8 1. INTRODUCTION

integral operators. Applications include the semiclassical Strichartz esti-

mates and

Lp

bounds on eigenfunction clusters.

Chapter 11: This chapter proves Egorov’s Theorem, characterizing propa-

gators 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 chap-

ter by showing that Egorov’s Theorem is in fact valid until times of order

log(h−1),

the so-called Ehrenfest time.

Chapter 12: Chapter 12 illustrates how methods from Chapter 11 provide

elegant and useful normal forms of differential and pseudodifferential oper-

ators. Among the applications, we build quasimodes for certain nonnormal

operators and discuss the implications for pseudospectra.

Chapter 13: We consider the question of how close semiclassical quantiza-

tion can get to multiplication. This leads to an alternative presentation of

the semiclassicall calculus based on Toeplitz quantization acting on spaces

of holomorphic functions. The FBI–Bargmann transform intertwines the

quantization of Chapter 4 with the quantization by operators acting on

holomorphic functions.

Chapter 14: Chapter 14 briefly discusses general manifolds and modifica-

tions to the symbol calculus to cover pseudodifferential operators on mani-

folds. Chapter 9 provides the change of variables formulas we need to work

with coordinate patches.

Chapter 15: 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

hn

a≤Ej ≤b

Auj,uj − −

{a≤p≤b}

σ(A) dxdξ

2

→ 0

as h → 0, for a wide class of pseudodifferential operators A. In this expres-

sion 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.