New classes of quasi-solvable potentials, their exactly-solvable limit and related orthogonal polynomials

TL;DR

This paper introduces new classes of quasi-solvable elliptic potentials using an sl(2,R) approach, which transition to exactly solvable forms, and provides explicit solutions and orthogonal polynomials related to these potentials.

Contribution

The paper develops novel quasi-solvable elliptic potentials via an sl(2,R) formalism and derives explicit solutions and orthogonal polynomials, expanding the understanding of solvable quantum models.

Findings

New classes of quasi-solvable elliptic potentials generated

Explicit solutions for spectral problems obtained

Orthogonal polynomials related to these potentials explicitly expressed

Abstract

We have generated, using an sl(2,R) formalism, several new classes of quasi-solvable elliptic potentials, which in the appropriate limit go over to the exactly solvable forms. We have obtained exact solutions of the corresponding spectral problems for some real values of the potential parameters. We have also given explicit expressions of the families of associated orthogonal polynomials in the energy variable.