Fractal Weyl law for open quantum chaotic maps

TL;DR

This paper proves a fractal Weyl upper bound for resonances in open quantum chaotic maps, linking fractal trapped sets with quantum scattering phenomena, applicable to multiple obstacle scattering problems.

Contribution

It establishes a fractal Weyl law for the distribution of resonances in open quantum systems with fractal trapped sets, advancing understanding of quantum chaos and scattering.

Findings

Proved a fractal Weyl upper bound for resonances near the real axis.

Applied results to scattering by multiple convex obstacles under no-eclipse condition.

Extended the law to systems with fractal hyperbolic trapped sets.

Abstract

We study the semiclassical quantization of Poincar\'e maps arising in scattering problems with fractal hyperbolic trapped sets. The main application is the proof of a fractal Weyl upper bound for the number of resonances/scattering poles in small domains near the real axis. This result encompasses the case of several convex (hard) obstacles satisfying a no-eclipse condition.