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Scattering Theory for dissipative and time-dependent systems has been intensively studied in the last fifteen years. The results in this field, based on various tools and techniques, may be found in many published papers. This monograph presents an approach which can be applied to spaces of both even and odd dimension. The ideas on which the approach is based are connected with the RAGE type theorem, with Enss' decomposition of the phase space and with a time-dependent proof of the existence of the operator W which exploits the decay of the local energy of the perturbed and free systems. Som

Boundary value problems. --- Scattering operator. --- Wave equation.

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Scattering (Mathematics) --- Dispersion (Mathématiques) --- Dispersion (Mathématiques) --- Inverse scattering transform --- Scattering theory (Mathematics) --- Boundary value problems --- Differential equations, Partial --- Scattering operator --- Scattering transform, Inverse --- Transform, Inverse scattering --- Transformations (Mathematics)

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Authored by two experts in the field who have been long-time collaborators, this monograph treats the scattering and inverse scattering problems for the matrix Schrödinger equation on the half line with the general selfadjoint boundary condition. The existence, uniqueness, construction, and characterization aspects are treated with mathematical rigor, and physical insight is provided to make the material accessible to mathematicians, physicists, engineers, and applied scientists with an interest in scattering and inverse scattering. The material presented is expected to be useful to beginners as well as experts in the field. The subject matter covered is expected to be interesting to a wide range of researchers including those working in quantum graphs and scattering on graphs. The theory presented is illustrated with various explicit examples to improve the understanding of scattering and inverse scattering problems. The monograph introduces a specific class of input data sets consisting of a potential and a boundary condition and a specific class of scattering data sets consisting of a scattering matrix and bound-state information. The important problem of the characterization is solved by establishing a one-to-one correspondence between the two aforementioned classes. The characterization result is formulated in various equivalent forms, providing insight and allowing a comparison of different techniques used to solve the inverse scattering problem. The past literature treated the type of boundary condition as a part of the scattering data used as input to recover the potential. This monograph provides a proper formulation of the inverse scattering problem where the type of boundary condition is no longer a part of the scattering data set, but rather both the potential and the type of boundary condition are recovered from the scattering data set.

Scattering (Mathematics) --- Scattering theory (Mathematics) --- Boundary value problems --- Differential equations, Partial --- Scattering operator --- Partial differential equations. --- Functional analysis. --- Quantum physics. --- Mathematical physics. --- Partial Differential Equations. --- Functional Analysis. --- Quantum Physics. --- Mathematical Physics. --- Physical mathematics --- Physics --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Mechanics --- Thermodynamics --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Partial differential equations --- Mathematics

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This book present the lecture notes used in two courses that the late Professor Kasra Barkeshli had offered at Sharif University of Technology, namely, Advanced Electromagnetics and Scattering Theory. The prerequisite for the sequence is vector calculus and electromagnetic fields and waves. Some familiarity with Green's functions and integral equations is desirable but not necessary. The book  provides a brief but concise introduction to classical topics in the field. It is divided into three parts including annexes. Part I covers principle of electromagnetic theory. The discussion starts with a review of the Maxwell's equations in differential and integral forms and basic boundary conditions. The solution of inhomogeneous wave equation and various field representations including Lorentz's potential functions and the Green's function method are discussed next. The solution of Helmholtz equation and wave harmonics follow. Next, the book presents plane wave propagation in dielectric and lossy media and various wave velocities. This part concludes with a general discussion of planar and circular waveguides. Part II presents basic concepts of electromagnetic scattering theory. After a brief discussion of radar equation and scattering cross section, the author reviews the canonical problems in scattering. These include the cylinder, the wedge and the sphere. The edge condition for the electromagnetic fields in the vicinity of geometric discontinuities are discussed. The author also presents the low frequency Rayleigh and Born approximations. The integral equation method for the formulation of scattering problems is presented next, followed by an introduction to scattering from periodic structures.        .

Engineering. --- Microwaves, RF and Optical Engineering. --- Optics and Electrodynamics. --- Microwaves. --- Ingénierie --- Micro-ondes --- Electromagnetic fields -- Mathematical models. --- Electromagnetic theory -- Data processing. --- Electromagnetism -- Data processing. --- Electromagnetism. --- Scattering (Mathematics) --- Scattering theory (Mathematics) --- Electromagnetics --- Optics. --- Electrodynamics. --- Optical engineering. --- Boundary value problems --- Differential equations, Partial --- Scattering operator --- Magnetic induction --- Magnetism --- Metamaterials --- Classical Electrodynamics. --- Hertzian waves --- Electric waves --- Electromagnetic waves --- Geomagnetic micropulsations --- Radio waves --- Shortwave radio --- Dynamics --- Physics --- Light --- Mechanical engineering

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Inverse scattering theory is an important area of applied mathematics due to its central role in such areas as medical imaging , nondestructive testing and geophysical exploration. Until recently all existing algorithms for solving inverse scattering problems were based on using either a weak scattering assumption or on the use of nonlinear optimization techniques. The limitations of these methods have led in recent years to an alternative approach to the inverse scattering problem which avoids the incorrect model assumptions inherent in the use of weak scattering approximations as well as the strong a priori information needed in order to implement nonlinear optimization techniques. These new methods come under the general title of qualitative methods in inverse scattering theory and seek to determine an approximation to the shape of the scattering object as well as estimates on its material properties without making any weak scattering assumption and using essentially no a priori information on the nature of the scattering object. This book is designed to be an introduction to this new approach in inverse scattering theory focusing on the use of sampling methods and transmission eigenvalues. In order to aid the reader coming from a discipline outside of mathematics we have included background material on functional analysis, Sobolev spaces, the theory of ill posed problems and certain topics in in the theory of entire functions of a complex variable. This book is an updated and expanded version of an earlier book by the authors published by Springer titled Qualitative Methods in Inverse Scattering Theory Review of Qualitative Methods in Inverse Scattering Theory All in all, the authors do exceptionally well in combining such a wide variety of mathematical material and in presenting it in a well-organized and easy-to-follow fashion. This text certainly complements the growing body of work in inverse scattering and should well suit both new researchers to the field as well as those who could benefit from such a nice codified collection of profitable results combined in one bound volume. SIAM Review, 2006.

Scattering (Mathematics) --- Inverse scattering transform. --- Scattering transform, Inverse --- Transform, Inverse scattering --- Scattering theory (Mathematics) --- Mathematics. --- Fourier analysis. --- Partial differential equations. --- Physics. --- Electrical engineering. --- Partial Differential Equations. --- Electrical Engineering. --- Theoretical, Mathematical and Computational Physics. --- Fourier Analysis. --- Electric engineering --- Engineering --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Partial differential equations --- Analysis, Fourier --- Mathematical analysis --- Math --- Science --- Transformations (Mathematics) --- Boundary value problems --- Differential equations, Partial --- Scattering operator --- Differential equations, partial. --- Computer engineering. --- Computers --- Design and construction --- Mathematical physics. --- Physical mathematics --- Physics --- Mathematics