Extended States For The Schr Dinger Operator With Quasi Periodic Potential In Dimension Two

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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 Mathematics 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.
Random Schr Dinger Operators
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Author : Margherita Disertori
language : en
Publisher: SMF
Release Date : 2008
Random Schr Dinger Operators written by Margherita Disertori and has been published by SMF this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.
During the last thirty years, random Schrodinger operators, which originated in condensed matter physics, have been studied intensively and very productively. The theory is at the crossroads of a number of mathematical fields: the theory of operators, partial differential equations, the theory of probabilities, in particular the study of stochastic processes and that of random walks and Brownian motion in a random environment. This monograph aims to give the reader a panorama of the subject, from the now-classic foundations to very recent developments.
Green S Functions In Quantum Physics
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Author : Eleftherios N. Economou
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-14
Green S Functions In Quantum Physics written by Eleftherios N. Economou 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 2013-03-14 with Science categories.
In this edition the second and main part of the book has been considerably expanded as to cover important applications of the formalism. In Chap.5 a section was added outlining the extensive role of the tight binding (or equivalently the linear combination of atomic-like orbitals) approach to many branches of solid-state physics. Some additional informa tion (including a table of numerical values) regarding square and cubic lattice Green's functions were incorporated. In Chap.6 the difficult subjects of superconductivity and the Kondo effect are examined by employing an appealingly simple connection to the question of the existence of a bound state in a very shallow potential well. The existence of such a bound state depends entirely on the form of the un perturbed density of states near the end of the spectrum: if the density of states blows up there is always at least one bound state. If the density of states approaches zero continuously, a critical depth (and/or width) of the well must be reached in order to have a bound state. The borderline case of a finite discontinuity (which is very important to superconductivity and the Kondo effect) always produces a bound state with an exponentially small binding energy.
Schr Dinger Operators
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Author : Hans L. Cycon
language : en
Publisher: Springer
Release Date : 2009-08-19
Schr Dinger Operators written by Hans L. Cycon and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-08-19 with Science categories.
A complete understanding of Schrödinger operators is a necessary prerequisite for unveiling the physics of nonrelativistic quanturn mechanics. Furthermore recent research shows that it also helps to deepen our insight into global differential geometry. This monograph written for both graduate students and researchers summarizes and synthesizes the theory of Schrödinger operators emphasizing the progress made in the last decade by Lieb, Enss, Witten and others. Besides general properties, the book covers, in particular, multiparticle quantum mechanics including bound states of Coulomb systems and scattering theory, quantum mechanics in constant electric and magnetic fields, Schrödinger operators with random and almost periodic potentials and, finally, Schrödinger operator methods in differential geometry to prove the Morse inequalities and the index theorem.
Substitution Dynamical Systems Spectral Analysis
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Author : Martine Queffélec
language : en
Publisher: Springer
Release Date : 2006-11-14
Substitution Dynamical Systems Spectral Analysis written by Martine Queffélec 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.
Random Operators
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Author : Michael Aizenman
language : en
Publisher: American Mathematical Soc.
Release Date : 2015-12-11
Random Operators written by Michael Aizenman 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 2015-12-11 with Mathematics categories.
This book provides an introduction to the mathematical theory of disorder effects on quantum spectra and dynamics. Topics covered range from the basic theory of spectra and dynamics of self-adjoint operators through Anderson localization--presented here via the fractional moment method, up to recent results on resonant delocalization. The subject's multifaceted presentation is organized into seventeen chapters, each focused on either a specific mathematical topic or on a demonstration of the theory's relevance to physics, e.g., its implications for the quantum Hall effect. The mathematical chapters include general relations of quantum spectra and dynamics, ergodicity and its implications, methods for establishing spectral and dynamical localization regimes, applications and properties of the Green function, its relation to the eigenfunction correlator, fractional moments of Herglotz-Pick functions, the phase diagram for tree graph operators, resonant delocalization, the spectral statistics conjecture, and related results. The text incorporates notes from courses that were presented at the authors' respective institutions and attended by graduate students and postdoctoral researchers.
Quantum Theory Groups And Representations
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Author : Peter Woit
language : en
Publisher: Springer
Release Date : 2017-11-01
Quantum Theory Groups And Representations written by Peter Woit and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-11-01 with Science categories.
This text systematically presents the basics of quantum mechanics, emphasizing the role of Lie groups, Lie algebras, and their unitary representations. The mathematical structure of the subject is brought to the fore, intentionally avoiding significant overlap with material from standard physics courses in quantum mechanics and quantum field theory. The level of presentation is attractive to mathematics students looking to learn about both quantum mechanics and representation theory, while also appealing to physics students who would like to know more about the mathematics underlying the subject. This text showcases the numerous differences between typical mathematical and physical treatments of the subject. The latter portions of the book focus on central mathematical objects that occur in the Standard Model of particle physics, underlining the deep and intimate connections between mathematics and the physical world. While an elementary physics course of some kind would be helpful to the reader, no specific background in physics is assumed, making this book accessible to students with a grounding in multivariable calculus and linear algebra. Many exercises are provided to develop the reader's understanding of and facility in quantum-theoretical concepts and calculations.
Spectral Theory Of Canonical Systems
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Author : Christian Remling
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2018-08-21
Spectral Theory Of Canonical Systems written by Christian Remling and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-08-21 with Mathematics categories.
Canonical systems occupy a central position in the spectral theory of second order differential operators. They may be used to realize arbitrary spectral data, and the classical operators such as Schrödinger, Jacobi, Dirac, and Sturm-Liouville equations can be written in this form. ‘Spectral Theory of Canonical Systems’ offers a selfcontained and detailed introduction to this theory. Techniques to construct self-adjoint realizations in suitable Hilbert spaces, a modern treatment of de Branges spaces, and direct and inverse spectral problems are discussed. Contents Basic definitions Symmetric and self-adjoint relations Spectral representation Transfer matrices and de Branges spaces Inverse spectral theory Some applications The absolutely continuous spectrum
Mathematical Scattering Theory
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Author : D. R. Yafaev
language : en
Publisher: American Mathematical Soc.
Release Date : 1992-09-09
Mathematical Scattering Theory written by D. R. Yafaev 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 1992-09-09 with Mathematics categories.
Preliminary facts Basic concepts of scattering theory Further properties of the WO Scattering for relatively smooth perturbations The general setup in stationary scattering theory Scattering for perturbations of trace class type Properties of the scattering matrix (SM) The spectral shift function (SSF) and the trace formula
Number Theory And Physics
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Author : Jean-Marc Luck
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Number Theory And Physics written by Jean-Marc Luck 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 2012-12-06 with Science categories.
7 Les Houches Number theory, or arithmetic, sometimes referred to as the queen of mathematics, is often considered as the purest branch of mathematics. It also has the false repu tation of being without any application to other areas of knowledge. Nevertheless, throughout their history, physical and natural sciences have experienced numerous unexpected relationships to number theory. The book entitled Number Theory in Science and Communication, by M.R. Schroeder (Springer Series in Information Sciences, Vol. 7, 1984) provides plenty of examples of cross-fertilization between number theory and a large variety of scientific topics. The most recent developments of theoretical physics have involved more and more questions related to number theory, and in an increasingly direct way. This new trend is especially visible in two broad families of physical problems. The first class, dynamical systems and quasiperiodicity, includes classical and quantum chaos, the stability of orbits in dynamical systems, K.A.M. theory, and problems with "small denominators", as well as the study of incommensurate structures, aperiodic tilings, and quasicrystals. The second class, which includes the string theory of fundamental interactions, completely integrable models, and conformally invariant two-dimensional field theories, seems to involve modular forms and p adic numbers in a remarkable way.