The Hermite–Krichever Ansatz for Fuchsian equations with applications to the sixth Painlevé equation and to finite-gap potentials
The Hermite–Krichever Ansatz for Fuchsian equations with applications to the sixth Painlevé equation and to finite-gap potentials
复制标题
DOI：
10.1007/s00209-008-0415-5
复制
发表时间：
2005-04
影响因子：
0.8
通讯作者：
K. Takemura
中科院分区：
文献类型：
--
作者：
K. Takemura
将解的积分表示和Hermite-Krichever Ancherton Heun方程等结果推广到一类Fuchsian微分方程，并将其应用到与物理有关的方程.研究了一类线性微分方程通过保单值变形产生Painlevé方程，得到了第六类Painlevé方程的解，其中包含Hitchin解.讨论了它与有限能隙势的关系。我们发现了新的有限能隙势。也就是说，我们证明了写为Treibich-Verdier势和指数为− 1和2的附加表观奇点之和的势是有限能隙的，这推广了Treibich以前得到的结果。我们还研究了薛定谔算子在我们势上的本征函数及其单值性。
Several results including integral representation of solutions and Hermite– Krichever Ansatz on Heun’s equation are generalized to a certain class of Fuchsian differential equations, and they are applied to equations which are related with physics. We investigate linear differential equations that produce Painlevé equation by monodromy preserving deformation and obtain solutions of the sixth Painlevé equation which include Hitchin’s solution. The relationship with finite-gap potential is also discussed. We find new finite-gap potentials. Namely, we show that the potential which is written as the sum of the Treibich–Verdier potential and additional apparent singularities of exponents − 1 and 2 is finite-gap, which extends the result obtained previously by Treibich. We also investigate the eigenfunctions and their monodromy of the Schrödinger operator on our potential.