Exact WKB analysis of microdifferential equations of infinite order

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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