Real-root property of the spectral polynomial of the Treibich-Verdier potential and related problems

Abstract

We study the spectral polynomial of the Treibich-Verdier potential. Such spectral polynomial, which is a generalization of the classical Lame polynomial, plays fundamental roles in both the finite-gap theory and the ODE theory of Heun's equation. In this paper, we prove that all the roots of such spectral polynomial are real and distinct under some assumptions. The proof uses the classical concept of Sturm sequence and isomonodromic theories. We also prove an analogous result for a polynomial associated with a generalized Lame equation. Differently, our new approach is based on the viewpoint of the monodromy data.

22 pages

References in corpus (5)

• The Heun equation and the Calogero-Moser-Sutherland system IV: the Hermite-Krichever Ansatz
• Root asymptotics of spectral polynomials for the Lame operator
• On reducible monodromy representations of some generalized Lamé equation
• Simple zero property of some holomorphic functions on the moduli space of tori
• Sharp nonexistence results for curvature equations with four singular sources on rectangular tori

Cited by in corpus (6)

• Polynomial solutions of $q$-Heun equation and ultradiscrete limit
• Sharp nonexistence results for curvature equations with four singular sources on rectangular tori
• A necessary and sufficient condition for the Darboux-Treibich-Verdier potential with its spectrum contained in $\mathbb{R}$
• Heun's differential equation and its q-deformation
• The geometry of generalized Lamé equation, I
• On number and evenness of solutions of the $SU(3)$ Toda system on flat tori with non-critical parameters