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Polynomial solutions to the generalized Lam\'e equation, the Stieltjespolynomials, and the associated Van Vleck polynomials have been studied sincethe 1830's in various contexts including the solution of Laplace equations onan ellipsoid. Recently there has been renewed interest in the distribution ofthe zeros of Van Vleck polynomials as the degree of the corresponding Stieltjespolynomials increases. In this paper we show that the zeros of Van Vleckpolynomials corresponding to Stieltjes polynomials of successive degreesinterlace. We also show that the spectral polynomials formed from the Van Vleckzeros are not orthogonal with respect to any weight. This furnishes acounterexample, coming from a second order differential equation, to theconverse of the well known theorem that the zeros of orthogonal polynomialsinterlace.

Interlacing and non-orthogonality of spectral polynomials for the Lam\'e operator