Physical applications of second-order linear differential equations that admit polynomial solutions

TL;DR

This paper establishes conditions for second-order linear differential equations to have polynomial solutions and explores their applications to quantum mechanics and special functions, providing explicit criteria for several important equations.

Contribution

It offers explicit conditions for polynomial solutions of second-order differential equations and applies these to key equations in physics and mathematics, including Heun equations.

Findings

Conditions for polynomial solutions of differential equations are derived.

Applications to Schrödinger's equation are discussed.

New classes of exactly solvable equations are identified.

Abstract

Conditions are given for the second-order linear differential equation P3 y" + P2 y'- P1 y = 0 to have polynomial solutions, where Pn is a polynomial of degree n. Several application of these results to Schroedinger's equation are discussed. Conditions under which the confluent, biconfluent, and the general Heun equation yield polynomial solutions are explicitly given. Some new classes of exactly solvable differential equation are also discussed. The results of this work are expressed in such way as to allow direct use, without preliminary analysis.