QUASI-EXACTLY SOLVABLE SCHRÖDINGER EQUATIONS, SYMMETRIC POLYNOMIALS AND FUNCTIONAL BETHE ANSATZ METHOD

For applications to quasi-exactly solvable Schrödinger 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.

Abstract licence: CC BY

24 more references, whose wording the paper’s licence keeps private.

A person counts once a day per paper. We keep no cookie and no address: a reader is a hash that changes every day. Any month under five reads shows no number.