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

h
a dx
of the sort
Ih =
(2πh)n/2|
det
∂2ϕ(x0)|−1/2e

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 arding inequality, learn when
aw
is a bounded
operator on
L2,
etc.
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