A remark on imaginary part of resonance points

TL;DR

This paper establishes a mathematical relationship between the imaginary part of resonance points and the scattering phase change in quantum scattering, aligning with the Breit-Wigner formula, and extends to general trace class perturbations.

Contribution

It proves a new formula linking resonance point properties with scattering phase change for rank one perturbations and generalizes to trace class perturbations.

Findings

Negative twice reciprocal of imaginary part equals rate of change of scattering phase

Formula aligns with Breit-Wigner resonance theory

Provides spectral shift function formula involving resonance points

Abstract

In this paper we prove for rank one perturbations that negative two times reciprocal of the imaginary part of resonance point is equal to the rate of change of the scattering phase as a function of the coupling constant, where the coupling constant is equal to the real part of the resonance point. This equality is in agreement with Breit-Wigner formula from quantum scattering theory. For general relatively trace class perturbations, we also give a formula for the spectral shift function in terms of resonance points, non-real and real.