# Quasi-exactly solvable Schr\"odinger equations, symmetric polynomials, and functional Bethe ansatz method. (arXiv:1802.02902v2 [math-ph] UPDATED)

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 $k=4$.