Widths of highly excited shape resonances

TL;DR

This paper investigates the widths of shape resonances in semiclassical multi-dimensional Schrödinger operators, providing an optimal lower bound under specific conditions on the resonant state within the well.

Contribution

It introduces a new lower bound for resonance widths in multi-dimensional semiclassical quantum systems, assuming certain conditions on the resonant state.

Findings

Established an optimal lower bound for resonance widths

Analyzed the behavior of resonant states inside the well

Focused on systems with frequencies above the well's bottom

Abstract

We study the widths of shape resonances for the semiclassical multi-dimensional Schr\"odinger operator, in the case where the frequency remains close to some value strictly larger than the bottom of the well. Under a condition on the behavior of the resonant state inside the well, we obtain an optimal lower bound for the widths.