Multidimensional periodic Schrödinger operator : perturbation theory and applications
著者
書誌事項
Multidimensional periodic Schrödinger operator : perturbation theory and applications
Springer, c2019
2nd ed
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注記
Includes bibliographical references and index
The first edition was originally published 2015 as vol. 263 in "Springer tracts in modern physics" --T.p. verso
内容説明・目次
内容説明
This book describes the direct and inverse problems of the multidimensional Schroedinger 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.
目次
Chapter 1 - Preliminary Facts.- Chapter 2- From One-dimensional to Multidimensional.- Chapter 3 - Asymptotic Formulas for the Bloch Eigenvalues and Bloch Functions.-Chapter 4 -Constructive Determination of the Spectral Invariants.- Chapter 5 - Periodic Potential from the Spectral Invariants.- Chapter 6 - Conclusions.
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