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Virtual Turning Points

  • Book
  • © 2015

Overview

Authors:
  1. Naofumi Honda

    1. Department of Mathematics, Hokkaido University, Sapporo, Japan

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  2. Takahiro Kawai

    1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

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  3. Yoshitsugu Takei

    1. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

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  • Is the first book that expounds a virtual turning point, a new and important notion in WKB analysis
  • Contains essential know-how in its concrete treatment of the theory of virtual turning points, written by the founders of that theory
  • Establishes an important bridge between pure mathematics and applied mathematics
  • Includes supplementary material: sn.pub/extras

Part of the book series: SpringerBriefs in Mathematical Physics (BRIEFSMAPHY, volume 4)

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Table of contents (3 chapters)

  1. Front Matter

    Pages i-xii
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  2. Definition and Basic Properties of Virtual Turning Points

    • Naofumi Honda, Takahiro Kawai, Yoshitsugu Takei
    Pages 1-49

  3. Application to the Noumi-Yamada System with a Large Parameter

    • Naofumi Honda, Takahiro Kawai, Yoshitsugu Takei
    Pages 51-77

  4. Exact WKB Analysis of Non-adiabatic Transition Problems for 3-Levels

    • Naofumi Honda, Takahiro Kawai, Yoshitsugu Takei
    Pages 79-110

  5. Back Matter

    Pages 111-126
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Keywords

About this book

The discovery of a virtual turning point truly is a breakthrough in WKB analysis of higher order differential equations. This monograph expounds the core part of its theory together with its application to the analysis of higher order Painlevé equations of the Noumi–Yamada type and to the analysis of non-adiabatic transition probability problems in three levels.

As M.V. Fedoryuk once lamented, global asymptotic analysis of higher order differential equations had been thought to be impossible to construct. In 1982, however, H.L. Berk, W.M. Nevins, and K.V. Roberts published a remarkable paper in the Journal of Mathematical Physics indicating that the traditional Stokes geometry cannot globally describe the Stokes phenomena of solutions of higher order equations; a new Stokes curve is necessary.

Reviews

“This monograph provides an introduction to the theory of virtual turning points and its applications as well as a historical view of the theory. … The monograph is written for researchers and students working in mathematical sciences.” (Takashi Aoki, zbMATH 1354.34003, 2017)

Authors and Affiliations

  • Department of Mathematics, Hokkaido University, Sapporo, Japan

    Naofumi Honda

  • Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan

    Takahiro Kawai, Yoshitsugu Takei

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