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。我们得到了以下结果：（1）引入了WKB型微微分算子的概念，并证明了WKB型算子可以构造精确的WKB解。转向点和Stokes曲线的概念可以定义为具有大参数的微分算子的情况。证明了在转向点的邻域内，这类算子可分解为两个算子的乘积，相应的方程可化为有限阶方程.因此，WKB解的局部理论与有限阶方程的情况完全相同。因此，如果转折点是简单的，那么方程被简化为艾里方程。我们发现，至少在局部，方程的阶是无关的， ...更多信息 对于WKB解的连接问题，转折点的e是必不可少的。(2)As应用微分方程的联络问题，我们得到了多个zeta值之间的新的关系族。有两种方法定义多个zeta值。两者都是由正整数的幂倒数的乘积的欧拉和定义的：一个是由具有严格不等式的幂指数上的和定义的，另一个是具有非严格不等式的。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个Fuchsian型非齐次常微分方程的唯一解。利用幂级数或积分直接求解该方程，得到了几类多重zeta值关系。也就是说，我们证明了具有非严格不等式的多个zeta值的和，我们称之为多个zeta-star值，具有固定的重量和高度，可以表示为黎曼zeta值的有理倍数。(3)To对给定的高阶微分方程的Stokes几何进行完整的描述，一般来说是一个非常困难的问题。我们发现，虚拟转折点的概念是至关重要的理解斯托克斯几何和方程的连接问题。例如，一个高阶方程的变形参数，我们有一个家庭的斯托克斯曲线。我们知道，不仅Stokes曲线，而且所谓的新Stokes曲线是不可缺少的描述Stokes几何。如果变形参数的变化，Stokes几何也发生变化，我们观察到，有时普通的Stokes曲线和新的Stokes曲线的作用相互交换。这一现象可以通过使用虚拟转折点的概念来很好地理解。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