Intertwining of exactly solvable generalized Schrodinger equations

TL;DR

This paper develops a method using Darboux transformations to generate and analyze exactly solvable generalized Schrödinger equations with position-dependent mass and energy-dependent potentials, linking supersymmetry and isospectral potentials.

Contribution

It introduces explicit intertwining operators and generalized Darboux transformations for arbitrary order, expanding the toolkit for solving complex quantum systems.

Findings

Constructed explicit intertwining operators for generalized Schrödinger equations.

Developed methods to generate isospectral and fully isospectral potentials.

Illustrated the approach with practical examples.

Abstract

The Darboux transformation operator technique in differential and integral forms is applied to the generalized Schrodinger equation with a position-dependent effective mass and with linearly energy-dependent potentials. Intertwining operators are obtained in an explicit form and used for constructing generalized Darboux transformations of an arbitrary order. A relation between supersymmetry and the generalized Darboux transformation is considered. The method is applied to generation of isospectral potentials with additional or removal bound states or construction of new partner potentials without changing the spectrum, i.e. fully isospectral potentials. The method is illustrated by some examples.