Communications in Mathematical Physics

, Volume 311, Issue 2, pp 277–300 | Cite as

On Spectral Polynomials of the Heun Equation. II

  • Boris Shapiro
  • Kouichi Takemura
  • Miloš Tater


The 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.


Singular Point Riemann Surface Positive Measure Quadratic Differential Real Measure 
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Authors and Affiliations

  1. 1.Department of MathematicsStockholm UniversityStockholmSweden
  2. 2.Department of Mathematics, Faculty of Science and TechnologyChuo UniversityTokyoJapan
  3. 3.Department of Theoretical PhysicsNuclear Physics Institute, Academy of SciencesŘež near PragueCzech Republic

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