Non-self-adjoint Schrödinger Operator with a Periodic Potential

Non-self-adjoint Schrödinger Operator with a Periodic Potential
Title Non-self-adjoint Schrödinger Operator with a Periodic Potential PDF eBook
Author Oktay Veliev
Publisher
Pages 0
Release 2021
Genre
ISBN 9783030726843

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This book gives a complete spectral analysis of the non-self-adjoint Schrödinger operator with a periodic complex-valued potential. Building from the investigation of the spectrum and spectral singularities and construction of the spectral expansion for the non-self-adjoint Schrödinger operator, the book features a complete spectral analysis of the Mathieu-Schrödinger operator and the Schrödinger operator with a parity-time (PT)-symmetric periodic optical potential. There currently exists no general spectral theorem for non-self-adjoint operators; the approaches in this book thus open up new possibilities for spectral analysis of some of the most important operators used in non-Hermitian quantum mechanics and optics. Featuring detailed proofs and a comprehensive treatment of the subject matter, the book is ideally suited for graduate students at the intersection of physics and mathematics.

Non-self-adjoint Schrödinger Operator with a Periodic Potential

Non-self-adjoint Schrödinger Operator with a Periodic Potential
Title Non-self-adjoint Schrödinger Operator with a Periodic Potential PDF eBook
Author Oktay Veliev
Publisher Springer Nature
Pages 301
Release 2021-06-19
Genre Science
ISBN 3030726835

Download Non-self-adjoint Schrödinger Operator with a Periodic Potential Book in PDF, Epub and Kindle

This book gives a complete spectral analysis of the non-self-adjoint Schrödinger operator with a periodic complex-valued potential. Building from the investigation of the spectrum and spectral singularities and construction of the spectral expansion for the non-self-adjoint Schrödinger operator, the book features a complete spectral analysis of the Mathieu-Schrödinger operator and the Schrödinger operator with a parity-time (PT)-symmetric periodic optical potential. There currently exists no general spectral theorem for non-self-adjoint operators; the approaches in this book thus open up new possibilities for spectral analysis of some of the most important operators used in non-Hermitian quantum mechanics and optics. Featuring detailed proofs and a comprehensive treatment of the subject matter, the book is ideally suited for graduate students at the intersection of physics and mathematics.

Multidimensional Periodic Schrödinger Operator

Multidimensional Periodic Schrödinger Operator
Title Multidimensional Periodic Schrödinger Operator PDF eBook
Author Oktay Veliev
Publisher Springer Nature
Pages 420
Release
Genre
ISBN 3031490355

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Perturbation Theory for the Schrödinger Operator with a Periodic Potential

Perturbation Theory for the Schrödinger Operator with a Periodic Potential
Title Perturbation Theory for the Schrödinger Operator with a Periodic Potential PDF eBook
Author Yulia E. Karpeshina
Publisher Springer
Pages 358
Release 2006-11-14
Genre Mathematics
ISBN 3540691561

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The book is devoted to perturbation theory for the Schrödinger operator with a periodic potential, describing motion of a particle in bulk matter. The Bloch eigenvalues of the operator are densely situated in a high energy region, so regular perturbation theory is ineffective. The mathematical difficulties have a physical nature - a complicated picture of diffraction inside the crystal. The author develops a new mathematical approach to this problem. It provides mathematical physicists with important results for this operator and a new technique that can be effective for other problems. The semiperiodic Schrödinger operator, describing a crystal with a surface, is studied. Solid-body theory specialists can find asymptotic formulae, which are necessary for calculating many physical values.

Non-Selfadjoint Operators in Quantum Physics

Non-Selfadjoint Operators in Quantum Physics
Title Non-Selfadjoint Operators in Quantum Physics PDF eBook
Author Fabio Bagarello
Publisher John Wiley & Sons
Pages 432
Release 2015-09-09
Genre Science
ISBN 1118855272

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A unique discussion of mathematical methods with applications to quantum mechanics Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects presents various mathematical constructions influenced by quantum mechanics and emphasizes the spectral theory of non-adjoint operators. Featuring coverage of functional analysis and algebraic methods in contemporary quantum physics, the book discusses the recent emergence of unboundedness of metric operators, which is a serious issue in the study of parity-time-symmetric quantum mechanics. The book also answers mathematical questions that are currently the subject of rigorous analysis with potentially significant physical consequences. In addition to prompting a discussion on the role of mathematical methods in the contemporary development of quantum physics, the book features: Chapter contributions written by well-known mathematical physicists who clarify numerous misunderstandings and misnomers while shedding light on new approaches in this growing area An overview of recent inventions and advances in understanding functional analytic and algebraic methods for non-selfadjoint operators as well as the use of Krein space theory and perturbation theory Rigorous support of the progress in theoretical physics of non-Hermitian systems in addition to mathematically justified applications in various domains of physics such as nuclear and particle physics and condensed matter physics An ideal reference, Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects is useful for researchers, professionals, and academics in applied mathematics and theoretical and/or applied physics who would like to expand their knowledge of classical applications of quantum tools to address problems in their research. Also a useful resource for recent and related trends, the book is appropriate as a graduate-level and/or PhD-level text for courses on quantum mechanics and mathematical models in physics.

Multidimensional Periodic Schrödinger Operator

Multidimensional Periodic Schrödinger Operator
Title Multidimensional Periodic Schrödinger Operator PDF eBook
Author Oktay Veliev
Publisher Springer
Pages 326
Release 2019-08-02
Genre Science
ISBN 3030245780

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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 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.

Spectral Theory of Random Schrödinger Operators

Spectral Theory of Random Schrödinger Operators
Title Spectral Theory of Random Schrödinger Operators PDF eBook
Author R. Carmona
Publisher Springer Science & Business Media
Pages 611
Release 2012-12-06
Genre Mathematics
ISBN 1461244889

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Since the seminal work of P. Anderson in 1958, localization in disordered systems has been the object of intense investigations. Mathematically speaking, the phenomenon can be described as follows: the self-adjoint operators which are used as Hamiltonians for these systems have a ten dency to have pure point spectrum, especially in low dimension or for large disorder. A lot of effort has been devoted to the mathematical study of the random self-adjoint operators relevant to the theory of localization for disordered systems. It is fair to say that progress has been made and that the un derstanding of the phenomenon has improved. This does not mean that the subject is closed. Indeed, the number of important problems actually solved is not larger than the number of those remaining. Let us mention some of the latter: • A proof of localization at all energies is still missing for two dimen sional systems, though it should be within reachable range. In the case of the two dimensional lattice, this problem has been approached by the investigation of a finite discrete band, but the limiting pro cedure necessary to reach the full two-dimensional lattice has never been controlled. • The smoothness properties of the density of states seem to escape all attempts in dimension larger than one. This problem is particularly serious in the continuous case where one does not even know if it is continuous.