Analytic results in the position-dependent mass Schrodinger problem

TL;DR

This paper derives analytical solutions for the Schrödinger equation with position-dependent mass in specific potential scenarios, revealing hypergeometric and Heun class solutions and eigenstates for hyperbolic potentials.

Contribution

It provides new analytical solutions for the position-dependent mass Schrödinger equation with hyperbolic potentials, including hypergeometric and Heun class functions.

Findings

Solutions involve hypergeometric functions for massless case

Eigenbasis found for hyperbolic-tangent potential

Eigenstates computed for sinh^2 potential

Abstract

We investigate the Schrodinger equation for a particle with a nonuniform solitonic mass density. First, we discuss in extent the (nontrivial) position-dependent mass $V(x)=0$ case whose solutions are hypergeometric functions in $\tanh^2(x)$. Then, we consider an external hyperbolic-tangent potential. We show that the effective quantum mechanical problem is given by a Heun class equation and find analytically an eigenbasis for the space of solutions. We also compute the eigenstates for a potential of the form $V(x)=V_0 \sinh^2(x)$.