2018/02/28 by C. Quesne
Physics and Astronomy · Mathematics · #math-ph #math.MP #nlin.SI #quant-ph
published as Acta Polytech. 58(2):118-127, 2018 · 20 pages, no figure, published version. arXiv admin note: substantial text overlap with arXiv:1704.01406
arxiv created 2018/05/09 · arxiv updated 2018/05/11
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 nth-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.