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E-book
Author Mochizuki, Kiyoshi, 1939- author.

Title Spectral and scattering theory for second order partial differential operators / Kiyoshi Mochizuki
Published Boca Raton, FL : CRC Press, [2017]
©2017

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Description 1 online resource (xvii, 231 pages)
Series Monographs and research notes in mathematics
Monographs and research notes in mathematics.
Contents 880-01 Introduction -- Second-order elliptic operators in exterior domain -- Essential spectrum of self-adjoint operators -- Statinary equations and functional identities -- Growth properties of generalized eigenfunctions -- Principle of limiting absorption and absolute continuity -- Spectral representations and scattering for short-range pertubations -- Spectral representations and scattering 2, "long-range" perturbations -- One dimensional schrödinger operators -- Uniform resolvent estimates and smoothing properties -- Scattering for time dependent perturbations -- Strichartz estimates for perturbed equations -- Another approach to growth properties of generalized Eigenfunctions
880-01/(S Machine generated contents note: 1. Second-Order Elliptic Operators in Exterior Domain -- 1.1. Self-adjoint realization of the operator ---Δa, b -- 1.2. Short-range perturbations of ---Δa, b -- 1.3. Cases of more general potentials -- 1.4. Operators with strongly singular potentials -- 1.5. Notes and remarks -- 2. Essential Spectrum of Self-Adjoint Operators -- 2.1. Stability of the essential spectrum -- 2.2. Essential spectrum of operators with exploding potentials -- 2.3. Notes and remarks -- 3. Statinary Equations and Functional Identities -- 3.1. Approximate phase for stationary equations -- 3.2. Assumptions and examples of electric potentials -- 3.3. Functional identities for stationary problems -- 3.4. Notes and remarks -- 4. Growth Properties of Generalized Eigenfunctions -- 4.1. Statements of the theorems -- 4.2. Proof of Theorem 4.1 when (K3.4)1 is required -- 4.3. Proof of Theorem 4.1 when (K3.4)2 is required -- 4.4. Notes and remarks -- 5. Principle of Limiting Absorption and Absolute Continuity -- 5.1. Radiation condition and unique existence of solutions -- 5.2. Absolute continuity of the continuous spectrum -- 5.3. modification of the radiation conditions -- 5.4. Notes and remarks -- 6. Spectral Representations and Scattering for Short-Range Pertubations -- 6.1. Fourier inversion formula and the Laplace operator in Rn -- 6.2. case of short-range perturbations of the Laplace operator -- 6.3. Stationary approach to the scattering theory -- 6.4. inverse scattering problem -- 6.5. Notes and remarks -- 7. Spectral Representations and Scattering 2, "Long-Range" Perturbations -- 7.1. Spectral representation of the operator L -- 7.2. Unitarity of F± and expression of F*± -- 7.3. Time dependent representations for the stationary wave operators -- 7.4. Proof of Propositions 7.4 and 7.5 -- 7.5. Notes and remarks -- 8. One Dimensional Schrodinger Operators -- 8.1. Schrodinger operators on a star graph -- 8.2. Expression of the resolvent kernel and spectral representations -- 8.3. Stationary approach to the Møller scattering theory -- 8.4. Marchenko equation and inverse scattering -- 8.5. Notes and remarks -- 9. Uniform Resolvent Estimates and Smoothing Properties -- 9.1. Magnetic Schrodinger operators in exterior domain -- 9.2. Laplace operator and its perturbations in R2 -- 9.3. Smoothing properties for Schrodinger evolution equations -- 9.4. Smoothing properties for relativistic Schrodinger equations -- 9.5. Notes and remarks -- 10. Scattering for Time Dependent Perturbations -- 10.1. Abstract setting for time dependent small perturbations -- 10.2. Applications to Schrodinger, Klein--Gordon, and wave equations -- 10.3. Space-time weighted energy methods for wave equations -- 10.4. Decay-nondecay problems for time dependent complex potential -- 10.5. Inverse scattering for small nonself-adjoint perturbation of wave equations -- 10.6. Notes and remarks -- 11. Strichartz Estimates for Perturbed Equations -- 11.1. framework of the problems -- 11.2. Perturbed Schrodinger equations -- 11.3. Perturbed Klein--Gordon equations -- 11.4. Perturbed wave equations -- 12. Another Approach to Growth Properties of Generalized Eigenfunctions -- 12.1. Assumptions and statement of results -- 12.2. Proof of Theorem 12.1 -- 12.3. Applications to the operator with homogeneous potentials -- 12.4. Notes and remarks
Summary "The book is intended for students of graduate and postgraduate level, researchers in mathematical sciences as well as those who want to apply the spectral theory of second order differential operators in exterior domains to their own field. In the first half of this book, the classical results of spectral and scattering theory: the selfadjointness, essential spectrum, absolute continuity of the continuous spectrum, spectral representations, short-range and long-range scattering are summarized. In the second half, recent results: scattering of Schrodinger operators on a star graph, uniform resolvent estimates, smoothing properties and Strichartz estimates, and some applications are discussed."--Provided by publisher
Bibliography Includes bibliographical references and index
Notes Online resource; title from electronic title page (EBSCOhost, viewed March 16, 2018)
Subject Spectral theory (Mathematics)
Differential equations.
MATHEMATICS -- Calculus.
MATHEMATICS -- Mathematical Analysis.
Differential equations
Spectral theory (Mathematics)
Form Electronic book
ISBN 9781351648943
1351648942
9781498756037
1498756034
1498756026
9781498756020
1315152908
9781315152905