Dispersion Decay and Scattering Theory E-Book

Alexander Komech

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 203

Veröffentlichungsjahr: 2014

Beschreibung

A simplified, yet rigorous treatment of scattering theory methods and their applications Dispersion Decay and Scattering Theory provides thorough, easy-to-understand guidance on the application of scattering theory methods to modern problems in mathematics, quantum physics, and mathematical physics. Introducing spectral methods with applications to dispersion time-decay and scattering theory, this book presents, for the first time, the Agmon-Jensen-Kato spectral theory for the Schr?dinger equation, extending the theory to the Klein-Gordon equation. The dispersion decay plays a crucial role in the modern application to asymptotic stability of solitons of nonlinear Schr?dinger and Klein-Gordon equations. The authors clearly explain the fundamental concepts and formulas of the Schr?dinger operators, discuss the basic properties of the Schr?dinger equation, and offer in-depth coverage of Agmon-Jensen-Kato theory of the dispersion decay in the weighted Sobolev norms. The book also details the application of dispersion decay to scattering and spectral theories, the scattering cross section, and the weighted energy decay for 3D Klein-Gordon and wave equations. Complete streamlined proofs for key areas of the Agmon-Jensen-Kato approach, such as the high-energy decay of the resolvent and the limiting absorption principle are also included. Dispersion Decay and Scattering Theory is a suitable book for courses on scattering theory, partial differential equations, and functional analysis at the graduate level. The book also serves as an excellent resource for researchers, professionals, and academics in the fields of mathematics, mathematical physics, and quantum physics who would like to better understand scattering theory and partial differential equations and gain problem-solving skills in diverse areas, from high-energy physics to wave propagation and hydrodynamics.

Leseprobe

CONTENTS

List of Figures

Foreword

Preface

Acknowledgments

Introduction

1 Basic Concepts and Formulas

1 Distributions and Fourier transform

2 Functional spaces

3 Free propagator

2 Nonstationary Schrödinger Equation

4 Definition of solution

5 Schrödinger operator

6 Dynamics for free Schrödinger equation

7 Perturbed Schrödinger equation

8 Wave and scattering operators

3 Stationary Schrödinger Equation

9 Free resolvent

10 Perturbed resolvent

4 Spectral Theory

11 Spectral representation

12 Analyticity of resolvent

13 Gohberg-Bleher theorem

14 Meromorphic continuation of resolvent

15 Absence of positive eigenvalues

5 High Energy Decay of Resolvent

16 High energy decay of free resolvent

17 High energy decay of perturbed resolvent

6 Limiting Absorption Principle

18 Free resolvent

19 Perturbed resolvent

20 Decay of eigenfunctions

7 Dispersion Decay

21 Proof of dispersion decay

22 Low energy asymptotics

8 Scattering Theory and Spectral Resolution

23 Scattering theory

24 Spectral resolution

25 T-Operator and S-Matrix

9 Scattering Cross Section

26 Introduction

27 Mainresults

28 Limiting amplitude principle

29 Spherical waves

30 Plane wave limit

31 Convergence of flux

32 Long range asymptotics

33 Cross section

10 Klein-Gordon Equation

34 Introduction

35 Free Klein-Gordon equation

36 Perturbed Klein-Gordon equation

37 Asymptotic completeness

11 Wave equation

38 Introduction

39 Free wave equation

40 Perturbed wave equation

41 Asymptotic completeness

42 Appendix: Sobolev Embedding Theorem

References

Index

Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New JerseyPublished simultaneously in Canada

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, [phone], fax [phone], or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, [phone], fax [phone], or online at http://www.wiley.com/go/permission.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your Situation. You should consult with a Professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at [phone], outside the United States at [phone] or fax [phone].

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com.

Library of Congress Cataloging-in-Publication Data:

Komech, A. I., 1946–

 Dispersion decay and scattering theory / Alexander Komech, Elena Kopylova.

   p. cm.

 Includes bibliographical references and index.

 ISBN 978-1-118-34182-7 (cloth)

1. Klein-Gordon equation. 2. Spectral theory (Mathematics) 3. Scattering (Mathematics) I. Kopylova, Elena, 1960- II. Title.

 QC174.26.W28K646 2012

 530.12’4–dc23

2012007292

10 9 8 7 6 5 4 3 2 1

LIST OF FIGURES

I.1 Scattering and wave operators for ψ(0) ∈ Xc.

1.1 The function e–z2 decays in gray sectors.

2.1 Function (4.6).

3.2 Application of the Cauchy Residue Theorem.

3.4 The cutoff function.

4.1 The gray half-plane is the region of analyticity of .

4.2 The contours of integration.

4.3 Analytic continuation.

4.4 The contour .

6.1 The cutoff function.

6.2 The contour of integration.

7.1 The cutoff function.

9.1 Incident flux and scatterer.

9.2 Incident plane wave and outgoing spherical wave.

9.3 Incident and outgoing spherical waves.

10.1 The cutoff function.

10.2 The spectral gap and the continuous spectrum.

10.3 The cutoff functions.

FOREWORD

The book is a concise introduction to the dispersion decay and its applications to the scattering and spectral theory for the Schrödinger, Klein-Gordon, and wave equations. We expose the Agmon, Jensen, and Kato results on analytical properties of the resolvent in weighted Sobolev norms and applications to the spectral and scattering theory. The course is intended for readers who have a nodding acquaintance with the Fourier transform of distributions, the Sobolev embedding theorems, and the Fredholm Theorem.

PREFACE

We present the extended lecture notes of the course delivered by one of the authors in the Faculty of Mathematics of Vienna University in the spring 2009 for graduate students IV–V years.

Our aim is to give an introduction to spectral methods for the Schrxödinger and Klein-Gordon equations with applications to a dispersion time-decay and scattering theory. This method relies on analytical properties of the resolvent: high energy decay and low energy asymptotics of the resolvent, and the limiting absorption principle (a smoothness of the resolvent in the continuous spectrum).

This strategy in the dispersion time-decay was introduced by Vainberg for general hyperbolic equations with constant coefficients outside a compact region, and initial functions with compact support. The approach was extended by Agmon, Jensen, Kato, Murata and others to the Schrödinger equation with generic potentials of algebraic decay, and initial functions from the weighted Sobolev spaces. These results play a crucial role in the study of asymptotic stability of solutions to nonlinear Schrödinger equations, see [7, 8, 11, 64, 65, 80, 81].

We present the Agmon, Jensen, and Kato results for the first time in the textbook literature. Then we apply them to a new dynamical justification of the scattering cross section via the limiting amplitude principle and convergence of the “spherical limit amplitudes” to the “plane limit amplitudes”. We also present an extension of the methods and results to the Klein-Gordon and wave equations obtained in [45, 48, 51]. Recently the results were successfully applied for proving asymptotic stability for the kinks of relativistic invariant nonlinear Ginzburg-Landau equations [43, 44].

The course is intended for readers who have a nodding acquaintance with the Fourier transform of distributions, the Sobolev embedding theorems, and the Fredholm Theorem.

We do not touch alternative approaches to the dispersion decay and scattering (Birman-Kato theory [70], Strichartz estimates [38], Mourre estimates [26], Hunziker-Sigal method of minimal escape velocity [28, 29], and other) not to over-burden the exposition.

In Sections 1 and 2 we collect basic concepts and facts which we need: the Fourier transform of distributions, the Sobolev embedding theorems, the Fredholm Theorem, and basic technique of pseudodifferential operators (everything is covered, e.g., by [77] or [40]). In Sections 3–15 we establish basic properties of the Schrödinger equation. In the central sections 16–22 we present the Agmon-Jensen-Kato spectral theory of the dispersion decay in the weighted Sobolev norms. In the remaining sections 23–41 we apply the dispersion decay to scattering and spectral theories, to a justification of scattering cross section, and to a weighted energy decay for 3D Klein-Gordon and wave equations with a potential.

One of the cornerstones of the Agmon-Jensen-Kato approach is the high energy decay of the resolvent in the weighted Sobolev norms, which was stated by Agmon in [1, (A.2’)]. We give a complete proof explaining all related details: the Sobolev Trace Theorem, the Hölder continuity of the traces, the Sokhotsky-Plemelj formula, etc. The next cornerstones are Kato’s theorem on the absence of embedded eigen-values and Agmon’s theorem on the decay of the eigenfunctions. We give complete streamlined proofs.

A. I. KOMECH AND E. A. KOPYLOVA

Moscow-ViennaJanuary, 2012

ACKNOWLEDGMENTS

The authors thank B. R. Vainberg for useful remarks. The authors are indebted to the Faculty of Mathematics of Vienna University and Institute for Information Transmission Problems of RAS for excellent conditions for the work. The book was written under the support of the Alexander von Humboldt Award of A. I. Komech, and by the Austrian Science Fund (FWF): P22198-N13 and M1329-N13.

K. A. I. and K. E. A.

Keywords: dispersion decay, Schrödinger and Klein-Gordon equations, Schrödinger operator, potential, resolvent, Fredholm Theorem, distribution, Fourier transform, Fourier-Laplace transform, pseudodifferential operator, Sobolev norm, Sobolev space, Sobolev Embedding Theorem, weighted spaces, continuous spectrum, convolution, Born series, limiting absorption principle, asymptotic completeness, wave operators, scattering operator, S-matrix, T-operator, limiting amplitude principle, plane limit amplitude, spherical limit amplitude, Wiener condition, scattering cross section, Lippmann-Schwinger equation.

2000 Mathematics Subject Classification: 35L10, 34L25, 47A40, 81U05.

INTRODUCTION

Dispersion decay and scattering The main subject of our book is the study of wave radiation and scattering for solutions to the Schrödinger and Klein-Gordon equations with a decaying potential

(0.1)

(0.2)

which are the basic wave equations of quantum mechanics, introduced in 1925–1926. The key peculiarity of the wave processes is the energy propagation and energy radiation to infinity known since Huygens' “Treatise on light” (1678).

(0.3)

if

(0.4)

In particular, (0.3) implies

(0.5)

for any R > 0. This wave divergence was widely recognized in theoretical physics in the nineteenth and twentieth centuries. In particular, it was one of the key inspirations for Bohr's theory of radiation induced by the quantum transitions.

However, a mathematical justification of this phenomenon was discovered only after 1960 by Lax, Morawetz, Phillips, and Vainberg for wave and Klein-Gordon equations and extended by Ginibre and Velo, Rauch, and others for the Schrödinger equation in the theory of local energy decay: