Scattering by Obstacles and Potentials (Mathematical Physics)

604 Pages·2017·3.399 MB·English

by  ALEXANDER G RAMM

Scattering by
Obstacles and
Potentials
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b2530 International Strategic Relations and China’s National Security: World at the Crossroads
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Scattering by
Obstacles and
Potentials
Alexander G Ramm
Kansas State University, USA
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
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Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Names: Ramm, A. G. (Alexander G.) author.
Title: Scattering by obstacles and potentials / by Alexander G. Ramm (Kansas State University, USA).
Description: New Jersey : World Scientific, [2017] | Includes bibliographical references and index.
Identifiers: LCCN 2017024304 | ISBN 9789813220966 (hc : alk. paper)
Subjects: LCSH: Inverse problems (Differential equations) | Scattering (Mathematics)
Classification: LCC QA371 .R336 2017 | DDC 515/.352--dc23
LC record available at https://lccn.loc.gov/2017024304
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2018 by World Scientific Publishing Co. Pte. Ltd.
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Desk Editor: V. Vishnu Mohan
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Contents
Preface xiii
I Scattering by Obstacles 1
Introduction 3
1 Scattering by an Obstacle 17
1.1 Statement of the Problem: Basic Integral
Equations . . . . . . . . . . . . . . . . . . . . . . 17
1.2 Existence and Uniqueness of the Solution
to the Scattering Problems . . . . . . . . . . . . . 30
1.3 Eigenfunction Expansion Theorem . . . . . . . . 51
1.4 Properties of the Scattering Amplitude . . . . . . 59
1.5 The S-matrix and Wave Operators . . . . . . . . 71
1.6 Inequalities for Solutions to Helmholtz’s Equation
for Large Frequencies . . . . . . . . . . . . . . . . 74
1.7 Representation of Solutions to Helmholtz’s
Equation . . . . . . . . . . . . . . . . . . . . . . . 80
2 The Inverse Scattering Problem 95
2.1 Statement of the Problem and Uniqueness
Theorems . . . . . . . . . . . . . . . . . . . . . . 95
vii
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viii Scattering by Obstacles and Potentials
2.2 Construction of Obstacles from the Scattering Data
at High Frequencies . . . . . . . . . . . . . . . . . 117
2.3 Stability of the Surface with Respect to Small
Perturbations of the Data . . . . . . . . . . . . . 129
3 Time-Dependent Problems 133
3.1 Statement of the Problems . . . . . . . . . . . . . 133
3.2 The Limiting Amplitude Principle
(Abstract Results). . . . . . . . . . . . . . . . . . 138
3.3 The Limiting Amplitude Principle
for the Laplacian in Exterior Domains . . . . . . 156
3.4 Decay of Energy . . . . . . . . . . . . . . . . . . . 162
3.5 Singularity and Eigenmode Expansion Methods . 165
4 The T-matrix Scheme 171
4.1 Statement of the Problem . . . . . . . . . . . . . 171
4.2 Justification of the T-matrix Scheme . . . . . . . 175
4.3 Numerical Results . . . . . . . . . . . . . . . . . . 204
4.4 Other Schemes . . . . . . . . . . . . . . . . . . . . 209
5 Scattering by Small Bodies 213
5.1 Scattering by a Single Small Body . . . . . . . . . 213
5.2 Scattering by Many Small Bodies . . . . . . . . . 220
5.3 Electromagnetic Wave Scattering by Small
Particles . . . . . . . . . . . . . . . . . . . . . . . 225
5.4 Behavior of the Solutions to the Exterior
Boundary Value Problems at Low
Frequencies . . . . . . . . . . . . . . . . . . . . . 230
6 Some Inverse Scattering Problems
of Geophysics 251
6.1 Inverse Scattering for Geophysical Problems . . . 251
6.2 Two-Parameter Inversion . . . . . . . . . . . . . . 282
6.3 An Inversion Formula in Scattering Theory . . . . 294
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Contents ix
6.4 A Model Inverse Problem of Induction
Logging . . . . . . . . . . . . . . . . . . . . . . . . 309
7 Scattering by Obstacles with Infinite Boundaries 321
7.1 Statement of the Problem . . . . . . . . . . . . . 321
7.2 Spectral Properties of the Laplacians . . . . . . . 322
7.3 Spectral Properties of the Dirichlet Laplacian
in Semi-infinite Tubes . . . . . . . . . . . . . . . . 330
7.4 Absence of Positive Eigenvalues for the Dirichlet
Laplacian under Local Assumptions
at Infinity . . . . . . . . . . . . . . . . . . . . . . 333
7.5 LAP and Compact Perturbations
of the Boundary . . . . . . . . . . . . . . . . . . . 347
7.6 Eigenfunction Expansion in Canonical
Domains . . . . . . . . . . . . . . . . . . . . . . . 356
II Scattering by Potentials 359
8 Scattering by Potentials 361
8.1 Introduction: Existence and Uniqueness
of the Scattering Solution — Uniqueness
of the Solution to Inverse Scattering
Problem . . . . . . . . . . . . . . . . . . . . . . . 361
8.2 Global Perturbation Formula: A New Definition
of the Scattering Solution — High-Frequency
Approximation. . . . . . . . . . . . . . . . . . . . 364
8.3 Completeness Properties of the Scattering
Solutions . . . . . . . . . . . . . . . . . . . . . . . 372
8.4 Property C as the Tool for Proving Uniqueness
Theorems in Inverse Scattering . . . . . . . . . . 374
8.5 Existence of Special Solutions . . . . . . . . . . . 377
8.6 Reconstruction Formula for the Potential . . . . . 383
8.7 Stability Estimate for the Inversion
Formulas . . . . . . . . . . . . . . . . . . . . . . . 391