A Class of Partially Solvable Two-Dimensional Quantum Models with Periodic Potentials

TL;DR

This paper employs supersymmetry to analyze a class of two-dimensional quantum models with periodic potentials, achieving partial solvability and revealing spectral properties for models generalizing classical potentials.

Contribution

It introduces a supersymmetrical method for partially solving 2D quantum models with periodic potentials that are not separable by conventional means.

Findings

Partial energy spectra and wave functions found for several models

Models generalize Lame and Razavy potentials in 2D

Lame potential exhibits self-isospectrality

Abstract

The supersymmetrical approach is used to analyse a class of two-dimensional quantum systems with periodic potentials. In particular, the method of SUSY-separation of variables allowed us to find a part of the energy spectra and the corresponding wave functions (partial solvability) for several models. These models are not amenable to conventional separation of variables, and they can be considered as two-dimensional generalizations of Lame, associated Lame, and trigonometric Razavy potentials. All these models have the symmetry operators of fourth order in momenta, and one of them (the Lame potential) obeys the property of self-isospectrality.