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001431603 020__ $$a9783030384319$$q(electronic bk.)
001431603 020__ $$a3030384314$$q(electronic bk.)
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001431603 020__ $$a9783030384333$$q(print)
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001431603 0243_ $$z9783030384302
001431603 0247_ $$a10.1007/978-3-030-38431-9$$2doi
001431603 0248_ $$a10.1007/978-3-030-38
001431603 035__ $$aSP(OCoLC)1155325454
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001431603 049__ $$aISEA
001431603 050_4 $$aQA329$$b.A38 2021
001431603 08204 $$a515/.353$$223
001431603 1001_ $$aAktosun, Tuncay,$$eauthor.
001431603 24510 $$aDirect and inverse scattering for the matrix Schrödinger equation /$$cTuncay Aktosun, Ricardo Weder.
001431603 264_1 $$aCham, Switzerland :$$bSpringer,$$c[2021]
001431603 264_4 $$c©2021
001431603 300__ $$a1 online resource (xiii, 624 pages) :$$billustrations
001431603 336__ $$atext$$btxt$$2rdacontent
001431603 337__ $$acomputer$$bc$$2rdamedia
001431603 338__ $$aonline resource$$bcr$$2rdacarrier
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001431603 347__ $$bPDF
001431603 4901_ $$aApplied mathematical sciences,$$x0066-5452 ;$$vvolume 203
001431603 504__ $$aIncludes bibliographical references (pages 607-618) and index.
001431603 5050_ $$aIntro -- Preface -- Acknowledgments -- Contents -- 1 Introduction -- 1.1 An Overview -- 1.2 Notes on the Bibliography -- 2 The Matrix Schrödinger Equation and the Characterization of the Scattering Data -- 2.1 Outline of the Chapter -- 2.2 The Matrix Schrödinger Equation on the Half Line -- 2.3 Star Graphs -- 2.4 The Schrödinger Equation on the Full Line -- 2.5 The Faddeev Class and the Marchenko Class -- 2.6 A First Characterization of the Scattering Data -- 2.7 Alternate Characterizations of the Scattering Data -- 2.8 Another Characterization of the Scattering Data -- 3 Direct Scattering I
001431603 5058_ $$a3.1 Outline of the Solution to the Direct Problem -- 3.2 Special Solutions to the Schrödinger Equation -- 3.3 The Hamiltonian -- 3.4 Equivalence of the Formulations of the Boundary Condition -- 3.5 The Quadratic Form of the Hamiltonian -- 3.6 Transformations of the Jost and Scattering Matrices -- 3.7 The Jost and Scattering Matrices with Zero Potential -- 3.8 Low-Energy Analysis with Potentials in the Faddeev Class -- 3.9 Low-Energy Analysis with Potentials of FiniteSecond Moment -- 3.10 High-Energy Analysis -- 3.11 Bound States -- 3.12 Levinson's Theorem
001431603 5058_ $$a3.13 Further Properties of the Scattering Data -- 3.14 The Marchenko Integral Equation -- 3.15 The Boundary Matrices -- 3.16 The Existence and Uniqueness in the Direct Problem -- 4 Direct Scattering II -- 4.1 Basic Principles of the Scattering Theory -- 4.2 The Limiting Absorption Principle -- 4.3 The Generalized Fourier Maps for the Absolutely Continuous Subspace -- 4.4 The Wave Operators -- 4.5 The Scattering Operator and the Scattering Matrix -- 4.6 The Spectral Shift Function -- 4.7 Trace Formulas -- 4.8 The Number of Bound States -- 5 Inverse Scattering
001431603 5058_ $$a5.1 Nonuniqueness Due to the Improperly Defined ScatteringMatrix -- 5.2 The Solution to the Inverse Problem -- 5.3 Bounds on the Constructed Solutions -- 5.4 Relations Among the Characterization Conditions -- 5.5 The Proof of the First Characterization Theorem -- 5.6 Equivalents for Some Characterization Conditions -- 5.7 Inverse Problem Using Only the Scattering Matrix as Input -- 5.8 Characterization via Levinson's Theorem -- 5.9 Parseval's Equality -- 5.10 The Generalized Fourier Map -- 5.11 An Alternate Method to Solve the Inverse Problem
001431603 5058_ $$a5.12 Characterization with Potentials of Stronger Decay -- 5.13 The Dirichlet Boundary Condition -- 6 Some Explicit Examples -- 6.1 Illustration of the Theory with Explicit Examples -- 6.2 Some Methods Yielding Explicit Examples -- 6.3 Explicit Examples in the Characterization of the Scattering Data -- 6.4 Explicit Examples of Particular Solutions -- Appendix A Mathematical Preliminaries -- A.1 Vectors, Matrices, and Functions -- A.2 Banach and Hilbert Spaces -- A.3 Inequalities -- A.4 Mollifiers -- A.5 Equicontinuity -- A.6 Distributions -- A.7 Absolute Continuity -- A.8 Sobolev Spaces
001431603 506__ $$aAccess limited to authorized users.
001431603 520__ $$aAuthored by two experts in the field who have been long-time collaborators, this monograph treats the scattering and inverse scattering problems for the matrix Schrödinger equation on the half line with the general selfadjoint boundary condition. The existence, uniqueness, construction, and characterization aspects are treated with mathematical rigor, and physical insight is provided to make the material accessible to mathematicians, physicists, engineers, and applied scientists with an interest in scattering and inverse scattering. The material presented is expected to be useful to beginners as well as experts in the field. The subject matter covered is expected to be interesting to a wide range of researchers including those working in quantum graphs and scattering on graphs. The theory presented is illustrated with various explicit examples to improve the understanding of scattering and inverse scattering problems. The monograph introduces a specific class of input data sets consisting of a potential and a boundary condition and a specific class of scattering data sets consisting of a scattering matrix and bound-state information. The important problem of the characterization is solved by establishing a one-to-one correspondence between the two aforementioned classes. The characterization result is formulated in various equivalent forms, providing insight and allowing a comparison of different techniques used to solve the inverse scattering problem. The past literature treated the type of boundary condition as a part of the scattering data used as input to recover the potential. This monograph provides a proper formulation of the inverse scattering problem where the type of boundary condition is no longer a part of the scattering data set, but rather both the potential and the type of boundary condition are recovered from the scattering data set.
001431603 588__ $$aDescription based on print version record.
001431603 650_0 $$aScattering (Mathematics)
001431603 650_0 $$aSchrödinger equation.
001431603 650_6 $$aDispersion (Mathématiques)
001431603 650_6 $$aÉquation de Schrödinger.
001431603 655_0 $$aElectronic books.
001431603 7001_ $$aWeder, Ricardo,$$eauthor.
001431603 77608 $$iPrint version:$$aAktosun, Tuncay.$$tDirect and inverse scattering for the matrix Schrödinger equation.$$dCham, Switzerland : Springer, [2021]$$z9783030384319$$w(OCoLC)1161312094
001431603 830_0 $$aApplied mathematical sciences (Springer-Verlag New York Inc.) ;$$vv. 203.
001431603 852__ $$bebk
001431603 85640 $$3Springer Nature$$uhttps://univsouthin.idm.oclc.org/login?url=https://link.springer.com/10.1007/978-3-030-38431-9$$zOnline Access$$91397441.1
001431603 909CO $$ooai:library.usi.edu:1431603$$pGLOBAL_SET
001431603 980__ $$aBIB
001431603 980__ $$aEBOOK
001431603 982__ $$aEbook
001431603 983__ $$aOnline
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