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Multidimensional Periodic Schrödinger Operator

Perturbation Theory and Applications

  • Book
  • © 2015

Overview

Authors:
  1. Oktay Veliev

    1. Department of Mathematics, Dogus University, Istanbul, Turkey

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  • 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
  • Includes supplementary material: sn.pub/extras

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

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-x
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  2. Preliminary Facts

    • Oktay Veliev
    Pages 1-30

  3. Asymptotic Formulas for the Bloch Eigenvalues and Bloch Functions

    • Oktay Veliev
    Pages 31-126

  4. Constructive Determination of the Spectral Invariants

    • Oktay Veliev
    Pages 127-169

  5. Periodic Potential from the Spectral Invariants

    • Oktay Veliev
    Pages 171-225

  6. Conclusions

    • Oktay Veliev
    Pages 227-239

  7. Back Matter

    Pages 241-242
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Keywords

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.

Authors and Affiliations

  • Department of Mathematics, Dogus University, Istanbul, Turkey

    Oktay Veliev

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