Three-Dimensional Solutions of Supersymmetrical Intertwining Relations and Pairs of Isospectral Hamiltonians

TL;DR

This paper derives general three-dimensional supersymmetric intertwining relations for Schrödinger operators, constructing models with non-separable variables that are partially integrable and can be extended to non-Hermitian cases.

Contribution

It provides the first comprehensive solution for 3D SUSY intertwining relations using second order supercharges with a nondegenerate metric, including models with arbitrary parameters.

Findings

Constructed Hamiltonians commute with fourth-order symmetry operators.

Models include non-separable quantum systems.

Extension to non-Hermitian isospectral Hamiltonians is possible.

Abstract

The general solution of SUSY intertwining relations for three-dimensional Schr\"odinger operators is built using the class of second order supercharges with nondegenerate constant metric. This solution includes several models with arbitrary parameters. We are interested only in quantum systems which are not amenable to separation of variables, i.e. can not be reduced to lower dimensional problems. All constructed Hamiltonians are partially integrable - each of them commutes with a symmetry operator of fourth order in momenta. The same models can be considered also for complex values of parameters leading to a class of non-Hermitian isospectral Hamiltonians.