On spectral polynomials of the Heun equation. I

The classical Heun equation has the form  Q(z) d2 dz2 + 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].

• Type: Journal
• Year: 2010
• Where: JAT
• Authors: Boris Shapiro, Milos Tater
• Keywords: Classical Heun Equation, Cubic Complex Polynomial, JAT 2010, Nineteenth Century E