Confluent Heun functions and the Coulomb problem for spin ½ particle in Minkowski space.

The quantum mechanical problem for a spin ½ particle in external Coulomb potential, reduced to a system of two first-order differential equations, is reconsidered from the point of view of solving this system by using the Heun function theory. It is shown that, besides the standard approach of solving the problem in terms of confluent hypergeometric functions, there are several other possibilities, which rely on using the confluent Heun functions. We consider two new methods to construct the solutions of the problem: the first implies that only one component of the pair of relevant functions is expressed in terms of the Heun functions, and in the second approach both functions of the system are expressed in terms of the Heun functions. In this context, certain relations between the two classes of involved functions are established. It is shown that all the considered cases lead to the same energy spectrum, which validates the correctness of the approaches. [ABSTRACT FROM AUTHOR]