"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 Content: Cover Title Page Copyright Page Dedication Table of Contents Preface Introduction 1: Second-Order Elliptic Operators in Exterior Domain 1.1 Self-adjoint realization of the operator -- ∆a,b 1.2 Short-range perturbations of -- ∆a,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 equations3.2 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 A modification of the radiation conditions5.4 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 The case of short-range perturbations of the Laplace operator 6.3 Stationary approach to the scattering theory 6.4 An 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 operators7.4 Proof of Propositions 7.4 and 7.5 7.5 Notes and remarks 8: One Dimensional Schrödinger Operators 8.1 Schrödinger 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 Schrödinger operators in exterior domain 9.2 Laplace operator and its perturbations in R2 9.3 Smoothing properties for Schrödinger evolution equations9.4 Smoothing properties for relativistic Schrödinger 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 Schrödinger, 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
