Exactly solvable models of supersymmetric quantum mechanics and connection to spectrum generating algebra

TL;DR

This paper demonstrates the equivalence between supersymmetric quantum mechanics and spectrum generating algebra methods for shape invariant Hamiltonians, developing an algebraic framework involving nonlinear Lie algebra extensions.

Contribution

It introduces a unified algebraic framework connecting supersymmetric and group theoretic solutions for shape invariant Hamiltonians.

Findings

Equivalence of supersymmetric and algebraic methods established

Development of nonlinear Lie algebra extensions for shape invariance

Analytic expressions for eigenvalues and eigenvectors derived

Abstract

For nonrelativistic Hamiltonians which are shape invariant, analytic expressions for the eigenvalues and eigenvectors can be derived using the well known method of supersymmetric quantum mechanics. Most of these Hamiltonians also possess spectrum generating algebras and are hence solvable by an independent group theoretic method. In this paper, we demonstrate the equivalence of the two methods of solution by developing an algebraic framework for shape invariant Hamiltonians with a general change of parameters, which involves nonlinear extensions of Lie algebras.