SUSY-Approach for Investigation of Two-Dimensional Quantum Mechanical Systems

TL;DR

This paper explores advanced supersymmetry methods to analyze complex two-dimensional quantum systems that cannot be separated by traditional means, introducing new techniques for spectral analysis and algebraic construction of eigenstates.

Contribution

It introduces a novel SUSY separation of variables method and generalizes shape invariance to two-dimensional models, expanding the analytical tools for quantum systems.

Findings

Supercharges enable separation of variables even when Hamiltonians do not.

Polynomial supersymmetry relates scalar Hamiltonians with nearly identical spectra.

New algebraic methods construct eigenstates for 2D Morse potentials.

Abstract

Different ways to incorporate two-dimensional systems, which are not amenable to separation of variables, into the framework of Supersymmetrical Quantum Mechanics (SUSY QM) are analyzed. In particular, the direct generalization of one-dimensional Witten's SUSY QM is based on the supercharges of first order in momenta and allows to connect the eigenvalues and eigenfunctions of two scalar and one matrix Schr\"odinger operators. The use of second order supercharges leads to polynomial supersymmetry and relates a pair of scalar Hamiltonians, giving a set of such partner systems with almost coinciding spectra. This class of systems can be studied by means of new method of $SUSY-$separation of variables, where supercharges {\bf allow} separation of variables, but Hamiltonians {\bf do not}. The method of shape invariance is generalized to two-dimensional models to construct purely algebraically a chain of eigenstates and eigenvalues for generalized Morse potential models in two dimensions.