Exact WKB analysis and microlocal analysis

1°对于高阶大参数线性常微分方程的Stokes几何，(1)我们首先借助普通最陡下降法（[AKT5]）对laplace型方程进行了具体而详细的研究；然后，从Borel恢复的角度反思所得结果的wkb理论意义，(2)我们在[AKT3]中提出了精确最陡下降法，该方法利用了新发明的“精确最陡下降路径”概念，以便我们可以描述一般算子的Stokes几何。(3) [AkoT]和[KoT]用最陡下降法对Stokes几何中的一些具体但微妙的问题进行了检验。鉴于本课题的精神目标，在精确WKB分析中引入精确最陡下降法是非常重要的，因为它清楚地体现了精确WKB分析和微局部分析的互补特征，它表明了量化的Legendre变换的全局方面可以用精确最陡下降路径来描述。基于多特征算子的微局部分析（[AKT1]）计算了Landau-Zener型问题的2°非绝热跃迁概率。重要的是其自身权利的具体算法检测虚拟拐点的算子的问题。当红外散度相关时，用[KS]研究了s -矩阵的微局部结构。[K]从WKB分析的角度研究了非线性常微分方程解的4°自然边界。[KStr]也对Dirichlet级数的自然边界进行了微局部研究。[AKKT]在微局部分析的基本概念之一——量化接触变换的帮助下，发展了具有大参数的无穷微分算子的精确WKB分析的局部理论。6°[T1]构建了微分方程系统的精确WKB分析，我们目前（2002）正试图将其应用于高阶Painlev方程（Noumi方程等）的研究。少

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