Semiclassical Dynamics and Exponential Asymptotics

We discuss various constructions of approximate solutions to the time–dependent Schrodinger equation i ~ ∂ψ ∂t = − ~ 2 2 ∆ψ + V ψ in IRd for small values of ~ by means of wave packets that are localized in position and momentum. After reviewing earlier constructions that yield approximations up to O(~) errors, we show that if V satisfies appropriate analyticity and growth hypotheses, similar constructions agree with exact solutions up to errors of order e γ/~ for some γ > 0, provided t belongs to some fixed, compact time interval. Under more restrictive hypotheses, we extend the validity of these exponential estimates up to the Ehrenfest time scale: For sufficiently small T , |t| ≤ T ′ | log(~)| implies that the norm of the error is of order e γ ′/~σ for some γ > 0, and σ > 0.