Some Properties of the Shape Invariant Two-Dimensional Scarf II Model

TL;DR

This paper analyzes the two-dimensional Scarf II quantum model within SUSY QM, revealing its exact and quasi-exact solvability, spectrum degeneracies, and symmetry properties across different parameter regions.

Contribution

It systematizes previous results and uncovers new properties, including spectrum degeneracies and symmetry operator actions, for the integrable Scarf II model.

Findings

Model is exactly or quasi-exactly solvable in various parameter regions.

Spectrum degeneracy occurs at specific parameter values.

Symmetry operators of fourth order in momenta are explicitly calculated.

Abstract

Two-dimensional Scarf~II quantum model is considered in the framework of Supersymmetrical Quantum Mechanics (SUSY QM). Previously obtained results for this integrable system are systematized, and some new properties are derived. In particular, it is shown that the model is exactly or quasi-exactly solvable in different regions of parameter of the system. The degeneracy of the spectrum is detected for some specific values of parameters. The action of the symmetry operators of fourth order in momenta is calculated for the arbitrary wave functions, obtained by means of double shape invariance.