1.2. CLASSICAL AND QUANTUM MECHANICS 3

unique continuation. These were developed independently in physics and in

mathematics and are unified nicely by semiclassical Carleman estimates; see

Chapter 7.

1.1.4. Other directions. This book is devoted to semiclassical analysis as

a branch of linear PDE theory. The ideas explored here are useful in other

areas. One is the study of quantum maps where symplectic transformations

on compact manifolds are quantized to give matrices. The semiclassical

parameter is then related to the size of the matrix. These are popular

models in physics partly due to the relative ease of numerical computations;

see Haake [Hak] and references in Chapter 13 of this text. Many other

large N limit problems enjoy semiclassical interpretation, in the sense of

connecting analysis to geometry. In this book we present one example: a

semiclassical proof of Quillen’s Theorem (Theorem 13.18) which is related

to Hilbert’s 17th problem.

Semiclassical concepts also appear in the study of nonlinear PDE. One

direction is provided by nonlinear equations with an asymptotic parameter

which in some physically motivated problems plays a role similar to h in

Section 1.1.1 above. One natural equation is the Gross-Pitaevskii nonlin-

ear Schr¨ odinger equation; see for instance the book by Carles [Car]. An

example of a numerical study is given in Potter [Po] where a semiclassical

approximation is used to describe solitons in an external field.

Another set of microlocal methods useful in nonlinear PDE is provided

by the paradifferential calculus of Bony, Coifman, and Meyer; see for instance

M´ etivier [Me], and for a brief introduction see B´enyi–Maldonado–Naibo

[B-M-N]. The semiclassical parameter appears in the Littlewood-Paley

decomposition just as it does in Chapter 7, while the pseudodifferential

classes are more exotic than the ones considered in Chapter 4.

1.2. CLASSICAL AND QUANTUM MECHANICS

We introduce and foreshadow a bit about quantum and classical correspon-

dences.

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 denotes momentum. We usually call a a symbol.

Let h 0 be given. We will associate with the observable a a correspond-

ing quantum observable

aw(x,

hD), an operator defined by the formula

aw(x,

hD)u(x) :=

1

(2πh)n

Rn Rn

e

i

h

x−y,ξ

a

(

x+y

2

, ξ

)

u(y) dξdy