On Schr\"odinger equation with potential U = - {\alpha}r^{-1} + {\beta}r + kr^{2} and the bi-confluent Heun functions theory

TL;DR

This paper demonstrates that the Schrödinger equation with a combined Coulomb, linear, and harmonic potential reduces to a bi-confluent Heun equation, and develops a polynomial solution method with specific parameter constraints, revealing a modified energy spectrum.

Contribution

It introduces a novel approach to solving the Schrödinger equation with a complex potential using bi-confluent Heun functions and constructs polynomial solutions under new parameter constraints.

Findings

Solution reduces to bi-confluent Heun equation

Polynomial solutions are constructed with parameter constraints

Energy spectrum combines oscillator and Coulomb characteristics

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.