On Spectral Polynomials of the Heun Equation. II

AbstractThe well-known Heun equation has the form $$\begin{array}{ll}\left\{Q(z)\frac {d^2}{dz^2}+P(z)\frac{d}{dz}+V(z)\right\}S(z)=0,\end{array}$$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 19th 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 Shapiro and Tater (JAT 162:766–781, 2010) 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.