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Multidimensional Periodic Schrödinger Operator

Perturbation Theories for High Energy Regions and Their Applications

  • Book
  • © 2024

Overview

Authors:
  1. Oktay Veliev

    1. Faculty of Mechanical Engineering, Dogus University, İstanbul, Türkiye

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  • Features a now complete theory of self-adjoint Schrödinger operators within periodic potentials valid in all dimensions
  • Opens up new horizons for spectral analysis of quantum mechanical operators
  • Represents a useful resource on mathematical problems in solid state quantum physics

Part of the book series: Springer Tracts in Modern Physics (STMP, volume 291)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Preliminary Facts

    • Oktay Veliev
    Pages 1-29

  3. From One-Dimensional to Multidimensional

    • Oktay Veliev
    Pages 31-177

  4. Asymptotic Formulas for the Bloch Eigenvalues and Bloch Functions

    • Oktay Veliev
    Pages 179-273

  5. Constructive Determination of the Spectral Invariants

    • Oktay Veliev
    Pages 275-318

  6. Periodic Potential From the Spectral Invariants

    • Oktay Veliev
    Pages 319-376

  7. Conclusions and Some Generalization

    • Oktay Veliev
    Pages 377-408

  8. Back Matter

    Pages 409-412
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Keywords

About this book

This book describes the direct and inverse problems of the multidimensional Schrödinger operator with a periodic potential, a topic that is especially important in perturbation theory, constructive determination of spectral invariants and finding the periodic potential from the given Bloch eigenvalues. It provides a detailed derivation of the asymptotic formulas for Bloch eigenvalues and Bloch functions in arbitrary dimensions while constructing and estimating the measure of the iso-energetic surfaces in the high-energy regime. Moreover, it presents a unique method proving the validity of the Bethe–Sommerfeld conjecture for arbitrary dimensions and arbitrary lattices. Using the perturbation theory constructed, it determines the spectral invariants of the multidimensional operator from the given Bloch eigenvalues. Some of these invariants are explicitly expressed by the Fourier coefficients of the potential, making it possible to determine the potential constructively using Bloch eigenvalues as input data. Lastly, the book presents an algorithm for the unique determination of the potential.

This updated and significantly expanded third edition features an extension of this framework to all dimensions, offering a now complete theory of self-adjoint Schrödinger operators within periodic potentials. Drawing from recent advancements in mathematical analysis, this edition delves even deeper into the intricacies of the subject. It explores the connections between the multidimensional Schrödinger operator, periodic potentials, and other fundamental areas of mathematical physics. The book's comprehensive approach equips both students and researchers with the tools to tackle complex problems and contribute to the ongoing exploration of quantum phenomena.

Authors and Affiliations

  • Faculty of Mechanical Engineering, Dogus University, İstanbul, Türkiye

    Oktay Veliev

About the author

Oktay Veliev received his B.S. degree in Mathematics in 1977 and Ph.D. degree in Mathematics in 1980 from Moscow State University, earning a Doctor of Sciences degree in 1989. From 1980 to 1983, he was a researcher and then senior researcher (1983–1988) at the Institute of Mathematics of the Academy of Sciences of Azerbaijan SSR. At Baku State University (Azerbaijan) he has been an Associate Professor (1988–1991), a Professor (1991–1992), and Head of the Department of Functional Analysis (1992–1997). Between 1993 and 1997, he was President of the Azerbaijan Mathematical Society. He was a visiting Professor at the University of Nantes (France), the Institute of Mathematics at the ETH (Switzerland), and Sussex University (England). From 1997 to 2002 he was a Professor at Dokuz Eylul University (Turkey), and since 2003 has been a Professor at Dogus University (Turkey). He has received grants from the American Mathematical Society and the International Science Foundation (GrantNo. MVVOOO).

Bibliographic Information

  • Book Title: Multidimensional Periodic Schrödinger Operator

  • Book Subtitle: Perturbation Theories for High Energy Regions and Their Applications

  • Authors: Oktay Veliev

  • Series Title: Springer Tracts in Modern Physics

  • DOI: https://doi.org/10.1007/978-3-031-49035-4

  • Publisher: Springer Cham

  • eBook Packages: Physics and Astronomy, Physics and Astronomy (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024

  • Hardcover ISBN: 978-3-031-49034-7Published: 29 February 2024

  • Softcover ISBN: 978-3-031-49037-8Due: 15 April 2024

  • eBook ISBN: 978-3-031-49035-4Published: 27 February 2024

  • Series ISSN: 0081-3869

  • Series E-ISSN: 1615-0430

  • Edition Number: 3

  • Number of Pages: XIII, 411

  • Number of Illustrations: 1 b/w illustrations

  • Topics: Quantum Physics, Operator Theory, Analysis, Condensed Matter Physics, Mathematical Physics

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