On Schrödinger equation with potential U = - αr^{-1} + βr + kr^{2} and the bi-confluent Heun functions theory

Abstract

It is shown that Schrödinger equation with combination of three potentials U = - α r^{-1} + β r + kr^{2}, Coulomb, linear and harmonic, the potential often used to describe quarkonium, is reduced to a bi-confluent Heun differential equation. The method to construct its solutions in the form of polynomials is developed, however with additional constraints in four parameters of the model, α, β, k, l. The energy spectrum looks as a modified combination of oscillator and Coulomb parts.

arXiv comment8 pages, 1 figure, Report to 18 Conference NPCS-2011, May 17-20, 2011, Minsk, Belarus