Counter-examples to the fractal Weyl law for semiclassical resonances

TL;DR

This paper presents counterexamples demonstrating that the known upper bounds on the number of semiclassical resonances, related to the fractal dimension of the trapped set, are not always tight.

Contribution

The authors provide explicit examples of operators with fewer resonances than predicted by the fractal Weyl law, challenging the sharpness of existing bounds.

Findings

Counterexamples with fewer resonances than the upper bounds

Shows the fractal Weyl law bounds are not always sharp

Highlights limitations of current semiclassical resonance estimates

Abstract

Under general assumptions, the numbers of semiclassical resonances is known to be bounded from above by a negative power of $h$ which is given by the fractal dimension of the trapped set. In this paper we provide examples of operators with much less resonances, showing that these upper bounds are not always sharp.