Remarks on the Solution of the Position Dependent Mass (PDM) Schr\"odinger Equation

TL;DR

This paper introduces an approximate analytical method using asymptotic Taylor expansion to solve the position-dependent mass Schrödinger equation, providing eigenfunctions and eigenvalues without potential-mass transformation.

Contribution

It presents a novel asymptotic Taylor expansion approach for solving PDM Schrödinger equations, avoiding potential-mass space transformation and revealing non-isospectrality.

Findings

Method yields approximate eigenfunctions and eigenvalues.

PDM and constant mass Schrödinger equations are not isospectral.

Algorithm implemented with symbolic and numerical computation.

Abstract

An approximate method is proposed to solve position dependent mass Schr\"odinger equation. The procedure suggested here leads to the solution of the PDM Schr\"odinger equation without transforming the potential function to the mass space or vice verse. The method based on asymptotic Taylor expansion of the function, produces an approximate analytical expression for eigenfunction and numerical results for eigenvalues of the PDM Schr\"odinger equation. The results show that PDM and constant mass Schr\"odinger equations are not isospectral. The calculations are carried out with the aid of a computer system of symbolic or numerical calculation by constructing a simple algorithm.