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Algebraic Analysis of Differential Equations T. Aoki· H. Majima- Y. Takei· N. Tose (Eds.) Algebraic Analysis of Differential Equations from Microlocal Analysis to Exponential Asymptotics Festschrift in Honor of Takahiro Kawai ~ Springer Editors Takashi Aoki Department of Mathematics Kinki University Higashi-Osaka 577-8502, Japan e-mail: [email protected] Hideyuki Majima Department of Mathematics Ochanomizu University Tokyo 112-8610, Japan e-mail: [email protected] Yoshitsugu Takei Research Institute for Mathematical Sciences Kyoto University Kyoto 606-8502, Japan e-mail: [email protected] Nobuyuki Tose Faculty of Economics Keio University Yokohama 223-8521, Japan e-mail: [email protected] Library of Congress Control Number: 2007939560 ISBN 978-4-431-73239-6 Springer Tokyo Berlin Heidelberg New York Springer is a part of Springer Science+Business Media springer.com c Springer 2008 This work is subject to copyright. All rights are reserved, whether the whole or a part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Camera-ready copy prepared from the authors’ LATEXfiles. Printed and bound by Shinano Co. Ltd., Japan. SPIN: 12081080 Printed on acid-free paper Dedicated to Professor Takahiro Kawai on the Occasion of His Sixtieth Birthday Preface This is a collection of articles on algebraic analysis of differential equations and related topics. The contributors (T. Kawai excepted) and we, the editors, dedicate this volume to Professor Takahiro KAWAI, who is one of the creators of microlocal analysis, a central object of the subjects. To respect his many substantial contributions to the subjects, we have included some explanatory notes about the work of Professor T. Kawai in Part I of the volume. Most of the contributed articles in Part II of this volume were read at the Conference “Algebraic Analysis of Differential Equations — from Microlo- cal Analysis to Exponential Asymptotics”, which we organized from July 7 through July 14, 2005 at the Research Institute for Mathematical Sciences, Kyoto University to celebrate Professor T. Kawai’s sixtieth birthday. Taking this opportunity, we would like to express once again our sincerest congratu- lations to Professor Kawai. Kyoto in Japan, Takashi Aoki June 27, 2007 Hideyuki Majima Yoshitsugu Takei Nobuyuki Tose Contents Preface ................................................................ IX Part I The work of T. Kawai Publications of Professor Takahiro Kawai ................................................................ 3 The work of T. Kawai on hyperfunction theory and microlocal analysis, Part 1 — Microlocal analysis and differential equations Nobuyuki Tose ................................................... 11 The work of T. Kawai on hyperfunction theory and microlocal analysis, Part 2 — Operators of infinite order and convolution equations Takashi Aoki .................................................... 15 The work of T. Kawai on exact WKB analysis Yoshitsugu Takei ................................................. 19 Part II Contributed papers Virtual turning points — A gift of microlocal analysis to the exact WKB analysis Takashi Aoki, Naofumi Honda, Takahiro Kawai, Tatsuya Koike, Yukihiro Nishikawa, Shunsuke Sasaki, Akira Shudo, Yoshitsugu Takei ... 29 Regular sequences associated with the Noumi-Yamada equations with a large parameter Takashi Aoki, Naofumi Honda ..................................... 45 XII Contents Ghost busting: Making sense of non-Hermitian Hamiltonians Carl M. Bender .................................................. 55 Vanishing of the logarithmic trace of generalized Szeg¨o projectors Louis Boutet de Monvel ........................................... 67 Nonlinear Stokes phenomena in first or second order differential equations Ovidiu Costin ................................................... 79 Reconstruction of inclusions for the inverse boundary value problem for non-stationary heat equation Yuki Daido, Hyeonbae Kang, Gen Nakamura ........................ 89 Exact WKB analysis near a simple turning point Eric Delabaere ...................................................101 The Borel transform Leon Ehrenpreis .................................................119 On the use of Z-transforms in the summation of transseries for partial differential equations Christopher J. Howls .............................................133 Some dynamical aspects of Painlev´eVI Katsunori Iwasaki ................................................143 An algebraic representation for correlation functions in integrable spin chains Michio Jimbo ....................................................157 Inverse image of D-modules and quasi-b-functions Yves Laurent ....................................................167 The hypoelliptic Laplacian of J.-M. Bismut Gilles Lebeau ....................................................179 Commuting differential operators with regular singularities Toshio Oshima ..................................................195 The behaviors of singular solutions of some partial differential equations in the complex domain Sunao Ouchi¯ ....................................................225 Observations on the JWKB treatment of the quadratic barrier Hujun Shen, Harris J. Silverstone ..................................237 Contents XIII A role of virtual turning points and new Stokes curves in Stokes geometry of the quantum H´enon map Akira Shudo .....................................................251 Spectral instability for non-selfadjoint operators Johannes Sj¨ostrand...............................................265 Boundary and lens rigidity, tensor tomography and analytic microlocal analysis Plamen Stefanov, Gunther Uhlmann ................................275 Coupling of two partial differential equations and its application Hidetoshi Tahara ................................................295 Instanton-type formal solutions for the first Painlev´e hierarchy Yoshitsugu Takei .................................................307 From exact-WKB toward singular quantum perturbation theory II Andr´e Voros.....................................................321 WKB analysis and Poincar´e theorem for vector fields Masafumi Yoshino ...............................................335 List of Contributors Takashi Aoki Eric Delabaere Department of Mathematics D´epartement de Math´ematiques Kinki University UMR CNRS 6093, Higashi-Osaka 577-8502, Japan Universit´ed’Angers [email protected] 2 Boulevard Lavoisier 49045 Angers Cedex 01, France Carl M. Bender [email protected] Physics Department Washington University Leon Ehrenpreis St. Louis, MO 63130, USA Temple University [email protected] USA Louis Boutet de Monvel Naofumi Honda Universit´e Pierre et Marie Curie Department of Mathematics –Paris6 Hokkaido University France Sapporo 060-0808, Japan [email protected] [email protected] Ovidiu Costin Mathematics Department Christopher J. Howls The Ohio State University University of Southampton Columbus, OH 43210, USA UK [email protected] Katsunori Iwasaki Yuki Daido Faculty of Mathematics Department of Mathematics Kyushu University Hokkaido University 6-10-1 Hakozaki, Higashi-ku Sapporo 060-0810, Japan Fukuoka 812-8581, Japan [email protected] [email protected] XVI List of Contributors Michio Jimbo Yukihiro Nishikawa Graduate School of Mathematical Hitachi Ltd. Sciences Asagaya-kita 2-13-2, Suginami-ku The University of Tokyo Tokyo 166-0001, Japan Komaba, Tokyo 153-8914, Japan Toshio Oshima [email protected] Graduate School of Mathematical Sciences Hyeonbae Kang University of Tokyo School of Mathematical Sciences 7-3-1, Komaba, Meguro-ku Seoul National University Tokyo 153-8914, Japan Seoul 151-747, Korea [email protected] [email protected] Sunao Ouchi¯ Takahiro Kawai Sophia University Research Institute for Mathematical Chiyoda-ku, Tokyo 102-8554, Japan Sciences [email protected] Kyoto University Shunsuke Sasaki Kyoto 606-8502, Japan Mitsubishi UFJ Securities Co. Ltd. Marunouchi 2-4-1, Chiyoda-ku Tatsuya Koike Tokyo 100-6317, Japan Department of Mathematics Graduate School of Science Hujun Shen Kyoto University Department of Chemistry Kyoto 606-8502, Japan The Johns Hopkins University 3400 N. Charles St. Yves Laurent Baltimore, MD 21218, USA Institut Fourier Math´ematiques Akira Shudo UMR 5582 CNRS/UJF, BP 74 Department of Physics 38402 St Martin d’H`eres Cedex Tokyo Metropolitan University France Minami-Ohsawa, Hachioji [email protected] Tokyo 192-0397, Japan [email protected] Gilles Lebeau D´epartement de Math´ematiques Harris J. Silverstone Universit´e de Nice Sophia-Antipolis Department of Chemistry Parc Valrose The Johns Hopkins University 06108 Nice Cedex 02, France 3400 N. Charles St. Baltimore, MD 21218, USA [email protected] [email protected] Gen Nakamura Johannes Sj¨ostrand Department of Mathematics CMLS Hokkaido University Ecole Polytechnique Sapporo 060-0810, Japan FR 91120 Palaiseau, France [email protected] [email protected] List of Contributors XVII Plamen Stefanov Gunther Uhlmann Department of Mathematics Department of Mathematics Purdue University University of Washington
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