ar X iv : 0 81 2 . 23 21 v 1 [ m at h - ph ] 1 2 D ec 2 00 8 ON SPECTRAL POLYNOMIALS OF THE HEUN EQUATION

The classical Heun equation has the form  Q(z) d 2 dz 2 + P (z) d dz + V (z) ff S(z) = 0, where Q(z) is a cubic complex polynomial, P (z) is a polynomial of degree at most 2 and V (z) is at most linear. In the second half of the nineteenth century E. Heine and T. Stieltjes in [5], [13] initiated the study of the set of all V (z) for which the above equation has a polynomial solution S(z) of a given degree n. The main goal of the present paper is to study the union of the roots of the latter set of V (z)'s when n → ∞. We formulate an intriguing conjecture of K. Takemura describing the limiting set and give a substantial amount of additional information obtained using some technique developed in [7].