Extension of Nikiforov-Uvarov Method for the Solution of Heun Equation

TL;DR

This paper introduces an extended Nikiforov-Uvarov method to solve second order differential equations with up to four singular points, successfully applying it to Heun equations and quantum mechanical problems.

Contribution

The paper develops an extended NU method by modifying polynomial degrees, enabling solutions to Heun equations and related quantum problems not previously accessible with standard NU.

Findings

Eigenvalues for Heun and confluent Heun equations obtained

Applied to quantum problems like Coulomb on a sphere and double-well potential

Demonstrates effectiveness of extended NU method in complex differential equations

Abstract

We report an alternative method to solve second order differential equations which have at most four singular points. This method is developed by changing the degrees of the polynomials in the basic equation of Nikiforov-Uvarov (NU) method. This is called extended NU method for this paper. The eigenvalue solutions of Heun equation and confluent Heun equation are obtained via extended NU method. Some quantum mechanical problems such as Coulomb problem on a 3-sphere, two Coulombically repelling electrons on a sphere and hyperbolic double-well potential are investigated by this method.