Extended Nikiforov-Uvarov method, roots of polynomial solutions, and functional Bethe ansatz method

TL;DR

This paper extends the Nikiforov-Uvarov method and compares it with the functional Bethe ansatz to analyze polynomial solutions of second-order differential equations relevant in quantum mechanics, revealing how integration constants relate to roots.

Contribution

It introduces an extended Nikiforov-Uvarov method for equations with multiple singular points and connects its parameters to roots via symmetric polynomials, enhancing solution techniques.

Findings

Extended the Nikiforov-Uvarov method to equations with more singular points.

Established the relationship between integration constants and roots of polynomial solutions.

Compared two approaches, showing their equivalence under certain conditions.

Abstract

For applications to quasi-exactly solvable Schr\"odinger equations in quantum mechanics, we establish the general conditions that have to be satisfied by the coefficients of a second-order differential equation with at most $k+1$ singular points in order that this equation has particular solutions which are $n$th-degree polynomials. In a first approach, we extend the Nikiforov-Uvarov method, which was devised to deal with hypergeometric-type equations (i.e., for $k=2$), and show that the conditions involve $k-2$ integration constants. In a second approach, we consider the functional Bethe ansatz method in its most general form. Comparing the two approaches, we prove that under the assumption that the roots of the polynomial solutions are real and distinct, the $k-2$ integration constants of the extended Nikiforov-Uvarov method can be expressed as linear combinations of monomial symmetric polynomials in those roots, corresponding to partitions into no more than two parts.