Quasi-exact solvability and intertwining relations

TL;DR

This paper explores the use of intertwining relations to construct and analyze one-dimensional and multidimensional quasi-exactly solvable operators, revealing connections to nonlinear supersymmetry and providing new examples beyond monomial subspaces.

Contribution

It introduces a generalized approach using intertwining relations for constructing quasi-exactly solvable operators, including multidimensional cases and operators with non-monomial invariant subspaces.

Findings

Intertwining relations relate quasi-exactly solvable operators to nonlinear parasupersymmetry.

Quantum systems with higher-order nonlinear supersymmetry are generally not quasi-exactly solvable.

New examples of quasi-exactly solvable operators with non-monomial invariant subspaces are constructed.

Abstract

We emphasize intertwining relations as a universal tool in constructing one-dimensional quasi-exactly solvable operators and offer their possible generalization to the multidimensional case. Considered examples include all quasi-exactly solvable operators with invariant subspaces of monomials. We show that the simplest case of generalized intertwining relations allows to naturally relate quasi-exactly solvable operators with two invariant monomial subspaces to a nonlinear parasupersymmetry of second order. Quantum-mechanical systems with linear and nonlinear supersymmetry are discussed from the viewpoint of quasi-exact solvability. We construct such a general system with a cubic supersymmetry and argue that quantum-mechanical systems with nonlinear supersymmetry of fourth order and higher are generally not quasi-exactly solvable. Besides, we construct two examples of quasi-exactly solvable operators with invariant subspaces which cannot be reduced to monomial spaces.