In the quantization of a rotating rigid body, a top, one is concerned with the Hamiltonian operator Lα = α 2 0 L 2An explicit formula is known for the eigenvalues of Lα in the case of the spherical top (α1 = α2 = α3) and symmetrical top (α1 = α2 = α3) [LL]. However, for the asymmetrical top, no such explicit expression exists, and the study of the spectrum is much more complex. In this paper, we compute the semiclassical density of states for the eigenvalues of the family of operators Lα = α 2 0 L 2 x +α 2 1 L 2 y +α 2 2 L 2 z for any α0 < α1 < α2.

a b s t r a c tPolynomial solutions to the generalized Lamé equation, the Stieltjes polynomials, and the associated Van Vleck polynomials, have been studied extensively in the case of real number parameters. In the complex case, relatively little is known. Numerical investigations of the location of the zeros of the Stieltjes and Van Vleck polynomials in special cases reveal intriguing patterns in the complex case, suggestive of a deeper structure. In this article we report on these investigations, with the main result being a proof of a theorem confirming that the zeros of the Van Vleck polynomials lie on special line segments in the case of the complex generalized Lamé equation having three free parameters. Furthermore, as a result of this proposition, we are able to obtain in this case a strengthening of a classical result of Heine on the number of possible Van Vleck polynomials associated with a given Stieltjes polynomial.

Polynomial solutions to the Heine-Stieltjes equation, the Stieltjes polynomials, and the associated Van Vleck polynomials have been studied since the 1830's in various contexts including the solution of the Laplace equation on an ellipsoid. Recently there has been renewed interest in the distribution of the zeros of Van Vleck polynomials as the degree of the corresponding Stieltjes polynomials increases. In this paper we show that the zeros of Van Vleck polynomials corresponding to Stieltjes polynomials of successive degrees interlace. We also show that the spectral polynomials formed from the Van Vleck zeros are not orthogonal with respect to any measure. This furnishes a counterexample, coming from a second order differential equation, to the converse of the well-known theorem that the zeros of orthogonal polynomials interlace.

Abstract. Polynomial solutions to the generalized Lamé equation, the Stieltjes polynomials, and the associated Van Vleck polynomials have been studied since the 1830's, beginning with Lamé in his studies of the Laplace equation on an ellipsoid, and in an ever widening variety of applications since. In this paper we show how the zeros of Stieltjes polynomials are distributed and present two new interlacing theorems. We arrange the Stieltjes polynomials according to their Van Vleck zeros and show, firstly, that the zeros of successive Stieltjes polynomials of the same degree interlace, and secondly, that the zeros of certain Stieltjes polynomials of successive degrees interlace.

In this paper, we compute the leading term in the asymptotic expansion of the trace of a family of Jacobi matrices as their order gets arbitrarily large. We then apply this formula to compute the density of states of the zeros of a class of polynomials associated with the Heun differential equation.