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Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations

  • Johannes Sjöstrand
Book

Part of the Pseudo-Differential Operators book series (PDO, volume 14)

Table of contents

  1. Front Matter
    Pages i-x
  2. Johannes Sjöstrand
    Pages 1-5
  3. Basic Notions, Differential Operators in One Dimension

    1. Front Matter
      Pages 7-7
    2. Johannes Sjöstrand
      Pages 9-28
    3. Johannes Sjöstrand
      Pages 135-155
    4. Johannes Sjöstrand
      Pages 157-181
  4. Some general results

    1. Front Matter
      Pages 183-183
    2. Johannes Sjöstrand
      Pages 185-189
    3. Johannes Sjöstrand
      Pages 211-217
    4. Johannes Sjöstrand
      Pages 219-244
    5. Johannes Sjöstrand
      Pages 245-293
  5. Spectral Asymptotics for Differential Operators in Higher Dimension

    1. Front Matter
      Pages 295-295
    2. Johannes Sjöstrand
      Pages 297-311
    3. Johannes Sjöstrand
      Pages 329-361
    4. Johannes Sjöstrand
      Pages 363-407
    5. Johannes Sjöstrand
      Pages 443-486
  6. Back Matter
    Pages 487-496

About this book

Introduction

The asymptotic distribution of eigenvalues of self-adjoint differential operators in the high-energy limit, or the semi-classical limit, is a classical subject going back to H. Weyl of more than a century ago.

In the last decades there has been a renewed interest in non-self-adjoint differential operators which have many subtle properties such as instability under small perturbations. Quite remarkably, when adding small random perturbations to such operators, the eigenvalues tend to distribute according to Weyl's law (quite differently from the distribution for the unperturbed operators in analytic cases). A first result in this direction was obtained by M. Hager in her thesis of 2005. Since then, further general results have been obtained, which are the main subject of the present book.

Additional themes from the theory of non-self-adjoint operators are also treated. The methods are very much based on microlocal analysis and especially on pseudodifferential operators. The reader will find a broad field with plenty of open problems.

Keywords

Determinant Eigenvalue Microlocal analysis Pseudodifferential operator Random perturbation WKB method

Authors and affiliations

  • Johannes Sjöstrand
    • 1
  1. 1.Université de Bourgogne Franche-Comté DijonFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-030-10819-9
  • Copyright Information Springer Nature Switzerland AG 2019
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-030-10818-2
  • Online ISBN 978-3-030-10819-9
  • Series Print ISSN 2297-0355
  • Series Online ISSN 2297-0363
  • Buy this book on publisher's site
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