On Schr\"odinger equation with potential U = - αr-1 + βr + kr2 and the bi-confluent Heun functions theory
Abstract
It is shown that Schr\"odinger equation with combination of three potentials U = - α r-1 + β r + kr2, 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.
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