Coulomb problem for a Dirac particle in flat Minkowski space and the Heun functions, extension to curved models

TL;DR

This paper explores solving the Coulomb problem for a Dirac particle using Heun functions in flat and curved spacetimes, extending methods to handle equations with more singular points in curved models.

Contribution

It introduces a method to extend the Heun function approach from flat Minkowski space to curved spaces of constant curvature, handling equations with additional singular points.

Findings

Multiple methods to treat the Coulomb problem with Heun functions in flat space.

Extension of the approach to curved spaces with constant curvature.

Development of a technique to reduce equations with six singular points to five.

Abstract

It is shown that there exist several ways to treat the quantum-mechanical Coulomb problem for a Dirac particle in flat Minkowski space with the help of the Heun differential equation, Fuchs's equation with four singular points. When extending the problem to curved spaces of constant curvature, Lobachevsky H_{3} and Rimann S_{3}, there arise 2-nd order differential equations of the Fuchs type with 6 singular points, a method to get relevant equations with five singular points has been elaborated.