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(x 0 ) 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.

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