Exact WKB analysis and microlocal analysis

1.关于高阶线性大参数常微分方程组的Stokes几何问题，(1)我们首先用普通最陡下降法([AKT5])对拉普拉斯方程进行了具体而详细的研究，然后从Borel恢复的观点考虑了所得结果的WKB理论意义，(2)我们在[AKT_3]中提出了精确最陡下降法，它利用了新发明的概念，使得我们可以用一般算子来描述Stokes几何。(3)我们在[AKT]和[Kot]中用精确最陡下降法研究了Stokes几何中的一些具体而微妙的问题。考虑到本项目的精神目标，将精确最陡下降法引入到精确WKB分析中是相当重要的，因为它清楚地例证了精确WKB分析和微局部分析的互补性，它表明了…的全局方面此外，量子化的勒让德变换可以用精确的最陡下降路径来描述。基于多特征算子的微域分析([AKT1])，计算了Landau-Zener型问题的2°非绝热跃迁几率。文[K]研究了红外线散度相关时S矩阵的3°微局部结构。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。Dirichlet级数自然边界的微局域研究也在[KSTR]中得到了。利用微局域分析中的基本概念之一--量子化接触变换，[AKKT]发展了大参数微分算子无穷级数的精确WKB分析的5°局域理论。6°[T1]构造了微分方程组的精确WKB分析，目前我们(2002)正试图将它应用于高阶Painlev方程(Noumi方程等)的研究。较少

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