Schrödinger Operators

Author : Gerald Teschl
Publisher : American Mathematical Soc.
Page : 322 pages
File Size : 44,5 Mb
Release : 2009
Category : Mathematics
ISBN : 9780821846605

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Mathematical Methods in Quantum Mechanics by Gerald Teschl Pdf

Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrodinger operators. Part 1 of the book is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are covered. The spectral theorem is a central topic in this approach and is introduced at an early stage. Part 2 starts with the free Schrodinger equation and computes the free resolvent and time evolution. Position, momentum, and angular momentum are discussed via algebraic methods. Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom. Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory. This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced calculus and a one-semester introduction to complex analysis are required. In particular, no functional analysis and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics. Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature. It is well suited for self-study and includes numerous exercises (many with hints).

Author : Hans L. Cycon,Barry Simon
Publisher : Springer Science & Business Media
Page : 337 pages
File Size : 40,5 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 : Hans L. Cycon,Richard G. Froese,Werner Kirsch,Barry Simon
Publisher : Springer
Page : 319 pages
File Size : 44,5 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 : R. Carmona,J. Lacroix
Publisher : Springer Science & Business Media
Page : 611 pages
File Size : 40,5 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.D. Hislop,I.M. Sigal
Publisher : Springer Science & Business Media
Page : 331 pages
File Size : 50,6 Mb
Release : 2012-12-06
Category : Technology & Engineering
ISBN : 9781461207412

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Introduction to Spectral Theory by P.D. Hislop,I.M. Sigal Pdf

The intention of this book is to introduce students to active areas of research in mathematical physics in a rather direct way minimizing the use of abstract mathematics. The main features are geometric methods in spectral analysis, exponential decay of eigenfunctions, semi-classical analysis of bound state problems, and semi-classical analysis of resonance. A new geometric point of view along with new techniques are brought out in this book which have both been discovered within the past decade. This book is designed to be used as a textbook, unlike the competitors which are either too fundamental in their approach or are too abstract in nature to be considered as texts. The authors' text fills a gap in the marketplace.

Author : Rafael del Río,Carlos Villegas-Blas
Publisher : American Mathematical Soc.
Page : 264 pages
File Size : 42,7 Mb
Release : 2004
Category : Schrödinger operator
ISBN : 9780821832974

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Spectral Theory of Schrodinger Operators by Rafael del Río,Carlos Villegas-Blas Pdf

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.

Author : Sandro Graffi
Publisher : Springer
Page : 282 pages
File Size : 48,7 Mb
Release : 2006-11-14
Category : Science
ISBN : 9783540397069

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Schrödinger Operators, Como 1984 by Sandro Graffi Pdf

Author : Huzihiro Araki,Hiroshi Ezawa
Publisher : World Scientific
Page : 288 pages
File Size : 53,7 Mb
Release : 2004-05-07
Category : Mathematics
ISBN : 9789814482981

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Topics in the Theory of Schrödinger Operators by Huzihiro Araki,Hiroshi Ezawa Pdf

This invaluable book presents reviews of some recent topics in the theory of Schrödinger operators. It includes a short introduction to the subject, a survey of the theory of the Schrödinger equation when the potential depends on the time periodically, an introduction to the so-called FBI transformation (also known as coherent state expansion) with application to the semi-classical limit of the S-matrix, an overview of inverse spectral and scattering problems, and a study of the ground state of the Pauli–Fierz model with the use of the functional integral. The material is accessible to graduate students and non-expert researchers. Contents: Time-Periodic Schrödinger Equations (K Yajima)An Application of Phase Space Tunneling to Multistate Scattering Theory (A Martinez et al.)Inverse Spectral Theory (H Isozaki)Analysis of Ground States of Atoms Interacting with a Quantized Radiation Field (F Hiroshima) Readership: Researchers and graduate students in mathematical physics, high energy physics, theoretical physics, and analysis and differential equations. Keywords:Schrödinger Operator;Time-Periodic Potential;Scattering Theory;Coherent State Expansion;Semi-Classical Limit of the S-Matrix;Inverse Problems;Pauli–Fierz Model;Functional Integral

Author : David Damanik,Jake Fillman
Publisher : American Mathematical Society
Page : 464 pages
File Size : 42,5 Mb
Release : 2022-08-19
Category : Mathematics
ISBN : 9781470470869

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One-Dimensional Ergodic Schrödinger Operators by David Damanik,Jake Fillman Pdf

The theory of one-dimensional ergodic operators involves a beautiful synthesis of ideas from dynamical systems, topology, and analysis. Additionally, this setting includes many models of physical interest, including those operators that model crystals, disordered media, or quasicrystals. This field has seen substantial progress in recent decades, much of which has yet to be discussed in textbooks. Beginning with a refresher on key topics in spectral theory, this volume presents the basic theory of discrete one-dimensional Schrödinger operators with dynamically defined potentials. It also includes a self-contained introduction to the relevant aspects of ergodic theory and topological dynamics. This text is accessible to graduate students who have completed one-semester courses in measure theory and complex analysis. It is intended to serve as an introduction to the field for junior researchers and beginning graduate students as well as a reference text for people already working in this area. It is well suited for self-study and contains numerous exercises (many with hints).

Author : Rupert L. Frank,Ari Laptev,Timo Weidl
Publisher : Cambridge University Press
Page : 524 pages
File Size : 55,5 Mb
Release : 2022-11-17
Category : Mathematics
ISBN : 9781009218443

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Schrödinger Operators: Eigenvalues and Lieb–Thirring Inequalities by Rupert L. Frank,Ari Laptev,Timo Weidl Pdf

The analysis of eigenvalues of Laplace and Schrödinger operators is an important and classical topic in mathematical physics with many applications. This book presents a thorough introduction to the area, suitable for masters and graduate students, and includes an ample amount of background material on the spectral theory of linear operators in Hilbert spaces and on Sobolev space theory. Of particular interest is a family of inequalities by Lieb and Thirring on eigenvalues of Schrödinger operators, which they used in their proof of stability of matter. The final part of this book is devoted to the active research on sharp constants in these inequalities and contains state-of-the-art results, serving as a reference for experts and as a starting point for further research.

Author : Friedrich Haslinger
Publisher : Walter de Gruyter GmbH & Co KG
Page : 348 pages
File Size : 42,6 Mb
Release : 2014-08-20
Category : Mathematics
ISBN : 9783110377835

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The d-bar Neumann Problem and Schrödinger Operators by Friedrich Haslinger Pdf

The topic of this book is located at the intersection of complex analysis, operator theory and partial differential equations. It begins with results on the canonical solution operator to restricted to Bergman spaces of holomorphic d-bar functions in one and several complex variables.These operators are Hankel operators of special type. In the following the general complex is investigated on d-bar spaces over bounded pseudoconvex domains and on weighted d-bar spaces. The main part is devoted to the spectral analysis of the complex Laplacian and to compactness of the Neumann operator. The last part contains a detailed account of the application of the methods to Schrödinger operators, Pauli and Dirac operators and to Witten-Laplacians. It is assumed that the reader has a basic knowledge of complex analysis, functional analysis and topology. With minimal prerequisites required, this book provides a systematic introduction to an active area of research for both students at a bachelor level and mathematicians.

Author : Jean Bourgain
Publisher : Princeton University Press
Page : 183 pages
File Size : 48,9 Mb
Release : 2005
Category : Mathematics
ISBN : 9780691120980

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Green's Function Estimates for Lattice Schrodinger Operators and Applications. (AM-158) by Jean Bourgain Pdf

This book presents an overview of recent developments in the area of localization for quasi-periodic lattice Schrödinger operators and the theory of quasi-periodicity in Hamiltonian evolution equations. The physical motivation of these models extends back to the works of Rudolph Peierls and Douglas R. Hofstadter, and the models themselves have been a focus of mathematical research for two decades. Jean Bourgain here sets forth the results and techniques that have been discovered in the last few years. He puts special emphasis on so-called "non-perturbative" methods and the important role of subharmonic function theory and semi-algebraic set methods. He describes various applications to the theory of differential equations and dynamical systems, in particular to the quantum kicked rotor and KAM theory for nonlinear Hamiltonian evolution equations. Intended primarily for graduate students and researchers in the general area of dynamical systems and mathematical physics, the book provides a coherent account of a large body of work that is presently scattered in the literature. It does so in a refreshingly contained manner that seeks to convey the present technological "state of the art."

Author : Joel S. Feldman,Richard Gerd Froese,Lon M. Rosen
Publisher : American Mathematical Soc.
Page : 314 pages
File Size : 47,9 Mb
Release : 1995
Category : Science
ISBN : 9780821803660

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Mathematical Quantum Theory II: Schrodinger Operators by Joel S. Feldman,Richard Gerd Froese,Lon M. Rosen Pdf

The articles in this collection constitute the proceedings of the Canadian Mathematical Society Annual Seminar on Mathematical Quantum Theory, held in Vancouver in August 1993. The meeting was run as a research-level summer school concentrating on two related areas of contemporary mathematical physics. The first area, quantum field theory and many-body theory, is covered in volume 1 of these proceedings. The second area, treated in the present volume, is Schrödinger operators. The meeting featured a series of four-hour mini-courses, designed to introduce students to the state of the art in particular areas, and thirty hour-long expository lectures. With contributions from some of the top experts in the field, this book is an important resource for those interested in activity at the frontiers of mathematical quantum theory.

Author : P. Bougerol,Lacroix
Publisher : Springer Science & Business Media
Page : 290 pages
File Size : 43,8 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 : Sergio Albeverio,Anindita Balslev,Ricardo Weder
Publisher : Springer Nature
Page : 316 pages
File Size : 44,7 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.