Novel exactly solvable Schr\"odinger equations with a position-dependent mass in multidimensional spaces obtained from duality

TL;DR

This paper introduces a new exactly solvable Schrödinger equation with a position-dependent mass in multiple dimensions, extending duality concepts between oscillator and Coulomb problems to curved spaces.

Contribution

It extends the duality between oscillator and Coulomb problems to include position-dependent mass in multidimensional curved spaces, providing new exactly solvable models.

Findings

Derived a mapping between nonlinear oscillator and Coulomb problems in curved spaces.

Extended duality to include position-dependent mass in multidimensional Schrödinger equations.

Provided explicit solutions for the new PDM Schrödinger equation.

Abstract

A novel exactly solvable Schr\"odinger equation with a position-dependent mass (PDM) describing a Coulomb problem in $D$ dimensions is obtained by extending the known duality relating the quantum $d$-dimensional oscillator and $D$-dimensional Coulomb problems in Euclidean spaces for $D = (d+2)/2$. As an intermediate step, a mapping between a quantum $d$-dimensional nonlinear oscillator of Mathews-Lakshmanan type (or oscillator in a space of constant curvature) and a quantum $D$-dimensional Coulomb-like problem in a space of nonconstant curvature is derived. It is finally reinterpreted in a PDM background.