Shape-Resonance in Spectral density, Scattering Cross-section, Time delay and Bound on Sojourn time

TL;DR

This paper revisits the Friedrichs model to derive precise asymptotic behaviors of resonances, spectral concentration, and related scattering properties, providing new insights into spectral analysis and quantum scattering phenomena.

Contribution

It offers exact asymptotic results for resonance behavior, spectral concentration, and scattering metrics within the Friedrichs model, extending to rank-one Laplacian perturbations.

Findings

Asymptotic formulas for resonances near embedded eigenvalues

Spectral concentration results derived from abstract analysis

Exact asymptotics for sojourn time, scattering amplitude, and time delay

Abstract

The Friedrichs model~\cite{Friedrichs} is revisited to obtain precise results about the asymptotic behaviour (the so-called Breit-Wigner formula~\cite{Breit}) of a resonance near an embedded eigenvalue and the ``spectral concentration" results as a corollary. Some of the abstract results involved can also be used to address similar questions about a rank-one perturbation of the Laplacian. Exact asymptotic properties are also obtained for the sojourn time, the scattering amplitude and time delay.