1.3. OVERVIEW 5 Quantum evolution. We can quantize the foregoing by putting P = pw(x, hD), A = aw(x, hD) and defining (1.2.4) A(t) := F −1 (t)AF(t) for F (t) := e− itP h . The operator A(t) represents, according to the so-called Heisenberg picture of quantum mechanics, the evolution of the quantum observable A under the flow (1.2.1). Then we have the evolution equation (1.2.5) ∂tA(t) = i h [P, A(t)], an obvious analogue of (1.2.3). Here then is a basic principle we will later work out in some detail: an assertion about Hamiltonian dynamics, and so the Poisson bracket {·, ·}, will involve at the quantum level the commutator [·, ·]. REMARK: h and ¯. In this book h denotes a dimensionless parameter and is consequently not immediately to be identified with the dimensional physical quantity = Planck’s constant/2π = 1.05457 × 10−34joule-sec. As the example of the damped wave equation (1.1.1) shows, the use of h → 0 asymptotics is not restricted to problems motivated by quantum mechanics. 1.3. OVERVIEW Chapters 2–4 develop the basic machinery, followed by applications to partial differential equations in Chapters 5–7. We develop more advanced theory and applications in Chapters 8–13, and in Chapters 14 and 15 we discuss semiclassical analysis on manifolds. The following diagram indicates the dependencies of the chapters and may help in selective reading of the book:

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