Random Schrödinger Operators

Author : R. Carmona,J. Lacroix
Publisher : Springer Science & Business Media
Page : 611 pages
File Size : 45,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461244882

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Spectral Theory of Random Schrödinger Operators by R. Carmona,J. Lacroix Pdf

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.

Author : P. Bougerol,Lacroix
Publisher : Springer Science & Business Media
Page : 290 pages
File Size : 40,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781468491722

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Products of Random Matrices with Applications to Schrödinger Operators by P. Bougerol,Lacroix Pdf

CHAPTER I THE DETERMINISTIC SCHRODINGER OPERATOR 187 1. The difference equation. Hyperbolic structures 187 2. Self adjointness of H. Spectral properties . 190 3. Slowly increasing generalized eigenfunctions 195 4. Approximations of the spectral measure 196 200 5. The pure point spectrum. A criterion 6. Singularity of the spectrum 202 CHAPTER II ERGODIC SCHRÖDINGER OPERATORS 205 1. Definition and examples 205 2. General spectral properties 206 3. The Lyapunov exponent in the general ergodie case 209 4. The Lyapunov exponent in the independent eas e 211 5. Absence of absolutely continuous spectrum 221 224 6. Distribution of states. Thouless formula 232 7. The pure point spectrum. Kotani's criterion 8. Asymptotic properties of the conductance in 234 the disordered wire CHAPTER III THE PURE POINT SPECTRUM 237 238 1. The pure point spectrum. First proof 240 2. The Laplace transform on SI(2,JR) 247 3. The pure point spectrum. Second proof 250 4. The density of states CHAPTER IV SCHRÖDINGER OPERATORS IN A STRIP 2';3 1. The deterministic Schrödinger operator in 253 a strip 259 2. Ergodie Schrödinger operators in a strip 3. Lyapunov exponents in the independent case. 262 The pure point spectrum (first proof) 267 4. The Laplace transform on Sp(~,JR) 272 5. The pure point spectrum, second proof vii APPENDIX 275 BIBLIOGRAPHY 277 viii PREFACE This book presents two elosely related series of leetures. Part A, due to P.

Author : Margherita Disertori,Werner Kirsch,Abel Klein
Publisher : SMF
Page : 244 pages
File Size : 48,9 Mb
Release : 2008
Category : Mathematics
ISBN : UOM:39015072684635

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Random Schrödinger Operators by Margherita Disertori,Werner Kirsch,Abel Klein Pdf

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.

Author : Reinhard Lang
Publisher : Unknown
Page : 136 pages
File Size : 50,6 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 3662191512

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Spectral Theory of Random Schrodinger Operators by Reinhard Lang Pdf

Author : Reinhard Lang
Publisher : Springer
Page : 133 pages
File Size : 51,6 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540466277

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Spectral Theory of Random Schrödinger Operators by Reinhard Lang Pdf

The interplay between the spectral theory of Schr|dinger operators and probabilistic considerations forms the main theme of these notes, written for the non-specialist reader and intended to provide a brief and elementaryintroduction to this field. An attempt is made to show basic ideas in statu nascendi and to follow their evaluation from simple beginnings through to more advanced results. The term "genetic" in the title refers to this proceedure. The author concentrates on 2 topics which, in the history of the subject, have been of major conceptual importance - on the one hand the Laplacian is a random medium and the left end of its spectrum (leading to large deviation problems for Brownian motion and the link to thenotion of entropy) and on the other, Schr|dinger operators with general ergodic potentials in one-dimensional space. Ideas and concepts are explained in the simplest, possible setting and by means of a few characteristic problems with heuristic arguments preceding rigorous proofs.

Author : Hans L. Cycon,Barry Simon
Publisher : Springer Science & Business Media
Page : 337 pages
File Size : 47,9 Mb
Release : 1987
Category : Computers
ISBN : 9783540167587

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Schrödinger Operators by Hans L. Cycon,Barry Simon Pdf

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.

Author : René Carmona,Jean Lacroix
Publisher : Unknown
Page : 587 pages
File Size : 50,9 Mb
Release : 1990
Category : Schrèodinger operator
ISBN : 376433486X

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Spectral Theory of Random Schrödinger Operators by René Carmona,Jean Lacroix Pdf

Author : Hans L. Cycon,Richard G. Froese,Werner Kirsch,Barry Simon
Publisher : Springer
Page : 319 pages
File Size : 52,5 Mb
Release : 2009-08-19
Category : Science
ISBN : 9783540775225

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Schrödinger Operators by Hans L. Cycon,Richard G. Froese,Werner Kirsch,Barry Simon Pdf

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.

Author : Ivan Veselic
Publisher : Springer Science & Business Media
Page : 151 pages
File Size : 45,5 Mb
Release : 2008-01-02
Category : Mathematics
ISBN : 9783540726890

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Existence and Regularity Properties of the Integrated Density of States of Random Schrödinger Operators by Ivan Veselic Pdf

This book describes in detail a quantity encoding spectral feature of random operators: the integrated density of states or spectral distribution function. It presents various approaches to the construction of the integrated density of states and the proof of its regularity properties. The book also includes references to and a discussion of other properties of the IDS as well as a variety of models beyond those treated in detail here.

Author : R. Carmona,J. LaCroix
Publisher : Unknown
Page : 620 pages
File Size : 47,6 Mb
Release : 1990-01-01
Category : Electronic
ISBN : 1461244897

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Spectral Theory of Random Schrodinger Operators by R. Carmona,J. LaCroix Pdf

Author : Sergio Albeverio,Anindita Balslev,Ricardo Weder
Publisher : Springer Nature
Page : 316 pages
File Size : 46,6 Mb
Release : 2021-06-03
Category : Mathematics
ISBN : 9783030684907

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Schrödinger Operators, Spectral Analysis and Number Theory by Sergio Albeverio,Anindita Balslev,Ricardo Weder Pdf

This book gives its readers a unique opportunity to get acquainted with new aspects of the fruitful interactions between Analysis, Geometry, Quantum Mechanics and Number Theory. The present book contains a number of contributions by specialists in these areas as an homage to the memory of the mathematician Erik Balslev and, at the same time, advancing a fascinating interdisciplinary area still full of potential. Erik Balslev has made original and important contributions to several areas of Mathematics and its applications. He belongs to the founders of complex scaling, one of the most important methods in the mathematical and physical study of eigenvalues and resonances of Schrödinger operators, which has been very essential in advancing the solution of fundamental problems in Quantum Mechanics and related areas. He was also a pioneer in making available and developing spectral methods in the study of important problems in Analytic Number Theory.

Author : P. Bougerol,LaCroix
Publisher : Unknown
Page : 300 pages
File Size : 43,6 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 1468491733

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Products of Random Matrices with Applications to Schrodinger Operators by P. Bougerol,LaCroix Pdf

Author : Philippe Bougerol,Jean Lacroix
Publisher : Unknown
Page : 283 pages
File Size : 53,5 Mb
Release : 2024-06-30
Category : Random matrices
ISBN : OCLC:1008596668

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Products of Random Matrices with Applications to Schrödinger Operators by Philippe Bougerol,Jean Lacroix Pdf

Author : Leonid Pastur,Alexander Figotin
Publisher : Springer
Page : 0 pages
File Size : 42,9 Mb
Release : 2011-12-10
Category : Science
ISBN : 364274348X

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Spectra of Random and Almost-Periodic Operators by Leonid Pastur,Alexander Figotin Pdf

In the last fifteen years the spectral properties of the Schrodinger equation and of other differential and finite-difference operators with random and almost-periodic coefficients have attracted considerable and ever increasing interest. This is so not only because of the subject's position at the in tersection of operator spectral theory, probability theory and mathematical physics, but also because of its importance to theoretical physics, and par ticularly to the theory of disordered condensed systems. It was the requirements of this theory that motivated the initial study of differential operators with random coefficients in the fifties and sixties, by the physicists Anderson, 1. Lifshitz and Mott; and today the same theory still exerts a strong influence on the discipline into which this study has evolved, and which will occupy us here. The theory of disordered condensed systems tries to describe, in the so-called one-particle approximation, the properties of condensed media whose atomic structure exhibits no long-range order. Examples of such media are crystals with chaotically distributed impurities, amorphous substances, biopolymers, and so on. It is natural to describe the location of atoms and other characteristics of such media probabilistically, in such a way that the characteristics of a region do not depend on the region's position, and the characteristics of regions far apart are correlated only very weakly. An appropriate model for such a medium is a homogeneous and ergodic, that is, metrically transitive, random field.

Author : Michael Aizenman,Simone Warzel
Publisher : American Mathematical Soc.
Page : 326 pages
File Size : 41,6 Mb
Release : 2015-12-11
Category : Functional analysis -- Miscellaneous applications of functional analysis -- Applications in quantum physics
ISBN : 9781470419134

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Random Operators by Michael Aizenman,Simone Warzel Pdf

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.