Lamé differential equations and electrostatics

Abstract:

The problem of existence and uniqueness of polynomial solutions of the Lamé differential equation \begin{equation*} A(x) y^{\prime \prime } + 2 B(x) y’ + C(x) y = 0, \end{equation*} where $A(x), B(x)$ and $C(x)$ are polynomials of degree $p+1, p$ and $p-1$, is under discussion. We concentrate on the case when $A(x)$ has only real zeros $a_{j}$ and, in contrast to a classical result of Heine and Stieltjes which concerns the case of positive coefficients $r_{j}$ in the partial fraction decomposition $B(x)/A(x) = \sum _{j=0}^{p} r_{j}/(x-a_{j})$, we allow the presence of both positive and negative coefficients $r_{j}$. The corresponding electrostatic interpretation of the zeros of the solution $y(x)$ as points of equilibrium in an electrostatic field generated by charges $r_{j}$ at $a_{j}$ is given. As an application we prove that the zeros of the Gegenbauer-Laurent polynomials are the points of unique equilibrium in a field generated by two positive and two negative charges.