The structure theory of differential equations by the algebraic analysis of singular perturbation theory

2016 Fiscal Year Final Research Report

• Project/Area Number: 24340026
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kyoto University
• Principal Investigator: KAWAI Takahiro 京都大学, 数理解析研究所, 名誉教授 (20027379)
• Co-Investigator(Kenkyū-buntansha): 竹井 義次 京都大学, 数理解析研究所, 准教授 (00212019) ; 小池 達也 神戸大学, 理学(系)研究科(研究院), 准教授 (80324599)
• Co-Investigator(Renkei-kenkyūsha): AOKI Takashi 近畿大学, 理工学部, 教授 (80159285)
• Project Period (FY): 2012-04-01 – 2017-03-31
• Keywords: 完全WKB解析 / 超局所解析学 / ボレル変換 / 動かない特異点 / alien derivative / 仮想的変わり点 / 無限階微分作用素 / 超局所解析的S-行列論

Outline of Final Research Achievements

We have succeeded in concretely describing the singularity structure of the Borel transformed WKB solutions of several basic one-dimensional Schroedinger equations near their fixed singular points. The main tool for the description is the so-called alien calculus, and the idea is based on the role of a bicharacteristic strip as a carrier of singularities of solutions of linear partial differential equations. Thus our result is an outcome of the complementality of microlocal analysis and the exact WKB analysis. We have also studied higher order ordinary differential equations with a large parameter through its Borel transform ; a virtual turning point is an indispensable ingredient in the exact WKB analysis of higher order equations.
Kawai and Takei, with the collaborator N. Honda(Hokkaido University) published a monograph titled “Virtual Turning Points”(Springer ; 2015), so that this novel notion may be widely appreciated by mathematical physicists and engineers.

Free Research Field

数物系科学