Spectral and scattering theory for second order partial differential operators / Kiyoshi Mochizuki.

Format:

Book

Author/Creator:

Mochizuki, Kiyoshi, 1939- author.

Series:

Monographs and research notes in mathematics

Language:

English

Subjects (All):

Spectral theory (Mathematics).

Differential equations.

Physical Description:

xvii, 231 pages ; 24 cm.

Place of Publication:

Boca Raton, FL : CRC Press, [2017]

Summary:

Spectral and Scattering Theory for Second-Order Partial Differential Operators is ideal for researchers and postgraduate students seeking to apply the spectral theory of second-order differential operators in exterior domains to their own field, and addresses the growing need for spectral transforms to deal with deep problems of analysis of partial differential equations. In additions to presenting the classical results of spectral and scattering theory, it also outlines recent work in the field including the scattering of Schrödinger operators on a star graph, uniform resolvent estimates, smoothing properties and Strichartz estimates, and other applications. Key Features, First book to bring together this material in an easily accessible and comprehensible form, Among the first texts devoted to the study of exterior problems for differential operators in terms of their spectral and scattering theory by non-abstract PDE related tools, Offers early-career researchers a useful guide to the essential techniques and structural properties of such spectral representations, Contains results with wide applications to different types of PDEs Book jacket.

Contents:

1 Second-Order Elliptic Operators in Exterior Domain 1

1.1 Self-adjoint realization of the operator -Δ _a,b 1

1.2 Short-range perturbations of-Δ _a,b 4

1.3 Cases of more general potentials 7

1.4 Operators with strongly singular potentials 9

1.5 Notes and remarks 13

2 Essential Spectrum of Self-Adjoint Operators 15

2.1 Stability of the essential spectrum 15

2.2 Essential spectrum of operators with exploding potentials 19

2.3 Notes and remarks 20

3 Statinary Equations and Functional Identities 21

3.1 Approximate phase for stationary equations 21

3.2 Assumptions and examples of electric potentials 23

3.3 Functional identities for stationary problems 31

3.4 Notes and remarks 34

4 Growth Properties of Generalized Eigenfunctions 35

4.1 Statements of the theorems 35

4.2 Proof of Theorem 4.1 when (K3.4)₁ is required 37

4.3 Proof of Theorem 4.1 when (K3.4)₂ is required 43

4.4 Notes and remarks 47

5 Principle of Limiting Absorption and Absolute Continuity 49

5.1 Radiation condition and unique existence of solutions 49

5.2 Absolute continuity of the continuous spectrum 58

5.3 A modification of the radiation conditions 59

5.4 Notes and remarks 62

6 Spectral Representations and Scattering for Short-Range Pertubations 65

6.1 Fourier inversion formula and the Laplace operator in Rⁿ 65

6.2 The case of short-range perturbations of the Laplace operator 72

6.3 Stationary approach to the scattering theory 78

6.4 An inverse scattering problem 84

6.5 Notes and remarks 87

7 Spectral Representations and Scattering 2, "Long-Range" Perturbations 89

7.1 Spectral representation of the operator L 89

7.2 Unitarity of F_± and expression of F^*_± 98

7.3 Time dependent representations for the stationary wave operators 99

7.4 Proof of Propositions 7.4 and 7.5 103

7.5 Notes and remarks 110

8 One Dimensional Schrödinger Operators 113

8.1 Schrodinger operators on a star graph 113

8.2 Expression of the resolvent kernel and spectral representations 121

8.3 Stationary approach to the Møller scattering theory 126

8.4 Marchenko equation and inverse scattering 129

8.5 Notes and remarks 134

9 Uniform Resolvent Estimates and Smoothing Properties 135

9.1 Magnetic Schrödinger operators in exterior domain 135

9.2 Laplace operator and its perturbations in R² 145

9.3 Smoothing properties for Schrödinger evolution equations 148

9.4 Smoothing properties for relativistic Schrödinger equations 149

9.5 Notes and remarks 153

10 Scattering for Time Dependent Perturbations 155

10.1 Abstract setting for time dependent small perturbations 155

10.2 Applications to Schrödinger, Klein-Gordon, and wave equations 161

10.3 Space-time weighted energy methods for wave equations 168

10.4 Decay-nondecay problems for time dependent complex potential 173

10.5 Inverse scattering for small nonself-adjoint perturbation of wave equations 182

10.6 Notes and remarks 186

11 Strichartz Estimates for Perturbed Equations 189

11.1 The framework of the problems 189

11.2 Perturbed Schrödinger equations 190

11.3 Perturbed Klein Gordon equations 193

11.4 Perturbed wave equations 197

12 Another Approach to Growth Properties of Generalized Eigenfunctions 201

12.1 Assumptions and statement of results 201

12.2 Proof of Theorem 12.1 206

12.3 Applications to the operator with homogeneous potentials 213

12.4 Notes and remarks 217.

Notes:

Includes bibliographical references and index.

ISBN:

9781498756020

1498756026

OCLC:

974673786