Deformed algebras, position-dependent effective masses and curved spaces: An exactly solvable Coulomb problem

TL;DR

This paper explores the deep connections between deformed algebras, position-dependent masses, and curved spaces in quantum mechanics, demonstrating their equivalence through a solvable Coulomb problem with unique bound state properties.

Contribution

It establishes the equivalence of three unconventional Schrödinger equations and solves a new Coulomb problem using supersymmetric methods, revealing finite bound states.

Findings

Connections between deformed algebras, masses, and curved spaces established

A new Coulomb problem with finite bound states solved

Supersymmetric techniques applied to unconventional quantum systems

Abstract

We show that there exist some intimate connections between three unconventional Schr\"odinger equations based on the use of deformed canonical commutation relations, of a position-dependent effective mass or of a curved space, respectively. This occurs whenever a specific relation between the deforming function, the position-dependent mass and the (diagonal) metric tensor holds true. We illustrate these three equivalent approaches by considering a new Coulomb problem and solving it by means of supersymmetric quantum mechanical and shape invariance techniques. We show that in contrast with the conventional Coulomb problem, the new one gives rise to only a finite number of bound states.