New Methods for Two-Dimensional Schr\"odinger Equation: SUSY-separation of Variables and Shape Invariance

TL;DR

This paper introduces two novel methods for solving two-dimensional quantum systems that are not separable, utilizing SUSY separation of variables and a generalized shape invariance framework, enabling partial analytical solutions.

Contribution

The paper develops a formal framework for applying SUSY separation of variables and shape invariance to two-dimensional quantum systems, extending these concepts beyond one dimension.

Findings

Partial explicit solvability of a 2D quantum system achieved.

New methods enable analytical construction of some eigenfunctions.

Framework extends shape invariance to higher dimensions.

Abstract

Two new methods for investigation of two-dimensional quantum systems, whose Hamiltonians are not amenable to separation of variables, are proposed. 1)The first one - $SUSY-$ separation of variables - is based on the intertwining relations of Higher order SUSY Quantum Mechanics (HSUSY QM) with supercharges allowing for separation of variables. 2)The second one is a generalization of shape invariance. While in one dimension shape invariance allows to solve algebraically a class of (exactly solvable) quantum problems, its generalization to higher dimensions has not been yet explored. Here we provide a formal framework in HSUSY QM for two-dimensional quantum mechanical systems for which shape invariance holds. Given the knowledge of one eigenvalue and eigenfunction, shape invariance allows to construct a chain of new eigenfunctions and eigenvalues. These methods are applied to a two-dimensional quantum system, and partial explicit solvability is achieved in the sense that only part of the spectrum is found analytically and a limited set of eigenfunctions is constructed explicitly.