Three-Dimensional Shape Invariant Non-Separable Model With Equidistant Spectrum

TL;DR

This paper constructs three-dimensional quantum models with shape invariance and equidistant spectra, which are not separable but have real spectra and solvable quadratic cases, revealing new symmetry properties.

Contribution

It introduces a new class of non-separable 3D models with shape invariance and real, equidistant spectra, including an analytically solvable quadratic potential case.

Findings

Models have real, equidistant spectra similar to harmonic oscillators.

The quadratic interaction case is solved analytically.

Hamiltonian is non-diagonalizable with explicit wave functions.

Abstract

A class of three-dimensional models which satisfy supersymmetric intertwining relations with the simplest - oscillator-like - variant of shape invariance is constructed. It is proved that the models are not amenable to conventional separation of variables for the complex potentials, but their spectra are real and equidistant (like for isotropic harmonic oscillator). The special case of such potential with quadratic interaction is solved completely. The Hamiltonian of the system is non-diagonalizable, and its wave functions and associated functions are built analytically. The symmetry properties of the model and degeneracy of energy levels are studied.