SUSY Method for the Three-Dimensional Schr\"odinger Equation with Effective Mass

TL;DR

This paper applies SUSY Quantum Mechanics to solve the three-dimensional Schrödinger equation with position-dependent mass, deriving analytical solutions and exploring spectral properties and symmetries.

Contribution

It provides the first general solution to SUSY intertwining relations with first order supercharges for this problem, including analytical expressions for mass functions and potentials.

Findings

Analytical expressions for mass functions and partner potentials.

Spectra of intertwined Hamiltonians coincide up to zero modes.

Models possess at least one second order symmetry operator.

Abstract

The three-dimensional Schr\"odinger equation with a position-dependent (effective) mass is studied in the framework of Supersymmetrical (SUSY) Quantum Mechanics. The general solution of SUSY intertwining relations with first order supercharges is obtained without any preliminary constraints. Several forms of coefficient functions of the supercharges are investigated and analytical expressions for the mass function and partner potentials are found. As usual for SUSY Quantum Mechanics with nonsingular superpotentials, the spectra of intertwined Hamiltonians coincide up to zero modes of supercharges, and the corresponding wave functions are connected by intertwining relations. All models are partially integrable by construction: each of them has at least one second order symmetry operator.