Semiclassical Dynamics¶with Exponentially Small Error Estimates

Semiclassical Dynamics with Exponentially Small Error Estimates

We construct approximate solutions to the time–dependent Schrödinger equation i h̄ ∂ψ ∂t = − h̄ 2 2 ∆ψ + V ψ for small values of h̄. If V satisfies appropriate analyticity and growth hypotheses and |t| ≤ T , these solutions agree with exact solutions up to errors whose norms are bounded by C exp {− γ/h̄ } , for some C and γ > 0. Under more restrictive hypotheses, we prove that for sufficiently smal...

Exponentially Small Estimates for Separatrix Splittings

ABSTRACT This paper reviews our previous estimates and gives an example exhibiting a new phenomenon. In problems involving asymptotics beyond all orders in a perturbation parameter ε, it is a common assumption that the quantity being studied (such as a separatrix splitting distance or angle, a solitary wave mismatch, etc.) can be “estimated” by an expression of the form aεbe−c/ε as ε → 0. Here,...

A Time–Dependent Born–Oppenheimer Approximation with Exponentially Small Error Estimates

We present the construction of an exponentially accurate time–dependent Born– Oppenheimer approximation for molecular quantum mechanics. We study molecular systems whose electron masses are held fixed and whose nuclear masses are proportional to ǫ−4, where ǫ is a small expansion parameter. By optimal truncation of an asymptotic expansion, we construct approximate solutions to the time–dependent...

Simplified Semiclassical Propagation Estimates

Semiclassical L Estimates

The purpose of this paper is to use semiclassical analysis to unify and generalize L estimates on high energy eigenfunctions and spectral clusters. In our approach these estimates do not depend on ellipticity and order, and apply to operators which are selfadjoint only at the principal level. They are estimates on weakly approximate solutions to semiclassical pseudodifferential equations. To mo...