Semiclassical resolvent estimates in chaotic scattering

TL;DR

This paper establishes polynomial bounds on the resolvent for semiclassical operators in chaotic scattering, linking classical chaos with quantum estimates, and applies these results to Schrödinger smoothing and wave decay.

Contribution

It provides new resolvent estimates in chaotic scattering scenarios, connecting classical chaotic dynamics with quantum resolvent bounds in a semiclassical setting.

Findings

Resolvent bounds are polynomial in the semiclassical parameter h.

The bounds depend on topological pressure related to classical flow.

Applications include improved local smoothing and energy decay results.

Abstract

We prove resolvent estimates for semiclassical operators such as $-h^2 \Delta+V(x)$ in scattering situations. Provided the set of trapped classical trajectories supports a chaotic flow and is sufficiently filamentary, the analytic continuation of the resolvent is bounded by $h^{-M}$ in a strip whose width is determined by a certain topological pressure associated with the classical flow. This polynomial estimate has applications to local smoothing in Schr\"odinger propagation and to energy decay of solutions to wave equations.