On spectral polynomials of the Heun equation. I

The classical Heun equation has the form {Q(z)d2dz2+P(z)ddz+V(z)}S(z)=0, where Q(z)Q(z) is a cubic complex polynomial, P(z)P(z) is a polynomial of degree at most 22 and V(z)V(z) is at most linear. In the second half of the nineteenth century E. Heine and T. Stieltjes initiated the study of the set of all V(z)V(z) for which the above equation has a polynomial solution S(z)S(z) of a given degree nn. The main goal of the present paper is to study the union of the roots of the latter set of V(z)V(z)’s when n→∞n→∞. We provide an explicit description of this limiting set and give a substantial amount of preliminary and additional information about it obtained using a certain technique developed by A.B.J. Kuijlaars and W. Van Assche.