Supersymmetrical Separation of Variables for Scarf II Model: Partial Solvability

TL;DR

This paper explores a supersymmetry-based method to partially solve a two-dimensional quantum model generalizing Scarf II for arbitrary parameters, revealing quasi-exact solvability and shape invariance properties.

Contribution

It introduces a novel SUSY-separation approach for the model with arbitrary parameters, extending previous solutions limited to integer values.

Findings

Partial analytical solutions for the spectrum and wave functions.

Identification of two variants of shape invariance in the model.

Demonstration of quasi-exact solvability through supersymmetry.

Abstract

Recently, a new quantum model - two-dimensional generalization of the Scarf II - was completely solved analytically by SUSY method for the integer values of parameter. Now, the same integrable model, but with arbitrary values of parameter, will be studied by means of supersymmetrical intertwining relations. The Hamiltonian does not allow the conventional separation of variables, but the supercharge operator does allow, leading to the partial solvability of the model. This approach, which can be called as the first variant of SUSY-separation, together with shape invariance of the model, provides analytical calculation of the part of spectrum and corresponding wave functions (quasi-exact-solvability). The model is shown to obey two different variants of shape invariance which can be combined effectively in construction of energy levels and wave functions.