© 2015

Multidimensional Periodic Schrödinger Operator

Perturbation Theory and Applications


  • Presents an introduction to the theoretical base for solving the inverse problems of the multidimensional Schrödinger operator

  • Opens new horizons for spectral analysis of quantum mechanical operators

  • Covers the constructive determination of spectral invariants

  • A useful resource for mathematical problems of solid state quantum physics


Part of the Springer Tracts in Modern Physics book series (STMP, volume 263)

Table of contents

  1. Front Matter
    Pages i-x
  2. Oktay Veliev
    Pages 1-30
  3. Oktay Veliev
    Pages 227-239
  4. Back Matter
    Pages 241-242

About this book


The book describes the direct problems and the inverse problem of the multidimensional Schrödinger operator with a periodic potential. This concerns perturbation theory and constructive determination of the spectral invariants and finding the periodic potential from the given Bloch eigenvalues. The unique method of this book derives the asymptotic formulas for Bloch eigenvalues and Bloch functions for arbitrary dimension. Moreover, the measure of the iso-energetic surfaces in the high energy region is construct and estimated. It implies the validity of the Bethe-Sommerfeld conjecture for arbitrary dimensions and arbitrary lattices. Using the perturbation theory constructed in this book, the spectral invariants of the multidimensional operator from the given Bloch eigenvalues are determined. Some of these invariants are explicitly expressed by the Fourier coefficients of the potential. This way the possibility to determine the potential constructively by using Bloch eigenvalues as input data is given. In the end an algorithm for the unique determination of the potential is given.


Bloch Eigenvalues Derivatives of Band Functions Derivatives of Band Functions Inverse Problem of Multidimensional Isoenergetic Surfaces Periodic Bloch Functions Schrödinger Operator Spectral Invariants

Authors and affiliations

  1. 1.Department of MathematicsDogus UniversityIstanbulTurkey

Bibliographic information

  • Book Title Multidimensional Periodic Schrödinger Operator
  • Book Subtitle Perturbation Theory and Applications
  • Authors Oktay Veliev
  • Series Title Springer Tracts in Modern Physics
  • Series Abbreviated Title Springer Tracts Mod.Physics
  • DOI
  • Copyright Information Springer International Publishing Switzerland 2015
  • Publisher Name Springer, Cham
  • eBook Packages Physics and Astronomy Physics and Astronomy (R0)
  • Hardcover ISBN 978-3-319-16642-1
  • Softcover ISBN 978-3-319-38671-3
  • eBook ISBN 978-3-319-16643-8
  • Series ISSN 0081-3869
  • Series E-ISSN 1615-0430
  • Edition Number 1
  • Number of Pages X, 242
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Quantum Physics
    Solid State Physics
    Mathematical Physics
  • Buy this book on publisher's site
Industry Sectors
Energy, Utilities & Environment
IT & Software