TY - JOUR
T1 - Real-root property of the spectral polynomial of the Treibich–Verdier potential and related problems
AU - Chen, Zhijie
AU - Kuo, Ting Jung
AU - Lin, Chang Shou
AU - Takemura, Kouichi
N1 - Publisher Copyright:
© 2018 Elsevier Inc.
PY - 2018/4/15
Y1 - 2018/4/15
N2 - 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.
AB - 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.
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U2 - 10.1016/j.jde.2018.01.005
DO - 10.1016/j.jde.2018.01.005
M3 - Article
AN - SCOPUS:85040604778
SN - 0022-0396
VL - 264
SP - 5408
EP - 5431
JO - Journal of Differential Equations
JF - Journal of Differential Equations
IS - 8
ER -