Series solutions of Bessel-type differential equation in terms of orthogonal polynomials and physical applications

TL;DR

This paper derives exact solutions for a complex Bessel-type differential equation using orthogonal polynomial series, and applies these solutions to quantum mechanics problems involving novel potentials.

Contribution

It introduces a new class of solutions expressed as orthogonal polynomial series for a six-parameter Bessel-type equation, with applications to Schrödinger equations.

Findings

Solutions expressed as bounded series of Bessel polynomials

Orthogonal polynomial coefficients depend on equation parameters

Application to Schrödinger equation with new potential functions

Abstract

We obtain a class of exact solutions of a Bessel-type differential equation, which is a six-parameter linear ordinary differential equation of the second order with irregular (essential) singularity at the origin. The solutions are obtained using the Tridiagonal Representation Approach (TRA) as bounded series of square integrable functions written in terms of the Bessel polynomial on the real line. The expansion coefficients of the series are orthogonal polynomials in the equation parameters space. We use our findings to obtain solutions of the Schr\"odinger equation for some novel potential functions.