Multidimensional Periodic Schrödinger Operator

Perturbation Theory and Applications

  • Oktay Veliev

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

  • Oktay Veliev
    • 1
  1. 1.Department of MathematicsDogus UniversityIstanbulTurkey

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