On the Structure of Higher Order Simple-Pole Type Operators in Exact WKB Analysis
On the Structure of Higher Order Simple-Pole Type Operators in Exact WKB Analysis
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DOI：
10.1619/fesi.53.249
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
T. Kawai;T. Koike;Yoshitsugu Takei
中科院分区：
文献类型：
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作者：
T. Kawai;T. Koike;Yoshitsugu Takei
引入了一类系数为简单极点的大参数高阶线性常微算子，并从精确WKB分析的角度研究了它的结构，即基于Borel恢复的WKB分析。我们的主要结果是，这类算子被表示为两个算子Q和R的乘积，其中Q与Stokes几何无关，R是Koike([Ko1]，[Ko2]，[KO_3])研究的二阶单极型算子。这个分解定理给出了WKB解在简单极点附近的一个联系公式，并用它来解释我们在研究一个特定的三阶算子时遇到的一些有趣现象的背景机制。并对该理论的未来发展方向进行了一些讨论。
We introduce an appropriate class of higher order linear ordinary differential operators with a large parameter and with simple poles in their coefficients, and study their structure from the viewpoint of exact WKB analysis, i.e., the WKB analysis based on the Borel resummation. Our main result is that an operator in the class is expressed as a product of two operators Q and R, with Q being irrelevant to the Stokes geometry and with R being a second order simple-pole type operator studied by Koike ([Ko1], [Ko2], [Ko3]). This decomposition theorem gives us a connection formula for WKB solutions near the simple pole in question, and it is used to explain the background mechanism of some intriguing phenomenon we encounter in the study of a particular third order operator. Some discussions on the scope of the future development of the theory are included.