An example of resonance instability

TL;DR

This paper constructs a semiclassical Schrödinger operator demonstrating that small, compactly supported potential perturbations can cause significant shifts in the imaginary parts of resonances near the real axis.

Contribution

It introduces a specific operator example showing how tiny potential changes can induce order-h shifts in resonance imaginary parts, revealing resonance instability.

Findings

Small potential perturbations can cause large shifts in resonance imaginary parts.

Resonance positions are highly sensitive to localized potential changes.

The constructed operator exemplifies resonance instability phenomena.

Abstract

We construct a semiclassical Schr\"{o}dinger operator such that the imaginary part of its resonances closest to the real axis changes by a term of size $h$ when a real compactly supported potential of size $o ( h )$ is added.