Fractal upper bounds on the density of semiclassical resonances

TL;DR

This paper establishes upper bounds on the number of semiclassical resonances near the real axis, relating them to the dimension of trapped trajectories using advanced microlocal analysis techniques.

Contribution

It introduces new upper bounds on resonance density in semiclassical problems based on the trapped set dimension, employing second microlocalization methods.

Findings

Resonance counts are bounded by the trapped set dimension.

Second microlocalization is effective for resonance analysis.

Provides a new link between classical dynamics and quantum resonances.

Abstract

For semiclassical problems we establish upper bounds on the number of resonances in boxes of size $h$ along the real axis, in terms of the dimension of the set of trapped trajectories. The proof uses second microlocalization.