Quantum decay rates in chaotic scattering

TL;DR

This paper establishes a link between the classical dynamics of chaotic systems and quantum decay rates, showing that a more filamentary trapped set leads to a spectral gap and faster quantum decay.

Contribution

It proves a lower bound on quantum decay rates based on the classical trapped set's dimension and topological pressure for a broad class of operators.

Findings

Existence of a spectral gap for operators with hyperbolic classical flows.

Quantum decay rate is bounded below by classical trapped set properties.

Resolvent estimates with logarithmic loss are established.

Abstract

In this article we prove that for a large class of operators, including Schroedinger operators, with hyperbolic classical flows, the smallness of dimension of the trapped set implies that there is a gap between the resonances and the real axis. In other words, the quantum decay rate is bounded from below if the classical repeller is sufficiently filamentary. The higher dimensional statement is given in terms of the topological pressure. Under the same assumptions we also prove a resolvent estimate with a logarithmic loss compared to nontrapping estimates.