Scattering resonances as viscosity limits

TL;DR

This paper demonstrates that scattering resonances can be approximated as limits of eigenvalues of a modified operator with an imaginary quadratic potential, providing a rigorous foundation for a computational chemistry method.

Contribution

It establishes a rigorous link between scattering resonances and eigenvalues of a perturbed operator using complex scaling, justifying a computational approach.

Findings

Resonances are limits of eigenvalues of the modified operator as epsilon approaches zero.

The method applies to potentials in L-infinity with compact support.

Provides a theoretical foundation for a computational chemistry technique.

Abstract

Using the method of complex scaling we show that scattering resonances of $ - \Delta + V $, $ V \in L^\infty_{\rm{c}} ( \mathbb R^n ) $, are limits of eigenvalues of $ - \Delta + V - i \epsilon x^2 $ as $ \epsilon \to 0+ $. That justifies a method proposed in computational chemistry and reflects a general principle for resonances in other settings.