Position Dependent Mass Schroedinger Equation and Isospectral Potentials : Intertwining Operator approach

TL;DR

This paper develops a systematic method using intertwining operators to generate isospectral potentials in the position-dependent mass Schrödinger equation, enabling spectral modifications like adding or removing bound states.

Contribution

It introduces a general second-order intertwining approach for arbitrary potentials and mass functions, extending the toolkit for spectral engineering in quantum systems.

Findings

Generated isospectral potentials with added, removed, or unchanged bound states.

Established consistency with N-fold supersymmetry for specific cases.

Provided a complete solution scheme applicable to any potential and mass function.

Abstract

Here we have studied first and second-order intertwining approach to generate isospectral partner potentials of position-dependent (effective) mass Schroedinger equation. The second-order intertwiner is constructed directly by taking it as second order linear differential operator with position depndent coefficients and the system of equations arising from the intertwining relationship is solved for the coefficients by taking an ansatz. A complete scheme for obtaining general solution is obtained which is valid for any arbitrary potential and mass function. The proposed technique allows us to generate isospectral potentials with the following spectral modifications: (i) to add new bound state(s), (ii) to remove bound state(s) and (iii) to leave the spectrum unaffected. To explain our findings with the help of an illustration, we have used point canonical transformation (PCT) to obtain the general solution of the position dependent mass Schrodinger equation corresponding to a potential and mass function. It is shown that our results are consistent with the formulation of type A N-fold supersymmetry [14,18] for the particular case N = 1 and N = 2 respectively.