Perturbation Theory For The Schrodinger Operator With A Periodic Potential
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Perturbation Theory For The Schr Dinger Operator With A Periodic Potential
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Author : Yulia E. Karpeshina
language : en
Publisher: Springer
Release Date : 2006-11-14
Perturbation Theory For The Schr Dinger Operator With A Periodic Potential written by Yulia E. Karpeshina and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-14 with Mathematics categories.
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.
Perturbation Theory For The Schrodinger Operator With A Periodic Potential
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Author : Yulia E. Karpeshina
language : en
Publisher:
Release Date : 2014-01-15
Perturbation Theory For The Schrodinger Operator With A Periodic Potential written by Yulia E. Karpeshina and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-01-15 with categories.
Multidimensional Periodic Schr Dinger Operator
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Author : Oktay Veliev
language : en
Publisher: Springer Nature
Release Date :
Multidimensional Periodic Schr Dinger Operator written by Oktay Veliev and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on with categories.
Multidimensional Periodic Schr Dinger Operator
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Author : Oktay Veliev
language : en
Publisher: Springer
Release Date : 2015-03-28
Multidimensional Periodic Schr Dinger Operator written by Oktay Veliev and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-03-28 with Science categories.
The book describes the direct problems and the inverse problem of the multidimensional Schrödinger operator with a periodic potential. This concerns perturbation theory and constructive determination of the spectral invariants and finding the periodic potential from the given Bloch eigenvalues. The unique method of this book derives the asymptotic formulas for Bloch eigenvalues and Bloch functions for arbitrary dimension. Moreover, the measure of the iso-energetic surfaces in the high energy region is construct and estimated. It implies the validity of the Bethe-Sommerfeld conjecture for arbitrary dimensions and arbitrary lattices. Using the perturbation theory constructed in this book, the spectral invariants of the multidimensional operator from the given Bloch eigenvalues are determined. Some of these invariants are explicitly expressed by the Fourier coefficients of the potential. This way the possibility to determine the potential constructively by using Bloch eigenvalues as input data is given. In the end an algorithm for the unique determination of the potential is given.
Gaps In The Dispersion Relation Of The One Dimensional Schr Dinger Operator With Periodic Potentials
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Author : Thomas Z. Dean
language : en
Publisher:
Release Date : 2022
Gaps In The Dispersion Relation Of The One Dimensional Schr Dinger Operator With Periodic Potentials written by Thomas Z. Dean and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022 with Schrödinger operator categories.
The self-adjoint Schrödinger operator is the difference of a kinetic (Laplacian operator)and potential energy (multiplication operator). The study of this operator continues to attract the interest of many mathematicians and physicists. A commonly used mathematical approach to understand quantum mechanics is through the use of spectral and perturbation theory of the Schrödinger operator. By understanding the spectrum of the Schrödinger operator, we can understand the allowed energy states of a quantum system corresponding to a specific potential. The choice of potential dictates the behavior of the spectrum of the Schrödinger operator which in return provides insight into the behavior of the corresponding quantum system. We study periodic potentials for the Schrödinger operator because of its relation to the phenomena of Anderson localization and semi-conductor theory. A new algorithm is developed to numerically approximate the spectrum of one-dimensional periodic Schrödinger operators. From this, the behavior of spectral gaps are understood when parameters of the potential are changed (e.g. period and amplitude).Moreover, the convergence properties and the behavior of the spectrum as continuous periodic potentials are approximated by their Fourier modes are studied. The behavior of the first spectral gap for such convergences are demonstrated. These results show that the first spectral gap is well-behaved in the strong and norm resolvent convergence.
Introduction To Quantum Mechanics
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Author : H. J. W. Mller-Kirsten
language : en
Publisher: World Scientific
Release Date : 2006
Introduction To Quantum Mechanics written by H. J. W. Mller-Kirsten and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with Science categories.
After a consideration of basic quantum mechanics, this introduction aims at a side by side treatment of fundamental applications of the Schrdinger equation on the one hand and the applications of the path integral on the other. Different from traditional texts and using a systematic perturbation method, the solution of Schrdinger equations includes also those with anharmonic oscillator potentials, periodic potentials, screened Coulomb potentials and a typical singular potential, as well as the investigation of the large order behavior of the perturbation series. On the path integral side, after introduction of the basic ideas, the expansion around classical configurations in Euclidean time, such as instantons, is considered, and the method is applied in particular to anharmonic oscillator and periodic potentials. Numerous other aspects are treated on the way, thus providing the reader an instructive overview over diverse quantum mechanical phenomena, e.g. many other potentials, Green's functions, comparison with WKB, calculation of lifetimes and sojourn times, derivation of generating functions, the Coulomb problem in various coordinates, etc. All calculations are given in detail, so that the reader can follow every step.
Introduction To Quantum Mechanics
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Author : Harald J W Müller-Kirsten
language : en
Publisher: World Scientific Publishing Company
Release Date : 2012-07-19
Introduction To Quantum Mechanics written by Harald J W Müller-Kirsten and has been published by World Scientific Publishing Company this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-07-19 with Science categories.
This text on quantum mechanics begins by covering all the main topics of an introduction to the subject. It then concentrates on newer developments. In particular it continues with the perturbative solution of the Schrödinger equation for various potentials and thereafter with the introduction and evaluation of their path integral counterparts. Considerations of the large order behavior of the perturbation expansions show that in most applications these are asymptotic expansions. The parallel consideration of path integrals requires the evaluation of these around periodic classical configurations, the fluctuation equations about which lead back to specific wave equations. The period of the classical configurations is related to temperature, and permits transitions to the thermal domain to be classified as phase transitions. In this second edition of the text important applications and numerous examples have been added. In particular, the chapter on the Coulomb potential has been extended to include an introduction to chemical bonds, the chapter on periodic potentials has been supplemented by a section on the band theory of metals and semiconductors, and in the chapter on large order behavior a section has been added illustrating the success of converging factors in the evaluation of asymptotic expansions. Detailed calculations permit the reader to follow every step.
Lecture Notes On Schr Dinger Equations
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Author : Aleksandr Andreevich Pankov
language : en
Publisher:
Release Date : 2007
Lecture Notes On Schr Dinger Equations written by Aleksandr Andreevich Pankov and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007 with Combinatorial analysis categories.
CONTENTS: Preface; A Bit of Quantum Mechanics; Operators in Hilbert Spaces; Spectral Theorem for Self-adjoint Operators; Compact Operators and the Hilbert-Schmidt Theorem; Elements of Perturbation Theory; Variational Principles; One-Dimensional Schrödinger Operator; Multidimensional Schrödinger Operator; Periodic Schrödinger Operator; Quantum Graphs; Non-linear Schrödinger Equation; References; Index.
Schr Dinger Operators
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Author : Hans L. Cycon
language : en
Publisher: Springer Science & Business Media
Release Date : 1987
Schr Dinger Operators written by Hans L. Cycon and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1987 with Computers categories.
Are you looking for a concise summary of the theory of Schrödinger operators? Here it is. Emphasizing the progress made in the last decade by Lieb, Enss, Witten and others, the three authors don’t just cover general properties, but also detail multiparticle quantum mechanics – including bound states of Coulomb systems and scattering theory. This corrected and extended reprint contains updated references as well as notes on the development in the field over the past twenty years.
Extended States For The Schr Dinger Operator With Quasi Periodic Potential In Dimension Two
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Author : Yulia Karpeshina
language : en
Publisher: American Mathematical Soc.
Release Date : 2019-04-10
Extended States For The Schr Dinger Operator With Quasi Periodic Potential In Dimension Two written by Yulia Karpeshina and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-04-10 with Schrödinger equation categories.
The authors consider a Schrödinger operator H=−Δ+V(x⃗ ) in dimension two with a quasi-periodic potential V(x⃗ ). They prove that the absolutely continuous spectrum of H contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves ei⟨ϰ⃗ ,x⃗ ⟩ in the high energy region. Second, the isoenergetic curves in the space of momenta ϰ⃗ corresponding to these eigenfunctions have the form of slightly distorted circles with holes (Cantor type structure). A new method of multiscale analysis in the momentum space is developed to prove these results. The result is based on a previous paper on the quasiperiodic polyharmonic operator (−Δ)l+V(x⃗ ), l>1. Here the authors address technical complications arising in the case l=1. However, this text is self-contained and can be read without familiarity with the previous paper.