Semiclassical analysis

SEMICLASSICAL ANALYSIS Lawrence C. Evans and Maciej Zworski Department of Mathematics University of California, Berkeley

PREFACE

This book originates with a course MZ taught at UC Berkeley during the spring semester of 2003, notes for which LCE took in class. In this presentation we have provided full details for many proofs only sketched in the original lectures. We have reworked the order of presentation, added many, many additional topics, and included more heuristic commentary. We have as well introduced consistent notation, recounted in Appendix A. Relevant functional analysis and other background mathematics have been consolidated into Appendices B–D. We should mention that several excellent treatments of mathematical semiclassical analysis have appeared recently. The book [D-S] by Dimassi and Sj¨ ostrand starts with the WKB-method, develops the general semiclassical calculus, and then provides high tech spectral asymptotics. The presentation of Martinez [M] is based on a systematic development of FBI (Fourier-Bros-Iagolnitzer) transform techniques, with applications to microlocal exponential estimates and to propagation estimates. Our text is intended as a more elementary, but broader, introduction. Except for the general symbol calculus, where we followed Chapter 7 of [D-S], there is little overlap with these other two texts, nor with the influential book by Robert [R]. Guillemin and Sternberg [G-St] offer yet another perspective on the subject, very much complementary to the one presented here. Their notes concentrate on global and functorial aspects of semiclassical analysis, in particular on the theory of Fourier integral operators and on trace formulas. We are especially grateful to Hans Christianson, Semyon Dyatlov, Justin Holmer, and St´ephane Nonnenmacher for their careful reading of earlier versions of these notes and for many valuable comments and corrections. 3

4

PREFACE

Our thanks also to Faye Yeager for typing a first draft and to Jonathan Dorfman for TEX advice. Stephen Moye at the AMS provided us with fantastic help on deeper TEX issues. In his study of semiclassical analysis MZ has been influenced by his long collaboration with Johannes Sj¨ostrand, whom he acknowledges with pleasure and gratitude. We will maintain on our websites at the UC Berkeley Mathematics Department a list of errata and typos. Please let us know about any errors you find. LCE has been supported in part by NSF grant DMS-1001724 and MZ by NSF grant DMS-0654436. LCE, MZ August, 2011 Berkeley

Contents

Preface

3

Chapter 1.

Introduction

9

§1.1.

Basic themes

§1.2.

Classical and quantum mechanics

10

§1.3.

Overview

12

§1.4.

Notes

14

9

Part 1. BASIC THEORY Chapter 2. §2.1.

Symplectic geometry and analysis

Flows

17 17

R2n

§2.2.

Symplectic structure on

§2.3.

Symplectic mappings

19

§2.4.

Hamiltonian vector fields

23

§2.5.

Lagrangian submanifolds

27

§2.6.

Notes

30

Chapter 3. §3.1.

Fourier transform, stationary phase

Fourier transform on S S0

18

31 31

§3.2.

Fourier transform on

§3.3.

Semiclassical Fourier transform

42

§3.4.

Stationary phase in one dimension

43

§3.5.

Stationary phase in higher dimensions

49

§3.6.

Notes

55

39

5

6

Chapter §4.1. §4.2. §4.3. §4.4. §4.5. §4.6. §4.7. §4.8.

Contents

4. Semiclassical quantization Definitions Quantization formulas Composition, asymptotic expansions Symbol classes Operators on L2 Compactness Inverses, G˚ arding inequalities Notes

57 58 61 67 73 82 87 90 95

Part 2. APPLICATIONS TO PARTIAL DIFFERENTIAL EQUATIONS Chapter §5.1. §5.2. §5.3. §5.4.

5. Semiclassical defect measures Construction, examples Defect measures and PDE Damped wave equation Notes

99 99 104 106 116

Chapter §6.1. §6.2. §6.3. §6.4. §6.5.

6. Eigenvalues and eigenfunctions The harmonic oscillator Symbols and eigenfunctions Spectrum and resolvents Weyl’s Law Notes

117 117 122 126 129 133

Chapter §7.1. §7.2. §7.3. §7.4. §7.5. §7.6.

7. Estimates for solutions of PDE Classically forbidden regions Tunneling Order of vanishing L∞ estimates for quasimodes Schauder estimates Notes

135 136 139 144 149 154 162

Part 3. ADVANCED THEORY Chapter 8. More on the symbol calculus §8.1. Beals’s Theorem §8.2. Real exponentiation of operators

165 165 171

Contents

§8.3. §8.4. §8.5. Chapter §9.1. §9.2. §9.3. §9.4.

7

Generalized Sobolev spaces Wavefront sets, essential support, microlocality Notes

175 180 189

9. Changing variables Invariance, half-densities Changing symbols Invariant symbol classes Notes

191 191 195 198 206

Chapter 10. Fourier integral operators §10.1. Operator dynamics §10.2. An integral representation formula §10.3. Strichartz estimates §10.4. Lp estimates for quasimodes §10.5. Notes

207 208 210 217 222 225

Chapter 11. Quantum and classical dynamics §11.1. Egorov’s Theorem §11.2. Quantizing symplectic mappings §11.3. Quantizing linear symplectic mappings §11.4. Egorov’s Theorem for longer times §11.5. Notes

227 227 232 237 245 252

Chapter 12. Normal forms §12.1. Overview §12.2. Normal forms: real symbols §12.3. Propagation of singularities §12.4. Normal forms: complex symbols §12.5. Quasimodes, pseudospectra §12.6. Notes

253 253 256 260 263 267 270

Part 4. SEMICLASSICAL ANALYSIS ON MANIFOLDS Chapter 13. Manifolds §13.1. Definitions, examples §13.2. Pseudodifferential operators on manifolds §13.3. Schr¨ odinger operators on manifolds §13.4. Notes

273 273 279 287 295

8

Contents

Chapter 14.

Quantum ergodicity

297

§14.1.

Classical ergodicity

298

§14.2.

A weak Egorov Theorem

300

§14.3.

Weyl’s Law generalized

302

§14.4.

Quantum ergodic theorems

304

§14.5.

Notes

310

Appendix A.

Notation

311

§A.1.

Basic notation

311

§A.2.

Functions, differentiation

312

§A.3.

Operators

315

§A.4.

Estimates

315

§A.5.

Symbol classes

316

Appendix B.

Differential forms

317

§B.1.

Definitions

317

§B.2.

Push-forwards and pull-backs

320

§B.3.

Poincar´e’s Lemma

322

§B.4.

Differential forms on manifolds

323

Appendix C.

Functional analysis

325

§C.1.

Operator theory

325

§C.2.

Spectral theory

329

§C.3.

Trace class operators

337

Appendix D.

Fredholm theory

341

§D.1.

Grushin problems

341

§D.2.

Fredholm operators

342

§D.3.

Meromorphic continuation

344

Bibliography

347

Index

351

Chapter 1

INTRODUCTION

1.1 1.2 1.3 1.4

Basic themes Classical and quantum mechanics Overview References and comments

1.1. BASIC THEMES One of our major goals in this book is understanding the relationships between dynamical systems and the behavior of solutions to various linear PDE and pseudodifferential equations containing a small positive parameter h. 1.1.1. PDE with small parameters. The principal realm of motivation is quantum mechanics, in which case we informally understand h as related to Planck’s constant. With this interpretation in mind, we break down our basic task into these two subquestions: (i) How and to what extent do classical dynamics determine the behavior as h → 0 of solutions to Schr¨ odinger’s equation ih∂t u = −h2 ∆u + V u and the relatedSchr¨ odinger eigenvalue equation −h2 ∆u + V u = Eu? The name “semiclassical” comes from this interpretation. 9

10

1. INTRODUCTION

(ii) Conversely, given various mathematical objects associated with classical mechanics, for instance symplectic transformations, how can we profitably “quantize” them? In fact the techniques of semiclassical analysis apply in many other settings and for many other sorts of PDE. For example we will later study the damped wave equation (1.1.1)

∂t2 u + a∂t u − ∆u = 0

for large times. A rescaling in time will introduce the requisite small parameter h. 1.1.2. Basic techniques. We will construct in Chapters 2–4 and 8–12 a wide variety of mathematical tools to address these issues, among them: • the apparatus of symplectic geometry (to record succintly the behavior of classical dynamical systems); • the Fourier transform (to display dependence upon both the position variables x and the momentum variables ξ); • stationary phase (to describe asymptotics as h → 0 of various expressions involving rescaled Fourier transforms); and • pseudodifferential operators (to localize or, as is said in the trade, to microlocalize functional behavior in phase space).

1.2. CLASSICAL AND QUANTUM MECHANICS In this section we introduce and foreshadow a bit about quantum and classical correspondences. 1.2.1. Observables. We can think of a given function a : Rn × Rn → C, a = a(x, ξ), as a classical observable on phase space, where as above x denotes position and ξ momentum. We will also call a a symbol. Let h > 0 be given. We will associate with the observable a, a corresponding quantum observable aw (x, hD), an operator defined by the formula Z Z
i 1 w a (x, hD)u(x) := e h hx−y,ξi a x+y , ξ u(y) dξdy 2 n (2πh) Rn Rn for appropriate smooth functions u. This is Weyl’s quantization formula.

1.2. CLASSICAL AND QUANTUM MECHANICS

11

1.2.2. Dynamics. We are concerned as well with the evolution in time of classical particles and quantum states. Classical evolution. Our most important example will concern the symbol p(x, ξ) := |ξ|2 + V (x), corresponding to the phase space flow ( x˙ = 2ξ ξ˙ = −∂V, where ˙ = ∂t . We generalize by introducing the arbitrary Hamiltonian p : Rn × Rn → R, p = p(x, ξ), and the corresponding Hamiltonian dynamics ( x˙ = ∂ξ p(x, ξ) (1.2.1) ξ˙ = −∂x p(x, ξ). It is instructive to change our viewpoint somewhat, by writing ϕt = exp(tHp ) for the solution of (1.2.1), where Hp q := {p, q} = h∂ξ p, ∂x qi − h∂x p, ∂ξ qi is the Poisson bracket. Select a symbol a and set at (x, ξ) := a(ϕt (x, ξ)). Then a˙ t = {p, at },

(1.2.2)

and this equation tells us how the symbol evolves in time. Quantum evolution. We next quantize the foregoing by putting P = pw (x, hD), A = aw (x, hD) and defining A(t) := F −1 (t)AF (t)

(1.2.3) for F (t) := e−

itP h

. Then we have the evolution equation

i [P, A(t)], h an obvious analog of (1.2.2). Here then is a basic principle we will later work out in some detail: an assertion about Hamiltonian dynamics, and so the Poisson bracket {·, ·}, will involve at the quantum level the commutator [·, ·]. (1.2.4)

∂t A(t) =

REMARK: h and ~. In this book h denotes a dimensionless parameter, and is consequently not immediately to be identified with the dimensional physical quantity ~ = Planck’s constant/2π = 1.05457×10−34 joule-sec. 

12

1. INTRODUCTION

1.3. OVERVIEW Chapters 2–4 develop the basic machinery, followed by applications to partial differential equations in Chapters 5–7. We develop more advanced theory and applications in Chapters 8–12, and in Chapters 13–14 discuss semiclassical analysis on manifolds. Here is a quick overview, with some of the highpoints: Chapter 2: We start with a quick introduction to symplectic analysis and geometry and their implications for classical Hamiltonian dynamical systems. Chapter 3: This chapter provides the basics of the Fourier transform and derives also important stationary phase asymptotic estimates for the oscillatory integral Z iϕ

Ih :=

e h a dx Rn

of the sort iπ

Ih = (2πh)n/2 | det ∂ 2 ϕ(x0 )|−1/2 e 4

sgn ∂ 2 ϕ(x0 )

e

iϕ(x0 ) h

 n+2  a(x0 ) + O h 2

as h → 0, provided the gradient of the phase ϕ vanishes only at the point x0 . Chapter 4: Next we introduce the Weyl quantization aw (x, hD) of the symbol a(x, ξ) and work out various properties, chief among them the composition formula aw (x, hD)bw (x, hD) = cw (x, hD), where the symbol c := a#b is computed explicitly in terms of a and b. We will prove as well the sharp G˚ arding inequality, learn when aw is a bounded 2 operator on L , etc. Chapter 5: This section introduces semiclassical defect measures, and uses them to derive decay estimates for the damped wave equation (1.1.1), where a ≥ 0 on the flat torus Tn . A theorem of Rauch and Taylor provides a beautiful example of classical/quantum correspondence: the waves decay exponentially if all classical trajectories within a certain fixed time intersect the region where positive damping occurs. Chapter 6: In Chapter 6 we begin our study of the eigenvalue problem P (h)u(h) = E(h)u(h), for the operator P (h) := −h2 ∆ + V (x).

1.3. OVERVIEW

13

We prove Weyl’s Law for the asymptotic distributions of eigenvalues as h → 0, stating for all a < b that #{E(h) | a ≤ E(h) ≤ b} =

1 (|{a ≤ |ξ|2 + V (x) ≤ b}| + o(1)) (2πh)n

as h → 0. Our proof is a semiclassical analog of the classical Dirichlet/Neumann bracketing argument of Courant. Chapter 7: Chapter 7 deepens our study of eigenfunctions, first establishing an exponential vanishing theorem in the “classically forbidden” region. We derive as well a Carleman-type estimate: if u(h) is an eigenfunction of a Schr¨ odinger operator, then for any open set U ⊂⊂ Rn , ku(h)kL2 (U ) ≥ e−c/h ku(h)kL2 (Rn ) . This provides a quantitative estimate for quantum mechanical tunneling. We also present a self-contained “semiclassical” derivation of interior Schauder estimates for the Laplacian. Chapter 8: We return in Chapter 8 to the symbol calculus, firstly proving semiclassical version of Beals’s Theorem, characterizing pseudodifferential operators. As an application we show how quantization commutes with exponentiation at the level of order functions, and then use these insights to define useful generalized Sobolev spaces. This chapter introduces also wavefront sets and the notion of microlocality. Chapter 9: We next introduce the useful formalism of half-densities and use them to see how changing variables in a symbol affects the Weyl quantization. This motivates our introducing the new class of Kohn–Nirenberg symbols, which behave well under coordinate changes and are consequently useful later when we investigate the semiclassical calculus on manifolds. Chapter 10: Chapter 10 discusses the local construction of propagators, using solutions of Hamilton–Jacobi PDE to build phase functions for Fourier integral operators. Applications include the semiclassical Strichartz estimates and Lp bounds on eigenfunction clusters. Chapter 11: This next chapter proves Egorov’s Theorem, characterizing propagators for bounded time intervals in terms of the classical dynamics applied to symbols, up to O(h) error terms. We then employ Egorov’s Theorem to quantize linear and nonlinear symplectic mappings, and conclude the chapter by showing that Egorov’s Theorem is in fact valid until times of order log(h−1 ), the so-called Ehrenfest time.

14

1. INTRODUCTION

Chapter 12: Chapter 12 illustrates how methods from Chapter 11 provide elegant and useful normal forms of differential operators. Among the applications, we build quasimodes for certain nonnormal operators and discuss the implications for pseudospectra. Chapter 13: Chapter 13 discusses briefly general manifolds and modifications to our the symbol calculus to cover pseudodifferential operators on manifolds. The earlier Chapter 9 provides the change of variables formulas we need to work on coordinate patches. Chapter 14: This chapter concerns the quantum implications of ergodicity for underlying dynamical systems on manifolds. A key assertion is that if the underlying dynamical system satisfies an appropriate ergodic condition, then 2 Z X hn σ(A) dxdξ → 0 hAuj , uj i − − {a≤p≤b} a≤Ej ≤b

as h → 0, for a wide class of pseudodifferential operators A. In this expression the classical observable σ(A) is the symbol of A. Appendices: Appendix A records our notation in one convenient location, and Appendix B is a very quick review of differential forms. Appendix C collects various useful functional analysis theorems (with selected proofs). Appendix D discusses Fredholm operators within the framework of Grushin problems.

1.4. NOTES The book of Griffiths [G] provides a nice elementary introduction to quantum mechanics. For a modern physical perspective, consult Heller–Tomsovic [H-T] or St¨ ockmann [Sto].

Part 1

BASIC THEORY

Chapter 2

SYMPLECTIC GEOMETRY AND ANALYSIS

2.1 2.2 2.3 2.4 2.5 2.6

Flows Symplectic structure on R2n Changing variables Hamiltonian vector fields Lagrangian submanifolds Notes

We provide in this chapter a quick discussion of the symplectic geometric structure on Rn × Rn = R2n and its interplay with Hamiltonian dynamics. These will be important for our later goals of understanding interrelationships between dynamics and PDE. The reader may wish first to review our basic notation and also the theory of differential forms, set forth respectively in Appendices A and B.

2.1. FLOWS Let V : RN → RN denote a smooth vector field. Fix a point z ∈ RN and solve the ODE ( z(t) ˙ = V (z(t)) (t ∈ R) (2.1.1) z(0) = z, 17

18

2. SYMPLECTIC GEOMETRY AND ANALYSIS

where ˙ = ∂t . We assume that the solution of the flow (2.1.1) exists and is unique for all times t ∈ R. NOTATION. We define ϕt z := z(t) and sometimes also write We call {ϕt }t∈R

ϕt =: exp(tV ). the flow map or the exponential map.

The following records some standard assertions from theory of ordinary differential equations: LEMMA 2.1 (Properties of flow map). (i) ϕ0 z = z for all z ∈ RN . (ii) ϕt+s = ϕt ϕs for all s, t ∈ R. (iii) For each time t ∈ R, the mapping ϕt : RN → RN is a diffeomorphism, with (ϕt )−1 = ϕ−t .

2.2. SYMPLECTIC STRUCTURE ON R2n We henceforth specialize to the even-dimensional space RN = R2n = Rn × Rn . NOTATION. We refine our previous notation and henceforth denote an element of R2n as z = (x, ξ), and interpret x ∈ Rn as denoting position, ξ ∈ Rn as momentum. We will likewise write w = (y, η) for another typical point of R2n . We let h·, ·i denote the usual inner product on Rn , and then define this new pairing on R2n : DEFINITION. Given z = (x, ξ), w = (y, η) on R2n = Rn × Rn , define their symplectic product (2.2.1)

σ(z, w) := hξ, yi − hx, ηi.

Note that (2.2.2)

σ(z, w) = hJz, wi

2.3. SYMPLECTIC MAPPINGS

19

for the 2n × 2n matrix  (2.2.3)

0 I −I 0

J :=

 .

Observe J 2 = −I, J T = −J = J −1 .

(2.2.4)

LEMMA 2.2 (Properties of σ). The bilinear form σ is antisymmetric: σ(z, w) = −σ(w, z)

(2.2.5) and nondegenerate: (2.2.6)

if σ(z, w) = 0 for all w, then z = 0.

These assertions are straightforward to check. We now bring in the terminology of differential forms, reviewed in Appendix B. NOTATION. We introduce for x = (x1 , . . . , xn ) and ξ = (ξ1 , . . . , ξn ) the 1-forms dxj and dξj for j = 1, . . . , n, and then write σ = dξ ∧ dx =

(2.2.7)

n X

dξj ∧ dxj .

j=1

Observe also (2.2.8)

σ = dω

for ω := ξdx =

n X

ξj dxj .

j=1

Since

d2

(2.2.9)

= 0, it follows that dσ = 0.

2.3. SYMPLECTIC MAPPINGS Suppose next that U, V ⊂ R2n are open sets and κ:U →V is a smooth mapping. We will write κ(x, ξ) = (y, η) = (y(x, ξ), η(x, ξ)). DEFINITION. We call κ a symplectic mapping, or a symplectomorphism, provided (2.3.1)

κ∗ σ = σ.

20

2. SYMPLECTIC GEOMETRY AND ANALYSIS

Here the pull-back κ∗ σ of the symplectic product σ is defined by (κ∗ σ)(z, w) := σ(κ∗ (z), κ∗ (w)), κ∗ denoting the push-forward of vectors: see Appendix B. NOTATION. We will usually write (2.3.1) in the more suggestive notation dη ∧ dy = dξ ∧ dx.

(2.3.2)

EXAMPLE 1: Linear symplectic mappings. Suppose κ : R2n → R2n is linear:    A B x κ(x, ξ) = = (Ax + Bξ, Cx + Dξ) = (y, η), C D ξ where A, B, C, D are n × n matrices. THEOREM 2.3 (Symplectic matrices). The linear mapping κ is symplectic if and only if the matrix   A B K := C D satisfies K T JK = J.

(2.3.3)

In particular the linear mapping (x, ξ) 7→ (ξ, −x) determined by J is symplectic. DEFINITION. We call a 2n × 2n matrix K symplectic if (2.3.3) holds. Proof. Let us compute dη ∧ dy = (Cdx + Ddξ) ∧ (Adx + Bdξ) = AT Cdx ∧ dx + B T Ddξ ∧ dξ + (AT D − C T B)dξ ∧ dx = dξ ∧ dx if and only if (2.3.4)

AT C and B T D are symmetric, AT D − C T B = I.

Therefore

if and only if (2.3.4) holds.



AT BT

CT DT

 A B K JK = C D  T  T T A C − C A A D − CT B = B T C − DT A B T D − DT B = J T



O I −I O





2.3. SYMPLECTIC MAPPINGS

21

We record some useful observations: THEOREM 2.4 (More on symplectic matrices). (i) The product of two symplectic matrices is symplectic. (ii) If K is a symplectic matrix, then (2.3.5)

σ(Kz, Kw) = σ(z, w)

(z, w ∈ R2n ).

(iii) A matrix K is symplectic if and only if K is invertible, K −1 = JK T J T .

(2.3.6) (iv) If

AT J + JA = 0, then Kt := exp(tA) is symplectic for each t ∈ R. Proof. Assertions (i), (ii) and (iii) follow directly from the definitions and J T = −J = J −1 . To prove (iv), write Wt := KtT JKt − J and compute ∂t Wt = AT Wt + Wt A + AT J + JA = AT Wt + Wt A. Since W0 = 0, we deduce from uniqueness that Wt = 0 for all t ∈ R.



EXAMPLE 2: Nonlinear symplectic mappings. Assume next that κ : R2n → R2n is nonlinear: κ(x, ξ) = (y, η) for smooth functions y = y(x, ξ), η = η(x, ξ). Its linearization is the 2n × 2n matrix   ∂x y ∂ξ y ∂κ = . ∂x η ∂ξ η THEOREM 2.5 (Symplectic transformations). The mapping κ is symplectic if and only if the matrix ∂κ is symplectic at each point. Proof. We have dη ∧ dy = (Cdx + Ddξ) ∧ (Adx + Bdξ) for A := ∂x y, B := ∂ξ y, C := ∂x η, D := ∂ξ η. Consequently, as in the previous proof, we have dη ∧ dy = dξ ∧ dx if and only if (2.3.4) is valid, which in turn is so if and only if ∂κ is a symplectic matrix. 

22

2. SYMPLECTIC GEOMETRY AND ANALYSIS

EXAMPLE 3: Lifting diffeomorphisms. Let γ : Rn → Rn be a diffeomorphism on Rn , with nondegenerate Jacobian matrix ∂γ = ∂x γ. We propose to extend γ to a symplectomorphism κ : R2n → R2n having the form (2.3.7)

κ(x, ξ) = (γ(x), η(x, ξ)) = (y, η),

by “lifting” γ to variables ξ. THEOREM 2.6 (Extending to a symplectic mapping). The transformation (2.3.7) is symplectic if  T (2.3.8) η(x, ξ) := ∂γ(x)−1 ξ. Proof. As the statement suggests, it will be easier to look for ξ as a function of x and η. We compute dy = A dx,

dξ = E dx + F dη,

for A := ∂x y,

E := ∂x ξ,

F := ∂η ξ.

Therefore dη ∧ dy = dη ∧ (A dx) and dξ ∧ dx = (Edx ∧ F dη) ∧ dx = Edx ∧ dx + dη ∧ F T dx. We would like to construct ξ = ξ(x, η) so that A = FT

and E is symmetric,

the latter condition implying that Edx ∧ dx = 0. To do so, let us define ξ(x, η) := (∂γ)T η. Then clearly F T = A, and E = E T = ∂ 2 γ, as required.



EXAMPLE 4: Generating functions. Our next example demonstrates that we can, locally at least, build a symplectic transformation from a realvalued generating function. Suppose ϕ : Rn × Rn → R, ϕ = ϕ(x, y), is smooth. Assume also that (2.3.9)

2 det(∂xy ϕ(x0 , y0 )) 6= 0.

Define (2.3.10)

ξ = ∂x ϕ, η = −∂y ϕ,

2.4. HAMILTONIAN VECTOR FIELDS

23

and observe that the Implicit Function Theorem implies (y, η) is a smooth function of (x, ξ) near (x0 , ∂x ϕ(x0 , y0 )). THEOREM 2.7 (Generating functions and symplectic maps). The mapping κ implicitly defined by (x, ∂x ϕ(x, y)) 7−→ (y, −∂y ϕ(x, y))

(2.3.11)

is a symplectomorphism near (x0 , ξ0 ). A simple example is ϕ(x, y) = hx, yi, which generates the symplectic mapping (x, ξ) 7→ J(x, ξ) = (ξ, −x). Proof. We compute dη ∧ dy = d(−∂y ϕ) ∧ dy 2 = [(−∂y2 ϕdy) ∧ dy] + [(−∂xy ϕdx) ∧ dy] 2 = −(∂xy ϕ)dx ∧ dy,

since ∂y2 ϕ is symmetric. Likewise, dξ ∧ dx = d(∂x ϕ) ∧ dx 2 = [(∂x2 ϕ dx) ∧ dx] + [(∂xy ϕ dy) ∧ dx] 2 = −(∂xy ϕ)dx ∧ dy = dη ∧ dy.



Section 2.5 will generalize this example and provide more geometric insight.

2.4. HAMILTONIAN VECTOR FIELDS DEFINITION. Given f ∈ C ∞ (R2n ), we define the corresponding Hamiltonian vector field by requiring (2.4.1)

σ(z, Hf ) = df (z)

for all z = (x, ξ).

We can write explicitly that (2.4.2)

Hf = h∂ξ f, ∂x i − h∂x f, ∂ξ i =

n X j=1

fξj ∂xj −

n X

fxj ∂ξj .

j=1

Another way to write the definition of Hf is by using the contraction defined in Appendix B: LEMMA 2.8 (Differentials and Hamiltonian vector fields). We have (2.4.3)

df = −(Hf

σ),

24

2. SYMPLECTIC GEOMETRY AND ANALYSIS

Proof. This follows directly from the definition, as we can calculate for each z that (Hf σ)(z) = σ(Hf , z) = −σ(z, Hf ) = −df (z).  DEFINITION. If f, g ∈ C ∞ (R2n ), we define their Poisson bracket {f, g} := Hf g = σ(∂f, ∂g).

(2.4.4) That is, (2.4.5)

{f, g} = h∂ξ f, ∂x gi − h∂x f, ∂ξ gi =

n X


fξj gxj − fxj gξj .

j=1

LEMMA 2.9 (Brackets, commutators). (i) We have Jacobi’s identity (2.4.6)

{f, {g, h}} + {g, {h, f }} + {h, {f, g}} = 0

for all functions f, g, h ∈ C ∞ (R2n ). (ii) Furthermore, (2.4.7)

H{f,g} = [Hf , Hg ].

Proof. A direct calculation verifies assertion (i); and we observe that H{f,g} h = [Hf , Hg ]h is a rewriting of (2.4.6).



REMARK: Another derivation of Jacobi’s identity. An alternative proof of (2.4.6) follows, this illustrating the essential property that dσ = 0. Lemma B.1 provides the identity 0 = dσ(Hf , Hg , Hh ) (2.4.8)

= Hf σ(Hg , Hh ) + Hg σ(Hh , Hf ) + Hh σ(Hf , Hg ) − σ([Hf , Hg ], Hh ) − σ([Hg , Hh ], Hf ) − σ([Hh , Hf ], Hg ).

Now (2.4.4) implies Hf σ(Hg , Hh ) = {f, {g, h}} and σ([Hf , Hg ], Hh ) = [Hf , Hg ]h = Hf Hg h − Hg Hf h = {f, {g, h}} − {g, {f, h}}. Similar identities hold for other terms. Substituting into (2.4.8) gives Jacobi’s identity. 

2.4. HAMILTONIAN VECTOR FIELDS

25

THEOREM 2.10 (Jacobi’s Theorem). If κ is a symplectomorphism, then (2.4.9)

Hf = κ∗ (Hκ∗ f ).

In other words, the pull-back of a Hamiltonian vector field generated by f , κ∗ Hf := (κ−1 )∗ Hf ,

(2.4.10)

is the Hamiltonian vector field generated by the pull-back of f . Proof. Using the notation of (2.4.10), κ∗ (Hf ) σ = κ∗ (Hf ) κ∗ σ = κ∗ (Hf ∗

σ)

∗

= −κ (df ) = −d(κ f ) = Hκ∗ f

σ.

Since σ is nondegenerate, (2.4.9) follows.



EXAMPLE. Define κ = J, so that κ(x, ξ) = (ξ, −x); and recall κ is a symplectomorphism. We have κ∗ f (x, ξ) = f (ξ, −x), and therefore Hκ∗ f = h∂x f (ξ, −x), ∂x i + h∂ξ f (ξ, −x), ∂ξ i. Then κ∗ Hf = h∂ξ f (ξ, −x), ∂ξ i − h∂x f (ξ, −x), ∂−x i = Hκ∗ f .



THEOREM 2.11 (Hamiltonian flows as symplectomorphisms). If f is smooth, then for each time t, the mapping (x, ξ) 7→ ϕt (x, ξ) = exp(tHf ) is a symplectomorphism. Proof. According to Cartan’s formula (Theorem B.3), we have ∂t (ϕ∗t σ) = LHf σ = d(Hf

σ) + (Hf

dσ).

Since dσ = 0, it follows that ∂t (ϕ∗t σ) = d(−df ) = −d2 f = 0. Thus (ϕt )∗ σ = σ for all times t.



The next result shows that locally all nondegenerate closed two forms are equivalent to the standard symplectic form σ on R2n .

26

2. SYMPLECTIC GEOMETRY AND ANALYSIS

THEOREM 2.12 (Darboux’s Theorem). Let U be a neighborhood of (x0 , ξ0 ) and suppose η is a nondegenerate 2-form defined on U , satisfying dη = 0. Then near (x0 , ξ0 ) there exists a diffeomorphism κ such that κ∗ η = σ.

(2.4.11)

INTERPRETATION. A symplectic structure on R2n is determined by a choice of nondegenerate, closed 2-form η. Darboux’s theorem states that all symplectic structures are identical locally, in the sense that all are equivalent to that given by σ. This is dramatic contrast to Riemannian geometry: there are no local invariants in symplectic geometry.  Proof. 1. Let us assume (x0 , ξ0 ) = (0, 0). We first find a linear mapping L so that L∗ η(0, 0) = σ(0, 0). This means that we find a basis {ek , fk }nk=1 of R2n such that η(fl , ek ) = δkl , η(ek , el ) = 0, η(fk , fl ) = 0 P P for all 1 ≤ k, l ≤ n. Then if u = ni=1 xi ei + ξi fi , v = nj=1 yj ej + ηj fj , we have n X η(u, v) = xi yj η(ei , ej ) + ξi ηj η(fi , fj ) + xi ηj σ(ei , fj ) + ξi yj σ(fi , ej ) i,j=1

= hξ, yi − hx, ηi = σ((x, ξ), (y, η)). We leave finding L as a linear algebra exercise. 2. Next, define ηt := tη + (1 − t)σ for 0 ≤ t ≤ 1. Our intention is to find κt so that κ∗t ηt = σ near (0, 0); then κ := κ1 solves our problem. We will construct κt by solving the flow ( z(t) ˙ = Vt (z(t)) (0 ≤ t ≤ 1) (2.4.12) z(0) = z, and setting κt := ϕt . For this to work, we must design the vector fields Vt in (2.4.12) so that ∂t (κ∗t ηt ) = 0. Let us therefore calculate ∂t (κ∗t ηt ) = κ∗t (∂t ηt ) + κ∗t LVt ηt = κ∗t [(η − σ) + d(Vt ηt ) + Vt dηt ] ,

2.5. LAGRANGIAN SUBMANIFOLDS

27

where we used Cartan’s formula, Theorem B.3. Now dηt = tdη + (1 − t)dσ, and hence (d/dt)(κ∗t ηt ) = 0 provided (η − σ) + d(Vt ηt ) = 0.

(2.4.13)

3. According to Poincar´e’s Theorem B.4, we can write η − σ = dα

near (0, 0).

So (2.4.13) will hold, if (2.4.14)

Vt ηt = −α

(0 ≤ t ≤ 1).

Since η = σ at (0, 0), ηt = σ at (0, 0). In particular, ηt is nondegenerate for 0 ≤ t ≤ 1 in a neighborhood of (0, 0), and hence we can solve (2.4.13) for the vector field Vt . 

2.5. LAGRANGIAN SUBMANIFOLDS This section provides some further geometric interpretations of generating functions, introduced earlier in Example 4 in Section 2.3. DEFINITION. A Lagrangian submanifold Λ in R2n is an n-dimensional submanifold for which (2.5.1)

σ|Λ = 0.

The meaning of (2.5.1) is that σ(u) = 0 for each point z ∈ Λ and for all u = (u1 , u2 ) with u1 , u2 ∈ Tz (Λ), the tangent space to Λ at z. THEOREM 2.13 (Lagrangian submanifolds). Let Λ be a Lagrangian submanifold of R2n . Then each point z ∈ Λ lies in a relatively open neighborhood U ⊂ Λ within which (2.5.2)

ω|Λ = dϕ

for some smooth function ϕ : U → R. Proof. Given z ∈ Λ, we find a relatively open neighborhood U ⊂ Λ and a smooth diffeomorphism γ : U → V , where V = B 0 (0, 1) is the open unit ball in Rn . Then ρ := γ −1 pulls back ω|Λ to the one-form α := ρ∗ (ω|Λ ), defined on V . According to (2.5.1), we have dα = d(ρ∗ ω|Λ ) = ρ∗ (dω|Λ ) = ρ∗ (σ|Λ ) = 0 within V . Poincar´e’s Theorem B.5 therefore implies α = dψ for some smooth function ψ : V → R. Set ϕ := ψ ◦ γ = γ ∗ ψ. Then dϕ = d(γ ∗ ψ) = γ ∗ dψ = γ ∗ α = ω|Λ .



28

2. SYMPLECTIC GEOMETRY AND ANALYSIS

We show next that a Lagrangian submanifold is locally determined by the graph of a generating function of appropriate coordinates: THEOREM 2.14 (Generating functions for Lagrangian submanifolds). (i) Suppose that Λ ⊂ Rn × Rn is a smooth Lagrangian submanifold and that (x0 , ξ0 ) ∈ Λ. Then there exist a neighborhood U ⊂ Rn × Rn of (x0 , ξ0 ), a splitting of coordinates (2.5.3)

x = (x0 , x00 ), ξ = (ξ 0 , ξ 00 ),

where k ∈ {0, . . . n} and x0 , ξ 0 ∈ Rk , x00 , ξ 00 ∈ Rn−k , and a smooth function (2.5.4)

ϕ = ϕ(x0 , ξ 00 )

such that (2.5.5)

Λ ∩ U = {(x0 , −∂ξ00 ϕ; ∂x0 ϕ, ξ 00 ) | x0 ∈ Rk , ξ 00 ∈ Rn−k } ∩ U.

We call ϕ = ϕ(x0 , ξ 00 ) a local generating function of Λ near (x0 , ξ0 ). Proof: 1. Let V ⊂ Rn be a coordinate chart for a neighborhood of (x0 , ξ0 ) in Λ: ρ : V → Λ ⊂ R2n , with ρ(0) = (x0 , ξ0 ). The Jacobian ∂ρ(0) has full rank and hence has n independent rows. We choose n such rows and call the corresponding coordinates x0 ∈ Rk and ξ 00 ∈ Rn−k . 2. Define p : R2n → Rk × Rn−k by p(x, ξ) := (x0 , ξ 00 ). Our choice of the coordinates (x0 , ξ 00 ) means that p ◦ ρ : V → Rk × Rn−k has an invertible Jacobian at 0 ∈ Rn . Hence the Implicit Function Theorem implies p ◦ γ is invertible in a neighborhood of 0. This means that we can use (x0 , ξ 00 ) as local coordinates on Λ, and so there exists a neighborhood U ⊂ R2n of (x0 , ξ0 ) and smooth functions f : Rk × Rn−k → Rn−k , g : Rk × Rn−k → Rk such that Λ ∩ U = {(x0 , f (x0 , ξ 00 ), g(x0 , ξ 00 ), ξ 00 ) | (x0 , ξ 00 ) ∈ Rk × Rn−k } ∩ U.

2.5. LAGRANGIAN SUBMANIFOLDS

29

3. Recalling that ω = ξdx, we use Theorem 2.13 to see that for some function ψ = ψ(x0 , ξ 00 ), ω|Λ = hg, dx0 i + hξ 00 , ∂x0 f dx0 + ∂ξ00 f dξ 00 i = hg + (∂x0 f )T ξ 00 , dx0 i + h(∂ξ00 f )T ξ 00 , dξ 00 i = h∂x0 ψ, dx0 i + h∂ξ00 ψ, dξ 00 i. That is, ψx0 = g + (∂x0 f )T ξ 00 = g + ∂x0 hf, ξ 00 i, ψξ00 = (∂ξ00 f )T ξ 00 . If we put ϕ(x0 , ξ 00 ) := ψ(x0 , ξ 00 ) − hf (x0 , ξ 00 ), ξ 00 i, then f = −∂ξ00 ϕ, g = ∂x0 ϕ.



EXAMPLES. (i) The simplest case is k = n, when Λ ∩ U = {(x, ∂x ϕ(x)}. Then (2.5.2) reads ω|Λ = dϕ = ∂ϕdx. (ii) Theorem 2.14 generalizes Theorem 2.7. To see this, consider the twisted graph of κ:  (2.5.6) Λκ = (x, y, ξ, −η) | (x, ξ) = κ(y, η), (y, η) ∈ R2n . We readily check that Λκ is a Lagrangian submanifold of R2n × R2n , with the symplectic form σ = dη ∧ dy + dξ ∧ dx. If the map (x, y, ξ, η) 7→ (y, x), has a nonvanishing differential on Λκ , we can employ (x, y) as coordinates in Theorem 2.14: Λκ = {(x, y, ∂x ϕ(x, y), ∂y ϕ(x, y)) | x, y ∈ Rn } . Then (2.5.6) shows this is equivalent to (2.3.11). (iii) Another interesting class of generating functions for symplectic maps is when (x, y, ξ, η) 7→ (x, η) has nonvanishing differential on Λκ . Then (2.5.7)

Λκ = {(x, −∂η ϕ(x, η), ∂x ϕ(x, η), η) | x, η ∈ Rn } = {(x, ∂η ψ(x, η), ∂x ψ(x, η), −η) | x, η ∈ Rn }

30

2. SYMPLECTIC GEOMETRY AND ANALYSIS

for ψ(x, η) := ϕ(x, −η). This means that κ : (∂η ψ(x, η), η) 7→ (x, ∂x ψ(x, η)).



2.6. NOTES The proof of Theorem 2.12 is from Moser [Mo]; see also Cannas da Silva [CdS]. A PDE oriented introduction to symplectic geometry may be found in H¨ ormander [H3, Chapter 21]. In Greek, the word “symplectic” means “intertwined”. This is consistent with Example 4, since the generating function ϕ = ϕ(x, y) is a function of a mixture of half of the original variables (x, ξ) and half of the new variables (y, η). “Symplectic” can also be interpreted as “complex”, mathematical usage due to H. Weyl who renamed “line complex group” the “symplectic group”: see Cannas da Silva [CdS].

Chapter 3

FOURIER TRANSFORM, STATIONARY PHASE

3.1 3.2 3.3 3.4 3.5 3.6

Fourier transform on S Fourier transform on S 0 Semiclassical Fourier transform Stationary phase in one dimension Stationary phase in higher dimensions Notes

We discuss in this chapter how to define the Fourier transform F and its inverse F −1 on various classes of smooth functions and nonsmooth distributions. We introduce also the rescaled semiclassical transforms Fh , Fh−1 depending on the small parameter h, and develop stationary phase asymptotics to help us understand various formulas involving Fh in the limit as h → 0. Be warned that our use of the symbols “D” and “Dα ” differs from that in first author’s textbook [E].

3.1. FOURIER TRANSFORM ON S We begin by defining and investigating the Fourier transform of smooth functions that decay rapidly as |x| → ∞. 31

32

3. FOURIER TRANSFORM, STATIONARY PHASE

DEFINITIONS. (i) The Schwartz space is (3.1.1) S = S (Rn ) := {ϕ ∈ C ∞ (Rn ) | sup |xα ∂ β ϕ| < ∞ for all multiindices α, β}. Rn

(ii) For each pair of multiindices α, β and each ϕ ∈ S , we define the seminorm |ϕ|α,β := sup |xα ∂ β ϕ|.

(3.1.2)

Rn

(iii) We say ϕj → ϕ

in S

provided |ϕj − ϕ|α,β → 0 for all multiindices α, β. In words, the Schwartz space consists of functions which are smooth and which, together with all their derivatives, decay faster than any power of |x|−1 . DEFINITION. If ϕ ∈ S , define the Fourier transform Z

e−ihx,ξi ϕ(x) dx

Fϕ(ξ) = ϕ(ξ) ˆ :=

(3.1.3)

Rn

(ξ ∈ Rn ).

The reader is warned that many other texts use slightly different definitions, entailing normalizing factors involving π. EXAMPLE: Exponential of a real quadratic form. THEOREM 3.1 (Transform of a real exponential). Let Q be a real, symmetric, positive definite n × n matrix. Then 1

F(e− 2 hQx,xi ) =

(3.1.4)

(2π)n/2 − 1 hQ−1 ξ,ξi e 2 . (det Q)1/2

Proof. Let us calculate F(e

− 12 hQx,xi

Z ) =

1

e− 2 hQx,xi−ihx,ξi dx

Rn

Z = Rn

1

e− 2 hQ(x+iQ

−1 ξ), x+iQ−1 ξi

1

e− 2 hQ

−1 ξ,ξi

dx

3.1. FOURIER TRANSFORM ON S

= e

− 21 hQ−1 ξ,ξi

33

Z

1

e− 2 hQy,yi dy.

Rn

We compute the last integral by making an orthogonal change of variables that converts Q into diagonal form diag(λ1 , . . . , λn ). Then Z Z n Z ∞ P Y λk 2 − 12 hQy,yi − 12 n λk wk2 k=1 e e dy = dw = e− 2 w dw Rn

Rn

= =

k=1 −∞

Z n Y 21/2

∞

2

1/2 −∞ k=1 λk (2π)n/2

(λ1 · · · λn )1/2

e−y dy =

(2π)n/2 . (det Q)1/2



The Fourier transform F lets us move from position variables x to momentum variables ξ, and we need to catalog how it converts various algebraic and analytic expressions in x into related expressions in ξ: THEOREM 3.2 (Properties of Fourier transform). (i) The mapping F : S → S is an isomorphism. (ii) We have the Fourier inversion formula (3.1.5)

F −1 =

1 RF, (2π)n

where Rf (x) := f (−x). In other words, Z 1 −1 eihx,ξi ψ(ξ) dξ; (3.1.6) F ψ(x) = (2π)n Rn and therefore (3.1.7)

ϕ(x) =

1 (2π)n

Z

eihx,ξi ϕ(ξ) ˆ dξ.

Rn

(iii) In addition, (3.1.8)

Dξα (Fϕ) = F((−x)α ϕ)

and (3.1.9)

F(Dxα ϕ) = ξ α Fϕ.

(iv) Furthermore, (3.1.10)

F(ϕψ) =

1 F(ϕ) ∗ F(ψ). (2π)n

34

3. FOURIER TRANSFORM, STATIONARY PHASE

REMARKS. (i) In these formulas we employ the notation from Appendix A: 1 Dα = |α| ∂ α . i In particular, Dxα e−ihx,ξi = (−ξ)α e−ihx,ξi ,

Dξα e−ihx,ξi = (−x)α e−ihx,ξi .

(ii) We will later interpret the Fourier inversion formula (3.1.6) as saying that Z 1 eihx−y,ξi dξ in the sense of distributions, (3.1.11) δy (x) = (2π)n Rn with δy = δ(· − y) denoting the Dirac measure.



Proof. 1. Let us calculate for ϕ ∈ S that Z α α e−ihx,ξi ϕ(x) dx dx Dξ (Fϕ) = Dξ n R Z = e−ihx,ξi (−x)α ϕ(x) dx = F((−x)α ϕ). Rn

Likewise, Z e−ihx,ξi Dxα ϕ dx = (−1)|α| Dxα (e−ihx,ξi )ϕ dx n n R R Z = (−1)|α| (−ξ)α e−ihx,ξi ϕ dx = ξ α (Fϕ).

F(Dxα ϕ) =

Z

Rn

This proves (iii). 1

2. Recall from Appendix A the useful notation hxi = (1 + |x|2 ) 2 . Then for all multiindices α, β, we have sup |ξ β Dξα ϕ| ˆ = sup |ξ β F((−x)α ϕ)| ξ

ξ

= sup |F(Dxβ ((−x)α ϕ)| ξ

1 n+1 β α e hxi Dx ((−x) ϕ) dx n+1 hxi ξ Rn Z n+1 β α ≤ sup |hxi Dx ((−x) ϕ)| hxi−n−1 dx < ∞. Z = sup

−ihx,ξi

x

Rn

Hence F : S → S , and a similar calculation shows that ϕi → ϕ in S implies F(ϕj ) → F(ϕ). 3. To show F is invertible, note that RFFDxj

= RFMξj F

3.1. FOURIER TRANSFORM ON S

35

= R(−Dxj )FF = Dxj RFF, where Mξj denotes multiplication by ξj . Thus RFF commutes with Dxj and it likewise commutes with the multiplication operators Mxj . According to Lemma 3.3, stated and proved below, RFF is a multiple of the identity operator: (3.1.12)

RFF = cI.

From the example above, we know that F(e− Thus F(e−

|ξ|2 2

) = (2π)n/2 e−

|x|2 2

|x|2 2

) = (2π)n/2 e−

|ξ|2 2

.

. Consequently c = (2π)n , and hence

F −1 =

1 RF. (2π)n

4. Lastly, since Z 1 ϕ(x) = eihx,ξi ϕ(ξ) ˆ dξ, (2π)n Rn

1 ψ(x) = (2π)n

Z

ˆ dη, eihx,ηi ψ(η)

Rn

we have ϕψ = = =

Z Z 1 ˆ dξdη eihx,ξ+ηi ϕ(ξ) ˆ ψ(η) (2π)2n Rn Rn Z  Z 1 ihx,ρi ˆ e ϕ(ξ) ˆ ψ(ρ − ξ) dρ dξ (2π)2n Rn Rn 1 ˆ F −1 (ϕˆ ∗ ψ). (2π)n

But ϕψ = F −1 F(ϕψ), and so assertion (iv) follows.



LEMMA 3.3 (Commutativity). Let Mf : g 7→ f g be the multiplication operator. Suppose that L : S → S is linear, and that (3.1.13)

LMxj = Mxj L,

LDxj = Dxj L

j = 1, . . . , n. Then L = cI for some constant c, where I denotes the identity operator. Proof. 1. Choose ϕ ∈ S , fix y ∈ Rn , and write ϕ(x) − ϕ(y) =

n X j=1

(xj − yj )ψj (x)

36

3. FOURIER TRANSFORM, STATIONARY PHASE

for 1

Z

ϕxj (y + t(x − y)) dt.

ψj (x) := 0

Since typically ψj ∈ / S , we select a smooth function χ with compact support such that χ ≡ 1 for x near y. Write ϕj (x) := χ(x)ψj (x) +

(xj − yj ) (1 − χ(x))ϕ(x). |x − y|2

Then (3.1.14)

ϕ(x) − ϕ(y) =

n X

(xj − yj )ϕj (x)

j=1

with ϕj ∈ S . 2. We claim next that if ϕ(y) = 0, then Lϕ(y) = 0. This follows from (3.1.14), since n X Lϕ(x) = (xj − yj )Lϕj = 0 j=1

at x = y. 2

Therefore Lϕ(x) = c(x)ϕ(x) for some function c. Taking ϕ(x) = e−|x| , we deduce that c ∈ C ∞ . Finally, since L commutes with differentiation, we conclude that c must be a constant.  THEOREM 3.4 (Integral identities). If ϕ, ψ ∈ S , then Z Z (3.1.15) ϕψ ˆ dx = ϕψˆ dy Rn

Rn

and Z (3.1.16)

ϕψ¯ dx =

Rn

1 (2π)n

Z

¯ ϕˆψˆ dξ.

Rn

In particular, kϕk2L2 =

(3.1.17)

1 kϕk ˆ 2L2 . (2π)n

Proof. Note first that  Z Z Z −ihx,yi ϕψ ˆ dx = e ϕ(y) dy ψ(x) dx Rn Rn Rn  Z Z Z −ihy,xi = e ψ(x) dx ϕ(y) dy = Rn

Rn

Rn

ˆ dy. ψϕ

3.1. FOURIER TRANSFORM ON S ¯ Replace ψ by ψˆ in (3.1.15): Z

¯ ϕˆψˆ dξ =

Rn

37

Z

¯ˆ ∧ ϕ(ψ) dx.

Rn

¯ R ¯ˆ ∧ ¯ ¯ and so (ψ) ¯ But ψˆ = Rn eihx,ξi ψ(x) dx = (2π)n F −1 (ψ) = (2π)n ψ.



We record next some elementary estimates that we will need later: LEMMA 3.5 (Useful estimates). (i) We have the bounds kˆ ukL∞ ≤ kukL1

(3.1.18) and

kukL∞ ≤

(3.1.19)

1 kˆ ukL1 . (2π)n

(ii) There exists a constant C such that kˆ ukL1 ≤ C sup k∂ α ukL1 .

(3.1.20)

|α|≤n+1

Proof. Estimates (3.1.18) and (3.1.19) follow easily from (3.1.3) and (3.1.7). Furthermore, Z |ˆ u|hξin+1 hξi−n−1 dξ ≤ Ckˆ uhξin+1 kL∞ kˆ ukL1 = Rn

≤ C sup kξ α u ˆkL∞ = C sup k(∂ α u)∧ kL∞ ≤ C sup k∂ α ukL1 . |α|≤n+1

|α|≤n+1

|α|≤n+1

This proves (3.1.20).



APPLICATION 1: Solving a PDE. Consider the initial-value problem ( ∂t u = x∂y u + ∂x2 u on R2 × (0, ∞) (3.1.21) u = δ(x0 ,y0 ) on R2 × {t = 0}. Let u ˆ := Fu denote the Fourier transform of u in the variables x, y (but not in t). Then (∂t + η∂ξ )ˆ u = −ξ 2 u ˆ. This is a linear first-order PDE we can solve by the method of characteristics: u ˆ(t, ξ + tη, η) = u ˆ(0, ξ, η)e−

Rt

2 0 (ξ+sη) ds

= u ˆ(0, ξ, η)e−ξ

2 3 2 t−ξηt2 − η t 3

1

= u ˆ(0, ξ, η)e− 2 hBt (ξ,η),(ξ,η)i ,

38

3. FOURIER TRANSFORM, STATIONARY PHASE

for   2t t2 Bt := 2 . t 2t3 /3 Furthermore, u ˆ(0, ξ, η) = δˆ(x0 ,y0 ) . Taking the inverse Fourier transform, F −1 , we find 1

u(t, x, y − tx) = δ(x0 ,y) ∗ F −1 (e− 2 hBt (ξ,η),(ξ,η)i ) √   (x − x0 )2 3(x − x0 )(y − y0 ) 3(y − y0 )2 3 exp − − ) ; + = 2πt3 t t2 t3 and hence

√

u(t, x, y) =

3 −Φ(t,x,x0 ,y−y0 ) e , 2πt3

where Φ(t, x, x0 , y) =

(x − x0 )2 3(x − x0 )(y + tx) 3(y + tx)2 − + . t t2 t3



APPLICATION 2: Almost analytic extensions. Let 1 ∂¯z := (∂x + i∂y ) 2 for z = x + iy denote the Cauchy-Riemann operator, and remember that g is analytic provided ∂¯z g ≡ 0 in the complex plane C. A function f ∈ S (R) need not be the restriction to R of an analytic function in C. But we can build an extension f˜ that is almost analytic in the sense that ∂¯z f˜ vanishes to infinite order near the real axis. We will use this almost analytic extension later. For the construction below, select a function χ such that χ ∈ Cc∞ ((−1, 1)), with χ ≡ 1 on [−1/2, 1/2]. THEOREM 3.6 (Almost analytic extension). If f ∈ S (R), then Z 1 (3.1.22) f˜(z) := χ(y) χ(yξ)fˆ(ξ)eiξ(x+iy) dξ 2π R is an almost analytic extension of f to the complex plane. This means fe ∈ C ∞ (C), fe|R = f, spt fe ⊂ {z | |Imz| ≤ 1} and (3.1.23)

∂¯z fe(z) = O(|Imz|∞ ).

3.2. FOURIER TRANSFORM ON S 0

39

The notation (3.1.23) means |∂¯z fe(z)| ≤ CN |Imz|N for each N . Proof. 1. The Fourier inversion formula shows that fe = f on R and the term χ(y) restricts the support of fe to {|Imz| ≤ 1}. 2. Let Z F (z) :=

χ(yξ)fˆ(ξ)eiξ(x+iy) dξ.

R

We calculate Z

ξχ0 (yξ)fˆ(ξ)eiξ(x+iy) dξ R Z 0 χ (t) N =y i ξ N +1 fˆ(ξ)eiξ(x+iy) dξ N t=yξ R t |χ0 (t)|e−t = O(|y|N )kξ N +1 fˆkL1 sup . tN t∈R

∂¯z F = i

Since χ(t) ≡ 1 near t = 0 and fˆ ∈ S , the right hand side is bounded for any N . Thus |∂¯z F (x + iy)| ≤ CN |y|N for each N , and therefore (3.1.23) holds.



3.2. FOURIER TRANSFORM ON S 0 Next we extend the Fourier transform to S 0 , the dual space of S . We will then be able to study the Fourier transforms of various important, but nonsmooth, expressions. DEFINITIONS. (i) We write S 0 = S 0 (Rn ) for the space of tempered distributions, which is the dual of S . That is, u ∈ S 0 provided u : S → C is linear and ϕj → ϕ in S implies u(ϕj ) → u(ϕ). (ii) We say uj → u in S 0 if uj (ϕ) → u(ϕ)

for all ϕ ∈ S .

DEFINITION. If u ∈ S 0 , we define Dα u, xα u, Fu ∈ S 0

40

3. FOURIER TRANSFORM, STATIONARY PHASE

by the rules Dα u(ϕ) := (−1)|α| u(Dα ϕ) (xα u)(ϕ) := u(xα ϕ) (Fu)(ϕ) := u(Fϕ) for ϕ ∈ S . EXAMPLE 1: Dirac measure. It follows from the definitions that Z ˆ δ0 (ϕ) = δ0 (ϕ) ˆ = ϕ(0) ˆ = ϕ dx. Rn

We interpret this calculation as saying that δˆ0 ≡ 1.



EXAMPLE 2: Exponential of an imaginary quadratic form. DEFINITION. The signature of a real, symmetric, nonsingular matrix Q is sgn Q := number of positive eigenvalues of Q (3.2.1) − number of negative eigenvalues of Q. THEOREM 3.7 (Transform of an imaginary exponential). Let Q be a real, symmetric, nonsingular n × n matrix. Then  i  (2π)n/2 e iπ4 sgn(Q) i −1 hQx,xi (3.2.2) F e2 = e− 2 hQ ξ,ξi . |det Q|1/2 Compare this carefully with the earlier formula (3.1.4). The extra phase iπ shift term e 4 sgn Q in (3.2.2) arises from the complex exponential. Proof. 1. Let  > 0, Q := Q + iI. Then Z  i  i hQ x,xi 2 F e = e 2 hQ x,xi−ihx,ξi dx n ZR −1 −1 −1 i i = e 2 hQ (x−Q ξ),x−Q ξi e− 2 hQ ξ,ξi dx Rn Z −1 i i = e− 2 hQ ξ,ξi e 2 hQ y,yi dy. Rn

Now change variables, to write Q in the form diag(λ1 , . . . , λn ), with λ1 , . . . , λr > 0 and λr+1 , . . . , λn < 0. Then Z Z n Z ∞ Pn 1 Y i 1 2 hQ y,yi (iλk −)wk2 k=1 2 2 e dy = e dw = e 2 (iλk −)w dw. Rn

Rn

k=1 −∞

3.2. FOURIER TRANSFORM ON S 0

41

2. If 1 ≤ k ≤ r, then λk > 0 and we set z = ( − iλk )1/2 w, and we take the branch of the square root so that Im( − iλk )1/2 < 0. Then Z ∞ Z 1 1 2 1 2 (iλ −)w k e2 e− 2 z dz, dw = 1/2 ( − iλk ) Γk −∞ for the contour Γk as shown in Fig.1 Since 

1 exp − z 2 2




= exp (y 2 − x2 )/2 − ixy ,

and x2 > y 2 on Γk , we can deform Γk into the real axis. Im z

Im

z=

−

Re

z

< Γ k, λ k

0

Γk , λ k > 0

Im

z=

Re

Re z

z

Figure 1. The contours used in the proof of Theorem 3.7.

Hence Z

1 2

e− 2 z dz =

Z

∞

1 2

e− 2 x dx =

√

2π.

−∞

Γk

Thus r Z Y

∞

1

2

e 2 (iλk −)w dw = (2π)r/2

k=1 −∞

r Y

1 . ( − iλk )1/2 k=1

Also for 1 ≤ k ≤ r: iπ

lim

→0+

1 1 e4 = = , 1/2 1/2 ( − iλk )1/2 (−i)1/2 λk λk

since we take the branch of the square root with (−i)1/2 = e−iπ/4 .

42

3. FOURIER TRANSFORM, STATIONARY PHASE

3. Similarly for r + 1 ≤ k ≤ n, we set z = ( − iλk )1/2 w, but now take the branch of square root with Im( − iλk )1/2 > 0. Hence Z ∞ n n Y Y n−r 1 1 (iλk −)w2 2 2 dw = (2π) e ; ( − iλk )1/2 k=r+1 k=r+1 −∞ and for r + 1 ≤ k ≤ n iπ

lim

→0+

1 e− 4 1 = = , ( − iλk )1/2 (−iλk )1/2 |λk |1/2 iπ

since we take the branch of the square root with i1/2 = e 4 . 4. Combining the foregoing calculations gives us   i   i = lim F e 2 hQ x,xi F e 2 hQx,xi →0

iπ

i

−1 ξ,ξi

= e− 2 hQ =

(2π)n/2 e 4 (r−(n−r)) |λ1 λ2 . . . λn |1/2

sgn Q n/2 e iπ 4 − 2i hQ−1 ξ,ξi (2π) . e | det Q|1/2



3.3. SEMICLASSICAL FOURIER TRANSFORM DEFINITION. The semiclassical Fourier transform for h > 0 is Z i (3.3.1) Fh ϕ(ξ) := e− h hx,ξi ϕ(x) dx Rn

and its inverse is Fh−1 ψ(x)

(3.3.2)

1 := (2πh)n

Z

i

e h hx,ξi ψ(ξ) dξ.

Rn

Consequently (3.3.3)

δ{y=x}

1 = (2πh)n

Z

i

e h hx−y,ξi dξ

Rn

in S 0 .

This is a rescaled version of (3.1.11). We record for future reference some formulas involving the parameter h: THEOREM 3.8 (Properties of Fh ). We have (3.3.4)

(hDξ )α Fh ϕ = Fh ((−x)α ϕ);

(3.3.5)

Fh ((hDx )α ϕ) = ξ α Fh ϕ;

3.4. STATIONARY PHASE IN ONE DIMENSION

43

and kϕkL2 =

(3.3.6)

1 kFh ϕkL2 ; (2πh)n/2

We present next a scaled version of the uncertainty principle, which in its various guises limits the extent to which we can simultaneously localize our calculations in both the x and ξ variables. THEOREM 3.9 (Uncertainty principle). We have (3.3.7)

h kf kL2 kFh f kL2 ≤ kxj f kL2 kξj Fh f kL2 2

(j = 1, · · · , n).

Proof. To see this, note first that ξj Fh f (ξ) = Fh (hDxj f ). Also observe that [xj , hDxj ]f =

h [hxj , ∂xj f i − ∂xj (xj f )] = ihf. i

Thus kxj f kL2 kξj Fh f kL2

= kxj f kL2 kFh (hDxj f )kL2 = (2πh)n/2 kxj f kL2 khDxj f kL2 ≥ (2πh)n/2 |hhDxj f, xj f i| ≥ (2πh)n/2 | ImhhDxj f, xj f i| = = =

(2πh)n/2 |h[xj , hDxj ]f, f i| 2 (2πh)n/2 hkf k2L2 2 h kf kL2 kFh f kL2 . 2



3.4. STATIONARY PHASE IN ONE DIMENSION Understanding the right hand side of (3.3.1) in the limit h → 0 requires our studying integral expressions with rapidly oscillating integrands. We begin with the one dimensional case. DEFINITION. Given functions a ∈ Cc∞ (R), ϕ ∈ C ∞ (R), we define for h > 0 the oscillatory integral Z ∞ iϕ Ih = Ih (a, ϕ) := e h a dx. −∞

44

3. FOURIER TRANSFORM, STATIONARY PHASE

LEMMA 3.10 (Rapid decay). If ϕ0 6= 0 on K := spt(a), then Ih = O(h∞ )

(3.4.1)

as h → 0.

NOTATION. As explained in Appendix A, the identity (3.4.1) means that for each positive integer N , there exists a constant CN such that |Ih | ≤ CN hN

for all 0 < h ≤ 1.

Proof. We will integrate by parts N times. For this, observe that the operator h 1 L := ∂x i ϕ0 is defined on K, since ϕ0 6= 0 there. Notice also that  iϕ  iϕ L eh =eh. Hence LN (eiϕ/h ) = eiϕ/h , for N = 1, 2, . . . . Consequently Z ∞ Z ∞  iϕ  N iϕ/h ∗ N e h a dx = L e (L ) a dx , |Ih | = −∞

L∗

−∞

denoting the adjoint of L. Since a is smooth,   h a ∗ L a = − ∂x i ϕ0

is of size h. We deduce that |Ih | ≤ CN hN .



Suppose next that ϕ0 vanishes at some point within K := spt(a), in which case the oscillatory integral is no longer of order h∞ . We instead want to expand Ih in an asymptotic expansion in powers of h: THEOREM 3.11 (Stationary phase). Let a ∈ Cc∞ (R). Suppose that x0 ∈ K = spt(a) and ϕ0 (x0 ) = 0, ϕ00 (x0 ) 6= 0. Assume further that ϕ0 (x) 6= 0 on K − {x0 }. (i) There exist for each k = 0, 1, . . . differential operators A2k (x, D), of order less than or equal to 2k, such that for all N ! N −1 X 1 i A2k (x, D)a(x0 )hk+ 2 e h ϕ(x0 ) Ih − k=0 (3.4.2) X 1 ≤ CN hN + 2 sup |a(m) |, 0≤m≤2N +2 R

where CN depends also on the set K.

3.4. STATIONARY PHASE IN ONE DIMENSION

45

(i) In particular, iπ

A0 = (2π)1/2 |ϕ00 (x0 )|−1/2 e 4

(3.4.3)

sgn ϕ00 (x0 )

;

and consequently iπ

Ih = (2πh)1/2 |ϕ00 (x0 )|−1/2 e 4

(3.4.4)

sgn ϕ00 (x0 )

e

iϕ(x0 ) h

a(x0 ) + O(h3/2 )

as h → 0. NOTATION. We will sometimes write (3.4.2) in the less precise form i

Ih ∼ e h ϕ(x0 )

(3.4.5)

∞ X

1

A2k (x, D)a(x0 )hk+ 2 .

k=0

We present two proofs of this important theorem. The second proof is more complicated, but provides us with explicit expressions for the terms of the expansion (3.4.5), see (3.4.8). First proof of Theorem 3.11. 1. We may without loss assume x0 = 0, ϕ(0) = 0. Then ϕ(x) = 12 ψ(x)x2 , for Z 1 ψ(x) := 2 (1 − t)ϕ00 (tx) dt. 0

Notice that ψ(0) =

ϕ00 (0)

6= 0. We change variables by writing y := |ψ(x)|1/2 x

for x near 0. Thus ∂y x = |ϕ00 (0)|−1/2

at x = y = 0.

Now select a smooth function χ : R → R such that 0 ≤ χ ≤ 1, χ ≡ 1 near 0, and sgn ϕ00 (x) = sgn ϕ00 (0) 6= 0 on the support of χ. Then Lemma 3.10 implies Z ∞ Z ∞ iϕ(x)/h Ih = e χ(x)a(x) dx + eiϕ(x)/h (1 − χ(x))a(x) dx −∞ −∞ Z ∞ i 2 = e 2h y u(y) dy + O(h∞ ), −∞

for  :=

sgn ϕ00 (0)

= ±1, u(y) := χ(x(y))a(x(y))| det ∂y x|.

2. The Fourier transform formula (3.2.2) tells us that   2 iπ ihξ2 − iy 2h = (2πh)1/2 e− 4 e 2 . F e

46

3. FOURIER TRANSFORM, STATIONARY PHASE

Applying (3.1.16), we see that consequently  1/2 Z ∞ ihξ2 iπ h Ih = e 4 e− 2 u ˆ(ξ) dξ + O(h∞ ). 2π −∞ The advantage is that the small parameter h, and not h−1 , occurs in the exponential. 3. Next, write Z

∞

J(h, u) :=

e−

ihξ2 2

u ˆ(ξ) dξ, J(0, u) = 2πu(0).

−∞

Then Z

∞

∂h J(h, u) =

e

− ihξ 2

2

−∞



 ξ 2 u ˆ(ξ) dξ = J(h, P u) 2i

for P := (/2i) ∂ 2 . Continuing, we discover ∂hk J(h, u) = J(h, P k u). Therefore J(h, u) =

N −1 X k=0

hk hN J(0, P k u) + RN (h, u), k! N!

for the remainder term Z RN (h, u) := N

1

(1 − t)N −1 J(th, P N u) dt.

0

Thus Lemma 3.5 implies [ N uk 1 ≤ C |RN | ≤ CN kP N L

X

sup |∂ k (P N u)|.

0≤k≤2 R

4. Since the definition of J gives hk J(0, P k u) = h2 P k u(0) = (h/2i)k u(2k) (0) and since u = χ(x(y))a(x(y))| det ∂y x|, the expansion follows.



The second proof of stationary phase asymptotics will employ a quantitative version of Lemma 3.10: LEMMA 3.12 (More on rapid decay). Suppose that a ∈ Cc∞ (R) and that ϕ ∈ C ∞ (R). For each positive integer m, there exists a constant Cm depending also spt a such that Z ∞ X iϕ/h ≤ Cm hm (3.4.6) e a dx sup(|a(k) ||ϕ0 |k−2m ). −∞

0≤k≤m R

3.4. STATIONARY PHASE IN ONE DIMENSION

47

This inequality will be useful at points where ϕ0 is small, provided a(m) is also small. Proof. The proof is an induction on m, the case m = 0 being obvious. Assume the assertion for m − 1. Then Z ∞ Z h ∞  iϕ/h 0 a iϕ/h e a dx = e dx i −∞ ϕ0 −∞   Z Z h ∞ iϕ/h a 0 h ∞ iϕ/h =− e dx = − e a ˜ dx, i −∞ ϕ0 i −∞ for a ˜ := (a/ϕ0 )0 . Observe that X

|˜ a(k) | = |(a/ϕ0 )(k+1) | ≤ C

|a(j) ||ϕ0 |j−k−2 .

0≤j≤k+1

The induction hypothesis therefore implies Z ∞ Z ∞ iϕ(x)/h iϕ(x)/h e a dx ≤ h e a ˜ dx −∞ −∞ X sup(|˜ a(k) ||ϕ0 |k−2(m−1) ) ≤ hCm−1 hm−1 0≤k≤m−1 R

≤ C m hm

X

sup(|a(j) ||ϕ0 |j−2m ).



0≤j≤m R

Second proof of Theorem 3.11. 1. As before, we may assume x0 = 0, ϕ(0) = ϕ0 (0) = 0, ϕ00 (0) 6= 0. To find the expansion in h of our integral Z ∞ Ih = eiϕ/h a dx, −∞

we write ϕs (x) := ϕ00 (0)x2 /2 + sg(x) for 0 ≤ s ≤ 1, where g(x) := ϕ(x) − ϕ00 (0)x2 /2. Then ϕ = ϕ1 and g = O(x3 ) as x → 0. Furthermore, ϕ0s (x) = ϕ00 (0)x + O(x2 ), and therefore |x| ≤ |ϕ00 (0)|−1 |ϕ0s (x) + O(x2 )| ≤ 2|ϕ00 (0)|−1 |ϕ0 (x)|

48

3. FOURIER TRANSFORM, STATIONARY PHASE

for sufficiently small x. Consequently, using a cutoff function χ as in the first proof, we may assume that x (3.4.7) is bounded on K = spt(a). ϕ0s (x) 2.We also write

Z

∞

eiϕs /h a dx.

Ih (s) := −∞

Let us calculate d2m Ih (s) = (i/h)2m ds2m

Z

∞

eiϕs /h g 2m a dx.

−∞

Lemma 3.12, with 3m replacing m, implies X C (2m) |Ih (s)| ≤ 2m h3m sup(|(ag 2m )(k) ||ϕ0s |k−6m ). h R 0≤k≤3m

Now the amplitude ag 2m vanishes to order 6m at x = 0. Consequently, for each 0 ≤ k ≤ 3m we recall (3.4.7) to estimate |(ag 2m )(k) ||ϕ0s |k−6m ≤ C|x|6m−k |x|k−6m ≤ C. Therefore (2m)

|Ih

(s)| ≤ M hm .

It follows that Ih = Ih (1) =

2m−1 X

(l)

Ih (0)/l! +

l=0

=

2m−1 X

1 (2m − 1)!

Z 0

1

(2m)

(1 − s)2m−1 Ih

(s) ds

(l)

Ih (0)/l! + O(hm ).

l=0

3. It remains to compute the expansions in h of the terms Z ∞ (l) l Ih (0) = (i/h) eiϕ0 /h g l a dx −∞

for l = 0, . . . , 2m − 1. But this follows as in the first proof, since the phase ϕ0 (x) = ϕ00 (0)x2 /2 is purely quadratic. Up to constants, the terms in the expansion are 1 h 2 +k−l (g l a)(2k) (0) for l < 2m and k = 0, 1, · · · . This at first first looks discouraging because of −l in the power of h. Recall however that g = O(x3 ) near 0; so that (g l a)(2k) (0) = 0 unless 2k ≥ 3l. Also, if k − l = j, then 3j = 3k − 3l ≥ k, 2j = 2k − 2l ≥ l.

3.5. STATIONARY PHASE IN HIGHER DIMENSIONS

49

This means that there are at most finitely many values of k and l in the 1 1 expansion corresponding to the term h 2 +j = h 2 +k−l .  REMARK. This second proof avoids the Morse Lemma (see Theorem 3.15 below), but at some considerable technical expense. However this proof in fact provides the explicit expansion 1  Z 2 iπ 2πh iϕ/h sgn ϕ00 (x0 ) 4 e a dx ∼ e |ϕ00 (0)| R (3.4.8)  k ∞ X ∞ X h 1 1 d2k ((i/h)l g l a)(0). 2iϕ00 (0) l! k! dx2k k=0 l=0

More complicated but in principle explicit expansions can be obtained in higher dimensions as well. 

3.5. STATIONARY PHASE IN HIGHER DIMENSIONS We turn next to n-dimensional integrals. DEFINITION. We introduce now the oscillatory integral Z Ih = Ih (a, ϕ) = eiϕ/h a dx, Rn

where a ∈ Cc∞ (Rn ), ϕ ∈ C ∞ (Rn ) are real-valued. 3.5.1. Quadratic phase function. We begin with the case of a quadratic phase 1 ϕ(x) = hQx, xi, 2 where Q is a nonsingular, symmetric matrix. THEOREM 3.13 (Quadratic phase asymptotics). For each postive integer N , we have the expansion (3.5.1) Ih = iπ

(2πh)

n 2

e4

sgn Q

N −1 X

1

| det Q| 2

k=0

hk k!



hQ−1 D, Di 2i

k

! N

a(0) + O(h ) .

Proof. 1. The Fourier transform formulas (3.2.2) and (3.1.16) imply  n/2 iπ sgn Q Z ih h e4 −1 Ih = e− 2 hQ ξ,ξi a ˆ(ξ) dξ. 1 2π | det Q| 2 Rn

50

3. FOURIER TRANSFORM, STATIONARY PHASE

Write

Z

ih

−1 ξ,ξi

e− 2 hQ

J(h, a) :=

a ˆ(ξ) dξ;

Rn

then Z e

∂h J(h, a) =

− ih hQ−1 ξ,ξi 2

Rn

for

  i −1 − hQ ξ, ξiˆ a(ξ) dξ = J(h, P a) 2

i P := − hQ−1 D, Di. 2

Therefore J(h, a) =

N −1 X k=0

hN hk J(0, P k a) + RN (h, a), k! N!

for the remainder term Z RN (h, a) := N

1

(1 − t)N −1 J(th, P N a) dt.

0

2. Now (3.1.7) gives k Z  i −1 k J(0, P a) = − hQ ξ, ξi a ˆ(ξ) dξ = (2π)n P k a(0). 2 Rn Furthermore, Lemma 3.5,(ii) implies [ N ak 1 ≤ C |RN | ≤ CN kP N L

sup

|∂ α a|.



|α|≤2N +n+1

3.5.2. General phase function. Assume next that the phase ϕ is an arbitrary smooth function. LEMMA 3.14 (Rapid decay again). If ∂ϕ 6= 0 on K := spt(a), then Ih = O(h∞ ). In particular, for each positive integer N X (3.5.2) |Ih | ≤ ChN sup |∂ α a|, |α|≤N

Rn

where C depends upon only K and n. Proof. Define the operator L :=

h 1 h∂ϕ, ∂i i |∂ϕ|2

for x ∈ K, and observe that   L eiϕ/h = eiϕ/h .

3.5. STATIONARY PHASE IN HIGHER DIMENSIONS


Hence LN eiϕ/h = eiϕ/h , and consequently Z Z   iϕ/h ∗ N N iϕ/h e (L ) a dx ≤ ChN . L e a dx = |Ih | = n n

51



R

R

DEFINITION. We say ϕ : Rn → R has a nondegenerate critical point at x0 if ∂ϕ(x0 ) = 0, det ∂ 2 ϕ(x0 ) 6= 0. We also write sgn ∂ 2 ϕ(x0 ) := number of postive eigenvalues of ∂ 2 ϕ(x0 ) − number of negative eigenvalues of ∂ 2 ϕ(x0 ). Next we change variables locally to convert the phase function ϕ into a quadratic: THEOREM 3.15 (Morse Lemma). Let ϕ : Rn → R be smooth, with a nondegenerate critical point at x0 . Then there exist neighborhoods U of 0 and V of x0 and a diffeomorphism γ:V →U such that 1 (ϕ ◦ γ −1 )(x) = ϕ(x0 ) + (x21 + · · · + x2r − x2r+1 · · · − x2n ), 2 where r is the number of positive eigenvalues of ∂ 2 ϕ(x0 ). (3.5.3)

Proof. 1. As usual, we suppose x0 = 0, ϕ(0) = 0. After a linear change of variables, we have 1 ϕ(x) = (x21 + · · · + x2r − x2r+1 · · · − x2n ) + O(|x|3 ); 2 and so the problem is to design a further change of variables that removes the cubic and higher terms. 2. Now Z ϕ(x) = 0

1

1 (1 − t)∂t2 ϕ(tx) dt = hx, Q(x)xi, 2

where



 Ir O . O −In−r In this expression the upper identity matrix is r × r and the lower identity matrix is (n − r) × (n − r). We want to find a smooth mapping A from Rn to Mn×n such that Q(0) = ∂ 2 ϕ(0) =

(3.5.4)

hA(x)x, Q(0)A(x)xi = hx, Q(x)xi.

52

3. FOURIER TRANSFORM, STATIONARY PHASE

Then γ(x) = A(x)x is the desired change of variable. Formula (3.5.4) will hold provided AT (x)Q(0)A(x) = Q(x).

(3.5.5)

Let F : Mn×n → Sn×n be defined by F (A) = AT Q(0)A. We want to find a right inverse G : Sn×n → Mn×n , so that FG = I

near Q(0).

Then A(x) := G(Q(x)) will solve (3.5.5). 3. We will apply a version of the Inverse Function Theorem (Theorem C.2). To do so, it suffices to find B ∈ L(Sn×n , Mn×n ) such that ∂F (I)B = I. Now ∂F (I)(C) = C T Q(0) + Q(0)C. Define 1 B(D) := Q(0)−1 D 2 for D ∈ Sn×n . Then 1 ∂F (I)(Q−1 (0)D) 2 1 [(Q(0)−1 D)T Q(0) + Q(0)(Q(0)−1 D)] = 2 = D.

∂F (I)B(D) =



Given now a general phase function ϕ, we apply the Morse Lemma to convert locally to a quadratic phase for which the asymptotics provided by Theorem 3.13 apply: THEOREM 3.16 (Stationary phase asymptotics). Assume that a ∈ Cc∞ (Rn ). Suppose x0 ∈ K := spt(a) and ∂ϕ(x0 ) = 0, det ∂ 2 ϕ(x0 ) 6= 0. Assume further that ∂ϕ(x) 6= 0 on K − {x0 }.

3.5. STATIONARY PHASE IN HIGHER DIMENSIONS

53

(i) Then there exist for k = 0, 1, . . . differential operators A2k (x, D) of order less than or equal to 2k, such that for each N ! N −1 X iϕ(x ) n 0 A2k (x, D)a(x0 )hk+ 2 e h Ih − k=0 (3.5.6) X n sup |∂ α a|. ≤ C N hN + 2 |α|≤2N +n+1

Rn

(ii) In particular, iπ

A0 = (2π)n/2 |det∂ 2 ϕ(x0 )|−1/2 e 4

(3.5.7)

sgn ∂ 2 ϕ(x0 )

;

and therefore Ih = (3.5.8)

iπ

(2πh)n/2 |det∂ 2 ϕ(x0 )|−1/2 e 4

sgn ∂ 2 ϕ(x0 )

e

iϕ(x0 ) h

 n+2  a(x0 ) + O h 2

as h → 0. Proof. Without loss x0 = 0, ϕ(0) = ∂ϕ(0) = 0. Introducing a cutoff function χ and applying the Morse Lemma, Theorem 3.15, and then Lemma 3.14, we can write Z Z i Ih = eiϕ(x)/h a dx = e 2h hQx,xi u dx + O(h∞ ), Rn

Rn

where  Q=

Ir O O −In−r



and 1

u(x) := a(κ−1 (x))| det ∂κ−1 (x)|, | det ∂κ−1 (0)| = | det ∂ϕ(0)|− 2 . and  Q=

Ir O O −In−r



Note that sgn Q = sgn ∂ 2 ϕ(x0 ) and |detQ| = 1. We invoke Theorem 3.13 to finish the proof.  3.5.3. Important Examples. In Chapter 4 we will consider asymptotic behaviour of various expressions involving the Fourier transform. These involve the particular phase function ϕ(x, y) = hx, yi

54

3. FOURIER TRANSFORM, STATIONARY PHASE

on Rn × Rn , corresponding to the Euclidean inner product. We will also encounter important applications with the phase ϕ(z, w) = σ(z, w) = hJz, wi R2n ×R2n ,

on corresponding to the symplectic structure. We therefore record in this section the stationary phase expansions corresponding to these special cases. THEOREM 3.17 (Important phase functions). (i) Assume that a ∈ Cc∞ (R2n ). Then for each postive integer N , Z Z i (3.5.9) e h hx,yi a(x, y) dxdy = Rn

Rn

(2πh)n

N −1 X k=0

hk k!



hDx , Dy i i

k

! a(0, 0) + O(hN )

as h → 0. (ii) Assume that a ∈ Cc∞ (R4n ). Then for each postive integer N , Z Z i (3.5.10) e h σ(z,w) a(z, w) dzdw = R2n

R2n

(2πh)2n

N −1 X k=0

hk k!



σ(Dx , Dξ , Dy , Dη ) i

k

! a(0, 0) + O(hN ) ,

where z = (x, ξ), w = (y, η), and σ(Dx , Dξ , Dy , Dη ) := hDξ , Dy i − hDx , Dη i. Proof. 1. We write (x, y) to denote a typical point of R2n , and let   O I Q := . I O Then Q is symmetric, Q−1 = Q, |detQ| = 1, sgn(Q) = 0 and Q(x, y) = (y, x). Consequently 21 hQ(x, y), (x, y)i = hx, yi. Furthermore, since D = (Dx , Dy ), 1 −1 hQ D, Di = hDx , Dy i. 2 Hence Theorem 3.13 gives (3.5.9). 2. We write (z, w) to denote a typical point of R4n , where z = (x, ξ), w = (y, η). Set   O −J Q := . J O

3.6. NOTES

55

Then Q is symmetric, Q−1 = Q, |det Q| = 1, sgn(Q) = 0 and Q(z, w) = (−Jw, Jz). Consequently 1 hQ(z, w), (z, w)i = hJz, wi = σ(z, w). 2 We have D = (Dz , Dw ) = (Dx , Dξ , Dy , Dη ), and therefore 1 −1 hQ D, Di = σ(Dx , Dξ , Dy , Dη ). 2 Theorem 3.13 now provides us with the expansion (3.5.10).



3.6. NOTES Good references are Friedlander–Joshi [F-J] and H¨ormander [H1]. The PDE example in Section 3.1 is from [H1, Section 7.6], and the second proof of one-dimensional stationary phase is a variant of [H1, Section 7.7].

Chapter 4

SEMICLASSICAL QUANTIZATION

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Definitions Quantization formulas Composition, asymptotic expansions Symbol classes Operators on L2 Compactness Inverses, G˚ arding inequalities Notes

The Fourier transform and its inverse allow us to move at will between the position x and momentum ξ variables, but what we really want is to deal with both sets of variables simultaneously. This chapter therefore introduces the quantization of symbols, that is, of appropriate functions of both x and ξ. The resulting operators applied to functions entail information in the full (x, ξ) phase space, and particular choices of the symbol will later prove very useful, allowing us for example to “localize” in phase space. From the physical point of view the symbols should be thought of classical observables and the corresponding operators as quantum observables: recall Section 1.2. The plan is to introduce quantization and then to work out the resulting symbol calculus, meaning the systematic rules for manipulating symbols and their associated operators. We will also establish criteria for the L2 boundedness, compactness and approximate positivity of operators in terms of their symbols. 57

58

4. SEMICLASSICAL QUANTIZATION

4.1. DEFINITIONS NOTATION. For this section we take h > 0 and a ∈ S (R2n ), a = a(x, ξ). We hereafter call a a symbol. To quantize this symbol means to associate with it an h-dependent linear operator acting on functions u = u(x). There are several standard ways to do so: DEFINITIONS. (i) We define the Weyl quantization to be the operator aw (x, hD) acting on u ∈ S (Rn ) by the formula Z Z i 1 x+y
w hx−y,ξi h (4.1.1) a (x, hD)u(x) := e a 2 , ξ u(y) dydξ. (2πh)n Rn Rn (ii) We define also the standard quantization Z Z i 1 (4.1.2) a(x, hD)u(x) := e h hx−y,ξi a(x, ξ)u(y) dydξ n (2πh) Rn Rn for u ∈ S . (iii) More generally, for u ∈ S and 0 ≤ t ≤ 1, we set (4.1.3) Opt (a)u(x) := 1 (2πh)n

Z Rn

Z

i

e h hx−y,ξi a(tx + (1 − t)y, ξ)u(y) dydξ.

Rn

Hence (4.1.4)

Op 1 (a) = aw (x, hD), Op1 (a) = a(x, hD). 2

We hereafter refer to any operator of the form Opt (a) as a semiclassical pseudodifferential operator. REMARKS. (i) Observe that (4.1.5)

a(x, hD)u = Fh−1 (a(x, ·)Fh u(·)).

This simple expression makes most of the subsequent calculations much easier for the standard quantization, as opposed to the Weyl quantization. However the latter has many better properties and will be our principal concern. (ii) We will only rarely be directly interested in the operators Opt for t 6= 21 , 1; but they will prove useful for interpolating between the Weyl and standard quantizations. 

4.1. DEFINITIONS

59

ELEMENTARY EXAMPLES. (i) If a(x, ξ) = ξ α , then Opt (a)u = (hD)α u

(4.1.6) (ii) If a(x, ξ) = (4.1.7)

P

(0 ≤ t ≤ 1).

aα (x)ξ α , X aα (x)(hD)α u. a(x, hD) =

|α|≤N

|α|≤N

(iii) If a(x, ξ) = hx, ξi, then Opt (a)u = (1 − t)hhD, xui + thx, hDui

(0 ≤ t ≤ 1).

In particular, hx, hDiw u =

(4.1.8)

h h hD, xui + hx, Dui. 2 2

The formulas above follow straightforwardly from the definitions. We will work out many more explicit quantization formulas in the next section.  REMARK: Rescaling in h. It is often convenient to rescale to the case h = 1, by changing to the new variables 1

x ˜ := h− 2 x,

(4.1.9)

1

y˜ := h− 2 y,

1 ξ˜ := h− 2 ξ.

Then aw (x, hD)u(x) Z Z 1 x+y
hi hx−y,ξi = a u(y) dydξ 2 ,ξ e (2πh)n Rn Rn Z Z   1 ˜ x ˜+˜ y ˜ ˜ = a , ξ eih˜x−˜y,ξi u ˜(˜ y ) d˜ y dξ; h 2 n (2π) Rn Rn and therefore (4.1.10)

aw (x, hD)u(x) = aw x, D)˜ u(˜ x), h (˜

for (4.1.11)

1 1 ˜ := a(x, ξ) = a(h 12 x ˜ ˜), ah (˜ x, ξ) ˜, h 2 ξ). u ˜(˜ x) := u(x) = u(h 2 x

We call (4.1.9)–(4.1.11) the standard rescaling. We now see how the quantizations act on S and S 0 :



60

4. SEMICLASSICAL QUANTIZATION

THEOREM 4.1 (Schwartz class symbols). Assume a ∈ S . (i) Then for each 0 ≤ t ≤ 1, Opt (a) can be defined as an operator mapping S 0 to S ; and furthermore Opt (a) : S 0 → S is continuous. (ii) The formal adjoint is given by Opt (a)∗ = Op1−t (¯ a)

(4.1.12)

(0 ≤ t ≤ 1);

and in particular the Weyl quantization of a real symbol is formally selfadjoint: aw (x, hD)∗ = aw (x, hD)

(4.1.13)

if a is real.

We will later learn that for very general class of symbols a, aw (x, hD) is bounded on L2 , in which case aw (x, hD) self-adjoint provided a is real. Proof. (i) We have Z Opt (a)u(x) =

Kt (x, y)u(y) dy Rn

for the kernel Kt (x, y) := =

1 (2πh)n

Z

Rn −1 Fh (a(tx +

i

e h hx−y,ξi a(tx + (1 − t)y, ξ) dξ (1 − t)y, ·))(x − y).

Thus Kt ∈ S , and so Opt (a)u(x) = u(Kt (x, ·)) maps S 0 continuously into S. (ii) The kernel of Opt (a)∗ is Kt∗ (x, y) := K t (y, x) = K 1−t (x, y), which is the kernel of Op1−t (¯ a).  We next observe that the formulas (4.1.1)–(4.1.3) make sense if a is merely a distribution: THEOREM 4.2 (Distributional symbols). If a ∈ S 0 , then Opt (a) can be defined as an operator mapping S to S 0 ; and furthermore Opt (a) : S → S 0 is continuous.

(0 ≤ t ≤ 1)

4.2. QUANTIZATION FORMULAS

61

Proof. The formula for the distributional kernel Kt of Opt (a) given in the proof of Theorem 4.1 can be interpreted in the distributional sense if a ∈ S 0 . This shows that Kt ∈ S 0 (Rn × Rn ). Hence Opt (a) is well defined as an operator from S to S 0 . So if u, v ∈ S , then (Opt (a)u)(v) := Kt (u ⊗ v).



4.2. QUANTIZATION FORMULAS Exact computations for quantization can be carried out only in certain cases, but these are important. For future reference, we collect in this section various explicit calculations of Opt (a), and especially aw (x, hD). 4.2.1. Symbols depending only on x. A first simple, but not entirely trivial, case is when a does not depend upon ξ: THEOREM 4.3 (Quantizing symbols of x only). If a(x, ξ) = a(x), then (4.2.1)

Opt (a) = a

(0 ≤ t ≤ 1).

Proof. Let u ∈ S and compute the derivative Z Z i 1 ∂t Opt (a)u = e h hx−y,ξi h∂a(tx + (1 − t)y), x − yiu(y) dydξ n (2πh) Rn Rn  Z Z i h hx−y,ξi h ∂a(tx + (1 − t)y)u(y) dy dξ divξ e = i(2πh)n Rn Rn Z  i  h hx,ξi h e = div α ˆ (ξ) dξ ξ i(2πh)n Rn for α(y) := ∂a(tx + (1 − t)y)u(y). Since α ˆ (ξ) → 0 rapidly as |ξ| → ∞, the last expression vanishes. Consequently for all 0 ≤ t ≤ 1, Opt (a)u = Op1 (a)u = au.  4.2.2. Linear symbols. The formulas (4.1.6) and (4.2.1) immediately imply THEOREM 4.4 (Quantizing linear symbols). Let l be a linear symbol of the form l(x, ξ) := hx∗ , xi + hξ ∗ , ξi

(4.2.2) for (x∗ , ξ ∗ ) ∈ R2n . Then (4.2.3)

Opt (l) = hx∗ , xi + hξ ∗ , hDi

(0 ≤ t ≤ 1).

62

4. SEMICLASSICAL QUANTIZATION

NOTATION. In view of this result, we hereafter write l(x, hD) = lw (x, hD) = hx∗ , xi + hξ ∗ , hDi.

(4.2.4)

We can also compute explicitly the quantization of symbols linear in ξ, but nonlinear in x: THEOREM 4.5 (Symbols linear in ξ). Assume that c = (c1 (x), . . . , cn (x)) does not depend on ξ. Then n

hX hc, hDi = (Dxj cj + cj Dxj ). 2 w

(4.2.5)

j=1

The notation means (Dxj cj )u = Dxj (cj u). Proof. We calculate that Z n Z X i 1 )ξj e h hx−y,ξi u(y) dξdy cj ( x+y hc, hDi u = 2 n (2πh) n Rn j=1 R Z n Z   i X 1 x+y
hx−y,ξi h =− hD e u(y) dξdy c y j j 2 (2πh)n n Rn j=1 R Z Z n X
i hx−y,ξi h 1 (∂xj cj ) x+y = eh u(y) dξdy 2 n (2πh) 2i Rn Rn j=1 Z n Z X
i 1 + , ξ e h hx−y,ξi hDxj u(y) dξdy cj x+y 2 n (2πh) Rn Rn w

j=1

h = 2i =−

n X

n X (cj )w Dxj u (∂xj cj ) u + h w

j=1

j=1

ih 2

n X

n X

j=1

j=1

(∂xj cj )u + h

cj Dxj u,

according (4.2.1). Consequently, w

hc, hDi = h

n X

n

(cj Dxj

j=1

i hX − ∂xj cj ) = (Dxj cj + cj Dxj ). 2 2 j=1

EXAMPLE. The case c(x) = x gives n

hx, hDiw =

hX (Dxj xj + xj Dxj ), 2 j=1



4.2. QUANTIZATION FORMULAS

63

in agreement with our previous calculation (4.1.8).



4.2.3. Commutators. The Weyl quantizations of derivatives of a symbol can be characterized as appropriate commutators: THEOREM 4.6 (Commutators and derivatives). (Dxj a)w = [Dxj , aw ]

(4.2.6) and

h(Dξj a)w = −[xj , aw ]

(4.2.7) for j = 1, . . . , n.

Proof. We compute for u ∈ S that Z Z
i 1 Dxj a x+y , ξ e h hx−y,ξi u(y) dξdy (Dxj a)w u = 2 n (2πh) Rn Rn Z Z

i 1 = (Dxj + Dyj ) a x+y , ξ e h hx−y,ξi u(y) dξdy 2 n (2πh) Rn Rn Z Z

i 1 Dxj a x+y , ξ e h hx−y,ξi u(y) dξdy = 2 n (2πh) Rn Rn   Z Z 1 x+y
hi hx−y,ξi ξj + a 2 ,ξ e − Dyj u(y) dξdy (2πh)n Rn Rn h = Dxj (aw u) − aw (Dxj u) = [Dxj , aw ]u. Similarly, Z Z
i h h(Dξj a) u = Dξj a x+y , ξ e h hx−y,ξi u(y) dξdy 2 n (2πh) Rn Rn Z Z  i  1 x+y
hx−y,ξi h =− a 2 , ξ hDξj e u(y) dξdy (2πh)n Rn Rn Z Z
i 1 =− a x+y , ξ e h hx−y,ξi (xj − yj )u(y) dξdy 2 n (2πh) Rn Rn = −[xj , aw ]u. w



4.2.4. Exponentials of linear symbols. We will later need the Weyl quantization of complex exponentials of linear symbols: THEOREM 4.7 (Quantizing exponentials of linear symbols). (i) For each linear symbol l of the form (4.2.2) we have the identity  i w i (4.2.8) e h l (x, hD) = e h l(x,hD) ,

64

4. SEMICLASSICAL QUANTIZATION

where i

i

e h l(x,hD) u(x) := e h hx

(4.2.9)

∗ ,xi+ i hx∗ ,ξ ∗ i 2h

u(x + ξ ∗ ).

(ii) If l, m ∈ R2n , then i

i

i

i

e h l(x,hD) e h m(x,hD) = e 2h σ(l,m) e h (l+m)(x,hD) .

(4.2.10)

Proof. 1. Consider for u ∈ S the PDE ( ih∂t v + l(x, hD)v = 0 v(0) = u.

(t ∈ R)

Its unique solution is denoted it

v(x, t) = e h l(x,hD) u, it

this formula defining the operators e h l(x,hD) for t ∈ R. But we can check by a direct calculation using (4.2.4) that it

v(x, t) = e h hx

∗ ,xi+ it2 hx∗ ,ξ ∗ i 2h

u(x + tξ ∗ );

and therefore (4.2.9) holds. 2. Furthermore, Z Z il i 1 i) w hx−y,ξi hi (hξ ∗ ,ξi+hx∗ , x+y 2 h h (e ) u = e e u(y) dydξ (2πh)n Rn Rn i Z Z ∗  i ∗  i e 2h hx ,xi hx−y+ξ ∗ ,ξi hx ,yi h 2h = e e u(y) dydξ (2πh)n Rn Rn i Z Z ∗  i ∗  i e 2h hx ,xi hx−y,ξi hx ,y+ξ ∗ i ∗ h 2h e e u(y + ξ ) dydξ = (2πh)n Rn Rn i

= e h hx

∗ ,xi+ i hx∗ ,ξ ∗ i 2h

u(x + ξ ∗ ),

since

Z i 1 e h hx−y,ξi dξ δ{y=x} = n (2πh) Rn according to (3.3.3). This proves (4.2.8).

in S 0 ,

3. Suppose l(x, ξ) = hx∗1 , xi + hξ1∗ , ξi and m(x, ξ) = hx∗2 , xi + hξ2∗ , ξi. According to (4.2.9), i

i

∗

i

∗

i

∗

∗

e h m(x,hD) u(x) = e h hx2 ,xi+ 2h hx2 ,ξ2 i u(x + ξ2∗ ); and consequently i

i

e h l(x,hD) e h m(x,hD) u(x) = i

∗

i

∗

∗

∗

i

∗

∗

e h hx1 ,xi+ 2h hx1 ,ξ1 i e h hx2 ,x+ξ1 i+ 2h hx2 ,ξ2 i u(x + ξ1∗ + ξ2∗ ).

4.2. QUANTIZATION FORMULAS

65

Furthermore, (4.2.9) implies also that i

i

∗

i

∗

∗

∗

∗

∗

e h (l+m)(x,hD) u(x) = e h hx1 +x2 ,xi+ 2h hx1 +x2 ,ξ1 +ξ2 i u(x + ξ1∗ + ξ2∗ ). Using the formula above, we therefore compute i

i

∗

∗

∗

∗

i

i

e h (l+m)(x,hD) u(x) = e 2h (hx1 ,ξ2 i−hx2 ,ξ1 i) e h l(x,hD) e h m(x,hD) u(x). This confirms (4.2.10), since σ(l, m) = hξ1∗ , x∗2 i − hx∗1 , ξ2∗ i.



4.2.5. Exponentials of quadratic symbols. We next record some useful integral representation formulas for the quantization of certain quadratic exponentials: THEOREM 4.8 (Quantizing quadratic exponentials). (i) Let Q denote a nonsingular, symmetric, n × n matrix. Then 1 Z i ih | det Q|− 2 iπ sgn Q −1 hQD,Di 2 4 e− 2h hQ y,yi u(x + y) dy u(x) = e (4.2.11) e n (2πh) 2 Rn for u ∈ S (Rn ). (ii) In particular, if u ∈ S (R2n ), u = u(x, y), then (4.2.12) eihhDx ,Dy i u(x, y) = 1 (2πh)n

Z Rn

Z

i

e− h hx1 ,y1 i u(x + x1 , y + y1 ) dx1 dy1 .

Rn

(iii) Suppose that u ∈ S (R4n ), u = u(z, w). Then (4.2.13) eihσ(Dz ,Dw ) u(z, w) = Z Z i 1 e− h σ(z1 ,w1 ) u(z + z1 , w + w1 ) dz1 dw1 . (2πh)2n R2n R2n Proof. 1. Observe first that Theorem 3.7 gives Z i i i 1 e h hw,ξi e 2h hQξ,ξi dξ = Fh−1 (e 2h hQξ,ξi )(w) n (2πh) Rn 1

=

| det Q|− 2 iπ sgn Q − i hQ−1 w,wi e4 e 2h . n (2πh) 2

Therefore ih

i

e 2 hQD,Di u(x) = e 2h hQhD,hDi u(x) Z Z i i 1 = e h hx−y,ξi e h hQξ,ξi u(y) dydξ n (2πh) Rn Rn

66

4. SEMICLASSICAL QUANTIZATION

1

| det Q|− 2 iπ sgn Q = e4 n (2πh) 2 1

| det Q|− 2 iπ sgn Q e4 = n (2πh) 2

Z

i

−1 (x−y),x−yi

i

−1 y,yi

e− 2h hQ

u(y) dy

Rn

Z

e− 2h hQ

u(x + y) dy.

Rn

2. Assertion (4.2.12) is a special case of (4.2.11), had by replacing n by 2n and taking   O I Q := . I O See the proof of Theorem 3.17,(i). 3. Similarly, assertion (4.2.13) is a special case of (4.2.11) obtained by replacing n by 4n and taking   O −J Q := . J O See the proof of Theorem 3.17,(ii).



4.2.6. Conjugation by Fourier transform. THEOREM 4.9 (Conjugation and Fourier transform). We have Fh−1 aw (x, hD)Fh = aw (hD, −x).

(4.2.14)

Note that a ˜(x, ξ) := a(ξ, −x) is the pull-back of a under the symplectic transformation J. In Section 11.3 we will generalize this insight and in particular interpret (4.2.14) as saying that the semiclassical Fourier transform Fh quantizes J. Proof. We observe that the Schwartz kernel of Fh−1 aw Fh is Kh (x, y) = 1 (2πh)2n

Z Rn

Z

Z

Rn

i

Rn

0

0

0

0

0

0

0 0 e h (hx ,xi+hx −y ,ζi−hy ,yi) a( x +y 2 , ζ) dy dx dζ. 0

0

The change of variables x0 = x0 , z = x +y shows that 2 Z Z Z i 1 1 0 Kh (x, y) = e h Φ(x ,z,ζ,y,x) a(z, ζ) dx0 dzdζ, 2n n (2πh) 2 Rn Rn Rn where Φ(x0 , z, ζ, y, x) := 2 hx0 , ζ +

x+y 2 i


− hz, y + ζi .

We note that 1 (2πh)n

Z Rn

2i

0

e h hx ,ζ+

x+y 2 i dx0

= 2n δ(ζ +

x+y 2 ).

4.3. COMPOSITION, ASYMPTOTIC EXPANSIONS

67

Hence Z i 1 e h hx−y,zi a(z, −( x+y 2 )))dz, n (2πh) Rn the Schwartz kernel of a ˜w (x, hD) for a ˜(x, ξ) := a(ξ, −x). Kh (x, y) =



4.3. COMPOSITION, ASYMPTOTIC EXPANSIONS We now commence a careful study of the properties of the quantized operators defined In Section 4.1, especially the Weyl quantization. 4.3.1. Composing symbols. We next establish the fundamental formula aw bw = (a#b)w , along with a recipe for computing the new symbol a#b. The plan will be to represent the Weyl quantization of a general symbol in terms of the quantizations of complex exponentials of linear symbols. Remember that linear symbols have the form l(x, ξ) := hx∗ , xi + hξ ∗ , ξi for (x∗ , ξ ∗ ) ∈ R2n . To simplify calculations, we will sometimes identify this linear symbol l with the point (x∗ , ξ ∗ ). LEMMA 4.10 (Fourier decomposition of aw ). (i) Define Z a ˆ(l) :=

i

e− h l(x,ξ) a(x, ξ) dxdξ

R2n

for a ∈ S and l ∈ R2n . Then (4.3.1)

1 a (x, hD) = (2πh)2n w

Z

i

a ˆ(l)e h l(x,hD) dl.

R2n

(ii) If a ∈ S 0 , then the decomposition formula (4.3.1) holds in the sense i of tempered distributions. This means that if u, v ∈ S , then he h l(x,hD) u, vi ∈ S as a function of l = (x∗ , ξ ∗ ) ∈ R2n and D E i 1 l(x,hD) h (4.3.2) haw (x, hD)u, vi = u, vi a ˆ (l), he . (2πh)2n Proof. 1. For a ∈ S , the Fourier inversion formula implies Z i 1 e h l(x,ξ) a ˆ(l) dl; (4.3.3) a(x, ξ) = 2n (2πh) R2n and therefore (4.3.1) follows from Theorem 4.7.

68

4. SEMICLASSICAL QUANTIZATION

2. To see the validity of (4.3.2) for a ∈ S 0 , we only need to check that Z Z Z i i l(x,hD) he h e h (l(x+y/2,ξ)+hx−y,ξi) u(y)v(x) dydξdx u, vi = Rn

Rn

Rn

lies in S as a function of l. We leave the verification as an exercise.



Now we show for the Weyl quantization that the product of two pseudodifferential operators is a pseudodifferential operator. THEOREM 4.11 (Composition for Weyl quantization). (i) Suppose that a, b ∈ S . Then aw (x, hD)bw (x, hD) = (a#b)w (x, hD)

(4.3.4) for the symbol


a#b(x, ξ) := eihA(D) a(x, ξ)b(y, η) y=x ,

(4.3.5)

η=ξ

where 1 A(D) := σ(Dx , Dξ , Dy , Dη ). 2

(4.3.6)

(ii) We have the integral representation formula a#b(x, ξ) = (4.3.7)

1 (πh)2n

Z

Z

R2n

2i

e− h σ(w1 ,w2 ) a(z + w1 )b(z + w2 ) dw1 dw2 ,

R2n

where z = (x, ξ). Proof. 1. We have the representation formula (4.3.1) and likewise Z 1 w ˆb(m)e hi m(x,hD) dm. b (x, hD) = 2n (2πh) R2n Theorem 4.7,(ii) lets us next compute aw (x, hD)bw (x, hD) Z Z i i 1 a ˆ(l)ˆb(m)e h l(x,hD) e h m(x,hD) dm dl 4n (2πh) 2n 2n ZR ZR i i 1 = a ˆ(l)ˆb(m)e 2h σ(l,m) e h (l+m)(x,hD) dm dl 4n (2πh) 2n R2n ZR i 1 = cˆ(r)e h r(x,hD) dr 2n (2πh) R2n

=

for (4.3.8)

cˆ1 (r) :=

1 (2πh)2n

Z {l+m=r}

a ˆ(l)ˆb(m)e

iσ(l,m) 2h

dl.

4.3. COMPOSITION, ASYMPTOTIC EXPANSIONS

69

To get this, we changed variables by setting r = m + l. 2. We will show that c1 = c, where cˆ1 defined by (4.3.8), and c is defined by the right hand side of (4.3.5). We first simplify notation by writing z = (x, ξ), w = (y, η). Then i

ih

c(z) = e 2 σ(Dz ,Dw ) a(z)b(w)|w=z = e 2h σ(hDz ,hDw ) a(z)b(w)|w=z and Z i 1 e h l(z) a ˆ(l) dl, (2πh)2n R2n Z i 1 e h m(w)ˆb(m) dm. 2n (2πh) R2n

a(z) = b(w) =

Furthermore, since l(z) = hl, zi and m(w) = hm, wi we have i

i

i

i

e 2h σ(hDz ,hDw ) e h (l(z)+m(w)) = e h (l(z)+m(w))+ 2h σ(l,m) . Consequently Z Z i i 1 σ(hD ,hD ) (l(z)+m(w)) z w a ˆ(l)ˆb(m) dldm c(z) = e 2h eh 4n (2πh) R2n R2n z=w Z Z i i 1 = e h (l(z)+m(z))+ 2h σ(l,m) a ˆ(l)ˆb(m) dldm. 4n (2πh) R2n R2n The semiclassical Fourier transform of c is therefore   Z Z Z i i 1 1 (l+m−r)(z) eh dz e 2h σ(l,m) a ˆ(l)ˆb(m) dldm. 2n 2n (2πh) (2πh) 2n 2n n R R R According to (3.3.3), the term inside the parentheses is δ{l+m=r} in S 0 . Thus the foregoing equals Z i 1 e 2h σ(l,m) a ˆ(l)ˆb(m) dl = cˆ1 (r), 2n (2πh) {l+m=r} in view of (4.3.8).

h.

3. Formula (4.3.7) follows from Theorem 4.8,(iii), with h/2 replacing 

4.3.2. Asymptotics. We next apply stationary phase to derive a useful asymptotic expansion of a#b. Remember the definition (4.3.6) of the operator A(D). THEOREM 4.12 (Semiclassical expansions). Assume a, b ∈ S .

70

4. SEMICLASSICAL QUANTIZATION

(i) We have for N = 0, 1, . . . , (4.3.9)

a#b(x, ξ) =

A(D) (a(x, ξ)b(y, η)) + OS (hN +1 ) y=x k!

N k k X i h k=0

k

η=ξ

as h → 0. (ii) In particular, (4.3.10)

a#b = ab +

h {a, b} + OS (h2 ); 2i

and [aw (x, hD), bw (x, hD)] =

(4.3.11)

h {a, b}w + OS (h2 ). i

(iii) If spt(a) ∩ spt(b) = ∅, then a#b = OS (h∞ ).

(4.3.12)

REMARKS. (i) The notation ϕ = OS (hk ) means that for all multiindices α, β |ϕ|α,β := sup |xα ∂ β ϕ| ≤ Cα,β hk Rn

as h → 0. (ii) The important formula (4.3.11) shows that the commutator of two pseudodifferential operators is of order h.  Proof. 1. To prove (4.3.9), we apply the stationary phase Theorem 3.17,(ii), with h/2 replacing h and −σ replacing σ, to the integral formula (4.3.7). 2. Next, compute a#b = =

ab + ihA(D)(a(x, ξ)b(y, η))|y=x + O(h2 ) η=ξ ih ab + (hDξ a, Dy bi − hDx a, Dη bi) + O(h2 ) y=x 2 η=ξ

h = ab + (h∂ξ a, ∂x bi − h∂x a, ∂ξ bi) + O(h2 ) 2i h = ab + {a, b} + O(h2 ). 2i Consequently, [aw , bw ] = aw bw − bw aw = (a#b − b#a)w    w h h 2 = ab + {a, b} − ba + {b, a} + O(h ) 2i 2i

4.3. COMPOSITION, ASYMPTOTIC EXPANSIONS

71

h {a, b}w + O(h2 ). i

=

3. If spt(a) ∩ spt(b) = ∅, each term in the expansion (4.3.9) vanishes.



EXAMPLE: Symbols linear in ξ. Let a = cj (x) and b = ξj . Then aw bw = (a#b)w = (ab)w +

h {a, b}w , 2i

since Dα b = 0 for |α| ≥ 2. Summing j = 1, . . . , n, we see that n n X i hX hc, hDi = h (cj Dxj − ∂xj cj ) = (Dxj cj + cj Dxj ), 2 2 w

j=1

j=1

where c = (c1 , . . . , cn ). This agrees with our previous calculation (4.2.5).  4.3.3. Transforming between different quantizations. We record an interesting conversion formula: THEOREM 4.13 (Changing quantizations). If A = Opt (at )

(0 ≤ t ≤ 1),

then at (x, ξ) = ei(t−s)hhDx ,Dξ i as (x, ξ).

(4.3.13)

Proof. The decomposition formula (4.3.1) implies Z i 1 a ˆt (l)Opt (e h l ) dl. Opt (at ) = 2n (2πh) R2n Denoting the Fourier transform used there by Fh , we have   i ∗ ∗ Fh ei(t−s)hhDx ,Dξ i as (x, ξ) (l) = e h (t−s)hx ,ξ i Fh as (l); and as before we identify l = (x∗ , ξ ∗ ) ∈ R2n with the linear function l(x, ξ) = hx∗ , xi + hξ ∗ , ξi. The theorem is a consequence of the identity  i   i  i ∗ ∗ Opt e h l(x,ξ) = e h (s−t)hx ,ξ i Ops e h l(x,ξ) , which can be checked by calculations similar to those in the proof of Theorem 4.7:   i

i

Opt e h l(x,ξ) u(x) = e h hx,x

∗ i+ i (1−t)hx∗ ,ξ ∗ i h

u(x + ξ ∗ ).



72

4. SEMICLASSICAL QUANTIZATION

4.3.4. Standard quantization. Next we replace Weyl (t = 12 ) by standard (t = 1) quantization in our formulas. The proofs are simpler. THEOREM 4.14 (Formulas for standard quantization). (i) Let a, b ∈ S . Then a(x, hD)b(x, hD) = c(x, hD) for the symbol c(x, ξ) = eihhDξ ,Dy i (a(x, ξ)b(y, η)) y=x .

(4.3.14)

η=ξ

(ii) We have the integral representation formula (4.3.15) c(x, ξ) := 1 (2πh)n

Z

Z

Rn

i

e− h hx1 ,ξ1 i a(x, ξ + ξ1 )b(x + x1 , ξ) dx1 dξ1 .

Rn

(iii) For each N = 0, 1, . . . , (4.3.16)

c(x, ξ) =

N X hk k=0

k!

(ihDξ , Dy i)k (a(x, ξ)b(y, η)) y=x + OS (hN +1 ) η=ξ

as h → 0. (iv) If a ∈ S , then

a(x, hD)∗ = b(x, hD),

for b(x, ξ) := eihhDx ,Dξ i a ¯(x, ξ).

(4.3.17)

Proof. 1. Let u ∈ S . Then a(x, hD)b(x, hD)u(x) Z Z Z i 1 e h (hx,ηi+hy,ξ−ηi) a(x, η)b(y, ξ)ˆ u(ξ) dydηdξ = 2n (2πh) Rn Rn Rn Z i 1 = c(x, ξ)e h hx,ξi u ˆ(ξ) dξ, n (2πh) Rn = c(x, hD)u(x) for

Z Z i 1 e− h hx−y,ξ−ηi a(x, η)b(y, ξ) dydη. (2πh)n Rn Rn Change variables by putting x1 = y − x, ξ1 = η − ξ, to rewrite c in the form (4.3.15). Then (4.3.14) is a consequence of Theorem 4.8,(ii). Finally, c(x, ξ) =

4.4. SYMBOL CLASSES

73

the stationary phase Theorem 3.17, (i) provides the asymptotic expansion (4.3.16)  2. We recall from (4.1.12) that a(x, hD)∗ = Op1 (a)∗ = Op0 (¯ a). Now invoke (4.3.13), to write Op0 (¯ a) = Op1 (b), the symbol b defined by (4.3.17). 

4.4. SYMBOL CLASSES We next extend our calculus to symbols which can depend on the parameter h and which can have varied behavior, in terms of growth and decay, as (x, ξ) → ∞. 4.4.1. Order functions and symbol classes. DEFINITION. A measurable function m : R2n → (0, ∞) is called an order function if there exist constants C, N such that m(w) ≤ Chz − wiN m(z)

(4.4.1) for all w, z ∈ R2n .

EXAMPLES. Standard examples are m(z) ≡ 1, m(z) = hzi = (1 + |z|2 )1/2 . We also check that for any a, b ∈ R m(z) = hxia hξib are order functions, where z = (x, ξ). Observe also that if m1 , m2 are order functions, so is m1 m2 .  DEFINITIONS. (i) Given an order function m on R2n , we define the corresponding class of symbols: (4.4.2)

S(m) := {a ∈ C ∞ | for each multiindex α there exists a constant Cα so that |∂ α a| ≤ Cα m}.

(ii) We as well define (4.4.3)

Sδ (m) := {a ∈ C ∞ | |∂ α a| ≤ Cα h−δ|α| m for all multiindices α}.

REMARKS. (i) Symbols a = a(x, ξ) in S(m) are allowed to depend upon h, although this dependence is usually not reflected in our notation. Symbols in Sδ (m) depend on h, although again our notation will mostly not show this explicitly.

74

4. SEMICLASSICAL QUANTIZATION

If a ∈ S(m) or Sδ (m) depends on h, we require that the constants Cα in the definitions be uniform for 0 < h ≤ h0 for some number h0 > 0. (ii) The spaces Sδ (m) will appear naturally in later applications, for example the sharp G˚ arding inequality (Section 4.6) and the Ehrenfest time theorem (Section 11.4). The index δ > 0 allows for increasing singularity of the higher derivatives.  NOTATION. If the order function is the constant function m ≡ 1, we will usually not write it. Thus S := S(1), Sδ := Sδ (1); that is, (4.4.4)

S = {a ∈ C ∞ (R2n ) | |∂ α a| ≤ Cα for all α},

(4.4.5)

Sδ = {a ∈ C ∞ (R2n ) | |∂ α a| ≤ Cα h−δ|α| for all α}.

REMARK: Critical and subcritical values of δ. Note that if a ∈ Sδ , then |α| 1 (4.4.6) |∂ α a | = h 2 |∂ α a| ≤ C h|α|( 2 −δ) α

h

for each multiindex α, where ah is given by the standard rescaling (4.1.11). If δ > 21 , the last term is unbounded as h → 0; and consequently we will henceforth always assume 0 ≤ δ ≤ 21 . We see also that the case δ = 12 is critical, in that we do not then get decay as h → 0 for the terms on the right hand side of (4.4.6) when |α| > 0.  4.4.2. Asymptotic series. Next we consider infinite sums of terms in various symbol classes. DEFINITION. P∞Let jaj ∈ Sδ (m) for j = 0, 1, . . . . We say that a ∈ Sδ (m) is asymptotic to j=0 h aj , and write (4.4.7)

a∼

∞ X

hj aj

in Sδ (m),

j=0

provided for each N = 1, 2, . . . (4.4.8)

a−

N −1 X

hj aj = OSδ (m) (hN ).

j=0

REMARKS. (i) The notation (4.4.8) means   PN −1 α ∂ a − j=0 hj aj ≤ Cα,N hN −δ|α| m for all multiindices α.

4.4. SYMBOL CLASSES

75

P j (ii) Observe that for each h > 0, the formal series ∞ j=0 h aj need not converge in any sense. We are requiring rather in (4.4.8) that for each N , P −1 j the difference a− N j=0 h aj , and its derivatives, vanish at appropriate rates as h → 0.

a.

(iii) If the expansion (4.4.7) holds, we call a0 the principal symbol of 

Perhaps surprisingly, we can always construct such an asymptotic sum of symbols: THEOREM 4.15 (Borel’s Theorem). (i) Assume aj ∈ Sδ (m) for j = 0, 1, . . . . Then there exists a symbol a ∈ Sδ (m) such that ∞ X a∼ hj aj in Sδ (m). j=0

(ii) If also a ˆ∼

P∞

j j=0 h aj ,

then

a−a ˆ = OS(m) (h∞ ). Proof. 1. Choose a C ∞ function χ such that 0 ≤ χ ≤ 1, χ ≡ 1 on [0, 1] and χ ≡ 0 on [2, ∞). Define (4.4.9)

a :=

∞ X

hj χ(λj h)aj ,

j=0

where the sequence λj → ∞ must be selected. Since λj → ∞, there are for each h > 0 at most finitely many nonzero terms in the sum (4.4.9). 2. Now for each multiindex α, with |α| ≤ j, we have

(4.4.10)

hj χ(λj h)|∂ α aj | ≤ Cj,α hj−δ|α| χ(λj h)m λj h = Cj,α hj−δ|α| χ(λj h) m λj h ≤ 2Cj,α

hj−1−δ|α| m λj

≤ hj−1−δ|α| 2−j m if λj is selected sufficiently large. We can accomplish this for all j and multiindices α with |α| ≤ j. We may assume also λj+1 ≥ λj , for all j.

76

4. SEMICLASSICAL QUANTIZATION

3. We have a−

N X

∞ X

hj aj =

j=0

hj aj χ(λj h) +

N X

hj aj (χ(λj h) − 1).

j=0

j=N +1

Fix any multiindex α. Then taking N ≥ |α|, we have ∞ N   X X PN α j α j h |(∂ aj )|χ(λj h) + hj |∂ α aj |(1 − χ(λj h)) ∂ a − j=0 h aj ≤ j=0

j=N +1

=: A + B. According to estimate (4.4.10), A≤

∞ X

hj−1−δ|α| 2−j m ≤ hN −δ|α| m.

j=N +1

Also B≤

N X

Cα,j hj−δ|α| m(1 − χ(λj h)).

j=0 −1 Since χ ≡ 1 on [0, 1], B = 0 if 0 < h ≤ λ−1 N . If λN ≤ h ≤ 1, we have 1 ≤ λN h and hence

B≤m

N X j=0

Cα,j h−δ|α| ≤ m

N X

N −δ|α| eα,N hN −δ|α| . Cα,j λN = mC Nh

j=0

Thus   PN α ∂ a − j=0 hj aj ≤ Cα,N hN −δ|α| m if N ≥ |α|. Therefore, for any N   PN −1 α ∂ a − j=0 hj aj ≤ Cα,N hN −δ|α| m.



4.4.3. Quantization. Next we discuss the Weyl quantization of symbols in the class Sδ (m). The mapping properties in the next theorem concern a fixed value of h and the main point is their validity for general order functions m. THEOREM 4.16 (Quantizing general symbols). If a ∈ Sδ (m), then (4.4.11)

aw (x, hD) : S → S

and (4.4.12)

aw (x, hD) : S 0 → S 0

are continuous linear transformations.

4.4. SYMBOL CLASSES

77

Proof. 1. We take h = 1 for simplicity; so that Z Z 1 eihx−y,ξi a aw (x, D)u(x) = (2π)n Rn Rn

x+y
2 , ξ u(y) dydξ

for u ∈ S . Observe next that L1 eihx−y,ξi = eihx−y,ξi , where L1 :=

1 + hx − y, Dξ i ; 1 + |x − y|2

and L2 eihx−y,ξi = eihx−y,ξi for L2 :=

1 − hξ, Dy i . 1 + |ξ|2

We employ these operators and an integration by parts argument, to show aw (x, D) : S → L∞ . Furthermore, 1 xj a (x, D)u = (2π)n w

Z Rn

Z Rn

(Dξj + yj )eihx−y,ξi a

x+y
2 , ξ u(y) dydξ.

We can again integrate by parts, to conclude that xα aw (x, D) : S → L∞ for each multinomial xα . 2. Using the Fourier conjugation formula (4.2.14) we see that Dβ aw (x, D) = F −1 h(ξ β aw (−D, ξ)i)F. Now Step 1 implies hxin+1 xβ aw (x, D) : S → L∞ for all β. Hence ∂ β aw (x, D) : S → F −1 (hξi−n−1 L∞ ) ⊂ L∞ , according to Lemma 3.5. We similarly show that xα ∂ β aw (x, D) : S → L∞ for all multiindices α, β. This proves (4.4.11). The continuity statement easily follows from similar arguments: if all seminorms of uj ∈ S tend to 0 with j, so do the seminorms of aw (x, D)uj . 3. To establish (4.4.12), we note that if u, v ∈ S we have the distributional pairing (aw (x, D)u) (v) = u (˜ aw (x, D)v) , where a ˜(x, ξ) = a(x, −ξ) ∈ S(m). According to (4.4.11) we have a ˜w (x, D)v ∈ S , and this means that aw (x, D)u is well defined for u ∈ S 0 . The continuity of aw on S 0 follows from the continuity of (4.4.11) and the definition of the topology on S 0 in Section 3.2. 

78

4. SEMICLASSICAL QUANTIZATION

4.4.4. Semiclassical expansions in Sδ . Next we need to reexamine some of our earlier asymptotic expansions, deriving improved estimates on the error terms. The following theorem will let us translate results derived for a ∈ S in Section 4.2 into assertions for a ∈ Sδ (m). THEOREM 4.17 (Semiclassical expansions in Sδ .). Let Q be symmetric, nonsingular matrix. (i) If 0 ≤ δ ≤ 12 , then ih

e 2 hQD,Di : Sδ (m) → Sδ (m). (ii) If 0 ≤ δ < expansion (4.4.13)

1 2,

e

we furthermore have for each symbol a ∈ Sδ (m) the

ih hQD,Di 2

  ∞ X hk hQD, Di k a∼ i a k! 2

in Sδ (m).

k=0

Proof. 1. First, let 0 ≤ δ < that

1 2

and a ∈ Sδ (m). Recall from Theorem 4.8,(i) 1

e

ih hQD,Di 2

| det Q|− 2 iπ sgn Q a(z) = e4 (2πh)n

Z

i

e h ϕ(w) a(z + w) dw

R2n

for the quadratic phase 1 ϕ(w) := − hQ−1 w, wi. 2 Let χ : Rn → R be a smooth function with χ ≡ 1 on B(0, 1), χ ≡ 0 on Rn \ B(0, 2). Then Z iϕ(w) ih C e 2 hQD,Di a(z) = e h a(z − w) dw n h R2n Z iϕ(w) C = e h χ(w)a(z − w) dw n h R2n Z iϕ(w) C + n e h (1 − χ(w))a(z − w) dw h R2n =: A + B, for the constant 1

| det Q|− 2 iπ sgn Q C := e4 . (2π)n

4.4. SYMBOL CLASSES

79

2. Estimate of A. Since χ(w)a(z − w) has compact support, the method of stationary phase, Theorem 4.8, gives  k ∞ X hk i A∼ hQD, Di a(z). k! 2 k=0

Furthermore, if |w| ≤ 2, we have m(z − w) ≤ Cm(z). The remainder estimate in (3.5.6) and the expansion above show that |∂ α A(z)| ≤ C0 h−|α| m(z) + C1 hN

sup

|∂ α+β a(z + w)|

0≤β≤N +n+1 |w|≤2

≤ C2 h−|α|δ m(z). Hence A ∈ Sδ (m). 3. Estimate of B. Let L :=

h∂ϕ, hDi ; |∂ϕ|2

then Leiϕ/h = eiϕ/h Furthermore, since |∂ϕ(w)| ≥ γ|w| for some positive constant γ, the operator L has smooth coefficients on the support of 1 − χ and M ∗ M (L ) ((1 − χ)a) ≤ CM h sup |∂ α a(z − w)| hwiM |α|≤M Consequently, Z   M iϕ/h |B| = (1 − χ(w))a(z − w) dw 2n L e ZR iϕ/h ∗ M = (L ) ((1 − χ)a) dw 2n e R Z hwi−M sup |∂ α a(z − w)| dw ≤ ChM −n C hn C hn

R2n

≤ ChM −n−δM

|α|≤M

Z

hwiN −M m(z) dw

R2n

= ChM −n−δM m(z), provided M > 2n + N . The number N is from the definition (4.4.1) of the order function m. We similarly check also the higher derivatives, to conclude that B = OSδ (m) (h∞ ). 4. Now assume δ = 1/2. In this case we can rescale, by setting w ˜ = wh−1/2 .

80

4. SEMICLASSICAL QUANTIZATION

Then e

i hhQD,Di 2

Z a(z) = C

˜ eiϕ(w) a(z − wh ˜ 1/2 ) dw. ˜

Rn

We use χ = χ(w) ˜ to break the integral into two pieces A and B, as above. Then |∂ α A| ≤ C sup |∂ α a(z + h1/2 w)| ˜ ≤ h−|α|/2 m. |w|≤2 ˜

Furthermore |∂ α B| ≤ Cα h−|α|/2 m. for each k and α. We leave the verification to the reader.



REMARKS. (i) Observe that since we can always rescale to the case h = 1, there cannot exist an expansion like (4.4.13) for δ = 1/2 . (ii) Theorem 4.17 is interesting and nontrivial for h = 1. It then states that if |∂ α a| ≤ Cα m for an order function m and all multiindices α, then   ∂ α eihQD,Di/2 a = O(m), for all α.



Recall from Theorem 4.11 that for a, b ∈ S , aw (x, hD)bw (x, hD) = (a#b)w (x, hD), where a#b is defined by (4.3.5). THEOREM 4.18 (Symbol class of a#b). (i) If a ∈ Sδ (m1 ) and b ∈ Sδ (m2 ), then a#b ∈ Sδ (m1 m2 ),

(4.4.14) and (4.4.15)

aw (x, hD)bw (x, hD) = (a#b)w (x, hD)

as operators mapping S to S . (ii) Furthermore, (4.4.16)

a#b − ab ∈ OSδ (m1 m2 ) (h1−2δ ).

Proof. 1. Clearly c(z, w) := a(z)b(w) ∈ Sδ (m1 (z)m2 (w)) in R4n . If we put D = (Dx , Dξ , Dy , Dη ) and hQD, Di = σ(Dx , Dξ ; Dy , Dη ) for z = (x, ξ) and w = (y, η), then Theorem 4.17 implies ih

e 2 hQD,Di c ∈ Sδ (m1 (z)m2 (w)).

4.4. SYMBOL CLASSES

81

Then (4.4.14) and (4.4.15) follow, since (4.3.5) and (4.3.6) say ih

a#b(z) = eihA(D) c(z, z) = e 2 hQD,Di c(z, z). The second statement of assertion (i) follows from the density of S in Sδ (m). 2. We leave the verification of (4.4.16) as an exercise.



4.4.5. More useful formulas. We describe next how to obtain the symbol from the operator, in the particularly nice case of the standard quantization. This is called oscillatory testing. THEOREM 4.19 (Constructing the symbol from the operator). Suppose a ∈ Sδ (m). Then (4.4.17)

i

i

a(x, ξ) = e h hx,ξi a(x, D)(e h h·,ξi ).

Proof. For a ∈ S we verify this formula using the inverse Fourier transform: Z Z i i 1 a(x, η)e h hx,η−ξi e− h hy,η−ξi dydξ = n (2πh) Rn Rn Z i a(x, η)δ0 (ξ − η)e h hx,η−ξi dη = a(x, ξ). Rn

Approximation of a by elements of S concludes the proof.



We record finally the following useful fact. Suppose m is an order function on R2n and put m(x, e y, ξ) := m(x, ξ) + m(y, ξ); this is an order function on R3n . THEOREM 4.20 (Another transformation formula). Suppose that 0 ≤ δ ≤ 1/2 and e a ∈ Sδ (m). e Define Z Z i 1 Au(x) := e a(x, y, ξ)e h hx−y,ξi u(y)dydξ n (2πh) Rn Rn for u ∈ S (Rn ). Then A = aw (x, hD) for the symbol a ∈ Sδ (m) given by (4.4.18)

 z z  a(x, ξ) = eihhDz ,Dξ i e a x − , x + , ξ z=0 . 2 2

82

4. SEMICLASSICAL QUANTIZATION

Proof. We outline the idea. The equality of the Schwartz kernels of A and aw (x, hD) implies Z Z i 1 a(x, ξ) = e a(x − z/2, x + z/2, ζ + ξ)e− h hζ,zi dzdζ n (2πh) Rn Rn for a ˜ ∈ S . We then use (4.2.12) to obtain (4.4.18). Theorem 4.17 implies the validity for e a ∈ Sδ (m). e 

4.5. OPERATORS ON L2 Thus far our symbol calculus has produced operators acting on either the Schwartz space S or its dual space S 0 . But for applications we would like also to handle functions in more convenient spaces, most notably L2 . 4.5.1. Symbols in S . We first observe that if a ∈ S , then aw (x, hD) is in fact a bounded linear operator on L2 : THEOREM 4.21 (L2 boundedness for symbols in S ). If the symbol a belongs to S , then aw (x, hD) : L2 (Rn ) → L2 (Rn ) is bounded. Proof. 1. We recall from the proof of Theorem 4.1 that aw (x, hD)u(x) =

(4.5.1)

Z K(x, y)u(y) dy Rn

for the kernel 1 K(x, y) := (2πh)n

Z Rn

i

x+y −1 e h hx−y,ξi a( x+y 2 , ξ) dξ = Fh (a( 2 , ·))(x − y).

Since a ∈ S , we have Z Z (4.5.2) C1 := sup |K(x, y)| dy < ∞, C2 := sup x

Rn

y

|K(x, y)| dx < ∞.

Rn

2. We estimate for u ∈ L2 that Z Z Z kaw uk2L2 ≤ |K(x, y)||K(x, z)||u(y)||u(z)| dxdydz Rn Rn Rn Z Z Z 1 |K(x, y)||K(x, z)|(|u(y)|2 + |u(z)|2 ) dxdydz. ≤ 2 Rn Rn Rn Now Z Z Z |K(x, y)||K(x, z)||u(y)|2 dxdydz Rn

Rn

Rn

4.5. OPERATORS ON L2

83

Z

Z

|K(x, y)||u(y)|2 dxdy

≤ C1 Rn

Rn

Z ≤ C1 C2

|u(y)|2 dy;

Rn

and a similar estimate holds with the roles of y and z reversed. Thus 1

kaw ukL2 ≤ (C1 C2 ) 2 kukL2 .



The second part of this proof is a special case of Schur’s inequality. 4.5.2. Symbols in S and Sδ . It is important for applications that we extend the foregoing to a wider class of symbols. Our next task therefore is showing that if a ∈ Sδ for some 0 ≤ δ ≤ 12 , then aw (x, hD) extends to become a bounded linear operator acting upon L2 . This is much harder than the calculations above for a ∈ S . For the time being, we take h = 1. Preliminaries. We select χ ∈ Cc∞ (R2n ) such that 0 ≤ χ ≤ 1, χ ≡ 0 on R2n \ B(0, 2), and X χα ≡ 1, α∈Z2n

where χα := χ(· − α) denotes χ shifted by the lattice point α ∈ Z2n . Write (4.5.3)

aα := χα a;

then a=

X

aα .

α∈Z2n

We also define (4.5.4)

bαβ := a ¯α #aβ

(α, β ∈ Z2n ).

THEOREM 4.22 (Decay of mixed terms). (i) For each N and each multiindex γ, we have the estimate (4.5.5)

|∂ γ bαβ (z)| ≤ Cγ,N hα − βi−N hz −

α+β −N 2 i

for z = (x, ξ) ∈ R2n . (ii) For each N , there exists a constant CN such that (4.5.6) for all α, β ∈ Z2n .

−N kbw αβ (x, D)kL2 →L2 ≤ CN hα − βi

84

4. SEMICLASSICAL QUANTIZATION

Proof. 1. We can rewrite formula (4.3.7) to read Z Z 1 bαβ (z) = 2n eiϕ(w1 ,w2 ) a ¯α (z − w1 )aβ (z − w2 ) dw1 dw2 , π 2n 2n R R for ϕ(w1 , w2 ) = −2σ(w1 , w2 ). Select ζ : R4n → R such that 0 ≤ ζ ≤ 1, ζ ≡ 1 on B(0, 1), ζ ≡ 0 on R4n \ B(0, 2). Then Z Z 1 bαβ (z) = eiϕ ζ(w)¯ aα (z − w1 )aβ (z − w2 ) dw1 dw2 π 2n R2n R2n Z Z 1 + 2n eiϕ (1 − ζ(w))¯ aα (z − w1 )aβ (z − w2 ) dw1 dw2 π R2n R2n =: A + B. 2. Estimate of A. We have ZZ |A| ≤ C |¯ aα (z − w1 )||aβ (z − w2 )| dw1 dw2 , {|w|≤2}

for w = (w1 , w2 ). The integrand equals χ(z − w1 − α)χ(z − w2 − β)|a(z − w1 )||a(z − w2 )| and thus vanishes, unless |z − w1 − α| ≤ 2 and |z − w2 − β| ≤ 2. But then |α − β| ≤ 4 + |w1 | + |w2 | ≤ 8 and z −



α+β 2

≤ 4 + |w1 | + |w2 | ≤ 8.

Hence |A| ≤ CN hα − βi−N hz −

α+β −N 2 i

for any N . Similarly, for each multiindex γ we can estimate (4.5.7)

|∂ γ A| ≤ CN,γ hα − βi−N hz −

3. Estimate of B. We have |∂ϕ(w)| = 2|w| and Leiϕ = eiϕ , for L :=

h∂ϕ, Di . |∂ϕ|2

α+β −N . 2 i

4.5. OPERATORS ON L2

85

Since the integrand of B vanishes unless |w| ≥ 1, the usual argument based on integration by parts shows that Z Z hwi−M c¯α (z − w1 )cβ (z − w2 ) dw1 dw2 |B| ≤ CM R2n

R2n

for appropriate functions cα , cβ , with spt cα ⊂ B(α, 2), spt cβ ⊂ B(β, 2). Thus the integrand vanishes unless hα − βi ≤ Chwi, hz −

α+β 2 i

≤ Chwi.

Hence |B| ≤ CM hα − βi−N hz − ≤ CM hα − βi−N hz −

α+β −N 2 i α+β −N 2 i

R

2N −M R2n R2n hwi

R

dw1 dw2

if M is large enough. Likewise, (4.5.8)

|∂ γ B| ≤ CN,γ hα − βi−N hz −

α+β −N . 2 i

This proves (4.5.5). 4. Recall next that 1 a (x, D) = (2π)2n w

Z

a ˆ(l)eil(x,D) dl

R2n

and that, owing to (4.2.9), eil(x,D) is a unitary operator on L2 . Consequently Z w ka (x, D)kL2 →L2 ≤ C |ˆ a(l)| dl. R2n

Therefore we can estimate kbw αβ (x, D)kL2 →L2

≤ Ckˆbαβ kL1 ≤ Ckhξi2n+1ˆbαβ kL∞ ≤ C

sup

γb k ∞ \ kD αβ L

|γ|≤2n+1

≤ C

sup

kDγ bαβ kL1

|γ|≤2n+1

≤ C

sup

khzi2n+1 Dγ bαβ kL1

|γ|≤2n+1

≤ Chα − βi−N , according to (4.5.5)



THEOREM 4.23 (L2 boundedness for symbols in S). (i) If the symbol a belongs to S, then aw (x, D) : L2 (Rn ) → L2 (Rn )

86

4. SEMICLASSICAL QUANTIZATION

is bounded, with the estimate X

kaw (x, D)kL2 →L2 ≤ C

(4.5.9)

|α|≤M n

sup |∂ α a|, Rn

where M is a universal constant. (ii) Furthermore, if a ∈ Sδ for some 0 ≤ δ ≤ 1/2, then X h|α|/2 sup |∂ α a|. (4.5.10) kaw (x, hD)kL2 →L2 ≤ C |α|≤M n

Rn

∗ w Proof. 1. We have bw αβ (x, D) = Aα Aβ , where Aα := aα (x, D). Thus Theorem 4.22,(ii) asserts

kA∗α Aβ kL2 →L2 ≤ Chα − βi−N . Therefore sup

X

α

kAα A∗β k1/2 ≤ C

X hα − βi−N/2 ≤ C; β

β

and similarly sup α

aw (x, D)

X

kA∗α Aβ k1/2 ≤ C.

β

P

Since = α Aα , we can apply the Cotlar–Stein Theorem C.5. The constants in the estimates in Theorem 4.22 depend only on a finite number of derivatives of a, growing linearly with the dimension. That proves (4.5.9). 2. Estimate (4.5.10) follows from a rescaling, the details of which for δ = 0 we will later provide in the proof of Theorem 5.1.  As a first application, we record the useful THEOREM 4.24 (Composition and multiplication). Suppose that a, b ∈ Sδ for 0 ≤ δ < 12 . Then (4.5.11)

kaw (x, hD)bw (x, hD) − (ab)w (x, hD)kL2 →L2 = O(h1−2δ )

as h → 0. Proof. 1. In light of (4.4.16), we have a#b − ab = OSδ (h1−2δ ). Hence Theorem 4.23 implies aw bw − (ab)w = (a#b − ab)w = OL2 →L2 (h1−2δ ). For the borderline case δ = 12 , we have this assertion:



4.6. COMPACTNESS

87

THEOREM 4.25 (Disjoint supports). Suppose that a, b ∈ S 1 , and and 2

dist(spt(a), spt(b)) ≥ γ > 0

(4.5.12)

for some constant γ. Assume also that spt(a) ⊂ K, where the compact set K and the constant γ are independent of h. Then kaw (x, hD)bw (x, hD)kL2 →L2 = O(h∞ ).

(4.5.13)

Proof. Remember from (4.3.7) that Z Z i 1 a#b(z) = e h ϕ(w1 ,w2 ) a(z − w1 )b(z − w2 ) dw1 dw2 , 2n (hπ) R2n R2n for z = (x, ξ) and ϕ(w1 , w2 ) = −2σ(w1 , w2 ). We proceed as in the proof of Theorem 4.22: |∂ϕ| = 2|w| and thus the operator h∂ϕ, hDi L := |∂ϕ|2 has smooth coefficients on the support of a(z − w1 )b(z − w2 ). From our assumption that a, b ∈ S 1 , we see that 2

∗ M

M

(L ) (a(z − w1 )b(z − w2 )) = O(h 2 hwi−M ). The uniform bound on the support shows that a#b = OS (h∞ ). Its Weyl quantization is therefore bounded on L2 , with norm of order O(h∞ ). 

4.6. COMPACTNESS In this section we modify the proof of Theorem 4.23 to show that if a ∈ S(m) and if m goes to zero as (x, ξ) → ∞, then aw (x, D) is a compact operator on L2 . A first observation is this: LEMMA 4.26 (Schwartz symbols and compactness). Suppose that a ∈ S . Then aw (x, D) : L2 (Rn ) → L2 (Rn ) is a compact operator. Proof. 1. The Schwartz kernel of aw (x, D) is  Z  1 x+y a , ξ eihξ,x−yi dξ; K(x, y) := (2π)n 2 and so K ∈ S (Rn × Rn ). Hence for any multiindices α and β, (4.6.1)

sup |xα ∂xβ (aw (x, D)u)|

x∈Rn

88

4. SEMICLASSICAL QUANTIZATION

≤

sup (x,y)∈R2n

|xα ∂xβ hyiN K(x, y)|

Z Rn

hyi−N |u(y)| dy ≤ Cαβ kukL2 ,

where for the last estimate we took N > n/2. 2. Given a bounded set F ⊂ L2 (Rn ) , we need to find a sequence w 2 {fk }∞ k=1 ⊂ F such that a (x, D)fk converges in L . Fix N > n/2. It will be enough to show that gk := hxiN aw (x, D)fk converges in L∞ (Rn ), since kaw (x, D)fk − aw (x, D)fl kL2 ≤ khxi−N kL2 kgk − gl kL∞ . 3. Since estimate (4.6.1) shows that |∂gk (x)| ≤ M, hxi|gk (x)| ≤ M for some constant M , it follows from the Arzela–Ascoli Theorem that the n w sequence {gk }∞ k=1 converges uniformly on R . The sequence a (x, D)fk −N 2 therefore converges to hxi g in L .  Next we revisit Theorem 4.22 for general symbol classes. Recall that aα and bαβ are defined in (4.5.4). THEOREM 4.27 (Decay of mixed terms for general symbols). Suppose that a ∈ S(m). Then for each N , there exists a constant CN such that (4.6.2)

−N kbw αβ (x, D)kL2 →L2 ≤ CN m(α)m(β)hα − βi

for all α, β ∈ Z2n . Proof. We observe that |∂ γ aα (w)| = |∂ γ (χ(w − α)a(w))| (4.6.3)

≤ Cγ sup |∂ ρ χ(w − α)|m(w) |ρ|≤|γ|

≤ Cγ m(α), since the support of χ(w − α) is contained in |w − α| ≤ 2. Given (4.6.3), the proof of (4.5.5) now shows (4.6.4)

|∂ γ bαβ (z)| ≤ Cγ,N m(α)m(β)hα − βi−N hz −

α+β −N 2 i

for all N ,γ and z = (x, ξ) ∈ R2n . We now apply (4.6.4) in the same way (4.5.5) was employed in the proof of Theorem 4.22. This gives (4.6.2). 

4.6. COMPACTNESS

89

THEOREM 4.28 (Compactness for decaying order functions). Suppose a ∈ S(m) and (4.6.5)

lim

m = 0.

(x,ξ)→∞

Then (4.6.6)

aw (x, D) : L2 (Rn ) → L2 (Rn ) is a compact operator.

REMARK. The same assertion holds for the other quantizations of a. A converse is also true: if for every a ∈ S(m) aw (x, D) is compact, then (4.6.5) holds.  Proof: 1. We recall the notation of the proof of Theorem 4.23 A := aw (x, D), Aα := aw α (x, D), and define AM :=

X

Aα .

|α|<M

According to Lemma 4.26, AM is a finite sum of compact operators and hence is compact. 2. The space of compact operators is closed in operator norm topology, and hence to prove A is compact it suffices to show (4.6.7)

lim kA − AM kL2 →L2 = 0.

M →∞

We write A − AM = to estimate: 

P

α≥M

Aα and use the Cotlar–Stein Theorem C.5 

kA − AM k ≤ max  sup

X

|α|≥M |β≥M

kAα A∗β k1/2 , sup

X

|α|≥M |β≥M

kA∗α Aβ k1/2  .

Since A∗α Aβ = (¯ aα #aβ )w = bw αβ , we can apply Theorem 4.27 to obtain X X p m(α)m(β)hα − βi−N/2 sup kA∗α Aβ k1/2 ≤ CN sup |α|≥M |β≥M

|α|≥M |β≥M

≤ C sup m(α). |α|≥M

A similar estimate applies in the case of Aα A∗β . Therefore kA − AM k ≤ C sup m(α) → 0 |α|≥M

as M → ∞, thanks to our hypothesis (4.6.5) on m. This shows (4.6.7).



90

4. SEMICLASSICAL QUANTIZATION

4.7. INVERSES, G˚ ARDING INEQUALITIES At this stage we have constructed in appropriate generality the quantizations aw (x, hD) of various symbols a. We turn therefore to the practical problem of understanding how the algebraic and analytic behavior of the function a dictates properties of the corresponding quantized operators. 4.7.1. Inverses. Suppose in particular that a : R2n → C is nonvanishing and so is pointwise invertible. Can we draw the same conclusion about aw (x, hD)? DEFINITIONS. (i) We say the symbol a is elliptic if there exists a constant γ > 0 such that |a| ≥ γ > 0

(4.7.1)

on R2n .

(ii) More generally, a is elliptic in S(m) if |a| ≥ γm for some constant γ > 0. THEOREM 4.29 (Inverses for elliptic symbols). Assume that a ∈ Sδ (m) for some 0 ≤ δ < 21 and that a is elliptic in S(m). (i) If m ≥ 1, there exist h0 > 0 and C such that (4.7.2)

kaw (x, hD)ukL2 ≥ CkukL2

for all u ∈ S and 0 < h < h0 . (ii) If m = 1, there exists h0 > 0, such that aw (x, hD)−1 exists as a bounded linear operator on L2 (Rn ) for 0 < h ≤ h0 . Proof. 1. Let b := 1/a, b ∈ Sδ (1/m). Then (4.4.16) gives a#b = 1 + r1 , with r1 ∈ h1−2δ Sδ . Likewise b#a = 1 + r2 , with r2 ∈ h1−2δ Sδ . Hence if A := aw (x, hD), B := bw (x, hD), R1 := r1w (x, hD) and R2 := r2w (x, hD), we have AB = I + R1 BA = I + R2 ,

4.7. INVERSES, G˚ ARDING INEQUALITIES

91

with kR1 kL2 →L2 , kR2 kL2 →L2 = O(h1−2δ ) ≤ if 0 < h ≤ h0 and h0 is small enough.

1 2

2. When m = 1, A = aw (x, hD) has an approximate left inverse and an approximate right inverse. Applying then Theorem C.3, we deduce that A−1 exists. 3. If m ≥ 1, we see that for u ∈ S kukL2 = k(I + R2 )−1 bw (x, hD)aw (x, hD)kL2 ≤ Ckaw (x, hD)kL2 , since b ∈ S(1/m) ⊂ S(1) is bounded on L2 , according to Theorem 4.23.



4.7.2. G˚ arding inequalities. We suppose next that a is real-valued and nonnegative, and ask the consequences for aw (x, hD). THEOREM 4.30 (Easy G˚ arding inequality). Assume a is a real-valued symbol in S and a≥γ>0

(4.7.3)

on R2n .

Then for each  > 0 there exists h0 = h0 () > 0 such that haw (x, hD)u, ui ≥ (γ − )kuk2L2 (Rn )

(4.7.4)

for all 0 < h ≤ h0 and u ∈ L2 (Rn ). Proof. We will show that (a − λ)−1 ∈ S

(4.7.5)

if λ < γ − .

Indeed if b := (a − λ)−1 , then h {a − λ, b} + OS (h2 ) = 1 + OS (h2 ), 2i the bracket term vanishing since b is a function of a − λ. Therefore (a − λ)#b = 1 +

(aw (x, hD) − λ)bw (x, hD) = I + OL2 →L2 (h2 ), and so bw (x, hD) is an approximate right inverse of aw (x, hD) − λ. Likewise bw (x, hD) is an approximate left inverse. Hence Theorem C.3 implies aw (x, hD)−λ is invertible for each λ < γ −. Consequently, Spec(aw (x, hD)) ⊂ [γ − , ∞). According then to Theorem C.8, haw (x, hD)u, ui ≥ (γ − )kuk2L2 for all u ∈ L2 .



92

4. SEMICLASSICAL QUANTIZATION

To improve the preceding estimate, we will need a simple calculus inequality: LEMMA 4.31 (Gradient estimate). Let f : Rn → R be C 2 , with |∂ 2 f | ≤ A. Suppose also f ≥ 0. Then |∂f | ≤ (2Af )1/2 . Proof. By Taylor’s Theorem, Z f (x + y) = f (x) + h∂f (x), yi +

1

(1 − t)h∂ 2 f (x + ty)y, yi dt.

0

Let y = −λ∂f (x), λ > 0 to be selected. Then since f ≥ 0, we have Z 1 2 2 λ|∂f (x)| ≤ f (x) + λ (1 − t)h∂ 2 f (x − λt∂f (x))∂f (x), ∂f (x)i dt 0

λ2 ≤ f (x) + A|∂f (x)|2 . 2 Putting λ = 1/A, we conclude |∂f (x)|2 ≤ 2Af (x).



We next sharpen Theorem 4.30: THEOREM 4.32 (Sharp G˚ arding inequality). Assume a ∈ S and a≥0

(4.7.6)

on R2n .

Then there exist constants C ≥ 0 and h0 > 0 such that (4.7.7)

haw (x, hD)u, ui ≥ −Chkuk2L2 (Rn )

for all 0 < h < h0 and u ∈ L2 (Rn ). REMARK. The estimate (4.7.7) is in fact true for each quantization Opt (a) (0 ≤ t ≤ 1). For the Weyl quantization, the stronger Fefferman–Phong inequality holds: haw (x, hD)u, ui ≥ −Ch2 kuk2L2 (Rn ) for 0 < h ≤ h0 , u ∈ L2 (Rn ).



˜ sufficiently small and write Proof. 1. Our goal is to show that if we fix h ˜ (4.7.8) λ = h/h, then (4.7.9)

˜ 1/2 , h(a + λ)−1 ∈ hS

4.7. INVERSES, G˚ ARDING INEQUALITIES

93

˜ We can then argue as in the proof of with estimates independent of h. Theorem 4.30. Our notation is that ˜ 1/2 b ∈ hS means ˜ |∂ α b| ≤ Cα h−|α|/2 h ˜ for all multiindices α, with Cα independendent of h and h. 2. We first claim that (4.7.10) ∂ α (a + λ)−1 = (a + λ)−1

|α| X

X

Cβ 1 ,...,β k

k=1 α=β 1 +···+β k |β j |≥1

k  Y

 j (a + λ)−1 ∂ β a ,

j=1

for appropriate constants Cβ 1 ,...,β k . To see this, observe that when we compute ∂ α (a + λ)−1 a typical term involves k differentiations of (a + λ)−1 with the remaining derivatives falling on a. For each k ≤ |α| we partition α into multiindices β 1 , . . . , β k , each of which corresponds to one derivative falling on (a + λ)−1 and the remaining derivatives falling on a. Summing over k gives (4.7.10). 3. Lemma 4.31 implies λ1/2 |∂a| ≤ Cλ1/2 a1/2 ≤ C(λ + a). Hence for |β| = 1 (4.7.11)

|∂ β a|(a + λ)−1 ≤ Cλ−1/2 ;

and furthermore (4.7.12)

|∂ β a|(a + λ)−1 ≤ Cλ−1

if |β| ≥ 2, since a ∈ S. Consequently, for each partition α = β 1 + · · · + β k and 0 < λ ≤ 1: k k Y Y Y |βj | Y |α| −1 β −1 −1/2 (a + λ) ∂ j a ≤ C λ λ ≤ C λ− 2 = Cλ− 2 . j=1 j=1 |βj |≥2 |βj |=1 Therefore (4.7.13)

|∂ α (a + λ)−1 | ≤ Cα (a + λ)−1 λ−

˜ this implies Because λ = h/h, (a + λ)−1 ∈

˜ h S ; h 1/2

|α| 2

.

94

4. SEMICLASSICAL QUANTIZATION

that is, ˜ 1/2 , h(a + λ)−1 ∈ hS

(4.7.14)

with estimates independent of λ. 4. Since a + λ ∈ S ⊂ S 1 , we can define (a + λ)#b, for b = (a + λ)−1 . 2 Using Taylor’s formula, we compute (a + λ)#b(z) = eihA(D) (a(z) + λ)b(w) w=z Z 1 (1 − t)eithA(D) (ihA(D))2 (a(z) + λ)b(w)|w=z dt =1+ 0

=: 1 + r(z), where used {a + λ, (a + λ)−1 } = 0. ˜ 1/2 and so h2 ∂ α b ∈ hS ˜ 1/2 for |α| = 2. Now according to (4.7.14), hb ∈ hS ihA(D) ˜ 1/2 . Consequently, An application of e preserves the symbol class hS ˜ ≤ 1, krw (x, hD)kL2 →L2 ≤ C h 2 ˜ is now fixed small enough. Thus bw (x, hD) is an approximate right if h inverse of aw (x, hD) + λ, and is similarly an approximate left inverse. 5. So (aw (x, hD) + λ)−1 exists. Likewise (aw (x, hD) + γ + λ)−1 exists for all γ ≥ 0. Therefore Spec(aw (x, hD)) ⊂ [−λ, ∞). According then to Theorem C.8, haw (x, hD)u, ui ≥ −λkuk2L2 ˜ this inequality finishes the proof. for all u ∈ L2 . Since λ = h/h,



REMARK: More on rescaling. The rescaling (4.1.9) can be generalized to (4.7.15)

1 ˜ 2 x, x ˜ := (h/h)

1 ˜ 2 y, y˜ := (h/h)

1 ˜ 2 ξ. ξ˜ := (h/h)

Then the calculation which lead to (4.1.10) gives (4.7.16)

˜ u(˜ aw (x, hD)u(x) = aw x, hD)˜ x), h (˜

for ˜ 21 x ˜ := a((h/h) ˜ 12 x ˜ 21 ξ). ˜ u ˜(˜ x) := u((h/h) ˜), ah (˜ x, ξ) ˜, (h/h) ˜ We have thus rescaled from the h-semiclassical calculus to the h-semiclassical calculus.

4.8. NOTES

95

Note in particular that if −|α|/2 ˜ ∂ α a = O((h/h) ),

then a ˜ ∈ S. The bound (4.7.13) precisely an estimate of this type. It is es˜ |α|/2 ); sential in the proof of Theorem 4.32 that if a ∈ S, then ∂ α a ˜ = O((h/h) that is, the derivative improves. 

4.8. NOTES Our presentation of semiclassical calculus is based upon Dimassi–Sj¨ostrand [D-S, Chapter 7]. See also Martinez [M], in particular for the FeffermanCordoba proof of the sharp G˚ arding inequality. The argument presented here followed the proof of [D-S, Theorem 7.12]. Good introductions to the theory of pseudodifferential operators include Alinhac–G´erard [A-G], Grigis–Sj¨ostrand [G-S], Martinez [M] and Saint Raymond [SR]. A major treatise is H¨ormander [H1]–[H4].

Part 2

APPLICATIONS TO PARTIAL DIFFERENTIAL EQUATIONS

Chapter 5

SEMICLASSICAL DEFECT MEASURES

5.1 5.2 5.3 5.4

Construction, examples Defect measures and PDE Damped wave equation Notes

One way to understand limits as h → 0 of a collection of functions u = {u(h)}0
5.1. CONSTRUCTION, EXAMPLES In the first two sections of this chapter, we consider a collection of functions u = {u(h)}0
sup ku(h)kL2 < ∞. 0
For the time being, we do not assume that u(h) solves any PDE. THEOREM 5.1 (An operator norm bound). Suppose a ∈ S. Then (5.1.2)

1

kaw (x, hD)kL2 →L2 ≤ C sup |a| + O(h 2 ) R2n

99

100

5. SEMICLASSICAL DEFECT MEASURES

as h → 0. Proof. We showed earlier in Theorem 4.23 that if a ∈ S and h = 1, then (5.1.3)

kaw (x, D)kL2 →L2 ≤ C

|∂ α a|.

sup |α|≤2n+1

Suppose now a ∈ S and u ∈ S . We rescale according to (4.1.9), but now write n n 1 u ˜(˜ x) := h 4 u(x) = h 4 u(h 2 x ˜). This is different from the standard rescaling (4.1.11) of u, the advantage being that u 7→ u ˜ is now a unitary transformation of L2 : kukL2 = k˜ ukL2 . Then Z Z
i 1 , ξ e h hx−y,ξi u(y) dydξ aw (x, hD)u(x) = a x+y 2 n (2πh) Rn Rn n Z Z   h− 4 ˜ (5.1.4) x ˜+˜ y ˜ , ξ eih˜x−˜y,ξi u ˜(˜ y ) d˜ y dξ˜ a = h 2 (2π)n Rn Rn n

= h− 4 aw x, D)˜ u(˜ x) h (˜ for ah as in the standard rescaling (4.1.11). We hence deduce from (5.1.4) and (5.1.3) that x, D)˜ ukL2 kaw (x, hD)ukL2 = kaw h (˜ ≤ kaw ukL2 h kL2 →L2 k˜ ≤C

sup

|∂ α ah |kukL2

|α|≤2n+1

≤C

sup

h

|α| 2

|∂ α a|kukL2 .

|α|≤2n+1

This implies (5.1.2).



THEOREM 5.2 (Existence of defect measure). There exists a Radon measure µ on R2n and a sequence hj → 0 such that Z (5.1.5) haw (x, hj D)u(hj ), u(hj )i → a(x, ξ) dµ R2n

for each symbol a ∈ Cc∞ (R2n ). DEFINITION. We call µ a microlocal defect measure associated with the family u = {u(h)}0
5.1. CONSTRUCTION, EXAMPLES

101

Choose next a further subsequence {h2j } ⊂ {h1j } such that 2 2 2 haw 2 (x, hj D)u(hj ), u(hj )i → α2 .

Continue, at the k th step extracting a subsequence {hkj } ⊂ {hk−1 } such that j k k k haw k (x, hj D)u(hj ), u(hj )i → αk .

A standard diagonal argument shows that for hj := hjj → 0 we have haw k (x, hj D)u(hj ), u(hj )i → αk for all k = 1, . . . . 2. Define Φ(ak ) := αk . Owing to Theorem 5.1, we see for each k that |Φ(ak )| = |αk | = lim |haw k u(hj ), u(hj )i| (5.1.6)

hj →0

≤ C lim sup kaw k kL2 →L2 →L2 ≤ C sup |ak |. hj →0

R2n

The mapping Φ is bounded, linear and densely defined. Hence it uniquely extends to a bounded linear functional on Cc (R2n ), with the estimate |Φ(a)| ≤ C sup |a| R2n

for all a ∈ Cc (R2n ). The Riesz Representation Theorem therefore implies the existence of a (possibly complex-valued) Radon measure on R2n such that Z Φ(a) =

a(x, ξ) dµ. R2n



REMARK. Theorem 5.2 is also valid if we replace the Weyl quantization aw = Op1/2 (a) by Opt (a) for any 0 ≤ t ≤ 1, since the error is then O(h).  THEOREM 5.3 (Positivity). The measure µ is real and nonnegative: (5.1.7)

µ ≥ 0.

Proof. We must show that a ≥ 0 implies Z a dµ ≥ 0. R2n

Now when a ≥ 0, the sharp G˚ arding inequality, Theorem 4.32, implies aw (x, hD) ≥ −Ch; that is, haw (x, hD)u(h), u(h)i ≥ −Chku(h)k2L2

102

5. SEMICLASSICAL DEFECT MEASURES

for sufficiently small h > 0. Let h = hj → 0, to deduce Z a dµ = lim haw (x, hj D)u(hj ), u(hj )i ≥ 0. hj →0

R2n

 EXAMPLE 1: Coherent states. Fix a point (x0 , ξ0 ) and define the corresponding coherent state n

i

1

2

u(h)(x) := (πh)− 4 e h hx−x0 ,ξ0 i− 2h |x−x0 | , where we have normalized so that ku(h)kL2 = 1. Then there exists precisely one associated semiclassical defect measure, namely µ := δ(x0 ,ξ0 ) . To confirm this statement, take t = 1 in the quantization and calculate ha(x, hD)u(h), u(h)i Z Z Z i 1 a(x, ξ)e h hx−y,ξi u(h)(y)u(h)(x) dydξdx = n (2πh) Rn Rn Rn n Z Z Z i 22 = a(x, ξ)e h (hx−y,ξi+hy−x0 ,ξ0 i−hx−x0 ,ξ0 i) 3n (2πh) 2 Rn Rn Rn 1

e− 2h (|y−x0 | 2

=

n 2

(2πh)

Z 3n 2

Z

Rn

Rn

Z

2 +|x−x

0|

2)

0|

2)

dydξdx

i

a(x, ξ)e h hx−y,ξ−ξ0 i

Rn 1

e− 2h (|y−x0 |

2 +|x−x

dydξdx.

For each fixed x and ξ, the integral in y is Z e Rn

i hx−y,ξ−ξ0 i h

e

1 − 2h |y−x0 |2

i hx−x0 ,ξ−ξ0 i h

Z

i

1

2

e− h hy,ξ−ξ0 i e− 2h |y| dy  1 2   i ξ−ξ0 = e h hx−x0 ,ξ−ξ0 i F e− 2h |y| h

dy = e

Rn

n

i

1

2

= (2πh) 2 e h hx−x0 ,ξ−ξ0 i e− 2h |ξ−ξ0 | , where we used formula (3.1.4) for the last equality. Therefore ha(x, hD)u(h), u(h)i n Z Z i 1 22 2 2 = a(x, ξ)e h hx−x0 ,ξ−ξ0 i e− 2h (|x−x0 | +|ξ−ξ0 | ) dxdξ n (2πh) Rn Rn n Z Z i 1 22 2 2 = a(x0 , ξ0 ) e h hx−x0 ,ξ−ξ0 i e− 2h (|x−x0 | +|ξ−ξ0 | ) dxdξ + o(1) (2πh)n Rn Rn = Ca(x0 , ξ0 ) + o(1),

5.1. CONSTRUCTION, EXAMPLES

103

for the constant n

Z

22 C := (2π)n

Z

Rn

1

eihx,ξi e− 2 (|x|

2 +|ξ|2 )

dxdξ.

Rn

Taking a ≡ 1 and recalling that ku(h)kL2 = 1, we deduce that C = 1.



EXAMPLE 2: Stationary phase and defect measures. For our next example, take u(h)(x) := e where ϕ, b ∈

C∞

iϕ(x) h

b(x),

and kbkL2 = 1. Then

ha(x, hD)u(h), u(h)i = Z Z Z i 1 a(x, ξ)e h (hx−y,ξi+ϕ(y)−ϕ(x)) b(y)b(x) dydξdx. n (2πh) Rn Rn Rn We assume a ∈ Cc∞ (R2n ) and apply stationary phase. For a given value of x, define ϕ(y, ξ) := hx − y, ξi + ϕ(y) − ϕ(x). Then ∂y ϕ = ∂ϕ(y) − ξ, ∂ξ ϕ = x − y; and the Hessian matrix of ϕ is 

2

∂ ϕ=

 ∂ 2 ϕ −I . −I O

The signature of a matrix is integer valued, and consequently is invariant if we move along a curve of nonsingular matrices. Since   O −I sgn = 0, −I O it follows that  sgn

t∂ 2 ϕ −I −I O

 =0

for 0 ≤ t ≤ 1; and therefore sgn(∂ 2 ϕ) = 0. In addition, |det ∂ 2 ϕ| = 1. Thus as h → 0 the stationary phase asymptotic expression (3.5.8) implies Z Z w 2 ha (x, hD)u(h), u(h)i → a(x, ∂ϕ(x))|b(x)| dx = a(x, ξ) dµ Rn

R2n

for the semiclassical defect measure µ := |b(x)|2 δ{ξ=∂ϕ(x)} Ln , Ln denoting n-dimensional Lebesgue measure in the x-variables.



104

5. SEMICLASSICAL DEFECT MEASURES

5.2. DEFECT MEASURES AND PDE We now assume more about the family {u(h)}0
(5.2.1)

if |ξ| ≥ C

for constants C, γ > 0. 5.2.1. Properties of semiclassical defect measures. First, let us suppose P (h)u(h) vanishes up to an o(1) error term and see what we can conclude about a corresponding semiclassical defect measure µ. THEOREM 5.4 (Support of defect measure). Suppose that u(h) satisfies ( kP (h)u(h)kL2 = o(1) as h → 0, (5.2.2) ku(h)kL2 = 1. Then spt µ ⊂ p−1 (0)

(5.2.3)

for any microlocal defect measure µ associated with the family {u(h)}0
R2n m S(hξi ) such

that spt(a) ∩ spt(q) = ∅ and

|p + iq| ≥ δhξim > 0

on R2n

for some δ > 0. We can for instance choose a function q ∈ C ∞ that is equal to one on p−1 (0), and then modify it near the compact support of a. Write Q(h) := q w (x, hD). Then Theorem 4.29 ensures us that for small enough h the operator hhDi−m (P (h) + iQ(h)) is invertible on L2 . Next, put A(h) := aw (x, hD). We observe that ap aq − a = −i p + iq p + iq

5.2. DEFECT MEASURES AND PDE

105

Since a and q have disjoint support, Theorems 4.24 and 4.25 imply kA(h)(P (h) + iQ(h))−1 P (h) − A(h)kL2 →L2 = O(h). Therefore (5.2.2) implies kA(h)u(h)kL2 = o(1); and thus hA(h)u(h), u(h)i → 0. But also Z

w

hA(hj )u(hj ), u(hj )i = ha (x, hj D)u(hj ), u(hj )i →

a dµ. R2n



Now we make the stronger assumption that the error term in (5.2.2) is o(h). THEOREM 5.5 (Flow invariance). Assume ( kP (h)u(h)kL2 = o(h) as h → 0, (5.2.4) ku(h)kL2 = 1. Then Z {p, a} dµ = 0

(5.2.5) R2n

for all a ∈ Cc∞ (R2n ). INTERPRETATION. Let ϕt be the flow generated by the Hamiltonian vector field Hp . Then Z Z Z ∗ ∂t ϕt a dµ = (Hp a)(ϕt ) dµ = {p, a} dµ. R2n

R2n

R2n

Conseqently (5.2.5) asserts that the semiclassical defect measure µ is flowinvariant.  The proof illustrates one of the basic principles mentioned in Chapter 1, that an assertion about Hamiltonian dynamics involving the Poisson bracket corresponds to a commutator argument at the quantum level. Proof. Since p is real, P (h) = pw (x, hD) is self-adjoint on L2 . Select a as above and write A(h) = aw (x, hD). Recall that A(h) = A(h)∗ . Then h[P (h), A(h)]u(h), u(h)i = h(P (h)A(h) − A(h)P (h))u(h), u(h)i = hA(h)u(h), P (h)u(h)i −hP (h)u(h), A(h)u(h)i = o(h)

106

5. SEMICLASSICAL DEFECT MEASURES

as h → 0. On the other hand, h [P (h), A(h)] = {p, a}w (x, hD) + OL2 →L2 (h2 ). i Hence h h[P (h), A(h)]u(h), u(h)i = h{p, a}w u(h), u(h)i + o(h). i Divide by h > 0 and let h = hj → 0: Z {p, a} dµ = 0. R2n

Note that even though p may not have compact support, {p, a} does.



5.3. DAMPED WAVE EQUATION 5.3.1. Quantization and semiclassical defect measures on the torus. The torus Tn is the simplest compact manifold for which we can consider semiclassical quantization. We will need the following analysis for an immediate application, but defer the general study of quantization on manifolds until Chapter 13. NOTATION. (i) We identify the torus Tn with the fundamental domain Tn ' {x | 0 ≤ xi < 1, 1 ≤ i ≤ n} ⊂ Rn . (ii) We likewise identify functions on Tn with periodic functions on Rn : (5.3.1)

R2n

u(x + k) = u(x)

(k ∈ Zn ).

(iii) Symbols a on Tn × Rn are similarly identified with symbols a on that are periodic in x:

(5.3.2)

a(x + k, ξ) = a(x, ξ)

(k ∈ Zn ).

Operators obtained by quantizing such symbols satisfy (aw (x, hD)u)(x + k) = (aw (x, hD)u(· + k))(x), and hence preserve periodicity. A key fact is that these operators are also bounded on L2 (Tn ): THEOREM 5.6. Suppose that a ∈ S satisfies (5.3.2). Then for u satisfying (5.3.1), we have the estimate Z Z w 2 (5.3.3) |a (x, hD)u| dx ≤ C |u|2 dx. Tn

Tn

5.3. DAMPED WAVE EQUATION

107

Proof. 1. We use the periodicity to write X aw (x, hD)u(x) = Ak u(x) k∈Zn

for

Z Z i 1 Ak u(x) := a( x+y , ξ)e h hx−y+k,ξi u(y) dξdy. 2 n (2πh) Rn Tn n Letting 1T denote the characteristic function of Tn ⊂ Rn , we can write Ak = 1Tn aw k (x + k, hD)1Tn

(5.3.4) for

ak (x, ξ) := a(x − k/2, ξ). Owing to the periodicity of a in x, ak ’s form a set of 2n symbols. 2. We next claim that for |k| > 2 kAk kL2 (Tn )→L2 (Tn ) = O(h∞ hki−∞ ).

(5.3.5)

To prove this, we note that for |k| > 2 eihx−y+k,ξi = h2N |x − y + k|−2N |Dξ |2N eihx−y+k,ξi for x, y ∈ Tn . Hence 1 Ak u(x) = (2πh)n

Z

Z

Rn

Tn

i

e ak (x + k, y, ξ)e h hx−y+k,ξi u(y) dξdy

for
k e ak (x, y, ξ) := χ(x − k)χ(y)h2N |x − y|−2N |Dξ |2N a ( x+y 2 − 2 , ξ). Here χ ∈ Cc∞ (Rn ) is equal to 1 near Tn . Theorem 4.20 then shows that Ak u(x) = h2N k −2N 1Tn bw k (x + k, hD)1Tn , for a symbol bk ∈ S, seminorms of which in S are bounded independently of k. Theorem 4.23 therefore implies (5.3.5). 3. Since (5.3.4) shows L2 boundedness for any fixed k, the estimate (5.3.3) follows.  THEOREM 5.7 (Defect measures on the torus). Let {u(h)}0
for each symbol a ∈

Cc∞ (Tn

×

Rn ).

108

5. SEMICLASSICAL DEFECT MEASURES

(ii) If kP (h)u(h)kL2 = o(1), then spt µ ⊂ p−1 (0). (iii) If kP (h)u(h)kL2 = o(h), then Z {p, a} dµ = 0 R2n

for all a ∈ Cc∞ (Tn × Rn ). 5.3.2. A damped wave equation. In this section Tn denotes the flat n-dimensional torus. We consider now the initial-value problem ( (∂t2 + a∂t − ∆)u = 0 on Tn × {t > 0} (5.3.7) u = 0, ut = f on Tn × {t = 0}, in which the smooth function a = a(x) is nonnegative, and thus represents a damping mechanism, as we will see. Our results extend with no difficulty if Tn is replaced by a general compact Riemannian manifold. DEFINITION. The energy at time t is Z 1 E(t) := (∂t u)2 + |∂x u|2 dx. 2 Tn LEMMA 5.8 (Elementary energy estimates). (i) If a ≡ 0, t 7→ E(t) is constant. (ii) If a ≥ 0, t 7→ E(t) is nonincreasing. Proof. These assertions follow easily from the calculation Z 0 2 E (t) = ∂t u∂t2 u + h∂x u, ∂xt ui dx Tn Z Z 2 = ∂t u(∂t u − ∆u) dx = − a(∂t u)2 dx ≤ 0.  Tn

Tn

Our eventual goal is showing that if the support of the damping term a is large enough, then we have exponential energy decay for our solution of the wave equation (5.3.7). Here is the key assumption:

5.3. DAMPED WAVE EQUATION

109

DYNAMICAL HYPOTHESIS.   There exists a time T > 0 such that any     trajectory of the Hamiltonian vector field of (5.3.8)  p(x, ξ) = |ξ|2 , starting at time 0 with |ξ| = 1,     intersects the set {a > 0} by the time T . Equivalently, for each initial point z = (x, ξ) ∈ Tn × Rn , with |ξ| = 1, we have Z ZT 1 T a(x + tξ) dt > 0, haiT = − a(x + tξ) dt := T 0 0 the slash through the first integral denoting an average. 5.3.3. Resolvent estimates. MOTIVATION. Since the damping term a in general depends upon x, we cannot use Fourier transform (or Fourier series) in x to solve (5.3.7). Instead we define u ≡ 0 for t < 0 and take the Fourier transform in t: Z ∞ u ˆ(x, τ ) := e−itτ u(x, t) dt (τ ∈ R). 0

Then Z

∞

∆ˆ u = Z0 ∞ =

e−itτ ∆u dt =

Z

∞

e−itτ (∂t2 u + a∂t u) dt

0

((iτ )2 + aiτ )e−itτ u dt − f = (−τ 2 + aiτ )ˆ u − f.

0

Consequently, (5.3.9)

P (τ )ˆ u := (−∆ + iτ a − τ 2 )ˆ u = f.

Now take τ to be complex, with Re τ ≥ 0, and define √ √ (5.3.10) P (z, h) := −h2 ∆ + i zha − z = h2 P (h−2 z) for the rescaled variable z = τ 2 h2 .

(5.3.11) Then (5.3.9) reads

P (z, h)ˆ u = h2 f ; and so, if P (z, h) is invertible, (5.3.12)

u ˆ = h2 P (z, h)−1 f.

We therefore need to study the inverse of P (z, h). We start with a general result about the inverse of P (τ ):

110

5. SEMICLASSICAL DEFECT MEASURES

THEOREM 5.9 (Meromorphy of the resolvent). The operator P (τ ) is invertible except at a discrete set of points in C. More precisely, τ 7→ P (τ )−1 : L2 (Tn ) → H 2 (Tn ) is a meromorphic family of operators with poles of finite rank. It has no poles for τ ∈ R \ {0}, a simple pole at τ = 0, and is holomorphic for Imτ > 0 and for Imτ < −kak∞ . Proof. 1. We note first that the pseudodifferential calculus and Theorem 5.6 show that (−h2 ∆ + 1)−1 : L2 → H 2 exists. That means that for τ0 = is and large |s|, we have (−∆ − τ02 )−1 : L2 → H 2 . Consequently, P (τ ) = (−∆ − τ02 )(I + (−∆ − τ02 )−1 (τ02 − τ 2 − iτ a(x))); and the existence of the inverse of P (τ ) is equivalent to invertibility of I + K(τ ) on

L2 (T2 ),

where K(τ ) := (−∆ − τ02 )−1 (τ02 − τ 2 − iτ a(x)).

2. Theorem 4.28 shows that K(τ ) is compact and hence I + K(τ ) is a holomorphic family of Fredholm operators. If |τ0 |  1, then kK(τ0 )kL2 →H 2 < 1, and therefore I + K(τ0 ) is invertible. Theorem D.4 gives the meromorphy of τ 7→ P (τ )−1 . For Imτ > 0 we use the fact that a ≥ 0 to see that kP (τ )ukL2 kukL2 ≥ |ImhP (τ )u, ui| ≥ Imτ kuk2L2 . This proves invertibility, since the meromorphy implies that the Fredholm alternative holds. A similar estimate is valid for Imτ < −kakL∞ . 3. Finally we exclude the possibility of a real non-zero τ satisfying (5.3.13)

(−∆ − τ 2 + iτ a)u = 0

for some u 6= 0. Multiplying by u ¯, integrating, and taking the imaginary part shows that Z a|u|2 dx = 0. Tn

Since a ≥ 0, this implies that u ≡ 0 on spt a. Hence (−∆ − τ 2 )u = 0. But this is impossible owing to unique continuation results which we will prove in Section 7.2, since spt a has a nonempty interior. The Fredholm alternative now guarantees that P (τ )−1 has no pole on the real axis. 

5.3. DAMPED WAVE EQUATION

111

THEOREM 5.10 (Resolvent bounds). Under the dynamical assumption (5.3.8), there exist constants α, C, h0 > 0 such that kP (z, h)−1 kL2 →L2 ≤

(5.3.14)

C h

for (5.3.15)

|Im z| ≤ αh, |z − 1| ≤ α, 0 < h ≤ h0 .

Proof. 1. It is enough to show that there exists a constant C such that C kukL2 ≤ kP (z, h)ukL2 h for all u ∈ L2 , provided z and h satisfy (5.3.15). We argue by contradiction. If the assertion were false, then for m = 1, 2, . . . there would exist zm ∈ C, 0 < hm ≤ 1/m and functions um in L2 such that hm 1 hm kum kL2 , |Im zm | ≤ , |zm − 1| ≤ . kP (zm , hm )um kL2 ≤ m m m We may assume kum kL2 = 1. Then (5.3.16)

P (zm , hm )um = o(hm ).

Also, (5.3.17)

zm → 1, Im(zm ) = o(hm ).

2. Let µ be a microlocal defect measure associated with {um }∞ m=1 . Then Theorem 5.4 implies for the symbol p := |ξ|2 − 1 that spt(µ) ⊂ p−1 (0) = {|ξ|2 = 1}. But hum , um i = 1, and so Z (5.3.18)

dµ = 1. Tn ×Rn

We will derive a contradiction to this. 3. Hereafter write Pm := P (zm , hm ). Then √ Pm = −h2m ∆ + i zm hm a − zm , √ ∗ Pm = −h2m ∆ − i z¯m hm a − z¯m ; and therefore √ √ ∗ Pm − Pm = i( zm + z¯m )hm a − zm + z¯m = 2ihm a + o(hm ), √ √ since (5.3.17) implies that zm + z¯m = 2 + o(1) and that −zm + z¯m = −2iIm(zm ) = o(hm ). (5.3.19)

112

5. SEMICLASSICAL DEFECT MEASURES

Now select b ∈ Cc∞ (Tn × Rn ) and set Bm := bw (x, hm D). Then Bm = Using (5.3.16) and (5.3.19), we calculate that

∗ . Bm

o(hm ) = 2i ImhBm Pm um , um i = hBm Pm um , um i − hum , Bm Pm um i ∗ = h(Bm Pm − Pm Bm )um , um i

= h[Bm , Pm ]um , um i ∗ +h(Pm − Pm )Bm um , um i hm = h{b, p}w um , um i i +2hm ih(ab)w um , um i + o(hm ).

Divide by hm and let hm → 0, through a subsequence if necessary, to discover that Z (5.3.20) {p, b} + 2ab dµ = 0. Tn ×Rn

RWe will build a function b so that {p, b} + 2ab > 0 on spt(µ). This will imply Tn ×Rn dµ = 0, a contradiction to (5.3.18). 4. For (x, ξ) ∈ Tn × Rn , with |ξ| = 1, define ZT c(x, ξ) := − (T − t)a(x + ξt) dt, 0

where T is the time from the dynamical hypothesis (5.3.8). Hence ZT hξ, ∂x ci = − (T − t)hξ, ∂a(x + ξt)i dt 0

ZT = − (T − t)∂t a(x + ξt) dt 0

ZT = − a(x + ξt) dt − a(x) 0

= haiT − a. Let b := ec χ(p), where χ ∈ Cc∞ (R) is equal to 1 near 0. Then hξ, ∂x bi = ec hξ, ∂x ciχ(p) = ec haiT χ(p) − aec χ(p) since Hp (χ(p)) = 0. Consequently {p, b} + 2ab = 2hξ, ∂x bi + 2ab = 2ec haiT χ(p) > 0 as desired.

on p−1 (0), 

5.3. DAMPED WAVE EQUATION

113

5.3.4. Energy decay. THEOREM 5.11 (Exponential energy decay). Assume the dynamic hypothesis (5.3.8) and suppose u solves the wave equation with damping (5.3.7). Then there exist constants C, β > 0 such that E(t) ≤ Ce−βt kf kL2

(5.3.21)

for all times t > 0.

Motivation. The following calculations are based upon this idea: to get decay estimates of g on the positive real axis, we estimate gˆ in a complex strip |Im z| ≤ α. Then if β < α, Z ∞ Z ∞ βt g(τ ) = ed eβt g(t)e−itτ dt = g(t)e−it(τ +iβ) dt = gˆ(τ + iβ). −∞

Hence our t → ∞.

L2

−∞

estimate of gˆ(· + iβ) will imply exponential decay of g(t) for

Proof. 1. Recall from (5.3.10), (5.3.11) that P (τ ) = h−2 P (z, h)

for τ 2 = h−2 z.

First we assert that there exists γ > 0 such that (5.3.22)

kP (τ )−1 kL2 →H 1 ≤ C

for |Im τ | ≤ γ, |τ | > 1/γ.

To prove (5.3.22), we note that provided the inequalities (5.3.15) hold, then √ C kh2 ∆P (z, h)−1 ukL2 = k(i zha − z)P (z, h)−1 u − ukL2 ≤ kukL2 , h the last inequality holding according to Theorem 5.10. Thus (5.3.23)

kh2 P (z, h)−1 ukH 2 ≤

C kukL2 . h

Recall next that z = h2 τ 2 . Write τ = λ + iµ, for λ > 0, and set h = λ−1 ; so that z = h2 (λ2 − µ2 ) + i(h2 2λµ). Thus |Im z| ≤ αh and |z − 1| ≤ α provided if |µ| ≤ γ and |λ| > 1/γ for some sufficiently small γ, and so the inequalities (5.3.15) hold. Hence (5.3.23) implies (5.3.24)

kP (τ )−1 ukH 2 ≤

for |µ| ≤ γ and |λ| > 1/γ.

C kukL2 |τ |

114

5. SEMICLASSICAL DEFECT MEASURES

Also C kukL2 . |τ | Interpolating between the last two inequalities demonstrates that kP (τ )−1 ukL2 ≤

kP (τ )−1 ukH 1 ≤ CkukL2 for |Im τ | ≤ γ and |τ | > 1/γ. This proves (5.3.22) except for a bounded range of τ ’s. Theorem 5.9 shows that the set of poles of P (τ )−1 is discrete and there are no poles on the real axis other than τ = 0. 2. We conclude that for 0 < β < α we have (5.3.25)

sup kP (τ + iβ)−1 kL2 →H 1 ≤ Cβ . τ ∈R

3. Next select χ : R → R, χ = χ(t), such that 0 ≤ χ ≤ 1, χ ≡ 1 on [1, ∞), χ ≡ 0 on (−∞, 0). Then if u1 := χu, we have (5.3.26)

(∂t2 + a∂t − ∆)u1 = g1 ,

for (5.3.27)

g1 := χ00 u + 2χ0 ∂t u + a(x)χ0 u.

Note that u1 (t) = 0 for t ≤ 0, and observe also that the support of g1 lies within Tn × [0, 1]. Furthermore, using energy estimates in Lemma 5.8, we see that kg1 kL2 (R+ ;L2 )
(5.3.28) ≤ C kukL2 ((0,1);L2 ) + k∂t ukL2 ((0,1);L2 ) ≤ Ckf kL2 . Now take the Fourier transform of (5.3.26) in time: P (τ )ˆ u1 (τ ) = gˆ1 (τ ). Then (5.3.29)

u ˆ1 (τ ) = P (τ )−1 gˆ1 (τ ),

where, in principle, we allow the left hand side to have a pole at τ = 0. 4. We now deduce exponential decay. Noting that u1 is supported in t > 0, we use Plancherel’s theorem to compute 1 βt u k 2 keβt u1 kL2 (R+ ;H 1 ) = (2π)− 2 ke[ 1 L (R;H 1 ) 1

= (2π)− 2 kˆ u1 (· + iβ)kL2 (R;H 1 )

5.3. DAMPED WAVE EQUATION

115

1

= (2π)− 2 kP (· + iβ)−1 gˆ1 (· + iβ)kL2 (R;H 1 ) ≤ Ckˆ g1 (· + iβ)kL2 (R;H 1 ) Since g1 is compactly supported in t we also see that βt g (·); gˆ1 (· + iβ) = e[ 1

and hence keβt u1 kL2 (R+ ;H 1 ) ≤ Ckˆ g1 kL2 (R;L2 ) ≤ Ckg1 kL2 (R+ ;L2 ) ≤ Ckf kL2 . Since u1 = χu, it follows that (5.3.30)

keβt ukL2 ((1,∞);H 1 ) ≤ Ckf kL2 .

5. Finally, fix T > 2 and χT := χ(t − T + 1), where χ is as in Step 2. Let u2 = χT u. Then (5.3.31)

(∂t2 + a∂t − ∆)u2 = g2 ,

for (5.3.32)

g2 := χ00T u + 2χ0T ∂t u + aχ0T u.

Therefore sptg ⊂ Tn × (T − 1, T ). Define E2 (t) :=

1 2

Z

(∂t u2 )2 + |∂x u2 |2 dx.

Tn

Modifying the calculations in the proof of Lemma 5.8, we use (5.3.31) and (5.3.32) to compute Z 2 E20 (t) = ∂t u2 ∂t2 u2 + h∂x u2 , ∂xt u2 i dx n T Z = ∂t u2 (∂t2 u2 − ∆u2 ) dx Tn Z Z 2 = − a(∂t u2 ) dx + ∂t u2 g2 dx Tn Tn Z ≤ C |∂t u2 |(|∂t u| + |u|) dx Tn Z ≤ CE2 (t) + C u2 + (∂t u)2 dx. Tn

116

5. SEMICLASSICAL DEFECT MEASURES

Since E2 (T − 1) = 0 and E2 (T ) = E(T ), Gronwall’s inequality implies that   (5.3.33) E(T ) ≤ C kuk2L2 ((T −1,T );L2 ) + k∂t uk2L2 ((T −1,T );L2 ) . 6. We need to control the right hand term in (5.3.33). For this, select χ : R → R, such that   0 ≤ χ ≤ 1, χ ≡ 0 for t ≤ T − 2 and t ≥ T + 1,   χ ≡ 1 for T − 1 ≤ t ≤ T. We multiply the wave equation (5.3.7) by χ2 u and integrate by parts, to find Z T +1 Z 0 = χ2 u(∂t2 u + a∂t u − ∆u) dxdt T −2 Tn T +1 Z

Z =

T −2

−χ2 (∂t u)2 − 2χχ0 u∂t u + χ2 au∂t u + χ2 |∂x u|2 dxdt.

Tn

From this identity we derive the estimate k∂t ukL2 ((T −1,T );L2 ) ≤ CkukL2 ((T −2,T +1);H 1 ) . This, (5.3.33) and (5.3.30) therefore imply E(T ) ≤ Ckuk2L2 ((T −2,T +1);H 1 ) ≤ Ce−βT kf kL2 , as asserted.



5.4. NOTES Semiclassical defect measures were introduced independently in G´erard [Ge] and Lions–Paul [L-P]; see also Tartar [T]. Theorem 5.11 is due to Rauch–Taylor [R-T], but the proof here follows Lebeau [L]. To pass from the resolvent estimate to the energy decay we use also some ideas of Morawetz, explained to us by N. Burq.

Chapter 6

EIGENVALUES AND EIGENFUNCTIONS

6.1 6.2 6.3 6.4 6.5

The harmonic oscillator Symbols and eigenfunctions Spectrum and resolvents Weyl’s Law Notes

In this chapter we are given the potential V : Rn → R, and investigate how the symbol (6.0.1)

p(x, ξ) = |ξ|2 + V (x)

provides interesting information about the corresponding operator (6.0.2)

P (h) := P (x, hD) = −h2 ∆ + V.

We will focus mostly upon learning how p controls the aysmptotic distribution of the eigenvalues of P (h) in the semiclassical limit h → 0.

6.1. THE HARMONIC OSCILLATOR Our plan is to consider first the simplest case, when the potential is quadratic; and to simplify even more, we begin in one dimension. So suppose that n = 1, h = 1 and V (x) = x2 . Thus we start with the one-dimensional quantum harmonic oscillator, meaning the operator P0 := −∂ 2 + x2 . 117

118

6. EIGENVALUES AND EIGENFUNCTIONS

6.1.1. Eigenvalues and eigenfunctions of P0 . We can as follows employ certain auxiliary first-order differential operators to compute explicitly the eigenvalues and eigenfunctions for P0 . NOTATION. Let us write A+ := Dx + ix, A− := Dx − ix, where Dx = 1i ∂x , and call A+ the creation operator and A− the annihilation operator. (This terminology is from particle physics.) LEMMA 6.1 (Properties of A± ). The creation and annihilation operators satisfy these identities: A∗+ = A− , A∗− = A+ , P0 = A+ A− + 1 = A− A+ − 1. Proof. It is easy to check that Dx∗ = Dx and (ix)∗ = −ix. Furthermore, A+ A− u = (Dx + ix)(Dx − ix)u    1 1 ∂x + ix ux − ixu = i i = −uxx − (xu)x + xux + x2 u = −uxx − u − xux + xux + x2 u = P0 u − u; and similarly, A− A+ u = (Dx − ix)(Dx + ix)u    1 1 = ∂x − ix ux + ixu i i = −uxx + (xu)x − xux + x2 u = P0 u + u.



We can now use A± to find all the eigenvalues and eigenfunctions of P0 :

THEOREM 6.2 (Eigenvalues and eigenfunctions). (i) We have hP0 u, ui ≥ kuk2L2 for all u ∈ Cc∞ (Rn ). That is, P0 ≥ 1.

6.1. THE HARMONIC OSCILLATOR

119

(ii) The function x2

v0 =: e− 2 is an eigenfunction corresponding to the smallest eigenvalue 1. (iii) Set vn := An+ v0 for n = 1, 2, . . . . Then (6.1.1)

P0 vn = (2n + 1)vn .

(iv) Define the normalized eigenfunctions vn un := . kvn kL2 Then un (x) = Hn (x)e−

(6.1.2)

x2 2

where Hn (x) = cn xn + · · · + c0 (cn 6= 0) is a polynomial of degree n. (v) We have hun , um i = δnm ; and furthermore, the collection of eigenfunctions {un }∞ n=0 is complete in L2 (Rn ). The functions Hn mentioned in assertion (iv) are the Hermite polynomials. Proof. 1. We note that 1 x u [Dx , x]u = (xu)x − ux = , i i i and consequently i[Dx , x] = 1. Therefore kuk2L2

= hi[Dx , x]u, ui ≤ 2kxukL2 kDx ukL2 ≤ kxuk2L2 + kDx uk2L2 = hP0 u, ui.

Next, observe   x2 1 − x2 A− v 0 = e 2 − ixe− 2 = 0; i x so that P0 v0 = (A+ A− + 1)v0 = v0 . 2. We can further calculate that P0 vn = (A+ A− + 1)A+ vn−1 = A+ (A− A+ − 1)vn−1 + 2A+ vn−1 = A+ P0 vn−1 + 2A+ vn−1

120

6. EIGENVALUES AND EIGENFUNCTIONS

= (2n − 1)A+ vn−1 + 2A+ vn−1

(by induction)

= (2n + 1)vn . The form (6.1.2) of vn , un follows by induction. 3. Note also that [A− , A+ ] = A− A+ − A+ A− = (P0 + 1) − (P0 − 1) = 2. Hence if m > n, hvn , vm i = hAn+ v0 , Am + v0 i n = hAm − A+ v0 , v0 i

(since A− = A∗+ )

n−1 = hAm−1 (A+ A− + 2)A+ v0 , v0 i. −

After finitely many steps, the foregoing equals h(. . . )A− v0 , v0 i = 0, since A− v0 = 0. 4. Lastly, we demonstrate that the collection of eigenfunctions we have found spans L2 . Suppose hun , gi = 0 for n = 0, 1, 2, . . . ; we must show g ≡ 0. Now since Hn (x) = cn xn + . . . , with cn 6= 0, we have Z ∞ x2 g(x)e− 2 p(x) dx = 0 −∞

for each polynomial p. Hence Z Z ∞ 2 − x2 −ixξ e dx = g(x)e and so F

2

− x2

ge

2

g(x)e

− x2

−∞

−∞



∞



≡ 0. This implies ge−

∞ X (−ixξ)k k=0

x2 2

k!

dx;

≡ 0 and consequently g ≡ 0. 

6.1.2. Higher dimensions, rescaling. Suppose now n > 1, and write P0 := −∆ + |x|2 ; this is the n-dimensional quantum harmonic oscillator. We define also n n Y Y |x|2 uα (x) := uαj (xj ) = Hαj (xj )e− 2 j=1

j=1

for each multiindex α = (α1 , . . . , αn ). Then P0 uα = (−∆ + |x|2 )uα = (2|α| + n)uα , for |α| = α1 + · · · + αn . Hence uα is an eigenfunction of P0 corresponding to the eigenvalue 2|α| + n.

6.1. THE HARMONIC OSCILLATOR

121

We next restore the parameter h > 0 by setting P0 (h) := −h2 ∆ + |x|2 ,

(6.1.3) (6.1.4)

−n 4

uα (h)(x) := h

n Y

 Hαj

j=1

xj √ h



e−

|x|2 2h

,

and (6.1.5)

Eα (h) := (2|α| + n)h.

Then P0 (h)uα (h) = Eα (h)uα (h); and upon reindexing, we can write these eigenfunction equations as (6.1.6)

P0 (h)uj (h) = Ej (h)uj (h)

(j = 1, . . . ).

6.1.3. Asymptotic distribution of eigenvalues. With these explicit formulas in hand, we can study the behavior in the semiclassical limit of the eigenvalues E(h) of the harmonic oscillator: THEOREM 6.3 (Weyl’s Law for harmonic oscillator). Assume that 0 ≤ a < b < ∞. Then (6.1.7) #{E(h) | a ≤ E(h) ≤ b} =

1 (|{a ≤ |ξ|2 + |x|2 ≤ b}| + o(1)). (2πh)n

as h → 0. Proof. We may assume that a = 0. Since E(h) = (2|α| + n)h for some multiindex α according to (6.1.5), we have   b #{E(h) | 0 ≤ E(h) ≤ b} = # α | 0 ≤ 2|α| + n ≤ h = # {α | α1 + · · · + αn ≤ R} , for R := (b − nh)/2h. Therefore #{E(h) | 0 ≤ E(h) ≤ b} = |{x | xi ≥ 0, x1 + · · · + xn ≤ R}| + o(Rn ) 1 n R + o(Rn ) as R → ∞ n!  n 1 b = + o(h−n ) as h → 0. n! 2h

=

Note that that the volume of {x | xi ≥ 0, x1 + · · · + xn ≤ 1} is (n!)−1 .

122

6. EIGENVALUES AND EIGENFUNCTIONS

Next we observe that |{|ξ|2 + |x|2 ≤ b}| = α(2n)bn , where α(k) := π (Γ( k2 + 1))−1 is the volume of the unit ball in Rk . Setting k = 2n, we compute that α(2n) = π n (n!)−1 . Hence  n b 1 + o(h−n ) #{E(h) | 0 ≤ E(h) ≤ b} = n! 2h 1 = |{|ξ|2 + |x|2 ≤ b}| + o(h−n ).  (2πh)n k 2

6.2. SYMBOLS AND EIGENFUNCTIONS For this section, we return to the general symbol (6.0.1) and the quantized operator (6.0.2). We assume that the potential V : Rn → R is smooth, and satisfies the growth conditions: ( |∂ α V (x)| ≤ Cα hxik , (6.2.1) V (x) ≥ chxik for |x| ≥ R, for appropriate constants k, c, Cα , R > 0. Our plan in the next section is to employ our detailed knowledge about the eigenvalues of the harmonic oscillator (6.1.3) to estimate the asymptotics of the eigenvalues of P (h). 6.2.1. Concentration in phase space. First, we make the important observation that in the semiclassical limit the eigenfunctions u(h) “are concentrated in phase space” on the energy surface {|ξ|2 + V (x) = E}. THEOREM 6.4 (h∞ estimates). Suppose that u(h) ∈ L2 (Rn ) solves (6.2.2)

P (h)u(h) = E(h)u(h).

Assume as well that a ∈ S is a symbol satisfying {|ξ|2 + V (x) = E} ∩ spt(a) = ∅. Then if |E(h) − E| < δ for some sufficiently small δ > 0, we have the estimate (6.2.3)

kaw (x, hD)u(h)kL2 = O(h∞ )ku(h)kL2 .

Proof. 1. The set K := {|ξ|2 + V (x) = E} ⊂ R2n is compact. Hence there exists χ ∈ Cc∞ (R2n ) such that 0 ≤ χ ≤ 1, χ ≡ 1 on K, χ ≡ 0 on spt(a).

6.2. SYMBOLS AND EIGENFUNCTIONS

123

Define the symbol b := |ξ|2 + V (x) − E(h) + iχ = p − E(h) + iχ and the order function m := hξi2 + hxik . Therefore if |E(h) − E| is small enough, |b| ≥ γm

on R2n

for some constant γ > 0. Consequently b ∈ S(m), with b−1 ∈ S(m−1 ). 2. Thus there exist c ∈ S(m−1 ), r1 , r2 ∈ S such that ( bw (x, hD)cw (x, hD) = I + r1w (x, hD) cw (x, hD)bw (x, hD) = I + r2w (x, hD). where r1w (x, hD), r2w (x, hD) are O(h∞ ). Then (6.2.4)

aw (x, hD)cw (x, hD)bw (x, hD) = aw (x, hD) + O(h∞ ),

and (6.2.5)

bw (x, hD) = P (h) − E(h) + iχw (x, hD).

Furthermore aw (x, hD)cw (x, hD)χw (x, hD) = O(h∞ ), since spt(a) ∩ spt(χ) = ∅. Since P (h)u(h) = E(h)u(h), (6.2.4) and (6.2.5) imply that aw (x, hD)u = aw (x, hD)cw (x, hD)(P (h) − E(h) + iχw )u + O(h∞ ) = O(h∞ ).



For the next result, we temporarily return to the case of the quantum harmonic oscillator, developing some sharper estimates: THEOREM 6.5 (Improved estimates for the harmonic oscillator). Suppose that u(h) ∈ L2 (Rn ) is an eigenfunction of the harmonic oscillator: (6.2.6)

P0 (h)u(h) = E(h)u(h).

Assume also that a ∈ Cc∞ . Then there exists R > 0, depending only on the support of a, such that for E(h) > R, ∞   h w (6.2.7) ka (x, hD)u(h)kL2 = O ku(h)kL2 . E(h)

124

6. EIGENVALUES AND EIGENFUNCTIONS

The precise form of the right hand side of (6.2.7) will later let us handle eigenvalues E(h) → ∞. Proof. 1. We rescale the harmonic oscillator so that we can work near a fixed energy level E. Set x

˜ := h , E(h) ˜ := E(h) , , h E E E where we choose E so that |E(h) − E| ≤ E/4. Then put y :=

1 2

˜ := −h ˜ 2 ∆y + |y|2 ; P0 (h) := −h2 ∆x + |x|2 , P0 (h) whence ˜ − E( ˜ e h)). P0 (h) − E(h) = E(P (h) We next introduce the unitary transformation n

1

U u(y) := E 2 u(E 2 y). Then ˜ U P0 (h)U −1 = EP0 (h); and more generally ˜ eb(y, η) := b(E 21 y, E 12 η). U bw (x, hD)U −1 = ebw (y, hD), ˜ by the symbol Seδ . We will denote the symbol classes defined using h 2. We now apply Theorem 6.4. If ˜ − E(h))e ˜ u(h) ˜ = 0, |E(h) ˜ − 1| < δ, (P0 (h) and eb(y, η) ∈ Se has its support contained in {|y|2 + |η|2 ≤ 1/2}, then ˜ ∞ )ke ˜ L2 . ˜ u(h)k ˜ L2 = O(h kebw (y, hD)e u(h)k Translated to the original h and x as above, this assertion provides us with the bound (6.2.8)

kbw (x, hD)u(h)kL2 = O((h/E)∞ )ku(h)kL2 ,

for b(x, ξ) = eb(E −1/2 x, E −1/2 ξ) ∈ S. Note that spt(b) ⊂ {|x|2 + |ξ|2 ≤ E/2}. 3. In view of (6.2.8), we only need to show that for a ∈ C ∞ (R2n ), spt(a) ⊂ {|x|2 + |ξ|2 ≤ 1/4}, we have k(aw (x, hD)(1 − bw (x, hD))kL2 →L2 = O((h/E)∞ ),

6.2. SYMBOLS AND EIGENFUNCTIONS

125

for E large enough, where b is as in (6.2.8). That is the same as showing ˜ ˜ ˜∞ ke aw (y, hD)(1 − ebw (y, hD))k L2 →L2 = O(h ),

(6.2.9) for

1

1

e a(y, η) = a(E 2 y, E 2 η). ˜ < 1/h ˜ and hence We first observe that E = h/h e a ∈ Se1 . 2

Since the support of a is compact, we see that for E large enough, dist(spt(e a), spt(1 − eb)) ≥ 1/C > 0, ˜ The estimate (6.2.9) is now a consequence of Theorem uniformly in h. 4.25.  6.2.2. Projections. We next study how projections onto the span of various eigenfunctions of the harmonic oscillator P0 (h) are related to our symbol calculus. THEOREM 6.6 (Projections and symbols). Suppose for the symbol a ∈ S that spt(a) ⊂ {|ξ|2 + |x|2 < R}. Let Π := projection in L2 onto span{u(h) | P0 (h)u(h) = E(h)u(h) for E(h) ≤ R}. Then kaw (x, hD)(I − Π)kL2 →L2 = O(h∞ )

(6.2.10) and

k(I − Π)aw (x, hD)kL2 →L2 = O(h∞ ).

(6.2.11)

Proof. First of all, observe (I − Π) =

X

uj (h) ⊗ uj (h),

Ej (h)>R

meaning that (I − Π)u =

X

huj (h), uiuj (h).

Ej (h)>R

Therefore aw (x, hD)(I − Π) =

X Ej (h)>R

(aw (x, hD)uj (h)) ⊗ uj (h);

126

6. EIGENVALUES AND EIGENFUNCTIONS

and so 1

 (6.2.12)

2

X

w

ka (x, hD)(I − Π)kL2 →L2 ≤ 

w

ka

(x, hD)uj (h)k2L2 

.

Ej (h)>R

Next, observe that Weyl’s Law for the harmonic oscillator, Theorem 6.3, implies that 1

Ej (h) ≥ γj n h for some constant γ > 0. According then to Theorem 6.5, for each M < N we have N  h w ka (x, hD)uj (h)kL2 ≤ CN Ej (h) N −M  h M ≤ Ch Ej (h) ≤ ChM j −

N −M n

.

Consequently, if we fix N − M > n, the sum on the right hand side of (6.2.12) is less than or equal to ChM . This proves (6.2.10), and the proof of (6.2.11) is similar. 

6.3. SPECTRUM AND RESOLVENTS We next show that the spectrum of P (h) consists entirely of eigenvalues. THEOREM 6.7 (Resolvents and spectrum). (i) There exists a constant h0 > 0 such that if 0 < h ≤ h0 , then the resolvent (P (h) − z)−1 : L2 (Rn ) → L2 (Rn ) is a normal compact operator. (ii) The mapping z 7→ (P (h)−z)−1 is meromorphic, with real and simple poles. (iii) The spectrum of P (h) is discrete. Furthermore L2 (Rn ) has a complete orthonormal basis of eigenfunctions {uj (h)}∞ j=1 : (6.3.1)

P (h)uj (h) = Ej (h)uj (h)

(j = 1, 2, . . . ).

Proof. 1. Let m(x, ξ) := 1 + |ξ|2 + |x|k . Then p ∈ S(m), C|p + i| ≥ m, and P (h) = pw (x, hD).

6.3. SPECTRUM AND RESOLVENTS

127

For h small enough, mw (x, hD) is invertible. We can therefore define the Hilbert space: H := {u ∈ S 0 | (I − h2 ∆ + hxik )u ∈ L2 } = mw (x, hD)−1 L2 . For small h, the inverse (P (h) − i)−1 : L2 → H is bounded; and the compactness of H as a subspace of L2 follows from Theorem 4.28. 4. We now write P (h) − z = (I − K(z, h))(P (h) − i), for K(z, h) := (z + i)(P (h) − i)−1 . Since I − K(−i, h) = I and K(z, h) is compact, Theorem D.4 shows that z → (I − K(z, h))−1 is a meromorphic family of operators, with poles of finite rank. Consequently, (P (h) − z)−1 = (P (h) − i)−1 (I − K(z, h))−1 is a meromorphic family of compact operators from L2 to L2 . 3. Since the poles of (P (h) − z)−1 are discrete, there exists λ ∈ R for which P (h) − λ : H → L2 is invertible. Hence for any vj ∈ L2 , there exist uj ∈ H such that (P (h) − λ)uj = vj and h(P (h) − λ)−1 v1 , v2 i = h(P (h) − λ)−1 (P (h) − λ)u1 , (P (h) − λ)u2 i = hu1 , (P (h) − λ)u2 i. We integrate by parts, to find h(P (h) − λ)−1 v1 , v2 i = h(P (h) − λ)u1 , u2 i = hv1 , (P (h) − λ)−1 v2 i. Hence (P (h) − λ)−1 is self-adjoint. 4. We now apply part (vi) of Theorem C.7 to obtain an orthonormal set {uj (h)}Jj=1 and a sequence of real numbers {Ej (h)}Jj=1 such that (6.3.2)

(P (h) − λ)

−1

v=

J X

(Ej (h) − λ)−1 uj (h)hv, uj (h)i,

j=1

for all v ∈ L2 , where J ∈ N or J = ∞.

128

6. EIGENVALUES AND EIGENFUNCTIONS

5. Taking v = uj and applying P (h) − λ to both sides (6.3.2), we deduce that P (h)uj (h) = Ej (h)uj . Applying P (h) − λ to both sides of (6.3.2) for an arbitrary v ∈ L2 , we discover that v=

J X

uj (h)hv, uj (h)i.

j=1

Consequently the eigenfunctions {uj (h)}Jj=1 form a complete orthonormal set, and in particular J = ∞.  REMARK. Using (6.3.1) and the fact that V ∈ C ∞ we can apply Lemma 7.1 iteratively to conclude that u ∈ Hhl (Rn ) for all l, and in particular that u ∈ C ∞ (Rn ). Similarly we can we can use V (x) ≥ chxik − C to obtain hxiN u ∈ Hhl (Rn ). Putting this together we obtain uj (h) ∈ S ,

(6.3.3)

with seminorms depending on h.



REMARK: An alternative proof of meromorphy. To illustrate further the semiclassical calculus, we provide a different proof of the meromorphy of z 7→ (P − z)−1 for h small. 1. Let |z| ≤ E, where E is fixed; and as before let P0 (h) = −h2 ∆ + |x|2 be the harmonic oscillator. As in Theorem 6.6 define Π := projection in L2 onto span{u | P0 (h)u = E(h)u for E(h) ≤ R + 1}. Suppose now spt(a) ⊂ {|x|2 + |ξ|2 ≤ R}. Owing to Theorem 6.6, we have kaw (x, hD) − aw (x, hD)ΠkL2 →L2 = O(h∞ ). and kaw (x, hD) − Πaw (x, hD)kL2 →L2 = O(h∞ ). 2. Fix R > 0 so large that {|ξ|2 + V (x) ≤ E} ⊂ {|x|2 + |ξ|2 < R}. Select χ ∈ C ∞ (R2n ) with spt(χ) ⊂ {|x|2 + |ξ|2 ≤ R} so that |ξ|2 + V (x) − z + χ ≥ γm for m = hξi2 + hxik and all |z| ≤ E. Then χ = ΠχΠ + O(h∞ ). Recall that the symbolic calculus guarantees that P (h) − z + χ is invertible, if h is small enough. Consequently, so is P (h) − z + ΠχΠ, since the two operators differ by an O(h∞ ) term. 3. Now write P (h) − z = P (h) − z + ΠχΠ − ΠχΠ

6.4. WEYL’S LAW

129

Consequently P (h) − z = (P (h) − z + ΠχΠ)(I − (P (h) − z + ΠχΠ)−1 ΠχΠ). Note that ΠχΠ is an operator of finite rank. So Theorem D.4 asserts that the family of operators (I − (P (h) − z + ΠχΠ)−1 ΠχΠ)−1 is meromorphic in z. It follows that (P (h) − z)−1 is meromorphic on L2 . The poles are the eigenvalues, and the self-adjointness of P (h) implies these eigenvalues are real and simple. 

6.4. WEYL’S LAW We are now ready for the main result of this chapter: THEOREM 6.8 (Weyl’s Law). Suppose that V satisfies the conditions (6.2.1) and that E(h) are the eigenvalues of P (h) = −h2 ∆ + V (x). Then for each a < b, we have (6.4.1) #{E(h) | a ≤ E(h) ≤ b} =

1 (|{a ≤ |ξ|2 + V (x) ≤ b}| + o(1)). (2πh)n

as h → 0. Proof. 1. Let Select χ ∈

Cc∞ (R2n )

N (λ) = #{E(h) | E(h) ≤ λ}. so that

χ ≡ 1 on {p ≤ λ + }, χ ≡ 0 on {p ≥ λ + 2}. Then a := p + (λ + )χ − λ ≥ γ m, for m = + and some constant γ > 0. Hence a is elliptic; and so for small h > 0, aw (x, hD) is invertible. hξi2

hxim

2. Claim #1: We have (6.4.2)

h(P (h) + (λ + )χw (x, hD) − λ)u, ui ≥ γkuk2L2

for some γ > 0. To see this, take b ∈ S(m1/2 ) so that b2 = a. Then b2 = b#b + r0 , where r0 ∈ hS(m). We also recall from Theorem 4.29, or rather its proof, that bw (x, hD)−1 exists and bw (x, hD)−1 r0w (x, hD)bw (x, hD)−1 = OL2 →L2 (1).

130

6. EIGENVALUES AND EIGENFUNCTIONS

Thus aw (x, hD) = bw (x, hD)bw (x, hD) + r0w (x, hD) = bw (x, hD) (1 + bw (x, hD)−1 r0w (x, hD)bw (x, hD)−1 )bw (x, hD) = bw (x, hD)(1 + OL2 →L2 (h))bw (x, hD). Hence for sufficiently small h > 0, h(P (h) + (λ + )χw − λ)u, ui = haw (x, hD)u, ui ≥ kbw (x, hD)uk2L2 (1 − O(h)) ≥ γkuk2L2 , for some γ > 0, since bw (x, hD)−1 exists. This proves (6.4.2). 3. Claim #2: For each δ > 0, there exists a bounded linear operator Q such that χw (x, hD) = Q + OL2 →L2 (h∞ )

(6.4.3) and

rank(Q) ≤

(6.4.4)

1 (|{p ≤ λ + 2}| + δ). (2πh)n

To prove this, cover the set {p ≤ λ + 2} with balls Bj := B((xj , ξj ), rj )

(j = 1, · · · , N )

such that N X j=1

δ |Bj | ≤ |{p ≤ λ + 2}| + . 2

We then d!efine the “shifted” harmonic oscillator Pj (h) := |hDx − ξj |2 + |x − xj |2 ; and set Π := orthogonal projection in L2 onto V , the span of {u | Pj (h)u = Ej (h)u, Ej (h) ≤ rj , j = 1, . . . , N }. We now claim that (I − Π)χw (x, hD) = OL2 →L2 (h∞ ). P To see this, let χ = N j=1 χj , where spt χj ⊂ B((xj , ξj ), rj ), and put

(6.4.5)

Πj

:= orthogonal projection in L2 onto the span of {u | Pj (h)u = Ej (h)u, Ej (h) ≤ rj }.

6.4. WEYL’S LAW

131

∞ Theorem 6.6 shows that (I − Πj )χw j (x, hD) = O(h ). We note that ΠΠj = Πj and hence w

(I − Π)χ (x, hD) =

N X

(I − Π)χw j (x, hD)

j=1

=

N X

(I − Π)(I − Πj )χw j (x, hD)

j=1

= OL2 →L2 (h∞ ). This proves (6.4.5). It now follows that χw (x, hD) = Πχw (x, hD) + (I − Π)χw (x, hD) = Q + O(h∞ ) for Q := Πχw (x, hD). Clearly Q has finite rank, since rank Q = dim(image of Q) ≤ dim(image of Π) ≤

N X

#{Ej (h) | Ej (h) ≤ rj }

j=1

 =

1  (2πh)n

N X

 |Bj | + o(1) ,

j=1

according to Weyl’s law for the harmonic oscillator, Theorem 6.3. Consequently   δ 1 (6.4.6) rank Q ≤ |{p ≤ λ + 2}| + + o(1) . (2πh)n 2 This proves Claim #2. 4. We next employ Claims #1,2 and Theorem C.15. We have hP (h)u, ui ≥ (λ + γ)kuk2L2 − (λ + )hQu, ui + hO(h∞ )u, ui ≥ λkuk2L2 − (λ + )hQu, ui, where the rank of Q is bound by (6.4.6). Theorem C.15,(i) implies then that 1 N (λ) ≤ (|{p ≤ λ + 2}| + δ + o(1)). (2πh)n This holds for all , δ > 0, and so 1 (6.4.7) N (λ) ≤ (|{p ≤ λ}| + o(1)) (2πh)n

132

6. EIGENVALUES AND EIGENFUNCTIONS

as h → 0. 5. We must prove the opposite inequality. Claim #3: Suppose Bj = B((xj , ξj ), rj ) ⊂ {p < λ} and put Vj := span{u | Pj (h)u = Ej (h)u, Ej (h) ≤ rj }. We claim that for u ∈ Vj . (6.4.8)

hP (h)u, ui ≤ (λ +  + O(h∞ ))kuk2L2 .

To prove this claim, select a symbol a ∈ Cc∞ (R2n ), with a ≡ 1 on {p ≤ λ}, spt(a) ⊂ {p ≤ λ + }. Let c := 1 − a. Then u − aw (x, hD)u = cw (x, hD)u = O(h∞ ) according to Theorem 6.6, since spt(1 − a) ∩ Bj = ∅. Define bw := P (h)aw (x, hD). Now p ∈ S(m) and a ∈ S(m−1 ). Thus b = pa + O(h) ∈ S and so bw is bounded in L2 . Observe also that b ≤ λ + 2 , and so 3 bw (x, hD) ≤ λ + . 4 Therefore   3 w w kuk2L2 . hP (h)a (x, hD)u, ui = hb (x, hD)u, ui ≤ λ + 4 Since aw (x, hD)u = u + O(h∞ ), we deduce hP (h)u, ui ≤ (λ +  + O(h∞ ))kuk2L2 . This proves Claim #3. 6. Now find disjoint balls Bj ⊂ {p < λ} such that |{p < λ}| ≤

N X

|Bj | + δ.

j=1

and denote V = V1 + V2 · · · + VN . The spaces Vi and Vj , i 6= j are not orthogonal; but because Bi and Bj are disjoint we see, as in Theorem 6.6, that (6.4.9)

hu, vi = O(h∞ )kukkvk

if u ∈ Vi , v ∈ Vj and i 6= j . Since each Vj has an orthonormal basis of eigenvectors (6.4.8) holds for u ∈ Vj . The approximate orthogonality (6.4.9) then gives hP u, ui ≤ (λ + δ)kuk2L2

6.5. NOTES

133

for all u ∈ V . Also, (6.4.9) and Theorem 6.3 imply that for h small enough dim V

=

N X

dim Vj

j=1

=

N X

#{Ej (h) ≤ rj }

j=1

=

  N X 1  |Bj | + o(1) (2πh)n j=1

≥

1 (|{p < λ}| − δ + o(1)). (2πh)n

According then to Theorem C.15,(ii), 1 N (λ) ≥ (|{p < λ}| − δ + o(1)). (2πh)n 

6.5. NOTES The proof of Weyl asymptotics is a semiclassical version of the classical Dirichlet-Neumann bracketing proof for the bounded domains. In Chapter 12 we will present a more general form of Weyl’s Law, proved using a functional calculus of pseudodifferential operators. That proof leads to many further improvements: see Dimassi-Sj¨ostrand [D-S]. The proof using min-max principle comparisons with harmonic oscillator has the advantage of providing upper bounds for the number of eigenvalues of nonself-adjoint operators.

Chapter 7

ESTIMATES FOR SOLUTIONS OF PDE

7.1 7.2 7.3 7.4 7.5 7.6

Classically forbidden regions Tunneling Order of vanishing L∞ estimates for quasimodes Schauder estimates Notes We continue our study of semiclassical behavior of eigenfunctions:

(7.0.1)

P (h)u(h) = E(h)u(h)

for the operator P (h) = −h2 ∆ + V (x) and corresponding symbol p(x, ξ) = |ξ|2 + V (x). We assume that V ∈ S(hxia ) for some a ∈ R such that p ∈ S(m), for m = hξi2 + hxia . We first demonstrate that if E(h) is close to the energy level E, then u(h) exponentially small within the classically forbidden region V −1 (E, ∞) = {x ∈ Rn | V (x) > E}. Then we show, conversely, that in any open set the L2 norm of u(h) is bounded from below by a quantity exponentially small in h. We conclude 135

136

7. ESTIMATES FOR SOLUTIONS OF PDE

with a discussion of the order of vanishing of eigenfunctions in the semiclassical limit.

7.1. CLASSICALLY FORBIDDEN REGIONS We first need an important definition and a useful Lemma: DEFINITION. Let U ⊂ Rn be an open set. The semiclassical Sobolev norms are defined as  1/2 X Z |(hD)α u|2 dx kukH k (U ) :=  h

U

|α|≤k

for u ∈ C ∞ (U ), k = 0, 1, . . . . If u ∈ Hhk (V ) for any V ⊂⊂ U we write k (U ). u ∈ Hh,loc These modified spaces differ from the standard Sobolev norms by the introduction of appropriate powers of h. LEMMA 7.1 (Semiclassical elliptic estimates). Suppose that Q(h) := −h2 ∆ + ha(x), hDui + b(x) where the coefficients a, b are smooth and complex-valued. Assume also that W ⊂⊂ U ⊂⊂ Rn are open sets. Then there exists a constant C such that kukH 2 (W ) ≤ C(kP (h)ukL2 (U ) + kukL2 (U ) )

(7.1.1) for all u ∈

h

C ∞ (U ).

Proof. 1. Let χ ∈ Cc∞ (U ), χ ≡ 1 on W . We multiply P (h)u by χ2 u ¯ and integrate by parts: Z Z 2 Re P (h)u¯ uχ dx = h2 h∂(χ2 u ¯), ∂ui + Reha, hDuiχ2 u ¯ U

U

+ Re b(x)|u|2 χ2 dx Z

χ2 |hDu|2 − Cχ|hDx u||u| − C|u|2 dx Z Z 1 2 2 0 ≥ χ |hDu| dx − C |u|2 dx 2 U U ≥

U

Therefore Z

2

Z

|hDu| dx ≤ C W

U

|P (h)u|2 + |u|2 dx.

7.1. CLASSICALLY FORBIDDEN REGIONS

137

2. We first note that for u ∈ Cc∞ (U ) integration by parts gives   ! Z Z n n X X  | − ∆u|2 dx = ∂x2j u ∂x2k u ¯ dx U

U

=

j=1

n X

Z

i,k=1 U

=

n Z X i,k=1 U

k=1

∂x2j u ∂x2k u ¯ dx |∂x2j xk u| dx

Z =

|D2 h|2 dx.

U

Hence we proceed as in part 1 of the proof by multiplying P (h)u by −χ2 h2 ∆¯ u and integrating by parts. This gives Z Z 2 2 |(hD) u| dx ≤ C |P (h)u|2 + |u|2 dx. U

W

 Before turning again to eigenfunctions, we present the following general estimates. Our primary tool will be properly designed conjugations of the operator P (h). DEFINITION Given ϕ ∈ C ∞ (Rn ), we define the conjugation of P (h) by eϕ/h : Pϕ (h) := eϕ/h P e−ϕ/h .

(7.1.2)

LEMMA 7.2 (Symbol of conjugation). We have Pϕ (h) = pw ϕ

(7.1.3) for the symbol

pϕ (x, ξ) := hξ + i∂ϕ(x), ξ + i∂ϕ(x)i + V (x).

(7.1.4)

Proof. We calculate for functions u ∈ C ∞ (Rn ) that Pϕ (h)u = eϕ/h (−h2 ∆ + V )(e−ϕ/h u) = −h2 ∆u + 2hh∂ϕ, ∂ui − |∂ϕ|2 u + V u + h∆ϕu. On the other hand, Theorem 4.5 shows that pw ϕ (x, hD)u = − h2 ∆ + i (h∂ϕ(x), hDx u(x)i + hhDx , ∂ϕ(x)u(x)i) + V (x). This proves (7.1.3).



138

7. ESTIMATES FOR SOLUTIONS OF PDE

THEOREM 7.3 (Exponential estimate from above). Suppose that U is an open set such that U ⊂⊂ V −1 (E, ∞). Then for each open set W ⊃⊃ U and for each λ near E, there exist constants h0 , δ, C > 0, such that (7.1.5)

kukL2 (U ) ≤ Ce−δ/h kukL2 (W ) + Ck(P (h) − λ)ukL2 (W )

for u ∈ Cc∞ (Rn ) and 0 < h ≤ h0 . We call (7.1.5) an Agmon estimate. Proof. 1. Select ψ, ϕ ∈ Cc∞ (W ) such that 0 ≤ ψ, ϕ ≤ 1, ψ ≡ 1 on U , and ϕ ≡ 1 on spt ψ. We may assume as well that W ⊂⊂ V −1 (E, ∞). As in Lemma 7.2, we observe that the symbol of A(h) := eδψ/h (P (h) − λ)e−δψ/h is hξ + iδ∂ψ, ξ + iδ∂ψi + V − λ. Now for λ close to E, x ∈ W and δ sufficiently small, we have (7.1.6) |hξ + iδ∂ψ, ξ + iδ∂ψi + V − λ|2 ≥ |ξ 2 + V − δ 2 |∂ϕ|2 − λ|2 ≥ γ 2 hξi2 > 0 for some positive constant γ. 2. Let ϕ1 have the same properties as ϕ and be equal to one on spt ϕ. The lower bound (7.1.6) implies that
B := hhDi−2 ϕ1 A(h)∗ A(h)ϕ1 − γ 2 ϕ21 hhDi−2 = bw (x, hD) for a symbol b ∈ S such that b(x, ξ) ≥ −OS (h). According to the sharp G˚ arding inequality, Theorem 4.32, we see that provided δ > 0 is sufficiently small (so that (7.1.6) holds), hBv, vi ≥ −Chkvk2L2 . Putting v = hhDi2 ϕw in this inequality gives kA(h)ϕwk2L2 ≥ γ 2 kϕwk2L2 − ChkhhDi2 ϕwk2L2 2 for w ∈ Hh,loc (Rn ).

Lemma 7.1 applied with W such that spt ϕ ⊂⊂ W shows that khhDi2 ϕwkL2 ≤ CkϕwkH 2 ≤ C 0 (kϕwkL2 + kA(h)ϕwkL2 ) . h

7.2. TUNNELING

139

Hence for h small enough, we have kA(h)ϕwkL2 ≥

(7.1.7)

γ kϕwkL2 2

2 when w ∈ Hh,loc (Rn ).

3. We now apply (7.1.7) with w = eδψ/h : keδψ/h ϕukL2 ≤ CkA(h)(eδψ/h ϕu)kL2 = Ckeδψ/h (P (h) − λ)ϕukL2 ≤ Ckeδψ/h ϕ(P (h) − λ)ukL2 + Ckeδψ/h [P (h), ϕ]ukL2 , for u ∈ Cc∞ (Rn ). Next is the key observation that since ϕ ≡ 1 on spt ψ, we have ψ ≡ 0 on spt [P (h), ϕ]u. Thus Lemma 7.1 implies
keδψ/h [P (h), ϕ]ukL2 = k[P (h), ϕ]ukL2 ≤ C khDx ukL2 (W ) + kukL2 (W ) ≤ CkukL2 (W ) + Ck(P (h) − λ)ukL2 (W ) . Combining these estimates, we conclude that eδ/h kukL2 (U ) ≤ keδψ/h ϕukL2 ≤ CkukL2 (W ) + C(eδ/h + 1)k(P (h) − λ)ukL2 (W ) . Mutliplying both sides by e−δ/h gives the estimate (7.1.5).



Specializing to eigenfunctions, we deduce THEOREM 7.4 (Exponential decay estimates). Suppose that U ⊂⊂ V −1 (E, ∞), and that u(h) ∈ L2 (Rn ) solves P (h)u(h) = E(h)u(h), where E(h) → E

as h → 0.

Then there exists a constant δ > 0 such that (7.1.8)

ku(h)kL2 (U ) ≤ e−δ/h ku(h)kL2 (Rn ) .

7.2. TUNNELING In this section we assume in this section u = u(h) solves the eigenvalue problem (7.0.1) and we assume that V ∈ S(hxik ) for some k > 0 and that for some constant C > 0, (7.2.1)

V (x) ≥ hxik /C for |x| ≥ C.

140

7. ESTIMATES FOR SOLUTIONS OF PDE

In the previous section we showed that u(h) is exponentially small in the physically forbidden region. In this section we will show that it can never be smaller than this: for small h > 0 and any bounded, open subset U of Rn , we have the lower bound C

kukL2 (U ) ≥ e− h kukL2 (Rn ) . This is a mathematical version of quantum mechanical “tunneling into the physically forbidden region”. It is closely related to unique continuation for solutions of second order elliptic equations. DEFINITION. H¨ ormander’s hypoellipticity condition is the requirement for the symbol pϕ , defined by (7.1.4), that (7.2.2)

if pϕ = 0,

then i{pϕ , pϕ } > 0.

Observe that for any complex function q = q(x, ξ), i{q, q¯} = i{Re q + iIm q, Re q − iIm q} = 2{Re q, Im q}. Hence the expression i{pϕ , pϕ } is real. THEOREM 7.5 (L2 -estimate for Pϕ (h)). Let W ⊂⊂ Rn be an open set. If H¨ ormander’s hypoellipticity condition (7.2.2) is valid within W , then h1/2 kukL2 (W ) ≤ CkPϕ (h)ukL2 (W )

(7.2.3)

for all u ∈ Cc∞ (W ), provided 0 < h ≤ h0 with h0 > 0 sufficiently small. Proof. We calculate kPϕ (h)uk2L2

= hPϕ (h)u, Pϕ (h)ui = hPϕ∗ (h)Pϕ u, ui = hPϕ (h)Pϕ∗ (h)u, ui + h[Pϕ∗ (h), Pϕ (h)]u, ui = kPϕ∗ (h)uk2L2 + h[Pϕ∗ (h), Pϕ (h)]u, ui.

The idea will be to use the positivity of the second term on the right hand side wherever Pϕ∗ (h) fails to be elliptic. More precisely, for any M > 1 and h small enough the calculation above gives kPϕ (h)uk2L2 ≥ M hkPϕ∗ (h)uk2L2 + h[Pϕ∗ (h), Pϕ (h)]u, ui = hh(M |pϕ |2 + i{pϕ , p¯ϕ })w u, ui − O(h2 )kuk2H 2 , h

the last term resulting from estimates of the lower order terms in p¯ϕ #pϕ and the commutator. H¨ ormander’s hypoellipticity condition (7.2.2) implies for M large enough that M |pϕ (x, ξ)|2 + i{pϕ , p¯ϕ }(x, ξ) ≥ γ 2 hξi2 > 0.

7.2. TUNNELING

141

¯ . Then the sharp G˚ for x ∈ W arding inequality (Theorem 4.32) applied as in the proof of Theorem 7.3, and Lemma 7.1 show us that kPϕ (h)uk2L2 ≥ Chkuk2L2 − O(h2 )(kPϕ (h)uk2L2 + kuk2L2 ).  Next we carefully design a weight ϕ, to ensure that Pϕ (h) satisfies the hypothesis of Theorem 7.5. LEMMA 7.6 (Constructing a weight). Let 0 < r < R. There exists a nonincreasing radial function ϕ ∈ C ∞ (Rn ) which satisfies the H¨ ormander hypoellipticity condition (7.2.2) in B(0, R) \ B(0, r). Proof. 1. Recall that pϕ = |ξ|2 + 2ihξ, ∂ϕi − |∂ϕ|2 + V − E. So pϕ = 0 implies both |ξ|2 − |∂ϕ|2 + V − E = 0

(7.2.4) and

hξ, ∂ϕi = 0.

(7.2.5) Furthermore,

i {pϕ , pϕ } = {Re pϕ , Im pϕ } 2 = h∂ξ (|ξ|2 − |∂ϕ|2 + V − E), 2∂x hξ, ∂ϕii −h∂x (|ξ|2 − |∂ϕ|2 + V − E), 2∂ξ hξ, ∂ϕii

(7.2.6)

= 4h∂ 2 ϕ ξ, ξi + 4h∂ 2 ϕ ∂ϕ, ∂ϕi − 2h∂V, ∂ϕi. 2. Assume now ϕ = eλψ , where λ > 0 will be selected and ψ : Rn → R is positive and radial, ψ = ψ(|x|). Then ∂ϕ = λ∂ψeλψ and ∂ 2 ϕ = (λ2 ∂ψ ⊗ ∂ψ + λ∂ 2 ψ)eλψ . Hence h∂ 2 ϕ ξ, ξi = (λ2 h∂ψ, ξi2 + λh∂ 2 ψ ξ, ξi)eλψ = λh∂ 2 ψ ξ, ξieλψ , since (7.2.5) implies h∂ψ, ξi = 0. Also h∂ 2 ϕ ∂ϕ, ∂ϕi = λ4 |∂ψ|4 e3λψ + λ3 h∂ 2 ψ ∂ψ, ∂ψie3λψ ,

142

7. ESTIMATES FOR SOLUTIONS OF PDE

and h∂V, ∂ϕi = λh∂V, ∂ψieλψ . According to (7.2.6), we have (7.2.7)

i {pϕ , pϕ } = 4λh∂ 2 ψ ξ, ξieλψ + 4λ4 |∂ψ|4 e3λψ 2 + 4λ3 h∂ 2 ψ ∂ψ, ∂ψie3λψ − 2λh∂V, ∂ψieλψ .

3. Now take ψ := µ − |x|, for a constant µ so large that ψ ≥ 1 on B(0, R). Then ϕ is radial and nonincreasing. Furthermore |∂ψ| = 1, |∂ 2 ψ| ≤ C

on B(0, R) \ B(0, r).

Owing to (7.2.4) we have |ξ|2 ≤ C + |∂ϕ|2 ≤ C + Cλ2 e2λψ

on B(0, R) \ B(0, r).

Inserting these estimates into (7.2.7), we compute i {pϕ , pϕ } ≥ 2λ4 e3λψ − Cλ3 e3λψ − C ≥ 1, 2 in B(0, R) \ B(0, r), if λ is selected large enough. Lastly, we modify ψ within B(0, r) to obtain a smooth function on B(0, R).  THEOREM 7.7 (Exponential estimate from below). Let a < b and suppose U ⊂⊂ Rn is an open set. There exist constants C, h0 > 0 such that if u(h) solves P (h)u = E(h)u(h) in Rn for E(h) ∈ [a, b] and 0 < h ≤ h0 , then (7.2.8)

C

ku(h)kL2 (U ) ≥ e− h ku(h)kL2 (Rn ) .

We call (7.2.8) a Carleman estimate. REMARK. The condition (7.2.1) can easily be relaxed to allow k = 0. In that case we have to assume that for V (x) − b ≥ 1/C for |x| ≥ C.  Proof. 1. We may assume without loss that U = B(0, 3r) for some 0 < r < 1 3 . Select R > 1 so large that p(x, ξ) − λ = |ξ|2 + V (x) − λ ≥ |ξ|2 + hxik /C

7.2. TUNNELING

143

for |x| ≥ R and a ≤ λ ≤ b. Since p−E(h) is therefore elliptic on Rn \B(0, R), we have the estimate kvkL2 (Rn \B(0,R)) ≤ Ck(P (h) − E(h))vkL2 (Rn \B(0,R))

(7.2.9)

for all v ∈ Cc∞ (Rn \ B(0, R)), and h small enough. In fact, we can consider P (h) + V0 (x) − λ where spt V0 ⊂ B(0, R) which is elliptic everywhere in S(hxi2 + hxik ). Then for all v ∈ Cc∞ (Rn ), and a ≤ λ ≤ b, k(P (h) + V0 (x) − λ)vkL2 ≥ kvkL2 by Theorem 4.29. The term V0 disappears when v ∈ Cc∞ (Rn \ B(0, R)). 2. Select two radial functions χ1 , χ2 : Rn → R such that 0 ≤ χ1 ≤ 1 and   χ1 ≡ 0 on B(0, r), χ1 ≡ 1 on B(0, R + 2) \ B(0, 2r),   χ1 ≡ 0 on R2n \ B(0, R + 3); and 0 ≤ χ2 ≤ 1, ( χ2 ≡ 0 χ2 ≡ 1

on B(0, R) on R2n \ B(0, R + 1).

Applying (7.2.9) to v = χ2 u gives kχ2 ukL2 ≤ Ck(P (h) − E(h))(χ2 u)kL2 = Ck[P (h), χ2 ]ukL2 . Now [P (h), χ2 ]u = −h2 u∆χ2 − 2h2 h∂χ2 , ∂ui, and consequently [P (h), χ2 ]u is supported within B(0, R+1)\B(0, R). Hence Lemma 7.1 implies k[P (h), χ2 ]ukL2 ≤ ChkukH 1 (B(0,R+1)\B(0,R)) h

≤ Ch(k(P (h) − E(h))ukL2 (B(0,R+2)\B(0,R−1)) + kukL2 (B(0,R+2)\B(0,R−1)) ) ≤ ChkukL2 (B(0,R+2)\B(0,R−1)) ≤ Chkχ1 ukL2 Therefore kχ2 ukL2 ≤ Chkχ1 ukL2 .

(7.2.10)

3. Next apply Theorem 7.5: ϕ

ϕ

ϕ

h1/2 ke h χ1 ukL2 ≤ Cke h (P (h) − E(h))(χ1 u)kL2 = Cke h [P (h), χ1 ]ukL2 Now [P (h), χ1 ] is supported within the union of B(0, 2r) \ B(0, r) and B(0, R + 3) \ B(0, R + 2). Since ϕ is nonincreasing, we therefore have

144

7. ESTIMATES FOR SOLUTIONS OF PDE

ϕ

ke h [P (h), χ1 ]ukL2 ≤ Che

ϕ(R+2) h

kχ2 ukH 1 (B(0,R+3)\B(0,R+2)))) h

+ Che

ϕ(0) h

kukH 1 (B(0,2r) . h

The right hand sides can be estimated by Lemma 7.1. This gives ϕ

ke h χ1 ukL2 ≤ Ch1/2 e

(7.2.11)

ϕ(R+2) h

kχ2 ukL2 + Ch1/2 e

ϕ(0) h

kukL2 (U ) .

4. Put A = ϕ(R + 2). We observe that e

2A h

χ21 ≤ 2(e

2ϕ h

χ21 + e

2A h

χ22 ).

Hence multiplying (7.2.10) by eA/h gives  ϕ  A A ke h χ2 ukL2 ≤ Ch ke h χ1 ukL2 + ke h χ2 ukL2 . Adding this to (7.2.11) results in ϕ

A

ke h χ2 ukL2 + ke h χ1 ukL2 ϕ

A

≤ Chke h χ1 ukL2 + Ch1/2 ke h χ2 ukL2 + Ch1/2 e

ϕ(0) h

kukL2 (U ) .

Take 0 < h ≤ h0 , for h0 sufficiently small, to deduce ϕ

A

ke h χ2 ukL2 + ke h χ1 ukL2 ≤ Ch1/2 e

ϕ(0) h

kukL2 (U ) .

This gives, upon our using one more time the fact that ϕ is non-increasing, that 2ϕ(0) kχ2 ukL2 + kχ1 ukL2 ≤ Ch1/2 e h kukL2 (U ) . Since χ21 +χ22 ≥ 1/2 on Rn \B(0, 2r) ⊃ Rn −U , the Theorem follows.



7.3. ORDER OF VANISHING Assume, as usual, that (7.3.1)

P (h)u(h) = E(h)u(h),

where E(h) ∈ [a, b]. To simplify notation, we will in this subsection write u for u(h). DEFINITION. We say u vanishes to order N at the point x0 if u(x) = O(|x − x0 |N )

as x → x0 .

We will consider potentials which are analytic in x and, to avoid technical difficulties, make a strong assumption on the growth of derivatives: (7.3.2)

C + V (x) ≥ hxim /C, |∂ α V (x)| ≤ C 1+|α| |α||α| hxim

for some m > 0, a constant C > 0 and all multiindices α.

7.3. ORDER OF VANISHING

145

We note that the second condition holds when V has a holomorphic extension bounded by |z|m into a conic neighborhood of Rn in Cn . THEOREM 7.8 (Semiclassical estimate on vanishing order). Suppose that u ∈ L2 solves (7.3.1) for a ≤ E(h) ≤ b and that V is a real analytic potential satisfying (7.3.2). Let K be compact subset of Rn . There exists a constant C such that if u vanishes to order N at a point x0 ∈ K, we have the estimate N ≤ Ch−1 .

(7.3.3)

The proof will be given later, after we first establish the analyticity of the solution in a semiclassically quantitative way. This provides an h-dependent estimate on the derivatives of solutions to (7.3.1). THEOREM 7.9 (Semiclassical derivative estimates). If u satisfies the assumptions of Theorem 7.8, then there exists a constant C1 such that for any positive integer k: kukH k (Rn ) ≤ C1k (1 + kh)k kukL2 (Rn ) .

(7.3.4)

h

Proof. 1. By adding C0 to V we can assume without loss that V (x) ≥ hxim /C0 . The Lemma will follow from the following stronger estimate, which we will prove by induction: khxim/2 (hD)α ukL2 + k(h∂)(hD)α ukL2

(7.3.5)

≤ C2k+2 (1 + kh)k+1 kukL2 .

for |α| = k. 2. To prove this inequality, we observe first that by multiplying (7.3.1) by u ¯ and integrating by parts, estimate (7.3.5) holds for |α| = 0. Next, note that 1

kV 2 (hD)α uk2L2 + k(h∂)(hD)α uk2L2 = h(−h2 ∆ + V − E(h))(hD)α u, (hD)α ui + E(h)k(hD)α uk2L2 1

1

= hV − 2 [V, (hD)α ]u, V 2 (hD)α ui + E(h)k(hD)α uk2L2 1 1 1 ≤ kV − 2 [V, (hD)α ]uk2L2 + kV 2 (hD)α uk2L2 + E(h)k(hD)α uk2L2 . 4

Hence (7.3.6)

1 1 kV 2 (hD)α uk2L2 + k(h∂)(hD)α uk2L2 2 1

≤ kV − 2 [V, (hD)α ]uk2L2 + E(h)k(hD)α uk2L2 .

146

7. ESTIMATES FOR SOLUTIONS OF PDE

3. We can now expand the commutator, and use (7.3.2) (with V replaced by V + C0 ) to obtain the following inequality: for |α| = k we have 1

(7.3.7)

kV − 2 [V, (hD)α ]ukL2 ≤ k−1   3 X k 2 C0k−l (h(k − l))k−l sup khxim/2 (hD)β ukL2 . C0 l |β|=l l=0

This follows from the Leibnitz rule k   X k k−l ∂ (wv) = ∂ w ∂ l v. l k

l=0

4. We prove (7.3.5) by induction, and thus assume that (7.3.5) is valid for |α| < k. Now Stirling’s formula implies   k kk ≤C l . l l (k − l)k−l Hence, in view of (7.3.6) and (7.3.7), it is enough to show that there exists a constant C2 such that 3 2

C0

k−1 X

hk−l k k l−l C0k−l C2l+2 (1 + lh)l+1 + C2k+1 (1 + hk)k

l=0

≤ C2k+2 (1 + hk)k+1 . This estimate we rewrite as  k−1  3 X C0 k−l 2 C0 (hl)−l (1 + hl)l (1 + hl) + C2−1 (hk)−k (1 + hk)k C2 l=1

≤ (hk)−k (1 + hk)k (1 + hk). Since we can choose C2 to be large and since we can estimate the (1 + hl) factor in the sum by (1 + hk), this will follow once we show that for  small enough, k−1 X k−l al ≤ ak for al := (1 + (hl)−1 )l . l=0

This is true by induction if ak−1 /ak is bounded: the induction hypothesis and ak−1 ≤ C3 ak imply k−1 X l=0

k−l



al = 

k−2 X l=0

and we need  ≤ 1/(2C3 ).

k−1−l al + ak−1 ≤ 2ak−1 ≤ 2C3 ak ,

7.3. ORDER OF VANISHING

147

For our ak = (1 + (hk)−1 )k , k−1  ak−1 1 + (h(k − 1))−1 (1 + (hk)−1 )−1 = ak 1 + (hk)−1  k−1 1 = 1+ (1 + (hk)−1 )−1 (k − 1)(1 + hk)   hk 1 ≤ exp < 1. 1 + hk 1 + hk This completes the proof of (7.3.5), a stronger statement than (7.3.4).



Proof of Theorem 7.8: 1. Assume that kukL2 = 1 and that u vanishes to order N at a point x0 ∈ K. Then Dα u(x0 ) = 0 for |α| < N and Taylor’s formula shows that |u(x)| ≤

(7.3.8)

N sup sup |Dα u(y)| N ! |α|=N y∈Rn

for |x − x0 | < .

Lemma 3.5 and Theorem 7.9 allow us to estimate the derivatives. If M = N + n, |α| = N, then sup |Dα u(y)| ≤ kukH M ≤ h−M kukH M ≤ h−M C1M (1 + hM )M . h

y∈Rn

Inserting this into (7.3.8) and using Stirling’s formula, we deduce that for |x − x0 | < ,  e N  C M 1 (1 + hM )M |u(x)| ≤ C N h !M 1/n e C  1 ≤ Nn (1 + hM )M . Nh 2. If we put A := M h,

1 := C1 1/n e < 1, K

then −1

|u(x)| ≤ (Ah−1 )n (KA)−Ah (1 + A)Ah = (Ah−1 )n (1 + 1/A)Ah

−1

−1

exp(−Ah−1 log K)

≤ exp −Ah−1 (log K − 1 − nh(log(1/h) + log A)/A) We can assume that A is large, as otherwise N < M = Ah−1 ,




148

7. ESTIMATES FOR SOLUTIONS OF PDE

in which case the conclusion of the Theorem holds. Hence, taking  small enough, depending only on C1 and n, |u(x)| ≤ exp(−αAh−1 ), for |x − x0 | < , and a fixed α > 0 depending on . 3. It follows that Z

|u|2 dx ≤ C2 n e−2αA/h ,

{|x−x0 |<}

uniformly in h. But according to Theorem 7.7, Z |u|2 dx > e−C3 /h . {|x−x0 |<}

Consequently A = M h = (N + n)h is bounded, and this means that N ≤ Ch−1 , as claimed.  EXAMPLE: Optimal order of vanishing. Theorem 7.8 is optimal in the semiclassical limit, meaning as regards the dependence on h in estimate (7.3.3). We can see this by considering the harmonic oscillator in dimension n = 2. In polar coordinates (r, θ) the harmonic oscillator for h = 1 takes the form P0 = r−2 ((rDr )2 + Dθ2 + r4 ). The eigenspace corresponding to the eigenvalue 2k + 2 has dimension k + 1, corresponding to the number of multiindices α = (α1 , α2 ), with |α| = α1 + α2 = k. Separating variables, we look for eigenfunctions of the form u = ukm (r)eimθ . Then r−2 ((rDr )2 + m2 + r4 − (2k + 2)r2 )ukm (r) = 0. Since the number of linearly independent eigenfunctions is k + 1, there must be solution for some integer m > k/2. Near r = 0, we have the asymptotics ukm ' r±m , and the case ukm ' r−m is impossible since u ∈ L2 . Therefore u ' rm has to vanish to order m. Rescaling to the semiclassical case, we see that for the eigenvalue E(h) = (2k + 2)h ' 1 we have an eigenfunction vanishing to order ' 1/h. 

7.4. L∞ ESTIMATES FOR QUASIMODES

149

7.4. L∞ ESTIMATES FOR QUASIMODES Next we show how a natural frequency localization condition on approximate solutions to pseudodifferential equations implies h-dependent L∞ bounds. As an application we will provide bounds on eigenfunction clusters for compact Riemannian manifolds. 7.4.1. Quasimodes. We consider again in this section families of approximate solutions {u(h)}0
P u(h) = OL2 (h),

and u(h) = OL2 (1). This bound is valid for eigenvalue clusters: see Theorem 7.13. Finer quasimodes require that (7.4.2)

P u(h) = oL2 (h),

and the finest quasimodes are defined by conditions such as (7.4.3)

P u(h) = OS (h∞ ).

Localization. Under some further assumptions, quasimodes can be localized. For instance, if (7.4.1) holds and χ ∈ S , then we have (7.4.4)

P χw (x, hD)u = χw (x, hD)P u + [P, χw (x, hD)]u = OL2 (h),

since we gain a factor h from the commutator, as noted after Theorem 4.24. This means that the condition (7.4.1) can be achieved without relying on global properties of P . On the other hand, we saw in Theorem 5.5 that the finer quasimodes (7.4.2) have semiclassical defect measure invariant under the flow of p. Their construction requires global considerations. 7.4.2. Nondegeneracy condition and L∞ bounds. We now pose a basic question: What pointwise bounds are valid for approximate, L2 normalized, solutions of P u(h) = 0 as h → 0 ? We start with the following semiclassical version of Lemma 3.5: THEOREM 7.10 (Basic L∞ bounds). Suppose that {u(h)}0
k(1 − ψ(hD))u(h)kH k (Rn ) = O(h∞ )ku(h)kL2 (Rn ) h

150

7. ESTIMATES FOR SOLUTIONS OF PDE

Then (7.4.6)

ku(h)kL∞ (Rn ) ≤ Ch−n/2 ku(h)kL2 (Rn ) .

We regard (7.4.5) as a frequency localization condition. Proof. We can assume that ku(h)kL2 = 1. Then condition (7.4.5) and Lemma 3.5 imply k(1 − ψ(hD))u(h)kL∞ = O(h∞ ). To estimate kψ(hD)ukL∞ , we use (3.3.2): kψ(hD)ukL∞ ≤

1 1 kψkL2 kFh ukL2 = kψkL2 kukL2 . n (2πh) (2πh)n/2 

We will later need the following LEMMA 7.11 (A simple L2 estimate). Suppose that {at }t∈R ⊂ S is a family of real-valued symbols, depending continuously on t, and that ( (hDt + aw t (x, hD))u = f u(·, 0) = u0 . Then (7.4.7)

ku(·, t)kL2 (Rn )

p |t| ≤ kf kL2 (R×Rn ) + ku0 kL2 (Rn ) . h

2 n Proof. Since aw t (x, hD) is family of bounded operators on L (R ), the existence of solutions follows from existence theory for linear ordinary differential equations in the variable t.

Suppose first that f ≡ 0. Then 1 ∂t ku(·, t)k2L2 (Rn ) = Reh∂t u(·, t), u(·, t)iL2 (Rn ) 2 1 = Rehiaw t u(·, t), u(·, t)i = 0. h Thus, if we set E(t)u0 := u(t), we have kE(t)u0 kL2 (Rn ) = ku0 kL2 (Rn ) . If f 6= 0, Duhamel’s formula (which the reader can simply verify) gives Z i t u(·, t) = E(t)u0 + E(t − s)f (·, s) ds. h 0 Hence ku(·, t)kL2 (Rn )

1 ≤ ku0 kL2 (Rn ) + h

Z

t

kf (·, s)kL2 (Rn ) ds. 0

7.4. L∞ ESTIMATES FOR QUASIMODES

The estimate (7.4.7) is an immediate consequence.

151



Nondegeneracy condition. Let us now suppose m is an order function and K ⊂ R2n is a compact set. We will suppose that a real-valued symbol p ∈ S(m) satisfies the following nondegeneracy condition: ∂ξ p 6= 0 on {p = 0} ∩ K.

(7.4.8)

THEOREM 7.12 (L∞ bounds for quasimodes). Assume p satisfies (7.4.8). Suppose further that u = {u(h)}0
(7.4.9) Then if (7.4.10)

kpw (x, hD)u(h)kL2 (Rn ) = O(h)ku(h)kL2 (Rn ) ,

we have the estimate (7.4.11)

ku(h)kL∞ (K) ≤ Ch−(n−1)/2 ku(h)kL2 (Rn ) .

So if u(h) is a quasimode in the sense of the estimate (7.4.10), we can 1 improve the earlier L∞ estimate (7.4.6) by a factor of h 2 . The localization condition (7.4.9) is not very restrictive in view of (7.4.10) and (7.4.4). Proof. 1. We can assume that ku(h)k = 1. In light of (7.4.4) it is enough to prove the Theorem for u(h) replaced by χw u(h), where χ is supported near a given point in K. A partition of unity argument will then give the bound. 2. Suppose that p 6= 0 on the support of χ. Then q = χ/p ∈ Cc∞ (R2n ) and the composition formula in Theorem 4.18 shows that q w (x, hD)pw (x, hD) = χw (x, hD) + hrw (x, hD) for r ∈ S. Hence P (h)χw u(h) = OL2 (h) implies that χw u(h) = OL2 (h). Theorem 7.10 then shows that kχw u(h)kL∞ ≤ Chh−n/2 ≤ Ch−(n−1)/2 . 3. Now suppose that p vanishes in the support of χ. By applying a linear change of variables we can assume that pξ1 6= 0 there. The Implicit Function Theorem shows that in a neighborhood of spt χ, we have (7.4.12)

p(x, ξ) = e(x, ξ)(ξ1 − a(x, ξ 0 )),

152

7. ESTIMATES FOR SOLUTIONS OF PDE

where ξ = (ξ1 , ξ 0 ) and e > 0. We extend e arbitrarily to e ∈ S so that e ≥ γ > 0 and extend a to a real-valued a ∈ S. Then ew (x, hD)(hDx1 − a(x, hDx0 ))χw u(h) = P (h)(χw u(h)) + OL2 (h) = OL2 (h). Since ew is elliptic, it follows, as in part 2 of the proof, that (hDx1 − a(x, hDx0 ))χw u(h) = OL2 (h).

(7.4.13)

4. The proof will be completed once we show k(χw u)(x1 , ·)kL2 (Rn−1 ) = O(1),

(7.4.14)

and for this we use (7.4.13) and Lemma 7.11. We now apply Theorem 7.10 in x0 variables only, that is with n − 1 replacing n and t replacing x1 . That is allowed since we have k(1 − ψ(hD0 ))χw u(h)(x1 , ·)kL2 (Rn−1 ) = O(h∞ ), uniformly in x1 .



REMARKS. (i) The bound provided by Theorem 7.12 is already optimal in the simplest case in which the assumptions are satisfied: p(x, ξ) = ξ1 . Indeed, write x = (x1 , x0 ) and let ϕ ∈ Cc∞ (R), and χ ∈ Cc∞ (Rn−1 ). Then u(h) := h−(n−1)/2 ϕ(x1 )χ(x0 /h) satisfies ku(h)kL2 = O(1), P (h)u(h) = hDx1 u(h) = OL2 (h); and for any non-trivial choices of ϕ and χ, ku(h)kL∞ ' h−(n−1)/2 . (ii) That condition (7.4.8) is in general necessary is shown by another simple example. Let p(x, ξ) = x1 , and u(h) = h−n/2 ϕ(x1 /h)χ(x0 /h). Then ku(h)kL2 = O(1), P (h)u(h) = hh−n/2 (tϕ(t))|t=x1 /h χ(x0 /h) = OL2 (h), and ku(h)kL∞ ' h−n/2 . This is the general bound of Lemma 7.10.



7.4. L∞ ESTIMATES FOR QUASIMODES

153

7.4.3. Bounds for spectral clusters. We provide next L∞ bounds on “spectral clusters”, that is, linear combinations of eigenfunctions for the Laplace–Beltrami operator on a compact manifold. (The proof requires the material presented in Chapter 12). Suppose that M is an n-dimensional compact Riemannian manifold and let ∆g be its Laplace-Beltrami operator . Assume that 0 = λ0 < λ1 ≤ · · · ≤ λj → ∞ are the eigenvalues of −∆g and that −∆g ϕj = λj ϕj

(j = 1, . . . )

are a corresponding orthonormal basis of eigenfunctions. THEOREM 7.13 (L∞ bounds on eigenfunction clusters). (i) There exists a constant C such that for any choices of constants cj ∈ C, we have the inequality kuµ kL∞ (M ) ≤ Cµ(n−1)/2 kuµ kL2 (M ) , P uµ := µ≤√λj ≤µ+1 cj ϕj

(7.4.15) for µ ≥ 1.

(ii) In particular, (n−1)/4

kϕj kL∞ (M ) ≤ Cλj

(7.4.16)

kϕj kL2 (M ) .

Proof. Put h = 1/µ, P (h) := −h2 ∆g − 1, and P u(h) := µ≤√λj ≤µ+1 cj ϕj . Then the assumption (7.4.8) holds everywhere. Also P kP (h)u(h)kL2 = µ≤√λj ≤µ+1 cj (h2 λj − 1)ϕj

L2

=

P µ≤

√

|c |2 (h2 λj − 1)2 kϕj k2L2 λj ≤µ+1 j

1 2

≤ 2hku(h)kL2 . Thus (7.4.10) holds. On a compact manifold the frequency localization condition (7.4.5) follows from k(1 − ϕ(−h2 ∆g ))u(h)kH k = O(h∞ )ku(h)kL2 . h

Cc∞ (R)

for ϕ ∈ satisfying ϕ(t) ≡ 1 for |t| ≤ 2. But this is a consequence of the Spectral Theorem. 

154

7. ESTIMATES FOR SOLUTIONS OF PDE

7.5. SCHAUDER ESTIMATES All of the estimates provided so far in this and previous chapters have been based upon calculations in L2 , related Sobolev spaces and L∞ . We next show that we can in fact use Fourier methods to characterize spaces of H¨older continuous functions, and as application provide a semiclassical proof of the interior Schauder estimates for the Laplacian. 7.5.1. Littlewood–Paley decomposition. This important tool is a decomposition of a given function into components with controlled Fourier frequencies. We start with LEMMA 7.14 (Dyadic partition of unity). There exist functions ψ0 ∈ Cc∞ (R) and ψ ∈ Cc∞ (R \ {0}) such that (7.5.1)

1 = ψ0 (|x|) +

∞ X

ψ(2−j |x|)

j=0

for each point x ∈

Rn .

Proof. Select ϕ0 ∈ Cc∞ ((−1, 1)) so that 0 ≤ ϕ0 ≤ 1 and ϕ0 (ρ) = 1 for |ρ| ≤ 1/2. Then ∞ X ϕ1 (ρ) := ϕ0 (ρ − j) j=−∞

is a smooth positive function satisfying ϕ1 (ρ) ≥ 1 and ϕ1 (ρ − j) = ϕ1 (ρ) for all j ∈ Z. Hence for ϕ := ϕ0 /ϕ1 we have ∞ X

ϕ(ρ − j) = 1

j=−∞

for all ρ ∈ R. Now define ψ(r) := ϕ(log r/ log 2). Then ψ ∈ Cc∞ (1/2, 2) and ∞ X

ψ(2−j r) = 1

j=−∞

for each r > 0. We next put ψ0 (r) := 1 −

∞ X

ψ(2−j r).

j=0

Notice that ψ0 (r) = 1 for r < 1/2 and ψ0 (r) = 0 for r > 1. Hence ψ0 ∈ Cc∞ ([0, ∞)). 

7.5. SCHAUDER ESTIMATES

155

We will hereafter identify ψ0 and ψ with smooth radial function on Rn : ψ0 (x) = ψ0 (|x|) and ψ(x) = ψ(|x|). DEFINITION. The Littlewood–Paley decomposition of u ∈ S 0 (Rn ) is (7.5.2)

u = ψ0 (D)u +

∞ X

ψ(2−j D)u,

j=0

the functions ψ0 , ψ from (7.5.1). The terms in the decomposition (7.5.2) are localized near momenta comparable to 2j . We can therefore think of h ∼ 2−j as the relevant semiclassical parameter. We record for future reference some useful estimates: LEMMA 7.15 (Multiplier estimates). (i) For each χ ∈ Cc∞ (Rn ), we have −n p

kχ(hD)ukL∞ (Rn ) ≤ Ch

(7.5.3)

kukLp (Rn ) .

(ii) Furthermore, kχ(hD)kLp (Rn )→Lp (Rn ) ≤

(7.5.4)

1 kb χkL1 (Rn ) (2π)n

for 1 ≤ p ≤ ∞. (iii) Suppose ϕ ∈ S (Rn ) and χ, χ e ∈ Cc∞ (Rn ), with χ e ≡ 1 on a neighborhood of spt χ. Then kχ(hD)ϕ(1 − χ e(hD))kLp (Rn )→Lq (Rn ) = O(h∞ ),

(7.5.5)

for all 1 ≤ p, q ≤ ∞. Proof. 1. We have Z Z i 1 χ(hD)u(x) = χ(ξ)e h hx−y,ξi u(y) dξdy n (2πh) Rn Rn   Z 1 x−y = χ ˆ u(y) dy. (2πh)n Rn h

(7.5.6)

Thus |χ(hD)u(x)| ≤ where

1 q

+

1 p

1 −n kb χkq ku(x − h ·)kp = Cχ h p kukp , n (2π)

= 1.

2. The bound (7.5.4) follows from (7.5.6) and Young’s inequality kf ∗ gkLp (Rn ) ≤ kf kLq (Rn ) kgkLr (Rn )

156

for

7. ESTIMATES FOR SOLUTIONS OF PDE

1 p

+

1 r

=

1 q

+ 1.

3. The estimate (7.5.5) is an immediate consequence of the composition rule for pseudodifferential operators (see Section 4.3) which shows that χ(hD)ϕ(1 − χ e(hD)) = OS 0 →S (h∞ ). To give a direct proof we write the operator in (7.5.5) using an integral kernel: Z Kh (x, y)u(y) dy, Kh u(x) := χ(hD)ϕ(1 − χ e(hD))u(x) = Rn

where Kh (x, y) Z Z Z i 1 = χ(ξ)ϕ(z)(1 − χ e(η))e h (hx,ξi−hy,ηi+hη−ξ,zi) dzdξdη 2n (2πh) Rn Rn Rn   Z Z i 1 η−ξ = χ(ξ)(1 − χ e(η))ϕ b e h (hx,ξi−hy,ηi) dξdη. 2n (2πh) h Rn Rn 1

4. Fix N and recall the notation hzi := (1 + |z|2 ) 2 . Since spt χ ∩ spt(1 − χ e) = ∅ and since ϕ ∈ S , we see that on the support of the integrand  
η−ξ α (∂ ϕ) b = O hN hξ − ηi−N h for each multiindex α. Now i

i

(1 − h2 ∆ξ )N (1 − h2 ∆η )N e h (hx,ξi−hy,ηi) = hxi2N hyi2N e h (hx,ξi−hy,ηi) , and so integration by parts shows that |Kh (x, y)| ≤ CN hN hxi−N hyi−N . Then kKh ukpLp

Np

Z

−N p

≤ CN h

Z

hxi Rn

−N

hyi

p |u(y)|dy

Rn

dx ≤ CN hN p kukpLq . 

7.5.2. H¨ older continuity. We now show how the Littlewood–Paley decomposition (7.5.2) provides a characterization of H¨older continuous functions. Let U ⊂ Rn be an open set. We write (7.5.7)

kukC k,γ (U¯ ) := max k∂ α ukL∞ (U ) + max sup |α|≤k

for k = 0, 1, . . . and 0 < γ ≤ 1

|α|=k x6=y x,y∈U

|∂ α u(x) − ∂ α u(y)| |x − y|γ

7.5. SCHAUDER ESTIMATES

157

THEOREM 7.16 (Characterization of H¨ older spaces). Suppose u ∈ Lp (Rn ) for some 1 ≤ p ≤ ∞. Then for k = 0, 1, . . . and 0 < γ < 1, we have u ∈ C k,γ (Rn )

(7.5.8) if and only if

kχ(hD)ukL∞ (Rn ) ≤ Cχ hk+γ

(7.5.9)

for each χ ∈ Cc∞ (Rn \ {0}) and all 0 < h < 1. When we assert here that u ∈ Lp in fact belongs to C k,γ , we mean that there exists a function u ¯ ∈ C k,γ such that u = u ¯ almost everywhere. Proof. In the proof we can assume that k = 0, as the modification in the case of higher derivatives is straightforward. 1. We start with the easier implication that (7.5.8) implies (7.5.9). For this, we use (7.5.6) to write Z 1 χ(hD)u(x) = χ b(y)u(x − hy) dy. (2π)n Rn R Since χ(0) = 0 and χ ∈ S , we have Rn χ b(y) dy = 0; and hence Z 1 χ b(y)(u(x − hy) − u(x)) dy. χ(hD)u(x) = (2π)n Rn Now |b χ(y)| ≤ CN (1 + |y|2 )−N ; and so since u ∈ C 0,γ (Rn ), we obtain Z 1 |χ(hD)u(x)| ≤ |b χ(y)||u(x − hy) − u(x)| dy (2π)n Rn Z ≤C (1 + |y|2 )−N |yh|γ dy ≤ Chγ . Rn

Consequently, kχ(hD)ukL∞ ≤ Chγ ; this is (7.5.9) for k = 0. 2. To prove the opposite implication, let us write   (7.5.10) Λγ (u) := sup h−γ kψ(hD)ukL∞ + max kψk (hD)ukL∞ , 1≤k≤n

0
where ψ is as in (7.5.1) and ψk (ξ) := ξk ψ(ξ). We are assuming now that Λγ (u) is finite.

158

7. ESTIMATES FOR SOLUTIONS OF PDE

Let us estimate kukL∞ ≤ kψ0 (D)ukL∞ +

∞ X

kψ(2−j D)ukL∞

j=0

≤ kψ0 (D)ukL∞ + Λγ (u)

∞ X

2−jγ

j=0

= kψ0 (D)ukL∞ + (2 − 1)−1 Λγ (u). γ

Since (7.5.11)

1 ψ0 (hD)u(x) = (2π)n

Z Rn

ψb0 (x − y)u(y) dy,

we see that kukL∞ ≤ C(kukLp + Λγ (u)).

(7.5.12)

3. We next claim that (7.5.13)

|u(x) − u(y)| ≤ C(kukLp + Λγ (u))rα

if |x − y| ≤ r. In view of the Littlewood–Paley decomposition (7.5.2) it is enough to show that (7.5.14)

|ψ0 (D)u(x) − ψ0 (D)u(y)| ≤ Crγ kukLp ,

and that (7.5.15)

∞ X

|ψ(2−j D)u(x) − ψ(2−j D)u(y)| ≤ Crγ Λγ (u).

j=0

Inequality (7.5.14) is immediate from the integral representation (7.5.11) b and the smoothness of ψ. 4. To establish (7.5.15) we exploit different estimates in regions with different rates of oscillations. The high oscillation estimate is simply (7.5.16)

kψ(hD)u(x) − ψ(hD)u(y)k ≤ 2kψ(hD)ukL∞ ≤ 2hγ Λγ (u);

and the low oscillation estimate has the form kψ(hD)u(x) − ψ(hD)u(y)k ≤ rC max kDxk ψ(hD)ukL∞ 1≤k≤n

(7.5.17)

−1

≤ rh

C max kψk (hD)ukL∞ 1≤k≤n

−1+γ

≤ rh for ψk (ξ) = ξk ψ(ξ) as above.

CΛγ (u),

7.5. SCHAUDER ESTIMATES

159

To prove (7.5.15) we divide up the sum and use (7.5.17) for h = 2−j ≥ r and (7.5.16) for h = 2−j < r: ∞ X

|ψ(2−j D)u(x) − ψ(2−j D)u(y)|

j=0

 =

 X

X

+

2−j ≥r

 |ψ(2−j D)u(x) − ψ(2−j D)u(y)|

2−j


 X

≤ Λγ (u)C r

X

2j(1−γ) +

2−jγ  ≤ CΛγ (u)rγ .

2j >r−1

2j ≤r−1

This establishes (7.5.15).



The proof provides the useful estimate that for u ∈ Lp (Rn ),
1 kukL∞ (Rn ) + Λγ (u) ≤ kukC γ (Rn ) C (7.5.18)
≤ Cp kukLp (Rn ) + Λγ (u) . As an immediate application, we have: THEOREM 7.17 (Morrey’s inequality). If u ∈ Lp (Rn ) and ∂u ∈ 1− n Lp (Rn ) for some p > n, then u ∈ C p (Rn ). Furthermore, (7.5.19)

kukC γ (Rn ) ≤ C kukLp (Rn ) + k∂ukLp (Rn )




where γ = 1 − np . The constant C depends on n and p only. For a real variables proof see [E, Section 5.6.2]. Proof. For ψ as in (7.5.10) we write ψ(ξ) =

n X k=1

Then ψek ∈

Cc∞ (Rn

ξk ψek (ξ)ξk , ψek (ξ) := 2 ψ(ξ). |ξ|

\ {0}. Estimate (7.5.3) now implies kψ(hD)uk∞ ≤

n X

kψek (hD)hDk uk∞

k=1

≤

n X

1− n p

h

k=1 1− n p

≤ Ch

kDk ukLp

k∂ukLp .

160

7. ESTIMATES FOR SOLUTIONS OF PDE

Similar estimates for valid for each functions ψk in (7.5.10). Hence Λ1− np (u) ≤ Ck∂ukLp .  7.5.3. Schauder estimates. In this section we derive an important estimate for distributional solutions u of the PDE −∆u = f

(7.5.20)

in U .

Here U ⊂ Rn is an open set. THEOREM 7.18 (Interior Schauder estimate). Suppose that u ∈ ¯ ) for some 0 < γ < 1 and k ∈ N, L1 (U ) solves (7.5.20), where f ∈ C k,γ (U Then for each compactly contained open subset V ⊂⊂ U , we have u ∈ C k+2,γ (V¯ ), with the estimate (7.5.21)

  kukC k+2,γ (V¯ ) ≤ C kf kC k,γ (U¯ ) + kukL1 (U ) .

REMARK. To illustrate the idea of the proof we first assume that u ∈ L1 (Rn ) and that −∆u = f in Rn , with f ∈ C k,γ (Rn ). In view of (7.5.18) we need to show for each χ ∈ Cc∞ (Rn \ {0}) that (7.5.22)

kχ(hD)ukL∞ ≤ Cχ hk+2+γ

for all 0 < h < 1. But this follows from our writing the equation in a way consistent with the semiclassical viewpoint of these notes: −h2 ∆u = h2 f ;

(7.5.23) from which it follows that (7.5.24)

χ(hD)u = χ0 (hD)h2 f,

where χ(ξ) ∈ Cc∞ (Rn \ {0}). |ξ|2 We now apply Theorem 7.16 to f . χ0 (ξ) :=

Proof. 1. Let {ψj }j=1,··· ,N be a sequence of functions satisfying spt ψ1 ⊂⊂ U, spt ψj+1 ⊂⊂ {ψj ≡ 1}, V ⊂⊂ {ψN = 1} for j = 1, . . . , N − 1. We will select N later. Then (7.5.25)

−h2 ∆(ψj+1 u) = −[h2 ∆, ψj+1 ]ψj u + ψj+1 h2 f,



7.5. SCHAUDER ESTIMATES

161

for the commutators −[h2 ∆, ψj+1 ] = h(h∆ψj+1 + 2hD · Dψj+1 ), where we use notation (D · Dϕ)u = D · ((Dϕ)u). 2. We assert now that for any functions χ, χ e ∈ Cc∞ (Rn \ {0}) with χ e≡1 on spt χ, we have the estimate (7.5.26)

kχ(hD)ψj+1 ukL∞ ≤ Chke χ(hD)ψj ukL∞ + h2 kχ(hD)ψ ˜ j+1 f kL∞ + O(h∞ ) (kψj ukL1 + kf kL∞ ) .

To prove (7.5.26) we use the notation of (7.5.24) and (7.5.25) and obtain
χ(hD)ψj+1 u = χ0 (hD) −[h2 ∆, ψj+1 ]ψj u (7.5.27)
+ χ0 (hD) ψj+1 h2 f . Since spt χ0 = spt χ, Lemma 7.15 shows that
kχ0 (hD) ψj+1 h2 f kL∞ ≤ h2 kχ0 (hD)e χ(hD) (ψj+1 f ) kL∞ (7.5.28) 2 ≤ Ch ke χ(hD) (ψj+1 f ) kL∞ . We expand the first term on right hand side of (7.5.27) using the expression for the commutator, h2 χ0 (hD) ((∆ψj+1 )ψj u) + 2h

n X

χ0 (hD)(hDxk ((Dxk ψj+1 )ψj u)) ,

k=1

and apply Lemma 7.15 to both terms: kh2 χ0 (hD)(∆ψj+1 )ψj ukL∞ = h2 kχ0 (hD)(∆ψj+1 )e χ(hD)(ψj u)kL∞ + O(h∞ )kψj ukL1 ≤ Ch2 ke χ(hD)(ψj u)kL∞ + O(h∞ )kψj ukL1 . Upon our writing ϕk (x) := Dxk ψj (x), χk (ξ) := ξk χ0 (ξ), this gives hkχk (hD)ϕk ψj ukL∞ = hkχk (hD)ϕk χ e(hD)ψj ukL∞ + O(h∞ )kψj ukL1 ≤ Chke χ(hD)(ψj u)kL∞ + O(h∞ )kψj ukL1 , Combining the foregoing estimates proves (7.5.26). 3. From (7.5.3) we see that kχ(hD)ψ1 ukL∞ ≤ Ch−n kukL1 (U ) .

162

7. ESTIMATES FOR SOLUTIONS OF PDE

Then (7.5.26) applied inductively shows that for 1 ≤ j ≤ N − 1,   kχ(hD)ψj+1 ukL∞ ≤ Cj hj−n kukL1 (U ) + h2+k+γ kf kC k,γ (U¯ ) . We used (7.5.18) to estimate ke χ(hD)f k∞ . Now we select N >n+2+k+γ and obtain   kχ(hD)ψN ukL∞ ≤ Ch2+k+γ kukL1 (U ) + kf kC k,γ (U¯ ) . According to Theorem 7.16, this shows that ψN u ∈ C 2+k,γ (Rn ).



7.6. NOTES Estimates in the classically forbidden region in Section 7.1 are known as Agmon or Lithner-Agmon estimates. These play a crucial role in the analysis of spectra of multiple well potentials and of the Witten complex: see Dimassi– Sj¨ ostrand [D-S, Chapter 6]. Here we followed an argument of Nakamura [N]. The presentation of Carleman estimates in Section 7.2 is based on discussions with N. Burq and D. Tataru, and Burq suggested the estimates for the order of vanishing. [K-T-Z] presents the semiclassical pointwise bounds reproduced here. The estimate (7.4.15) is essentially optimal, whereas the optimality of (7.4.16) is rare. See [S-Z] for a recent discussion. H. Smith suggested the application to Schauder estimates. For an indepth discussion of Schauder estimates based on real analysis methods, see Gilbarg–Trudinger [G-T], and for developments of Littlewood–Paley techniques, consult Stein [St].

Part 3

ADVANCED THEORY

Chapter 8

MORE ON THE SYMBOL CALCULUS

8.1 8.2 8.3 8.4 8.5

Beals’s Theorem Real exponentiation of operators Generalized Sobolev spaces Wavefront sets, essential support, microlocality Notes

This chapter collects various more advanced topics concerning the symbol calculus. Subsequent chapters will provide many applications.

8.1. BEALS’S THEOREM We present next a semiclassical version of Beals’s Theorem, a characterization of pseudodifferential operators in terms of h-dependent bounds on commutators. This important theorem answers a fundamental question: When can a given linear operator be represented using the symbol calculus? We start with h = 1: THEOREM 8.1 (Estimating a symbol by operator norms). There exist constants C, M > 0 such that X (8.1.1) kbkL∞ (Rn ) ≤ C k(∂ γ b)w (x, D)kL2 (Rn )→L2 (Rn ) |γ|≤M

for all b ∈ S 0 . 165

166

8. MORE ON THE SYMBOL CALCULUS

Proof. 1. We will first consider the classical quantization Z 1 b(x, D)u(x) = b(x, ξ)eihx,ξi u ˆ(ξ) dξ, (2π)n Rn where by the integration we mean the Fourier transform in S 0 . Then if ϕ = ϕ(x), ψ = ψ(ξ) are functions in the S , we consider the Fourier transform in Rn × Rn , b ihx,ξi )(x∗ , ξ ∗ ), (x∗ , ξ ∗ ) 7→ F(b ϕ ψe as a function of the dual variables (x∗ , ξ ∗ ) ∈ R2n . We have Z Z ihx,ξi ihx,ξi b b |F(b ϕ ψe )(0, 0)| = dxdξ b(x, ξ)ϕ(x) ψ(ξ)e n n R

R

= (2π)n |hb(x, D)ψ, ϕi| ≤ (2π)n kbkL2 →L2 kϕkL2 kψkL2 . ∗

Fix (x∗ , ξ ∗ ) ∈ R2n and rewrite this inequality with ϕ(x)eihx ,xi replacing ∗ ϕ(x) and ψ(ξ)e−ihξ ,ξi replacing ψ(ξ), a procedure which does not change the L2 norms. It follows that 1 b ihx,ξi )(x∗ , ξ ∗ )| ≤ kbkL2 →L2 kϕkL2 kψkL2 . (8.1.2) |F(b ϕ ψe (2π)n 2. Now take χ ∈ Cc∞ (R2n ), and select ϕ, ψ ∈ S so that ϕ(x) = 1 if b = 1 if (x, ξ) ∈ sptχ. Write (x, ξ) ∈ sptχ and ψ(ξ) (8.1.3)

χ e = χe−ihx,ξi ,

Fχ e(ξ ∗ ) = Fχ(ξ ∗ + ξ).

According to (3.1.20), (8.1.4)

kF χ ekL1 = kFχkL1 ≤ C

X

k∂ α χkL1 .

|α|≤2n+1

Also, ihx,ξi b χ(x, ξ) = χ e(x, ξ)ϕ(x) ψ(ξ)e . ∗ ∗ Thus (8.1.2) and (8.1.4) show that for any (x , ξ ) ∈ R2n

b ihx,ξi )kL∞ |F (χ b)(x∗ , ξ ∗ )| ≤ kF (e χb ϕ ψe 1 b ihx,ξi )kL∞ = kF χ e ∗ F(b ϕ ψe (2π)n 1 b ihx,ξi )kL∞ kF χ ≤ kF(b ϕ ψe ekL1 (2π)n ≤ CkbkL2 →L2 , the constant C depending on ϕ, ψ and χ, but not (x∗ , ξ ∗ ). Hence (8.1.5)

kF(χ b)kL∞ ≤ CkbkL2 →L2

8.1. BEALS’S THEOREM

167

with the same constant for any translate of χ. 3. Next, we assert that X

(8.1.6) |F(χ b)(x∗ , ξ ∗ )| ≤ Ch(x∗ , ξ ∗ )i−2n−1

k(∂ α b)(x, hD)kL2 →L2 .

|α|≤2n+1

To see this, compute (x∗ )α (ξ ∗ )β F(χ b)(x∗ , ξ ∗ ) =

Z

|α|+|β|

= (−1) Z Z = Rn

Z

∗ , xi+hξ ∗ , ξi)

(x∗ )α (ξ ∗ )β e−i(hx

Rn

Rn

Z

Z

Rn

Rn

χ b(x, ξ) dxdξ

  ∗ ∗ Dxα Dξβ e−i(hx , xi+hξ , ξi) χ b dxdξ

−i(hx∗ , xi+hξ ∗ , ξi)

e

Rn

Dxα Dξβ (χ b) dxdξ.

Summing absolute values of the left hand side over all (α, β) with |α| + |β| ≤ 2n + 1 and using the estimate (8.1.5), we obtain the bound X kh(x∗ , ξ ∗ )i2n+1 F(χ b)kL∞ ≤ C kF(Dxα Dξβ (χ b))kL∞ |α|+|β|≤2n+1

≤ C

X

k(∂ γ b)(x, hD)kL2 →L2 .

|γ|≤2n+1

This gives (8.1.6). Consequently, kχ bkL∞ ≤ CkF(χ b)kL1 ≤ C

X

k(∂ γ b)(x, hD)kL2 →L2 .

|γ|≤2n+1

4. This implies the desired inequality (8.1.1), except that we used the classical (t = 1), and not the Weyl (t = 1/2) quantization. To remedy this, recall from Theorem 4.13 that if i

b = e 2 hDx ,Dξ i˜b, then bw (x, D) = ˜b(x, D), (∂ α b)w (x, D) = (∂ α˜b)(x, D). The continuity statement in Theorem 4.17 shows that X kbkL∞ ≤ C k∂ α˜bkL∞ , |α|≤K

and reduces the argument to the classical quantization.  The following notation will be useful, if slightly odd-looking, in expressions involving multiple commutators:

168

8. MORE ON THE SYMBOL CALCULUS

NOTATION. We henceforth write (8.1.7)

adB A := [B, A];

“ad” is called the adjoint action. Easy calculations show LEMMA 8.2 (Properties of ad). The adjoint action ad satisfies the derivation property (8.1.8)

adA (BC) = (adA B)C + B(adA C)

and therefore adA B = −B(adA B −1 )B.

(8.1.9)

Remember that we identify a pair (x∗ , ξ ∗ ) ∈ R2n with the linear operator l(x, ξ) = hx∗ , xi + hξ ∗ , ξi. Recall also from Theorem 4.4 that lw (x, hD) = l(x, hD) = hx∗ , xi + hξ ∗ , hDi. THEOREM 8.3 (Semiclassical Beals’s Theorem). Let A : S → S 0 be a continuous linear operator. Then (i) A = aw (x, hD) for a symbol a ∈ S if and only if (ii) for all N = 0, 1, 2, . . . and all linear functions l1 , . . . , lN , we have (8.1.10)

kadl1 (x,hD) · · · adlN (x,hD) AkL2 (Rn )→L2 (Rn ) = O(hN ).

APPLICATION: Resolvents as pseudodifferential operators. Suppose a ∈ S is real-valued, so that A = aw (x, hD) is a self-adjoint operator on L2 . If λ does not lie in the spectrum of A, the resolvent B = (A + λ)−1 is a bounded operator on L2 . Can we represent B as a pseudodifferential operator? To see that we can, first calculate using (8.1.9) that adl(x,hD) B = −B(adl(x,hD) (A + λ))B = −B(adl(x,hD) A)B for each linear l. Therefore kadl(x,hD) BkL2 →L2 ≤ Ckadl(x,hD) AkL2 →L2 = O(h), according to (8.1.10). A similar computation shows for each N that kadl1 (x,hD) · · · adlN (x,hD) BkL2 →L2 = O(hN ), and so the assumptions of Beals’s Theorem are satisfied. Consequently B = (A + λ)−1 = bw (x, hD) for some symbol b ∈ S. 

8.1. BEALS’S THEOREM

169

Proof. 1. That (i) implies (ii) follows from the symbol calculus developed in Chapter 4. Indeed, kAkL2 →L2 = O(1) according to Theorem 4.23 and formula (4.3.11) shows each commutator with lj (x, hD) yields a bounded operator of order h. Observe that although lj ∈ / S, we can still apply the composition formula since ∂ α lj ∈ S for |α| ≥ 1. 2. That (ii) implies (i) is harder and we will first prove the implication for h = 1. The Schwartz Kernel Theorem (Theorem C.1) asserts that we can write Z KA (x, y)u(y) dy (8.1.11) Au(x) = Rn

for KA ∈ S 0 (Rn × Rn ). We call KA the kernel of A. We now claim that if we define a ∈ S 0 (R2n ) by Z (8.1.12) a(x, ξ) := e−ihw,ξi KA x + w2 , x − Rn

w 2




dw,

then (8.1.13)

1 KA (x, y) = (2π)n

Z a Rn

x+y
ihx−y,ξi dξ, 2 ,ξ e

where the integrals are a shorthand for the Fourier transforms defined on S 0 . To confirm this, we calculate using (8.1.12) and the Fourier inversion formula that Z
1 a x+y , ξ eihx−y,ξi dξ 2 n (2π) Rn Z Z
1 x+y ihx−y−w,ξi w x+y w = e K + , − A 2 2 2 2 dwdξ (2π)n Rn Rn Z
w x+y w = δ(x − y − w)KA x+y 2 + 2 , 2 − 2 dw Rn

= KA (x, y). In view of (8.1.11) and (8.1.13), we see that A = aw (x, D), for a defined by (8.1.12). 3. Now we must show that a belongs to the symbol class S; that is, (8.1.14)

sup |∂ α a| ≤ Cα R2n

for each multiindex α. To do so we will make use of our hypothesis (8.1.10) with l = xj , ξj , that is, with l(x, hD) = xj , Dj . We recall the commutator formulas (4.2.7) and

170

8. MORE ON THE SYMBOL CALCULUS

(4.2.6), which imply for j = 1, . . . , n that ( adxj A = [xj , aw ] = −(Dξj a)w (8.1.15) adDxj A = [Dxj , aw ] = (Dxj a)w . This and the hypothesis (8.1.10) with h = 1 imply that k(∂ α a)w kL2 →L2 ≤ Cα , for all multiindices α. The estimate (8.1.14) now follows from Theorem 8.1. 4. Next we convert the case with arbitrary h to the case of h = 1 by rescaling (4.1.9). For this, define Uh u(x) := hn/4 u(h1/2 x) and check that Uh : L2 → L2 is unitary. Then Uh aw (x, hD)Uh−1 = aw (h1/2 x, h1/2 D) = aw h (x, D) for ah (x, ξ) := a(h1/2 x, h1/2 ξ).

(8.1.16)

Our hypothesis (8.1.10) is invariant under conjugation by Uh , and is consequently equivalent to (8.1.17)

N adl1 (h1/2 x,h1/2 D) · · · alN (h1/2 x,h1/2 D) aw h = OL2 →L2 (h ).

But since lj is linear, lj (h1/2 x, h1/2 D) = h1/2 l(x, D). Thus (8.1.17) is equivalent to (8.1.18)

N/2 ). adl1 (x,D) · · · alN (x,D) aw h = OL2 →L2 (h

Taking lk (x, ξ) = xj or ξj , it follows from (8.1.18) that k(∂ β ah )w kL2 →L2 ≤ Ch

(8.1.19)

|β| 2

for all multiindices β. 5. Finally, we claim that (8.1.20)

|∂ α ah | ≤ Cα h|α|/2 for each multiindex α.

But this follows from Theorem 8.1, owing to estimate (8.1.19): X k∂ α ah kL∞ ≤ C k(∂ α+β ah )w kL2 →L2 ≤ Cα h|α| . |β|≤n+1

Recalling (8.1.16), we rescale to derive the desired inequality (8.1.14). REMARK: Beals’s Theorem for Sδ . Similar arguments show that (8.1.21)

A = aw (x, hD) for a symbol a ∈ Sδ



8.2. REAL EXPONENTIATION OF OPERATORS

171

if and only if (8.1.22)

kadl1 (x,hD) · · · adlN (x,hD) AkL2 (Rn )→L2 (Rn ) = O(hN (1−δ) ).

for all N = 0, 1, 2, . . . and all linear functions l1 , . . . , lN .



8.2. REAL EXPONENTIATION OF OPERATORS In this section we will consider families of operators which give real exponentials of certain pseudodifferential operators. As we have seen in Theorem 4.7, quantization and exponentiation commute for linear symbols. This is certainly not true for nonlinear symbols, but is in a certain sense valid at the level of order functions, as we will see in this section. We henceforth assume m = m(x, ξ) is an order function. Set (8.2.1)

g := log m.

We assume also (8.2.2)

|∂ α g| ≤ Cα for all multiindices |α| ≥ 1.

Then etg = mt ∈ S(mt )

(8.2.3)

(t ∈ R).

We will discuss in Section 8.3 how to find order functions m for which these conditions hold. LEMMA 8.4 (Inverting exponentials). Consider U (t) := (exp tg)w (x, D) as a mapping from S to itself. There exists t0 > 0 such that the operator U (t) is invertible for |t| < t0 and U (t)−1 = bw t (x, D)

(8.2.4) for a symbol

bt ∈ S(m−t ).

(8.2.5)

Proof. 1. Owing to (8.2.3) U (t) is the quantization of an element of S(mt ). We assert that (8.2.6)

U (−t)U (t) = I + ew t (x, D)

for a symbol et ∈ S.

172

8. MORE ON THE SYMBOL CALCULUS

To see this, we employ the composition formula (4.3.5) with h = 1, to write 1

(8.2.7)

1

2

2

et (x, ξ) = eA(D) (e−tg(x ,ξ )+tg(x ,ξ ) )|x1,x2 =x, ξ1,ξ2 =ξ − 1 Z 1 d sA(D) −tg(x1 ,ξ1 )+tg(x2 ,ξ2 ) = e (e )|x1,x2 =x, ξ1,ξ2 =ξ ds 0 ds Z 1 1 1 2 2 = esA(D) A(D)(e−tg(x ,ξ )+tg(x ,ξ ) )|x1,x2 =x, ξ1,ξ2 =ξ ds 0

it = 2

1

Z

  1 1 2 2 esA(D) F e−tg(x ,ξ )+tg(x ,ξ ) |x1,x2 =x, ξ1,ξ2 =ξ ds,

0

where A(D) = 2i σ(Dx1 , Dξ1 ; Dx2 , Dξ2 ) and F = ∂x1 g(x1 , ξ 1 ) · ∂ξ2 g(x2 , ξ 2 ) − ∂ξ1 g(x1 , ξ 1 ) · ∂x2 g(x2 , ξ 2 ). Our assumptions imply that F ∈ S and that exp(−tg(x1 , ξ 1 ) + tg(x2 , ξ 2 )) ∈ S(m ˜ t ). for m(x ˜ 1 , x2 , ξ 1 , ξ 2 ) := m(x2 , ξ 2 )/m(x1 , ξ 1 ). Thus Theorem 4.17 shows that esA(D) : S(m ˜ t ) → S(m ˜ t ). Furthermore the restriction to x1 = x2 , ξ 1 = ξ 2 shows that et ∈ S, since m(x ˜ 1 , x1 , ξ 1 , ξ 1 ) ≡ 1. This proves (8.2.6). 2. It follows from (8.2.7) that et = te et for eet ∈ S. Therefore Theorem 4.23 implies kew t (x, D)kL2 →L2 = O(t), and so I + ew t (x, D) is invertible for |t| small enough. Then the application of Beals’s Theorem 8.3 to resolvents presented on page 168 implies −1 (I + ew = cw t (x, D)) t (x, D)

for a symbol ct ∈ S. Hence bt = ct # exp(−tg(x, ξ)) ∈ S(m−t ), according to Theorem 4.18.  We record for later reference: LEMMA 8.5 (Solving an operator equation). Suppose that C(t) = cw t (x, D), where the symbols ct ∈ S depend continuously on t for |t| ≤ t0 . Assume also q ∈ S.

8.2. REAL EXPONENTIATION OF OPERATORS

173

Then the equation ( (∂t + C(t))Q(t) = 0, (8.2.8) Q(0) = q w (x, D) has a unique solution Q(t) : S → S given by Q(t) = qtw (x, D), the symbols qt ∈ S depending continuously on t for |t| ≤ t0 . Proof. 1. The Picard Theorem for ODE shows that there exists a unique solution Q(t), which is bounded on L2 . 2. We assert next that for any choice of lj0 s and any N adl1 (x,D) · · · adlN (x,D) Q(t) : L2 → L2 .

(8.2.9)

We prove this by induction on N . Observe from the derivation property (8.1.8) of adl that (8.2.10) adl1 (x,D) · · · adlN (x,D) (C(t)Q(t)) = C(t)adl1 (x,D) · · · adlN (x,D) Q(t) + R(t), where R(t) is the sum of terms of the form Ak (t)adl1 (x,D) · · · adlk (x,D) Q(t) with k < N , for Ak (t) = (akt )w and symbols akt ∈ S depending continuously on t. Then the induction hypothesis implies R(t) is bounded on L2 . Now
∂t adl1 (x,D) · · · adlN (x,D) Q(t) + adl1 (x,D) · · · adlN (x,D) (C(t)Q(t)) = 0, and consequently (∂t + C(t)) (adl1 (x,D) · · · adlN (x,D) Q(t)) = R(t). Since R(t) is bounded on L2 and the assertion (8.2.9) is clearly valid at t = 0, it holds also for all |t| < t0 . 3. In view of (8.2.9) and Beals’s Theorem for h = 1, the unique solution bounded on L2 is a pseudodifferential operator, and hence maps S to S ⊂ L2 . As such, it is also unique.  Our next theorem identifies exp(tg w (x, hD)) as the quantization of an element of S(mt ).

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THEOREM 8.6 (Exponentials and order functions). Assume for the order function m and for g = log m that conditions (8.2.1) and (8.2.2) hold. (i) Then the equation ( ∂t B(t) = g w (x, hD)B(t), (8.2.11) B(0) = I has a unique solution B(t) : S → S for t ∈ R. (ii) Furthermore, we have B(t) = bw t (x, hD)

(8.2.12) for a symbol

bt ∈ S(mt ).

(8.2.13)

Using the rescaling given in (4.1.9), we only need to prove the result for the case h = 1. Proof: 1. To begin, let us assume that a solution of (8.2.11) exists, with B(t) : S → S . We assert that (8.2.14)

∂t (U (−t)B(t)) = V (t)B(t)

in the notation of Lemma 8.4, where V (t) = aw t (x, D)

(8.2.15)

for at ∈ S(m−t ).

In fact ∂t U (−t) = −(g exp(−tg))w (x, D)

(8.2.16) and

U (−t)g w (x, D) = (exp(−tg)#g)w (x, D).

(8.2.17)

Hence (8.2.14) holds with V (t) = (exp(−tg)#g − (g exp(−tg)))w (x, D). 2. To analyze V (t), we note that Z 1 exp(iA(D)) = 1 + exp(isA(D))A(D) ds, 0

as in (8.2.7). Consequently (4.3.5) gives exp(−tg)#g − exp(−tg)g Z 1   1 1 = exp(sA(D))A(D) e−tg(x ,ξ ) g(x2 , ξ 2 ) |x1 =x2 =x,ξ1 =ξ2 =ξ ds. 0

8.3. GENERALIZED SOBOLEV SPACES

175

From the hypothesis on g we see that A(D) exp(tg(x1 , ξ 1 ))g(x2 , ξ 2 ) is a sum of terms of the form a(x1 , ξ 1 )b(x2 , ξ 2 ), where a ∈ S(m−t ) and b ∈ S. The continuity of exp(A(D)) on the spaces of symbols in Theorem 4.17 now gives (8.2.15). 3. Set C(t) := −V (t)U (−t)−1 . Then Lemma 8.4 implies C(t) = cw t where ct ∈ S. The symbolic calculus shows that ct depends smoothly on t and (∂t + C(t))(U (−t)B(t)) = 0. 4. The existence part of Theorem 8.5 imply that B(t) = U (−t)−1 Q(t) and Q(0) = I. This shows that B(t) exists and that is unique. Since Q(t) quantizes qt ∈ S, Lemma 8.4 gives the statement of Theorem 8.6 for small times. Because the solution of (8.2.11) has the group property B(t)B(s) = B(t + s), the assertion for small times and the pseudodifferential calculus imply the assertion for all times t ∈ R.  REMARK: Real and complex exponentials. The foregoing Theorems 8.4, 8.5 and 8.6 concern real exponential expressions arising from operator dynamics of the form (∂t + C(t))Q(t) = 0. Quantum dynamics like (hDt + C(t))Q(t) = 0 yield instead complex exponential expressions, and these we will study more in Chapters 10, 11 and 14. 

8.3. GENERALIZED SOBOLEV SPACES 8.3.1. Sobolev spaces compatible with symbols. The quantization of real exponentials developed in the previous section allows us now to define generalized Sobolov spaces Hh (m) on which operators with symbols in S(m) naturally act. We first record LEMMA 8.7 (Logarithms of order functions). (i) Suppose that m is an order function and that (8.3.1)

m ∈ S(m).

Then (8.3.2)

m−1 ∈ S(m−1 ),

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and g = log m satisfies the assumptions (8.2.1) and (8.2.2). (ii) Given an arbitrary order function m, there exists another order function m e such that S(m) = S(m) e and m e ∈ S(m). e Proof. 1. The statement (8.3.2) follows from the formula (4.7.10) applied with a = m and λ = 0. That g satisfies (8.2.1) and (8.2.2) follows from (8.3.2). 2. For an arbitrary order function m define m e := m ∗ η, where η ∈ R Cc∞ (R2n ), η ≥ 0, η dw = 1. According to the definition (4.4.1) of an order function, m(z − w) ≤ ChwiN C −1 hwi−N ≤ m(z) for all w, z ∈ R2n . Consequently C −1 m ≤ m e ≤ Cm and |∂ α m| e ≤ Cα m ≤ Cα m e for all multiindices α. Hence S(m) = S(m) e and m e ∈ S(m). e



Hereafter m denotes an order function satisfying m ∈ S(m) and, as above, set g := log m. DEFINITION. We define the generalized Sobolev space associated to m as (8.3.3)

Hh (m) := {u ∈ S 0 (Rn ) | exp(g w (x, hD))u ∈ L2 (Rn )} = exp(−g w (x, hD))L2 (Rn ) ⊂ S 0

where exp(±g w ) : S 0 → S 0 by Theorems 4.16 and 8.6. The Hilbert space norm on Hh (m) is defined by kukHh (m) := k exp(g w (x, hD))ukL2 .

(8.3.4)

When m and thus g are functions of both x and ξ, we call Hh (m) a microlocally weighted space. EXAMPLES. (i) If m = hξis for s ≥ 0, then (8.3.5)

Hh (m) = Hhs (Rn ) = {u ∈ L2 (Rn ) | (1 + |ξ|2 )s/2 Fh u ∈ L2 (Rn )}

are the usual semiclassical Sobolev spaces.

8.3. GENERALIZED SOBOLEV SPACES

177

(ii) When m depends only on x, the space Hh (m) corresponds to changing Lebesgue measure Ln in the definition of L2 (Rn ) to exp(−2g(x))Ln . So kukHh (m(x)) = kukL2 (exp(−2g(x))Ln ) .

(8.3.6) In particular,

Hh (m) = L2 (Rn )

if m ≡ 1.

(iii) If m depends only on ξ, then the measure is changed on the semiclassical Fourier transform side to exp(−2g(ξ))Ln : (8.3.7)

n

kukHh (m(ξ)) = (2πh)− 2 kFh ukL2 (exp(−2g(ξ))Ln ) ,

where the prefactor is explained by Theorem 3.8



THEOREM 8.8 (Properties of Hh (m) spaces). (i) Suppose that m ∈ S(m), m e ∈ S(m) e are two order functions satisfying c−1 m ≤ m e ≤ cm, where c > 0. Then (8.3.8)

Hh (m) = Hh (m) e

and (8.3.9)

C −1 kukHh (m) ≤ kukHh (m) e ≤ CkukHh (m)

for a constant C > 0 and all u ∈ Hh (m). (ii) We can use the L2 -inner product to identify the dual space of Hh (m) with Hh (1/m): (Hh (m))0 = Hh (1/m)

(8.3.10)

REMARKS. (i) So given any order function m, we write Hh (m) := Hh (m), e where m e is any order function satisfying S(m) = S(m) e and m e ∈ S(m). e (ii) The precise identification abbreviated by (8.3.10) will be explained in the proof.  Proof. 1. Let g = log m and ge = log m. e To prove (8.3.9), we note that Theorem 8.6 implies exp(g w (x, hD)) exp(−e g w (x, hD)) = aw (x, hD) for a symbol a ∈ S. By Theorem 4.23, aw (x, hD) = OL2 →L2 (1); so that w

w

w

kukHh (m) = keg ukL2 = kaw eg˜ ukL2 ≤ Ckeg˜ ukL2 = CkukHh (m) e .

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This proves the first inequality in (8.3.9)and the second one follows as m and m e are exchangeable. 2. The definition shows that exp(±g w ) : Hh (m±1 ) → L2 are R Hilbert space isometries. Since L2 is its own dual under the pairing u(v) = Rn v¯ u dx, we identify the dual of Hh (m) with Hh (1/m) using these isometries. Explicitly, if v ∈ Hh (m) and u ∈ Hh (1/m), then w

w

u(v) = hv, ui = heg v, e−g ui.  THEOREM 8.9 (Generalized Sobolev spaces and Schwartz space). For each fixed h > 0, we have \ [ (8.3.11) S = Hh (m), S 0 = Hh (m), m∈M

m∈M

where M denotes the set of all order functions on R2n . We also see from (8.3.11) that S0


0

= S,

a standard result in functional analysis. Proof. 1. If g = log m for m ∈ S(m), then Theorem 8.6 shows exp(g w ) = aw for a symbol a ∈ S(m). Hence Theorem 4.16 implies that if u ∈ S , then w )u ∈ S ⊂ L2 . Consequently S ⊆ H (m) for all m and consequently exp(gT h S ⊆ m∈M Hh (m). 2. Next, put m(x, ξ) = hxi|α| hξi−2n+|β| , g := log m. Then sup |Dxα xβ u| ≤ Ch−2n−|α| k(I − h2 ∆)n (hDx )α xβ ukL2

x∈Rn

w

(8.3.12)

w

≤ Ch−2n−|α| k(I − h2 ∆)n (hDx )α xβ e−g eg ukL2 w

≤ C1 h−2n−|α| keg ukL2 = C1 h−2n−|α| kukHh (m) The last inequality holds since (I − h2 ∆)n (hDx )α xβ = bw w

for a symbol b ∈ S(m); and e−g = cw for a symbol c ∈ S(1/m). Consequently their composition is bounded on L2 . This proves that S ⊇ T  m∈M Hh (m).

8.3. GENERALIZED SOBOLEV SPACES

179

THEOREM 8.10 (Pseudodifferential operators on generalized Sobolev spaces). Suppose that m1 and m2 are two order functions and that a ∈ S(m1 ). (i) Then aw (x, hD) : Hh (m2 ) → Hh (m2 /m1 )

(8.3.13)

is a bounded operator, with norm bounded independently of h. (ii) If lim (x,ξ)→∞

m1 = 0.

then aw (x, hD) : Hh (m2 ) → Hh (m2 ) ,

(8.3.14)

is a compact operator. Proof. 1. Following Theorem 8.8 we can take mj ∈ S(mj ). Lemma 8.7 also implies that m2 /m1 ∈ S(m2 /m1 ). We restrict ourselves to the case h = 1 as we can again use the rescaling (4.1.9). 2. In view of the definition of H(m) = H1 (m), the theorem is equivalent to showing the boundedness of w

w

w

A := e−g1 (x,D)+g2 (x,D) aw (x, D)e−g2 (x,D)

(8.3.15)

on L2 , where gj := log mj . Theorem 8.6 tells us that w

w

w

e−g2 (x,D) = bw (x, D), e−g1 (x,D)+g2 (x,D) = cw (x, D), for symbols b ∈ S(1/m2 ), c ∈ S(m2 /m1 ). Hence the composition rule in Theorem 4.11 implies w

w

w

e−g1 +g2 aw e−g2 = cw aw bw = e aw , where e a ∈ S(m2 /m1 × m1 × 1/m2 ) = S. So Theorem 4.23 implies A = e aw (x, D) is bounded on L2 . 3. Assertion (ii) is equivalent to our showing that w

w

B := eg2 (x,D) aw (x, D)e−g2 (x,D) is a compact operator on L2 . As above, we observe that B = bw (x, D) for a symbol b ∈ S(m1 ). We then apply Theorem 4.28. 

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8. MORE ON THE SYMBOL CALCULUS

8.3.2. Application: estimates for eigenfunctions. The next theorem provides a general regularity assertion for L2 -eigenfunctions of pseudifferential operators. Let m ≥ 1 be an order function. Suppose that a ∈ S(m) is real and that C + a ≥ cm for constants C ≥ 0, c > 0. THEOREM 8.11 (Eigenfunctions and Hh (m)). Assume there exist h0 > 0 and constants α < β such that for 0 < h < h0 we have aw (x, hD)u(h) = E(h)u(h), with u(h) ∈ L2 (Rn ) and α ≤ E(h) ≤ β. Then there exist 0 ≤ h1 ≤ h0 and constants Ck , such that (8.3.16)

ku(h)kHh (mk ) ≤ Ck ku(h)kL2 (Rn ) ,

for all k = 0, 1, . . . and 0 < h < h1 . Proof: Replacing a by a + C if necessary, we may assume that a ≥ cm and 0∈ / [α, β]. Hence for h < h1 , with h1 small enough, we have aw (x, hD)−1 = for b ∈ S(1/m). This implies that Hh (m) = (bw (x, hD))k L2 . Since u(h) = E(h)−k (bw (x, hD))k u(h), we obtain the estimate (8.3.16). 

bw (x, hD)

8.4. WAVEFRONT SETS, ESSENTIAL SUPPORT, MICROLOCALITY We introduce in this section some precise ways to describe asymptotic properties of a family of functions and operators in phase space as h → 0. 8.4.1. Tempered functions and operators, localization. We begin by identifying some convenient classes of h-dependent distributions, which can deteriorate as h → 0, but for which we have some uniform control: DEFINITION. We call u = {u(h)}0
ku(h)kHh (m) = O(h−N ).

From (8.3.11) we know that each u(h) belongs to Hh (m) for some m. The purpose of the definition is to have m and N independent of h.

8.4. WAVEFRONT SETS, ESSENTIAL SUPPORT, MICROLOCALITY 181

We likewise introduce some useful classes of operators: DEFINITION. A family T = {T (h)}0
kT (h)kHh (m)→Hh (mh(x,ξ)i−N ) = O(h−N ).

The point is that although the operators T (h) may become increasingly singular as h → 0, they do so at a controlled rate. REMARKS. (i) Most of the operators we encounter in this book are tempered. According to Theorem 8.10, if a ∈ hk S(m) for any k ∈ R and any order function m, then T = aw (x, hD) is tempered. Theorem 10.1 will show that the operators discussed later in Theorems 11.1 and 11.5 are also tempered. The precise definitions of tempered functions and operators vary in the literature, and we have taken formulations most convenient for our purposes. (ii) Condition (8.4.2) implies that for any seminorm | · |1 on S , there exists another seminorm | · |2 and a constant N such that (8.4.3)

|T (h)u|1 = O(h−N )|u|2

for all u ∈ S .



Since tempered operators deteriorate only algebraically as h → 0, they behave well under composition with O(h∞ ) operators: LEMMA 8.12 (Tempered operators and h∞ ). If T is tempered and if a ∈ h∞ S , then (8.4.4)

T aw (x, hD) = OS 0 →S (h∞ ), aw (x, hD)T = OS 0 →S (h∞ ).

This follows from Theorem 8.10. We next introduce the very useful concept of asymptotic localization in phase space: DEFINITION. We call a tempered family u = {u(h)}0
(I − χw (x, hD))u(h) = OS (h∞ ).

The interpretation is that u is negligible off the support of χ as h → 0.

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8.4.2. Semiclassical wavefront sets. We next introduce the semiclassical wavefront set WFh (u) for a family u of tempered distribution. This wavefront set records where u is asymptotically localized in phase space as h → 0. Being localized to a neighborhood of x0 means that away from that neighborhood u is O(h∞ ). Being localized near a frequency ξ0 means the same thing for the semiclassical Fourier transform of u. We now want to understand what simultaneous localization means. DEFINITION. The semiclassical wavefront set WFh (u) of a tempered family u = {u(h)}0
a(x0 , ξ0 ) 6= 0

and (8.4.7)

kaw (x, hD)u(h)kHh (m) = O(h∞ ),

where the order function m is the same as in (8.4.1). REMARKS. (i) The name “wavefront set” is motivated by the example u(h) = eiϕ(x)/h with ∂ϕ 6= 0. Then W Fh (u) = {(x, ∂ϕ(x)) | x ∈ Rn }, which is indeed the wavefront of the highly oscillatory wave. For example, if ϕ(x) = hx, ωi, with |ω| = 1, then u is a plane wave with wavefront Rn × {ω}. (ii) The wavefront set provides a crude but robust measure of semiclassical regularity of a function. For instance a smooth function u not depending on h has WFh (u) = spt u × {0}. Observe also that if WFh (u) = K × {0}, we cannot conclude that the partial derivatives ∂ α u away from K are uniformly bounded in h, only that (h∂)α u = o(1) as h → 0. (iii) An alternative definition of the wavefront set is this: (x0 , ξ0 ) ∈ / W Fh (u) if there exist ϕ, ψ ∈ Cc∞ such that ϕ = 1 near x0 , ψ = 1 near ξ0 , and ψFh (ϕu) = O(h∞ ). Because of the compactly supported localizations, the O term can be taken in any form one wishes: L2 , L∞ , C ∞ . They are all equivalent.  EXAMPLES.

8.4. WAVEFRONT SETS, ESSENTIAL SUPPORT, MICROLOCALITY 183

(i) Fix a point (x0 , ξ0 ) and recall from Section 5.1 the corresponding coherent state: n

i

1

2

u(h)(x) := (πh)− 4 e h hx−x0 ,ξ0 i− 2h |x−x0 | . Then WFh (u) = {(x0 , ξ0 )}, as can be seen from the argument in Example 1 of Section 5.1. (ii) Suppose that b ∈ S (Rn ) and ϕ ∈ C ∞ (Rn ) is real. Then i

WFh (be h ϕ ) = {(x, ∂x ϕ(x)) | x ∈ spt(b)}. This corresponds to Example 2 in Section 5.1. (iii) This example generalizes the first two. Suppose that b ∈ S(Rn ) and that ϕ ∈ C ∞ (Rn ) is complex, satisfying Imϕ(x) ≥ 0, Imϕ(x) ≥ chxi2 − C, ϕ ∈ S(hxi2 ). Then i

WFh (b(x)e h ϕ(x) ) = {(x, ∂x ϕ(x)) | x ∈ spt(b), Im ϕ(x) = 0}. The verification is an exercise. (iv) Suppose that f ∈ L2 (R) and that spt f is compact. Define
u(h)(x) := f x − h1 . Then WFh (u) = ∅. This simple example shows that our wavefront set is measuring behavior only in compact sets in phase space and may not be senstitive to C ∞ regularity. (v) The Fourier conjugation formula (4.2.14) implies WFh (Fh u) = {(x, ξ) | (−ξ, x) ∈ WFh (u)}. (vi) Example (ii) states that if u ∈ S and u independent of h, then spt(u) × {0} = WFh (u). More generally, if u ∈ S 0 is independent of h, we have spt(u) × {0} ⊆ WFh (u). REMARKS: Classical wavefront set. (i) For readers familiar with H¨ ormander’s classical wavefront sets WF(u) (H¨ormander [H1, Chapter 8]), we suggest checking the following statement: (8.4.8) If u ∈ S 0 is independent of h, then

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WFh (u) = (spt(u) × {0}) ∪ WF(u). (ii) There is no unique way of designing a wavefront set which simultaneously captures C ∞ and semiclassical properties. To illustrate these complications, suppose u ∈ L2 (R) and |ˆ u(ξ)| ≤ Chξi− for some  > 0. Then WFh (u(x) exp(ie1/h x)) = ∅. But WF(u exp(ie1/h x)) = WF(u) for every h, and this is nonempty if u is not a smooth function.  Next we show that we can replace the given operator aw (x, hD) in the definition of wavefront set with others: THEOREM 8.13 (Localization and wavefront sets). Suppose that u is a tempered family of functions and (x0 , ξ0 ) ∈ / WFh (u). Then for any b ∈ Cc∞ (R2n ) with support sufficiently close to (x0 , ξ0 ), we have kbw (x, hD)u(h)kL2 = O(h∞ ).

(8.4.9)

Proof. 1. Suppose a ∈ S, a(x0 , ξ0 ) 6= 0. There exists χ ∈ C ∞ (R2n ) supported near (x0 , ξ0 ) such that |χ(x, ξ)(a(x, ξ) − a(x0 , ξ0 )) + a(x0 , ξ0 )| ≥ γ > 0 for (x, ξ) ∈ which

R2n .

Then according to Theorem 4.29 there exists c ∈ S for

cw (x, hD)(χw (x, hD)aw (x, hD) + a(x0 , ξ0 )(I − χw (x, hD))) = I, provided that h is small enough. 2. Now consider (8.4.10) bw (x, hD)u(h) =bw (x, hD)cw (x, hD)χw (x, hD)aw (x, hD)u(h) + a(x0 , ξ0 )bw (x, hD)cw (x, hD)(I − χw (x, hD)))u(h). If we choose a to be the symbol appearing in (8.4.7), then the first term on the right hand side is bounded by O(h∞ ) in L2 . If the support of b is sufficiently close to (x0 , ξ0 ), then spt(b) ∩ spt(1 − χ) = ∅ and according to Theorem 4.12, kbw (x, hD)cw (x, hD)(I − χw (x, hD)))kS 0 →S = O(h∞ ). Hence the second term in (8.4.10) is also O(h∞ ) in L2 .



8.4. WAVEFRONT SETS, ESSENTIAL SUPPORT, MICROLOCALITY 185

In Chapter 9 we will study how pseuodifferential operators vary under coordinate changes. Anticipating Theorem 9.3 from that chapter, we record here the invariance property of the wavefront set: THEOREM 8.14 (Invariance of wavefront sets). Suppose γ : Rn → Rn is a diffeomorphism equal to the identity outside of a compact set. Suppose also u is a tempered family of functions. Then (8.4.11)

WFh (γ ∗ u) = {(x, (∂γ(x))T ξ) | (γ(x), ξ) ∈ WFh (u)}.

8.4.3. Essential support. Since compactness of the support of a symbol is not preserved operations such as composition, we introduce the more flexible notion of the essential support. It will be useful for studying phenomena localized in phase space. DEFINITIONS. (i) Let a = {a(x, ξ, h)}0
spt χ ∩ K = ∅ implies χa ∈ h∞ S

for all χ ∈ S. (ii) The smallest such compact set K ⊂ Rn × Rn is called the essential support of a, denoted ess-spt(a). The essential support ess-spt(a) is defined only when we know that a has compact essential support. REMARK. The assumption on the family of symbols a can be replaced by a = {a(x, ξ, h)}0
WFh (aw u) ⊆ WFh (u).

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(ii) If a = {a(x, ξ, h)}0
(8.4.14)

Proof. For part (i) we need to show that if (x0 , ξ0 ) ∈ / WFh (u) then (x0 , ξ0 ) ∈ / WFh (aw u). Similarly for part (ii) we need the same conclusion when (x0 , ξ0 ) ∈ / ess-spt(a). 1. Suppose first that (x0 , ξ0 ) ∈ / WFh (u). Choose b ∈ Cc∞ (R2n ), with b(x0 , ξ0 ) 6= 0 and bw (x, hD)u(h) = OL2 (h∞ ). The existence of such b follows from Theorem 8.13. The composition formula in Theorem 4.11 shows that bw (x, hD)aw (x, hD, h) = cw (x, hD, h) + rw (x, hD, h), where spt c ⊂ spt b and r ∈ h∞ S . Theorem 8.13 implies that kbw (x, hD)aw (x, hD, h)u(h)kL2 ≤ kcw (x, hD, h)u(h)k + O(h∞ )kukL2 = O(h∞ ). This shows that (x0 , ξ0 ) ∈ / WFh (aw u). 2. Now assume that a has compact essential support and that (x0 , ξ0 ) ∈ / ess-spt(a). If the support of a symbol b is sufficiently close to (x0 , ξ0 ), (8.4.12) implies that bw (x, hD)aw (x, hD, h) = cw (x, hD, h), where c ∈ h∞ S . Consequently kbw (x, hD)aw (x, hD, h)u(h)kL2 = O(h∞ ).  THEOREM 8.16 (Alternative characterization of ess-spt(a)). Suppose that a = {a(x, ξ, h)}0
(x, ξ) ∈ / ess-spt(a)

if and only if (8.4.16)

(x, ξ) ∈ / WFh (aw u)

for each family u = {u(h)}0
8.4. WAVEFRONT SETS, ESSENTIAL SUPPORT, MICROLOCALITY 187

THEOREM 8.17 (Invariance of essential support). Suppose γ : Rn → Rn is a diffeomorphism equal to the identity outside of a compact set. Suppose that a = {a(x, ξ, h)}0
ess-spt(aγ ) = {(x, (∂γ(x))T ξ) | (γ(x), ξ) ∈ ess-spt(a)}.

Proof. We employ Theorems 8.14 and 8.16: (x, ξ) ∈ / ess-spt(aγ ) if and only if (x, ξ) ∈ / WFh ((γ ∗ )−1 Aγ ∗ u) for all u if and only if (γ(x), (∂γ(x)T )−1 ξ) ∈ / WFh (Aγ ∗ u) for all u if and only if (γ(x), (∂γ(x)T )−1 ξ) ∈ / WFh (Av) for all v if and only if (γ(x), (∂γ(x)T )−1 ξ) ∈ / ess-spt(a). Here the families of functions u and v are assumed to satisfy (8.4.1).



REMARK. Operators A = aw (x, hD, h) where a has compact essential support can be characterized as pseudodifferential operators such that (8.4.18)

(1 − χw (x, hD))A = OS 0 →S (h∞ )

for some χ ∈ Cc∞ (R2n ). This follows from the composition formula since χ#a satisfies (8.4.12) with K = spt χ ⊂⊂ R2n .  The remark above motivate the following DEFINITION. If A = aw (x, hD, h) and a has a compact essential support, then (8.4.19)

WFh (A) := ess-spt(a).

Roughly speaking, WFh (A) describes the compact set in the phase space R2n on which the operator A is localized.

188

8. MORE ON THE SYMBOL CALCULUS

8.4.4. Wavefront sets of localized functions. We conclude this section a stronger notion of the wavefront set for a family of functions u = {u(h)}0
(8.4.21)

Hence WF∗h serves only as a useful shorthand notation indicating that the localization (8.4.5) holds. 8.4.5. Microlocality. We introduce finally the concept of microlocal identity of operators: DEFINITIONS. Let U, V be open, bounded subsets of R2n . (i) Given two tempered operators T and S, we say that (8.4.22)

T ≡S

microlocally on U × V

e ⊃ U and Ve ⊃ V such that if there exist open sets U A(T − S)B = OS →S (h∞ ) e. for all A, B with WFh (A) ⊂ Ve , WFh (B) ⊂ U (ii) In particular, we say (8.4.23)

T ≡ I microlocally near U × U

e ⊃ U such that if there exists an open set U A − T A = A − AT = OS →S (h∞ ) e. for all A with WFh (A) ⊂ U (iii) We say that T is microlocally invertible near U × V if there exists an operator S such that T S ≡ I and ST ≡ I microlocally near V × V and U × U , respectively. When confusion is unlikely, we will write S = T −1 and call S a microlocal inverse of T .

8.5. NOTES

189

EXAMPLE. Suppose that a ∈ S and a(x0 , ξ0 ) 6= 0. Then aw (x, hD) is microlocally invertible in a neighbourhood of (x0 , ξ0 ).  INTERPRETATION. The notion of microlocality gives us a quick way to refer to phenomena occuring in phase space. Owing to the uncertainty principle localization in phase space is more subtle than localization in space, but in many ways it has the same properties. As shown in Theorem 8.15 pseudodifferential operators, that is quantum observables, do not “move things” in phase space. That however will not be so for other natural operators introduced later, in Chapters 10 and 11. We can best and most conveniently understand their properties geometrically using the notion of microlocality. It will be particularly useful in Chapter 12 when we study microlocal normal forms. 

8.5. NOTES Our presentation of Beals’s Theorem follows Dimassi–Sj¨ostrand [D-S, Chapter 8] and Helffer–Sj¨ ostrand [H-S]. Section 8.2 provides a special case of a general result in Bony–Chemin [B-C, Th´eoreme 6.4]. The self-contained proof in the simple case considered here comes from the appendix to [S-Z3]. See [D-S-Z], [S-Z3] for examples of conjugation techniques, and Martinez [M] for a different perspective based on the FBI transform. The semiclassical wavefront set is an analog of the usual wavefront set in microlocal analysis (see H¨ormander [H1, Chapter 8], [H3, Chapter 18], and Grigis-Sj¨ ostrand [G-S] and is closely related to the frequency set introduced in Guillemin–Sternberg [G-St]. See Alexandrova [A] for more about semiclassical wavefront sets.

Chapter 9

CHANGING VARIABLES

9.1 9.2 9.3 9.4

Invariance, half-densities Changing symbols Invariant symbol classes Notes

9.1. INVARIANCE, HALF-DENSITIES 9.1.1. Motivation, definitions. We begin with a general discussion concerning the invariance of various quantities under the change of variables (9.1.1)

x ˜ = γ(x),

where γ : Rn → Rn is a diffeomorphism that equals the identity outside some bounded set. Functions. Recall first that functions transform under (9.1.1) by pull-back. This means that we change u into a function u ˜ of the new variables x ˜ by the rule (9.1.2)

u ˜(˜ x) = u ˜(γ(x)) := u(x)

for x ∈ Rn . In the notation of pull-backs acting on functions, this is written: (9.1.3)

γ∗u ˜ = u. 191

192

9. CHANGING VARIABLES

Notice however that in general the integral of |u|2 over a Borel set E is not then invariant: Z Z 2 |˜ u(˜ x)| d˜ x 6= |u(x)|2 dx. γ(E)

E

We introduce the formalism of half-densities to fix this defect. Half-densities. In quantum mechanics the probability of “finding our state in the set E” is given by Z |u(x)|2 dx.

E

This probability density should be invariantly defined, and so should not depend on the choice of coordinates x. This means that it is not the function u which should be defined invariantly but rather the density |u|2 dx, or, up to the phase information, the half-density 1

(9.1.4)

u|dx| 2 .

For half-densities we therefore demand that 1

1

“u ˜(˜ x)|d˜ x| 2 = u(x)|dx| 2 ”, meaning that integrals of the squares should be invariantly defined. To accomplish this, we as follows modify our earlier definitions (9.1.2), (9.1.3): DEFINITION. The pull-back of γ acting on half-densities is defined by the rule   1 1 ˜|d˜ x| 2 = u|dx| 2 , (9.1.5) γ∗ u for (9.1.6)

1

u ˜(˜ x) = u ˜(γ(x)) := u(x)| det(∂γ(x))|− 2 .

With this modified definition we have Z Z 2 |˜ u(˜ x)| d˜ x= |u(x)|2 dx. γ(E)

for all Borel sets E ⊆

E

Rn .

NOTATION. We will hereafter write 1

1

L2 (Rn ; Ω 2 ) := {u|dx| 2 | u ∈ L2 (Rn )} for the collection of half-densities on Rn , with the obvious norm 1

ku|dx| 2 k

1

L2 (Rn ;Ω 2 )

:= kukL2 (Rn ) .

Then (9.1.7)

1

1

γ ∗ , (γ −1 )∗ : L2 (Rn ; Ω 2 ) → L2 (Rn ; Ω 2 ).

9.1. INVARIANCE, HALF-DENSITIES

193

DISCUSSION. The foregoing formalism is at first rather unintuitive, but turns out later to be useful in the rigorous semiclassical calculus, in particular in the theory of Fourier integral operators, which we will touch upon later. Section 9.2 will show how the half-density viewpoint fits naturally within the Weyl calculus, and Section 10.2 will explain how half-densities simplify some related calculations for a propagator.  9.1.2. Operators on half-densities. We continue to suppose γ : Rn → Rn is a smooth diffeomorphism equally the identity outside a bounded set. DEFINITION. If 1

1

A : L2 (Rn ; Ω 2 ) → L2 (Rn ; Ω 2 ), we define the pull-back of A to be the operator e := (γ −1 )∗ Aγ ∗ , A

(9.1.8)

the pull-backs acting on half-densities. In half-density notation, this is written (9.1.9)

1 1 1 1 ˜ u|d˜ A(˜ x| 2 )|d˜ x| 2 = A(u|dx| 2 )|dx| 2 .

Half-densities also elegantly appear when we use operator kernels: DEFINITION. Let K ∈ S (R2n ). Then the formal expression 1

1

K(x, y)|dx| 2 |dy| 2 acts as an integral kernel, defining as follows a map on half-densities:  Z  1 1 1 1 A u|dx| 2 = K(x, y)|dx| 2 |dy| 2 u(y)|dy| 2 n RZ  (9.1.10) 1 := K(x, y)u(y)dy |dx| 2 . Rn

THEOREM 9.1 (Operator kernels and half-densities). Let K be the kernel for an operator A acting on half-densities, as above. Then the kernel e is for the pull-back A 1 1 e x, y˜)|d˜ K(˜ x| 2 |d˜ y| 2 , where (9.1.11)

e x, y˜) = K(x, y))| det(∂γ(x))|− 12 | det(∂γ(y))|− 12 K(˜

for x ˜ = γ(x), y˜ = γ(y).

194

9. CHANGING VARIABLES

Proof. REMARK: Numerical interpretation. The half-density formalism is also a surprisingly useful bookkeeping technique, appearing implicitly in some numerical schemes. Suppose that an operator A is given by by a symmetric integral kernel K(x, y). A numerical implementation converts A into a matrix AN = ((K(xi , xj )))1≤i,j≤N , which acts on vectors v ∈ CN intepreted as v = (u(xi ))1≤i≤N . Integration of u involves multiplication of each entry by a weight µj > 0 and summation: Z u(x) dx ≈

N X

µj v j .

j=1

Hence Z (Au)(xi ) =

K(xi , y)u(y) dy ≈

N X

AN ij µj vj .

j=1

In other words the application of A is where  µ1 0  0 µ2  M :=  . ..  .. . 0

0

approximated by the matrix AN M ... ... .. .

0 0 .. .

   . 

. . . µN

What is lost is the symmetry of the matrix and this may be numerically inconvenient. A simple remedy is to consider 1

(Au)(xi )µi2 ≈

N X 1 1 1 2 2 (µi2 AN ij µj )(µj vj ), j=1 1

which acts on, and returns, discretized functions multiplied by M 2 . The application of A corresponds to the symmetric matrix 1

1

M 2 AN M 2 . Since the weights correspond to integration, this is a discrete version of the half-density formalism.  9.1.3. Quantization and half-densities. We can also interpret the Weyl quantization of a symbol a ∈ S as an operator acting on half-densities: DEFINITIONS.

9.2. CHANGING SYMBOLS

195

(i) If a ∈ S , we write 1

1

Ka (x, y)|dx| 2 |dy| 2 (9.1.12)

1 := (2πh)n

Z a Rn

1 1 x+y
hi hx−y,ξi dξ|dx| 2 |dy| 2 . 2 ,ξ e

(ii) We define aw (x, hD) as an operator acting on half-densities by the formula Z 1 1 w Ka (x, y)u(y) dy|dx| 2 . (9.1.13) a (x, hD)(u|dx| 2 ) := Rn

THEOREM 9.2 (Quantization acting on half-densities). If a ∈ S , then 1 1 aw (x, hD) : L2 (Rn , Ω 2 ) → L2 (Rn , Ω 2 ). is a bounded operator. This is shown by arguments as in Chapter 4, details of which we omit.

9.2. CHANGING SYMBOLS In this section we investigate the invariance properties of quantization under changes of variables. 9.2.1. Changing variables and changing symbols. Next, let γ : Rn → Rn be a smooth diffeomorphism which as before we assume to be the identity outside a bounded set. As above, we write x ˜ = γ(x) and 1

1

u ˜(˜ x)|d˜ x| 2 = u(x)|dx| 2 . THEOREM 9.3 (Quantization and change of variables). Assume a ∈ S. (i) Then (γ −1 )∗ aw (x, hD)γ ∗ = a ˜w (x, hD)

(9.2.1)

as operators on half-densities, for the symbol (9.2.2)

a ˜(x, ξ) := a(γ −1 (x), ∂γ(x)T ξ) + OS (h2 ).

(ii) Furthermore, (9.2.3)

(γ −1 )∗ aw (x, hD)γ ∗ = a ˜w 1 (x, hD)

as operators on functions, for the symbol (9.2.4)

a ˜1 (x, ξ) := a(γ −1 (x), ∂γ(x)T ξ) + OS (h).

196

9. CHANGING VARIABLES

DEFINITION. We call b(x, ξ) := a(γ −1 (x), ∂γ(x)T ξ)

(9.2.5)

the pull-back of the symbol a under the mapping γ. (It is more precisely pull-back by the lift of the diffeomorphism to the cotangent bundle: recall Theorem 2.6.) DISCUSSION. Yet another motivation for half-densities is that assertion (i) for half-densities (with error term of order OS (h2 )) is more precise than the assertion (ii) for functions (with error term OS (h)). Remember that the notation ϕ = OS (hk ) means |xα ξ β ∂ γ ϕ| ≤ Cαβγ hk for all multiindices α, β, γ.



Proof. 1. Write A = aw (x, hD). Since a ∈ S , we have Ka ∈ S . Take a ˜ ∈ S (R2n ) for which (γ −1 )∗ Aγ ∗ = a ˜w . 2. Remember that Z

1 2

1

1

1

Ka (x, y)|dx| 2 |dy| 2 u(y)|dy| 2

A(u|dx| ) = Rn

for

Likewise

Z
i 1 Ka (x, y) := a x+y , ξ e h hx−y,ξi dξ. 2 n (2πh) Rn Z 1 1 1 1 ˜ u|d˜ Ka˜ (˜ x, y˜)|d˜ x| 2 |d˜ y| 2 u ˜(˜ y ) |d˜ y| 2 A(˜ x| 2 ) = Rn

for

1 Ka˜ (˜ x, y˜) := (2πh)n 1 2

Z a ˜ Rn





x ˜+˜ y ˜ 2 ,ξ

i ˜ ˜ e h h˜x−˜y,ξi dξ.

1 2

y = | det ∂γ(y)|dy, the invariance (9.1.9) Since u ˜(˜ y )|d˜ y | = u(y)|dy| and d˜ implies (9.2.6)

1

1

Ka (x, y) = Ka˜ (˜ x, y˜)| det ∂γ(y)| 2 | det ∂γ(x)| 2 .

3. We need to compute a ˜ in terms of a and γ, up to O(h2 ) error terms. Now Z   i 1 ˜ x ˜+˜ y ˜ Ka˜ (˜ x, y˜) = a ˜ , ξ e h h˜x−˜y,ξi dξ˜ 2 (2πh)n Rn Z   i 1 ˜ ˜ γ(x)+γ(y) ˜ = a ˜ , ξ e h hγ(x)−γ(y),ξi dξ. 2 n (2πh) Rn We have (9.2.7)

γ(x) − γ(y) = g(x, y)(x − y),

9.2. CHANGING SYMBOLS

197

where g(x, y) is a matrix satisfying   x+y + O(|x − y|2 ). (9.2.8) g(x, y) = ∂γ 2 Also  (9.2.9)

γ(x) + γ(y) = 2γ

x+y 2



+ O(|x − y|2 ).

Let us also write ξ˜ = (g(x, y)T )−1 ξ.

(9.2.10)

Substituting above, we deduce that Ka˜ (˜ x, y˜) =

1 (2πh)n

Z



Rn


 T −1 2 a ˜ γ( x+y 2 ), (g(x, y) ) ξ + O(|x − y| ) i

T )−1 ξi

e h hγ(x)−γ(y),(g(x,y)

˜ dξ.

We now use the “Kuranishi trick” to rewrite this expression as a pseudodifferential operator. First, hγ(x) − γ(y), (g(x, y)T )−1 ξi = hg(x, y)−1 (γ(x) − γ(y)), ξi = hx − y, ξi, according to (9.2.7). Remembering also (9.2.8), we compute Ka˜ (˜ x, y˜) = Z 
 i 1 a ˜ γ( x+y ), (∂γ( x+y )T )−1 ξ + O(|x − y|2 ) e h hx−y,ξi dξ˜ 2 2 n (2πh) Rn Z  i  x+y 1 ˜ = a( 2 , ξ) + O(|x − y|2 ) e h hx−y,ξi dξ. n (2πh) Rn Furthermore dξ˜ = | det g(x, y)|−1 dξ and 2 det g(x, y) = det ∂γ( x+y 2 ) + O(|x − y| ).

Also, we claim that det ∂γ( x+y ) 2 = | det ∂γ(x)|| det ∂γ(y)| + O(|x − y|2 ). 2 This identity is clear if we add a term hA( x+y 2 ), x − yi on the right hand side. But the symmetry under switching x and y shows that A ≡ 0. 4. Finally we observe that (9.2.11)

i

i

(x − y)α e h hx−y,ξi = (hDξ )α e h hx−y,ξi .

198

9. CHANGING VARIABLES

Hence integrating by parts in the terms with O(|x − y|2 ) gives us terms of order O(h2 ). So Ka˜ (˜ x, y˜) = Z
i 1 , ξ) + O(h2 ) e h hx−y,ξi dξ a( x+y 2 n (2πh) Rn | det ∂γ(x)|−1/2 | det ∂γ(y)|−1/2 . This proves (9.2.6) with a ˜ satisfying (9.2.2). 5. When A acts on functions, then Ka has to transform as a density. In other words, we need to show (9.2.12)

Ka (x, y) = Ka1 (˜ x, y˜)| det ∂γ(y)| + O(h),

instead of (9.2.6). Since | det ∂γ(y)| = | det ∂γ(y)|1/2 | det ∂γ(x)|1/2 + O(|x − y|), we see from (9.2.11) that (9.2.12) follows from (9.2.6) with a1 = a ˜ +O(h). 

9.3. INVARIANT SYMBOL CLASSES We now introduce more general classes of symbols a than those discussed in Chapter 4, and explain their invariance under mappings γ : Rn → Rn . 9.3.1. Classical symbols. To explain the need for these new classes of symbols, let m be an order function and recall the class S(m) introduced in Chapter 4: S(m) := {a ∈ C ∞ (R2n ) | |∂ α a| ≤ Cα m for all α}. In view of Theorem 9.3, for a symbol a in S(m) to be invariant under γ, we would need for its pull-back b(x, ξ) := a(γ −1 (x), ∂γ(x)T ξ) that (9.3.1)

|∂ α b| ≤ Cα m

for all multiindices α. But this bound is in general false except for a very restrictive class of order functions m, that in particular would exclude all differential operators. However an estimate of the type (9.3.1) would hold if differentiation in ξ improves the decay in ξ. This observation leads us to the following definition, in which we restrict to the order functions hξim . DEFINITIONS. (i) The Kohn-Nirenberg symbols for m ∈ Z are (9.3.2)

S m := {a ∈ C ∞ (R2n ) | |∂xα ∂ξβ a| ≤ Cαβ hξim−|β| for all α, β}.

9.3. INVARIANT SYMBOL CLASSES

199

(ii) We also write (9.3.3)

Ψm := {aw (x, hD) | a ∈ S m }

and (9.3.4)

Ψ−∞ :=

\

Ψm .

m∈Z

As for the classes introduced in Chapter 4, symbols in S m are allowed to depend upon h, although this dependence is usually not displayed in our notation. If a ∈ S m depends on h, we require that the constants Cαβ are uniform for 0 < h ≤ h0 . THEOREM 9.4 (Invariance of S m ). Assume γ : Rn → Rn is a smooth diffeomorphism, with |∂ α γ|, |∂ α γ −1 | ≤ Cα for all multiindices α. Then for each symbol a ∈ S m , its pull-back b(x, ξ) := a(γ −1 (x), ∂γ(x)T ξ) also belongs to S m . Proof. We have b(x, ξ) = c(x, γ(x)T ξ) for c(x, ξ) := a(γ −1 (x), ξ). Now ∂xα b has the form X ∂xα b = gγσρ (∂xγ ∂ξσ c)ξ ρ , the sum over multiindices satisfying |γ| + |σ| ≤ |α| and |σ| = |ρ|. Therefore X gγσρνκλ (∂xγ ∂ξσ+ν−κ c)ξ ρ−λ , ∂xα ∂ξβ b = where |γ| + |σ| ≤ |α|, |σ| = |ρ|, |κ| = |λ|, |ν| = |β|, ν ≥ κ, ρ ≥ λ. Since clearly c ∈ S m , we can estimate X |∂xα ∂ξβ b| ≤ Cγσρνκλ hξim−|σ|−|ν|+|κ| hξi|ρ|−|λ| ≤ Cαβ hξim−|β| .  9.3.2. Symbol calculus for S m . Since S m ⊂ S(hξim ), the results of Chapter 4 are all applicable; but due to the improvement under differentiation in ξ there are many new features, important in the study of partial differential equations. THEOREM 9.5 (Composition for Ψm ). (i) Assume a ∈ S m1 and b ∈ S m2 . Then aw (x, hD)bw (x, hD) = cw (x, hD) where the symbol c ∈ S m1 +m2 is given by (4.3.5).

200

9. CHANGING VARIABLES

(ii) Moreover, (9.3.5) N k k X i h c(x, ξ) = A(D)k a(x, ξ)b(y, η)|x=ξ,y=η + OS m1 +m2 −N −1 (hN +1 ) k! k=0

where A(D) = 21 σ(Dx , Dξ ; Dy , Dη ). (iii) Similar statements hold for the usual quantization: a(x, hD)b(x, hD) = c1 (x, hD) = eihhDξ ,Dy i a(x, ξ)b(y, η)|y=x,η=ξ and c1 (x, ξ) =

X 1 ∂ α a(x, ξ)(hDx )α b(x, ξ) + OS m1 +m2 −N −1 (hN +1 ). α! ξ

|α|≤N

Proof. 1. Since S mj ⊂ S(hξimj ), (4.3.5) follows from Theorem 4.11, but we must show that c ∈ S m1 +m2 . Similarly, the expansion (9.3.5) is valid, but we need to prove the stated error bounds. 2. We first recall from (4.3.4) that c(x, ξ) = exp(hA(D))a(x, ξ)b(y, η)|x=y,η=ξ ; and observe also that ik hk A(D)k a(x, ξ)b(y, η)|x=ξ,y=η ∈ hk S m1 +m2 −k , since the differential operator A(D)k entails k derivatives in the variables ξ, η and k derivatives in x, y. We claim that the remainder satisfies (9.3.6) c(x, ξ) −

N k k X i h k=0

k!

A(D)k a(x, ξ)b(y, η)|x=ξ,y=η ∈ hN +1 S(hξim1 +m2 −N −1 ).

Since N is arbitrary, it will follow from (9.3.6) that |∂xα ∂ξβ c(x, ξ)| ≤ Cα,β hξim1 +m2 −|β| , uniformly for 0 < h ≤ 1; and consequently, c ∈ S m1 +m2 . 3. The left hand side of (9.3.6) is a constant times Z 1 (9.3.7) (1 − t)N exp(thA(D))(hA(D))N +1 (a(x, ξ)b(y, η))|x=y,ξ=η dt. 0

9.3. INVARIANT SYMBOL CLASSES

201

As a ∈ S m1 and b ∈ S m2 , we see that (hA(D))N +1 (a(x, ξ)b(y, η)) ∈ hN +1

N +1 X

S(hξim1 −k hηim2 −N −1+k ).

k=0

Theorem 4.17 shows that exp(thA(D)) : S(hξim1 −k hηim2 −N −1+k ) → S(hξim1 −k hηim2 −N −1+k ). Observing that S(R4n , hξim1 −k hηim2 −N −1+k )|x=y,ξ=η = S(R2n , hξim1 +m2 −N −1 ), we use the foregoing to estimate (9.3.7) and thereby establish the claim (9.3.6).  Let us also record the following useful result: THEOREM 9.6 (Schwartz kernels of operators in Ψm ). (i) Suppose that a ∈ S m and that Ka ∈ S 0 (Rn × Rn ) is the Schwartz kernel of aw (x, hD). Then Ka (x, y) ∈ C ∞ (Rn × Rn \ ∆)

(9.3.8)

for the diagonal ∆ := {(x, x) | x ∈ Rn }. (ii) Furthermore, we have the estimates (9.3.9)

α



β

|(hDx ) (hDy ) Ka (x, y)| ≤ CN

h |x − y|

N

for (x, y) ∈ Rn × Rn \ ∆ and N > |α| + |β| + m + n. (iii) If K satisfies (9.3.10)

|∂xα ∂yβ K(x, y)|

 ≤ CN

h hx − yi

N

for all (x, y) ∈ Rn ×Rn , then K is the Schwartz kernel of aw for some symbol a ∈ h∞ S −∞ . This illustrates one of the many advantages of symbol classes S m , since we now have smoothness and rapid decay away from the diagonal. Proof. 1. We can consider either the Weyl quantization or the standard quantization. This follows from Theorem 4.13, since (9.3.11)

exp (i(t − s)hhDx , Dξ i) : S m → S m ,

as in the proof of Theorem 9.5. So for simplicity of notation, we take a(x, hD).

202

9. CHANGING VARIABLES

2. Suppose first that a ∈ S . Then the kernel of a(x, hD) is Z 1 Ka (x, y) = a(x, ξ)eihx−y,ξi/h dξ. (2πh)n Rn Since (9.3.12)

1 (x − y) Ka (x, y) = (2πh)n γ

Z

(−hDξ )γ a(x, ξ)eihx−y,ξi/h dξ,

Rn

we have |γ|

γ

|(x − y) Ka (x, y)| ≤ Cγ h

khξin+1 ∂ξγ akL∞

Z

hξi−n−1 dξ.

Rn

Observe next that N

sup |(x − y)γ | ≥ n− 2 |x − y|N , |γ|=N

and therefore  (9.3.13)

|Ka (x, y)| ≤ CN

h |x − y|

N

sup khξin+1 ∂ξγ akL∞ .

|γ|=N

Going back to (9.3.12) gives a similar estimate for derivatives: |(hDx )α (hDy )β ((x − y)γ Ka (x, y))| ≤ CN,αβ hN

sup |ρ|≤|α|,|γ|=N

khξin+1+|α|+|β| ∂ξγ ∂xρ akL∞ .

Since (hDx )α (hDy )β ((x − y)γ Ka (x, y)) = (x − y)γ (hDx )α (hDy )β Ka (x, y) X 0 0 0 + cα0 β 0 γ 0 (x − y)γ (hDx )α (hDy )β Ka (x, y), |α0 |+|β 0 |<|α|+|β| |γ 0 |<|γ|

induction on |α| + |β| shows that (9.3.14) |(hDx )α (hDy )β Ka (x, y)|  N h ≤ CN,αβ sup khξin+1+|α|+|β| ∂xρ ∂ξγ akL∞ . |x − y| |ρ|≤|α|,|γ|=N If a ∈ S m , we observe that seminorms appearing on the right hand side of (9.3.14) are finite and bounded if N > |α| + |β| + m + n. Approximation by symbols in S concludes the proof of (9.3.9). 4. Assertion (iii) follows from the inverse Fourier transform. Since |∂xα ∂zβ Ka (x, x − z)| ≤ CN

hN , hziN

9.3. INVARIANT SYMBOL CLASSES

203

for all N and Z

Ka (x, x − z)eihz,ξi/h dz,

a(x, ξ) = Rn

we have a ∈ h∞ S −∞ .



As an immediate consequence we obtain the following THEOREM 9.7 (More on disjoint support). Let b ∈ S m and suppose ϕ, ψ ∈ Cc∞ (Rn ). If spt(ϕ) ∩ spt(ψ) = ∅,

(9.3.15) then

kϕ bw (x, hD) ψkH −N (Rn )→H N (Rn ) = O(h∞ )

(9.3.16)

h

h

for all N . 9.3.3. Changing variables for S m . Before restating Theorem 9.3 in this more general setting, we will discuss usual quantization acting on functions. LEMMA 9.8 (Changing variables and exponentials). Suppose that a ∈ S . Then (γ −1 )∗ a(x, hD)γ ∗ = aγ (x, hD),

(9.3.17) where

i

i

aγ (γ(x), η) = e− h hγ(x),ηi a(x, hD)e h hγ(·),ηi .

(9.3.18)

Proof. Theorem 4.19 shows i

i

i

i

aγ (y, η) = e− h hy,ηi aγ (x, hD)e h hy,ηi = e− h hy,ηi (γ −1 )∗ a(x, hD)γ ∗ (e− h h·,ηi ), and this gives (9.3.18).



THEOREM 9.9 (Changing variables I). Suppose that a ∈ S m . (i) Then the formula (9.3.17) defines a symbol aγ ∈ S m for which (9.3.18) holds. (ii) Moreover, aγ (γ(x), η) = (9.3.19)

X 1 i ∂ξα a(x, ∂γ(x)T η)(hDy )α e h hρx (y),ηi |y=x α!

|α|≤N

+ OS m−N −1 (hN +1 ), where ρx (y) = γ(y) − γ(x) − ∂γ(x)(y − x).

204

9. CHANGING VARIABLES

(iii) In particular, aγ (γ(x), η) = a(x, ∂γ(x)T η) + OS m−1 (h).

(9.3.20)

Proof. 1. We need to show that aγ defined by (9.3.18) is in S m . This will imply (γ −1 )∗ a(x, hD)γ ∗ u = aγ (x, hD)u for u(x) = exp(ihx, ηi/h). The operator identity (9.3.17) will follow, since such functions u are dense in S 0 . 2. Select a function χ ∈ Cc∞ (R, [0, 1]) satisfying χ ≡ 1 on [−1, 1] and χ ≡ 0 on R \ [−2, 2]. We assert that (9.3.21) aγ (γ(x), η) = Z Z 1 a(x, ξ)χx (y)χη (ξ)ei(hx−y,ξi+hγ(y)−γ(x),ηi)/h dydξ (2πh)n Rn Rn + OS (hηi−∞ h∞ ) where χx (y) := χ(x − y), χη (ξ) := χ((ξ − ∂γ(x)T η)/hηi). To prove this, observe that on the support of 1 − χx (y)χη (ξ) the phase is not stationary, since dy,ξ (hx − y, ξi + hγ(y) − γ(x), ηi) = 0 if and only if x = y and ξ = ∂γ(x)T η. Consequently a standard integration-by-parts argument, which we leave to the reader, gives (9.3.21). 3. We rewrite the integral term in (9.3.21) as Z Z 1 ˜ ˜ h ˜ x (y)χ ˜ i(hx−y,ξi+hγ(y)−γ(x),η/hηii)/ aη (x, ξ)χ ˜η (ξ)e dydξ˜ n ˜ n n (2π h) R R for ˜ := a(x, hηiξ), ˜ χ ˜ := χη (ξhηi), ˜ ˜ := h/hηi. aη (x, ξ) ˜η (ξ) h The support of the integrand is contained in a fixed compact set, and hηi−m aη χη ∈ S, uniformly in h and η. Hence the stationary phase method (Theorem 3.16) ˜ = h/hηi. Computing the leading term, yields an expansion in powers of h we easily check that (9.3.20) holds. 4. Since we will not use (9.3.19), we refer to H¨ormander [H2, Theorem 18.1.17] for the proof. 

9.3. INVARIANT SYMBOL CLASSES

205

The proof given in Section 9.6 can be adapted to the present setting using the integration by parts arguments from the proof of Theorem 9.6. This gives THEOREM 9.10 (Changing variables II). Let a ∈ S m . (i) Then (9.3.22)

(γ −1 )∗ aw (x, hD)γ ∗ = a ˜w (x, hD),

acting on half-densities, for (9.3.23)

a ˜(x, ξ) := a(γ −1 (x), ∂γ(x)T ξ) + OS m−2 (h2 ).

(ii) Also, (γ −1 )∗ aw (x, hD)γ ∗ = a ˜w 1 (x, hD), acting on functions, for (9.3.24)

a ˜1 (x, ξ) := a(γ −1 (x), ∂γ(x)T ξ) + OS m−1 (h).

The invariance presented in the two theorems above leads to a definition of a symbol of a pseudodifferential operator: DEFINITION. We define the principal symbol of the pseudodifferential operator A = aw (x, hD) ∈ Ψm to be the equivalence class: (9.3.25)

σ(A) = σm (A) := a(x, ξ) mod (hS m−1 ).

Theorem 9.9 shows that this is defined independently of the choice of coordinates, and Theorem 9.10 demonstrates that the principal symbol does not depend upon the particular quantization nor upon whether we act on functions or on half-densities. For the Weyl quantization and for action on half-densities, the same theorem shows that the symbol can be defined mod(h2 S m−2 ). 9.3.4. Beals’s Theorem again. THEOREM 9.11 (Beals’s Theorem for Ψm ). Let A : S → S 0 be a continuous linear operator. Then A ∈ Ψm if and only if kadxi1 · · · adxiN adDxj · · · adDxj AkH l+m (Rn )→H l+N (Rn ) = O(hN ) 1

M

h

h

for some l and all N, M = 0, 1, 2, . . . and all sequences ik , jl ∈ {1, · · · , n}, 1 ≤ k ≤ N, 1 ≤ l ≤ M. See Grigis and Sj¨ ostrand [G-S] for the modifications needed in the proof of Theorem 8.3.

206

9. CHANGING VARIABLES

9.4. NOTES For more on invariance, half-densities, and a functorial point of view towards semiclassical analysis see Guillemin–Sternberg [G-St1]. J. Strain suggested the numerical interpretation of half-densities. The proof of symbol invariance is from the appendix to [S-Z1]. For h = 1, Ψm is the class of Kohn-Nirenberg pseudodifferential operators: see H¨ ormander [H2, Section 18.1] or Grigis–Sj¨ostrand [G-S] for a thorough presentation. When h = 1, the calculus presented in this section becomes the classical pseudodifferential calculus, going back to the works of Bokobza-Unterberger, Kohn-Nirenberg, and H¨ormander in the 1960s.

Chapter 10

FOURIER INTEGRAL OPERATORS

10.1 10.2 10.3 10.4 10.5

Operator dynamics An integral representation formula Strichartz estimates Lp estimates for quasimodes Notes

Let m be an order function and suppose that {pt }t∈R ⊂ S(m) is a family of real-valued symbols that depend smoothly on t ∈ R. Set P (t) := pw t (x, hD). We investigate in this chapter and the next the corresponding quantum operator dynamics: (10.0.1)

( hDt F (t) + F (t)P (t) = 0 F (0) = I.

(t ∈ R)

This chapter mostly concerns explicit but local representations of approximate solutions of (10.0.1) for small times |t| < t0 , whereas Chapter 11 addresses global properties of solutions for large times. 207

208

10. FOURIER INTEGRAL OPERATORS

10.1. OPERATOR DYNAMICS We start by identifying several circumstances under which the quantum dynamics (10.0.1) are uniquely solved by a family {F (t)}t∈R of bounded, unitary operators on L2 (Rn ). 10.1.1. Symbols in S. We consider first the case that m ≡ 1: THEOREM 10.1 (Quantum dynamics in S). (i) If {pt }t∈R ⊂ S is a family of real-valued symbols that depend smoothly on t, then the operator equation (10.0.1) is uniquely solved by a family {F (t)}t∈R of bounded, unitary operators on L2 (Rn ). (ii) Furthermore, (10.1.1)

F (t) : Hh (m) → Hh (m).

for each order function m. Proof. 1. Since pt ∈ S, Theorem 4.23 implies that the operator P (t) = pw t is bounded from L2 to L2 , with P (t)∗ = P (t). As P (t) depends smoothly on t as an operator, equation (11.1.2) consequently has a unique solution. Furthermore, ( hDt F (t)∗ − P (t)F (t)∗ = 0 (10.1.2) F (0)∗ = I. 2. We assert that F (t) is unitary on L2 . To confirm this, let us calculate using (10.0.1) and (10.1.2) that hDt (F (t)F (t)∗ ) = hDt F (t)F (t)∗ + F (t)hDt F (t)∗ = −F (t)P (t)F (t)∗ + F (t)P (t)F (t)∗ = 0. Since F (0)F (0)∗ = I, it follows that F (t)F (t)∗ ≡ I. On the other hand, hDt (F (t)∗ F (t) − I) = P (t)F (t)∗ F (t) − F (t)∗ F (t)P (t) = [P (t), F (t)∗ F (t) − I]. with F (0)∗ F (0) − I = 0. Since this equation for F (t)∗ F (t) − I is homogeneous, it follows that F (t)∗ F (t) ≡ I. 3. To prove (10.1.1) we can assume that m ∈ S(m). We then define g := log m, write G := g w (x, hD), and introduce the operator FG (t) := eG F (t)e−G .

10.1. OPERATOR DYNAMICS

209

Then FG (t) : Hh (m) → Hh (1/m). Consider now the equation satisfied by FG (t): (10.1.3)

hDt FG (t) = eG F (t)P (t)e−G = FG (t)PG (t),

for PG (t) := eG P (t)e−G . Theorem 8.6 implies e±G = cw ± (x, hD) for symbols c± ∈ S(m±1 ); and hence PG (t) = bt (x, hD) for a symbol bt ∈ S. Since FG (0) = I, equation (10.1.3) implies Z t kPG (s)kL2 →L2 ds < ∞.  kF (t)kHh (m)→Hh (m) = kFG (t)kL2 →L2 ≤ exp 0

REMARKS. (i) According to (8.3.10), we can use the L2 inner product to identify the dual space of Hh (m) with Hh (1/m). Hence (10.1.1) implies that (10.1.4)

F (t)−1 = F (t)∗ : Hh (m) → Hh (m)

for all order functions m. (ii) We recall from (8.3.11) that \ S = Hh (m). m≥1

Then (10.1.1) and (10.1.4) imply (10.1.5)

F (t), F (t)−1 : S → S . 

10.1.2. Time independent, elliptic symbols. Assume next that m ≥ 1 is an order function and the real, time-independent symbol p belongs to S(m). We next establish under an ellipticity condition the self-adjointness of P = pw (x, hD) for small h and hence the solvability of (10.0.1) for all times. THEOREM 10.2 (Ellipticity and self-adjointness). Assume that m is an order function satisfying m ≥ 1. Let P = pw (x, hD) for a real-valued symbol p ∈ S(m), and assume p is elliptic in the sense that C + p ≥ cm for constants C ≥ 0, c > 0. (i) Then for h small enough P is essentially self-adjoint on L2 (Rn ) with the domain Hh (m).

210

10. FOURIER INTEGRAL OPERATORS

(ii) Consequently, F (t) := e−itP /h

(10.1.6)

(t ∈ R)

is a family of unitary operators on L2 (Rn ) solving (10.0.1). DEFINITION. We call {e−itP /h }t∈R the propagators generated by P . Proof. 1. The operator P with domain S is symmetric, since hP u, vi = hu, P vi for all u, v ∈ S . We claim that for h small the inverses (P ± i)−1 : L2 → Hh (m)

(10.1.7)

exist and are surjective. It then follows from Theorem C.12 that P with domain Hh (m) is essentially self-adjoint on L2 . 2. We have (p ± i)−1 ∈ S(1/m) and hence
w
w R w (P ± i) (p ± i)−1 = (p ± i)w (p ± i)−1 = I + h2 (r± ) , R ∈ S. Also for r±

(p ± i)−1


w

(P ± i) = (p ± i)−1


w

L w (p ± i)w = I + h2 (r± )

L ∈ S. Since (r R )w and (r L )w are bounded on L2 , uniformly with for r± ± ± respect to h, Theorem C.3 implies (P ± i)−1 exist when h is small enough.

The symbolic calculus also shows that
w hhDis (p ± i)−1 = aw (x, hD)
w for a symbol a ∈ S, and thus is bounded on L2 . Consequently (p ± i)−1 = OL2 →Hh (m) (1). Combining these two statements we see that the operators P ± i : Hh (m) → L2 have two-sided inverses, and consequently (10.1.7) holds. 3. The second assertion follows from Stone’s Theorem C.13.



10.2. AN INTEGRAL REPRESENTATION FORMULA Since the propagators F (t) := e−itP /h given by Theorem 10.2 commute with P , we have ( (hDt + P )F (t)u = 0 (t ∈ R) F (0)u = u. for each u ∈ L2 .

10.2. AN INTEGRAL REPRESENTATION FORMULA

211

Given an order function m, we consider next a local, time-dependent version of these dynamics: ( (hDt + P (t))F (t)u = 0 (−t0 < t < t0 ) (10.2.1) F (0)u = u where P (t) = pw t (x, hD) for a family of symbols {pt }|t| 0, a phase function ϕ and an amplitude b defined near (x0 , ξ0 ) for times |t| < t0 such that 1 (10.2.3) U (t)u(x) := (2πh)n

Z

Z

Rn

i

e h (ϕ(t,x,η)−hy,ηi) b(t, x, η; h)u(y) dydη

Rn

satisfies (10.2.2) for each family of functions u = {u(h)}0
U (t) = F (t)

microlocally near (x0 , ξ0 )

for times |t| ≤ t0 , where F (t) is the unitary operator given by Theorem 10.1. DEFINITION. The right hand side of (10.2.3) is called a Fourier integral operator.

212

10. FOURIER INTEGRAL OPERATORS

The proof will appear after the following constructions of the phase and amplitude. For a construction in the special case of a quadratic time independent p see Theorem 11.8 below. 10.2.2. Construction of the phase function. We start by identifying the phase function ϕ for (10.2.3) as a local generating function (see Section 2.3) associated with the symplectomorphisms generated by the flow (10.2.6)

∂t κt = Hpt (κt ).

on R2n . We can assume that (x0 , ξ0 ) = (0, 0) and let U denote a bounded open set containing (0, 0). LEMMA 10.4 (Hamilton–Jacobi equation). (i) If t0 > 0 is small enough, there exists a smooth function ϕ = ϕ(t, x, η) defined in (−t0 , t0 ) × U × U , such that κt given by (10.2.6) locally satisfies κt (y, η) = (x, ξ) if and only if (10.2.7)

ξ = ∂x ϕ(t, x, η), y = ∂η ϕ(t, x, η).

(ii) Furthermore, ϕ = ϕ(t, x, η) solves the Hamilton–Jacobi equation ( ∂t ϕ + pt (x, ∂x ϕ) = 0 (−t0 ≤ t ≤ t0 ) (10.2.8) ϕ(0, x, η) = hx, ηi. Proof. 1. For points (y, η) lying in a compact subset of R2n , the map (10.2.9)

(x, ξ, y, η) = (κt (y, η), (y, η)) 7→ (x, η)

is surjective near (0, 0, 0, 0) for times |t| ≤ t0 , provided t0 is small enough. This is so since κ0 (y, η) = (y, η). 2. To show the existence of ϕ, consider the 2n + 1 dimensional submanifold of R2 × R2n × R2n Λ := {(t, pt (y, η); κt (y, η); y, η) | t ∈ R, (y, η) ∈ R2n }. We write a typical point of Λ as (t, τ, x, ξ, y, η). Introduce now the one-form ν := −τ dt +

n X j=1

ξj dxj +

n X j=1

yj dηj .

10.2. AN INTEGRAL REPRESENTATION FORMULA

213

That κt is a symplectic implies dν|Λ = 0. By Poincar´e’s Lemma (Theorem B.4), there exists a smooth function ϕ such that dϕ = ν. In view of (10.2.9) we can use (t, x, η) as coordinates on Λ ∩ ((−t0 , t0 ) × U × U ); and hence −τ dt +

n X j=1

ξj dxj +

n X

yj dηj = ∂t ϕdt +

j=1

n X

∂xj ϕdxj +

j=1

n X

∂ηj ϕdηj .

j=1

Comparing terms on the two sides shows that (10.2.7) holds, as well as the Hamilton–Jacobi PDE. Since κ0 (x, η) = (x, η), we see as well that dϕt=0 = d(hx, ηi). Since ϕ is determined up to an additive constant, we can therefore ensure that ϕ = hx, ηi at t = 0.  10.2.3. Construction of the amplitude. The amplitude b in (10.2.3) must satisfy iϕ/h (hDt + pw b) = OS (h∞ ), t (x, hD))(e and so (10.2.10)

iϕ/h (∂t ϕ + hDt + e−iϕ/h pw )b = OS (h∞ ), t e

for (x, η) ∈ U, |t| ≤ t0 . In this equation η appears only as a parameter. We will construct b in the form (10.2.11)

b(t, x, η; h) ∼

∞ X

hk bk (t, x, η).

k=0

Once all the terms bk have been computed, Borel’s Theorem 4.15 produces the amplitude b, defined up to terms in OS (h∞ ). We assume that b is supported in a neighborhood of U and the equation holds for (x, η) ∈ U . LEMMA 10.5 (Calculation of b0 ). For (x, η) ∈ U the leading term of the expansion (10.2.11) is (10.2.12)

1

2 b0 (t, x, η) = (det ∂xη ϕ(t, x, η)) 2 .

2 ϕ = 1 and hence det ∂ 2 ϕ > 0 for small |t|. Note that at t = 0, det ∂ηx ηx

Proof. 1. We first claim that for compactly supported ϕ = ϕ(x), we have (10.2.13)

iϕ/h e−iϕ/h pw = qtw (x, hD), t (x, hD)e

where (10.2.14)

qt (x, ξ; h) = pt (x, ∂x ϕ + ξ) + OS(m) (h2 ).

214

10. FOURIER INTEGRAL OPERATORS

To see this, write ϕ(x) − ϕ(y) = hF (x, y), x − yi, where 2 F (x, y) = ∂x ϕ( x+y 2 ) + O(|x − y| ).

We check that iϕ/h (10.2.15) e−iϕ/h pw u= t e Z Z

pt Rn

Rn

x+y 2 ,ξ


+ F (x, y) eihx−y,ξi/h u(y) dydξ.

Hence iϕ/h e−iϕ/h pw u t e

Z

Z

= Rn

Rn

x+y pt ( x+y 2 , ξ + ∂x ϕ( 2 ))

+ hEt (x, y, ξ)(x − y), (x − y)i)eihx−y,ξi/h u(y) dydξ, the entries of the matrix valued function Et belonging to S(m). Integration by parts based on (9.2.11) and Theorem 4.20 give (10.2.14). 2. Recalling from Lemma 10.4 that ∂t ϕ = −pt (x, ∂x ϕ), we deduce from (10.2.10) that for (x, η) in a neighborhood of U , (10.2.16)

(hDt + ftw (x, hD, η))b(t, x, η) = OS (h2 ),

where ft (x, ξ) := pt (x, ∂x ϕ(t, x, η) + ξ) − pt (x, ∂x ϕ(t, x, η)), and where η is again considered as a parameter. So (10.2.17)

ft (x, ξ, η) =

n X

ξj ∂ξj pt (x, ∂x ϕ(t, x, η)) + et (x, ξ, η)

j=1

for a symbol et of the form et (x, ξ, η) = hFt (x, ξ, η)ξ, ξi. Remember that if g ∈ S, then (hDx )2 g ∈ OS (h2 ). Hence if g is compactly supported in x, we 2 have ew t g ∈ OS (h ). Since ∂ξj pt = ∂ξj pt (x, ∂x ϕ) does not depend on ξ, (10.2.17) and the quantization formula (4.2.5) for symbols linear in ξ imply n

ftw (x, hD, η)b

1X = hDxj (∂ξj pt b) + (∂ξj pt )hDxj b + OS (h2 ). 2 j=1

Consequently the leading term in the expansion b0 satisfies n

hDt b0 +

1X hDxj (∂ξj pt b0 ) + (∂ξj pt )hDxj b0 = 0. 2 j=1

Rewrite this equation as follows: (10.2.18)

1 (∂t + Vt + div Vt )b0 = 0 2

10.2. AN INTEGRAL REPRESENTATION FORMULA

215

for (10.2.19)

Vt :=

n X

∂ξj pt (x, ∂x ϕ)∂xj .

j=1

We require also that b0 (0, x, η) = 1 in U . 3. Next observe from (10.2.18) that σ := b20 solves (10.2.20)

∂t σ + divx (σ∂ξ pt (x, ∂x ϕ)) = 0,

with σ(0, x, η) = 1 in U . We claim that a solution is 2 σ = det ∂xη ϕ.

(10.2.21)

To see this, first differentiate the Hamilton–Jacobi equation (10.2.8) with respect to η:
T 2 ∂t ∂η ϕ + ∂xη ϕ ∂ξ pt = 0.
2 ϕ) = det ∂ 2 ϕ (∂ 2 ϕ)T −1 , to deduce Apply the cofactor matrix cof(∂xη xη xη 2 cof(∂xη ϕ)∂t ∂η ϕ + σ∂ξ pt = 0.


2 ϕ) = 0 according to [E, Section ??], we discover conseSince divx cof(∂xη quently that 2 2 cof(∂xη ϕ)∂t ∂xη ϕ + div(σ∂ξ pt (x, ∂x ϕ)) = 0.

But ∂(det P ) = cof(P ) and therefore 2 2 cof(∂xη ϕ)∂t ∂xη ϕ = ∂t σ.

This proves (10.2.20); and then (10.2.12) follows, since 1

1

2 ϕ) 2 . b0 = σ 2 = (det ∂xη



Proof of Theorem 10.3: 1. Using the same argument for the higher order terms in b, we can find its full expansion with all the equations valid in (−t0 , t0 ) × U , with b(0, x, η) = 1. Furthermore b(t, x, η) is supported within some neighborhood of U in the (x, η) variables. Thus (10.2.4) holds for A for which WFh (A) contained in U ; and so (10.2.2) is valid. 2. When pt ∈ S, the operators F (t) are well defined according to Theorem 10.1. We can now use Duhamel’s formula (see for instance [E, Chapter 2]) to show that U (t) = F (t) microlocally. For this, consider ( (hDt + P (t))F (t, r) = 0 F (r, r) = I.

216

10. FOURIER INTEGRAL OPERATORS

If ( (hDt U (t) + P (t)U (t))A = −E(t), U (0)A = A − R, where E(t), R = OS 0 →S (h∞ ), then Z

t

F (t, s)E(s) ds

F (t)A = U (t)A + F (t)R + 0 ∞

= U (t)A + OS 0 →S (h ).  REMARK: Half-densities, geometric interpretation of the amplitude. We provide an alternative derivation of the formula for b0 given in (10.2.12) that yields an invariant interpretation of the evolution. To understand the equation (10.2.18) geometrically we consider b0 (t, ·, η) as a function on Λt,η := {(x, ∂x ϕ(t, x, η)) | x ∈ Rn }. Although we write x ∈ Rn , it should be understood that Λt,η is defined for (x, η) ∈ U . Since Λ0,η = {(y, η) | y ∈ Rn } and κt (∂η ϕ(t, x, η), η) = (x, ∂x ϕ(t, x, η)), we have κt (Λ0,η ) = Λt,η . We parametrize Λt,η by x 7→ (x, ∂x ϕ(t, x, η)), and consider the function κt,η := π ◦ κt mapping Λ0,η →

Rn ,

where π(x, ξ) := x Then (10.2.6) implies

∂t κ∗t,η u = ∂t κ∗t π ∗ u = κ∗t (Hpt π ∗ u) = κ∗t π ∗ (π∗ Hpt u) = κ∗t,η (Vt u), Vt defined by (10.2.18). In other words, ∂t κt,η (x) = Vt (κt,η (x). We now introduce 1

a := u|dx| 2 , a half-density on Λt,η . We claim that (10.2.22)

LVt a :=

∂t κ∗t,η a

=

κ∗t,η



  1 1 2 Vt u + divVt u |dx| . 2

Indeed, 1

1

κ∗t,η a = κ∗t,η u(det ∂κt,η ) 2 |dx| 2 ,

10.3. STRICHARTZ ESTIMATES

217

and (10.2.22) follows from 1 1 1 ∂t (det ∂κt,η ) 2 = κ∗t,η (div Vt ) (det ∂κt,η ) 2 . 2 This in turn follows from (A.2.1) .

(10.2.23)

1

So if we consider b0 (t, x, η)|dx| 2 as a half-density on Λt,η , then (10.2.18) becomes   1 1 ∂t κ∗t,η (b0 |dx| 2 ) = (∂t + LVt ) b0 (t, x, η)|dx| 2 = 0. This is the same as   1 1 κ∗t,η b0 (t, x, η)|dx| 2 = |dx| 2 as half-densities on Λt,η and Λ0,η , with κt,η (Λ0,η ) = Λt,η . It follows that 1

κ∗t,η b0 = (det ∂κt,η )− 2 . We now recall that κ−1 t,η maps x to ∂η ϕ(t, x, η), and thus 2 ∂κ−1 t,η = ∂xη .

This completes our alternative derivation of (10.2.12). It is appealing that the amplitude, interpreted as a half-density, is invariant under the flow. When coordinates change, and in particular when we move to larger times at which (10.2.7) and (10.2.8) are no longer valid, the statement about the amplitude as a half-density remains simple. 

10.3. STRICHARTZ ESTIMATES In this section we will employ Theorem 10.3 to obtain Lp bounds on approximate solutions to the quasimode equations considered earlier, in Section 9.2. These supplement the L∞ estimates we derived before. 10.3.1. Strichartz estimates. We first reference the following general result of Keel-Tao [K-T]: THEOREM 10.6 (Strichartz estimates). Let {U (t)}t∈R be a family of linear operators satisfying the bounds ( kU (t)kL2 (Rn )→L2 (Rn ) ≤ C, (10.3.1) kU (t)U (s)∗ kL1 (Rn )→L∞ (Rn ) ≤ Ch−γ |t − s|−σ , for all t, s ∈ R, where C, σ, γ > 0 are fixed constants. Then we have the estimate Z 1 p − γ p ≤ Ch pσ kf kL2 (Rn ) (10.3.2) kU (t)f kLq (Rn ) dt R

218

10. FOURIER INTEGRAL OPERATORS

for every pair p, q satisfying 2 2σ + = σ, 2 ≤ p ≤ ∞, 1 ≤ q ≤ ∞, (p, q) 6= (2, ∞). p q We will need this assertion only for p = q, and for the reader’s convenience present the proof for that case. Proof: 1. A rescaling in time converts h = 1, and inequality (10.3.2) for p = q then reads kU (t)f kLp (R×Rn ) ≤ Ckf kL2 (Rn )

(10.3.3) where p =

2(1+σ) . σ

2. We apply the Riesz-Thorin Interpolation Theorem (H¨ormander [H1, Theorem 7.1.12]) to the operator U (t)U (s)∗ , for fixed times t, s ∈ R. Interpolating between the two estimates provided by the hypothesis (10.3.1) gives 0 kU (t)U (s)∗ kLp0 →Lp ≤ C|t − s|−σ(2/p −1) for 1 ≤ p0 ≤ 2, where 1/p + 1/p0 = 1 In particular, we have (10.3.4) |hU (t)∗ G(t, ·), U (s)∗ F (s, ·)i| 0

≤ C|t − s|−σ(2/p −1) kG(t, ·)kLp0 kF (s, ·)kLp0 for functions F = F (t, x), G = G(t, x). 3. Next, we recall the Hardy-Littlewood-Sobolev inequality (H¨ormander [H1, Theorem 4.5.3]), which says for Ka (t) := |t|−1/a and 1 < a < ∞ that kKa ∗ ukLr (R) ≤ CkukLp0 (R)

(10.3.5) for

1 1 1 + = , 1 < p0 < r. p r a   We apply (10.3.5) to (10.3.4) with a1 = σ p20 − 1 and p = r, in which case p = (10.3.6)

2(1+σ) . σ

The resulting inequality is Z Z ∗ ∗ hU (t) G(t, ·), U (s) F (s, ·)i dtds R

R

≤ CkGkLp0 (R×Rn ) kF kLp0 (R×Rn ) . Taking F = G, we deduce that Z ∗ U (t) G(t, x) dt R

L2 (Rn )

≤ C ||G||Lp0 (R×Rn ) ;

10.3. STRICHARTZ ESTIMATES

and therefore Z

R×Rn

219

U (t)f (x) G(t, x) dxdt ≤ Ckf kL2 (Rn ) kGkLp0 (R×Rn )

0

for all G ∈ Lp (R × Rn ). This implies (10.3.3).



10.3.2. Semiclassical Strichartz estimates. Let {at }t∈R be a smooth family of real-valued symbols in S. We introduce at a point (t, x, ξ) the nondegeneracy condition that ∂ξ2 at (x, ξ) is nonsingular.

(10.3.7)

Put A(t) := aw t (x, hD) and consider then the initial value problem ( (hDt + A(t))F (t, r) = 0 (10.3.8) F (r, r) = I for some fixed time r ∈ R. THEOREM 10.7 (Semiclassical Strichartz estimates). Select a function χ ∈ Cc∞ (R2k ) and assume the nondegeneracy condition that (10.3.7) holds for all (x, ξ) ∈ spt(χ) and t ∈ R. Then given ψ ∈ Cc∞ (R) with support sufficiently close to 0 and a compact interval I ⊂ R, we have the estimate 1 Z p −1 p (10.3.9) sup kU (t, r)f kLq (Rk ) dt ≤ Ch p kf kL2 (Rk ) , r∈I

R

for U (t, r) := ψ(t − r)χw (x, hD)F (t, r), where

2 k k + = , 2 ≤ p ≤ ∞, 1 ≤ q ≤ ∞, (p, q) 6= (2, ∞). p q 2

Proof. 1. In view of Theorem 10.6 we need to show that (10.3.10)

kU (t, r)U (s, r)∗ kL1 →L∞ ≤ Ch−k/2 |t − s|−k/2

for s, t ∈ R, with constants independent of r ∈ I. We can for simplicity put r = 0 in the argument and drop the dependence on r in U and F . 2. Since at ∈ S, Theorem 10.3 provides a microlocal description of U (t) for small values of |t|. We have e (t) + E(t), U (t) = U where E(t) = OS 0 →S (h∞ )

220

10. FOURIER INTEGRAL OPERATORS

e (t) is and the Schwartz kernel of U Z i 1 e (t, x, y) = (10.3.11) U e h (ϕ(t,x,η)−hy,ηi)˜b(t, y, x, η; h) dη, k (2πh) Rk for ˜b ∈ S ∩ Cc∞ (R1+3k ) and ϕ = ϕ(t, x, η) solving the Hamilton–Jacobi equation ( ∂t ϕ + a(t, x, ∂x ϕ) = 0 (10.3.12) ϕ(0, x, η) = hx, ηi. All this is true since the local construction of Section 10.2 works on the support of χ. e , and for this 3. Hence we need to prove (10.3.10) with U replaced by U e (t)U e (s)∗ , which need an L∞ bound on the Schwartz kernel of W (t, s) := U is Z Z Z i 1 e h (ϕ(t,x,η)−ϕ(s,y,ζ)−hz,η−ζi) B dzdζdη, W (t, s, x, y) = 2k (2πh) Rk Rk Rk where B = B(t, s, x, y, z, η, ζ; h) ∈ S ∩ Cc∞ (R2+6k ). The phase ϕ is nondegenerate in the (z, ζ) variables and stationary at ζ = η, z = ∂ζ ϕ(s, y, ζ). Hence Theorem 3.16 implies Z i 1 e h (ϕ(t,x,η)−ϕ(s,y,η)) B1 (t, s, x, y, η; h) dη, W (t, s, x, y) = k (2πh) Rk where B1 ∈ S ∩ Cc∞ (R2+3k ). We now rewrite the phase: ϕ e := ϕ(t, x, η) − ϕ(s, y, η) = (t − s) (a (0, x, η) + O(|t| + |s|)) + hx − y, η + sF (s, x, y, η)i for F ∈ C ∞ (R1+3k ). Here we used (10.3.12) to write ϕ(s, x, η) − ϕ(s, y, η) = hx − y, ηi + hx − y, sF (s, x, y, η)i. The phase is stationary when ∂η ϕ e = (I + s∂η F )(x − y) + (t − s)(∂η a + O(|t| + |s|)) = 0. In particular, for s small, having a stationary point implies x − y = O(t − s), and then (I + s∂η F ) is invertible. The Hessian is

∂η2 ϕ e = s ∂η2 F (x − y) + (t − s) ∂η2 a + O(|t| + |s|)
= (t − s) ∂η2 a + O(|t| + |s|) , where ∂η2 a = ∂η2 a(0, x, η).

10.3. STRICHARTZ ESTIMATES

221

4. Hence for t and s sufficiently small, that is, for a suitable choice of the support of ψ in the definition of U (t), the nondegeneracy assumption (10.3.7) implies that ∂η2 ϕ e = (t − s)ψ(x, y) at the critical point. If |t − s| > M h, where M is a large constant, we can use the stationary phase estimate in Theorem 3.16 to deduce |W (t, s, x, y)| ≤ Ch−k/2 |t − s|−k/2 . When |t − s| < M h, |W (t, s, x, y)| ≤

1 (2πh)2k

Z |B(t, s, x, y, z, η, ζ; h)| dzdζdη R3k −k/2

≤ Ch−k ≤ Ch

|t − s|−k/2 .

The last two estimates let us apply Theorem 10.6.



We will later need this corollary: THEOREM 10.8. In the notation of Theorem 10.9, we have (10.3.13)

Z t U (t, s)1I (s)f (s, x) ds 0

Lp (R×Rk ) −1/p

Z

≤ Ch

kf (s, x)kL2 (Rk ) ds. R

Proof. We apply the integral version of Minkowski’s inequality: Z t U (t, s)1I (s)f (s, x) ds p 0 L (R×Rk ) Z ≤C k1[s,∞) (t)U (t, s)f (s, x)kLp (R×R) ds I∩R+

Z ≤C

kU (t, s)f (s, x)kLp (R×Rk ) ds. I∩R+

Now we use the estimate (10.3.9) with p = q = 2(n + 1)/(n − 1): kU (t, s)f (s, x)kLp (R×Rk ) ≤ Ch−1/p kf (s, x)kL2 (Rk ) . Since I is compact, (10.3.13) follows.



222

10. FOURIER INTEGRAL OPERATORS

10.4. Lp ESTIMATES FOR QUASIMODES 10.4.1. Nondegeneracy conditions and Lp bounds. Let us now suppose K ⊂ Rn is compact set, and the real-valued symbol p ∈ S(m) satisfies ∂ξ p 6= 0 on {p = 0} ∩ K.

(10.4.1)

Thus if p(x0 , ξ0 ) = 0 and (x0 , ξ0 ) ∈ K, then {ξ | p(x0 , ξ) = 0} is a smooth hypersurface in Rn near ξ0 . After a linear change of variables, we have ∂ξ p(x0 , ξ0 ) = (ρ, 0, · · · , 0) where ρ 6= 0. Then near (x0 , ξ0 ), p(x, ξ) = e(x, ξ)(ξ1 − a(x, ξ 0 ))

(10.4.2)

where ξ = (ξ1 , ξ 0 ) and e(x, ξ) > 0. We make the further assumption that for each (x0 , ξ0 ) this surface has nondegenerate second fundamental form, meaning that ∂ξ20 a(x0 , ξ00 ) is nondegenerate.

(10.4.3)

THEOREM 10.9 (Lp bounds for quasimodes). Assume p satisfies the nondegeneracy conditions (10.4.1) and (10.4.3). Suppose further that u = {u(h)}0
kpw (x, hD)u(h)kL2 (Rn ) = O(h)ku(h)kL2 (Rn ) ,

we have the estimate (10.4.5)

ku(h)kLp (K) ≤ Ch−1/p ku(h)kL2 (Rn )

for p=

2(n + 1) . n−1

The localization assumption on u is the same as the frequency localization condition (7.4.5) in Theorem 7.12. REMARK. Estimates of the form have proved useful in linear and nonlinear PDE theory. The improvement from estimate (10.4.1), as compared to interpolation using the L∞ bound provided by Theorem 7.12, is most pronounced in lower dimensions. See Figure ??  Proof: 1. We may assume ku(h)kL2 = 1. We follow the same procedure as in the proof of Theorem 7.12. 1. The condition (10.4.4) implies for each χ ∈ Cc∞ (R2n ) that kpw (x, hD)χw (x, hD)u(h)kL2 = O(h).

10.4. Lp ESTIMATES FOR QUASIMODES

223

We factor p as in (10.4.2) and conclude since e is elliptic that k(hDx1 − a(x, hDx0 ))χw (x, hD)u(h)kL2 = O(h) for χ with sufficiently small support. Let f (x1 , x0 ) := (hDx1 − a(x, hDx0 ))χw u(h); then Z kf (x1 , ·)kL2 (Rn−1 ) dx1 ≤ Ckf kL2 (Rn ) = O(h).

(10.4.6) R

3. We now apply Theorem 10.7 with t = x1 and x replaced by x0 ∈ Rn−1 ; that is, k = n−1. The assumption (10.4.3) shows that ∂ξ20 a is nondegenerate in the support of χ. We can choose ψ and χ in the definition of U (t, s) in the statement of Theorem 10.7; so that Z i x1 w 0 χ (x, hD)u(x1 , x , h) = U (t, s)f (s, x0 ) ds + OS (h∞ ). h 0 Let us choose p = q, k = n − 1 in (10.3.9); that is, 2(n + 1) . n−1 Then using (10.3.9), (10.4.6) and (10.3.13), we deduce Z 1 kf (s, ·)kL2 (Rn−1 ) ds + O(h∞ ) = O(h−1/p ). kχw (x, hD)ukLp ≤ h−1/p h R p=q=

A partition of unity argument finishes the proof.



REMARK. The first example after Theorem 7.12 shows that the curvature condition (10.4.3) is in general necessary. In fact, if P (h) = hDx1 and u(h) = h−(n−1)/2 χ(x1 )χ(x0 /h), then for p = 2(n + 1)/(n − 1), kukLp ' h(n−1)(1/p−1/2) = h−(n−1)/(n+1) 6= O(h−1/p ). For the simplest case in which (10.4.3) holds, p(x, ξ) = ξ1 − ξ22 − · · · − ξn2 , the estimate (10.4.1) is optimal. To see this, put u(h) := h−(n−1)/4 χ0 (x1 ) exp(−|x0 |2 /2h), where x = (x1 , x0 ), χ0 ∈ Cc∞ (R). Then (−h2 ∆x0 + |x0 |2 )u(h) = (n − 1)h u(h), ku(h)kL2 ' 1, |x0 |2k u(h) = OL2 (hk ). Hence pw (x, hD)u(h) = OL2 (h), and ku(h)kLp (Rn ) ' h(n−1)(2/p−1)/4 = h−1/p for p = 2(n + 1)/(n − 1) .



224

10. FOURIER INTEGRAL OPERATORS

n=3

n−1 2

1 2

n−1 2(n+1)

0

n=7

n−1 2

n−1 2(n+1)

1 2

n−1 2(n+1)

1 2

0

n−1 2(n+1)

1 2

Figure 1. The horizontal axis is 1/p and the vertical axis is α in the estimate kukLp ≤ Ch−α kukL2 .

10.4.2. Bounds for spectral clusters. Suppose that M is a compact Riemannian manifold of dimension n and let ∆g be its Laplace-Beltrami operator. Assume 0 = λ0 < λ1 ≤ · · · λj → ∞ is the complete set of eigenvalues of −∆g , and −∆g ϕj = λj ϕj

10.5. NOTES

225

are the corresponding eigenfunctions. THEOREM 10.10 (Lp bounds on eigenfunction clusters). (i) There exists a constant C such that for any choices of constants cj ∈ C, we have the inequality P P (10.4.7) µ≤√λj ≤µ+1 cj ϕj ≤ Cµσ(p) µ≤√λj ≤µ+1 cj ϕj , p 2 L (M )

L (M )

for (10.4.8)

σ(p) :=

  

n−1 2



 

n−1 2

−

1 2

−

1 p



n p

if 2 ≤ p ≤ if

2(n+1) n−1

2(n+1) n−1 ,

≤ p ≤ ∞.

(ii) In particular, (10.4.9)

σ(p)/2

kϕj kLp (M ) ≤ Cλj

kϕj kL2 (M ) .

Proof. 1. We argue as in the proof of Theorem 7.13, but need to check the curvature assumption (10.4.3). But at any point (x0 , ξ0 ), we have in suitable coordinates p(x0 , ξ) = |ξ|2 − 1, ξ0 = (1, 0, · · · , 0). The hypersurface p(x0 , ξ) = 0 is the unit sphere in Rn and clearly has a nondegenerate second fundamental form. 2. Complex interpolation (see H¨ormander [H1, Theorem 7.1.12]) between the estimate in Theorem 7.13, the trivial L2 estimate, and the estimate in Theorem 10.9 gives the full result. 

10.5. NOTES Quantization of a general real-valued symbol p ∈ S(m) might not be selfadjoint on L2 (Rn ): see Dimassi–Sj¨ostrand [D-S]. The construction of U (t) borrows from the presentations in Helffer– Sj¨ ostrand [H-S1, Appendix A] and [S-Z1, Section 7]. For a more general problem of p depending on h we refer to [S-Z1, Section 7] and references given there. Here we note that the proof works for P (t) = pw (t, x, hD) + h2 p w 2 (t, x, hD) and that form of operators acting on half-densities is invariant (see Theorem 9.3). See Duistermaat [D] and H¨ormander [H2, Chapter 25] for more on Fourier integral operators. Semiclassical Strichartz estimates for P = −h2 ∆g −1 appeared in Burq– G´erard–Tzvetkov [B-G-T]. The adaptation of Sogge’s Lp estimates to the semiclassical setting is from [K-T-Z] and was inspired by discussions with

226

10. FOURIER INTEGRAL OPERATORS

N. Burq, H. Koch, C. Sogge, and D. Tataru. See also Koch–Tataru [Ko-T] and Sogge [S].

Chapter 11

QUANTUM AND CLASSICAL DYNAMICS

11.1 11.2 11.3 11.4 11.5

Egorov’s Theorem Quantizing symplectic mappings Quantizing linear symplectic mappings Egorov’s Theorem for longer times Notes

Chapter 10’s discussion of Fourier integral operators provided an explicit, but only local in time, representation for solutions of quantum dynamics. This chapter switches attention to dynamics over longer time intervals and to the corresponding classical evolution.

11.1. EGOROV’S THEOREM We assume for this section that {qt }0≤t≤T denotes a family of smooth symbols that vanish outside some fixed, bounded open set U0 . Then the flow ϕt = exp(tHqt ) induces a smooth family {κt }0≤t≤T of symplectomorphisms of R2n , which equal the identity outside U0 . These satisfy ( ∂t κt = (κt )∗ Hqt (0 ≤ t ≤ T ) (11.1.1) κ0 = I. 227

228

11. QUANTUM AND CLASSICAL DYNAMICS

Write Q(t) := qtw (x, hD). Since {qt }0≤t≤T ⊂ S, Theorem 10.1 implies that the operator equation ( hDt F (t) + F (t)Q(t) = 0 (0 ≤ t ≤ T ) (11.1.2) F (0) = I has a unique solution of unitary operators {F (t)}0≤t≤T on L2 (Rn ). Observe for later use that (11.1.2) implies (11.1.3)

hDt (F (t)−1 ) − Q(t)F (t)−1 = 0.

THEOREM 11.1 (Egorov’s Theorem). Let m be an order function and suppose a ∈ S(m). Then for 0 ≤ t ≤ T , we have (11.1.4)

F (t)−1 aw (x, hD)F (t) = bw t (x, hD)

for a symbol bt ∈ S(m) having the form (11.1.5)

bt = κ∗t a + OS (h).

Theorem 10.1 asserts F (t) : Hh (m) → Hh (m) and consequently the formula (11.1.4) makes sense on Hh (m). INTERPRETATION. Quantization given by the conjugation (11.1.4) with F (t) follows the Heisenberg picture of quantum mechanics. The assertion is that the quantum evolution of A = aw (x, hD) is well approximated up to time T by the classical evolution, in the sense that the quantum operator is a quantization of the classical observable κ∗t a ∈ S(m), up to an error of order h in S.  Proof. 1. Write A := aw (x, hD), define (11.1.6)

B(t) := F (t)−1 AF (t).

Our aim is to prove (11.1.7)

B(t) = bw t (x, hD)

for a symbol bt ∈ S(m) having the form (11.1.5). 2. To show this, we first note that κ∗t a ∈ S(m) since the vector field defining κt is smooth and has compact support. Hence we can define the pseudodifferential operators B0 (t) := (κ∗t a)w (x, hD).

11.1. EGOROV’S THEOREM

229

The dynamics (11.1.1) imply ∂t κ∗t a = Hqt κ∗t a. We can therefore calculate hDt B0 (t) = =

h h (∂t κ∗t a)w = (Hqt κ∗t a)w i i h {qt , κ∗t a}w = [Q(t), B0 (t)] + E(t), i

where 2 E(t) = ew t (x, hD) for et ∈ h S

To see this, remember that the expansion formulas in Theorem 4.12 show that et is computed in terms of expressions involving the product of qt and κ∗t a, thus has compact support, and thus lies in S . It follows from Theorem 4.21 that E(t) = OL2 →L2 (h2 ). Then the operator equations (11.1.2), (11.1.3) let us calculate hDt (F (t)B0 (t)F (t)−1 ) = hDt F (t)B0 (t)F (t)−1 + F (t)hDt B0 (t)F (t)−1 + F (t)B0 (t)hDt (F (t)−1 ) = −F (t)Q(t)B0 (t)F (t)−1 + F (t)([Q(t), B0 (t)] + E(t))F (t)−1 + F (t)B0 (t)Q(t)F (t)−1 = F (t)E(t)F (t)−1 = OL2 →L2 (h2 ). Integrating and dividing by h gives Z i t F (s)E(s)F (s)−1 ds h 0 = A + OL2 →L2 (h)

F (t)B0 (t)F (t)−1 = A +

for 0 ≤ t ≤ T . According then to (11.1.6) Z t i −1 B(t) := B0 (t) − F (t) F (s)E(s)F (s)−1 ds F (t); h 0 so that B0 (t) − B(t) = OL2 →L2 (h).

3. We next construct families of pseudodifferential operators Bk (t) so that for each k = 1, 2, . . . ( hDt Bk (t) = [Q(t), Bk (t)] + Ek (t), (11.1.8) Bk (0) = A,

230

11. QUANTUM AND CLASSICAL DYNAMICS

and Bk+1 (t) − Bk (t) = OL2 →L2 (hk+1 )

(11.1.9)

k+2 Ek (t) = ew S. k,t (x, hD), ek,t ∈ h

We proceed by induction. Assuming (11.1.9), define Z t i ∗ ∗ ck+1,t := (κt ) (κ−1 s ) ek,s ds h 0 Then k+1 Ck+1 (t) := cw Ψ. k+1,t (x, hD) ∈ h

Therefore hDt Ck+1 (t) = (hDt ck+1,t )w   w Z t h i ∗ = ∂t (κt )∗ (κ−1 ) e ds k,s s i h 0   w h i = Hqt ck+1,t + ek,t i h = [Q(t), Ck+1 (t)] + Ek (t) − Ek+1 (t), k+3 S. Hence where Ek+1 = ew k+1,t for ek+1,t ∈ h

Bk+1 (t) := Bk (t) − Ck+1 (t) satisfies (11.1.8) and (11.1.9) with k replaced by k + 1. 4. To compare Bk (t) with B(t) we proceed as in Step 1 above to obtain
hDt F (t)Bk (t)F (t)−1 = F (t)Ek (t)F (t)−1 . Integrating in t gives (11.1.10)

B(t) − Bk (t) = F (t)−1 (A − F (t)Bk (t)F (t)−1 )F (t) Z i t = F (t)−1 F (s)Ek (s)F (s)−1 F (t) ds. h 0

In particular, (11.1.11)

B(t) = Bk (t) + OL2 →L2 (hk+1 ).

5. It remains to show that B(t) is a pseudodifferential operator and that B(t) − B0 (t) = OL2 →L2 (h). To do so, we invoke Beals’s Theorem 8.3 by showing that for any linear l1 , · · · , lM , we have the estimate (11.1.12)

adl1 (x,hD) · · · adlM (x,hD) (B(t) − B0 (t)) = OL2 →L2 (hM +1 ).

11.1. EGOROV’S THEOREM

231

It suffices to show that (11.1.13)

adl1 (x,hD) · · · adlM (x,hD) (B(t) − Bk (t)) = OL2 →L2 (hk+1 ).

Then in light of (11.1.11) it will follow that adl1 · · · adlM (B(t) − B0 (t)) = adl1 · · · adlM (Bk (t) − B0 (t)) + adl1 · · · adlM (B(t) − Bk (t)) = OL2 →L2 (hM +1 ) + OL2 →L2 (hk+1 ) = OL2 →L2 (hM +1 ), if k ≥ M . 6. To establish (11.1.13), we select a linear l and use (11.1.10): Z i t adl (F (t)−1 F (s)Ek (s)F (s)−1 F (t)) ds. adl (B(k) − Bk (t)) = h 0 In view of (11.1.9), we have i adl Ek (s) = OL2 →L2 (hk+2 ) = OL2 →L2 (hk+1 ); h and a similar calculation holds for additional application of adlj . Recalling the formulas (8.1.8), (8.1.9) we see that we need now to show (11.1.14)

adl1 (x,hD) · · · adlM (x,hD) F (t) = OL2 →L2 (1)

for linear lj on R2n . For this, observe from the derivation property (8.1.8) that adl F (t) for a linear l satisfies ( hDt (adl F (t)) + (adl F (t))Q(t) = −F (t)adl Q(t) adl F (0) = 0. Therefore hDt (adl F (t)F (t)−1 ) = −F (t)adl Q(t)F (t)−1 , according to (11.1.3). Since Q(t) is a pseudodifferential operator, we have adl Q(t) = hQ1 (t), where Q1 = OL2 →L2 (1). Hence Z t adl F (t) = i F (s)Q1 (s)F (s)−1 F (t) ds = OL2 →L2 (1). 0

This argument can be iterated for additional applications of ad, thereby proving (11.1.14).  REMARKS. (i) The proof shows that if a∼

∞ X k=0

hk ak

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for ak ∈ S(m), then bt ∼

κ∗t a0

+

∞ X

hk bkt .

k=1

However, the higher order terms are difficult to compute. (ii) In Theorem 11.1 we assumed that qt vanish outside of a fixed compact set and that a ∈ S(m). The proof applies directly when qt ∈ S and a ∈ S. We are then in the setting of Theorem 10.1 and the statements (11.1.4) and (11.1.5) are valid. 

11.2. QUANTIZING SYMPLECTIC MAPPINGS Suppose now we are given a symplectomorphism κ : R2n → R2n , with κ(0, 0) = (0, 0). Our next task is to quantize κ locally, which means to construct a unitary operator F : L2 → L2 such that if a ∈ S(m), then F −1 aw (x, hD)F = bw (x, hD)

“microlocally near” (0, 0),

for a symbol b ∈ S(m) of the form b = κ∗ a + OS (h). This assertion can be very useful in practice, since sometimes we can design κ so that κ∗ a is more tractable than a. See Chapter 12 for applications. 11.2.1. More on symplectic matrices. We will first of all require some more precise information about symplectic matrices. Remember from Example 1 in Section 2.3 that a real 2n × 2n matrix K is symplectic provided (11.2.1)

K T JK = J.

THEOREM 11.2 (Decomposing symplectic matrices). Let K be a symplectic matrix. Then we can write (11.2.2)

K = exp(B) exp(A),

where A, B are real 2n × 2n matrices satisfying (11.2.3)

AT = A,

AT J + JA = 0,

and (11.2.4)

B T = −B,

B T J + JB = 0.

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233

REMARK. It is in general impossible to write an arbitrary symplectic matrix K as exp A, with AT J + JA = 0. We leave it as an exercise to show this for   −1 1 K= . 0 −1  Proof. 1. Recall from (2.3.6) that K’s being symplectic is equivalent to K −1 = JK T J T .

(11.2.5)

As K is invertible, we have the unique polar decomposition K = QP, where Q is orthogonal and P is symmetric, positive definite: see for instance [E-G, Section ??]. From (11.2.5) we deduce that (QT )−1 (P T )−1 = (K T )−1 = JQJ T JP J T ; whence the uniqueness of the polar decomposition implies (QT )−1 = JQJ T , (P T )−1 = JP J T . Consequently (2.3.6) implies both Q and P are symplectic. 2. Since P is symmetric and positive definite, we can write P = exp A, where A = AT . We assert further that AJ + JA = 0. To see this, compute exp(−A) = P −1 = JP T J T = JP J T = J exp(A)J T = exp(JAJ T ); whence −A = JAJ T . 3. We next identify R2n with Cn , using the relation (x, ξ) = x+iξ. Since hx + iξ, x0 + iξ 0 iCn = h(x, ξ), (x0 , ξ 0 )iRn + iσ((x, ξ), (x0 , ξ 0 )), that Q is orthogonal and symplectic implies that it is unitary: hQ(x + iξ), Q(x0 + iξ 0 )iCn = hQ(x, ξ), Q(x0 , ξ 0 )iRn + iσ(Q(x, ξ), Q(x0 , ξ 0 )) = h(x, ξ), (x0 , ξ 0 )iRn + iσ((x, ξ), (x0 , ξ 0 )) = hx + iξ, x0 + iξ 0 iCn . So Q is normal and thus diagonalizable on Cn : there exists V so that Q = V −1 diag(λ1 , . . . , λn )V for complex eigenvalues λk satisfying |λk | = 1 for k = 1, . . . , n. We can write Q = exp B −1 for B = V diag(log λ1 , . . . , log λn )V . In particular B = −B ∗ is antiHermitian on Cn .

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11. QUANTUM AND CLASSICAL DYNAMICS

4. We note next that the action of Q on Cn is given by Q1 + iQ2 , where Q1 and Q2 are real: (Q1 + iQ2 )(x + iξ) = (Q1 x − Q2 ξ) + i(Q2 x + Q1 ξ). Likewise the action of B is B1 + iB2 for real B1 and B2 . So the action of Q and B on R2n are represented by the matrices     B1 −B2 Q1 −Q2 . , B= Q= B2 B1 Q2 Q1 Since B is anti-Hermitian acting on Cn , B1T = −B1 , B2T = B2 ;

(11.2.6)

and hence B T = −B acting on R2n . The properties (11.2.6) imply also that −BJ + JB = 0.  11.2.2. Deformation of symplectomorphisms. The plan is to locally embed κ into a one parameter family of symplectomorphisms, generated by a family {qt }0≤t≤1 to which Egorov’s Theorem 11.1 applies. So assume that U0 ⊂ R2n = Rn × Rn is an open set star-shaped with respect to (0, 0), meaning that tU0 := {tw | w ∈ U } ⊆ U0 for all 0 ≤ t ≤ 1. Suppose further that κ : U0 → U1 := κ(U0 ) is a symplectomorphism with κ(0, 0) = (0, 0). THEOREM 11.3 (Deforming symplectomorphisms I). There exists a continuous, piecewise smooth family {κt }0≤t≤1 of symplectomorphisms κt : U0 → Ut := κt (U0 ) such that (i) κt (0, 0) = 0

(0 ≤ t ≤ 1)

(ii) κ1 = κ, κ0 = I. (iii) Also, there exists a family of smooth symbols {qt }0≤t≤1 ⊂ C ∞ (U0 ), which is piecewise smooth in t, such that (11.2.7)

∂t κt = (κt )∗ Hqt

(0 ≤ t ≤ 1).

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235

Proof. 1. Assume first that κ is linear, given by the symplectic matrix K. Theorem 11.2 asserts that we can write K = exp B exp A for matrices A, B satisfying AT J + JA = 0, B T J + JB = 0. According to Theorem 2.4, exp(tA) and exp(tB) are both linear symplectomorphisms; and so also is their composition. A smooth deformation to the identity is now clear: Kt := exp(tB) exp(tA)

(0 ≤ t ≤ 1).

2. For the general case that κ is nonlinear, set K := ∂κ(0, 0). Then for 1/2 ≤ t ≤ 1, −1 κt := K2−2t ◦κ

is a piecewise smooth family of symplectomorphisms satisfying κ1 = κ, ∂κ1/2 (0, 0) = I. For 0 ≤ t ≤ 1/2, we set κt z :=

1 κ (2tz). 2t 1/2+t

4. Define Vt := ∂t κt ; we must show Vt = (κt )∗ Hqt for some function qt . According to Cartan’s formula (Theorem B.3): LVt σ = dσ Vt + d(σ Vt ). d But LVt σ = ∂t κ∗t σ = dt σ = 0, since κ∗t σ = σ. Furthermore, dσ = 0, and consequently d(σ Vt ) = 0. Owing to Poincar´e’s Lemma (Theorem B.4), we have

κ∗t (σ Vt ) = dqt for a function qt ; and this means that Vt = (κt )∗ Hqt .



Next we extend symplectomorphisms defined only locally to to be globally defined R2n . That will allow us to apply the deformation methods developed above.

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11. QUANTUM AND CLASSICAL DYNAMICS

THEOREM 11.4 (Deforming symplectomorphisms II). Suppose that κ : U0 → U1 is a symplectomorphism and that κ extends as a symplectomorphism to an e0 ⊃⊃ U0 , where U e0 is star-shaped with respect to (0, 0). open set U (i) Then κ extends to a symplectomorphism κ e : R2n → R2n which equals the identity outside a compact set. (ii) There exists a continuous, piecewise smooth family {e κt }0≤t≤1 of symplectomorphisms κ et : R2n → R2n such that κ e1 = κ e, κ e0 = I and (11.2.8)

∂t κ et = (e κt )∗ Hqt

(0 ≤ t ≤ 1)

e0 ; R) for a family of smooth compactly supported functions {qt }0≤t≤1 ⊂ Cc∞ (U that is piecewise smooth in t. e0 . Proof. Let κ denote the extension of κ to the star shaped neighborhood U e0 ). We then apply Theorem 11.3 to obtain a family of symbols qet ∈ C ∞ (U ∞ Let χ ∈ Cc (U0 ) satisfy 0 ≤ χ ≤ 1 with χ ≡ 1 in U0 , and put qt := χe qt ∈ Cc∞ (U0 ). We then define κ et by (11.2.8). Since qt and qet are the same within U0 , we have κt |U0 = κ et |U0 . In particular κ|U0 = κ e|U0 , κ e := κ e1 . Since qt has compact support, κ et equals the identity outside a compact set.  11.2.3. Locally quantizing symplectomorphisms. We now apply Theorem 11.1 to the deformations provided by Theorem 11.4: THEOREM 11.5 (Local quantization of symplectomorphisms). Let κ : U0 → U1 be a symplectomorphism satisfying the assumptions of Theorem 11.4. Then there exists a unitary operator F : L2 (Rn ) → L2 (Rn ) such that for all a ∈ S(m), we have (11.2.9)

F −1 aw (x, hD)F = bw (x, hD)

for a symbol b ∈ S(m) satisfying (11.2.10)

b|U0 := κ∗ (a|U1 ) + OS (h).

11.3. QUANTIZING LINEAR SYMPLECTIC MAPPINGS

237

Proof. According to Theorem 11.4, there exists a piecewise smooth family of symplectomorphisms κt : U0 → Ut for 0 ≤ t ≤ 1, such that κ = κ1 , κ0 = I, and κt extends to κ et : R2n → R2n satisfying ∂t κt = (κt )∗ Hqt

(0 ≤ t ≤ 1).

Furthermore, qt ∈ Cc∞ (U ) for some fixed U0 ⊂ R2n . We invoke Theorem 11.1, to obtain the family of unitary operators {F (t)}0≤t≤1 and put F = F (1). Then (11.1.5) shows that (11.2.9) and (11.2.10) hold.  11.2.4. Microlocal reformulation. It is useful to restate Theorem 11.5 without reference to the global properties of the operator F . Remember from Section 8.4 the notion of microlocality. THEOREM 11.6 (Local Egorov’s Theorem). Let κ : U0 → U1 be a symplectomorphism satisfying the assumptions of Theorem 11.4. Then there exists a unitary operator F : L2 (Rn ) → L2 (Rn ) such that F is microlocally invertible near U1 × U0 ; and for all a ∈ S(m), we have (11.2.11)

F −1 aw (x, hD)F = bw (x, hD)

microlocally near U1 × U0 ,

where b ∈ S(m) has the form (11.2.12)

b := κ∗ a + OS (h).

In (11.2.12) we do not specify any neighborhoods, as we did in (11.2.10), since the statement needs to make sense only microlocally near U1 × U0 .

11.3. QUANTIZING LINEAR SYMPLECTIC MAPPINGS Linear symplectic maps are not, strictly speaking, covered in Section 11.2. The problem is that the deformation given in the proof of Theorem 11.3 requires that κ be the identity outside some compact set. We therefore devote this section to a direct study of linear symplectic functions, deriving in particular an exact form of Egoroff’s Theorem.

238

11. QUANTUM AND CLASSICAL DYNAMICS

11.3.1. Quantizing J. As motivation, we discuss first how to quantize J. The Fourier conjugation formula (4.2.14) shows that for F = Fh we have F −1 aw (x, hD)F = (J ∗ a)w (x, hD),

(11.3.1)

and thus (11.1.5) holds without any error terms. Consequently we can say that the semiclassical Fourier transform F = Fh quantizes J. Let us now reconsider this quantization of κ = J from the perspective of deformations introduced in Section 11.2. Since κ(x, ξ) = (ξ, −x), we can take           tπ tπ tπ tπ κt (x, ξ) = cos x + sin ξ , − sin x + cos ξ 2 2 2 2 for 0 ≤ t ≤ 1. Then dκt = (κt )∗ Hq , dt for (11.3.2)

q :=

π (|x|2 + |ξ|2 ). 4

THEOREM 11.7 (J quantized). Suppose that an operator F is associated with the transformation J as in Theorem 11.5. That is, F = F (1), where F (t) solves (11.1.2) with Q(t) = q w (x, hD) and q is given by (11.3.2). Then π

(11.3.3)

e− 4 i F u(x) := n (2πh) 2

π

Z e Rn

−

ihx,yi h

e− 4 i u(y) dy = n Fh u. (2πh) 2

Proof. 1. Theorem 10.2 shows that F (t) = exp(−itQ/h) solves (11.1.2) and is unitary. Since F (t)Q = QF (t) and \ S = hQi−N L2 , N ∈N

we see that F (t) : S → S . 2. To show that F = F (1) verifies (11.3.3), we first show that for any order function m and any symbol a ∈ S(m), we have (11.3.4)

F −1 aw (x, hD)F = aw (−hD, x),

the operators acting on S . Write A(t) := F (t)−1 aw (x, hD)F (t). As in the proof of Theorem 11.1, π hDt A(t) = [−h2 ∆ + |x|2 , A(t)]. 4

11.3. QUANTIZING LINEAR SYMPLECTIC MAPPINGS

239

Let l = l(x, ξ) be a linear function on R2n and consider as in Theorem 4.7 the exponential symbol at (x, ξ) := exp(iκ∗t l(x, ξ)/h). An explicit computation shows that π hDt aw [−h2 ∆ + |x|2 , aw t (x, hD) = t (x, hD)]. 4 Since (4.3.1) and (4.3.2) show that a Weyl operator is a superposition of exponentials of linear symbols, assertion (11.3.4) follows. This also shows that (11.3.4) remains valid for a ∈ S(m). 3. Suppose now that Fe is another unitary operator for which (11.3.4) holds. Then F˜ = cF for c ∈ C, |c| = 1, as follows from applying Lemma 3.3 to L = F ∗ Fe. Since by Theorem 4.9 the Fourier transform satisfies (11.3.4) and since (2πh)−n/2 Fh is unitary, we deduce that c F = n Fh . (2πh) 2 4. Thus it remains to compute the constant c. For this, let us put u0 = exp(−|x|2 /2) and consider the ODE ( hDt u(t) = π4 (−h2 ∆ + |x|2 )u(t), u(0) = u0 . Then u(t) = F (t)∗ u0 . Since u0 is the ground state of the harmonic oscillator and its eigenvalue is h, we deduce that u(t) = a(t)u0 , where a solves the ODE ( ∂t a = πi 4a a(0) = 1. Therefore a(t) = exp(πit/4). Finally, we note that eπi/4 u0 = F (1)∗ u0 = c¯(2πh)−n/2 Fh u0 = c¯u0 ; whence c = exp(−πi/4). This proves (11.3.3)



11.3.2. Quantizing linear symplectic mappings. Suppose now that κ : R2n → R2n is linear and symplectic, and is given by the symplectic matrix K. Preliminaries. To quantize K we deform it to the identity. Assume first that K has the form (11.3.5)

K = exp A,

JA + AT J = 0.

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11. QUANTUM AND CLASSICAL DYNAMICS

We write  A=

A1 A2 A3 −AT1

 ,

AT2 = A2 , AT3 = A3 .

We will define FK := F (1) where F (t) is a family of unitary operators associated to Kt := exp(tA). To proceed in the spirit of Section 11.1, we want to find qt such that ∂t Kt = Kt (Hqt ), where (Kt )∗ = Kt since Kt is linear. We have ∂t Kt = exp(tA)A; and therefore That is, Hqt (x, ξ) = (A1 x + A2 ξ)∂x + (A3 x − AT1 ξ)∂ξ . Hence we can take qt = p(x, ξ) for (11.3.6)

1 1 p(x, ξ) := hA2 ξ, ξi + hA1 x, ξi − hA3 x, xi. 2 2

LEMMA 11.8 (Propagators for quadratic Hamiltonians). Suppose that P := pw (x, hD), where p is given by (11.3.6). Then the equation ( hDt F (t) + F (t)P = 0 (11.3.7) F (0) = I

(t ∈ R)

is uniquely solved by a family {F (t)}t∈R of bounded, unitary operators on L2 (Rn ). We remark that Theorem 10.2 cannot be invoked here, as P is not elliptic if the quadratic form p(x, ξ) is not positive definite. Instead, motivated by Theorem 10.3, we propose an explicit formula for F (t) for small values of t. Proof. 1. Let us solve the Hamilton–Jacobi equation ( ∂t ϕ + p(x, ∂x ϕ) = 0 ϕ(0, x, η) = hx, ηi. As shown in the proof of Lemma 10.4, the solution ϕ = ϕ(x, η) generates Kt = exp(tA) in the sense that Kt : (∂η ϕ(x, η), η) 7→ (x, ∂x ϕ(x, η)). If  Kt =

At Bt Ct D t

 ,

11.3. QUANTIZING LINEAR SYMPLECTIC MAPPINGS

241

where At = I + O(t), Dt = I + O(t), Bt , Ct = O(t), then an explicit calculation shows 1 1 −1 −1 ϕ(t, x, η) = hCt A−1 t x, xi + hAt x, ηi − hAt Bt η, ηi. 2 2 2. Motivated by the amplitude computation in Lemma 10.5, we define 1 Z Z i (det At )− 2 F (t)u := e h (ϕ(t,x,η)−hy,ηi) u(y) dydη n (2πh) Rn Rn for u ∈ S . The square root is well defined as At is close to the identity. A calculation like that in the proof of Theorem 10.3 shows that the equation (11.3.7) holds exactly for small times t. We can now define F (t) : S → S for all times using the group property F (t + s) = F (t)F (s). We similarly show that F (t)∗ = F (−t) is well defined on S ; and hence F (t)∗ F (t) = F (−t)F (t) = I.  THEOREM 11.9 (Quantizing linear symplectic maps). Let K : R2n → R2n be a linear symplectic mapping. (i) There exists a unitary mapping FK : L2 (Rn ) → L2 (Rn ) such that (11.3.8)

−1 w a (x, hD)FK = (K ∗ a)w (x, hD). FK

for all symbols a ∈ S(m), m denoting any order function. (ii) The operator FK is unique up to a multiplicative factor. Hence for two linear symplectic maps K and L, we have (11.3.9)

FK FL = cFKL , |c| = 1.

Remember that the pull-back of a under K is defined by K ∗ a(x, ξ) = a(K(x, ξ)). INTERPRETATION. The quantization formula (11.3.8) for linear symplectic mappings generalizes (11.3.1), is a counterpart of the nonlinear formulas (11.2.9) and (11.2.11) from Section 11.2, and is exact in that there are no error terms.  Proof: 1. We only need to consider the case K = exp A, where AT J + JA, as Theorem 11.2 shows that general linear symplectic K can be written as a product of two such linear maps. Define FK = F (1) where F (t) is provided by Lemma 11.8.

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11. QUANTUM AND CLASSICAL DYNAMICS

2. We need to show that (11.3.10)

F (t)−1 aw (x, hD)F (t) = (Kt∗ a)w (x, hD)

for a ∈ S(m). We proceed as in the proof of Theorem 11.7 and consider a(x, ξ) := exp(il/h) where l = l(x, ξ) is a real-valued and linear. We recall from Theorem 4.7 that aw (x, hD) = exp (il(x, hD)/h) . This operator is bounded on L2 and hence both sides of (11.3.10) are well defined. The superposition described in Lemma 4.10 and then Lemma 11.8 provide the general case. 3. Let A(t) = F (t)−1 aw (x, hD)F (t); so that ( hDt A(t) = [P, A(t)] (11.3.11) A(0) = aw (x, hD). Define at (x, ξ) := exp(iKt∗ l(x, ξ)/h). We claim that aw t (x, hD) solves the same equation (11.3.11) as A(t), and therefore aw t (x, hD) = A(t). Let us therefore calculate that  w  w ∗ iKt∗ l/h ∗ iKt∗ l/h (11.3.12) hDt aw = (∂ K l) e = H (K l)e . t t p t t On the other hand, we see that w ∗ ∗ [P, aw t ] = (p# exp(iKt l/h) − exp(iKt l/h)#p) .

Since p is quadratic, (4.3.5) and (4.3.9) show that ∗

p# exp(iKt∗ l/h) − exp(iKt∗ l/h)#p = Hp (Kt∗ l)eiKt l/h . This is the right hand side of (11.3.12). 4. To see the uniqueness up to a multiplicative constant, we apply ∗ where both F and F eK satisfy (11.3.10). The concluLemma 3.3 to FK FeK K sion that K 7→ FK gives a projective representation of the group of linear symplectic maps (that is, a representation up to a multiplicative constant) is immediate from the uniqueness. 

11.3. QUANTIZING LINEAR SYMPLECTIC MAPPINGS

243

11.3.3. An explicit formula. We next provide a representation for FK in a special case. THEOREM 11.10 (An explicit quantization rule). Suppose that  K=

A B C D



is symplectic and det A 6= 0. Then we have formula 1

(11.3.13)

| det A|− 2 FK u(x) = (2πh)n

Z Rn

Z

i

e h (ϕ(x,η)−hy,ηi) u(y) dydη

Rn

for u ∈ S , where (11.3.14)

1 1 ϕ(x, η) := hCA−1 x, xi + hA−1 x, ηi − hA−1 Bη, ηi. 2 2

Proof. 1. Let ΛK be the twisted graph of K: ΛK := {(x, y, ξ, −η) | (x, ξ) = K(y, η)}. The condition det A 6= 0 means that linear map ΛK → Rn × Rn given by (11.3.15)

(x, y, ξ, η) 7→ (x, η)

is surjective: (Ay + Bη, Cy + Dη, x, −η) 7→ (Ay + Bη, −η). From (2.5.7) we see that there exists a generating function ϕ = ϕ(x, η) such that K is implicitely given by (x, ∂x ϕ(x, η)) 7→ (∂η ϕ(y, η), η). By following the proof of Theorem 2.14 or by direct inspection we see that ϕ is given by (11.3.14), up to an additive constant. 2. As suggested by the proof of Theorem 10.3, we arrive at the definition (11.3.13). It remains to check that FK is unitary and that (11.3.16)

aw (x, hD)FK = FK (K ∗ a)w (x, hD).

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11. QUANTUM AND CLASSICAL DYNAMICS

3. The unitarity of FK follows once we calculate the Schwartz kernel of ∗F : FK K Z Z Z i det A ∗ e h (hy,ηi−ϕ(x,η)+ϕ(z,ζ)−hy,ζi) dz dζ dη (FK FK )(x, y) = 2n (2πh) n n n ZR R R i det A = e h (ϕ(y,η)−ϕ(x,η) dη n (2πh) Rn Z i det A i (hCy,yi−hCx,xi) −1 2h e h h(η,A (x−y)i dη = e n (2πh) n R = δ(x − y), where in the last integral (interpreted in the S 0 sense) we made a change of variables η = Aξ. 4. To obtain (11.3.16) we check it first for the linear function a(x, ξ) = l(x, ξ) = hx∗ , xi + hξ ∗ , ξi. Using the properties of A, B, C, D we see that i

i

i

i

i

i

xe h (ϕ(x,η)−hy,ηi) = AhDη e h (ϕ(x,η)−hy,ηi) + (Ay + Bη)e h (ϕ(x,η)−hy,ηi) , hDx e h (ϕ(x,η)−hy,ηi) = ChDη e h (ϕ(x,η)−hy,ηi) + (Cy + Dη)e h (ϕ(x,η)−hy,ηi) . Since the variable η appears only in the phase, the terms with Dη disappear after integration. Also for any n × n matrix M , Z Z i (M η)e h (ϕ(x,η)−hy,ηi) u(y) dydη Rn Rn Z Z i = e h (ϕ(x,η)−hy,ηi) (M hDy )u(y) dydη. Rn

Rn

Hence 1 Z Z i | det A|− 2 (ϕ(x,η)−hy,ηi) h e (K ∗ l)(y, hDy )u(y) dydη (2πh)n n n R R = FK (K ∗ l)(x, hD)u.

l(x, hD)FK u =

This implies that eil(x,hD)/h FK = FK eiK

∗ l(x,hD)/h

.

The decomposition formula in Theorem 4.7 shows that (11.3.10) is therefore valid for all a ∈ S(m).  REMARK. The construction in the proof of Theorem 11.8 is a special case of the formula (11.3.13). A representation of F (t) as an oscillatory integral for larger times would involve linear symplectic transformations for which the map (11.3.15) is no longer surjective. The change of the generating function of ΛK produces phase factors of the form exp(imπ/4). This leads

11.4. EGOROV’S THEOREM FOR LONGER TIMES

245

to the study of the metaplectic group and of the Maslov index. See the references noted in Section 11.5 for more. 

11.4. EGOROV’S THEOREM FOR LONGER TIMES In Section 11.1 we proved Egorov’s Theorem relating the quantum and classical evolutions of time dependent Hamiltonians. In this section we show that for time independent Hamiltonians Egorov’s Theorem 11.1 in fact holds for times 0 ≤ t ≤ C log h−1 . That means that the classical/quantum correspondence described by (11.4.8) and (11.4.9) below is valid up to times comparable to log(h−1 ), known as the Ehrenfest time. 11.4.1. Estimates for flows. Suppose for the following that m ≥ 1 is an order function and p ∈ S(m). We assume as well that C + p ≥ cm for constants c > 0, C ≥ 0. We introduce the corresponding Hamiltonian flow ϕt = exp tHp . and define finally (11.4.1)

ΓR :=

sup

{|∂ 2 p(x, ξ)|}

{(x,ξ)|p0 ≤R}

for R > 0. LEMMA 11.11 (Exponential estimates for flows). For each γ0 > ΓR and each point (x, ξ) satisfying p(x, ξ) ≤ R we have the estimate (11.4.2)

|∂ α ϕt (x, ξ)| ≤ Cα eγ0 |α|t , α ∈ N2n , |α| > 0.

Proof. 1. The set {p ≤ R} is preserved by ϕt . Consequently on this set |∂ α p| ≤ Cα m ≤ Cα (1 + p) ≤ Cα for all multiindices α. 2. The proof of (11.4.2) is an induction on |α|. For α = 0, we have a weaker estimate on |ϕt (x, ξ)| ≤ Ct, which is valid since d ϕt = Hp (ϕt ), dt and Hp is bounded on {p ≤ R}.

(11.4.3)

246

11. QUANTUM AND CLASSICAL DYNAMICS

2. Now assume |α| = m and suppose the estimate (11.4.2) is valid for all multiindices of order less than or equal to m − 1. We differentiate (11.4.3), to find d α (11.4.4) (∂ ϕt ) = ∂Hp (ϕt )∂ α ϕt + ρt , dt where ρt is a sum of terms having the form g∂ α1 ϕt · · · ∂ αk ϕt for α1 + · · · αk = α and 0 < |αj | < |α| = m (j = 1, . . . , k). The induction hypothesis implies |ρt | ≤ Ceγ0 |α||t| . From (11.4.4) it follows that d α |∂ ϕt | ≤ ΓR |∂ α ϕt | + Ceγ0 |α||t| . dt We use Gronwall’s inequality to finish the proof of (11.4.2).



REMARK. The flows can also be estimated using a bound on Lyapunov exponents: (11.4.5)

e R := lim 1 sup log |∂ϕt |. Γ t→∞ t p0 ≤R

The limit exists since the function t 7→ supp≤R log |∂ϕt | is subbaditive. (A function f is subadditive if f (t+s) ≤ f (t)+f (s), in which case limt→∞ f (t)/t exists.) e R , as shown in Anantharaman– Then Lemma 11.11 holds for γ0 > Γ Nonnenmacher [A-N, Section 5.2]. A striking example is provided by p = e R = 0. However, ΓR is much easier |ξ|2 + |x|2 , for which ΓR = 2 whereas Γ to compute.  11.4.2. Egorov’s Theorem for long times. We will hereafter write (11.4.6)

F (t) := e−itP/h

(t ∈ R)

to denote the propagators generated by the operator P = pw (x, hD), as discussed in Theorem 10.2. We will also use the notation that b ∈ Sδ (m−∞ ) provided |∂ α b| ≤ CαN h−δ|α| m−N . for all multiindices α and all N ≥ 0.

11.4. EGOROV’S THEOREM FOR LONGER TIMES

247

THEOREM 11.12 (Egorov’s Theorem up to Ehrenfest time). Suppose a ∈ S and spt a ⊂ {p ≤ R} for some R > 0. For any γ > ΓR , T ≥ 0 and δ ∈ [0, 1/2), if |t| ≤ T +

(11.4.7)

δ log(h−1 ), γ

then (11.4.8)

F (t)−1 aw (x, hD)F (t) = bw t (x, hD)

for a symbol bt ∈ Sδ (m−∞ ) having the form (11.4.9)

bt = ϕ∗t a + OSδ (m−∞ ) (h2−3δ ).

INTERPRETATION. The statement bt −ϕ∗t a = O(h2−3δ ) means that the quantum evolution of aw (x, hD) given by the conjugation with exp(−itP/h) is well approximated by the classical evolution up to the time δ/γ log(h−1 ). Until that time we also know that the quantum evolved operator is a quantization of the (slightly exotic) classical observable bt . We also note that when δ = 0 we have an improved estimate valid up to any fixed time T .  We will mimic the proof of Theorem 11.1, paying careful attention to the dependence on t. Proof. 1. Select γ0 so that γ > γ 0 > ΓR . We always assume that the time t satisfies (11.4.10)

0≤t≤T+

1 δ log , γ h

as the case of negative times is the same. Consequently, (11.4.11)

tk eγ0 qt ≤ Ckp h−qδ

for all k, q ≥ 0, As noted previously, |∂ α p| ≤ Cα within {p ≤ R}. Since spt a ⊂ {p ≤ R}, Lemma 11.11 shows (11.4.12)

|∂ α ϕ∗t a| ≤ Cα eγ0 |α|t .

In particular, we have spt ϕ∗t a ⊂ {p ≤ R} and (11.4.13)

ϕ∗t a ∈ Sδ .

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11. QUANTUM AND CLASSICAL DYNAMICS

2. Put A := aw (x, hD) and define B(t) := F (t)−1 AF (t). As useful approximations, we construct a family of pseudodifferential operators Bk (t) = bw k,t (x, hD),

Ek (t) = ew k,t (x, hD)

(k = 0, 1, . . . )

such that (11.4.14)

( hDt Bk (t) = [P, Bk (t)] + Ek (t) + OS 0 →S (h∞ ) Bk (0) = A,

where  α k k(2−3δ) eγ0 |α|t )  ∂ (bk,t − bk−1,t ) = O(t h ∂ α ek,t = O(tk h1+(k+1)(2−3δ) eγ0 |α|t )   spt ek,t ⊂ {p ≤ R},

(11.4.15)

with the convention that b−1,t ≡ 0. 3. We will build Bk (t) and Ek (t) by induction. For k = 0, define B0 (t) := (ϕ∗t a)w (x, hD). Then using the definition of the flow (11.4.3), we calculate that h h (∂t ϕ∗t a)w = (Hp ϕ∗t a)w i i h ∗ w = {p, ϕt a} i = [P, B0 (t)] + E0 (t) + OS 0 →S (h∞ ).

hDt B0 (t) =

In this expression P = pw (x, hD),

E0 (t) = ew 0,t (x, hD)

for a symbol  2l+1 ∞ X h2l+1 i p(z)ϕ∗t a(w)|z=w=(x,ξ) e0,t (x, ξ) ∼ 2 σ(Dz , Dw ) (2l + 1)! 2 l=1

satisfying spt e0,t ⊂ {p ≤ R}. We used the composition formula (4.3.9) to define e0,t . This is justified in view of (11.4.12) and the assumption (11.4.10). According to (11.4.13) each term al in the expansion belongs h(2l+1)(1−δ) Sδ and so we can apply Theorem 4.15. Since we also have ∂ α al = O(h2l+1 eγ0 |α|t ), it follows that ∂ α e0,t = O(h3 eγ0 (3+|α|)t ) = O(h3−3δ eγ0 |α|t ).

11.4. EGOROV’S THEOREM FOR LONGER TIMES

249

This confirms (11.4.14) and (11.4.15) for k = 0. 4. Next, assume by induction that (11.4.14) and (11.4.15) hold for some k, and put Z i t ∗ ϕ ek,s ds, Ck+1 (t) := cw ck+1,t := k+1,t (x, hD). h 0 t−s Lemma 11.11 and (11.4.15) imply α

(k+1)(2−3δ)

Z

t

|∂ ck+1,t | ≤ Cα h

(11.4.16)

sk eγ0 |α|)s ds

0

≤ Cα t(k+1)(2−3δ) h(k+1)/2 eγ0 |α|t . Therefore hDt Ck+1 (t) = (hDt Ck+1 (t))w   Z t w h i ∗ = ∂t ϕ ek,s ds i h 0 t−s   w h i = Hp ck+1,t + ek,t i h = [P, Ck+1 (t)] + Ek (t) − Ek+1 (t) + OS 0 →S (h∞ ), where Ek+1 = ew k+1,t (x, hD), with spt ek+1,t ⊂ {p ≤ R} and  2l+1 ∞ X h2l+1 i ek+1,t (x, ξ) ∼ 2 σ(Dz , Dw ) p(z)ck+1,t (w)|z=w=(x,ξ) . (2l + 1)! 2 l=1

In view of (11.4.16) and (11.4.11) we obtain ∂ α ek+1,t = O(tk+1 h(k+1)(2−3δ)+3 eγ0 (3+|α|)t ) = O(tk+1 h1+(k+2)(2−3δ) eγ0 |α|t ) Then Bk+1 (t) := Bk (t) − Ck+1 (t) bw k+1,t (x, hD)

where Bk+1 (t) = for a symbol bk+1,t ∈ Sδ provides the operator satisfying (11.4.14) and (11.4.15), with k replaced by k + 1. 5. We next compare Bk (t) with B(t). To do so, we proceed again as in the proof of Theorem 11.1. A calculation from that proof gives hDt (F (t)Bk (t)F (t)−1 ) = F (t) (Ek (t) + OS 0 →S (h∞ )) F (t)−1 .

250

11. QUANTUM AND CLASSICAL DYNAMICS

Integrating in t and conjugating by F (t)−1 , we see that B(t) − Bk (t) = F (t)−1 (A − F (t)Bk (t)F (t)−1 )F (t) Z i t = F (s − t) (Ek (s) + OS 0 →S (h∞ )) F (t − s)ds (11.4.17) h 0   = OL2 →L2 h(k+1)(2−3δ) tk+1 . In the last estimate we used unitarity of F (t) and the bound on the L2 norm in terms of derivatives of ek,t given in part (ii) of Theorem 4.23. 6. Next, we assert that for any linear l1 , · · · , lM , we have (11.4.18)

adl1 · · · adlM (B(t) − Bk (t)) = OL2 →L2 (h(k+1)/2 ).

As we will see in Step 9 of the proof the exact power of h is not important, so long as it goes to infinity with k. To establish (11.4.18), use (11.4.17). We have Z i t adl (F (s − t)Ek (s)F (t − s)) ds adl (B(t) − Bk (t)) = h 0 Z i t + adl (F (s − t)OS 0 →S (h∞ )F (t − s)) ds, h 0 and a similar expression is valid for M > 1. We will focus upon the first term on the right hand side, the second term being handled by the same methods. Since Ek (s) = ew k,s for a symbol ek,s ∈ Sδ with spt ek,s ⊂ {p ≤ R}, we can write ek (s)(P + i)−N , E ek (s) := (P + i)N Ek (s)(P + i)N , Ek (s) = (P + i)−N E where ek (s) = e˜w (x, hD), E k,s

e˜k,s ∈ h1+(k+1)(2−3δ) Sδ .

Since δ < 1/2, Remark at the end of Section 8.1 shows that i ek (s) = OL2 →L2 (h(k+1)(2−3δ)+M (1−δ) ), adl · · · adl1 E h M Recalling (8.1.8) and (8.1.9), to obtain (11.4.18) we only need to show that
adlM · · · adl1 (P + i)−N F (t) = OL2 →L2 (tM ), (11.4.19)
adlM · · · adl1 (P + i)−N F (t)−1 = OL2 →L2 (tM ) for lj linear on R2n and for N ≥ 2M . (Recall the assumption (11.4.10) which gives control of powers of t.)

11.4. EGOROV’S THEOREM FOR LONGER TIMES

251

7. To make the argument clear we first consider
the case of M = 1 and look the at equation satisfied by adl (P + i)−N F (t) :

hDt adl (P + i)−N F (t)
= adl (P + i)−N hDt F (t) (11.4.20)
= −adl F (t)(P + i)−N P
= −adl F (t)(P + i)−N P − F (t)(P + i)−N adl P,
with adl F (0)(P + i)−N = adl (P + i)−N = OL2 →L2 (h). We also note that for N ≥ 2, the pseudodifferential calculus gives QN := (P + i)−N adl P

= (P + i)−1 adl (P + i)−N +1 P + (P + i)−1 adl (P + i)−N +1 P = OL2 →L2 (h). Applying Duhamel’s formula to (11.4.20) gives
adl (P + i)−N F (t) Z
i t −N F (s)QN F (t − s)ds = F (t)adl (P + i) F (0) − h 0 = OL2 →L2 (t). This argument applies to F (t)−1 and shows (11.4.19) for M = 1. 8. To check the first claim in (11.4.19) for M > 1 we modify it to
(11.4.21) adlM · · · adl1 (P + i)−N F (t) (P + i)K = OL2 →L2 (tM ), for N ≥ 2M + K and proceed by induction on M . We note that (11.4.21) holds for M = 0 and N ≥ K. Put
S(t) := adlM +1 · · · adl1 (P + i)−N F (t) (P + i)K . Then, arguing as in Step 7 we get hDt S(t) = S(t)P + R(t), with
R(t) = adlM · · · adl1 (P + i)−N F (t) (P + i)K+2 (P + i)−2 adlM +1 P = OL2 →L2 (htM ). We note that, as required by the induction hypothesis, we need to take N ≥ 2M + K + 2 = 2(M + 1) + K. As in Step 7 above an application of the Duhamel formula gives (11.4.21).

252

11. QUANTUM AND CLASSICAL DYNAMICS

9. It remains to show that B(t) is a pseudodifferential operator, and to do so, we invoke the Remark after Beals’s Theorem 8.3 and hence need to show for any linear l1 , · · · , lM that (11.4.22)

adlM · · · adl1 B(t) = OL2 →L2 (hM (1−δ) ).

Using (11.4.18), we have adlM · · · adl1 B(t) = adlM · · · adl1 Bk (t) + adlM · · · adl1 (B(t) − Bk (t)) = OL2 →L2 (hM (1−δ) ) + OL2 →L2 (h(k+1)/2 ) = OL2 →L2 (hM (1−δ) ), if (k + 1)/2 ≥ M . Retracing our steps we see that B(t) = bw t (x, hD), where bt ∈ Sδ . Since the arguments in Step 6 above apply also to (P + i)N B(t) for any N , we see furthermore that bt ∈ Sδ (m−∞ ).  REMARK. The Theorem is also valid P = pw (x, hD) for a real-valued symbol p having the form p = p 0 + h2 p 1 where p0 , p1 ∈ S(m) and p0 has the property p0 ≥ cm − C. 

11.5. NOTES The definition of quantization of symplectomorphisms using deformation follows the Heisenberg picture of quantum mechanics. The essence of the proof of Theorem 11.1 comes from Taylor [Ta, §Section 8.1]. For a semiclassical version, consult Christiansen [Chr, Section 3]. See Bony [B] and references therein for generalizations of Theorem 11.1, allowing the symbols {qt }0≤t≤T to belong to more general classes. See Dimassi–Sj¨ ostrand [D-S, Appendix to Chapter 7] for another selfcontained presentation of semiclassical quantization of linear symplectic relations and Folland [F, Chapter 4] for more and for references. For the discussion of the Maslov index we refer to H¨ormander [H3, Section 21.6], Guillemin–Sternberg [G-St], [G-St1], and Lion–Vergne [L-V]. See Robert [R1] for the closely related theory of coherent states. The term Ehrenfest time has origins in the physics literature (Chirikov [Ch], Zaslavsky[Za]), and a standard reference is Bouzouina-Robert [B-R]. The precise dependence upon the dynamics of the constant in front of log(1/h) is not clear. In [B-R] such a finer version is given for a class of Schr¨ odinger operators. An interesting application to scattering theory is Bony–Burq–Ramond [B-B-R]. For a physics perspective and references, see Zurek [Zu].

Chapter 12

NORMAL FORMS

12.1 12.2 12.3 12.4 12.5 12.6

Overview Normal forms: real symbols Propagation of singularities Normal forms: complex symbols Quasimodes, pseudospectra Notes

12.1. OVERVIEW We devote this chapter to deriving microlocal normal forms for principal type operators, that is, operators P whose symbols p satisfy (12.1.1)

∂p 6= 0 on {p = 0}.

The main endeavor will be to find ways to microlocally convert such operators P into much simpler forms, either hDx1 for real p (Section 12.2) or hDx1 ± ix1 for complex p (Section 12.4). MOTIVATION: WKB method. It is indeed remarkable that entire classes of differential and pseudodifferential equations can be microlocally transformed into the same simple transport operator hDx1 = 0, but we should be careful to understand that this is meaningful only for highly oscillatory solutions the limit h → 0. We provide some motivation by discussing briefly the WKB (= Wentzel– Kramers–Brillouin) approximation for solutions of a linear PDE, say of the 253

254

12. NORMAL FORMS

form (12.1.2)

P (h)u(h) =

X

aα (x)(hD)α u(h) = 0.

|α|≤N

Let us look for a solution having the so-called WKB form (12.1.3)

i

u(h) = ae h ϕ ,

where ϕ is independent of h and where the amplitude a has the expansion (12.1.4)

a∼

∞ X

ak (x)hk ,

k=0

If we formally plug (12.1.3), (12.1.4) into (12.1.2) and compare like powers of h, we discover that ϕ solves the nonlinear PDE (12.1.5)

p(x, ∂ϕ) = 0

for (12.1.6)

p(x, ξ) =

X

aα (x)ξ α .

|α|≤N

As explained in [E, Section 3.2], we can locally study the PDE (12.1.5) by introducing the characteristic equations ( x˙ = ∂ξ p(x, ξ) ξ˙ = −∂x p(x, ξ). It is easy to plausible that we can sometimes change variables to make this ODE flow translation in the x1 direction, which is related to Dx1 . This motivation for the following theory only provides heuristics, since we may not to be able to write the solution in WKB form. For instance, take n = 1, p = ξ 2 − x and consider a neighborhood of x = 0. In this example near x = 0 the characteristic variety is x = ξ 2 and so is not a graph of a function of x. Hence p(x, ϕ0 ) = 0 has no smooth solutions. So we cannot represent a solution near 0 in the simple WKB form (12.1.3). Note that in Section 10.2 we have represented solutions of evolution equations using superposition of WKB states (10.2.3).  We conclude this section with an example illustrating the applications of the theory. The point of the general theory of normal forms is that the approach described below for constant coefficient equations works for operators with nonconstant coefficients. EXAMPLE: Helmholtz’s equation. Let us first consider Helmholtz’s equation (12.1.7)

(−h2 ∆ − 1)u = 0,

12.1. OVERVIEW

255

the symbol of which is p = |ξ|2 − 1. Changing variables. We introduce the change of variables η = γ(ξ), defined by η1 = |ξ|2 − 1, η2 = ξ2 , . . . , ηn = ξn . Proceeding as in Theorem 2.6 we extend this to a symplectic transformation (x, ξ) 7→ (y, η) by putting y = ((∂γ(ξ))−1 )T x. That is, y1 = x1 /2ξ1 , y2 = x2 − x1 ξ2 /ξ1 , . . . , yn = xn − x1 ξn /ξ1 . This is well defined near any point (x0 , ξ0 ) ∈ R2n at which ξ1 6= 0. The symplectic transformation κ given by the inverse of the above transformation then gives κ∗ p = η1 . In this sense κ “simplifies” the symbol p near (x0 , ξ0 ) . Quantizing κ. We next quantize κ and likewise “simplify” Helmholtz’s equation. We accomplish this by defining the operator T u := Fh−1 (χγ ∗ (Fh u)) , where u ∈ L2 and χ ∈ Cc∞ (B(0, 1/2)) equals one near 0. We check directly that for functions u with Fh u supported in a sufficiently small neighborhood of ξ0 , we have hDy1 T u = Fh−1 (χη1 γ ∗ (Fh u))
= Fh−1 χγ ∗ ((|ξ|2 − 1)Fh u)
= T (−h2 ∆ − 1)u . So T has “intertwined” the Helmholtz operator and the much simpler operator hDy1 . Flows and wavefront sets. We stress that this simplification holds only locally in phase space, but nevertheless allows us to draw interesting conclusions. For instance, it is immediate that if hDy1 v = f , then WFh (v)\WFh (f ) is invariant under the Hamiltonian flow of η1 , which is (y, η) 7→ (y + te1 , η) for e1 = (1, 0, · · · , 0). We can therefore deduce that if (12.1.8)

(−h2 ∆ − 1)u = g,

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12. NORMAL FORMS

then WFh (u)\WFh (g) is invariant under the Hamiltonian flow of p = |ξ|2 −1, which is (x, ξ) 7→ (x + 2tξ, ξ). This tells us in which regions of phase space the solution is oscillating or decaying. See Section 12.3 for details of all these assertions.  The example of P == −h2 ∆ − 1 illustrates that solutions of P u = 0 that are highly oscillatory as h → 0 exist owing to the characteristic variety {p = 0}. We want to understand the geometric structure of these oscillations as measured by the notion of the wavefront set.

12.2. NORMAL FORMS: REAL SYMBOLS In this section we show that microlocally near a point an operator P quantizing a real symbol with a non-vanishing differential is equivalent to hDx1 . This is important since we can then transplant various mathematical objects associated to P to those for hDx1 , which are much easier to study. 12.2.1. More symplectic geometry. To apply the local theory of quantized symplectic transformations from Chapter 11 to the study of semiclassical operators we will need two results from symplectic geometry. The first is a stronger form of Darboux’s Theorem 2.12: THEOREM 12.1 (Variant of Darboux’s Theorem). Let A and B be two subsets of {1, · · · , n}, and suppose that pj (x, ξ) (j ∈ A), qk (x, ξ) (k ∈ B) are smooth, real-valued functions defined in a neighborhood of (0, 0) ∈ R2n , with linearly independent gradients at (0, 0). If (12.2.1)

{qi , qj } = 0 (i, j ∈ A), {pk , pl } = 0 (k, l ∈ B), {pk , qj } = δkj (j ∈ A, k ∈ B),

then there exists a symplectomorphism κ, locally defined near (0, 0), such that κ(0, 0) = (0, 0) and (12.2.2)

κ∗ qj = xj (j ∈ A), κ∗ pk = ξk (k ∈ B).

We will also need

12.2. NORMAL FORMS: REAL SYMBOLS

257

THEOREM 12.2 (Symplectic integrating factor). Let p and q be smooth, real-valued functions defined near (0, 0) ∈ R2n , satisfying p(0, 0) = q(0, 0) = 0, {p, q}(0, 0) > 0.

(12.2.3)

Then there exists a smooth, positive function u for which {up, uq} ≡ 1

(12.2.4) in a neighborhood of (0, 0).

Consult H¨ ormander [H2, Theorem 21.1.6] and [H2, Lemma 21.3.4] for proofs. 12.2.2. Symbols of real principal type. Now let m be an order function and set P = pw (x, hD), where p(x, ξ) ∼

∞ X

hk pk (x, ξ)

k=0

for symbols pj ∈ S(m). DEFINITIONS. (i) The operator P is of real principal type if p0 is real-valued and ∂p0 6= 0 on {p0 = 0}.

(12.2.5)

(ii) If the principal symbol p0 is real-valued and satisfies p0 (0, 0) = 0, ∂p0 (0, 0) 6= 0,

(12.2.6)

we say that P is of real principal type at the point (0, 0). THEOREM 12.3 (Normal form for real principal type operators). Suppose that P = pw (x, hD) is of real principal type at (0, 0). Then there exist (i) a local canonical transformation κ defined near (0, 0), such that κ(0, 0) = (0, 0) and κ∗ p0 = ξ1 ;

(12.2.7)

and (ii) an operator T , quantizing κ in the sense of Theorem 11.6, such that (12.2.8)

T −1 exists microlocally near ((0, 0), (0, 0))

and (12.2.9)

T P T −1 = hDx1

microlocally near ((0, 0), (0, 0)).

258

12. NORMAL FORMS

Proof. 1. Theorem 12.1 with A = ∅ and B = {1}, provides κ satisfying (12.2.7) near (0, 0). By taking a sufficiently small neighborhood of (0, 0) we can assume that the assumptions of Theorem 11.5 are satisfied, and consequently we have a quantization T0 of κ. Thus (12.2.10)

T0 P T0−1 = hDx1 + E microlocally near (0, 0) ,

where E = ew for a symbol e ∈ hS. 2. We want to transform away the term E in (12.2.10). To do so, we look for a symbol a ∈ S so that a is elliptic at (0, 0) and (12.2.11)

hDx1 + E = AhDx1 A−1 microlocally near (0, 0),

where A := aw . This is the same as solving (12.2.12)

[hDx1 , A] + EA = 0 microlocally near (0, 0) .

w 2 w Since P = pw 0 + hp1 + h p2 + · · · , the remark after the proof of Theorem 11.1 shows that

e(x, ξ) = he0 (x, ξ) + h2 e1 (x, ξ) + · · · . We can solve the differential equation 1 {ξ1 , a0 } + e0 a0 = 0 i near (0, 0) for a0 ∈ S such that a0 (0, 0) 6= 0. Define A0 := aw 0 ; then [hDx1 , A0 ] + EA0 = r0w for a symbol r0 ∈ h2 S. 3. We now inductively find Ak = aw k for ak ∈ S, satisfying w [hDx1 , A0 + hA1 + · · · + hN AN ] + E(A0 + hA1 + · · · hN AN ) = rN ,

where rN ∈ hN +2 S. We invoke Borel’s Theorem 4.15 to put a∼

∞ X

k

w

h ak , A = a ∼

k=0

∞ X

hk Ak .

k=0

Then A solves (12.2.12), and a is elliptic near (0, 0). Finally, define T := A−1 T0 . According to (12.2.10) and (12.2.11), this operator verifies (12.2.9).



12.2. NORMAL FORMS: REAL SYMBOLS

259

12.2.3. L2 estimates and principal type. Here is a first application of Theorem 12.3: THEOREM 12.4 (Principal type and estimates). Suppose that P = pw is of real principal type at (0, 0) Assume for u = u(h) ∈ L2 (Rn ) that WF∗h (u) is contained in a small neighborhood of (0, 0). Then we have the estimate C (12.2.13) kukL2 (Rn ) ≤ kP ukL2 (Rn ) . h Proof: Let T be the operator given in Theorem 12.3 and T −1 be its microlocal inverse near (0, 0). The localization property of u implies kP ukL2 = kT −1 (hDx1 )T ukL2 + O(h∞ )kukL2 ≥ ckT T −1 hDx1 T ukL2 + O(h∞ )kukL2 = ckhDx1 T ukL2 + O(h∞ )kukL2 . Localization of u and the local Egorov Theorem 11.6 imply that T u = ψT u + OS (h∞ )kukL2 for ψ ∈ Cc∞ (Rn ). Also i v(x) = h where x = (x1

, x0 ),

Z

x1

hDx1 v(y, x0 ) dy,

−∞

for v := ψT u. Therefore kvkL2 ≤

C khDx1 vkL2 ; h

and hence kP ukL2 ≥ chkT ukL2 + O(h∞ )kukL2 ≥ chkT −1 T χw ukL2 + O(h∞ )kukL2 ≥ chkχw ukL2 + O(h∞ )kukL2 . This proves (12.2.13), since WF∗h (u) lies within a small neighborhood of (0, 0).  REMARK. The operators used in the microlocal transformations have good mapping properties on L2 -based spaces, such as our generalized Sobolev spaces Hh (m). However, as we have seen in Section 10.3 and 10.4 the mapping properties on Lp spaces are complicated and depend on finer geometric aspects of the canonical transformation. Hence the normal forms cannot be directly used for the study of such phenomena as dispersions or multiplicative properties of solutions. 

260

12. NORMAL FORMS

12.3. PROPAGATION OF SINGULARITIES 12.3.1. Propagation of wavefront sets. As a further application of Theorem 12.3 we next prove a propagation theorem that refines our earlierTheorem 5.5. Suppose that P (h) = pw (x, hD) is an operator of real principal type and let p0 (x, ξ) be its principal symbol: p∼

∞ X

hk pk ,

k=0

where p, pk ∈ S(m) for some order function m. THEOREM 12.5 (Flow invariance of wavefront sets). Assume that ∂p|p−1 (0) 6= 0. Then for each tempered family u = {u(h)}0
WFh (T u) ∩ V = κ(W Fh (u) ∩ V ),

when V is a small neighborhood of (0, 0). We need therefore to check that if (12.3.3)

hDx1 v = OL2 (h∞ )

12.3. PROPAGATION OF SINGULARITIES

261

microlocally near (0, 0) and if (0, 0; 0, 0) ∈ WFh (v), then (t, 0; 0, 0) ∈ W Fh (v) for small |t|. Here and hereafter we write (x, ξ) = (x1 , x0 ; ξ1 , ξ 0 ) for x0 , ξ 0 ∈ Rn−1 . 3. The equation (12.3.3) shows v(x1 , x0 ) = v(0, x0 ) + OL2 (h∞ )

(12.3.4) for |x1 | < .

Then if (t, 0; 0, 0) ∈ / WFh (v), it follows that (0, 0; 0, 0) ∈ / WFh (v) for ∞ small |t|. To see this, suppose that a ∈ Cc , a(t, 0; 0, 0) 6= 0, and aw (x, hD)v = OL2 (h∞ ). Define b(x, ξ) := a(x1 − t, x0 , ξ). Then b(0, 0) 6= 0 if |t| is small enough, and (12.3.4) implies bw (x, hD)v = aw (x1 − t, x0 , hD)v = aw (x, hD)v + OL2 (h∞ ) = OL2 (h∞ ).



EXAMPLE: Wave equation. Consider the semiclassical wave equation ((h∂t )2 − h2 ∆)u = f,

(12.3.5)

the right side of which is highly oscillatory: i

f = be h ϕ

(12.3.6)

for smooth functions b, ϕ, where ϕ is real. The symbol of the wave operator is p = |ξ|2 − τ 2 . According to [E, Section 2.4]) we have a unique solution to (12.3.5) with the property that u ≡ 0 for t ≤ 0. Example (ii) in Section 8.4 shows that WFh (f ) = {(t, x, ∂t ϕ, ∂x ϕ) | (t, x) ∈ spt b}. Then (12.3.7)

WFh (u) \ (WFh (f ) ∪ {ξ 2 = τ 2 = 0})

is invariant under the flow Φs : (t, x, τ, ξ) 7→ (t − 2sτ, x + 2sξ, τ, ξ),

ξ 2 = τ 2 6= 0.

The restriction that τ 6= 0 in (12.3.7) comes from the fact that (12.2.5) fails when τ = 0. Since u ≡ 0 for negative times, we see that [ [ WFh (u) \ {τ = 0} ⊂ Φs (WFh (f ) ∩ {p = 0, ∓τ > 0}) ∪ WFh (f ). ± ±s>0

262

12. NORMAL FORMS

Suppose next that ϕ = ϕ(x) is independent of t and that ∂ϕ 6= 0 in the support of b. Suppose that U ⊂ Rn+1 is an open bounded subset satisfying ¯ ∩ {(t + 2|∂ϕ(x)|s, x + 2s∂ϕ(x)) | (x, t) ∈ spt b, s ≥ 0} = ∅. U Then u|U = OC ∞ (h∞ ).

(12.3.8)

We should note that the wavefront set analysis gives us only k(1 − χ(hDt ))χ((hDt )2 − |hDx |2 ))ukHhs (U ) = O(h∞ ), where s ∈ R and χ ∈ Cc∞ (R) equals 1 near 0. The assumption that ∂ϕ 6= 0 on the support of b shows kχ(hDt )χ((hDt )2 − |hDx |2 )f kL2 = O(h∞ ), and we can use energy estimates for the wave equation to conclude that kχ(hDt )χ((hDt )2 − |hDx |2 )ukL2 (U ) = O(h∞ ) for any bounded set U . This implies the global estimate (12.3.8). This example illustrates that at places where (12.2.5) fails other methods can sometimes be used to obtain stronger conclusions. For another take on the wave equation with highly oscillatory initial data, see H¨ormander [H2, Section 12.2].  REMARK: Propagation of C ∞ singularities. Using Theorem 12.5 and the relation (8.4.8) between the semiclassical wavefront set and the wavefront set measuring C ∞ singularities we can recover the now classical theorem of H¨ ormander on propagation of singularities for operators of real principal type. Suppose that P (x, D) :=

X

aα (x)Dα

|α|≤m

is a differential operator with coefficients aα ∈ C ∞ , and that u ∈ S 0 solves P (x, D)u = f. We reformulate this semiclassically by putting X Pe(x, hD) := hm P (x, D) = aα (x)hm−|α| (hD)α . |α|≤m

In the notation of Theorem 12.5, Pe(x, hD) has the principal symbol X p(x, ξ) = aα (x)ξ α . |α|=m

12.4. NORMAL FORMS: COMPLEX SYMBOLS

263

This is the same as the classical principal symbol obtained by taking the highest order derivatives. If we assume that dp|p−1 (0) 6= 0 in a conic neighborhood (x0 , ξ0 ) ∈ R2n \ 0, then (8.4.8) and Theorem 12.5 show that in a conic neighborhood of (x0 , ξ0 ) WF(u) \ WF(f ) is invariant under the flow of Hp . EXAMPLE. Suppose that u ∈ S 0 solves the inhomogeneous wave equation in R × Rn : (∂t2 − ∆)u = f where f ∈ S 0 . Then WF(u) \ WF(f ) ⊂ {(t, x, τ, ξ) | τ 2 = |ξ|2 } is invariant under the flow ξ t˙ = 1, x˙ = , τ˙ = 0, ξ˙ = 0. |ξ| So singularities of solutions to the wave equation, as measured by the wavefront set, propagate along light rays. 

12.4. NORMAL FORMS: COMPLEX SYMBOLS Motivated by the study of pseudospectra in Section 12.5, we now consider the case of complex p. In that case we need a stronger non-degeneracy condition stating that Re∂p and Im∂p are independent. Once that is true then p = 0 is a codimension two submanifold and it can have different symplectic properties, determining possible normal forms. Operators of complex principal type. Assume as before that P = pw (x, hD) has the symbol ∞ X p∼ hk pk k=0

with pj ∈ S(m). We now allow p(x, ξ) to be complex-valued, and still say that P is principal type at (0, 0) provided p0 (0, 0) = 0, ∂p0 (0, 0) 6= 0. THEOREM 12.6 (Normal form for the complex symplectic case). Suppose that P = pw (x, hD; h) is a semiclassical principal type operator at (0, 0), with principal symbol p0 satisfying (12.4.1)

p0 (0, 0) = 0, {Re p0 , Im p0 }(0, 0) 6= 0.

Then there exist

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(i) a local canonical transformation κ defined near (0, 0) and a smooth function u such that κ(0, 0) = (0, 0), u(0, 0) > 0, and κ∗ (ξ1 ± ix1 ) = up0 , and (ii) an operator T , quantizing κ in the sense of Theorem 11.6, and a pseudodifferential operator A, elliptic at (0, 0), such that (12.4.2)

T −1 exists microlocally near ((0, 0), (0, 0))

and (12.4.3)

T P T −1 = A(hDx1 ± ix1 ) microlocally near ((0, 0), (0, 0)).

INTERPRETATION. If ∂(Re p0 ) and ∂(Im p0 ) are linearly independent, then the submanifold p−1 (0) ⊂ R2n , where P is not elliptic, has codimension two, as opposed to codimension one in the real-valued case. The symplectic form restricted to that submanifold is nondegenerate if {Re p0 , Im p0 } = 6 0. Under this assumption a combination of Theorems 12.1 and 12.2 shows that there exists a canonical transformation κ, defined near (0, 0), and a smooth positive function u such that κ∗ (ξ1 ± ix1 ) = up0 . That is, after a multiplication by a function we obtain the symbol of the creation or annihilation operator for the harmonic oscillator in the (x1 , ξ1 ) variables.  Proof. 1. Let us assume {Re p0 , Im p0 } > 0. As noted above, using Theorems 12.1 and 12.2 we can find a smooth function u, with u(0, 0) > 0, and a local canonical transformation κ such that κ(0, 0) = (0, 0) and κ∗ (ξ1 + ix1 ) = up0 . Quantizing as before, we obtain an operator T0 satisfying (12.4.4)

T0 P T0−1 = Q(hDx1 + ix1 + E)

microlocally near (0, 0), where Q = q w for a function q satisfying κ∗ q = 1/u and E = ew for some e ∈ hS. As in the proof of Theorem 12.3, we have e = he0 (x, ξ) + h2 e1 (x, ξ) + · · · .

12.4. NORMAL FORMS: COMPLEX SYMBOLS

265

2. To remove the term E from (12.4.4), we now need to find pseudodifferential operators B and C, elliptic at (0, 0), such that (hDx1 + ix1 + E)B = C(hDx1 + ix1 )

(12.4.5)

microlocally.

We will find the symbols of B and C by computing successive terms in their expansions: ∞ ∞ X X b∼ hk bk , c ∼ hk ck . k=0

k=0

We use Theorem 4.11 to write (hDx1 + ix1 + E)B − C(hDx1 + ix1 ) = rw , where r=

∞ X

hk rk

k=0

with r0 = (ξ1 + ix1 )(b0 − c0 ), r1 = (ξ1 + ix1 )(b1 − c1 ) + e0 b0 + {ξ1 + ix1 , b0 + c0 }/2i. 3. We intend to select b and c so that rj ≡ 0 for all j. For r0 = 0 we simply need b0 = c0 . To force r1 = 0, we must solve −i(∂x1 − i∂ξ1 )b0 + e0 b0 + (ξ1 + ix1 )(b1 − c1 ) = 0. We first find b0 such that ( −i(∂x1 − i∂ξ1 )b0 + e0 b0 = O(x∞ 1 ) (12.4.6) b0 |x1 =0 = 1. The notation means that the left hand side of the first equation vanishes to infinite order at x1 = 0. We compute ∂xk+1 b0 |x1 =0 inductively: 1   k k ∂xk+1 b = i ∂ ∂ b − ∂ (e b ) . 0 0 0 0 ξ x1 1 x1 1 We see that the derivatives ∂xk1 e0 |x1 =0 therefore determine ∂xk1 b0 |x1 =0 . Then Borel’s Theorem 4.15 produces a smooth function b0 with these prescribed derivatives, that therefore solves (12.4.6). With b0 = c0 chosen as above, we see that t1 := (−i(∂x1 − i∂ξ1 )b0 + e0 b0 )/(ξ1 + ix1 ) is a smooth function, since the numerator vanishes to infinite order on the zero set of the denominator. So if we put (12.4.7) then r1 = 0.

c1 = b1 + t1 ,

266

12. NORMAL FORMS

4. Now using (12.4.7) the same calculation as before, we see that r3 = (ξ1 + ix1 )(b2 − c2 ) + e0 b1 − i{ξ1 + ix1 , b1 } + r˜3 , where r˜3 depends only on b0 = c0 , t1 , and e. Hence r˜3 is already determined. We proceed as in Step 4 and first solve ( −i(∂x1 − i∂ξ1 )b1 + e0 b1 + r˜3 = O(x∞ 1 ) b1 |x=1 = 0. This determines b1 and hence c1 . We continue in the same way to determine b2 , and thus c2 . An iteration of the argument completes the construction of b and c, for which (12.4.5) holds microlocally near (0, 0). 5. Finally, we put T = B −1 T0 , where B −1 is the microlocal inverse of B near (0, 0), and A = B −1 QC.  Applications of Theorem 12.6 are given in Section 12.5. The next example shows that other phenomena occur when symbols are complex but the assumptions on the theorem do not hold. EXAMPLE: Heat equation. Consider now the semiclassical heat equation h∂t u − h2 ∆u = 0.

(12.4.8)

The symbol is p = τ − iξ 2 is complex, p = {τ = 0, ξ = 0} ⊂ T ∗ Rn+1 , and for any χ ∈ Cc∞ (Rn+1 ), WFh (χu) ∩ Rn+1 × {(0, 0)}. The conclusion about the wavefront set corresponds to the fact that solutions to (12.4.8) have no strong oscillations in h. We can have a propagation result as well. Using the fact that ∂Rep 6= 0 and Imp ≤ 0 we can prove the following result: suppose that u is tempered and solves h∂t u − h2 ∆u = f. Then WFh (u) \ WFh (f ) ⊂ {p = 0} is closed under backward time propagation. That means that if (t, x, 0, 0) is not in WFh (u) \ WFh (f ) and (t − s, x, 0, 0) ∈ / WFh (f ) for 0 ≤ t ≤ T , then (t − s, x, 0, 0) ∈ / WFh (u).

12.5. QUASIMODES, PSEUDOSPECTRA

A similar statement can be made for illustrate it with a simple example. Let u ( h∂t u − h2 ∆u = 0 (12.4.9) u=g

267

the initial value problem and we solve (t > 0) (t = 0),

where i

g = e h ϕb for smooth functions b, ϕ, the phase ϕ being real. A tempered solution to (12.4.9) is (see [E, Section 2.2]) : Z |x−y|2 1 + hi ϕ(y) 4ht e u(t, x) = b(y) dy. n/2 (4πht) Rn For t >  > 0 we have the pointwise estimate (hDx )α (hDt )k u(t, x) = O(h|α|/2+k/2 ).

(12.4.10)

This shows that τ = ξ = 0 belongs to the wavefront set of u. When x ∈ / spt b we also have (hDx )α (hDt )k u(t, x) = O(h∞ ).

(12.4.11)

When ∂ϕ(x) 6= 0, integration by parts based upon the identity 2t

|x−y|2 h2t∂ϕ(y) − i(x − y), hDy i |x−y|2 + i ϕ(y) + hi ϕ(y) 4ht h 4ht = e e 2 0 2 2 4t |ϕ (y)| + |x − y|

shows that (12.4.11) also holds. It follows therefore that WFh (u) ∩ {t > 0} = ((0, ∞) × {x ∈ spt b | ∂ϕ(x) = 0}) × {(0, 0)}. We should note that the statement that WFh (u) is contained in the zero section means that we do not have oscillations on the scale h but not that the solution is actually smooth uniformly with respect to h – see (12.4.10).  REMARK. These example again shows that solutions to P u = 0 that are highly oscillatory as h → 0 are due to the presence of the characteristic variety {p = 0}. The heat equation example also shows that the situation is richer for complex symbols, as we can have different conditions on Rep and Imp. 

12.5. QUASIMODES, PSEUDOSPECTRA We present in this concluding section an application of Theorem 12.6 to the construction of quasimodes for nonnormal operators.

268

12. NORMAL FORMS

12.5.1. Quasimodes and eigenvalues. Recall from Theorem 6.7 that if P = P (h) = −h2 ∆ + V (x) and V is real-valued, satisfying (12.5.1)

V (x) ≥ chxik for |x| ≥ R, |∂ α V (x)| ≤ Cα hxik ,

then the spectrum of P is discrete. Since V is real and consequently P is self-adjoint, the approximate location of eigenvalues can be determined by the existence of quasimodes, that is, approximate eigenfunctions: THEOREM 12.7 (Quasimodes and eigenvalues). Suppose under the foregoing assumptions that (12.5.2)

k(P − z(h))u(h)kL2 = O(h∞ ), ku(h)kL2 = 1.

Then there exist an eigenvalue E(h) and eigenfunction v(h) such that ( (P − E(h))v(h) = 0, kv(h)kL2 = 1, (12.5.3) |E(h) − z(h)| = O(h∞ ). In other words, if we can construct a quasimode solving (12.5.2), then the approximate eigenvalue z(h) is in fact close to a true eigenvalue E(h), although the quasimode u(h) need not be close to a true eigenfunction v(h). Proof. If z(h) ∈ / Spec(P ), we can use (C.2.17) to compute 1 = ku(h)kL2 = k(P − z(h))−1 (P − z(h))u(h)kL2 ≤ k(P − z(h))−1 kk(P − z(h))u(h)kL2 = O(h∞ )/dist(z(h), Spec(P )). As noted above, Spec(P ) consists solely of eigenvalues .



12.5.2. Quasimodes for nonnormal operators. But Theorem 12.7 is in general false for nonnormal operators P , those for which the commutator [P ∗ , P ] does not vanish. In particular, if p = |ξ|2 +V (x), where V is complex, the symbol of this commutator is 1 {¯ p, p} = 2{Re p, Im p}; i and when this is nonzero we are in the situation discussed in Theorem 12.6. (12.5.4)

This discussion leads us to THEOREM 12.8 (Constructing quasimodes). Suppose P = −h2 ∆ + V (x) and (12.5.5)

z0 = ξ02 + V (x0 ), Imhξ0 , ∂V (x0 )i = 6 0.

12.5. QUASIMODES, PSEUDOSPECTRA

269

(i) Then there exists a family of functions u(h) ∈ Cc∞ (Rn ) such that (12.5.6)

k(P − z0 )u(h)kL2 = O(h∞ ), ku(h)kL2 = 1.

(ii) Moreover, we can choose u(h) and ξ0 so that (12.5.7)

WFh (u(h)) = {(x0 , ξ0 )}, Imhξ0 , ∂V (x0 )i < 0.

INTERPRETATION. The point is that, although we can solve the quasimode equation (12.5.6) under the hypotheses (12.5.5), the point z0 may in fact be far from the spectrum of P . We informally say that z0 belongs to the pseudospectrum of P .  Proof. Our function u(h) will be constructed with support near x0 , and therefore with no loss we can assume that V is compactly supported. By changing the sign of ξ0 if necessary, but without changing z0 , we can also assume that {Re p, Im p}(x0 , ξ0 ) = 2Imhξ0 , ∂V (x0 )i < 0. Now according to Theorem 12.6, P − z0 is microlocally conjugate to A(hDx1 − ix1 ) near ((x0 , ξ0 ), (0, 0)). Let u0 (x, h) := exp(−|x|2 /2h); so that (hDx1 − ix1 )u0 (h) = 0, WFh (u0 (h)) = {(0, 0)}. Following the notation of Theorem 12.6, we define u(h) := T −1 u0 (h). Then WFh (u(h)) = {(x0 , ξ0 )} and (P − z0 )u(h) = T −1 A(hDx1 − ix1 )T (T −1 u0 ) = 0 microlocally.



EXAMPLE 1. If p(x, ξ) = |ξ|2 + V (x), the potential V satisfies (12.5.1) and {p(x, ξ) | (x, ξ) ∈ R2n } = 6 C, then the operator P has a discrete spectrum. This follows from the proof of Theorem 6.7, once we have found a point z at which P − z is elliptic. But this is so for any z not in the set of values of p(x, ξ). However, the hypotheses of Theorem 12.8 hold in a dense open subset of the interior of the closure of the range of p.  EXAMPLE 2. It is also clear that more general operators can be considered. As a simple one-dimensional example, take P = (hDx )2 + ihDx + x2 with p(x, ξ) = ξ 2 + iξ + x2 , {Re p, Im p} = −2x.

270

12. NORMAL FORMS

Then there exists a quasimode corresponding to any point in the interior of the range of p, namely {z | Re z ≥ (Im z)2 }. This set is thus contained in the pseudospectrum of P . But since 1 ex/2h P e−x/2h = (hD)2 + x2 + , 4 P has the discrete spectrum {1/4 + nh | n ∈ N}. 

12.6. NOTES See H¨ ormander [H2, Theorem 21.1.6] and [H2, Lemma 21.3.4] for proofs of the theorems cited in Section 12.2. Theorem 12.3 is a semiclassical analog of the standard result of Duistermaat-H¨ormander: [H2, Proposition 26.1.30 ]. Theorem 12.6 is a semiclassical adaptation of a microlocal result of Duistermaat-Sj¨ ostrand: [H2, Proposition 26.3.1]. We consider the case in which {p = 0} is symplectic, that is the symplectic form restricted to {p = 0} is non-degenerate. The opposite case is the involutive case in which {Rep, Imp}|p−1 (0) = 0. The normal form in that case is given by the Cauchy-Riemann operator: consult H¨ormander [H4, Section 26.2]. When n = 2 then the operator is elliptic, but in higher dimensions we have interesting propagation phenomena. Davies [Da] proved Theorem 12.8 in one dimension. See Embree–Trefethen [E-T] for background on quasimodes and pseudospectra and for further references, and see [D-S-Z] for more on semiclassical pseudospectra.

Part 4

SEMICLASSICAL ANALYSIS ON MANIFOLDS

Chapter 13

MANIFOLDS

13.1 13.2 13.3 13.4

Definitions, examples Pseudodifferential operators on manifolds Schr¨ odinger operators on manifolds Notes

13.1. DEFINITIONS, EXAMPLES 13.1.1. Manifolds. DEFINITION. Let M be a Hausdorff topological space with a countable basis. We say that M is a smooth n-dimensional manifold if there exists a family F of homeomorphisms between open sets: γ : Uγ → Vγ , Uγ ⊂ M, Vγ ⊂ Rn satisfying the following properties: (i) (Smooth overlaps) If γ1 , γ2 ∈ F, then γ2 ◦ γ1−1 ∈ C ∞ (γ1 (Uγ1 ∩ Uγ2 ); γ2 (Uγ1 ∩ Uγ2 )). (ii) (Covering) The open sets Uγ cover M : [ Uγ = M. γ∈F

We call {(γ, Uγ ) | γ ∈ F} an atlas for M . The open set Uγ ⊂ M is a coordinate patch. 273

274

13. MANIFOLDS

Assume next that M and N are smooth manifolds of dimensions n and m respectively. Let F be an atlas for M and G an atlas for N . DEFINITION. (i) A function F : M → N is said to be smooth if for every γ ∈ F and ρ ∈ G,


ρ◦ F |Uγ ◦ γ −1 |Uρ ∈ C ∞ (Vγ ; Vρ ), whenever F (Uγ ) ∩ Uρ 6= ∅. (ii) If F : M → N is invertible and both F and F −1 are smooth, F is called a diffeomorphism. 13.1.2. Vector bundles. DEFINITION. A C ∞ complex vector bundle over M with fiber dimension N consists of a C ∞ manifold V and a smooth map π : V → M, defining the fibers Vx := π −1 ({x}) for x ∈ M , each of which has the structure of complex N dimensional vector space. Furthermore, for each point x ∈ M there exists an open neighborhood U of x and a diffeomorphism ψ such that (13.1.1)

ψ : π −1 (U ) → U × CN

and for each y ∈ U , ψ maps Vy onto {y} × CN as a linear isomorphism: (13.1.2)

ψ(Vy ) = {y} × CN .

A real vector bundle is defined analogously. REMARKS: Transition matrices. (i) We can choose a covering {Xi }i∈I of M such that for each index i there exists ψi : π −1 (Xi ) → Xi × CN with the properties (13.1.1), (13.1.2) in the definition of a vector bundle. Then gij := ψi ◦ ψj−1 ∈ C ∞ (Xi ∩ Xj ; GL(N, C)), are the associated transition matrices, where GL(N, C) denotes the group of invertible, complex N × N matrices.

13.1. DEFINITIONS, EXAMPLES

275

(ii) We can recover the vector bundle V from the transition matrices. To see this, suppose that we are given functions gij satisfying the identities ( gij (x)gji (x) = I for x ∈ Xi ∩ Xj , gij (x)gjk (x)gki (x) = I for x ∈ Xi ∩ Xj ∩ Xk . Now define on the set W := I × M × CN , the equivalence relation (i, x, t) ∼ (i0 , x0 , t0 ) if and only if x = x0 and t0 = gi0 i (x)t. Then one can check that V = W/ ∼,



is vector bundle over M with the given transitions functions. DEFINITION. A section of the vector bundle V is a smooth map u:M →V such that π ◦ u(x) = x

(x ∈ M ).

We write u ∈ C ∞ (M, V ). EXAMPLE 1: Tangent bundle. Let M be an n-dimensional smooth manifold. We define the tangent bundle of M , denoted T M, as a real vector bundle with fibers given by Rn and with the transition functions (13.1.3)

gij (x) := ∂(γi ◦ γj−1 )(γj (x)) ∈ GL(n, R)

for x ∈ Uγi ∩ Uγj . Its sections C ∞ (M, T M ) are the smooth vector fields on M . The fiber (13.1.4)

Tx M := π −1 (x)

is the tangent space at the point x ∈ M . In local coordinates, sections of T M correspond to vector fields on Rn and hence act on smooth functions. Thus if V ∈ C ∞ (M, T M ), then (13.1.5)

V : C ∞ (M ) → C ∞ (M ). 

EXAMPLE 2: Cotangent bundle. For any vector bundle V we can define its dual [ V ∗ := (Vx )∗ , x∈X

with transition functions g˜ij = ((gij )−1 )T ,

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13. MANIFOLDS

where gij ’s are defined in (13.1.3). If V = T M , we obtain the cotangent bundle, denoted T ∗ M. Its sections C ∞ (M, T ∗ M ) are the differential one-forms on M . The fiber Tx∗ M := π −1 (x)

(13.1.6) is the cotangent space at x.

Canonical symplectic form on T ∗ M . In Chapter 2 we introduced the symplectic form on R2n . The cotangent bundle is likewise equipped with a canonical symplectic form: DEFINITIONS. (i) The canonical one-form η on T ∗ M is given by the rule ηρ (X) = ρ(π∗ X) for ρ ∈ T ∗ M and tangent vectors X ∈ Tρ (T ∗ M ). We note that ρ ∈ T ∗ M defines a linear form on Tπ(ρ) M . The informal but persuasive way to write the canonical one-form is (13.1.7)

η = ξdx.

(ii) The canonical symplectic form on T ∗ M is σ = dη. The form is obviously closed and a calculation in local coordinates shows that it is nondegenerate. We informally write (13.1.8)

σ = dξ ∧ dx.

NOTATION. If f : T ∗ M → R is Lebesgue measurable, we will write Z f dxdξ (13.1.9) T ∗M

to denote the integral of f over M with respect to the measure induced by symplectic form dxdξ = dx1 ∧ · · · ∧ dxn ∧ dξ1 ∧ · · · ∧ dξn =

σn . n!



EXAMPLE 3: s-density bundles. Let M be an n-dimensional manifold and let {(γ, Uγ ) | γ ∈ F} be an atlas. We define the s-density bundle over X, denoted Ωs (M ),

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277

by choosing the transition functions gij (x) := | det ∂(γj ◦ γi−1 )|s ◦ γi (x), for x ∈ Uγi ∩ Uγj . This is a line bundle over M , that is, a bundle with fibers of complex dimension one. As in Section 9.1 we can use an informal notation for sections of s-density bundles: u(x)|dx|s ∈ C ∞ (M, Ωs (M )).

(13.1.10)

REMARK. If u ∈ Ω1 (M ), th en for any open W ⊂⊂ M , we can invariantly define Z u W

the integral of u over W . To see this, let χ ∈ Cc∞ (M ) be supported in U = Uγ and let ψ : → U × C be as in (13.1.1), ψ(u(x)) = (x, ψ2 (x)). Then Z Z χu := χ(γ −1 (y))ψ2 (γ −1 (y)) dy.

π −1 (U )

Uγ

Vγ

Ω1 (M )

The definition of shows that this is independent of the choice R of γ ∈ F; and a partition of unity argument then gives the definition of W u.  EXAMPLE 4: pull backs. If F : M → N is a C ∞ mapping between two manifolds then we defined the pull back F ∗ : C ∞ (N, C) → C ∞ (M ; C)

u 7→ F ∗ u := u ◦ F,

u ∈ C ∞ (N ; C). If M and N are manifolds of the same dimension we also define the pull back for sections of the s-density bundles: F ∗ : C ∞ (N ; Ωs (N )) → C ∞ (M, Ωs (M )). In the informal notation of (13.1.10), when F (x) = y, x ∈ M , y ∈ N , F ∗ (u(y)|dy|s ) = u(F (x))| det ∂F (x)/∂x|s |dx|s . 13.1.3. Riemannian manifolds. DEFINITION. A n-dimensional Riemannian manifold (M, g) comprises a smooth, n-dimensional manifold M and a metric defined using a smooth inner product g on fibers of T M . This means that g ∈ C ∞ (M, T ∗ M ⊗ T ∗ M ) is symmetric and positive definite.

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13. MANIFOLDS

Recall that an element of V ∗ ⊗ V ∗ defines a bilinear form on V × V . A local construction and a partition of unity argument show that any manifold can be equipped with a Riemannian metric. We will use this observation in the case of compact manifolds. NOTATION. (i) If (M, g) is a Riemannian manifold, we put ((g ij )) := ((gij ))−1 ,

g¯ := det((gij )).

(ii) If f : M → R is measurable, we will write Z f dx (13.1.11) M

to denote the integral of f over M with respect to the density induced by the metric g. In local coordinates, we write √ dx = g¯|dx1 ∧ · · · ∧ dxn |. REMARKS. (i) An informal way of writing the metric g is g :=

n X

gij (x)dxi dxj ,

i,j=1

meaning that for X, Y ∈ Tx M , gx (X, Y ) =

n X

gij (x)Xi Yj .

i,j=1

(ii) We can use the metric to identify ξ ∈ Tx∗ M with X ∈ Tx M , written ξ ∼ X,

(13.1.12) provided

ξ(Y ) = gx (Y, X) for all Y ∈ Tx M . (iii) Under the identification (13.1.12) the flow of Hp on T ∗ M generated by the symbol n n X X 2 g ij (x)ξi ξj = gij (x)Xi Xj = gx (X, X) (13.1.13) p(x, ξ) := |ξ|gx = i,j=1

i,j=1

is the geodesic flow on T M . (iv) Half-densities on M can be identified with functions using the Riemannian density:  1 1 1 2 2 ˜(x) g¯ 2 dx .  u = u(x)|dx| = u

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13.2. PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS 13.2.1. Differential operators on manifolds. To define the action of vector fields, and differential operators, on L2 we need a notion of distributions on M : DEFINITION. We say that a linear map u : Cc∞ (M ) → C is a distribution on M if for every γ ∈ F and every χ ∈ Cc∞ (Vγ ), the mapping uγ,χ : ϕ 7→ u(γ ∗ (χϕ)) for ϕ ∈ Cc∞ (Rn ) is a distribution on Rn . We denote the space of distributions on M by D0 (M ). For V ∈ C ∞ (M, T M ), V : D0 (M ) → D0 (M ). See Friedlander–Joshi [F-J] or H¨ ormander [H1] for more. DEFINITION. A differential operator P of order at most m is a finite linear combination of compositions of at most m vector fields. Thus X P = Vj1 Vj2 · · · Vjk , where Vji ∈ C ∞ (M, T M ) and 1 ≤ k ≤ m. As in (13.1.5), we can interpret P as a linear mapping P : C ∞ (M ) → C ∞ (M ), P : Cc∞ (M ) → Cc∞ (M ),

(13.2.1)

P : D0 (M ) → D0 (M ). DEFINITIONS. (i) For a manifold M we can define L2 sections of the half-density bundle: 1

L2 (M, Ω 2 (M )) 1

as the completion of Cc∞ (M, Ω 2 (M )) with respect to the norm Z 1 2 2 kukL2 := |u| , M

where

|u|2

∈

Ω1 (M )

is a density on M and hence can be integrated.

(ii) For a Riemannian manifold (M, g) we define Z 2 L (M ) := {u : M → C | u is measurable, |u|2 dx < ∞.} M

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13. MANIFOLDS

(iii) If k is a nonnegative integer, we define k Hloc (M ) := {u ∈ L2loc (M ) | for all 1 ≤ l ≤ k, V1 · · · Vl u ∈ L2loc (M )

for all vectorfields V1 , . . . , Vl }; k (M ) is a Sobolev space. Hloc

(iv) We define as well negative order Sobolev spaces of distributions with compact support: Hc−k (M ) := span{u ∈ D 0 (M ) | u = V1 · · · Vl f, where f ∈ L2c (M ), 0 ≤ l ≤ k, and V1 , . . . , Vl ∈ Cc∞ (M, T M )}. with the convention that for l = 0, V1 · · · Vl u = u. (v) For U ⊂⊂ M , an open pre-compact subset of M , a Hilbert space norm on H k (U ) is defined by choosing a family of vector fields {Wj }1≤j≤J ¯ and putting that spans Tx M at every x ∈ U  1 2 k X X 2   (13.2.2) kukH k (U ) := kWα1 · · · Wαl ukL2 (U ) . l=0 |α|=l

We likewise define the semiclassical norms  1 2 k X X 2l 2   (13.2.3) kukH k (U ) := h kWα1 · · · Wαl ukL2 (U ) . h

l=0 |α|=l

(vi) For negative orders the space H −k (U ) is defined using the norm (13.2.4)   J J X  X kukH −k (U ) := inf kfj kL2 (U ) | u = Vlj · · · Vlpj fj , Vljk , 0 ≤ lpj ≤ k , 1   j=1

j=1

where Vlj are vector fields. The norm kukH −k (U ) is obtained by replacing h

Vlj ’s by hVlj ’s. REMARK. If M is not compact, the space L2 (M ) depends on the choice k (M ) and H −k (M ) for k ≥ 1 do not depend of the metric. The spaces Hloc c on the choice of metric, as on compact sets all densities are equivalent. In particular, if M is compact, the definitions of H −k (M ) are independent of the choice of the metric.  When P is a differential operator of order m, then for k ≥ 0 we have k+m k P : Hloc (M ) → Hloc (M ).

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13.2.2. Pseudodifferential operators on manifolds. Recall from Section 9.3 that a symbol class for which we have invariance under coordinate chages is S m = S m (Rn ) = {a ∈ C ∞ (R2n ) | |∂xα ∂ξβ a| ≤ Cαβ hξim−|β| for all α, β}. Remember that symbols in S m can depend upon h, in which case the constants Cαβ are uniform for 0 < h ≤ h0 . DEFINITION. A linear operator A : C ∞ (M ) → C ∞ (M ) is called a pseudodifferential operator on M if (i) there exists an integer m such that for each coordinate patch Uγ , there exists a symbol aγ ∈ S m such that (13.2.5)

−1 ∗ ϕA(ψu) = ϕγ ∗ aw γ (x, hD)(γ ) (ψu)

for any ϕ, ψ ∈ Cc∞ (Uγ ) and u ∈ C ∞ (M ); and (ii) for all ϕ1 , ϕ2 ∈ Cc∞ (M ) satisfying spt ϕ1 ∩ spt ϕ2 = ∅, we have (13.2.6)

kϕ1 Aϕ2 kH −N (U2 )→H N (U1 ) = O(h∞ )

for all N , where Uj ⊂⊂ M are open neighborhoods of spt ϕj , j = 1, 2. The norms on H N (U1 and H −N (U2 ) were defined in (13.2.2) and (13.2.4) respectively. Condition (13.2.6) is motivated by Theorem 9.7: away from the diagonal the operator A is regularizing and negligible in h. The regularizing effect means that ϕ1 Aϕ2 improves regularity by any order, in particular, maps distributions to smooth functions. NOTATION. For a pseudodifferential operator defined above we write A ∈ Ψm (M ) and sometimes call A a quantum observable. We will also use the notation \ Ψ(M ) := Ψ0 (M ), Ψ−∞ (M ) := Ψm (M ).  m∈Z

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13. MANIFOLDS

13.2.3. Symbols of pseudodifferential operators. We can also define symbols on T ∗ M : DEFINITION. We say a ∈ S m (T ∗ M ) if a ∈ C ∞ (T ∗ M ) and for any γ ∈ F, the pull-back of a under the identification Vγ × Rn 7→ T ∗ (Uγ ) belongs to S m (Vγ × Rn ). This definition does not depend on the choice of the atlas. Our goal next is to associate with a pseudodifferential operator A a symbol a defined on T ∗ M . THEOREM 13.1 (Symbols and quantizations). There exist linear maps (13.2.7)

σ : Ψm (M ) → S m (T ∗ M )/hS m−1 (T ∗ M )

and Op : S m (T ∗ M ) → Ψm (M )

(13.2.8) such that (13.2.9)

σ(A1 A2 ) = σ(A1 )σ(A2 )

and (13.2.10)

σ(Op(a)) = [a] ∈ S m (T ∗ M )/hS m−1 (T ∗ M ).

In (13.2.10) we use [a] to denote the equivalence class in S m /hS m−1 : [a] = [ˆ a]

if and only if a − a ˆ ∈ hS m−1 (T ∗ M ).

for a, a ˆ ∈ hS m (T ∗ M ). NOTATION. We call a = σ(A) the symbol of the pseudodifferential operator A. We will often later write (13.2.11)

Op(a) = aw (x, hD).

The symbol is uniquely defined on S m (T ∗ M ), up to a lower order term which is less singular and of order O(h) as h → 0. Proof. 1. Let U be an open subset of Rn . Suppose that B : Cc∞ (U ) → C ∞ (U ) and that for all ϕ, ψ ∈ Cc∞ (U ) the mapping u 7→ ϕBψu belongs to Ψm (Rn ).

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283

m (U × Rn ) the class of functions for which the esLet us denote by Sloc timates in the definition of S m are uniformly valid on compact subsets of U. m (U × Rn ) such that 2. We claim that there exists a symbol a ∈ Sloc

B = aw (x, hD) + B0 ,

(13.2.12) where for all N

B0 = OH −N (V )→H N (V ) (h∞ ).

(13.2.13)

for each open set V ⊂⊂ U . To see this, first choose a smooth, locally finite partition of unity {ψj }j∈J , X ψj (x) ≡ 1, x ∈ U, ψj ∈ Cc∞ (U ). j∈J

Then ψj Bψk = aw jk (x, hD), where ajk ∈ S m and ajk (x, ξ) = 0 if x ∈ / sptψj . Now put X0 m (U × Rn ), a := ajk (x, ξ) ∈ Sloc j,k

where we are sum over those indices j, k for which sptψj ∩ sptψk 6= ∅. We must verify (13.2.13) for X00 B0 := B − aw (x, hD) = ψj Bψk , j,k

the sum now over indices j, k for which sptψj ∩ sptψk = ∅. Since we assumed that ψj Bψk ∈ Ψm , Theorem 9.7 and the the local finiteness in the partition of unity give the global mapping property from H −N (V ) to H N (V ), with norm bounded by O(h∞ ). 3. For each coordinate chart (γ, Uγ ), we can now use (13.2.12) with B = (γ −1 )∗ Aγ ∗ , to define aγ ∈ T ∗ (Uγ ). The second part of Theorem 9.10 shows that if Uγ1 ∩ Uγ2 6= ∅, then (13.2.14)

(aγ1 − aγ2 )|Uγ1 ∩Uγ2 ∈ hS m−1 (T ∗ (Uγ1 ∩ Uγ2 )).

To define the symbol map we first choose a covering of M by coordinate charts {Uα }α∈J and a corresponding locally finite partition of unity {ϕα }α∈J , with sptϕα ⊂ Uγ . We then put X a := ϕα aα . α∈J

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13. MANIFOLDS

We see from (13.2.14) that a ∈ S m (T ∗ M ) is invariantly defined up to terms in hS m−1 (T ∗ M ). We consequently can define σ(A) := [a] ∈ S m (T ∗ M )/hS m−1 (T ∗ M ). 4. It remains to show the existence of Op : S m (T ∗ M ) → Ψm (M ), σ(Op(a)) = [a]. Suppose that for our covering {Uα }α∈J of M by coordinate charts, we choose {ψα }α∈J such that X sptψα ⊂ Uα , ψj2 (x) ≡ 1. α∈J

Define A :=

X

ψα γα∗ Op(˜ aα )(γα−1 )∗ ψα ,

α∈J

where a ˜α (x, ξ) := a(γα−1 (x), (∂γα (x)T )−1 ξ). Theorem 9.10 demonstrates that σ(A) equals [a].



REMARK. Theorem 13.1 is valid also when h = 1 in which case it is the classical result about a symbol of a pseudodifferential operator on a manifold. In that case the symbol is defined in S m (T ∗ M )/S m−1 (T ∗ M ) and the only gain is in regularity.  EXAMPLE: Dirichlet to Neumann map. Consider the boundary value problem −∆u = 0 in U ,

u|∂U = g,

where U ⊂ Rn has a smooth boundary. We define the Dirichlet to Neumann map N : g → ∂u ∂ν ∂U

where ν is the outward unit normal vector to ∂U. Then N is a pseudodifferential operator on ∂U and its symbol is p σ(N ) = σ( −∆∂U + 1), where ∆∂U is the Laplace-Beltrami operator on ∂U for the metric induced by the Euclidean metric: see Section 13.2.6. 

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13.2.4. Properties of pseudodifferential operators on manifolds. The mapping properties of operators in Ψm (M ) are easiest to establish when M is compact. Remember from Section 13.2.1 that the Sobolev spaces H k (M ) are independent of the choice of a Riemmanian metric. THEOREM 13.2 (Mapping properties for a compact manifold). Suppose that M is a compact manifold. Then (i) If A ∈ Ψ0 (M ), then A : L2 (M ) → L2 (M ) is bounded.. (ii) If A ∈ Ψm (M ) for m < 0, then A : L2 (M ) → L2 (M ) is a compact. Proof. Since M is compact we can find a finite partition of unity subordinate to a cover by elements of an atlas. Hence (i) and (ii) follow from Theorems 4.23 and 4.28.  REMARK. Sobolev spaces can be characterized using pseudodifferential operators: H m (M ) = {u ∈ D0 (M )| Au ∈ L2 (M ) for all A ∈ Ψm (M )}. This immediately provides a definition of H m (M ) for m ∈ R and shows that if A ∈ Ψm (M ), then A : H s (M ) → H s−m (M ) for each s ∈ R.  13.2.5. Pseudodifferential operators and half-densities. Pseudodifferential operators on manifolds also act on half-densities: DEFINITION. A linear operator 1

1

A : C ∞ (M ; Ω 2 (M )) → C ∞ (M ; Ω 2 (M )) is called a semiclassical pseudodifferential operator on half-densities if there exists an integer m such that for each coordinate patch Uγ there exists a symbol aγ ∈ S m such that for any ϕ, ψ ∈ Cc∞ (Uγ ) (13.2.15)

−1 ∗ ϕA(ψu) = ϕ γ ∗ aw γ (x, hD)(γ ) (ψu) 1

for each u ∈ C ∞ (M ; Ω 2 (M )). Here 1

1

(γ −1 )∗ : C ∞ (Uγ , Ω 2 (Uγ )) → C ∞ (Vγ , Ω 2 (Vγ )). with a similar statement for γ ∗ . We also demand that the analog of (13.2.6) holds.

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13. MANIFOLDS

NOTATION. In this case, we write 1

A ∈ Ψm (M ; Ω 2 (M )). By adapting the proof of Theorem 13.1 to the case of half-densities and using the first part of Theorem 9.3 we obtain THEOREM 13.3 (Symbol on half-densities). There exist linear maps (13.2.16)

σ : Ψm (M ; Ω1/2 (M )) → S m (T ∗ M )/h2 S m−2 (T ∗ M )

and (13.2.17)

Op : S m (T ∗ M ) → Ψm (M ; Ω1/2 (M )))

such that (13.2.18)

σ(A1 A2 ) = σ(A1 )σ(A2 )

and (13.2.19)

σ(Op(a)) = [a] ∈ S m (T ∗ M )/h2 S m−2 (T ∗ M ).

13.2.6. PDE on manifolds. In this subsection we will consider examples of linear PDE on manifolds. Here and in Section 13.3.4 we will revisit certain facts established earlier in simpler settings. The Laplace-Beltrami operator. The operator ∆g on a Riemannian manifold M is defined in local coordinates by   n ∂ 1 X ∂ ij √ g g¯ . (13.2.20) ∆g := √ g¯ ∂xj ∂xj i,j=1

One checks that (13.2.20) is independent of the choice of coordinates. The function p defined by (13.1.13) is the symbol of the Laplace-Beltrami operator −h2 ∆g . The Schr¨ odinger operator. Given a potential V ∈ C ∞ (M ), we define also (13.2.21)

P (h) := −h2 ∆g + V (x).

When M is compact then P (h) has properties similar to the Schr¨odinger operator with a confining potential on Rn : −h2 ∆ + V (x), V ∈ S(hxim ), V ≥ hxim /C − C. Section 13.3 is devoted to the study of (13.2.21) on a compact manifold. When V = 0 we can always rescale h to 1 so that results apply to the Laplace-Beltrami operator −∆g .

¨ 13.3. SCHRODINGER OPERATORS ON MANIFOLDS

287

The damped wave equation. We consider this initial-value problem for the wave equation: ( (∂t2 + a(x)∂t − ∆g )u = 0 on M × R (13.2.22) u = 0, ∂t u = f on M × {t = 0}, where a ≥ 0. As in Section 5.3 we define the energy of a solution at time t to be Z 1 (∂t u)2 + |∂x u|2 dx. E(t) := 2 M Using the pseudodifferential calculus developed in Section 13.2.3 it is straightforward to adapt the proofs in Section 5.3 to establish the following theorem. THEOREM 13.4 (Exponential decay on manifold). Suppose u solves the damped wave equation (13.2.22) with the initial condition f ∈ L2 (M ). Assume also that there exists a time T > 0 such that each geodesic of length greater than or equal to T intersects the set {a > 0}. Then there exist constants C, β > 0 such that E(t) ≤ Ce−βt kf kL2

(13.2.23) for all times t ≥ 0.

¨ 13.3. SCHRODINGER OPERATORS ON MANIFOLDS In this section we present basic theory of operators (13.2.21) on compact Riemannian manifolds. In particular, we will prove the analog of the Weyl’s law from Section 6.4, using as a different approach a functional calculus developed in Section 13.3.2. 13.3.1. Spectral theory. LEMMA 13.5 (Smoothness of eigenfunctions). Suppose that z ∈ C and that u ∈ L2 (M ) satisfies (P (h) − z)u = 0 in the sense of distributions. Then u ∈ C ∞ (M ). In particular the eigenfunctions of P (h) are smooth. Proof. 1. Choose χ ∈ Cc∞ (R) equal to 1 on for |t| ≤ T . Then for T sufficiently large, we have q(x, ξ) := (1 − χ(|ξ|g ))(|ξ|2g + V (x) − z)−1 ∈ S −2 (T ∗ M )

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13. MANIFOLDS

and p(x, ξ)q(x, ξ) = 1 + r1 (x, ξ),

r1 ∈ S −∞ (T ∗ M ).

2. Theorem 13.1 shows that Q0 (P (h) − z) = I − R1 , where Q0 := Op(q) and R1 ∈ Ψ−1 (M ). It follows that (13.3.1)

QN (P (h) − z) = I − RN +1 ,

for RN +1 ∈ Ψ−N −1 (M ) and QN :=

N X

R1k Q0 .

k=0

3. If u solves (P (h) − z)u = 0, then (13.3.1) gives u − RN +1 u = QN (P (h) − z)u = 0. For vectorfields Vj and for 1 ≤ j ≤ l ≤ N + 1, we then have hl V1 · · · Vl RN +1 ∈ Ψ−N −1+l (M ) ⊂ Ψ0 (M ). The definition of H N (M ) and Theorem 13.2 show that RN +1 u ∈ H N +1 (M ). Hence u ∈ H N +1 . This conclusion is valid for any N , and so u ∈ C ∞ (M ).  The next lemma provides a resolvent estimate which will also be useful in the section. LEMMA 13.6 (Resolvents on manifold). For z ∈ C \ R, the operator P (h) − z : Hh2 (M ) → L2 (M ) is invertible and (13.3.2)

k(P (h) − z)−1 kL2 (M )→H 2 (M ) = O(|Imz|−1 ). h

The semiclassical Sobolev spaces, Hhk (M ) are defined in (13.2.3) where, as M is compact, we can take U = M . Proof. 1. We first observe that if u ∈ Hh2 (M ) and (P (h) − z)u = 0 then u = 0. In fact, Lemma 13.5 shows that u ∈ C ∞ (M ) and the symmetry of P (h) gives (13.3.3)

0 = h(P (h) − z)u, ui = −Imzkuk2L2 .

Since Imz 6= 0 we obtain a contradiction. 2. To show injectivity we suppose that u ∈ L2 (M ) is orthogonal to the image of P (h) − z on C ∞ (M ) ⊂ Hh2 (M ). The symmetry of P (h) shows again that, in the sense of distributions, (P (h) − z¯)u = 0. As in part 1 of the proof we first see that u ∈ C ∞ and then obtain a contradiction.

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289

3. It remains to prove the estimate (13.3.2). The calculation in (13.3.3) shows that for u ∈ C ∞ (M ), k(P (h) − z)ukL2 ≥ |Imz|kukL2 , and all we need to show is (13.3.4)

kukH 2 (M ) ≤ Ck(P (h) − z)ukL2 + CkukL2 . h

For that we use (13.3.1) with N = 1, so that for any vector fields V1 and V2 : kh2 V1 V2 ukL2 = kh2 V1 V2 R2 ukL2 + kh2 V1 V2 Q1 (P (h) − z)ukL2 . Since h2 V1 V2 R2 , h2 V1 V2 Q1 ∈ Ψ0 (M ) the estimate (13.3.4) follows from Theorem 13.2 and definition (13.2.3).  REMARK. Since for a vector field V we have hV (P (h) − z)−1 = (P (h) − z)−1 hV + (P (h) − z)−1 [P (h), hV ](P (h) − z)−1 , if also follows that for k ≥ 0, (13.3.5)

(P (h) − z)−1 = OH k →H k+2 (|Imz|−1−k ). h

h

 The operator (13.2.21) has nice properties as an unbounded operator on L2 (M ): THEOREM 13.7 (Eigenvalues and eigenfunctions on manifolds). Let M be a compact Riemannian manifold and let V ∈ C ∞ (M ; R). (i) The operator P (h) := −h2 ∆g + V with the domain C ∞ (M ) is essentially self-adjoint; the domain of the closure is given by H 2 (M ). (ii) For h > 0 there exists an orthonormal basis, ∞ {uj (h)}∞ j=1 ⊂ C (M ),

of L2 (M ) comprised of eigenfunctions of P (h): (13.3.6)

P (h)uj (h) = Ej (h)uj (h)

(j = 1, . . . )

where Ej (h) → ∞ as j → ∞. REMARK. In the special case of V = 0 we can take h = 1 as the eigenvalues can be rescaled. We then see that the Laplace–Beltrami operator −∆g has eigenvalues 0 = λ0 < λ1 ≤ · · · λj → ∞ 2 and there existes an orthonormal basis {ϕj }∞ j=1 of L (M ) consisting of eigenfunctions: (13.3.7)

−∆g ϕj = λj ϕj

(j = 1, . . . ).

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13. MANIFOLDS

This is setting of Theorems 7.13,10.10, and of Theorem 14.6 below.



Proof. 1. The local coordinate definition of ∆g in (13.2.20) and the local formula for integration on M given in (13.1.11) show that ∆g is symmetric: for u, v ∈ Cc∞ (Uγ ), Z Z n   X g¯−1/2 ∆g u v¯ dx = ∂yj g ij g¯1/2 ∂yj (γ −1 )∗ u (γ −1 )∗ v¯ g¯1/2 dy Vγ

M

i,j=1 n X

Z =

  ∂yj g ij g¯1/2 ∂yj (γ −1 )∗ u (γ −1 )∗ v¯ dy

Vγ i,j=1 n X

Z =

u Vγ

  ∂yj g ij g¯1/2 ∂yj ((γ −1 )∗ v¯ dy

i,j=1

Z u ∆g v dx.

= M

Essential self-adjointness now follows from Lemma 13.6 and Theorem C.12. 2. For Imz 6= 0, (13.3.1) shows that (P (h) − z)−1 = Q1 + R2 (P (h) − z)−1 . Since Q1 , R2 ∈ Ψ−2 (M ), part (ii) of Theorem 13.2 implies that (P (h) − z)−1 is a compact operator. According to Theorem C.7, the spectrum of (P (h) + i)−1 is discrete, with an accumulation point at 0. An application of the Spectral Theorem C.10 gives the orthonormal basis of eigenvectors corresponding to Ej (h) → ∞. By Lemma 13.5 the eigenvectors are smooth.  13.3.2. A functional calculus. Using the basis of eigenfunctions given in (13.3.6), we can write (13.3.8)

P (h) =

∞ X

Ej (h) uj (h) ⊗ uj (h),

j=1

where u ⊗ v(ϕ)(x) := u

R M

v ϕ dx.

DEFINITION. For each f ∈ L∞ (R), we define the bounded operator f (P (h)) : L2 (M ) → L2 (M ) by (13.3.9)

f (P (h)) :=

∞ X j=1

f (Ej (h))uj (h) ⊗ uj (h).

¨ 13.3. SCHRODINGER OPERATORS ON MANIFOLDS

291

REMARK. This definition is a special case of functional calculus for unbounded self-adjoint operators, see [D-S, Chapter 4]. We need to rewrite (13.3.9) into more useful form, and for this we assume now f ∈ S (R) and recall the almost analytic extension fe discussed in Theorem 3.6. THEOREM 13.8 (Helffer-Sj¨ ostrand formula). We have Z 1 (13.3.10) f (P (h)) = ∂¯z fe(z)(P (h) − z)−1 dm, πi C where m denotes Lebesgue measure on C. We call (13.3.10) the Helffer-Sj¨ ostrand formula and stress that it is valid in much greater generality. Proof. Let B(t, ) denote the disk in C with center t and radius . Using Green’s formul we calculate for t ∈ R that Z Z 1 1 −1 ¯ e ∂z f (z)(t − z) dm = lim ∂¯z fe(z)(t − z)−1 dm πi C πi →0 C\B(t,) Z   1 lim ∂¯z fe(z)(t − z)−1 dm = πi →0 C\B(t,) I 1 = lim fe(z)(t − z)−1 dz 2πi →0 ∂B(t,) I 1 = lim (f (t) + O())(t − z)−1 dz 2πi →0 ∂B(t,) = f (t). We now put t = Ej (h) and apply (13.3.9) to obtain (13.3.10).



We next use the Helffer-Sj¨ostrand formula (13.3.10) to learn that f (P (h)) is a pseudodifferential operator: THEOREM 13.9 (Symbols and functional calculus). If f ∈ S (R), then (13.3.11)

f (P (h)) ∈ Ψ−∞ (M )

and has the symbol (13.3.12)

σ(f (P (h))) = f (|ξ|2g + V (x)).

Proof. 1. Write P = P (h). We first demonstrate that (13.3.13)

f (P ) ∈ Ψ(M ).

292

13. MANIFOLDS

We first check part (ii) of the definition in Section 13.2.2. Let ϕ, ψ ∈ C ∞ (M ) satisfy spt ϕ ∩ spt ψ = ∅. We need to show that ϕf (P )ψ = OH −N →H N (hN ),

(13.3.14) for any N . Let

Q0 (z) := (|ξ|2g + V (x) − z)−1


w

Imz 6= 0,

,

Ψ−2 (M )

so that Q0 (z) ∈ but with symbolic estimates blowing up polynomially as |Imz| → 0. Pseudodifferential calculus shows that (P − z)Q0 (z) = I − R0 (z),

(13.3.15)

where R1 (z) ∈ hΨ−1 (M ), and R1 (z) = OH −N →H −N +1 (h|Imz|−K ), h

h

for some K. Iteration of (13.3.15) as in part 2 of the proof of Lemma 13.5 shows that for any M we have (P − z)QM (z) = I − RM +1 (z),

(13.3.16) where

RM +1 (z) = OH −N →H −N +M +1 (hM |Imz|−KM ), h

h

Ψ−2 (M ),

and QM (z) ∈ with symbolic estimates blowing up polynomially as Imz → 0. The support condition on ϕ and ψ and the pseudodifferential calculus show that ϕ QM (z) ψ = OH −N →H N (hN |Imz|−LN ). From (13.3.16) we see that (P − z)−1 = QM (z) − (P − z)−1 RM +1 (z). Combing the above with the estimate (13.3.5) shows that for M large enough, ϕ (P − z)−1 ψ 0

= OH −N →H N (hN |Imz|−LN ) + OH −N →H M −N (hM |Imz|−KM ) h

N

−MN

= OH −N →H N (h |Imz|

h

),

for some MN (we replaced the semiclassical Sobolev spaces with the usual ones using the fact that M is large enough). Since the almost analytic extension satisfies ∂¯z fe = O(|Imz|∞ hzi−∞ ) and spt fe ⊂ {|Imz| ≤ 1}, (13.3.10) shows that (13.3.14) holds. 2. Once we have part (ii) of the definition of Ψ(M ) we need to check that for ϕ ∈ Cc∞ (M ) with spt ϕ ⊂ Uγ , (13.3.17)

(γ −1 )∗ ϕf (P )ϕγ ∗ ∈ Ψ(Rn ).

¨ 13.3. SCHRODINGER OPERATORS ON MANIFOLDS

293

A partition of unity argument and (13.3.14) then show that part (i) of the definition of Ψ(M ) holds. In view of Theorem 9.11, to prove (13.3.17) it suffices to check that for any N and any choices of linear symbols lj we have (13.3.18)

kadl1 (x,hD) · · · adlN (x,hD) (γ −1 )∗ ϕf (P )ϕγ ∗ kL2 →L2 = O(hN ).

Note that for a linear function l on R2n , we have adl(x,hD) ((γ −1 )∗ ϕ(P − z)−1 ϕγ ∗ ) = (γ −1 )∗ adL ϕ(P − z)−1 ϕ





γ∗,

where L := γ ∗ ϕ˜ l(x, hD) ϕ˜ (γ −1 )∗ ∈ Ψ1 (M ) for a function ϕ e ∈ C ∞ (Rn ) such that spt ϕ˜ ⊂ γ(Uγ ), γ ∗ ϕ| ˜ spt ϕ = 1. Using the derivation property (8.1.9) of adL and Lemma 13.6, we deduce adL (P − z)−1 = −(P − z)−1 (adL P )(P − z)−1 = OL2 →L2 (h|Imz|−2 ). Therefore kadl(x,hD) (γ −1 )∗ ϕf (P )ϕγ ∗ kL2 →L2 Z ≤ |∂¯z fe(z)|kadl(x,hD) (γ −1 )∗ ϕ(P − z)−1 ϕγ ∗ kL2 →L2 dm C Z

≤ |∂¯z fe(z)|k(γ −1 )∗ adL ϕ(P − z)−1 ϕ γ ∗ kL2 →L2 dm C Z ≤ Ch |∂¯z fe(z)||Imz|−2 dm = O(h), C

where we used again the properties of fe. This proves (13.3.18) for N = 1, the case N > 1 following similarly. We conclude that (13.3.17) holds, concluding the proof of (13.3.13). 3. Since (P + i)k f (P ) = gk (P ) for gk := (t+i)k f (t) ∈ S (R), we see using (13.3.5) that f (P (h)) : L2 (M ) → HhN (M ) for any N ; and hence f (P (h)) ∈ Ψ−∞ (M ). 4. To compute the symbol of f (P ) we recall that (13.3.15) gives
w (|ξ|2g + V (x) − z)−1 (P − z) = I + OL2 →L2 (h|Imz|−K ) for some K. Hence it follows that
w (|ξ|2g + V (x) − z)−1 = (P − z)−1 + OL2 →L2 (h|Imz|−K−1 ).

294

13. MANIFOLDS

The Helffer-Sj¨ ostrand formula (13.3.10) now shows that Z
w 1 f (P ) = |ξ|2g + V (x) − z)−1 ∂¯z fe(z) dm + OL2 →L2 (h) πi C w  Z 1 (|ξ|2g + V (x) − z)−1 ∂¯z fe(z) dm = + OL2 →L2 (h) πi C = (f (|ξ|2g + V (x)))w + OL2 →L2 (h). As we already know f (P ) ∈ Ψ−∞ (M ), this calculation implies that principal symbol of f (P ) is f (|ξ|2g + V (x)).  13.3.3. Trace class operators. Trace class properties of f (P ) will be useful in the next section and in Section 14.3. See Section C.3 for the basic definitions and facts. THEOREM 13.10 (Trace class properties of f (P (h))). Assume V ∈ C ∞ (M, R) and that P (h) = −h2 ∆g + V (x) is a Schr¨ odinger operator on a compact Riemannian manifold. Then if f ∈ S (R), we have (13.3.19)

f (P (h)) ∈ L1 (L2 (M )).

Proof. This is a statement for a fixed h. Since Theorem 13.9 gave f (P ) ∈ Ψ−∞ (M ), f (P ) is a finite sum of operators of the form ϕγ1∗ aw (x, D)(γ2−1 )∗ ψ, with ϕ, ψ ∈ C ∞ (M ) and a ∈ S (R2n ). According to (C.3.4) each of these operators is in L1 and hence so is f (P (h)).  13.3.4. Weyl’s Law for compact manifolds. The symbolic calculus we developed in Section 13.3.2 gives us a quick proof of the Weyl Law for Schr¨ odinger operators. THEOREM 13.11 (Weyl’s asymptotics on compact manifolds). (i) For any a < b, we have #{E(h) | a ≤ E(h) ≤ b} = (13.3.20) as h → 0.

1 (VolT ∗ M {a ≤ |ξ|2g + V (x) ≤ b} + o(1)) (2πh)n

13.4. NOTES

295

(ii) If V ≡ 0 and 0 = λ0 < λ1 ≤ λ2 ≤ · · · ≤ λj → ∞ is the complete set of eigenvalues of the Laplace–Beltrami operator −∆g on M , then #{j | λj ≤ r} ∼

(13.3.21)

α(n) n/2 r Vol(M)

as r → ∞, where α(n) is the volume of the unit ball in Rn . Proof. 1. Let f1 , f2 be two functions satisfying the assumptions of Theorem 13.9, such that for real x f1 (x) ≤ 1[a,b] (x) ≤ f2 (x),

(13.3.22)

where 1[a,b] (x) is the characteristic function of the interval [a, b]. It follows that trf1 (P ) ≤ #{E(h) | a ≤ E(h) ≤ b} ≤ trf2 (P ). The operators are of trace class by Theorem 13.10. 2. Theorem C.18 now shows that for j = 1, 2 Z  1 2 trfj (P ) = fj (|ξ|g + V (x)) dxdξ + O(h) . (2πh)n T ∗M We note that since fj (P ) ∈ h∞ Ψ(M ), the errors in the symbolic computations are all O(hhξi−∞ ), and hence can be integrated. 3. The next step is to construct f1 and f2 satisfying the hypotheses of Theorem 13.9 and (13.3.22), and such that for j = 1, 2, we have Z fj (|ξ|2g + V (x)) dxdξ → VolT ∗ M {a ≤ |ξ|2g + V (x) ≤ b}, T ∗M

as  → 0. That is easily done by regularizing the characteristic function of a ≤ E ≤ b. √ 4. To prove (13.3.21), we take a = 0, b = 1, h = 1/ r, and apply Theorem 13.11. The eigenvalues −∆g are the rescaled eigenvalues of −h2 ∆g and the α(n) term comes from integrating out the ξ variables. 

13.4. NOTES Pseudodifferential calculus on manifolds for h = 1 is presented in H¨ormander [H3, Section 18.2] and we adapt that presentation to the semiclassical case. The notes [G-St1] of Guillemin–Sternberg provide a more detailed and functorial exposition of invariance issues. For more about Sobolev spaces and mapping properties of pseudodifferential operators, we refer to H¨ormander [H3, Chapter 18]. For these results h can be considered as a fixed parameter.

296

13. MANIFOLDS

The Helffer-Sj¨ ostrand formula is also called the Dynkin-Droste-HelfferSj¨ ostrand formula. See Dimassi–Sj¨ostrand [D-S] for references. A proof of the spectral theorem based on that formula is provided in Davies [Da1]. Much finer asymptotic formulas for the eigenvalue counting function are known: see Dimassi–Sj¨ ostrand [D-S], Ivrii [I], and Safarov–Vassiliev [Sa-V].

Chapter 14

QUANTUM ERGODICITY

14.1 14.2 14.3 14.4 14.5

Classical ergodicity A weak Egorov Theorem Weyl’s Law generalized Quantum ergodic theorems Notes

In this chapter we are given a smooth real-valued potential V on a compact Riemannian manifold (M, g). We consider the classical Hamiltonian (14.0.1)

p(x, ξ) = |ξ|2g + V (x)

for (x, ξ) ∈ T ∗ M , the Hamiltonian flow of which is denoted ϕt = exp(tHp )

(t ∈ R).

When V ≡ 0 and T ∗ M is identified with T M using (13.1.12), ϕt gives the geodesic flow on T M . We devote this chapter to proving quantum ergodicity, meaning that ergodicity for the classical evolution {ϕt }t∈R implies a type of equidistribution of eigenfunctions of the associated quantum operator (14.0.2)

P (h) = −h2 ∆g + V.

297

298

14. QUANTUM ERGODICITY

14.1. CLASSICAL ERGODICITY We hereafter select a < b, and assume that (14.1.1)

|∂p| ≥ γ > 0 on {a ≤ p ≤ b}.

Then, according to the Implicit Function Theorem, for each a ≤ c ≤ b, the set Σc := p−1 (c) is a smooth, 2n − 1 dimensional hypersurface in the cotangent space T ∗ M . We can interpret Σc as an energy surface. DEFINITION. For each c ∈ [a, b], we denote by µc Liouville measure on the hypersurface Σc = p−1 (c) corresponding to p. This measure is characterized by the formula ZZ Z bZ (14.1.2) f dxdξ = f dµc dc p−1 [a,b]

a

Σc

for all a < b and each continuous function f : T ∗ M → R. Theorem 2.11 shows that the measure µc is invariant under the flow ϕt . EXAMPLE. If M = Tn := Rn /Z2 is the n-dimensional torus and T ∗ M = Tn × Rn , we have 1 µc = H 2n−1 Σc , |∂p| H 2n−1 denoting 2n − 1 dimensional Hausdorff measure. Then (14.1.2) follows from the Coarea Formula: see [E-G].  DEFINITIONS. (i) Let z = (x, ξ) ∈ Σc and f : T ∗ M → C. For T > 0 we define the time average Z ZT 1 T (14.1.3) hf iT := f (ϕt (z)) dt = − f (ϕt (z)) dt, T 0 0 the slash through the second integral denoting an average. Note carefully that hf iT depends upon the starting point z ∈ T ∗ M . (ii) We say the flow ϕt is ergodic on p−1 [a, b] if for each c ∈ [a, b], ( if E ⊂ Σc is flow invariant, then (14.1.4) either µc (E) = 0 or else µc (E) = µc (Σc ). In other words, we are requiring that each flow invariant subset of the energy level Σc have either zero measure or full measure.

14.1. CLASSICAL ERGODICITY

299

THEOREM 14.1 (Mean Ergodic Theorem). Suppose the flow is ergodic on Σc := p−1 (c). Then for each f ∈ L2 (Σc , µc ) we have 2 Z  Z hf iT − f dµc dµc = 0. (14.1.5) lim T →∞ Σc

Σc

REMARK. According to Birkhoff’s Ergodic Theorem, in fact Z (14.1.6) hf iT → f dµc as T → ∞, Σc

for µc –a.e. point m belonging to Σc , But we will only need the weaker statement of Theorem 8.1.  Proof. 1. Define A := {f ∈ L2 (Σc , µc ) | ϕ∗t f = f for all times t}, ¯0 ⊂ L2 (Σc , µc ). B0 := {Hp g | g ∈ C ∞ (Σc )}, B := B We claim that B0⊥ = A,

(14.1.7)

where ⊥ denotes the L2 -orthogonal complement. To see this, first let h ∈ A and f = Hp g ∈ B0 . Then, using invariance of µc under the flow, we see that Z Z Z ¯ hf dµc = hHp g dµc = ∂t hϕ∗t g dµc |t=0 Σc Σc Σc Z Z ∗ hg dµc |t=0 = 0, = ∂t ϕ−t hg dµc |t=0 = ∂t Σc

Σc

and thus h ∈

B0⊥ .

Conversely, suppose h ∈ B0⊥ . Then for any g ∈ C ∞ , we have Z Z Z ∗ ∗ 0= h Hp ϕ−t g dµc = ∂t h ϕ−t g dµc = ∂t ϕ∗t h g dµc . Σc

Σc

Σc

C ∞,

Therefore for all times t and all functions g ∈ Z Z ∗ ϕt h g dµc = h g dµc . Σc

Hence

ϕ∗t h

Σc

≡ h, and so h ∈ A.

2. It follows from (14.1.7) that we have the orthogonal decomposition L2 (Σc , µc ) = A ⊕ B. Thus if we write f = fA + fB , for fA ∈ A, fB ∈ B, then hfA iT ≡ fA

300

14. QUANTUM ERGODICITY

for all T . For g ∈ C ∞ we compute 2 Z Z Z T 1 2 ∗ |hHp giT | dµc = 2 (d/dt)ϕt gdt dµc T Σc 0 Σc Z 1 = 2 |ϕ∗ g − g|2 dµc T Σc T Z 4 |g|2 dµc → 0, ≤ 2 T Σc as T → ∞. Hence for fB ∈ B := B 0 , we have hfB iT → 0 in L2 (Σc , dµc ). It follows that hf iT → fA in L2 (Σc , dµc ). 3. The ergodicity hypothesis is equivalent to saying that A consists of constant functions. Indeed, for any h ∈ A, the set h−1 ([α, ∞)) is invariant under the flow, and hence has either full measure or measure zero. Since the functions in L2 (Σc , dµc ) are defined up to sets of measure zero, h is equivalent to a constant function. Lastly, observe that the orthogonal projection f 7→ fA is just the space average with respect to µc . This proves (14.1.5). 

14.2. A WEAK EGOROV THEOREM We next estimate the difference between the classical and quantum evolutions governed by our symbol p(x, ξ) = |ξ|2g + V (x). For this we present a simple weak variant of Egorov’s Theorem 11.12 in the setting of compact manifolds. NOTATION. (i) We write (14.2.1)

F (t) = e−itP (h)/h

(t ∈ R)

for the unitary group on L2 (M ) generated by the self-adjoint operator P (h). See Theorems 13.7 and C.13. (ii) If A is a pseudodifferential operator in Ψ(M ), we define its quantum evolution as (14.2.2)

A(t) := F −1 (t)AF (t)

(t ∈ R).

This accords with the Heisenberg picture of quantum mechanics.

14.2. A WEAK EGOROV THEOREM

301

THEOREM 14.2 (Weak form of Egorov’s Theorem). Fix a time T > 0 and define ˜ := Op(at ) A(t)

(14.2.3)

(0 ≤ t ≤ T ),

where (14.2.4)

at (x, ξ) := a(ϕt (x, ξ)).

Then (14.2.5)

˜ kA(t) − A(t)k L2 →L2 = O(h)

uniformly for 0 ≤ t ≤ T.

We are using the notation of Theorem 13.1. Proof. We have ∂t at = {p, at }, where {p, at } = Hp at is the Poisson bracket on T ∗ M . If σ denotes the symbol of a pseudodifferential operator as defined in Section 13.2.3,   i σ [P (h), B] = {p, σ(B)}. h This is checked in local coordinates using Theorem 9.5 and then follows from the invariance of both sides. Theorem 13.1 shows that ˜ = i [P (h), A(t)] ˜ (14.2.6) ∂t A(t) + E(t), E(t) ∈ hΨ(M ), h and by Theorem 13.2, kE(t)kL2 →L2 = O(h). Hence   itP (h) itP (h) ˜ h ∂t e− h A(t)e   itP (h) itP (h) i − h ˜ ˜ ∂t A(t) − [P (h), A(t)] e h =e h   itP (h) itP (h) i i − h ˜ ˜ =e [P (h), A(t)] + E(t) − [P (h), A(t)] e h h h = e−

itP (h) h

E(t)e

itP (h) h

= OL2 →L2 (h).

Integrating, we deduce ke−

itP (h) h

˜ A(t)e

itP (h) h

− AkL2 →L2 = O(h);

and so ˜ − A(t)kL2 →L2 = kA(t) ˜ −e kA(t) uniformly for 0 ≤ t ≤ T .

itP (h) h

Ae−

itP (h) h

kL2 →L2 = O(h), 

302

14. QUANTUM ERGODICITY

14.3. WEYL’S LAW GENERALIZED NOTATION. We hereafter turn our attention to the eigenvalue problems (14.3.1)

P (h)uj (h) = Ej (h)uj (h)

(j = 1, . . . ).

To simplify notation, we write uj = uj (h) and Ej = Ej (h). We assume as well the normalization (14.3.2)

kuj kL2 (M ) = 1

(j = 1, . . . ).

We have (14.3.3)

e−itP (h)/h uj (h) = e−itEj /h uj (h)

(t ∈ R)

for j = 1, . . . . The following result generalizes Theorem 13.11, showing that we can localize the asymptotics using a quantum observable. THEOREM 14.3 (Weyl’s Theorem generalized). Let B ∈ Ψ(M ). Then ZZ X n (14.3.4) (2πh) hBuj , uj i → σ(B) dxdξ a≤Ej ≤b

{a≤p≤b}

as h → 0. REMARK. If B = I, and so σ(B) ≡ 1, (14.3.4) reads (2πh)n #{a ≤ Ej ≤ b} → Vol({a ≤ p ≤ b}). This is the usual form of Weyl’s Law, Theorem 13.11.



Proof. The proof uses the functional calculus developed in Section 13.3.2. 1. As shown in the proof of Theorem 13.10 A ∈ Ψ−∞ (M ) is of trace class. For a ∈ S (R2n ), the formula for the Schwartz kernel (4.5.1) and Theorem C.18 show that Z Z 1 w tr a (x, hD) = a(x, ξ) dx dξ. (2πh)n Rn Rn This and the invariance of the principal symbol of A give Z Z  1 (14.3.5) tr A = σ(A) dxdξ + O(h) . (2πh)n T ∗M 2. Fix a small number  > 0 and construct f , g ∈ Cc∞ (R) satisfying 0 ≤ f , g ≤ 1, and spt f ⊂ [a + , b − ], spt g ⊂ [a − 2, b + 2],

f ≡ 1 on [a + 2, b − 2], g ≡ 1 on [a − , b + ].

14.3. WEYL’S LAW GENERALIZED

303

Theorem 13.9 then shows that f (P ) ∈ Ψ−∞ (M ) and that (14.3.6)

σ(f (P ))(x, ξ) = f (p(x, ξ))).

Similar statements hold for g . Define Π := projection onto the span of {uj | a ≤ Ej ≤ b}, so that (14.3.7)

f (P )Π = f (P ),

g (P )Π = Π.

3. We now write X hBuj , uj i = tr(ΠBΠ) a≤Ej ≤b (14.3.8) = tr(f (P )B) + tr(Πg (P )(1 − f (P ))BΠ). The operator g (P )(1 − f (P )) is self adjoint and, as 0 ≤ g , f ≤ 1, ∞ X

sj (g (P )(1 − f (P ))) =

j=1

∞ X

g (Ej (h))(1 − f (Ej (h))).

j=1

The definition of the trace class norm (C.3.3) and Theorem 13.11 imply that kg (P )(1 − f (P ))kL1 ≤ #{a − 2 ≤ Ej (h) ≤ a + 2} + #{b − 2 ≤ Ej (h) ≤ b + 2} ≤ C(α() + β(h, ))h−n , where lim β(h, ) = 0,

h→0

and α() := Vol ({a − 2 ≤ p ≤ a + 2} ∪ {b − 2 ≤ p ≤ b + 2}) satisfies lim α() = 0.

→0

We use (14.3.8) and (14.3.5) with A = f (P ) to deduce (2πh)n tr(ΠBΠ) = tr(f (P )B) + O (hn kBkkg (P )(1 − f (P ))kL1 ) ZZ = f (p)σ(B) dxdξ + O (h) + O(α() + β(h, )). T ∗M

Here O (h) indicates that the bound depends on , since the error term in (14.3.5) depends on A.

304

14. QUANTUM ERGODICITY

4. We conclude that X lim sup(2πh)n hBuj , uj i = lim sup(2πh)n tr(ΠBΠ) h→0

h→0

a≤Ej ≤b

ZZ = T ∗M

f (p)σ(B) dxdξ + O(α()),

and similarly for lim inf h→0 . Since the left hand side does not depend on  and since ZZ ZZ σ(B) dxdξ, f (p)σ(B) dxdξ + O(α()) = lim →0

T ∗M

{a≤p≤b}

the Theorem follows.



14.4. QUANTUM ERGODIC THEOREMS Assume now that A ∈ Ψ(M ) has the symbol σ(A), satisfying the condition that Z (14.4.1) α := − σ(A) dµc is the same for all c ∈ [a, b], Σc

the slash through the integral denoting the average. In other words, we are requiring that the averages of the symbol of A over each level surface p−1 (c) are equal. REMARK: Symbol averages. The condition (14.4.1) on averages is easy to obtain. Since we assume that |∂p| > γ on p−1 ([a, b]), we also have |∂p| > γ/2 on for some small δ > 0. Any point (x, ξ) ∈ p−1 ([a − δ, b + δ]) has a neighborhood U ⊂ T ∗ M such that p−1 ([a − δ, b + δ])

(p, ρ) = (p(x, ξ), ρ(x, ξ)) are local coordinates in U . If b ∈ Cc∞ (U ), then in these coordiates we define Z T b(c, ρ) := b(c, ρ) − − b dµc . p−1 (c)

Then (14.4.2)

Z − p−1 (c)

T b dµc = 0

for c ∈ [a, b]. The set p−1 ([a−δ, b+δ]) is compact and we can use a partition of unity to construct an operator T : Cc∞ (T ∗ M ) → Cc∞ (T ∗ M ) such that T2 = T and (14.4.2) holds for any b. This means that T is a projection onto the space of functions satisfying (14.4.1) with α = 0.

14.4. QUANTUM ERGODIC THEOREMS

305

To obtain a large class of operators A satisfying (14.4.1) with an arbtrary α take any B ∈ Ψ(M ), χ ∈ S(T ∗ M ) equal to 1 near p−1 ([a, b]), and define A := Op( T σ(B)) + αOp(χ).



We now establish our first connection between the classical ergodicity of the flow ϕt and the distribution of the eigenfunctions satisfying (14.3.1): THEOREM 14.4 (Quantum ergodicity 1). Assume the ergodic condition (14.1.4) and that A ∈ Ψ(M ) satisfies the condition (14.4.1). Then (14.4.3)

(2πh)n

2 Z X σ(A) dxdξ → 0 hAuj , uj i − − {a≤p≤b}

a≤Ej ≤b

as h → 0. Proof. 1. Write B := A − αI,

(14.4.4)

α defined by (14.4.1). In view of our hypothesis (14.4.1) Z (14.4.5) σ(B) dµc = 0 for each c ∈ [a, b]. Σc

Define X

(h) := (2πh)n

hBuj , uj i2 ;

a≤Ej ≤b

we must show (h) → 0. Now hBuj , uj i = hBe−

itEj h

uj , e−

itEj h

uj i = hBe−

itP (h) h

uj , e−

itP (h) h

uj i

according to (14.3.3). Consequently (14.4.6)

hBuj , uj i = he

itP (h) h

Be−

itP (h) h

uj , uj i = hB(t)uj , uj i

in the notation of (14.2.2). This identity is valid for each time t ∈ R. We can therefore average: ZT (14.4.7) hBuj , uj i = h − B(t) dt uj , uj i = hhBiT uj , uj i, 0

for

Z ZT 1 T hBiT := B(t) dt = − B(t) dt. T 0 0 2 Now since kuj k = 1, (14.4.7) implies hBuj , uj i2 = hhBiT uj , uj i2 ≤ khBiT uj k2 = hhB ∗ iT hBiT uj , uj i.

306

14. QUANTUM ERGODICITY

Hence X

(h) ≤ (2πh)n

(14.4.8)

hhB ∗ iT hBiT uj , uj i

a≤Ej ≤b

2. Theorem 14.2 tells us that ZT ˜ T + OL2 →L2 (h), hBi ˜ T := − B(t) ˜ dt, hBiT = hBi 0

˜ ∈ Ψ(M ), σ(B(t)) ˜ where B(t) = ϕ∗t σ(B), and ZT ˜ σ(hBiT ) = − σ(B) ◦ ϕt dt = hσ(B)iT . 0

We note that the error OL2 →L2 (h) depends on T . This means that modulo OT (h) errors we can replace eitP (h)/h Be−itP (h)/h ˜ by B(t). Theorem 14.3 and (14.4.8) show that (14.4.9) 



lim sup (h) ≤ lim sup (2πh)n h→0

h→0

X

˜ ∗ iT hBi ˜ T uj , uj i + OT (h) hhB

a≤Ej ≤b

ZZ

˜ ∗ iT hBi ˜ T ) dxdξ σ(h(B

= {a≤p≤b}

ZZ

|σ(hBiT ))|2 dxdξ,

= {a≤p≤b}

since the symbol map is multiplicative and the symbol of an adjoint is given by the complex conjugate. 3. We can now apply Theorem 14.1 with f = σ(B), to conclude that Z |hσ(B)iT |2 dxdξ → 0, p−1 [a,b]

as T → ∞. Since the left hand side of (14.4.9) is independent of T , this calculation shows that the limit must in fact be zero.  The pointwise limit along a sequence of density one is obtained using a density argument: THEOREM 14.5 (Quantum ergodicity 2). Assume the ergodic condition (14.1.4). Then there exists a family of subsets Λ(h) ⊂ {a ≤ Ej ≤ b} such that (14.4.10)

#Λ(h) = 1; h→0 #{a ≤ Ej ≤ b} lim

14.4. QUANTUM ERGODIC THEOREMS

and for each A ∈ Ψ(M ) satisfying (14.4.1), we have Z (14.4.11) hAuj , uj i → − σ(A) dxdξ

307

as h → 0

{a≤p≤b}

for Ej ∈ Λ(h).

Proof. 1. We first show that assertion (14.4.3) implies (14.4.10) with Λ(h) depending on A. For this, let B be given by (14.4.4) so that again Z σ(B) dxdξ = 0. {a≤p≤b}

According to (14.4.3), (2πh)n

X

hBuj , uj i2 =: (h) → 0.

a≤Ej ≤b

Define Γ(h) := {a ≤ Ej ≤ b | hBuj , uj i2 ≥ (h)1/2 }; so that (2πh)n #Γ(h) ≤ (h)1/2 . Next, write Λ(h) := {a ≤ Ej ≤ b} \ Γ(h). Then if Ej ∈ Λ(h), |hBuj , uj i| ≤ (h)1/4 ; and so |hAuj , uj i − α| ≤ (h)1/4 . Also, #Λ(h) #Γ(h) =1− . #{a ≤ Ej ≤ b} #{a ≤ Ej < b} But according to Weyl’s law, #Γ(h) (2πh)n #Γ(h) = ≤ C(h)1/2 → 0. #{a ≤ Ej ≤ b} Vol({a ≤ p ≤ b}) + o(1) 2. Now let {Ak }∞ k=1 ⊂ Ψ be a countable family of pseudodifferential operators each satisfying (14.4.1) for some αk . For each k we have Λk (h) ⊂ {a ≤ Ej ≤ b} such that (14.4.10) and (14.4.11) hold for Ak and Λk (h). Since (14.4.10) for Λ(h) = Λk (h) and Λ(h) = Λl (h) implies (14.4.10) for Λ(h) = Λk (h) ∩ Λl (h) (immediate by considering the zero density complements), we can assume that (14.4.12)

Λk+1 (h) ⊂ Λk (h).

308

14. QUANTUM ERGODICITY

For each k, let h(k) > 0 be small enough so that #Λk (h) 1 ≥1− , #{a ≤ Ej ≤ b} k

for 0 < h < h(k),

which is possible in view of (14.4.10). We can take h(k) > h(k + 1) → 0, k → ∞. and define Λ∞ (h) := Λk (h), h(k + 1) ≤ h < h(k). We then have 1 #Λ∞ (h) ≥1− , #{a ≤ Ej ≤ b} k

for 0 < h < h(k),

and since h(k) → 0, #Λ∞ (h) = 0. h→0 #{a ≤ Ej ≤ b} lim

3. We now claim that for any Ak Z (14.4.13) hAk uj , uj i → − σ(Ak ) dxdξ, h → 0, Ej ∈ Λ∞ (h). {a≤p≤b}

This is true since (14.4.12) and the definition of Λ∞ show that Λ∞ (h) ⊂ Λk (h) for h < h(k), and since we assumed that (14.4.13) held for Ej ∈ Λk (h). 4. In the last step of the proof we will choose a set {Ak }∞ k=1 dense in (14.4.14)

P :={A ∈ Ψ−∞ (M ) | Z − σ(A) dµc = α, independently of c ∈ [a, b]}. Σc

The density is meant in the following sense, that given A ∈ Ψ, (14.4.15)

for any  > 0 there exist k and h0 such that for 0 < h < h0 , Z − |σ(Ak − A)|dxdξ <  and kAk − AkL2 →L2 < . {a≤p≤b}

This and (14.4.13) show that for Ej ∈ Λ∞ (h), ! Z σ(A) dxdξ < 2, lim sup hAuj , uj i − − h→0 {a≤p≤b} ! Z σ(A) dxdξ < 2, lim inf hAuj , uj i − − h→0 {a≤p≤b}

14.4. QUANTUM ERGODIC THEOREMS

309

proving (14.4.11) for A ∈ Ψ−∞ (M ). As in the proof of Theorem 14.3 we see that we only need to consider that case as A can be replaced by f (P )A, where f ∈ Cc∞ (R) is equal to 1 near [a, b]. 5. It remains to find {Ak }∞ k=1 ⊂ P, where P is given by (14.4.14), so that (14.4.15) holds. Theorems 5.1 and 13.1 show that 1

kA − Ak kL2 →L2 ≤ kσ(A) − σ(Ak )kL∞ (T ∗ M ) + C(a, ak )h 2 , and we also immediately have Z − |σ(A − Ak )| dxdξ ≤ Ckσ(A) − σ(Ak )kL∞ (T ∗ M ) . a≤p(x,ξ)≤b −∞ (T ∗ M ) satisfying Thus we only need to find the set {ak }∞ k=1 ⊂ S Z (14.4.16) − ak (x, ξ) dµc = α, Σc

independently of c ∈ [a, b], such that for every a ∈ S −∞ (T ∗ M ) an every  > 0 there exists k such that ka − ak kL∞ (T ∗ M ) < . Hence we need to find ak ∈ Cc∞ (T ∗ M ) satisfying (14.4.16) and dense in the space C0 (T ∗ M ) of continuous functions vanishing at infinity and satisfying (14.4.16). We can assume that α = 0 since adding αχ, χ ∈ Cc∞ (R2n ), χ = 1 near p−1 ([a, b]) to a produces averages α. We can then take α ∈ Q. Since T in (14.4.2) is continuous on C0 (R2n ), we need only select a dense set of bk ∈ Cc∞ (R2n ) in C0 (R2n ), as then we can take ak = T bk .  REMARK. It is important that we consider A ∈ Ψ−∞ (M ) in the proof above, even though the theorem applies to A ∈ Ψ(M ). This is because the space Ψ(M ) is not separable, just as the space of symbols S is not separable.  APPLICATION. The simplest and most striking application concerns the complete set of eigenfunctions of the Laplace–Beltrami operator on a compact Riemannian manifold: −∆g uj = λj uj

(j = 1, . . . ),

normalized so that kuj kL2 (M ) = 1.

310

14. QUANTUM ERGODICITY

THEOREM 14.6 (Equidistribution of eigenfunctions). Suppose that (M, g) is a compact Riemannian manifold with an ergodic geodesic flow. Then there exists a sequence jk → ∞ of density one, lim

m→∞

#{k | jk ≤ m} = 1, m

such that Z

2

Z

|ujk | f dx →

(14.4.17) M

f dx M

for each f ∈ C(M ).

14.5. NOTES The Quantum Ergodicity Theorem 14.5 is from a 1974 paper of Shnirelman. The first complete proof was provided by Zelditch [Ze]. We have followed his more recent proof in [Z-Z] for Theorem 14.4, and the argument from Colin de Verdi`ere [CdV] for Theorem 14.5. A refined version of semiclassical ergodicity can be found in Helffer–Martinez–Robert [H-M-R]. Hopf showed that the geodesic flow on a negatively curved surface is ergodic, and Anosov and Sinai proved this for general manifolds: see Brin [Ba, Appendix] for a self contained presentation. This provides a large class of examples to which Theorem 14.6 applies. A more examples are provided by metrics with Anosov geodesic flows. Since these are structurally stable, ergodicity is also valid for the flow of the Hamiltonian p = |ξ|2g +V (x) on {p = 1} provided that V is small. Examples of Hamiltonians p = |ξ|2 + V (x) on T2 for which the flow is ergodic have been constructed recently by Donnay– Liverani [Do-Li].

Appendix A

NOTATION

A.1. BASIC NOTATION Z = integers, N = nonnegative integers R = real line, R+ = (0, ∞) Rn = n-dimensional Euclidean space x, y denote typical points in Rn : x = (x1 , . . . , xn ), y = (y1 , . . . , yn ). R2n = Rn × Rn z = (x, ξ), w = (y, η) denote typical points in Rn × Rn : z = (x1 , . . . , xn , ξ1 , . . . , ξn ), w = (y1 , . . . , yn , η1 , . . . , ηn ). We usually write γ : Rn → Rn to denote a smooth diffeomorphism, and κ : R2n → R2n to denote a smooth symplectomorphism.  Λκ = (x, y, ξ, −η) | (x, ξ) = κ(y, η), (y, η) ∈ R2n = twisted graph of κ Tn = n-dimensional flat torus = Rn /Zn C = complex plane, Cn = n-dimensional complex space GL(n, C) = the group of invertible linear transformations on Cn P hx, yi = ni=1 xi y¯i = inner product on Cn |x| = hx, xi1/2 , hxi = (1 + |x|2 )1/2 σ(z, w) = hJz, wi = symplectic inner product on R2n 311

312

A. NOTATION

• Sets: #S = cardinality of the set S L n = n-dimensional Lebesgue measure |E| = L n (E) = Lebesgue measure of the measurable set E ⊂ Rn H s = s-dimensional Hausdorff measure U, V usually denote open subsets ¯ is a compact subset of V U ⊂⊂ V means U • Matrices: Mm×n = m × n-matrices Sn = n × n real symmetric matrices tr = trace, det = determinant AT = transpose of the matrix A |A|2 = tr(AT A) [A, B] = AB − BA = commutator of matrices A and B exp A = eA =

Ak k=0 k!

P∞

F −1 AF = conjugation of A by F sgn Q = signature of the symmetric matrix Q I denotes both the identity matrix and the identity mapping   O I J= −I O

A.2. FUNCTIONS, DIFFERENTIATION The support of a function is denoted “spt”, and a subscript “c” on a space of functions means those with compact support. • Composition: f ◦ g = composition of the functions f and g

A.2. FUNCTIONS, DIFFERENTIATION

313

• Averages: ZT Z 1 T f dt − f dt := T 0 0 • Partial derivatives: ∂xj :=

∂ , ∂xj

Dxj :=

1 ∂ , i ∂xj

∂t =

∂ ∂t

We will also write

d dt applied to functions of the single variable t. ∂t =

• Multiindex notation: A multiindex is a vector α = (α1 , . . . , αn ), the entries of which are nonnegative integers: α ∈ Nn . The size of α is |α| := α1 + · · · + αn . We define for x ∈ Rn : xα := x1 α1 . . . xn αn , where x = (x1 , . . . , xn ). Also ∂ α := ∂xα11 . . . ∂xαnn and Dα :=

1 α1 ∂x1 |α| i

. . . ∂xαnn .

(WARNING: Our use of the symbols “D” and “Dα ” differs from that in first author’s PDE textbook [E].) • If ϕ : Rn → R, then we write ∂ϕ := (ϕx1 , . . . , ϕxn ) = gradient, and



ϕ x1 x1

 ∂ 2 ϕ :=  ϕxn x1 Also

... .. . ...

ϕx1 xn

   = Hessian matrix

ϕ xn xn

1 Dϕ := ∂ϕ. i

• If ϕ depends on both the variables (x, y) ∈ Rn × Rn , we put     ϕx1 x1 . . . ϕx1 xn ϕy1 y1 . . . ϕy1 yn     2 .. .. ∂x2 ϕ :=   , ∂y ϕ :=   . . ϕxn x1 . . . ϕxn xn ϕyn y1 . . . ϕyn yn

314

A. NOTATION

and



ϕx1 y1

 2 ∂x,y ϕ :=  ϕxn y1

... .. . ...

ϕx1 yn

  .

ϕxn yn

• Jacobians: Let x 7→ y = y(x) be a diffeomorphism, y = The Jacobian matrix is  1  ∂y ∂y 1 . . . ∂xn   ∂x1 ..  ∂y = ∂x y :=  . .   ∂y n ∂y n . . . ∂xn ∂x1 (y 1 , . . . , y n ).

n×n

• Differentiation of determinants: Suppose A(t) is a function from R to invertible m × m matrices. Then ∂t det A(t) = tr(A(t)−1 ∂t A(t)) det A(t).

(A.2.1)

• Poisson bracket: If f, g : Rn → R are C 1 functions, {f, g} := h∂ξ f, ∂x gi − h∂x f, ∂ξ gi =

n X ∂f ∂g ∂f ∂g − . ∂ξj ∂xj ∂xj ∂ξj j=1

• The Schwartz space is S = S (Rn ) := {ϕ ∈ C ∞ (Rn ) | sup |xα ∂ β ϕ| < ∞ for all multiindices α, β}. Rn

We say ϕj → ϕ

in S

provided sup |xα Dβ (ϕj − ϕ)| → 0 Rn

for all multiindices α, β We write S 0 = S 0 (Rn ) for the space of tempered distributions, which is the dual of S = S (Rn ). That is, u ∈ S 0 provided u : S → C is linear and ϕj → ϕ in S implies u(ϕj ) → u(ϕ). We say uj → u in S 0 provided uj (ϕ) → u(ϕ)

for all ϕ ∈ S .

A.4. ESTIMATES

315

A.3. OPERATORS Multiplication operator: Mλ f (x) = λf (x) Translation operator: Tξ f (x) = f (x − ξ) Reflection operator: Rf (x) := f (−x) A∗ = adjoint of the operator A [A, B] = AB − BA = commutator of operators A and B adA B = [A, B] σ(A) = symbol of the pseudodifferential operator A Spec(A) = spectrum of A tr(A) = trace of A • If A : X → Y is a bounded linear operator, we define the operator norm kAk := sup{kAukY | kukX ≤ 1}. We will usually write this norm as kAkX→Y , to emphasize the spaces between which A maps. • L(X, Y ) = the space of bounded linear operators from X to Y L(X) = L(X, X).

A.4. ESTIMATES A.4.1. Use of constants. • We use “C” to denote a general positive constant appearing to the right of the inequality sign ≤ in various estimates throughout the text. The letter C will in general stand for different constants in different lines of the calculations. • We will also employ a lower case “c” to denotes a positive constant occurring in various estimates to the left of the inequality sign ≤ or to the right of ≥.

316

A. NOTATION

A.4.2. Order estimates. • We write f = O(h∞ ) as h → 0 if there exist h0 > 0 and for each positive integer N a constant CN such that |f | ≤ CN hN for all 0 < h < h0 . • We write f = OX (hN ) to mean kf kX = O(hN ). • If A is a bounded linear operator between the spaces X, Y , we will often write A = OX→Y (hN ) to mean kAkX→Y = O(hN ). • We will sometimes write a = OS (hN ), to mean that for all α |∂ α a| ≤ Cα hN . We use similar notation for other spaces with seminorms.

A.5. SYMBOL CLASSES We record from Chapters 4 and 9 various classes of symbols: • Given an order function m on R2n , we define the corresponding class of symbols: S(m) := {a ∈ C ∞ | |∂ α a| ≤ Cα m for all multiindices α}, Sδ (m) := {a ∈ C ∞ | |∂ α a| ≤ Cα h−δ|α| m for all multiindices α}, S := S(1), Sδ = Sδ (1). • Kohn-Nirenberg symbols: S m := {a ∈ C ∞ | |∂xα ∂ξβ a| ≤ Cαβ hξim−|β| for all α and β}, Ψm := {aw (x, hD) | a ∈ S m }.

Appendix B

DIFFERENTIAL FORMS

In this section we provide a minimalist review of differential forms on RN .

B.1. DEFINITIONS We start with a long list of algebraic and analytic concepts: DEFINITIONS. (i) An m-form on RN is a multilinear mapping
m ω : RN = RN × · · · × RN → R, which is alternating: if ui = uj for some 1 ≤ i < j ≤ m then ω(u1 , · · · , ui , · · · , uj , · · · um ) = 0. We will write ω(u) where u = (u1 , · · · , um ) for u1 , . . . um ∈ RN (ii) If α, β are one-forms, their wedge product is the two-form (α ∧ β)(u) := α(u1 )β(u2 ) − α(u2 )β(u1 ) for u = (u1 , u2 ). (iii) More generally, if {αj }j=1,··· ,m are one-forms and m ≤ N , we define their wedge product to be the m-form (B.1.1)

(α1 ∧ · · · ∧ αm )(u) = det((αj (uk ))),

where u = (u1 , · · · , um ). 317

318

B. DIFFERENTIAL FORMS

(iv) The one-forms dxj on RN are defined by the rule (B.1.2)

dxj (u) = xj

for j = 1, . . . , N and u = (x1 , . . . , xN ) ∈ RN . (v) A differential m-form on RN is an expression X (B.1.3) ω= fi1 ···im (x)dxi1 ∧ · · · dxim , i1
with fi1 ···im ∈ C ∞ (RN ). Its action at a point x ∈ RN on an m-tuple of vectors u is defined using (B.1.1) and (B.1.2). (vi) If f : RN → R, the differential of f is the differential 1-form n X ∂f df = dxi . ∂xi

(B.1.4)

j=1

(vii) The differential of an m-form is defined by induction using (B.1.3) and the rule d(f η) = df ∧ η + f dη,

(B.1.5)

where f is a function and η an (m − 1)-form. The operator d satisfies d2 = 0.

(B.1.6) NOTATION. We write

Ωm (U ) to denote the space of m-forms defined on the open set U ⊆ RN with smooth coefficients.  THEOREM B.1 (Alternative definition of d). Suppose w is a differential 2-form, and u ∈ C ∞ (Rn , R3 ), u = (u1 , u3 , u3 ) is a 3-tuple of vector fields. Then (B.1.7)

dw(u) = u1 (w(u2 , u3 )) + u2 (w(u3 , u1 )) + u3 (w(u1 , u2 )) − w([u1 , u2 ], u3 ) − w([u2 , u3 ], u1 ) − w([u3 , u1 ], u2 ).

1. Both sides of (B.1.7) are linear in w and trilinear in u. 2. When, say, u1 is multiplied by f ∈ C ∞ (Rn ), then dw(f u1 , u2 , u3 ) = f dw(u). The right hand side of (B.1.7) with u1 replaced by f u1 is equal to f u1 (w(u2 , u3 )) + u2 (f w(u3 , u1 )) + u3 (f w(u1 , u2 )) − w([f u1 , u2 ], u3 ) − w([u2 , u3 ], f u1 ) − w([u3 , f u1 ], u2 ),

B.1. DEFINITIONS

319

and this is equal to the right hand side of (B.1.7) multiplied by f . In fact, [f u1 , u2 ] = f [u1 , u2 ] − (u2 f )u1 , [u3 , f u1 ] = f [u3 , u1 ] + (u3 f )u1 , and u2 (f w(u3 , u1 )) = f u2 (w(u3 , u1 )) + (u2 f )w(u3 , u1 ), u3 (f w(u1 , u2 )) = f u3 (w(u1 , u2 )) + (u3 f )w(u1 , u2 ). 3. Hence we only need to check (B.1.7) for u constant in x, and for w = w1 dw2 ∧ dw3 , where w1 ∈ C ∞ , and w2 , w3 are coordinate functions (that is are among x1 , · · · xn ). Then, using (B.1.1), dw(u) = (dw1 ∧ dw2 ∧ dw3 )(u1 , u2 , u3 ) = det ((dwi (uj ))1≤i,j≤3 ) = det ((uj wi )1≤i,j≤3 ) , and the right hand side of (B.1.7) is given by (remember that now uj wi , i = 2, 3 are constant) by the expansion of this determinant with respect to the first row, (u1 w1 , u2 w1 , u3 w1 ).  NOTATION. We will often take N = 2n, in which case we write a point z ∈ R2n = Rn × Rn as z = (x, ξ) for x = (x1 , . . . , xn ), ξ = (ξ1 , . . . , ξn ). The one-forms dxj , dξj on R2n are then defined by the rules (B.1.8)

dxj (z) = xj , dξj (z) = ξj

for j = 1, . . . , n.



We next explain how to contract an m-form with a vector field: DEFINITION. If η is a differential m-form and V a vector field, then the contraction of η by V , denoted V

η,

is the (m − 1)-form defined by (B.1.9)

(V

η)(u) = η(V, u),

where u = (u1 , · · · , um−1 ) for u1 , . . . um−1 ∈ RN . We also will write V 0, if f is a function. It follows from (B.1.1) that (B.1.10)

V

(α ∧ β) = (V

if α is a k-form and β is an m-form.

α) ∧ β + (−1)k α ∧ (V

β)

f=

320

B. DIFFERENTIAL FORMS

B.2. PUSH-FORWARDS AND PULL-BACKS We record next how forms transform under mappings. Let κ : RN → RN be a smooth mapping. We write κ = (κ1 , . . . , κN ) and y = κ(x). DEFINITIONS. (i) If V is a vector field on RN , its push-forward by κ is κ∗ V = ∂κ(V ). (ii) If η is an m-form on RN , its pull-back by κ is the m-form (κ∗ η)(u) = η(κ∗ u),

(B.2.1)

where κ∗ u = (∂κ u1 , . . . , ∂κ um ) for u = (u1 , . . . , um ). EXAMPLES. (i) We easily check that κ∗ dyi =

N X ∂yi dxj . ∂xj j=1

(ii) For an m-form η and f ∈ C ∞ , κ∗ (f η) = κ∗ f κ∗ η. The differential commutes with pull-back: THEOREM B.2 (Differentials and pull-backs). Let ω be a differential m-form. Then d(κ∗ ω) = κ∗ (dω).

(B.2.2)

Proof. We first prove this for functions, by observing N N X X ∂ ∂yi d(κ f ) = f (κ(x))dxj = fy (κ(x))dxj ∂xj ∂xj i j=1 i,j=1 ! N N X X = κ∗ fyi κ∗ dyi = κ∗ fyi dyi ∗

i=1 ∗

i=1

= κ (df ). The proof now follows by induction on the order of the differential form. Note that any m-form can be written as a linear combination of terms f dη, where f is a function and η is an (m − 1)-form. 

B.2. PUSH-FORWARDS AND PULL-BACKS

321

DEFINITION. If V is a vector field generating the flow ϕt then the Lie derivative of w is LV w := ∂t (ϕ∗t w)|t=0 . Here w denotes a function, a vector field or a form and ϕt = exp(tV ) is the flow generated by the vector field V : see see Section 2.1. We have d(LV ω) = LV (dω)

(B.2.3) and

LV (f ω) = (LV f )ω + f LV ω,

(B.2.4)

where f is a function and ω a differential form. EXAMPLES. (i) If f is a function, LV f = V (f ). (ii) If W is a vector field, LV W = [V, W ].  THEOREM B.3 (Cartan’s formula). If ω is a differential form, (B.2.5)

LV ω = d(V

ω) + V

dω.

Proof. 1. We proceed by induction on the order of differential forms. For 0-forms, that is for functions, we have LV f = V f = V by our convention that V

df = d(V

f ) + (V

df ),

f = 0.

2. Any m-form is a linear combination of terms f dη where f is a function and η is an (m − 1)-form. Using (B.2.3), (B.2.4), d2 = 0 and the induction hypothesis, we see that LV (f dη) = (LV f )dη + f LV dη (B.2.6)

= (V f )dη + f d(LV η) = (V f )dη + f d(d(V = (V f )dη + f d(V

η) + V

dη)

dη).

3. The right hand side of (B.2.5) for ω = f dη is equal to (B.2.7) d(V

(f dη)) + V

d(f dη) =

322

B. DIFFERENTIAL FORMS

df ∧ (V

dη) + f d(V

dη) + V

(df ∧ dη).

We now use (B.1.10) to obtain V

(df ∧ dη) = (V f )dη − df ∧ (V

dη).

Inserting this into (B.2.7) and comparing with (B.2.6) gives Cartan’s formula (B.2.5) for ω = f dη and hence for all differential m-forms. 

´ LEMMA B.3. POINCARE’S On all of RN , or more generally on star-shaped regions, all closed forms are exact. This means that if dα = 0, then there exists ω such that α = dω. The construction of ω can be made explicit: THEOREM B.4 (Poincar´ e’s Lemma). If α is a k-form defined in the 0 open ball V = B (0, R) and if dα = 0, then there exists a (k − 1)-form ω in V such that dω = α. Proof. 1. Let Ωk (V ) denote the space of k-forms on V . Define A : Ωk (V ) → Ωk (V ) by Z 1  k−1 A(f (x) dxi1 ∧ · · · ∧ dxik ) = t f (tx) dt dxi1 ∧ · · · ∧ dxik . 0

Set X := hx, ∂x i. We claim that (B.3.1)

ALX = I

on Ωk (V ).

and that d ◦ A = A ◦ d.

(B.3.2)

2. To prove (B.3.1), we compute using (B.2.4):    n X ALX (f (x) dxi1 ∧ · · · ∧ dxik ) = A kf (x) + xj fxj (x) (dxi1 ∧ · · · ∧ dxik ) j=1

Z

1

ktk−1 f (tx) +

= 0

Z = 0



n X

 tk xj fxj (tx) dt dxi1 ∧ · · · ∧ dxik

j=1 1

∂t (tk f (tx)) dt dxi1 ∧ · · · ∧ dxik = f dxi1 ∧ · · · ∧ dxik .

B.4. DIFFERENTIAL FORMS ON MANIFOLDS

323

To verify (B.3.2), note  A ◦ d(f (x) dxi1 ∧ · · · ∧ dxik ) = A 

n X

 fxj (x)dxj ∧ dxi1 ∧ · · · ∧ dxik 

j=1

 Z =

1

tk

0

n X

 fxj (tx)dxj dt dxi1 ∧ · · · ∧ dxik

j=1 1

Z =d

  tk−1 f (tx) dt dxi1 ∧ · · · ∧ dxik

0

= d ◦ A(f (x)dxi1 ∧ · · · ∧ dxik ). 3. Now define H : Ωk (V ) → Ωk−1 (V ) by H := A ◦ X . By Cartan’s formula (B.2.5), LX = d ◦ (X

)+X

◦d.

Thus I = ALX

= A ◦ d ◦ (X = d(A ◦ X

)+A◦X ) + (A ◦ X

◦d )◦d

= d ◦ H + H ◦ d. Consequently d ◦ H + H ◦ d = I; and therefore d(Hα) + Hdα = α. So dω = α for ω := Hα.



B.4. DIFFERENTIAL FORMS ON MANIFOLDS Let M be a smooth N -dimensional manifold: consult Section 13.1.1 for definitions and notation. DEFINITION. A linear functional ω on m-tuples of vectorfields, ω : (C ∞ (M, T M ))m → C ∞ (M ) is called a differential form on M if for every γ in an atlas we have (γ −1 )∗ ω|Uγ ∈ Ωm (Vγ ). NOTATION. We denote by Ωm (M )

324

B. DIFFERENTIAL FORMS

the space of m-forms defined on M with smooth coefficients.



REMARKS. (i) According to Example 2 in Section 13.1.2, Ω1 (M ) = C ∞ (M, T ∗ M ). (ii) The restriction of a form ω to the chart Uγ is defined by ω|Uγ := ω|Cc∞ (Uγ ,T Uγ )m . (iii) The pull-back of a form is defined using the natural operations on vector fields γ∗−1 : C ∞ (Vγ , T Vγ ) → C ∞ (Uγ , T Uγ ). (See Example 1 in Section 13.1.2 for the definition of C ∞ (M, T M ). ) (iv) Theorem B.2 shows that the differential is well defined, d

d

d

d

C ∞ (M ) = Ω0 (M ) − → Ω1 (M ) − → ··· − → ΩN (M ) − → 0, and d2 = 0.  A big difference with manifolds is that closed forms need not be exact; as Poincar´e Lemma holds only locally. THEOREM B.5 (Poincar´ e’s Lemma on manifolds). Suppose that M is a smooth manifold. Then every x ∈ M has a neighbourhood U such that if α ∈ Ωm (M ) is closed: dα = 0, m−1 then there exists ω ∈ Ω (U ) such that (B.4.1)

α|U = dω.

Proof. Every point x belongs to Uγ for some γ and Vγ = γ(Uγ ) ⊃ B(γ(x), R) for some R. We can assume that γ(x) = 0. We then apply Theorem B.4 to αγ := (γ −1 )∗ (α|Uγ ) to obtain ωγ ∈ satisfying dωγ = αγ in B(0, R). The theorem follows from our putting U = γ −1 (B(0, R)) and

Ωm−1 (B(0, R))

ω := γ ∗ ωγ ∈ Ωm−1 (U ).



Appendix C

FUNCTIONAL ANALYSIS

C.1. OPERATOR THEORY We group here some results about invertibility of operators and operator norms, providing some proofs for the reader’s convenience. THEOREM C.1 (Schwartz Kernel Theorem). Let A : S → S 0 be a continuous linear operator. Then there exists a distribution KA ∈ S 0 (Rn × Rn ) such that for all u, v ∈ S Au(v) = KA (u ⊗ v),

(u ⊗ v)(x, y) := u(x)v(y) ∈ S (Rn × Rn ).

We write this as Z Au(x) =

KA (x, y)u(y) dy Rn

We call KA the kernel, or Schwartz kernel, of A. THEOREM C.2 (Inverse Function Theorem). Let X, Y denote Banach spaces and assume f :X→Y is C 1 . Select a point x0 ∈ X and write y0 := f (x0 ). (i) (Right inverse) If there exists A ∈ L(Y, X) such that ∂f (x0 )A = I, 325

326

C. FUNCTIONAL ANALYSIS

then there exists g ∈ C 1 (Y, X) such that f ◦g =I

near y0 .

(ii) (Left inverse) If there exists B ∈ L(Y, X) such that B∂f (x0 ) = I, then there exists g ∈

C 1 (Y, X)

such that

g◦f =I

near x0 .

THEOREM C.3 (Approximate inverses). Let X, Y be Banach spaces and suppose A : X → Y is a bounded linear operator. Suppose there exist bounded linear operators B1 , B2 : Y → X such that ( AB1 = I + R1 on Y (C.1.1) B2 A = I + R2 on X, where kR1 k < 1, kR2 k < 1. Then A is invertible. Proof. The operator I + R1 is invertible, with (I + R1 )

−1

=

∞ X

(−1)k R1k ,

k=0

this series converging since kR1 k < 1. Hence AC1 = I

for C1 := B1 (I + R1 )−1 .

Likewise, (I + R2 )−1 =

∞ X

(−1)k R2k ;

k=0

and C2 A = I

for C2 := (I + R2 )−1 B2 .

So A has a left and a right inverse, and is consequently invertible, with A−1 = C1 = C2 .  THEOREM C.4 (Norms of powers of operators). For H1 , H2 , complex Hilbert spaces, let A ∈ L(H1 , H2 ) be a bounded operator. (i) Then kAk =

sup kuk,kvk=1

| hAu, vi |,

kAk = kA∗ k,

kAk2 = kA∗ Ak.

C.1. OPERATOR THEORY

327

(ii) If A is self-adjoint, kAkm = kAm k for all m ∈ N. Proof. 1. We may assume kAk > 0. Note that | hAu, vi | ≤ kAukkvk ≤ kAkkukkvk = kAk for any two unit vectors u, v. Thus supkuk,kvk=1 | hAu, vi | ≤ kAk. Now if u 6∈ ker(A), we can put v = Au/kAuk. Consequently, 1 |hAu, Aui| = sup kAuk = kAk; u6∈ker(A) kAuk u6∈ker(A)

|hAu, vi| ≥ sup

sup kuk,kvk=1

kuk=1

kuk=1

and therefore kAk2 = sup kAuk2 = sup |hA∗ Au, ui| kuk=1

≤

kuk=1 ∗

|hA Au, vi| = kA∗ Ak.

sup kuk,kvk=1

Also, for any u, v with norm one we have |hA∗ Au, vi| = |hAu, Avi| ≤ kAukkAvk ≤ kAk2 . Taking the supremum over u, v gives us the inequality kA∗ Ak ≤ kAk2 . k

2. For A self-adjoint, A = A∗ , a simple induction now yields kAk2 = kA k for all natural numbers k. For a general m, find an n such that m + n is a power of 2. Then kAn+m k = kAkn+m , and so 2k

kAkm kAkn = kAkm+n = kAm+n k = kAm An k ≤ kAkm kAkn . Therefore the inequality signs above must be equalities, and this implies kAm k = kAkm .  THEOREM C.5 (Cotlar–Stein Theorem). Let H1 , H2 be Hilbert spaces and Aj ∈ L(H1 , H2 ) for j = 1, . . . . Assume sup j

∞ X

kA∗j Ak k1/2

≤ C, sup j

k=1

Then the series A :=

∞ X

∞ X k=1

Aj

j=1

converges in strong operator topology and kAk ≤ C.

kAj A∗k k1/2 ≤ C.

328

C. FUNCTIONAL ANALYSIS

Proof. 1. Let us first assume that Aj = 0 for j > J so that A is well defined. Since A∗ A is self-adjoint, the previous theorem implies kAk2m = k(A∗ A)m k. In addition, ∗

m

(A A)

∞ X

=

A∗j1 Aj2 . . . A∗j2m−1 Aj2m =:

X

aj1 ,...,j2m .

j1 ,...,j2m

j1 ,...,j2m =1

Now kaj1 ,...,j2m k ≤ kA∗j1 Aj2 kkA∗j3 Aj4 k . . . kA∗j2m−1 Aj2m k, and also kaj1 ,...,j2m k ≤ kAj1 kkAj2 A∗j3 k . . . kAj2m−2 A∗j2m−1 kkAj2m k. 1

Note that kAj k = kA∗j Aj k 2 ≤ C. Multiply these estimates and take square roots: kaj1 ,...,j2m k ≤ CkA∗j1 Aj2 k1/2 kAj2 A∗jm k1/2 . . . kA∗j2m−1 Aj2m k1/2 . Consequently, 2m

kAk

∗

∞ X

m

= k(A A) k ≤

kaj1 ,...,j2m k

j1 ,...,j2m =1

≤ C

∞ X

kAj1 A∗j2 k1/2 . . . kA∗j2m−1 Aj2m k1/2

j1 ,...,j2m =1

≤ JCC 2m , where the J factor came from having 2m sums and only 2m − 1 factors in the summands. Hence 1

kAk ≤ J 2m C

2m+1 2m

→C

as m → ∞.

2. To consider the general case, take u ∈ E, and suppose u = A∗k v for some k. Then ∞ ∞ X X k Aj uk = k Aj A∗k vk j=1

j=1

≤

∞ X

kAj A∗k k1/2 kAj A∗k k1/2 kvk

j=1

≤ C 2 kvk.

C.2. SPECTRAL THEORY

329

P ∗ Thus ∞ j=1 Aj u converges for u ∈ Σ := span{Ak (E) | k = 1, . . . , n} and so ¯ ¯ also for P∞u ∈ Σ. If u is orthogonal to Σ, then u ∈ ker(Ak ) for all k; in which case j=1 Aj u = 0. P P 3. We have proved that k Jj=1 Aj k ≤ C for any J and that ∞ j=1 Aj u converges for any u ∈ H. Hence, k

∞ X

Aj uk ≤ Ckuk.



j=1

C.2. SPECTRAL THEORY Henceforth H denotes a complex, separable Hilbert space with inner product h·, ·i. DEFINITIONS. Suppose that A : H → H is a bounded linear operator. (i) The spectrum of A is Spec(A) := C \ {λ ∈ C | (A − λ)−1 : H → H is bounded}. (ii) We say that λ ∈ Spec(A) is an eigenvalue of A if there exists u 6= 0 such that (C.2.1)

Au = λu.

(iii) The operator A is called compact if the image of {u | kuk ≤ 1} under A is a precompact subset of H. We recall that compact operators form a closed ideal in the space of bounded operators with operator norm topology. That means that for A compact and B bounded, AB and BA are compact, and that if kCj −Ck → 0 and Cj ’s are compact, then C is compact. DEFINITION. For a bounded linear operator A : H → H, we define the adjoint A∗ : H → H by the formula hAu, vi = hu, A∗ vi. A bounded operator A is self-adjoint if A∗ = A. THEOREM C.6 (Spectral Theorem). Let A be a bounded self-adjoint operator on H. Then there exist a measure space (X, M, µ), a real-valued function f ∈ L∞ (X, µ) and a unitary operator U : H → L2 (X, µ)

330

C. FUNCTIONAL ANALYSIS

such that U ∗ Mf U = A.

(C.2.2)

Here Mf is the multiplication operator: Mf u = f u for u ∈ L2 (X, µ). THEOREM C.7 (Spectra of compact operators). Suppose A is a compact operator on H. Then (i) Every λ ∈ Spec(A) \ {0} is an eigenvalue of A. (ii) For all nonzero λ ∈ Spec(A) \ {0}, there exist N such that ker(A − λ)N = ker(A − λ)N +1 . (iii) The eigenvalues can only accumulate at 0. In particular, Spec(A) is countable. (iv) Every λ ∈ Spec(A)\{0} is a finite rank pole of the resolvent operator ζ 7→ (A − ζ)−1 More precisely, −1

(A − ζ)

=

N X j=1

Aj + Q(ζ, λ), (ζ − λ)j

where dim Aj H < ∞, Aj = (P − λ)j−1 A1 , and ζ 7→ Q(ζ, λ) is holomorphic near ζ = λ. (v) Suppose in addition that A is self-adjoint. Then there exists an orthonormal set {uk }k∈K ⊂ H such that X (C.2.3) Au(x) = λk uk (x)hu, uk i, k∈K

where λ0 ≥ λ1 ≥ · · · are the non-zero eigenvalues of A. Here the index set K is either {0, 1, 2, · · · , N } or N. (vii) Conversely, if (C.2.3) holds and λj → 0, then A is compact.

C.2. SPECTRAL THEORY

331

THEOREM C.8 (Spectra of self-adjoint operators). Suppose A : H → H is a bounded self-adjoint operator. (i) Then (A − λ)−1 exists and is a bounded linear operator on H for λ ∈ C \ Spec(A). (ii) If Spec(A) ⊂ [a, ∞), then hAu, ui ≥ akuk2

(C.2.4)

(u ∈ A).

We next review the more complicated theory for unbounded operators. DEFINITIONS. (i) An unbounded operator A : H → H is given by a subspace D(A) ⊂ H and a linear operator A : D(A) → H. We call D(A) the domain of A, and say that A is densely defined if D(A) is dense in A. (ii) The graph of A is graph(A) := {(u, Au) | u ∈ D(A)} ⊂ H × H. (iii) If A, B are unbounded operators on H, we say that A ⊆ B if D(A) ⊆ D(B) and Au = Bu for all u ∈ D(A). (iv) The operator A is closed if graph(A) is a closed subspace of H × H equipped with the norm k(u, v)k2 = kuk2 + kvk2 . (v) An unbounded operator A is closable if there exists a closed un¯ The operator A¯ is unique and is bounded operator A¯ such that A ⊆ A. called the closure of A. THEOREM C.9 (Adjoint operator). Suppose A : H → H is an unbounded, densely defined operator. Then there exists an unbounded operator A∗ : H → H defined by the rule hA∗ v, ui := hv, Aui

(C.2.5)

for all v ∈ D(A∗ ), u ∈ D(A), where (C.2.6)

D(A∗ ) := {v ∈ H | |hAu, vi| ≤ C(v)kuk for all u ∈ D(A)}.

Here C(v) is a constant depending on v. The unbounded operator A∗ is always closed. If A∗ is densely defined, then A is closable and A¯ = (A∗ )∗ , A¯∗ = A∗ . DEFINITIONS. (i) An unbounded densely defined operator A is called symmetric if (C.2.7)

A ⊆ A∗ .

332

C. FUNCTIONAL ANALYSIS

Equivalently, hAu, vi = hu, Avi for all u, v ∈ D(A). (ii) An unbounded densely defined operator A is called self-adjoint if A = A∗ .

(C.2.8)

(iii) A symmetric operator is called essentially self-adjoint if A¯ = A∗ .

(C.2.9)

EXAMPLE: Quadratic symbols. Suppose 1 1 p(x, ξ) := hAx, xi + hBx, ξi + hDξ, ξi, 2 2 where A, B, and D are complex n × n matrices and AT = A, DT = D. We define a corresponding unbounded operator Np on L2 (Rn ) by the rule Np u := pw (x, D)u for u ∈ S =: D(Np ). Since hpw (x, D)u, vi = hu, p¯w (x, D)vi for u, v ∈ S , we see that Np is symmetric if and only if p is real. We next define an extension Mp of Np , by first defining D(Mp ) to be the set of u ∈ L2 such that pw (x, D)u ∈ L2 , where pw (x, D) is meant in the sense of S 0 . Then let Mp u := pw (x, D)u for u ∈ D(Mp ). We assert that (C.2.10)

Mp is closed;

(C.2.11)

¯p ; Mp = N

and (C.2.12)

Np∗ = Mp∗ = Mp¯.

In particular, Mp is self-adjoint if and only if p is real-valued, in which case Np is essentially self-adjoint. Proof of (C.2.10). Remember that pw (x, D) : S 0 → S 0 is continuous. If uj → u and pw (x, D)uj → v in L2 , then uj → u in S 0 and so v = pw (x, D)u ∈ L2 . Therefore u ∈ D(Mp ) and Mp u = v.  Proof of (C.2.11). To show that Mp is the closure of Np . we have to show that for any u ∈ D(Mp ) we can find u ∈ S such that u → u and

C.2. SPECTRAL THEORY

333

pw (x, D)u → pw (x, D)u in L2 as  → 0. So take χ ∈ Cc∞ (R2n ) equal to one in B(0, 1) and write χ (x, ξ) := χ(x, ξ). Set 2 u := χw  (x, D)u ∈ S , u → u in L .

Observe that w w w pw (x, D)u = χw  (x, D)p (x, D)u + [p (x, D), χ (x, D)]u w w 2 and χw  (x, D)p (x, D)u → p (x, D)u in L . We therefore need to show

(C.2.13)

2 [pw (x, D), χw  (x, D)]u → 0 in L

as  → 0. Since p is quadratic, Theorem 4.11 implies w [pw (x, D), χw  (x, D)] = {χ , p} (x, D).

Now {χ , p}(x, ξ) = 

n X


∂xj p(x, ξ)(∂ξj χ)(x, ξ) − ∂ξj p(x, ξ)(∂xj χ)(x, ξ) .

j=1

Hence {χ , p} is bounded in S(1), uniformly with respect to . Consequently Theorem 4.23 implies that the commutator is uniformly bounded on L2 . For ψ ∈ Cc∞ (Rn ) supported in B(0, 1) and equal to one near 0 we put ψ (x) = ψ(x). This function vanishes on the support of {χ , p}. It follows, again using Theorems 4.11 and 4.23, that [pw (x, D), χw  (x, D)]ψ (x) = OL2 →L2 (). Since ψ u → u in L2 , we obtain (C.2.13).



Proof of (C.2.12). Suppose v ∈ D(Mp∗ ). Then there exists C(v) such that (C.2.14)

hMp u, vi ≤ C(v)kuk

for all u ∈ D(Mp ) ⊃ S . For u ∈ S we have hMp u, vi = hu, p¯w (x, D)vi, where pw (x, D)v ∈ S 0 . But then (C.2.14) implies that p¯w (x, D)v ∈ L2 . Hence Mp∗ ⊂ Mp¯. Also, since Mp∗ is closed, ¯p∗ = Mp∗ . Np¯ ⊂ Np∗ = N ¯p¯ ⊂ M ∗ and consequently M ∗ = Mp¯. Therefore Mp¯ = N p p



THEOREM C.10 (Spectral Theorem for unbounded operators). Let A be an unbounded self-adjoint operator on H. Then there exist a measure space (X, M, µ), a real-valued measurable function f and a unitary operator U : H → L2 (X, µ)

334

C. FUNCTIONAL ANALYSIS

such that (C.2.15)

x ∈ D(A) if and only if Mf (U x) ∈ L2 (X, µ)

and (C.2.16)

x ∈ D(A) implies U (Ax) = Mf (U x).

Here Mf : x 7→ f x denotes the unbounded multiplication operator on X. As an immediate consequence we obtain this useful result valid for both bounded and unbounded operators: THEOREM C.11 (Distance to spectrum). (i) If A is a self-adjoint operator, then Spec(A) = ess-image (f ) ⊂ R, where f is given in the Spectral Theorems C.10, C.6 and ess-image (f ) := {t | µ(f −1 ((t − , t + )) > 0 for all  > 0}. (ii) Furthermore, if λ ∈ C \ Spec(A), then 1 (C.2.17) k(A − λ)−1 k = . dist (λ, Spec(A)) There are many criteria determining if an operator is essentially selfadjoint and there are many subtleties in the subject. Here we only need the simplest one: THEOREM C.12 (Criteria for essential self-adjointness). Suppose that A : H → H is symmetric. Then the following conditions are equivalent: (i) A is essentially self-adjoint. (ii) For both signs, (A∗ ± i)x = 0, x ∈ D(A∗ ), implies x = 0. (iii) For both signs, {(A ± i)x | x ∈ D(A)} is dense in H. In quantum dynamics the Schr¨odinger propagators of self-adjoint operators play a crucial role. THEOREM C.13 (Stone’s Theorem). Suppose that P : D(A) ⊆ H → H is a (possibly unbounded) self-adjoint operator. (i) Then (C.2.18)

U (t) = exp(−itP )

C.2. SPECTRAL THEORY

335

defines a strongly continuous unitary group: ( U (t)U (s) = U (t + s), U (t)∗ = U (−t), (C.2.19) limt→0 kU (t)u − ukH = 0 (u ∈ H). In addition, (C.2.20)

Dt (U (t)u) + U (t)P u = 0

for all u ∈ D(P ). (ii) Conversely, if U (t) satisfies (C.2.19), then there exists a self-adjoint operator P such that (C.2.18) and (C.2.20) hold. EXAMPLE. Lemma 11.8 shows that for real quadratic functions p = p(x, ξ), we can define exp(−itpw (x, D)) : S → S and that it extends to a unitary group on L2 . The continuity property in (C.2.19) is easily checked for u ∈ S , and owing to the density of S in L2 it follows for u ∈ L2 . According to Stone’s Theorem, pw (x, D) is essentially self-adjoint, as we have already seen in the first example of this section.  When the spectrum of a self-adjoint operator is discrete, tools from linear algebra are applicable. THEOREM C.14 (Maximin and minimax principles). Suppose that A : H → H is self-adjoint and semibounded, meaning A ≥ −c0 . Assume also that (A + 2c0 )−1 : H → H is a compact operator. Then the spectrum of A is discrete: λ1 ≤ λ2 ≤ λ3 · · · ; and furthermore (i) (C.2.21)

λj =

max

min

V ⊂H v∈V codimV <j v6=0

hAv, vi , kvk2

(ii) (C.2.22)

λj = min max V ⊂H v∈V dimV ≤j v6=0

hAv, vi . kvk2

In these formulas, V denotes a linear subspace of H. DEFINITIONS. (i) Let Q : H → H be a bounded linear operator. We define the rank of Q to be the dimension of the range Q(H).

336

C. FUNCTIONAL ANALYSIS

(ii)If A is an operator with real and discrete spectrum, we set N (λ) := #{λj | λj ≤ λ} to count the number of eigenvalues less than or equal to λ. THEOREM C.15 (Estimating N (λ)). Let A satisfy the assumptions of Theorem C.14. (i) If

(C.2.23)

  there exist δ > 0 and a self-adjoint operator Q, with rank Q ≤ k, such that   hAu, ui ≥ (λ + δ)kuk2 − hQu, ui for u ∈ H,

then N (λ) ≤ k. (ii) If

(C.2.24)

  for each δ > 0, there exists a subspace V with dim V ≥ k, such that   hAu, ui ≤ (λ + δ)kuk2 for u ∈ V,

then N (λ) ≥ k. Proof. 1. Set W be the orthogonal complement of Q(H), W := Q(H)⊥ . Thus codim W = rank Q ≤ k. Therefore the maximin formula (C.2.21) implies hAv, vi hAv, vi ≥ min v∈W kvk2 kvk2 v6=0   hQv, vi = min λ + δ − = λ + δ, v∈W kvk2

λk+1 =

max

min

V ⊂H v∈V codimV ≤k v6=0

v6=0

since hQv, vi = 0 if v ∈ Q(H)⊥ . Hence λ < λ + δ ≤ λk+1 , and so N (λ) = max{j | λj ≤ λ} ≤ k. This proves assertion (i). 2. The minimax formula (C.2.22) directly implies that λk ≤ max v∈V v6=0

hAv, vi ≤ λ + δ. kvk2

C.3. TRACE CLASS OPERATORS

337

Hence λk ≤ λ + δ. This is valid for all δ > 0, and so N (λ) = max{j | λj ≤ λ} ≥ k. This is assertion (ii).



C.3. TRACE CLASS OPERATORS Let A : H → H be a compact operator on a complex separable Hilbert space H. Then A∗ A : H → H is a self-adjoint semidefinite compact operator, and hence it has discrete spectrum kAk2 = s0 (A)2 ≥ s1 (A)2 ≥ · · · ≥ sk (A)2 → 0. DEFINITION. The singular values of A are the nonnegative square roots of these eigenvalues: sj (A) (j = 0, 1, . . . ). We note that we obtain the same singular values by considering eigenvalues of AA∗ . Singular values have many interesting properties but here we will only need the fact that (C.3.1)

sj (AB) ≤ kBksj (A)

for A compact and B bounded. DEFINITIONS. (i) A compact operator A : H → H is said to be of trace class, written A ∈ L1 (H), if (C.3.2)

∞ X

sj (A) < ∞.

j=1

(ii) The trace class norm is (C.3.3)

kAkL1 :=

∞ X

sj (A).

j=1

It is not immediate that the right hand side of (C.3.3) in fact defines a norm. medskip THEOREM C.16 (Quantization and trace class). Suppose that a ∈ S (R2n ). Then (C.3.4)

aw (x, D) ∈ L1 (L2 (Rn )).

338

C. FUNCTIONAL ANALYSIS

Proof. Using (4.5.1) we see that a ∈ S (R2n ) is equivalent to Z w a (x, D)u = Au(x) := K(x, y)dy, K ∈ S (Rn × Rn ). Rn

From this we see that A∗ Au(x) =

Z K1 (x, y)dy, Rn

Z K(z, x)K(z, y)dz,

K1 (x, y) := Rn

K1 ∈ S (Rn × Rn ).

Let P0 = −∆ + |x|2 be the quantum harmonic oscillator analyzed in Section 6.1. Theorem 6.2 and the discussion in Section 6.1.2 show that P0−1 : L2 → L2 exists and that the eigevalues of the self-adjoint compact operator P0−1 are given by (2|α| + n)−1 , α ∈ N. In particular, we check that for N ≥ 1, (C.3.5)

sj (P0−N ) ≤ Cj −N/n .

Since (−∆+|x|2 )N K1 (x, y) ∈ S (R2n ) we see that P0N A∗ A is bounded on L2 for any N . We can now use (C.3.1) and (C.3.5) to estimate the singular values of A: 1

1

sj (A) = sj (A∗ A) 2 = sj (P0−N P0N A∗ A) 2 1

1

≤ sj (P0−N ) 2 kP0N A∗ AkL2 2 →L2 ≤ Cj −N/2n . By taking N ≥ 2n + 1 we obtain (C.3.2) and hence (C.3.4).



REMARK. Much finer conditions on a are possible to guarantee that aw (x, D) ∈ L1 . Consult Dimassi-Sj¨ostrand [D-S, Chapter 8]. In particular, (C.3.6)

if a ∈ S(m) and m ∈ L1 (R2n ), then aw (x, D) ∈ L1 (L2 (Rn )). 

THEOREM C.17 (Definition of the trace). Suppose A is of trace class on a Hilbert space H. (i) Let {ej }∞ j=0 be any orthonormal basis of H. Then (C.3.7)

trA :=

∞ X hAej , ej iH . j=0

is finite and independent of the choice of {ej }∞ j=0 . (ii) Suppose that B is a bounded operator on H. Then AB and BA are of trace class and (C.3.8)

tr(AB) = tr(BA).

C.3. TRACE CLASS OPERATORS

339

DEFINITION. We call trA defined by (C.3.7) the trace of A. THEOREM C.18 (Traces of integral operators). Suppose that B is 1 an operator of trace class on L2 (M ; Ω 2 (M )), given by the integral kernel 1

K ∈ C ∞ (M × M ; Ω 2 (M × M )). Then K∆ , the restriction to the diagonal ∆ := {(m, m) | m ∈ M }, has a well-defined density; and Z (C.3.9) tr B = K∆ . ∆

A next theorem relates the trace to the eigenvalues of A. It is easy for self-adjoint operators. THEOREM C.19 (Lidskii’s Theorem). Suppose that A is of trace class and that Spec(A) = {λj }∞ j=0 ⊂ C, |λ0 | ≥ |λ1 | ≥ · · · ≥ |λj | → ∞. Then (C.3.10)

trA =

∞ X j=0

λj .

Appendix D

FREDHOLM THEORY

This appendix describes the role of the Schur complement formula in spectral theory, in particular in analytic Fredholm theory.

D.1. GRUSHIN PROBLEMS Linear algebra. The Schur complement formula states for two-by-two systems of matrices that if  −1   P R− E E+ = , R+ R0 E− E0 then P is invertible if and only if E0 is invertible, with (D.1.1)

P −1 = E − E+ E0−1 E− ,

E0−1 = R0 − R+ P −1 R− .

Grushin problem. The Schur complement formula can be used in infinite dimensions. We will apply it to problems of the form      P R− u v (D.1.2) = R+ O u− v+ where P : X1 → X2 , R+ : X1 → X+ , R− : X− → X2 , for appropriate Banach spaces X1 , X2 , X+ , X− . We call (D.1.2) a Grushin problem. It is useful for reducing problems of infinite dimension to problems of finite dimension. In practice, we start 341

342

D. FREDHOLM THEORY

with an operator P and build a Grushin problem by choosing R± , in which case it is normally sufficient to take R0 = O, as above. If the Grushin problem (D.1.2) is invertible, we call it well-posed and we write its inverse as follows:      v u E E+ (D.1.3) = E− E0 v+ u− for operators E : X2 → X1 , E0 : X+ → X− , E+ : X+ → X1 , E− : X2 → X− . LEMMA D.1 (The operators in a Grushin problem). If (D.1.2) is well-posed, then the operators R+ , E− are surjective and the operators E+ , R− are injective.

D.2. FREDHOLM OPERATORS DEFINITIONS. (i) A bounded linear operator P : X1 → X2 is called a Fredholm operator if the kernel of P , ker P := {u ∈ X1 | P u = 0}, and the cokernel of P , coker P := X2 /P X 1 , where P X1 := {P u | u ∈ X1 }, are both finite dimensional. (The quotient is defined algebraically as the set of equivalence classes for the relation ∼ on X2 : v ∼ v 0 if v − v 0 ∈ P X1 .) (ii) The index of a Fredholm operator is ind P := dim ker P − dim coker P. EXAMPLE. Many important Fredholm operators have the form (D.2.1)

P = I + K,

where K a compact operator mapping a Banach space X to itself. Theorem D.3 below shows that the index does not change under continuous deformations of Fredholm operators (with respect to operator norm topology). Hence for operators of the form (D.2.1) the index is 0: ind P = ind(I + tK) = ind I = 0

(0 ≤ t ≤ 1). 

The connection between Grushin problems and Fredholm operators is this:

D.2. FREDHOLM OPERATORS

343

THEOREM D.2 (Grushin problem for Fredholm operators). (i) Suppose that P : X1 → X2 is a Fredholm operator. Then there exist finite dimensional spaces X± and operators R− : X− → X2 , R+ : X1 → X+ , for which the Grushin problem (D.1.2) is well posed. As a consequence P X1 ⊂ X2 is closed. (ii) Conversely, suppose that that for some choice of spaces X± and operators R± , the Grushin problem (D.1.2) is well posed. Then P : X1 → X2 is a Fredholm operator if and only if E0 : X+ → X− is a Fredholm operator; in which case (D.2.2)

ind P = ind E0 .

Assertion (ii) is particularly useful when the spaces X± are finite dimensional. Proof. 1. Assume P : X1 → X2 is Fredholm. Let n+ := dim ker P and n− := dim coker P , and write X+ := Cn+ , X− := Cn− . Select then linear operators R− : X− → X2 , R+ : X1 → X+ , of ranks n∓ respectively, such that R− X− ∩ P X1 = {0},

ker(R+ |ker P ) = {0}.

Then the operator 

 P R− R+ O has a trivial kernel and is onto. Hence it is invertible, and by the Open Mapping Theorem the inverse is continuous. In particular, consider P acting on the quotient space X1 / ker P , which is a Banach space since ker P is closed. We have n+ = 0, and  
X1 / ker P P X1 = P (X1 / ker P ) = P R− {0} is a closed subspace. 2. Conversely, suppose that Grushin problem (D.1.2) is well-posed. According to Lemma D.1, the operators R+ , E− are surjective, and the operators E+ , R− are injective. We take u− = 0. Then ( the equation P u = v is equivalent to (D.2.3) u = Ev + E+ v+ , 0 = E− v + E0 v+ . This means that E− : Im P → Im E0 ,

344

D. FREDHOLM THEORY

and so we can define the induced map E # : X2 / Im P → X− / Im E0 . Since E− is surjective, so is E # . Also, ker E # = {0}. This follows since if E− v ∈ Im E0 , we can use (D.2.3) to deduce that v ∈ Im P . Hence E # is a bijection of the cokernels X2 / Im P and X− / Im E0 . 3. Next, we claim that E+ : ker E0 → ker P is a bijection. Indeed, if u ∈ ker P , then u = E+ v+ and E0 v+ = 0. Therefore E+ is onto; and this is all we need check, since E+ injective. We conclude that dim ker P = dim ker E0 , dim coker P = dim coker E0 . In particular, the indices of P and E0 are equal.



THEOREM D.3 (Invariance of the index under deformations). The set of Fredholm operators is open in L(X1 , X2 ), and the index is constant in each component. Proof. When P is a Fredholm operator, we can use Theorem D.2 to obtain E0 : Cn+ → Cn− , with (D.2.4)

ind E0 = n+ − n−

according to the Rank-Nullity Theorem of linear algebra. The Grushin problem remains well-posed with the same operators R± , if P is replaced by P 0 , provided kP − P 0 k <  for some sufficiently small  > 0. Hence the set of Fredholm operators is open. Using (D.2.4) we see that the index of P 0 is the same as the index of P . Consequently it remains constant in each connected component of the set of Fredholm operators. 

D.3. MEROMORPHIC CONTINUATION The Grushin problem framework also provides an elegant proof of the following standard result: THEOREM D.4 (Analytic Fredholm Theory). Suppose U ⊂ C is a connected open set and {A(z)}z∈U is a family of Fredholm operators depending holomorphically on z. Then if A(z0 )−1 exists at some point z0 ∈ U , the mapping z 7→ A(z)−1 is a meromorphic family of operators on U .

D.3. MEROMORPHIC CONTINUATION

345

Proof. 1. Fix z1 ∈ U . We form a Grushin problem for P = A(z1 ), as z1 also provide described in the proof of Theorem D.2. The same operators R± a well-posed Grushin problem for P = A(z) for z in some sufficiently small neighborhood V (z1 ) of z1 . According to Theorem D.3 ind A(z) = ind A(z0 ) = 0. Consequently n+ = n− = n, z1 and E0 (z) is an n×n matrix with holomorphic coefficients. of E0z1 (z) is equivalent to the invertibility of A(z).

The invertibility

2. This shows that there exists a locally finite covering {Uj }j∈J of U and a family of functions fj , holomorphic in Uj , such that if z ∈ Uj , then A(z) is invertible precisely when fj (z) 6= 0. Indeed, we can define fj := det E0z , where E0z exists for z ∈ Uj by the construction in Step 1. Since U is connected and since A(z0 ) is invertible for at least one z0 ∈ U , none of the functions fj is identically zero. So det E0 (z) is a non-trivial holomorphic function in V (z1 ); and consequently E0 (z)−1 is a meromorphic family of matrices. Applying (D.1.1), we conclude that A(z)−1 = E(z) − E+ (z)E−+ (z)−1 E− (z) is a meromorphic family of operators in the neighborhood V (z1 ). Since z1 was arbitrary, A(z)−1 is in fact meromorphic in all of U . 

346

D. FREDHOLM THEORY

Notes for the Appendices: The short book [dC] by do Carmo is a good introduction to differential forms. See also Warner [W, Chapter 2]. For a quick and elegant presentation of the Inverse Function Theorem, see H¨ ormander [H1, Chapter 1]; and for the Schwartz Kernel Theorem, consult H¨ ormander [H1, Chapter 5], or Friedlander–Joshi [F-J, Section 6.1]. For proofs of other theorems in Appendix C, and in particular the Spectral Theorem, see Reed–Simon [R-S]. For trace class operators and properties of singular values see Simon [Si]. A review of spectral theory, with semiclassical problems in mind, is in Dimassi–Sj¨ostrand [D-S, Chapter 4]. The first example in Section C.2 comes from H¨ormander [H] and was suggested to us by M. Hitrik. I. Hirshberg provided us with the proof of Theorem C.4. Appendix D follows the presentation in [S-Z2]. We refer to H¨ormander [H2, Sect.19.1] for a comprehensive introduction to Fredholm operators. Consult Zhang [Zh] for the Schur complement formula and its applications.

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Index

Beals’s Theorem, 13, 165–173, 189, 205, 230, 252 Birkhoff’s Ergodic Theorem, 299 Borel’s Theorem, 75, 213, 258, 265

defect measure, semiclassical, 99–116 definition, 100 examples, 102–103 on torus, 107 properties, 101, 104–106 diffeomorphism, 274 differential forms, 19, 317–324 differential operator, 279 distribution, 39, 60, 77, 278, 279, 314, 325 dynamics classical, 11, 17–18 quantum, 11

Cartan’s formula, 25, 27, 235, 321, 323 Cauchy-Riemann operator, 38, 270 characteristic variety, 104 Coarea Formula, 298 commutator, 11, 63, 70, 105 composition formula, 12 Weyl quantization, 68 conjugation, 137, 257, 264 and symbols, 137 by Fourier transform, 58, 66 by unitary operators, 228, 232, 236–238, 241, 247, 300 coordinate patch, 273 cotangent bundle, 275 canonical symplectic form on, 276 integral over, 276 cotangent space, 276 Cotlar–Stein Theorem, 86, 89, 327

Egorov’s Theorem, 13, 227–232 for long times, 13, 245–252 weak, 300–301 Ehrenfest time, 13, 245–252 eigenfunctions, 13, 117–133, 139, 148 basis of, 126, 153, 289, 290 clusters of, 153, 225 concentration in phase space, 122 equidistribution of, 297, 310 exponential decay estimates, 139 for harmonic oscillator, 118–122, 125 for Laplace–Beltrami operator, 289, 310 for pseudodifferential operator, 180 on manifolds, 289 order of vanishing, 148 regularity, 287 eigenvalues, 51, 118, 121, 268, 289, 295, 302, 330, 337, 338

adjoint, 60 almost analytic extension, 38, 290, 292 asymptotic sum, 74 atlas, 273 average in time, 109, 298 of symbols, 304

351

352

Index

and trace, 339 counting, 336 for harmonic oscillator, 118–122 for Laplace–Beltrami operator, 153, 224, 289, 294 for Schr¨ odinger’s equation, 9, 12, 117, 267, 289 minimax formulas for, 335 of matrix, 40 of operator, 329 elliptic estimates, 136 symbol, 90, 129, 140, 143, 152, 209, 223, 240, 258, 264, 269, 270 energy decay, 113–116 surface, 298 wave equation, 108 ergodicity, 14 classical, 298–300 quantum, 297–310 essential support, 185–187 estimates Agmon, 138 Carleman, 13, 142 elliptic, 136 notation for, 315–316 Schauder, 13 Strichartz, 13, 217–221 exponential map, 18

dual space of, 177 examples, 176 pseudodifferential operators and, 179, 180 geodesic flow, 297 Grushin problems, 341–342

flow map, 18 Fourier decomposition, 67 integral operator, 13, 193, 211, 225, 227 Fourier transform, 31–43 exponential of imaginary quadratic form, 40 exponential of real quadratic form, 32 on S , 32 on S 0 , 39 semiclassical, 42–43 Fredholm operator, 342 theory, 341–345 functional calculus, 133, 287, 290–294, 302

kernel Schwartz, 61, 302

generalized Sobolev space Hh (m), 175–180 definition, 176

half-density, 191–198, 216–217, 278, 279 Hamilton–Jacobi equation, 13, 212–215, 220, 240 Hamiltonian dynamics, 11 harmonic oscillator, 117–123, 131 Weyl’s Law for, 121 heat equation, 266 Helffer-Sj¨ ostrand formula, 290, 291, 293, 295 Helmholtz’s equation, 254 hypoellipticity condition, 140, 141 estimate, 140 Implicit Function Theorem, 298 index of Fredholm operator, 342 inequality Fefferman–Phong, 92 G˚ arding, 12, 74, 91–95, 101, 138, 141 Hardy-Littlewood-Sobolev, 218 interpolation, 218, 222, 225 inverse, 90–91 Inverse Function Theorem, 52, 325, 346

Laplace–Beltrami operator, 153, 224, 284, 286, 289, 294, 309 Lidskii’s Theorem, 339 Liouville measure, 298 Littlewood–Paley theory, 154–156, 158, 162 localization, 149, 181, 188 manifolds, 273–295 definition of, 273 PDE on, 286–295 pseudodifferential operators on, 278–285 Riemannian, 277–278 smooth functions on, 274 matrices J, 19 notation for, 312

Index

symplectic, 232, 243 transition, 274 Mean Ergodic Theorem, 299 meromorphic family of operators, 110, 344 resolvents, 126, 128 microlocality, 188 microlocally invertible, 188 Morse Lemma, 49, 51–53 nondegeneracy condition, 19, 51, 151, 152, 219, 221, 222, 225, 264 normal forms, 253–270 complex symbols, 263–267 real symbols, 256–259 notation, 311–316 basic, 311 for estimates, 315–316 for functions, 312–315 for matrices, 312 for operators, 315 for sets, 312 multiindex, 313 observables, 10, 57 order functions, 74 change of, 176, 177 definition, 73 examples, 73 log of, 175 order of vanishing, 144–148 oscillatory integral, 12, 43, 49 Planck’s constant, 9, 11 Poincar´e’s Lemma, 213, 235, 322 on manifolds, 324 Poisson bracket, 11, 24, 105, 301, 314 principal type, 257, 258, 263 projection, 125 propagation of singularities, 259–263 pseudodifferential operators, 57–95 on manifold, 280 symbol of, 281, 282, 284 pseudospectrum, 269 quantization and commutators, 63 composition, 68 Fourier decomposition, 67 general, 58 linear symbols, 61, 62 on torus, 106–108 standard, 58

353

symbols exponentials of linear symbols, 63 exponentials of quadratic symbols, 65 symbols depending on x only, 61 symbols linear in x, 62 Weyl, 10, 12, 58 quantum mechanics, 9, 192 Heisenberg picture, 228, 252, 300 quantum observable, 281 quasimode, 149–152, 222–223, 267–269 rescaling, 10, 59, 94, 124 standard, 59 Riemannian manifold, 277–278 Riesz Representation Theorem, 101 Riesz-Thorin Theorem, 218 s-density bundles, 276 Schr¨ odinger’s equation, 9, 13 Schwartz space S , 32 section, 275 self-adjoint operator, 60 seminorm, 32 signature of matrix, 40 singular values, 337 Sobolev space, 136, 176, 279, 284, 288 generalized, 13, 175–180, 259 spectral clusters, 153, 224 spectrum, 126–129, 168, 267–269, 290, 329, 331, 334–337 stationary phase, 43–55 higher dimensional, 49–55 one dimensional, 43–49 Stone’s Theorem, 210, 334, 335 symbol calculus, 57 symbols, 10, 58, 316 depending only on x, 61 distributional, 60 exponentials of linear symbols, 63 exponentials of quadratic symbols, 65 Kohn-Nirenberg, 13, 198–206, 316 linear, 61, 62 linear in x, 62 symplectic mapping, 19–23 matrix, 20–21, 232, 243 product σ, 18, 19, 54 symplectic geometry, 17–30 tangent bundle, 275 tangent space, 275

354

tempered distributions, 39 family of distributions, 180 family of operators, 181 torus, 298 trace, 339 integral operators, 339 trace class, 294, 337, 338 norm, 337 tunneling, 13, 139–144 uncertainty principle, 43, 189 unitary matrix, 233 operators, 85, 100, 124, 170, 208, 210, 211, 228, 232, 236, 237, 240, 241, 243, 300, 329, 333, 335 vector bundles, 274–277 fibers of, 274 sections of, 275 transition matrices, 274 wave equation, 261, 263 damped, 10, 12, 108–116 wavefront set classical, 183 for operators, 187 semiclassical, 182, 184, 185, 189 weight, 141 Weyl’s Law, 13, 129–133, 302 for harmonic oscillator, 121 on manifolds, 294–295 WKB approximation, 253–254

Index