On Spectral Polynomials of the Heun

The well-known Heun equation has the form { Q(z) d2 dz2 + P (z) d dz + V (z) } S(z) = 0, where Q(z) is a cubic complex polynomial, P (z) and V (z) are polynomials of degree at most 2 and 1 respectively. One of the classical problems about the Heun equation suggested by E. Heine and T. Stieltjes in the late 19-th century is for a given positive integer n to find all possible polynomials V (z) such that the above equation has a polynomial solution S(z) of degree n. Below we prove a conjecture of the second author, see [17] claiming that the union of the roots of such V (z)’s for a given n tends when n → ∞ to a certain compact connecting the three roots of Q(z) which is given by a condition that a certain natural abelian integral is real-valued, see Theorem 2. In particular, we prove several new results of independent interest about rational Strebel differentials.