Spectral Theory of Random Schr dinger Operators

Since the seminal work of P. Anderson in 1958, localization in disordered systems has been the object of intense investigations.

Author: R. Carmona

Publisher: Springer Science & Business Media

ISBN: 9781461244882

Category: Mathematics

Page: 589

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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.
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Spectral Theory of Random Schr dinger Operators

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 ...

Author: Reinhard Lang

Publisher: Springer

ISBN: 9783540466277

Category: Mathematics

Page: 126

View: 831

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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.
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Spectral Theory of Schr dinger Operators

This volume gathers the articles based on a series of lectures from a workshop held at the Institute of Applied Mathematics of the National University of Mexico.

Author: Rafael del Río

Publisher: American Mathematical Soc.

ISBN: 9780821832974

Category: Mathematics

Page: 249

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This volume gathers the articles based on a series of lectures from a workshop held at the Institute of Applied Mathematics of the National University of Mexico. The aim of the book is to present to a non-specialized audience the basic tools needed to understand and appreciate new trends of research on Schrodinger operator theory. Topics discussed include various aspects of the spectral theory of differential operators, the theory of self-adjoint operators, finite rank perturbations, spectral properties of random Schrodinger operators, and scattering theory for Schrodinger operators. The material is suitable for graduate students and research mathematicians interested in differential operators, in particular, spectral theory of Schrodinger operators.
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Spectral Theory of Random Schr dinger Operators

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 ...

Author: Reinhard Lang

Publisher: Springer

ISBN: 3540549757

Category: Mathematics

Page: 126

View: 506

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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.
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Existence and Regularity Properties of the Integrated Density of States of Random Schr dinger Operators

The present text describes in detail a quantity encoding spectral features of random operators: the integrated density of states or spectral distribution function.

Author: Ivan Veselic

Publisher: Springer Science & Business Media

ISBN: 9783540726890

Category: Mathematics

Page: 142

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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.
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Spectral Theory and Mathematical Physics Ergodic Schr dinger operators singular spectrum orthogonal polynomials and inverse spectral theory

In several cases, this collection of surveys portrays both the history of a subject and its current state of the art. Exhaustive lists of references enhance the presentation offered in these surveys.

Author: Barry Simon

Publisher: American Mathematical Soc.

ISBN: 9780821842492

Category: Mathematics

Page: 948

View: 825

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This Festschrift had its origins in a conference called SimonFest held at Caltech, March 27-31, 2006, to honor Barry Simon's 60th birthday. It is not a proceedings volume in the usual sense since the emphasis of the majority of the contributions is on reviews of the state of the art of certain fields, with particular focus on recent developments and open problems. The bulk of the articles in this Festschrift are of this survey form, and a few review Simon's contributions to a particular area. Part 1 contains surveys in the areas of Quantum Field Theory, Statistical Mechanics, Nonrelativistic Two-Body and $N$-Body Quantum Systems, Resonances, Quantum Mechanics with Electric and Magnetic Fields, and the Semiclassical Limit. Part 2 contains surveys in the areas of Random and Ergodic Schrodinger Operators, Singular Continuous Spectrum, Orthogonal Polynomials, and Inverse Spectral Theory. In several cases, this collection of surveys portrays both the history of a subject and its current state of the art. A substantial part of the contributions to this Festschrift are survey articles on the state of the art of certain areas with special emphasis on open problems. This will benefit graduate students as well as researchers who want to get a quick, yet comprehensive introduction into an area covered in this volume.
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Equivariant Sheaves and Functors

1514 : U. Krengel , K. Richter , V. Warstat ( Eds . ) , Ergodic Theory and Related Topics III , Proceedings , 1990 . VIII , 236 pages . 1992 . ... 1498 : R. Lang , Spectral Theory of Random Schrödinger Operators . X , 125 pages . 1991 .

Author: Joseph Bernstein

Publisher: Springer

ISBN: 3540580719

Category: Mathematics

Page: 146

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The equivariant derived category of sheaves is introduced. All usual functors on sheaves are extended to the equivariant situation. Some applications to the equivariant intersection cohomology are given. The theory may be useful to specialists in representation theory, algebraic geometry or topology.
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Spectral Theory and Mathematical Physics

This proceedings volume contains peer-reviewed, selected papers and surveys presented at the conference Spectral Theory and Mathematical Physics (STMP) 2018 which was held in Santiago, Chile, at the Pontifical Catholic University of Chile ...

Author: Pablo Miranda

Publisher: Springer Nature

ISBN: 9783030555566

Category: Mathematics

Page: 272

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This proceedings volume contains peer-reviewed, selected papers and surveys presented at the conference Spectral Theory and Mathematical Physics (STMP) 2018 which was held in Santiago, Chile, at the Pontifical Catholic University of Chile in December 2018. The original works gathered in this volume reveal the state of the art in the area and reflect the intense cooperation between young researchers in spectral theoryand mathematical physics and established specialists in this field. The list of topics covered includes: eigenvalues and resonances for quantum Hamiltonians; spectral shift function and quantum scattering; spectral properties of random operators; magnetic quantum Hamiltonians; microlocal analysis and its applications in mathematical physics. This volume can be of interest both to senior researchers and graduate students pursuing new research topics in Mathematical Physics.
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Fourth Summer School in Analysis and Mathematical Physics

This book consists of three expository articles written by outstanding researchers in Mathematical Physics: Rafael Benguria, Peter Hislop, and Elliott Lieb.

Author: Carlos Villegas-Blas

Publisher: American Mathematical Soc.

ISBN: 9780821840641

Category: Mathematics

Page: 148

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This book consists of three expository articles written by outstanding researchers in Mathematical Physics: Rafael Benguria, Peter Hislop, and Elliott Lieb. The articles are based on their lectures at the Fourth Summer School in Analysis and Mathematical Physics, held at the Institute of Mathematics, Universidad Nacional Autonoma de Mexico, Cuernavaca in May 2005. The main goal of the articles is to link the basic knowledge of a graduate student in Mathematics with three current research topics in Mathematical Physics: Isoperimetric inequalities for eigenvalues of the Laplace Operator, Random Schrodinger Operators, and Stability of Matter, respectively. These well written articles will guide and introduce the reader to current research topics and will also provide information on recent progress in some areas of Mathematical Physics.
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Stochastic Spectral Theory for Selfadjoint Feller Operators

[40] M. Bramanti, Potential theory for stationary Schrödinger operators: a survey of results obtained with non-probabilistic ... [47] R. Carmona and J. Lacroix, Spectral theory of random Schrödinger operators, Probability Theory and its ...

Author: Michael Demuth

Publisher: Birkhäuser

ISBN: 9783034884600

Category: Mathematics

Page: 463

View: 798

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In this book, a beautiful interplay between probability theory (Markov processes, martingale theory) on the one hand and operator and spectral theory on the other yields a uniform treatment of several kinds of Hamiltonians such as the Laplace operator, relativistic Hamiltonian, Laplace-Beltrami operator, and generators of Ornstein-Uhlenbeck processes. The unified approach provides a new viewpoint of and a deeper insight into the subject.
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Orthogonal Polynomials on the Unit Circle Spectral theory

[ 188 ] R. Carmona , One - dimensional Schrödinger operators with random or deterministic potentials : New spectral types , J. Funct . Anal . 51 ( 1983 ) , 229-258 . ( Cited on 630 , 706 , 972. ) ( 189 ) R. Carmona and J. Lacroix ...

Author: Barry Simon

Publisher: American Mathematical Soc.

ISBN: 0821836757

Category:

Page: 578

View: 937

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Determining Spectra in Quantum Theory

This work focuses on various known criteria in the spectral theory of selfadjoint operators.

Author: Michael Demuth

Publisher: Springer Science & Business Media

ISBN: 9780817644390

Category: Mathematics

Page: 219

View: 102

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This work focuses on various known criteria in the spectral theory of selfadjoint operators. The concise, unified presentation is aimed at graduate students and researchers working in the spectral theory of Schrodinger operators with either fixed or random potentials. But given the large gap this book fills in the literature, it will serve a wider audience of mathematical physicists in its contribution to works in spectral theory.
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Limit Operators Collective Compactness and the Spectral Theory of Infinite Matrices

MR0105611 (21:4350) R. Carmona and J. Lacroix: Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, 1990. MR1102675 (92k:47143) M. Capinski and E. P. Kopp: Measure, Integral and Probability, Springer-Verlag, 2004.

Author: Simon N. Chandler-Wilde

Publisher: American Mathematical Soc.

ISBN: 9780821852439

Category: Mathematics

Page: 111

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In the first half of this memoir the authors explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner (2006)) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang (2002). They build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator $A$ (its operator spectrum). In the second half of this memoir the authors study bounded linear operators on the generalised sequence space $\ell^p(\mathbb{Z}^N,U)$, where $p\in [1,\infty]$ and $U$ is some complex Banach space. They make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator $A$ is a locally compact perturbation of the identity. Especially, they obtain stronger results than previously known for the subtle limiting cases of $p=1$ and $\infty$.
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Schr dinger Operators

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 ...

Author: Hans L. Cycon

Publisher: Springer

ISBN: 9783540775225

Category: Science

Page: 319

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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.
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Spectral Theory and Mathematical Physics Ergodic Schr dinger operators singular spectrum orthogonal polynomials and inverse spectral theory

In several cases, this collection of surveys portrays both the history of a subject and its current state of the art.

Author: Fritz Gesztesy

Publisher: American Mathematical Soc.

ISBN: 0821868535

Category: Mathematics

Page: 948

View: 961

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This Festschrift had its origins in a conference called SimonFest held at Caltech, March 27-31, 2006, to honor Barry Simon's 60th birthday. It is not a proceedings volume in the usual sense since the emphasis of the majority of the contributions is on reviews of the state of the art of certain fields, with particular focus on recent developments and open problems. The bulk of the articles in this Festschrift are of this survey form, and a few review Simon's contributions to aparticular area. Part 1 contains surveys in the areas of Quantum Field Theory, Statistical Mechanics, Nonrelativistic Two-Body and $N$-Body Quantum Systems, Resonances, Quantum Mechanics with Electric and Magnetic Fields, and the Semiclassical Limit. Part 2 contains surveys in the areas of Random andErgodic Schrodinger Operators, Singular Continuous Spectrum, Orthogonal Polynomials, and Inverse Spectral Theory. In several cases, this collection of surveys portrays both the history of a subject and its current state of the art. A substantial part of the contributions to this Festschrift are survey articles on the state of the art of certain areas with special emphasis on open problems. This will benefit graduate students as well as researchers who want to get a quick, yet comprehensiveintroduction into an area covered in this volume.
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Spectral Theory and Mathematical Physics

This survey is based on a series of lectures given during the School on Random Schrödinger Operators and the International Conference on Spectral Theory and Mathematical Physics at the Pontificia Universidad Catolica de Chile, ...

Author: Marius Mantoiu

Publisher: Birkhäuser

ISBN: 9783319299921

Category: Mathematics

Page: 255

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The present volume contains the Proceedings of the International Conference on Spectral Theory and Mathematical Physics held in Santiago de Chile in November 2014. Main topics are: Ergodic Quantum Hamiltonians, Magnetic Schrödinger Operators, Quantum Field Theory, Quantum Integrable Systems, Scattering Theory, Semiclassical and Microlocal Analysis, Spectral Shift Function and Quantum Resonances. The book presents survey articles as well as original research papers on these topics. It will be of interest to researchers and graduate students in Mathematics and Mathematical Physics.
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Intermediate Spectral Theory and Quantum Dynamics

[CaL90] [CFGM90] [Ch68] [CoL55] [Con85] [Coo57] [Cor89] [CouH53] [CrHM02] [Cw77] [CyFKS87] [DaL03] [DaT05] [Dav80] [Dav95] [DeBF98] [deO90] R. Carmona and J. Lacroix, Spectral Theory of Random Schrödinger Operators.

Author: César R. de Oliveira

Publisher: Springer Science & Business Media

ISBN: 3764387955

Category: Science

Page: 410

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The spectral theory of linear operators plays a key role in the mathematical formulation of quantum theory. This textbook provides a concise and comprehensible introduction to the spectral theory of (unbounded) self-adjoint operators and its application in quantum dynamics. Many examples and exercises are included that focus on quantum mechanics.
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