Perturbations of eigenvalues embedded at threshold: one, two and three dimensional solvable models

TL;DR

This paper studies how eigenvalues and resonances at the spectrum threshold of multi-channel quantum models are affected by perturbations, providing algorithms for series expansions of spectral singularities in 1D, 2D, and 3D.

Contribution

It introduces algorithms for convergent series expansions of spectral singularities caused by perturbations in multi-channel quantum Hamiltonians across different dimensions.

Findings

Eigenvalues and resonances at the spectrum threshold are sensitive to perturbations.

Series expansion algorithms for spectral singularities are developed and validated.

The analysis covers one, two, and three-dimensional models.

Abstract

We examine perturbations of eigenvalues and resonances for a class of multi-channel quantum mechanical model-Hamiltonians describing a particle interacting with a localized spin in dimension $d=1,2,3$. We consider unperturbed Hamiltonians showing eigenvalues and resonances at the threshold of the continuous spectrum and we analyze the effect of various type of perturbations on the spectral singularities. We provide algorithms to obtain convergent series expansions for the coordinates of the singularities.