Exact WKB analysis of microdifferential equations of infinite order

本项目的目的是建立无限阶微微分方程精确WKB分析的基本理论并发展其应用。得到了以下结果：(1)引入了WKB型微微分算子的概念，并证明了对于WKB型算子，可以构造精确的WKB解。拐点和Stokes曲线的概念以及具有大参数的微分算子的情况都可以被定义。证明了在拐点的邻域内，这样的算子可以分解为两个算子的乘积，其对应的方程可以简化为有限阶方程。因此，WKB解的局部理论与有限阶方程的局部理论完全相同。因此，如果拐点是简单的，则将方程简化为艾里方程。我们发现，至少在局部情况下，方程的阶数是无关的，而转折点的度对于WKB解的连接问题是必不可少的。(2)作为微分方程连接问题的一个应用，我们得到了多个zeta值之间存在的新的关系族。有两种方法可以定义多个zeta值。两者都是由正整数幂倒数乘积的欧拉和定义的：一个是由严格不等式和另一个是非严格不等式的幂指数和定义的。我们构造了一个由后者构成的生成函数，并证明了该函数是一类Fuchsian型非齐次常微分方程的唯一解。直接用幂级数或积分法求解该方程，得到了多个zeta值的关系式。也就是说，我们证明了具有固定权重和高度的非严格不等式的多个zeta值的和，我们称之为多个zeta-star值，可以表示为Riemann zeta值的有理倍数。(3)对给定高阶微分方程的Stokes几何给出完整的描述是一个非常困难的问题。我们发现虚拟拐点的概念对于理解Stokes几何和方程的连接问题是至关重要的。例如，一个具有变形参数的高阶方程，我们有斯托克斯曲线族。我们知道，不仅斯托克斯曲线，而且所谓的新斯托克斯曲线对于描述斯托克斯几何都是不可或缺的。当变形参数改变时，Stokes几何形状也会发生变化，我们观察到有时普通Stokes曲线和新Stokes曲线的作用会相互交换。使用虚拟转折点的概念可以很好地理解这种现象。少

The purpose of this project was to establish the fundamental theory of the exact WKB analysis for microdifferential equations of infinite order and develop its applications. We have obtained the following results :(1)We introduced the notion of microdifferential operators of WKB type and we showed that for operators of WKB type, we can construct exact WKB solutions. The notions of turning points and Stokes curves can be defined as well as the case of differential operators with a large parameter. We proved that in a neighborhood of a turning point, such an operator can be decomposed into the product of two operators and the equation corresponding to the operator is reduced to an equation of finite order. Thus local theory for WKB solutions is exactly the same as in the case of equations of finite order. Thus, if the turning point is simple, then the equation is reduced to the Airy equation. We have found that, at least locally, the order of the equation is irrelevant and that the degre … More e of the turning point is essential for the connection problem of the WKB solutions.(2)As an application of connection problems of differential equations, we have obtained new families of relations that hold among multiple zeta values. There are two ways of defining multiple zeta values. Both are defined by the Euler sum of products of reciprocals of powers of positive integers : one is defined by sums over indices of powers with strict inequalities and another with non-strict inequalities. We constructed a generating function made of the latter and showed that the function is a unique solution of an inhomogeneous ordinary differential equation of Fuchsian type. Solving this equation directly by using power series or integration, we obtained some families of relations of multiple zeta values. That is, we showed that sums of multiple zeta values with non-strict inequalities, which we call multiple zeta-star values, with fixed weight and height can be expressed as a rational multiple of Riemann zeta values.(3)To have the complete description of the Stokes geometry of a given higher-order differential equation is very difficult problem in general. We found that the notion of virtual turning points is crucial to understand the Stokes geometry and the connection problem for the equation. For example, an equation of higher order with a deformation parameter, we have a family of Stokes curves. We know that not only the Stokes curves but also the so called new Stokes curves are indispensable to describe the Stokes geometry. If the deformation parameter changes, the Stokes geometry also changes and we observed that sometimes the role of ordinary Stokes curves and new Stokes curves interchange each other. This phenomenon can be well understood by using the notion of virtual turning points. Less