Quasi-exactly solvable Schr\"odinger equations, symmetric polynomials, and functional Bethe ansatz method

TL;DR

This paper explores conditions for second-order differential equations to have polynomial solutions, linking symmetric polynomials and the functional Bethe ansatz, with applications to quasi-exactly solvable quantum systems.

Contribution

It establishes a connection between integration constants, symmetric polynomials, and the functional Bethe ansatz in the context of quasi-exactly solvable Schrödinger equations.

Findings

Conditions involve $k-2$ integration constants satisfying linear equations.

Constants can be expressed as linear combinations of monomial symmetric polynomials.

Application to a quasi-exactly solvable extension of the Mathews-Lakshmanan oscillator.

Abstract

For applications to quasi-exactly solvable Schr\"odinger equations in quantum mechanics, we consider 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 that are $n$th-degree polynomials. In a first approach, we show that such conditions involve $k-2$ integration constants, which satisfy a system of linear equations whose coefficients can be written in terms of elementary symmetric polynomials in the polynomial solution roots whenver such roots are all real and distinct. In a second approach, we consider the functional Bethe ansatz method in its most general form under the same assumption. Comparing the two approaches, we prove that the above-mentioned $k-2$ integration constants can be expressed as linear combinations of monomial symmetric polynomials in the roots, associated with partitions into no more than two parts. We illustrate these results by considering a quasi-exactly solvable extension of the Mathews-Lakshmanan nonlinear oscillator corresponding to $k=4$.