Real-root property of the spectral polynomial of the Treibich-Verdier potential and related problems
Real-root property of the spectral polynomial of the Treibich-Verdier potential and related problems
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Treibich-Verdier势谱多项式的实根性质及相关问题
DOI：
10.1016/j.jde.2018.01.005
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发表时间：
2018
影响因子：
2.4
通讯作者：
Kouichi Takemura
中科院分区：
文献类型：
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作者：
Zhijie Chen;Ting-Jung Kuo;Chang-Shou Lin;Kouichi Takemura
我们研究了Treibich-Verdier势的谱多项式。这种谱多项式是经典的LAMé多项式的推广，在Heun方程的有限间隙理论和常微分方程组理论中都起着重要的作用。在本文中，我们证明了在一定的假设下，这种谱多项式的所有根都是实数且不同的。这一证明使用了经典的Sturm序列概念和同位投射理论。我们还证明了一个与广义Lamé方程相关的多项式的类似结果，其中我们应用了一种基于单调数据观点的新方法。
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.