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

Abstract: It is shown that Schr\"odinger equation with combination of three potentials U = - {\alpha} r^{-1} + {\beta} 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, {\alpha}, {\beta}, k, l. The energy spectrum looks as a modified combination of oscillator and Coulomb parts.