2 1. INTRODUCTION

(ii) Conversely, given various mathematical objects associated with clas-

sical mechanics, for instance symplectic transformations, how can we prof-

itably “quantize” them?

In fact the techniques of semiclassical analysis apply in many other set-

tings and for many other sorts of PDE. For example we will later study the

damped wave equation

(1.1.1) ∂t

2u

+ a∂tu − Δu = 0

for large times. A rescaling in time will introduce the requisite small pa-

rameter h.

1.1.2. Basic techniques. We will construct, mostly in Chapters 2–4, 8–9,

and 14, a wide variety of mathematical tools to address these issues, among

them:

• the apparatus of symplectic geometry (to record succinctly the behav-

ior 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 expres-

sions involving rescaled Fourier transforms); and

• pseudodifferential operators (to localize or, as is said in the trade, to

microlocalize functional behavior in phase space).

1.1.3. Microlocal analysis. There is a close relation between asymptotic

properties of PDE with a small parameter and regularity of solutions to

PDE. Asymptotic properties of ˆ(ξ) u as 1/|ξ| =: h → 0 are related to

C∞

regularity of u. For instance, we will see in Chapter 12 how to obtain

results about propagation of singularities for general classes of equations.

Answering questions about propagation of singularities has been one of the

motivations of microlocal analysis, and most of the techniques presented

in this book, such as pseudodifferential operators, come from that subject.

Roughly speaking, in standard microlocal analysis 1/|∂x| plays the role of

h. These ideas are behind the study of the damped wave equation (1.1.1).

Some techniques developed for pure PDE questions, such as local solv-

ability, have acquired a new life when translated to the semiclassical setting.

An example is the study of pseudospectra of nonselfadjoint operators; see

Chapter 12. Another example is the connection between tunneling and