2009/04/03 by Boris Shapiro, Shapiro, Boris, Kouichi Takemura +4 · 1 citation
Mathematics · Physics and Astronomy · #30C15 #33E05 (Secondary) #34L20 (Primary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #math-ph #math.CA #math.MP #msc:30C15 #msc:33E05 #msc:34L20
paper · pdf · doi:10.48550/arxiv.0904.0650
23 pages, 9 figures
arxiv created 2009/04/03 · openalex publication_date 2009/04/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The well-known Heun equation has the form: Q(z)S''(z)+P(z)S'(z)+V(z)S(z)=0 where Q(z) is a cubic complex polynomial, P(z) and V(z) are polynomials of degrees at most 2 and 1 resp. One of the classical problems about the Heun equation is for a given positive integer N to find all possible linear polynomials V(z) such that the latter equation has a polynomial solution S(z) of degree N. Below we prove a conjecture of the 2nd author claiming that the union of roots of such V(z)'s for a given N tends when N->oo to a certain compact connecting the three roots of Q(z) and given by the condition that a certain natural abelian integral is real-valued.