6 1. INTRODUCTION

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8 Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13 Chapter 14

Chapter 15

Here is a quick overview of the book, with some of the highlights:

Chapter 2: We start with a quick introduction to symplectic analysis and

geometry and their implications for classical Hamiltonian dynamical sys-

tems.

Chapter 3: This chapter provides the basics of the Fourier transform and

derives also important stationary phase asymptotic estimates for the oscil-

latory integral

Ih :=

Rn

e

iϕ

h

a dx

of the sort

Ih =

(2πh)n/2|

det

∂2ϕ(x0)|−1/2e

iπ

4

sgn

∂2ϕ(x0)e

iϕ(x0)

h

a(x0) + O h

n+2

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 com-

position 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

operator on

L2,

etc.