4 1. INTRODUCTION for appropriate smooth functions u. This is Weyl’s quantization formula, and aw(x, hD) is a pseudodifferential operator. One major task will be to understand how the analytic properties of the symbol a dictate the functional analytic properties of its quantization aw(x, hD). We will in fact build up a symbol calculus, meaning systematic rules for manipulating pseudodifferential operators. 1.2.2. Dynamics. We will be concerned as well with the evolution in time of classical particles and quantum states. Classical evolution. Our most important example will concern the symbol p(x, ξ) := |ξ|2 + V (x), corresponding to the phase space flow ˙ = 2ξ ˙ = −∂V, where ˙ = ∂t. We generalize by introducing the arbitrary Hamiltonian p : R2n → R, p = p(x, ξ), and the corresponding Hamiltonian dynamics (1.2.1) ˙ = ∂ξp(x, ξ) ˙ = −∂xp(x, ξ). It is instructive to change our viewpoint somewhat, by writing ϕt = exp(tHp) for the solution of (1.2.1), where Hpq := {p, q} = ∂ξp, ∂xq− ∂xp, ∂ξq is the Poisson bracket. Select a symbol a and define (1.2.2) at(x, ξ) := a(ϕt(x, ξ)). Then (1.2.3) ˙ t = {p, at}, and this equation tells us how the symbol evolves in time, as dictated by the classical dynamics (1.2.1).

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