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

Zhijie Chen, Ting Jung Kuo, Chang Shou Lin, Kouichi Takemura

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3 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)5408-5431
Number of pages24
JournalJournal of Differential Equations
Issue number8
Publication statusPublished - 2018 Apr 15


ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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