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Scattering Theory of Classical and Quantum N-Particle Systems by Jan Derezinski,Christian Gerard Pdf
This monograph addresses researchers and students. It is a modern presentation of time-dependent methods for studying problems of scattering theory in the classical and quantum mechanics of N-particle systems. Particular attention is paid to long-range potentials. For a large class of interactions the existence of the asymptotic velocity and the asymptotic completeness of the wave operators is shown. The book is self-contained and explains in detail concepts that deepen the understanding. As a special feature of the book, the beautiful analogy between classical and quantum scattering theory (e.g., for N-body Hamiltonians) is presented with deep insight into the physical and mathematical problems.
Scattering Theory in Mathematical Physics by J.A. Lavita,J.P. Marchand Pdf
These proceedings contain lectures given at the N.A.T.O. Advanced Study Institute entitled "Scattering Theory in Mathematics and Physics" held in Denver, Colorado, June 11-29, 1973. We have assembled the main series of lectures and some presented by other participants that seemed naturally to complement them. Unfortunately the size of this volume does not allow for a full account of all the contributions made at the Conference; however, all present were pleased by the number and breadth of those topics covered in the informal afternoon sessions. The purpose of the meeting, as reflected in its title, was to examine the single topic of scattering theory in as many of its manifestations as possible, i.e. as a hub of concepts and techniques from both mathematics and physics. The format of all the topics presented here is mathematical. The physical content embraces classical and quantum mechanical scattering, N-body systems and quantum field theoretical models. Left out are such subjects as the so-called analytic S-matrix theory and phenomeno logical models for high energy scattering. We would like to thank the main lecturers for their excellent presentations and written summaries. They provided a focus for the exceptionally strong interaction among the participants and we hope that some of the coherence achieved is reflected in these published notes. We have made no attempt to unify notation.
Quantum Scattering Theory for Several Particle Systems by L.D. Faddeev,S.P. Merkuriev Pdf
The last decade witnessed an increasing interest of mathematicians in prob lems originated in mathematical physics. As a result of this effort, the scope of traditional mathematical physics changed considerably. New problems es pecially those connected with quantum physics make use of new ideas and methods. Together with classical and functional analysis, methods from dif ferential geometry and Lie algebras, the theory of group representation, and even topology and algebraic geometry became efficient tools of mathematical physics. On the other hand, the problems tackled in mathematical physics helped to formulate new, purely mathematical, theorems. This important development must obviously influence the contemporary mathematical literature, especially the review articles and monographs. A considerable number of books and articles appeared, reflecting to some extend this trend. In our view, however, an adequate language and appropriate methodology has not been developed yet. Nowadays, the current literature includes either mathematical monographs occasionally using physical terms, or books on theoretical physics focused on the mathematical apparatus. We hold the opinion that the traditional mathematical language of lem mas and theorems is not appropriate for the contemporary writing on mathe matical physics. In such literature, in contrast to the standard approaches of theoretical physics, the mathematical ideology must be utmost emphasized and the reference to physical ideas must be supported by appropriate mathe matical statements. Of special importance are the results and methods that have been developed in this way for the first time.
Principles of Quantum Scattering Theory by Dzevad Belkic Pdf
Scattering is one of the most powerful methods used to study the structure of matter, and many of the most important breakthroughs in physics have been made by means of scattering. Nearly a century has passed since the first investigations in this field, and the work undertaken since then has resulted in a rich literature encompassing both experimental and theoretical results. In scattering, one customarily studies collisions among nuclear, sub-nuclear, atomic or molecular particles, and as these are intrinsically quantum systems, it is logical that quantum mechanics is used as the basis for modern scattering theory. In Principles of Quantum Scattering Theory, the author judiciously combines physical intuition and mathematical rigour to present various selected principles of quantum scattering theory. As always in physics, experiment should be used to ultimately validate physical and mathematical modelling, and the author presents a number of exemplary illustrations, comparing theoretical and experimental cross sections in a selection of major inelastic ion-atom collisions at high non-relativistic energies. Quantum scattering theory, one of the most beautiful theories in physics, is also very rich in mathematics. Principles of Quantum Scattering Theory is intended primarily for graduate physics students, but also for non-specialist physicists for whom the clarity of exposition should aid comprehension of these mathematical complexities.
Scattering Theory in Mathematical Physics by James LaVita,J.P. Marchand Pdf
These proceedings contain lectures given at the N.A.T.O. Advanced Study Institute entitled "Scattering Theory in Mathematics and Physics" held in Denver, Colorado, June 11-29, 1973. We have assembled the main series of lectures and some presented by other participants that seemed naturally to complement them. Unfortunately the size of this volume does not allow for a full account of all the contributions made at the Conference; however, all present were pleased by the number and breadth of those topics covered in the informal afternoon sessions. The purpose of the meeting, as reflected in its title, was to examine the single topic of scattering theory in as many of its manifestations as possible, i.e. as a hub of concepts and techniques from both mathematics and physics. The format of all the topics presented here is mathematical. The physical content embraces classical and quantum mechanical scattering, N-body systems and quantum field theoretical models. Left out are such subjects as the so-called analytic S-matrix theory and phenomeno logical models for high energy scattering. We would like to thank the main lecturers for their excellent presentations and written summaries. They provided a focus for the exceptionally strong interaction among the participants and we hope that some of the coherence achieved is reflected in these published notes. We have made no attempt to unify notation.
Scattering Theory of Waves and Particles by Roger G. Newton Pdf
This volume crosses the boundaries of physics' traditional subdivisions to treat scattering theory within the context of classical electromagnetic radiation, classical particle mechanics, and quantum mechanics. Includes updates on developments in three-particle collisions, scattering by noncentral potentials, and inverse scattering problems. 1982 edition.
Inverse Problems in Quantum Scattering Theory by Khosrow Chadan,Pierre C. Sabatier Pdf
The normal business of physicists may be schematically thought of as predic ting the motions of particles on the basis of known forces, or the propagation of radiation on the basis of a known constitution of matter. The inverse problem is to conclude what the forces or constitutions are on the basis of the observed motion. A large part of our sensory contact with the world around us depends on an intuitive solution of such an inverse problem: We infer the shape, size, and surface texture of external objects from their scattering and absorption of light as detected by our eyes. When we use scattering experiments to learn the size or shape of particles, or the forces they exert upon each other, the nature of the problem is similar, if more refined. The kinematics, the equations of motion, are usually assumed to be known. It is the forces that are sought, and how they vary from point to point. As with so many other physical ideas, the first one we know of to have touched upon the kind of inverse problem discussed in this book was Lord Rayleigh (1877). In the course of describing the vibrations of strings of variable density he briefly discusses the possibility of inferring the density distribution from the frequencies of vibration. This passage may be regarded as a precursor of the mathematical study of the inverse spectral problem some seventy years later.
Inverse Spectral and Scattering Theory by Hiroshi Isozaki Pdf
The aim of this book is to provide basic knowledge of the inverse problems arising in various areas in mathematics, physics, engineering, and medical science. These practical problems boil down to the mathematical question in which one tries to recover the operator (coefficients) or the domain (manifolds) from spectral data. The characteristic properties of the operators in question are often reduced to those of Schrödinger operators. We start from the 1-dimensional theory to observe the main features of inverse spectral problems and then proceed to multi-dimensions. The first milestone is the Borg–Levinson theorem in the inverse Dirichlet problem in a bounded domain elucidating basic motivation of the inverse problem as well as the difference between 1-dimension and multi-dimension. The main theme is the inverse scattering, in which the spectral data is Heisenberg’s S-matrix defined through the observation of the asymptotic behavior at infinity of solutions. Significant progress has been made in the past 30 years by using the Faddeev–Green function or the complex geometrical optics solution by Sylvester and Uhlmann, which made it possible to reconstruct the potential from the S-matrix of one fixed energy. One can also prove the equivalence of the knowledge of S-matrix and that of the Dirichlet-to-Neumann map for boundary value problems in bounded domains. We apply this idea also to the Dirac equation, the Maxwell equation, and discrete Schrödinger operators on perturbed lattices. Our final topic is the boundary control method introduced by Belishev and Kurylev, which is for the moment the only systematic method for the reconstruction of the Riemannian metric from the boundary observation, which we apply to the inverse scattering on non-compact manifolds. We stress that this book focuses on the lucid exposition of these problems and mathematical backgrounds by explaining the basic knowledge of functional analysis and spectral theory, omitting the technical details in order to make the book accessible to graduate students as an introduction to partial differential equations (PDEs) and functional analysis.
Mathematical Scattering Theory by Dmitri_ Rauel_evich I_Afaev Pdf
The main subject of this book is applications of methods of scattering theory to differential operators, primarily the Schrodinger operator. There are two different trends in scattering theory for differential operators. The first one relies on the abstract scattering theory. The second one is almost independent of it. In this approach the abstract theory is replaced by a concrete investigation of the corresponding differential equation. In this book both of these trends are presented. The first half of this book begins with the summary of the main results of the general scattering theory of the previous book by the author, Mathematical Scattering Theory: General Theory, American Mathematical Society, 1992. The next three chapters illustrate basic theorems of abstract scattering theory, presenting, in particular, their applications to scattering theory of perturbations of differential operators with constant coefficients and to the analysis of the trace class method. In the second half of the book direct methods of scattering theory for differential operators are presented. After considering the one-dimensional case, the author returns to the multi-dimensional problem and discusses various analytical methods and tools appropriate for the analysis of differential operators, including, among others, high- and low-energy asymptotics of the Green function, the scattering matrix, ray and eikonal expansions. The book is based on graduate courses taught by the author at Saint-Petersburg (Russia) and Rennes (France) Universities and is oriented towards a reader interested in studying deep aspects of scattering theory (for example, a graduate student in mathematical physics).
The Inverse Problem of Scattering Theory by Z.S. Agranovich,V. A.. Marchenko Pdf
This monograph by two Soviet experts in mathematical physics was a major contribution to inverse scattering theory. The two-part treatment examines the boundary-value problem with and without singularities. 1963 edition.
Scattering Theory for Transport Phenomena by Hassan Emamirad Pdf
The scattering theory for transport phenomena was initiated by P. Lax and R. Phillips in 1967. Since then, great progress has been made in the field and the work has been ongoing for more than half a century. This book shows part of that progress. The book is divided into 7 chapters, the first of which deals with preliminaries of the theory of semigroups and C*-algebra, different types of semigroups, Schatten–von Neuman classes of operators, and facts about ultraweak operator topology, with examples using wavelet theory. Chapter 2 goes into abstract scattering theory in a general Banach space. The wave and scattering operators and their basic properties are defined. Some abstract methods such as smooth perturbation and the limiting absorption principle are also presented. Chapter 3 is devoted to the transport or linearized Boltzmann equation, and in Chapter 4 the Lax and Phillips formalism is introduced in scattering theory for the transport equation. In their seminal book, Lax and Phillips introduced the incoming and outgoing subspaces, which verify their representation theorem for a dissipative hyperbolic system initially and also matches for the transport problem. By means of these subspaces, the Lax and Phillips semigroup is defined and it is proved that this semigroup is eventually compact, hence hyperbolic. Balanced equations give rise to two transport equations, one of which can satisfy an advection equation and one of which will be nonautonomous. For generating, the Howland semigroup and Howland’s formalism must be used, as shown in Chapter 5. Chapter 6 is the highlight of the book, in which it is explained how the scattering operator for the transport problem by using the albedo operator can lead to recovery of the functionality of computerized tomography in medical science. The final chapter introduces the Wigner function, which connects the Schrödinger equation to statistical physics and the Husimi distribution function. Here, the relationship between the Wigner function and the quantum dynamical semigroup (QDS) can be seen.
Scattering Theory: Some Old and New Problems by Dmitri R. Yafaev Pdf
Scattering theory is, roughly speaking, perturbation theory of self-adjoint operators on the (absolutely) continuous spectrum. It has its origin in mathematical problems of quantum mechanics and is intimately related to the theory of partial differential equations. Some recently solved problems, such as asymptotic completeness for the Schrödinger operator with long-range and multiparticle potentials, as well as open problems, are discussed. Potentials for which asymptotic completeness is violated are also constructed. This corresponds to a new class of asymptotic solutions of the time-dependent Schrödinger equation. Special attention is paid to the properties of the scattering matrix, which is the main observable of the theory. The book is addressed to readers interested in a deeper study of the subject.
Dispersion Decay and Scattering Theory by Alexander Komech,Elena Kopylova Pdf
A simplified, yet rigorous treatment of scattering theory methods and their applications Dispersion Decay and Scattering Theory provides thorough, easy-to-understand guidance on the application of scattering theory methods to modern problems in mathematics, quantum physics, and mathematical physics. Introducing spectral methods with applications to dispersion time-decay and scattering theory, this book presents, for the first time, the Agmon-Jensen-Kato spectral theory for the Schr?dinger equation, extending the theory to the Klein-Gordon equation. The dispersion decay plays a crucial role in the modern application to asymptotic stability of solitons of nonlinear Schr?dinger and Klein-Gordon equations. The authors clearly explain the fundamental concepts and formulas of the Schr?dinger operators, discuss the basic properties of the Schr?dinger equation, and offer in-depth coverage of Agmon-Jensen-Kato theory of the dispersion decay in the weighted Sobolev norms. The book also details the application of dispersion decay to scattering and spectral theories, the scattering cross section, and the weighted energy decay for 3D Klein-Gordon and wave equations. Complete streamlined proofs for key areas of the Agmon-Jensen-Kato approach, such as the high-energy decay of the resolvent and the limiting absorption principle are also included. Dispersion Decay and Scattering Theory is a suitable book for courses on scattering theory, partial differential equations, and functional analysis at the graduate level. The book also serves as an excellent resource for researchers, professionals, and academics in the fields of mathematics, mathematical physics, and quantum physics who would like to better understand scattering theory and partial differential equations and gain problem-solving skills in diverse areas, from high-energy physics to wave propagation and hydrodynamics.
