Resonances in two-electron atoms below the critical charge

TL;DR

This paper investigates the transition of bound states into shape resonances in two-electron atoms as the nuclear charge approaches the critical value, using advanced variational and complex rotation methods.

Contribution

It provides the first direct calculation of resonance energies and widths below the critical charge, confirming the smooth transition hypothesis with a new combined variational and complex rotation approach.

Findings

Resonance widths vary smoothly with nuclear charge Z.

The resonance behavior agrees with 1/Z perturbation series analysis.

The study confirms the transition from bound state to shape resonance at Zc.

Abstract

The critical nuclear charge Zc required for a heliumlike atom to have at least one bound state was recently determined with high accuracy from variational calculations. Analysis of the wave functions further suggested that the bound state changes smoothly into a shape resonance as Z crosses the critical value. Using variational calculations combined with the complex coordinate rotation method, we study the energy and width of the resonance for Z \textless{} Zc, thus providing direct evidence of the validity of this hypothesis. The variation of the resonance width with Z is found to be in good agreement with a model derived from analysis of the 1/Z perturbation series.