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

Abstract

The object of study in this paper is the on-shell scattering matrix $S(E)$ of the Schrödinger 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 $E_{\text{res}}-\epsilon$ to $E_{\text{res}}+\epsilon$, where $E_{\text{res}}$ is a real part of a resonance, and $\epsilon$ 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.

arXiv comment22 pages, 1 figure