Real-root property of the spectral polynomial of the Treibich?�Verdier potential and related problems
Journal
Journal of Differential Equations
Journal Volume
264
Journal Issue
8
Pages
5408-5431
Date Issued
2018
Author(s)
Abstract
We study the spectral polynomial of the Treibich–Verdier potential. Such spectral polynomial, which is a generalization of the classical Lamé 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 Lamé equation, where we apply a new approach based on the viewpoint of the monodromy data.
Type
journal article