Exact WKB analysis and microlocal analysis

精确的 WKB 分析和微局部分析

基本信息

批准号：
11440042
负责人：
KAWAI Takahiro
金额：
$ 4.67万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440042/
关键词：
exact WKB analysis microlocal analysis Stokes geometry exact steepest descent path exact steepest descent method virtual turning point infra-red divergence natural boundaries ストークス曲線ストークス幾何ボレル和非断熱近似 Borel変換鞍点 Landau-Zener /

项目摘要

1°Concerning the Stokes geometry for higher order linear ordinary differential equations with a large parameter,(1) we first made a concrete and detailed study of Laplace-type equations with the help of the ordinary steepest descent method ([AKT5]), and then by musing on the WKB-theoretic meaning of the obtained results reflectively from the viewpoint of the Borel resummation,(2) we proposed in [AKT3] the exact steepest descent method that makes use of the newly invented notion "exact steepest descent paths" so that we may describe the Stokes geometry for general operators.(3) Some concrete but delicate issues in the Stokes geometry are examined by the exact steepest descent method in [AkoT] and [KoT].In view of the spiritual target of this project, the introduction of the exact steepest descent method into the exact WKB analysis is quite important, as it clearly exemplifies the complementary character of the exact WKB analysis and microlocal analysis, it shows that the global aspect o … More f the quantized Legendre transformation can be described in terms of the exact steepest descent paths.2° Non-adiabatic transition probabilities for Landau-Zener type problems are calculated on the basis of microlocal analysis of operators with multiple characteristics ([AKT1]). Important in its own right is the concrete algorithm for detecting virtual turning points for the operators in question.3° Microlocal structure of the S-matrix is studied ia [KS] when infra-red divergence is relevant.4° Natural boundaries of solution of non-liner ordinary differential equations are studied in [K] from the viewpoint of WKB analysis. Microlocal study of natural boundaries of Dirichlet series was also made in [KStr].5° Local theory of the exact WKB analysis for the infinite series of differential operators with a large parameter was developed in [AKKT] with the help of a quantized contact transformation, one of the basic notions in microlocal analysis.6° [T1] constructed the exact WKB analysis for systems of differential equations, and we are currently (2002) trying to apply it to the study of higher order Painlev」 equations (Noumi equation etc.). Less

1°Concerning the Stokes geometry for higher order linear ordinary differential equations with a large parameter,(1) we first made a concrete and detailed study of Laplace-type equations with the help of the ordinary steelepest descent method ([AKT5]), and then by musing on the WKB-theoretic meaning of the obtained results reflectively from the viewpoint of the Borel reservation,(2) we proposed in [AKT3]使用新发明的概念“确切的陡峭下降路径”的确切确切的最尖锐的下降方法，以便我们可以描述普通操作员的Stokes几何形状。（3）Stokes几何学中的某些具体但精致的问题在[Akot]和Spirt of Spirt of Invory of Incort of the of Interial of the of the of [akot]中的确切钢的精确词。 WKB分析非常重要，因为它清楚地说明了确切的WKB分析和微局部分析的完整特征，它表明，可以用确切的钢植物下降路径来描述全球方面。2°非绝热过渡可能性是根据Landau-Zener类型问题的基础来定量的。重要的是，重要的是用于检测有关操作员的虚拟转折点的混凝土算法。当infra-red Divergence是相关时，S-Matrix的3°微置结构是研究的。4°无线电差异方程式的自然界限是在[K]中研究的。在[KSTR]中，还对Dirichlet系列的自然界限进行了微局部。5°局部理论的局部理论的WKB分析的无限级别差异操作员的精确WKB分析在[AKKT]中开发了较大的参数，该量化了量化的接触转换，在微邻域分析中进行的基本注释之一，当前的基本注释之一[t1]构建的基本注释之一。它是针对高阶pachlev“方程（noumi方程等）的研究。