A limit of the confluent Heun equation and the Schroedinger equation for an inverted potential and for an electric dipole

TL;DR

This paper extends solutions of the confluent Heun equation and applies them to solve the Schrödinger equation for an inverted potential and an electric dipole, providing convergent series solutions for these quantum problems.

Contribution

It introduces new series solutions for the confluent Heun equation and applies them to specific quantum mechanical problems involving inverted potentials and electric dipoles.

Findings

Finite- and infinite-series solutions are convergent and bounded for all variables.

Series expansions in Jacobi polynomials are obtained for the angular equation.

The solutions extend the class of solvable models in quantum mechanics.

Abstract

We reexamine and extend a group of solutions in series of Bessel functions for a limiting case of the confluent Heun equation and, then, apply such solutions to the one-dimensional Schr\"odinger equation with an inverted quasi-exactly solvable potential as well as to the angular equation for an electron in the field of a point electric dipole. For the first problem we find finite- and infinite-series solutions which are convergent and bounded for any value of the independent variable. For the angular equation, we also find expansions in series of Jacobi polynomials.