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arXiv:2002.06438

Symmetries and Supersymmetries of Generalized Schrödinger equations

A. G. Nikitin

Published Feb 15, 2020• 0 citations
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Abstract

In this survey the contemporary results concerning supersymmetries in generalized Schrödinger equations are presented. Namely, position dependent mass Schödinger equations are discussed as well as the equations with matrix potentials. An extended number of realistic quantum mechanical problems admitting extended supersymmetries is described, an extended class of matrix potentials is classified.

DOI
10.48550/arxiv.2002.06438

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In this survey the contemporary results concerning supersymmetries in generalized Schrödinger equations are presented. Namely, position dependent mass Schödinger equations are discussed as well as the equations with matrix potentials. An extended number of realistic quantum mechanical problems admitting extended supersymmetries is described, an extended class of matrix potentials is classified.
A new method for generating exactly solvable Schrödinger equations with a position-dependent mass is proposed. It is based on a relation with some deformed Schrödinger equations, which can be dealt with by using a supersymmetric quantum mechanical approach combined with a deformed shape-invariance condition. The solvability of the latter is shown to impose the form of both the deformed superpotential and the position-dependent mass. The conditions for the existence of bound states are determined. A lot of examples are provided and the corresponding bound-state spectrum and wavefunctions are reviewed.
An algebraic method of constructing potentials for which the Schrödinger equation with position dependent mass can be solved exactly is presented. A general form of the generators of su (1,1) algebra has been employed with a unified approach to the problem. Our systematic approach reproduces a number of earlier results and also leads to some novelties. We show that the solutions of the Schrödinger equation with position dependent mass are free from the choice of parameters for position dependent mass. Two classes of potentials are constructed that include almost all exactly solvable potentials.