oscillator We have now the following fundamental theorem about the spectrum of the harmonic oscillator when n = 1. The higher dimensional case will easily follow. Theorem 2.2.1. Let H : D(H) ⊂ L2 (R) −→ L2 (R), i.e. we think of H as an unbounded operator on L2 with dense domain D(H) given in (2.4), Hu = pw 0 (x, D)u in the sense of distributions. Then we have the following facts. 1. Self-adjointness: H = H ∗ , that is, D(H) = D(H ∗ ) and H ∗ u = pw 0 (x, D)u for all u ∈ D(H). 2. Spec(H) = {n + 1/2; n ∈ Z+ } (where Z+ = {0, 1, 2, . . .}), with multiplicity 1. 2 3. Define v0 = e−x /2 and vn = Ψ+n v0 , n ∈ Z+ . Then Ψ− v0 = 0, all the functions vn 2 belong to S and are of the form vn = fn e−x /2 , where fn ∈ R[x] is a polynomial of degree n (i.e., of the form an xn + an−1xn−1 + . . . + a0, with non-zero leading coefficient an ). Moreover, the vn form a complete orthogonal system of L2 .

2.2 The Spectrum of the Harmonic Oscillator

11

Hence, upon defining un = vn /||vn ||, the system {un }n∈Z+ is an orthonormal basis of L2 . The functions vn are called Hermite functions. This theorem is well-known. However, we give the proof for the sake of completeness. Proof. We shall denote throughout by (·, ·) the canonical inner product in L2 , and by || · ||0 the induced L2 -norm. Note, in the first place, that for any given u1 , u2 ∈ S we have (Hu1 , u2 ) = (u1 , Hu2 ) (i.e. H is symmetric, or, equivalently, pw 0 (x, D) is formally self-adjoint), and 1 1 1 (Hu, u) = (Ψ+Ψ− + )u, u = ||Ψ− u||20 + ||u||20 ≥ ||u||20 , ∀u ∈ S . 2 2 2 On the other hand, since Ψ− v0 = 0 and v0 ∈ S , we indeed have that (Hu, u) 1 = . 2 u∈S \{0} ||u||20 min

Next, it is clear that the functions vn = Ψ+n v0 ∈ S , for all n ∈ Z+ . Hence we may compute by induction 1 1 Hvn = (Ψ+Ψ− + )vn = (Ψ+Ψ− + )Ψ+ vn−1 2 2 1 = Ψ+ (Ψ−Ψ+ − )vn−1 + Ψ+vn−1 2 (using (2.5) and the induction hypothesis Hvn−1 = (n − 1 + 1/2)vn−1) 1 = Ψ+ Hvn−1 + Ψ+vn−1 = Ψ+ (n − 1 + )vn−1 + Ψ+vn−1 2 1 1 = (n + )Ψ+ vn−1 = (n + )vn . 2 2 Of course, we have to make sure that vn ≡ 0 for all n. Hence, we compute by induction ||vn ||20 = (vn , vn ) = (Ψ+n v0 , Ψ+n v0 ) = (Ψ−nΨ+n v0 , v0 ) = Ψ−n−1 (Ψ+Ψ− + [Ψ− , Ψ+ ])Ψ+n−1 v0 , v0 = Ψ−n−1 (Ψ+Ψ− + 1)Ψ+n−1 v0 , v0 = n(Ψ−n−1Ψ+n−1 v0 , v0 ) = n!||v0||20 > 0, whence vn ≡ 0 for all n ∈ Z+ . Now, since Ψ− v0 = 0, a similar computation also shows that m > n =⇒ (vn , vm ) = 0,

12


that yields the orthogonality of the system {vn }n∈Z+ . To prove its completeness, we note in the first place that it is clear that vn = fn v0 , for a real polynomial fn of degree n (i.e., fn = an xn + an−1 xn−1 + . . . + a0 , with the a j ∈ R and non-zero leading coefficient an ). Hence, let g ∈ L2 (R) be such that +∞ −∞

g(x)vn (x)dx = 0, ∀n ∈ Z+ .

It then follows that, since any polynomial f ∈ R[x] can be written as a linear combination of the fn (by virtue of the fact that their leading coefficients are not zero), we have +∞ −∞

Using e−ixξ =

g(x) f (x)e−x

2 /2

dx = 0, ∀ f ∈ R[x].

(2.7)

(−ixξ ) j and (2.7) yields j! j≥0

∑

+∞ −∞

e−ixξ g(x)e−x

2 /2

dx = Fx→ξ (ge−x

2 /2

)(ξ ) = 0, ∀ξ ∈ R.

But an L2 -function which has zero Fourier-transform must be zero, that is ge−x

2 /2

≡ 0 =⇒ g ≡ 0 in L2 (R).

Hence {vn }n∈Z+ is an orthogonal basis of L2 (R) and {un }n∈Z+ , where un := vn /||vn ||0 , is an orthonormal basis of L2 (R). Using this, one then sees that Im(H ± i) = L2 (R), whence, by general arguments (see, for instance, Dunford-Schwartz [13] or Kato [35]), H = H ∗ . This therefore shows that 1 Spec(H) = {n + ; n ∈ Z+ }, 2 with multiplicity 1, and concludes the proof.

It is worth noting that the non-commutativity of the operators Ψ± allowed the operator H to be positive (i.e. the quadratic form (H·, ·) in L2 is bounded from below by a positive constant) although its symbol, the function p0 (x, ξ ), is merely ≥ 0. We have this by virtue of the uncertainty principle, or equivalently, putting things geometrically, by virtue of the presence of symplectic variables (the pair (x, ξ ) ∈ R2 satisfies {ξ , x} = 1) in the symbol, that makes p−1 0 (0)(= {0} in this case) too small to support an eigenfunction.

2.2 The Spectrum of the Harmonic Oscillator

13

How about higher dimensions? We start by putting ( j)

( j)w

( j)

Ψ± := ψ± (x, D), where ψ± (x, ξ ) =

x j ∓ iξ j √ , 1≤ j≤n 2

( j)

(note that when n > 1 we only have |ψ± (x, ξ )| ≈ p0 (x j , ξ j )1/2 and no longer that ( j) |ψ± (x, ξ )| ≈ p0,n (x, ξ )1/2 ). Then, on S (Rn ) or S (Rn ), H=

n

1 ( j) Ψ− + ) = 2

∑ (Ψ+

( j)

j=1 ( j)

( j )

( j)

n

∑ (Ψ−

1 ( j) Ψ+ − ), 2

( j)

j=1

( j )

( j)

( j )

[Ψ+ , Ψ+ ] = [Ψ− , Ψ− ] = 0, and [Ψ+ , Ψ− ] = −δ j j . Hence, if we define for α ∈ Zn+ n

uα (x) := ∏ uα j (x j ), j=1

where uα j is given by Theorem 2.2.1, we have the following theorem. Theorem 2.2.2. 1. As an unbounded operator in L2 (Rn ) with domain B2 (Rn ) the operator H, the (maximal) realization of pw 0,n (x, D), is self-adjoint; 2. {uα }α ∈Zn+ is a complete orthonormal system of L2 (Rn ), made of eigenfunctions of H; 3. Spec(H) = {|α | + n/2; α ∈ Zn+ } (no longer with multiplicity 1 for n > 1; we shall examine the multiplicity later on, see Section 4.3).

Chapter 3

The Weyl–Hörmander Calculus

In this chapter we shall review the Weyl-Hörmander pseudodifferential calculus (see Hörmander [29]; see also Hörmander [27]), in which we shall “embed” the “global” one (see Helffer [17] and Shubin [67]), to be recalled in Section 3.2. In section 3.3 we shall describe a few spectral properties of globally elliptic pseudodifferential operators.

3.1 Review of the Weyl–Hörmander Calculus We shall denote by X = (x, ξ ), Y = (y, η ) and Z = (z, ζ ) the points of R2n = Rn ×Rn . Definition 3.1.1. An admissible metric in R2n is a function R2n X −→ gX where gX is a positive-definite quadratic form on R2n such that: • Slowness: There exists C0 > 0 (the constant of slowness) such that for any given

X , Y ∈ R2n one has

gX (Y − X) ≤ C0−1 =⇒ C0−1 gY ≤ gX ≤ C0 gY ; • Uncertainty: For any given X ∈ R2n one has

gX ≤ gσX , where gσX is the dual metric (with respect to the symplectic form σ , see (2.1)) defined by

σ (Y, Z)2 ; Z=0 gX (Z)

gσX (Y ) = sup

• Temperateness: There exists C1 > 0 and N0 ∈ Z+ such that for all X , Y ∈ R2n

one has

N0 gX ≤ C1 gY 1 + gσX (X − Y ) .


15

16

3 The Weyl–Hörmander Calculus

The Planck function associated with g is by definition gX (Z) . σ Z=0 gX (Z)

h(X)2 = sup

It is important to note that the uncertainty of g yields that h(X ) ≤ 1 for all X ∈ R2n . Remark 3.1.2. One easily sees (exercise for the reader) that the metric |d ξ |2 , (x, ξ ) ∈ R2n , 1 + |ξ |2

gx,ξ = |dx|2 +

is an admissible metric. In this case one has h(x, ξ )−1 = (1 + |ξ |2 )1/2 , which is the usual pseudodifferential weight (which is then g-admissible; see Definition 3.1.4 below). Remark 3.1.3. The dual metric gσ appears naturally to keep into account localization of Fourier transforms when dealing with the composition of symbols (see (3.5) below). Notice that the uncertainty principle may also be rephrased as the basic observation that for the function uA (x) = e− Ax,x , with A symmetric n × n and positive, the −1 Fourier transform is uÂ (ξ ) = c(detA)−1/2 e− A ξ ,ξ , whence uA is as much “concentrated” within the ellipsoid {x; Ax, x ≤ 1}, as uÂ (ξ ) is “concentrated” within the ellipsoid {ξ ; A−1 ξ , ξ ≤ 1}. As an example of computation of gσ and h, consider the constant metric g=

n

∑ (λ j dx2j + μ j d ξ j2 ),

λ j , μ j > 0, 1 ≤ j ≤ n.

j=1

Then (exercise for the reader) gσ =

n

dx2j

∑( μj

j=1

+

d ξ j2

λj

),

and h(X ) = max (λ j μ j )1/2 , ∀X ∈ R2n . 1≤ j≤n

3.1 Review of the Weyl–Hörmander Calculus

17

Definition 3.1.4. Given an admissible metric g, a g-admissible weight is a positive function m on R2n (i.e. m(X ) > 0 for all X ∈ R2n ) for which there exist constants c,C,C > 0 and N1 ∈ Z+ such that for all X,Y ∈ R2n , gX (X − Y ) ≤ c =⇒ C−1 ≤ and

m(X ) ≤ C, m(Y )

N1 m(X) ≤ C 1 + gσX (Y − X ) . m(Y )

Remark 3.1.5. It is then seen (exercise for the reader) that the Planck function h is a g-admissible weight. Definition 3.1.6. Let g be an admissible metric and m be a g-admissible weight. Let a ∈ C∞ (R2n ). Denote by a(k) (X; v1 , . . . , vk ) the k-th differential of a at X in the directions v1 , . . . , vk of R2n (thought of as tangential directions in TX R2n , the tangent space of R2n at the point X ). Define |a|gk (X ) :=

|a(k) (X ; v1 , . . . , vk )| . k 1/2 0=v1 ,...,vk ∈R2n ∏ j=1 gX (v j ) sup

We say that a ∈ S(m, g) if for any given integer k ∈ Z+ the following seminorms are finite: ||a||k;S(m,g) :=

|a|g (X ) < +∞. ≤k, X∈R2n m(X ) sup

(3.1)

Given μ ∈ R, we shall say that a ∈ S μ (g) if a ∈ S(h−μ , g) (that is, we use the Planck function h as a weight for measuring the growth of the symbols). With BgX0 ,r = {X ; gX0 (X − X0 ) < r2 }, following Bony and Lerner [3] we say that a ∈ C∞ (R2n ) is a symbol (of weight m) confined to the ball BgX0 ,r , and write a ∈ Conf(m, g, X0 , r), if for all k ∈ Z+ gX

||a||k,Conf(m,g,X0 ,r) :=

|a| 0 (X) 1 + gσX0 (X − BX0,r ))k/2 < +∞, m(X ) 2n 0 ≤k, X∈R sup

(3.2)

where gYσ (X − B) = inf gYσ (X − Z). Hence the space of symbols confined to the Z∈B

ball BgX0 ,r coincides with S (R2n ) endowed with the seminorms (3.2). Any given ϕ ∈ C0∞ (BgX0 ,r ) is automatically confined (of weight 1) to the ball BgX0 ,r . Remark 3.1.7. For the standard pseudodifferential metric gx,ξ = |dx|2 +

|d ξ |2 , (x, ξ ) ∈ R2n , 1 + |ξ |2 μ

with h(x, ξ )−1 = (1 + |ξ |2 )1/2 , one has S μ (g) = S1,0 (Rn × Rn ), the familiar class of symbols of order μ .

18


Given a symbol a ∈ S(m, g) one then defines its Weyl-quantization by the formula aw (x, D)u(x) = (2π )−n

ei x−y,ξ a(

x+y , ξ )u(y)dyd ξ , u ∈ S (Rn ). 2

It is then seen (Hörmander [29]) that the expression makes sense, and that A = aw (x, D) : S (Rn ) −→ S (Rn ) and A = aw (x, D) : S (Rn ) −→ S (Rn ) is continuous, with Schwartz kernel given by the tempered distribution KA (x, y) = (2π )−n

ei x−y,ξ a(

that is, KA (x + t/2, x − t/2) = (2π )−n

x+y , ξ )d ξ , 2

ei t,ξ a(x, ξ )d ξ

is the inverse Fourier transform of a with respect to ξ , whence a(x, ξ ) =

e−i t,ξ KA (x + t/2, x − t/2)dt.

For example, in the case of a ∈ S μ (g), gx,ξ = |dx|2 + |d ξ |2 /(1 + |ξ |2), the kernel KA is interpreted as an oscillatory integral, that is KA (x, y) = S - lim (2π )−n ε →0+

ei x−y,ξ a(

x+y , ξ )χ (εξ )d ξ , 2

where χ is any given Schwartz function with χ (0) = 1, the limit being independent of χ . Recall that the expression for the Schwartz kernel comes from considering the “weak” form

x+y , ξ )u(y)v(x)dyd ξ dx 2 = KA |v¯ ⊗ u S ,S , ∀u, v ∈ S (Rn ),

(aw (x, D)u, v) = (2π )−n

ei x−y,ξ a(

where (v ⊗ u)(x, y) = v(x)u(y) and ·|· S ,S denotes the S -S duality. One has the following composition theorem (see Hörmander [29] or [27]). Theorem 3.1.8. Given a ∈ S(m1 , g), b ∈ S(m2 , g) then aw (x, D) bw (x, D) = (ab)w (x, D),


19

where for any given N ∈ Z+ (ab)(X) =

N

∑

j=0

j 1i σ (DX ; DY ) a(X )b(Y )X=Y +rN+1 (X ), j! 2

(3.3)

with rN+1 ∈ S(hN+1 m1 m2 , g). In particular, for any given N ∈ Z+ , σ (DX ; DY )N a(X)b(Y )X=Y ∈ S(hN m1 m2 , g).

(3.4)

Here we write, by (2.2) with X = (x, ξ ) and Y = (y, η ),

σ (DX ; DY ) := σ (Dx , Dξ ; Dy , Dη ) := Dξ , Dy − Dη , Dx . cw

This formula comes from considering in the first place the composition aw bw = for symbols a, b ∈ S (R2n ) and, by using the relation

f (x, y)e2ixy dxdy =

1 4π

fˆ(ξ , η )e−iξ η /2 d ξ d η , f ∈ S (R2 ),

through the formula c(X) = π −2n

e−2iσ (X−Y,X−Z) a(Y )b(Z)dY dZ

e−2iσ (T,Z) a(X + T )b(X + Z)dT dZ = eiσ (DX ;DY )/2 (a(X)b(Y ))X=Y , = π −2n

(3.5)

where (recall) X = (x, ξ ), Y = (y, η ), Z = (z, ζ ). This is then extended to general symbols a and b of the kind considered in the theorem. Remark 3.1.9. It follows from (3.3) that i aw (x, D)bw (x, D) = (ab)w (x, D) − {a, b}w (x, D) + . . . 2 Hence, for scalar symbols we have for the commutator 1 [aw (x, D), bw (x, D)] = {a, b}w (x, D) + . . . . i This is no longer true for matrix-valued symbols, simply because matrices do not commute. What one can say in this case is just that [aw (x, D), bw (x, D)] = [a, b]w (x, D) −

w i {a, b} − {b, a} (x, D) + . . ., 2

where [a, b] = ab − ba denotes the usual matrix-commutator.

20


The next important result is the existence of a partition of unity associated with the metric g (see Hörmander [29] and Bony-Lerner [3]). Notice that no smoothness assumption on either the metric g or the g-admissible weights m was made. The existence of the partition of unity makes it possible to regularize the metric and the g-admissible weights. Lemma 3.1.10. Let g be an admissible metric, and let r2 < C0−1 . Then there exists a sequence of centers {Xν }ν ∈Z+ , a covering of R2n made of g-balls Bgν ,r = {X; gXν (X − Xν ) < r2 } centered at Xν and radius r, and a sequence of functions {ϕν } uniformly in S(1, g), with supp ϕν ⊂ Bgν ,r , such that ∑ν ∈Z+ ϕν2 = 1. Moreover, for any given r∗ such that r2 ≤ r∗2 < C0−1 , there exists an integer Nr∗ such that no more than Nr∗ balls Bgν ,r∗ can intersect at each time, i.e. one has an a-priori finite number of overlappings of the dilates by r∗ /r of the Bgν ,r ; hence ∀E ⊂ N, E > Nr∗ =⇒

ν ∈E

Bgν ,r∗ = 0. /

In addition, with gσX (B − B) := and

inf

Y ∈B,Y ∈B

gσX (Y − Y ), B, B ⊂ R2n ,

1/2 Δ μν (r∗ ) := max 1, gσXμ (Bgμ ,r∗ − Bgν ,r∗ ), gσXν (Bgμ ,r∗ − Bgν ,r∗ ) ,

there exist constants N˜ and C˜ such that ˜ ˜ sup ∑ Δ μν (r∗ )−N < C.

μ

ν

Moreover, for all k ∈ Z+ there exist C > 0 and ∈ Z+ such that for any given a ∈ S(m, g) and b ∈ Conf(1, g, X, r) one has ||ab||k,Conf(1,g,X,r) ≤ Cm(X )||a||,S(m,g) ||b||,Conf(1,g,X,r) .

(3.6)

Finally, for all k, N ∈ Z+ there exist C = Ck,N > 0 and ∈ Z+ such that for every μ , ν ∈ N and every a ∈ Conf(1, g, Xμ , r) and b ∈ Conf(1, g, Xν , r) one has ||ab||k,Conf(1,g,Xμ ,r) + ||ab||k,Conf(1,g,Xν ,r) ≤ C||a||,Conf(1,g,Xμ ,r) ||b||,Conf(1,g,Xν ,r) Δ μν (r)−N .

(3.7)

One has also the following useful lemma, due to Bony and Lerner [3]. Lemma 3.1.11. Let g be an admissible metric, and let m be a g-admissible weight. Let Bν be a g-ball as in Lemma 3.1.10. Let gν = gXν and mν = m(Xν ). Let {aν }ν ∈Z+ be a sequence of symbols with aν ∈ S(mν , gν ), such that for any given integer k ∈ Z+


21

sup ||aν ||k,Conf(mν ,gν ,Xν ,r) < +∞.

ν ∈Z+

Then a := ∑ν ∈Z+ aν belongs to S(m, g). The sequence {aν }ν ∈Z+ is said to be uniformly confined in S(m, g). When m = 1 we have from the Cotlar-Stein Lemma (see Hörmander [29], Lemma 18.6.5; see also Lemma 3.1.14 below) that aw = 2 ∑ν aw ν is a bounded operator in L . We next come to the action of the affine symplectic group. We have the following theorem (see Hörmander [29]). Theorem 3.1.12 (Symplectic invariance). For every affine symplectic transformation χ : Rn × Rn −→ Rn × Rn (i.e. χ ∗ σ = σ , where χ ∗ σ is the pull-back of the 2-form σ by χ , defined by (χ ∗ σ )(v, w) = σ (χ v, χ w) where χ : R2n −→ R2n is the tangent map associated with χ ) there exists a unitary Uχ : L2 (Rn ) −→ L2 (Rn ), uniquely determined apart from a complex constant factor of modulus 1, such that for all a ∈ S(m, g) one has Uχ−1 aw (x, D)Uχ = (a ◦ χ )w(x, D). One has that Uχ is also an automorphism of S (Rn ) and S (Rn ). The proof of the theorem is obtained by considering the generators of the affine symplectic group of Rn × Rn and the associated metaplectic operator Uχ . One has that • when χ : (x, ξ ) −→ (x + x0 , ξ ), then (Uχ f )(x) = f (x − x0 );

• when χ : (x, ξ ) −→ (x, ξ + ξ0 ), then (Uχ f )(x) = ei x,ξ0 f (x); • when χ maps (x j , ξ j ) −→ (ξ j , −x j ) (for some j) keeping the other variables

fixed, then Uχ is the normalized partial Fourier transform in the x j variable Uχ f = (2π )−1/2Fx j →ξ j f ; • when χ : (x, ξ ) −→ (T x, t T −1 ξ ), where T : Rn −→ Rn is a linear isomorphism, then (Uχ f )(x) = |detT |−1/2 f (T −1 x); • when χ : (x, ξ ) −→ (x, ξ −Ax), for some A = tA, then (Uχ f )(x) = e−i Ax,x /2 f (x). We close this section by recalling the following fundamental result about the L2 -continuity. Theorem 3.1.13. Let a ∈ S(m, g). Then aw (x, D) : L2 (Rn ) −→ L2 (Rn ) is bounded iff m is bounded. Moreover, the operator aw (x, D) is compact in L2 (Rn ) iff m → 0 at ∞. Finally, there exist constants C > 0 and k ∈ Z+ (depending only on the dimension and the “structural” constants relative to g and m) such that ||aw ||L2 →L2 ≤ C||a||k;S(m,g) . When a ∈ S (R2n ) we have ||aw ||L2 →L2 ≤ (2π )−2n ||a|| ˆ L1 .

22


The proof of this theorem uses Lemma 3.1.10 and the very important Cotlar-Stein Lemma (see [29]), of which we give a proof for the sake of completeness. Lemma 3.1.14. Let H1 , H2 be Hilbert spaces. Let {A j } j=1,...,+∞ be linear bounded operators from H1 to H2 (i.e. they belong to L (H1 , H2 )). Suppose that there exists a constant M > 0 such that +∞

+∞

j∈N k=1

j∈N k=1

sup ∑ ||A∗j Ak ||1/2 ≤ M, sup ∑ ||A j A∗k ||1/2 ≤ M.

(3.8)

+∞

Then Au :=

∑ (Ak u) exists for any given u ∈ H1 , with strong convergence in H2 ,

k=1

and for the norm of A we have ||A|| ≤ M. Proof. We remark in the first place that if T ∈ L (H1 , H2 ), then ||T ∗ || = ||T ||, and ||T ∗ T || = ||T ||2 , so that, when T = T ∗ ,

(3.9)

||T 2 || = ||T ||2 .

In order to prove (3.9), one first notices that (with obvious notation) ||T || :=

sup || f ||H1 =||g||H2 =1

|(T f , g)H2 |,

(3.10)

for on the one hand, for any given f˜ ∈ H1 and g˜ ∈ H2 with || f˜||H1 = ||g|| ˜ H2 = 1, one has |(T f˜, g) ˜ H2 | ≤ sup |(T f , g)H2 |, || f ||H1 =||g||H2 =1

whence ||T || ≤

sup || f ||H1 =||g||H2 =1

|(T f , g)H2 |,

because ||T f˜||H2 = sup |(T f˜, g)H2 | ≤ ||g||H2 =1

sup || f ||H1 =||g||H2 =1

|(T f , g)H2 |,

and on the other sup || f ||H1 =||g||H2 =1

|(T f , g)H2 | ≤ ||T ||.

Now, from (3.10) it follows that ||T || :=

sup || f ||H1 =||g||H2 =1

|(T f , g)H2 | =

sup || f ||H1 =||g||H2 =1

||T ∗ T || ≤ ||T ∗ ||||T || = ||T ||2 ,

|( f , T ∗ g)H1 | = ||T ∗ ||,


23

and ||T ||2 = sup (T f , T f )H2 = sup |(T ∗ T f , f )H1 | ≤ ||T ∗ T || || f ||H1 =1

|| f ||H1 =1

which proves (3.9). The above remark yields ||T ||2m = ||(T ∗ T )m ||. Define now, for N ∈ N, AN :=

N

∑ Ak .

k=1

For the general term in (A∗N AN )m we have

||A∗j1 A j2 . . . A∗j2m−1 A j2m || ≤ min{||A∗j1 A j2 || . . . ||A∗j2m−1 A j2m ||, ||A∗j1 ||||A j2 A∗j3 || . . . ||A j2m−2 A∗j2m−1 ||||A j2m ||}. Since ||A∗j1 ||, ||A j2m || ≤ M by hypothesis, taking the geometric mean of the above bounds gives N

∑

||AN ||2m = ||(A∗N AN )m || ≤

j1 , j2 ,..., j2m =1

||A∗j1 A j2 . . . A∗j2m−1 A j2m ||

N

≤M

∑

≤M

∑ M2m−1 = NM2m ,

||A∗j1 A j2 ||1/2 ||A j2 A∗j3 ||1/2 . . . ||A∗j2m−1 A j2m ||1/2 j1 , j2 ,..., j2m =1 N

j1 =1

where in the last inequality we have used (3.8). Thus ||AN || ≤ N 1/(2m) M −→ M as m → +∞, ∀N ∈ N. We now show that the linear operator A : H1 u −→ Au :=

(3.11)

+∞

∑ (Ak u) ∈ H2 is well-

k=1

defined and bounded. Suppose that u = A∗j v for some j and some v ∈ H2 . Then, since ||Ak A∗j || = ||A j A∗k || ≤ M||A j A∗k ||1/2 , we get that for N1 + 1 ≤ N2 ||AN2 u − AN1 u||H2 = ||

as N1 , N2 → +∞, whence also defined when u ∈

N2

∑

k=N1 +1

Ak A∗j v||H2 ≤ M

N2

∑

k=N1 +1

||Ak A∗j ||1/2 ||v||H2 −→ 0

+∞

∑ (Ak u) exists in this case. It thus also follows that Au is

k=1

∑ Im(A∗j ). Define now

finite +∞

W :=

∑ Im(A∗j ) := {u ∈ H1 ; ∃{v j } j∈N ⊂ H2 , u = lim

j=1

N→+∞

N

∑ A∗j v j }.

j=1

24

3 The Weyl–Hörmander Calculus N

Define uN :=

∑ A∗j v j . Then {uN }N is a Cauchy-sequence in H1. Moreover, from

j=1

(3.11) one obtains that for any given m ≥ 1, m

|| ∑ Ak (uN1 − uN2 )||H2 ≤ M||uN1 − uN2 ||H1 k=1

uniformly in m, whence since AuN1 and AuN2 are defined (all N1 , N2 ) one gets ||AuN1 − AuN2 ||H2 ≤ M||uN1 − uN2 ||H1 −→ 0 as N1 , N2 → +∞. Hence there exists +∞

∑ (Ak uN ), N→+∞

Au = lim

k=1

+∞

and || ∑ Ak u||H2 ≤ M||u||H1 , ∀u ∈ W. k=1

It follows that A is a bounded operator in W, whence it is also bounded in W . Since ⊥

W⊥ = W =

Ker(Ak ), one has u ∈ W ⊥ =⇒ Au = 0,

k∈N

whence, as H1 = W ⊕ W ⊥ , we get A ∈ L (H1 , H2 ), with ||A|| ≤ M.

In the case of matrix-valued and vector-valued symbols, Definitions 3.1.4 and 3.1.6, the composition formula (3.3) (keeping track, where necessary, of the order of the terms), Theorem 3.1.12 and the continuity theorem Theorem 3.1.13 all hold true. Upon denoting by MN the set of N × N complex matrices, we shall write S(m, g; MN ) = S(m, g) ⊗ MN , and S μ (g; MN ) = S μ (g) ⊗ MN , for the matrix-valued analogue of the symbol spaces S(m, g) and S μ (g) considered above. In general, analogous notation will be used for the spaces S(m, g; V) = S(m, g) ⊗ V (where V is some real or complex finite-dimensional vector space) etc. Remark 3.1.15. Note that for a ∈ S(m, g) (or in general a ∈ S(m, g; V)), the formal adjoint (i.e. on S and S ) of aw (x, D) is aw (x, D)∗ = (a∗ )w (x, D), a∗ being the complex conjugate a¯ of a when a is scalar, or the linear form t a¯ ∈ V∗ when a is a vector in V, or the adjoint matrix a∗ = t a¯ in case V = MN . It is also useful to have the rule for passing from the Weyl-quantization to the ordinary quantization “to the right” (the so-called left-quantization: “taking derivatives before taking multiplications”). Theorem 3.1.16. Let g be an admissible metric, and suppose that gx,ξ (y, η ) = gx,ξ (y, −η ), for all (x, ξ ), (y, η ) ∈ R2n . Let m be a g-admissible weight. Then exp iκ Dx , Dξ is a weakly continuous isomorphism of S(m, g) for every κ ∈ R,

3.2 Global Metrics and Global Pseudodifferential Operators

25

iκ Dx , Dξ j a(x, ξ ) ∈ S(hN+1 m, g), j! j=0 N

eiκ Dx ,Dξ a(x, ξ ) − ∑

for all N ∈ Z+ . Hence, given a, b ∈ S(m, g) we have aw (x, D) = b(x, D), with b(x, D)u(x) = (2π )−n

ei x−y,ξ b(x, ξ )u(y)dyd ξ , u ∈ S ,

where b(x, ξ ) = ei Dx ,Dξ /2 a(x, ξ ), a(x, ξ ) = e−i Dx ,Dξ /2 b(x, ξ ).

(3.12)

Equalities (3.12) follow in a way similar to that of the composition formula (3.3). Remark 3.1.17. In Hörmander [29, p. 159], Theorem 3.1.16 is stated in a more general form, and takes care also of the relation of the Weyl-quantization to the so-called right-quantization (“taking derivatives after taking multiplications”).

3.2 Global Metrics and Global Pseudodifferential Operators We now consider the Weyl-Hörmander pseudodifferential calculus modelled after the harmonic oscillator. Let us hence consider, for X ∈ R2n , m(X ) := (1 + |X|2)1/2 , gX = so that

|dX|2 , m(X )2

(3.13)

gσX = m(X )2 |dX|2 , h(X ) = m(X )−2 .

The metric g is called the global metric. Let us check that g is indeed admissible, and that m and m−1 are g-admissible. • The metric g is slowly varying. Let c0 ∈ (0, 1/2) to be picked, and suppose

gX (Y − X ) ≤ c0 , that is, |Y − X|2 ≤ c0 m(X)2 . Then, on the one hand

m(Y )2 = 1 + |Y |2 ≤ 1 + (|X | + |X − Y |)2 ≤ 1 + 2|X |2 + 2|X − Y |2 ≤ 2(c0 + 1)m(X )2, and on the other m(X)2 = 1 + |X |2 ≤ 1 + 2|Y|2 + 2|X − Y |2 ≤ 2m(Y )2 + 2c0m(X )2 ,

26


that is, in the end, 1 − 2c0 m(X )2 ≤ m(Y )2 ≤ 2(c0 + 1)m(X )2 , 2 or, equivalently, 1 − 2c0 gY ≤ gX ≤ 2(c0 + 1)gY . 2 We may hence pick c0 = 1/4 (say), and get that g is slowly varying with slowness constant C0−1 = 1/10. • Uncertainty. It is trivial: gX =

|dX|2 ≤ m(X )2 |dX|2 = gσX . m(X)2

• Temperateness. We must find universal constants C1 > 0 and N0 ∈ Z+ such that

N0 gX (Z) ≤ C1 gY (Z) 1 + gσX (X − Y ) , ∀X ,Y, Z ∈ R2n , that is to say, N0 |Z|2 |Z|2 2 2 1 + m(X ) ≤ C |X − Y | . 1 m(X )2 m(Y )2 We hence consider m(Y )2 /m(X)2 . Since m(Y )2 1 + 2|X |2 + 2|X − Y |2 m(X )2 + |X − Y |2 ≤ ≤ 2 m(X )2 m(X)2 m(X )2 ≤ 2 1 + m(X)2|X − Y |2 ,

(3.14)

as m(X ) ≥ 1 for all X, the temperateness follows. Hence g is admissible. • To prove that m and m−1 are g-admissible, note that the first part of the above argument shows also that gX (Y − X) ≤ C0−1 =⇒ m(X ) ≈ m(Y ), that m(Y ) 1 + gσX (X − Y ), m(X) and that m(X)2 1 + 2|Y|2 + 2|X − Y |2 |X − Y |2 ≤ ≤ 2 1+ 2 2 m(Y ) m(Y ) m(Y )2 ≤ 2 1 + m(X)2|X − Y |2 ,

(3.15)


27

for one has m(Y )−1 ≤ m(X) for all X,Y. Inequalities (3.14) and (3.15) prove that m and m−1 are g-admissible weights. The weight m is called the global weight. This shows that we may consider the class S(mμ , g), μ ∈ R. Hence a ∈ S(mμ , g) if for all α ∈ Z2n + there exists Cα > 0 such that |∂Xα a(X )| ≤ Cα m(X)μ −|α | , ∀X ∈ R2n . Definition 3.2.1. When a ∈ S(mμ , g) we shall say that a has order μ . Next, we introduce the basic notion of elliptic elements. Definition 3.2.2. A symbol a ∈ S(mμ , g) is said to be globally elliptic when there exist constants c,C > 0 such that |X | ≥ C =⇒ |a(X )| ≥ c m(X )μ . Hence, we may say that not only does the symbol a(X ) grow at most like m(X )μ , but that for large X it is equivalent to the weight m(X )μ . The next definition is introduced for keeping track of the important class of differential operators with polynomial coefficients, that are Weyl-quantizations of polynomials of the kind d

∑

∑

j=0 |α |+|β |=2d−2 j

aαβ xα ξ β , aαβ ∈ C.

The grading 2d − 2 j, in place of the more general one 2d − j, is natural, when considering spectral properties of Weyl-quantizations of degree 2 polynomials in (x, ξ ) ∈ R2n . Definition 3.2.3. We say that a symbol a ∈ S(mμ , g) is classical, and write a ∈ Scl (mμ , g), if there exists a sequence {a μ −2 j } j≥0 ⊂ C∞ (R2n \ {0}) such that for all j ≥ 0 aμ −2 j (tX) = t μ −2 j aμ −2 j (X), ∀t > 0, ∀X = 0 (that is, the a μ −2 j are positively homogeneous of degree μ − 2 j), and for any given N ∈ Z+ N

a(X ) − χ (X) ∑ aμ −2 j (X) ∈ S(mμ −2(N+1) , g), j=0

χ being some excision function, that is a function 0 ≤ χ ≤ 1 that is supported away from 0, for instance χ ≡ 0 for |X| ≤ 1/2, and χ ≡ 1 for |X | ≥ 1. We shall write a∼

∑ aμ −2 j .

j≥0

28


Remark 3.2.4. More generally, one may consider also semi-regular classical symbols a ∈ S(mμ , g), that is symbols for which there exists a sequence {a μ − j } j≥0 ⊂ C∞ (R2n \ {0}) such that for all j ≥ 0 a μ − j (tX) = t μ − j aμ − j (X), ∀t > 0, ∀X = 0, and a∼

∑ aμ − j .

j≥0

We shall write a ∈ Ssrcl (mμ , g). However, our interest will rest mainly on classical symbols. Remark 3.2.5. One may also define classical symbols by requiring the sequence {a μ −2 j } j≥0 to be smooth on the whole R2n , but with the positive homogeneity property holding for all t > 1 and |X| > 1 only. This does not make any difference in the theory (it merely avoids the use of excision functions) and it is just a matter of taste. Note hence that, with d ∈ Z+ , d

∑

∑

aαβ xα ξ β ∈ Scl (m2d , g),

∑

aαβ xα ξ β ∈ Ssrcl (m2d , g).

j=0 |α |+|β |=2d−2 j

whereas

d

∑

j=0 |α |+|β |=2d− j

Definition 3.2.6. Let μ ∈ Z+ . A classical symbol a ∈ Scl (mμ , g) is a global polynomial differential (GPD for short) symbol of order μ if a=

[ μ /2]

∑ aμ −2 j

j=0

(with [μ /2] denoting, as usual, the integer part of μ /2), where the entries of the a μ −2 j are homogeneous polynomials in X ∈ R2n of degree μ − 2 j. A global polynomial differential operator (GPDO for short) of order μ is the Weyl-quantization of a GPD symbol of order μ . To recognize globally elliptic classical symbols, it is very useful to have the following lemma. Lemma 3.2.7. Let a ∈ Scl (mμ , g). Then min |a μ (X)| > 0,

|X|=1

iff a is globally elliptic.

(3.16)


29

Proof. Let χ be an excision function as in Definition 3.2.3. Then a(X ) = χ (X)a μ (X) + O(m(X )μ −2). We may hence take |X | ≥ 1, so that in the above identity χ = 1. Now, for |X | ≥ 1 we also have m(X) ≈ |X |, whence, using the homogeneity of aμ we get a(X) = m(X )μ Hence

|X |μ −2 a (X/|X|) + O(m(X ) ) , |X | ≥ 1. μ m(X)μ

a(X) ≈ m(X)μ , ∀X with |X | 1,

iff (3.16) holds. Lemma 3.2.7 then induces the following definition.

Definition 3.2.8. We say that a ∈ Scl (mμ , g) is globally elliptic when (3.16) holds. In the vector-valued case, we will use the following definition of globally elliptic symbols. Definition 3.2.9. Let V be a finite-dimensional vector space (complex or real). We denote by S(mμ , g; V), resp. Scl (mμ , g; V), the class of vector-valued symbols, resp. vector-valued classical symbols. In other words, we have S(mμ , g; V) = S(mμ , g) ⊗ V, and likewise for Scl (mμ , g; V). In particular, when V = MN , having that a ∈ S(mμ , g; MN ), resp. Scl (mμ , g; MN ), means that each entry a jk of the matrix a belongs to S(mμ , g), resp. Scl (mμ , g), and having an N × N global polynomial differential system of order μ means having a symbol a ∈ Scl (mμ , g; MN ) such that each entry is a GPD symbol of order μ (see Definition 3.2.6). We shall say that a symbol a ∈ S(mμ , g; MN ) is globally elliptic if there exist constants c,C > 0 such that |det a(X )| ≥ c m(X)μ N , whenever |X | ≥ C. Equivalently, when a ∈ S(mμ , g; L (V, W)), where W is another finite-dimensional vector space, we require (for some norms | · |V in V and | · |W in W) |a(X)v|W ≈ m(X )μ |v|V , ∀v ∈ V, ∀X with |X | 1, and when a = a∗ ∈ S(mμ , g; MN ) is positive-definite or negative-definite as an Hermitian matrix, we may equivalently require | a(X )v, v CN | ≈ m(X)μ |v|2CN , ∀v ∈ CN , ∀X with |X | 1. When a ∈ Scl (mμ , g; MN ) is classical, we shall then require min |det a μ (X)| > 0,

|X|=1

30


or, equivalently, |a μ (X )v|CN ≈ |v|CN , ∀v ∈ CN , ∀X with |X | = 1, and when aμ = a∗μ is positive/negative-definite as an Hermitian matrix, | a μ (X )v, v CN | ≈ |v|2CN , ∀v ∈ CN , ∀X with |X | = 1. Analogous requirements for classical symbols are possible in the case one has V = L (CN1 , CN2 ) etc. (of course det can be used only where meaningful). We shall say that a symbol a = a∗ ∈ S(mμ , g; MN ) is globally positive elliptic if a(X )v, v CN ≈ m(X)μ |v|2CN , ∀v ∈ CN , ∀X with |X | 1. When a = a∗ ∈ Scl (mμ , g; MN ), we shall say that a is globally positive elliptic if aμ (X )v, v CN ≈ |v|2CN , ∀v ∈ CN , ∀X with |X | = 1.

(3.17)

We shall say that a matrix-valued GPDO of order μ is elliptic (resp. positive elliptic) if its principal symbol aμ is globally elliptic (resp. positive elliptic). Remark 3.2.10. Recall that a(X)−1 =

1 co a(X ), deta(X)

where coa ∈ S(m(N−1)μ , g; MN ) is the cofactor matrix (i.e. the transpose of the the matrix whose i j-entry is the algebraic cofactor of the i j-entry of a) associated with a(X). One then sees that, equivalently: • a ∈ S(mμ , g; MN ) is globally elliptic if a(X)−1 exists for all X ∈ R2n with |X | 1

and χ a−1 ∈ S(m−μ , g; MN ), where χ is some excision function; • a ∈ Scl (mμ , g; MN ) is globally elliptic if a μ (X )−1 exists for all X ∈ R2n with |X| = 1. Definition 3.2.11. A non-commutative harmonic oscillator (NCHO for short) is the Weyl-quantization of any given 2nd-order N × N GPD system a ∈ Scl (m2 , g; MN ). Hence a = a2 + a0 , where a2 is a matrix whose entries are homogeneous quadratic forms in X = (x, ξ ) ∈ R2n , and a0 is a constant matrix. We shall be particularly interested in the following two-parameter family of 0 −1 NCHOs. Let α , β ∈ C, J = , and 1 0 α 0 x2 + ξ 2 + iJxξ , x, ξ ∈ R. Q(α ,β ) (x, ξ ) = 2 0 β

(3.18)


31

Hence, Q(α ,β ) ∈ Scl (m2 , g; M2 ) and the NCHO Qw (α ,β ) (x, D) is then the system of GPDOs of order 2

1 α 0 ∂x2 x2 + + J x . (3.19) − (x, D) = ∂ + Qw x (α ,β ) 2 2 2 0 β A NCHO is elliptic (resp. positive elliptic) when it is elliptic (resp. positive elliptic) as a GPDO. Remark 3.2.12. The NCHO Q(α ,β ) (x, ξ ), for α , β > 0 and αβ > 1 is positive elliptic. (We leave it to the reader to check this claim.) It is important to make the following observation on formally self-adjoint positive elliptic N × N systems of GPDOs. Lemma 3.2.13. Let a = a∗ ∈ Scl (mμ , g; MN ) be a globally positive elliptic GPD system. Then μ ∈ Z+ is even. Hence a = a μ + aμ −2 + . . . + a0 , where a0 is an N × N constant Hermitian matrix, and all the a μ −2 j , 0 ≤ j ≤ (μ − 2)/2, are N × N Hermitian matrices. Proof. By (3.17), taking v = e1 (the first vector of the canonical basis of CN ), we have for the 11-entry of a μ aμ ,11 (X) ≈ |X |μ , ∀X ∈ R2n . Hence a μ ,11 is a nonnegative homogeneous polynomial in X ∈ R2n of degree μ , which is positive when X = 0. Hence μ must be an even (nonnegative) integer. It is also important to define the smoothing elements. Definition 3.2.14. A symbol a∈

S(mμ , g) =: S(m−∞ , g) = S (Rn × Rn )

μ ∈R

is said to be a smoothing symbol. In fact, in this case aw (x, D) : S (Rn ) −→ S (Rn ) is continuous, and for its Schwartz kernel Kaw one has Kaw ∈ S (Rn × Rn ). Analogously in the matrix-valued case. In general, any continuous map R : S (Rn ; CN ) −→ S (Rn ; CN ) is called a smoothing operator. Equivalently, for the Schwartz kernel KR of a smoothing operator R we have KR ∈ S (Rn × Rn ; MN ). We next have the following useful result (of basic importance when constructing a parametrix of an elliptic operator in these global classes).

32


Proposition 3.2.15. Let μ j −∞, μ j > μ j+1 , j ∈ N, be a monotone strictly decreasing sequence of real numbers. Let a j ∈ S(mμ j , g), j ∈ N. Then there exists a ∈ S(mμ1 , g) such that a ∼ ∑ a j, j≥1

that is, for all r ∈ N we have r

a − ∑ a j ∈ S(mμr+1 , g). j=1

If another a has the same property, then a − a ∈ S (R2n ). Proof. Let χ be an excision function, with 0 ≤ χ ≤ 1, such that χ (X ) = 0 if |X| ≤ 1/2 and χ (X ) = 1 if |X | ≥ 1. In the first place we show that we can choose a monotone strictly increasing sequence of positive R j → +∞, increasing so quickly as j → +∞ that for any given j ≥ 2 and for all α ∈ Z2n + with |α | ≤ j, α ∂X χ (X /R j )a j (X) ≤ 2− j m(X )μ j +1−|α | .

(3.20)

To see this, note that |∂Xα (χ (X /R))| ≤ Cα m(X)−|α | , for R ≥ 1. In fact,

(3.21)

∂Xα (χ (X/R)) = R−|α | (∂Xα χ )(X/R),

and |α | ≥ 1, X ∈ supp(∂Xα χ )(·/R) =⇒ R/2 ≤ |X | ≤ R, from which (3.21) follows, and shows that χ (·/R) ∈ S(1, g) uniformly in R ≥ 1. Then for all α ∈ Z2n + α ∂X χ (X/R)a j (X) ≤ C j,α m(X )μ j −|α | , if R ≥ 1, whence

(3.22)

χ (·/R)a j ∈ S(mμ j , g), j ∈ N, R ≥ 1.

Of course, this is seen also by noting that χ (X/R)a j (X ) = a j (X ) for X large, for supp χ (·/R)a j ⊂ {X ∈ R2n ; |X| ≥ R/2}, ∀ j ∈ N.

(3.23)

Now, given any j ≥ 1, if R ≥ 1 and |α | ≤ j we have α ∂X χ (X /R)a j (X ) ≤ max {C j,α } m(X )μ j −|α | =: C j m(X )μ j −|α | . |α |≤ j

(3.24)


33

On the other hand, m(X )μ j −|α | ≤ ε m(X)μ j +1−|α | , when X is such that

1 m(X) = (1 + |X|2)1/2 ≥ . ε

So, to satisfy (3.20), it suffices to choose ε = 1/(2 jC j ), and take (1 + R2j /4)1/2 ≥ 2 jC j , j ≥ 2. That is, it suffices to take R j ≥ 2 j+1C j , j ≥ 2. Hence we may choose R1 = 1, R j = 2 j+1 (C j + 1) + R j−1, j ≥ 2. Now, (3.23) and R j +∞ yield that the sum a(X ) :=

∑ χ (X/R j )a j (X )

j≥1

is locally finite, hence a ∈ C∞ . On the other hand, given any r ∈ N and any α ∈ Z2n +, we may find N ∈ N so large that |α | ≤ N + 1 and μN+1 + 1 ≤ μr . Hence N α ∂X a(X ) − ∑ χ (X /R j )a j (X) =

α ∂X

j=1

≤

χ (X /R j )a j (X )

∞

∑

j=N+1 ∞

1 m(X )μ j +1−|α | j 2 j=N+1

∑

≤ 2−N m(X )μr −|α |

(3.25)

(recall that μ j > μ j+1 ). Therefore, by choosing r = 1 we obtain that for any given α ∈ Z2n + (with N large depending on α as above) N N ∂Xα a(X ) = ∂Xα a(X ) − ∑ χ (X/R j )a j (X) + ∂Xα ∑ χ (X /R j )a j (X ) , j=1

j=1

whence, by (3.22) and (3.25), |∂Xα a(X )| ≤

N 1 μ1 −|α | m(X ) + ∑ C j,α m(X)μ j −|α | ≤ Cα m(X )μ1 −|α |, 2N j=1

that is a ∈ S(mμ1 , g).

34


On the other hand, we also obtain from (3.25) that for any given r ∈ N and any given α ∈ Z2n + with N ≥ r + 1 so large depending on α and r that |α | ≤ N + 1 and μN+1 + 1 ≤ μr+1 , r N α ∂X a(X ) − ∑ a j (X ) ≤ ∂Xα a(X) − ∑ χ (X /R j )a j (X ) j=1

j=1

+ ∑ ∂Xα (1 − χ (X /R j ))a j (X ) r

j=1

+

N

∑

α ∂X χ (X /R j )a j (X )

j=r+1

≤ Cα ,r m(X)μr+1 −|α | , for we have (1 − χ (·/R j ))a j ∈ S(m−∞ , g), ∀ j ∈ N, and

χ (·/R j )a j ∈ S(mμr+1 , g), ∀ j ≥ r + 1. This shows that a ∼ ∑ j≥1 a j . Finally, if a ∈ S(mμ1 , g) has this last property, then for all r ∈ N, r r a − a = a − ∑ a j − a − ∑ a j ∈ S(mμr+1 , g), j=1

j=1

that is a − a ∈ S(m−∞ , g) = S (R2n ), which concludes the proof.

The following proposition (that we state without proof, for which we address the reader to Helffer [17] or Shubin [67]) is also useful. Proposition 3.2.16. Let a j ∈ S(mμ j , g), j ∈ N, where μ j is monotone strictly decreasing to −∞ as j → +∞. Let a ∈ C∞ (R2n ) be such that for any given α ∈ Z2n + there are constants να and Cα such that |∂Xα a(X)| ≤ Cα m(X)να . Finally, let there exist j and C j with j monotone strictly decreasing to −∞ as j → +∞, and suppose the following estimate holds r−1

|a(X ) − ∑ a j (X)| ≤ Cr m(X )r , j=1

for any given r ∈ N. Then a ∼ ∑ j≥1 a j .


35

In these global classes we have that for any given Au(x) := aw (x, D)u(x) = lim (2π )−n

ε →0+

ei x−y,ξ a(

x+y , ξ )χ (ε x, ε y, εξ )u(y)dyd ξ , 2

the limit being independent of χ ∈ S (R3n ) such that χ (0) = 1. In this case we have, using the relations (1 + |x − y|2)−M (1 − Δξ )M ei x−y,ξ = ei x−y,ξ , (1 + |ξ |2)−N (1 − Δy)N ei x−y,ξ = ei x−y,ξ , and integrating by parts, that −n

Au(x) = (2π )

ei x−y,ξ (1 + |x − y|2)−M

×(1 − Δξ )M (1 − Δy )N

1 x+y , a( ξ )u(y) dyd ξ . (1 + |ξ |2)N 2

In fact, for large M and N the integral becomes absolutely convergent, for if a ∈ S(mμ , g) then (see e.g. Shubin [67]), for b(x, y, ξ ) = a((x + y)/2, ξ ) we have γ

|∂xα ∂yβ ∂ξ b(x, y, ξ )| ≤ Cαβ γ (1 + |x| + |y| + |ξ |)μ −|α |−|β |−|γ |(1 + |x − y|)|μ |+|α |+|β |+|γ |, for all (x, y, ξ ) ∈ R3n , which allows us to conclude the convergence. In this case the Schwartz kernel KA ∈ S (Rn × Rn ) is given by KA |ψ S ,S = lim (2π )−n ε →0+

ei x−y,ξ a(

x+y , ξ )χ (ε x, ε y, εξ )ψ (x, y)dydxd ξ , 2

for all ψ ∈ S (R2n ). Also in the case of these global classes we may pass from the Weyl-quantization to the ordinary quantization (left-quantization). Theorem 3.1.16 is thus stated in the following form. Theorem 3.2.17. Given a ∈ S(mμ , g) we have aw (x, D) = b(x, D), where b(x, ξ ) = ei Dx ,Dξ /2 a(x, ξ ), a(x, ξ ) = e−i Dx ,Dξ /2 b(x, ξ ). We next summarize some properties of aw (x, D).

(3.26)

36


Theorem 3.2.18. 1. Given any a ∈ S(mμ , g) we have: • aw (x, D) : S (Rn ) −→ S (Rn ) and aw (x, D) : S (Rn ) −→ S (Rn ) is

continuous; • When a is real-valued, then aw (x, D) is formally self-adjoint.

The same holds true for a ∈ S(mμ , g; MN ). 2. If a ∈ S(1, g) then aw (x, D) : L2 (Rn ) −→ L2 (Rn ) is continuous. The same holds true for a ∈ S(1, g; MN ). 3. If a ∈ S(m−s , g), with s > 0, then aw (x, D) : L2 (Rn ) −→ L2 (Rn ) is compact. The same holds true for a ∈ S(m−s , g; MN ), s > 0. We next define general global pseudodifferential operators. Definition 3.2.19. 1. We say that A ∈ OPS(mμ , g) (i.e., that A is a global pseudodifferential operator of order μ ) if there exists a ∈ S(mμ , g) such that A = aw (x, D) + R, with KR ∈ S (Rn × Rn). 2. We say that A ∈ OPScl (mμ , g) if a ∈ Scl (mμ , g). 3. We say that A ∈ OPS(mμ , g) is elliptic when a is globally elliptic. 4. If A ∈ OPScl (mμ , g), a ∼ ∑ j≥0 aμ −2 j , we call aμ the principal symbol of A and aμ −2 the subprincipal symbol of A. Analogously for N × N systems. Example 3.2.20. For example, for the “translated” harmonic oscillator p0,n (x, ξ)+C (C a real constant), which is a classical symbol of order 2 (and also a GPD symbol of order 2), we have that p0,n (x, ξ ) is the principal symbol and the constant C is the subprincipal one. As another example, consider the symbol of the NCHO

0 −i α 0 x2 + ξ 2 Q(α ,β ) (x, ξ ) = + xξ . 2 i 0 0 β

The principal part is Q(α ,β ) (x, ξ ) itself, and there is no subprincipal part here. Observe hence that in the corresponding Weyl-quantization

Qw (α ,β ) (x, D) =

the part

1 2

1 0 −1 α 0 −∂x2 + x2 + (x∂x + ), 2 1 0 2 0 β

0 −1 is not the subprincipal term. 1 0


37

Theorem 3.2.21. Let a ∈ Scl (mμ1 , g) and b ∈ Scl (mμ2 , g). Then aw (x, D)bw (x, D) ∈ OPScl (mμ1 +μ2 , g), and for the symbol composition ab we have (using the fact that the derivative of a function which is homogeneous of degree μ is a function homogeneous of degree μ − 1) ab ∼ ∑ (ab)μ1 +μ2 −2 j , (3.27) j≥0

so that the principal symbol of the composition is (ab)μ1 +μ2 = aμ1 bμ2 ,

(3.28)

i (ab)μ1 +μ2 −2 = aμ1 bμ2 −2 + a μ1 −2 bμ2 − {a μ1 , b μ2 }. 2

(3.29)

and the subprincipal symbol is

Exercise 3.2.22. Write down the composition formula for semiregular classical symbols. We now come to a central construction in the theory of elliptic global pseudodifferential operators: the existence of a (two-sided) parametrix. Theorem 3.2.23. Let A ∈ OPS(mμ , g) be elliptic. Then there exists B ∈ OPS(m−μ , g) such that BA = I + R, AB = I + R , (3.30) where R, R are smoothing operators. Furthermore, if A ∈ OPScl (mμ , g) then B ∈ OPScl (m−μ , g). One calls B a two-sided parametrix. Proof. We shall give the proof when A is “classical”. ˜ for some B In the first place, it suffices to see that BA = I + R and AB = I + R, and B . In fact, we then have ˜ = (I + R)B, BAB = B(I + R) that is

˜ . B = B + (RB − BR) smoothing

We hence prove that we may find B as in the statement, such that BA = I + R (i.e. B is a left-parametrix). The construction of a right-parametrix is completely analogous. Let χ be an excision function. Let b− μ =

χ ∈ Scl (m−μ , g). aμ

38


Then

B1 = (b−μ )w (x, D) ∈ OPScl (m−μ , g),

and

B1 A = I + R1, R1 ∈ OPScl (m−1 , g),

and actually R1 ∈ OPScl (m−2 , g), because we are working with classical symbols. We hence “Neumann-invert” I + R1 as follows: for any given N ∈ Z+ we have

N

∑ (−1) j R1j

B1 A = I − (−1)N+1 R1N+1 .

j=0 ( j)

(0)

j

( j)

If r1 is the symbol of R1 (which is classical), that is r1 = 1 and r1 = r1 . . . r1 , then by Proposition 3.2.15 we may find a symbol s ∈ Scl (1, g) such that s∼

∞

∑ (−1) j r1

( j)

j times

.

j=0

Hence S = sw (x, D) satisfies SB1 A = I + R, with R smoothing. We thus set B := SB1 . This concludes the proof of the theorem.

Remark 3.2.24. 1. Theorem 3.2.21 also holds for matrix-valued operators. 2. Theorem 3.2.23 also holds for matrix-valued elliptic global pseudodifferential operators. 3. An equivalent proof may be given by using the composition formula (3.3) and by considering, at each degree of homogeneity, the equation ba = 1 + r, and finally by using an excision function to “re-sum” the homogeneous terms. Hence, for instance, we have the equations i b−μ aμ = 1, b−μ aμ −2 + b−μ −2aμ − {b−μ , a μ } = 0, . . . , 2 so that we take b−μ = a−1 μ , b− μ −2 =

i 2

−1 −1 {a−1 μ , a μ } − a μ a μ −2 a μ , . . . .

Of course, in the matrix-valued case we must be careful in keeping the right order of the factors.


39

We are now ready to define the natural spaces on which global operators act. Let Λ 2 = 1 + |x|2 + |D|2 (where |D|2 is a pseudodifferential notation for −Δx ) be the harmonic oscillator in n-dimensions, translated by 1. Let s ∈ R. By virtue of our knowledge of the spectrum of the harmonic oscillator (see Theorem 2.2.1 and its n-dimensional extension Theorem 2.2.2) we may define the s/2-power Λ s = (1 + |x|2 + |D|2 )s/2 (see also Section 5.3 of Chapter 5 below, where a few facts of the functional calculus are recalled). Using the proof of Theorem 5.5.1 below (see Helffer [17, pp. 46–52]) in the special case of Λ 2 , one has that Λ s = w s (x, D) ∈ OPScl (ms , g) is globally elliptic, with principal symbol (|x|2 + |ξ |2 )s/2 . Hence for all s, s ∈ R, one has that on S (Rn ) (and S (Rn ))

Λ s Λ s = Λ s+s , Λ −s = (Λ s )−1 , and Λ s = (Λ s )∗ also as global pseudodifferential operators. Definition 3.2.25. Let s ∈ R. Define the spaces Bs (Rn ) = {u ∈ S (Rn ); Λ s u ∈ L2 (Rn )}, endowed with the norm and inner-product respectively given by ||u||Bs = ||Λ s u||0 , (u, v)s := (Λ s u, Λ s v)0 , u, v ∈ Bs (Rn ). We define the CN -valued case as follows: Bs (Rn ; CN ) := Bs (Rn ) ⊗ CN , endowed with the natural norms and inner-products induced by the norm and innerproduct of Bs and CN . Here are some important properties of the Bs , that are derived by the continuity properties and the existence of a parametrix of the operators Λ s (see Helffer [17]). Proposition 3.2.26. 1. For each s ∈ R, Bs (Rn ) is a Hilbert space, and S (Rn ) ⊂ Bs (Rn ) is densely embedded. 2. For s ∈ Z+ , Bs (Rn ) = {u ∈ L2 (Rn ); and

∑

|α |+|β |≤s

∑

|α |+|β |≤s

||xα ∂xβ u||20 < ∞},

||xα ∂xβ u||20 ≈ ||u||2Bs ,

that is, the two norms are equivalent.

40


3. For any given s ∈ R, the L2 inner-product defined for u, v ∈ S extends to a sesquilinear form which is continuous on Bs (Rn ) × B−s (Rn ), and one has that B−s (Rn ) is identified with (Bs (Rn ))∗ (the dual of Bs (Rn )). 4. For any given s < s we have that Bs (Rn ) ⊂ Bs (Rn ) with compact and dense range. 5. We have S (Rn ) =

Bs (Rn ), S (Rn ) =

s∈R

Bs (Rn ).

s∈R

Hence, given any p, q, ∈ Z+ , if |u| p,q = sup sup (1 + |x|) p|∂xα u(x)| |α |≤q x∈Rn

denotes a seminorm in S , there is s0 = s0 (p, q, n) such that for all s ≥ s0 there exists Cs > 0 for which we have |u| p,q ≤ Cs ||u||Bs , ∀u ∈ S (Rn ).

(3.31)

6. For any given s ∈ R, any given A ∈ OPS(mμ , g) is bounded as an operator A : Bs (Rn ) −→ Bs−μ (Rn ). 7. The spaces Bs (Rn ) do not depend on Λ s , that is, for any given globally elliptic A ∈ OPScl (ms , g) one has Bs (Rn ) = {u ∈ S (Rn ); Au ∈ L2 (Rn )}. 8. Properties 1–7 above hold true also in the vector-valued and matrix-valued cases. It is worth noting that Property 5 of Proposition 3.2.26 does not hold for the more usual Sobolev spaces H s (Rn ), for in this case we have, upon setting H +∞ (Rn ) :=

H s (Rn ), H −∞ (Rn ) :=

s∈R

H s (Rn ),

s∈R

that S (Rn ) H +∞ (Rn ) H −∞ (Rn ) S (Rn ). Notice also that S (Rn ) =

k∈Z

Bk (Rn ).

3.3 Spectral Properties of Globally Elliptic Pseudodifferential Operators

41

3.3 Spectral Properties of Globally Elliptic Pseudodifferential Operators We now want to study how the spectrum of an elliptic global psedudodifferential operator depends on the spaces Bs . Remark that all the results stated in this section hold true also for matrix-valued operators. We have the following very important lemma, which shows that the kernel of an elliptic global pseudodifferential operator does not depend on the spaces Bs in which the operator is realized (in fact, it consists exclusively of Schwartz functions). Lemma 3.3.1. Let A ∈ OPS(mμ , g) be elliptic. Then Ker(A : Bs+μ → Bs ) = Ker(A : S → S ) = Ker(A : S → S ), ∀s ∈ R. Proof. Take ψ ∈ S such that Aψ = 0. Then, by the existence of a parametrix Q, 0 = QAψ = ψ + Rψ , i.e. ψ = −Rψ ∈ S , since R is smoothing. Hence Ker(A : S → S ) ⊂ Ker(A : S → S ) ⊂ Ker(A : S → S ). Since it is trivial that Ker(A : S → S ) ⊂ Ker(A : Bs+μ → Bs ) ⊂ Ker(A : S → S ),

we obtain the claim.

Next we show that the kernel of an elliptic global pseudodifferential operator has a finite dimension. Lemma 3.3.2. Let A ∈ OPS(mμ , g) be elliptic. Then dim Ker A < +∞. Proof. Let Q be a two-sided parametrix of A, with QA = I + R, R smoothing. Since Ker(A : Bμ → L2 ) ⊂ Ker(QA : Bμ → Bμ ) = Ker(I + R : Bμ → Bμ ), and since R : Bμ −→ S → Bμ +1 →→ Bμ , where →→ denotes compact embedding, we get that R : Bμ −→ Bμ is compact, whence the claim.

42


The range of an elliptic global pseudodifferential operator enjoys some nice properties as well. They are stated in the following results. Lemma 3.3.3. Let A ∈ OPS(mμ , g) be elliptic. For all s ∈ R one has: (i) Im(A : Bμ +s → Bs ) is closed, with finite codimension; (ii) codimIm(A : Bμ +s → Bs ) = dim Ker A∗ (hence independent of s). Proof. Let Q be a two-sided parametrix of A, with AQ = I + R , R smoothing. One has Im(A : Bμ +s → Bs ) ⊃ Im(AQ : Bs → Bs ) = Im(I + R : Bs → Bs ), and since R : Bs → Bs is compact, Im(I + R : Bs → Bs ) is closed, with finite codimension. This proves (i). We prove (ii). By the Closed Range Theorem we have dim(Bs /A(Bμ +s )) = dim Ker(A+ : Bs → Bμ +s ), where the operator A+ is defined by (Au, g)s = (u, A+ g)μ +s , u ∈ Bμ +s , g ∈ Bs . (Recall that (Au, g)s = (Λ s Au, Λ s g)0 .) Taking sequences {u j } j , {g j } j ⊂ S with Bμ +s

Bs

u j → u and g j → g, as j → +∞, yields (Λ s Au, Λ s g)0 = lim (Λ s Au j , Λ s g j )0 = lim (u j , A∗Λ 2s g j )0 j→+∞

j→+∞

= lim (Λ

μ +s

j→+∞

u j, Λ

−( μ +s) ∗

A Λ 2s g j )0 = (Λ μ +s u, Λ −(μ +s) A∗Λ 2s g)0 .

We thus have (Λ μ +s u, Λ −(μ +s) A∗Λ 2s g)0 = (u, A+ g)μ +s = (Λ μ +s u, Λ μ +s A+ g)0 . Then, by comparison, Λ μ +s A+ g = Λ −(μ +s) A∗Λ 2s g, whence A+ g = Λ −2(μ +s) A∗Λ 2s g, ∀g ∈ Bs , and, finally,

dim Ker(A+ : Bs → Bμ +s ) = dim Ker A∗ ,

which concludes the proof. Lemma 3.3.4. Let A ∈ OPS(mμ , g) be elliptic. One has: (i) Im(A : S → S ) is closed, with finite codimension equal to dim Ker A∗ ; (ii) Im(A : S → S ) is closed, with finite codimension equal to dim Ker A∗ .


43

S

Proof. To see (i), take {u j } j ⊂ S such that Au j → g ∈ S , as j → +∞. Then, in L2

particular, Au j → g, whence g ∈ Im(A : Bμ −→ L2 ). Therefore there exists u ∈ Bμ such that Au = g and, by the existence of a parametrix, we have that u ∈ S , since g ∈ S . This proves that Im(A : S → S ) is closed. We now claim that Im(A : S → S ) = {u ∈ S ; (u, g)0 = 0, ∀g ∈ Ker A∗ } =: V. (Recall that KerA∗ ⊂ S .) To prove the claim, observe that the inclusion ⊂ is trivial. To prove ⊃, suppose in the first place that μ = 0. Then, since L2 = Im A ⊕ Ker A∗ (A : L2 → L2 is now continuous), we have that if u ∈ S , then u = Au +g, with g ∈ S and u ∈ L2 , whence Au ∈ S . Thus, by the existence of a parametrix, u ∈ S . When u ∈ V , then u = Au and u ∈ A(S ), which proves the claim for μ = 0. When μ = 0, the assertion follows by observing that Im(A : S → S ) = Im(AΛ −μ : S → S ), and that Ker A∗ = Ker (Λ −μ A∗ ). This reduces matters to the case μ = 0 and proves (i). To prove (ii), we put, for ψ ∈ S and ϕ ∈ S , (ψ , ϕ )S ,S := ψ |ϕ¯ S ,S , so that

(Aψ , ϕ )S ,S = (ψ , A∗ ϕ )S ,S , ∀ψ ∈ S , ∀ϕ ∈ S .

We claim that Im(A : S → S ) = {ψ ∈ S ; (ψ , g)S ,S = 0, ∀g ∈ Ker A∗ } =: V . The inclusion ⊂ is trivial. To prove ⊃, take ψ ∈ V . Since ψ ∈ Bs0 for some s0 ∈ R, we get (recalling that the dual of Bs is B−s ) (ψ , g)S ,S = (ψ , g)Bs0 ,B−s0 . Suppose at first that μ = 0. Then A : Bs0 → Bs0 , A∗ : B−s0 → B−s0 , and, by construction, (Aψ , g)Bs0 ,B−s0 = (ψ , A∗ g)Bs0 ,B−s0 . Since A∗ has closed range, the Closed Range Theorem yields that if (ψ , g)Bs0 ,B−s0 = 0, ∀g ∈ Ker A∗ ,

44


then ψ ∈ Im(A : Bs0 → Bs0 ), whence ψ ∈ Im(A : S → S ). This proves the claim and also that Im(A : S → S ) is closed with codimension equal to dim KerA∗ , when μ = 0. When μ = 0, we reduce matters to the case μ = 0, by observing that Im(A : S → S ) = Im(AΛ −μ : S → S ), and that Ker A∗ = Ker (Λ −μ A∗ ).

This concludes the proof. We next have the following useful proposition.

Proposition 3.3.5. Let A ∈ OPS(mμ , g) be elliptic. The following assertions are equivalent: (i) (ii) (iii) (iv) (v)

Ker A = {0}; A∗ : S → S is onto; A∗ : S → S is onto; A∗ : Bμ +s → Bs is onto, for all s ∈ R; A∗ : Bμ +s → Bs is onto, for some s ∈ R.

Proof. We start by observing that (iv)⇒(iii), and that, in turn, by the existence of a parametrix, (iii)⇒(ii). Of course, (iv) implies (v). We now prove that (i) implies any of the (ii) to (v). Since dim(Bs /A∗ (Bμ +s )) = dim Ker A∗∗ = dim Ker A = 0, we immediately get that A∗ : Bμ +s → Bs is onto, for any given s ∈ R (since we already know that Im(A∗ : Bμ +s → Bs ) is closed). It follows that for any given u ∈ S ⊂ L2 there exists g ∈ Bμ such that A∗ g = u, whence, as Q∗ A∗ = I + R with R smoothing, S Q∗ u = Q∗ A∗ g = g + R g =⇒ g ∈ S . ∈S

A∗ (S )

=S. Thus That A∗ (S ) = S is trivial. In fact, u ∈ S implies u ∈ Bs0 for some s0 ∈ R, whence the existence of g ∈ Bμ +s0 ⊂ S (as we already know that (i)⇒(iv)) such that A∗ g = u. This concludes the proof that (i)⇒(ii), (iii), (iv) and (v). We finally prove that (v)⇒(i). If A∗ : Bμ +s → Bs is onto for some s, it follows that dim Ker A = codim A∗ (Bμ +s ) = 0. This concludes the proof.

Corollary 3.3.6. Let A ∈ OPS(mμ , g) be elliptic. If A = A∗ then the injectivity of A is equivalent to the surjectivity of A, which is in turn equivalent to the invertibility of A (on any of the spaces S , S , or Bs ). Recall that (see Helffer [17], or Hörmander [29], or Shubin [67]) a linear continuous operator T : B1 → B2 , B1 and B2 Banach spaces, is a Fredholm operator if


45

• dim Ker T < +∞, • dim Coker T < +∞, where Coker T = B2 /T (B1 ) ,

so that one may define the index of T as ind T = dim Ker T − dim Coker T. Recall that in this situation T (B1 ) is then closed. From the preceding results one has the following Fredholm property of a globally elliptic psedudodifferential operator (see [17] or [67]). Theorem 3.3.7. Let A ∈ OPS(mμ , g) be elliptic. Then: • A : B μ +s → Bs is a Fredholm operator, for any given s ∈ R; • ind A is independent of s and is expressed by the formula

ind A = dim Ker A − dimKer A∗ . In particular, if A = A∗ , that is, if A is formally self-adjoint, then ind A = 0. Here is another important consequence of global ellipticity, and of the previous results. Theorem 3.3.8. Let A ∈ OPScl (mμ , g) be elliptic and suppose there exists s0 ∈ R such that A Bs0 : Bs0 (Rn ) −→ Bs0 −μ (Rn ) is an isomorphism (that is, it is invertible with linear and continuous inverse). Then A−1 : Bs (Rn ) −→ Bs+μ (Rn ) is continuous ∀s ∈ R, and (ABs )−1 is the restriction to Bs−μ (Rn ) of an operator in OPScl (m−μ , g). The same holds true in the case of N × N systems. Proof. We give a proof in the scalar case. From Lemma 3.3.1 and Lemma 3.3.3 we have that ABs : Bs (Rn ) −→ Bs−μ (Rn ) is an isomorphism for all s ∈ R. By the global ellipticity, we know that there is Q ∈ OPScl (m−μ , g) such that QA = I + R, AQ = I + R , R, R smoothing. By hypothesis, upon setting A0 := ABs0 , we have −1 s A0 A−1 0 = IBs0 −μ , A0 A0 = IB 0 .

Then

−1 QBs0 −μ = A−1 0 A0 Q Bs0 −μ = A0 (IBs0 −μ + R Bs0 −μ ),

whence it follows that −1 A−1 0 = Q Bs0 −μ −A0 R Bs0 −μ .

(3.32)

46


On the other hand, we also have −1 −1 −1 s QBs0 −μ = QBs0 −μ A0 A−1 0 = (IB 0 + R Bs0 )A0 = A0 + R Bs0 A0 , whence it follows, using (3.32), that −1 −1 A−1 0 = Q Bs0 −μ −R Bs0 A0 = Q Bs0 −μ −R Bs0 Q Bs0 −μ +R Bs0 A0 R Bs0 −μ .

(3.33)

But RQ is smoothing, as well as RA−1 0 R , for A−1

R

0 Bs0 (Rn ) −→ S (Rn ) S (Rn ) −→ S (Rn ) → Bs0 −μ (Rn ) −→

R

is continuous. Hence from (3.33) it follows that A−1 ∈ OPScl (m−μ , g) and this concludes the proof. Let us now suppose that μ > 0 (the case μ = 0 is trivial). Consider PA : D(PA ) ⊂ L2 (Rn ) −→ L2 (Rn ), D(PA ) = {u ∈ L2 ; PA u = Au ∈ L2 }, where Au is understood in the sense of distributions. The unbounded operator PA is called the maximal operator (associated with A). When A is an elliptic global pseudodifferential operator, it is straightforward to see that PA is a closed operator, that is (see also Chapter 2) ⎫ ⎪ D(PA ) u j → u, as j → +∞ ⎪ ⎬ L2

L2

PA u j → v, as j → +∞

⎪ ⎪ ⎭

=⇒ u ∈ D(PA ) and PA u = v.

S

In fact, from the hypothesis we have Au j → Au, so that Au = v ∈ L2 whence u ∈ D(PA ), which proves that PA is a closed operator. Moreover,we have the following lemma (see also point 7 of Proposition 3.2.26). Lemma 3.3.9. One has D(PA ) = Bμ . Proof. To see this, just note that we obviously have Bμ ⊂ D(PA ). If now u ∈ L2 with Au ∈ L2 , by the existence of a parametrix we get Bμ QAu = u + Ru =⇒ u ∈ Bμ , which gives D(PA ) ⊂ Bμ .

This immediately gives back the fact that D(H) (see (2.4)), where H is the har2 2 2 n monic oscillator pw 0,n (x, D) = (|D| + |x| )/2, is exactly the Hilbert space B (R ). w Hence, in the case of the NCHO Q(α ,β ) (x, D), for α , β > 0 and αβ > 1, we have


47

2 2 D(Qw (α ,β ) (x, D)) = B (R; C ).

The next proposition gives that when A = A∗ , i.e. when A is formally self-adjoint, then PA = PA∗ , i.e. PA is self-adjoint. Proposition 3.3.10. One has PA∗ = PA∗ . Hence if A is formally self-adjoint then PA is self-adjoint. Proof. Let g ∈ D(PA∗ ), that is g ∈ L2 and there exists h ∈ L2 such that (Au, g)0 = (u, h)0 , ∀u ∈ D(PA ) = Bμ .

(3.34)

Hence (3.34) holds in particular for all u ∈ S . Now, from t ¯ S ,S , ∀u ∈ S , ¯ ¯ S ,S = A∗ g|u S ,S = h|u

(Au, g)0 = g|Au

S ,S = Ag|u

S

we obtain A∗ g = h ∈ L2 . The existence of a parametrix then yields g ∈ Bμ . Thus D(PA∗ ) ⊂ Bμ , and PA∗ g = A∗ g when g ∈ D(PA∗ ). On the other hand, Bμ ⊂ D(PA∗ ). In fact, if g ∈ Bμ take u ∈ Bμ and sequences {u j } j , {g j } j ⊂ S such that Bμ

Bμ

u j −→ u, g j −→ g, as j → +∞. L2

Since A∗ ∈ OPS(mμ , g), we thus have that A∗ g j −→ A∗ g, as j → +∞, and (Au, g)0 = lim (Au j , g j )0 = lim (u j , A∗ g j )0 = (u, A∗ g)0 , j→+∞

j→+∞

whence g ∈ D(PA∗ ), and this concludes the proof.

One defines the minimal operator PA by PA : D(PA ) = S (Rn ) ⊂ L2 (Rn ) −→ L2 (Rn ), PA u = Au, ∀u ∈ D(PA ). By the same method used in the case of the maximal operator, one sees (exercise for the reader!) that PA is closable, i.e. that D(PA )

⎫ ⎪

u j −→ 0, as j → +∞ ⎪ ⎬

L2 PA u j −→ v,

L2

as j → +∞

Moreover, we have the following proposition.

⎪ ⎪ ⎭

=⇒ v = 0.

48


Proposition 3.3.11. One has (PA )∗ = PA∗ Proof. As before, D((PA )∗ ) ⊂ Bμ , and (PA )∗ g = A∗ g, for all g ∈ D((PA )∗ ). Take Bμ

now g ∈ Bμ and u ∈ S , and consider a sequence {g j } j ⊂ S such that g j −→ g, as j → +∞. Then, as before, (Au, g)0 = lim (Au, g j )0 = lim (u, A∗ g j )0 = (u, A∗ g)0 , j→+∞

j→+∞

whence g ∈ D((PA )∗ ) and (PA )∗ g = A∗ g.

Corollary 3.3.12. A = A∗ implies PA = PA∗ = (PA )∗ . We may now give the following theorem about the spectrum of a formally self-adjoint positive elliptic global pseudodifferential operator (or system) of order μ ≥ 1. Theorem 3.3.13. Let A ∈ OPScl (mμ , g), μ ≥ 1, be elliptic, with real symbol and positive principal symbol aμ . Then A : D(A) = Bμ ⊂ L2 −→ L2 , thought of as an unbounded operator, is self-adjoint, with a discrete spectrum, made of an increasing to +∞ and bounded from below sequence {λ j } j∈N of eigenvalues with finite multiplicities, −∞ < λ1 ≤ λ2 ≤ . . . −→ +∞, with repetitions according to the multiplicity. The corresponding eigenfunctions belong to S and, possibly after an orthonormalization procedure, form a basis of L2 . Proof. The fact that A is a self-adjoint unbounded operator with a discrete spectrum and Schwartz eigenfunctions follows from the arguments above (Bμ is compactly embedded in L2 , and if λ ∈ Spec(A) then Ker(A − λ ) ⊂ S with finite dimension). We have to prove that the spectrum is bounded from below. Since aμ > 0, using the calculus of global operators we may find Q ∈ OPScl (mμ /2 , g) such that A = Q∗ Q + R, R ∈ OPScl (mμ −2 , g).

(3.35)

Then (Au, u) = ||Qu||20 + (Ru, u), ∀u ∈ S . 1/2

Since Q is elliptic, with principal symbol q μ /2 = aμ , we have, by the existence of a parametrix M for Q, that for some c > 0 ||Qu||20 + ||u||20 ≥ c||u||2Bμ /2 , ∀u ∈ S .

(3.36)

˜ so that In fact, from MQ = I + R˜ (R˜ smoothing) it follows u = MQu − Ru, ˜ 20 ||Qu||20 + ||u||20, ∀u ∈ S , ||u||2Bμ /2 = ||Λ μ /2 u||20 = ||Λ μ /2 MQu − Λ μ /2 Ru||


49

since, M : L2 −→ Bμ /2 being continuous, the operator Λ μ /2 M has order zero and Λ μ /2 R˜ is smoothing, and hence they are both L2 → L2 bounded. This proves (3.36). Now, as the inverse (Λ μ /2 )−1 of Λ μ /2 belongs to S(m−μ /2 , g), we have |(Ru, u)| = (((Λ

μ −1 2

)−1 )∗ R(Λ

μ −1 2

)−1Λ

μ −1 2

u, Λ

μ −1 2

u)| ≤ C||u||2B(μ −1)/2 , ∀u ∈ S , (3.37)

for ((Λ (μ −1)/2 )−1 )∗ R(Λ (μ −1)/2 )−1 has order −1 and hence, in particular, is L2 → L2 bounded. On the other hand, for any given ε > 0 (to be picked later) there exists Cε > 0 such that (note that μ − 1 ≥ 0) ||u||2B(μ −1)/2 ≤ ε ||u||2Bμ /2 + Cε ||u||20 , ∀u ∈ S .

(3.38)

From (3.36), (3.37) and (3.38) we finally get (Au, u) ≥ (c − ε C)||u||2Bμ /2 + O(||u||20) ≥ −C0 ||u||20 , ∀u ∈ S , by choosing 0 < ε < c/C.

Exercise 3.3.14. Let μ ≥ 1. Prove inequality (3.38).

Corollary 3.3.15. In the hypotheses of Theorem 3.3.13 we have that Spec(A) does not depend on s ∈ R. More precisely, if we think of A as an unbounded operator As : D(As ) = Bμ +s ⊂ Bs −→ Bs , As u = Au, u ∈ D(As ), then it follows that A∗s = As = A∗ = A and, by Lemma 3.3.1, Spec(As ) = Spec(A0 ) =: Spec(A). Remark 3.3.16. Theorem 3.3.13 and Corollary 3.3.15 hold true also in the matrixvalued case. The only delicate point is to obtain (3.35) in the case of systems, by constructing the expansion of the symbol of Q. One can actually find Q = Q∗ ∈ OPScl (mμ /2 , g; MN ) such that A = Q2 + R, R ∈ OPS(m−∞ , g; MN ), for at the level of the principal symbol one has to find a matrix qμ /2 = q∗μ /2 > 0 such that q2μ /2 = aμ , X = 0, and at the level of lower degrees of homogeneity one has to solve j − th term of (q∗μ /2 q μ /2 ) = a μ −2 j , X = 0, ∀ j ∈ Z+ .

(3.39)

50


In particular, the subprincipal part is the solution of i qμ /2 qμ /2−2 + q μ /2−2qμ /2 = aμ −2 + {q μ /2, q μ /2 }, X = 0, 2

(3.40)

with q∗μ /2−2 = qμ /2−2. Lemma 3.3.17 below takes care of (3.39) and Lemma 3.3.19 below takes care of (3.40). Lemma 3.3.17. Let 0 < a μ = a∗μ ∈ C∞ (R2n \ {0}; MN ) be positively homogenous of degree μ , and let 0 < c1 < c2 be such that c1 |v|2CN ≤ a μ (ω )v, v CN ≤ c2 |v|2 , ∀v ∈ CN , ∀ω ∈ S2n−1 . Then there exists a unique 0 < qμ /2 = q∗μ /2 ∈ C∞ (R2n \ {0}; MN ) positively homogeneous of degree μ /2 such that qμ /2 (X)∗ qμ /2 (X) = a μ (X ) for all X = 0. Proof. We construct q μ /2 on S2n−1 , for then it suffices to put q μ /2 (X ) = |X |μ /2 qμ /2 (

X ), ∀X = 0. |X |

From the hypothesis we have that Spec(a μ (ω )) ⊂ [c1 , c2 ] ⊂ R+ , for all ω ∈ S2n−1 . Let hence γ ⊂ C be a closed, counter-clockwise oriented curve contained in Re ζ > 0 enclosing [c1 , c2 ] such that γ¯ = γ , and define q μ /2 (ω ) =

1 2π i

γ

ζ 1/2 (ζ − a μ (ω ))−1 d ζ ,

where ζ 1/2 is the branch of the square-root that is positive when ζ is positive. It is then clear that q μ /2 is smooth and that q∗μ /2 = qμ /2 . Let γ be another curve (closed, counter-clockwise oriented) contained in Re ζ > 0 which encloses γ , with γ ∩ γ = 0. / It follows from the Cauchy formula that qμ /2 (ω ) =

1 2π i

γ

ζ 1/2 (ζ − a μ (ω ))−1 d ζ ,

so that, using the resolvent identity (ζ − ζ )(ζ − a μ (ω ))−1 (ζ − a μ (ω ))−1 = (ζ − a μ (ω ))−1 − (ζ − a μ (ω ))−1 , we get

1 ζ 1/2 (ζ )1/2 (ζ − a μ (ω ))−1 (ζ − a μ (ω ))−1 d ζ d ζ (2π i)2 γ γ (ζ )1/2 1/2 1 1 = d ζ ζ (ζ − a μ (ω ))−1 d ζ 2π i γ 2π i γ ζ − ζ

qμ /2 (ω )2 =


51

ζ 1/2 1 1 (ζ )1/2 (ζ − a μ (ω ))−1 d ζ d ζ 2π i γ 2π i γ ζ − ζ 1 = ζ (ζ − a μ (ω ))−1 d ζ = aμ (ω ), 2π i γ

−

for 1 2π i since

ζ

∈

γ

γ

ζ 1/2 d ζ = 0, ζ −ζ

is “outside” γ . This concludes the proof.

Remark 3.3.18. When working with a symbol 0 < a = a∗ ∈ S(mμ , g; MM ) such that for 0 < c1 < c2 c1 |v|2CN ≤

a(X )v, v CN ≤ c2 |v|2CN , ∀v ∈ CN , ∀X ∈ R2n |X | ≥ c > 0, m(X )μ

one has the analogue of Lemma 3.3.17, by working with

a(X ) , |X | ≥ c. m(X )μ

Lemma 3.3.19. The equation i qμ /2 qμ /2−2 + q μ /2−2qμ /2 = aμ −2 + {q μ /2, q μ /2 }, X = 0, 2

(3.41)

has a unique smooth solution q μ /2−2 = q∗μ /2−2 defined for X = 0, positively homogeneous of degree μ /2 − 2. Proof. Call cμ −2 the right-hand side of (3.41). Notice that c∗μ −2 = cμ −2 . Again, we work on S2n−1 and then extend the result by homogeneity of degree μ /2 − 2 as before. Since by Lemma 3.3.17 Spec(qμ /2 (ω )) ⊂ [c1 , c2 ] ⊂ R+ , we have inf

ζ ∈Spec(qμ /2 (ω )) ζ ∈Spec(−qμ /2 (ω ))

|ζ − ζ | ≥ c0 > 0, ∀ω ∈ S2n−1 .

We therefore readily see that, with γ ⊂ {Re ζ > 0} a closed, counter-clockwise oriented curve encircling [c1 , c2 ] with γ = γ¯, q μ /2−2(ω ) =

1 2π i

γ

(ζ − q μ /2(ω ))−1 cμ −2 (ω )(ζ + q μ /2 (ω ))−1 d ζ .

Hence q μ /2−2 is smooth, with q μ /2−2(ω )∗ = qμ /2−2(ω ) for all ω ∈ S2n−1 , and defining qμ /2−2(X ) = |X|μ /2−2qμ /2−2 (X/|X|) for X = 0 proves the lemma.

52


Remark 3.3.20. Given an operator A as in Theorem 3.3.13, it is useful to introduce a calculus with parameters tailored to A, for understanding, for example, the resolvent (A − λ )−1 . For X = (x, ξ ) ∈ R2n , λ ∈ C, consider (for reasons of homogeneity, as suggested by the example |x|2 + |ξ |2 − λ ) the weight mλ and the metric gλ defined by mλ (X ) := (1 + |X |2 + |λ |2/μ )1/2 , gλ ,X :=

|dX|2 . mλ (X)2

Then one sees that the metric gλ is admissible (with structural constants independent of λ ) and that mλ (X )±1 is a g-admissible function for all λ ∈ C (uniformly μ in λ ). One may then work with the class of symbol S(mλ , gλ ; λ ∈ Ω ) of functions a ∈ C∞ (R2n × Ω ) satisfying the following estimates: For any given α ∈ Z2n + there is Cα > 0 such that |∂Xα a(X , λ )| ≤ Cα mλ (X)μ −|α | , ∀X ∈ R2n , ∀λ ∈ Ω . In this framework, the classical symbols are defined to be those symbols a ∈ μ S(mλ , gλ ; λ ∈ Ω ), Ω now a cone of C, for which there exists {aμ −2 j } j≥0 ⊂ C∞ (R2n × Ω \ {(0, 0)}) such that a(X, λ ) ∼

∑ aμ −2 j (X, λ ),

j≥0

where

aμ −2 j (tX,t μ λ ) = t μ −2 j aμ −2 j (X, λ ), ∀t > 0, |X | + |λ | = 0.

In this case the excision function χ to be used is a function constructed by taking ω ∈ C∞ (Rt ), with 0 ≤ ω ≤ 1, ω ≡ 0 for |t| ≤ 1/2 and ω ≡ 1 for |t| ≥ 1, and then by putting χ (X , λ ) := ω (|X |2 + |λ |2/μ ). (See Helffer [17] and Shubin [67] for more on this.) Remark 3.3.21. It is interesting to note that, when μ = 2, in Theorem 3.3.13 above it is not necessary to assume that a2 > 0 for deducing the semi-boundedness from below of the operator A. It suffices that a2 ≥ 0. One has in fact the following result of Hörmander’s, his celebrated Sharp-G˚arding inequality, which is valid even for infinite-size systems (see Hörmander [29, Theorem 18.6.14]). Theorem 3.3.22 (Hörmander’s Sharp-G˚arding inequality). Let g be admissible (hence with uncertainty Planck’s function h ≤ 1 everywhere). Suppose that 0 ≤ a = a∗ ∈ S(h−1, g; L (H, H)), where H is a Hilbert space. Then there exists C > 0 such that (aw (x, D)u, u) ≥ −C||u||20, ∀u ∈ S (Rn ; H). Since, in the case of the global metric, saying that a ∈ S(h−1 , g) is equivalent to saying that a ∈ S(m2 , g), we conclude from this theorem that any given “second

3.4 Notes

53

order” (in the sense of the growth measured by m) global system, even infinitedimensional, is semi-bounded from below in L2 . The “fourth order” case (that is, the case S(h−2, g) = S(m4 , g)) for nonnegative global symbols is completely different, and is the celebrated Fefferman-Phong inequality. It is known that in general this inequality does not hold for systems (see the references to Brummelhuis, Hörmander and Parmeggiani in [53]). However, we have recently proved that it is indeed valid for large classes of systems of PDEs (see Parmeggiani [53, 54, 56, 57]).

3.4 Notes For the Weyl-Hörmander calculus, the reader is addressed to the original paper by Hörmander [27] and Hörmander’s book [29]. For the global calculus, the reader is addressed to Helffer’s book [17] and to Shubin’s book [67]. m The L2 -continuity theorem was first proved in the class S1/2,1/2 by Calderón and Vaillancourt and then generalized by many people (Beals and Fefferman, Cotlar and Stein, Unterberger and others; see the references in Hörmander [27] or [29]) to more general pseudodifferential calculi.

Chapter 4

The Spectral Counting Function N(λ ) and the Behavior of the Eigenvalues: Part 1

To understand the eigenvalues of an elliptic global operator, a very useful, basic and general tool is the Minimax Principle. After recalling it, we shall use the Minimax Principle to study the first properties of the spectral counting function, and of the behavior of the large eigenvalues, of an elliptic global operator. Remark that everything we say in this section holds also for matrix-valued operators.

4.1 The Minimax Principle In the first place, it is very useful to recall the Minimax Principle in the abstract setting. Theorem 4.1.1 (Minimax Principle). Let A : D(A) ⊂ H −→ H be an unbounded, densely defined and closed self-adjoint operator in the (infinite-dimensional) Hilbert space H, semi-bounded from below (that is, A ≥ −CI on D(A) for some constant C > 0), and with compact resolvent. Then, with Spec(A) given by the sequence −∞ < λ1 ≤ λ2 ≤ . . . → +∞, where the eigenvalues are repeated according to multiplicity, for the j-th eigenvalue one has ⎤ ⎡

λj =

sup

u1 ,...,u j−1 lin.ind.

⎢ ⎣

inf

u∈Span{u1 ,...,u j−1 ||u||=1

}⊥ ∩D(A)

⎥ (Au, u)⎦ ,

(4.1)

or, in other words, ⎤

⎡

λj =

sup

codimV = j−1 V closed

⎢ ⎣

inf

⎥ (Au, u)⎦ .

u∈V ∩D(A) ||u||=1

Proof. (We follow Dimassi-Sjöstrand’s book [7].) The first step is to see that D(A) ∩ E is always dense in E, where E = Span{u1 , . . . , u j−1 }⊥ and where A. Parmeggiani, Spectral Theory of Non-Commutative Harmonic Oscillators: An Introduction, Lecture Notes in Mathematics 1992, c Springer-Verlag Berlin Heidelberg 2010 DOI 10.1007/978-3-642-11922-4 4,

55

56

4 The Spectral Counting Function N(λ ) and the Behavior of the Eigenvalues: Part 1

u1 , . . . , u j−1 are linearly independent. Let v1 , . . . , v j−1 ∈ D(A) be close to u1 , . . . , u j−1 in norm. Then the space F := Span{v1 , . . . , v j−1 } ⊂ D(A) has dimension j − 1 and is transversal to E, whence it follows that there exists a unique bounded projection πE : H −→ H with Im πE = E, Ker πE = F. Then u = u +πE (u) =⇒ πE (u) = u − u ∈ D(A). D(A) u = u + ∈F

∈E

∈D(A)

Thus πE D(A) : D(A) −→ D(A), and we also have that πE (D(A)) = D(A)∩E is dense in E. The next step is to prove that

Λ j (u1 , . . . , u j−1 ) :=

inf

(Au, u) ≤ λ j .

u∈E∩D(A) ||u||=1

(4.2)

Let in fact e1 , . . . , e j ∈ D(A) be an orthonormal (ON) system of eigenfunctions of A corresponding to the eigenvalues λ1 ≤ . . . ≤ λ j . Then Span{e1 , . . . , e j } ∩ E = {0} (notice that the first space has dimension j and the second one has codimension j − 1). We then let u := ∑1j ck ek be a normalized vector in the intersection, so that j ∑1 |ck |2 = 1. It is clear that u ∈ D(A) ∩ E, and that (Au, u) =

j

j

k=1

k=1

∑ λk |ck |2 ≤ λ j ∑ |ck |2 = λ j ,

from which we deduce (4.2). We finally prove (4.1). We have just seen that the right-hand side is ≤ λ j . But if we take uk = ek , 1 ≤ k ≤ j − 1, and consider the sequence {e j } j≥1 ⊂ D(A) of the ON eigenvectors of A, we may then take u ∈ Span{e1 , . . . , e j−1 }⊥ ∩ D(A) of the form u = ∑k≥ j ck ek with

∑ |λk ck |2 < +∞,

and

k≥ j

so that (Au, u) =

k≥ j

∑ λ j |ck |2 ≥ λ j ∑ |ck |2 = λ j ,

k≥ j

which yields inf

u∈Span{e1 ,...,e j−1 }⊥ ∩D(A) ||u||=1

This concludes the proof.

∑ |ck |2 = 1,

k≥ j

(Au, u) = λ j .

4.1 The Minimax Principle

57

Corollary 4.1.2. Formula (4.1) is equivalent to ⎤

⎡

λj =

sup

u1 ,...,u j−1

⎢ ⎣

inf

⎥ (Au, u)⎦ ,

u∈Span{u1 ,...,u j−1 }⊥ ∩D(A) ||u||=1

(4.3)

where it is important to notice that the u1 , . . . , u j−1 may be taken not necessarily linearly independent. Proof. Denote by λ j the right-hand side of (4.3). Then it is clear that λ j ≤ λ j . Suppose that λ j > λ j , that is, λ j is obtained by using vectors u1 , . . . , u j−1 which are not linearly independent. Hence for any given ε > 0 there exist uε1 , . . . , uεj−1 with dim Span{uε1 , . . . , uεj−1 } ≤ j − 2, such that (with the notation of (4.2), where E = Span{uε1 , . . . , uεj−1 }⊥ )

λ j − ε < Λ j (uε1 , . . . , uεj−1 ) ≤ λ j−1 ≤ λ j , by Theorem 4.1.1, whence a contradiction.

Remark 4.1.3. Formula (4.3) also follows by observing that the suprema of the Λ j (u1 , . . . , u j−1 ) in (4.1) and (4.3) are equal, because j − 1 vectors can be always ε -approximated by j − 1 linearly independent vectors (in other words, systems of j − 1 vectors that are linearly independent are always dense in systems of j − 1 vectors). To prove this claim, it suffices to show that given j vectors u1 , . . . , u j and given any ε > 0 we may always find vectors v1 , . . . , v j that are linearly independent and such that ||vk − uk || < ε , 1 ≤ k ≤ j. To see this we first consider Span{u1 , . . . , u j } =: V = Span{ui1 , . . . , ui }, where I := {i1 , . . . , i } ⊂ {1, . . . , j} =: J and ui1 , . . . , ui are linearly independent. We order the elements in J \I = I (that we suppose non-empty, for otherwise there is nothing to prove) in increasing order, so that I = {i1 , . . . , ij− }, with i1 < . . . < ij− . Recall that uk ∈ V for all k ∈ I . Consider then V ⊥ , so that H = V ⊕V ⊥ , with orthogonal sum. Let {ek }k≥1 ⊂ V ⊥ be an ON basis of V ⊥ . Define

ε vi = ui + ek , k = 1, . . . , j − , k k j and vik = uik , 1 ≤ k ≤ . It is then clear that the v1 , . . . , v j are now linearly independent and that j

∑ ||ui − vi|| < ε .

i=1

This gives the claim of the remark.

58


We conclude this section with the following important application (called “the Rayleigh-Ritz technique”) of Corollary 4.1.2 (see Reed-Simon [61, p. 82]). Theorem 4.1.4. Let A be as in Theorem 4.1.1. Let V ⊂ D(A) be a j-dimensional subspace, and let ΠV : H −→ H be the orthogonal projection of H onto V . Write AV := ΠV AΠV , and denote by λˆ 1 ≤ . . . ≤ λˆ j the eigenvalues of the j × j matrix AV V . Then λk ≤ λˆ k , 1 ≤ k ≤ j. Proof. By the Minimax Principle (in the form (4.3)), AV V has eigenvalues given by ⎤

⎡

λˆ k =

sup

u1 ,...,uk−1 ∈V

⎢ ⎣

inf

u∈V ∩Span{u1 ,...,uk−1 }⊥ ||u||=1

⎥ (Au, u)⎦ ⎤

⎡ =

sup

u1 ,...,uk−1 ∈H

⎢ ⎣

inf

u∈V ∩Span{ΠV (u1 ),...,ΠV (uk−1 )}⊥ ||u||=1

⎥ (Au, u)⎦

(using the fact that for u ∈ V one has (u, ΠV (u )) = (u, u )) ⎤ ⎡ =

sup

u1 ,...,uk−1 ∈H

⎡ ≥

sup

u1 ,...,uk−1 ∈H

⎢ ⎣

⎢ ⎣

inf

u∈V ∩Span{u1 ,...,uk−1 }⊥ ||u||=1

inf

u∈D(A)∩Span{u1 ,...,uk−1 }⊥ ||u||=1

⎥ (Au, u)⎦ ⎤ ⎥ (Au, u)⎦ = λk ,

which concludes the proof.

4.2 The Spectral Counting Function Fix an elliptic A ∈ OPScl (mμ , g) (possibly matrix-valued), μ ≥ 1, with real-valued symbol and principal symbol aμ > 0. Then we know that A = A∗ . It is no restriction, by Theorem 3.3.13 and Remark 3.3.16, to assume that A > 0, for it suffices to consider A + c for a suitable constant c ∈ R. Then the spectrum Spec(A) of A is made by a sequence of eigenvalues 0 < λ1 ≤ λ2 ≤ . . . −→ +∞, where the eigenvalues are repeated according to their multiplicities. Remark 4.2.1. Notice that if A ∈ OPScl (mμ , g), μ > 0, and c is a constant, then A + c ∈ OPS(mμ , g), and it is classical only when μ ∈ 2N. However, since c times

4.2 The Spectral Counting Function

59

the identity operator commutes with everything and is continuous on any desired space and, in case A = A∗ has a discrete spectrum, adding c amounts to shifting each eigenvalue of A by the same constant c, this won’t be causing any problems throughout these notes, and we shall keep thinking of A + c as a classical operator. Let λ ∈ [0, +∞) =: R+ (recall also that R+ = (0, +∞)), and define the spectral counting functions (associated with A) N(λ ) = { j ∈ N; λ j < λ }, and N0 (λ ) = { j ∈ N; λ j ≤ λ }.

(4.4)

When comparing two elliptic global pseudodifferential operators A and B, we shall at times write NA and NB to denote the respective counting functions N(λ ). We shall study in this chapter the first basic properties of the function N. The function N0 is also introduced (some authors prefer using it in place of N). As far as the asymptotic properties of N and N0 are concerned, there is no difference, and it is just a matter of taste. We next list a few immediate properties of the spectral counting functions: • N(λ ) and N0 (λ ) are non-decreasing functions, with limit +∞ at +∞; • One has

N(λ j ) < N0 (λ j ), ∀λ j ∈ Spec(A),

and

N(λ ) = N0 (λ ), ∀λ ∈ Spec(A);

• N(λ j ) ≤ j, for all λ j ∈ Spec(A).

Notice that this might not be true for N0 (λ ). One may just think of a case in which λ j = λ j+1 = . . . = λ j+k j , for some j and k j , so as to have N0 (λ j ) = j + k j .

The next proposition (see Helffer [17]) shows how a-priori inequalities are used to compare the behavior of eigenvalues related to different elliptic global pseudodifferential operators (this also holds for elliptic operators on a compact boundaryless manifold). This will then be used (see Theorem 4.3.4, Theorem 4.4.1, and Corollary 4.4.3 below) to obtain a first information on the behavior of the eigenvalues λ j (A) as j → +∞ of an elliptic global pseudodifferential operator, by using the harmonic oscillator as a reference operator B. Proposition 4.2.2. Suppose that A, B : D ⊂ L2 (Rn ; CN ) −→ L2 (Rn ; CN ) be positive, self-adjoint operators with compact resolvent, with same domain D such that S (Rn ; CN ) ⊂ D. Suppose there exist C1 ,C2 > 0 with ||Au||2 ≤ C1 ||Bu||2 + ||u||2 , ∀u ∈ S (Rn ; CN ), ||Bu||2 ≤ C2 ||Au||2 + ||u||2 , ∀u ∈ S (Rn ; CN ).


60

Then

λ j (A) ≈ λ j (B), as j → +∞,

that is, one can find c1 , c2 > 0 and j0 ∈ N such that c1 λ j (A) ≤ λ j (B) ≤ c2 λ j (A), ∀ j ≥ j0 . Hence, by possibly changing the constants, we may always suppose that c1 λ j (A) ≤ λ j (B) ≤ c2 λ j (A), ∀ j ∈ N. Proof. One rewrites the above inequalities as (A2 u, u) ≤ C1 (B2 + I)u, u , (B2 u, u) ≤ C2 (A2 + I)u, u , ∀u ∈ S (Rn ; CN ), whence from the Minimax Principle (see (4.1))

λ j (A)2 ≤ C1 (λ j (B)2 + 1), and λ j (B)2 ≤ C2 (λ j (A)2 + 1), for all j ∈ N. Since λ j (A), λ j (B) → +∞ as j → +∞, there exists j0 ∈ N such that λ j (A), λ j (B) ≥ 1 for all j ≥ j0 , whence the desired conclusion. To see how this preliminary information influences the behavior of the function N(λ ) we show the following corollary. Corollary 4.2.3. Let A, B be as in Proposition 4.2.2. Suppose there exist C1 ,C2 > 0 such that C1 λ j (A) ≤ λ j (B) ≤ C2 λ j (A), ∀ j ∈ N. (4.5) Then, upon writing NA (resp. NB ) for the counting-fuction associated with A (resp. B), we have NA (C2−1 λ ) ≤ NB (λ ) ≤ NA (C1−1 λ ), ∀λ ∈ [0, +∞).

(4.6)

Proof. Let τ ≥ 0 and let JA (τ ) := { j ∈ N; λ j (A) < τ }, JB (τ ) := { j ∈ N; λ j (B) < τ }. Then from (4.5) we have that JA (τ ) ⊂ JB (C2 τ ) ⊂ JA (

C2 τ ), C1

whence NA (τ ) ≤ NB (C2 τ ) ≤ NA ( for all τ ≥ 0. Setting τ = λ /C2 gives (4.6).

C2 τ ), C1

4.3 The Spectral Counting Function of the (Scalar) Harmonic Oscillator

61

Remark 4.2.4. From Corollary 4.2.3 we have that if we know that (say) NA (λ ) = cA λ m (1 + o(1)) as λ → +∞, then we may conclude only that NB (λ ) ≈ λ m as λ → +∞, that is, there are constants C1 ,C2 , λ0 > 0 such that C1 λ m ≤ NB (λ ) ≤ C2 λ m , ∀λ ≥ λ0 . We now prepare the ground for exploiting Proposition 4.2.2 in the case in which A is an elliptic global pseudodifferential system (with positive-definite principal symbol) in Rn and B = pw 0 (x, D)I, where I is the 2 × 2, or N × N, identity matrix, and where p0 (x, ξ ) = (|x|2 + |ξ |2 )/2 (for simplicity, we shall drop the dependence on n in the notation for the harmonic oscillator). In particular, the results of the next section apply to the case in which A is our elliptic NCHO Qw (α ,β ) (x, D) (for α , β > 0, αβ > 1). In order to deduce the growth-rate of the eigenvalues λ j (A) of A, we have to understand NB (λ ) for B and deduce the growth of the eigenvalues λ j (B) as jpower . This is done in the next section.

4.3 The Spectral Counting Function of the (Scalar) Harmonic Oscillator We have the following well-known lemmas. Lemma 4.3.1. Let p(x, ξ ) = |x|2 + |ξ |2 . For the spectral counting function N(λ ) associated with pw (x, D) we have N(λ ) ∼ (2π )−n

p(x,ξ )≤1

dxd ξ λ n , λ → +∞.

Proof. It is well-known that the eigenvalues of pw (x, D) are

λ=

n

n

j=1

j=1

∑ (2α j + 1) = n + 2 ∑ α j ,

α = (α1 , . . . , αn ) ∈ Zn+ .

Hence, if μ = α1 + . . . + αn (= |α |), one has λ = n + 2 μ . Now, {α ∈ Zn+ ; |α | < μ + 1} = {α ∈ Zn+ ; |α | ≤ μ } =

μ

∑ {α ∈

j=0

=

Zn+ ; |α |

= j} =

μ

∑

j=0

μ +n j+n−1 = n n−1

(μ + n)(μ + n − 1) . . .(μ + 1) μ n ∼ , n! n!

(4.7)

62


as μ → +∞, so that, for λ = n + 2 + 2 μ , N(n + 2 + 2 μ ) ∼

μn , as μ → +∞. n!

On the other hand, (2π )−n

p(x,ξ )≤λ

dxd ξ = λ n (2π )−n

|x|2 +|ξ |2 ≤1

dxd ξ =

λn , 2n n!

and when λ = n + 2 + 2 μ ,

λn μn ∼ , as μ → +∞, n 2 n! n!

which proves the claim. For the sake of completeness we prove formula (4.7), that is, we show that

j+n = {α ∈ Zn+ ; |α | ≤ j}. n

(4.8)

Proof (of (4.8)). We proceed by induction on the dimension n. The case n = 1 is obviously true. Now, we have {α ∈ Zn+ ; |α | ≤ j} =

j

∑ {α ∈ Zn+ ; |α | = k},

k=0

and, writing α = (α , αn ) ∈ Zn−1 + × Z+ , {α ∈ Zn+ ; |α | = k} =

k

∑

αn =0

{α ∈ Zn−1 + ; |α | = k − αn }

= {α ∈ Zn−1 + ; |α | ≤ k} =

by the induction hypothesis. We must thus prove that k+n−1 j+n ∑ n−1 = n . k=0 j

But this follows from the formula n n−1 n−1 = + , k k k−1

k+n−1 , n−1

4.3 The Spectral Counting Function of the (Scalar) Harmonic Oscillator

for (with the convention that

n−1 n

63

= 0)

j k+n−1 k+n k+n−1 j+n n−1 − = − , ∑ n−1 = ∑ n n n n k=0 k=0 j

which proves the claim.

We may hence prove the following lemma, which gives the behavior of the eigenvalues of pw 0 (x, D) as j → +∞. Note that having the asymptotic behavior of N(λ ) given by Lemma 4.3.1 would suffice in case the eigenvalues of pw 0 (x, D) were all simple, for N(λ j+1 ) = j then, but this is not the case for pw (x, D) when n ≥ 2. 0 Note that a harmonic oscillator with simple eigenvalues is given, for example, by the Weyl-quantization of n

∑ μ j (x2j + ξ j2),

where μ1 , . . . , μn > 0 are Q-independent.

j=1

Lemma 4.3.2. For the eigenvalues λ j (repeated according to the multiplicity) of pw 0 (x, D) we have

λ j ≈ j1/n , as j → +∞.

(4.9)

Proof. The case n = 1 is trivial, for the eigenvalues of pw 0 (x, D) are then all simple and, by Lemma 4.3.1, we have j = N(λ j+1 ) ≈ λ j , as j → +∞, where λ j = j − 1/2, j ≥ 1. Therefore we consider the case n > 1. Let mk+1 be the multiplicity of the eigenvalues λ j = |α | + n/2, when α ∈ Zn+ and |α | = k. Then mk+1 =

k+n−1 , n−1

and, for k ≥ 0, k k k+n +n−1 = ∑ m+1 = ∑ =: Mk+1 . n n−1 =0 =0 We now consider the intervals of integers Ik := N ∩ (Mk , Mk+1 ], k ≥ 1, and write N = {M1 } ∪

k≥1

Ik , M1 = m1 = 1,


64

the union being disjoint. Since λ1 ≤ λ2 ≤ . . . are the eigenvalues of pw 0 (x, D), we have that for every j > 1 there exists k = k j ∈ N such that j ∈ Ik and λ j = λMk+1 , and that

N(λMk+1 ) = Mk ≈ (λMk+1 )n , as k → +∞,

by Lemma 4.3.1. As Mk =

k

k−1

k−1

=1

=0

=0

∑ m = ∑ m+1 = ∑

+n−1 k−1+n = , n−1 n

we have that

(k − 1)n kn ∼ , k → +∞. n! n! Since Mk < j ≤ Mk+1 , we get that j → +∞ iff k → +∞ and Mk ∼

(k − 1)n j kn , as j → +∞, whence j ≈ kn and finally

λ j = λMk+1 ≈ k ≈ j1/n , as j → +∞.

This concludes the proof of the lemma.

Remark 4.3.3. The above argument applies also in the case of pw 0 (x, D)IN . In this case we have an increase of multiplicity due to the presence of the N × N identity matrix IN . μ /2

Theorem 4.3.4. Let μ > 0, and let P0

μ /2

pw 0 (x, D). By Theorem 5.5.1 below, P0

be the μ /2-functional power of P0 = μ /2

∈ OPScl (mμ , g), that is P0

is an elliptic μ /2

classical global pseudodifferential operator of order μ . Denoting by {λ j (P0 )} j≥1 μ /2 the non-decreasing sequence of the eigenvalues of P0 , repeated according to multiplicity, we have μ /2 λ j (P0 ) ≈ j μ /2n , as j → +∞. It is important to note the power of j: it is the order of the operator divided by twice the dimension. Proof. Since

μ /2 2/ μ

λ j (P0

)

= λ j , j ∈ N,

the proof follows immediately from (4.9).

Remark 4.3.5. Of course, when μ ∈ 2N Theorem 4.3.4 holds without making use of Theorem 5.5.1.

4.4 Consequences on the Spectral Counting Function of an Elliptic Global ψ do

65

4.4 Consequences on the Spectral Counting Function of an Elliptic Global ψ do Using Proposition 4.2.2 we may now obtain the following behavior of the large eigenvalues of an elliptic global pseudodifferential operator, and in particular of our NCHO Qw (α ,β ) (x, D) when α , β > 0 and αβ > 1. The first result we state is the following. Theorem 4.4.1. Let 0 < A = A∗ ∈ OPScl (mμ , g; MN ), μ > 0, be an elliptic, selfadjoint and positive global pseudodifferential operator. Then, upon denoting by λ j (A), j ≥ 1, the sequence of the repeated eigenvalues of A we have

λ j (A) ≈ j μ /2n , as j → +∞. Proof. We start by remarking that, by virtue of the results of Section 3.3, A has a discrete spectrum, made of eigenvalues with finite multiplicities which, due to the positivity of A, diverge to +∞. μ /2 Now, by Theorem 5.5.1 below, the μ /2-power P0 of P0 = pw 0 (x, D) is an elliptic μ /2

classical global pseudodifferential operator of order μ . Put, for short, B = P0 IN , where IN is the N × N identity matrix. Let therefore QA and QB be two-sided parametrices (of order −μ ) for A and B, respectively, so that Q A A = I + R A , QB B = I + R B , where RA and RB are smoothing operators. Then, for all u ∈ S (Rn ; CN ), Au = A(QB B − RB)u = (AQB )Bu − ARBu, Bu = B(QA A − RA)u = (BQA )Au − BRAu, so that, by the

L2 -boundedness of

AQB , ARB , BQA and of BRA , we get

||Au||20 ||Bu||20 + ||u||20, ∀u ∈ S (Rn ; CN ), and, analogously, ||Bu||20 ||Au||20 + ||u||20, ∀u ∈ S (Rn ; CN ). The result therefore follows from Proposition 4.2.2 and Theorem 4.3.4.

Remark 4.4.2. Of course, when μ ∈ 2N Theorem 4.4.1 holds without making use of Theorem 5.5.1. In particular, for our elliptic NCHO Qw (α ,β ) (x, D), α , β > 0 and αβ > 1, we therefore have the following result.

66


Corollary 4.4.3. Let A = Qw (α ,β ) (x, D), with α , β > 0 and αβ > 1. Hence A is a positive elliptic system of GPDOs of order two, to which we may apply our results. Then λ j (A) ≈ j, as j → +∞. Recall that, given a self-adjoint operator A > 0 with a discrete spectrum {λ j } j≥1 , the spectral zeta function associated with A is by definition the series

ζA (s) =

1

∑ λs,

j≥1

s ∈ C with Re s sufficiently large.

(4.10)

j

From Theorem 4.4.1 and Corollary 4.4.3 we get the following immediate information about a first region of convergence of the spectral zeta functions ζA and ζQ associated with an elliptic A and an elliptic Qw (α ,β ) (x, D), respectively. We have in fact the following corollary. Corollary 4.4.4. The spectral zeta function ζA is holomorphic for Re s > 2n/μ . In particular, for α , β > 0 and αβ > 1 the spectral zeta function ζQ is holomorphic for Re s > 1. Proof. The proof follows immediately from the behaviors

λ j (A) ≈ j μ /2n , and λ j (Q) ≈ j, as j → +∞, and the fact that the harmonic series ∑ j≥1 j−s converges for Re s > 1.

Remark 4.4.5. It is well-known that for the spectral zeta function of the harmonic oscillator H = pw 0 (x, D) (n = 1) one has (exercise for the reader)

ζH (s) =

1

∑ ( j + 1/2)s = (2s − 1)ζ (s),

(4.11)

j≥0

where ζ (s) is the Riemann zeta function.

Chapter 5

The Heat-Semigroup, Functional Calculus and Kernels

We shall review in this chapter some elementary properties of the heat-semigroup associated with a globally elliptic operator, along with its functional calculus and some properties of the Schwartz kernels involved that give rise to the definition of the “trace” of the operator.

5.1 Elementary Properties of the Heat-Semigroup Let a ∈ Scl (mμ , g; MN ) be globally elliptic with μ ∈ N and principal symbol a μ = a∗μ > 0 (as an Hermitian matrix), and suppose that A∗ = A = aw (x, D) be positive on S (Rn ; CN ), that is there exists c > 0 such that (Au, u) ≥ c||u||20, ∀u ∈ S (Rn ; CN ). Let s ∈ R and let also As : D(As ) = Bs+μ (Rn ; CN ) ⊂ Bs (Rn ; CN ) −→ Bs (Rn ; CN ), As := ABs , be a realization of A as an unbounded operator in Bs (Rn ; CN ). Then we know that As = A∗s , Spec(As ) = Spec(A0 ) =: Spec(A), that the eigenfunctions of As are the eigenfunctions of A, and that they belong to S , by the elliptic regularity (i.e., the existence of the parametrix). We have that the resolvent Rs (λ ) = (As − λ )−1 : Bs (Rn ; CN ) −→ Bs (Rn ; CN ) is defined, bounded and holomorphic for all λ ∈ C \ Spec(A). It is important to note that s ≤ s =⇒ Rs (λ )Bs = Rs (λ ), ∀λ ∈ C \ Spec(A), (5.1)

for if λ ∈ Spec(A), then (As − λ )u = f ∈ Bs is solvable in Bs +μ and, by the elliptic regularity, for the solution u we have u ∈ Bs+μ ⊂ Bs +μ .


67

68

5 The Heat-Semigroup, Functional Calculus and Kernels

By virtue of (5.1), since

Bs = S and

s∈R

Bs = S , we may hence define, as a

s∈R

continuous map, R(λ ) : S (Rn ; CN ) −→ S (Rn ; CN ), R(λ )Bs := Rs (λ ), ∀λ ∈ C\ Spec(A). (5.2) Note that R(λ ) : S (Rn ; CN ) −→ S (Rn ; CN ) is also continuous. We may then simply write R(λ ) for the resolvent of A, regardless the domain used to realize A as an unbounded operator. It is hence well-known that for any given s ∈ R there is Cs > 0 such that Cs ||R(λ )||Bs →Bs ≤ . dist(λ , Spec(A)) Let δA := min Spec(A). We may therefore choose δ ∈ (0, δA ) such that Dδ := {λ ∈ C; Re λ ≤ δ } ∪ {λ = ρ eiφ ; ρ ≥

δ π 7 , φ ∈ [ , π ]} ⊂ C \ Spec(A). 2 4 4

Hence there exists Cs > 0 such that

λ ∈ Dδ =⇒ ||R(λ )||Bs →Bs ≤

Cs . 1 + |λ |

Let ∂ Dδ be oriented in such a way that Dδ is kept to the right-hand side. Then (see Kato’s book [35]) we may define ⎧ 1 ⎪ e−λ t R(λ )d λ , t > 0, ⎨ e−tA = 2π i ∂ Dδ ⎪ ⎩ −tA = I. e t=0

Hence, for any fixed s ∈ R, {e−tA }t≥0 is a strongly continuous semigroup in Bs with generator −As . Furthermore, for every δ ∈ (0, δA ), we have sup eδ t ||e−tA ||Bs →Bs < +∞. t≥1

Using the same arguments as in Chazarain-Piriou [6], one can prove the following lemma. (Recall that R+ = [0, +∞).) Lemma 5.1.1. Let f ∈ Bs (Rn ; CN ). Put F(t) = e−tA f . Then for every p, j ∈ Z+ , t p(

d j ) F ∈ C0 (R+ ; Bs+(p− j)μ (Rn ; CN )), dt

and sup ||t p ( t≥0

d j ) F||Bs+(p− j)μ ≤ C p, j || f ||Bs . dt

5.2 Direct Definition of Tr e−tA

69

From the lemma we may therefore think of e−tA as a map e−tA : S (Rn ; CN ) −→ S (R+ ; S (Rn ; CN )), and as a map e−tA : S (Rn ; CN ) −→ S (R+ ; S (Rn ; CN )). We also have the following fact about the heat-equation associated with −A. Lemma 5.1.2. For any given f ∈ S (Rn ; CN ) and g ∈ S (R+ ; S (Rn ; CN )), there exists a unique u(t) = u ∈ S (R+ ; S (Rn ; CN )) such that ⎧ du ⎪ ⎪ + Au(t) = g(t), t > 0, ⎨ dt ⎪ ⎪ ⎩ u(0) = f , with u(t) = e−tA f +

t 0

e−(t−t )A g(t )dt .

Moreover, f ∈ S (Rn ; CN ), g ∈ S (R+ ; S (Rn ; CN )) =⇒ u ∈ S (R+ ; S (Rn ; CN )).

5.2 Direct Definition of Tr e−tA In this section we give a direct proof that the kernel e−tA (x, y), t > 0, of e−tA (the heat-kernel of A) is a rapidly decreasing function of (x, y), and that Tr e−tA =

∑ e−t λ j ,

t > 0.

j≥1

Let {ϕ j } ⊂ S (Rn ; CN ) be an ON systems of L2 (Rn ; CN ) made of eigenfunctions of A. To start with, note that if u =

∞

∞

j=1

j=1

∑ u j ϕ j ∈ L2 , i.e. ∑ |u j |2 < +∞, then

e−tA u =

∞

∑ e−t λ j (u, ϕ j )ϕ j ,

j=1

so that, e−tA : S −→ S → S being continuous (and recalling that (v∗ ⊗ w)ζ = v∗ (ζ )w), we have by the Schwartz-kernel Theorem that e−tA (x, y) =

∑ e−t λ j ϕ j (y)∗ ⊗ ϕ j (x) ∈ S ,

j≥1

t > 0.

70


If we proceed formally, we have Tr e−tA =

Rn

Tr e−tA (x, x)dx =

∑ e−t λ j

j≥1

Rn

|ϕ j (x)|2CN dx =

∑ e−t λ j ,

j≥1

=1

where Tr denotes the matrix-trace. We next show that the formal procedure is actually correct. Since Aϕ j = λ j ϕ j , we get Ar ϕ j = λ jr ϕ j , for all r ∈ N, and ||ϕ j ||2Bμ r ||ϕ j ||20 + ||Ar ϕ j ||20 = (1 + λ j2r )||ϕ j ||20 .

(5.3)

(The reader may prove (5.3) as an exercise by using a parametrix of A, or else look at Lemma 5.3.1 below.) From (5.3) and (3.31) we thus get that given any S -seminorm | · | p,q, p, q ∈ Z+ , there exists r so large that |ϕ j | p,q ≤ C pq (1 + λ j2r )||ϕ j ||20 = C pq (1 + λ j2r ).

(5.4)

We now notice that (t λ j )k e−t λ j ≤ Ce−t λ j /2 , t > 0,

(5.5)

where C = (2k/e)k . Indeed, it is easy to see that

τ a e− τ ≤

a a e

, ∀τ , a > 0,

(5.6)

so that

τ a e−τ = τ a e−τ /2 e−τ /2 =

τ a 2

e−τ /2 2a e−τ /2 ≤

a a e

2a e−τ /2 =

2a a e

e−τ /2 .

Hence we obtain from (5.4) and (5.5), recalling that λ j +∞ as j → +∞, |e−tA (·, ·)| p,q ≤ Cpqt −k ∑ e−t λ j /2 < +∞, ∀t > 0, j≥1

for we already know that λ j ≈ j μ /2n as j → +∞. Hence, t → e−tA (·, ·) ∈ C∞ (R+ ; S (Rn × Rn ; MN )), and

Tr e−tA =

∑ e−t λ j ,

j≥1

as we wanted.

t > 0,

5.3 Abstract Functional Calculus

71

The next step, to be carried out in Chapter 6, will be to study the singularity as t → 0+ of the trace of the heat-kernel. We shall accomplish this by constructing a parametrix approximation of e−tA (the parametrix we are referring to is a parametrix for d/dt + A), that will give the sought information. After that, we shall relate the singularity of Tr e−tA as t → 0+ to the counting function N(λ ) through the Karamata theorem. But before doing that, we continue with a section about the abstract functional calculus of an elliptic global system, a section about kernels, and finally close the chapter with a section about f (A) as a global pseudodifferential operator.

5.3 Abstract Functional Calculus We recall in this section how to obtain an abstract functional calculus for an elliptic 0 < A = A∗ ∈ OPScl (mμ , g; MN ), μ > 0, with a discrete spectrum, Spec(A), made of a sequence 0 < λ1 ≤ λ2 ≤ . . . → +∞ of eigenvalues repeated according to multiplicity, and with eigenfunctions {ϕ j } j∈N ⊂ S (Rn ; CN ) that form an orthonormal basis of L2 (Rn ; CN ). Let f : Spec(A) −→ C. Define the unbounded operator f (A) : D( f (A)) ⊂ L2 (Rn ; CN ) −→ L2 (Rn ; CN ) by ! D( f (A)) := u =

∑ u j ϕ j ∈ L2 (Rn ; CN );

j≥1

D( f (A)) u =

" 2 2 | f ( λ )| |u | < +∞ , j j ∑

j≥1

∑ u j ϕ j −→ f (A)u := ∑ f (λ j )u j ϕ j .

j≥1

j≥1

Then f (A) is a closed operator with dense domain, since the ϕ j ∈ D( f (A)). If f is real-valued, then f (A) = f (A)∗ . Now we want to have conditions on f which ensure that f (A) : S (Rn ; CN ) −→ S (Rn ; CN ) be continuous. The first step is to understand D(A p ), for p ∈ N. Lemma 5.3.1. Define, for p ∈ N, the unbounded operator A p : D(A p ) ⊂ L2 −→ L2 , by D(A p ) = {u ∈ D(A p−1 ); A p−1 u ∈ D(A)}, and A p u = A(A p−1u). Remark that the domain is dense in L2 since S ⊂ D(A p ). Using the eigenfunctions {ϕ j } j∈N , we then have ! " D(A p ) = u = ∑ u j ϕ j ∈ L2 ; ∑ λ j2p |u j |2 < +∞ = B pμ (Rn ; CN ), j≥1

j≥1

and Au =

∑ λ ju jϕ j.

j≥1

Proof. Note in the first place that ϕ j ∈ S (Rn ; CN ) ⊂ Bs (Rn ; CN ), for all s ∈ R. If u ∈ D(A p ) we want then to prove that Λ pμ u ∈ L2 , where, recall, Λ pμ u is taken in the sense of distributions. Using a parametrix Q of A p such that QA p = I + R gives

72

5 The Heat-Semigroup, Functional Calculus and Kernels pμ 2 u ∈ D(A p ) =⇒ Λ pμ u = Λ pμ (QA p − R)u = (Λ pμ Q) A pu − Λ Ru ∈ L . order 0

∈L2

∈L2

And conversely, supposing u ∈ B pμ , and using a parametrix E of Λ pμ such that ˜ yields E Λ pμ = I + R, ˜ = (A p E) Λ pμ u − A pRu ˜ ∈ L2 , S A p u = A p (E Λ pμ − R)u order 0

∈L2

∈L2

and this concludes the proof. Remark 5.3.2. Note hence that for p ∈ N, ||u||2B pμ ≈ ||u||2D(A p ) := ||u||20 + ||A pu||20 ≈ ||A p u||20 ,

(5.7)

for, by our assumption, Spec(A) ⊂ (0, +∞).

When f is slowly increasing on the spectrum of A one has that f (A) is continuous from S into itself. One has in fact the following lemma. Lemma 5.3.3. Suppose f is slowly increasing on Spec(A), i.e. there exists C > 0 and p ∈ Z+ such that | f (λ j )| ≤ C(1 + λ j ) p , ∀ j ∈ N.

(5.8)

Then f (A) : S (Rn ; CN ) −→ S (Rn ; CN ) is continuous. Proof. For all r ∈ Z+ we have |(1 + λ j )r f (λ j )u j | ≤ C(1 + λ j )r+p |u j |, ∀ j ∈ N.

(5.9)

By Lemma 5.3.1 we have D(A p ) = B pμ , whence (5.9) implies the continuity of f (A)B(r+p)μ : D(Ar+p ) = B(r+p)μ −→ D(Ar ) = Br μ , ∀r ∈ Z+ , the spaces being endowed with the respective Hilbert-space structures (and making use of (5.7)). This proves the lemma. Remark 5.3.4. One has the following immediate consequences. 1. If f is bounded on Spec(A) (e.g. p = 0 in (5.8)) then f (A) : L2 −→ L2 is continuous. 2. Since A > 0, we may consider f (A) when f (λ ) = λ s , λ > 0, s ∈ R. 3. If f ∈ C0∞ (R) then f (A) is smoothing and compact (and actually of finite rank). 4. If f (λ ) = eit λ , t ∈ R, then unitary

eitA : L2 −→ L2 , (eitA )∗ = e−itA .

5.4 Kernels

73

5. If f ∈ S ([0, +∞)), then f (A) is smoothing. This is in particular the case when f (λ ) = e−t λ , for t, λ > 0. It is useful to have also the following proposition about the Schrödinger group. Proposition 5.3.5. The function t −→ e−itA ∈ C∞ (Rt ; L (S , S )). Moreover, with Dt = −i∂t , it solves the Cauchy problem ⎧ ⎨ (Dt + A)e−itA = 0, in C∞ (Rt ; L (S , S )), ⎩

e−itA t=0 = I.

Proof. One directly sees, from the very definition, that for all m, p ∈ N t −→ e−itA ∈ Cm−1 (Rt ; L (D(Am+p ), D(A p ))),

and that the equation is satisfied.

5.4 Kernels We now come to the study of the Schwartz kernel of f (A). Let hence f be slowly increasing on Spec(A). Then, by Lemma 5.3.3 f (A) : S −→ S is continuous so that, using the embedding S → S , f (A) : S −→ S is also continuous. Hence, by the Schwartz-kernel theorem, f (A) has a distribution kernel: for any given u, v ∈ S we have ( f (A)u, v) = K|v¯ ⊗ u S ,S . We have the following proposition. Proposition 5.4.1. If f is slowly increasing on Spec(A), then the distribution kernel K of f (A) belongs to S (Rn × Rn ; MN ), and one has K = S - lim Km , where Km (x, y) = m→+∞

m

∑ f (λ j )ϕ j (y)∗ ⊗ ϕ j (x),

j=1

where ϕ j (y)∗ ⊗ ϕ j (x)v = v, ϕ j (y) CN ϕ j (x), for all v ∈ CN (that is, ϕ j (y)∗ ⊗ ϕ j (x) = ϕ j (x) t ϕ j (y), colum-times-row).

74


Proof. Let k0 ∈ N be such that f (λ j ) −→ 0, as j → +∞. (1 + λ j )2k0

(5.10)

Let Pm be the L2 -orthogonal projection onto Span{ϕ j }1≤ j≤m . Then Rm := f (A)Pm is the operator whose kernel is Km . In addition, Rm can be extended as an operator belonging to L (D(Ak0 ), D(Ak0 )∗ ), where, recall, D(Ak0 )∗ is the dual space of D(Ak0 ). By (5.10), {Rm }m∈N is a Cauchy sequence in L (D(Ak0 ), D(Ak0 )∗ ). Hence there exists lim Rm = R ∈ L (D(Ak0 ), D(Ak0 )∗ ). m→+∞

But for all j ≥ 1 Rϕ j = lim Rm ϕ j = lim f (A)ϕ j = f (A)ϕ j . m→+∞

m→+∞

Since S → D(Ak0 ), and D(Ak0 )∗ → S , we have R ∈ L (S , S ), with

Rϕ j = f (A)ϕ j , ∀ j ≥ 1.

Hence R = f (A) in L (S , S ), so that f (A) = lim Rm .

m→+∞

Proposition 5.4.2. If f is rapidly decreasing on Spec(A) then S

Km −→ K, as m → +∞. L2

Proof. From Proposition 5.4.1 with k0 = 0 we obtain that Km −→ K as m → +∞. Let Ax be the operator A acting on x-functions (or distributions). Then, with (A∗y ⊗ Ax )(v∗ ⊗ u) = (Ay v)∗ ⊗ (Ax u), we have, for r ∈ N, ((A∗y )r ⊗ Arx )Km (x, y) =

m

∑ λ j2r f (λ j )ϕ j (y)∗ ⊗ ϕ j (x).

j=1

Hence, on the one hand, by Proposition 5.4.1, ((A∗y )r ⊗ Arx )Km −→ ((A∗y )r ⊗ Arx )K, as m → +∞, in the topology of S (Rn × Rn ; MN ), and on the other hand λ j2r f (λ j ) is rapidly decreasing for j → +∞, whence it also follows that ((A∗y )r ⊗ Arx )Km −→ ((A∗y )r ⊗ Arx )K, as m → +∞,

5.4 Kernels

75

in the topology of L2 (Rn × Rn ; MN ), for all r ∈ Z+ . Since A∗y and Ax are elliptic we obtain, by a natural vector-valued regularity theorem for Ax or A∗y starting from L2x,y , that Km −→ K, as m → +∞, ˆ ry , for all r ∈ Z+ . Since Brx ⊗B ˆ ry ⊂ Brx,y and in the topology of Brx ⊗B

r∈Z+

the claim follows.

Brx,y = Sx,y ,

Corollary 5.4.3. If f is rapidly decreasing on Spec(A) then Tr f (A) :=

∑

f (λ j ) =

j≥1

Rn

Tr K(x, x) dx,

where, recall, Tr denotes the matrix-trace. To make this section self-contained, we prove the following “rough” information on the behavior of the eigenvalues of A. Proposition 5.4.4. There exists k0 ∈ N such that ∞

1

∑ λ 2k0

j=1

< +∞.

j

Proof. For simplicity we consider the scalar case. We know that the operator A−k ∈ OPScl (m−k μ , g), k ∈ N. Using Theorem 3.1.16 (or Theorem 3.2.17), write A−k = b(x, D) + R, where R is smoothing. Then for the Schwartz-kernel K of A−k we have K(x, y) = K0 (x, y) + K1 (x, y), where K1 ∈ S (Rn × Rn ) is the Schwartz-kernel of R, and K0 is the Schwartz-kernel of b(x, D), given by K0 (x, y) = (2π )−n

ei x−y,ξ b(x, ξ )d ξ ∈ S (Rn × Rn )

(in the sense of oscillatory integrals, see Helffer [17] or Shubin [67]). Let us choose in the first place the integer k so that k μ > n. Then K0 ∈ C(Rn × Rn ) ∩ L∞ (Rn × Rn ). If k is picked so that k μ > n + 1, we have in addition that y j K0 (x, y) = (2π )−n

y j ei x−y,ξ b(x, ξ )d ξ

Dξ j (ei x−y,ξ )b(x, ξ )d ξ + x j (2π )−n ei x−y,ξ b(x, ξ )d ξ = (2π )−n ei x−y,ξ (Dξ j b)(x, ξ ) + x j b(x, ξ ) d ξ , = −(2π )−n

76


from which we deduce that y j K0 ∈ L∞ (Rn × Rn ). By the same token, we may finally conclude that there is k0 = k0 (n) such that picking k = k0 gives xα yβ K0 ∈ C(Rn × Rn ) ∩ L∞ (Rn × Rn), ∀α , β ∈ Zn+ , |α | + |β | ≤ 2(n + 1), whence K0 ∈ L2 (Rn × Rn ), and therefore also K ∈ L2 (Rn × Rn ). Since {ϕ j } j≥1 and {ϕ¯ j } j≥1 are orthonormal systems in L2x and L2y , respectively, we get that ϕ j (y)∗ ⊗ ϕ j (x)(= ϕ j (x)ϕ j (y) in the scalar case) is an orthonormal system in L2x,y . The Fourier coefficients of K with respect to this basis are given by

a j j :=

K(x, y)ϕ j (y)ϕ j (x)dxdy = (A−k0 ϕ j , ϕ j )

= λ j−k0 (ϕ j , ϕ j ) = λ j−k0 δ j j . Hence

|K(x, y)|2 dxdy =

∞

∑

|a j j |2 =

j, j =1

∞

1

∑ λ 2k0

j=1

< +∞,

j

and this concludes the proof.

It is useful to define also the trace of t −→ e−itA , that is the trace of the Schrödinger group (which is clearly important in its own right, and basic when studying Poisson-like relations). We have the following proposition. Proposition 5.4.5. The map S (R) φ − → Tr φˆ (A), φˆ (t) =

e−itx φ (x)dx,

defines a distribution in S (R), denoted by Tr e−itA . Proof. The map φ −→ φˆ is continuous in S . It suffices therefore to prove that for some C > 0 we have |Tr φˆ (A)| ≤ C|φˆ | p,q , (5.11) where |φˆ | p,q is some suitable S -seminorm of φˆ . By Proposition 5.4.4 (alternatively, in case μ = 2ν by Theorem 4.3.4, Remark 4.3.5, Theorem 4.4.1 and Remark 4.4.2, we may also use the behavior λ j ≈ jν /n as j → +∞), there exists r ∈ N sufficiently large such that ∑ (1 + λ j2)−r < +∞. Hence j≥1

∑ φˆ (λ j ) = ∑ (1 + λ j2)−r (1 + λ j2)r φˆ (λ j ) j≥1

j≥1

≤

∑ (1 + λ j2)−r

j≥1

which proves (5.11) and the proposition.

sup |(1 + t 2)r φˆ (t)|, t∈R

5.6 Notes

77

5.5 f (A) as a Pseudodifferential Operator The following theorem, which concludes the chapter, is also useful (see Helffer [17] and Robert [63–65]). It is obtained through the calculus with parameters. Theorem 5.5.1. Let 0 < A = A∗ ∈ OPScl (mμ , g), μ ∈ N, be a scalar elliptic GPDO of order μ . Let f : (−c, +∞) −→ C satisfy, for some r ∈ R, the inequalities ∀k ∈ Z+ , ∃Ck > 0, such that | f (k) (λ )| ≤ Ck (1 + |λ |)r−k . Then the operator-function f (A) = F w (x, D) ∈ OPS(mr μ , g), with F∼

μ

∑ Frμ − j ,

Fr μ − j ∈ S(mr μ − j , g),

j=0

and • Fr μ = f (a μ ); • Fr μ −1 = a μ −1 f (a μ ) (and hence it is equal to 0, for in our case a μ −1 = 0);

d jk (a) (k) f (a μ ), j ≥ 2, where the d jk ∈ S(mk μ − j , g) and depend only k=1 k! on the symbol a. j

• Fr μ − j =

∑

Furthermore, if f is classic, that is f (λ ) ∼ ∑ j≥0 c j λ r− j , λ > 0, then f (A) ∈ OPScl (mr μ , g). Remark 5.5.2. In the matrix-valued case we shall need the result of Theorem 5.5.1 only for the complex power Az , z ∈ C, of an elliptic system of GPDOs 0 < A = A∗ ∈ OPScl (mμ , g; MN ). By a result due to Robert [63], one has that Az belongs to the class OPScl (mμ Re z , g; MN ). Corollary 5.5.3. In particular, for Q(α ,β ) ∈ Scl (m2 , g; M2 ), which is globally positive elliptic for α , β > 0 and αβ > 1, Remark 5.5.2 holds true.

5.6 Notes The reader is addressed to Helffer’s book [17] for the functional calculus of global pseudodifferential operators (see also the paper by Helffer and Robert [19]).

Chapter 6

The Spectral Counting Function N(λ ) and the Behavior of the Eigenvalues: Part 2

In this chapter we shall describe the parametrix approximation of e−tA , A a positive elliptic global polynomial differential system, which will then be used, through Karamata’s Tauberian theorem (proved in Section 6.2), to compute the leading coefficient of the asymptotic behavior for large eigenvalues of the spectral counting function, in terms of the symbol of the system.

6.1 A Parametrix Approximation of e–tA In this section, following Parenti-Parmeggiani [50], we consider an N × N elliptic system of GPDOs 0 < A = A∗ ∈ OPScl (mμ , g; MN ), μ ∈ N, and introduce a class of symbols which is suitable for constructing a pseudodifferential approximation of e−tA . Recall that R+ = [0, +∞). Definition 6.1.1. Let r ∈ R. By S(μ , r) we denote the set of all smooth maps b: R+ × Rn × Rn −→ MN satisfying the following estimates: For any given α ∈ Z2n + and any given p, j ∈ Z+ there exists C > 0 such that d sup t p ( ) j ∂Xα b(t, X ) ≤ Cm(X )r−|α |+( j−p)μ . dt t∈R+

(6.1)

For b ∈ S(μ , r) we then consider the pseudodifferential operator bw (t, x, D)u(x) = (2π )−n

ei x−y,ξ b(t,

x+y , ξ )u(y)dyd ξ , u ∈ S (Rn ; CN ), 2

and we shall say that B ∈ OPS(μ , r) if B = bw (t, x, D) + R, where R is smoothing. In this setting, a smoothing operator R is any continuous map R : S (Rn ; CN ) −→ S (R+ ; S (Rn ; CN )).


(6.2)

79

80


Notice in particular that for the time-dependent seminorms of b ∈ S(μ , r) we have m(X)|α | |∂t j ∂Xα b(t, X)| ≤ Cα , j,p m(X)r+( j−p)μ t −p , ∀X ∈ R2n , ∀t ≥ 1, that is, for any given k, j, p ∈ Z+ there exists C = Ck, j,p > 0 such that (see (3.1)) ||∂t j b(t, ·)||k,S(mr+( j−p)μ ,g) ≤ Ct −p , ∀t ≥ 1.

(6.3)

We now introduce the “classical operators”. The key point here is to consider the correct homogeneity properties. The basic example to bear in mind is the matrix e−taμ (x,ξ ) . For instance, the bound for t p e−taμ (X) appearing in (6.1) follows from writing t p e−taμ (X) = aμ (X)−p (ta μ (X)) p e−taμ (X) , and from the inequality τ p e−τ ≤ (p/e) p , all τ > 0 (see (5.6)). The other bounds follow by the same considerations. Definition 6.1.2. We say that the operator B ∈ OPS(μ , r), B = bw + R is classical, and write B ∈ OPScl (μ , r), if there exists a sequence of functions br−2 j = br−2 j (t, X), j ≥ 0, t ≥ 0 and X = 0, such that: (i) One has the homogeneity br−2 j (t, τ X ) = τ r−2 j br−2 j (τ μ t, X), ∀τ > 0, ∀ j ≥ 0;

(6.4)

(ii) The function R2n \ {0} X −→ br−2 j (·, X) ∈ S (R+ ; MN ) is smooth for all j ≥ 0; (iii) For all ν ≥ 1 b(t, X ) −

ν −1

∑ χ (X)br−2 j (t, X ) ∈ S(μ , r − 2ν ),

j=0

where χ is an excision function. We call br = σr (B) the principal symbol of B. Of course, semi-regular classical symbols are defined accordingly (see Remark 3.2.4). Note that

σr (B) ≡ 0 =⇒ B ∈ OPScl (μ , r − 2),

6.1 A Parametrix Approximation of e−tA

81

and that given any smooth map b : R2n \ {0} −→ S (R+ ; MN ), with the homogeneity property b(t, τ X) = τ r b(τ μ t, X), ∀τ > 0, then there exists B ∈ OPScl (μ , r) such that σr (B) = b. One has the following results (whose proofs are left as an exercise to the reader) that are useful to the parametrix construction. Lemma 6.1.3. • One has

B ∈ OPScl (μ , r), C ∈ OPScl (μ , r ) =⇒ BC ∈ OPScl (μ , r + r ), with

σr+r (BC)(t, X ) = σr (B)(t, X )σr (C)(t, X ).

• One has

B ∈ OPScl (μ , r), P ∈ OPScl (m , g; MN ) =⇒ PB, BP ∈ OPScl (μ , r + ), with

σr+ (PB)(t, X) = p (X)σr (B)(t, X ),

and likewise for σr+ (BP).

• For B ∈ OPScl (μ , r) consider

d d B : S (Rn ; CN ) −→ S (R+ ; S (Rn ; CN )), u −→ (Bu). dt dt Then d d d B ∈ OPScl (μ , μ + r), with σμ +r ( B)(t, X ) = σr (B) (t, X ). dt dt dt • For B ∈ OPScl (μ , r) consider

∞ 0

Bdt : S (Rn ; CN ) −→ S (Rn ; CN ), u −→

Then

∞ 0

∞ 0

(Bu)dt.

Bdt ∈ OPScl (mr−μ , g; MN ),

with principal symbol X −→

∞ 0

σr (B)(t, X )dt.

We are now in a position to prove the following theorem about the existence of a parametrix approximation of e−tA . The parametrix we are interested in is that of the operator d/dt + A.

82


Theorem 6.1.4. There exists UA ∈ OPScl (μ , 0) such that (i)

d UA + AUA : S (Rn ; CN ) −→ S (R+ ; S (Rn ; CN )) dt

is smoothing, and (ii)

UA t=0 −I : S (Rn ; CN ) −→ S (Rn ; CN )

is smoothing. One has

σ0 (UA )(t, X ) = e−taμ (X) .

Proof. Let b0 (t, X) = e−taμ (X) , and let B0 ∈ OPScl (μ , 0) with σ0 (B0 ) = b0 . By Lemma 6.1.3 we have d B0 + AB0 ∈ OPScl (μ , μ − 2), dt with principal symbol rμ −2 (t, X ), and B0 t=0 −I ∈ OPScl (m−2 , g; MN ), with principal symbol p−2 (X ). We next look for a symbol b−2 (t, X ), positively homogeneous of degree −2 (in the sense of (6.4)), such that ⎧ d ⎪ ⎪ ⎨ b−2 + a μ b−2 = −rμ −2 , dt ⎪ ⎪ ⎩ b = −p . −2 t=0 −2

(6.5)

The solution of (6.5), b−2 (t, X) = −e−taμ (X) p−2 (X) −

t 0

e−(t−t )aμ (X) rμ −2 (t , X )dt ,

(6.6)

is easily seen to be smooth and have the required homogeneity properties. Taking B−2 ∈ OPScl (μ , −2) with σ−2 (B−2 ) = b−2 , gives d (B0 + B−2) + A(B0 + B−2 ) ∈ OPScl (μ , μ − 4), dt and

(B0 + B−2)t=0 −I ∈ OPScl (m−4 , g; MN ).

Iterating the above procedure gives a formal series

∑ B−2k ,

k≥0

B−2k ∈ OPScl (μ , −2k).

6.2 The Karamata Theorem

83

We may hence take, by an adaptation of Proposition 3.2.15, an operator UA ∈ OPScl (μ , 0) for which UA −

ν −1

∑ B−2k ∈ OPS(μ , −2ν ),

∀ν ≥ 1,

k=0

and therefore obtain the required parametrix. Hence, from Lemma 5.1.1 and Theorem 6.1.4 we have that

R(t) := e−tA − UA(t) : S (Rn ; CN ) −→ S (R+ ; S (Rn ; CN )) is continuous (6.7) (i.e., it is a smoothing operator), and

e−tA − UA (t)

t=0

: S (Rn ; CN ) −→ S (Rn ; CN ) is smoothing.

(6.8)

Remark 6.1.5. In the applications of Theorem 6.1.4 we shall always consider a parametrix approximation of e−tA where b−2 j t=0 = 0 for j ≥ 1, B−2 j (t) = (χ b−2 j )w (t, x, D), where χ is our usual excision function, and hence consider the symbol cA (t, X ) of UA (t), i.e. UA (t) = cw A (t, x, D), given by cA (t, X ) =

∑ χ j (X)b−2 j (t, X ),

j≥0

where χ0 (X ) = χ (X ) and χ j (X) = χ (X/R j ), j ≥ 1, with R j +∞, as j → +∞, sufficiently fast (in analogy with the R j constructed in Proposition 3.2.15). We shall write UA (t) ∼ ∑ B−2 j (t). (6.9) j≥0

In order to exploit the parametrix approximation UA (t) of e−tA to study NA (λ ) as λ → +∞, we have to recall the Karamata theorem.

6.2 The Karamata Theorem Recall that the Euler Gamma function Γ (x) =

∞ 0

e−t t x−1 dt, x > 0, fulfills the rela-

tion Γ (x + 1) = xΓ (x). Also, following Halmos’ book on Measure Theory [16], given a Borel measure μ over a measure space X , a measurable space Y and a measurable transformation

84


T : X −→ Y , the T -image measure μT of Y is defined for any Borel subset F ⊂ Y by

μT (F) = μ (T −1 F). Hence, for any given μT -measurable function f : Y −→ [0, +∞]

f (y)d μT (y) =

F

T −1 F

f (T x)d μ (x).

(6.10)

Theorem 6.2.1 (Karamata). Let μ be a positive (locally finite) Borel measure on [0, +∞), and let α > 0. Then ∞

e−t λ d μ (λ ) ∼ at −α , as t → 0+,

0

implies

x 0

d μ (λ ) ∼

a xα , as x → +∞. Γ (α + 1)

(6.11)

(6.12)

Proof. For t > 0 define the measure d μt (λ ) by μt (A) = t α μ (t −1 A). Let d ν (λ ) = αλ α −1 d λ , λ > 0. Then νt = ν . In fact, in general we have by definition that, given any Borel set A ⊂ [0, +∞),

μt (A) =

A

d μt (λ ) = t α μ (t −1 A) = t α

t −1 A

d μ (λ ),

and that, by (6.10), for any given measurable function f : [0, +∞) −→ [0, +∞] A

f (λ )d μt (λ ) = t α

t −1 A

f (t σ )d μ (σ ).

(6.13)

Hence, specializing to the case of νt gives

νt (A) =

A

d νt (λ ) = t α ν (t −1 A) = α t α

t −1 A

λ α −1 d λ = ν (A).

With b = a/Γ (α + 1), we have that, by virtue of (6.13), hypothesis (6.11) can be rewritten as ∞ ∞ lim e−λ d μt (λ ) = b e−λ d ν (λ ), (6.14) t→0+ 0

and also, since tα

0

1/t 0

d μ (λ ) =

1 0

d μt (λ ),

that the desired conclusion can be rewritten (just by putting x = 1/t) as 1

lim

t→0+ 0

d μt (λ ) = b

1 0

d ν (λ ).

(6.15)

6.2 The Karamata Theorem

85

Suppose we know that ∞

lim

t→0+ 0

f (λ )d μt (λ ) = b

∞ 0

f (λ )d ν (λ ), ∀ f ∈ C0 ([0, +∞)),

(6.16)

where C0 ([0, +∞)) denotes the space of continuous functions compactly supported in [0, +∞). Since ν and the μt , t ∈ (0, 1], are regular measures, we have that for any given open subset V of [0, +∞), ! ν (V ) = sup

[0,+∞)

" f (λ )d ν (λ ); f ∈ C0 ([0, +∞)), 0 ≤ f ≤ 1V ,

(6.17)

where 1V is the characteristic function of V (hence in particular supp f ⊂ V ), and for any given compact K ⊂ [0, +∞), ! ν (K) = inf

[0,+∞)

" f (λ )d ν (λ ); f ∈ C0 ([0, +∞)), 1K ≤ f ≤ 1 ,

(6.18)

hence f K ≡ 1, and likewise for the measures μt . Since (6.16) says that μt → ν weakly as measures on [0, +∞) as t → 0+, we have from (6.17) that ! b ν ((0, 1)) = sup b

[0,+∞)

! ≤ lim inf sup t→0+

f d ν ; f ∈ C0 ([0, +∞)), 0 ≤ f ≤ 1(0,1)

[0,+∞)

"

f d μt ; f ∈ C0 ([0, +∞)), 0 ≤ f ≤ 1(0,1)

"

= lim inf μt ((0, 1)) ≤ lim inf μt ([0, 1]), t→0+

t→0+

and likewise, from (6.18), that b ν ([0, 1]) ≥ lim sup μt ([0, 1]). t→0+

Since ν ((0, 1)) = ν ([0, 1]) (for ν ({0}) = ν ({1}) = 0) we have (a particular case of Prohorov’s theorem) lim μt ([0, 1]) = b ν ([0, 1]), t→0+

and this proves that (6.16) implies (6.15). Next, by (6.14) the measures e−λ d μt (λ ) are uniformly bounded, for t ∈ (0, 1], so that (6.16) follows from ∞

lim

t→0+ 0

g(λ )e−λ d μt (λ ) = b

∞ 0

g(λ )e−λ d ν (λ ),

(6.19)

for a dense (in the sup-norm) subset of g in C∞ ([0, +∞)), the space of continuous functions that vanish at infinity. But (6.19) holds for g(λ ) = e−nλ by hypothesis (6.11). Since polynomials in e−λ are dense in C∞ ([0, +∞)) by the Stone-Weierstrass Theorem, (6.19) holds and the theorem is proven.


86

6.3 Use of the Parametrix Approximation of e–tA for Obtaining the Weyl Asymptotics of N(λ ) We may now exploit the existence of the parametrix approximation of e−tA to study, through Karamata’s theorem, the Weyl-asymptotics of N(λ ) as λ → +∞. Hence, let Spec(A) = {λ j } j≥1 , 0 < λ1 ≤ λ2 ≤ . . . → +∞, repeated according to multiplicity, and let N(λ ) = { j ∈ N; λ j < λ }. Then

∑ e−t λ j =

j≥1

∞ 0

e−t λ dN(λ ) = Tr e−tA .

Actually, from Theorem 6.1.4, (6.7) and Remark 6.1.5 we have Tr e−tA = TrUA (t) + TrR(t), t > 0, where TrUA (t) =

∑ Tr B−2 j (t)

(6.20)

(6.21)

j≥0

(we shall specify in a moment in what sense the = sign holds). Note that since the operator R(t) is smoothing (see (6.2)), we have 0 ≤ t −→ Tr R(t) ∈ S (R+ ; C). Now, Tr B−2 j (t) = (2π )−n

R2n

(6.22)

χ (X)Tr b−2 j (t, X )dX.

Hence, upon denoting by S2n−1 the unit sphere in R2n , using polar coordinates 0 = X = ρω , where ρ ∈ R+ and ω ∈ S2n−1 , gives X = |X| |X| Tr B−2 j (t) = (2π )−n

+∞ 0

S2n−1

χ (ρω )Tr b−2 j (t, ρω )ρ 2n−1d ρ d ω ,

(6.23)

where d ω is the induced Riemannian measure on S2n−1 . In general, if b ∈ S(μ , −2ν ), with ν > n, then by the estimates (6.1) we have that uniformly in t ≥ 0, (2π )−n

+∞ 0

S2n−1

b(t, ρω )ρ 2n−1d ρ d ω

0

+∞

ρ 2n−1 d ρ < +∞, (1 + ρ )2ν

from which it follows that 0 ≤ t −→ fb (t) := (2π )−n

+∞ 0

S2n−1

b(t, ρω )ρ 2n−1d ρ d ω

(6.24)

6.3 Use of the Parametrix Approximation of e−tA for Obtaining the Weyl Asymptotics

87

is continuous and bounded on R+ . Hence ν

0 ≤ t −→ TrUA (t) − ∑ Tr B−2 j (t) ∈ C(R+ ) ∩ L∞ (R+ ), ∀ν > n.

(6.25)

j=0

More generally, given any p, k ∈ Z+ , we have |t p ∂tk fb (t)| < +∞, ∀t ≥ 0, provided b ∈ S(μ , −2ν ) with 2ν > 2n − (p − k)μ . Hence, on defining S p,k (R+ ) := { f ∈ Ck (R+ ); t p f (r) ∈ L∞ (R+ ), r = 0, 1, . . . , k},

(6.26)

for any given p, k ∈ Z+ there exists ν with 2ν > 2n − (p − k)μ such that ν

0 ≤ t −→ TrUA (t) − ∑ Tr B−2 j (t) ∈ S p,k (R+ ),

(6.27)

j=0

and this is the meaning of the = sign in (6.21). Next, by the homogeneity of the b−2 j we may write (supp χ ⊂ {|X | ≥ 1/2}, say) Tr B−2 j (t) = (2π )−n

+∞ 1/2

S2n−1

χ (ρω )Tr b−2 j (ρ μ t, ω )ρ 2(n− j)−1d ρ d ω .

(6.28)

Since Tr b−2 j (·, ω ) ∈ S (R+ , C) (which is also the case for the remainder in the asymptotic expansion of σ0 (UA ), by virtue of (6.1)), the integral in ρ converges for all t > 0, and hence the whole integral in d ρ d ω converges absolutely. Thus |Tr B−2 j (t)| < +∞, ∀t > 0, j ≥ 0, and (2π )−n

+∞ 0

S2n−1

Tr b−2 j (ρ μ t, ω )ρ 2(n− j)−1d ρ d ω < +∞, ∀t > 0, 0 ≤ j < n.

Furthermore, it follows from (6.23) and (6.24) that if j ≥ n + 1 in Tr B−2 j (t) the whole integral in d ρ d ω converges also for t = 0. We therefore have from (6.22), (6.25) and the fact (that we have just seen) that for j ≥ n + 1 the terms Tr B−2 j (t) are non-singular at t = 0, that the singularity of Tr e−tA at t = 0, which of course must be present, is necessarily contained in n

∑ Tr B−2 j (t).

j=0

(6.29)


88

Hence, to study the singularity at t = 0 of Tr e−tA , let us at first consider a term Tr B−2 j (t) of (6.29) with 0 ≤ j < n. Write for t > 0 Tr B−2 j (t) = β−2 j (t) − β−2 j (t), where −n β−2 j (t) := (2π )

+∞ S2n−1

0

Tr b−2 j (ρ μ t, ω )ρ 2(n− j)−1d ρ d ω

and −n β−2 j (t) := (2π )

+∞ S2n−1

0

1 − χ (ρω ) Tr b−2 j (ρ μ t, ω )ρ 2(n− j)−1d ρ d ω .

Since ρω ∈ supp(1 − χ ) =⇒ ρ ∈ [0, 1], it follows that we may rewrite −n β−2 j (t) = (2π )

1 0

S2n−1

whence it follows that

1 − χ (ρω ) Tr b−2 j (ρ μ t, ω )ρ 2(n− j)−1d ρ d ω ,

∞ ∞ β−2 j ∈ C (R+ ) ∩ L (R+ ),

with the derivatives of all orders bounded on R+ . , switching to the variable s = t 1/ μ ρ gives As regards β−2 j β−2 j (t)

=t

−2(n− j)/ μ

−n

(2π )

= c−2 j,n (μ )t

+∞

0 −2(n− j)/ μ

S2n−1

Tr b−2 j (sμ , ω )s2(n− j)−1 dsd ω

,

where c−2 j,n (μ ) := (2π )−n

+∞ 0

S2n−1

Tr b−2 j (sμ , ω )s2(n− j)−1 dsd ω

(6.30)

is such that |c−2 j,n (μ )| < +∞, because of the fact that Tr b−2 j (·, ω ) ∈ S (R+ ) and 0 ≤ j ≤ n − 1. Notice that, by the homogeneity, we may also write c−2 j,n (μ ) = (2π )−n

+∞ 0

S2n−1

Tr b−2 j (1, sω )s2n−1 dsd ω , 0 ≤ j ≤ n − 1. (6.31)

It therefore follows that Tr B−2 j (t) = t −2(n− j)/μ c−2 j,n (μ ) + o(1) , as t → 0+, 0 ≤ j ≤ n − 1.

6.3 Use of the Parametrix Approximation of e−tA for Obtaining the Weyl Asymptotics

89

As regards the term Tr B−2n (t), again by virtue of the fact that Tr b−2 j (·, ω ) ∈ S (R+ ), we have that for any given ε > 0 we may find Cε > 0 such that for all t > 0 |Tr B−2n (t)| = (2π )−n

+∞

1/2

= t −ε (2π )−n

S2n−1 +∞

χ (ρω )Tr b−2n (ρ μ t, ω )ρ −1 d ρ d ω

S2n−1

1/2

ε

χ (ρω )ρ ε μ t ε Tr b−2n (ρ μ t, ω )ρ −1−ε μ d ρ d ω

≤ Cε /t .

(6.32)

It follows that as t → 0+ n

# n−1

j=0

j=0

∑ Tr B−2 j (t) = t −2n/μ ∑

$ c−2 j,n (μ )t 2 j/μ + o(t 2 j/μ ) + t 2n/μ Tr B−2n (t) ,

so that, by (6.32), n

∑ Tr B−2 j (t) = t −2n/μ

c0,n (μ ) + o(1) , as t → 0 + .

j=0

This shows (using (6.24) to control the remainder term) that the leading singularity at t = 0 of Tr e−tA is exactly given by the term t −2n/μ with coefficient c0,n (μ ) = (2π )−n

+∞ S2n−1

0

Tr b0 (sμ , ω )s2n−1 dsd ω .

Since Tr b0 (sμ , ω ) = Tr e−s

μa

μ (ω )

=

N

∑ e−s

μ

j (ω )

(6.33)

,

j=1

where 0 < 1 (ω ), . . . , N (ω ), ω ∈ S2n−1 , are the (possibly repeated) eigenvalues of the N × N Hermitian matrix aμ (ω ), labelled in such a way to be represented by continuous functions on S2n−1 . Therefore N

c0,n (μ ) = (2π )−n ∑

+∞ S2n−1

j=1 0

= (2π )−n where we have used

+∞ 0

Γ ( 2n μ) μ

N

∑

j=1

e−s

S2n−1

μ

e−s s2n−1 ds = Γ (

We have thus proved the following theorem.

μ

j (ω )

s2n−1 dsd ω

1 dω , j (ω )2n/μ 2n )/ μ . μ


90

Theorem 6.3.1. Let 0 < A = A∗ ∈ OPScl (mμ , g; MN ) be an elliptic N × N system of GPDOs of order μ ∈ N in Rn . Then Tr e−tA ∼

Γ ( 2n μ )/ μ (2π )n

t −2n/μ

N

∑

S2n−1

j=1

1 d ω , as t → 0+, j (ω )2n/μ

where 0 < 1 (ω ), . . . , N (ω ), ω ∈ S2n−1 , are the (possibly repeated) eigenvalues of the N × N Hermitian matrix a μ (ω ), labelled in such a way to be represented by continuous functions on S2n−1 . Hence, Karamata’s theorem gives for the spectral counting function N(λ ) ∼

%

Γ ( 2n μ )/ μ

N

∑

(2π )nΓ ( 2n μ + 1)

2n−1 j=1 S

& 1 d ω λ 2n/μ , as λ → +∞. (6.34) j (ω )2n/μ

In particular, when A is an elliptic NCHO, we have N(λ ) ∼

1 2n(2π )n

S2n−1

Tr a2 (ω )−n d ω λ n , as λ → +∞.

(6.35)

Corollary 6.3.2. For the NCHO Qw (x, D), α , β > 0 and αβ > 1, we have (α ,β ) 1 N(λ ) ∼ 4π

α +β −1 Tr Q(α ,β ) (ω ) d ω λ = ' λ , as λ → +∞. 1 S αβ (αβ − 1) (6.36)

Remark 6.3.3. Given a positive function aμ , positively homogeneous of degree μ > 0 in R2n \ {0}, we have aμ (x,ξ )≤1

dxd ξ =

(2π )−n 2n

S2n−1

1 dω . aμ (ω )2n/μ

Hence, using the homogeneity of degree μ of the eigenvalues j , the main term in (6.34) (i.e. the coefficient of λ 2n/μ ) may also be written as 2n 2n Γ ( μ ) μ Γ ( 2n μ + 1)

N

∑

j=1

j (x,ξ )≤1

dxd ξ ,

(6.37)

and the one in (6.36) as ! " detQ(α ,β ) (x, ξ ) Vol (x, ξ ) ∈ R × R; ≤1 . Tr Q(α ,β ) (x, ξ )

(6.38)

6.4 Remarks on the Heuristics on N(λ ) and ζA (s)

91

6.4 Remarks on the Heuristics on N(λ ) and ζ A (s) It is worth noting that information on N(λ ) can be obtained from information on ζA (s) through Ikehara’s theorem. Here we follow Shubin [67, p. 120]. Theorem 6.4.1. Let N(λ ) be non-decreasing and equal to 0 for λ < λ1 . Suppose that

ζA (s) =

+∞

λ1

λ −s dN(λ )

converges for Re s > k0 , for some k0 > 0, and that

ζA (s) −

A s − k0

can be extended as a continuous function to the closed half-plane Re s ≥ k0 . Then N(λ ) ∼

A k0 λ , as λ → +∞. k0

What we have behind scenes is the following. Since the spectrum of A is made of eigenvalues 0 < λ1 ≤ λ2 ≤ . . . → +∞, considering the Stiltjes integral

ζA (s) =

+∞ 0

λ −s dN(λ )

gives the spectral zeta function ζA (s) = ∑ j≥1 λ j−s , which we already know to be holomorphic for Re s > 2n/μ . Suppose now that N(λ ) = c1 λ α1 + c2 λ α2 + . . . + ck λ αk + O(λ αk+1 ), as λ → +∞, where Re α1 > Re α2 > . . . > Re αk > Re αk+1 . Then

ζA (s) =

k

+∞

j=1

λ1

∑ cj

λ −s dN(λ ) + fk (s),

where fk is holomorphic for Re s > Re αk+1 . Since for Re s > Re α j +∞ λ1

λ

−s

αj

d(λ ) = α j

+∞ λ1

λ α j −s−1 d λ =

αj αj α j −s [λ α j −s ]+∞ , λ1 = s − α λ 1 αj − s j

we obtain, for Re s > Re αk+1 ,

ζA (s) =

k

∑ cj

j=1

α −s

λ1 j α j + fk (s), s − αj

92


so that for s = α j we have simple poles with residue c j α j , 1 ≤ j ≤ k. Conversely, knowing the poles of ζA gives information on the aymptotic behavior of N(λ ), as λ → +∞. However, things are very delicate and difficult even in the scalar case (be it on a compact boundaryless Riemannian manifold or the whole Rn × Rn ; see Ivrii [34] and references therein).

6.5 Notes The proof of Karamata’s Theorem 6.2.1 follows an unpublished idea of M. Aizenman. Corollary 6.3.2 was obtained by Ichinose and Wakayama in [31] as a consequence, through Ikehara’s theorem, of their result about the meromorphic continuation of the spectral ζ -function associated with Qw (α ,β ) (x, D) (where α , β > 0, αβ > 1), and later also by Parmeggiani in [53] (by Fourier integral operator methods) as a consequence of a theorem about the singularity at t = 0 of the Fourier transform of the spectral density ∑ j≥1 δ (λ − λ j ). Formula (6.38) for the main coefficient in the asymptotics for N(λ ) was observed by Parmeggiani in [53], where the relation between that coefficient and the periods of the bicharacteristics curves associated with the eigenvalues of Q(α ,β ) (x, ξ ) (they are all periodic, as it will be seen below) was also shown. It is worth mentioning also that Taniguchi in [68] approaches the construction of the heat-semigroup associated with Qw (α ,β ) (x, D), for α , β > 0, αβ > 1, by probabilistic methods. This method is deep and powerful, and should be extended to general NCHOs, to provide further geometrical understanding of the behavior of the eigenvalues of a NCHO in terms of “classical” quantities (i.e. related to the classical dynamics associated with the bicharacteristics).

Chapter 7

The Spectral Zeta Function

In the first section of this chapter we will briefly recall, addressing the reader to Robert’s paper [63] (see also Aramaki [1]), the construction of the spectral zeta function ζA (s) = Tr A−s , where 0 < A = A∗ ∈ OPScl (mμ , g; MN ) is an elliptic N × N system of GPDOs in Rn of order μ ∈ N. We will then prove a theorem about the meromorphic continuation of the spectral zeta function of an elliptic NCHO A = A∗ > 0 in Rn (Theorem 7.2.1) by using the parametrix approximation of the heat-semigroup e−tA constructed in Chapter 6, Section 6.1, from which we immediately deduce a corollary (Corollary 7.2.8) for our NCHO Qw (α ,β ) (x, D) (α , β > 0 and αβ > 1) which gives part of the result of Ichinose and Wakayama (Theorem 7.3.1 below; see [31]), whose proof we will sketch in the final section.

7.1 Robert’s Construction of ζ A by Complex Powers Let 0 < A = A∗ ∈ OPScl (mμ , g; MN ) be an elliptic N × N system of GPDOs in Rn of order μ ∈ N. Recall from Remark 3.3.20 that the construction of the complex powers of A may be cast in the Weyl-Hörmander calculus with parameters. So, suppose the principal symbol aμ of A be positive as an Hermitian matrix, that is a μ = a∗μ > 0, and let Λ = {z ∈ C; arg z ∈ [θ , 2π − θ ]}, for some θ ∈ (0, π /4). Recalling the weight mλ (X ) = (1 + |X |2 + |λ |2/μ )1/2 and the metric gλ ,X = mλ (X )−2 |dX|2 , we have that μ

A − λ ∈ OPScl (mλ , gλ ; λ ∈ Λ ; MN ), and A − λ is elliptic for λ ∈ Λ : |a μ (X ) − λ | ≥ Cmλ (X)μ , |X|2 + |λ |2/μ 1, λ ∈ Λ , X ∈ R2n . Note that

aμ (tX) − (t μ λ ) = t μ (a μ (X) − λ ).


93

94

7 The Spectral Zeta Function

Hence it suffices to check ellipticity only when |X|2 + |λ |2/μ = 1 (recall that we are dealing with systems of GPDOs in Rn ). −μ One next constructs a parametrix Bλ ∈ OPScl (mλ , gλ ; λ ∈ Λ ; MN ) of Aλ = A − λ , such that Bλ Aλ = I + Rλ , Aλ Bλ = I + Rλ , Rλ , Rλ ∈ OPS(m−∞ λ , gλ ; λ ∈ Λ ; MN ). Starting from the symbol b(X; λ ) ∼

∑ b−μ −2 j (X ; λ )

j≥0

of Bλ one then takes az (X ) ∼

1 2π i

γ

λ z b(X; λ )d λ ,

where γ ⊂ C is the curve {z ∈ C; |z| = c, |argz| ∈ [θ , 2π − θ ]} ∪ {z ∈ C; argz = θ , |z| ≥ c} ∪{z ∈ C; arg z = 2π − θ , |z| ≥ c}, for some fixed θ ∈ (θ , π /4) and c > 0, oriented in such a way that the circle-part is clockwise oriented. If az,N is any truncation of the asymptotics of az , with remainder rz,N , one defines w Az = a w z,N (x, D) + rz,N (x, D), when Re z < 0,

and for Re z < , ∈ Z+ , Az = Az− A . It follows that Az ∈ OPScl (mμ Re z , g; MN ). Robert then proved the following result (later generalized by Aramaki to some infinite-dimensional situations), that we state for classical symbols (Robert’s statement holds for more general polyhomogenous classes). Theorem 7.1.1. Let K(z) be the Schwartz kernel of Az . Then

ζA (s) = Tr A−s =

Rn

Tr K(−s) (x, x)dx,

is holomorphic in {s ∈ C; Re s > 2n/ μ }, and can be extended as a meromorphic 2n 2 j function in C, with at most simple poles belonging to the sequence s j = − , μ μ j ∈ Z+ , with residue

7.1 Robert’s Construction of ζA by Complex Powers

Res(ζA , s j ) =

μ i(2π )n+1

S2n−1 γ

95

λ −s j Tr b−μ −2 j (ω , λ )d λ d ω .

The function ζA is holomorphic in 0 with value

ζA (0) =

1 μ (2π )n

+∞ S2n−1 0

Tr b−μ −2n (ω , −λ )d λ d ω ,

and since in our case A is a system of GPDOs, we have that ζA is holomorphic in − j, j ∈ N, with value

ζA (− j) =

(−1) j μ (2π )n

+∞

S2n−1 0

λ j Tr b−μ −2n− j μ (ω , −λ )d λ d ω ,

hence it is surely 0 when j μ is not even, for in this case b−μ −2n− j μ = 0. Note that ζA (− j) = Tr A j , so that, since in our case A is a system of GPDOs, the value ζA (− j) is the trace of a local operator. In the case A = Qw (α ,β ) (x, D), we have μ = 2 and n = 1, so that ζQ is meromorphic in C, with at most simple poles that belong to the sequence s j = 1 − j, j ∈ Z+ . But, as we shall see below, by the theorem of Ichinose and Wakayama, or Corollary 7.2.8 to Theorem 7.2.1, we have that ζQ has only one simple pole at s = 1. Moreover, Ichinose and Wakayama prove in addition the remarkable result that ζQ (−2 j) = 0, j ∈ Z+ , i.e. that ζQ has, beyond the zero at s = 0, the same “trivial zeros” of the Riemann zeta function (the negative even integers −2k, k ∈ N). The method of Robert to construct the complex powers of a NCHO (i.e. through a parametrix of the resolvent operator) seems to be not well-suited if one is willing to keep the expressions of the terms of the parametrix Bλ as much explicit as possible, as one can see already in the basic case of Qw (α ,β ) (x, D). Since in a NCHO many symmetries are present (some of them are hidden because of the non-commutativity of the variables x and ξ , and because of the matrix-nature of the system), one should find the way to exploit them to simplify computations (we made some computations, and we could only prove that ζQ (0) = 0). Exercise 7.1.2. Upon computing a few terms in the symbol of the parametrix for H − λ , H = pw 0 (x, D), n = 1, calculate the residues of ζH and its value at − j, j ∈ Z+ , using Robert’s theorem. (See Remark 4.4.5.) In the next section we shall give a proof of the meromorphic continuation of the spectral zeta function ζA of an elliptic N × N NCHO 0 < A = A∗ ∈ OPScl (m2 , g; MN ), with positive principal symbol a2 = a∗2 > 0, by using the parametrix approximation of e−tA constructed in Section 6.1, which will yield Robert’s result in the special, yet meaningful, case of a second order system of GPDOs.

96


7.2 The Meromorphic Continuation of ζ A via the Parametrix Approximation of e–tA Let 0 < A = A∗ ∈ OPScl (m2 , g; MN ) be an elliptic N × N NCHO (i.e. an elliptic N × N second order GPD system) with positive principal symbol a2 = a∗2 > 0. Let ζA (s) be the corresponding spectral zeta function. We have the following theorem. Theorem 7.2.1. There exist constants c−2 j,n , 0 ≤ j ≤ n − 1, and constants C j , j ≥ n, such that, for any given integer ν ∈ Z+ with ν ≥ n,

ζA (s) =

$ ν Cj 1 #n−1 c−2 j,n + + H (s) , ν ∑ s−n+ j ∑ s−n+ j Γ (s) j=0 j=n

(7.1)

where Γ (s) is the Euler gamma function, and Hν is holomorphic in the region Re s > −(ν − n) − 1. Consequently, the spectral zeta function ζA (s) is meromorphic in the whole complex plane C with at most simple poles at s = n, n − 1, n − 2, . . . , 1. In particular, when n = 1, ζA has only one simple pole at s = 1. One has (recall (6.31)) −n

c−2 j,n = (2π )

+∞ 0

S2n−1

Tr b−2 j (1, ρω )ρ 2n−1d ρ d ω , 0 ≤ j ≤ n − 1,

(7.2)

where the b−2 j are the terms in the symbol of the parametrix UA ∈ OPScl (2, 0) of Theorem 6.1.4, Remark 6.1.5 and (6.9), UA ∼

∑ B−2 j .

j≥0

Moreover, c0,n = (2π )−n

+∞ 0

S2n−1

Tr(e−a2 (ρω ) )ρ 2n−1 d ρ d ω

is the leading coefficient in the Weyl asymptotics for NA (λ ) (see (6.33) and Theorem 6.3.1). Proof. By the properties of the semigroup 0 ≤ t → e−tA we may use the Mellin transform and write (recall Corollary 4.4.4 with μ = 2) A−s =

1 Γ (s)

+∞ 0

t s−1 e−tA dt, Re s > 2n/2 = n,

so that

ζA (s) = Tr A−s = Let hence UA ∼

1 Γ (s)

+∞

t s−1 Tr e−tA dt.

(7.3)

(7.4)

0

∑ B−2 j ∈ OPScl (2, 0) be the parametrix approximation of e−tA

j≥0

constructed in Theorem 6.1.4 and Remark 6.1.5.

7.2 The Meromorphic Continuation of ζA via the Parametrix Approximation of e−tA

97

Next write

ζA (s) =

1

1 Γ (s)

0

+

+∞

t s−1 Tr e−tA dt =: Z0 (s) + Z∞ (s).

1

(7.5)

In the first place we claim that Z∞ (s) is holomorphic in C. In fact, on the one hand, since t → Tr R(t) is rapidly decreasing for t → +∞ (where R(t) = e−tA − UA(t), see (6.7)), we have that for all p ∈ N and for all t ≥ 1 |Tr R(t)| t −p . On the other, given any ν ≥ 0 and any symbol b ∈ S(2, −2ν ), we have by (6.1) that for all t ≥ 1 and all p ∈ N −n π ) (2

+∞ S2n−1

0

Tr b(t, ρω )ρ 2n−1d ρ d ω

= t −p (2π )−n

t −p

+∞ 0

+∞

S2n−1 2n−1 ρ

t p Tr b(t, ρω )ρ 2n−1d ρ d ω

0

(1 + ρ )2ν +2p

d ρ t −p .

It thus follows that for all p ∈ N and for all t ≥ 1 |TrUA (t)| t −p . In conclusion, since

Tr e−tA = TrUA (t) + TrR(t),

for every p ≥ 1 there exists C p > 0 such that |Tr e−tA | ≤ C pt −p , ∀t ≥ 1, which proves the claim, since the term 1/Γ (s) is already holomorphic in C. Therefore the crux of the matter lies in the study of the function Z0 (s). It is important to be a little more precise on the form of the b−2 j . First of all, let us rewrite the composition formula (3.3) as (ab)(X) ∼ a(X )b(X ) + ∑

j≥1

1 −i j {a, b}( j)(X ), j! 2

where {·, ·}(1) = {·, ·} is the Poisson bracket. Then, from the construction of the parametrix approximation of e−tA , and using the fact that the term a0 in the symbol of A is a constant N × N Hermitian matrix, we have r−2 j = a0 b−2 j + (where we set b2 ≡ 0).

1 −i 2 i {a2 , b−2( j−1)}(2) − {a2 , b−2 j }, j ≥ 0 2 2 2

(7.6)

98


Recall hence, by (6.6), that ⎧ −ta (X) ⎪ ⎨ b0 (t, X) = e 2 , ⎪ ⎩ b−2( j+1)(t, X) = −

t 0

e−(t−t )a2 (X) r−2 j (t , X )dt , j ≥ 0.

(7.7)

To have a better understanding of the terms (6.28), j ≥ 0, we need to control the behavior of the b−2 j as t → 0 + . This is provided by the following technical proposition, whose proof is postponed to the end of the section. Proposition 7.2.2. For any given j ≥ 0 we have b−2 j (t, ω ) = O(t j ), t → 0+, and for all α , β ∈ Zn+ , with |α | = 2k + 1, k ≥ 0, and |β | ≤ 1 we have

∂Xα +β b−2 j (t, ω ) = O(t j+k+1 ), t → 0+, where the constants in O(·) do not depend on ω ∈ S2n−1 . Now, taking the proposition for granted, recall that by the homogeneity of the b−2 j (see (6.28)), for t > 0 Tr B−2 j (t) = (2π )−n −n

= (2π )

+∞ S2n−1

0

+∞

1/2 S2n−1

χ (ρω )Tr b−2 j (t, ρω )ρ 2n−1d ρ d ω χ (ρω )Tr b−2 j (ρ 2t, ω )ρ 2(n− j)−1d ρ d ω .

Consider c−2 j,n = (2π )−n

+∞ 0

S2n−1

Tr b−2 j (ρ 2 , ω )ρ 2(n− j)−1d ρ d ω .

We claim that |c−2 j,n | < +∞, ∀ j ∈ Z+ . In fact, the integral is convergent at ρ = +∞ for all j since Tr b−2 j (·, ω ) is a Schwartz function, it is clearly convergent at ρ = 0 for 0 ≤ j ≤ n − 1, and finally it is convergent at ρ = 0 also when j ≥ n, for the singularity at 0 of the factor ρ 2(n− j)−1 is compensated by Tr b−2 j (t, ω ) = O(t j ) as t → 0 + . Define now the functions f j (t) := −(2π )−n

1 0

S2n−1

(1 − χ (ρω ))Trb−2 j (t, ρω )ρ 2n−1d ρ d ω , j ∈ Z+ . (7.8)


99

Then f j ∈ C∞ ([0, +∞); C), for all j ∈ Z+ , and by Proposition 7.2.2 f j (t) = O(t j ), t → 0 + .

(7.9)

It follows that Tr B−2 j (t) = c−2 j,nt −(n− j) + f j (t) = c−2 j,nt −(n− j) + O(t j ), t → 0+,

(7.10)

for all j ≥ 0, and that (by the proof of Proposition 3.2.15 adapted to the present setting), ν

TrUA (t) − ∑ Tr B−2 j (t) =: Tr Rν +1 (t) = O(t ν +1 ), t → 0+, ∀ν ∈ Z+ .

(7.11)

j=0

However, the information contained in (7.10) alone is not yet sufficient to obtain the meromorphic continuation of ζA , and we need a better control of f j . Notice that for (k) all j, k ∈ Z+ , writing ∂tk f j (t) = f j (t), (k)

f j (t) = −(2π )−n

1 0

S2n−1

(1 − χ (ρω ))Tr ∂tk b−2 j (t, ρω )ρ 2n−1d ρ d ω ,

(k)

so that f j (0) is finite and can be computed through (7.6), and through the differential equations (7.20) and (7.24) below used to construct the b−2 j . Note, in particular, that (k)

f0 (0) = (−1)k+1 (2π )−n

1 S2n−1

0

(1 − χ (ρω ))Tra2 (ρω )k ρ 2n−1d ρ d ω .

It is useful now to have the following elementary result. Lemma 7.2.3. Let f ∈ C∞ ([0, 1]; C). Define, for s ∈ C, Re s > 0, F(s) =

1 0

t s−1 f (t)dt.

Then, for any given ν ∈ Z+ , F(s) =

ν

∑

k=0

f (k) (0) 1 + Fν (s), k! s + k

(7.12)

where Fν (s) =

1 ν!

1 0

t s+ν

1 0

(1 − τ )ν f (ν +1) (t τ )d τ dt

(7.13)

100


is holomorphic for Re s > −ν − 1. Hence F can be meromorphically continued to the whole complex plane, with at most simple poles at the non-positive integers s = −k, k ∈ Z+ , and f (k) (0) . (7.14) Res(F, −k) = k! In particular, if f (t) = O(t j ) as t → 0+ for some j ∈ Z+ , then F(s) is meromorphic on C with at most simple poles at s = − j − k, k ∈ Z+ . Proof (of Lemma 7.2.3). By Taylor’s formula, for any given ν ∈ Z+ , f (t) =

ν

∑

k=0

1

f (k) (0) k t ν +1 t + k! ν!

0

(1 − τ )ν f (ν +1) (t τ )d τ .

Hence, if ν = 0 and Re s > 0 we write f (t) = f (0) + t F(s) = where

f (0) + s

1 1

ts

0

1 1

ts

0

0

0

1 0

f (t τ )d τ and have

f (t τ )d τ dt,

f (t τ )d τ dt

is holomorphic for Re s > −1. This shows that F can be meromorphically extended for Re s > −1, with possibly a simple pole at s = 0 with residue f (0). Iterating the process using Taylor’s formula gives (7.12) to (7.14). If now f (t) = O(t j ) as t → 0+ for some j ∈ Z+ , we have that F(s) is holomorphic for Re s > − j, and meromorphic on C with at most simple poles at s = − j − k, k ∈ Z+ . This completes the proof of the lemma. We next apply Lemma 7.2.3 to the functions f j , so that for any given ν ∈ Z+ we may write, by (7.9), Fj (s) :=

1

t 0

s−1

f j (t)dt =

ν

∑

k=0

( j+k)

(0) 1 + Fj,ν (s), ( j + k)! s + j + k

fj

where Fj,ν is holomorphic for Re s > − j − ν − 1. Using this in (7.10) gives the following lemma. Lemma 7.2.4. For each j ≥ 0, for any given ν ∈ Z+ , s −→

1

( j+k)

t 0

s−1

ν f (0) c−2 j,n 1 j +∑ + Fj,ν (s), Tr B−2 j (t)dt = s − (n − j) k=0 ( j + k)! s + j + k

where Fj,k is holomorphic for Re s > − j − ν − 1.


101

Analogously, from (6.22) and Lemma 7.2.3 we also have, with fR (t) := Tr R(t), that for any given ν ∈ Z+ 1 0

t s−1 fR (t)dt =

ν

∑

k=0

(k)

fR (0) 1 + FR,ν (s), k! s + k

where FR,ν is holomorphic for Re s > −ν − 1. We therefore obtain that for any given ν ∈ Z+ Z0 (s) =

$ 1 1 1 # ν 1s−1 s−1 s−1 t Tr B (t)dt + t Tr R (t)dt + t Tr R(t)dt . −2 j ν +1 ∑ 0 Γ (s) j=0 0 0

Since the function s →

1 0

t s−1 Tr Rν +1 (t)dt =: Fν +1 (s) is holomorphic for Re s >

−ν − 1, we thus obtain that, for any given ν ∈ Z+ with ν ≥ n, ( j+k)

ν fj (0) 1 1 # ν c−2 j,n + ∑ Z0 (s) = ∑ Γ (s) j=0 s − n + j j,k=0 ( j + k)! s + j + k ν

+∑

k=0

(k) $ ν fR (0) 1 + ∑ Fj,ν (s) + FR,ν (s) + Fν +1 (s) k! s + k j=0

$ ν Cj 1 #n−1 c−2 j,n +∑ + H˜ ν (s) , = ∑ Γ (s) j=0 s − n + j j=n s − n + j

(7.15)

with H˜ ν (s) holomorphic for Re s > −(ν − n) − 1. Since the function 1/Γ (s) is holomorphic in C and has zeros at the non-positive integers −k, k ∈ Z+ , this proves the theorem. As a corollary of the proof of Theorem 7.2.1 one has the following expression of the coefficients C j appearing in (7.15). Corollary 7.2.5. With the notation of the proof of Theorem 7.2.1 one has, for all ≥ 0, C+n = c−2(+n),n +

1 () () f j (0) + fR (0) . ∑ ! j=0

(7.16)

As another corollary of the proof of Theorem 7.2.1, we have the following result on ζA (0). Corollary 7.2.6. We have

ζA (0) = c−2n,n = (2π )−n

+∞ 0

S2n−1

Tr b−2n(ρ 2 , ω )ρ −1 d ρ d ω .

102


Proof. In the first place, due to the (simple) pole at s = 0 of Γ (s), we have from (7.5), (7.15) and (7.16) that

ζA (0) = Res(Γ ζA , 0) = c−2n,n + f0 (0) + fR (0). Now, recalling that (see Remark 6.1.5) cA t=0 = χ ICN , we have R(0) = (e−tA − UA(t))

t=0

= (1 − χ )w (x, D) ICN .

Hence fR (0) = Tr R(0) = (2π )−n

+∞ S2n−1

0

(1 − χ (ρω ))Tr(ICN )ρ 2n−1d ρ d ω .

But we also have f0 (0) = −(2π )−n

+∞ 0

S2n−1

(1 − χ (ρω ))Tr(ICN )ρ 2n−1 d ρ d ω ,

whence c−2n,n + f0 (0) + fR (0) = c−2n,n,

which proves the corollary.

Analogously, by looking into the proof of Theorem 7.2.1, one may also compute the values ζA (−k), k ∈ Z+ , by using

ζA (−k) = Res(Γ ζA , −k), (k)

and formulas (7.15) and (7.16). One needs to control fR (0). This may be achieved as follows. For any given ν ∈ Z+ , write again UA (t) =

ν

∑ B−2 j (t) + Rν +1(t).

j=0

Then (

d d + A)R(t) = −( + A)Rν +1(t), dt dt

whence R(t) = e−tA R(0) −

t 0

e−(t−t )A (

d + A)Rν +1(t )dt , dt

(7.17)


103

and for any given k ≥ 1 one may compute from (7.17) by successive differentiations a formula for (k)

fR (0) =

dk Tr R . t=0 dt k

Remark 7.2.7. To obtain explicitly the terms {a2 , b−2 j }() in order to compute the coefficients c−2 j,n , it is covenient to make use of the following formula (see, e.g., Rossmann’s book [66, p. 15]): for α ∈ Zn+ with |α | = 1,

∂Xα ea2 (X) = ea2 (X)

1 − exp(−ad a2 (X )) α ∂X a2 (X ), ad a2 (X)

(7.18)

where ∞ 1 − exp(−ad a2 (X )) (−1)k := ∑ (ad a2 (X))k , ad a2 (X )b = [a2 (X ), b]. ad a2 (X ) (k + 1)! k=0

Recently, M. Hitrik and I. Polterovich have given in [24] (see also [23]), nice formulas for resolvent expansions and trace regularizations for Schrödinger operators. A nice problem is that of using these formulas to compute the coefficients c−2 j,n and therefore re-obtain the full result by Wakayama and Ichinose, Theorem 7.3.1 below. When the order of the system is μ , which then must be an even integer by Lemma 3.2.13, one can prove the analog of Theorem 7.2.1, that is, that ζA has a meromorphic continuation to the whole complex plane with at most simple poles at s = 2(n − j)/ μ , j ∈ Z+ . However, in this case the analog of Proposition 7.2.2 is much more involved, and we leave the (non trivial) details to the interested reader. We just give the following formula for the terms rμ −2ν , ν ≥ 1: rμ −2ν =

ν −1

∑

j=max(ν − μ2 ,0)

+

μ 2 −1

∑

k=0

aμ −2(ν − j)b−2 j

−i ν −k− j 1 {aμ −2k , b−2 j }(ν −k− j) . (ν − k − j)! 2 j=max(ν − μ +k,0) ν −1−k

∑

(7.19) Notice that ζA is still continuous at s = 0, and that the zero of the factor 1/Γ (s) at s = −2k now cancels the pole at s = 2(n − j)/ μ of Γ ζA only when j is such that 2n − 2 j = −2 μ k, i.e. j = n + μ k. We hence expect, for μ > 2, a sequence of simple poles also for Re s < 0. Specializing to the case A = Qw (α ,β ) (x, D), we have the following corollary of Theorem 7.2.1.

104


Corollary 7.2.8. Consider an elliptic NCHO Qw (α ,β ) (x, D) (α , β > 0 and αβ > 1). Then the associated spectral zeta function ζQ can be meromorphically continued to the whole complex place C with only one simple pole at s = 1. We close this section by providing the proof of Proposition 7.2.2. Proof (of Proposition 7.2.2). We won’t be writing the dependence on ω , and we () will write b−2 j for a generic ∂Xα b−2 j with |α | = . (2)

(2)

(1)

(1)

Denote by E(a2 , b−2 j ), resp. E(a2 , b−2 j ), a generic expression obtained by taking the (matrix) product of derivatives of order 2, resp. order 1, of a2 with derivatives of order 2, resp. order 1, of b−2 j . Hence for all j ≥ 0 we may generically write (1)

(1)

(2)

(2)

{a2 , b−2 j } = E(a2 , b−2 j ), and {a2 , b−2 j }(2) = E(a2 , b−2 j ). Notice, therefore, that in {a2, b−2 j }(2) we have a constant-coefficient matrix (given by partial derivatives of order 2 of a2 ) times partial derivatives of order 2 of b−2 j . We proceed by induction. We start with the case j = 0. In this case b0 is the solution of ⎧ ⎨ ∂t b0 + a2 b0 = 0, (7.20) ⎩ b0 t=0 = I, whence b0 (t) = O(1) as t → 0 + . () Next, by induction on we show that b0 has the claimed property. For = 1 we take a 1st-order partial derivative with respect to X of (7.20) and have ⎧ (1) (1) (1) ⎪ ⎨ ∂t b0 + a2b0 = −a2 b0 , (7.21) ⎪ ⎩ (1) b0 t=0 = 0, whence (1)

b0 (t) = −

t 0

(1)

e−(t−t )a2 a2 b0 (t )dt = O(t), t → 0 + .

(7.22)

For = 2 we take a 1st-order partial derivative with respect to X of (7.22) (of course, we may equivalently use again (7.20) by taking an extra derivative of (7.21)) and have (2)

b0 (t) = − −

t 0

t 0

(1)

(e−(t−t )a2 )(1) a2 b0 (t )dt −

t 0

(2)

e−(t−t )a2 a2 b0 (t )dt

(1) (1)

e−(t−t )a2 a2 b0 (t )dt

= O(t 2 ) + O(t) + O(t 2) = O(t), t → 0 + . (2k−1+)

Next, suppose b0

(t) = O(t k ) as t → 0+, for = 0, 1 and all k ≥ 0 We want to

(2k+1+) (t) = O(t k+1 ), as t prove that b0

→ 0+, for = 0, 1. Using (7.20) and taking a


105

()

2k + 1-st partial derivative with respect to X we get (recall that a2 = 0 for all ≥ 3, as a2 is a polynomial in X of degree 2) ⎧ (2k+1) (2k+1) (1) (2k) (2) (2k−1) ⎪ + a2 b0 = −a2 b0 − a2 b0 = O(t k ) + O(t k ) = O(t k ), ⎨ ∂t b0 ⎪ ⎩

(2k+1) = t=0

b0

(2k+1)

whence b0

(2k+2)

b0

0,

(t) = O(t k+1 ) as t → 0 + . Then, as before,

(t) = − −

t 0

t

= O(t

(1) (2k)

(2) (2k−1)

(∂X (e−(t−t )a2 )) (a2 b0 (t ) + a2 b0

(t ))dt

(1) (2k) (2) (2k−1) e−(t−t )a2 ∂X a2 b0 (t ) + a2 b0 (t ) dt

0 k+2

) + O(t k+1 ) = O(t k+1 ), t → 0 + .

Hence the result is proved for b0 . So suppose, by induction, that (2k+1+)

b−2 j (t) = O(t j ), b−2 j

(t) = O(t j+k+1 ), t → 0+, = 0, 1.

We prove the result for b−2( j+1). To do so, we have to examine r−2 j (see (7.7)). In the first place we have from (7.6) r−2 j (t) = O(t j ) + O(t j−1+1) + O(t j+1) = O(t j ), t → 0 + . Consider next, keeping into account the fact that a0 is a constant matrix, (2k+1)

r−2 j

(2k+1)

= a0 b−2 j

(2)

(2+2k+1)

(2)

(2k+1)

+ E(a2 , b−2( j−1) ) + E(a2 , b−2 j

(1)

(2k+2)

) + E(a2 , b−2 j

)

= O(t j+k+1 ) + O(t j−1+k+2) + O(t j+k+1) + O(t j+k+1) = O(t j+k+1 ), t → 0 + . Taking an extra derivative, one immediately sees also that (2k+2)

r−2 j

= O(t j+k+1 ), t → 0 + .

Hence (2k+1+)

r−2 j

= O(t j+k+1 ), t → 0+, = 0, 1, ∀k ≥ −1

(7.23)

(when k = −1 we take = 1). Since ⎧ ⎨ ∂t b−2( j+1) + a2b−2( j+1) = −r−2 j , ⎩

b−2( j+1)t=0 = 0,

(7.24)

106


we obtain b−2( j+1)(t) = O(t j+1 ) as t → 0 + . As before, taking one partial derivative with respect to X yields ⎧ (1) (1) (1) (1) (1) j+1 j+1 ⎪ ⎨ ∂t b−2( j+1) + a2 b−2( j+1) = −E(a2 , b−2( j+1)) − r−2 j = O(t ) + O(t ), ⎪ ⎩

(1) b−2( j+1)t=0 = 0, (1)

whence it follows that b−2( j+1)(t) = O(t j+2 ), and, taking an extra derivative, also that, as t → 0+, (2)

b−2( j+1)(t) = −∂X

t 0

(1) (1) (1) e−(t−t )a2 E(a2 , b−2( j+1)) + r−2 j dt = O(t j+2 ).

Supposing then by induction the estimates up to order 2k − 1, using ⎧ (2k+1) (2k+1) (1) (2k) (2) (2k−1) (2k+1) ∂t b−2( j+1) + a2b−2( j+1) = −E(a2 , b−2( j+1) ) − E(a2 , b−2( j+1)) − r−2 j ⎪ ⎪ ⎪ ⎪ ⎨ = O(t j+1+k ) + O(t j+1+k ) + O(t j+1+k ), ⎪ ⎪ ⎪ ⎪ ⎩ (2k+1) b−2( j+1) t=0 = 0, (2k+1)

we get b−2( j+1)(t) = O(t j+1+k+1 ), as t → 0+, and using (2k+2)

b−2( j+1)(t) t (1) (2k) (2) (2k−1) (2k+1) dt , e−(t−t )a2 E(a2 , b−2( j+1)) + E(a2 , b−2( j+1)) + r−2 j = −∂X 0

also that (2k+2)

b−2( j+1)(t) = O(t j+1+k+1 ), t → 0+, which proves the proposition.

7.3 The Ichinose–Wakayama Theorem In this section we want to outline the proof of Ichinose-Wakayama’s theorem on the spectral zeta function ζQ of our elliptic NCHO Qw (α ,β ) (x, D) (for α , β > 0 and αβ > 1). It is worth noting that, by Corollary 4.1 of Parmeggiani-Wakayama [59, p. 555], when α = β > 1 one has that Qw (x, D) is unitarily equivalent to the (α ,α ) scalar harmonic oscillator ∂ 2 x2

' 1 0 x 2 , α −1 − + 2 2 0 1

7.3 The Ichinose–Wakayama Theorem

107

√ which has eigenvalues α 2 − 1( j + 1/2), j ≥ 0, all of multiplicity 2. We therefore have that when α = β > 1

ζQ (s) = 2

2s − 1 ζ (s), (α 2 − 1)s/2

(7.25)

where ζ is the Riemann zeta function. Hence, when α = β > 1, ζQ has a meromorphic extension to the whole complex plane C with one single simple pole at s = 1 and it is zero at the non-positive integers −2k, k ∈ Z+ . Notice that the negative integers −2k, k ∈ N, are the so-called “trivial zeros” of ζ . We may hence say that ζQ has trivial zeros at the non-positive even integers. The fact that all this holds true also when α = β is a remarkable and elegant theorem due to Ichinose and Wakayama [31]. Theorem 7.3.1. There exist constants CQ, j , j ∈ N, such that ζQ (s) is represented, for every integer ν ∈ N, as

ζQ (s) =

$ ν CQ, j 1 α +β 1 # ' +∑ + HQ,ν (s) , Γ (s) αβ (αβ − 1) s − 1 j=1 s + 2 j − 1

(7.26)

where Γ (s) is the Euler gamma function, and HQ,ν is holomorphic in Re s > −2ν . Consequently, the spectral zeta function ζQ (s) is meromorphic in the whole complex plane C with a simple pole at s = 1, and has zeros (the so-called “trivial zeros”) for 0, −2, −4, . . . (the non-positive even integers 2Z− ). The non-positive even integers 2Z− , the “trivial zeros” of ζQ (s), are due to the presence of the function 1/Γ (s), which is holomorphic in C with zeros at the non-positive integers Z− = {0, −1, −2, . . .}, in the expression of ζQ (s). As already observed above, the negative trivial zeros are the same as those of Riemann’s ζ . As already mentioned in Chapter 6, Section 6.4, the use of Ikehara’s theorem gives the following corollary of Theorem 7.3.1 (Weyl’s law of the asymptotics of large eigenvalues; see Theorem 6.3.1, Corollary 6.3.2 and Remark 6.3.3; see also Theorem 5.2 in Parmeggiani [52]).

Corollary 7.3.2. N(λ ) =

α +β 1∼ ' λ , as λ → +∞. αβ (αβ − 1) λ j 1, (7.27) Γ (s) 0 so that

ζQ (s) = Tr Q Write

ζQ (s) =

1 Γ (s)

−s

1 = Γ (s)

1 0

+

+∞ 0

t s−1 Tr e−tQ dt, Re s > 1.

(7.28)

+∞

t s−1 Tr e−tQ dt =: Z0 (s) + Z∞ (s).

1

One then readily sees (as before) that Z∞ (s) is holomorphic in C, so that the crux of the matter lies, as before, in the study of the function Z0 (s), and for this, Ichinose and Wakayama need an asymptotic expansion for t > 0 of the kind Tr e−tQ ∼ c−1t −1 + c0 + c1t + c2t 2 + . . . , 0 < t → 0 + .

α 0 Put A = , and define the pseudodifferential operator P1 (t) (a quantization 0 β “to the right”) by P1 (t, x, D)u(x) = (2π )−1

ei(x−y)ξ e−t

A(y2 +ξ 2 )/2+iJyξ

u(y)dyd ξ , u ∈ S (R; C2 ).

Next put R2 (t) = e−tQ − P1(t). Ichinose and Wakayama then prove that 1 α +β 1 1 + Z0 (s) = ' αβ (αβ − 1) Γ (s) s − 1 Γ (s)

1 0

t s−1 Tr R2 (t) dt,

(7.29)

whence it follows that it suffices to treat the second term on the right-hand side of (7.29). We now use the heat-equation: 0 = (∂t + Q)e−tQ = (∂t + Q)P1(t) + (∂t + Q)R2(t), so that (∂t + Q)R2 (t) = −(∂t + Q)P1 (t) =: S(t).

(7.30)

By the Duhamel principle we may solve (7.30) obtaining R2 (t) = whence

t 0

e−(t−t )Q S(t )dt =

t 0

P1 (t − t ) + R2 (t − t ) S(t )dt =: P2 (t) + R3(t),

e−tQ = P1 (t) + P2(t) + R3 (t),

7.3 The Ichinose–Wakayama Theorem

109

and by repeating the process e−tQ = P1 (t) + P2(t) + . . . + Pν (t) + Rν +1(t). One has therefore to study Tr Pk (t) and Tr Rν +1 (t) for 0 < t → 0 + . One has that for any given (sufficiently small) ε > 0 there exist positive constants Cε (independent of t) and C (independent of t and ν ) such that |Tr R2 (t)| ≤ Cε t −ε , and |Tr Rν +1 (t)| ≤ Cν

Γ (1/2)ν ν /2 t , ν ≥ 1. Γ (1 + ν /2)

(7.31)

This yields that the Mellin transform of Tr R2 (t) is holomorphic in Re s > ε , and hence in Re s > 0, for ε is arbitrary, and that the Mellin tranform of TrRν +1 (t) (ν ≥ 1) is holomorphic in the region Re s > −ν /2. As for the Pk (t) terms, Ichinose and Wakayama then prove, by highly non-trivial computations, that Tr Pk (t) ∼

∑ ck, j t j ,

for k = 2, 3, . . . , and for t → 0+,

(7.32)

j≥0

with ck, j = 0 for 2 ≤ j < k − 2 and j ∈ 2N. It follows that

α +β Tr P1 (t) = ' t −1 , αβ (αβ − 1) and that Tr P2 (t) Tr P3 (t) Tr P4 (t) Tr P5 (t) Tr P6 (t) ... Tr Rν +1 (t)

∼ ∼ ∼ ∼ ∼

c2,1t + c2,3t 3 + c2,5t 5 + c2,7t 7 + c2,9t 9 + . . . c3,1t + c3,3t 3 + c3,5t 5 + c3,7t 7 + c3,9t 9 + . . . c4,3t 3 + c4,5t 5 + c4,7t 7 + c4,9t 9 + . . . c5,3t 3 + c5,5t 5 + c5,7t 7 + c5,9t 9 + . . . c6,5t 5 + c6,7t 7 + c6,9t 9 + . . .

= O(t ν /2 ).

Via the Mellin transform we thus obtain

ζQ (s) =

1 α +β 1 1 1 ' + (c2,1 + c3,1) Γ (s) αβ (αβ − 1) s − 1 Γ (s) s + 1 1 1 +(c2,3 + c3,3 + c4,3 + c5,3) Γ (s) s + 3 1 1 1 + ...+ HQ,ν (s), +(c2,5 + c3,5 + c4,5 + c5,5 + c6,5) Γ (s) s + 5 Γ (s)


110


Problem 7.3.4. Compute the constants CQ, j appearing in the statement of Theorem 7.3.1 by using the parametrix approximation of e−tA , as in the proof of Theorem 7.2.1 (see also Corollary 7.2.5).

7.4 Notes In the outline of the proof Theorem 7.3.1 and its consequences we followed Ichinose and Wakayama’s paper [33]. We address the reader to the papers by Ichinose and Wakayama [32], by Kimoto and Wakayama [36] (see also the references therein), and by Ochiai [48] for important work on the special values of the spectral ζQ . This direction of problems is very deep, interesting and promising.

Chapter 8

Some Properties of the Eigenvalues of Qw (α ,β ) (x, D)

In this chapter we shall show some properties of the eigenvalues of a NCHO Q(α ,β ) (when αβ > 1 and α , β > 0), and in particular give some upper and lower bounds to the lowest eigenvalue. In the first place, we establish the following simple consequence of Theorem 3.1.12, which is useful when one is willing to study the behavior of the eigenvalues of the NCHO Q(α ,β ) with respect to α , β ∈ R. Lemma 8.0.1. We have that, for any given α , β ∈ R, the symbols Q(α ,β )(x, ξ ) and Q(β ,α ) (x, ξ ) have the same eigenvalues, for all (x, ξ ) ∈ R × R, and, at the operator w level, Qw (α ,β ) (x, D) is unitarily equivalent to Q(β ,α ) (x, D), that is, there exists a unitary transformation U : L2 (R; C2 ) −→ L2 (R; C2 ) such that w U ∗ Qw (α ,β ) (x, D)U = Q(β ,α ) (x, D).

Moreover, U is an automorphism of S (R; C2 ) and S (R; C2 ), so that the operators w Qw (α ,β ) (x, D) and Q(β ,α ) (x, D) are also equivalent in S and S .

0 −1 0 1 α 0 ,J= , and K = . Consider the system Proof. Let A = 1 0 1 0 0 β

KQ(α ,β ) (x, ξ )K = KAK p0 (x, ξ ) − iJxξ . β 0 , by considering the symplectic transformation κ : R × R 0 α (x, ξ ) −→ (ξ , −x) ∈ R × R, we have that

As KAK =

Q(β ,α ) (x, ξ ) = KQ(α ,β ) K ◦ κ (x, ξ ). It thus follows, since Tr Q(α ,β ) (x, ξ ) = Tr Q(α ,β ) (ξ , −x),


111

8 Some Properties of the Eigenvalues of Qw (α ,β ) (x, D)

112

and

detQ(α ,β ) (x, ξ ) = det Q(α ,β )(ξ , −x),

that Q(α ,β ) (x, ξ ) and Q(β ,α )(x, ξ ) have exactly the same eigenvalues, and that by Theorem 3.1.12, with Uκ the metaplectic operator (the normalized Fourier transform) associated with κ , −1 w w Qw (β ,α ) (x, D) = Uκ KQ(α ,β ) (x, D)KUκ = (KQ(α ,β ) K ◦ κ ) (x, D)

are unitarily equivalent, for U := KUκ : L2 (R; C2 ) −→ L2 (R; C2 ) is unitary. Since clearly U is also an automorphism of S (R; C2 ) and S (R; C2 ), this concludes the proof of the lemma. From now on, unless otherwise specified, we shall assume α , β > 0 and αβ > 1, so that Q = Qw (α ,β ) is elliptic and positive (as a differential operator), and hence has a discrete spectrum contained in R+ .

8.1 The Ichinose and Wakayama Bounds In [33] Ichinose and Wakayama proved the following theorem about estimates on upper and lower bounds for the eigenvalues of Q. Theorem 8.1.1. Let λ2 j−1 , λ2 j , j = 1, 2, . . . , be the (2 j − 1)-st and 2 j-th eigenvalues of Q. Then ( 1 ( j − ) min{α , β } 2

αβ − 1 1 ≤ λ2 j−1 ≤ λ2 j ≤ ( j − ) max{α , β } αβ 2

(

αβ − 1 . αβ (8.1)

Proof. Put K(t) = K(t, x, y) = e−tQ (x, y), t > 0. Then (∂t + Q)K(t) = 0, K t=0 = δ (x − y). Now, following Parmeggiani-Wakayama [58, 59], define −∂x2 + x2 1 ) + γ J(x∂x + ) 2 2 1 1 − γ2 2 x , = (−i∂x + iγ Jx)2 + 2 2

Q := A−1/2 QA−1/2 = I(

' where γ := 1/ αβ . One then has that K (t) = e−tQ (x, y) solves (∂t + Q )K (t) = 0, K t=0 = δ (x − y),

8.1 The Ichinose and Wakayama Bounds

113

and K (t, x, y) = (1 − γ 2)1/4 eγ (x

2 −y2 )J/2

where

' ' ' pH ( 1 − γ 2 t, 1 − γ 2 x, 1 − γ 2 y),

pH (t, x, y) = e−tH (x, y)

is the heat-kernel of the harmonic oscillator H = pw 0 . This follows from [58] (see also [59]), for Q is unitarily equivalent to 1 1 Q0 := γ I −∂x2 + ( 2 − 1)x2 , 2 γ so that

" 1 1 − γ 2( j − ); j ≥ 1 , 2 with each eigenvalue of multiplicity 2. Therefore, it also follows that Spec(Q ) =

ζQ (s) = Tr Q

−s

!'

= (1 − γ 2)−s/2 ζQ0 (s) = 2

2s − 1 ζ (s). (1 − γ 2)s/2

Now, since each eigenvalue λ j of Q has multiplicity 2, we have that the (2 j − 1)-st eigenvalue λ2 j−1 and the 2 j-th eigenvalue λ2 j coincide and ( ' αβ − 1 1 1 λ2 j−1 = λ2 j = ( j − ) 1 − γ 2 = ( j − ) . 2 2 αβ We now use the fact that

Q = A1/2 Q A1/2 ,

and the Minimax Principle (4.1). For n = 2 j − 1 or n = 2 j, with [u1 , . . . , un−1 ]⊥ := Span(u1 , . . . , un−1 )⊥ ∩ D(Q), D(Q) being the domain of Q, we thus have that %

λn = =

sup

u1 ,...,un−1 lin. ind.

sup


%

(Qu, u) inf 2 0=u∈[u1 ,...,un−1 ]⊥ ||u||0

&

(Q A1/2 u, A1/2 u) inf ||u||20 0=u∈[u1 ,...,un−1 ]⊥

&

(note that the vectors u1 , . . . , un−1 are linearly independent if and only if the vectors A−1/2 u1 , . . . , A−1/2 un−1 are linearly independent, for A1/2 : L2 (R; C2 ) −→ L2 (R; C2 ) is an isomorphism, i.e. bounded with bounded inverse, and that u ∈ D(Q) or u ∈ D(Q ) if and only if A−1/2 u does)


114

%

(Q A1/2 u, A1/2 u) inf = sup ||u||20 u1 ,...,un−1 lin. ind. 0=u∈[A−1/2 u1 ,...,A−1/2 un−1 ]⊥ % & (Q A1/2 u, A1/2 u) inf = sup ||u||20 u1 ,...,un−1 lin. ind. 0=A1/2 u∈[u1 ,...,un−1 ]⊥

&

(putting v = A1/2 u) % = =

sup


%

sup


(Q v, v) inf 2 0=v∈[u1 ,...,un−1 ]⊥ ||A−1/2 v||0

&

(Q v, v) ||v||20 inf 2 2 0=v∈[u1 ,...,un−1 ]⊥ ||v||0 ||A−1/2 v||0

& .

Put m = min{α , β } and M = max{α , β }. Then M −1 ||v||20 ≤ (A−1 v, v) = ||A−1/2 v||20 ≤ m−1 ||v||20 , ∀v ∈ L2 (R; C2 ), and m≤

||v||20 ≤ M, ∀v = 0. ||A−1/2 v||20

We thus have %

λ2 j−1 ≤ λ2 j ≤ M

sup

u1 ,...,u2 j−1 lin.ind.

(Q v, v) inf 2 0=v∈[u1 ,...,u2 j−1 ]⊥ ||v||0

&

= M λ2 j = M λ2 j−1 , and %

λ2 j ≥ λ2 j−1 ≥ m

sup

u1 ,...,u2 j−2 lin. ind.

(Q v, v) inf 2 0=v∈[u1 ,...,u2 j−2 ]⊥ ||v||0

& = mλ2 j−1 ,

which concludes the proof. The bounds in (8.1) make up an interval #

(

1 I j := ( j − ) min{α , β } 2

αβ − 1 1 , ( j − ) max{α , β } αβ 2

When j < k we have I j ∩ Ik = 0/ ⇐⇒

2k − 1 max{α , β } > . 2j−1 min{α , β }

(

αβ − 1 $ . αβ

8.2 A Better Upper-Bound for the Lowest Eigenvalue

115

It follows that the eigenvalue λ2 j−1 or λ2 j has a multiplicity less than or equal to 2 if j
0. Then, as is well-known, √ √ π −( αβ −1−i)x2 e dx = , ω whence

cos(x2 )|ϕ0 (x)|2 dx =

(αβ − 1)1/4 √ 1 (αβ − 1)1/4 √ ' π Re = Re ω , ω π αβ

and ||A−1/2 f ||20 =

1 (αβ − 1)1/4 α + β − (α − β ) ' Re ω > 0. αβ αβ

We therefore have that √ 1 λ1 ≤ ' αβ =

αβ −1 2

√

αβ −1 2 (αβ −1)1/4 − (α − β ) √ Re ω αβ

+

α +β ' ' αβ αβ − 1

1 αβ

α + β − (α − β ) (αβ√−1)

1/4

αβ

Re ω

.

w Since, by Lemma 8.0.1, Qw (α ,β ) (x, D) and Q(β ,α ) (x, D) are unitarily equivalent, they have the same lowest eigenvalue. Thus (with the same ω )

λ1 ≤ min

⎧ ⎪ ⎨

' ' αβ αβ − 1

' ' αβ αβ − 1

αβ

αβ

⎫ ⎪ ⎬

, , 1/4 1/4 ⎪ ⎩ α + β − (α − β ) (αβ√−1) Re ω α + β − (β − α ) (αβ√−1) Re ω ⎪ ⎭

that is

λ1 ≤

' ' αβ αβ − 1

α + β + |α − β | (αβ√−1)

1/4

αβ

Re ω

.

This shows that inequality (8.2) holds and concludes the proof of the theorem.

120


8.3 Notes The problem of determining the lowest eigenvalue, and its multiplicity, of a generic elliptic positive NCHO is an open, and important, problem, which should be explored in depth.

Chapter 9

Some Tools from the Semiclassical Calculus

In this chapter we shall describe some tools from Semiclassical Analysis, that will be used in the subsequent chapters to obtain localization properties of the large eigenvalues of an elliptic positive NCHO Qw (α ,β ) (x, D), and more generally of an elliptic positive NCHO (with no sub-principal term), relating them to properties of the periods of the bicharacteristics of the (principal) symbol.

9.1 The Semiclassical Calculus We recall here some basic properties of semiclassical the Weyl-quantization and the semiclassical calculus. (In general, for Semiclassical Analysis, we address the reader to Dimassi-Sjöstrand [7], Evans-Zworski [15], Ivrii [34], Martinez [40], Robert [65], Shubin [67] and Voros [71].) Definition 9.1.1 (Semiclassical symbols). We shall say that a function a(X ; h) = a(·; h) ∈ C∞ (R2n X ), possibly depending on a parameter h ∈ (0, h0 ], h0 ∈ (0, 1], belongs to the symbol class Sδk (mμ , g), k, μ ∈ R and δ ∈ [0, 1/2], if for all α ∈ Z2n + there exists Cα > 0 such that |∂Xα a(X ; h)| ≤ Cα m(X)μ −|α | h−k−|α |δ , ∀X ∈ R2n , ∀h ∈ (0, h0 ].

(9.1)

As before, Sδk (mμ , g; MN ) = Sδk (mμ , g) ⊗ MN , and Sδk (mμ , g; V) = Sδk (mμ , g) ⊗ V, for any given finite-dimensional complex vector space V. Given a ∈ Sδk (mμ , g; V), we shall consider its h-Weyl quantization aw (x, hD)u(x) = (2π h)−n

eih

−1 x−y,ξ

a(

x+y , ξ ; h)u(y)dyd ξ , u ∈ S . 2

It is easy to see that aw (x, hD) is in fact the Weyl-quantization of the symbol a(x, hξ ; h). We shall also need more general classes, defined as follows (see DimassiSjöstrand [7]). We say that a smooth function a(·; h) ∈ C∞ (R2n X ), possibly depending


121

122

9 Some Tools from the Semiclassical Calculus

on a parameter h ∈ (0, h0 ], h0 ∈ (0, 1], belongs to the symbol class Sδk (mμ ), k, μ ∈ R and δ ∈ [0, 1/2], if for all α ∈ Zn+ there exists Cα > 0 such that |∂Xα a(X ; h)| ≤ Cα m(X)μ h−k−|α |δ , ∀X ∈ R2n , ∀h ∈ (0, h0 ].

(9.2)

Notice that, compared to (9.1), no further decay in m is obtained by taking derivatives of the symbol. The h-Weyl quantization of a symbol a ∈ Sδk (mμ ) is defined as before. The vector and matrix-valued symbol classes are defined as before. Notice that Sδk (mμ ) = Sδk (mμ , |dX|2 ), where |dX|2 is the Euclidean metric in R2n X , and that S(mμ , g) ⊂ S00 (mμ , g) ⊂ Sδk (mμ , g) ⊂ Sδk (mμ ).

(9.3)

Remark 9.1.2. Let E > 0. Define the L2 -isometry (also automorphism of S and S ), √ UE : u(x) −→ E −n/4u(x/ E). (9.4) Then, given any symbol a ∈ S0k (mμ ; MN ) one has from Theorem 3.1.12

In particular

√ √ h ˜ UE−1 aw (x, hD)UE = aw ( E x, E hD), where h˜ = . E

(9.5)

√ √ Uh−1 aw (x, hD)Uh = aw ( h x, h D).

(9.6)

Notice that since Uh corresponds to the symplectic transformation

κh : (x, ξ ) −→ (h1/2 x, h−1/2 ξ ), by using h1/2 m(X ) ≤ m(h1/2 X ) ≤ m(X), we have that if a ∈ S(mμ , g), then √ √ − min{ μ ,0}/2 (mμ , g). (a ◦ κh)(·, h·) = a( h ·, h ·) ∈ S0 By Proposition 3.2.15 we have the following result, that we shall frequently use k when dealing with “classical semiclassical symbols” in the class S0,cl (mμ , g) (see Definition 9.1.9 below). Proposition 9.1.3. Let μ ∈ R. Let a j ∈ S(mμ −2 j , g), j ∈ Z+ (in particular, the a j are independent of h). Then there exists a ∈ S00 (mμ , g) such that a∼

∑ h ja j,

j≥0

9.1 The Semiclassical Calculus

123

that is, for all r ∈ Z+ we have r

a − ∑ h j a j ∈ S0

−(r+1)

(mμ −2(r+1) , g).

j=0

If another symbol a has the same property, then

a − a ∈ S−∞ (m−∞ , g) :=

S0−k (mμ −2k , g).

(9.7)

k∈Z+

Proof. One takes an excision function χ ∈ C∞ (Rt ; [0, 1]), such that χ (t) = 0 for |t| ≤ 2 and χ (t) = 1 for |t| ≥ 4, and defines ∞

a(X; h) :=

∑ h jχ

j=0

m(X) a j (X ), hR j

where {R j } j∈Z+ ⊂ [1, +∞) is an increasing sequence with R j +∞ as j → +∞ sufficiently fast. Following the steps of the proof of Proposition 3.2.15, using the fact that h ∈ (0, 1] and that on the support of 1 − χ (m(X )/hR j ) we have |X | ≤ 4R j and 1 ≤ h−1 ≤ 4R j (so that we may divide and multiply by hr , for any given r ∈ N), one sees that the sequence R j can indeed be so chosen that a(X ; h) has the required properties. Using a Borel-summation argument one also has the following proposition (see Evans-Zworski [15]). Proposition 9.1.4. Let k j −∞, k j > k j+1 , j ∈ Z+ , be a monotone decreasing k

sequence of real numbers. Let a j ∈ Sδ j (mμ ). Then there exists a ∈ Sδ0 (mμ ) such that a ∼ ∑ j≥0 a j , that is, for all r ∈ Z+ r

k

a − ∑ a j ∈ Sδr+1 (mμ ). k

j=0

If another symbol a has the same property, then a − a ∈ S−∞ (mμ ) :=

Sδk (mμ ).

k∈R

Proof. One chooses χ ∈ C∞ ([0, +∞); R) such that 0 ≤ χ ≤ 1, χ [0,1] = 1, χ [2,+∞) ≡ 0, and then defines a(X) :=

∑ χ (λ j h)a j (X),

j≥0

X ∈ R2n ,

(9.8)

124


where the sequence 1 ≤ λ1 ≤ . . . ≤ λ j ≤ λ j+1 ≤ . . . → +∞ must be picked. The choice of the λ j is therefore made as follows: for each multiindex α with |α | ≤ j, we have |∂Xα χ (λ j h)a j (X ) | = |χ (λ j h)∂Xα a j (X)| ≤ C j,α h−k j −δ |α | χ (λ j h)m(X )μ λ jh = C j,α h−k j −δ |α | χ (λ j h)m(X )μ λ jh (since λ j h ≤ 2 in the support of χ ) 1 ≤ 2C j,α h−k j −1−δ |α | m(X )μ λj ≤ h−k j −1−δ |α | 2− j m(X)μ , ∀X ∈ R2n , if λ j is picked sufficiently large. Since we can accomplish this for all j and all α with |α | ≤ j, and trivially make it possible to have λ j+1 ≥ λ j , the sequence {λ j } j is therefore determined. One then concludes in a way similar to that used in the proof of Proposition 3.2.15. We leave it as an exercise for the reader to fill in the details of the proofs of Proposition 9.1.3 and of Proposition 9.1.4. From Evans-Zworski [15] (see also Dimassi-Sjöstrand [7]) we have the following important result. Theorem 9.1.5. We have that for a ∈ Sδ0 (mμ ; MN ), the operator aw (x, hD) as a linear map aw (x, hD) : S (Rn ; CN ) −→ S (Rn ; CN ) and as a linear map aw (x, hD) : S (Rn ; CN ) −→ S (Rn ; CN ) is continuous. In particular, by virtue of (9.3), this is the case also for a ∈ Sδ0 (mμ , g; MN ). As regards the L2 -continuity we have the following theorem. Theorem 9.1.6. Let a ∈ Sδ0 (1; MN ), 0 ≤ δ ≤ 1/2. Then aw (x, hD) : L2 (Rn ; CN ) −→ L2 (Rn ; CN ) is bounded and there is a constant C > 0, independent of h, such that ||aw (x, hD)||L2 →L2 ≤ C, ∀h ∈ (0, 1]. As regards the composition one has the following theorems. For the classes Sδk (mμ ; MN ) we have the following result (see Dimassi-Sjöstrand [7] and Evans-Zworski [15]).


125

Theorem 9.1.7. Given a ∈ Sδ0 (mμ1 ; MN ) and b ∈ Sδ0 (mμ2 ; MN ), one has aw (x, hD)bw (x, hD) = (ah b)w (x, hD), where

ah b = eihσ (DX ;DY )/2 (a(X)b(Y ))X=Y ∈ Sδ0 (mμ1 +μ2 ; MN ).

Here, recall, σ (DX ; DY ) = σ (Dx , Dξ ; Dy , Dη ). Furthermore, when δ ∈ [0, 1/2) one has k ∞ 1 ih ah b ∼ ∑ σ (DX ; DY ) a(X )b(Y )X=Y . (9.9) k=0 k! 2 0 (1; M ) with When δ = 1/2 one has (see [15]) that if a, b ∈ S1/2 N

supp a ⊂ K, and dist(supp a, supp b) ≥ γ > 0, where the compact K and the constant γ are independent of h, then ||aw (x, hD)bw (x, hD)||L2 →L2 = O(h∞ ).

(9.10)

Recall that f (h) = O(hN0 ) for some N0 ∈ Z+ if there exists CN0 > 0 such that | f (h)| ≤ CN0 hN0 . We say that f (h) = O(h∞ ) if for any given N0 ∈ Z+ one has f (h) = O(hN0 ). For the classes Sδk (mμ , g; MN ) we have the following result (see Shubin [67, p. 245]). Theorem 9.1.8. Let δ ∈ [0, 1/2). For any given symbols a ∈ Sδk1 (mμ1 , g; MN ) and b ∈ Sδk2 (mμ2 , g; MN ) we have aw (x, hD)bw (x, hD) = (ah b)w (x, hD) where ah b = eihσ (DX ;DY )/2 (a(X)b(Y ))X=Y ∈ Sδk1 +k2 (mμ1 +μ2 , g; MN ), and for all N0 ∈ Z+ (ah b) =

k 1 ih σ (D ; D ) a(X)b(Y )X=Y +hN0 +1 rN0 +1 , X Y ∑ k! 2 k=0 N0

k +k2 +2(N0 +1)δ

where rN0 +1 ∈ Sδ1

(mμ1 +μ2 −2(N0 +1) , g; MN ).

We next define the classical symbols in the semiclassical setting.

(9.11)

126


Definition 9.1.9 (Classical semiclassical symbols). 1. We shall say that a semiclassical symbol a ∈ S0k (mμ ; MN ) is classical and write k (m μ ; M ) if a ∈ Scl N a(X ; h) ∼ h−k ∑ h j a j (X) in S0k (mμ ; MN ), j≥0

where the a j ∈ S00 (mμ ; MN ) are independent of h, j ≥ 0. 2. We shall say that a semiclassical symbol a ∈ S0k (mμ , g; MN ) is classical and write k (m μ , g; M ) if there exists a sequence {a a ∈ S0,cl N μ −2 j } j≥0 of symbols a μ −2 j ∈ μ −2 j , g; M ), j ≥ 0, with the a S(m independent of h, such that for any given N μ −2 j N0 ∈ Z+ N0

a(X ; h) − h−k ∑ h j aμ −2 j (X) ∈ S0

k−(N0 +1)

(mμ −2(N0 +1) , g; MN ).

j=0

We shall write a(X; h) ∼ h−k ∑ h j aμ −2 j (X) in S0k (mμ , g; MN ). j≥0

For a classical semiclassical symbol a ∼ aμ + ha μ −2 + . . ., one calls a μ the principal symbol of a, and a μ −2 the subprincipal symbol of a. 0 (m μ , g; M ) is elliptic 3. We shall say that a classical semiclassical symbol a ∈ S0,cl N μ if its principal symbol a μ belongs to S(m , g; MN ) and aμ (X )−1 exists for all X ∈ R2n and belongs to S(m−μ , g; MN ). Equivalently, one requires that det a μ ∈ S(mN μ , g) be such that |det a μ (X)| m(X)N μ for all X ∈ R2n . 0 (m μ , g; M ) is 4. We shall say that a classical semiclassical symbol a = a∗ ∈ S0,cl N ∗ positive elliptic if its principal symbol aμ = aμ belongs to S(mμ , g; MN ) and there are 0 < c1 < c2 such that c1 m(X )μ |v|2CN ≤ a μ (X)v, v CN ≤ c2 m(X )μ |v|2CN , for all v ∈ CN and all X ∈ R2n . Remark 9.1.10. From Theorem 9.1.7 and Theorem 9.1.8, respectively, we have that k (m μ ; M ) and Sk (m μ , g; M ), respectively, are well-behaved also the classes Scl N N 0,cl under composition. We now wish to consider inverses in the semiclassical calculus. We need the following result, due to R. Beals, which characterizes pseudodifferential operators in the semiclassical setting (see Dimassi-Sjöstrand [7] or EvansZworski [15]; we give a statement in the scalar case for simplicity).


127

Theorem 9.1.11. Let A = Ah : S (Rn ) −→ S (Rn ) be a continuous linear operator, 0 < h ≤ 1. Then the following statements are equivalent: 1. A = aw (x, hD) for some a(x, ξ ; h) = a ∈ S00 (1); 2. For every N ∈ N and for every family 1 (x, ξ ), . . . , N (x, ξ ) of linear forms on R2n , the operator ad1 (x,hD) ◦ . . . ◦ adN (x,hD) Ah is continuous in L2 (Rn ), i.e. it belongs to L (L2 , L2 ), with norm O(hN ) in that space. (Recall that adA B = [A, B].) One may then prove the following proposition about “elliptic” elements in S00 (mμ ; MN ). Theorem 9.1.12. Let a(X ; h) = a ∈ S00 (mμ ; MN ) be elliptic, that is, by definition, a(X; h)−1 = a−1 ∈ S00 (m−μ ; MN ), for 0 < h ≤ h0 . Then, by possibly shrinking h0 , there exists a symbol b ∈ S00 (m−μ ; MN ) such that bw (x, hD)aw (x, hD) = I = aw (x, hD)bw (x, hD), 0 < h ≤ h0 , and, furthermore, bw (x, hD) possesses an asymptotic expansion, that is, b ∼ a−1 + h(a−1h r) + h2 (a−1 h rh r) + . . . , where r ∈ S00 (1; MN ) is such that (a−1 )w (x, hD)aw (x, hD) = I − hrw (x, hD). Proof. Let b˜ = a−1 ∈ S00 (m−μ ; MN ). Then b˜ w (x, hD)aw (x, hD) = I − hrLw (x, hD), rL ∈ S00 (1; MN ), and aw (x, hD)b˜ w (x, hD) = I − hrRw (x, hD), rR ∈ S00 (1; MN ). Then by Theorem 9.1.6 I − hrLw (x, hD) : L2 −→ L2 and I − hrRw (x, hD) : L2 −→ L2 are both invertible, provided h ∈ (0, h0 ], if h0 ∈ (0, 1] is taken sufficiently small. By the Beals Theorem 9.1.11 it follows that there exists cL , cR ∈ S00 (1; MN ) such that −1 −1 I − hrLw (x, hD) = cL (x, hD), I − hrRw (x, hD) = cR (x, hD), so that

and

−1 0 −μ ; MN ), b˜ w (x, hD) =: bw I − hrLw (x, hD) L (x, hD), bL ∈ S0 (m

−1 0 −μ =: bw ; MN ), b˜ w (x, hD) I − hrRw (x, hD) R (x, hD), bR ∈ S0 (m

128


whence we get the existence of a left-inverse bL (x, ξ ; h) = bL and of a right-inverse bR (x, ξ ; h) = bR , both belonging to the symbol class S00 (m−μ ; MN ), such that w w w bw L (x, hD)a (x, hD) = I = a (x, hD)bR (x, hD), 0 < h ≤ h0 ,

and, finally, w w w w bw L (x, hD) = bL (x, hD)a (x, hD)bR (x, hD) = bR (x, hD).

Put therefore b = bL = bR . In addition, we have that b possesses an asymptotic expansion. In fact, for any given N0 ∈ Z+ , let bw N0 (x, hD)

:= b˜ w (x, hD) I + hrRw (x, hD) + h2rRw (x, hD)2 + . . . + hN0 rRw (x, hD)N0 .

Then N0 +1 rR (x, hD)N0 +1 , aw (x, hD)bw N0 (x, hD) = I − h

and it follows that w w w bw N0 (x, hD) = b (x, hD)a (x, hD)bN0 (x, hD)

= bw (x, hD) − hN0 +1 bw (x, hD)rRw (x, hD)N0 +1 = bw (x, hD) + hN0 +1 rNw0 +1 (x, hD),

where we have put rNw0 +1 (x, hD) := −bw (x, hD)rRw (x, hD)N0 +1 . By Theorem 9.1.7 we have rN0 +1 ∈ S00 (m−μ ; MN ), and this concludes the proof. It will be also useful to have the following variation of Theorem 9.1.12 for 0 elliptic classical semiclassical symbols a ∈ S0,cl (mμ , g; MN ). The usual parametrix construction (using h as “parameter of homogeneity”) gives the following theorem. 0 Theorem 9.1.13. Let a ∈ S0,cl (mμ , g; MN ) be elliptic, that is, with a ∼ ∑ j≥0 h j aμ −2 j , −1 − μ let aμ ∈ S(m , g; MN ). Then there exists a classical semiclassical symbol 0 (m− μ , g; M ) such that b ∈ S0,cl N

bw (x, hD)aw (x, hD) = I + rLw (x, hD), aw (x, hD)bw (x, hD) = I + rRw (x, hD), where rL , rR ∈ S−∞ (m−∞ , g; MN ) (see (9.7)).

9.2 Decoupling a System

129

9.2 Decoupling a System In this section, using an adaptation taken from Parenti-Parmeggiani [49] of the classical decoupling argument of Taylor [69], we prove the following result. (See also Helffer-Sjöstrand [21].) 0 (m μ , g; M ). Theorem 9.2.1. Let μ > 0 and let a = a∗ ∼ ∑ j≥0 h j aμ −2 j ∈ S0,cl N ∗ ∗ Suppose there exists e0 ∈ S(1, g; MN ) such that e0 e0 = e0 e0 = I and

e∗0 aμ e0

λ1,μ 0 , = bμ = 0 λ2,μ

(9.12)

where λ j,μ = λ j,∗ μ ∈ S(mμ , g; MN j ), j = 1, 2, N1 + N2 = N, and dλ1 ,λ2 (X) m(X )μ , ∀X ∈ R2n ,

(9.13)

where, for each X ∈ R2n , ! " dλ1 ,λ2 (X ) = inf |μ1 − μ2|; μ1 ∈ Spec(λ1,μ (X)), μ2 ∈ Spec(λ2,μ (X )) .

(9.14)

0 (1, g; M ) with principal symbol e such that: Then there exists e ∈ S0,cl N 0

1. One has ew (x, hD)∗ ew (x, hD) − I, ew (x, hD)ew (x, hD)∗ − I ∈ S−∞ (m−∞ , g; MN ); 2. ew (x, hD)∗ aw (x, hD)ew (x, hD) − bw (x, hD) ∈ S−∞ (m−∞ , g; MN ), where the 0 (m μ , g; M ) is blockwise diagonal, with symbol b ∼ ∑ j≥0 h j bμ −2 j ∈ S0,cl N 0 b1,μ −2 j (X) , ∀X ∈ R2n , ∀ j ≥ 0, b μ −2 j (X ) = 0 b2,μ −2 j (X)

with blocks b j,μ of sizes N j , j = 1, 2, respectively, and with principal symbol

b μ (X ) =

λ1,μ (X) 0 , ∀X ∈ R2n . λ2,μ (X) 0

We shall call b an h∞ -(block)-diagonalization of a. Notice that b depends on a and e0 . Proof. We immediately observe that once ew (x, hD) has been found with the property that its principal symbol is e0 and ew (x, hD)ew (x, hD)∗ = I + rw (x, hD), with r ∈ S−∞ (m−∞ , g; MN ),

130


then by the ellipticity (using Theorem 9.1.13) we also get ew (x, hD)∗ ew (x, hD) = I + sw (x, hD), with s ∈ S−∞ (m−∞ , g; MN ). Hence it suffices to prove the existence of ew (x, hD) and b with the required properties. We show that for every integer k ∈ Z+ there exist e−2k ∈ S(m−2k , g; MN ),

(i) and

b j,μ −2k ∈ S(mμ −2k , g; MN j ), j = 1, 2,

(ii)

0 such that, with EN0 (X ) := ∑Nk=0 hk e−2k (X),

0 (m−2(N0 +1) , g; MN ), EN0 h EN∗0 = I + hN0 +1 S0,cl

and EN∗0 h ah EN0 =

N0

0 (mμ −2(N0 +1) , g; MN ), ∑ bμ −2k + hN0+1S0,cl

k=0

0 b1,μ −2k are in block-diagonal form. We shall then where the b μ −2k = 0 b2,μ −2k take e ∼ ∑k≥0 hk e−2k . Hence we proceed by induction. So, suppose we have already constructed symbols e0 , e−2 , . . . , e−2N0 , and bμ , b μ −2 , . . . , b μ −2N0 , independent of h, with the required properties. Put hence

SNw0 (x, hD) := ENw0 (x, hD)ENw0 (x, hD)∗ − I,

(9.15)

0 (m−2(N0 +1) , g; M ). We shall write, for short, E w , Sw and where SN0 ∈ hN0 +1 S0,cl N N0 N0 w w e−2k in place of EN0 (x, hD) etc. We look for a matrix-symbol e−2(N0 +1) ∈ S(m−2(N0 +1) , g; MN ) such that

∗ (ENw0 )∗ + hN0 +1 (ew − I = hN0 +2 rw (x, hD), ENw0 + hN0 +1 ew −2(N0 +1) −2(N0 +1) ) 0 (m−2(N0 +2) , g; M ), that is, we look for e where r ∈ S0,cl N −2(N0 +1) such that

w ∗ w w ∗ = hN0 +2 r˜w (x, hD), SNw0 + hN0 +1 ew (e ) + e (e ) 0 −2(N0 +1) −2(N0 +1) 0 0 (m−2(N0 +2) , g; M ). Using the composition formula (9.11) we thus look with r˜ ∈ S0,cl N at the coefficient of hN0 +1 and require that it be zero, obtaining the equation

s−2(N0 +1) + e0 (e−2(N0 +1) )∗ + e−2(N0 +1) e∗0 = 0.

(9.16)


131

Notice that s∗−2(N0 +1) = s−2(N0 +1) (this follows from (9.15)). Equation (9.16) has general solution 1 e−2(N0 +1) = − s−2(N0 +1) e0 + φ−2(N0 +1) , (9.17) 2 where φ−2(N0 +1) solves ∗ e0 φ−2(N + φ−2(N0 +1) e∗0 = 0, 0 +1)

which in turn gives that φ−2(N0 +1) has the form

φ−2(N0 +1) = α−2(N0 +1) e0 ,

(9.18)

where α−2(N0 +1) ∈ S(m−2(N0 +1) , g; MN ) and ∗ α−2(N + α−2(N0 +1) = 0. 0 +1)

We next perform a choice of α−2(N0 +1) for obtaining the blocks b j,μ −2(N0 +1) , j = 1, 2. Since on the one hand (ENw0 +1 )aw ENw0 +1 =

N0 +1

∑

k=0

N0 +2 w hk bw r1 , μ −2k + h

0 (mμ −2(N0 +2) , g; MN ), and on the other with r1 ∈ S0,cl

∗ w w (ENw0 +1 )aw ENw0 +1 = (ENw0 )∗ aw ENw0 + hN0 +1 (ew −2(N0 +1) ) a e0 ∗ w w N0 +2 w +(ew r2 0 ) a e−2(N0 +1) + h =: TNw0 + hN0 +2 r2w ,

(9.19)

0 (m μ −2(N0 +2) , g; M ), the conditions for the blocks b with r2 ∈ S0,cl N j, μ −2k are already satisfied for 0 ≤ k ≤ N0 , independently of e−2(N0 +1) . Let q μ −2(N0 +1) be the coefficient of hN0 +1 in EN∗0 h ah EN0 . Then the coefficient of hN0 +1 in TN0 is

qμ −2(N0 +1) + e∗−2(N0 +1) aμ e0 + e∗0 aμ e−2(N0 +1)

= qμ −2(N0 +1) + (e∗−2(N0 +1) e0 )e∗0 aμ e0 + e∗0 aμ e0 (e∗0 e−2(N0 +1) ).

(9.20)

Using (9.17) and (9.18), we write ⎧ 1 ∗ ⎪ ∗ ∗ ⎪ ⎨ e0 e−2(N0 +1) = − e0 s−2(N0 +1) e0 + e0 α−2(N0 +1) e0 =: τ + β , 2 ⎪ ⎪ ⎩ e∗ −2(N0 +1) e0 = τ − β ,

(9.21)

132


where 1 τ = − e∗0 s−2(N0 +1) e0 = τ ∗ , and β = e∗0 α−2(N0 +1) e0 = −β ∗ . 2 By (9.21), the term (9.20) goes over to qμ −2(N0 +1) + (e∗0 aμ e0 )τ + τ (e∗0 aμ e0 ) + (e∗0 aμ e0 )β − β (e∗0aμ e0 ).

(9.22)

We now show that β , hence in turn

α−2(N0 +1) = e0 β e∗0 ,

(9.23)

can be so chosen as to kill the off-diagonal terms in (9.22). In fact, upon writing qμ −2(N0 +1) + (e∗0 aμ e0 )τ + τ (e∗0 aμ e0 ) =

u1 γ , γ ∗ u2

where the u j = u∗j are N j × N j blocks, j = 1, 2, we look for β in the form

β=

0 δ , −δ ∗ 0

and, using (9.12), we are therefore led to the matrix equation

λ1,μ δ − δ λ2,μ = −γ .

(9.24)

By Lemma 9.2.2 below, hypothesis (9.13) yields that equation (9.24) has a unique smooth N1 × N2 matrix-valued solution δ ∈ S(m−2(N0 +1) , g; MatN1 ×N2 (C)). Since this fixes β , and hence α−2(N0 +1) , the terms b j,μ −2(N0 +1) are then the block-diagonal terms in (9.22). This concludes the inductive step and the proof of the theorem. Lemma 9.2.2. Let E = E ∗ ∈ S(mμ , g; MN1 ) and F = F ∗ ∈ S(mμ , g; MN2 ) be such that (recall (9.14)) dE,F (X) ≥ c0 m(X)μ , ∀X ∈ R2n . Then for each X ∈ R2n the map ⎧ ⎨ ΦE,F (X ) : MatN1 ×N2 (C) −→ MatN1 ×N2 (C), ⎩

(9.25)

ΦE,F (X )T = E(X)T − T F(X ),

is an isomorphism. Moreover, ||ΦE,F (X)−1 || ≤

C , ∀X ∈ R2n , m(X)μ

(9.26)


133

for a universal constant C > 0. Hence, if S ∈ S(mμ −2k , g; MatN1 ×N2 (C)), for some k ∈ Z+ , we have that X −→ T (X ) := ΦE,F (X)−1 S(X) ∈ S(m−2k , g; MatN1 ×N2 (C)).

(9.27)

Proof. For each fixed X ∈ R2n , let Spec(E(X )) = {e1 (X ), . . . , eνX (X )}. Hence e j (X ) = e j (X ) if j = j . Consider the contour in C

γ (X ) =

νX

γ j (X ), with γ j (X) ∩ γ j (X ) = 0/ for j = j ,

j=1

counter-clockwise oriented, where each γ j (X) = ∂ D j (X ) is a small circle that encloses only the eigenvalue e j (X ) of E(X ), 1 ≤ j ≤ νX , and Spec(F(X)) ⊂ C \

νX

D j (X ).

j=1

Then, on considering the equation E(X )T −T F(X) = S(X ) we have that the solution can be written as T = T (X ) = =

1 2π i νX

γ (X)

(ζ − E(X))−1 S(X )(ζ − F(X ))−1 d ζ −1 e j (X) − F(X ) ,

∑ PE, j (X)S(X)

(9.28)

j=1

where PE, j (X ) : CN1 → CN1 is the orthogonal projection associated with the eigenvalue e j (X ). This shows that for each fixed X the map ΦE,F (X ) is an isomorphism (for it is injective and linear). Moreover, using the fact that the norm of the resolvent of a normal operator equals the spectral radius, we also obtain from (9.28) that |T (X)| = |ΦE,F (X )−1 S(X) | ≤

C |S(X )|, ∀X ∈ R2n , m(X)μ

where C = N1 /c0 . This shows (9.26). Now, (9.26) gives that T (X ) is continuous and differentiable to all orders, for we have that for any given X1 , X2 ∈ R2n ΦE,F (X1 ) T (X1 ) − T (X2 ) = E(X2 ) − E(X1) T (X2 ) + T (X2 ) F(X1 ) − F(X2 ) + S(X1) − S(X2), (9.29) from which, since (9.26) yields |T (X2 )| |S(X2 )| for all X2 (recall that m(X ) ≥ 1 for all X), we obtain the continuity of T by considering the map ΦE,F (X1 )−1 , taking

134


the limit as X2 → X1 , and using the continuity of E, F and S. The differentiability claim follows from (9.29) by induction. It remains to control the growth of |∂Xα T (X)|. For any given α ∈ Z2n + , the matrix ∂Xα T is a solution to E ∂Xα T − ∂Xα T F + Sα = ∂Xα S, where, by the Leibniz rule, Sα :=

α α −β β β α −β ∂ E ∂ T − ∂ T ∂ F . ∑ β X X X X β ≤α

β =α

Hence

∂Xα T (X ) = ΦE,F (X)−1 ∂Xα S(X ) − Sα (X ) , β

so that assuming by induction on α that |∂X T | m−2k−|β | , gives by (9.26) |∂Xα T (X )| ≤ Cα m(X)−2k−|α | , ∀X ∈ R2n ,


As a consequence of the proof of Theorem 9.2.1 we have the following analogue for classical symbols (which is used in Parmeggiani [52]). Theorem 9.2.3. Let μ > 0 and let a = a∗ ∼ ∑ j≥0 aμ −2 j ∈ Scl (mμ , g; MN ). Suppose there exists e0 ∈ C∞ (R2n \ {0}; MN ), positively homogeneous of degree 0, such that e∗0 e0

=

e0 e∗0

= I, and

e∗0 aμ e0

λ1,μ 0 , X = 0, = bμ = 0 λ2,μ

where λ j,μ = λ j,∗ μ ∈ C∞ (R2n \ {0}; MN j ) are positively homogeneous of degree μ , j = 1, 2, and Spec(λ1,μ (X )) ∩ Spec(λ2,μ (X)) = 0, / ∀X ∈ R2n , |X | = 1.

(9.30)

Then there exists e ∈ Scl (1, g; MN ) with principal symbol e0 such that 1. ew (x, D)∗ ew (x, D) − I, ew (x, D)ew (x, D)∗ − I ∈ S(m−∞ , g; MN ); 2. ew (x, D)∗ aw (x, D)ew (x, D) − bw (x, D) ∈ S(m−∞ , g; MN ), where the symbol b ∼ ∑ j≥0 bμ −2 j ∈ Scl (mμ , g; MN ) is blockwise diagonal, with

b μ −2 j (X ) =

0 b1,μ −2 j (X) , ∀X ∈ R2n \ {0}, ∀ j ≥ 0, 0 b2,μ −2 j (X)


135

with blocks b j,μ of sizes N j , j = 1, 2, respectively, and with principal symbol

b μ (X ) =

λ1,μ (X) 0 , ∀X ∈ R2n \ {0}. λ2,μ (X) 0

We shall call such a symbol b a (block)-diagonalization of a. Notice that b depends on a and e0 . Remark 9.2.4. Given N × N matrices a and b, a straightforward computation gives the following very useful formula: {a, b}∗ = −{b∗, a∗ }

(9.31)

which yields in particular that i i {a, a∗} and {a∗ , a} are Hermitian matrices. 2 2 It will be useful to compute the subprincipal part bμ −2 (that is, the coefficient of the h term) of the block-diagonal system obtained in Theorem 9.2.1. The same formula holds for the version given in Theorem 9.2.3 for classical operators. We have the following proposition. Proposition 9.2.5. For the subprincipal part bμ −2 of the h∞ -diagonalization given in Theorem 9.2.1 one has, by (9.20), the formula b μ −2 = e∗−2 e0 bμ + b μ e∗0 e−2 + e∗0 aμ −2 e0 −

i ∗ e0 {aμ , e0 } + {e∗0, a μ e0 } , 2

(9.32)

where

i e−2 = {e0 , e∗0 }e0 + α−2 e0 , 4 ∗ = −α ∗ with α−2 −2 determined by equation (9.24) through β−2 = e0 α−2 e0 . In the case N = 2, supposing that aμ = a∗μ > 0 and that there exist positive smooth functions λ1 , λ2 ∈ S(mμ , g) (where we now write λ j , j = 1, 2, for the eigenvalues of a μ ) such that (9.33) |λ1 (X ) − λ2(X)| m(X)μ , ∀X ∈ R2n , whence the existence of a smooth unitary matrix e0 such that e0 (X )∗ aμ (X )e0 (X) = we have the following corollary.

λ1 (X) 0 , ∀X ∈ R2n , λ2 (X ) 0

(9.34)

136


Corollary 9.2.6. Suppose a μ = a∗μ > 0 possesses smooth eigenvalues λ1 , λ2

1 0 2 satisfying (9.33). Let {w1 , w2 } be the canonical basis of C , w1 = , w2 = , 0 1 so that b μ (X )w j = λ j (X )w j , j = 1, 2, for all X ∈ R2n . Then the symbol of the h∞ -diagonalization b is a diagonal 2 × 2 matrix. Moreover, for the subprincipal + * (11) bμ −2 0 we have symbol bμ −2 = (22) 0 b μ −2 ( j j)

b μ −2 = b μ −2 w j , w j

1 = e∗0 aμ −2 e0 w j , w j + Im {e∗0 , λ j }e0 w j , w j

2 1 ∗ + Im e0 {aμ , e0 }w j , w j , j = 1, 2. 2

(9.35)

Proof. From the proof of Theorem 9.2.1 we have that the matrix β = β−2 is skewadjoint, whence ∗ β−2 bμ w j , w j + b μ β−2 w j , w j = −λ j β−2 w j , w j + λ j β−2 w j , w j = 0. (9.36)

Then, from (9.32), recalling that i e∗0 e−2 = e∗0 {e0 , e∗0 }e0 + β−2, 4 using aμ e0 = e0 bμ and (9.36), and since (∂ bμ )w j = (∂ λ j )w j , we obtain bμ −2 w j , w j = Re bμ −2 w j , w j

i = e∗0 aμ −2 e0 w j , w j + 2 Re e∗0 {e0 , e∗0 }e0 bμ w j , w j

4 i i ∗ −Re e0 {aμ , e0 }w j , w j − Re {e∗0 , e0 bμ }w j , w j

2 2 = e∗0 aμ −2 e0 w j , w j

1 1 − Im e∗0 {e0 , e∗0 }e0 bμ w j , w j + Im e∗0 {aμ , e0 }w j , w j

2 2 1 ∗ + Im {e0 , e0 }bμ w j , w j

2 ∂ e∗ ∂ λ ∂ e∗ ∂ λ j 1 n j − 0 e0 )w j , w j . + ∑ Im ( 0 e0 2 =1 ∂ ξ ∂ x ∂ x ∂ ξ

Now, since e0 e∗0 = I = e∗0 e0 , we have that (∂ e0 )e∗0 + e0 (∂ e∗0 ) = 0

(9.37)


137

whence ∂e

0 ∗ ∂ e0 e0 ∂ ξ ∂ x =1 n

{e0 , e∗0 e0 } = 0 = {e0 , e∗0 }e0 + ∑

−

∂ e0 ∗ ∂ e0 e ∂ x 0 ∂ ξ

(using (9.37)) = {e0 , e∗0 }e0 − e0{e∗0 , e0 }, that is

e∗0 {e0 , e∗0 }e0 = {e∗0 , e0 }.

(9.38)

Using (9.38) in the above expression for bμ −2 w j , w j and noting that

∂ e∗0 ∂ λ j ∂ e∗0 ∂ λ j ∂ e∗ ∂ λ j ∂ e∗ ∂ λ j e0 − e0 = 0 e0 − 0 e0 = {e∗0 , λ j }e0 ∂ ξ ∂ x ∂ x ∂ ξ ∂ ξ ∂ x ∂ x ∂ ξ (∂ λ j being a scalar), proves (9.35).

We must now study the “transformation properties” (we are interested just in the 2 × 2 case) of the subprincipal terms depending on the choice of e0 . More precisely, we have the following proposition. Proposition 9.2.7. Let 0 < aμ = a∗μ ∈ S(mμ , g; M2 ) satisfy (9.33). Let e0 and e˜0 be smooth, unitary 2 × 2 matrices in S(1, g; M2 ) such that e∗0 aμ e0 = e˜∗0 aμ e˜0 = bμ =

λ1 0 . 0 λ2

Denote by bμ −2 and b˜ μ −2 the subprincipal terms given in Corollary 9.2.6, associated respectively with e0 and e˜0 . Let hence f ∈ S(1, g; M2 ) be the unitary matrix

f=

f1 0 , such that f ∗ f = f f ∗ = I and e0 = e˜0 f , 0 f2

so that the f j ∈ C∞ (R2n ; C) belong to S(1, g) and | f j (X )| = 1, for all X ∈ R2n , j = 1, 2. Then, with {w1 , w2 } the canonical basis of C2 as before, ( j j) bμ −2 = b μ −2 w j , w j = b˜ μ −2 w j , w j + Im f j { f¯j , λ j } , j = 1, 2.

(9.39)

Proof. In the first place we have, since f w j = f j w j and | f j | = 1, j = 1, 2, e∗0 aμ −2 e0 w j , w j = f ∗ e˜∗0 aμ −2 e˜0 f w j , w j

= f j aμ −2 e˜0 w j , f j e˜0 w j = a μ −2 e˜0 w j , e˜0 w j .

(9.40)

138


Next, using (∂ f )w j = (∂ f j )w j , we compute {e∗0 , λ j }e0 w j , w j = { f ∗ e˜∗0 , λ j }e˜0 f j w j , w j

= f j {e˜∗0 , λ j }e˜0 w j , f j w j

n ∂ f∗ ∗ ∂λj ∂ f∗ ∗ ∂λj + f j ∑ e˜0 w j , w j

e˜0 − e˜ ∂ x ∂ x 0 ∂ ξ =1 ∂ ξ

= {e˜∗0 , λ j }e˜0 w j , w j

n ∂λj ∂ fj ∂λj ∂ fj + ∑ f j e˜∗0 e˜0 w j , w j − f j e˜∗0 e˜0 w j , w j

∂ x ∂ ξ ∂ ξ ∂ x =1 ∗ ¯ = {e˜0 , λ j }e˜0 w j , w j + f j { f j , λ j }w j , w j

= {e˜∗0 , λ j }e˜0 w j , w j + f j { f¯j , λ j }, j = 1, 2,

that is {e∗0 , λ j }e0 w j , w j = {e˜∗0 , λ j }e˜0 w j , w j + f j { f¯j , λ j }, j = 1, 2. Hence, for j = 1, 2, 1 1 1 ∗ Im {e0 , λ j }e0 w j , w j = Im {e˜∗0 , λ j }e˜0 w j , w j + Im f j { f¯j , λ j } . (9.41) 2 2 2 We now consider e∗0 {aμ , e0 }w j , w j = f¯j {a μ , e˜0 f }w j , e˜0 w j

= {a μ , e˜0 }w j , e˜0 w j + f¯j {a μ , f j }e˜0 w j , e˜0 w j

= e˜∗0 {aμ , e˜0 }w j , w j + f¯j {a μ , f j }e˜0 w j , e˜0 w j . Since (∂ aμ )e˜0 + a μ (∂ e˜0 ) = e˜0 (∂ bμ ) + (∂ e˜0 )b μ , we get

∂ (aμ − λ j )e˜0 w j = −(aμ − λ j )(∂ e˜0 )w j , j = 1, 2. It hence follows, with ∂ and

∂

(9.42)

generic first-order derivatives,

(∂ aμ )(∂ f j )e˜0 w j = (∂ f j )(∂ aμ )e˜0 w j = (∂ f j ) (∂ λ j )e˜0 w j − (a μ − λ j )(∂ e˜0 )w j , whence {a μ , f j }e˜0 w j = {λ j , f j }e˜0 w j − (a μ − λ j ){e˜0 , f j }w j , j = 1, 2,


139

so that, by using (a μ − λ j ){e˜0 , f j }w j , e˜0 w j = {e˜0 , f j }w j , (a μ − λ j )e˜0 w j = 0, we obtain f¯j {a μ , f j }e˜0 w j , e˜0 w j = f¯j {λ j , f j }e˜0 w j , e˜0 w j = f¯j {λ j , f j }, j = 1, 2. Hence e˜∗0 {aμ , e˜0 }w j , w j + f¯j {a μ , f j }e˜0 w j , e˜0 w j = e˜∗0 {aμ , e˜0 }w j , w j + f¯j {λ j , f j }, for j = 1, 2, and thus 1 1 1 ∗ Im e0 {aμ , e0 }w j , w j = Im e˜∗0 {aμ , e˜0 }w j , w j + Im f¯j {λ j , f j } , 2 2 2 (9.43) for j = 1, 2. We now observe that 0 = {1, λ j } = {| f j |2 , λ j } = f j { f¯j , λ j } + f¯j { f j , λ j }, so that

f¯j {λ j , f j } = f j { f¯j , λ j }, j = 1, 2.

(9.44)

Plugging (9.40), (9.41), (9.43) and (9.44) in (9.35) gives 1 1 bμ −2 w j , w j = b˜ μ −2 w j , w j + Im f j { f¯j , λ j } + Im f¯j {λ j , f j } 2 2 = b˜ μ −2 w j , w j + Im f j { f¯j , λ j } , j = 1, 2,

which proves the proposition.

Remark 9.2.8. It is useful to remark that Proposition 9.2.5, Corollary 9.2.6 and Proposition 9.2.7 all hold true in the case of classical symbols (with the usual Weylquantization). Remark 9.2.9. Notice that when a = a∗ is an N × N globally positive elliptic differential system of order μ (hence μ is even), by virtue of the homogeneity

λ j,μ (X ) = |X|μ λ j,μ (

X ), ∀X ∈ R2n \ {0}, j = 1, . . . , N, |X |

we have that condition (9.33) on the eigenvalues of aμ becomes |λ j,μ (ω ) − λ j,μ (ω )| 1, ∀ω ∈ S2n−1 , j = j .

(9.45)

140


9.3 Some Estimates for Semiclassical Operators Recall from Definition 3.2.25 and Proposition 3.2.26 that for s ∈ Z+ , Bs (Rn ) = {u ∈ L2 ; xα ∂xβ u ∈ L2 , |α | + |β | ≤ s}, and B−s = (Bs )∗ . It is also useful to recall that B2 (Rn ) = D(pw 0 (x, D)), 2 2 2 n where pw 0 is the usual harmonic oscillator (|x| + |D| )/2 = (|x| − Δ )/2 in R . Recall also that Bs (Rn ; CN ) = Bs (Rn ) ⊗ CN ,

and that, from (5.7), on B2s we have the equivalent norms ||u||2B2s ,1 :=

∑

|α |+|β |≤2s

s 2 ||xα ∂xβ u||20 , and ||u||2B2s := ||u||20 + ||pw 0 (x, D) u||0 .

We next wish to introduce the semiclassical parameter h ∈ (0, 1] into the game. Consider the L2 -isometry, also automorphism of S and S , √ Uh : u −→ (Uh u)(x) = h−n/4 u(x/ h). From (9.6) we have √ √ w w Uh−1 pw 0 (x, hD)Uh = p0 ( h x, h D) = h p0 (x, D).

(9.46)

Since √ ∂xβ (Uh−1 u)(x) = hn/4 h|β |/2 (∂xβ u)( h x) = h|β |/2 (Uh−1 (∂xβ u))(x), we get ||xα ∂xβ (Uh−1 u)||20 = h|β |−|α |||Uh−1 (xα ∂xβ u)||20 = h|β |−|α | ||xα ∂xβ u||20 , since Uh−1 is an L2 -isometry as well. Consider now the h-dependent norm ||u||2B2s ,h :=

∑

|α |+|β |≤2s

h2|β |||xα ∂xβ u||20 ,

which is an equivalent norm of B2s , for one readily has h4s ||u||2B2s ,1 ≤ ||u||2B2s ,h ≤ ||u||2B2s ,1 , ∀u ∈ B2s .

(9.47)

9.3 Some Estimates for Semiclassical Operators

141

Using (9.47) we get ||u||2B2s ,h =

∑

|α |+|β |≤2s

h2|β | ||Uh−1 (xα ∂xβ u)||20 =

∑

|α |+|β |≤2s

h|α |+|β | ||xα ∂xβ (Uh−1 u)||20 ,

from which it follows, with constants independent of h, that s −1 2 2 −1 2 w s −1 2 h2s ||Uh−1 u||20 + ||pw 0 (x, D) Uh u||0 ||u||B2s ,h ||Uh u||0 + ||p0 (x, D) Uh u||0 , and since s −1 2 −1 2 w s −1 2 ||Uh−1 u||20 + ||pw 0 (x, D) Uh u||0 = ||Uh u||0 + ||Uh p0 (x, D) Uh u||0 =

(by (9.46)) s 2 −2s s 2 h2s ||u||20 + ||pw = ||u||20 + h−2s||pw 0 (x, hD) u||0 = h 0 (x, hD) u||0 , we get, with constants independent of h, ⎧ 2s 2 w s 2 2 −2s ||u||2 + ||pw (x, hD)s u||2 , ⎪ ⎨ h ||u||0 + ||p0 (x, hD) u||0 ||u||B2s ,h h 0 0 0 ⎪ ⎩

h4s ||u||2B2s ,1 ≤ ||u||2B2s ,h ≤ ||u||2B2s ,1 ,

(9.48)

for all u ∈ B2s . We may hence prove the following useful fact. Proposition 9.3.1. Let r ∈ S00 (m−2N0 ; MN ), for some N0 ∈ N. Then, given any integer k0 with 0 ≤ k0 ≤ N0 , rw (x, hD) : L2 (Rn ; CN ) −→ B2k0 (Rn ; CN ) is continuous, satisfying the following estimates: there exists C > 0 (dependent on k0 but independent of h) such that ||rw (x, hD)u||B2k0 ,1 ≤ Ch−3k0 ||u||0 , ∀u ∈ L2 (Rn ; CN ), and ||rw (x, hD)u||B2k0 ,h ≤ Ch−k0 ||u||0 , ∀u ∈ L2 (Rn ; CN ), for all h ∈ (0, 1]. In particular, when 0 < k0 ≤ N0 , rw (x, hD) : L2 (Rn ; CN ) −→ B2k0 (Rn ; CN ) →→ L2 (Rn ; CN ) is compact, for all h ∈ (0, 1].

142


Proof. We want to estimate ||rw (x, hD)u||B2k0 ,h (of course, we may suppose k0 > 0, otherwise there is nothing to prove). Now, rw (x, hD) is also a bounded operator L2 −→ L2 with norm bounded independently of h ∈ (0, 1]. Take a sequence {u j } j∈Z+ ⊂ S (Rn ; CN ) such that u j → u in L2 as j → +∞. Since (p0 h . . . h p0 )h r ∈ S00 (1; MN ), we have, by the L2 -boundedness, with constants k0

independent of h, k0 w ||pw 0 (x, hD) r (x, hD)(u j − u j )||0 ||u j − u j ||0 −→ 0, as j, j → +∞.

Hence, by (9.48), with constants independent of h, h4k0 ||rw (x, hD)(u j − u j )||2B2k0 ,1 ≤ ||rw (x, hD)(u j − u j )||2B2k0 ,h k0 w 2 h−2k0 ||rw (x, hD)(u j − u j )||20 + ||pw 0 (x, hD) r (x, hD)(u j − u j )||0 h−2k0 ||u j − u j ||20 −→ 0, as j, j → +∞. B2k0

Hence rw (x, hD)u j −→ v ∈ B2k0 as j → +∞. But on the other hand we also have S

rw (x, hD)u j −→ rw (x, hD)u as j → +∞. It therefore follows that rw (x, hD)u ∈ B2k0 , ∀u ∈ L2 , and

||rw (x, hD)u||B2k0 ,1 ≤ Ch−3k0 ||u||0 , ∀u ∈ L2 , ||rw (x, hD)u||B2k0 ,h ≤ Ch−k0 ||u||0 , ∀u ∈ L2 ,


Corollary 9.3.2. Let r ∈ S00 (m−2N0 ; MN ), for some N0 ∈ N. Then, given any integers k0 , k1 with 0 ≤ k0 ≤ k1 ≤ N0 , rw (x, hD) : B2k0 (Rn ; CN ) −→ B2k1 (Rn ; CN ) is continuous, satisfying the following estimates: there exists C > 0 (independent of h) such that ||rw (x, hD)u||B2k1 ,1 ≤ Ch−3k1 ||u||B2k0 ,1 , ∀u ∈ B2k0 (Rn ; CN ), and

||rw (x, hD)u||B2k1 ,h ≤ Ch−k1 ||u||B2k0 ,h , ∀u ∈ B2k0 (Rn ; CN ),

for all h ∈ (0, 1]. In particular, when 0 ≤ k0 < k1 ≤ N0 , rw (x, hD) : B2k0 (Rn ; CN ) −→ B2k1 (Rn ; CN ) →→ B2k0 (Rn ; CN ) is compact, for all h ∈ (0, 1].

9.4 Some Spectral Properties of Semiclassical GPDOs

143

Proof. Given any u ∈ S (Rn ; CN ), we have that rw (x, hD)u ∈ S (Rn ; CN ). Hence it suffices to prove the inequalities for Schwartz functions u, with constants independent of u and h. Since B2k0 (Rn ; CN ) ⊂ L2 (Rn ; CN ) with ||u||20 ≤ ||u||2B2k0 ,h ,

and

||u||20 ≤ ||u||2B2k0 ,1 ,

from Proposition 9.3.1 we have ||rw (x, hD)u||2B2k1 ,h ≤ Ch−2k1 ||u||20 ≤ Ch−2k1 ||u||2B2k0 ,h , and ||rw (x, hD)u||2B2k1 ,1 ≤ Ch−6k1 ||u||20 ≤ h−6k1 ||u||2B2k0 ,1 , for a constant C > 0 independent of u and of h, which concludes the proof.

9.4 Some Spectral Properties of Semiclassical GPDOs We establish in this section a few useful results about spectral properties of h-Weyl quantizations of N × N semiclassical GPD systems which are positive elliptic. We start by giving the definition of semiclassical global polynomial differential system. Definition 9.4.1 (Semiclassical GPD). We shall say that a classical semiclassical 0 (m μ , g; M ) is a semiclassical GPD system of order μ if μ ∈ N symbol a ∈ S0,cl N and a=

[ μ /2]

∑ h j aμ −2 j ,

a μ −2 j ∈ S(mμ −2 j , g; MN )

j=0

(with [μ /2] denoting, as usual, the integer part of μ /2), where the entries of the a μ −2 j are homogeneous polynomials in X ∈ R2n of degree μ − 2 j. 0 (m μ , g; M ) of order μ is ellipWe say that a semiclassical GPD system a ∈ S0,cl N ∗ tic (resp. positive elliptic, when a = a ) if the principal part aμ is a homogeneous globally elliptic (resp. globally positive elliptic) symbol. The h-Weyl quantization of a semiclassical GPD will be called an h-GPDO. Suppose a=

[ μ /2]

∑ h j aμ −2 j

j=0

is an N × N semiclassical GPD symbol of order μ ∈ N in Rn . Since (9.5) holds also on S (Rn ; CN ) and S (Rn ; CN ), we have the following lemma.

144


0 (m μ , g; M ) be an N × N semiclassical Lemma 9.4.2. Let E > 0 and let a ∈ S0,cl N GPD system of order μ ∈ N. Let UE be the isometry introduced in Remark 9.1.2. Then ˜ UE−1 aw (x, hD)UE = E μ /2 aw (x, hD), (9.49)

where h˜ = h/E and ˜ = aw (x, hD)

[ μ /2]

˜ ∑ h˜ j awμ −2 j (x, hD).

j=0

In particular, when E = h, we have Uh−1 aw (x, hD)Uh = hμ /2 aw (x, D),

(9.50)

where aw (x, D) =

[ μ /2]

∑ awμ −2 j (x, D).

j=0

Proof. The proof follows immediately from Remark 9.1.2. In fact, it suffices to observe that for h˜ = h/E we have √ √ ˜ = U −1 aw (x, hD)UE aw ( E x, E hD) E =

[ μ /2]

∑

h jUE−1 aw μ −2 j (x, hD)UE =

j=0

=

[ μ /2]

√ √ ˜ E x, E hD)

∑ h˜ j E j awμ −2 j (

j=0

[ μ /2]

˜ = E μ /2 aw (x, hD), ˜ ∑ h˜ j E j E μ /2− j awμ −2 j (x, hD)

(9.51)

j=0

in view of the fact that h = E h˜ and that the aμ −2 j have entries which are all homogeneous of degree μ − 2 j. We therefore get the following scaling properties of eigenvalues. 0 (m μ , g; M ) be an N × N semiclassical Lemma 9.4.3. Let E > 0 and let a ∈ S0,cl N GPD system of order μ ∈ N. Let UE be the isometry introduced in Remark 9.1.2. Then aw (x, hD)u(h) = λ (h)u(h), u(h) ∈ L2 (Rn ; CN ), u(h) = 0,

that is u(h) is an eigenfunction of aw (x, hD) belonging to the eigenvalue λ (h), iff, ˜ := U −1 u(h) ∈ L2 (Rn ; CN ), with u( ˜ h) E ˜ u( ˜ = ˜ h) aw (x, hD)

˜ λ (E h) ˜ where h˜ = h , u( ˜ h), μ /2 E E


145

˜ λ (E h) ˜ is an eigenfunction of aw (x, hD) ˜ that is, u( ˜ h) belonging to the eigenvalue μ /2 . E Hence,

λ (h) ∈ Spec(aw (x, hD)) ⇐⇒

˜ λ (E h) h ˜ ∈ Spec(aw (x, hD)), h˜ = . μ /2 E E

In particular, when E = h, one obtains that

λ (h) ∈ Spec(aw (x, hD)) ⇐⇒

λ (h) ∈ Spec(aw (x, D)). hμ /2

Proof. The proof follows immediately from Lemma 9.4.2.

As a (by now elementary) consequence of the results of Section 3.2, namely Remark 9.1.2, Lemma 9.4.2 and Theorem 3.3.13 we have the following fact concerning the spectrum of semiclassical GPD systems. 0 (m μ , g; M ) be an N × N positive elliptic semiProposition 9.4.4. Let A = A∗ ∈ S0,cl N classical GPD system of order μ ∈ 2N. Consider the unbounded operator A(h) defined by A(h) : Bμ (Rn ; CN ) ⊂ L2 (Rn ; CN ) −→ L2 (Rn ; CN ),

A(h)u = Aw (x, hD)u, ∀u ∈ Bμ (Rn ; CN ). Then A(h) is semi-bounded from below for all h ∈ (0, 1]. Hence, Spec(A(h)) is made of a sequence of eigenvalues {λ j (h)} j≥1 ⊂ R with finite multiplicities, such that −∞ < λ1 (h) ≤ λ2 (h) ≤ . . . ≤ λ j (h) ≤ . . . −→ +∞, with repetitions according to the multiplicity. As before, the eigenfunctions of A(h) all belong to the Schwartz space and form, possibly after an orthonormalization procedure, a basis of L2 (Rn ; CN ). Proof. In fact, by Lemma 9.4.2 we have Uh−1 A(h)Uh = hμ /2 A(1), so that (A(h)u, u) = (Uh−1 A(h)UhUh−1 u,Uh−1 u) = hμ /2 (A(1)Uh−1 u,Uh−1 u) ≥ −hμ /2C||Uh−1 u||20 = −hμ /2C||u||20 , ∀u ∈ S (Rn ; CN ). This concludes the proof of the lemma.

Hence, when A is an N × N positive elliptic semiclassical GPD system of order μ we immediately obtain from the Minimax Principle, Lemma 9.4.3 and Proposition 9.4.4 the following corollary.

146


0 (m μ , g; M ) be an N × N positive elliptic semiCorollary 9.4.5. Let A = A∗ ∈ S0,cl N classical GPD system of order μ ∈ 2N. Let φ j ∈ S (Rn ; CN ), j ∈ N, be an eigenfunction of Aw (x, D) (i.e. with h = 1) belonging to the eigenvalue λ j . Then

x ϕ j (h; x) := (Uh φ j )(x) = h−n/4φ j ( √ ) h

(9.52)

belongs to the eigenvalue λ j (h) := h μ /2 λ j of Aw (x, hD). In particular √ h ϕ j ( ; x) = E n/4 ϕ j (h; E x), j ≥ 1, E belongs to the eigenvalue λ j ( Eh ) = imax, for every j ∈ N

μ /2 h E

(9.53)

λ j of Aw (x, Eh D). Hence, using the Min-

λ j ∈ Spec(Aw (x, D)) ⇐⇒ hμ /2 λ j = λ j (h) ∈ Spec(Aw (x, hD)). We finally consider the following situation, that will be very useful later on. Let 0 (m μ , g; M ) be an N × N semiclassical positive elliptic GPD system. A = A∗ ∈ S0,cl N Let R = R∗ ∈ S00 (1, g; MN ) with supp(R) ⊂ K, where K is a compact set independent of h. Then Rw (x, hD) : S (Rn ; CN ) −→ S (Rn ; CN ) is continuous and, as an operator in L2 , ||Rw (x, hD)||L2 →L2 = O(1) for all h ∈ (0, 1] (see Dimassi-Sjöstrand [7]). We have the following important result. 0 (m μ , g; M ) be an N × N semiclassical posiProposition 9.4.6. Let A = A∗ ∈ S0,cl N ∗ 0 tive elliptic GPD system, and let R = R ∈ S0 (1, g; MN ) with supp(R) ⊂ K, where w K is a compact set independent of h. Let us consider Aw 0 (x, hD) = A (x, hD) + w R (x, hD). Then for all h ∈ (0, 1] the unbounded operator A0 (h) defined by

A0 (h) : Bμ (Rn ; CN ) ⊂ L2 (Rn ; CN ) −→ L2 (Rn ; CN ), μ n N A0 (h)u = Aw 0 (x, hD)u, ∀u ∈ B (R ; C ),

is self-adjoint with a discrete spectrum bounded from below, made of a sequence of eigenvalues {λ j (h)} j≥1 ⊂ R with finite multiplicities, such that −∞ < λ1 (h) ≤ λ2 (h) ≤ . . . ≤ λ j (h) ≤ . . . −→ +∞, with repetitions according to the multiplicity. Proof. That A0 (h) = A0 (h)∗ with the same domain Bμ of A(h) is trivial, for Rw (x, hD) is bounded in L2 and symmetric. By Theorem 10.1.1 below the resolvent set of A0 (h) is non-empty, and since Bμ is compactly embedded into L2 , A0 (h)


147

also has a discrete spectrum, that must be bounded from below, for the spectrum of A(h) is bounded from below and Rw (x, hD) is bounded in L2 . Remark 9.4.7. One may prove the discreteness of Spec(A0 (h)) also as follows. Since the Schwartz-kernel

x+y , hξ )d ξ 2 −1 x+y , ξ )d ξ = (2π h)−n eih x−y,ξ R( 2 x + y ,t) t=h−1 (x−y) = h−n (Fξ →t R)( 2

KR (x, y; h) = (2π )−n

ei x−y,ξ R(

of Rw (x, hD) belongs to S (Rn × Rn ; MN ), the operator Rw (x, hD) is actually Hilbert-Schmidt. Hence A0 (h) and A(h) have the same essential spectrum. Since the essential spectrum of A(h) is empty, the same is true for that of A0 (h). Remark 9.4.8. More generally, one may prove that if a classical semiclassical sym0 bol A = A∗ ∈ S0,cl (mμ , g; MN ), with μ ≥ 1, has globally positive elliptic principal symbol, then there exists h0 ∈ (0, 1] such that the unbounded operator A(h) defined by A(h) : Bμ (Rn ; CN ) ⊂ L2 (Rn ; CN ) −→ L2 (Rn ; CN ), A(h)u = Aw (x, hD)u, ∀u ∈ Bμ (Rn ; CN ), is self-adjoint, semi-bounded from below and with a discrete spectrum, for all h ∈ (0, h0 ]. In addition, from the elliptic regularity we have that its eigenfunctions are in S (Rn ; CN ), and, possibly after an orthonormalization procedure, they form a basis of L2 (Rn ; CN ). In fact, one uses (9.5) of Remark 9.1.2 (with E = h), namely √ √ Uh−1 Aw (x, hD)Uh = Aw ( h x, h D) to reduce matters to Theorem 3.3.13.

Chapter 10

On Operators Induced by General Finite-Rank Orthogonal Projections

In this chapter we further prepare the ground for the eigenvalue localization of elliptic global systems. Namely, in the following chapters we shall have to control the sandwitch (I − Π )B(I − Π ) of an operator B, semi-bounded from below, by the orthogonal projectors (I − Π ) relative to another operator A, semi-bounded from below. We present things in an abstract setting, for this is a useful machinery. Throughout this chapter H will always stand for a separable (infinite-dimensional) Hilbert space endowed with the scalar product (·, ·) = (·, ·)H .

10.1 Reductions by a Finite-Rank Orthogonal Projection The first result we need is the following theorem (see Kato’s book [35, Theorem 4.10, p. 291]). Theorem 10.1.1. Let T : D(T ) ⊂ H −→ H be self-adjoint (hence it is densely defined), and let B = B∗ ∈ L (H, H) (where, recall, L (H, H) denotes the space of bounded linear operators on H). Then S = T + B : D(T ) ⊂ H −→ H is self-adjoint and dist Spec(T ), Spec(S) ≤ ||B||, that is, sup

ζ ∈Spec(S)

dist(ζ , Spec(T )) ≤ ||B||,

sup

ζ ∈Spec(T )

dist(ζ , Spec(S)) ≤ ||B||,

(10.1)

where ||B|| = ||B||H→H . Let next T : D(T ) ⊂ H −→ H be a densely defined semi-bounded from below self-adjoint operator with compact resolvent. Hence T has a discrete spectrum made of a sequence {λ j } j≥1 of eigenvalues −∞ < λ1 ≤ λ2 ≤ . . . ≤ λ j ≤ . . . → +∞, repeated according to the multiplicity. Let {u j } j∈Z+ ⊂ D(T ) be any fixed orthonormal basis of H.


149

150

10 On Operators Induced by General Finite-Rank Orthogonal Projections

Remark 10.1.2. It is important to note that we do not assume that the u j be eigenfunctions of T (otherwise the result we wish to prove, namely Theorem 10.1.3 below, is a trivial consequence of the Spectral Theorem). Fix N0 ∈ N and let Π = ΠN0 : H −→ H be the finite-rank orthogonal projector onto Span{u1 , . . . , uN0 },

Π=

N0

∑ u∗j ⊗ u j : u −→

j=1

Note that

N0

∑ (u, u j )u j .

j=1

Π u ∈ D(T ), ∀u ∈ H,

and that (I − Π )u ∈ D(T ) ⇐⇒ u ∈ D(T ). We have the following consequence of Theorem 10.1.1. Theorem 10.1.3. The operator T+ := (I − Π )T (I − Π ) = T − Π T + T Π + Π T Π : D(T ) ⊂ H −→ H is self-adjoint, semi-bounded from below, with compact resolvent. Hence its spectrum is discrete and real. Proof. Consider the operators Π T, T Π , Π T Π : D(T ) ⊂ H −→ H. It is immediately seen that T− := Π T Π ∈ L (H, H) is symmetric and compact. Now, since the u j belong to D(T ) and T is self-adjoint, we have D(T ) u −→ Π Tu =

N0

N0

j=1

j=1

∑ (Tu, u j )u j = ∑ (u, Tu j )u j ,

which shows that Π T extends to a bounded operator T1 ∈ L (H, H) such that T1 D(T ) = Π T. Similarly, D(T ) u −→ T Π u =

N0

∑ (u, u j )Tu j ,

j=1

shows that T Π = T2 ∈ L (H, H). We now have that T1 + T2 is symmetric. In fact, (T1 u, v) = (Π Tu, v) = (u, T Π v) = (u, T2 v), ∀u, v ∈ D(T ), from which, T1 and T2 being bounded operators, the claim follows. Hence, the boundedness of T1 , T2 , T− and Theorem 10.1.1 give that T+ = T − (T1 + T2 ) + T− is self-adjoint and semi-bounded from below.

10.1 Reductions by a Finite-Rank Orthogonal Projection

151

Pick now ζ0 ∈ C such that dist(ζ0 , Spec(T )) > ||B||,

(10.2)

where B = −T1 − T2 + T− . Then, again by Theorem 10.1.1, ζ0 ∈ Spec(T+ ). Let R(ζ ) = (ζ − T )−1 be the resolvent operator of T and R+ (ζ ) the one of T+ . Then, since 1 ||R(ζ )|| = , dist(ζ , Spec(T )) (10.2) gives that the Neumann series relative to (I − BR(ζ0 ))−1 converges in L (H, H), whence −1 R+ (ζ0 ) = R(ζ0 ) I − BR(ζ0) . Since R(ζ0 ) is a compact operator, this completes the proof of the theorem.

Corollary 10.1.4. The operator T˜ := T− + T+ : D(T ) ⊂ H −→ H

(10.3)

is self-adjoint, semi-bounded from below, with compact resolvent. Hence its spectrum is discrete and real. Proof. It is immediate from Theorem 10.1.3 and Theorem 10.1.1.

Remark 10.1.5. It is important to notice that T− u ∈ D(T ) = D(T+ ) for all u ∈ H, so that it makes always sense to consider T+ T− , and that T+ T− u = 0, ∀u ∈ H, and T− T+ u = 0, ∀u ∈ D(T ).

(10.4)

[T+ , T− ] = 0 on D(T ).

(10.5)

In particular Define now the operators T+− := (I − Π )T Π : H −→ H, T+− ∈ L (H, H), T−+ := Π T (I − Π ) : D(T ) ⊂ H −→ H. It is readily seen that T−+ = Π T − T− can be extended to a bounded operator in H, that we keep denoting by T−+ , so that we also have ∗ T−+ = T+− .

Moreover, on setting H+ := (I − Π )H, H− := Π H,

152


we have that the H± are closed subspaces of H with H±⊥ = H∓ , so that H = H− ⊕ H+ with orthogonal direct-sum. Notice that, since H− ⊂ D(T ), it makes sense to con sider T+ H . Notice also that −

T+ H = 0, and T− H = 0. −

+

(10.6)

Take now an ON basis {e j } j≥1 ⊂ D(T ) of H which diagonalizes T+ , such that Span{e j }1≤ j≤N0 = H− , Span{e j } j≥N0 +1 = H+ . (This is possible, for [Π , T+ ] = 0 on D(T ).) Rotate next the basis {e j }1≤ j≤N0 into a basis that diagonalizes the N0 × N0 Hermitian matrix T− H . Call {e˜ j }1≤ j≤N0 the − resulting basis, and notice that T+ e˜ j = 0, j = 1, . . . , N0 . Set e˜ j = e j for j ≥ N0 + 1. We thus have that {e˜ j } j≥1 ⊂ D(T ) is an ON basis of H which diagonalizes T˜ (see (10.3)). This proves that Spec(T˜ ) = Spec(T− H− ) Spec(T+ H+ ),

where

T+ H+ : D(T ) ∩ H+ ⊂ H+ −→ H+

is the part of T+ in H+ (i.e. T+ is reduced by H+ ; see Kato [35, pp. 172, 178 and 278]). Notice that D(T ) ∩ H+ = (I − Π )D(T ), for on the one hand (I − Π )D(T ) ⊂ D(T ) and (I − Π )D(T ) ⊂ (I − Π )H = H+ , and on the other u ∈ D(T ) ∩ H+ =⇒ u = Π u + (I − Π )u = (I − Π )u ∈ (I − Π )D(T ). Since T = T˜ + T0, where T0 = T+− + T−+ = T0∗ ∈ L (H, H), once more from Theorem 10.1.1 we have dist Spec(T ), Spec(T˜ ) = dist Spec(T ), Spec(T± H± ) ≤ ||T0 ||. ±

(10.7)

10.1 Reductions by a Finite-Rank Orthogonal Projection

153

Since (T− being symmetric and compact) Spec(T− ) = Spec(T− H− ) ∪ {0} ⊂ [−||T− ||, ||T− ||], and since T+ has a discrete spectrum which is positively diverging to +∞ (T+ is semi-bounded from below, for T is so, and it has a discrete spectrum by Theorem 10.1.3) one obtains the following theorem. Theorem 10.1.6. In the hypotheses of Theorem 10.1.3, upon denoting δ := ||T0 || and choosing E > ||T− || + 3δ , we have that Spec(T ) ∩ (E, +∞) ⊂

[λ+ − δ , λ+ + δ ].

(10.8)

λ+ ∈Spec(T+ )∩(E−δ ,+∞)

Remark 10.1.7. Notice that when the u j are eigenfunctions of T, T+− = 0 and T−+ = 0 then, so that δ = ||T0 || = 0. Notice that one could have obtained (10.7) and the result of Theorem 10.1.6 by writing T as the (unbounded) system *

+ T− H− T−+ H+ : H− ⊕ (D(T ) ∩ H+ ) ⊂ H− ⊕ H+ −→ H− ⊕ H+, T+− H T+ H −

+

by using Proposition 10.1.8 below that shows that T+ H+ is self-adjoint in H+ (semi-bounded from below) with compact resolvent, and finally by using once more Theorem 10.1.1. Proposition 10.1.8. The operator T+ H+ : D(T ) ∩ H+ ⊂ H+ −→ H+ is self-adjoint in H+ with compact resolvent (and is obviously semi-bounded from below). Proof. Put, for short, T(+) = T+ H+ . Recall that (u, v)H+ := ((I − Π )u, (I − Π )v)H = (u, v)H , ∀u, v ∈ H+ . Then ∗ ) = {u ∈ H+ ; ∃v ∈ H+ , (T(+) u , u)H+ = (u , v)H+ , ∀u ∈ D(T(+) )}. D(T(+)

Since for all u, v ∈ D(T(+) ) = D(T ) ∩ H+ (T(+) u, v)H+ = (T+ u, v)H = (u, T+ v)H = (u, T(+) v)H+ , we always have

∗ D(T(+) ) ⊂ D(T(+) ).

154


∗ ). Then, for some v ∈ H , Take u ∈ D(T(+) +

(T(+) u , u)H+ = (u , v)H+ , ∀u ∈ D(T(+) ), that is, (T+ u , u)H = (u , v)H , ∀u ∈ D(T ) ∩ H+ .

Since T+ H− = 0 and H− is orthogonal to H+ , it also follows that (T+ u , u)H = (u , v)H , ∀u ∈ D(T ), for, given any u ∈ D(T ), (T+ u , u)H = (T+ [Π u + (I − Π )u ], u)H = (T+ (I − Π )u, u)H = ((I − Π )u , v)H = (Π u + (I − Π )u , v)H = (u , v)H . Hence u ∈ D(T ∗ ) = D(T ), and since u ∈ H+ , we thus obtain u ∈ D(T ) ∩ H+ = D(T(+) ). It follows that ∗ D(T(+) ) ⊂ D(T(+) ) and this proves the self-adjointness claim. To prove the compactness of the resolvent we observe, in the first place, that if ζ belongs to the resolvent set of T+ (which is non-empty by the semi-boundedness assumption) we have that the operator H+ u −→ (I − Π )(ζ − T+ )−1 (I − Π )u ∈ D(T ) ∩ H+ ⊂ H+ is continuous and compact, for (ζ − T+ )−1 is a compact operator and (I − Π ) is bounded. On the other hand, for all u ∈ H+ , (ζ − T(+))(I − Π )(ζ − T+ )−1 (I − Π )u = (ζ − T+ )(I − Π )(ζ − T+ )−1 (I − Π )u = (I − Π )(ζ − T+ )(ζ − T+ )−1 (I − Π )u = (I − Π )u = u, and for all u ∈ D(T ) ∩ H+ , (I − Π )(ζ − T+)−1 (I − Π )(ζ − T(+) )u = (I − Π )(ζ − T+ )−1 (I − Π )(ζ − T+ )u = (I − Π )(ζ − T+ )−1 (ζ − T+ )(I − Π )u = (I − Π )u = u. This proves that (ζ − T(+) )−1 = (I − Π )(ζ − T+ )−1 (I − Π ) for all ζ in the resolvent set of T+ , and concludes the proof.

10.2 Semiclassical Reduction by a Finite-Rank Orthogonal Projection

155

10.2 Semiclassical Reduction by a Finite-Rank Orthogonal Projection We now make things more precise, and introduce also the dependence on the semiclassical parameter h ∈ (0, 1]. Let D ⊂ H be independent of h, dense and compactly embedded in H, and let A = A(h) : D(A) = D ⊂ H −→ H, be a selfadjoint operator with A ≥ −CI on D. Let therefore {λ j (h)} j≥1 be the sequence −∞ < λ1 (h) ≤ λ2 (h) ≤ . . . ≤ λ j (h) ≤ . . . → +∞ of its eigenvalues, repeated according to their multiplicities. Let {u j (h)} j≥1 ⊂ D be a corresponding ON basis of eigenfunctions. Let E0 ≥ 10 be fixed (independent of h), and let us consider the orthogonal projector

Π = Π (h) = ∑ u j (h)∗ ⊗ u j (h) = λ j (h)≤E0

N0

∑ u∗j ⊗ u j : H −→ Span{u1, . . . , uN0 } ⊂ D,

j=1

where N0 = N0 (h) is the number of eigenvalues λ j (h) of A, repeated according to multiplicity, which satisfy λ j (h) ≤ E0 (in the sequel we will often drop the explicit dependence on h). Hence [Π , A] = 0 on D, and, by the Spectral Theorem, Spec((I − Π )A(I − Π )) ∩ (E0 , +∞) = Spec(A) ∩ (E0 , +∞), or, equivalently, Spec((I − Π )A(I − Π )) \ {0} = Spec(A) ∩ (E0 , +∞). Let R = R(h) = R∗ : H −→ H be in L (H, H) with ||R||H→H ≤ cR , ∀h ∈ (0, 1],

(10.9)

where cR > 0 is independent of h, and suppose that ||(I − Π )R||H→H = ||R(I − Π )||H→H = O(h∞ )

(10.10)

(we shall write (I − Π )R = O(h∞ ) and R(I − Π ) = O(h∞ ), respectively). Consider B = B(h) = A + R : D ⊂ H −→ H.

(10.11)

By Theorem 10.1.1 (and the proof of Theorem 10.1.3), we have that B = B∗ with compact resolvent for all h ∈ (0, 1]. Moreover, B ≥ −C I on D. Let hence { μ j (h)} j≥1 be the sequence −∞ < μ1 ≤ μ2 ≤ . . . → +∞ of its eigenvalues, repeated according to multiplicity. Define, as before, B− := Π BΠ , B+ = (I − Π )B(I − Π ), and B˜ = B− + B+.

156


The operator B− is bounded and symmetric (in fact, a finite-rank operator), and it follows from Theorems 10.1.1 and 10.1.3 that B+ and B˜ are self-adjoint with the same domain D, semi-bounded from below, with a discrete spectrum. Let ˜ = {μ˜ j } j≥1 , −∞ < μ˜ 1 ≤ μ˜ 2 ≤ . . . → +∞, Spec(B) + + Spec(B+ ) = { μ + j } j≥1 , −∞ < μ1 ≤ μ2 ≤ . . . → +∞,

with repetitions according to multiplicity. Notice, moreover, that since B− u ∈ D for all u ∈ H, it always makes sense to consider B+ B− . Let, as before, H− := Π H, H+ := (I − Π )H, so that H− ⊂ D, H+ = H−⊥ , and H = H− ⊕ H+ . We have B+ H− = 0, B− H+ = 0, and B+ B− = B− B+ = 0 on D

(10.12)

(hence, in particular, [B+ , B− ] = 0 on D). It is readily seen, by (10.10), that (I − Π )A(I − Π ) = B+ + R1 , where R1 = O(h∞ )

(10.13)

and that (since (I − Π )AΠ = Π A(I − Π ) = 0 by the Spectral Theorem) B = B˜ + R2 , where R2 = O(h∞ ).

(10.14)

Notice that there exists h0 ∈ (0, 1] such that for any given h ∈ (0, h0 ] we have B+ H+ > 0. (10.15) This follows from (10.13). In fact, let us fix an arbitrary integer N ≥ 1. Then we have from (10.13) that there exists CN+1 > 0 such that ||R j ||H→H ≤ CN+1 hN+1 , j = 1, 2, ∀h ∈ (0, 1]. We may therefore fix h0 ∈ (0, 1], such that CN+1 h < 1/102 for all h ∈ (0, h0 ], whence ||R j ||H→H < 10−2 hN , j = 1, 2, ∀h ∈ (0, h0 ]. Then, for all u ∈ D ∩ H+ , (B+ u, u) = ((I − Π )A(I − Π )u, u) − (R1u, u) ≥ ≥ (E0 − 10−2hN )||u||2 > (E0 − 1)||u||2, ∀h ∈ (0, h0 ], which proves the claim.

(10.16)


157

Hence, from (10.15) we have that 0 ∈ Spec(B+ ) with multiplicity N0

(10.17)

(notice that N0 = dim H− ). From (10.13) and (10.16) we have that for all u ∈ D with ||u|| = 1 −10−2hN + (B+ u, u) ≤ ((I − Π )A(I − Π )u, u) ≤ (B+ u, u) + 10−2 hN , ∀h ∈ (0, h0 ]. (10.18) Since for each j ≥ N0 + 1 E0 < λ j = (Au j , u j ) = ((I − Π )A(I − Π )u j , u j ) ⎡ =

sup

v1 ,...,v j−1 lin. ind.

⎢ ⎣

inf

v∈Span{v1 ,...,v j−1 ||v||=1

and since

⎤

⎥ ((I − Π )A(I − Π )v, v)⎦ ,

}⊥ ∩D

⎤

⎡

μ+ j =

sup

v1 ,...,v j−1 lin.ind.

⎢ ⎣

inf

v∈Span{v1 ,...,v j−1 ||v||=1

⎥ (B+ v, v)⎦ ,

}⊥ ∩D

it thus follows from (10.18) and the Minimax Principle that for any given h ∈ (0, h0 ] −2 N + |λ j (h) − μ + j (h)| < 10 h , and 0 < E0 − 1 < μ j (h), ∀ j ≥ N0 (h) + 1. (10.19)

Hence (10.19) says that for any given h ∈ (0, h0 ], for j ≥ N0 (h) + 1 the j-th eigenvalue of A(h) is within distance 10−2 hN to the j-th eigenvalue of B+ (h). Therefore, for the eigenvalues of B+ we have

μ1+ = . . . = μN+0 = 0, 0 < μN+0 +1 ≤ μN+0 +2 ≤ . . . .

(10.20)

Notice also that B− H− is represented in the basis {u1, . . . , uN0 } by the N0 × N0 matrix acting on N0 -dimensional vectors ⎡ ⎤ (u, u1 ) N0 ⎢ ⎥ .. u = ∑ (u, u j )u j $ ⎣ ⎦, . j =1

(u, uN0 )

as B− H u = −

N0

∑

j, j =1

(u, u j )(Bu j , u j )u j ,

158


i.e. by the matrix whose j j -entry is given by

and that

B− H−

j j

= (Bu j , u j ), 1 ≤ j, j ≤ N0 ,

Spec(B− ) = Spec(B− H− ) ∪ {0} ⊂ [−c0 , c0 ],

(10.21)

where c0 := cR + E0 . In fact, B− is a compact self-adjoint operator, so that its spectrum is contained in the interval [−||B− ||H→H , ||B− ||H→H ]. Since ||B− ||H→H = ||Π (A + R)Π ||H→H ≤ E0 + ||Π RΠ ||H→H ≤ E0 + cR = c0 , (10.21) follows. Put Spec(B− H− ) = { μ1− , . . . , μN−0 }. We next consider an ON basis {e j } j≥1 ⊂ D of H, where Span{e1 , . . . , eN0 } = H− , and Span{e j } j≥N0 +1 = H+ , which diagonalizes B+ (this is possible, for [Π , B+ ] = 0 on D): B+ e j = 0, 1 ≤ j ≤ N0 , B+ e j = μ + j e j , j ≥ N0 + 1.

(10.22)

Notice then that, from (10.12), B− e j = 0, ∀ j ≥ N0 + 1. Let us then rotate the e1 , . . . , eN0 to obtain a new ON basis e˜1 , . . . , eÑ0 of H− which diagonalizes the Hermitian matrix B− H− . Defining e˜ j = e j for j ≥ N0 + 1 gives ˜ B− therefore an ON basis {e˜ j } j≥1 ⊂ D of H which simultaneously diagonalizes B, and B+ . Hence, in particular, + ˜ B˜ e˜ j = μ − j e˜ j , 1 ≤ j ≤ N0 , Be˜ j = μ j e˜ j , j ≥ N0 + 1.

Notice that a-priori we do not know whether or not μN−0 ≤ μN+0 +1 , so that we do not have, as yet, that the sequence {μ1− , . . . , μN−0 , μN+0 +1 , . . .} coincides with the ˜ We have therefore to non-decreasing sequence {μ˜ j } j≥1 of the eigenvalues of B. rearrange the μ ± in a non-decreasing order. Let j + S+ = { μ + j ; 0 < μ j ≤ c0 }.

Then { j ∈ Z+ ; μ + j ∈ S+ } = {N0 + 1, . . ., N0 + ν+ }, / Since for some ν+ = ν+ (h) ∈ Z+ (where ν+ = 0 if S+ = 0). {μ1− , . . . , μN−0 } ∪ S+


159

is a finite set, we may arrange, in a non-decreasing order, with repetitions (according + − + to multiplicity among the μ − j , resp. the μ j , and equalities between the μ j and μ j ), the elements + μ− j , 1 ≤ j ≤ N0 , and μ j , N0 + 1 ≤ j ≤ N0 + ν+ ,

the resulting sequence being μ˜ j , 1 ≤ j ≤ N0 + ν+ , and call u˜ j the associated eigenvectors. Put finally u˜ j = e˜ j when j ≥ N0 + ν+ + 1. We therefore obtain the ON ˜ and have that sequence {u˜ j } j≥1 of eigenvectors of B, j ≥ N0 + ν+ + 1 ⇐⇒ μ + j > c0 . Hence, since for j ≥ N + 1 + ν+

μ+ j = (B+ u˜ j , u˜ j ) =

⎤

⎡ sup

v1 ,...,v j−1 lin. ind.

⎢ ⎣

inf

v∈Span{v1 ,...,v j−1 }⊥ ∩D ||v||=1

⎤

⎡

since

μj =

sup

v1 ,...,v j−1 lin. ind.

⎢ ⎣

˜ v)⎥ (Bv, ⎦,

inf

v∈Span{v1 ,...,v j−1 ||v||=1

⎥ (Bv, v)⎦ ,

}⊥ ∩D

and since by (10.14) and (10.16) we have that for all u ∈ D with ||u|| = 1 ˜ u) ≤ (Bu, u) ≤ (Bu, ˜ u) + 10−2hN , ∀h ∈ (0, h0 ], −10−2hN + (Bu, it therefore follows that for any given h ∈ (0, h0 ], for j ≥ N + 1 + ν+ −2 N |μ + j (h) − μ j (h)| < 10 h .

(10.23)

Hence (10.23) says that for any given h ∈ (0, h0 ], for j ≥ N0 (h) + ν+ (h) + 1 the j-th ˜ eigenvalue of B+ (h) is the j-th eigenvalue of B(h) and is within distance 10−2hN to the j-th eigenvalue of B(h). Therefore, combining (10.19) and (10.23) yields the following theorem. Theorem 10.2.1. Let E, E > 0 with E > E ≥ E0 + cR + 102 be fixed. Given any fixed N ≥ 1, there exists h0 ∈ (0, 1] such that for any given h ∈ (0, h0 ] the following holds: , Spec(A) ∩ (E, +∞) ⊂ (10.24) μ (h) − hN , μ (h) + hN , μ (h)∈Spec(B)∩(E ,+∞)

and, more precisely, given any j ≥ 1 so large that λ j (h) > E, then, with the same j, |λ j (h) − μ j (h)| ≤ hN ,

(10.25)

where, recall, Spec(A(h)) = {−∞ < λ1 (h) ≤ λ2 (h) ≤ . . . → +∞} and Spec(B(h)) = {−∞ < μ1 (h) ≤ μ2 (h) ≤ . . . → +∞}.

Chapter 11

Energy-Levels, Dynamics, and the Maslov Index

Our aim here is to study periodicity properties of the integral trajectories of the Hamilton vector field of a given pseudodifferential symbol p, lying in energy-level sets of the kind p−1 (E), E ∈ [E1 , E2 ], with which we shall associate the action integral. This will be done in the next section. In section 11.2 we shall then give a crash introduction to the Maslov index of a periodic trajectory, with the aim of enabling the reader to compute it in the cases of interest for us. In the notes to the chapter we shall also give a rapid overview of the reason why one needs this symplectic invariant, that will systematically appear in Chapter 12.

11.1 Introducing the Dynamics We now introduce the Dynamics into our considerations. Recall that, given a smooth real-valued function p defined on phase-space Rnx × Rnξ = R2n X , X = (x, ξ ), we may consider the associated Hamilton vector-field defined by Hp =

∂p ∂ ∂p ∂ . − ∂xj ∂ξj j=1 ∂ ξ j ∂ x j n

∑

Notice that, using the canonical symplectic 2-form σ on R2n and its non-degeneracy, one has that H p is uniquely defined, at (x, ξ ) ∈ R2n , by the relation d p(v) = σ (v, H p ), ∀v ∈ R2n ,

(11.1)

where v is thought of as a tangent vector to R2n at (x, ξ ). The integral trajectories γ of H p , called the bicharacteristic curves of p (or simply bicharacteristics of p), are the solutions of dγ (t) = H p (γ (t)). dt


161

162

11 Energy-Levels, Dynamics, and the Maslov Index

We shall write

γX0 (t) = exp(tH p )(x0 , ξ0 ) = exp(tH p )(X0 ), γX0 (0) = (x0 , ξ0 ) = X0 , for the integral trajectory issued from X0 . It is easy to see that if X0 ∈ p−1 (E) then γX0 (t) ∈ p−1 (E) for all t in the interval of existence of γX0 . When p−1 (E) is compact, γX0 exists for all times. One calls p−1 (E) an energy surface (or energy level). Following Dimassi-Sjöstrand [7] and Helffer-Robert [20], we make the following assumptions on p. Assumption 11.1.1 (Dynamical assumptions (H1)-(H3)). Let E1 < E2 , and let 0 < ε be sufficiently small. We assume that: • (H1) d p = 0 for all X ∈ p−1 ([E1 − ε , E2 + ε ]); • (H2) p−1 (E) is connected for all E ∈ [E1 − ε , E2 + ε ]; • (H3) There exists a smooth function T =T (X) > 0 defined in p−1 ([E1 −ε , E2 +ε ])

such that exp(T (X)H p )(X) = X, ∀X ∈ p−1 ([E1 − ε , E2 + ε ]).

(11.2)

We have the following result. Lemma 11.1.2. Assume (H1), (H2) and (H3). Let X ∈ p−1 ([E1 − ε , E2 + ε ]) and let γX : [0, T (X)] t −→ exp(tH p )(X). Hence γX is a closed curve in p−1 ([E1 − ε , E2 + ε ]) (a periodic bicharacteristic of period T (X)). The functions X −→ T (X ), and X −→ J(X) :=

γX

ξ dx

are smooth and depend only on the value p(X ) : if X ∈ p−1 (E), for E ∈ [E1 −ε , E2 +ε ], we have T (X ) = Tp (E), J(X ) = J p (E), for some functions Tp and J p . Moreover, with J p = dJ p /dE, one has J p (E) = Tp (E). Recall that ξ dx = ∑nj=1 ξ j dx j is the canonical 1-form on R2n , such that d(ξ dx) = σ . Proof. Let [0, 1] s −→ X(s) ∈ p−1 ([E1 − ε , E2 + ε ]) be a C1 curve and put

ϕ (t, s) := exp(tH p )(X(s)), 0 ≤ s ≤ 1, 0 ≤ t ≤ T (X (s)),

11.1 Introducing the Dynamics

163

so that ϕ (T (X (s)), s) = ϕ (0, s) by (H3). Put γs := ϕ (t, s); t ∈ [0, T (X (s))] . Hence γs is a closed curve, for all s ∈ [0, 1]. We consider now the pull-back ϕ ∗ σ of the symplectic form σ by the map ϕ . Then

ϕ ∗ σ = α (t, s)dt ∧ ds, where, writing ϕ∗ for the tangent map associated with ϕ , ∂ ∂ ∂ (by (11.1)) α (t, s) = σ ϕ∗ ( ), ϕ∗ ( ) = σ H p , ϕ∗ ( ) ∂t ∂s ∂s ∂ ∂ = −d p(ϕ∗ ( )) = − p(ϕ (t, s)) . ∂s ∂s Since d(ξ dx) = σ , by the Stokes formula we have γs

ξ dx −

γ0

ξ dx =

s T (X(s )) ∂ 0

∂ s

0

p(ϕ (t, s )) dt ds .

(11.3)

But by the definition of ϕ (t, s) we have that p(ϕ (t, s)) = p(ϕ (0, s)) =: p(s), ¯ so that (11.3) reduces to γs

ξ dx −

γ0

ξ dx =

s 0

T (X(s ))

s ∂ p( ϕ (0, s )) ds = T (X (s ))d p(s ¯ ). (11.4) ∂ s 0

If all the γs are contained in the same energy surface p−1 (E), we get that γs

ξ dx −

γ0

ξ dx = 0,

that is, J(X (s)) is independent of s. Hence J(X) = J p (E) and (11.4) shows that J p (E) = Tp (E), where Tp (E) = T (X(0)).

We next show that assumptions (H1) and (H2) are fulfilled by smooth functions that are positively homogeneous of some positive degree (we will be interested in the case 2k, for some k ∈ N). This follows from Proposition 11.1.4 below. First of all, we have to recall Euler’s identity. Lemma 11.1.3. Let p ∈ C∞ (R2n \ {0}) be positively homogeneous of degree 0 = α ∈ R, that is p(τ X ) = τ α p(X), ∀τ > 0, ∀X = 0. (11.5) Then p(X) =

1 X, ∇X p(X) , ∀X = 0. α

(11.6)

164


Proof. Given any X = 0, we have on the one hand d p(τ X) = X, ∇X p(X ) , dτ τ =1 and on the other d d α p(τ X ) = τ p(X) = α p(X ). dτ τ =1 dτ τ =1

This concludes the proof.

Proposition 11.1.4. Let p ∈ C∞ (R2n \ {0}; R+ ) be positively homogeneous of degree 2k, for some k ∈ N, that is p(τ X ) = τ 2k p(X), ∀τ > 0, ∀X = 0. Let I be any given closed interval of R, with I ⊂ (0, +∞). Then • p−1 (I) is a connected set, which is compact when I is bounded; • For all E > 0, the set p−1 (E) is a smooth compact and connected hypersurface

of R2n \ {0}.

Hence hypotheses (H1) and (H2) are always satisfied in this case, for every closed energy interval I ⊂ (0, +∞). Proof. Write any given X = 0 using polar coordinates as X = |X |

ρ > 0 and ω ∈ S2n−1 . Then

X = ρω , with |X |

p(X) = ρ 2k p(ω ), where 0 < c1 = min p(ω ) ≤ c2 = max p(ω ). ω ∈S2n−1

ω ∈S2n−1

Now, given any X ∈ p−1 (I), we have that p(X ) = ρ 2k p(ω ) = E for some E ∈ I, whence ρ = (E/p(ω ))1/2k . This shows that the map

γ : S2n−1 × I −→ p−1 (I), γ (ω , E) =

E 1 2k ω, p(ω )

is smooth and onto. Since S2n−1 × I is connected, this proves that p−1 (I) is also connected. When I is in addition also bounded, hence compact, then S2n−1 × I is compact, and therefore the same holds for p−1 (I). This proves the first claim in the statement. To prove the second claim, let E > 0 and take any X0 ∈ p−1 (E) (then, necessarily, X0 = 0). Hence X0 satisfies the equation p(X0 ) = E. By Euler’s relation (11.6) we have 1 0 < E = p(X0 ) = X0 , ∇X p(X0 ) , 2k which implies that ∇X p(X0 ) = 0, and concludes the proof.


165

Let us now define the averaged action-integral associated with a trajectory [0, Tp (E)] t −→ γX (t) = exp(tH p )(X) contained in p−1 (E). Definition 11.1.5. In the dynamical assumptions (H1), (H2) and (H3) we define the averaged action-integral A p (E) = A p,γX (E) of an integral curve γX ⊂ p−1 (E), γX : [0, T (X) = Tp (E)] t −→ exp(tH p )(X), by A p (E) :=

1 T (X)

γX

ξ dx − E =

J p (E) − E. Tp (E)

(11.7)

Notice that, equivalently, since p(γX (t)) = E for all t ∈ [0, T (X )], A p (E) =

1 T (X)

γX

ξ dx −

1 T (X)

T (X) 0

p ◦ exp(tH p )(X)dt.

(11.8)

By Lemma 11.1.2 the definition of A p does not depend on γX , but only on E. When p satisfies (H1), (H2), (H3) and is such that the period Tp (E) is independent of E, for E belonging to some energy-interval I, then A p (E) is constant, as shown by the next lemma. Lemma 11.1.6. Assume (H1), (H2), (H3). Suppose that p is such that for some energy-interval I = [E1 , E2 ] the period Tp is independent of E. Then A p (E) is constant for all E ∈ I. Proof. Let Tp (E) = T0 for all E ∈ I. Since J p (E) = Tp (E) = T0 for all E ∈ [E1 , E2 ], we have that J p (E) = T0 E + J p (E1 ) on I. Hence A p (E) =

J p (E) J p (E1 ) J p (E1 ) T0 E −E = −E + = , ∀E ∈ I. Tp (E) T0 T0 T0

It will be useful also to have the following resut, about the behavior of the period Tp (E) and the action A p (E) when one changes the symbol p to α p, where α ∈ R+ is a constant. Lemma 11.1.7. Suppose p satisfies hypotheses (H1), (H2), (H3). Let α ∈ R+ be a constant. Put q=α p. Then q satisfies (H1), (H2), (H3) on the interval [α (E1 − ε ), α (E2 + ε )] with Tq (α E) = Furthermore,

Tp (E) , E ∈ [E1 − ε , E2 + ε ]. α

Aq (α E) = α A p (E), ∀E ∈ [E1 − ε , E2 + ε ].

(11.9)

166


Proof. We have p−1 (E) = q−1 (α E), ∀E ∈ [E1 − ε , E2 + ε ].

Hq = α H p ,

Given any X ∈ p−1 (E) = q−1 (α E), E ∈ [E1 − ε , E2 + ε ], consider the curve

γ˜X : [0, Tp (E)/α ] t −→ exp(α tH p )(X) = exp(tHq )(X). Then, for all X ∈ q−1 (α E) = p−1 (E), with E ∈ [E1 − ε , E2 + ε ],

γ˜X (t) = (xq (t), ξq (t)) = exp(tHq )(X) = exp(t α H p )(X) = (x p (α t), ξ p (α t)) = γX (α t), ∀t ∈ [0, Tq (α E)], where γX : [0, Tp (E)] s −→ exp(sH p )(X). It follows that exp(

Tp (E) Hq )(X) = exp(Tp (E)H p )(X) = X , α

which shows that the function q−1 ([α (E1 − ε ), α (E2 + ε )] X −→ T (X) = Tq (q(X )) =

Tp (p(X )) >0 α

is smooth. Hence hypotheses (H1), (H2) and (H3) hold for q. Next, one has Jq (α E) = =

ξ dx =

γ˜X Tq (α E) 0

=α

Tq (α E) 0

ξq (t), x˙q (t) dt

ξq (t), (∂ξ q)(exp(tHq )(X)) dt

Tq (α E) 0

ξ p (α t), (∂ξ p)(exp(t α H p )(X)) dt

(setting α t = s) =

α Tq (α E) 0

ξ p (s), x˙p (s) ds =

γX

ξ dx = J p (E).

It thus follows that Aq (α E) =

Jq (α E) J p (E) − αE = − α E = α A p (E), Tq (α E) Tp (E)/α

for all E ∈ [E1 − ε , E2 + ε ], which concludes the proof.


167

We next show that whenever there is T = Tp (E) for which hypothesis (H3) holds for a p positively homogeneous of degree 2, then T is independent of E, and A p (E) = 0. Lemma 11.1.8. Suppose p : R2n \ {0} −→ R+ is smooth, positively homogeneous of degree 2 and that (H3) is satisfied on [E1 − ε , E2 + ε ] ⊂ R+ . Then Tp does not depend on E ∈ [E1 − ε , E2 + ε ], and A p (E) = 0 for all E ∈ [E1 − ε , E2 + ε ]. Proof. Consider γX (t) = exp(tH p )(X), where X ∈ p−1 (E). In the first place, we notice that γX∗ (ξ dx) = ξ (t), x(t)

˙ dt. Now, on the one hand by virtue of the periodicity we have Tp (E) d

dt

0

ξ (t), x(t) dt = ξ (Tp (E)), x(Tp (E)) − ξ (0), x(0) = 0,

and on the other Tp (E) d

dt

0

ξ (t), x(t) dt =

Tp (E) 0

ξ˙ (t), x(t) + ξ (t), x(t)

˙ dt,

whence J p (E) =

γX

ξ dx =

Tp (E) 0

ξ (t), x(t) dt ˙ =−

Tp (E) 0

ξ˙ (t), x(t) dt.

Since, by Hamilton’s equations, Tp (E) 0

and

Tp (E) 0

ξ (t), x(t) dt ˙ =

Tp (E) .

ξ˙ (t), x(t) dt = −

it follows that

0

ξ (t),

Tp (E) . ∂p 0

∂x

/ ∂p (x(t), ξ (t)) dt ∂ξ

/ (x(t), ξ (t)), x(t) dt,

1 Tp (E) γX (t), (∇X p)(γX (t)) dt. 2 0 But Euler’s relation (11.6) says that J p (E) =

2p(X ) = X , (∇X p)(X) , so that, since p(γX (t)) = E for all t, J p (E) =

1 2

Tp (E) 0

γX (t), (∇X p)(γX (t)) dt =

Tp (E) 0

p(γX (t))dt = ETp (E). (11.10)

168


Using Lemma 11.1.2 and Proposition 11.1.4, we therefore get J p (E) = Tp (E) = ETp (E) + Tp(E), ∀E ∈ [E1 − ε , E2 + ε ], whence

Tp (E) = 0, ∀E ∈ [E1 − ε , E2 + ε ].

On the other hand, from (11.10) we have A p (E) =

ETp (E) J p (E) −E = − E = 0, Tp (E) Tp (E)


Remark 11.1.9. If p : R2n \ {0} −→ R+ is positively homogeneous of degree 2k, k ≥ 2, then we have J p (E) = kETp (E), ∀E > 0, whence, with Tp = dTp /dE, the equation Tp (E) = kETp (E) + kTp (E), i.e. ETp (E) =

1−k Tp (E), E > 0. k

Hence, fixing E0 ∈ [E1 − ε , E2 + ε ] ⊂ R+ (thus E0 > 0) gives E (1−k)/k Tp (E) = Tp (E0 ) , E ∈ [E1 − ε , E2 + ε ]. E0 When n = 1 and 0 < p is positively homogeneous of degree 2 we have that p−1 (E) is a periodic trajectory of H p of period T > 0, independent of E, for all E > 0. In fact, we have the following proposition. Proposition 11.1.10. Let n = 1, and let p : R2 \ {0} −→ R+ be smooth, positively homogenoeus of degree 2. Then all the integral trajectories of H p lying in p−1 (0, +∞) are periodic with least period Tp independent of the energy and given by the formula 2π dθ . (11.11) Tp = 2p(sin θ , cos θ ) 0 Proof. Let E > 0, and take X0 ∈ p−1 (E). We use polar coordinates

ϕ : (0, +∞) × [0, 2π ] −→ R2 \ {0}, ϕ (ρ , θ ) = (ρ sin θ , ρ cos θ ) = X . Then

ϕ ∗ σ = ρ d ρ ∧ d θ , ϕ ∗ { f , g} =

1 ∗ {ϕ f , ϕ ∗ g}ρ ,θ , ρ


169

where ϕ ∗ f = f ◦ ϕ , { f , g} is the Poisson bracket of f and g, and, finally, { f˜, g} ˜ ρ ,θ := ∂ρ f˜∂θ g˜ − ∂θ f˜∂ρ g. ˜ Hence exp(tH p )(X0 ) = ϕ (ρ (t), θ (t)), X0 = ϕ (ρ0 , θ0 ), where, with θ˙ = d θ /dt and ρ˙ = d ρ /dt, ρ and θ satisfy the equations ⎧ 1 ⎪ ∗ ⎪ ⎨ θ˙ = ∂ρ (ϕ p) ρ 1 ⎪ ⎪ ⎩ ρ˙ = − ∂θ (ϕ ∗ p). ρ

(11.12)

(ϕ ∗ p)(ρ , θ ) = p(ρ sin θ , ρ cos θ ) = ρ 2 p(sin θ , cos θ ),

(11.13)

By the homogeneity

where for p(sin θ , cos θ ) = p(ω ) we have 0 < c1 = min p(ω ) ≤ c2 = max p(ω ). ω ∈S1

ω ∈S1

It follows from (11.13) and the first equation in (11.12) that the map t −→ θ (t) exists for all t ∈ R (the derivative being uniformly bounded), and that

θ˙ = 2p(sin θ , cos θ ) ≥ 2c1 , that is,

R t −→ θ (t) ∈ R is an increasing diffeomorphism,

(11.14)

and t −→ exp(tH p )(X0 ) is therefore the closed curve p−1 (E). By Lemma 11.1.8 we thus have that the least period, given by the relations exp(Tp (E)H p )(X0 ) = X0 , and exp(tH p )(X0 ) = X0 , ∀t ∈ (0, Tp (E)), i.e. given by the condition θ (Tp ) = θ0 + 2π , is independent of the energy. The formula for the least period Tp follows again from the homogeneity of p, for 1 ∂ θ˙ = (ϕ ∗ p) = 2p(sin θ , cos θ ), ρ ∂ρ whence (by the periodicity of p(sin θ , cos θ )) Tp =

θ0 +2π θ0

dθ = 2p(sin θ , cos θ )

2π 0

dθ . 2p(sin θ , cos θ )

170


It is important to notice that from Proposition 11.1.10 it follows that the closed curve p−1 (E) is clockwise oriented. It will also be important, when considering the Maslov index of a curve in R × R, to be able to establish a-priori the orientation of a periodic trajectory p−1 (E) = γE = {exp(tH p )(X); 0 ≤ t ≤ T (E)} ⊂ R × R for p satisfying Assuption 11.1.1. In this case, one observes that

σ (∇X p(X ), H p (X )) = d p(X )(∇X p(X)) = |∇X p(X )|2 > 0, ∀X ∈ γE , so that

σ (∇X p, H p ) = − det[∇x p|H p ] > 0, ∀X ∈ γE ,

whence, in R × R, • whenever ∇x p(X ) points outwards from γE at X ∈ γE , then this is true at all

points of γE , and the couple {∇X p, H p } forms on γE a reference frame with the orientation opposite to the canonical one, that is, γE is clockwise oriented; • whereas, whenever ∇x p(X ) points inwards from γE at X ∈ γE , then this is true at all points of γE , and the couple {∇X p, H p } forms on γE a reference frame which is canonically oriented, that is, γE is counter-clockwise oriented.

Lemma 11.1.11. Let p : R2n \ {0} −→ R+ be smooth, positively homogeneous of degree 2k, k ∈ N. Let E > 0 be given, and consider the compact (connected) set p−1 (E) = γE . Then ∇X p(X ) points outwards from γE , for all X ∈ γE . Hence, in particular, when n = 1 and γE is a periodic trajectory of H p , it is clockwise oriented. Proof. We know that γE ⊂ R2n \ {0} is a smooth compact and connected hypersurface of R2n , whence it is orientable. The vector-field γE X −→ ∇X p(X ) is a nowhere-zero normal to γE , whence it either points outwards from γE at all points of γE , or else it points inwards from γE at all points of γE . Consider the compact set DE = {X = 0; p(X) ≤ E} ∪ {0}. Then ∂ DE = γE , and if E > E, by virtue of the homogeneity, we have DE DE . Hence γE is contained in the exterior of γE . Since moving from γE to γE increases the value of p from E to E , and since ∇X p gives the direction of maximal growth of p, we get that ∇X pγ points outwards from γE . E This concludes the proof. Of course, the above lemma also holds when the homogeneity degree of p is a real α > 0. Remark 11.1.12. Consider the “Schrödinger” non-homogeneous Hamiltonian in R2 given by p(x, ξ ) = ξ 2 +V (x), where = (x2 −1)2 . Consider the equation V (x) = ' V (x) √ E, E ∈ (0, 1). Let x±,b (E) 1 + E the outer solutions (“big”, in absolute ' =± √ value), and x±,s (E) = ± 1 − E the inner solutions (“small”, in absolute value). It is well known that we then have two periodic motions associated with H p , whose x-projections lie in [x−,b (E), x−,s (E)] and [x+,s (E), x+,b (E)], respectively, which are possible due to the conditions


171

V (x−,b (E)) < 0 < V (x−,s (E)), V (x+,s (E)) < 0 < V (x+,b (E)), which yield that on the periodic trajectories the vector-field ∇X p is pointing outwards. Hence the periodic trajectories are clockwise oriented. Let us now use Lemma 11.1.2 to compute the periods of the periodic trajectories associated with the eigenvalues of our NCHO Q(α ,β ) , for α , β > 0, αβ > 1 and α = β . Of course, one may equivalently use (11.11). We have the following result. Lemma 11.1.13. Recall that α , β > 0, αβ > 1, and that p0 (x, ξ ) = (x2 + ξ 2 )/2, (x, ξ ) ∈ R2 . Put μ± = (α ± β )/2. For E > 0, the integral curves γ± (E) = λ±−1 (E) of the Hamilton vector-fields associated with the eigenvalues

λ± (x, ξ ) = μ+ p0 (x, ξ ) ±

)

μ−2 p0 (x, ξ )2 + x2 ξ 2

of Q(α ,β ) (x, ξ ), (x, ξ ) ∈ R2 \ {0}, are all periodic, with least periods

α +β π∓ T± = T± (α , β ) = ' αβ (αβ − 1)

2π 0

)

μ−2 + sin2 (2θ )

αβ − sin2 (2θ )

dθ ,

(11.15)

all independent of the energy E. (When E = 0, γ± (0) = {(0, 0)}.) Proof. We have to compute Jλ ± (E). We use polar coordinates to parametrize the curves γ± (E). For θ ∈ [0, 2π ], consider √ √ (x(θ ), ξ (θ )) = ( 2 ρ± (θ ) sin θ , 2 ρ± (θ ) cos θ ), where

⎛

ρ± (θ ) = ⎝

⎞1/2

μ+ ±

)

E

μ−2 + sin2 (2θ )

⎠

.

Now, we have that on λ±−1 (E) ξ dxλ −1 (E) = 2ρ± (θ ) cos θ d ρ± (θ ) sin θ ± dρ ± = (θ )ρ± (θ ) sin(2θ ) + 2ρ±(θ )2 cos2 θ d θ . dθ Hence γ± (E)

ξ dx =

2π d ρ± (θ )2 0

dθ

2

sin(2θ )d θ + 2E

2π 0

cos2 θ ) dθ μ+ ± μ−2 + sin2 (2θ )

172


2π ρ± (θ )2

= −2 = −E

2π

+2E =E

2

0

0

⎛

2π 0

cos2 θ ) dθ μ+ ± μ−2 + sin2 (2θ )

0

cos2 θ ) dθ μ+ ± μ−2 + sin2 (2θ )

μ+ ±

= E ⎝ μ+

2π

cos2 θ − sin2 θ ) dθ μ+ ± μ−2 + sin2 (2θ )

2π 0

cos(2θ )d θ + 2E

2π 0

)

dθ

μ−2 + sin2 (2θ )

dθ ∓ αβ − sin2 (2θ )

)

2π

μ−2 + sin2 (2θ )


0

⎞ dθ ⎠ .

Using the well-known formula 2π 0

dθ 2π =' , 2 αβ − sin (θ ) αβ (αβ − 1)

we thus obtain ⎛

α +β π∓ Jλ± (E) = E ⎝ ' αβ (αβ − 1)

2π

)

μ−2 + sin2 (2θ )


0

⎞ dθ ⎠ ,

whence, finally,

α +β π∓ T± = T± (α , β ) = ' αβ (αβ − 1)

2π 0

)

μ−2 + sin2 (2θ )


dθ ,

by Lemma 11.1.2. That the T± are the smallest periods follows at once from Proposition 11.1.10 and (11.14). We close this section by giving an example of symbol p whose (non-trivial) trajectories are all periodic, with period dependent on the energy, and whose trajectories are counter-clockwise oriented. Consider in R2 the symbol p(x, ξ ) = 2 2 e−(x +ξ ) . Let E ∈ (0, 1) and consider 2 +ξ 2 )

p−1 (E) = {(x, ξ ); e−(x

= E} = {(x, ξ ); x2 + ξ 2 = ln(1/E)} =: γE . ' Observe that ∇X p = 0 iff X = 0. One computes, at X0 = ( ln(1/E), 0) ∈ γE , ∇X p(X0 ) = −2EX0,

11.2 The Maslov Index

173

which is opposite to the direction (1, 0), whence it points inwards from γE . By the above discussion, we have that γE is counter-clockwise oriented. We now compute the period of the trajectory. We use polar coordinates (x, ξ ) = (ρ cos θ , ρ sin θ ). Then Hamilton’s equations become, using ρ 2 ≡ ln(1/E),

ρ˙ = 0, θ˙ = 2E > 0. The latter equation proves, once more, that γE is counter-clockwise oriented. Hence x=

' ' ln(1/E) cos(2Et), ξ = ln(1/E) sin(2Et),

with least period T (E) = π /E. Hence γE = {exp(tH p )(X0 ); 0 ≤ t ≤ T (E)}. Notice, finally, that in this case J(E) = π ln E < 0, E ∈ (0, 1).

11.2 The Maslov Index The classical approach to the Maslov index is due to V. Arnold (see [2] and also Hörmander [26]). We shall follow here (with some differences, though) Duistermaat’s book [12], Hörmander’s book [29], McDuff-Salamon’s book [43], Treves’ book [70], and the paper by Robbin and Salamon [62]. We address the reader to those books for a deep study of Symplectic Geometry. We will content ourselves here with giving the statements in a form suited to our purposes, and prove only those statements whose proofs will not carry us “too far away”. The main difference with the approaches in the above-mentioned books is that we shall give the Maslov cycle a different orientation, so as to have that the closed trajectories of the harmonic oscillator, oriented in the clockwise direction (which is natural to the symplectic form d ξ ∧ dx in R × R), have Maslov index 2 (our choice is consistent with Arnold’s one). Recall that we are working in phase-space T ∗ Rn = Rn × Rn , endowed with the canonical symplectic form (see (2.1))

σ=

n

∑ d ξ j ∧ dx j ,

j=1

where (x, ξ ) are symplectic coordinates in T ∗ Rn . Recall we shall write a tangent

that δ x . Hence vector v ∈ T(x,ξ ) T ∗ Rn = Rn × Rn at a point (x, ξ ) as δξ

σ (v, v ) = δ ξ , δ x − δ ξ , δ x , v =

δx δx , v = . δξ δξ

174


The map Rn × Rn (x, ξ ) −→ x + iξ ∈ Cn allows us to identify T ∗ Rn with Cn . The Hermitian scalar product in Cn is defined by v, w Cn =

n

∑ (δ x j + iδ ξ j )(δ y j − iδ η j ),

j=1

where

v=

δx δy $ δ x + iδ ξ , w = $ δ y + iδ η , δξ δη

so that Re v, w Cn = v, w R2n = is the Euclidean scalar product in

Rn × R n ,

n

∑ (δ x j δ y j + δ ξ j δ η j )

j=1

and

Im v, w Cn = σ (v, w). Throughout this section we shall put Sn := T ∗ Rn = Rn × Rn . Definition 11.2.1. Define Λ (n) to be the set of n-dimensional real subspaces λ of Rn × Rn such that σ (v, v ) = 0, ∀v, v ∈ λ . One calls λ a Lagrangian subspace. In other words, denoting by λ σ the σ -orthogonal of λ , one has that λ is a Lagrangian subspace if λ = λ σ . Notice hence that, when Rn × Rn is identified with Cn , having a Lagrangian subspace λ means that λ is a real subspace of Cn (of real dimension n) and that λ and iλ are orthogonal with respect to the Eucliden scalar product ·, · R2n . Remark 11.2.2. In general, given a subspace λ ⊂ Sn , one says that λ is • isotropic when λ ⊂ λ σ , • Lagrangian when λ = λ σ , • involutive when λ σ ⊂ λ .

The following result gives that Λ (n) is a smooth manifold. Lemma 11.2.3. The unitary group U(n) acts on Λ (n) transitively, with stabilizer the orthogonal group O(n). Hence Λ (n) = U(n)/O(n) is a smooth manifold of dimension n2 − n(n − 1)/2 = n(n + 1)/2. Proof. It is clear that if U ∈ U(n), then λ ∈ Λ (n) implies U λ ∈ Λ (n). To see that U(n) acts transitively, take λ ∈ Λ (n) and let w1 , . . . , wn be a ·, · R2n -orthonormal basis of λ . Thus w j , wk R2n = δ jk , σ (w j , wk ) = 0, so that w j , wk Cn = δ jk , and we have a unitary basis of Cn . We have therefore the unitary matrix U = [w1 |w2 | . . . |wn ] (the j-th column of U is the vector w j ) such that URn = λ , which shows the transitive action. We now have that URn = Rn if and only if U has real coefficients, that is U ∈ O(n), which concludes the proof.


175

We next study (following Duistermaat [12]) the tangent spaces Tλ Λ (n) at λ ∈ Λ (n). We have first to show that any given Lagrangian subspace λ possesses a Lagrangian complement μ , that is, an element μ ∈ Λ (n) such that λ ∩ μ = {0} and λ + μ = Sn . Lemma 11.2.4. Given any λ ∈ Λ (n), there is a Lagrangian complement μ . Notice that a Lagrangian complement is not unique. Proof. We start with a subspace μ ⊂ Sn , with dim μ < n, such that λ ∩ μ = {0} and μ μ σ . Then μ σ cannot be contained in λ + μ , since μ σ ⊂ λ + μ yields μ ⊃ λ σ ∩ μ σ = λ ∩ μ σ . Hence λ ∩ μ σ = {0}, and since dim μ σ > n and dim λ = n, this is impossible. Thus we may choose e ∈ μ σ \ (λ + μ ). It follows that μ + Span{e} ⊂ (μ + Span{e})σ (as the reader may easily check) and (μ + Span{e}) ∩ λ = {0}. We can therefore construct by induction a Lagrangian subspace μ such that μ ∩ λ = {0}, and hence conclude the proof. Remark 11.2.5. The proof of Lemma 11.2.4 is valid in any given symplectic vector space. Using Lemma 11.2.3, namely the transitivity of the U(n)-action, one may prove Lemma 11.2.4 also as follows. We have that λ = URn , for some U ∈ U(n). One may then take μ = eiθ URn , for any fixed θ = kπ , k ∈ Z, which concludes the proof, and shows also the non-uniqueness of the Lagrangian complement. Theorem 11.2.6. Given any λ ∈ Λ (n), let Symm(λ ) be the vector space of all symmetric bilinear forms on λ . Then Tλ Λ (n) $ Symm(λ ), in a canonical way. Proof. Let λ , μ ∈ Λ (n) with λ ∩ μ = {0}. Each n-dimensional subspace λ which is transversal to μ (that is, λ + μ = Sn ) is of the form λ = {v + Av; v ∈ λ }, for a linear map A : λ −→ μ . Then qλ (v, w) := σ (v, Aw), v, w ∈ λ , defines a bilinear form on λ which is symmetric iff λ ∈ Λ (n). The latter follows immediately by observing that for v, w ∈ λ

σ (v + Av, w + Aw) = σ (v, Aw) − σ (w, Av) = 0 ⇐⇒ λ ∈ Λ (n). (Notice that λ → qλ differs from the map constructed by Duistermaat in [12] by a minus sign). This map defines therefore a bijective map q : {λ ∈ Λ (n); λ ∩ μ = {0}} =: Λ0 (μ ) −→ Symm(λ ), q : λ −→ qλ , (11.16) which depends on λ and μ (notice that q(λ ) = qλ = 0). However, its differential at λ does not depend on μ . In fact, let μ˜ ∈ Λ (n) be such that μ˜ ∩ λ = {0}. Then

176


μ˜ = {v + Bv; v ∈ μ }, for some linear map B : μ −→ λ . If λ is close enough to λ , then λ is still transversal to both μ and μ˜ , and we may write ˜ u ∈ λ }, λ = {w + Aw; w ∈ λ } = {u + Au; for certain linear maps A : λ −→ μ and A˜ : λ −→ μ˜ . It follows that ˜ ˜ = v + Bv, w + Aw = u + Au, and Au for some v ∈ μ . Hence

w + Aw = u + v + Bv,

from which it follows that w − Bv − u = v − Aw, ∈λ

∈μ

which yields, since λ ∩ μ = {0}, that w − Bv = u, and v = Aw. Hence ˜ − BAw), w + Aw = (w − BAw) + A(w and taking the symplectic product with z ∈ λ gives ˜ − BAw)) = σ (z, Aw) ˜ − σ (z, ABAw). ˜ σ (z, Aw) = σ (z, A(w Now, λ → λ iff A → 0 and A˜ → 0, which gives that ˜ ˜ σ (z, ABAw) = O(AA), i.e. ˜ + term vanishing to 2nd order when λ → λ , ∀w, z ∈ λ , σ (z, Aw) = σ (z, Aw) which concludes the proof.

Remark 11.2.7. From the proof of the above theorem it follows that the pairs (Λ0 (μ ), q), as μ varies in Λ (n), form an atlas of Λ (n). One sees that each Λ0 (μ ) is dense and open in Λ (n), and that, since the latter is compact, 2n of the former sets cover Λ (n). Moreover, in the notation of Treves’ book [70] one has qλ (v, w) = −βμλ,λ (v, w) = σ (Pμλ,λ v, w), v, w ∈ λ ,


177

where Pμλ,λ is the projection μ ⊕ λ → μ restricted to λ . In fact, if λ = {v+Av; v ∈ λ } where A : λ −→ μ , with λ , λ , μ as in the theorem, then Av = −Pμλ,λ v, for all v ∈ λ , and the claim follows from the symmetry of qλ . It will be convenient to have the following lemma, which is essentially contained in the previous theorem. Lemma 11.2.8. Let λ , μ ∈ Λ (n), with λ ∩ μ = {0}. Then any other given Lagrangian complement α of λ is the graph of a symmetric quadratic form on μ , with radical given by μ ∩ α . Proof. We shall write μ ∗ for the (real) dual space of μ . Consider thus a Lagrangian complement α ⊂ Sn of λ . Then α = {v + Av; v ∈ μ } for a linear map A : μ −→ λ . Note that Ker A = μ ∩ α . Since λ $ μ ∗ , where the isomorphism is realized through ∼σ the symplectic form σ by λ v −→ (μ w −→ σ (w, v)) ∈ μ ∗ (the proof is left to ∼σ A μ ∗ ∈ μ ∗ ⊗ μ ∗, the reader), we may consider the composite map A˜ : μ −→ λ −→ ˜ = σ (·, Av). Consider now the bilinear form Av ˜ )v ∈ R. qα : μ × μ (v, v ) −→ σ (v, Av ) = (Av As α ∈ Λ (n), the form qα is symmetric, which precisely means that A˜ ∈ S2 μ ∗ . Moreover, it is clear that Rad qα = μ ∩ α . This concludes the proof. From Lemma 11.2.8 it follows in particular that when, say, λ = Rn × {0}, choosing μ = {0} × Rn gives that every Lagrangian complement α ⊂ Sn of λ is therefore of the form

α = {(Av, v); v ∈ Rn }, for a symmetric n × n real matrix A.

(11.17)

We next describe (following [43,62]) Lagrangian subspaces of Sn in more explicit terms. One has the following lemma. Lemma

11.2.9. Let X ,Y be real n × n matrices, let Z be the 2n × n real matrix X Z= , and define λ ⊂ Sn by Y

λ = Im Z = {Zu; u ∈ Rn }. Then λ ∈ Λ (n) iff rank Z = n and XY = tY X.

t

Every element of Λ (n) can be written in this form.

(11.18)

178


Proof. Let v = Zu, v = Zu ∈ λ , for u, u ∈ Rn . Then

σ (v, v ) = Yu, Xu − Yu, Xu = (tXY − tY X)u, u = 0, ∀u, u ∈ Rn , iff (11.18) holds.

X , with rank n and We now prove that given any λ ∈ Λ (n) there exists Z = Y (11.18) fulfilled, such that λ = Im Z. In case λ = λo := Rn × {0} we may take X = I and Y = 0, and in case λ = λv := {0} × Rn we may take X = 0 and Y = I. So, we may suppose λ = λo , λv . Since Sn = λo ⊕ λv , we have that each w ∈ λ is uniquely decomposed as w = w + w , where w ∈λo and w ∈ λv . Choose a basis X by taking X = [v1 | . . . |vn ] and {w1 , . . . , wn } of λ and define the matrix Z = Y Y = [v1 | . . . |vn ]. This concludes the proof. Notice that Lemma 11.2.9 gives back (11.17), for it suffices to take X = A, Y = I. The matrix Z of Lemma 11.2.9 is called a Lagrangian frame. It is useful to recall now that a linear map χ : Sn −→ Sn is symplectic if

σ (χ (v), χ (v )) = σ (v, v ), ∀v, v ∈ Sn , which is equivalent to saying that

t

0 −I 0 −I χ χ= , I 0 I 0

which in turn is equivalent to saying that the inverse χ −1 of χ = block-form

χ −1 =

−tB . −t C tA tD

A B has the C D (11.19)

Notice that if χ : Sn −→ Sn is a linear symplectic map, then χ (λ ) ∈ Λ (n) if λ ∈ Λ (n). It is now possible to make the quadratic forms in Tλ Λ (n), λ ∈ Λ (n), given by Theorem 11.2.6, more explicit. We have the following result, due to Robbin and Salamon [62] (adapted to our choice of symplectic form and quadratic forms qλ ). Theorem 11.2.10. Let [a, b] t −→ λ (t) ∈ Λ (n) be a smooth curve of Lagrangian subspaces, with λ (0) = λ and λ˙ (0) = λ0 ∈ Tλ Λ (n). (i) Let μ ∈ Λ (n) be a fixed Lagrangian complement of λ , and for v ∈ λ and t small, define w(t) ∈ μ by v + w(t) ∈ λ (t). Then the quadratic form on λ , qλ ,λ0 (v) = is independent of the choice of μ .

d σ (v, w(t)), dt t=0


179

(ii) Let [a, b] t −→ Z(t) =

X (t) be a Lagrangian frame of λ (t) as in (i). Then Y (t)

˙ − X(0)u, Y˙ (0)u , v = Z(0)u, u ∈ Rn . qλ ,λ0 (v) = Y (0)u, X(0)u

(11.20)

(iii) The form qλ ,λ0 is natural, in the sense that q χ (λ ),χ (λ0) ◦ χ = qλ ,λ0 , for any given symplectic matrix χ . Proof. We may choose coordinates in Sn in such a way that λ (0) = λ = Rn × {0}. We start by proving (i). By Lemma 11.2.8 we then have that any given Lagrangian complement of λ (0) is the graph of a real n × n symmetric matrix A. Hence μ = {(Au, u); u ∈ Rn }, and for small t the Lagrangian subspace λ (t) is the graph of n a real n × n symmetric

matrix B(t), λ (t) = {(z, B(t)z); z ∈ R }. We therefore have z Au(t) that v = , w(t) = , and by construction u(t) = B(t)(z + Au(t)). Thus 0 u(t) σ (v, w(t)) = − u(t), z , and ˙ ˙ z = − B(0)z, z , qλ ,λ0 (v) = − u(0), which is independent of A and proves (i). To prove (ii), assume that μ = {0} × Rn. Let Z(t) be a Lagrangian frame for X(0)u 0 λ (t). Then v = , and w(t) = , where, by construction, Y (0)u z(t) Y (0)u + z(t) = Y (t)X(t)−1 X (0)u (since λ (0) = λ = Rn × {0}, X (0) = I and therefore X (t) is invertible for small t). Thus σ (v, w(t)) = − z(t), X (0)u and, since ˙ + X (0)−1 X(0)

dX(t)−1 X (0) = 0, dt t=0

this yields qλ ,λ0 (v) = − ˙z(0), X (0)u = (Y (0)X(0)−1 X˙ (0) − Y˙ (0))u, X (0)u = (using tXY = t Y X) = Y (0)u, X˙ (0)u − Y˙ (0)u, X (0)u , ∀u ∈ Rn . This proves (ii). The proof of (iii) is an immediate consquence of the definition.

180


Remark 11.2.11. It is useful to “spell out” the construction of w(t) in terms of Theorem 11.2.6. We take a Lagrangian subspace μ transversal to λ , and λ (t) transversal to μ (for small t). We then write λ (t) = {v + A(t)v; v ∈ λ } for A(t) : λ −→ μ , so that, since μ ⊕ λ (t) = Sn ,

λ v = v1 (t) + v2 (t) =⇒ v2 (t) = v − v1(t), ∈μ

that is

∈λ (t)

w(t) = A(t)v = −v1 (t) = −Pμλ,λ (t) v.

Hence qλ ,λ0 (v) =

d d d σ (v, w(t)) = − βμλ,λ (t) (v, v) = σ (v, A(t)v), v ∈ λ . dt t=0 dt t=0 dt t=0

= {z ∈ C; |z| = 1} takes values ±1 on O(n), Now, since det : U(n) −→ we have that det2 : Λ (n) −→ S1 is well-defined (where, recall, det2 : Mn A −→ det(A2 )). One has the following theorem (see [2]; see also [43]). S1

Theorem 11.2.12. The map det2 : Λ (n) −→ S1 induces an isomorphism of fundamental groups π1 (Λ (n)) $ π1 (S1 ) = Z, called the Maslov index of Rn × Rn . As a consequence, H 1 (Λ (n); Z) $ Hom(π1 (Λ (n)); Z) $ Z, with generator the Maslov class, that is, the pull-back by det2 of the generator of H 1 (S1 ; Z). The Maslov class depends only on the chosen symplectic structure (see Hörmander [26, p. 156]). Following Robbin-Salamon [62], but with the opposite choice of symplectic form, we now explain a way to compute the Maslov index of a curve in Λ (n) (their discussion is much more general, but we shall content ourselves with considering only the “vertical” case). Let λv = {0} × Rn be the “vertical” Lagrangian subspace of Sn . One has that

Λ (n) =

n

Λk (λv ),

k=0

where Λk (λv ) := {λ ∈ Λ (n); dim(λ ∩ λv ) = k}. It turns out (see, e.g., [12] or [62]) that Λk (λv ) is a submanifold of Λ (n) of codimension k(k + 1)/2, which is connected. The Maslov cycle (determined by λv ) is the algebraic variety (a singular hypersurface of Λ (n))

Σ (λv ) = Λ1 (λv ) =

n

Λk (λv ),

k=1

whose regular part is Λ1 (λv ), and whose tangent space at a point λ ∈ Λk (λv ) (k ≥ 1, of course) is given by (see Theorem 11.2.6, Lemma 11.2.8 and Theorem 11.2.10)


181

Tλ Λk (λv ) = {λ˜ ∈ Tλ Λ (n); qλ ,λ˜ λ ∩λv = 0}. The cycle Σ (λv ) carries a natural orientation. If λ ∈ Λ1 (λv ), we have that Tλ Λ1 (λv ) is identified with the space of all symmetric bilinear forms on λ which vanish on the (one-dimensional) line λ ∩ λv , which is a hyperplane in Symm(λ ) $ Tλ Λ (n). The positive side of Σ (λv ) at λ is then defined to be the half tangent space made out of those symmetric bilinear forms on λ whose restriction to λ ∩ λv is positive definite. Notice that the orientation introduced here is opposite to the usually chosen one in [12, 29, 70]. Example 11.2.13. As a matter of example, let us consider Λ (1) = U(1)/O(1). Since U(1) = S1 ⊂ C and O(1) = {−1, +1}, the map z −→ z2 realizes the quotient as S1 = P1 (R). The point corresponding to the subspace λv = iR gives the Maslov cycle. Hence, given a path λ : [a, b] −→ Λ (1), the Maslov index μM (λ ) of the path λ (see Definition 11.2.14 below) counts the number of intersections of the path λ with this point (with the appropriate correction at the endpoints). Let λ : [a, b] −→ Λ (n) be a smooth path of Lagrangian subspaces. Robbin and Salamon define a crossing for λ some t ∈ [a, b] such that dim(

λ (t)∩ λv ) ≥ 1, that X (t) is, a time t ∈ [a, b] such that λ (t) ∈ Σ (λv ). Hence, if Z(t) = is a frame for Y (t) λ (t), t is a crossing iff det X (t) = 0. The set of crossings is obviously compact. At each crossing t ∈ [a, b], one defines the crossing form ˙ Γ (λ ,t) : Ker X(t) u −→ X(t)u,Y (t)u ∈ R.

(11.21)

By Theorem 11.2.10 one has that the crossing form is natural, in the sense that

Γ (χ (λ ),t) ◦ χ = Γ (λ ,t),

for all symplectic matrices χ such that χ (λv ) = λv

(notice that, in the block-form (11.19) of χ , the condition χ (λv ) = λv corresponds to having B = 0). A crossing t is regular when Γ (λ ,t) is nonsingular. It can be shown that regular crossings are isolated. Definition 11.2.14 (Robbin and Salamon). Let λ : [a, b] −→ Λ (n) be a smooth curve with only regular crossings. One defines the Maslov index of the curve λ by 1 μM (λ ) = sign Γ (λ , a) + 2

∑

t∈(a,b) t is a crossing

1 sign Γ (λ ,t) + sign Γ (λ , b), 2

(11.22)

where sign is the signature of the form (the number of positive eigenvalues minus the number of negative eigenvalues).

182


The following lemmas (due to Robbin and Salamon; see [62]) are fundamental. Lemma 11.2.15. Suppose that λ j : [a, b] −→ Λ (n), j = 1, 2, with λ1 (a) = λ2 (a) and λ1 (b) = λ2 (b), have only regular crossings. If λ1 and λ2 are homotopic with fixed endpoints then they have the same Maslov index. Lemma 11.2.16. Every Lagrangian path λ : [a, b] −→ Λ (n) is homotopic with fixed endpoints to one having only regular crossings. The lemmas yield the following theorem, which characterizes μM . Theorem 11.2.17 (Robbin and Salamon). The Maslov index μM is defined for every continuous path in Λ (n) and has the following properties. • (Naturality) For all symplectic matrices χ such that χ (λv ) = λv ,

μM (χ (λ )) = μM (λ ). • (Catenation) For c ∈ (a, b),

μM (λ ) = μM (λ [a,c] ) + μM (λ [c,b] ). • (Direct sum) If n = n + n , we identify Λ (n ) × Λ (n ) with a submanifold

X ∈ Λ (n ) and of Λ (n) in the following natural way: if λ = Im Z = Im Y

X ∈ Λ (n ), then λ ⊕ λ is identified with λ = Im Z, where λ = Im Z = Im Y ⎤ ⎡ X 0 ⎢ 0 X ⎥ ⎥ ⎢ Z=⎢ ⎥. Then, identifying λv ⊕ λv with λv , where λv , resp. λv , is the ⎣Y 0 ⎦ 0 Y “vertical” Lagrangian subspace of Rn × Rn , resp. Rn × Rn , one has

μM (λ ⊕ λ ) = μM (λ ) + μM (λ ). • (Homotopy) Two paths λ j : [a, b] −→ Λ (n), j = 1, 2, with λ1 (a) = λ2 (a) and

λ1 (b) = λ2 (b) are homotopic with fixed endpoints iff they have the same Maslov index. • (Zero) For every path λ : [a, b] −→ Λk (λv ) one has μM (λ ) = 0. • (Normalization) For the loop λ : [0, π ] −→ Λ (1) defined by

cost sint λ (t) = R × {0} ⊂ S1 = R × R, − sint cost one has μM (λ ) = 1.


183

Remark 11.2.18. Theorem 11.2.17 is actually a particular case of a more generale theorem proved in [62, Theorem 2.4, p. 831]. Robbin and Salamon define a Maslov index of a curve λ : [a, b] −→ Λ (n) with respect to any fixed λ0 ∈ Λ (n), denoted by μM (λ ; λ0 ) (hence μM (λ ) = μM (λ ; λv )). In our framework, we may define μM (λ ; λ0 ) as follows. Let χ : Sn −→ Sn be linear symplectic such that χ (λv ) = λ0 . We define

μM (λ ; λ0 ) := μM (χ −1 (λ )). To show that this definition is consistent, take two linear symplectic maps χ j : Sn −→ Sn , such that χ j (λv ) = λ0 , j = 1, 2. Then (χ2−1 ◦ χ1)(λv ) = λv , whence, by the naturality property of Theorem 11.2.17,

μM (χ1−1 (λ )) = μM ((χ2−1 ◦ χ1)(χ1−1 (λ ))) = μM (χ2−1 (λ )), which proves the consistency of the definition.

Remark 11.2.19. It is important to notice that with this choice of Maslov index, the Hörmander index s(μ1 , μ2 , λ1 , λ2 ) (see [12], or [29], or [70]) of Lagrangian subspaces λ1 , λ2 and μ1 , μ2 , such that λ j is transverse to μk , j, k = 1, 2, we have s(μ1 , μ2 , λ1 , λ2 ) = −μM (γ ), where γ is a closed curve in Λ (n) consisting of an arc of Lagrangian subspaces from λ1 to λ2 transversal to μ1 , followed by an arc of Lagrangian subspaces from λ2 to λ1 transversal to μ2 . As a matter of example, let us now compute μM (λ ) where λ is the Lagrangian loop given in the last item of the previous theorem. In this case Λ (1) $ S1 and the Maslov cycle is a point. Hence

u cost λ (t) = { ; u ∈ R}, t ∈ [0, π ], −u sint

λ (0) = λ (π ), and the Maslov cycle is therefore given by {λ (π /2)}. We have to compute the crossing forms at the crossing times in [0, π ]. One has that X (t) = cost and Y (t) = − sint, so that det X(t) = 0, t ∈ [0, π ] ⇐⇒ t = π /2, and, according to (11.21),

Γ (λ , π /2)(u) = (− sin(π /2))2 u2 = 0 ⇐⇒ u = 0, that is, t = π /2 is a regular crossing, with signature 1, whence μM (λ ) = 1.

184


Remark 11.2.20. The choice of the normalization condition in Theorem 11.2.17 above is the natural one. In fact, we are considering S1 = Rx × Rξ , with symplectic form σ = d ξ ∧ dx, which is the opposite of the area form. Hence, the induced orientation on the circle x2 + ξ 2 = 1 is the clockwise one. Notice, moreover, that

Q(t, X ) cost sint x x = ,X= , ξ ξ P(t, X ) − sint cost is the integral trajectory issuing from X of the Hamiltonian vector field H p0 associated with the harmonic oscillator p0 (x, ξ ) = (ξ 2 + x2 )/2 in R2 . The tangent lines to these curves are automatically curves of Lagrangian subspaces, with Maslov index 2. In fact, let X0 = (1, 0). With γX0 (t) = (Q(t, X0 ), P(t, X0 )), consider

sint } ∈ Λ (1). [0, 2π ] t −→ λ (t) = Span{γ˙X0 (t)} = Span{ cost Then the crossing points, that are all regular, occur for t0 = 0, π , 2π , and there we have Γ (λ ,t0 )(u) = (cost0 )2 u2 = u2 , whence the claim. The case X0 = (1, 0) follows using the transitivity of the action of linear symplectomorphisms. We next compute the Maslov index of the (periodic) bicharacteristics, suitably lifted to a loop in Λ (2n), of the harmonic oscillator p0 (X ) = |X |2 /2, X = (x, ξ ) ∈ Sn = Rn × Rn . Recall that the integral trajectories t −→ Φ t (X ) := exp(tH p0 )(X) of H p0 exist for all times, and they are given by

Φ t (X ) = (x cost + ξ sint, ξ cost − x sint), t ∈ R, i.e. by

(cost)I (sint)I (− sint)I (cost)I

Q(t, X ) x =: , ξ P(t, X )

where I is the n × n identity matrix. Hence they are all periodic (with period 2π ). In the first place we have now to lift them so as to give rise to a loop of Lagrangian manifolds, where, recall, a Lagrangian manifold is a smooth manifold whose tangent spaces are all Lagrangian. To this purpose we note that the map X −→ Φ t (X ), for every fixed t, is a symplectomorphism of Sn , that is, it is a diffeomorphism and its tangent map is a linear symplectic transformation at each point (the proof is left to the reader). Hence the set L(t) constructed out of its graph, L(t) := {(Q(t, X), x, P(t, X), −ξ ); X ∈ Sn } ⊂ S2n , is for all t a Lagrangian manifold of S2n endowed with the symplectic form σ ⊕ σ . We lift t −→ Φ t (X ) to the curve γˆX : [0, 2π ] t → γˆX (t) = (Q(t, X ), x, P(t, X ), −ξ ) ∈ L(t). Since Q(t, X) and P(t, X) are linear in X, the tangent space to L(t) at γˆX (t) ∈ L(t) is given by

λ (t) = {(Q(t, δ X ), δ x, P(t, δ X), −δ ξ ); δ X =

δx ∈ Sn } ∈ Λ (2n), δξ


185

so that [0, 2π ] t −→ λ (t) ∈ Λ (2n) is a loop in Λ (2n). A corresponding Lagrangian frame for λ (t) is given by

Z(t) =

X(t) (cost)I (sint)I (− sint)I (cost)I , X(t) = , Y(t) = . Y(t) I 0 0 −I

(Since

t

X(t)Y(t) =

(− sint cost)I (cos2 t − 1)I = t Y(t)X(t), (sint cost)I (− sin2 t)I

λ (t) is indeed Lagrangian, for all t.) We nex take the canonical symplectic basis (e1 , . . . , en , ε1 , . . . , εn ) of Sn , where e j has all coordinates zero except for the j-th one which is 1, and ε j has all the coordinates zero except for the n + j-th one which is 1, j = 1, . . . , n. Now, for t ∈ [0, 2π ], det X(t) = 0 ⇐⇒ sint = 0 ⇐⇒ t = 0, π , 2π , with

Ker X(t0 ) = Span{ε1 , . . . , εn }, t0 = 0, π , 2π . (− sint)I (cost)I ˙ Since X(t) = , we have that the crossing form at t0 = 0, π , 2π 0 0 is

˙ 0 )u, Y(t0 )u = 0 0 u, u = |u |2 , u = 0 ∈ Ker X(t0 ), Γ (λ ,t0 )(u) = X(t 0 I u

where u ∈ Rn . It follows that 0, π , 2π are regular crossings with signature 1, and therefore that 1 1 (11.23) μM (λ ) = + 1 + = 2. 2 2 In some cases one takes a lift of the periodic curve as a periodic integral curve of ∂ /∂ t + H p (where p is a symbol of the kind we are interested in; in the preceding case we considered p = p0 ). Thus, the problem is: do we obtain the same value for the Maslov index? The next proposition gives a positive answer to the problem. Proposition 11.2.21. Let p be a smooth positive symbol such that its bicharacteristic flow (t, X) −→ exp(tH p )(X) exists for all (t, X ) ∈ R × Sn. Let X0 = (x0 , ξ0 ) ∈ p−1 (E), and suppose that γ : t −→ Φ t (X0 ) = exp(tH p )(X0 ) = (Q(t, X0 ), P(t, X0 )) be periodic with least period T (E) > 0. Consider for every t ∈ [0, T (E)] the lift γˆ(t) = (Q(t, X0 ), x0 , P(t, X0 ), −ξ0 ) of γ to L(t) := {(Q(t, X), x, P(t, X ), −ξ ); X ∈ Sn } ⊂ S2n , and the lift γ˜(t) = (t, Q(t, X0 ), x0 , τ = −E, P(t, X0 ), −ξ0 ) of γ as an integral trajectory of ∂ /∂ t + H p (the Hamiltonian vector field of τ + p(X )) issuing from (t0 = 0, x0 , τ0 = −p(X0 ) = −E, ξ0 ), contained in the Lagrangian manifold

186


L˜ := {(t, Q(t, X ), x, τ , P(t, X), −ξ ); τ = −p(X ), (t, X ) ∈ R × Sn} ⊂ S2n+1 . Let [0, T (E)] t → λ (t) = Tγˆ(t) L(t) and [0, T (E)] t → λ˜ (t) = Tγ˜(t) L˜ be the corresponding loops in Λ (2n) and Λ (2n + 1) (for the latter we identify T (E) with 0). Then μM (λ ) = μM (λ˜ ). Proof. It is clear that L(t) and L˜ are Lagrangian submanifolds of S2n and S2n+1, respectively (in the case of L˜ one uses the fact that p(Φ t (X )) = p(X ), for all t ∈ R). We rewrite L˜ = {(t, Q(t, X ), x, −p(X), P(t, X), −ξ ); (t, X ) ∈ R × Sn}. At first we consider λ (t). In this case (denoting by Qx the Jacobian matrix of Q with respect to x, etc.) *

+ * + Qx (t, X0 ) Qξ (t, X0 ) Px (t, X0 ) Pξ (t, X0 ) X(t) = , Y(t) = . I 0 0 −I Hence

δx = 0 Ker X(t) = {u = ; δ ξ ∈ Ker Qξ (t, X0 )}, δξ

and for the crossing points, i.e. those times t ∈ [0, T (E)] for which Ker X(t) = {0}, i.e. Ker Qξ (t, X0 ) = {0}, the crossing form is

Γ (λ ,t)(u) = ∂t Qξ (t, X0 )δ ξ , Pξ (t, X0 )δ ξ , u =

0 δξ

∈ Ker X(t).

We now consider λ˜ (t). In this case, by writing pξ for the row-vector of ξ -derivatives of p etc., ⎤ 1 0 0 ⎥ ˜ =⎢ X(t) ⎣ t pξ (Φ t (X0 )) Qx (t, X0 ) Qξ (t, X0 ) ⎦ , 0 I 0 ⎡

⎤ 0 −px (X0 ) −pξ (X0 ) ⎥ ˜ =⎢ Y(t) ⎣ t px (Φ t (X0 )) Px (t, X0 ) Pξ (t, X0 ) ⎦ . 0 0 −I ⎡

Hence

⎡

⎤ δt = 0 ˜ = {u˜ = ⎣ δ x = 0 ⎦ ; δ ξ ∈ Ker Q (t, X0 )}, Ker X(t) ξ δξ


187

˜ and for the crossing points, i.e. those t ∈ [0, T (E)] for which Ker X(t) = {0}, the crossing form is ⎡

⎤ 0 ˜ Γ˜ (λ˜ ,t)(u) ˜ = ∂t Qξ (t, X0 )δ ξ , Pξ (t, X0 )δ ξ , u˜ = ⎣ 0 ⎦ ∈ Ker X(t). δξ We therefore have that ˜; • t ∈ [0, T (E)] is a crossing point for λ iff it is a crossing point for λ • the respective crossing forms there are equal; ˜. • the crossing point is regular for λ iff it is regular for λ Hence, if λ does not have any crossings with λv , then neither has λ˜ with λ˜ v = {0} × R2n+1, and μM (λ ) = μM (λ˜ ) = 0. So we may suppose there is at least one crossing time t0 , necessarily the same for both λ and λ˜ . Define, for each s ∈ [0, 1], the

loop in Λ (2n + 1) defined by X˜ (t) , with [0, T (E)] t −→ λ˜ s (t) = ImZ˜ s (t), where Z˜ s (t) = ˜ s Ys (t) ⎤ 1 0 0 ⎥ ⎢ X˜ s (t) = ⎣ s tpξ (Φ t (X0 )) Qx (t, X0 ) Qξ (t, X0 ) ⎦ , 0 I 0 ⎡

⎡

⎤ 0 −spx (X0 ) −spξ (X0 ) ⎢ ⎥ Y˜ s (t) = ⎣ s tpx (Φ t (X0 )) Px (t, X0 ) Pξ (t, X0 ) ⎦ . 0 0 −I It is then clear from the above discussion that ˜ s iff it is a crossing for λ˜ = λ˜ 1 , and • for all s ∈ [0, 1], t ∈ [0, T (E)] is a crossing for λ

the respective crossing forms there are all equal (hence independent of s ∈ [0, 1]).

We now show that

μM (λ˜ ) = μM (λ˜ 0 ).

In the first place we observe that the curve [0, 1] s −→ λ˜ s (t0 ) ⊂ Λk (λ˜ v ) where k = dim Ker Qξ (t0 , X0 ) (since λ˜ s (t0 ) ∩ λ˜ v is the injective image of KerQξ (t0 , X0 )), and therefore, by the zero property, has Maslov index 0. Next, one constructs a loop λˆ that starts at λ˜ 0 (t0 ), follows λ˜ s (t0 ), s ∈ [0, 1], to reach λ˜ 1 (t0 ) = λ˜ (t0 ), then follows λ˜ (t) (with its orientation) till it comes back to λ˜ (t0 ), and finally follows λ˜ s (t0 ), s ∈ [0, 1], backward from λ˜ (t0 ) to λ˜ 0 (t0 ). Hence, by the catenation and zero properties, we have that μM (λ˜ ) = μM (λˆ ). Since λ˜ 0 is homotopic to λˆ with fixed endpoints (by using a riparametrization of the family (t, s) → λ˜ s (t)), we have, by the homotopy property, that μM (λˆ ) = μM (λ˜ 0 ), which proves the claim.

188


Xs (t) , be a homoLet now [0, T (E)] × [0, 1] (t, s) → λs (t) = Im Zs (t), Zs (t) = Ys (t) topy in Λ (2n) of the loop λ = λ0 to a loop λ1 (with fixed endpoints) with only regular crossings. Then μM (λ ) = μM (λ1 ). Now,

⎡

⎤ 1 0 ⎢ 0 Xs (t) ⎥ ⎥ [0, T (E)] × [0, 1] (t, s) −→ Z(t, s) := ⎢ ⎣0 0 ⎦ 0 Ys (t) defines a homotopy (with fixed endpoints) between λ˜ 0 and the lift Im Z(t, 1) of λ1 to Λ (2n + 1). Hence μM (λ˜ 0 ) = μM (Im Z(·, 1)). Since Im Z(·, 1) has only regular crossings, which, by the above considerations, are the same as those of λ1 , with equal respective crossing forms there, we have that μM (Im Z(·, 1)) = μM (λ1 ), and this concludes the proof. Proposition 11.2.21 and the previous discussion thus justify the following definition of Maslov index of a periodic bicharacteristic of a smooth, real-valued function p. Definition 11.2.22. Let the bicharacteristic flow of p be defined for all (t, X ) ∈ R × Sn . Let X0 ∈ p−1 (E), and let γX0 = {exp(tH p )(X0 ) = (Q(t, X0 ), P(t, X0 )); 0 ≤ t ≤ T (X0 )} be a periodic trajectory, with least period T (X0 ) > 0. Consider the lifts [0, T (X0 )] t −→ λ (t) ∈ Λ (2n) and [0, T (X0 )] t −→ λ˜ (t) ∈ Λ (2n + 1) of γX0 to Λ (2n) and Λ (2n + 1), respectively, given in Proposition 11.2.21. One defines the Maslov index of γX0 by

μM (γX0 ) = μM (λ ) = μM (λ˜ ).

(11.24)

Remark 11.2.23. Suppose p is a smooth, real-valued function of which we only know that it satisfies Assumption 11.1.1 for an energy interval [E1 − ε , E2 + ε ], with no other information on its bicharacteristic flow for other energy values. Take E ∈ [E1 , E2 ]. Let X0 ∈ p−1 (E) and let γX0 ⊂ p−1 (E) be a periodic trajectory through X0 , with least period T (E) > 0. Since the bicharacteristics of p, when X ∈ p−1 ([E1 −ε , E2 + ε ]), exist for all times, as done in Proposition 11.2.21 one may consider the Lagrangian submanifold of S2n , resp. S2n+1 , L(t) := {(Q(t, X ), x, P(t, X ), −ξ ); X ∈ p−1 (E1 − ε , E2 + ε )}, resp. L˜ := {(t, Q(t, X)x, τ , P(t, X ), −ξ ); τ = −p(X ), (t, X ) ∈ R × p−1(E1 − ε , E2 + ε )}, and hence may construct, on recalling the lifts γˆX0 and γ˜X0 of γX0 considered in Proposition 11.2.21, the loops [0, T (E)] t −→ λ (t) = TγˆX (t) L(t) ∈ Λ (2n) and 0

11.3 Notes

189

[0, T (E)] t −→ λ˜ (t) = Tγ˜X

0

˜ ∈ Λ (2n + 1) (where, again, for the latter we iden-

(t) L

tify T (E) with 0). Hence the Maslov index of a periodic trajectory γX0 ⊂ p−1 (E) is defined according to Definition 11.2.22. From Lemma 11.1.11 and the foregoing discussion we immediately have the following fundamental result. Proposition 11.2.24. Let n = 1. Let p be a smooth, real-valued function satisfying Assumption 11.1.1 for an energy interval [E1 − ε , E2 + ε ]. Let E ∈ [E1 , E2 ]. Then −1 γX0 := p−1 (E), X0 ∈ p (E), is homotopic to a circle, with orientation depending on whether ∇X p γ points outwards from γX0 (clockwise orientation) or not (counterX0

clockwise orientation), and μM (γX0 ) = 2 in the former case, μM (γX0 ) = −2 in the latter. When n ≥ 1, p : R2n \ 0 −→ R+ is smooth, positively homogeneous of positive degree, and γX0 ⊂ p−1 (E) is a periodic trajectory, then μM (γX0 ) = 2.

Remark 11.2.25. As shown by Duistermaat in [11], using the fact that for fixed t the map X −→ Φ t (X ) = exp(tH p )(X) = (Q(t; X ), P(t; X )) is a symplectomorphism (hence its Jacobian dX Φ t (X0 ) at any given X0 is a linear symplectic map), one may compute the Maslov index of a periodic curve γX0 as in Definition 11.2.22 (or Remark 11.2.23) by computing the Maslov index of the curve [0, T+(X0 )] t −→ * (t; X ) Q (t; X ) Q 0 0 x ξ , we have (dX Φ t (X0 ))−1 (λv ) ∈ Λ (n). Since dX Φ t (X0 ) = Px (t; X0 ) Pξ (t; X0 ) * (dX Φ t (X0 ))−1 =

t Pξ (t; X0 ) −tPx (t; X0 )

+ −tQξ (t; X0 ) , tQ (t; X ) 0 x

+ −tQξ (t; X0 )δ ξ ; δ ξ ∈ Rn }. λ (t) = { t Qx (t; X0 )δ ξ *

so that

Notice that

λ (t) ∩ λv = {0} ⇐⇒ Ker tQξ (t; X0 ) = {0} ⇐⇒ det Qξ (t; X0 ) = 0, i.e. iff Ker Qξ (t; X0 ) = {0}.

11.3 Notes Maslov introduced his index (see Maslov [42]) for studying problems related to quantum mechanics (high-frequency approximation and semiclassical approximation). It was Lax who noted that the asymptotic expansions of geometrical optics

190


could be used to construct solutions (at least locally) to some PDEs. The point is to approximate solutions to a particular PDE by formal solutions of the kind eiωφ

∑ ω − ja j,

ω → +∞,

j≥0

or more generally of the kind m

∑(2π )−n

ei(φk (x,y,ξ )− y,ξ ) ak (x, y, ξ ) f (y)dyd ξ ,

1

with ak ∼ ∑ j≥0 ak, j , where φ , φ j are (in general) real phase-functions and where the terms a j and ak, j are the transport terms. This gave rise to the modern theory of Fourier Integral Operators, which is due to Hörmander. The theory necessarily requires for the definition of the principal symbol the use of the Keller-Maslov bundle: a complex line bundle, whose trivializations are constructed in terms of the Maslov index. We address the reader to the fundamental “FIO” paper by Hörmander [26] (and the second fundamental “FIO” paper by Duistermaat and Hörmander [9]) and the books about FIO theory by Hörmander [29, 30], Duistermaat [12], Treves [70], Helffer [17] (for the theory in the framework of global symbols), and Robert [65] (for the theory in the semiclassical setting), and to the nice papers of EckmannSénéor [14] and Marsden-Weinstein [41] for elementary applications of the Maslov index to Schrödinger equations. The geometric discussion of caustics can by found in Duistermaat [10]. Constructions of semiclassical approximations can also be found in the book by Mishchenko, Shatalov and Sternin [44].

Chapter 12

Localization and Multiplicity of a Self-Adjoint Elliptic 2×2 Positive NCHO in Rn

Using the machinery developed in the past chapters, we can now describe the beautiful connection between the spectrum of a self-adjoint elliptic 2 × 2 positive NCHO in n variables, and the periods of the bicharacteristics of the principal symbol (i.e., in the case of systems, the periods of the bicharacteristic curves associated with the eigenvalues of the principal symbol). This connection is a very deep result and its history, in the scalar case, is based on fundamental papers by Chazarain, Coline de Verdière, Duistermaat and Guillemin, Ivrii, Helffer and Robert (see also Cardoso and Mendoza [4]), and many others (for such results in the semiclassical setting, see Chazarain [5], Dimassi-Sjöstrand [7], Dozias [8], Helffer-Robert [18, 20], Ivrii [34], Robert [65]). However, very few are the results for systems (see Ivrii [34], Parmeggiani [52, 55]). Our results here (see Section 12.3) for elliptic NCHOs in Rn , whose symbols have distinct eigenvalues, are based on the approach followed in Parmeggiani [55]. In Section 12.4 we shall specialize these results to NCHOs of the kind Qw (α ,β ) (x, D) (with α , β > 0 and αβ > 1), and in the final Section 12.5 we shall give a slightly more general result (namely, when the subprincipal part of the diagonalization has constant average on periodic bicharacteristics). We now briefly describe the strategy we will be using. Since the eigenvalues of a diagonalizable second order GPD system are not smooth at (0, 0) ∈ R2n , we shall consider a special set of symbols (which is still meaningful, for its Weylquantization contains our preferred NCHOs Qw (x, D) for α , β > 0, α = β and (α ,β ) αβ > 1), whose eigenvalues are distinct and can be regularized keeping them distinct (so as not to destroy the diagonalizability property). This will allow us to construct a reference operator that can be h-Weyl quantized, and whose spectral properties, for large energies, will be analyzed by semiclassical techniques through a full decoupling modulo O(h∞ ) and by using the well-known and very precise results for scalar operators (recalled in Section 12.2) due (mainly) to Helffer and Robert. The spectral properties of the reference operator, such as clustering and multiplicity of the large eigenvalues, approximate in a precise way (by the results of Chapter 10, Section 10.2) those of the h-quantization of the system under study. Finally, we will use Lemma 9.4.3 to get rid of the semiclassical parameter h and obtain the result for the NCHO under study in the usual Weyl-quantization.


191

192

12 Localization and Multiplicity of a Self-Adjoint Elliptic 2 × 2 Positive NCHO in Rn

12.1 The Set Q2 and Its Semiclassical Deformation In this section, we consider a particular set of symbols of 2× 2 NCHOs that contains our Qw (x, D) (for α , β > 0, α = β and αβ > 1), and show how to construct a (α ,β ) reference operator, whose spectral properties (for large energies) can be studied by means of Semiclassical Analysis. + * a11 a12 ∈ Scl (m2 , g; M2 ) belongs to Q2 if Definition 12.1.1. We say that A2 = a21 a22 ⎧ ⎪ X −→ A2 (X ) is an M2 -valued homogeneous polynomial in X of degree 2, ⎪ ⎪ ⎪ ⎪ ⎨ A2 (X ) = A2 (X )∗ > 0 and detA2 (X) ≈ |X|4 , ∀X ∈ R2n \ {0}, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ a (X ) = a (X ), ∀X ∈ R2n \ {0}. 11 22 (12.1) 2n Notice that since R \ {0} is connected for all n ≥ 1, the last condition may be rephrased by saying that either a11 (X ) > a22 (X ), or a22 (X) > a11 (X ), ∀X ∈ R2n \ {0}. We shall be interested mainly in the following subclass of Q2 . Definition 12.1.2. We denote by Q2s the subset of all A2 ∈ Q2 such that either a12 (X ) ∈ R for all X ∈ R2n , or a12 (X) ∈ iR for all X ∈ R2n .

(12.2)

Remark 12.1.3. For all α = β with α , β > 0 and αβ > 1 we have that Q(α ,β ) ∈ Q2s ⊂ Q2 . (x, D) is completely understood: Notice that when α = β > 1 the operator Qw (α ,α ) it is unitarily equivalent to a scalar harmonic oscillator. Also, when A2 = A∗2 ∈ Scl (m2 , g; M2 ) fulfills condition (12.2) with a12 (X) = γ a˜12 (X ), where either γ ∈ R or γ ∈ iR and where a˜12 is real-valued, and with, in addition, a11 (X ) = a22 (X ) for all X, we may find a constant unitary transformation U : C2 → C2 such that * ∗

U A2 (X )U =

a11 (X) − a˜ 12(X)

0

0

a11 (X) + a˜12(X )

+ , ∀X ∈ R2n ,

whence the spectral properties of its Weyl-quantization may be studied by scalar techniques. This is the reason why we are interested in generalizations of the case α = β , which gives rise to the “strange” requirement in the definition of Q2 that a11 (X ) = a22 (X ) for all X = 0.

12.1 The Set Q2 and Its Semiclassical Deformation

193

Of course, it is a very interesting problem to understand the case in which a11 (X ) = a22 (X ) for some 0 = X ∈ R2n . We remark that the case in which condition (12.2) is not fulfilled will be (partially) studied in Section 12.5 below. Let G2 := subgroup of GL2 (C) generated by I, J, K and S1 , where, recall,

I=

1 0 0 −1 0 1 ,J= ,K= , 0 1 1 0 1 0

and where

S1 = {ω ∈ C; |ω | = 1}

is embedded in GL2 (C) by the map ω −→ ω I. Note that U ∈ G2 implies that UU ∗ = U ∗U = I. It is then immediate to prove the following lemma. Lemma 12.1.4. We have the following “invariance” property of the set Q2 : given any A2 ∈ Q2 U ∗ A2U ∈ Q2 , A2 ◦ κ ∈ Q2 , (12.3) for all U ∈ G2 and all κ : R2n −→ R2n linear symplectic transformations. The same holds true for Q2s . Remark 12.1.5. Notice that in Lemma 12.1.4 we consider only linear symplectic maps, instead of general affine symplectic transformations. This is due to the fact that we wish to preserve homogeneous polynomials of degree 2. + * a11 (X) a12 (X ) Let hence A2 ∈ Q2 with A2 (X) = . Put a12 (X) a22 (X ) a± (X) := and let

a11 (X) ± a22(X )) , 2

) λ± (X ) := a+ (X ) ± a− (X)2 + |a12(X )|2 , X ∈ R2n ,

(12.4)

be the eigenvalues of A2 (X ). Lemma 12.1.6. Let A2 ∈ Q2 . Then for the discriminant we have

δ (X ) := a− (X )2 + |a12(X)|2 ≈ p0 (X )2 , ∀X ∈ R2n ,

(12.5)

where, recall, p0 (X ) = |X |2 /2. Hence, since A2 belongs to Q2 , we have that λ± are smooth in R2n \ {0}, positively homogeneous of degree 2 and 0 < λ− (X) < λ+ (X), ∀X ∈ R2n \ {0}.

194


By first restricting to S2n−1 , we therefore get by the homogeneity that for 0 < c1 < c2 c1 p0 (X ) ≤ λ− (X ) < λ+ (X) ≤ c2 p0 (X), ∀X ∈ R2n \ {0}.

(12.6)

Proof. We need only prove (12.5). By the third condition in (12.1) we have C0 c0 ≤ a− (ω )2 = (a11 (ω ) − a22(ω ))2 ≤ , ∀ω ∈ S2n−1 , 4 4 and

|a12 (ω )|2 ≤ C0 , ∀ω ∈ S2n−1 ,

for suitable c0 ,C0 ,C0 > 0. The result therefore follows by homogeneity.

Define now, for an energy E ≥ 0, the sets

Σ± (E) := {X ∈ R2n ; λ± (X ) = E}. Since λ± are continuous and λ± (X) −→ +∞ as |X | → +∞, the sets Σ± (E) are compact for all E ≥ 0 and they both reduce to {0} when E = 0 (by 12.6)). Consider next, for A2 ∈ Q2 , the operator Aw 2 (x, hD) thought of as an unbounded self-adjoint operator 2 n 2 2 n 2 2 n 2 Aw 2 (x, hD) : B (R ; C ) ⊂ L (R ; C ) −→ L (R ; C ),

(12.7)

which thus has a discrete spectrum, bounded from below, made of a diverging (to +∞) sequence of eigenvalues with finite multiplicities. We want to understand the large eigenvalues of Aw 2 (x, D) (i.e. with h = 1). Our strategy will be as follows: • Step 1: We study the large eigenvalues of the h-Weyl quantization Aw 2 (x, hD)

given in (12.7), which has the same domain B2 (Rn ; C2 ) and properties as w Aw 2 (x, D), and whose spectrum is linked to that of A2 (x, D) through Corollary 9.4.5 by the relation w λ j ∈ Spec(Aw 2 (x, D)) ⇐⇒ hλ j = λ j (h) ∈ Spec(A2 (x, hD)), j ≥ 1.

w • Step 2: We approximate Aw 2 (x, hD) by a reference operator Ar (x, hD), whose

symbol Ar (X ) has the same eigenvalues of A2 (X ) for large X , that can be h∞ -diagonalized into a couple of scalar operators whose principal symbols are the eigenvalues of A2 (X ) for large X . • Step 3: We use the precise, well-known, theorems, which we will recall in Section 12.2, that describe the location (and in some cases the multiplicity) of the spectrum of scalar h-pseudodifferential operators in terms of the dynamical quantities, related to the principal and subprincipal symbols, that were discussed in Chapter 11.


195

• Step 4: We use the results of Chapter 10, Section 10.2 to transfer the information

about the spectrum of the diagonalization of Aw r (x, hD) to information about the spectrum of Aw (x, hD). By Step 1 we hence obtain information about the spec2 trum of Aw (x, D). 2

Following Evans-Zworski [15] and Parmeggiani [55], we now prepare the ground for the localization of large eigenvalues. The first result concerns uniform h∞ -estimates. Proposition 12.1.7. Let E > 0. Let a ∈ Sδ0 (1; M2 ) with supp a ⊂ K, where K ⊂ R2n is a compact set independent of h such that / K ∩ Σ+ (E) ∪ Σ− (E) = 0, and let γ := dist(K, Σ+ (E) ∪ Σ− (E)). Let u(h) ∈ L2 (Rn ; C2 ) solve the eigenvalue equation Aw 2 (x, hD)u(h) = λ (h)u(h). If |λ (h) − E| < ε , for some ε ∈ (0, 1/2] sufficiently small (compared to E and to γ ), then ||aw (x, hD)u(h)||0 = O(h∞ )||u(h)||0 (where the constants in O(h∞ ) are allowed to depend on E and γ ). Proof. The set Σ+ (E) ∪ Σ− (E) =: K is compact. We may therefore find χ ∈ C0∞ (R2n ), with 0 ≤ χ ≤ 1, such that χ ≡ 1 near K and χ ≡ 0 near K. Define b(X ) = bh (X ) = A2 (X) − λ (h)I + iχ (X )I ∈ S00 (m2 ; M2 ). If ε is sufficiently small (with respect to E and γ ) we then have that |detb(X)|2 = |λ+ (X ) − λ (h)|2 + χ (X)2 |λ− (X ) − λ (h)|2 + χ (X )2 > 0, for all X ∈ R2n , and C−1 ≤

|detb(X )|2 ≤ C, m(X)4

for some C = CE,γ > 0. Hence we may find c ∈ S00 (m−2 ; M2 ) and r ∈ S00 (1; M2 ) such that cw (x, hD)bw (x, hD) = I + rw (x, hD), and rw = O(h∞ ), where by rw = O(h∞ ) we mean (as before in Section 10.2) ||rw ||L2 →L2 = O(h∞ ). Then aw (x, hD)cw (x, hD)bw (x, hD) = aw (x, hD) + O(h∞ ), where

w bw (x, hD) = Aw 2 (x, hD) − λ (h) + i χ (x, hD).

196


We now claim that aw (x, hD)cw (x, hD)χ w (x, hD) = O(h∞ ).

(12.8)

When δ ∈ [0, 1/2) the claim immediately follows, for we have supp a ∩ supp χ = 0/ uniformly in h. When δ = 1/2, we have that for any given N ∈ Z+ , N+1 cw (x, hD)χ w (x, hD) = (cχ )w ), N (x, hD) + O(h

where (cχ )N (X ) :=

N

k 1 ih σ (DX ; DY ) c(X )χ (Y )X=Y 2

∑ k!

k=0

has compact support (in fact supp(cχ )N ⊂ supp χ ), uniformly disjoint from supp a. We are hence in a position to use (9.10) and obtain (by the continuity) the claim also when δ = 1/2. In all cases, on the other hand, from Aw 2 (x, D)u(h) = λ (h)u(h) it follows aw (x, hD)u(h) = aw (x, hD)cw (x, hD)bw (x, hD)u(h) + O(h∞)u(h) = iaw (x, hD)cw (x, hD)χ w (x, hD)u(h) + O(h∞)u(h) = O(h∞ )u(h),

which concludes the proof of the proposition. We next prove the following estimate.

Proposition 12.1.8. Let u(h) ∈ L2 (Rn ; C2 ) be an eigenfunction of the operator 0 Aw 2 (x, hD) belonging to λ (h). Let a ∈ S0 (1; M2 ) be independent of h. Suppose that, for R > 0, supp a ⊂ {X ∈ R2n ; |X|2 ≤ R/c2 } and λ (h) > R + 1. Then ||a (x, hD)u(h)||0 = O w

h λ (h)

∞ ||u(h)||0 .

Proof. Let E > R+ 1 be such that |λ (h)− E| ≤ ε E, where ε ∈ (0, 1/2] is sufficiently small, so as to have (1 − ε )E > R + 1. √ Since A2 ( E X ) = EA2 (X ), we get from (9.5) that w ˜ ˜ UE−1 Aw 2 (x, hD)UE = EA2 (x, hD), where h =

h , E

√ and where UE is the L2 -isometry given in (9.4). Now, writing a(X ˜ ) := a( E X ), we have |∂Xα a(X)| ˜ ≤ Cα E |α |/2 ≤ Cα h˜ −|α |/2,


197

0 (1; M ) with respect to the parameter h, ˜ and that is, a˜ ∈ S1/2 2

supp a˜ ⊂ {X ∈ R2n ; |X |2 ≤ R/(c2 E)} ⊂ {X ∈ R2n ; |X |2 ≤ (1 − ε )/c2} =: K, ˜ − 1| < ε , where λ (h) ˜ = λ (h)/E ∈ Spec(Aw (x, hD)) ˜ (see while |λ (h) 2 Corollary 9.4.5). Hence K ∩ Σ+ (1) ∪ Σ− (1) = 0, / for, by (12.6), we have that

Σ± (1) ⊂ {X ∈ R2n ; |X|2 ≥ 2/c2 }. ˜ belonging to the eigen˜ := U −1 u(h) is an eigenfunction of Aw (x, hD) Therefore u( ˜ h) E 2 ˜ and by Proposition 12.1.7, applied to the h-Weyl ˜ ˜ value λ (h), quantization a˜w (x, hD) of a˜ (by shrinking ε if necessary), we get that ˜ u( ˜ 0 = O(h˜ ∞ )||u( ˜ 0 ||a˜w (x, hD) ˜ h)|| ˜ h)|| (now the constants in O(h˜ ∞ ) are universal constants), which is, through the L2 -isometry UE (see (9.4)), equivalent to ∞ h ||u(h)||0 . ||a (x, hD)u(h)||0 = O E w

Since E/2 ≤ (1 − ε )E ≤ λ (h) ≤ (1 + ε )E ≤ 3E/2 (and 1/2 ≤ 1 − ε < 1 + ε ≤ 3/2), we finally obtain ||a (x, hD)u(h)||0 = O w

h λ (h)

∞ ||u(h)||0 ,


We hence obtain the following crucial corollary, that will make it possible to w study the large eigenvalues of Aw 2 (x, hD), and hence those of A2 (x, D). Corollary 12.1.9. Let R > 0. Consider the orthogonal projection

Π : L2 (Rn ; C2 ) −→ Span{u(h); Aw 2 (x, hD)u(h) = λ (h)u(h), λ (h) ≤ R + 1}. Let a = a∗ ∈ S00 (1; M2 ) be independent of h, with supp a ⊂ {X ∈ R2n ; |X |2 < R/c2 }. Then ||aw (x, hD)(I − Π )||L2 →L2 = O(h∞ ), ||(I − Π )aw (x, hD)||L2 →L2 = O(h∞ ).

198


Proof. Let Spec(Aw 2 (x, hD)) = {−∞ < λ1 (h) ≤ λ2 (h) ≤ . . . → +∞}, with repetitions according to multiplicity, and let {u j (h)} j≥1 be an othonormal basis of L2 (Rn ; C2 ) made of eigenfunctions of Aw 2 (x, hD), where u j (h) belongs to λ j (h), j ≥ 1. We may write I−Π = ∑ u j (h)∗ ⊗ u j (h), λ j (h)>R+1

where, recall, u j (h)∗ ⊗ u j (h) v = (v, u j (h))u j (h). Then aw (x, hD)(I − Π ) =

∑

λ j (h)>R+1

u j (h)∗ ⊗ aw (x, hD)u j (h) ,

and ⎛ ||aw (x, hD)(I − Π )||L2 →L2 ≤ ⎝

⎞1/2

∑

λ j (h)>R+1

||aw (x, hD)u j (h)||20 ⎠

.

(12.9)

By Theorem 4.4.1, and Corollary 9.4.5, we have for the eigenvalues of Aw 2 (x, hD) that for some c > 0, λ j (h) ≥ c j1/n h, whence by Proposition 12.1.8 (and recalling that we consider the eigenvalues λ j (h) for which λ j (h) > R + 1) we obtain, for each M < N, ||a (x, hD)u j (h)||0 ≤ CN w

h λ j (h)

≤ CN hM

N

h λ j (h)

N−M ≤

CN M −(N−M)/n h j . N−M c

Hence we choose N = M + n + 1, so that the sum on the right-hand side of (12.9) > 0 such that converges. We therefore get that for any given M ∈ Z+ there exists CM M ||aw (x, hD)(I − Π )||L2 →L2 ≤ CM h .

Since aw (x, hD) = aw (x, hD)∗ , (I − Π )∗ = I − Π , and ||A||L2 →L2 = ||A∗ ||L2 →L2 (for any given L2 -bounded operator A), we also obtain the second desired inequality. Let us now construct the reference operator Aw r (x, hD), with symbol Ar (X ) whose eigenvalues are distinct and smooth for all X ∈ R2n and coincide with those of A2 (X ) for large X . Notice that by Theorem 9.2.1, Aw r (x, hD) thus possesses an h∞ -diagonalization. Recall from (12.6) that (λ± being continuous)

λ± (X) −→ +∞, as |X| → +∞,


and

λ+ (X) ≤

199

c2 λ− (X), ∀X ∈ R2n . c1

(12.10)

It thus follows that, for ε0 > 0 fixed, {X ∈ R2n ; λ+ (X ) ≤ ε0 } ⊂ {X ∈ R2n ; c1 p0 (X ) ≤ ε0 },

(12.11)

{X ∈ R2n ; c1 p0 (X ) ≤ ε0 } ∩ {X ∈ R2n ; c1 p0 (X ) ≥ 2ε0 } = 0, /

(12.12)

and {X ∈ R2n ; λ− (X ) ≥ 2

c2 ε0 =: ε1 } ⊂ {X ∈ R2n ; c1 p0 (X ) ≥ 2ε0 }. c1

(12.13)

Notice that ε1 ≥ 2ε0 . Hence, by (12.11), (12.12) and (12.13), we may take χ ∈ C∞ (R2n ), with 0 ≤ χ ≤ 1, such that

χ ≡ 0 when λ+ ≤ ε0 , χ ≡ 1 when λ− ≥ ε1 . Put χ1 = χ and χ2 = 1 − χ , so that χ1 + χ2 = 1 and χ2 ∈ C0∞ (R2n ), with supp χ2 ⊂ {X ∈ R2n ; λ− (X) ≤ ε1 }, and 0 ≤ χ2 ≤ 1.

(12.14)

We are now ready to define the reference operator. Definition 12.1.10. Let A2 ∈ Q2 . When a11 (X) > a22 (X ) for all X = 0, define the reference operator Aw r (x, hD) by the h-Weyl quantization of

Ar (X ) = χ1 (X )A2 (X) + χ2(X)

1 0 , X ∈ R2n . 0 1/2

(12.15)

In case a11 (X ) < a22 (X ) for all X = 0, define Aw r (x, hD) by the h-Weyl quantization of

1/2 0 Ar (X ) = χ1 (X)A2 (X) + χ2(X) . 0 1 Hence Ar ∈ S(m2 , g; M2 ). Remark 12.1.11. It is worth remarking once more that we need to use Ar in order to regularize the eigenvalues of A2 , that are not smooth at the origin. As shown below, our construction yields that the eigenvalues of Ar still do not cross, and that they are smooth everywhere. Hence, we shall be in a position to use the h∞ -diagonalization granted by Theorem 9.2.1. To fix ideas we shall assume from now on, with no loss of generality by Lemma 12.1.4 and Lemma 8.0.1, that a11 (X) > a22 (X ) for all X = 0. Recall that a11 (X) ± a22(X ) a± (X) = . 2

200


Then the eigenvalues of A2 (X ) are given by

λ± (X ) = a+ (X) ±

) a− (X)2 + |a12(X )|2 ,

and those of Ar (X ) are given by 3 l± (X ) = a+ (X )χ1 (X) + χ2 (X) 4 4 2 1 ± a− (X )χ1 (X) + χ2 (X) + χ1(X )2 |a12 (X )|2 , 4 for, in our case, the discriminant is given by

χ12 a2+ +

9 2 3 1 1 χ2 + χ1 χ2 a+ − χ12 a11 a22 + χ1 |a12 |2 − χ1 χ2 a22 + a11 − χ22 16 2 2 2

a11 − a22 1 2 1 2 + χ12|a12 |2 = χ1 a− + χ2 + χ12 |a12 |2 . χ2 + χ1 χ2 16 4 4 We have the following proposition. = χ12 a2− +

Proposition 12.1.12. We have Ar = A∗r ∈ S(m2 , g; M2 ) is globally positive elliptic,

(12.16)

i.e. Ar (X )v, v C2 ≈ m(X )2 |v|C2 , ∀v ∈ C2 , ∀X ∈ R2n . Moreover, upon denoting by l± (X) the eigenvalues of Ar (X ), we have l± ∈ C∞ (R2n ), 0 < l− (X) < l+ (X ), ∀X ∈ R2n , ) l± λ− ≥ε = a+ (X ) ± a− (X)2 + |a12 (X)|2 1

λ − ≥ ε1

(12.17)

= λ ± λ − ≥ε , 1

3 1 l± λ+ ≤ε = ± , 0 4 4 l± (X ) ≈ p0 (X), for |X | ≥ c > 0. Hence the l± ∈

S(m2 , g)

(12.19) (12.20)

are globally positive elliptic symbols and, in particular, l± (X ) −→ +∞, as |X | → +∞.

Proof. We have, for all v ∈ C2 , Ar (X )v, v C2

(12.18)

1 0 = χ1 (X) A2 (X)v, v C2 + χ2(X ) v, v C2 . 0 1/2


201

Since 0 ≤ χ1 , χ2 ≤ 1, and χ1 + χ2 ≡ 1, and since λ± (X ) ≈ p0 (X ), we immediately obtain the existence of C > 0 such that on the one hand Ar (X )v, v C2 ≤ χ1 (X )λ+ (X) + χ2 (X) |v|2C2 ≤ Cm(X )2 |v|2C2 , (12.21) and on the other 1 Ar (X )v, v C2 ≥ χ1 (X )λ− (X) + χ2 (X) |v|2C2 ≥ C−1 m(X )2 |v|2C2 , 2

(12.22)

for all X ∈ R2n , and all v ∈ C2 . This proves (12.16). Now, the eigenvalues of Ar (X) are given by ' 3 l± (X ) = a+ (X)χ1 (X) + χ2 (X) ± δ0 (X ), 4

(12.23)

where the discriminant δ0 has the form 2 1 δ0 (X ) = a− (X )χ1 (X ) + χ2 (X) + χ1(X )2 |a12 (X )|2 > 0 4

(12.24)

by construction. In fact, we have that when X = 0, δ0 (0) = χ2 (0)2 /16 > 0, and when X = 0 (on recalling that we are supposing a11 (X ) − a22(X ) = 2a− (X ) > 0 for all X = 0) 1 δ0 (X ) = 0 =⇒ a− (X )χ1 (X) + χ2 (X) = 0 =⇒ χ1 (X ) = χ2 (X ) = 0, 4 which is impossible, for χ1 (X ) + χ2 (X) = 1. Hence, from (12.23), (12.24), (12.21) and (12.22), we obtain (12.17), (12.18), (12.19) and (12.20). This concludes the proof of the proposition. By Proposition 9.4.6 we hence have the following fact concerning the reference operator Aw r (x, hD). Lemma 12.1.13. For all h ∈ (0, 1], the operator Aw r (x, hD), as an unbounded operator in L2 (Rn ; C2 ) with domain B2 (Rn ; C2 ), is self-adjoint with a discrete spectrum bounded from below, made of a diverging (to +∞) sequence of eigenvalues with finite multiplicities. w w Proof. Since Aw r (x, hD) = A2 (x, hD) + R (x, hD), where

1 0 − A2(X ) R(X) = χ2 (X) 0 1/2

(12.25)

has compact support (equal to the support of χ2 ), it follows that we may apply Proposition 9.4.6. This concludes the proof.

202


Remark 12.1.14. Note that Lemma 12.1.13 may also be proved directly by Proposition 12.1.12 using a parametrix construction for Aw r (x, hD). In addition, from (12.18) and the previous considerations, we also obtain the following basic lemma. Lemma 12.1.15. Put " ! c2 E0 := max 1, 2 ε1 , max l± . c1 λ− ≤2ε1

(12.26)

Then the subset Ω of T ∗ Rn ,

Ω := {X; λ− (X ) ≥ E0 } ⊂ {X ; λ± (X) ≥ E0 } ⊂ {X ; λ− (X ) ≥ 2ε1 }, is foliated by energy-surfaces

Σ± (E) = {X ; λ± (X) = E} = {X ; l± (X ) = E}, E ≥ E0 . At this point, we are in a position to measure the distance of Spec(Aw 2 (x, hD)) to = E + 10, and consider the orthogonal projecSpec(Aw (x, hD)). Fix the energy E 0 r 0 tor Π associated with Aw 2 (x, hD), considered in Corollary 12.1.9, with R + 1 = E0 . Since c 2 c2 2 E0 − 1 > E0 ≥ 2 ε1 = 4 ε0 , c1 c1 we get in particular that supp χ2 ⊂ {X; λ− (X) ≤ ε1 } ⊂ {X ; |X |2 < (E0 − 1)/c2}, whence the same is true for the symbol R(X) in (12.25). Hence, we have the existence of a constant cχ > 0 independent of h ∈ (0, 1], such that ||Rw (x, hD)||L2 →L2 ≤ cχ , ∀h ∈ (0, 1].

(12.27)

w w Π Aw r (x, hD)Π = Π A2 (x, hD)Π + Π R (x, hD)Π ,

(12.28)

Since and Π commutes with Aw 2 (x, hD) (for it is associated with the eigenfunctions of Aw (x, hD)), by the Spectral Theorem we get 2 ||Π Aw r (x, hD)Π ||L2 →L2 ≤ E0 + c χ .

(12.29)

Furthermore, from Corollary 12.1.9 we have that w ∞ (I − Π )Aw 2 (x, hD)(I − Π ) = (I − Π )Ar (x, hD)(I − Π ) + O(h ),

(12.30)


203

and w ∞ ||Π Aw r (x, hD)(I − Π )||L2 →L2 = ||(I − Π )Ar (x, hD)Π ||L2 →L2 = O(h ).

(12.31)

We may therefore use Theorem 10.2.1 from Section 10.2, and obtain the following result. Theorem 12.1.16. Let A2 ∈ Q2 , and let Spec(Aw 2 (x, hD)) = {−∞ < λ1 (h) ≤ λ2 (h) ≤ . . . → +∞} and

r r Spec(Aw r (x, hD)) = {−∞ < λ1 (h) ≤ λ2 (h) ≤ . . . → +∞},

with repetitions according to multiplicity. Let E, E > 0 with E > E ≥ E0 + 102 + cχ . For any given integer N ≥ 1 there exists h0 = h(N, E, E ) ∈ (0, 1] such that Spec(Aw 2 (x, hD)) ∩ (E, +∞) ⊂

[λ r (h) − hN , λ r (h) + hN ], (12.32)

E E, then, with the same j, |λ j (h) − λ jr (h)| ≤ hN . (12.33) We shall use the theorem with N = 10. By the h∞ -decoupling Theorem 9.2.1 we next have the following result for Aw r (x, hD). Proposition 12.1.17. For any given j ∈ Z+ we may find h-independent symbols ± 2−2 j E−2 j ∈ S(m−2 j , g; M2 ) and Λ2−2 , g) such that j ∈ S(m

Λ2± (X) = l± (X), ∀X ∈ R2n ,

(12.34)

and such that, for any given N ∈ Z+ , upon writing EN (h) :=

N

∑ h j E−2 j ,

j=0

Λ±,N (h) :=

N

± ∑ h j Λ2−2 j,

j=0

we have * w Aw r (x, hD) = EN (h)

where

+ w Λ+,N (h) 0 E w (h)∗ + hN+1 R˜ w N (h), w 0 Λ−,N (h) N

w ∗ w ENw (h)ENw (h)∗ = I + hN+1 Rw N (h) = EN (h) EN (h),

(12.35)

(12.36)


204

with ˜w ||Rw N (h)||L2 →L2 ≤ CN , ||RN (h)||L2 →L2 ≤ CN , ∀h ∈ (0, 1], where

(12.37)

R˜ N ∈ S00 (m2−2(N+1) , g; M2 ), and RN ∈ S00 (m−2(N+1) , g; M2 ).

Hence, in particular, by Proposition 9.3.1, when N ≥ 1, 2 n 2 2 n 2 Rw N (x, hD) : L (R ; C ) −→ B (R ; C ) is continuous, ∀h ∈ (0, 1].

(12.38)

We are now in a position to prove the following result, that allows us to approxiw mate the spectrum of Aw r (x, hD) by the spectrum of the scalar operators Λ±,N (h). Theorem 12.1.18. There exists h0 ∈ (0, 1] sufficiently small, and for all h ∈ (0, h0 ] an L2 -bounded operator E(h) and scalar positive elliptic symbols Λ± (h) ∈ 0 (m2 , g) for which (12.34) holds, such that S0,cl

Aw r (x, hD) = E(h)

Λ+w (h) 0 E(h)∗ + h5 R˜ w (h), ∀h ∈ (0, h0 ], Λ−w (h) 0

(12.39)

where ||R˜ w (h)||L2 →L2 ≤ C for all h ∈ (0, h0 ], and such that E(h)E(h)∗ = I = E(h)∗ E(h), ∀h ∈ (0, h0 ],

(12.40)

and E(h)u, E(h)∗ u ∈ B2 (Rn ; C2 ), ∀u ∈ B2 (Rn ; C2 ), ∀h ∈ (0, h0 ].

(12.41)

In particular Λ w (h) 0 w ∗ ∗ + E(h) Ar (x, hD)u, u = u, E(h) u + O(h5 )||u||20 , Λ−w (h) 0

(12.42)

for all u ∈ B2 , for all h ∈ (0, h0 ]. Proof. Take N = 4 in Proposition 12.1.17. Hence (12.36) gives the existence of h0 ∈ (0, 1] such that 1 h||Rw 4 (x, hD)||L2 →L2 ≤ , ∀h ∈ (0, h0 ], 2 whence the operator I + h 5 Rw 4 (x, hD) > 0, ∀h ∈ (0, h0 ], and we are allowed to take the inverse of its positive square root, i.e. the h-dependent operator in L2 (Rn ; C2 )


205

−1/2 I + h 5 Rw 4 (x, hD) −1/2 5k w h R4 (x, hD)k =: I + h5 R(h), h ∈ (0, h0 ]. = ∑ k k≥0

S(h) :=

It is important to notice that w R(h) = Rw 4 (x, hD)R0 (h) = R0 (h)R4 (x, hD),

where k−1 −1/2 5 w R0 (h) := ∑ h R4 (x, hD) : L2 (Rn ; C2 ) −→ L2 (Rn ; C2 ) k k≥1 is continuous with ||R0 (h)||L2 →L2 ≤ C for all h ∈ (0, h0 ]. By (12.38) we therefore have that R(h)u ∈ B2 , for all u ∈ L2 . Setting w (h) E(h) := S(h)E4w (h) and Λ±w (h) = Λ±,4

completes the proof. Set now and let

Λ w (h) = diag(Λ+w (h), Λ−w (h)), Spec(Λ w (h)) = {−∞ < λ˜ 1 (h) ≤ λ˜ 2 (h) ≤ . . . → +∞},

with repetitions according to multiplicity (that the spectrum of Λ w (h) is discrete is granted by Remark 9.4.8). By the Minimax Principle (Theorem 4.1.1) we thus obtain the following result, which is interesting in its own right, since it represents the crucial, intermediate step in achieving the result for Aw 2 (x, D) (which is Theorem 12.4.1 below). Theorem 12.1.19. There exists h0 ∈ (0, 1] and a constant C > 0 such that dist Spec(Aw Spec(Λ±w (h)) ≤ Ch5 , r (x, hD)),

(12.43)

±

for all h ∈ (0, h0 ]. More precisely, for each j ≥ 1, |λ jr (h) − λ˜ j (h)| ≤ Ch5 , ∀h ∈ (0, h0 ].

(12.44)

Remark 12.1.20. The error O(h5 ) in the approximation (12.44) is sufficient because, as it will be seen below, the knowledge of the large eigenvalues of Λ±w (h) is within an error O(h2 ). Of course, any fixed N ≥ 4 can be taken, yielding in (12.43) and (12.44) an error O(hN ).

206


Remark 12.1.21. Using (9.35) of Corollary 9.2.6 we have, for the subprincipal terms Λ0± of the operators Λ±w (h), 1 ∗ Im {e0 , l± }(X)e0 (X )w± , w±

2 1 + Im e∗0 (X){Ar , e0 }(X )w± , w± , (12.45) 2

1 0 2n 2 X ∈ R , where w± is the canonical basis of C (w+ = , w− = ), and 0 1 e∗0 Ar e0 = diag(l+ , l− ), e0 being a matrix whose columns are given by the normalized eigenvectors of Ar . In particular, when X ∈ {X ∈ R2n ; λ− (X) ≥ ε1 } we have A2 (X ) = Ar (X ), whence

Λ0± (X ) =

Λ0± (X ) =

1 ∗ Im {e0 , λ± }(X)e0 (X )w± , w±

2 1 + Im e∗0 (X){A2 , e0 }(X )w± , w± , 2

(12.46)

where now e∗0 A2 e0 = diag(λ+ , λ− ). (Recall that no subprincipal terms are present w in the symbols of Aw 2 (x, hD) and of Ar (x, hD).) We have thus seen how the spectrum of Aw r (x, hD) is approximated by the spectrum of the diagonal system Λ w (h), and thus by the spectra of the scalar operators Λ±w (h). In the next section we recall some results about the spectrum of scalar h-pseudodifferential operators, which will then be used in Section 12.3 to obtain a description of the spectrum of the 2 × 2 systems we are interested in.

12.2 Localization and Multiplicity of the Spectrum in the Scalar Case We now recall, in a form tailored to our purposes, some results about the localization, and multiplicity, of the spectrum for scalar h-pseudodifferential operators. In the first place we recall the Dyn’kin-Helffer-Sjöstrand formula (see DimassiSjöstrand [7] and also the Appendix). Given f ∈ C0∞ (R) one can always find an almost-analytic extension f˜ ∈ C0∞ (C), that is a function f˜ such that f˜R = f , and ∀N ∈ Z+ , ∃CN > 0,

∂ f˜ (z) ≤ CN |Im z|N , ∂ z¯

(12.47)

where ∂ /∂ z¯ = 2−1 (∂ /∂ x + i∂ /∂ y). One has the following theorem (the Dyn’kin-Helffer-Sjöstrand formula, a proof of which will be given in the Appendix).

12.2 Localization and Multiplicity of the Spectrum in the Scalar Case

207

Theorem 12.2.1. Let P be a self-adjoint operator (possibly unbounded) on a Hilbert space H. Let f ∈ C02 (R) and let f˜ ∈ C01 (C) be an extension of f for which ∂ f˜ (z) = O(|Im z|). Then ∂ z¯ f (P) = −

1 π

∂ f˜ (z)(z − P)−1 L(dz). C ∂ z¯

(12.48)

Here L(dz) = dxdy is the Lebesgue measure on C, and the integral in (12.48) converges as a Riemann integral for functions with values in L (H, H). Let now p ∈ S00 (mμ ) be real-valued. We are interested in the case μ > 0, and therefore shall assume that p + i be elliptic (that is, recall, |p(X ) + i| m(X )μ ). Let hence P = pw (x, hD; h) be the corresponding operator acting as an unbounded operator (since μ > 0) on L2 . By using Beals’ characterization of h-pseudodifferential operators (see Theorem 9.1.11), from Theorem 12.2.1 one has the following result (see [7]). Theorem 12.2.2. Let f ∈ C0∞ (R). Then there exists h0 ∈ (0, 1] such that f (P) = F w (x, hD; h), where F ∈ S00 (m−k ), h ∈ (0, h0 ], for every k ∈ N. Moreover, when p ∼ pμ + hp μ −1 + h2 pμ −2 + . . . in S00 (mμ ), with the p μ − j independent of h, we get F ∼ F0 + hF1 + . . . in S00 (m−1 ), with the Fj independent of h, where F0 (X ) = f (pμ (X )), F1 (X) = p μ −1 (X)F (pμ (X )), ∀X ∈ R2n . It is also useful to have the following result for positive elliptic classical semiclassical symbols, due to Robert [64]. 0 (m μ , g) be positive elliptic, μ > 0. Theorem 12.2.3. Let p ∼ pμ +hp μ −2 +. . . ∈ S0,cl ∞ Let f ∈ C (R) fulfill, for some r ∈ R, the inequalities

∀k ∈ Z+ , ∃Ck > 0, such that | f (k) (λ )| ≤ Ck (1 + |λ |)r−k . Let P(h) = pw (x, hD; h). Then f (P(h)) = F w (x, hD; h), where the symbol F ∈ 0 (mr μ , g), and S0,cl Fr μ = f (pμ ), Fr μ −2 = pμ −2 f (pμ ). One has also a formula for the successive terms (see Robert [64, p. 77]). We now have the following fundamental theorem, that we prove for classical 0 (m μ , g), due (in a much more general form) to Helffer semiclassical symbols in S0,cl and Robert [20] (see also Dimassi-Sjöstrand [7] and Dozias [8]). 0 (m μ , g), μ > 0, p ∼ p + hp Theorem 12.2.4. Let p ∈ S0,cl μ μ −2 + . . . , be real valued, with principal symbol pμ ≈ mμ satisfying hypotheses (H1), (H2) and (H3)

208


(i.e. Assumption 11.1.1) for an energy interval [E1 − ε , E2 + ε ], some ε ∈ (0, 1]. Suppose in addition that: • (H4) The period Tpμ (E) is a constant T for all E ∈ [E1 − ε , E2 + ε ];

• (H5) The subprincipal symbol p μ −2 is identically zero on p−1 μ ([E1 − ε , E2 + ε ]).

From hypothesis (H4) it follows by Lemma 11.1.6 that the action A pμ is constant on [E1 , E2 ]. Let hence A pμ (E) = δ , for all E ∈ [E1 , E2 ]. Let α ∈ Z be the Maslov index of the trajectories γX = exp(tH pμ )(X); t ∈ [0, T ] , X ∈ p−1 μ (E1 − ε , E2 + ε ). (The Maslov index is a constant α for it is locally constant and p−1 μ (E1 − ε , E2 + ε ) is connected.) Let P(h) = pw (x, hD; h). Then there exist C0 > 0 and h0 ∈ (0, 1] such that Spec(P(h)) ∩ [E1 , E2 ] ⊂ Ik (h), (12.49) k∈Z

for all h ∈ (0, h0 ], where Ik (h) =

$ # 2π α α 2π k+ h + δ − C0 h2 , k+ h + δ + C0 h2 . T 4 T 4

Proof. We start off by remarking that, since pμ (X ) −→ +∞, as |X| → +∞, p−1 μ ([E1 − ε , E2 + ε ]) is a compact set. Furthermore, by a result of Helffer and Robert (see [65]), there exists h0 ∈ (0, 1] such that P(h), as an unbounded operator in L2 with domain S , is essentially selfadjoint, with compact resolvent for all h ∈ (0, h0 ] (alternatively, when μ ≥ 1 one may use Remark 9.4.8 to conclude that P(h), as an unbounded operator in L2 with domain Bμ , is self-adjoint with a discrete spectrum bounded from below, for all h ∈ (0, h0 ]). Define the set ! Ψε (E1 , E2 ) := ψ ∈ C0∞ (R); supp ψ ⊂ (E1 − ε , E2 + ε ), 0 ≤ ψ ≤ 1, " ψ ≡ 1 in a neighborhood of [E1 , E2 ] . (12.50) Take hence ψ ∈ Ψε (E1 , E2 ). Then by Theorem 12.2.2, Ψ (h) := ψ (P(h)) is an h-pseudodifferential operator with principal symbol ψ (pμ ) and subprincipal symbol pμ −2 ψ (pμ ) ≡ 0, in view of (H5). Take now ψ , ψ˜ ∈ Ψε (E1 , E2 ), with ψ ψ˜ = ψ , and consider (following Chazarain [5] and Helffer and Robert [20]) Uψ (t) := e−ith

−1 P(h)

Ψ (h) = e−ith

−1 P(h)Ψ ˜ (h)

Ψ (h).

(12.51)


209

Notice that P(h), Ψ (h) and Ψ˜ (h) all commute. Equation (12.51) follows from the Spectral Theorem, for e−ith

−1 P(h)

Ψ (h) = =

e−ith

−1 λ

e−ith

−1 λ ψ ˜ (λ )

Spec(P(h))

ψ (λ )dE(λ )

Spec(P(h))

ψ (λ )dE(λ ) = e−ith

−1 P(h)Ψ ˜ (h)

Ψ (h),

since ψ˜ ≡ 1 on the support of ψ . The importance of formula (12.51) lies in the fact that as long as we are interested in the spectral properties of P(h) in the interval [E1 , E2 ], we may replace P(h) by the L2 -bounded operator P(h)Ψ˜ (h). It is well-known after Helffer and Robert (see, e.g., Dimassi-Sjöstrand [7], Ivrii [34], and Robert [65]) that for a time T0 > 0 sufficiently small, one can construct an h−1 ˜ Fourier integral operator (h-Fio) F(t) which approximates e−ih P(h)Ψ (h) for |t| ≤ T0 . The Schwartz-kernel F(t; x, y) of F(t) is written as −n

F(t; x, y) = (2π h)

eih

−1 (S(t,x,η )− y,η )

a(t, x, y, η ; h)d η ,

(12.52)

where the phase-function S is the solution of the Hamilton-Jacobi equation ∂t S(t, x, η ) + p μ ψ˜ (pμ ) (x, ∇x S(t, x, η )) = 0, St=0 = x, η , and amplitude

a(t, x, y, η ; h) ∼

∑ h j a j (t, x, y, η ),

j≥0

where the a j satisfy the transport equations with initial conditions a0 t=0 = 1, and a j t=0 = 0, j ≥ 1. Hence, an h-Fio approximation of Uψ (t) for |t| ≤ T0 has a kernel of the kind (12.52) (with the same phase-function S), and ||F(t)Ψ (h) − Uψ (t)||L2 →L2 = O(h∞ ), uniformly in |t| ≤ T0 . Next, for t ∈ [kT0 , (k + 1)T0 ], one writes k −1 ˜ Uψ (t) = e−ih T0 P(h)Ψ (h)Ψ˜ (h) Uψ (t − kT0 ) = Uψ˜ (T0 )kUψ (t − kT0 ), One then h-Fio-approximates Uψ (t) on the intervals [kT0 , (k + 1)T0 ] by using k Fk (t)Ψ (h) := F(T0 )Ψ˜ (h) F(t − kT0 )Ψ (h),


210

and has that for any given N ∈ Z+ ||Fk (t)Ψ (h) − Uψ (t)||L2 →L2 = O(hN+1 ), uniformly in t ∈ [kT0 , (k + 1)T0 ]. Considering Uψ (T ), and using the h-Fio Fk (t)Ψ (h) one now has the following lemma, due to Chazarain, Helffer and Robert. Lemma 12.2.5. The operator Uψ (T ) is an h-pseudodifferential operator with sym0 (1), where the u are supported in the bol u ∼ u0 + hu1 + h2 u2 + . . . belonging to Scl j −1 bounded set p μ (E1 − ε , E2 + ε ). Its principal symbol is given by u0 (X ) = ψ (pμ (X))e−2π iσ0 (h) , X ∈ R2n , with

α T β + , 4 2π h where α ∈ Z is the Maslov index of the closed trajectories σ0 (h) =

(12.53)

γX := exp(tH pμ )(X); t ∈ [0, T ]}, X ∈ p−1 μ ([E1 − ε , E2 + ε ]), and

β = A pμ (E) + h

1 T

T 0

pμ −2 ◦ exp(tH pμ )(X)dt = A pμ (E) =: δ .

(12.54)

Recall that the action A pμ is constant on [E1 − ε , E2 + ε ], and that we assumed pμ −2 p−1 ([E −ε ,E +ε ]) = 0. (Recall that the Maslov index is a constant α because it μ

1

2

is locally constant and the set p−1 μ ([E1 − ε , E2 + ε ]) is connected.)

Lemma 12.2.5 is a consequence of the following lemma for operators for which T = 2π , due to Helffer and Robert [20] (see also Dimassi-Sjöstrand [7, Theorem 15.4, p. 203]). Lemma 12.2.6. Suppose Q(h) is an operator satisfying the same hypotheses as ˜ = 2π , P(h), with respect to the energy interval [E˜ 1 − ε˜ , E˜2 + ε˜ ]. Suppose that Tq (E) for all E˜ ∈ [E˜1 − ε˜ , E˜2 + ε˜ ]. Let ψ0 ∈ Ψε˜ (E˜1 , E˜2 ), and consider Ψ0 (h) = ψ0 (Q(h)) −1

and Uψ0 (t) = e−ith Q(h)Ψ0 (h). Then the operator Uψ0 (2π ) is an h-pseudodif0 (1), where the ferential operator with symbol v ∼ v0 + hv1 + . . . belonging to Scl −1 ˜ ˜ ˜ ˜ v j are supported in the bounded set q μ (E1 − ε , E2 + ε ). Its principal symbol is given by v0 (X ) = ψ0 (q μ (X))e−2π iσ0 (h) , X ∈ R2n , with

σ0 (h) =

α β˜ + , 4 h

(12.55)


211

where α ∈ Z is the Maslov index of the closed trajectories ˜ ˜ γ˜X := exp(tHqμ )(X); t ∈ [0, 2π ] , X ∈ q−1 μ ([E1 − ε˜ , E2 + ε˜ ]), and

˜ +h 1 β˜ = Aqμ (E) 2π

2π 0

˜ =: δ˜ . qμ −2 ◦ exp(tHqμ )(X)dt = Aqμ (E)

˜ ˜ ˜ ˜ Recall that the action Aqμ is constant on [E1 − ε , E2 + ε ], that by assumption q μ −2 q−1 ([E˜ −ε˜ ,E˜ +ε˜ ]) = 0, and that the Maslov index is a constant α (it is locally 1 2 μ ˜ ˜ constant and q−1 μ ([E1 − ε˜ , E2 + ε˜ ]) is connected). Proof (that Lemma 12.2.5 is a consequence of Lemma 12.2.6). One considers the operator Q(h) := 2Tπ P(h). Then by Lemma 11.1.7 the principal and subprincipal symbols of Q(h), which are q μ = 2Tπ pμ and q μ −2 = 2Tπ pμ −2 , respectively, satisfy the same assumptions with respect to the energy interval [E˜ 1 − ε˜ , E˜2 + ε˜ ] = T ˜ ˜ ˜ ˜ ˜ ˜ 2π [E1 − ε , E2 + ε ], with constant period Tq (E) = 2π for all E ∈ [E1 − ε , E2 + ε ]. If 2π ˜ ˜ ˜ ˜ ˜ ˜ ˜ we consider ψ0 (λ ) := ψ ( T λ ), for λ ∈ [E1 − ε , E2 + ε ] and ψ ∈ Ψε (E1 , E2 ), we then have that ψ0 ∈ Ψε˜ (E˜1 , E˜2 ). Therefore

Ψ0 (h) = ψ0 (Q(h)) = ψ ( and Uψ0 (t) = e−ith

−1 Q(h)

Thus

2π Q(h)) = ψ (P(h)) = Ψ (h), T

Ψ0 (h) = e−ith

−1 T P(h) 2π

Ψ (h) = Uψ (

T t). 2π

Uψ0 (2π ) = Uψ (T ),

and by Lemma 12.2.6 we have v0 (X ) = ψ0 (q μ (X ))e−2πσ0 (h) = ψ (pμ (X))e−2πσ0 (h) = u0 (X ), X ∈ R2n , where σ0 (h) is defined as in (12.55), and the u j are supported in p−1 μ (E1 − ε , E2 + ε ). That α is also the Maslov index of the closed trajectories γX ⊂ p−1 μ ([E1 − ε , E2 + ε ]) follows immediately from the fact that the Maslov index is locally constant, that the set −1 ˜ ˜ p−1 μ ([E1 − ε , E2 + ε ]) = q μ ([E1 − ε˜ , E2 + ε˜ ]) is connected by assumption, and that for any given X ∈ p−1 μ ([E1 − ε , E2 + ε ]), T γ˜X = exp(tHqμ )(X); t ∈ [0, 2π ] = exp( tH pμ )(X); t ∈ [0, 2π ] 2π = exp(sH pμ )(X); s ∈ [0, T ] = γX , that is, any given closed integral curve γ˜X (as above) of Hqμ contained in q−1 μ ([E˜1 − ε˜ , E˜2 + ε˜ ]) is a riparametrization by [0, 2π ] t −→ s = (T /2π )t ∈ [0, T ] of an


212

−1 ˜ integral curve γX (as above) of H pμ contained in p−1 μ ([E1 −ε , E2 +ε ])=q μ ([E1 −ε˜ , ˜ E2 +ε˜ ]). Now, on the one hand

1 2π

2π 0

1 2π T T pμ −2 ◦ exp(t H pμ )(X)dt 2π 0 2π 2π 1 T pμ −2 ◦ exp(sH pμ )(X)ds = 2π 0 T T 1 pμ −2 ◦ exp(sH pμ )(X)ds, = 2π T 0

qμ −2 ◦ exp(tHqμ )(X)dt =

and on the other, again by Lemma 11.1.7, Aq μ (

T T E) = A p (E). 2π 2π μ

We therefore get

T 1 2π β˜ = Aqμ ( E) + h qμ −2 ◦ exp(tHqμ )(X)dt 2π 2π 0 T T 1 T T β, A pμ (E) + h pμ −2 ◦ exp(sH pμ )(X)ds = = 2π 2π T 0 2π from which we also conclude for the constants δ˜ and δ that δ˜ = 2Tπ δ . This completes the proof of the lemma. It therefore follows that −1 T e−2π ih 2π P(h)−hσ0(h) Ψ (h) = Ψ (h) + hW(h), where W (h) is a 0th-order h-pseudodifferential operator whose symbol belongs 0 (1) (and has compact support), with ||W (h)|| to Scl L2 →L2 = O(1) uniformly in h ∈ (0, h0 ]. Take now another φ ∈ Ψε (E1 , E2 ) such that ψφ = φ . Then, by composing with Φ (h) := φ (P(h)), and by noting that [Φ (h),W (h)] = 0, we obtain −1 T e−2π ih 2π P(h)−hσ0(h) Φ (h) = Φ (h) I + hW (h) . Hence, for h0 sufficiently small, we have h||W (h)||L2 →L2 < 1/2 for all h ∈ (0, h0 ], and we can consider −1 I + hW (h) = I + hW (h), where W (h) is an h-pseudodifferential operator of order 0. It follows that we may take (by possibly shrinking h0 ) the logarithm R(h) =

(−1)k k+1 k+1 1 1 log I + hW (h) = h W (h) , ∑ 2π i 2π i k≥0 k + 1


213

which turns out to be a 0th-order h-pseudodifferential operator (with symbol 0 (1)). We have belonging to Scl [R(h), P(h)Ψ˜ (h)] = 0, and ||R(h)||L2 →L2 = O(1), ∀h ∈ (0, h0 ]. It thus follows that −1 T 2 e−2π ih 2π P(h)−hσ0(h)−h R(h) Φ (h) = Φ (h), ∀h ∈ (0, h0 ]. Put now

(12.56)

2π 2π ˜ P(h) := P(h) − h σ0 (h) − h2 R(h), T T

˜ Ψ (h). Since and consider the compact (self-adjoint) operators P(h)Ψ (h) and P(h) they commute, they possess a common basis of eigenfunctions, that we denote by {u j (h)} j∈N , where ˜ Ψ (h)u j (h) = ν j (h)u j (h), j ∈ N. P(h)Ψ (h)u j (h) = μ j (h)u j (h), P(h) ˜ Ψ (h) we have that there exists C0 > 0 such that, using From the definition of P(h) (12.53) and (12.54), |μ j (h) − ν j (h) −

2π α h − δ | ≤ C0 h2 , ∀ j ∈ N. T 4

(12.57)

On the other hand, from (12.56) it follows that for h0 sufficiently small and for all h ∈ (0, h0 ], 2π hZ. (12.58) ν j (h) ∈ T Using (12.58) in (12.57) concludes the proof. Remark 12.2.7. It is important to notice that if h0 is sufficiently small, then / ∀h ∈ (0, h0 ], ∀k = k . Ik (h) ∩ Ik (h) = 0, Remark 12.2.8. Notice in particular that the spectrum of P(h) inside [E1 , E2 ] is discrete. One may find in Robert [65, Theorem III-4, p. 129, and Proposition III-13, p. 145] more general conditions that ensure the discreteness of the spectrum inside an energy band [E1 , E2 ]. One has the following corollary, that deals with the case when the subprincipal symbol pμ −2 is a non-zero constant in a region p−1 μ ([E1 − ε , E2 + ε ]). Corollary 12.2.9. Under the assumptions of Theorem 12.2.4, with μ ≥ 2 and with hypothesis (H5) replaced by the condition:


214

• the subprincipal symbol p μ −2 is a non-zero constant c0 in p−1 μ ([E1 − ε , E2 + ε ]),

one has the same conclusion of Theorem 12.2.4, that is Spec(P(h)) ∩ [E1 , E2 ] ⊂

Ik (h), ∀h ∈ (0, h0 ],

k∈Z

where this time $ # 2π α α 2π k+ h + c0h + δ − C0 h2 , k+ h + c0h + δ + C0 h2 . Ik (h) = T 4 T 4 Notice that in this case β = A pμ (E) + hc0 = δ + hc0 , and that, again, if h0 is chosen sufficiently small, then Ik (h) ∩ Ik (h) = 0, / for all h ∈ (0, h0 ] and all k = k . Proof. Let P(h) = pw (x, hD; h), and let P0 (h) := P(h) − hpw μ −2(x, hD). Notice that P(h) and P0 (h) have the same principal symbol p μ . Assume, in the first place, that pμ −2 (X ) ≡ c0 for all X . Then P(h) = P0 (h) + hc0 , and they have the same spectral family. Using the proof of the theorem (recall that A pμ (E) = δ ), one therefore obtains, with φ ∈ Ψε (E1 , E2 ), that e−2π ih

−1 ( T P (h)−h( α + T δ h−1 )−h2 R(h)) 2π 0 4 2π

φ (P0 (h)) = φ (P0 (h)), ∀h ∈ (0, h0 ],

that is, e−2π ih

−1 ( T P(h)− T hc −h( α + T δ h−1 )−h2 R(h)) 2π 2π 0 4 2π

φ (P0 (h)) = φ (P0 (h)), ∀h ∈ (0, h0 ],

which (by the proof of Theorem 12.2.4, since every operator which commutes with P0 (h) also commutes with P(h)) proves the corollary in case the subprincipal part is a constant everywhere. In case we only know that p μ −2 is a constant c0 in p−1 μ ([E1 − ε , E2 + ε ]) we proceed as follows. We write P(h) = P0 (h) + hc0 + h(pw μ −2 (x, hD) − c0 ) and put −1 b(X ) := pμ −2 (X ) − c0 . Then supp(b) ∩ p μ ([E1 − ε , E2 + ε ]) = 0. / Hence, from Theorem 12.2.2 it follows that bw (x, hD)ψ (P(h)) = O(h∞ ), and ψ (P(h)) = ψ (P0 (h) + hc0) + h2rw (x, hD; h), 0 (m−k , g), for all k ≥ 1. We thus obtain where, by Robert’s Theorem 12.2.3, r ∈ S0,cl that P(h)ψ (P(h)) = P0 (h) + hc0 + hbw(x, hD) ψ (P(h)) = P0 (h) + hc0 ψ (P(h)) + O(h∞) = P0 (h) + hc0 ψ (P0 (h) + hc0) + O(h2),

which, by virtue of the minimax principle for compact self-adjoint operators and the first part of the proof, gives the claim. One has also the following corollary, due to Helffer and Robert [20], that deals with the case when the period T is not a constant in [E1 − ε , E2 + ε ].


215

0 (m μ , g), μ > 0, p ∼ p + hp Corollary 12.2.10. Let p ∈ S0,cl μ μ −2 + . . . , be real μ valued, with principal symbol pμ ≈ m satisfying hypotheses (H1), (H2), (H3) (see Assumption 11.1.1) and (H5) (see Theorem 12.2.4) for an energy interval [E1 − ε , E2 + ε ], for some ε ∈ (0, 1]. Let α ∈ Z be the Maslov index of the trajectories γX ⊂ p−1 μ (E1 − ε , E2 + ε ). Then there exist constants δ ∈ R, C0 > 0 and h0 ∈ (0, 1] such that, with P(h) = pw (x, hD; h),

$ # α α h + δ − C0 h2 , k + h + δ + C0 h2 , J Spec(P(h)) ∩ [E1 , E2 ] ⊂ k+ 4 4 k∈Z (12.59) for all h ∈ (0, h0 ], where J = J pμ is the map defined in Lemma 11.1.2. Proof. In the first place we reduce matters to the case T (E) = 2π for all E ∈ [E1 − ε , E2 + ε ]. Consider the function J = J pμ and extend it to a map J˜: R −→ R such that 5 a1 E + b1, for E ≤ E1 − 2ε , ˜ J˜ is strictly increasing, J(E) = a2 E + b2, for E ≥ E2 + 2ε . Define then the operator 1 ˜ J(P(h)). 2π Using Robert’s Theorem 12.2.3 we have that Q(h) = qw (x, hD; h), where q ∈ 0 S0,cl (mμ , g), q ∼ q μ + hq μ −2 + . . . , with the q μ −2 j independent of h, and Q(h) =

˜ μ ), q μ −2 = pμ −2 J˜ (pμ ). q μ = J(p Hence

q μ −2 (X ) = 0, ∀X ∈ p−1 μ ([E1 − ε , E2 + ε ]).

Since Hqμ =

J˜ (pμ ) H pμ , 2π

it follows that ˜ exp(tHqμ ) is periodic with period 2π in a neighborhood of q−1 μ (J([E1 , E2 ])). It thus follows from Lemma 11.1.6 that the (averaged) action Aqμ of the integral trajectories of Hqμ is a constant δ (the one that appears in the statement of the theorem) for all energies belonging to such a neighborhood. Remark that, J˜ being a smooth diffeomorphism from a neighborhood of [E1 , E2 ] ˜ 1 , E2 ]) =: [E˜ 1 , E˜2 ], the properties of the point spectrum of to a neighborhood of J([E P(h) in [E1 , E2 ] are immediately deduced from those of Q(h) in [E˜1 , E˜2 ] : in fact,

λ ∈ Spec(P(h)) ∩ [E1 , E2 ] ⇐⇒

J(λ ) ∈ Spec(Q(h)) ∩ [E˜1 , E˜2 ]. 2π

Hence the corollary follows from Theorem 12.2.4.

216


If one assumes that the trajectories in p−1 μ ([E1 − ε , E2 + ε ]) are all periodic with least period T, independent of E ∈ [E1 − ε , E2 + ε ], that is • t −→ exp(tH pμ )(X) is periodic with period T > 0 for all X ∈ p−1 μ ([E1 −ε , E2 +ε ]) • and

exp(tH pμ )(X) = X , ∀t ∈ (0, T ), ∀X ∈ p−1 μ ([E1 − ε , E2 + ε ]),

then one can obtain information on the number NP(h) (Ik (h)) = card λ (h); λ (h) ∈ Spec(P(h)) ∩ Ik (h) with multiplicity , (12.60) that is, the number of the eigenvalues of P(h), repeated according to multiplicity, which belong to Ik (h). In order to obtain an asymptotic formula for the behavior of NP(h) (Ik (h)) as h → 0+, we have to review what the Leray-Liouville measure is. Definition 12.2.11. Let f : Ω ⊂ R2n −→ R be a smooth function (Ω an open set) such that, for some λ ∈ R, the set f −1 (λ ) = 0/ is compact, and d f (ρ ) = 0 for all ρ ∈ f −1 (λ ). One defines the Leray-Liouville (2n − 1)-form L f ,λ by the equation L f ,λ ∧ d f = dX1 ∧ . . . ∧ dX2n , on f −1 (λ ),

(12.61)

where dX1 ∧ . . . ∧ dX2n is the positive volume-form in R2n X associated with the coordinates X1 , . . . , X2n . By L f ,λ (dX) we shall denote the associated (positive) Leray-Liouville measure. Notice that (12.61) determines L f ,λ up to multiples of d f , for if L f ,λ ∧ d f = dX1 ∧ . . . ∧ dX2n then also

L f ,λ + ω ∧ d f ∧ d f =

dX1 ∧ . . . ∧ dX2n , for any given (2n − 2)-form ω . We shall hence always normalize L f ,λ so as to have that the (2n − 2)-form L f ,λ (∇X f , ·) = i∇X f L f ,λ = 0.

(12.62)

Here i∇X f is the contraction operator by ∇X f , where ∇X f denotes the gradient vector-field associated with d f through the Euclidean scalar product: d f (ρ )(v) = ∇X f (ρ ), v , for all ρ ∈ R2n and all v ∈ Tρ R2n . Lemma 12.2.12. The normalization condition (12.62) fixes L f ,λ uniquely. Proof. We work at any given point of f −1 (λ ). Let L1 , L2 be such that L1 ∧ d f = L2 ∧ d f = dX1 ∧ . . . ∧ dX2n, and i∇X f L j = 0, j = 1, 2. Then (L1 − L2 ) ∧ d f = 0. Hence, L1 and L2 being normalized, we get 0 = i∇X f (L1 − L2 ) ∧ d f = i∇X f (L1 − L2 ) ∧ d f + (−1)2n−1(L1 − L2 ) ∧ i∇X f d f = (−1)2n−1|∇X f |2 (L1 − L2 ), which proves the claim, for by hypothesis ∇X f = 0.


217

The following lemma will be useful in the proof of Theorem 12.2.16 below. Lemma 12.2.13. Let α ∈ R+ . Given f and λ as in Definition 12.2.11 above, let g := α f . Then for the respective Leray-Liouville (2n − 1)-forms associated with g and f we have on g−1 (αλ ) = f −1 (λ ) the scaling property Lα f ,αλ =

1 L . α f ,λ

Proof. Consider the smooth hypersurface S = f −1 (λ ) = g−1 (αλ ) of R2n . At any given point ρ ∈ S we have dg = α d f (and ∇X g = α ∇X f ), whence L f ,λ ∧ d f = dX1 ∧ . . . ∧ dX2n = Lg,αλ ∧ dg = α Lg,αλ ∧ d f . Since L f ,λ and α Lg,αλ are both normalized, that is i∇X f L f ,λ = 0, i∇X f (α Lg,αλ ) = i∇X g Lg,αλ = 0,

using Lemma 12.2.12 completes the proof.

We may also write a more explicit expression of L f ,λ . To this aim, recall first of all that for a finite dimensional vectore space V , if ω1 , . . . , ωk ∈ V ∗ and v1 , . . . , vk ∈ V, then , (ω1 ∧ . . . ∧ ωk (v1 , v2 , . . . , vk ) = det ω j (v j ) 1≤ j, j ≤k . (12.63) We have the following formula for L f ,λ (which yields in particular another proof of Lemma 12.2.13). Proposition 12.2.14. Let f and λ be as in Definition 12.2.11. Put 6j ∧ . . . ∧ dX2n, 1 ≤ j ≤ 2n, ω ( j) := dX1 ∧ . . . ∧ dX where the “hat” means that that term has been omitted. For all ρ ∈ S = f −1 (λ ) and all v1 , v2 , . . . , v2n−1 ∈ Tρ R2n one has L f ,λ (v1 , . . . , v2n−1 ) =

2n

1 |∇X

f |2

∂f

∑ (−1) j ∂ X j ω ( j) (v1 , . . . , v2n−1),

(12.64)

j=1

or, equivalently, L f ,λ = −

1 i (dX1 ∧ . . . ∧ dX2n ). |∇X f | ∇X f /|∇X f |

(12.65)

Recall that we take the normalization i∇X f L f ,λ = 0. In particular, when n = 1 we have 1 ∂f ∂f L f ,λ = (12.66) dX1 − dX2 , 2 |∇X f | ∂ X2 ∂ X1 where now the normalization condition can be written as L f ,λ (∇X f ) = 0.


218

Proof. We put v() = (v1 , . . . , v7 , . . . , v2n−1 ) ∈ R2n × . . . × R2n , 1 ≤ ≤ 2n − 1, where 2n−2 times

the “hat” means that that term has been omitted. Since {ω ( j) }1≤ j≤2n is a basis of the space of (2n − 1)-forms, we may write L f ,λ =

2n

∑ α j ω ( j).

(12.67)

j=1

We now use (12.63). We have on the one hand (L f ,λ ∧ d f )(∇X f , v1 , . . . , v2n−1 ) = =

∑ αj

∑ αj

( j) ω ∧ d f (∇X f , v1 , . . . , v2n−1 )

j=1

%

2n

2n

−|∇X f |2 ω ( j) (v1 , . . . , v2n−1 ) +

j=1

&

2n−1

∑ (−1)+1ω ( j) (∇X f , v() )d f (v )

=1

2n

= −|∇X f |2 ∑ α j ω ( j) (v1 , . . . , v2n−1 ) j=1

+

2n−1

%

∑ (−1)

+1

=1

&

2n

∑ α jω

( j)

()

(∇X f , v ) d f (v )

j=1

(recalling (12.67)) = −|∇X f |2 L f ,λ (v1 , . . . , v2n−1 ) +

2n−1

∑ (−1)+1L f ,λ (∇X f , v() )d f (v )

=1

(by the normalization condition) = −|∇X f |2 L f ,λ (v1 , . . . , v2n−1 ). On the other hand, (dX1 ∧ . . . ∧ dX2n )(∇X f , v1 , . . . , v2n−1 ) =

∂f

2n

∑ (−1) j+1 ∂ X j ω ( j)(v1 , . . . , v2n−1).

j=1

Therefore L f ,λ (v1 , . . . , v2n−1 ) = −

2n

1 |∇X

f |2

∂f

∑ (−1) j+1 ∂ X j ω ( j)(v1 , . . . , v2n−1).

(12.68)

j=1

Notice that if L˜ f ,λ denotes the right-hand side of (12.68), then, for any given v1 , . . . , v2n−2 ∈ R2n , L˜ f ,λ (∇X f , v1 , . . . , v2n−2 ) =−

1 |∇X f |2

(dX1 ∧ . . . ∧ dX2n)(∇X f , ∇X f , v1 , . . . , v2n−2 ) = 0,


219

that is, the normalization condition is indeed fulfilled. Specializing to the case n = 1 gives (12.66) from (12.64). This concludes the proof. As regards the Leray-Liouville measure associated with f −1 (λ ) we have the following corollary. Corollary 12.2.15. Let f and λ be as in Definition 12.2.11. Recall that the canonical volume form associated with { f = λ } is (for every ρ ∈ S) given by vol f =λ = i∇X f /|∇X f | (dX1 ∧ . . . ∧ dX2n )

f =λ

(12.69)

(the restriction is the pull-back by the natural inclusion map). Then, for the LerayLiouville measure we have L f ,λ (dX) = In particular,

f =λ

1 vol . |∇X f | f =λ

(12.70)

L f ,λ (dX) > 0.

Proof. The proof follows immediately from (12.65).

We are ready for the “multiplicity” theorem, due to Helffer and Robert (see [20]). Theorem 12.2.16. With the notation and assumptions of Theorem 12.2.4, suppose (H6)

exp(tH pμ )(X) = X , ∀t ∈ (0, T ), ∀X ∈ p−1 μ ([E1 − ε , E2 + ε ]),

that is, T > 0 is the least period in p−1 μ ([E1 − ε , E2 + ε ]). Let ψ ∈ Ψε (E1 , E2 ) (see (12.50)). Then there exist real-valued functions Γj ∈ C0∞ (E1 − ε , E2 + ε ) such that for 2Tπ (k + α4 )h + δ ∈ [E1 , E2 ] NP(h) (Ik (h)) ∼

∑ Γj

j≥1

2π T

(k +

α )h + δ h j−n (h → 0+), 4

(12.71)

where, for j = 1,

Γ1 (λ ) = (2π )−n ψ (λ )

2π T

pμ (X)=λ

L pμ ,λ (dX).

(12.72)

When n = 1 one has that p−1 μ (E) is a smooth closed curve γ (E) with Maslov index α = ±2 (for γ (E) is homotopic to a circle then), the sign depending on the orientation of γ (E). Since −2 ≡ 2 mod 4, we may take α = 2, so that for h0 sufficiently small NP(h) (Ik (h)) = 1, ∀(k, h) ∈ Z × (0, h0] with

1 2π (k + )h + δ ∈ [E1 , E2 ]. (12.73) T 2

220


Proof (sketch; see Helffer-Robert [20] and Dimassi-Sjöstrand [7]). Observe in the first place that if 2Tπ (k + α4 )h + δ ∈ [E1 , E2 ], then for h0 sufficiently small, NP(h) (Ik (h)) =

∑

λ ∈Spec(P(h))∩Ik (h)

ψ (λ ) =: Nk,ψ (h), ∀h ∈ (0, h0 ].

Following a by now classical idea of Colin De Verdière, take a real valued χ ∈ C0∞ (−3π /2, 3π /2) such that (12.74) ∑ χ (t − 2π j) = 1. j∈Z

Recall from the proof of Theorem 12.2.4 that, with Q(h) =

T T ˜ P(h), Q(h) = P(h) − hσ0(h) − h2R(h), 2π 2π

and ψ0 (λ˜ ) = ψ ( 2Tπ λ˜ ), one has

∑


and ˜ Spec(Q(h)) ∩

∑

ψ (λ ) =

ψ0 (λ˜ ),

λ˜ ∈Spec(Q(h))∩ 2Tπ Ik (h)

T [E1 − ε , E2 − ε ] ⊂ hZ, ∀h ∈ (0, h0 ]. 2π

Hence −1 ˜ Tr ψ0 (Q(h))e−ith Q(h) =

∑⎝

k∈Z

=

⎞

⎛

∑

k∈Z

%

∑

ψ0 (λ˜ )⎠ e−itk

λ˜ ∈Spec(Q(h))∩ 2Tπ Ik (h)

∑


&

ψ (λ ) e−itk ,

(12.75)

whence the Nk,ψ (h) are the Fourier-coefficients of (12.75), that is, −1 ˜ 1 2π ikt e Tr ψ0 (Q(h))e−ith Q(h) dt (using (12.74)) Nk,ψ (h) = 2π 0 −1 −1 1 (12.76) eih t τ χ (t)Tr ψ0 (Q(h))eithR(h) e−ith Q(h) dt, = 2π R

where

α T T )h + δ ∈ (E1 − ε , E2 + ε ). 4 2π 2π At this point, hypothesis (H6) ensures that t = 0 is the unique period of the integral curves of H pμ in p−1 μ (E1 − ε , E2 + ε ), and stationary-phase arguments (see Dimassi-Sjöstrand [7]) show that −1 ˜ Tr ψ0 (Q(h))e−ith Q(h) = O(h∞ ) τ := (k +


221

when t ∈ supp χ \ Ω , where Ω is any fixed neighborhood of 0, and that (12.71) holds, where

T λ) T L T p, T λ (dX) T 2π 2π 2π 2π p μ = 2π λ 2π L (dX), = (2π )−n ψ (λ ) T pμ =λ pμ ,λ

Γ1 (λ ) = (2π )−n ψ0 (

by virtue of Lemma 12.2.13. When n = 1, one has from the Stokes formula (keeping into account orientations) that for any given E ∈ (E1 − ε , E2 + ε ) J p (E) − J p(E1 − ε ) = =

γ (E)

ξ dx −

γ (E1 −ε )

E1 −ε ≤p(X)≤E

ξ dx

d ξ ∧ dx =

E1 −ε ≤p(X)≤E

dX,

from which it follows that J p (E) =

E1 −ε ≤p(X)≤E

dX + const.

Using the fact that ∇X pμ = 0 at the points of p−1 μ (E) gives dX = L pμ ,E (dX)dE, so that Tp = J p (E) =

pμ =E

L pμ ,E (dX),

which then gives NP(h) (Ik (h)) = 1 + O(h) for all (k, h) ∈ Z × (0, h0 ] with 2π 1 T (k + 2 )h + δ ∈ [E1 , E2 ], whence (12.73). When the subprincipal symbol pμ −2 (X) is a non-zero constant c0 for X ∈ p−1 μ ([E1 − ε , E2 + ε ]), one uses the definition (12.54) of β accordingly. Following Dozias’ adaptation of Hörmander [30] to the semiclassical case (see Dozias [8]), we next show how one can replace the condition that pμ −2 be constant with the following condition: • (H5 ) There exists c0 ∈ R such that for all X ∈ p−1 μ ([E1 − ε , E2 + ε ]) one has

1 T

T 0

pμ −2 ◦ exp(tH pμ )(X)dt = c0 .

0 (m μ , g) be real-valued, μ ≥ 2 and p ≈ m μ . Since we are Let hence p ∈ S0,cl μ interested in the spectral properties of pw (x, hD) in the interval [E1 − ε , E2 + ε ], we may replace pw (x, hD) with χ (pw (x, hD)), where χ is smooth, bounded, stricly

222


increasing and χ (t) = t for t ∈ [E1 − ε , E2 + ε ], and therefore may assume that p ∈ 0 (1), with p ∼ p + hp + . . . . The choice of χ ensures that p−1 ([E − ε , E + ε ]) = Scl 0 1 1 2 μ (χ ◦ pμ )−1 ([E1 − ε , E2 + ε ]) = p−1 ([E − ε , E + ε ]). The crucial result is contained 1 2 0 in the following proposition. 0 (1), p = p +hp +. . . . Assume that all the integral Proposition 12.2.17. Let p ∈ Scl 0 1 −1 trajectories of H p0 in p0 ([E1 − ε , E2 + ε ]) are periodic with the same period T > 0 and that (H5 ) holds for the subprincipal term p1 . Write w 2 w 2 P(h) = pw 0 (x, hD) + hp1 (x, hD) + h r (x, hD) =: P0 (h) + hP1(h) + h R(h),

where R(h) = O(1), for all h ∈ (0, h0 ]. Define also 1 Pˆ1 (h) = T

T

eith

−1 P (h) 0

0

P1 (h)e−ith

−1 P (h) 0

dt.

Then P0 (h) + hP1 (h) and P0 (h) + hPˆ1 (h) are unitarily equivalent modulo O(h2 ) in the energy band [E1 − ε , E2 + ε ]. Proof. Put P(t) := eith

−1 P

P1 e−ith

−1 P

, and S := −

1 T

T t 0

(

P(s)ds)dt. 0

Then, by the results in Robert’s book [65] (namely, Egorov’s Theorem) they are self0 (1). We now compare adjoint h-pseudodifferential operators with symbols in Scl eiS P0 (h) + hP1(h) e−iS with P0 (h) + hPˆ1(h), by computing their principal and subprincipal symbols, which we denote by σ0 (·) and sub(·), respectively. To this end we need the following lemma. 0 (1). Then eiS Ae−iS is an h-pseudodifLemma 12.2.18. Let A = A∗ with symbol in Scl ferential operator such that

symbol(eiS Ae−iS ) ∼

∑

j≥0

1 symbol (ad iS) j A , j!

(12.77)

where (ad iS)A = [iS, A]. Proof (of the lemma). Since S = S∗ , by Beals’ characterization of h-pseudodifferential operators (Theorem 9.1.11), we have that eitS is an h-pseudodifferential operator with symbol in S00 (1), and is unitary. Put A(t) = eitS Ae−itS .


Then

223

A (t) = eitS [iS, A]e−itS , A( j) (t) = eitS (ad iS) j A e−itS .

Hence

A( j) (0) = (ad iS) j A, ∀ j ∈ Z+ ,

and therefore, formally, A(1) = Now, for all j and all t

A( j) (0) . j! j≥0

∑

||A( j) (t)||L2 →L2 ≤ C j h j ,

for one sees that the first j terms in the symbol are 0. Using A(1) =

N

∑

j=0

1 1 (ad iS) j A + j! N!

1 0

(1 − t)N A(N+1) (t)dt,

we get N

||A(1) − ∑

j=0

1 (ad iS) j A||L2 →L2 ≤ CN hN+1 , j!

and this proves the lemma. We now use (12.77) and have that on the one hand σ0 eiS (P0 + hP1)e−iS = p0 , sub eiS (P0 + hP1)e−iS = {σ0 (S), p0 } + p1, and, on the other, σ0 P0 + hPˆ1 = p0 , 1 T ˆ sub P0 + hP1 = p1 ◦ exp(tH p0 )dt. T 0

Hence, equality at the operator-level modulo O(h2 ) occurs for X ∈ p−1 0 ([E1 − ε , E2 + ε ]) if 1 T p1 ◦ exp(tH p0 )dt. (12.78) {σ0 (S), p0 } + p1 = T 0 Notice that since p0 ∈ S00 (1), the Hamilton vector field H p0 is uniformly bounded on R2n , whence for every X ∈ R2n the integral trajectories t → exp(tH p0 )(X) exist for all t ∈ R. It follows that C∞

R × R2n (t, X ) −→ exp(tH p0 )(X),

224


and that R2n X −→

1 T

T 0

p1 ◦ exp(tH p0 )(X)dt

is well-defined and smooth. Now,

1 T t ( p1 ◦ exp(sH p0 )(X)ds)dt T 0 0 1 T T ( p1 ◦ exp(sH p0 )(X)dt)ds =− T 0 s 1 T (T − s)p1 ◦ exp(sH p0 )(X)ds =− T 0 T 1 T p1 ◦ exp(sH p0 )(X)ds + s p1 ◦ exp(sH p0 )(X)ds =− T 0 0 1 T s p1 ◦ exp(sH p0 )(X)ds, ∀X ∈ p−1 = −T c0 + 0 ([E1 − ε , E2 + ε ]). T 0

σ0 (S)(X) = −

Then, for all X ∈ p−1 0 ([E1 − ε , E2 + ε ]) we have d σ0 (S) ◦ exp(tH p0 )(X) dt t=0 d 1 T =− s p1 ◦ exp((s + t)H p0 )(X)ds dt T 0 t=0 1 T d =− s p1 ◦ exp((s + t)H p0 )(X) ds T 0 dt t=0 1 T d p1 ◦ exp(sH p0 )(X) ds s =− T 0 ds 1 T d s p1 ◦ exp(sH p0 )(X) ds + c0 =− T 0 ds = −p1 ◦ exp(T H p0 )(X) + c0 = −p1 (X ) + c0,

{σ0 (S), p0 }(X ) = −

which shows that (12.78) indeed holds. This concludes the proof of the proposition. Proposition 12.2.17 and Corollary 12.2.9 thus yield the following result. Corollary 12.2.19. Under the assumptions of Theorem 12.2.4, with μ ≥ 2 and with the subprincipal symbol p μ −2 fulfilling hypothesis (H5 ) instead of hypothesis (H5), one has the same conclusion of Theorem 12.2.4, that is Spec(P(h)) ∩ [E1 , E2 ] ⊂

k∈Z

Ik (h), ∀h ∈ (0, h0 ],


225

where, this time, Ik (h) =

$ # 2π 2π α α k+ h + c0h + δ − C0 h2 , k+ h + c0h + δ + C0 h2 , T 4 T 4

and

β = A pμ (E) + h

1 T

T 0

pμ −2 ◦ exp(tH pμ )(X)dt = δ + hc0.

Notice that, again, if h0 is chosen sufficiently small, then Ik (h) ∩ Ik (h) = 0, / for all h ∈ (0, h0 ] and all k = k . Furthermore, in Theorem 12.2.16, with (H5 ) replacing (H5), one has the same conclusion, with β as above. As a consequence of Corollary 12.2.9 and Corollary 12.2.19 we have the following result, which is useful when dealing with h∞ -diagonalizations. 0 (mμ , g), μ ≥ 2, p ∼ pμ + hp μ −2 + . . . , be real valued, Lemma 12.2.20. Let p ∈ S0,cl with principal symbol p μ ≈ mμ satisfying hypotheses (H1), (H2) and (H3) for an energy interval [E1 − ε , E2 + ε ], for some ε ∈ (0, 1]. Suppose in addition that:

• (H4) The period Tpμ (E) is a constant T for all E ∈ [E1 − ε , E2 + ε ];

• (H5 ) There exists c0 ∈ R such that for all X ∈ p−1 μ ([E1 − ε , E2 + ε ]) one has

1 T

T 0

pμ −2 ◦ exp(tH pμ )(X)dt = c0 .

Let α ∈ Z be the Maslov index of the trajectories γX ⊂ p−1 μ (E1 − ε , E2 + ε ). Consider 0 μ p˜ ∈ S0,cl (m , g), μ ≥ 2, such that p˜ = p + h Im( f { f¯, p μ }), where f : R2n −→ S1 ⊂ C belongs to S(1, g). We know from Corollary 12.2.19 that there exist C0 > 0 and h0 ∈ (0, 1] such that

Spec(P(h)) ∩ [E1 , E2 ] ⊂

Ik (h),

k∈Z

for all h ∈ (0, h0 ], where P(h) = pw (x, hD; h) and Ik (h) =

$ # 2π α α 2π k+ h + c0h + δ − C0 h2 , k+ h + c0h + δ + C0 h2 . T 4 T 4

˜ Put P(h) = p˜w (x, hD; h). Then also ˜ Spec(P(h)) ∩ [E1 , E2 ] ⊂

k∈Z

Ik (h), ∀h ∈ (0, h0 ].


226

Proof. We need only show that T 0

( f { f¯, pμ }) exp(tH pμ )(X) dt = 2π ik0 ,

(12.79)

where k0 ∈ Z is independent of X ∈ p−1 μ ([E1 − ε , E2 + ε ]). In fact, one then has 1 T

T 0

2π k0 , Im ( f { f¯, pμ }) exp(tH pμ )(X) dt = T

and through Corollary 12.2.9 and Corollary 12.2.19 this shows that

˜ Spec(P(h)) ∩ [E1 , E2 ] ⊂

k ∈Z

I˜k (h), where I˜k (h) = Ik +k0 (h),

thus concluding the proof modulo (12.79), that we now prove. Since f f¯ = 1 we have that 0 = { f f¯, pμ } = f¯{ f , p μ } + f { f¯, pμ }, whence f { f¯, pμ } ∈ iR. ¯ ¯ Let γX (t) = exp(tH pμ )(X) for X ∈ p−1 μ ([E1 − ε , E2 + ε ]). As f { f , p μ } = f {p μ , f }, we get T 0

T ( f { f¯, pμ }) γX (t) dt = ( f¯{pμ , f }) γX (t) dt 0

= =

T {p μ , f } 0

T 0

◦ γX )(t)dt f 1 dz d ( f ◦ γX )(t)dt = ∈ 2π iZ. f (γX (t)) dt f ◦γX z

(

Since p−1 μ ([E1 − ε , E2 + ε ]) is a connected set and p−1 μ ([E1 − ε , E2 + ε ]) X −→

f ◦γX

dz is continuous, z

we obtain (12.79) and complete the proof of the corollary.

Remark 12.2.21. Notice that when f (X) = eiϕ (X) , with ϕ ∈ S(1, g), then in (12.79) one has k0 = 0. ˜ Since P(h) and P(h) in Lemma 12.2.20 have the same principal symbol, they have in particular the same Leray-Liouville measure associated with p−1 μ (E), E ∈ [E1 − ε , E2 + ε ]. Hence from Theorem 12.2.16 and Corollary 12.2.19 we obtain the following consequence.

s 12.3 Localization and Multiplicity of Spec(Aw 2 (x, D)), with A2 ∈ Q2

227

Corollary 12.2.22. Let k ∈ Z and h ∈ (0, h0 ] be such that 2Tπ (k + α4 )h + c0 h + δ ∈ [E1 , E2 ]. Let k ∈ Z be such that k = k + k0 , where k0 is given in (12.79), whence I˜k (h) = Ik (h). Then ˜ / and Spec(P(h)) ∩ Ik (h) = 0, / Spec(P(h)) ∩ Ik (h) = 0, so that

˜ dist Spec(P(h)) ∩ Ik (h), Spec(P(h)) ∩ Ik (h) = O(h2 ).

12.3 Localization and Multiplicity of Spec(Aw 2 (x, D)), with A2 ∈ Q s2 We are finally in a position to describe in this section some properties of the large eigenvalues of a general 2 × 2 elliptic positive NCHO in Rn with symbol A2 ∈ Q2s , that are linked to properties of the periods of the bicharacteristics of the eigenvalues of the symbol A2 (X ). As already mentioned, we shall go through the semiclassical construction. In the first place we show that assumptions (H1) and (H2) are fulfilled by the eigenvalues of a system A2 ∈ Q2 . Proposition 12.3.1. Let λ± be the eigenvalues of a system A2 ∈ Q2 . Then hypotheses (H1) and (H2) hold for λ± . Proof. It is immediate using Proposition 11.1.4, for λ± : R2n \ {0} −→ R+ are positively homogeneous of degree 2. We next show that when A2 ∈ Q2s , we may find h∞ -diagonalizations of the reference operator Aw r (x, hD) for which the subprincipal terms in the diagonal form vanish in the region {X ∈ R2n ; λ− (X) ≥ ε1 } (see (12.4), (12.13) and Remark 12.1.21). Proposition 12.3.2. Suppose that A2 ∈ Q2s . Then there exists a unitary symbol e0 ∈ S(1, g; M2 ), e0 e∗0 = e∗0 e0 = I, such that for the subprincipal symbol Λ0 = diag(Λ0+ , Λ0− ) of the h∞ -diagonalization of the reference operator Aw r (x, hD) we have Λ0± (X ) = 0, ∀X ∈ {X ∈ R2n ; λ− (X ) ≥ ε1 }. By Remark 12.1.21, the proposition is a consequence of the following lemma (related to diagonalizations of systems of classical symbols, see Theorem 9.2.3). Lemma 12.3.3. When A2 ∈ Q2s we may always find a diagonalization whose resulting subprincipal symbol Λ0 = diag(Λ0+ , Λ0− ) satisfies

Λ0± (X) = 0, ∀X = 0.

228


Proof. We work for X = 0. Consider A2 ∈ Q2s and write * A2 (X) =

a11 (X) γ b(X )

+

γ¯ b(X) a22 (X )

,

where b is a real-valued homogeneous polynomial of degree 2 in X , γ = 1 or γ = i, and (say) a11 (X ) > a22 (X ). Recall that λ± are the eigenvalues of A 2 (see

(12.4)), 1 0 2 and that {w+ , w− } is the canonical basis of C , w+ = , w− = . Put also 0 1 (as before) a11 ± a22 . a± = 2 Then ) √ λ± = a+ ± a2− + b2 = a+ ± δ , where (as before) δ = a2− + b2. Consider therefore (always for X = 0) the eigenvectors belonging to λ± , ±-respectively, * v˜+ (X ) :=

a− (X ) +

' + δ (X)

γ¯ b(X )

+ −γ b(X ) , v˜− (X) := , ' a− (X ) + δ (X ) *

and hence the unitary matrix (positively homogeneous of degree 0) whose columns are, in the order, the normalized eigenvectors v+ and v− : e0 (X ) = [v+ (X)|v− (X)] , v± (X) =

v˜± (X ) . |v˜± (X )|C2

It is now crucial to notice that e0 (X )w+ = v+ (X ) ∈ R × γ R, e0 (X)w− = v− (X) ∈ γ R × R, ∀X = 0, whence It follows that so that

(12.80)

∂ e0 w+ ∈ R × γ R, ∂ e0 w− ∈ γ R × R, ∀X = 0. ∂ e0 w± , e0 w± ∈ R, ∂ e0 w± , e0 w± = e0 w± , ∂ e0 w± ,

whence, since |v± (X )|C2 = 1 for all X = 0, ∂ e0 (X )w± , e0 (X)w± = 0, ∀X = 0.

(12.81)


229

We thus have

∂ e0 (X )w± ∈ SpanC {e0 (X)w± }⊥ = SpanC {e0 (X )w∓ }, ∀X = 0.

(12.82)

Recall (see (12.46)) that ±-respectively for all X = 0 1 1 Λ0± = Im {e∗0 , λ± }e0 w± , w± + Im e∗0 {A2 , e0 }w± , w± =: I1 + I2 . 2 2 Consider I1 . By (12.81) we have ∂λ

∂ e∗0 ∂ λ± ∂ e∗0 e0 w± , w± − e0 w± , w±

∂ ξ ∂ x =1 ∂ x ∂ ξ n ∂ λ± ∂ e0 ∂ λ± ∂ e0 e0 w± , w± − e0 w± , w± ∈ R, = ∑ ∂ ξ ∂ ξ ∂ x =1 ∂ x

{e∗0 , λ± }e0 w± , w± =

n

∑

±

whence I1 = 0, ∀X = 0. We next consider I2 . Using (9.42) we have ∂ A2 ∂ e0 w± , e0 w± = ∂ e0 w± , (∂ A2 )e0 w±

= ∂ e0 w± , (∂ λ± )e0 w± − (A2 − λ±)∂ e0 w± , which, along with (12.81) and the fact that A2 − λ± is Hermitian, gives {A2 , e0 }w± , e0 w± =

n

∑

∂λ

∂ ξ

=1 n

−∑

±

=1

∂ e0 ∂ λ ± ∂ e0 w± , e0 w± − w± , e0 w±

∂ x ∂ x ∂ ξ ∈R

∂ e0 ∂ e0 (A2 − λ±) w± , w±

∂ x ∂ ξ − (A2 − λ±)

∂ e0 ∂ e0 w± , w± . ∂ ξ ∂ x

Therefore 1 n ∂ e0 ∂ e0 I2 = − Im ∑ (A2 − λ± ) w± , w±

2 ∂ x ∂ ξ =1 n 1 ∂e ∂e + Im ∑ (A2 − λ± ) 0 w± , 0 w±

2 ∂ ξ ∂ x =1 n ∂e ∂e = Im ∑ (A2 − λ± ) 0 w± , 0 w± . ∂ ξ ∂ x =1

Now it is important to notice that from (12.80) it also follows ∂ e0 w± , e0 w∓ ∈ γ R.

230


By (12.82) we have

∂ e0 w± = ∂ e0 w± , e0 w∓ e0 w∓ ,

whence, for 1 ≤ ≤ n and ±-respectively, (A2 − λ± )

∂ e0 ∂ e0 ∂ e0 ∂ e0 w± , w± = (λ∓ − λ± ) w± , e0 w∓ w± , e0 w∓ ∈ R. ∂ ξ ∂ x ∂ ξ ∂x ∈γ R

∈γ R

Thus also I2 = 0, and this concludes the proof of the lemma.

We now exploit the results of Section 12.2 to obtain at first information on the w spectrum of the reference operator Aw r (x, hD), and then on that of A (x, D). In the first place we give the following definition regarding the periods of the eigenvalues of A2 (X ), and afterwards define two important sets of integers that we shall use for localizing the eigenvalues. Definition 12.3.4. Let A2 ∈ Q2 and let λ± be its eigenvalues. Let 0 < E1 < E2 . We shall say that A2 ∈ Q2 satisfies hypothesis (T± ) on the energy interval [E1 , E2 ] if all the integral trajectories of the Hamilton vector-fields Hλ± contained in λ±−1 ([E1 , E2 ]), ±-respectively, are periodic with periods T± , necessarily independent of E ∈ [E1 , E2 ] by Lemma 11.1.8. Remark that when n = 1, by Proposition 11.1.10, hypothesis (T± ) is automatically satisfied on every energy interval [E1 , E2 ] ⊂ (0, +∞). Notice that since for the eigenvalues λ± (X) of A2 (X ) we have λ− (X ) < λ+ (X ) for all X = 0, it follows from (11.11) of Proposition 11.1.10 that T− /T+ > 1.

(12.83)

Definition 12.3.5. Let A2 ∈ Q2 satisfy hypothesis (T± ) on an energy interval [E1 − ε , E2 + ε ], for some ε ∈ (0, 1]. Let α± be the Maslov indices of the closed trajectories, ±-respectively, γX± := exp(tHλ± )(X); t ∈ [0, T± ] , X ∈ λ±−1 ([E1 , E2 ]). We define, ±-respectively, ! " α± 2π )h ∈ [E1 , E2 ] , (k + Z± E1 ,E2 (h) := k ∈ Z+ ; T± 4

(12.84)

and, when n = 1 (we have α± = 2 in this case), ! T+ " 2k + 1 − = . (h) × Z (h); QE1 ,E2 (h) := (k, k ) ∈ Z+ E1 ,E2 E1 ,E2 2k + 1 T−

(12.85)

In the first place we have the following result, interesting in its own right, about the large eigenvalue of the reference operator Aw r (x, hD). As already remarked, this is a fundamental step, to obtain information on the large eigenvalues of Aw 2 (x, hD) and hence on those of Aw (x, D). 2


231

Theorem 12.3.6. Let A2 ∈ Q2s satisfy hypothesis (T± ) on an energy interval [E1 − ε , E2 + ε ], for some ε ∈ (0, 1], where E0 + 102 + cχ < E1 < E2 (see Theorem 12.1.16). Let Ar be the symbol of the corresponding reference operator constructed in Section 12.1. We have: • There exists C0 > 0 and h0 ∈ (0, 1] such that

Spec(Aw r (x, hD)) ∩ [E1 , E2 ] ⊂ where Ik± (h) :=

# 2π T±

(k +

± k∈Z+

Ik± (h), ∀h ∈ (0, h0 ],

(12.86)

$ α± α± 2π )h − C0h2 , (k + )h + C0 h2 . 4 T± 4

We may choose h0 > 0 so small that Ik± (h) ∩ Ik± (h) = 0, / ±-respectively, for all k = k . • Suppose that T± are the least periods, and suppose there exists C∗ > 0 such that 2π (k + α+ ) − 2π (k + α− ) ≥ C∗ , ∀k, k ∈ Z, (12.87) T+ 4 T− 4 whence necessarily we must have α+ /4 ∈ Z or α− /4 ∈ Z. It is important to notice that by virtue of (12.87) there is h0 ∈ (0, 1] such that j

/ ∀k, k ∈ Z, ∀h ∈ (0, h0 ], ∀ j, j = ±. Ik (h) ∩ Ik (h) = 0, j

(12.88)

Then for h0 > 0 sufficiently small ± / ∀k ∈ Z± Spec(Aw r (x, hD)) ∩ Ik (h) = 0, E1 ,E2 (h), ∀h ∈ (0, h0 ],

(12.89)

±-respectively. When n = 1 we have that α± = 2, and for any fixed h ∈ (0, h0 ], NAwr (x,hD) (Ik± (h)) = 1, ∀k ∈ Z± E1 ,E2 (h), ±-respectively.

(12.90)

Proof. Take an h∞ -diagonalization Λ w (h) = diag(Λ+w (h), Λ−w (h)) of the reference operator Aw r (x, hD) with subprincipal part satisfying Proposition 12.3.2. Hence the action is 0 on λ±−1 ([E1 − ε , E2 + ε ]). From Proposition 11.1.4, Proposition 12.3.1 and the hypotheses, we have that we may use the spectral results of Section 12.2 applied to the scalar operators Λ±w (h). Using (12.43) of Theorem 12.1.19 (and possibly shrinking h0 , depending also on ε ) we thus have

Spec(Aw r (x, hD)) ∩ [E1 , E2 ] ⊂

, λ˜ (h) − h4, λ˜ (h) + h4

λ˜ (h)∈Spec(Λ w (h))∩[E1 −ε /2,E2 +ε /2]

=

# $ λ˜ ± (h) − h4, λ˜ ± (h) + h4 , ∀h ∈ (0, h0 ],

± λ˜ ± (h)∈Spec(Λ w (h))∩[E −ε /2,E +ε /2] 1 2 ±

which, in view of Theorem 12.2.4, proves (12.86).

232


The proof of (12.89) follows from (12.42) and (12.44) of Theorem 12.1.19, from Theorem 12.2.16 and the Minimax. In fact, (12.87) yields, for h0 sufficiently small and any given h ∈ (0, h0 ], − / ∀k ∈ Z+ Ik+ (h) ∩ Ik− (h) = 0, E1 ,E2 (h), ∀k ∈ ZE1 ,E2 (h).

(12.91)

We set # 2π α± α± C0 C0 $ 2π )h − h2 , (k + )h + h2 = middle half of Ik± (h). (k + Jk± (h) = T± 4 2 T± 4 2 (12.92) Take k ∈ Z± (h), ±-respectively. Then by Theorem 12.2.16 for such k E1 ,E2 / Spec(Λ±w (h)) ∩ Jk± (h) = 0. Let λ˜ (h) ∈ Spec(Λ w (h)) belong to Jk+ (h), resp. Jk− (h). Since λ˜ (h) must be the j-th eigenvalue λ˜ j (h) ∈ Spec(Λ w (h)), for some j sufficiently large, we have by (12.91) that λ˜ j (h) ∈ Jk+ (h), resp. λ˜ j (h) ∈ Jk− (h), and when n = 1 this happens exactly with multiplicity 1. Hence from (12.44) and (12.91) we get that, with the same index j,

λ jr (h) ∈ Ik+ (h), resp. λ jr (h) ∈ Ik− (h), which gives (12.89), and when n = 1 this must happen (by the Minimax and using (12.91)) with multiplicity 1, which gives (12.90) and concludes the proof. We next deal with a case in which condition (12.87) does not hold. We consider cases in which n = 1 and QE1 ,E2 (h) = 0. / We have the following result. Theorem 12.3.7. Suppose n = 1. There exists h0 ∈ (0, 1] so small that for any given h ∈ (0, h0 ], whenever QE1 ,E2 (h) = 0/ one has NAwr (x,hD) (Ik+ (h)) = 2, ∀k ∈ proj1 QE1 ,E2 (h) , where proj1 denotes the projection onto the first factor. Proof. We immediately observe that when QE1 ,E2 (h) = 0/ (k, k ), (k, k˜ ) ∈ QE1 ,E2 (h) =⇒ k = k˜ , ˜ k ) ∈ QE ,E (h) =⇒ k = k, ˜ (k, k ), (k, 1 2 and (k, k ) ∈ QE1 ,E2 (h) =⇒ Ik+ (h) = Ik− (h), k =

1 2

T− (2k + 1) − 1 . T+

(12.93)


233

It follows from (12.83) that (k, k ) ∈ QE1 ,E2 (h) =⇒ k = k , and, as Ik+ (h) = Ik− (h), Ik+ (h) ∩ Ik+ (h) = Ik+ (h) ∩ Ik− (h) = Ik+ (h) ∩ Ik− (h) = 0, / ∀k = k, k .

(12.94)

The proof is thus a consequence of (12.42), (12.44), the Minimax, and the fact that the scalar operators Λ±w (h) have spectrum of multiplicity 1 inside each interval Jk± (h), ±-respectively (see (12.92)). In fact, take (k, k ) ∈ QE1 ,E2 (h). We must then have exactly two eigenvalues λ˜ j (h) and λ˜ j (h) of Λ w (h), with λ˜ j (h) ∈ Spec(Λ+w (h)) and λ˜ j (h) ∈ Spec(Λ−w (h)), that are contained in Jk+ (h) = Jk− (h). Hence the eigen+ − values λ jr (h) and λ jr (h) of Aw r (x, hD) must be contained in Ik (h) = Ik (h), and in w no other interval, for the other eigenvalues of Ar (x, hD) must (by the Minimax) be close to eigenvalues of Λ w (h) that are “far away”, by (12.94), from λ˜ j (h) and λ˜ j (h) (in fact, they must be contained in intervals Jk± (h) with k = k , k). This concludes the proof. w We now use Theorem 12.1.16 and the fact that Aw 2 (x, hD) = hA2 (x, D) to obtain from Theorems 12.3.6 and 12.3.7 the following spectral information about Aw 2 (x, D).

Theorem 12.3.8. Let A2 ∈ Q2s satisfy hypothesis (T± ) on an energy interval [E1 − ε , E2 + ε ], for some ε ∈ (0, 1], where 10(E0 + 102 + cχ ) < E1 < E2 (see Theorem 12.1.16). We have: • There exists C0 > 0 and h0 ∈ (0, 1] so small that, for (all) fixed h ∈ (0, h0 ] one

has −1 −1 Spec(Aw 2 (x, D)) ∩ [E1 h , E2 h ] ⊂

where Ik± (h) :=

# 2π T±

(k +

± k∈Z

h−1 Ik± (h),

(12.95)

$ α± α± 2π )h − C0h2 , (k + )h + C0 h2 , 4 T± 4

and, for h0 chosen sufficiently small, Ik± (h) ∩ Ik± (h) = 0, / for all h ∈ (0, h0 ], and for all k = k , ±-respectively. • Suppose there exists C∗ > 0 such that (12.87) holds. Then there exists h0 ∈ (0, 1] so small that, for (all) fixed h ∈ (0, h0 ] one has −1 ± / ∀k ∈ Z± Spec(Aw 2 (x, D)) ∩ h Ik (h) = 0, E1 ,E2 (h), ±-respectively,

(12.96)

and when n = 1 (so that α± = 2) NAw2 (x,D) (h−1 Ik± (h)) = 1, ∀k ∈ Z± E1 ,E2 (h), ±-respectively.

(12.97)

234


Hence, when n = 1, ±-respectively, −1 ± (x, D)) ∩ h I (h) = 1, ∀k ∈ ZE1 ,E2 (h), multiplicity Spec(Aw 2 k ± −1 ± that is, the eigenvalues of Aw 2 (x, D) belonging to the h Ik (h), k ∈ ZE1 ,E2 (h), are all simple. • Let n = 1. There exists h0 ∈ (0, 1] so small that, for (all) fixed h ∈ (0, h0 ], whenever QE1 ,E2 (h) = 0/ one has

NAw2 (x,D) (h−1 Ik+ (h)) = 2, ∀k ∈ proj1 QE1 ,E2 (h) . Proof. Since, by Theorem 12.1.16 and Theorem 12.1.19, Spec(Aw 2 (x, hD)) ∩ [E1 , E2 ] ⊂

,

λ˜ (h) − h4, λ˜ (h) + h4 ,

λ˜ (h)∈Spec(Λ w (h))∩[E1 −ε /2,E2 +ε /2]

for all h ∈ (0, h0 ], the proof is a consequence of (12.33), (12.44), the use of Minimax arguments as in the proofs of Theorems 12.3.6 and 12.3.7, and of the fact w Spec(Aw 2 (x, hD)) = λ (h) = hλ ; λ ∈ Spec(A2 (x, D)) . Remark 12.3.9. By Proposition 9.2.7, Lemma 12.2.20 and Corollary 12.2.22 we have that Theorems 12.3.6, 12.3.7 and 12.3.8 hold regardless the choice of the h∞ diagonalization of the reference operator Aw r (x, hD).

12.4 Localization and Multiplicity of Spec(Qw (α ,β ) (x, D)) In this section, we specialize the results of the previous Section 12.3 to finally give some properties of the large eigenvalues of a NCHO Q(α ,β ) , α = β , α , β > 0 and αβ > 1, that are linked to properties of the periods of the bicharacteristics of the eigenvalues of the symbol Q(α ,β ) (x, ξ ). Since n = 1, we have that T± are the least periods of the eigenvalues of the symbol of Qw (α ,β ) (x, D), that do not depend on the energy on the whole energy interval (0, +∞). Moreover, the Maslov indices α± are both equal to 2. We decided to explicitly state Theorem 12.4.1 below, because the system Qw (α ,β ) (x, D) is the basic reference model that led us in our explorations. Theorem 12.4.1. Let α = β , with α , β > 0 and αβ > 1. Let 10(E0 + 102 + cχ ) ≤ E1 < E2 . With the notation of Theorems 12.3.6, 12.3.7 and 12.3.8, we have:

12.4 Localization and Multiplicity of Spec(Qw (α ,β ) (x, D))

235

• There exists C0 > 0 and h0 ∈ (0, 1] so small that, for (all) fixed h ∈ (0, h0 ] one

has −1 −1 Spec(Qw (α ,β ) (x, D)) ∩ [E1 h , E2 h ] ⊂

where Ik± (h) :=

± k∈Z+

h−1 Ik± (h),

(12.98)

$ 1 2π 1 (k + )h − C0h2 , (k + )h + C0h2 . T± 2 T± 2

# 2π

and, for h0 chosen sufficiently small, Ik± (h) ∩ Ik± (h) = 0, / for all h ∈ (0, h0 ], and for all k = k , ±-respectively. • Suppose there exists C∗ > 0 such that (12.87) holds. In this case (12.87) may be rewritten as T− 2k + 1 C∗ − T+ 2k + 1 ≥ 2k + 1 , ∀k, k ∈ Z+ . Then there exists h0 ∈ (0, 1] so small that, for (all) fixed h ∈ (0, h0 ] one has NQw

(α ,β )

(x,D) (h

−1 ± Ik (h))

= 1, ∀k ∈ Z± E1 ,E2 (h), ±-respectively.

Hence, ±-respectively, −1 ± ± multiplicity Spec(Qw (α ,β ) (x, D)) ∩ h Ik (h) = 1, ∀k ∈ ZE1 ,E2 (h),

(12.99)

−1 ± that is, the eigenvalues of Qw (α ,β ) (x, D) belonging to the h Ik (h), for k ∈ ± ZE1 ,E2 (h), are all simple. • There exists h0 ∈ (0, 1] so small that, for (all) fixed h ∈ (0, h0 ], whenever QE1 ,E2 (h) = 0/ one has NQw (x,D) (h−1 Ik+ (h)) = 2, ∀k ∈ proj1 QE1 ,E2 (h) . (α ,β )

Corollary 12.4.2. Let α = β , with α , β > 0, αβ > 1, and let 10(E0 + 102 + cχ ) ≤ E1 < E2 . Suppose there exists C∗ > 0 such that (12.87) holds. Then there exists h0 sufficiently small such that for (all) h fixed in (0, h0 ] one has

Σ0± ∩ h−1 Ikj (h) = Σ∞+ ∩ Σ∞− ∩ h−1 Ikj (h) = 0, / ∀k ∈ ZEj 1 ,E2 (h), j = ±, where Σ0± and Σ∞± are the sets introduced in Parmeggiani-Wakayama [58, 59]. Proof. This follows from the second point in Theorem 12.4.1 above, for in this case we have multiplicity Spec(Qw (x, D)) ∩ h−1 Ik± (h) = 1, ∀k ∈ Z± E1 ,E2 (h), ±-resp. Since higher multiplicity eigenvalues must lie in Σ0± or Σ∞+ ∩ Σ∞− (see [59]), the result follows.

236


As a byproduct of the foregoing corollary, recalling Theorems 2 and 3 of Ochiai [47] we immediately have the following result. Corollary 12.4.3. Following Ochiai [46, 47], for suitable real numbers a , b , c , d and complex numbers a, e with |a| > 1, let Hλ (z, ∂z ) = ∂z2 +

a z

+

d z − e b c ∂z + + , z−1 z−a z(z − 1)(z − a)

(x, D), with associbe the Heun operator associated with the eigenvalue λ of Qw (α ,β )

ated odd eigenfunctions. Let α = β , with α , β > 0, αβ > 1, and let 10(E0 + 102 + cχ ) ≤ E1 < E2 . Suppose there exists C∗ > 0 such that (12.87) holds. Then there exists h0 sufficiently small such that for (all) h fixed in (0, h0 ], for all k ∈ Z± E1 ,E2 (h), ±± w −1 resp., and for λ ∈ Spec(Q(α ,β ) (x, D))∩h Ik (h), one cannot find non-zero rational functions f1 (z), f2 (z) such that √ f1 (z), f2 (z) z − a ∈ Ker Hλ (z, ∂z ) at the origin. Proof. By Corollary 12.4.2 we have that Σ0− ∩ h−1 Ik± (h) = 0, / for all k ∈ Z± E1 ,E2 (h), ±-resp. Hence the result follows at once from Theorems 2 and 3 of [47]. Remark 12.4.4. Notice that (12.99) and Corollary 12.4.2 complement Proposition 3.14 of Parmeggiani [51]. Hence either when condition (12.87) is fulfilled or when α /β = 1 is not a ratio of positive odd integers, and αβ is sufficienly large, then the conclusions of Corollaries 12.4.2 and 12.4.3 hold. Notice that for α = β , upon putting 4 4 !4 α !4 α β" β" , m− := max , m+ := min , , β α β α one has

2 π m± T± = ' (1 + o(1)), as αβ

8

αβ → +∞, α /β constant = 1,

so that T+ /T− −→ m+ /m− . As shown in Proposition 3.14 of Parmeggiani [51], the eigenvalues of Qw (α ,β ) (x, D) smaller than or equal to any fixed E > 0 are still simple for all αβ sufficiently large with α /β fixed (= 1), even when m+ /m− is a ratio of certain positive odd integers (that is, in the case α < β , say, when m+ /m− = α /β = (2m0 + 1)/(2n0 + 1) with m0 = n0 − 2 − 4k, any given k ∈ Z+ ). It would therefore be ' interesting to carry out a refined study of the periods T± as functions of 1/ αβ and α /β in order to get more precise spectral information also in the case Q± / E1 ,E2 (h) = 0 or, more generally, when condition (12.87) does not hold.

s 12.5 Localization and Multiplicity of Spec(Aw 2 (x, D)), with A2 ∈ Q2 \ Q2

237

12.5 Localization and Multiplicity of Spec(Aw 2 (x, D)), s with A2 ∈ Q 2 \ Q2 When A2 ∈ Q2 \ Q2s we no longer have, in general, that for some e0 the h∞ diagonalization has no subprincipal term. However, the results at the end of Section 12.2 (for scalar operators with a subprincipal part whose average is constant on the periodic trajectories) and the methods of proof of the last section yield the following theorem. Theorem 12.5.1. Let A2 ∈ Q2 \ Q2s satisfy hypothesis (T± ) on an energy interval [E1 − ε , E2 + ε ], for some ε ∈ (0, 1], where 10(E0 + 102 + cχ ) < E1 < E2 (see Theorem 12.1.16). Suppose that there exists an h∞ -diagonalization Λ w (h) of the + − reference operator Aw r (x, hD) which has subprincipal symbol Λ0 = diag(Λ0 , Λ0 ) satisfying, ±-respectively, 1 T±

T± 0

−1 Λ0± ◦ exp(tHλ± )(X)dt = c± 0 , ∀X ∈ λ± ([E1 − ε , E2 + ε ]),

(12.100)

where c± 0 ∈ R are constants. In this setting define, ±-respectively, ! " 2π α± )h + c± Z± (k + E1 ,E2 (h) := k ∈ Z+ ; 0 h ∈ [E1 , E2 ] , T± 4 and, when n = 1, ! 2k + 1 + T+c+ T+ " − 0 /π . (h) × Z (h); = QE1 ,E2 (h) := (k, k ) ∈ Z+ E1 ,E2 E1 ,E2 T− 2k + 1 + T−c− 0 /π We have: • There exists C0 > 0 and h0 ∈ (0, 1] so small that, for (all) fixed h ∈ (0, h0 ] one

has

−1 −1 Spec(Aw 2 (x, D)) ∩ [E1 h , E2 h ] ⊂

± k∈Z

h−1 Ik± (h),

(12.101)

where Ik± (h) =

# 2π T±

(k +

$ α± α± 2 2π ± 2 , h − C h , (k + h + C h )h + c± )h + c 0 0 0 0 4 T± 4

and, for h0 chosen sufficiently small, Ik± (h) ∩ Ik± (h) = 0, / for all h ∈ (0, h0 ], and for all k = k , ±-respectively. • Suppose there exists C∗ > 0 such that the analog of (12.87) holds: 2π (k + α+ ) + c+ − 2π (k + α− ) + c− ≥ C∗ , ∀k, k ∈ Z. 0 0 T+ 4 T− 4

238


Then there exists h0 ∈ (0, 1] so small that, for (all) fixed h ∈ (0, h0 ] one has −1 ± Spec(Aw / ∀k ∈ Z± 2 (x, D)) ∩ h Ik (h) = 0, E1 ,E2 (h), ±-respectively,

(12.102)

and when n = 1 (so that α± = 2) NAw2 (x,D) (h−1 Ik± (h)) = 1, ∀k ∈ Z± E1 ,E2 (h), ±-respectively.

(12.103)

Hence, when n = 1, ±-respectively, −1 ± (x, D)) ∩ h I (h) = 1, ∀k ∈ Z± multiplicity Spec(Aw 2 E1 ,E2 (h), k ± −1 ± that is, the eigenvalues of Aw 2 (x, D) belonging to the h Ik (h), k ∈ ZE1 ,E2 (h), are all simple. − • Let n = 1. Suppose that c+ 0 = c0 . Then there exists h0 ∈ (0, 1] so small that, for (all) fixed h ∈ (0, h0 ], whenever QE1 ,E2 (h) = 0/ one has NAw2 (x,D) (h−1 Ik+ (h)) = 2, ∀k ∈ proj1 QE1 ,E2 (h) .

Proof. The proof is carried out by following the same lines of the proof of Theorem 12.3.8, and by using Corollary 12.2.19 and Lemma 12.2.20. Remark 12.5.2. By Proposition 9.2.7 and the proof of Lemma 12.2.20 we have that if condition (12.100) holds for an h∞ -diagonalization, then it holds for all 2π ± h∞ -diagonalizations (upon shifting the constants c± k , with k0± ∈ Z, 0 by T± 0 ±-respectively, see (12.79)). Hence Theorem 12.5.1 holds regardless the choice of the h∞ -diagonalization of the reference operator Aw r (x, hD), provided that (12.100) be satisfied by the subprincipal symbol Λ0 = diag(Λ0+ , Λ0− ) on the energy interval [E1 − ε , E2 + ε ], for at least one h∞ -diagonalization Λ w (h). Remark 12.5.3. Recalling that 1 1 Λ0± = Im {e∗0 , λ± }e0 w± , w± + Im e∗0 {A2 , e0 }w± , w± , 2 2 ±-respectively, where {w+ , w− } is the canonical basis of C2 , condition (12.100) may be thought of as a monodromy condition on the principal symbol e0 of the operator ew (x, hD) that h∞ -diagonalizes Aw r (x, hD).

12.6 Notes The results of Section 12.4 were proved in Parmeggiani [55]. In these notes we adapted the method of the proof there, to obtain the localization and multiplicity theorems for the more general case of 2 × 2 systems belonging to the class Q2 .

Appendix

In this Appendix we collect some useful facts related to the almost-analytic extension of a smooth function and give, following Dimassi-Sjöstrand [7], the proof of the Dyn’kin-Helffer-Sjostrand formula (12.48).

A.1 Almost-Analytic Extension and the Dyn’kin–Helffer–Sjöstrand Formula Let x, y ∈ R, and let z = x + iy ∈ C. We denote by L(dz) = dxdy the Lebesgue measure of C, so that dz ∧ d z¯ = −2idx ∧ dy = −2iL(dz), and write ∂ /∂ z¯ = (∂ /∂ x + i∂ /∂ y)/2. In the first place we recall the complex Gauss-Green formula, whose proof is easily deduced from the classical Gauss-Green formula when applied to the real and imaginary part of a complex-valued function. Lemma A.1.1 (Complex Gauss-Green formula). Let D ⊂ Ω ⊂ C be open sets such that D has a C1 boundary ∂ D in Ω . Let f ∈ C01 (Ω ). Then with the proper orientations (that is, with ∂ D oriented in such a way that D is kept to the left) one has ∂f ∂f (z)dz ∧ d z¯ = 2i (z)L(dz). (A.1) f (z)dz = − ∂D D ∂ z¯ D ∂ z¯ Notice that (A.1) may also be rewritten as

∂ 1D i |f = ∂ z¯ 2

∂D

f (z)dz, f ∈ C01 (Ω ),

(A.2)

where the left-hand side denotes the distribution duality, and 1D is the characteristic function of the set D. Next, following Hörmander [28], we recall the following result about the fundamental solution of the ∂ /∂ z¯-operator.

239

240

Appendix

Lemma A.1.2. With the notation of Lemma A.1.1, let ζ ∈ D. Then, with the proper orientations, 1 f (ζ ) = − π

1 ∂f (z)(z − ζ )−1 L(dz) + ∂ z ¯ 2 πi D

∂D

f (z)(z − ζ )−1 dz, ∀ f ∈ C01 (Ω ). (A.3)

In particular, when D = Ω there is no curvilinear integral in (A.3), that is, f (ζ ) = −

1 π

Ω

∂f (z)(z − ζ )−1 L(dz), ∀ f ∈ C01 (Ω ), ∂ z¯

(A.4)

and, furthermore, considering (x, y) −→ Eζ (x, y) = π −1 (z − ζ )−1 , which is a locally integrable function, gives ∂ Eζ = δζ , (A.5) ∂ z¯ that is, Eζ is a fundamental solution of ∂ /∂ z¯. Proof. For ζ ∈ D, we apply the complex Gauss-Green formula (A.1) to the function f (z)/(z − ζ ), with D replaced by D \ Bε , where Bε is a disc of radius ε with center at ζ , and ε is picked small. Hence

2i D\Bε

∂f (z)(z − ζ )−1 L(dz) = ∂ z¯

∂D

f (z)(z − ζ )−1 dz −

∂ Bε

f (z)(z − ζ )−1 dz,

with the proper orientations (i.e. ∂ Bε is oriented in the formula as the boundary of Bε ). Since ∂ Bε

f (z)(z − ζ )−1 dz = f (ζ )

∂ Bε

(z − ζ )−1 dz + O(ε ) −→ 2π i f (ζ ), as ε → 0+,

letting ε → 0+ gives (A.3).

We now pass to the proof of Theorem 12.2.1. In the first place we show that given a function f ∈ C02 (R), it is always possible to find an extension f˜ ∈ C01 (C) such that ∂ f˜/∂ z¯ = O(|Im z|), as in the statement of Theorem 12.2.1, and for which formula (A.4) holds true. We have in fact the following lemma. Lemma A.1.3. Given any f ∈ C02 (R) there exists f˜ ∈ C01 (C) such that f˜R = f , and

∂ f˜ (z) = O(|Im z|). ∂ z¯

(A.6)

∂f (z)(z − t)−1 L(dz), ∀t ∈ R. ∂ z¯

(A.7)

Moreover, one has f (t) = −

1 π

C

A.1 Almost-Analytic Extension and the Dyn’kin–Helffer–Sjöstrand Formula

241

Proof. We denote by g(k) the kth-derivative with respect to x or y of a function g. Take χ ∈ C0∞ (R) with 0 ≤ χ ≤ 1, χ |y|≤1 = 1, χ |y|≥2 = 0. Then define f˜(x + iy) := f (x) + iy f (1) (x) χ (y). It is clear that f˜ ∈ C01 (C). One then computes ∂ f˜ (x + iy) = f (1) (x) + iy f (2) (x) χ (y), ∂x ˜ ∂f (x + iy) = i f (1) (x)χ (y) + f (x) + iy f (1) (x) χ (1) (y), ∂y so that

∂ f˜ y (2) (z) = i f (x)χ (y) − f (1) (x)χ (1) (y) + i f (x)χ (1) (y). ∂ z¯ 2

(A.8)

Since supp χ (1) ⊂ {y; 1 ≤ |y| ≤ 2}, we notice that

χ (1) (y) ≤ |χ (1) (y)| ≤ C, y

whence we may write ∂ f˜ χ (1) (y) y (2) (z) = i f (x)χ (y) − f (1) (x)χ (1) (y) + iy f (x) . ∂ z¯ 2 y which proves (A.6). The fact that formula (A.7) holds, follows immediately from Lemma A.1.2 by taking Ω = C and ζ = x ∈ R. We next prove Theorem 12.2.1. Proof (of Theorem 12.2.1). Denote by Q ∈ L (H, H) the right-hand side of (12.48). For u, v ∈ H consider ((z − P)−1 u, v) =

(z − t)−1(dE(t)u, v),

where t → E(t) is the spectral family associated with P. It follows that (Qu, v) = −

1 π

∂ f˜ (z) (z − t)−1(dE(t)u, v) L(dz) = C ∂ z¯

242

Appendix

(by Fubini’s theorem, using the fact that f˜ is compactly supported and that 9 dE(t) = IdH ) =

−

1 π

∂ f˜ (z)(z − t)−1 L(dz) (dE(t)u, v) = f (t)(dE(t)u, v) = f (P), C ∂ z¯


However, when P is a pseudodifferential operator and when dealing with the pseudodifferential nature of f (P), as in the proof of Theorem 12.2.2, one does need to consider almost-analytic extensions of a given f ∈ C0∞ (R). The next lemma grants the existence of such almost-analytic extension. (This result also holds, by using a locally-finite partition of unity of R, for functions belonging to C∞ (R). See Treves [70, Chapter X, Section 2], for more on this.) Lemma A.1.4. Let f ∈ C0∞ (R). There exists f˜ ∈ C0∞ (C), called an almost-analytic extension of f , such that ∂ f˜ f˜R = f , and ∀N ≥ 1 ∃CN > 0 such that (z) ≤ CN |Im z|N , ∀z ∈ C, ∂ z¯

(A.9)

supp f˜ ⊂ {z = x + iy ∈ C; x ∈ supp f , |y| ≤ C}, where, recall, ∂ /∂ z¯ = (∂ /∂ x + i∂ /∂ y)/2. In addition one has f (t) = −

1 π

−1 ∂ f˜ (z) z − t L(dz), t ∈ R. ∂ z ¯ C

(A.10)

Proof. Take χ ∈ C0∞ (R) as in the proof of Lemma A.1.3. One can then choose a monotone increasing sequence Rk +∞ growing sufficiently fast so as to have that the series f (k) (x) (iy)k χ (Rk y) f˜(x + iy) := ∑ k! k≥0 is uniformly convergent, along with the series of the derivatives to all orders. One now computes ∂ f˜ f (k+1) (x) (z) = ∑ (iy)k χ (Rk y), ∂x k! k≥0 and i

∂ f˜ f (k+1) (x) k+2 k f (k) (x) (z) = ∑ i y χ (Rk+1 y) + ∑ (iy)k Rk χ (1) (Rk y), ∂y k! k! k≥0 k≥0

A.1 Almost-Analytic Extension and the Dyn’kin–Helffer–Sjöstrand Formula

243

obtaining 1 ∂ f˜ f (k+1) (x) (z) = ∑ (iy)k χ (Rk y) − χ (Rk+1y) ∂ z¯ 2 k≥0 k! + Since and

f (k) (x) i (iy)k Rk χ (1) (Rk y). ∑ 2 k≥0 k!

−1 supp χ (1) (Rk ·) ⊂ {y; R−1 k ≤ |y| ≤ 2Rk }, −1 supp(χ (Rk ·) − χ (Rk+1·)) ⊂ {y; R−1 k+1 ≤ |y| ≤ Rk },

one sees that (A.9) holds. Formula (A.10) follows as before from Lemma A.1.2. This completes the proof.

Main Notation

B.1 General Notation • N = {1, 2, . . .}, Z+ = {0, 1, 2, . . .}, R+ = (0, +∞), and R+ = [0, +∞). • We denote by A or by card A the cardinality of the set A.

f (x) = 1. g(x) • Let A, B > 0. We write A B (or equivalently B A) when there is a universal constant C > 0 such that A ≤ CB. We therefore write A ≈ B whenever A B and B A. • Given sequences {A j } j , {B j } j ⊂ R, we write • As usual, given functions f and g, we write f ∼ g as x → x0 when lim

x→x0

A j B j , as j → +∞ (or equivalently B j A j as j → +∞) if there are constants C > 0 and j0 ∈ N such that A j ≤ CB j , ∀ j ≥ j0 . We hence write A j ≈ B j , as j → +∞, when A j B j and B j A j as j → +∞.

• The N × N identity matrix is denoted by I, regardless N, or by IN , or by ICN . • The norm of a vector v belonging to RN , resp. CN , is denoted by |v|, resp. |v|CN .

• • •

•

The inner product in RN , resp. the Hermitian product in CN , of vectors v, v , is denoted by v, v , resp. v, v CN . The canonical inner product in L2 is denoted by (·, ·), the induced norm by || · ||0 . The ring of N × N complex matrices is always denoted by MN . An excision function is a C∞ function χ with 0 ≤ χ ≤ 1, supported away from the origin and ≡ 1 outside a large compact set. The standard one used in these notes is such that χ ≡ 0 for |X| ≤ 1/2 and χ ≡ 1 for |X | ≥ 1. The duality between S and S is denoted by u|ϕ S ,S , where u ∈ S and ϕ ∈S.

245

246

Main Notation

• The set of linear continuous maps between Hilbert (or Banach, or Fréchet) spaces

H1 and H2 is denoted by L (H1 , H2 ).

• We write Tr for the operator-trace, and Tr for the N × N matrix-trace. • The symbol of the standard harmonic oscillator p0 (x, ξ ) = (|ξ |2 + |x|2 )/2, also

written p0 (X ) = |X|2 /2, with X = (x, ξ ).

• The matrices J and K:

0 −1 0 1 J= , K= . 1 0 1 0 • We say that f (h) = O(hN0 ) for some N0 ∈ Z+ if there exists CN0 > 0 such that

| f (h)| ≤ CN0 hN0 . We say that f (h) = O(h∞ ) if for any given N0 ∈ Z+ one has f (h) = O(hN0 ). • The characteristic function of a set V is denoted by 1V . • → → denotes compact embedding.

B.2 Symbol, Function and Operator Spaces • The symbol space S(m, g) in the Weyl-Hörmander calculus, see Definition 3.1.6. • The global weight-function m(X) = (1 + |X|2 )1/2 and the global metric gX = • • • • • • • • • • • • • •

|dX|2 /m(X )2 , X ∈ R2n , see (3.13). The symbol space Scl (mμ , g), see Definition 3.2.3. The global polynomial differential (GPD) symbols, see Definition 3.2.6 and also Definition 3.2.9. The smoothing symbol class S(m−∞ , g), see Definition 3.2.14. The function space Bs , see Definition 3.2.25 and Proposition 3.2.26. The symbol space S(μ , r), see Definition 6.1.1. The set OPScl (μ , r) of pseudodifferential operators, see Definition 6.1.2. The semiclassical symbol space Sδk (mμ , g), see (9.1) of Definition 9.1.1. The semiclassical symbol space Sδk (mμ ), see (9.2) of Definition 9.1.1. The smoothing semiclassical symbol class S−∞ (m−∞ , g), see (9.7). The smoothing semiclassical symbol class S−∞ (mμ ), see (9.8). k (m μ ), see Point 1. of Definition The class of classical semiclassical symbols Scl 9.1.9. k The class of classical semiclassical symbols S0,cl (mμ , g), see Point 2. of Definition 9.1.9. The class of semiclassical GPD symbols, see Definition 9.4.1. The isometry of L2 , also automorphism of S and S (E > 0) √ UE : u(x) −→ E −n/4 u(x/ E),

see (9.4).

B.5 Classes of Systems

247

B.3 The Spectral Counting Function and the Spectral ζ -Function • The spectral counting functions N(λ ) and N0 (λ ), see (4.4). • The spectral ζ -function ζA (λ ) associated with a positive operator A with a dis-

crete spectrum, see (4.10).

B.4 Dynamical Quantities and Assumptions Hypotheses (H1)-(H3), see Assumption 11.1.1. Hypotheses (H4) and (H5), see Theorem 12.2.4. Hypothesis (H5 ), see Proposition 12.2.17 (and just a few lines above it). Hypothesis (H6), see Theorem 12.2.16. The functions J p (E) and Tp (E) associated with a periodic Hamiltonian trajectory of a symbol p at energy E, see Lemma 11.1.2. • The averaged action-integral A p (E), see Definition 11.1.5. • The Leray-Liouville measure L f ,λ (dX) associated with the hypersurface f −1 (λ ), see Definition 12.2.11, Proposition 12.2.14 and Corollary 12.2.15. • • • • •

B.5 Classes of Systems • Non-commutative harmonic oscillator (for short NCHO), see Definition 3.2.11. • The class Q2 , see Definition 12.1.1. • The class Q2s , see Definition 12.1.2.

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Index

h∞ -(block)-diagonalization, 129 h-Fourier integral operator, 209 (Block)-diagonalization, 135 A Almost-analytic extension, 206, 242 Averaged action-integral, 165 B Beals characterization, 126 Bicharacteristic curve, 161 C Canonical 1-form, 7 symplectic form, 7 volume form, 219 Classical semiclassical symbol, 126 Classical symbol, 28 Closed operator, 8 Complex Gauss-Green formula, 239 Composition theorem (and formula), 18, 19 Confined symbol, 17 Cotlar-Stein Lemma, 22 Creation/annihilation operators, 9 Crossing form, 181

Euler’s identity, 163 Excision function, 27, 245

F Finite-rank orthogonal projection, 150 Formal adjoint, 24 Fourier Integral Operator, 190 Fredholm operator, 44 Functional calculus, 71

G Global polynomial differential operator, 28 symbol, 28 system, 29 Global pseudodifferential operators, 36 Globally elliptic classical symbol, 28, 29 Globally elliptic symbol, 27, 29 Globally positive elliptic symbol, 30

H Hamilton vector-field, 161 Harmonic oscillator, 8 Heat-kernel, 69 Hermite functions, 11 Heun operator, 236 Hypothesis (T± ), 230

D Decoupling argument, 129 Dyn’kin-Helffer-Sjöstrand formula, 206, 239 Dynamical assumptions, 162

K Karamata’s Theorem, 84

E Energy level, 162 surface, 162

L Lagrangian frame, 178 Lagrangian manifold, 184 Lagrangian subspace, 174

253

254 Least period, 168, 169, 219 Leray-Liouville form and measure, 216

M Maslov cycle, 180 index, 180, 181 Maximal operator, 8, 46 Metaplectic operator, 21 Metric admissible, 15 dual, 15 global, 25, 246 Minimal operator, 47 Minimax principle, 55

N Non-commutative harmonic oscillator, 30

P Parametrix, 37 Parametrix approximation of e−tA , 81 Period, 162, 165, 168 Periodic bicharacteristic, 162 Planck function, 16 Poisson bracket, 9 Principal symbol, 36, 126

Index R Reference operator, 198, 199

S Schwartz kernel, 18 Self-adjoint, 47 Semiclassical global polynomial differential system, 143 Semiclassical symbol, 121 Semi-regular classical symbol, 28 Smoothing operator, 31 Spectral counting function, 59 zeta function, 66, 95 Subprincipal symbol, 36, 126 Symplectic form, 7 invariance, 21 transformation, 21 Symplectomorphism, 184

W Weight g-admissible, 17 global, 27, 246 Weyl-quantization, 7, 8, 18

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Edited by J.-M. Morel, F. Takens, B. Teissier, P.K. Maini Editorial Policy (for the publication of monographs) 1. Lecture Notes aim to report new developments in all areas of mathematics and their applications - quickly, informally and at a high level. Mathematical texts analysing new developments in modelling and numerical simulation are welcome. Monograph manuscripts should be reasonably self-contained and rounded off. Thus they may, and often will, present not only results of the author but also related work by other people. They may be based on specialised lecture courses. Furthermore, the manuscripts should provide sufficient motivation, examples and applications. This clearly distinguishes Lecture Notes from journal articles or technical reports which normally are very concise. Articles intended for a journal but too long to be accepted by most journals, usually do not have this “lecture notes” character. For similar reasons it is unusual for doctoral theses to be accepted for the Lecture Notes series, though habilitation theses may be appropriate. 2. Manuscripts should be submitted either online at www.editorialmanager.com/lnm to Springer’s mathematics editorial in Heidelberg, or to one of the series editors. In general, manuscripts will be sent out to 2 external referees for evaluation. If a decision cannot yet be reached on the basis of the first 2 reports, further referees may be contacted: The author will be informed of this. A final decision to publish can be made only on the basis of the complete manuscript, however a refereeing process leading to a preliminary decision can be based on a pre-final or incomplete manuscript. The strict minimum amount of material that will be considered should include a detailed outline describing the planned contents of each chapter, a bibliography and several sample chapters. Authors should be aware that incomplete or insufficiently close to final manuscripts almost always result in longer refereeing times and nevertheless unclear referees’ recommendations, making further refereeing of a final draft necessary. Authors should also be aware that parallel submission of their manuscript to another publisher while under consideration for LNM will in general lead to immediate rejection. 3. Manuscripts should in general be submitted in English. Final manuscripts should contain at least 100 pages of mathematical text and should always include – a table of contents; – an informative introduction, with adequate motivation and perhaps some historical remarks: it should be accessible to a reader not intimately familiar with the topic treated; – a subject index: as a rule this is genuinely helpful for the reader. For evaluation purposes, manuscripts may be submitted in print or electronic form (print form is still preferred by most referees), in the latter case preferably as pdf- or zipped ps-files. Lecture Notes volumes are, as a rule, printed digitally from the authors’ files. To ensure best results, authors are asked to use the LaTeX2e style files available from Springer’s web-server at: ftp://ftp.springer.de/pub/tex/latex/svmonot1/ (for monographs) and ftp://ftp.springer.de/pub/tex/latex/svmultt1/ (for summer schools/tutorials).

Additional technical instructions, if necessary, are available on request from: [email protected]. 4. Careful preparation of the manuscripts will help keep production time short besides ensuring satisfactory appearance of the finished book in print and online. After acceptance of the manuscript authors will be asked to prepare the final LaTeX source files and also the corresponding dvi-, pdf- or zipped ps-file. The LaTeX source files are essential for producing the full-text online version of the book (see http://www.springerlink.com/openurl.asp?genre=journal&issn=0075-8434 for the existing online volumes of LNM). The actual production of a Lecture Notes volume takes approximately 12 weeks. 5. Authors receive a total of 50 free copies of their volume, but no royalties. They are entitled to a discount of 33.3% on the price of Springer books purchased for their personal use, if ordering directly from Springer. 6. Commitment to publish is made by letter of intent rather than by signing a formal contract. Springer-Verlag secures the copyright for each volume. Authors are free to reuse material contained in their LNM volumes in later publications: a brief written (or e-mail) request for formal permission is sufficient. Addresses: Professor J.-M. Morel, CMLA, ´ Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan Cedex, France E-mail: [email protected] Professor F. Takens, Mathematisch Instituut, Rijksuniversiteit Groningen, Postbus 800, 9700 AV Groningen, The Netherlands E-mail: [email protected] Professor B. Teissier, Institut Mathématique de Jussieu, ´ UMR 7586 du CNRS, Equipe “Géométrie et Dynamique”, 175 rue du Chevaleret, 75013 Paris, France E-mail: [email protected] For the “Mathematical Biosciences Subseries” of LNM: Professor P.K. Maini, Center for Mathematical Biology, Mathematical Institute, 24-29 St Giles, Oxford OX1 3LP, UK E-mail: [email protected] Springer, Mathematics Editorial, Tiergartenstr. 17, 69121 Heidelberg, Germany, Tel.: +49 (6221) 487-259 Fax: +49 (6221) 4876-8259 E-mail: [email protected]

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