On the singularity structure of wkb solution of the boosted Whittaker equation: its relevance to resurgent functions with essential singularities
On the singularity structure of wkb solution of the boosted Whittaker equation: its relevance to resurgent functions with essential singularities
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提升惠特克方程的 wkb 解的奇点结构：其与具有本质奇点的复兴函数的相关性
DOI：
10.1007/s11005-016-0887-x
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
T. Koike
中科院分区：
文献类型：
--
作者：
S. Kamimoto;T. Kawai;T. Koike
受无穷阶线性微分算子的符号演算应用于单极型方程的Borel变换的WKB解的启发[Kamimoto et al.(Rims KôkyúRoku Bessatsu B 52：127-146,2014)]在第一节中，我们在第二节中引入了依赖于参数的简单再生函数空间，然后我们定义了作用于该空间并产生具有本质奇性的再生函数的分配算子。在第三节中，我们将算符应用于大参数Whittaker方程的Voros系数及其指数的Borel变换，从而可以得到大参数的Boost Whittaker方程的Voros系数及其指数的Borel变换。在第四节中，我们利用这些结果得到了大参数增强的Whittaker方程的Borel变换的WKB解的外来导数的显式形式。本文的结果表明了具有本质奇异性的再生函数在发展精确的WKB分析--基于再生函数理论的WKB分析中的重要性。还值得强调的是，我们遇到的本质奇点的具体形式是由无穷级线性微分算子表示的。
Inspired by the symbol calculus of linear differential operators of infinite order applied to the Borel transformed WKB solutions of simple-pole type equation [Kamimoto et al. (RIMS Kôkyûroku Bessatsu B 52:127–146, 2014)], which is summarized in Section 1, we introduce in Section 2 the space of simple resurgent functions depending on a parameter with an infra-exponential type growth order, and then we define the assigning operatorwhich acts on the space and produces resurgent functions with essential singularities. In Section 3, we apply the operatorto the Borel transforms of the Voros coefficient and its exponentiation for the Whittaker equation with a large parameter so that we may find the Borel transforms of the Voros coefficient and its exponentiation for the boosted Whittaker equation with a large parameter. In Section 4, we use these results to find the explicit form of the alien derivatives of the Borel transformed WKB solutions of the boosted Whittaker equation with a large parameter. The results in this paper manifest the importance of resurgent functions with essential singularities in developing the exact WKB analysis, the WKB analysis based on the resurgent function theory. It is also worth emphasizing that the concrete form of essential singularities we encounter is expressed by the linear differential operators of infinite order.