Real stabilization of resonance states employing two parameters: basis set size and coordinate scaling

TL;DR

This paper introduces a method to accurately compute resonance energies and widths for quantum systems using variational expansions with real basis functions, leveraging basis set size and coordinate scaling for improved convergence.

Contribution

It presents a novel recipe for selecting variational solutions and demonstrates regular scaling behavior, enabling precise approximation of resonance parameters.

Findings

Resonance energies and widths can be accurately computed using the proposed variational approach.

The method shows regular convergence with basis set size, N.

Scaling functions effectively guide the approximation of resonance widths.

Abstract

The resonance states of one- and two-particle Hamiltonians are studied using variational expansions with real basis-set functions. The resonance energies, $E_r$, and widths, $\Gamma$, are calculated using the density of states and an ${\mathcal L}^2$ golden rule-like formula. We present a recipe to select adequately some solutions of the variational problem. The set of approximate energies obtained shows a very regular behaviour with the basis-set size, $N$. Indeed, these particular variational eigenvalues show a quite simple scaling behaviour and convergence when $N\rightarrow \infty$. Following the same prescription to choose particular solutions of the variational problem we obtain a set of approximate widths. Using the scaling function that characterizes the behaviour of the approximate energies as a guide, it is possible to find a very good approximation to the actual value of the resonance width.