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Multidimensional Periodic Schrödinger Operator

Perturbation Theory and Applications

Oktay Veliev

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Springer International Publishing img Link Publisher

Naturwissenschaften, Medizin, Informatik, Technik / Theoretische Physik

Beschreibung

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.


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Schlagwörter

Derivatives of Band Functions, Inverse Problem of Multidimensional, Isoenergetic Surfaces, Periodic Bloch Functions, Spectral Invariants, Schrödinger Operator, Bloch Eigenvalues