Semiclassical wave propagation for large times

TL;DR

This paper investigates the accuracy duration of semiclassical wave approximations for the Schrödinger equation on negatively curved surfaces, especially in chaotic classical systems, extending understanding beyond the Ehrenfest time.

Contribution

It demonstrates that on surfaces of constant negative curvature, semiclassical approximations remain accurate for times up to h^(-1/2) when associated with unstable manifolds.

Findings

Semiclassical approximations are valid up to time ~ h^(-1/2) for certain states.

Accuracy extends beyond the Ehrenfest time in specific chaotic settings.

Results apply to surfaces of constant negative curvature.

Abstract

We study solutions of the time dependent Schr\"odinger equation on Riemannian manifolds with oscillatory initial conditions given by Lagrangian states. Semiclassical approximations describe these solutions for small h (where h is the semiclassical parameter), but their accuracy for large times is in general only understood up to the Ehrenfest time T ~ ln(1/h), and the most difficult case is the one where the underlying classical system is chaotic. We show that on surfaces of constant negative curvature semiclassical approximations remain accurate for times at least up to T ~ h^(-1/2) in the case that the Lagrangian state is associated with an unstable manifold of the geodesic flow.