Multidimensional Periodic Schrödinger Operator

Perturbation Theory and Applications

  • Oktay Veliev

Table of contents

  1. Front Matter
    Pages i-xii
  2. Oktay Veliev
    Pages 1-29
  3. Oktay Veliev
    Pages 31-111
  4. Oktay Veliev
    Pages 313-324
  5. Back Matter
    Pages 325-326

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 second edition includes an additional chapter that specifically focuses on lower-dimensional cases, providing the basis for the higher-dimensional considerations of the chapters that follow.


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 MathematicsDoğuş UniversityIstanbulTurkey

Bibliographic information

Industry Sectors
IT & Software
Energy, Utilities & Environment