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Multidimensional periodic Schrödinger operator : perturbation theory and applications / Oktay Veliev.

Second edition.

9783030245788 (electronic book)
3030245780 (electronic book)

Cham, Switzerland : Springer, [2019]

English

1 online resource (333 pages)

10.1007/978-3-030-24

QC174.17.S3 V45 2019

530.12/4

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.

Includes bibliographical references and index.

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Description based on online resource; title from digital title page (viewed on September 13, 2019).

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