Semiclassical analysis

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

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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 Stéphane 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 Sjöstrand, 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.

Poincaré’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


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


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


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


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 Poincaré’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 . Poincaré’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


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


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


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


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


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


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. 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


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.


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


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


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


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


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 .


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


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 }.


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


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 Hörmander [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


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


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


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


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∗ ).


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


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


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.


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


(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


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 ∗ ).

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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


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


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


4.4.4. 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


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


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


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


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


≤

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


4.7. 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


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


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–Sjöstrand [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–Gérard [A-G], Grigis–Sjöstrand [G-S], Martinez [M] and Saint Raymond [SR]. A major treatise is Hörmander [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 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 Hölder 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

0n+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


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


(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


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 ).

174


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 ),

176


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.


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 .

178


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).


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.

180


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

Semiclassical analysis

Semiclassical Analysis

Semiclassical Analysis

Lectures on semiclassical analysis