Multidimensional Periodic Schrödinger Operator

Perturbation Theories for High Energy Regions and Their Applications

Oktay Veliev (Autor)

Buch | Hardcover

XIII, 411 Seiten

2024 | 3rd ed. 2024
Springer International Publishing (Verlag)
978-3-031-49034-7 (ISBN)

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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 and significantly expanded third edition features an extension of this framework to all dimensions, offering a now complete theory of self-adjoint Schrödinger operators within periodic potentials. Drawing from recent advancements in mathematical analysis, this edition delves even deeper into the intricacies of the subject. It explores the connections between the multidimensional Schrödinger operator, periodic potentials, and other fundamental areas of mathematical physics. The book's comprehensive approach equips both students and researchers with the tools to tackle complex problems and contribute to the ongoing exploration of quantum phenomena.

lt;b>Oktay Veliev received his B.S. degree in Mathematics in 1977 and Ph.D. degree in Mathematics in 1980 from Moscow State University, earning a Doctor of Sciences degree in 1989. From 1980 to 1983, he was a researcher and then senior researcher (1983-1988) at the Institute of Mathematics of the Academy of Sciences of Azerbaijan SSR. At Baku State University (Azerbaijan) he has been an Associate Professor (1988-1991), a Professor (1991-1992), and Head of the Department of Functional Analysis (1992-1997). Between 1993 and 1997, he was President of the Azerbaijan Mathematical Society. He was a visiting Professor at the University of Nantes (France), the Institute of Mathematics at the ETH (Switzerland), and Sussex University (England). From 1997 to 2002 he was a Professor at Dokuz Eylul University (Turkey), and since 2003 has been a Professor at Dogus University (Turkey). He has received grants from the American Mathematical Society and the International Science Foundation (Grant No. MVVOOO).

Preliminary Facts.- From One-dimensional to Multidimensional.- Asymptotic Formulas for the Bloch Eigenvalues and Bloch Functions.- Constructive Determination of the Spectral Invariants.- Periodic Potential from the Spectral Invariants.- Conclusions and Some Generalization.

Erscheinungsdatum	02.03.2024
Reihe/Serie	Springer Tracts in Modern Physics
Zusatzinfo	XIII, 411 p. 1 illus.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Naturwissenschaften ► Physik / Astronomie ► Quantenphysik
Themenwelt	Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik
Schlagworte	asymptotic formulas • Bethe-Sommerfeld Conjecture • Bethe–Sommerfeld Conjecture • Bloch Eigenvalues • Constructive Potential Determination • Derivatives of Band Functions • Fourier coefficients • Inverse Problem of Multidimensional • Isoenergetic Surfaces • Periodic Bloch Functions • Quantum Exploration Tools • Schrödinger Operator Problems • self-adjoint operators • Spectral Invariants Determination
ISBN-10	3-031-49034-7 / 3031490347
ISBN-13	978-3-031-49034-7 / 9783031490347
Zustand	Neuware