New Exactly Solvable Two-Dimensional Quantum Model Not Amenable to Separation of Variables

TL;DR

This paper introduces a new two-dimensional quantum model that is exactly solvable despite not allowing standard separation of variables, expanding the class of solvable models in quantum mechanics.

Contribution

It presents a novel two-dimensional quantum model based on supersymmetric intertwining relations, which is exactly solvable and generalizes the P"oschl-Teller potential without separation of variables.

Findings

All bound state energy eigenvalues are analytically determined.

An algorithm for calculating all wave functions is provided.

The model exhibits shape invariance and integrability.

Abstract

The supersymmetric intertwining relations with second order supercharges allow to investigate new two-dimensional model which is not amenable to standard separation of variables. The corresponding potential being the two-dimensional generalization of well known one-dimensional P\"oschl-Teller model is proven to be exactly solvable for arbitrary integer value of parameter $p:$ all its bound state energy eigenvalues are found analytically, and the algorithm for analytical calculation of all wave functions is given. The shape invariance of the model and its integrability are of essential importance to obtain these results.