Abstract
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 <em>k </em>+ 1 singular points in order that this equation has particular solutions that are <em>n</em>th-degree polynomials. In a first approach, we show that such conditions involve <em>k </em>- 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 <em>k </em>- 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 <em>k </em>= 4.
Highlights
In quantum mechanics, solving the Schrödinger equation is a fundamental problem for understanding physical systems
Direct comparison between equations (2.10), (2.11) and equations (3.16), (3.17) shows that ck−2 is given by the same expression in both approaches, while for l = 0, 1, . . . , k − 3, cl is written in terms of al+2 and bl+1, as well as the integration constant Ck−l−2,n in the first one or a linear combination of monomial symmetric polynomials in z1, z2, . . . , zn in the second one
We have reconsidered 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 the equation has particular solutions that are nth-degree polynomials yn(z) and we have expressed them in terms of symmetric polynomials in the polynomial solution roots
Summary
In quantum mechanics, solving the Schrödinger equation is a fundamental problem for understanding physical systems. 2. Second-order differential equations with polynomial solutions and integration constants. Ck−2,n satisfy a system of linear equations whose coefficients can be expressed in terms of elementary symmetric polynomials in z1, z2, . After substituting the right-hand sides of equations (2.10) and (2.11) for ck−2 and ck−r+p in these relations, we obtain a system of k − 2 linear equations for the k − 2 integration constants C1,n, C2,n, . Whose coefficients are expressed in terms of elementary symmetric polynomials (2.13) in the roots of the polynomial solutions of equation (2.1) This is the second main result of this paper.
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