The spectrum of the scattering matrix near resonant energies in the semiclassical limit

TL;DR

This paper analyzes how the spectrum of the scattering matrix for a Schrödinger operator behaves near resonant energies in the semiclassical limit, revealing spectral flow related to the Breit-Wigner effect.

Contribution

It provides a detailed description of the spectral flow of the scattering matrix near resonances in the semiclassical regime, connecting it to the Breit-Wigner effect.

Findings

Spectral flow of the scattering matrix near resonances is characterized.

The results relate the spectral flow to the Breit-Wigner effect.

Analysis is conducted in the semiclassical limit for energies close to a resonance.

Abstract

The object of study in this paper is the on-shell scattering matrix $S(E)$ of the Schr\"odinger operator with the potential satisfying assumptions typical in the theory of shape resonances. We study the spectrum of $S(E)$ in the semiclassical limit when the energy parameter $E$ varies from $E_\text{res}-\varepsilon$ to $E_\text{res}+\varepsilon$, where $E_\text{res}$ is a real part of a resonance, and $\varepsilon$ is sufficiently small. The main result of our work describes the spectral flow of the scattering matrix through a given point on the unit circle. This result is closely related to the Breit-Wigner effect.