P.D. Hislop I.M. Sigal clied
Mathema al Sciences 113
Introduction to Spectral Theory With Applications to Schrodinger Operators
Springer
Applied Mathematical Sciences Volume 113 Editors J.E. Marsden L. Sirovich F. John (deceased) Advisors M. Ghil J.K. Hale T. Kambe J. Keller K. Kirchgassner B.J. Matkowsky C.S. Peskin J.T. Stuart
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6. Freiberger/Grenander: A Short Course in Computational Probability and Statistics. 7. Pipkin: Lectures on Viscoelasticity Theory.
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26. Kushner/Clark: Stochastic Approximation Methods for Constrained and Unconstrained Systems.
27. de Boor: A Practical Guide to Splines. 28. Keilson: Markov Chain ModelsRarity and Exponentiality. 29. de Veubeke: A Course in Elasticity. 30. Shiatycki: Geometric Quantization and Quantum Mechanics.
31. Reid: Sturmian Theory for Ordinary Differential Equations.
32. Meis/Markowitz: Numerical Solution of Partial Differential Equations. 33. Grenander Regular Structures: Lectures in Pattern Theory, Vol. Ill.
34. Kevorkian/Cole: Perturbation Methods in Applied Mathematics. 35. Carr: Applications of Centre Manifold Theory. 36. Bengtsson/Ghil/Kallen: Dynamic Meteorology: Data Assimilation Methods. 37. Saperstone: Semidynamical Systems in Infinite Dimensional Spaces. 38. Lichtenberg/Lieberman: Regular and Chaotic Dynamics, 2nd ed. 39. Piccini/Stampacchia/Vidossich: Ordinary Differential Equations in R. 40. Naylor/Sell: Linear Operator Theory in Engineering and Science. 41. Sparrow: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. 42, Guckenheimer/Holmes: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. 43. Ockendon/Taylor: Inviscid Fluid Flows. 44. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. 45. Glashof/Gustafson: Linear Operations and Approximation: An Introduction to the Theoretical Analysis and Numerical Treatment of SemiInfinite Programs. 46. Wilcox: Scattering Theory for Diffraction Gratings. 47. Hale et al: An Introduction to Infinite Dimensional Dynamical SystemsGeometric Theory. 48. Murray: Asymptotic Analysis. 49. Ladyzhenskaya: The BoundaryValue Problems of Mathematical Physics. 50. Wilcox: Sound Propagation in Stratified Fluids. 51. Golubitsky/Schaeffer: Bifurcation and Groups in Bifurcation Theory, Vol. 1. 52. Chipot: Variational Inequalities and Flow in Porous Media. 53. Majda: Compressible Fluid Flow and System of Conservation Laws in Several Space Variables. 54. Wasow: Linear Turning Point Theory. 55. Yosida: Operational Calculus: A Theory of Hyperfunctions.
56. Chang/Howes: Nonlinear Singular Perturbation Phenomena: Theory and Applications. 57. Reinhardt: Analysis of Approximation Methods for Differential and Integral Equations. 58. Dwoyer/Hussaini/Voigt (eds): Theoretical Approaches to Turbulence. 59. Sanders/Verhulst: Averaging Methods in Nonlinear Dynamical Systems. 60. Ghil/Childress: Topics in Geophysical Dynamics: Atmospheric Dynamics, Dynamo Theory and Climate Dynamics.
(continued following index)
P.D. Hislop
I.M. Sigal
Introduction to Spectral Theory With Applications to Schrodinger Operators
Springer
P.D. Hislop Department of Mathematics University of Kentucky Lexington, KY 405060027 USA
I.M. Sigal Department of Mathematics University of Toronto Toronto, Ontario M5S 1A1 Canada
Editors J.E. Marsden
Control and Dynamical Systems 10444 California Institute of Technology Pasadena, CA 91125
USA
L. Sirovich Division of Applied Mathematics Brown University Providence, RI 02912 USA
Mathematics Subject Classification (1991): 81Q05, 35J10, 35Q55
Library of Congress CataloginginPublication Data
Hislop, P.D., 1955Introduction to spectral theory : with applications to Schrodinger operators / P.D. Hislop, I.M. Sigal. p. cm. (Applied mathematical sciences ; v. 113) Includes bibliographical references (p.  ) and index. ISBN 0387945016 (hardcover : alk. paper) 1. Schrodinger operators. 2. Spectral theory (Mathematics) 1. Sigal, Israel Michael, 1945. II. Title. III. Series: Applied mathematical sciences (SpringerVeriag New York Inc.) ; v. 113.
QAI.A647
vol. 113 [QC174.17.S3]
510 sdc20 [515'72231
9512926
Printed on acidfree paper.
© 1996 SpringerVerlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (SpringerVeriag New York, Inc., 175 Fifth Avenue, New York, NY 10010 USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks act, may accordingly be used freely by anyone.
Production managed by Frank Ganz; manufacturing supervised by Jacqui Ashri. Photocomposed pages prepared from the author's LATEX file using SpringerVerlag's "svsing.sty" macro. Printed and bound by R.R. Donnelley & Sons, Harrisonburg, VA. Printed in the United States of America.
987654321 ISBN 0387945016 SpringerVerlag New York Berlin Heidelberg
Contents
Introduction and Overview I
The Spectrum of Linear Operators and Hilbert Spaces 1.1
1.2 1.3 2
The Spectrum . . . . Properties of the Resolvent Hilbert Space . . . . . .
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2.2 2.3
Subspaces . . . . . . . . Linear Functionals and the Riesz Theorem Orthonormal Bases . . . . . . . .
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Exponential Decay of Eigenfunctions 3.1
3.2 3.3 3.4 3.5 3.6
Introduction . . . . . . . . . Agmon Metric . . . . . . . The Main Theorem . . . Proof of Theorem 3.4 . Pointwise Exponential Bounds Notes . . . . . . . .
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The Geometry of a Hilbert Space and Its Subspaces 2.1
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27 30
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Contents
Operators on Hilbert Spaces 4.1
4.2 4.3 5
5.2 5.3 5.4
6.2 6.3 6.4
7.2 7.3
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49 49
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Definitions . . . . . . . General Properties of SelfAdjoint Operators . Determining the Spectrum of SelfAdjoint Operators Projections . . . . . . . . . . . . .
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Riesz Projections . . . . . . . . . . Isolated Points of the Spectrum . . . . . . More Properties of Riesz Projections . . Embedded Eigenvalues of SelfAdjoint Operators .
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. The Weyl Criterion . . . . Proof of Wevl's Criterion: First Part . Proof of Weyl's Criterion: Second Part .
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8.1
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65 67
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77 79
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82 84 85
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Compact Operators 9.1
9.2 9.3 9.4
Compact and FiniteRank Operators . The Structure of the Set of Compact Operators Spectral Theory of Compact Operators . . . . Applications of the General Theory . . . . .
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10 Locally Compact Operators and Their Application to Schrodinger Operators . . . 10.1 Locally Compact Operators . . . . . . 10.2 Spectral Properties of Locally Compact Operators 10.3 Essential Spectrum and Weyl's Criterion for Certain Closed Operators . . . . . . . . . .
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11 Semiclassical Analysis of Schrodinger Operators I: The Harmonic Approximation 1 1.1
Introduction
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59 59 64
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. Symmetric Operators . . . . Fundamental Criteria for SelfAdjointness The Kato Inequality for Smooth Functions Technical Approximation Tools . . The Kato Inequality . . . . . . . Application to Positive Potentials . . .
53 56
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SelfAdjointness: Part 1. The Kato Inequality 8.2 8.3 8.4 8.5 8.6
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The Essential Spectrum: Weyl's Criterion 7.1
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Riesz Projections and Isolated Points of the Spectrum 6.1
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SelfAdjoint Operators 5.1
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Remarks on the Operator Norm and Graphs The Adjoint of an Operator . . . . Unitary Operators . .
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11.2 Preliminary: The Harmonic Oscillator 11.3 Semiclassical Limit of Eigenvalues . . . . 11.4 Notes . . . . . . . . .
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12 Semiclassical Analysis of Schrodinger Operators II: The Splitting of Eigenvalues
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12.1 More Spectral Analysis: Variational Inequalities 12.2 DoubleWell Potentials and Tunneling . . . . 12.3 Proof of Theorem 12.3 . . . . . . . . . . 12.4 Appendix: Exponential Decay of Eigenfunctions . . . for DoubleWell Hamiltonians . . . . . 12.5 Notes . . . . . . . . . . . . .
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13 SelfAdjointness: Part 2. The KatoRellich Theorem .
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14 Relatively Compact Operators and the Weyl Theorem
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14.1 Relatively Compact Operators . . . . . . . . . . . . 14.2 Weyl's Theorem: Stability of the Essential Spectrum . . . 14.3 Applications to the Spectral Theory of Schrodinger Operators . 14.4 Persson's Theorem: The Bottom of the Essential Spectrum . . .
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15 Perturbation Theory: Relatively Bounded Perturbations .
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164 168
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16 Theory of Quantum Resonances I: The AguilarBalslevCombesSimon Theorem .
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17 Spectral Deformation Theory 17.1 Introduction to Spectral Deformation . . 17.2 Vector Fields and Diffeomorphisms . . . 17.3 Induced Unitary Operators . . . . . . . . 17.4 Complex Extensions and Analytic Vectors 17.5 Notes . . . . . . . . . . . . . . . . . . .
153 156 157
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16.1 Introduction to Quantum Resonance Theory . . . . . . 16.2 AguilarBalslevCombesSimon Theory of Resonances 16.3 Proof of the AguilarBalslevCombes Theorem . . . . 16.4 Examples of the Generalized Semiclassical Regime . . . 16.5 Notes . . . . . . . . . . . . . . . . . . . . . . . .
143 145
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. . . . . . . . . . 15.1 Introduction and Motivation 15.2 Analytic Perturbation Theory for the Discrete Spectrum . . 15.3 Criteria for Eigenvalue Stability: A Simple Case . 15.4 TypeA Families of Operators and Eigenvalue Stability: General Results . . . . . . . . . . . . . . . . . 15.5 Remarks on Perturbation Expansions . . . . . . . 15.6 Appendix: A Technical Lemma . . . . . . . . . . .
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13.1 Relatively Bounded Operators . . . . . . . . . . . 13.2 Schrodinger Operators with Relatively Bounded Potentials
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177 178 180
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viii
Contents
18 Spectral Deformation of Schrodinger Operators
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The Deformed Family of Schrodinger Operators . 18.2 The Spectrum of the Deformed Laplacian . . . 18.3 Admissible Potentials . . . . 18.4 The Spectrum of Deformed Schrodinger Operators 18.5 Notes . . . . . . . . . 18. I
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19 The General Theory of Spectral Stability
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19.1 Examples of Nonanalytic Perturbations . . 19.2 Strong Resolvent Convergence . . . . . . . . . 19.3 The General Notion of Stability . . 19.4 A Criterion for Stability . . . . . . 19.5 Proof of the Stability Criteria . . 19.6 Geometric Techniques and Applications to Stability 19.7 Example: A Simple Shape Resonance Model . . .
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20 Theory of Quantum Resonances II: The Shape Resonance Model 20.1 Introduction: The Gamow Model of Alpha Decay . . . 20.2 The Shape Resonance Model . . . . . . . . . 20.3 The Semiclassical Regime and Scaling . . . . . 20.4 Analyticity Conditions on the Potential . . . . . . . 20.5 Spectral Stability for Shape Resonances: The Main Results 20.6 The Proof of Spectral Stability for Shape Resonances . 20.7 Resolvent Estimates for H1(?., 0) and H(%, 0) . . . . . 20.8 Notes . . . . . . . . . . . . . . . .
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221
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198
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215 215 216 218
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187 190 192 193 195
21 Quantum Nontrapping Estimates
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235 235 238
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23 Other Topics in the Theory of Quantum Resonances 23.1 Stark and Stark Ladder Resonances . . . . . . .
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263 272 275
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21.1 Introduction to Quantum Nontrapping . 21.2 The Classical Nontrapping Condition . . 21.3 The Nontrapping Resolvent Estimate . . 21.4 Some Lxamples of Nontrapping Potentials 21.5 Notes . . . . . . . . . . . . . .
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22 Theory of Quantum Resonances III: Resonance Width
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22.1 Introduction and Geometric Preliminaries . . 22.2 Exponential Decay of Eigenfunctions of H0(A) 22.3 The Proof of Estimates on Resonance Positions .
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23.2 Resonances and the Zeeman Effect . . . . . . . . 23.3 Resonances of the Helmholtz Resonator . . . . . . 23.4 Comments on More General Potentials, Exponential Decay, and Lower Bounds . . . . . . . . . . . . . . . . .
247 249
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Contents
Appendix 1. Introduction to Banach Spaces A 1.1 Linear Vector Spaces and Norms . . . A 1.2 Elementary Topology in Normed Vector Spaces A 1.3 Banach Spaces . . . . . . . A 1.4 Compactness . . . . . . . . . . .
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Appendix 2. The Banach Spaces L"(IR" ), I < p < oc A2.1 The Definition of LP(!R"), I < p < oc A2.2 Important Properties of L"Spaces . .
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Appendix 4. The Fourier Transform, Sobolev Spaces, and Convolutions A4.1 Fourier Transform . . . . . . . . . . A4.2 Sobolev Spaces . . . . . . . A4.3 Convolutions . . . . . . . . . . . . . .
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A3.2 Continuity and Boundedness of Linear Operators A3.3 The Graph of an Operator and Closure . A3.4 Inverses of Linear Operators . . . A3.5 Different Topologies on £(X) . . . .
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. 1. Density results . . 2. The Holder Inequality . 3. The Minkowski Inequality . 4. Lebesgue Dominated Convergence
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ix
301
303 307 309 312
313 313 316 317
References
319
Index
333
Introduction and Overview
This book presents some basic geometric methods in the spectral analysis of linear operators. The techniques, applicable to many questions in the theory of partial differential operators, are developed so that we can address several central problems in the mathematical analysis of Schrodinger operators. Many of the ingredients of this analysis, such as the notion of spectral stability, localization and geometric
resolvent equations, spectral deformation, and resonances have applications in other areas of mathematical physics and partial differential equations. Important examples of these applications include spectral geometry [His], the theory of the wave equation [DeBHS], random differential operators [CH 1, FrSp], quantum field theory [BFS, JP, OY], and nonlinear equations [Si4]. One of the subjects occupying
an important place in this book is the semiclassical analysis of eigenvalues and resonances. This area has seen rapid development since the early 1970's, and here we present many of the key results in textbook form. Our approach to spectral analysis differs from the standard one. We emphasize the use of geometric methods that take advantage of the local action of a partial differential operator. This leads naturally to the geometric characterization of the discrete and essential parts of the spectrum. The spectrum of a linear operator is studied through properties of the resolvent, the Riesz projections, and Weyl and Zhislin sequences. These objects form the basis of what is sometimes called geometric spectral analysis. Broadly stated, the idea is that spectral properties of the operator can be obtained from an analysis of the behavior of the operator acting on families of functions whose supports satisfy certain geometric conditions. The methods developed from this approach apply to many differential operators appearing in quantum mechanics and geometric analysis. Although we do not study directly the continuous spectrum of selfadjoint operators and scattering theory in
2
Introduction and Overview
this book, many of the techniques developed here are applicable to these questions as well. Our goal is to develop the necessary mathematics in a selfcontained way with minimal reference to the general theory. The background mathematics have been chosen so that we can discuss the applications as soon as possible. This approach reflects our general view that the heart of modern analysis lies in its applications to other disciplines.
In order to illustrate the geometric techniques presented in the text, we have chosen our examples from the theory of Schrodinger operators. This theory, originating with quantum mechanics, is now becoming an area of mathematical analysis in its own right, with applications to problems in nonlinear partial differential equations and geometric analysis. A certain amount of physical intuition is useful in the study of Schrodinger operators. The applications chosen in this text provide a gradual introduction to the physical concepts in a mathematical context.
A basic tenet of quantum theory is that quantum systems, such as atoms, molecules, solids, and to some extent nuclei and even stars, are described by linear differential operators called Schrodinger operators. We refer the reader to the standard text by Landau and Lifshitz [LL] for an account of this theory. The Schrodinger operator for a quantum system is the linear partial differential operator h'
H =_ A + V,
(0.1)
acting on the Hilbert space L2(W"). The Laplace operator A is the secondorder differential operator that, in Cartesian coordinates on R", is given by "
0
a2
i=1 axt2
(0.2)
The constant m is the reduced mass of the system, and the constant h = 27rh is called Planck's constant. The realvalued function V is called the potential. This family of linear operators, for various potentials V, describes the different quantum systems mentioned above. Elements of the Hilbert space L2(W ), called wave functions, represent various states of the system. The timeevolution of a wave function for a quantum system with Schrodinger operator H is controlled by Schrodinger's equation,
ihiat
tlir = H't
(0.3)
One of the principal goals of quantum mechanics is to describe the spacetime evolution of wave functions. For any reasonable initial state of the system *o, the solution of (0.3) is given by the formula tl/t = UH(t)+ljo.
(0.4)
The mapping UH (t): Vo * 1/it is called the evolution operator for the Schrodinger equation. For a solution to exist, this operator must remain bounded for all time. In
Introduction and Overview
3
quantum theory, the probability density of a wave function is given by It/r12. The corresponding probability is conserved under the time evolution. This requires that I'011 = IIUH(t)foll = II01II
(0.5)
Furthermore, in order to have a unique solution to (0.3), the evolution operator must satisfy the condition that for all s, t E R, UH(s)UH(t) = UH(S +t).
(0.6)
Notice that if we set s = t in (0.6), then we obtain the relation
UH(t) = UH(t)1,
(0.7)
which reflects the principle of time reversal invariance. These conditions on the evolution operator Utt(t) are, in fact, equivalent to the fundamental property, called .selfadjointness, of the Schrodinger operator H in (0.3). The condition of selfadjointness of H is a necessary and sufficient condition
for the existence and uniqueness of a solution to the Cauchy problem for the Schrodinger equation (0.3) satisfying (0.5) and (0.6). The selfadjointness of the Schrodinger operator H guarantees that the evolution operator UH(t) forms a oneparameter group of unitary operators.
The theory of the Schrodinger equation began with Schrodinger's groundbreaking papers [Sch], published in 1926. In these papers, he formulated the basic equation of quantum mechanics, computed the bound states of the hydrogen atom, and developed a simple form of perturbation theory, which he then applied to study the Stark and Zeeman effects. The mathematical theory of quantum mechanics and the Schrodinger operator can be traced to the book of J. von Neumann [vN], first published in 1932. In this book, von Neumann presented the Hilbert space framework of quantum mechanics and proved the equivalence of the matrix approach of Heisenberg and the partial differential equation approach of Schrodinger. One of the central mathematical contributions of the book is the basic spectral theory of unbounded selfadjoint operators. von Neumann emphasized the importance of selfadjointness for solving the eigenvalue problem for the Schrodinger operator. The framework presented in von Neumann's book was developed and applied to specific quantum systems in the 1950's. One of the main, early achievements was the the pioneering work of T. Kato, who initiated the rigorous study of individual Schrodinger operators. Kato proved the selfadjointness of atomic Hamiltonians, thus establishing the existence of the evolution operator for atoms and molecules [K]. Because of the connection between the spectral properties of a Schrodinger
operator H and the dynamics of the quantum system, Kato's results generated much research into the spectral theory of the linear operators describing quantum mechanical systems. The next fundamental step, understanding the spectrum of Nelectron atomic systems, was made in the works of Hunziker, Van Winter, and Zhislin in the 1960's. This opened the way to the study of Nbody Schrodinger operators.
Several themes dominated the mathematical physics of quantum mechanics throughout the next decades. These include scattering theory for quantum me
4
Introduction and Overview
chanical particles, the stability of matter, and the theory of bound states. Scattering theory concerns the description of the spacetime asymptotic behavior of scattered particles and the recovery of characteristics of the scatterer from this information. We refer to the books by Amrein, Jauch, and Sinha [AJS], Reed and Simon [RS3],
and Sigal [Sig] for a description of quantum mechanical scattering theory and other references. The 1960's and the 1970's saw the resolution of problems in the quantum description of hulk matter. We refer to the review articles of Lieb [Li ] and Fefferman [Fef]. Two of the main questions, the existence of the thermodynamic limit and the stability of matter, were resolved on the basis of two basic principles of quantum mechanics, the Uncertainty Principle and the Pauli Principle. Hence, it was shown that quantum theory alone accounts for the stability of matter around us.
The 1970's and the 1980's was a period of intense research on the Schrodinger equation. This period witnessed a sharpening of our understanding of the relation between classes of potentials and selfadjointness. the discovery of estimates on the numberof eigenvalues in terms of the potential, and the proof of exponential bounds on eigenfunctions. The properties of Schrodinger operators for other quantum
systems, such as quantum particles in external electric and magnetic fields or under the effect of random potentials, were also explored. We refer the reader to review articles by rlerbst [He] l, Hunziker [Hull, and Simon [Sim131. Scattering theory for two and manybody systems was extensively studied and many of the major open problems were resolved. This period also saw the beginning of rigorous understanding of two basic concepts of modern physics, quantum tunneling and resonances, which are dealt with in this book. One of the principal tools for the investigation of these characteristic properties of Schrodinger operators is the semiclassical approximation. The semiclassical approximation has been used by theoretical physicists from the beginning of quantum theory (see [LL]). It has also been extensively developed by mathematicians, especially in the context of highfrequency wave phenomena. The investigations into resonances and tunneling have brought about many advances and refinements in this method. The semiclassical approximation addresses the fundamental issue of the relation between quantum and classical mechanics. It is based on the observation that Planck's constant h, appearing as a coefficient of the differential operator in (0.1), controls the quantum effects. If h could be taken arbitrarily small, the quantum effects would reveal themselves as small fluctuations about wellknown classical behavior and hence would be easier to compute. The semiclassical approximation consists of replacing h by a parameter and studying the asymptotic behavior of the spectrum of the Schrodinger operator and the solutions of the Schrodinger equation as this parameter tends to zero. The quantum system in this approximation is considered as a perturbation of a classical system. One of our goals for this book is to emphasize the importance of quantum resonances. Although resonances in quantum mechanical systems play an important role in determining the spacetime behavior of the system, the general, mathematical theory has been developed only in recent years. In addition to their importance in quantum phenomena, it has also become clear that, in the theory of linear and
Introduction and Overview
5
nonlinear partial differential equations, eigenvalues and resonances must be considered together to form a stable set with respect to perturbations. It is interesting to note the similarities between quantum and classical resonances. Originally introduced by Poincare, the theory of classical resonances was developed through the works of Kolmogorov, Arnold, and Moser. Resonances in classical systems are known to play an important role in the dynamical behavior of the systems. The concept of resonance contributes greatly to the understanding of nonintegrable systems that are close to integrable ones. An interesting characteristic of classical resonance theory is the occurrence of exponentially small separatrix splittings, first discussed by Poincare (see, for example, IST]). As we will see, exponentially small splittings are also a facet of quantum resonance theory. These similarities are quite intriguing and have not yet been fully investigated. This book emphasizes the geometric approach to the spectral analysis of linear differential operators on Hilbert space. The applications to Schrodinger operators are centered on the theory of bound states and quantum resonances. Although we restrict our examples to twobody potentials, many of the ideas described here apply to Nbody quantum systems. Some of the specific topics presented in the text include the exponential decay of eigenfunctions (Chapter 3): the semiclassical behavior of eigenvalues (Chapter 11);
the relation between quantum tunneling and the spectrum of Schrodinger operators (Chapter 12); the AguilarBalslevCombesSimon theory of quantum resonances (Chapter 16); the general theory of spectral stability (Chapter 19); the existence of quantum resonances in the semiclassical regime (Chapters 2021); exponential bounds on resonance widths (Chapter 22).
In addition, many recent developments in the theory of quantum resonances are discussed.
The book is roughly divided into three parts. The first part concerns the decomposition of the spectrum of a linear operator into the essential spectrum and the discrete spectrum. Basic properties of the spectrum, the resolvent, and Hilbert spaces are discussed in Chapters I and 2. Our first main application to Schrodinger operators occurs in Chapter 3. There, we study Schrodinger operators with potentials that are larger than a constant Mo at infinity and that have an eigenvalue Eo less than Mo. We present the method of Agmon (Ag I ] to prove that the eigenfunctions corresponding to the eigenvalue E0 decay at least exponentially fast as oo Physically, this decay results from the fact that in the region where Ilx II
6
Introduction and Overview
Et < Mo, the conservation of energy is violated. On the classical level, the orbits of a particle with energy E0 do not enter this region. Consequently, it is called the classically forbidden region. The quantum mechanical wave function, however, is not zero there. This effect is called quantum tunneling. The main result of Chapter 3 is that any wave function corresponding to the energy E0 tunnels into the classically forbidden region but its amplitude decays exponentially there. Chapter 3 also serves to introduce many of the techniques of estimation, which are used repeatedly in the applications. After this application, we continue to study the properties and characterizations
of the discrete and essential spectrum for general closed operators and for the special, but important, case of selfadjoint operators on a Hilbert space. Selfadjoint operators are discussed in general in Chapter 5, where characterization of their spectrum is given. Our main tool for the analysis of the discrete spectrum, the Riesz projections, is detailed in Chapter 6. We emphasize the characterization of the essential spectrum in terms of the behavior of the given operator on certain sequences of functions. This leads to the Weyl criterion for the essential spectrum, which is presented in Chapter 7. For partial differential operators, one can take
advantage of the local action of the operator, and the geometric ideas of local compactness and Zhislin sequences play a central role in the identification of the essential spectrum. These notions are discussed in Chapter 10. The underlying idea is that the essential spectrum is determined by the behavior of the operator acting on functions supported near infinity. A nice application of these geometric notions
is Persson's theorem, which provides a formula for the bottom of the essential spectrum. This is proved in Chapter 14 after some sufficient conditions for the selfadjointness of Schrodinger operators are developed. The second part of the text is devoted to understanding various notions of stability for the discrete and essential spectrum. The question of stability for discrete eigenvalues under various types of perturbations will occupy us in several chapters. Both the usual perturbation theory framework, analytic and asymptotic, and a more geometric framework, are presented. Geometric perturbation theory is developed in the context of semiclassical analysis. This theory of perturbations is based on localization; operators are small perturbations because they are localized in regions of configuration space or phase space that are in some sense forbidden. The most common reason that certain regions of space are forbidden is due to energy conservation. These methods are first applied in Chapter 11 to study the semiclassical behavior of the lowlying eigenvalues of Schrodinger operators with potentials that grow at infinity. We then present, in Chapter 12, a more detailed computation of the difference of two eigenvalues in a situation dominated by quantum tunneling. These two chapters follow the work of Simon [Sim5, Sim6].
The characteristic property of the essential spectrum, from which its name is derived, is its robustness under various perturbations. We first discuss relatively bounded perturbations, which, according to the fundamental KatoRellich theorem, preserve the selfadjointness of the unperturbed operator. We then refine this to the notion of relatively compact perturbations. Weyl's theorem, which we prove in Chapter 14, states the invariance of the essential spectrum under relatively
Introduction and Overview
7
compact perturbations. We apply this to Schrodinger operators with potentials vanishing at infinity in order to compute the essential spectrum. Spectral stability for the discrete spectrum is introduced in Chapter 15. There we discuss the standard theory of stability of an isolated eigenvalue with respect to analytic perturbations. The notion of a type A analytic family of operators is also presented in this chapter.
The main themes of spectral deformation, spectral stability, and nontrapping estimates are developed in the third part of the book. These are illustrated by the theory of quantum resonances. The basic theory of quantum resonances, as developed by Aguilar, Balslev, Combes. and Simon, is presented in detail. This theory, dating from the early 1970's, provides the basis of most recent studies of resonances in quantum mechanical systems. This theory is applied in Chapter 20 to the shape resonance model in the semiclassical regime. This provides a good example of geometric perturbation theory and other methods discussed earlier. Quantum resonances also provide examples of asymptotic. nonanalytic perturbation theory. In order to deal with these situations, we must broaden the notion of spectral stability introduced in Chapter 15. We do this in Chapter 19, in which the general notion of stability is presented, along with some wellknown examples from quantum mechanics such as the anharmonic oscillator and the Zeeman effect. The main result of this chapter is a stability criterion for discrete eigenvalues due to Vock and Hunziker [VHJ. Some geometric techniques, including localization formulas and geometric resolvent equations, that are useful in the verification of the stability criteria are presented in the last section of Chapter 19. The AguilarBalslevCombesSimon theory of resonances provides the initial motivation for the theory of spectral deformation. The general theory is described in Chapter 17 in the form developed by Hunziker [Hu2]. The details of the application to Schrodinger operators is presented in Chapter 18. This method is often useful for dealing with problems involving embedded eigenvalues. The shape resonance model in the theory of quantum resonances is perhaps the oldest in quantum mechanics, dating back to the late 1920's. It was introduced to model the emission of an alpha particle from an atomic nucleus. This model is described in detail in Chapter 20. In order to establish the criteria for spectral stability presented in Chapter 19, we need the technique of geometric perturbation theory. In this technique, one uses localization. formulas, involving the resolvent operators, to localize the perturbations in regions for which a priori estimates can be computed. These estimates follow either from quantum tunneling results, as described in Chapters 3 and 12, or from quantum nontrapping results. Nontrapping estimates imply the absence of bound states or resonances for quantum systems, and the decay of energy for waves. In Chapter 21, we show how certain nontrapping assumptions on the potential imply the resolvent estimates necessary for the stability criteria. Many of the objects of study in geometric perturbation theory involve exponentially small quantities (such as the splitting of eigenvalues discussed in Chapter l2). A technique for proving the exponential smallness of resonance widths is given in Chapter 22. Finally, we conclude with a chapter giving other examples of geometric perturbation theory and other topics in the quantum theory of resonances.
8
Introduction and Overview
The guiding idea of this text is to develop the essential mathematics as directly as possible and then to illustrate the mathematics with examples drawn from research articles. To keep the book selfcontained, we give background material on basic functional analysis in four appendices. The reader should be familiar with the basics of real analysis and point set topology. Measure theory and Lebesgue integration theory (with the exception of the Lebesgue dominated convergence theorem) are not explicitly used in the text. However, we assume that the reader is familiar with the idea of the completion of a normed linear space and will accept the fact that
L2(W?) is the completion of Co (W ). Various properties of the Banach spaces LP(R"), such as the density of certain sets of "nice" functions, basic inequalities, and the Lebesgue dominated convergence theorem, are reviewed in Appendix 2. Certain facts about Sobolev spaces over R" and the Fourier transform are given in Appendix 4. Exercises are intertwined with the text in each chapter. These problems range from routine verification of statements and immediate extensions of the ideas presented in the text, to more challenging and involved problems, especially in the later chapters. We certainly encourage the reader to work the problems while studying the text. References to original research papers related to the problems discussed here are given. We have limited ourselves to works published after 1980, except in certain cases of direct relevance. We refer the reader to the book by Cycon, Froese, Kirsch, and Simon [CFKS] and the books by Reed and Simon [RS1, RS2, RS3, RS4] for many of the earlier references. For more information concerning functional analysis and perturbation theory,
we refer the reader to the books of Akhiezer and Glazman [AG], the book of Kato [K], the fourvolume series by Reed and Simon [RS1, RS2, RS3, RS4], and Volumes 3 and 4 of the work by Thirring [Th3, Th4]. Other aspects of the semiclassical analysis of the Schrodinger equation can be found in the books of Heifer [Hel], Maslov [Mast, and Robert [R1].
Acknowledgements This book developed out of a course on mathematical physics given at the University of California, Irvine, during the academic year 19851986. Versions of it were used at the University of Kentucky; at the Centre de Physique Theorique in Marseille, France, for a DEA course; and for a mathematical physics course at the University de Paris VII. We would like to thank J. M. Combes, S. DeBievre, W. Hunziker, J. Marsden, and B. Simon for their comments. This book is based on research of PDH, partially supported by NSF grants DMS 9106479 and 9307438, and of IMS, partially supported by NSERC Grant NA 7901. The authors gratefully acknowledge this support.
1
The Spectrum of Linear Operators and Hilbert Spaces
We assume that the reader has the basic understanding of Banach spaces and linear operator theory contained in Appendices 1 and 3. We suggest that the reader review these appendices before beginning this chapter. We begin with the main topic of
our interest: the spectrum of a linear operator. We introduce the resolvent of a linear operator on a Banach space. This operator is crucial to the definition of the spectrum. We define the spectrum and give some of its properties. We then specialize to Hilbert spaces and develop their basic characteristics in this and the following chapter.
1.1
The Spectrum
Let A be a linear operator on a Banach space X with domain D(A) C X. Let us recall from Appendix 3 that A is invertible if there is a bounded operator, which
we call A, such that A'
:
X > D(A), AA' = Ix, and A'A = 1D(A)
(where 1 X is the identity on X). We distinguish the invertibility of the operator A from the existence of inverses for A, which may be unbounded. By considering the invertibility of the operator A  A. % E (here, % means >< 1 X), we obtain a disjoint decomposition of C into two sets that characterize many properties of A.
Definition 1.1. Let A be a linear operator on X with domain D(A). (1) The spectrum of A, Q(A), is the set of all points ),. E C for which A  >< is not invertible.
1. The Spectrum of Linear Operators and Hilbert Spaces
10
(2) The resolvent set of A, p(A), is the set of all points A E C for which A  ,l is invertible.
(3) If), E p( A), then the inverse of A  A is called the resolvent of A at % and is written as RA(n.)  (A Let us note that by definition, or(A) U p(A) = C
(1.1)
a(A)np(A)=0.
(1.2)
and
Theorem 1.2. The resolvent set p(A) is an open subset of C (and hence a(A) is closed), and RA(A) is an analytic operatorvalued function of ;. on p(A). A(A) E L(X) is said to be (norm) analytic at Remark 1.3. A map A E p C C Ao E p if A().) has a power series expansion in () Ao) (with coefficients in L(X )) that converges in the norm with nonzero radius of convergence; that is, there exist bounded operators An such that (A  n.o)nAn.
A(A) _ n=0
Proof of Theorem 1.2.
(1) To show that p(A) is open, we show that there is an cball about each point Ao E p(A) which is contained in p(A). Let .l E C be such that II1 I),  Aol < II(A . Then, by decomposing A  A as Ao)1
AA=(AAo)[1 (A;.0)1(X A0)],
(1.3)
we see that it is invertible by Theorems A3.29 and A3.30. Thus, it suffices 1 to take E < II (A Ao)1
II
(2) To show analyticity at A0 E p(A), it follows from (1.3) that (A  ),)1 = (A  ).0)1(1  (A  Xo)(A  A.0)1)1. Now ),O)1II1
< 1, so using the Neumann series for (1  T)1 IA  AoIII(A (see Theorem A3.30), we get (A
(A

0C
A0)1 T(), ;.o)k(A

Xo)k
k=0 00
_
T(A 
ko)k(A

ko)k1.
(1.4)
k=0
One can easily check that the expansion is normconvergent, and so we get analyticity.
1.1 The Spectrum
11
Let us examine a(A), which are those A E C for which A  A is not invertible. There are basically three reasons why A  ) fails to be invertible: ( 1 ) ker(A ),.) ¢ {0};
(2) ker(A ),.) _ {0}, and Ran(A  A) is dense so that (A  n.) has a densely defined inverse but is unbounded;
(3) ker(A  )) = (0), but Ran(A  A) is not dense; in this case (A  A)I exists and may be bounded on Ran(A  %) but is not densely defined; therefore, it cannot be uniquely extended to a bounded operator on X. According to these three situations, we classify a(A).
Definition 1.4.
(1) If a. E cr(A) is such that ker(A  %) :' {0}, then A is an eigenvalue of A and any u E ker(A  A), u ¢ 0, is an eigenvector of A for A and satisfies Au = Au. Moreover, dim(ker(A  a.)) is called the (geometric) multiplicity of A and ker(A  ),) is the (geometric) eigenspace of A at X. (Note that ker(A  ),) is a linear subspace of X.) (2) The discrete spectrum of A. ad (A), is the set of all eigenvalues of A with finite (algebraic) multiplicity and which are isolated points of a(A). (3) The essential spectrum of A is defined as the complement of ad(A) in a(A): Cress(A) = a(A) \ ad(A).
Remark 1.5. (1) If dim X < oo, then the only reason why (A),) is not invertible is if (A A) is not onetoone, that is, a(A) = ad(A). Of course, all multiplicities are finite, but the geometric multiplicity need not equal the algebraic multiplicity for any eigenvalue. Recall that A is diagonalizable if and only if the algebraic multiplicity equals the geometric multiplicity for each eigenvalue (see, for example, [HiSm]).
Problem 1.1. Verify Remark 1.5; show that if dim X < oc, the above cases (2) and (3) for lack of invertibility of A  k do not occur. (2) aess(A) and ad(A) provide a disjoint and complete decomposition of a(A). Note that creSS(A) may contain eigenvalues of A. For example, an eigenvalue of infinite multiplicity or one that is a limit of a sequence of eigenvalues (so it is not isolated) belongs to aess(A).
(3) The set of all points k E a (A) such that A is not an eigenvalue but Ran(A X) is not dense is called the residual spectrum of A. We will see later that for a wide class of operators (including many that occur in mathematical physics) this set is empty.
12
1. The Spectrum of Linear Operators and Hilbert Spaces
1.2
Properties of the Resolvent
We want to collect in this section some basic properties of the resolvent of A, RA(k) = (A  AY', A E p(A). Proposition 1.6. Let A be a linear operator on X. Then for ie.A E p(A), (1) RA(A)RAW)= RA(p)RA(A); (2) RA(A)  RA(A) = (A  µ)RA(A)RA(p ), which is the first resolvent identity. Proof. We first prove (2) by writing the identity:
RA(s)  RA(a)
 lt)RA(l1) 
=
RA(A)(A
=
RA(X)RA(u)(A  µ).
RA(A)(A  A)RA(h)
which holds on all X. If we interchange µ and A and compare the result, we get the commutativity of RA (; ,) and RA (/I)
El
Although we have defined a(A), it remains to show that a(A) is nonempty. This can be done using the resolvent and some elementary complex analysis if A is bounded. There is some subtlety here, for there are examples of unbounded operators with empty spectrum (as there are examples of unbounded operators with empty resolvent sets)!
Theorem 1.7. Let A E £(X). Then a(A) is nonempty and is a closed subset of {zl Izl < IIAII}.
Proof. We consider a formal expansion of RA(s), A E p(A):
RA(s) = (A  X)1 = X1
1 + t XkAk
.
(1.5)
k=0
If IA ' I II A II < I (i.e., if IAI is sufficiently large), the power series converges and the
right side of (1.5) defines a bounded operator. As in the proof of Theorem A3.30,
one shows that this is the inverse of (A  ),). Hence, {zl Izl > IIAII} C p(A). Now suppose a(A) is empty. This would mean that RA(A) is an entire analytic function by Theorem 1.2. Furthermore, from (1.5), it follows that limIxH IIRA(A)ll = 0, and hence RA(A) would be a bounded entire function. Therefore, by Liouville's theorem (see, for example, [Ma]), RA(A) = 0 for all A. This implies that (A  k)RA(A) = I = 0, a contradiction. Proposition 1.8. Let SZ C C contain only isolated eigenvalues of A. Then RA (Z) is a meromorphic function on Q with poles exactly at the eigenvalues of A in Q. Proof. RAW is analytic on SZ \ {eigenvalues of A in S2}. If * is an eigenfunction with eigenvalue >` E S2, then
1.3 Hilbert Space
13
RA(z)1 = (z  ))I>Ai, and so RA(z) has a pole (or an essential singularity) at z = . E Q. It is also appropriate to mention here the second resolvent identity. Whereas the first identity, Proposition 1.6, compares the resolvent of a fixed operator at two points in its resolvent set, the second compares the resolvent of two operators at a common point in their resolvent sets.
Proposition 1.9. (Second resolvent identity). Let A and B be two closed operators with z e p(A) f1 p(B). Then RA(z)  RB(Z)
=
RA(z)(A  B)RB(Z)
=
RB(z)(B  A)RA(z).
(1.6)
Proof. One applies the identity
a[  b t =a'(ba)b' taking note that the right side of (1.6) defines a bounded operator.
1.3
Hilbert Space
All of our discussion so far has concerned linear operators on a Banach space X. To go further, we must introduce more structure on X. We use the abbreviations LV S for linear vector space and N L V S for normed linear vector space, as introduced in Appendix 1.
Definition 1.10. Let V be an LVS. An inner product V x V + C satisfying, for any x, y, z E V, a E C:
on V is a map from
(1) (ax + y, z) = a(x, z) + (y, z) (linear in the first entry); (2) (x, y) = (y, x) (complex symmetric); (3) (x, x) > 0 and (x, x) = 0 if and only if x = 0 (positive definiteness). Any LVS with an inner product is called an inner product space (IPS).
Examples 1.11. (1) X =R" and define, for x, y E R": (x, y) = y,"_f x; y; .Then this is an inner product (note that it is real, so condition (2) is replaced by (x, y) = (y, x)). Moreover, this inner product depends on the choice of the basis for R". Here we have taken the components of x and y relative to the same fixed basis.
(2) X = C([0, 1]) and define for f, g E X: (f, g) = fo f(t)g(t)dt; then this defines an inner product on X.
1. The Spectrum of Linear Operators and Hilbert Spaces
14
Eti ttx; \q. This sum always (3) X = 12 and define, for any x. v E 12: (x, converges, and is an inner product of 12. Problem 1.2. Prove the assertions in Examples 1.1 1. Let V be an IPS with inner product Then for each t' E V, define 11V11 = (L. v)
Then II
II
(1.7)
is a norm on V. We say that it is the norm induced by the IP.
Definition 1.12. An inner product space that is complete in the norm induced hr the inner product (as in 1.7) is called a Hilbert space. Consequently, each Hilbert space is a Banach space with additional structure coming from the inner product.
Examples 1.13. (1) X = /22 is a Hilbert space as follows from Proposition Al. 18.
(2) X = R" or C" are trivially Hilbert spaces. (3) It follows from Remark A 1.14 that C([0, 11) equipped with the norm given in part (2) of Examples 1.11 is not a Hilbert space: there are Cauchy sequences in the norm induced by the IP with no limit in C([0, 1]). We will, however, later show how to "complete" this space so that it becomes a Hilbert space.
(4) X = l ", p ¢ 2, is not a Hilbert space.
(5) X = C([0, 1]) with the sup norm IIfIlax = suppa,
0,
jxI < a.
V(x) _
Vo
a
0
a
FIGURE 3.2. A square well potential in one dimension.
3.1 Introduction
29
The Schrodinger equations in each region are
Ixl a is
'E(x)
coe
h10
% I x'
(3.6)
which generalizes (3.5). Our task is to show that under the conditions described here, eigenfunctions are bounded above by exponential factors of this type. To do so, we must first find an appropriate generalization of the WKB factor occurring in (3.6). A function f E L2(R") satisfies an L2 upper bound if eFr E L2(W1), for some function F : IIl;" * R. This function measures the decay rate of r as Ilx II  oc If F = F(Ilxll), then F gives an isotropic decay rate for >(i. The Agmon method provides an anisotropic decay rate. Note that the L2 upper bound is a priori weaker
30
3. Exponential Decay of Eigenfunctions
than a pointwise bound on 0. It suggests that as llx II > oc, i/i * 0 faster than e"(". In many cases, this pointwise bound can be proved: we will give one such theorem ahead.
3.2
Agmon Metric
We introduce the necessary geometrical object, very much like a metric, which we will use to measure the exponential decay of eigenfunctions. Let us recall that for
a real, ndimensional manifold A4, the tangent space at a point x E M. denoted by T,,(A4), can be considered to be the real linear vector space R". A metric is an assignment of an inner product to the tangent space T, (M) for each x E A4. If this inner product is not positivedefinite, we say that it is degenerate. We consider R" with a degenerate Riemannian metric defined as follows. Definition 3.1. Let x c R", and let product on T,(IR") by
,
>7 E TT (R"). We define a (degenerate) inner
17).r = (V(x) 
t7) E,
(3.7)
where(', ')E is the usual Euclidean inner product and f (x),  max{ f (x), 0}. We call this the Agmon metric on R", even though it may vanish at some points x. Note that it depends on V and E. We want to use this structure to induce a metric on 118". Let y : [0, 11 > R" be a differentiable path in R". For any Riemannian
structure, the length of y is
L(y)
II Y(t )II Y(r)dt.
(3.8)
0
Note that ll IIx = (', x)11: y (3.8) is
.
In the Agmon structure (3.7), the length of the curve
(V(y(t)) 
LA(Y)
Y(t)II Edt,
(3.9)
0
where II
lIE denotes the usual Euclidean norm. Recall that a path y is a geodesic
if it minimizes the energy functional E(y)  z J II Y(t)Il y(f)dt. Definition 3.2. Given a continuous potential V and energy E, the distance between x, y E R" in the Agmon metric is
PE(x. V) = inf LA(Y),
(3.10)
YEP,.,
where P.,,. _ {y
:
[0, 11 > R"
I
y(O) = x, y(l) = y, y E AC[0, l]}. Here
AC[O, I ] is the space of all absolutely continuous functions on [0, 1 ]. Hence the distance between x, y E R" with the Agmon metric is the length of the shortest geodesic connecting x toy. Note that PE in (3.10) reduces to the WKB factor in (3.6) for the onedimensional case.
3.3 The Main Theorem
31
Problem 3.1. Show that PE satisfies the triangle inequality. (Hint: For any given path connecting x to y, PE(X, Y) < LA(y).)
Proposition 3.3. (1) The distance function pE(x, v) is local/v Lipschitz continuous and hence is differentiable almost everywhere in x and in y;
(2) at the points where it is differentiable: IV,,PE(x, y)I2 < (V(y)  E)+.
Proof. (1) Letx, y, z E R",and fix x. By the triangle inequality: IpE(x. y)pE(x, z)I pE(y, z). Let yo be a straight line from V to z: yo(t) = (1 1) Y+Iz, t E [0, 1], and evaluate LA(yo),
LA(Yo) = J (V(yo(t))  E)+llz  ylldt = Iz 
(3.11)
o
where C,.. is the constant fo (V(yo(t))  E)+ d t. For any R > 0, let BR(y) be the ball of radius R centered at y and let CR(y) maxU,EBR(I)(V(w) E)+/2. Consequently, for fixed x,
IPE(X, w)  pE(x, z)I
pE(w, z) 0 such that V(x) > E for all Ilxll > R. As we will see in Chapter 8, this condition implies that E < inf Qess(H). To prove Theorem 3.4, we give two preliminary lemmas. We will make use of the Sobolev spaces HS(R" ), for positive integers s. These are reviewed in Appendix 4.
Lemma 3.6. For any c > 0, let f  (1  O PE. Let 0 E D(V) fl H' (R") be such that supp(O) C .EE,s = {x I V (x)  E > S}. Then there exists 31 > 0 such that
Re(ef0, (H
 E)ef0) > 31 II0II2.
(3.14)
Proof. (1) We compute the left side of (3.14). Consider the gauge transformed H : H f
of Hef. To compute this, we have for any u E C (R"), ef Def u =
(ef V ef)(ef Def )u and of Def u = (V f + V )u. It follows that
ef Def u = (V  V f )2U = (p + iV f )2u (where p  iV) and H f = (p + i V f )2 + V. Writing this out, we have H f = p2  I V f I2 + (V V f + V f V) + V. (Note: This is to be understood in the sense of quadratic forms, because f is differentiable almost everywhere.)
(2) Returning to (3.14), the left side is
Re(t, (Hf  E)O) ? (0, (V  lV f I2  E)O),
(3.15)
where we used p2 > 0 and the fact that V V f + V f V is antisymmetric so that the real part of its contribution to the quadratic form vanishes. Now to compute (3.15), we use the definition of f, Proposition 3.3, and recall
supp(O) C .FE,S, Iof12 = (1  E)2IVpE(x)12 < (1  E)(V  E)+, SO ES II0II2 and the lemma is proved by taking (0, (V  I V f I2  E)O) Si  ES. We mention that the left side of (3.15) might be infinite because the function f is unbounded in general. It will be convenient to introduce the notation [A, B], which is called the commutator of the operators A and B and denotes AB  BA.
3.4 Proof of Theorem 3.4
33
Lemma 3.7. Let f =(I  E)pE, and let fa  f(1 +af)' (which is bounded). Suppose rl is a smooth, bounded function with supp Ioril compact. Set 0
rl
HVj=E,tr.Then Re(efa0. (H where
=
IV 1712
 E)efa0) = (Ee2j'' /r, 0),
(3.16)
+2(V,7 Ofa)77
Proof. Observe that ofO E L22 as fa is bounded by 1. We compute the left side of (3.16):
(e1"0, (H' E)e
0) =
(,7e2.1aVf,
(H  E)r
)=
[p2, 1710 ),
since (H  E)i(r = 0. Now the commutator [p22, rl] = AU  2Vr1 V, and so by integration by parts, we compute (2(V,7)x7e'I° ilr, V r) _ ([2(O77)rl + 2IVqI2 + 4(Vi7 . V fa ),71e2f° fir,
) + (2(o)?)qe2f° or. v,),
so that
Re(2(vq)i7e2fa Vf, V r) _ ([(0,7),7 + Io1112 + 2(V
V fa),7]e2
fa
r).
Combining this with the (A q)term from the commutator, we get
Re(ef0, (H
 E)ef0) = ([,o,712 + 2(Vt7 V fa),7]e2f° , )
which is the result, (3.16).
3.4
11
Proof of Theorem 3.4
Consider the following regions associated with a discrete eigenvalue E E ad(H): 17E. 2S a {xI V(x)  E > 23} and
AE,s  {xIV(x)E ) E CO0 be such that t7(x) = 1, x E .TE,26, and q(x) = 0, x E AE,s. By our assumption on V, such an rl has suppIVi I compact. Let f = (I  E)pE and fa = f (I + a f)' as before. Then 0  t7 exp(fa) >r satisfies the hypotheses of Lemma 3.6. Consequently, 331 > 0 such that 31110112
<
n/2. This condition and the fact that s = k + 2 imply that we must have k > min{0, (n  4)/2}. It follows from (3.19) and the Schwarz Inequality that there exists cs > 0 such that max
xEB(x0,1 /2)
I*(x)I
cs [f
I(o+ 1)'(X*)(x)12]
.
(3.20)
The right side is finite since t/r X E HS (1[8") fork as above by Theorem 3.8. Now
we consider the integral in the square brackets in (3.20). Since x has compact support, we can express this as the inner product (X,/i, (A + 1)SX,/ff)
The idea is to evaluate this by successively computing the action of (A + 1) on x i using the eigenvalue equation. We have ( A + 1)(X,/i)(x)
= (H + l)(X,f)(x)  (V X,li)(x) _
(E + 1)(X,/')(x) + [A, XI,Gi(x)  (V X,r)(x) (3.21)
We write the commutator in (3.21) as
[A,Xh,/'=(AX)*2Vx V*, Let j be a smooth function such that XX = x and suppX C B(xo, 1). Then
VX
VX 0(H + M)1(H + M)X,lj (3.22)
= VX 0(H + M)1 [X'(E + M),I
+(Aj + 2V Vj)i/il.
36
3. Exponential Decay of Eigenfunctions
Note that V(H + M)1 and V(H + M)1V are bounded and that a characteristic function or its derivative sits next to t'. Hence, 3CE,M.V such that IIVX
VIII  0 such that max
XEB(xo, 1/2)
I *(x)e(I
E)PE(x)I
< cE
Since xo is arbitrary, this proves the result.
Remark 3.11. The pointwise exponential bound of Theorem 3.10 can be obtained under weaker conditions on V, but this requires much more work.
3.6 Notes
3.6
37
Notes
There is a large amount of literature on the decay rate of eigenfunctions of secondorder differential operators. The methods of this chapter are taken from the book by Agmon [Ag I], which also contains many references to earlier work. Isotropic exponential bounds on eigenfunctions for Schrodinger operators were considered before the anisotropic estimates presented here. These isotropic estimates have the form l
(x)I < caea114
for some a > 0. We mention the papers of Combes and Thomas [CT] and of O'Connor [OC] because their proofs employ the method of complex scaling (see Chapter 17) to prove bounds of this type. R. Carmona, P. Deift, W. Hunziker, B. Simon, and E. Vock undertook a systematic study of the decay rates of eigenfunctions for 2 and Nbody Schrodinger operators in [Siml], [Sim2], [Sim3], [DHSV], and [CS]. There is also a body of work concerning the relation between the rate of decay of an eigenfunction for eigenvalues of Schrodinger operators and the positions of the thresholds for the operator. We refer the reader to the papers by Froese, Herbst, M. HoffmannOstenhof, and T. HoffmannOstenhoff [FHHO I], and Froese and Herbst [FrHe] and to the book [CFKS] for an account of this theory and other references. We will mention a simple case of this in Chapter 16 for twobody operators. The theory is much richer in the Nbody case because there may be many thresholds. For other results on exponential decay of wave functions, we refer to [ROAM] and references therein. Upper bounds are among the weakest estimates one can obtain on an eigenfunction. One might ask if there are lower bounds for an eigenfunction. In the case that the eigenfunction is positive, which occurs when it corresponds to the lowest eigenvalue for Schrodinger operators with reasonable potentials (see [RS4]), one can obtain isotropic lower bounds [FHHO2] and lower bounds in terms of the Agmon metric [CS]. It is still an open problem to determine in what cases the Agmon metric gives the actual rate of decay of an eigenfunction, even in the twobody case; see, for example, [He2] and the references therein.
4 Operators on Hilbert Spaces
We now continue to develop additional properties of linear operators on Hilbert spaces. The inner product structure of a Hilbert space has many consequences for the structure of operators mapping the space into itself. The most important of these is the existence of an adjoins operator acting on the same space. Although it is possible to define an adjoint operator corresponding to an operator on a Banach space, the operator acts on a different space in general. Because of the Riesz representation theorem, the adjoint of a Hilbert space operator can be taken to act on the same space. We conclude the chapter with a discussion of the resolvent of the Laplacian on JR3. This important partial differential operator is another example of a Schrodinger operator and will play a central role throughout the remainder of the book.
4.1
Remarks on the Operator Norm and Graphs
Let 7l be a (separable) Hilbert space, and recall that L(1l) denotes the set of bounded operators on R. This set G(7() is a Banach space with the norm 11A ! = sup{ 11Ax11
1
x E 7l, Ilxll = 1).
(4.1)
We first derive a useful formula for this norm.
Theorem 4.1. For any A E L (R),
IIAII = sup( I (Ax, y)I I x. y E 7l, Ilxll = Ilyll = 1). In fact, the supremum can be taken over a dense subset of R.
(4.2)
40
4. Operators on Hilbert Spaces
Proof. For fixed x E 71, consider the map
Y E 7{ H (y, Ax).
It is easy to check that this is a bounded linear functional on R. By the Riesz lemma (Theorem 2.13), the norm of this functional, sup{ I(y, Ax) I I y e 7l, Ilyll = 1},
is equal to the norm of the defining vector, namely, Ax. Consequently, IIA11 = sup{ IIAxl1 x
I
I1x11 = I} = sup{ 1(y, Ax)I I IlxII = Ilyll = 1). x.v
which proves the first part of the theorem. Problem 4.1. Prove the second statement in Theorem 4.1, that is, that the supremum can be taken over a dense set.
Let us recall from Appendix 3 the notion of the graph of an operator. Let A be an operator on 7l with domain D(A). The graph of A, T(A), is the subset of
71x71{(x, y)Ix,yE71)defined by I'(A) = {(x, Ax)lx E D(A)}.
(4.3)
The set 7{ x 71 can be made into a Hilbert space with the inner product
((x, y), (w, z))xx7{ = (X, ON + (y, z)7{
(4.4)
Problem 4.2. Prove that 7{ x 71, with the inner product given in (4.4), is a Hilbert space. The norm induced by the inner product in (4.4) is given by II(x, y)11nxn = [IIXII;. + IlyllH}7,
(4.5)
and is sometimes called the graph norm. Finally, let us recall the notion of a closed operator. An operator A with domain D(A) is said to be closed if r(A) is a closed
subset of 7t: x 71. If I'(A) is not closed but its closure, 1'(A), is the graph of an operator, we call this operator the closure of A. We refer the reader to Appendix 3 for a more complete discussion.
4.2
The Adjoint of an Operator
We use the inner product structure of a Hilbert space 7t: to associate with each operator A another operator on 7{, called its adjoins and denoted by A*. The adjoint operator provides a powerful tool for the study of A and its spectrum.
4.2 The Adjoint of an Operator
41
Definition 4.2. Let A be an operator on R with domain D(A). The adjoins of A, A*, is defined on the domain
D(A*)  {x E 7(I I(AY,x)I < CxIIYllforsome (4.6)
constant Cx (independent of y) and for ally E D(A)}.
as a map A* : D(A*) + N satisfying (Ay, x) = (y, A*x),
(4.7)
for ally E D(A) and X E D(A*). Note that if A E G(7l), then D(A) = 7l and
I(Ay. x)I < IIAII IlYll Ilxll, which implies that D(A*) = H, since we can take C, = I I A I I Ilx II The adjoint A* can be constructed explicitly as follows. For anyx E D(A*)andy E D(A), (Av,x) is a bounded linear functional on D(A), by definition. Any bounded linear func
tional f can be extended to all of R. We first extend it to D(A) as follows. If x E D(A), let {xn} C D(A) be a sequence converging to x. We define the value of the extended linear functional on x to be limn,,; f (x,,). If D(A) is dense, this H, we use the projection theorem defines a linear functional on R. If D(A) (Theorem 2.8) and write
H = D(A) ®D(A)= .
We can then define an extension of the functional to D(A)1 by f (w) = 0, for example, for any w E D(A)L. We note that the extension is unique if D(A) is dense. Otherwise, it is not unique and, consequently, the adjoint will not be uniquely defined. To avoid this problem, we will always assume that D(A) is dense. So we extend (Ay, x) to R. By the Riesz lemma, for each x E D(A*) there is a
unique z E N such that
(Ay, x) = (y. Z), y E D(A). The adjoint A* : D(A*) * N is then given by A*x = z.
Problem 4.3. Show that A* defined above is a linear operator. We can summarize the construction of A*, given a densely defined operator A on D(A), as follows. We first define D(A*) as in (4.6). Then, for Y E D(A), the map
x E D(A*)  fx(Y) = (Ay, x) can be extended by the Riesz lemma to obtain (Ay, x) = (Y, z(x))
42
4. Operators on Hilbert Spaces
We then define
A*x = z(x). By construction, A* is uniquely defined and satisfies
(Ay, x) = (y, z(x)) = (y, A*x)
forx c D(A*), y c D(A). Proposition 4.3. Let A, B be operators defined on a common, dense domain D, and let), it E C. Then (AA+AB)* = fiA* + AB*. Problem 4.4. Prove Proposition 4.3.
Proposition 4.4. If A E £(N), then A* E L(N), IIA*II = IIAII, and A** = A. If A, B E L (7i), then (AB)* = B*A*. Proof. As noted above, D(A*) = H. By the Riesz lemma, for any x E N: IIA*xll
= sup{I(A*x, y)I I Ilyll = 11 = sup( I(x, Ay) I
I
Ilyll = 1}
(4.8)
< IIAII IlxII, by (4.7) and the Schwarz inequality. Hence A* E £(H) and IIA* II < IIAII. Next, by (4.7), (A*x, y) = (x, Ay),
thus (A*)* = A; and by (4.8),
IIAII = II(A*)*II
IIA*II,
whence IIAII = IIA * II Finally, we compute (AB)*. For any x, y E N:
((AB)*x,y)
=
(x,ABy)
=
(A*x, By)
=
(B*A*x, y),
and so (AB)* = B*A*.
Remark 4.5. If A E £(R), then we can give an equivalent definition of A* as follows. A* is the unique bounded operator satisfying
(A*u, v) = (u, Av)
4.2 The Adjoint of an Operator
43
for all u, v E R. The existence of A* follows from the Riesz lemma as above. We used this fact in the last part of the proof of Proposition 4.4.
Proposition 4.6. For any densely defined linear operator A,
Ran A® kerA*=7l. Proof. It suffices to prove that ker A* is the orthogonal complement of Ran A, as the result then follows by use of the projection theorem. Let u e Ran A and v E ker A*. Then there exists f E D(A) such that u = Af. We compute
(u,v)=(Af,v)=(f,A*v)=0, and thus ker A* C (Ran A)'. Now let w E (Ran A)t. For any u = Af E Ran A, we have
0 = (u, w) = (Af, w) = (f, A*w),
(4.9)
(note that (Af, w) = 0 implies that w c D(A*) ). As D(A) is dense, it follows that A*w = 0 (by Proposition 2.9), that is, (Ran A)1 C ker A*.
Proposition 4.7. Let A be densely defined with dense range. If A has an inverse, then so does A* and (A*)' _ (A ')*. Proof. Since Ran A is dense, it follows from Proposition 4.5 that ker A* = {0}, and so by Theorem A3.26, A* has an inverse. Now AI = AI A = id, so by carefully considering domains, we get the second statement.
Problem 4.5. Fill in the details of the proof of (A*)' = (A')*. Corollary 4.8. If A is invertible, then so is A* and (A*)I = (A')*. It is a peculiar fact about unbounded operators that even if A is densely defined, A* may not be. However, we can make the following proposition about A and A* in general.
Proposition 4.9. Let A be a densely defined operator. Then
(1) A* is always closed; (2) if A is closed, then D(A*) is dense (i.e., A* is densely defined); (3) if A is closed, then A** = A.
Proof. (1) We will prove (2). The proofs of (1) and (3) will follow by a slight extension of the method given here. Suppose A is closed but D(A*) is not dense, that is, there exists a nonzero vector x perpendicular to D(A*). Let I'(A) be the graph of A defined in (4.3).
Claim 1. (x, 0) E 1'(A*)1. For any (u, v) E T(A*), we compute
((x, 0), (u, v)) = (x, u) = 0,
(4.10)
since u E D(A*) and x 1 D(A*). In (4.10), the first inner product is in the Hilbert space formed from 7l x 7l as in Problem 4.2.
44
4. Operators on Hilbert Spaces
(2) Our goal is to show that (0, x) E F (A). Since A is linear, this is a contradiction. We introduce an important operator V on 1i x 7l by
V(x, y = (Y, x)
(4.11)
It follows that V2 = 1 and V *(X' )') = (Y, x),
so that VV*=I=V*V. Claim 2. (VF(A))1 C r'(A*). Let x E D(A), so that (x, Ax) E F(A). Then if (u, v) is orthogonal to VF(A), we have
0 = ((u, v). (Ax, x)) = (u. Ax) + (v, x) or
(u, Ax) = (v, x),
thus, by Definition 4.2, u E D(A*) and A*u = v. Hence (u, A*u) E r(A*), and the claim is proved.
Problem 4.6. Let V be a unitary operator, that is V * = V1. Show that for any subspace M,
V(M) = (VM)l.
(4.12)
(3) By applying Problem 4.6 to Claim 2, we have
V(I(A)1) C r'(A*),
and as V2=1: r(A)1 C Vr'(A*).
(4.13)
Now, we use the facts that (M1)' = M and that M C N implies N1 C M1, together with (4.13), to arrive at
(Vr(A*))1 C I'(A) = r'(A), since A is closed. Finally, using (4.12) again, we get
V(P(A*)1) C I(A).
(4.14)
By claim 1, (x, 0) E r'(A*)1, x E D(A*)1, and so
(0, x) = V(x, 0) E V(r(A*)1) C ['(A) by (4.14). This shows that (0, x) E r'(A), which is a contradiction unless x = 0. Hence D(A*) is dense. Problem 4.7. Prove that if D(A*) is dense, then A** extends A. Then conclude that A is closable if and only if D(A*) is dense.
4.3 Unitary Operators
45
As a final aspect of the adjoint operator, we apply Proposition 4.9 to obtain a spectral relation between A and A*. Corollary 4.10. Let A be closed. Then
a(A*) = a(A) (complex conjugate). Proof. Since A is closed, Proposition 4.9 implies that D(A*) is dense, and so a (A* ) is well defined. By Corollary4.8, if AA is invertible, then (AA)* = A* X. is also.
This implies that p(A) C p(A*) or, equivalently, a(A*) C a(A). Furthermore, as A is closed, A** = A. Again, if A*  A is invertible, Corollary 4.8 shows that (A*  A)* = A**  . = A  A is also. Hence, p(A*) C p(A) or a(A) C a(A*). These two relations on the spectra imply that a(A) = a(A*).
4.3
Unitary Operators
We next turn to the study of another important class of operators on a Hilbert space.
Definition 4.11. A bounded operator U on a Hilbert space 7l is called an isometry if and only if II U f II = II .f II for all f E R.
Isometric transformations preserve the norms of vectors and as such play an important role in quantum mechanics. Note that by the parallelogram law (1.8), if U is an isometry, then
(Uf, Ug) _ (f, g), for all f, g E 7l. Furthermore, from (4.1), II U II = 1. We could equally well define an isometry by the relation (4.15) U*U = 1.
Problem 4.8. Prove that (1) for U E £(7l), (4.15) is equivalent to Definition 4.11; (2) if 7l is finitedimensional, (4.15) implies that UU* = 1, and so U1 = U*.
Definition 4.12. A bounded operator U on a Hilbert space 7l is called a unitary operator if (a) U is an isometry and (b) Ran U = 7l. Problem 4.9. Prove that U is unitary if and only if U is invertible and U* = U'*. If U is a unitary operator and A is a closed operator, the conjugation of A by U, AU =_ UAU', is often useful in the spectral analysis of A. Proposition 4.13. Let A be a closed operator and U a unitary operator. Let AU
UAU' on D(AU)  UD(A). (1) AU is closed on D(AU) and a(AU) = a(A); (2) if A is selfadjoint, then AU is selfadjoint.
46
4. Operators on Hilbert Spaces
Proof.
(1) The closure of Au on D(AU) is easy to check. As for the spectrum, note that RAE, (z) = U RA(z)U 1 for any z E p(A) and hence IIRAL,(z)ll = IIRA(z)II
Thus p(A) = p(AU ), and the result follows.
(2) AU is clearly symmetric (see Section 8.1). To prove A is selfadjoint (see
Chapters 5 and 8), it suffices to show that Ran(AU + i) = H. But U is invertible, so Ran(UAU1 + i) = U Ran(A + i) = H. Consequently, each unitary operator generates an isospectral mapping on operators. This may be very useful for calculations of the spectrum, for many times a unitary operator can be found such that the spectral analysis of the transformed operator is very transparent. An example of this is provided by the Laplacian A and the unitary operator given by the Fourier transform. We use material concerning the Fourier transform summarized in Appendix 4. With regard to spectral analysis of constantcoefficient differential operators, we note the following properties of the Fourier transform:
(1) F(8f /axi)(k) = iki F(f)(k), that is, F takes a/axi to multiplication by
iki; (2) F(xi f)(k) = i(a(Ff)/aki)(k). We now apply these properties to the Laplacian, a secondorder partial differential operator on 1R" defined by n
a2
`=ix2 For any f E S(][8" ), where Ilk 112
El
F(Af)(k) = Ilk 112(Ff)(k),
(4.16)
2
Problem 4.10. Prove that the Laplacian A, defined in (4.16), is a closed operator on the domain H2(R') (see Appendix 4 for a review of Sobolev spaces).
It follows from (4.16) and the comments after Proposition 4.13 that (z  A) is invertible for any z with Im z 0. Let RA (Z) = (z  A)' be the resolvent of A. From (4.16) we have for any z with Im z 0,
F(Ro(z)f)(k) = (z + IlkIl2)'(Ff)(k).
(4.17)
Our goal here is to compute an xspace representation for Ro(z) for dimension n = 3. The general case is considered in Problem 16.1. To do this, we use the inversion formula (see Appendix 4) and the following property: Let f * g be the convolution of f and g(see Section A4.3):
(f * g)(x)
f f(x  v)g(y)dy;
4.3 Unitary Operators
47
then
F(fg)(k) = (2ir); f * k(k) and
F'(fg)(k) _ (27r)2(F ' f) * (F1 g)(k).
We now see that from the convolution formula and (4.17),
(Ro(z)f)(x) = (G * f)(x), where
G(x) _ (27r)'
J
erk .e(z + Ilkll') 'dk.
Indeed, we can compute this result as follows:
Ro(z)f
=
F' FRo(z)f
=
F'((z + Ilk 112) '(Ff)
=
F'(z+Ilkil2)* f.
When n = 3, it follows by elementary contour integration that Gz(x) = [47r
llx1111 ez1 llxII
(4.18)
where the principal branch of the square root is taken. The formula for general n is given in Problem 16.1.
Problem 4.11. Derive (4.18). Hence, the resolvent of A in R3 is given by an integral operator:
(Ro(z)f)(x) = J [47t IIx 
YII]'ec2Ilx>YII f(v)dv,
and this operator is bounded for all z E C \ (oo, 0]. We will verify in Chapter 8 that 0 is selfadjoint (Example 8.4). This result and our calculation of RA(z) show that Q(0) = [0, oo).
5
SelfAdjoint Operators
In this chapter, we make a preliminary study of one of the most important classes of operators on a Hilbert space, the selfadjoint operators. These operators are the infinitedimensional analogues of symmetric matrices. They play an essential role in quantum mechanics as they determine the time evolution of quantum states. The goal of this chapter is to describe symmetric and selfadjoint operators and to understand some characteristics of the spectrum of a selfadjoint operator. In Chapter 8, we will present the fundamental criteria for selfadjointness. This will be applied in Chapters 8 and 13 to various families of Schrodinger operators.
5.1
Definitions
Let A be an operator on a Hilbert space 71 with domain D(A). Unless explicitly stated to the contrary, we always assume that D(A) is dense in N. Recall that if D(A*), the domain of the adjoint of A, is dense, then A is closable. The closability of A means that the closure of the graph of A, I(A), is the graph of an operator. This closed operator has an important relation to the original closable operator A. By an extension of an operator A with domain D(A), we mean an operator A with domain D(A), such that D(A) C D(A) and AID(A) (the restriction of A to the domain of A) is equal to A. Consequently, it is not hard to check that for a closable operator A, the closure of A is the unique smallest closed extension of A. Definition 5.1. An operator A with domain D(A) is called symmetric (orhermitian) if A* is an extension of A or equivalently, if (Ax, y) = (x, Ay) forall x, y E D(A).
Problem 5.1. Verify that the two definitions are indeed equivalent.
50
5. SelfAdjoint Operators
Example 5.2. Let 7( = L([0, 11), and let A I be the operator A i  d'/dx2 with
D(AI)={u E7llu,u' E AC[0, 11 andu(II(0)=0=ut't1(1), fork =0, 1}. (5.1) Recall that AC[0, 1 ] are precisely those functions that are antiderivatives, and so they can be integrated by parts. Then D(A i) is dense and A Ion D(A i) is symmetric.
To show that D(AI) is dense, note that D(A) contains Co [0, 11, which is dense in H. The symmetry of A i is easily verified by two integrations by parts. Let u, v E D(A i ); then 2
(u. A1v)
_ fo u(x)
xU(x) = u(x)dx(x)I0+fo (x)(x)
dxdx
2
(5.2)
_ x (x)U(x)I0  fo dxu WV W _ (A1u. v).
Definition 5.3. An operator A is called selfadjoint if A = A*, that is, if (a) A is symmetric and (b) D(A) = D(A*). Note that the property of selfadjointness depends both on the form of the operator A (i.e., what A does to a permissible vector) and on the domain D(A). The same symbol may define a selfadjoint operator on a domain DI, but it might not represent a selfadjoint operator on a domain D2. As the adjoint A** is always closed, a selfadjoint operator is closed. On the other hand, a symmetric operator need not be closed, and a closed, symmetric operator need not be selfadjoint. Problem 5.2. Let A be symmetric on D(A). Prove that A*** is also symmetric. Example 5.4. We continue Example 5.2. The operator A i defined there is not selfadjoint on D(A 1). To see this, note that u E D(A *) if and only if u, u' E AC[0, 1 ] and for all v E D(A 1)
 u(x)dx (x)l
+
du
()v()Io =
0.
(5.3)
o
The conditions on v mean that (5.3) is satisfied provided u, u' E AC[0, 1 } with no other conditions. Hence
D(Ai) = {v E NIv, V' E AM, M. Clearly, D(A *) D D(A i ). The problem is that A, is "too small" for selfadjointness; as a consequence, D(AB) is much bigger than D(A 1). We can search for an extension of A i that is selfadjoint by relaxing the requirements on the domain while preserving (5.3).
Problem 5.3. First, show that if u E D(A i ), then u, u' E AC[0, I]. Second, relaxing the constraints on the domain, find other dense domains on which the
5.2 General Properties of SelfAdjoint Operators
51
symbol A  d'/dx2 is symmetric. Try to find a domain on which the operator is selfadjoint. (Hint: Consider linear boundary conditions of the form u(0)+au(1) 0.) As Example 5.4 shows, a symmetric operator need not be selfadjoint, but it follows from the definition that every selfadjoint operator is symmetric.
5.2
General Properties of SelfAdjoint Operators
The property of selfadjointness is strong enough to enable us to make some rather specific statements about the spectrum and to obtain some bounds on the resolvent.
These resolvent bounds are crucial for the study of the discrete and essential spectrum. In later chapters, we will see how to extend some of these bounds to various closed, nonselfadjoint operators. Although we will not discuss it in this book, the property of selfadjointness of the Schrodinger operator is necessary and sufficient for the existence of quantum dynamics; see [RS 1). Let us recall from Chapter 2 the definition of the residual spectrum of an operator A:
Qres(A) = {X E C1 ker (A  A) = {0} and Ran(A  A) is not dense}. Our first theorem presents some results about the spectrum of selfadjoint operators.
Theorem 5.5. Let A be selfadjoint. Then
(1) a(A)CR; (2) ares(A) = 0;
(3) eigenvectors corresponding to distinct eigenvalues are orthogonal.
Proof. (1) We will prove this after we obtain a preliminary estimate on the resolvent in Theorem 5.6. (2) Let A be such that ker(A  A) = {0}. From Chapter 4, we have
Ran(A  A)1 = ker(A*  A) = ker(A  X) = {0), so that Ran(A  A) is dense in 7L and A ¢ areS(A).
(3) Let ii', ¢ be eigenvectors of A such that
AO=.10, A*=A*, and E.c¢A.
(5.4)
5. SelfAdjoint Operators
52
Then, from the eigenvalue equation,
A(4, ,)
_ (A (P,
(0. A>/i)
_ A (0, Vi),
and the fact that µ = /t, which follows from (1), we obtain
(Aµ)(0,t/i) =0. As
=0.
µ,we get
Theorem 5.6. Let A be selfadjoins. If for some M > 0 and, for all u E D(A), II (),  A)ull > MlluII.
(5.5)
then n. E p(A). Moreover, we have
{zCCCI Iz  Al < M}Cp(A). Proof. (1) We first show that equation (5.5) implies
(a) ker(A  A.) _ {0};
(b) Ran(A ,k) = R. As for statement (a), if Ai/i = ,lt/i, then inserting * into (5.5) implies that II Vi II = 0, so Vi = 0. To prove (b), we note by (5.4) that Ran(A  A) is dense,
and so it suffices to show that it is closed. Let X, E Ran(A  A) form a Cauchy sequence. Then there exists {y,) in D(A) such that xn = (A  A)yn. We claim {y, } is Cauchy. Indeed, by (5.5),
IIYn  Ymll < M'II(A A)(yn Ym)ll = M'Ilxn  xmll, so that as {x,) is Cauchy, the sequence {y,7} is also. Hence there is y E 7{ such that y = limn ....,w yn. Since y, + y and x, = (A  A)yn is a Cauchy sequence, it follows from the fact that A is closed that x (A  A)y E Ran(A  A); thus Ran(A  A) is closed.
(2) It is a consequence of (1) that (A  A) has an everywheredefined inverse
(A  X)1, and we must show that it is bounded. Let x E N and y (A  ,L)tx E D(A). Then x = (A  A)y and Iixll = II(A  ),)yll
M' IIYII = M' II(A  X)'xll,
so for all x E 71, (5.6) II(A  A)'xll Mllxll. which shows that (A  %) I E £(H) and hence that ,l E p(A). We leave the
proof of the remaining part of the theorem as a problem.
53
5.3 Determining the Spectrum of SelfAdjoint Operators
Problem 5.4. Prove the remaining part of Theorem 5.6. (Hint: Write A  z ()L  z) + (A  ),), and use the second resolvent identity. Proposition 1.9.)
Proof of part (1) of Theorem 5.5. Let Z = A + i u with tt 0. Then, by the selfadjointness of A, 11 (A  z)ull2
=
((A  z)u, (A  z)u) II(A  A,.)uI12+I 2I2llull2 42I1u112,
(5.7)
for any u E D(A). By Theorem 5.6, this implies that z e p(A), and so a(A) C R. Corollary 5.7. Let A be selfadjoint and z e C, Im z ¢ 0. Then IIRA(z)II < IIm zl'
Proof. This follows directly from (5.6) and (5.7). We can sharpen Corollary 5.7 in the following form.
Theorem 5.8. Let A be selfadjoint, and let;` E p(A). Then IIRA (1)II < [dist(A, a(A))]i.
(5.9)
We will not prove this here, although we will use the result in this book. The standard proof of this theorem uses the functional calculus; see [RS I]. We leave it as an exercise to prove the following simple extension of Corollary 5.7. Suppose that A is selfadjoint and a(A) C [A0, oc) Then for >< E IR and;, < n,p, we have IIRA(A)II < IA  Aol  '.
We want to emphasize that, despite the simplicity of condition (5.5), the notion that lower bounds on II (A z)u 11, for u in various subsets, leads to spectral estimates is a very powerful one. We will encounter variations on this idea in our discussion
of spectral stability in Chapter 19 and in the discussion of nontrapping estimates in Chapter 21.
5.3
Determining the Spectrum of SelfAdjoint Operators
The estimates on the resolvent of a selfadjoint operator A given in the preceding section allow us to obtain some detailed information about the location of a(A) in R. We give two such results here.
Theorem 5.9. Let A be selfadjoint. If for some e > 0 there exists some u E D(A) such that
5. SelfAdjoint Operators
54
then or (A) fl [  e, x + E ]
(5.10) II(A  A)ull < Ellull 0, that is, A has spectrum inside [A  e, ti + E ].
Proof. Assume to the contrary that [A  E, A + E ] C p(A). By Theorem 5.8, (5.11)
IIRA(A)II < E'
since p(A) is open, and therefore dist(n., Q(A)) > e. For any v E N, set u = RA(A)v. Then, u E D(A), and (5.11) implies Ilull < E111V11'
so that
(lull < e ' ll(A  ),)u 11,
for all u E D(A). This contradicts (5.10).
We have seen that, in general, the spectrum of an operator on an infinitedimensional space consists of much more than eigenvalues (see the end of Chapter 4). In fact, in many cases there are no eigenvalues at all. This is in stark contrast
to the case of operators between finitedimensional spaces. One may ask for a characterization of the elements of the spectrum which are not eigenvalues. We present one such characterization here for selfadjoint operators, and we will return to a similar characterization when we discuss the essential spectrum, in general, in Chapter 7. The following theorem, which states a version of Weyl's criterion for the spectrum of selfadjoint operators, may be interpreted as stating that any A E Q(A) is an approximate eigenvalue: Given any e > 0, there exists u E D(A) such that 11(A  X )u 11 < Ellull
Theorem 5.10. Let A be selfadjoint. Then A E Q(A) if and only if there exists a sequence {un }, un E D(A), such that 11 un II = 1 and 11(A A)un 11 + 0 as n + oc.
Proof. (1) Let A E Q (A ). Two cases arise:
(a) ker (A  A) 7' {0} (i.e., A is an eigenvalue). Then let un = f for any f E ker (A  A) with II f 11 = I I.
(b) ker (A  A) = {0}. Then Ran(A  A) is dense but not equal to 71, so (A  ),)' exists but is unbounded. Consequently, there exists a sequence { vn }, vn E D((A  i)
1 ).
II vn II = I such that
11(A Ay vn11 + CO.
Define un = [(A  X)'un]II(A  A)'vnll'. Then u E D(A), 11un11 = 1, and
11(A ),.)un11 = Ilvnll 11(A ),.)'vnil' > 0,
as n * oc.
55
5.3 Determining the Spectrum of SelfAdjoint Operators
(2) Conversely, let ,l E p(A). Then there exists M > 0 such that for any u E N, IIRA(A)ull < M11u11.
Let u = RA(A) 'v for v E D(A) so that (lull < M11(A  ;,)vll, and thus no sequence having the properties described can exist. (Remark: We proved the contrapositive of the "if" part of the theorem.) We finish this section with two results about specific types of selfadjoint operators.
Definition 5.11. An operator A is positive, A > 0, if (u, Au) > O forall u E D(A).
Proposition 5.12. Let A be a selfadjoint operator. Then A > 0 if and only if a(A) C [0, oo). Problem 5.5. Prove the "only if' part of Proposition 5.11.
Completion of proof of Proposition 5.12.
If a(A) C [0, oo), then for any a > 0, a E p(A) and dist (a, a(A)) > a. Hence, by Theorem 5.8,
II(A+a)ull ? allull. This implies that a211u1122 < II(A+a)u112 = 11 Au 112+2a(Au, u) +a211u1122,
(5.12)
(Au, u) ? (2a) ' I[Au112.
(5.13)
or
In deriving (5.12), we used the symmetry of A. The result, (Au, u) > 0, follows from (5.13) since a > 0 is arbitrary. For bounded selfadjoint operators, the norm of the operator can be expressed in terms of the "size" of the spectrum. Definition 5.13. Let A be a bounded, selfadjoint operator. The spectral radius of A, r(A), is defined by r(A) __ sup (IAI IA E a(A)}.
Theorem 5.14. Let A be a bounded, selfadjoint operator. Then 11A II = r(A) = sup ([Al IA E a(A)}.
56
5. SelfAdjoint Operators
Proof. (1) We first show that r(A) < IIAII If IzI > II All, then 11 (A  z)ull >(IzI  IIAII)llull, and so z E p(A) and Q(A) C {A.I ICI < IIAII). Thus, r(A) < IIAII. (2) Conversely, let Xo = II All. By an extension of (4.2) valid for selfadjoint oper(Au,,, un) = ators, there exists a sequence {un}, llunll = 1, such that We then have
2
2),,0
 2a.o(Aun, un) * 0,
and so II(A  a.o)un II ± 0. By Weyl's criterion, Theorem 5.10, Xo E Q(A). Hence, 11 All < r(A).
5.4
Projections
Definition 5.15. A bounded operator P is called a projection if P2 = P. If in addition, P* = P (i.e., P is selfadjoint), then P is called an orthogonal projection. Projections are the building blocks of selfadjoint operators. The spectral theorem (which we will not discuss in these chapters) associates with each selfadjoint operator a family of projections that completely determines the operator (see, for example, [RS I]). More basically, projections are intimately related to the subspaces of a Hilbert space, as we describe in the next theorem. Let us note that u E Ran P, a projection P, if and only if Pu = u.
Proposition 5.16. Let P be an orthogonal projection.
(1) Ran P is a closed subspace of 7l, ker P is orthogonal to Ran P, and
1l=ker P® Ran P. (2) If M is a closed subspace of N, then there exists a unique orthogonal projection P such that Ran P = M. Thus, there is a onetoone correspondence between closed subspaces of 7l and orthogonal projections. The geometry of the family of closed subspaces can be stated in terms of properties of orthogonal projections.
Proof of Proposition 5.16. (1) To show that Ran P is closed, suppose {yn = Pxn } is a sequence in Ran P and yn > y. Then by the remark preceding the proposition, Py, = y,,, and so Py = y and y E Ran P. The rest of the statement follows from the result Ran A ® ker A* = H, but it is also easy to prove directly using the facts that
if P  1  P, then P is an orthogonal projection, P + P = 1, and P P = 0.
5.4 Projections
57
(2) If M is a closed subspace of ?C, let M1 be its orthogonal complement so that M ® M1 = R. Hence any x E N has a unique representation x = xi + x2
with x I E M, X E M'. Define an operator P by
Px =x1. Then P is bounded as II Px II = Ilx111, and P2 = P is obvious. To show that P is symmetric, let x = xi + X2, let y = yi + Y2, and compute
(Px,Y) = (XI, Y1 +Y2) = (XI, Yl) = (XI, PY) = (x, Py), using the orthogonality of x1, y1 with x2, Y2.
6
Riesz Projections and Isolated Points of the Spectrum
In this and the following chapter, we will discuss the spectrum of a closed operator A. We know that we can form a disjoint decomposition of a(A) as ad(A) Uaess(A). The discrete spectrum of A, ad(A), consists of isolated eigenvalues of finite algebraic multiplicity, and aess(A), the essential spectrum of A, is the remaining part of the spectrum. We will study ad(A) in this chapter through the projection operators that can be obtained from the resolvent of A for each distinct eigenvalue in ad(A). We develop the basic theory of these projections, called Riesz projections. These operators provide a powerful tool for the study of the discrete spectrum of closed operators. The Riesz projections have the additional property that when the closed operator
A is selfadjoint, they are orthogonal and project onto the subspace spanned by the eigenfunctions of A for a given discrete eigenvalue. In the last section of this chapter, we discuss the projection onto the eigenspace of a selfadjoint operator corresponding to an eigenvalue embedded in the essential spectrum. Although a Riesz projection cannot be constructed in this case, we still obtain a representation of the projection in terms of the resolvent. We will discuss aess(A) in the next chapter.
6.1
Riesz Projections
Let A be a closed operator on a Banach space. Let .lo E a(A) be an isolated point of the spectrum, and let 1';,o be a simple closed contour around .lo such that the closure of the region bounded by i'XO and containing Xo intersects a(A) only at ao (see Figure 6.1). We refer to such a contour as admissible for Xo and A.
60
6. Riesz Projections and Isolated Points of the Spectrum
C
FIGURE 6.1. An admissible contour
for a Riesz projection.
Consider the following contour integral:
ti
2Tr i
(6.1)
r
Since RA(s) is analytic in a neighborhood of I'xo, the integral exists as a uniform limit of Riemann sums. Approximating the contour by a union of straightline segments Di so that Ur_] Ai is a closed polygonal contour containing k0, the Nth Riemann sum is N
Pxo =
RA(Aj)O;,
(27ri)i
where ;,i E Ai. One can then show that n  limN_.+ Pxo exists as a bounded operator.
Problem 6.1. Define the Riemann sums Pxo corresponding to the contour integral (6.1) and show that limN,,,,, P" exists. (Hint: Show that { Pxo) is a norm Cauchy sequence.) Alternately, we can use the analyticity of RA(A) to first define the integral in the weak sense. For any u E X and I E X *, define
I(Px0u) _ (27ri)l
(6.2) ray
Note that l(RA(,k)u) is analytic on the resolvent set of A. By a simple calculation, there exists a constant M > 0 such that II(Pxou)I
If a. E ad(A), then by Proposition 6.6, (A  A) acting on [ker(A  %)[1 has a bounded inverse. As ; is an isolated point in a(A), by Proposition 6.4, A is an eigenvalue, and so ker(A  A) contains more than the zero function. The finite dimensionality of ker(A A) follows from the definition of ad(A). (2)
The subspace ker(A  k) is a closed subspace of 7l because it is finitedimensional. Let P be the orthogonal projection onto ker(A  A), and let & _ (A  n.)I[ker(A  A)}1. Then A;, has a bounded inverse (on [ker(A A)]1) if and only if 0 E p(A;,). Since the resolvent set is open, there exists a neighborhood W of 0 such that W C p(AA). For z ' A but close to ,l such that i  z E W, (Ax  (A  z)) has a bounded inverse. Hence we define an operator on 7l, via the decomposition (6.11), by
R(z) = (,  z)1 P ® (A), + (z 
(6.12)
where the first component acts on ker (A  A), and the second component acts on [ker (A  A)]1. Note that R(z) is a bounded operator. We claim that R(z) is an inverse for (A  z). Once we establish this, the theorem will be proved. To prove this claim, we compute
(A  z)R(z) = (),.  z)`' (A  z)P ® (A  z)(A), + z 
= P ®(A  z)(A  z)'(1  P)
= P®(1P)=1, where we used the invariance of P7l and the definition of P. We conclude from this that z E p(A), so (W + A) \ (Al C p(A). This means that % is an isolated point of a(A). By Proposition 6.4, A is an eigenvalue of A, and so A E ad (A).
Corollary 6.8. Let ker(A  A) be finitedimensional. Then a. E aess(A) if and only if (A  A)I[ker(A  ).)]1 has an unbounded inverse.
66
6. Riesz Projections and Isolated Points of the Spectrum
FIGURE 6.3. Contour for PQ, where a(A) = a, U ay.
6.3
More Properties of Riesz Projections
We now develop some more properties of the Riesz integrals defined earlier. Let us suppose that for a closed operator A, a(A) decomposes into two disconnected components of and o2 (both nonempty). Let us suppose that or, is bounded. Because these are closed and disconnected, we can enclose or,, say, by a simple closed curve I', with positive orientation. Hence we can associate a Riesz integral with or, Pa, _ (2.7ri)1
ir
(6.13)
R A(z)dz ,
(see Figure 6.3).
Proposition 6.9. Let A be closed and as described above. Let Pa, be as defined in (6.13). Then
(1) Pa, commutes with A on D(A); (2) o(APa1) = o, and o(A(1  Pa, )) = Q(A) \ Q,.
Proof. The first statement follows from the fact that A and RA(z) commute on D(A). To prove the second, we first show that o(AP,,,) C o,. Since APa1 is a restriction of A to an Ainvariant subspace, p(AP0,) D p(A). To see this, note that the restriction of RA (z) to Ran Pa1 is the resolvent of the restricted operator A Pa1, and so the inclusion follows. Now to get a better estimate on p(AP0, ), take any z
outside I', (see Figure 6.3) (z may be in a2) and compute RA(Z)P0, =
(27ri),
ir R A(z)RA(w)dw ,
= (2ni)l [#(z  w)1 RA(z)dw ,
irI
(z  w)RA(w)dw]
,
(6.14)
where we used the first resolvent identity. The first integral in (6.14) vanishes since z is outside of I',. It is easy to check that the second integral is analytic in z outside
6.4 Embedded Eigenvalues of SelfAdjoint Operators
67
of F1. Hence RA(Z)P,, = (z is analytic on C \ (Int I71) (the closure of the region bounded by I'1). By shrinking FI close to aal, the boundary of Q1, and APPS)1
using the openness of the resolvent set, we see that RA (z)PQ, is analytic on C \ at.
In a similar way one shows that a(A(1  P 1)) C a(A) \ at. Problem 6.4. Complete the proof of Proposition 6.9.
6.4
El
Embedded Eigenvalues of SelfAdjoint Operators
Let A be a selfadjoint operator on a Hilbert space 11. An eigenvalue ), of A with multiplicity mx > I is called an embedded eigenvalue of A if), is not isolated in the spectrum of A. In this case, the projection Px for the mxdimensional eigenspace £x, spanned by the eigenvectors {t4r; i = 1, ... , mx) for A and A, cannot be obtained as a Riesz projection as discussed above for isolated eigenvalues. This is I
because there is no simple closed contour r C p(A) such that the only point of the spectrum of A in the interior of F is X. We can, however, obtain an expression for the projection onto the eigenspace in terms of the resolvent of A. We will use this representation in Chapter 16 when we discuss the AguilarBalslevCombesSimon theory of resonances. As we will see in Chapters 1623, embedded eigenvalues play an important role in the theory of quantum resonances. Unlike isolated eigenvalues, which are relatively stable under perturbations, as we will see in Chapter 15, embedded eigenvalues generically disappear from the spectrum of a selfadjoint operator under perturbations (see [FrHe] and [AgHeSk] for additional discussion and references). Actually, they do not disappear but, as we will see, they become resonances of the operator. Since the operator A is selfadjoint, the projection Px, corresponding to an embedded eigenvalue of A, is an orthogonal projection. We will denote the projection
orthogonal to Px by Qx = 1  Px. Theorem 6.10. Let A be a selfadjoint operator on the Hilbert space 11 with an embedded eigenvalue X. The projection Px onto the eigenspace £, is given by
Px = s  lim c (ie)(A  .  iE)1. E
(6.15)
Proof. Let us define PE  (ie)(AAie)1.ByCorollary 5.7,this isabounded operator as long as c
0 and, IIPEII
_< I.
For any c ' 0, we have (6.16)
68
6. Riesz Projections and Isolated Points of the Spectrum
since £x is an Ainvariant subspace and A I £x = A. For any * E N, a simple calculation, using the selfadjointness of A, shows that II PC Qx
112
=
(Q),i,E2[(AA)2+E21'QXf)
(6.17)
;,)2 [(A  A)2 +e2] Qxi) IIQA'I(2  (Qxi, (A 
Since ker(A  ),)I QAN is empty, we know that for all* ¢ £x, (A  .)QA 5` 0 (i.e., (A  ),) I QxH is invertible). It is clear that the vector (A  ,l) I Qxi E D(((A  .) I QxN) 1), and so it follows that lim 11(A
11 = IIQxfII,
Eo
(6.18)
for any r ' E. Thus, equations (6.18) and (6.17) imply that Jim
IIPEQX
E>o
II =0,
(6.19)
for any >G E 7l. By the orthogonality of the projections Px and Qx, it follows from (6.16) and (6.19) that
s  E 0
JimPE=fI
0
on &, on Qx7L.
(6.20)
This shows that the strong limit is an orthogonal projection and equal to Px. Problem 6.5. Show that the formula for Px is independent of the sign of E.
El
The Essential Spectrum: Weyl's Criterion
We continue to discuss the general properties of the spectrum of a closed operator
A. Here we will require that A act on a Hilbert space R. In the general Banach space setting, recall that we decomposed or (A) into two disjoint parts: ad(A) __ the "discrete spectrum " of A, which is the set of all isolated eigenvalues of A with finite algebraic multiplicity; cress(A) = the "essential spectrum" of A, which is simply given by a(A)\ad(A). Properties of ad(A) were studied in Chapter 6. The set a(A) \ ad(A) is called the essential spectrum because, unlike other subsets of a(A), it is stable under relatively compact perturbations of A. This result, called Weyl's theorem, is discussed in Chapter 14. The main goal of this chapter is to develop a convenient characterization of QeSS(A). In particular, given ), E a(A), we want to know whether
A E aess(A). A clue for such a characterization in the selfadjoint case can be found in Theorem 5.10. Elements in a(A) are approximate eigenvalues and can be characterized by the existence of sequences of approximating eigenfunctions.
7.1
The Weyl Criterion
Let us recall the following characterization of points in a(A), where A is a selfadjoint operator on a Hilbert space 7(:
(a) ,l E a(A) if and only if 31u,) c D(A), Ilu, II = 1, such that
II(A 
A)u,, II = 0;
(b) A is an eigenvalue of A with finite multiplicity if and only if there exists a finite number of linearly independent functions u; E D(A) such that (A  A)u; = 0;
70
7. The Essential Spectrum: Weyl's Criterion (c) ;, is an eigenvalue of A with infinite multiplicity if and only if there exists
a sequence of linearly independent functions {ui } C D(A) such that (A .k)ui = 0.
Of course, both (b) and (c) are of the form in (a). In (b), if we form an infinite sequence, the functions will not be linearly independent. We will use this fact to separate the ad(A) from Q(A). In case (c) it is possible, by the GramSchmidt procedure, to choose the elements of the sequence to be orthonormal:
M. uj) = Sij. Consequently, if v c H. then Bessel's inequality implies that
a Y, I(V,ui)I2 N, II(A  A)u II < c. This idea is familiar from quantum mechanics. The operator p = i(d/dx) on L2(R) represents momentum and is selfadjoint (we will show this later). Physicists commonly say that the plane
wave e'kx, k E IR, is an eigenfunction of p with eigenvalue k since we have the eigenvalue equation pe`kx = ke`kx. However, the function e`k' V L2(R). We can form wave packets Vi, (x) from functions f(q) E L2(18) which are supported in intervals about k of decreasing length using the Fourier transform: je1fn(q)dq.x
(7.2)
These functions are in L2(R) and approximate e'k' as n  oo. In fact, it is not too hard to show that these form a Weyl sequence for p and k. One might wonder about functions of the form e''x for z E C, Im z 0, since pe'`x = ze'zx also. But p is L2(R) and, because of the exponential selfadjoint, so z ` a(p). Luckily, e'?a increase of these functions, it is impossible to construct a Weyl sequence as in (7.2). In fact, no Weyl sequence exists for p and z, with Im z 0. `
7.2
Proof of Weyl's Criterion: First Part
(=) Let % E aeSs(A). If ker(A  ),) is infinitedimensional, then form a Weyl sequence from any infinite orthonormal basis for ker(A  A.). By Remark 7.1, any infinite orthonormal set converges weakly to zero. If ker(A  A) is finitedimensional (including the case that ker(A  A) = {0}), we consider [ker(A  A)]1 and use Theorem 6.7, which characterizes ad(A). First we need a lemma.
Lemma 7.3. Let A be selfadjoint. Then
(i) (ker A)1 is an Ainvariant subspace, that is, A : (ker A)1 fl D(A) > (ker A)1; (ii) the restriction of A to (ker A )L has trivial kernel {0} and is selfadjoint.
Problem 7.2. Prove Lemma 7.3. For part (ii), use the fundamental criterion for selfadjointness: K, a closed, symmetric operator, is selfadjoint if and only if Ran(K f i) = R. (See Chapter 8, Theorem 8.3.)
Now let Ax denote (A  A)I[ker(A  ).)]1. The operator Ax is selfadjoint because .1 is real, and hence it has a trivial kernel by Lemma 7.3. By the results of Section 4 of Appendix 3, the operator (A  A) has an inverse. It follows from Theorem 6.7 that if ker(A  A) is finitedimensional, then A E aess(A) if AX' is unbounded. Hence (using part) there is a sequence {vn} C D(Az ') such that Athen IIvnII = 1 and IIAj 1vnII  cc as n  oc. Set un IIunII = l and
IIAxunIl = Ilvnll IIA_'vnll' , 0.
72
7. The Essential Spectrum: Weyl's Criterion
It remains to show that u w> 0. Now let f E 1l; then we have the decomposition
f = f l (Df2, with fi E ker(A )) and f2 E [ker(A ))]1 Since A' : RanA). D(AA) n [ker (A ,k)]L, we have
(f, un) _ (f2, un), and so it suffices to take f E [ker(A ,k)]L. Moreover, it is sufficient to prove that
(f, u,) + 0 as n ) oc for f in a dense subset of [ker (A  a.)]1. Problem 7.3. Prove that if (f, un) + 0 for all f in a dense set, then un
 0.
I)*) is dense in [ker(A .k)]L. Clearly, any element of the We claim that D((A,form A;, f, f E D(A;,), belongs to D((A 1)*) since, for any u E D(AA 1),
(Axf, A, 1u) = (f, u). Now Ran Ax is dense, for suppose 3u E [ker(A  A)]1 such that for all v E D(AA)
(u, Axv) = 0. This implies that u E D(Ax) (since & is selfadjoint) and that A),u = 0. (This fact also follows from the fundamental criterion for selfadjointness, Theorem 8.3.)
But by Lemma 7.3(b), ker & = {0}, so u = 0. Hence Ran A;,, and therefore D((Aj 1)*), is dense. So, letting f E D((Aj I)*), we compute
I(f,un)I
=
I((Ax')*f vn)IIIAX'vnIl1
0, and so Q(A) C [0, oo) by Proposition 5.12. We use Weyl's criterion to show that if A > 0, then k E oess(A). To this end, we construct a sequence of approximate eigenfunctions, called Gaussian wave packets, concentrated at A. This construction is similar to the construction of approximate eigenfunctions discussed in Section 1 of this chapter. For example, we consider functions u,,, defined by um(x) _ (2jT ) 2 f um(k)eik xdk,
where k x
(7.6)
k;x;, and
um(k) =
(27rm)2e_2kkol2,
11k0ll2 = A.
(7.7)
This integral is absolutely convergent and defines a function in S(1W ). Notice
that u,,,(k) is a Gaussian function strongly localized around ko and that the root mean width decreases as m  I2. (2) We claim that {u", } in. (7.6) and (7.7) is a Weyl sequence for A and k. We compute the normalization: IIum II2 =
p. 112
= (2.7rm)"
=2in"S2"
f
f
e"tun'du
e2m211kkoll2dk
= 1,
where Q" is the area of a unit sphere in n dimensions. Next, we show that u" 0. Let f E S(111"), the dense set of Schwarz functions. Then by the Plancherel theorem, Theorem A4.2, and (7.7), we have (27rm)z f em2Ilkko112 f(k)dk
(um, f)
(?i
l
n
/f
eIlkll2 f\ k +k0 1 dk,
where we changed variables from k to m(k  k0). Now f E S also (see Appendix 4), so limm.x, f (k/m + k0) = f (k0). Consequently, we find
(um, f)I 0, independent of f. By Problem 7.3, [um } converges weakly to zero.
75
7.3 Proof of Weyl's Criterion: Second Part
(3) Finally, we show that (A  ),)um 4 0. By the Fourier transform and the smoothness of um,
((0  A)um)(X) _ (270) , f (Ilkil2 
),)um(k)e,k'.rdk
and by (7.7):
ll(oA)um112
=
II(IIk112A)um112 A)2e2,n2jIkkoll2
dk
=
(2)T m)" f (Ilk112 
=
(22 n)" f e11k112 (l(21m)t k+ko12),)'`dk
=
(22 n)" f
e11k112
((2m 2)1
+22mtk ko)2dk, 
Ilk
112
(7.8)
where we used the new variable k  21/2m(k  ko) and the fact that X = ko. It follows directly by the superexponential convergence of the integral in
(7.8), that the integral in (7.8) converges to zero like 1/m as m + 00. Hence, {um [ is a Weyl sequence for A and .l, and by the Weyl criterion, a. E oess(A). Thus (0, oo) C cress(A), so as the spectrum is a closed set, [0, oo) C a(A). These imply that
v(A) = Qess(A) = [0, 00), because zero is not an isolated point of the spectrum.
Remark 7.7. It follows from the previous calculations that the Laplacian, as a selfadjoint operator on the Hilbert space L2(R'), has no eigenvalues. Suppose that A has an eigenvalue at A > 0. Any corresponding eigenfunction' has to satisfy
(Ilk 112))f=0. This means that supp iff = {k I IIkII = the sphere of radius k. But, >%i E LZ(Rn), and so i = 0. This follows from the following simple version of elliptic regularity (see Theorem 3.8).
Proposition 7.8. If u E L2(W') satisfies Au = Au, for some k > 0, then u is infinitely differentiable.
Proof. It follows from the eigenvalue equation that if u E H2(RI), then u E H4(R" ). Iterating in this manner, it follows that u is in any positive indexed Sobolev space. So, by Theorem A4.6, the function u is infinitely differentiable.
Problem 7.5. Show that Proposition 7.8 and the preceding analysis imply that any eigenfunction * of the Laplacian must be zero.
8
SelfAdjointness: Part 1. The Kato Inequality
We will now concentrate on the class of selfadjoint operators in a Hilbert space R. Our first task will be to develop criteria that will allow us to determine which operators occurring in applications are selfadjoint. Then we will apply this to prove that Schrodinger operators with positive potentials are selfadjoint. After discussing in Chapters 11 and 12 the semiclassical analysis of eigenvalues for Schrodinger operators with positive, growing potentials, we will return to the question of selfadjointness in Chapter 13 and present the KatoRellich theory.
8.1
Symmetric Operators
Let us recall from Chapter 5 that an operator is selfadjoint if it is equal to its adjoint. A necessary condition is that the operator be symmetric. In Definition 5.1, we gave two characterizations of a symmetric operator. We recall here the most computationally convenient one.
Definition 8.1. An operator A on a dense domain D(A) is symmetric if for all u, v E D(A), (Au, v) = (u, Av). It is usually easy to check whether a given operator is symmetric.
Examples 8.2.
(1) Multiplication operators. Let V E L2(R") + L°O(R") and be real, that is, V can be decomposed as V = V1 + V2, with V1 E L2 and V2 E L. We
78
8. SelfAdjointness: Part 1. The Kato Inequality
denote by the same letter V the linear operator "multiplication by V," that is, (V f)(x)  V (x) f (x). Let Co(li8") be the space of all continuous functions
on W with compact (bounded) support. We define an operator V on the dense domain Co(R") in L(R") by V : f E Co(R'1) > V f.
(8.1)
Note that V f E L2(W1) as IIVfII2
< IIVif112+IlV2fII2 (8.2)
IIfIIcc 11V1112+IIVZIIocIIf112 Re[(sgn u)Du],
We then have
(8.8)
except where I u I is not differentiable.
Proof. Observe that uE > Iu1. Then from (8.7), if we differentiate uE = Iul2 + E2, we get u, V uE = Re u V u. (8.9) Squaring this and using the observation gives IV uEI < uE' Iul IV ul < IV UI.
(8.10)
82
8. SelfAdjointness: Part 1. The Kato Inequality
Next, take the divergence of (8.9), to obtain
IVuEI2+ufouE=IVu12+Reuou. By (8.10), this is equivalent to
u, Au, > Re uAu.
(8.11)
Let sgnEu  uuE so that (8.11) is Au, > Re[(sgnEu)Du].
(8.12)
Since Au, +A lu I pointwise and sgnEu > sgn u pointwise, we obtain the result by taking c > 0 in (8.12).
Problem 8.5. Verify the calculations in the proof of Lemma 8.7, especially the claims concerning the pointwise convergence at the end of the proof. We remark that one can also prove (8.8) by straightforward differentiation of Jul and by using the positivity of resulting terms to get a lower bound. Our next goal is to extend (8.8) to a more general class of functions. For this we need a few technical tools.
8.4
Technical Approximation Tools
Let us recall some notions from Appendix 4 about operations in the distributional sense. For this we need to introduce some new function spaces.
Definition 8.8. For any I < p < oo, we define a local LPspace by
L oC(R") = S .f I f If (x)IIdx < oc, for any bounded S2 C R" l
}
.
lJJ
We have LP(R") C L c(R"). We do not need any other structure on L OC(1R"). Note that if f E Lj1o,(11 ) and g E Co (R"), then f f g converges by the Holder inequality, Theorem A2.4. Definition 8.9. Let 0 E L11.C(R" ), and let (f, g)  f f g.
(1) A function V E LL is the distributional derivative of 0 with respect to x;
(formally, i = a0lax;) if (
for all f E Co'(R" ).
,f)=(0,(ax;)
8.4 Technical Approximation Tools
83
(2) Let 0n, 0 E LiOC(1R" ). Then 0n converges to 0 in the distributional sense if
(0., f)  (0, f) for all f E Co (I[8" ). (3) Let (p, i/i E L,1OC(R"). Then 0 > iti in the distributional sense if ((P, f) >_ (>[i, f)for all positive f e CO0°(II8" ).
Problem 8.6. Prove that distributional derivatives and distributional limits are unique. Next we need the notion of an approximate identity.
Definition 8.10. Let w E C°°(118"), w > 0, and f w(x)dx = 1. For S > 0, we define ws(x)  3"w(8 1x). Note that then f ws(x)dx = 1. We define a map Is by
Isu ws*U,
(8.13)
whenever the right side exists, and where (f * g)(x) = f f (x  y)g(y)dy is the convolution of g and f. The map Is is called an approximation of the identity, or simply, an approximate identity. Lemma 8.11. Let Is be an approximation of the identity.
(i) If u E Li.C(W'), then Isu E C°O(W'). (ii) For any differentiable function u,
ax; /
ax;
that is, the map 1s commutes with a/ax,.
(iii) The map Is : LP(R")  . LP(Rn) is bounded, and II Is II < 1. (iv) For any u E LP(W1), lims,o 11 16u  u11 p = 0. (v) For any u E LIOC(Rn), 18u + u in the distributional sense as 8 * 0.
Problem 8.7. Prove (i)(iii). (Hint: Use Young's inequality, Theorem A4.7, for (iii).) Proof of (iv) and W. We will show that sup,,R'l I (Isu)(x)  u(x)I > 0 as S > 0 for any u E Co (1[8" ). Statement (iv) then follows from (iii), the density of Co (R") in LP(II8"), and the inequality
Illsu ullp < (supl(lsu)(x)u(x)I)V(u), x
where V(u) = fSUPp(u) d"x. Furthermore, result (v) follows from the identity
84
8. SelfAdjointness: Part 1. The Kato Inequality (.f, 16 U) =
(1),x(1sf, u)
and the bound
I(f (1)"I sf,u)I
f),
x
where C(u, f) = fSupp(f) lu(x)Idx, for any f E Co'(R"). Now, turning to the proof, since f w(y)dy = 1, we have for any u E Co (l8"),
(Isu)(x)  u(x) = J wa(x  y)fu(y)  u(x)]dy. Divide the region of integration in (8.14) into two parts: IIx  yll
vr3. For the first, we have
IlxYII Re [(sgn u)Du] in the distributional sense.
(8.17)
8.6 Application to Positive Potentials
85
Proof. Let U E Lioj][l;"). By Lemma 8.11, Isu is smooth for any 8 > 0. Inserting Isu into (8.12) in place of u, we obtain for any E > 0,
A(Isu)E > Re[sgn, (Isu)A(Isu)].
(8.18)
We must remove the two cutoffs in equation (8.18). We leave it as Problem 8.9 to show that there exists a subsequence of sgnE(Isu)A(1su) which converges pointwise to sgne(u)Au, except possibly on a set of measure zero. Since Au E L110(W'), it follows from Lemma 8.11 that the limit in LioC(R") as 8 + 0+ of A(Isu) is Au. It follows from this and the boundedness of sgnE(1su) that the distribdtional limit as 8 + 0+ of sgnE(1su)[A(1su)  Au] is zero. These two facts, and the Lebesgue dominated convergence theorem, Theorem A2.7, suffice to prove that there is a subsequence such that sgnE (18u)A(Isu) converges to sgnE uAu in the distributional sense. Taking this subsequential limit in (8.18) yields
Au, > Re[(sgn, u)Au].
(8.19)
Finally, we take E + 0 in this equation to obtain the result.
Problem 8.9. Use the Li c(1 0. Then, the Schrodinger operator H = A + V is essentially selfadjoins on Co (W' ). Proof. We will prove in Lemma 8.15 a slight extension of the fundamental criteria,
Theorem 8.3, by which it is sufficient to show that ker (H* + 1) _ {0}. Since D(H*) C L2, the triviality of the kernel is implied by the statement
If  Au + Vu +u = 0, u E L2, then u = 0. We prove (8.20) by Kato's inequality. We note that u E L2 and V E by the Schwarz inequality, that uV E LL . Since we have the inclusion,
(8.20) imply,
2 L2CL10CCLL, 1
which follows from the estimate
f2 lu(x)I. I < V(2)[ f Iu(x)12]', where V(S2) is the volume of 0, we have u E L oc. Hence by (8.20), Au E Ll." where the derivative is a distributional derivative. From (8.17), we obtain Re [(sgn u)Du] Re [(sgn u)(V + 1)u I
lul(V+1)>0.
(8.21)
Hence, the function A Iu I > 0, and by Lemma 8.10,
Alslul = /soIul > 0.
(8.22)
On the other hand, 16 1 ut E D(A), and therefore
(DUalul), (/slut)) = IIo(Ialul)II2 < 0.
(8.23)
By (8.22), the left side of (8.23) is nonnegative, and so V(/slul) = 0 (in the L2sense) and hence Ia l u l = c > 0. But lu l E L2 and !a lu l  lu l in the L2sense (by Lemma 8.10), and so c = 0. Hence, /a l u I = 0, so J u I = 0 and u = 0.
Lemma 8.15. Let H be a closed, positive, symmetric operator. Then H is selfadjoint if and only if ker (H* + b) = {0} for some b > 0. Similarly, if H is a positive, symmetric operator, then the closure of H, H**, is selfadjoint if and only if this condition holds.
8.6 Application to Positive Potentials
87
Proof. (1) We can, without loss of generality, take b = 1. We repeat the same arguments as in the proof of Theorem 8.3. (a) Ran(H + 1) is closed. Let un E Ran(H+ 1) form a convergent sequence. There exists a sequence If, } C I)(H) such that un = (H + l) f,,. Then IIfftI2,
and so by the Schwarz inequality,
Ilfnli  IIunV.
(8.24)
Since u, * u, the set {un } is uniformly bounded (i.e. sup,, Ilun II < oo) and so (8.24) implies that sup,, II fn II < oo. Now, by positivity, II
fn
 fm II2
<
0, then ker(H + b) = {0}, El for any b > 0, since o(H) C R+; see Proposition 5.11.
9
Compact Operators
A compact operator is, loosely speaking, an operator that behaves as if it were almost an operator on a finitedimensional space. These operators form a very important class of operators, and a great deal of spectral analysis is based on them. We shall see that compact operators have a very transparent canonical form. Consequently, the spectral properties of a selfadjoint compact operator mimic those of a symmetric matrix as closely as possible. Moreover, the Fredholm alternative,
Theorem 9.12, is a fundamental tool in the solution of partial differential equations and in solving operator equations. Compact operators also enter the theory of Schrodinger operators and, more generally, partial differential operators, through the notions of local compactness and relative compactness. These important tools are discussed in Chapters 10 and 14, respectively.
9.1
Compact and FiniteRank Operators
We will use a streamlined notion of a compact operator. The standard definition is that an operator K on a Banach space X is compact if for every bounded set
N c X, KM has compact closure. We refer the reader to Kato [K] for more details. The following definition is equivalent to the usual one for reflexive Banach spaces and, in particular, for Hilbert spaces. Definition 9.1.A bounded linear operator Kon a reflexive Banach space X is called
compact if it maps any weakly convergent sequence into a strongly convergent sequence.
9. Compact Operators
90
This means that if {xn } is a sequence and x w* 0, then Kx II Kxn 11
0, that is,
* 0. Recall that x  0 if for all bounded linear functionals f on
X, f(xn)  0. Example 9.2. An integral operator K defined on C([0, 1]) is determined by a kernel function K(x, y) by
(Kf)(x) =
f
1
K(x, y)f(y)dy,
0
for all f such that the integral exists. This operator can be extended to a bounded operator on L22[0, 1] when K(x, y) is continuous in both variables. This follows from the boundedness of the function K and the Schwarz inequality.
Proposition 9.3. If the kernel K(x, y) is continuous on R  [0, 1] x [0, 1], then K is compact on L2[0, 1].
Proof. Let f,  0. Then, by continuity of the kernel, 1
ZI(x) = (Kfn)(x) = fo K(x, y)fn(y)dy  0,
(9.1)
that is, it converges pointwise to zero on [0, 1], since, for each x, the integral defines a bounded linear functional on L2[0, 1]. Since [0, 1] is compact and K is continuous, it is uniformly continuous in x. Thus, given E > 0, there exists S > 0 such that max I K(x, y)  K(x', y)I < E,
yE[o,i]
(9.2)
whenever Ix  x'I < 3. Next, divide [0, ]]into [31]+ 1 (greatest integer less than 31) intervals Ai of length < 3. Let x, be the middle of i, . Because of (9.1), we can choose n such that maxi IZn(xt)I < e. Then, sup lIn(x)I
0, and similarly for KB since Bun i 0.
92
9. Compact Operators
To prove part (c), we need a technical device, called the polar decomposition, which is extremely important in its own right. We refer the reader to the standard literature, for example [RS I ], for the proof. Recall from Chapter 5 the notion of a positive operator:
Definition5.11.AnoperatorAispositive(A > 0)if(Af, f) > Oforallf E D(A). In connection with positive operators, we have the next definition. Definition 9.6. A positive operator B is called the square root of a positive operator A if B2 = A. In this case, we write B = A112. If A is bounded, the absolute value of A is defined by Al I= (A*A)1/2. The main result we need is the following.
Theorem 9.7 (The polar decomposition). For any bounded operator A, there exists a bounded operator U such that A = U I Al. The operator U is a partial isometrv, that is, U is an isometry when restricted to (kerA)1.
Proof of Theorem 9.5 (c). We prove that K is compact if and only if IKI is compact. To see this, note that 11
1 K I f II
= II Kf II. Thus if K is compact, then I K I
is compact. Let K = UI KI be the polar decomposition of K. Taking the adjoint of this equation, we get K* = I K I U*. It follows from part (b) that K* is compact since U* is bounded and IKI is compact. We remark that there is a form of the polar decomposition theorem for a closed operator. It follows from Theorem 9.5 that k(H) is a twosided ideal (over (C) in the algebra C(N) and that it is closed under the operation of taking the adjoint. Next, we will show that this ideal is, in fact, normclosed.
Theorem 9.8. The norm limit of a convergent sequence of compact operators in G(7l, 7C) is compact.
Proof. Let { K, j be a normconvergent sequence of compact operators with limit K E G(7(, 7C). Let {xn } be a sequence weakly convergent to zero. We have
IIKxnll
0, choose M > 0 such that m > M implies 11K  Kn, II < E. Fix some mp > M. Now Kmoxn 4. 0, so choose N > 0 such that n > N implies II Kmoxn II
< E. Thus, for n > N and m = mp in (9.5), we get IIKxn11 <e(SUpIIxn11+ I),
(9.6)
and since xn 0, the sequence { ll xn 111 is uniformly bounded and so the result follows from (9.6).
Corollary 9.9. The set of all compact operators K(7l) is a normclosed ideal in L(7L) which is closed under the taking of the adjoint K  K*.
9.3 Spectral Theory of Compact Operators
93
We remark that 1C(H) is an example of C*algebra without identity since 1 1C(H) if dim N = oo. C*algebras play an important role in many areas of mathematical physics, notably in quantum field theory and statistical mechanics. One might wonder what role the finiterank operators play in determining IC(N, N'). We will see that they actually determine IC(7l, 71') (Theorem 9.15), but to prove this we must first develop some spectral properties of compact operators.
9.3
Spectral Theory of Compact Operators
The spectral theory for compact operators reflects the fact that these operators behave almost as if they were acting on a finitedimensional space. The following theorem is true for an arbitrary compact operator, but we prove it here for selfadjoint compact operators. Theorem 9.10 (RieszSchauder theorem). The spectrum of a compact operator consists of nonzero isolated eigenvalues of finite multiplicity with the only possible accumulation point at zero, and, possibly, the point zero (which may have infinite multiplicity).
Proof (Selfadjoint case). We prove that aess(K) C {0}. The theorem then follows from the disjoint decomposition a(K) = a l(K) U aess(K) and the following observation. If ),.o 0 is an accumulation point of eigenvalues of K, then Ap E aess(K) Let A E aeSs(K). Then by Weyl's criterion, Theorem 7.2, there is a Weyl sequence
(un} for K and k, that is, un  0, Ilunll = 1, and (K  ),.)u  0. Since K is compact, Kun .4 0, and so Aun  0. But Ilun II = 1, and so A = 0 and cress(K) C {01.
Corollary 9.11. If K is compact and 0 V a (K), then K is a finiterank operator.
Problem 9.3. Prove Corollary 9.11. (Hint: Use the boundedness of KI to prove that K has finitely many eigenvalues.) Compact operators have the useful property that one can state precisely when equations of the form
f+Kf=g
have a unique solution. We now state and prove the simplest of many variants of what is known as the Fredholm alternative.
Theorem 9.12 (Fredholm Alternative). Let K be a compact operator. (i) The equation f + K f = g has a unique solution for every g E N if and only
if 1 ' a(K) (i.e., if and only if f + Kf = 0, or f + K* f = 0, has no nontrivial solutions).
(ii) If l E a(K), then f + Kf = g has a unique solution if and only if g E [ker(l+K*)]l.
9. Compact Operators
94
Problem 9.4. Prove part (i) of Theorem 9.12.
Proof of Theorem 9.12(ii). Suppose  l E Q(K). (1) Suppose that g E [ker (K* + I)]l. Since a(K) is discrete (except, possibly at 0), the restriction (K + 1)I[ker(K + 1)]1  A has a bounded inverse by Theorem 6.7. Now a simple calculation shows that Ran A is closed and that
RanA = [ker(1+K*)]1.Consequently, the operator A' : [ker(1+K*)]1 + Iker(K + 1)]1 is abijection. Thus, if g E [ker(1 + K*)]L, then f  A1g is the unique solution off + Kf = g.
(2) Conversely, if g E ker(1 + K*), then from f + Kf = g, we have
IIgIl2 = ((I+K)f,g) =0, which is a contradiction, so f + K f = g cannot have a solution. There is an important class of integral operators that are compact and that afford a calculable characterization. Recall that an integral operator is determined by a kernel K(x, y). We may think of this as a function on 1[82".
Definition 9.13. An integral operator K with kernel K(x, y) is in the HilbertSchmidt class if and only if
[f IK(x, y)I2dxdy] < 00.
(9.7)
We remark that there is a more general characterization of a HilbertSchmidt operator using the notion of trace. The family of all HilbertSchmidt operators forms a closed, twosided ideal in C(H) like the compact operators. The HilbertSchmidt operators are one example of what are known as trace ideals. We refer to the excellent book by Simon [Sim4] for a discussion of these ideals and their many applications. All HilbertSchmidt operators are compact. We prove this for integral operators of the HilbertSchmidt type.
Theorem 9.14. Let K be an integral operator on L2(Rn) with kernel K(x, 11) E L2(R2n) (i.e., satisfying (9.7)). Then K is compact and IIKII
[fKx,Y2dxdY].
(9.8)
Proof. (1) By the Schwarz inequality for any x and f ,
f IK(x, y)f(y)Idy 5[
f I K(x, s)I2ds] 2
Ilf II,
from which (9.8) easily follows. This proves that K is bounded.
9.3 Spectral Theory of Compact Operators
95
(2) To prove that K is compact, we show that K is the norm limit of a sequence of finiterank operators. Let {0i } be an orthonormal basis for H. Then for
any f E L(R"),
f = (f Oi)Oi
(9.9)
andd
Kf
(9.10)
Let tlii  Klfii. Then we claim that E;_°i 11 *,112 < oc for
Y [f dxl KOi(x)IZJ fdxI(K(x .)2
fdx(I(K(x..))I2)
N+1
I(f,oi) I2 i>N+l
i?N+1
by the Schwarz inequality. Since by the Plancherel theorem 00
II.f 112
1(f,Oi)I,, i=1
we get
?N+1
IIKf  KN!II
II!II T Ilill2
96
9. Compact Operators
Now, as N * oo, ENJ II*tIl`' converges, and so the sum + 0 as N + oo. Consequently, given c > 0, there exists NE IIt II2 such that N > NE IIK  KNIT = supll(K  KN)fII IIfII f 'o
and so limNgoo II K  KN II = 0. By Theorem 9.8, K is compact.
9.4
F
Applications of the General Theory
We now apply the results obtained in the previous sections and prove some properties of compact operators. The first theorem states that the finiterank operators are normdense in the set of compact operators, a result anticipated at the end of Section 9.2. Theorem 9.15. The finiterank operators form a normdense subset of the compact operators (i.e., any compact operator can be approximated arbitrarily closely by a finiterank operator).
Proof in the selfadjoint case. Let K be a selfadjoint compact operator. If 0 V or (K), K is finiterank by Corollary 9.11. Hence we assume 0 E a(K). Fix any e > 0. Let aE = or (K) n {.l IXI > E}. By the RieszSchauder theorem, Theorem 9.10, aE consists of a finite number of isolated eigenvalues of K with finite multiplicities; see Figure 9.1. Let PE be the Riesz projection onto the eigenspace corresponding I
to aE,
PE = (27ri)'
RK(z)dz,
where 1'E is the union of two simple closed curves shown in Figure 9.1. Since K is selfadjoint, it follows from Theorem 6.3 that Ran PE is the span of the eigenspaces of K corresponding to all the eigenvalues in aE. This space is finitedimensional, and thus PE is a finiterank operator. Now K PE = PE K is a finiterank operator since it has finite range. We write K = PE K + (1  PE )K and show that II (1  PE )K II < e. But, by Theorem 5.14,
II (1  PE)KII < spectral radius (I  PE)K < E. Thus, K is approximated to within a by the finiterank operator PE K. By Definition 9.4, the finiterank operators have a particularly transparent form. In light of Theorem 9.15, compact operators are almost finiterank, so one may ask if there is a canonical form for compact operators similar to that for finiterank operators. Although we do not need this result for our later work, we include it for completeness.
9.4 Applications of the General Theory
97
(2)
("
FF
rF
rF=
FM
V
1F2)
FIGURE 9.1. Spectrum of K in Theorem 9.15.
Theorem 9.16. Let K be a compact, selfadjoint operator. Then there exists a complete orthonormal basis {0n } of 7l consisting of eigenvectors of K, that is, K(Pn = ),no, Problem 9.5. Prove Theorem 9.16. (Hint: Write 7(= 7l) ® N,, where N) is the span of all the eigenspaces of K and 712 = 7l .)
Theorem 9.17 (Canonical form). Let K be a compact operator on a Hilbert space R. Then there exist two orthonormal families {t/in}, {on} of vectors in H and positive real numbers {ltj }, with A j + 0, such that for any f E 71,
x Kf =T, ltj(f,
'j)O,.
j=)
and the sum converges in norm.
Proof. Since K is compact, so is K*K by Theorem 9.5. The operator K*K is a nonnegative, selfadjoint, compact operator, and so by Theorem 9.16, there exists a complete orthonormal basis { t/i j } for 7.1 consisting of eigenvectors for K * K such that
K*K*j with X j > 0 and lim j.. ,l j = 0. For each nonzero Xj, set A j =
A
1/2
and
K *j. Then, for any f = J:i (f, t/r) i/r; E 7l, we have
Kf = J:(.f, ii)Ki i
57, µi(.f. since K i/i j = 0 for any j such that K * K 1/i j = 0.
The numbers 14j) appearing in Theorem 9.17 are called the singular values of K. The final result we wish to discuss is the regularizing effect compact operators have upon the convergence of sequences of bounded operators. Let us recall that the family £(X) of all bounded operators on a Banach space X has three primary notions of convergence for sequences of operators.
98
9. Compact Operators
Definition 9.18. Let { B } be a sequence of bounded operators on a Banach space X. Then
(1) B '+ B, norm or uniform convergence, if II B  B II  0, (2) B  B, strong convergence, if for each f E X, II Bn f  Bf II
(3) B
* 0;
B, weak convergence, if for each l E X* (linear functionals on X)
and each f E X, II(Bnf)1(Bf)I * 0. Problem 9.6. Prove that uniform convergence implies strong convergence and that these both imply weak convergence.
Theorem 9.19. Let K be compact and let (B ) be a sequence of bounded operators
with B  B. Then K B,, _'+ K B and B K n* B K. Proof. For any e > 0, we decompose K as K = F + LE, where F is finiterank and II LE II < e. For any u, let N
Fu =
(u, Oi) fi i=1
be the canonical form of F. We first show that IIBnF  BFII we write
0. To see this,
N
IIBnF  BFII = sup IluII=1
BF)ujI
K B, and so that theorem is proved.
10
Locally Compact Operators and Their Application to Schrodinger Operators
10.1
Locally Compact Operators
The resolvent RH(z), Im z 0, of a Schrodinger operator H =A+ V on L2(R") is typically not compact (however, it usually is on L2(X), when X is compact). If RH(Z) is compact, then a(RH(z)) is discrete with zero the only possible point in the essential spectrum. Hence, one would expect that H has discrete spectrum with the only possible accumulation point at infinity (i.e., aess(H)) = 0). In this way, the a(H) reflects the compactness of RH(Z). It turns out that these properties are basically preserved if, instead of RH(z) being compact, it is compact only when restricted to any compact subset of R'. This is the notion of local compactness. From an analysis of this notion we will see that the discrete spectrum of H is determined by the behavior of H on bounded subsets of 1R" and the essential spectrum of H is determined by the behavior of H (in particular, V) in a neighborhood of infinity.
Definition 10.1. Let A be a closed operator on L2(R") with p(A) and let XB be the characteristic function for a set B C R". Then A is locally compact if for each bounded set B, XB(A z)1 is compact for some (and hence all) z E p(A).
Problem 10.1. Prove that if XB(A  z)I is compact for some z E p(A), then it is compact for all z E p(A).
Examples 10.2. (1) 0 is locally compact on L2(1R3). Note that XB(1  0)' has kernel
XB(x)[4nllx 
yll)' e11y11
100
10. Locally Compact Operators
which follows from (4.18) and is easily seen to belong to L2(1R3 x R3). By Theorem 9.14, it follows that XB(1  AY' is compact. We mention that the same compactness result holds in n dimensions (see [RS3] and [Sim4]).
Problem 10.2. Use the Fourier transform to prove that (1  A)
is not compact.
(Hint: Compute Q((]  A)').) (2) (A)'/2, the positive square root of A > 0, is locally compact. We offer two proofs of this fact.
Proof 1. Note that it suffices to show that XB(i + (A)1/2)' = A* is compact.
As A = (i +(A)'/')'XB, we have A*A = XB(1  A)'XB, and by (1) above, A*A is compact. Now we claim that this implies that A is
compact, for if u  0, IIAu,1112 = (un, A*Aun) < llunll
and as the sequence [ Ilun II } is uniformly bounded and A*Aun s+ 0, we have
Aun 2). 0. Hence, A is compact. Proof 2. Let XR(r) be 1 on the interval [0. R] and supp (XR) C [r I 0 < r < R+ 1) with XR > 0. Define the bounded operator XR((A)1/') through the Fourier
transform by
(FXR((A)'12)f)(k) = XR(IIkII)(Ff)(k).
(10.1)
where F is the Fourier transform. First, note that II
(1XRl(A)2))(1+(A)')
II
R_'.
(10.2)
This follows since in the Fourier representation (as in (10.1)) the operator on the left side of (10.2) is multiplication by the function
9R(k) = (1  XR(IIklI))(1 + Ilkll)',
and IIgRIloo < R'. On theotherhand,XB(1+(A)'/2)'XR((A)1/2)is compact because it can be written as
XB(1 
A)'(1
+ A) (i
+(_A)j)
i
XR
((A)I)
(10.3)
where the first factor XB(1  A)' is compact by (1) and the remaining factors form a bounded operator, as is easily checked in the Fourier representation.
Hence, it follows from (10.2) and (10.3) that XB(1 + (A)1/2)' is uniformly
10.2 Spectral Properties of Locally Compact Operators
101
approximated by the compact operators X,(1 +(A)112)1 XR((A)112), and so, by Theorem 9.8, it is compact.
We now show that certain classes of Schrodinger operators H = A + V are locally compact.
Theorem 10.3. Let V be continuous (or V E L2 (R n)), V > 0, and V > oc as 11x11 ). oo. Then H = A + V is locally compact. Proof. Note that H is selfadjoint by the Kato inequality, Theorem 8.12, and that H > 0. We first make the following claim:
(A); is H'bounded and ((A); + 1) (H + 1)2 is bounded.
(10.4)
Given the claim, we have XB(I + H)7 = XB (i
+(A)') (1 +(A)z) (H + 1);
(10.5)
and, by Example 10.2 (2), the first factor on the right in (10.5) is compact, the second is bounded, and so XB(1 + H)1/2 is compact. To prove the theorem, simply write XB(1 + H)1 = XB(1 + H)2(1 +H)7, and observe that the right side is the product of a compact and a bounded operator and is hence compact.
Proof of the claim. Since A > 0 and H > 0, all of the operators ( A )1 /2 H 1/2, and (H + 1)1/2 are well defined. We have a simple estimate for any u E Co (1R! ) II(A),u112
(u, Au) < (u, Hu) < (u, (H + 1)u) II(H + 1)zu112.
(10.6)
This estimate extends to all u E D(H 1/2). Consequently, relation (10.6) shows that
(A)1/2 is (H + 1)1/2bounded. Also, as we have (u, Hu) < follows from the Schwarz inequality, it follows from this and the third term of (10.6) that (A)1/2 is IIH1/2u112, which
H1/2bounded.
10.2
Spectral Properties of Locally Compact Operators
We now come to the main properties of locally compact operators. Roughly speaking, this family of operators enjoys the spectral properties described in the Intro
duction: The essential spectrum is determined by the action of the operator on states supported in a neighborhood of infinity. This idea provides an extremely
10. Locally Compact Operators
102
useful tool for the calculation of the essential spectrum. We already know from Chapter 7 that aess of a selfadjoint operator is determined by Weyl sequences. We now introduce a specific family of sequences, called Zhislin sequences, which will allow us to characterize the cress of locally compact, selfadjoint operators. Definition 10.4. Let Bk.  {x E 118" IIx 11 < k , k E N}. A sequence fun) is a Zhislin sequence for a closed operator A and), E C if u E D(A), I
I 1 u n 1 1 = 1 , suppu,, C{xIxE118"\Bn}, and11(AA)un11 *0asn>oc. If {un} is a Zhislin sequence, then un is supported on the complement of B,,. Note that because of this, un  0. We call these sequences Zhislin sequences after Zhislin [Z], who was one of the first mathematicians to introduce them into the study of Schrodinger operators. By Weyl's criterion, Theorem 7.2, it is clear that if A is selfadjoint and there exists a Zhislin sequence for A and ;,. then A E aess(A).
Definition 10.5. Let A be a closed operator. The set of all ), E C such that there exists a Zhislin sequence for A and k is called the Zhislin spectrum of A, which we denote by Z(A).
Let us recall from Chapter 3 that the commutator of two linear operators A and B is defined formally by [A, B]  AB  BA. As earlier, let BR(x) denote the ball of radius R centered at the point x c 118". Our main theorem states that the essential spectrum is equal to the Zhislin spectrum of a selfadjoint, locally compact operator that is also local in the sense of (10.7) ahead. Theorem 10.6. Let A be a selfadjoint and locally compact operator on LZ(II8" ). Suppose that A also satisfies 111A, On(x)](A 
i)111
> 0 as n > oo,
(10.7)
where cn(x) = q(x/n) for some 0 E Co (]18n), supp 0 C B2(0), 0 > 0, and 01B1(0) = 1. Then aeSS(A) = Z(A).
Proof.
(1) It is immediate that Z(A) C aess(A), by Weyl's criterion. To prove the converse, suppose A E aeSs(A). Then there exists a Weyl sequence fun) for A and A: 11un 11 = 1, un "i 0, and 11(A  k)un 11
the statement of the theorem, and let n (i  A)un 20'. 0, because
* 0. Let 'n be as in
1  ¢n. We first observe that
(i  A)un = ().  A)un + (i  ,k)un,
(10.8)
and the first term goes strongly to zero whereas the second goes weakly to zero. Next, note that by local compactness, for any fixed n, Onum s O as m > oe. This can be seen by writing
On um = On(i  A) 1(i  A)u,,,,
(10.9)
10.2 Spectral Properties of Locally Compact Operators
103
and noting that by (10.8), (i  A)um  0 and (Pn(i  A)' is compact, and so the result follows by Theorem 9.19. Consequently, Ilopnum II > 0 and Il0,,um II * I for any fixed n as m + oo.
(2) We want to construct a Zhislin sequence from to consider
To this end, it remains
(10.10)
ll(X  A)OnumIi < 110, 11Il().  A)umII + II[A, cbn]UmII.
The commutator term is analyzed using (10.7):
Il[A, Onlumll < II[A, 0nl(i 
A)l
Ii(II(),  A)umll + ii  XI),
since IIu,,, II = 1. This converges to zero as n > oo uniformly in m because the sequence {(X  A)um} is uniformly bounded, say by M, so II[A, onlum II
II[A, 0nl(i  A)' II(M + li  Al) > 0,
as n + oc. (3)
To construct the sequence, it follows from (10.10) that for each k there exists n(k) and m(k) such that n(k) * oc and m(k) > oo, ask > oc, and
Ilon(k)um(k)II > 1  k'
(10.11)
II(X  A)&(k)um(k)II < k',
(10.12)
and
as k oo. We define vk  0n(k)um(k) Il0n(k)um(k) II ' It then follows that { vk } is a Zhislin sequence for A and A by (10.11)(10.12) and the fact that supp vk C 1Rn \ B2k. Hence, X E Z(A) and aeSs(A) C Z(A).
Problem 10.3. Verify in detail the existence of functions n(k) and m(k) with the properties stated in the proof. Theorem 10.6 is our main result about locally compact, selfadjoint operators. It states that if A is locally compact and local in the sense that (10.7) holds, then QeSS(A) is determined by the behavior of A in a neighborhood of infinity.
Problem 10.4. Prove that the Laplacian A on L2(Rn) satisfies the assumptions of Theorem 10.6. Compute the Zhislin spectrum Z(A), and conclude that Qess(A) = [0, 00). We will now apply these ideas to compute 6eSs(H) of the locally compact Schrodinger operators H = A + V studied in Theorem 10.3. We expect that if V + oo as Ilxll  oo, then creSS(H) = {oo}, that is, is empty. We will consider the complementary case, V , 0 as llxll > oo, in Chapter 14.
Theorem 10.7. Assume that V > 0, V is continuous (or V E L Oe(Rn)), and V (x) ). oo as llx II 4 oo. Then H = A + V has purely discrete spectrum.
104
10. Locally Compact Operators
Proof. By Theorem 10.3, the selfadjoint Schrodinger operator H is locally compact. Let 46n be as in Theorem 10.6. We must verify (10.7). A simple calculation gives
[H,
2 On]=on.V_ On,
n
n2
(10.13)
where On and 0n are uniformly bounded in n. For any u E D(H), it follows as in (10.6) that
I!Vu112 < (u, Du) < (u, (H + 1)u), by the positivity of V. Taking u = (H + I )l v, for any v c L2(1P ), it follows that V(H + 1)1 and, consequently, V(H  i)I are bounded. This result and (10.13) verify (10.7). Hence, it follows by Theorem 10.6 that Z(H) = aess(H). We show that Z(H) _ {oc). If A E Z(H), then there exists a Zhislin sequence {un} for H and k. By the Schwarz inequality, we compute a lower bound,
II(a  H)unll > j(un, (a.  H)u )I
>
VUn112+(un, VUn)  Al
inf V(x) Lx
oc, the left side of (10.14) converges to zero whereas the right side As n O diverges to +oo unless A _ +oo. We will offer another proof of this theorem after we have discussed relatively compact operators in Chapter 13.
Theorem 10.6 is the starting point of a group of techniques for studying the essential spectra of operators which are known collectively as "geometric spectral analysis"; see [Ag I] and [DHSV]. It has become a powerful tool for studying elliptic and degenerate elliptic operators and can be extended to include the geometry of configuration and momentum space (i.e., phase space). We will give another application of these ideas in Chapter 14, where we discuss Persson's formula for the bottom of the essential spectrum of a Schrodinger operator. In the next section here, we show that the definition of Z(A) applies to closed, nonselfadjoint
operators and that, under additional technical assumptions, one can show that aess(A) = Z(A). This will be important for applications to spectral deformation in Chapters 17 and 18.
10.3
Essential Spectrum and Weyl's Criterion for Certain Closed Operators
In this section, we extend the results of Sections 10.1 and 10.2 to certain closed operators. This material is based on [DHSV], to which we refer for further information. For the first part of the discussion, concerning the Weyl spectrum, we will work in a general (separable) Hilbert space H. Then, we will limit ourselves to
10.3 Essential Spectrum and Weyl's Criterion for Certain Closed Operators
105
7I = L2(R") when we discuss the Zhislin spectrum. We will need these results in Chapters 17 and 18 when we discuss spectral deformation. Let us recall that the decomposition of a(A), the spectrum of A, introduced in Definition 1.4, is valid for any closed operator.
Definition 10.8. Let A be a closed operator on a Hilbert space 7l. The discrete spectrum of A, ad(A), consists of all isolated eigenvalues of A with finite algebraic multiplicity. The essential spectrum of A, aess(A), is the complement of ad(A) in
a(A): aess(A) = a(A) \ ad(A) As in Definition 7.1, we define Weyl sequences and a Weyl spectrum for A by
W(A) {A E C 13
E D(A), I1 un11 = 1, un  0, and II(A  A)unI+ 0).
Problem 10.5: Prove that aess(A) and W (A) are closed subsets of C. (Hint: To prove
that W(A) contains all its accumulation points, construct a new Weyl sequence using a diagonal argument.) Lemma 10.9. For a closed operator A, W(A) c or (A). Proof. Suppose ,l E W (A). Let (u } be a corresponding Weyl sequence. We define another sequence {vn } by (A  A)un
vn
II(Ak)u,, II
Then II v, II = 1 and v E D((A  A)1) (which, of course, may not be dense). Since II(A  ,.)un II  0, given any N > 0 3no such that n > no implies
II(A Hence, (A 
),)'vn
II = II(A  X)un II^' > N.
is an unbounded operator, and so X E a(A), by definition.
Unlike the selfadjoint case, W (A) is not necessarily equal to aess(A), in general. However, these two sets are related as follows.
Theorem 10.10. Let A be a closed operator on 7l with p(A) ¢ 0. Then W (A) C aess(A) and the boundary of aeSs(A) is contained in W(A). If in addition, each connected component of the complement of W(A) in C contains a point of p(A), then W(A) = aess(A). The converse of this last statement also holds. To explain the second statement, recall that aess(A) is some closed, nonempty subset of C which may have a nonempty interior. For example, in L2(1R2), multiplication by (x + iy)XD, where XD is the characteristic function on the unit disk, has spectrum equal to the closed unit disk. Hence, the boundary of aeS is S' I. The
theorem says that among the points of aes, determined by Weyl sequences are exactly those on the boundary (S! in our example). Clearly, if Int aes, ¢ 0, the complement of the boundary contains Int ae,,, which might not contain a point of p(A).
10. Locally Compact Operators
106
Proof of Theorem 10.10. (1) Let), E W (A), and let {un } be a corresponding Weyl sequence. If A E ad(A),
then there exists an Nodimensional, Ainvariant subspace where No < oo. Let P be the corresponding Riesz projection. Since u, 4'. 0, it must eventually leave
Ran P, that is,(1P)un = w,  Oand11wn11 > 1. Consequently, (1P)(AA) is not invertible. But this means k E a((1  P)A), which contradicts the fact that A E ad(A). (2) Let k E aess(A) and lie on the boundary. There exists a sequence zn E p(A) such that Zn * A (recall that p(A) is open and that p(A) z/ 0 by assumption). We
can choose f E 71 such that u, = (A  zn)l f does not vanish for all n. Define wn = un 11 un 11 1. Then II wn II = I and wn  0. For given s > 0, choose N such that IZn  ZN I < e for all n > N. Then by the first resolvent formula,
(h,wn)=IIun11'{(h,(AZN)lf)+(ZnZN)((AZN)l*h un)), for any h E 7l, so
I(h, wn)I < collunlll +scl < cc. Finally, we compute (A  a,)wn = (A  Zn )un II un 11 l + (Zn  A)wn
and find IRA  k)wn 11 < II f 11 Il un 11l + Izn  AI,
which vanishes as n f oo. Hence there exists a Weyl sequence for A, proving the second statement.
(3) If W(A) = aess(A), then we show that Int aess(A) = 0. Let C \ W(A) _ Ut=, C,, where C; is a connected, open set. If C; n p(A) = 0, then C, C a(A), which, as C, is open, means C, n aeSS(A) 0, a contradiction. Thus C, n p(A) ¢ 0 for all i. Conversely, if each C, n p(A) ¢ 0, since C \ W(A) C C \ aaess(A) (where aK denotes the boundary of K) by part (2), the set {C,} covers Int aeSS(A). Now
let (J, C, = C \ aaeSS(A), so Cf C C, for some i. Hence each C, contains a point of p(A). But C, consists entirely of Int aess(A). Hence, Int aess(A) = 0, and so W(A) = aess(A)
The computation of W(A) is not too easy. We would like a result, similar to Theorem 10.6 for the selfadjoint case, that tells us geometrically how to compute W(A). To do this, we take 71 = L2(R"). We introduce the notion of the Zhislin spectrum, Z(A), which is determined by the behavior of A at infinity, in analogy with Definition 10.4. Definition 10.11. A Zhislin sequence {un} for A and k E C is a sequence such that supp un, n K = 0 for each compact K C R" and for all m large, and such that II(A  ),)un II  0. The set of all A such that a Zhislin sequence exists for A and .k is called Z(A). Note that {un} necessarily converges weakly to zero and that Z(A) C W(A). Our main result is the following theorem, which extends the methods of geometric
107
10.3 Essential Spectrum and Weyl's Criterion for Certain Closed Operators
spectral analysis to certain nonselfadjoint operators. Combined with Theorem 10.10, this theorem tells us when we can determine Qess(A) by the behavior of A at infinity.
Theorem 10.12. Let A be a closed operator on L2(R") such that p(A) ¢ 0 and is a core. Let X E Co (R") be such that x BE(0) = 1, for some,, > 0. Cp We define Xd(x)  X(x!d). Suppose that for each d, Xd(A  z)1 is compact for some z E p(A), and that 3 E(d) * 0 as d + cc such that Vu E C' (W7), I
II[A, Xdlull < E(d)(IlAull + Ilull).
(10.15)
Then W(A) = Z(A). Proof. Let a. E W(A) and {un } be a corresponding Weyl sequence. Since Co (R") is a core, we cah assume u" E Co (1W ). For Z E p(A), Il Xdunll
= IIXd(A  z)1(A  z)unll  0,
(10.16)
since Xd(A  z)1 is compact for each d and u, w*' 0. We can also compute
II(AX)(1 Xd)unII < II(AX)unII+II[A.Xd]unII < (I +E(d))II(A  X)unll +E(d)(IXI + 1).
(10.17)
For each d = 1, 2, ... , we choose a subsequence n (d) > oc such that 11(1  Xd)un(d)II * 1,
which is possible by (10.16). Consequently, by (10.17), 11 (A 
coE(d) + 0,
where l n(d) = (1  Xd)un(d). Since supp t n(d) leaves every compact as d + oc, it follows that X E Z(A).
If we combine Theorems 10.10 and 10.12, we obtain a useful result for the computation of Qess(A) in certain circumstances (see Theorem 16.19).
Corollary 10.13. Let A be a locally compact, closed operator on Lz(Rn) such that p(A) ¢ 0 and Cp (R") is a core. Suppose that (10.15) holds and that each connected component of the complement of Z(A) contains a point of p(A). Then Z(A) = W(A) = Qess(A).
11
Semiclassical Analysis of Schrodinger Operators I: The Harmonic Approximation
11.1
Introduction
In this and the next chapter, we begin a study of the behavior of the spectrum of Schrodinger operators in the semiclassical regime. There is much literature on these topics, and we provide an outline in the Notes to this chapter. The material here follows a part of the work of Simon [Sim5]. In quantum mechanics, the Laplacian plays the role of the energy of a free particle. But, the Laplacian apparently has the dimensions of (length)2. This is because in our discussions of the Schrodinger operator, we have chosen to work in simple units in which the mass m = 1/2 and Planck's constant h is taken to be 2,r. In these units, the coefficient of the kinetic energy Ho = A is 1. If we restore these constants, the actual differential operator in quantum mechanics representing the energy of a free particle is Ho = (h2/87r2m)A. The Planck constant h is approximately 6.624 x 1027 ergs/sec. This is very small except on atomic scales. That is, the length scales over which quantum effects are important depend on h (for example, the Compton wavelength of an electron or the Bohr radius of an atom). This observation provides us with one way to understand the transition from classical to quantum phenomena. We consider quantum theory in which Planck's constant has been replaced by a small parameter. We then try to understand quantum mechanics in terms of the classical theory obtained as the limit of this quantum theory as the parameter is taken to zero. We call this limit h = 0 the classical limit since all quantum effects have been suppressed. The regime in which this parameter is nonzero, but taken arbitrarily small, is called the semiclassical regime. In this regime, the behavior of the quantum
mechanical system should be dominated by the limiting h = 0 classical system.
1 10
11. Semiclassical Analysis of Schrodinger Operators 1.
Quantum effects appear as small fluctuations about classical behavior and many times can be calculated when the exact quantum mechanical solution is unknown. We will study various aspects of this regime, beginning with the behavior of eigenvalues and eigenfunctions. In Chapter 16, we will begin a study of another aspect of quantum systemsresonancesin the semiclassical regime. We now restore Planck's constant in the Schrodinger equation and write fi for h/27r. We will continue to set the mass m = 1/2. The Schrodinger operator is H(1) _ he A+ V, on L2(R" ). We treat It as an adjustable parameter of the theory. We will study the semiclassical approximation to the eigenvalues and eigenfunctions of H(h) = h2A + V for potentials V with in particular V (x) = oo, for some R > 0. By this we mean that we will identify when limp,rll c the leading behavior of eigenvalues e(h) and eigenfunctions t/rh(x) of H(h), as h becomes arbitrarily small. Because the small parameter h appears in front of the differential operator A, it may not be clear what is happening as h is taken to be small. It is more convenient, and perhaps more illuminating, to change the scaling. Letting A  1 /h, we rewrite the Schrodinger operator as
H(A) _ 0 +,12V = h 2H(h). Looking at H(A), we see that the semiclassical approximation involves ,L * 00. This limit has the physical interpretation of the potential becoming very large, at least near infinity. The tunneling analysis of Chapter 3. and the work with Zhislin sequences in Chapter 10, suggests that the conditions infllxll>R V(x) > 0 and A large imply (1) that the infimum of the essential spectrum is greater than zero, and (2) that there may exist lowlying eigenvalues of H(A) below the bottom of the essential spectrum. It is the behavior of these eigenvalues as A  oo which we now study.
11.2
Preliminary: The Harmonic Oscillator
By a harmonic oscillator Hamiltonian, we mean the following. Let A E M"(R), a real n x n matrix, and let A > 0, that is, A is a positive, definite, and therefore symmetric, matrix. We define
K(k)
+),2(x,
Ax),
(11.1)
where (x, Ax)  En0=j A;jx;xj. Note that the Euclidean quadratic form (, ) is
bounded from below by
(x, Ax) > AminIIxlj2.
Here Ami is the smallest eigenvalue of A and is strictly positive. The lower bound is, therefore, strictly positive for x 0. As a consequence, we see that K(X) is positive with a lower bound strictly greater than zero. Since the harmonic oscillator
1 1.2 Preliminary: The Harmonic Oscillator
111
is continuous and Vhar(x) = (x, Ax) + oc, as IIx11 + oo, the harmonic oscillator Hamiltonian (11.1) is selfadjoint by Theorem 8.14. Moreover, the spectrum of K(l), a(K(;,.)), is purely discrete by Theorem 10.7. We would like to find out how the eigenvalues of K(),.) depend on A. We do this by performing a similarity transformation on K(A), (11.1). First, some general comments.
Definition 11.1. Two operators A and B, with D(A) = D(B) = D, are called similar if 3 a bounded, invertible operator C such that CD C D and A= C BC 1. Problem 11.1. If A and B are similar, then or (A) = a(B). We now introduce a representation of the multiplicative group 18+ on L2(R"). This is called the dilation group. For O E 18+, we define a map on any >1' E Co (18" ) by
U(8) : 1/i(x) +9 1i(Bx).
(11.2)
In Problem 11.2, we ask the reader to verify that this map is well defined and extends to a bounded operator on L2(lR ). Furthermore, for each 8 E 18+, U(8) is unitary on L2(1R") and
U(8)* = U(8)I = U(81). The map 8 E 18+ * U(8) enjoys another very important property. For 8, 8' E 18+, one can check from (11.2) that
U(8)U(8') = U(88'). That is, the map from the multiplicative group ]I8+ into the unitary operators U(8) preserves all the group structure. We call such a map a unitary representation of the group R+.
Problem 11.2. Verify that the map 8 E R[
+ U(8) is a unitary representation
of R, In addition, prove that the map is strongly continuous, that is, for any 8, 8' E R+, we have s  limee U(8') = U(8). We now claim that U(A'12) implements a similarity transformation on K(A) by
U /X`
\1 = i.K,
(11.3)
where
KA+(x,Ax). That is, the transformation x * .l112X is implemented by U is to scale A out of K(A).
(11.4) 1/2
), and its effect
Problem 11.3. Carefully prove (11.3). As a consequence of Problems 1 1.1 and 11.3, Aa(K ), where a(K) is independent of X. Hence the eigenvalues of K(),.) depend linearly on X. Moreover, the multiplicities of the related eigenvalues are the same. Now we let (e,') denote the eigenvalues of K, so that cr(K(A)) = (n.en}. The eigenfunctions are related through
112
1 1. Semiclassical Analysis of Schrodinger Operators I.
the unitary operator
then it follows from (11.3)
1/2). If K(.l)i/i,, =
that U(A'/2)*" = " is an eigenfunction of K with eigenvalue e, We remark that the eigenvalues of K are
EZ+U{0}},
(11.5)
where {cot}"_, are the eigenvalues of A.
11.3
Semiclassical Limit of Eigenvalues
We study the behavior of the eigenvalues of H(,l) A + oo. Here, we assume V satisfies
A + A2 V on L2(R") as
(VI) V E C3(R ), V > 0, limllxll, V(x) = oc; (V2) V has a single, nondegenerate zero at xo = 0 : V (O) = 0, V'(0) = 0 and 1
a'` y
2
ax; axe
A=
xo=o
Remarks 11.2. (1) The potential V need not be globally C3, but C3 only in a neighborhood of xo and, by modifying the following proof, we only need (limllx11,00
V(x)) > 0, where x = Ilxllw, for W E S"1. (2) The potential V can have several zeros, but they must he nondegenerate.
By Theorem 10.7, a(H(a.)) is discrete. All of the eigenvalues are positive, and infinity is the only possible accumulation point of eigenvalues. We label these eigenvalues by en(A), where we list the eigenvalues in increasing size, including multiplicity: eI (A) < e2(A) < e3(A) < e4(A) < ...
We label the distinct values by a"(A), with multiplicity m"(A), in strict increasing order. Ask becomes large, the "well region" about xo = 0 becomes more prominent and we expect the quadratic part of the potential V to dominate (see Figure 11.1). Consequently, we introduce a comparison harmonic oscillator Hamiltonian
K=A+(x,Ax), where A is given in (V2). Let e" be the eigenvalues of K listed in increasing size, counting multiplicity,
el < e2 <e3 < e4 0. It follows from this estimate and (1 1.14) that limx_«; IIlj II2 = 1. The result of all of this is that ek(n)  e I < C
and the upper bound part of result (11.6) follows upon multiplying (1 1.15) by X.
Lemma 11.4. In the notation of Theorem 11.3, there exists a constant D" such that IIP
VJx0nll < D
eb)' vio
Proof. Let X E Co°(W') be such that Xlsupp(OJ) = I and X = 0 for IIxII > 5/4 and IIxII < 3/4. Then it suffices to prove the result for pjX;,On, where Xx(x) _ X(,, )110x); see Figure 11.2. We have for j = 1, ... , n, pix.!On
=
PJXA(K + 1)1(K + 1)cn
=
pj(K +
=
pj(K + 1)1(en + 1)XxOn + pj(K + 1)1(P2Xx)On
1)'XA(en
+ 1)0n + pj(K + 1))[P2, X;, ]0,
+2pj(K + I)1 pi(pixA)On, so that (using Problem 11.4 ahead), IlpjXxOnll _ 1 (1 2
o2)2` O(etis). fg=t
for some S > 0. By (A3) and the fact that {xlg(x) = 1) is a neighborhood of xb, we have
02 > C,lN, and so the result (12.17) follows.
12.4
Appendix: Exponential Decay of Eigenfunctions for DoubleWell Hamiltonians
We discuss the exponential decay formula (12.15) and its relation to the decay described in Chapter 3. Recall that p(x) = min(P(xa, x), P(xb, x)),
(12.21)
where
P(x, y) = inf fo [V(Y(t))]'1 IY(t)Idt Y I
,
(12.22)
128
12. Semiclassical Analysis of Schrodinger Operators 11.
with y E AC[O, 11, y(O) = x, and y(l) = y. Since V > 0, this is well defined for all x, y E I(8". On the other hand, for an energy eigenvalue E(A) of distance in the Agmon metric introduced in Chapter 3 is pF(x, y) = inf Y
[(A2V(y(t))  E(A))+] IY(t)Idt
.
the
(12.23)
[fo
Le t SE(A) be the set A2
SE(A)  {x I
(12.24)
V(x) = E(A)},
that is, the boundary of the classically forbidden region .FE for energy E. The decay result of Chapter 3 for the eigenfunction tGE, HI/iF = El/IF, is Ile(p(:)lEII
(12.25)
< C,
where
p(x) =
inf PE(X, y).
(12.26)
VESE(i.)
To see the relation between (12.15) and (12.25), we first note that by the harmonic approximation (Chapter 11), E(k) = 0(),) and, consequently, (12.22) is the leading asymptotic contribution to A.i PE(X, y) as . f oo, that is,
hm
A+oc
PE(X, Y)
= p(x, y).
A
Next, note that if y E SE(;,), then y = 0(.t112), as follows from (12.24) and the harmonic approximation. This implies that for A sufficiently large, SE(;,) _ SE(A)USE(k),whereS' (A) are disjoint surfaces around xi, i = a, b.Consequently, in the large A limit,
p(x) 
min
i=a.b
inf PE(X, y) YES'
and
lim
.+oo
p(x) A
/
l
= min C lim infE I p(x' y) i=a,b
yES'
\\\
I
= p(x),
(12.27)
A
where p is given in (12.21) and (12.22). As a consequence of (12.27), we see that (12.15) is indeed the leading asymptotic contribution to the decay rate of l/'E. We remark that result (12.15) can also be derived directly using the Agmon method described in Chapter 3.
12.5
Notes
The behavior of the eigenvalues and eigenfunctions for double and multiplewell problems in the semiclassical regime has been studied by many authors. In the
12.5 Notes
129
onedimensional case, techniques of ordinary differential equations can be used quite effectively to obtain sharp estimates. We mention the work of Gerard and Grigis [GG], Harrell [HI], Kirsch and Simon [KS 1 J. and Nakamura [N 11. A typical result [N I ] in one dimension for the lower bound on the difference of the first two eigenvalues is
lim inf)' log[Ei(X)  E_tt(%)] >  J
V(x) dx.
Combined with upper bounds of the type obtained here, one sees that this result is optimal. In multiple dimensions, we mention, in addition to the work of Simon [Sim6] on which this discussion is based, the papers of Helffer and Robert [HR], which give the asymptotic expansions of the eigenvalues; Heifer and Sjostrand [HSjI] [HSj3]; Kirsch and Simon [KS2]; Martinez [M I]; and, for the case of degenerate wells, Martinez and Rouleux [MR]. All of these papers deal with the semiclassical regime. There are some related problems concerning estimates of eigenvalue differences for the Laplacian and Schrodinger operators on bounded domains with Dirichlet boundary conditions. For the Dirichlet Laplacian with dumbbell domains, which consist of two regions connected by a tube of width e, the eigenvalue difference can be estimated from above and below for e sufficiently small (see [BHM 1 ] and
references therein). The upper bounds are obtained in a manner similar to that discussed here. We will also examine a related problem in Chapter 23. There are some results available when there is no semiclassical parameter. Singer, Wong, Yau, and Yau [SWYY] obtained lower bounds for the eigenvalue difference for a Schrodinger operator on a convex domain with Dirichlet boundary conditions when the potential is nonnegative and convex.
13
SelfAdjointness: Part 2. The KatoRellich Theorem
The KatoRellich theorem and the Kato inequality form the basic tools for proving selfadjointness of Schrodinger operators H = A + V. The Kato inequality allows us to consider positive potentials that grow at infinity. These Schrodinger operators typically have a spectrum consisting only of eigenvalues. The semiclassical behavior of these eigenvalues near the bottom of the spectrum was studied in Chapters 11 and 12. The KatoRellich theorem will permit us to consider the
other extreme: V > 0 as IIxjj  oo. Families of such potentials, which might not necessarily be small in operator norm, will be shown to be small perturbations
relative to the Laplacian A, in a sense to be defined ahead. We will see in the next chapter that the essential spectrum of such operators H = A + V is usually 1R+ = Qess(0). that is, completely characterized by the essential spectrum of the Laplacian. We will first discuss the general theory of relatively bounded perturbations and then its application to Schrodinger operators.
13.1
Relatively Bounded Operators
We now study the following question: Suppose A is selfadjoint and B is a closed,
symmetric operator such that D(A) n D(B) is dense. What conditions must B satisfy in order that A + B be selfadjoint? This question will be answered in a very satisfactory manner by the KatoRellich theorem. We will apply this to study perturbations of the Laplacian, which is selfadjoint, by real potential functions V.
In the following, we assume that A and B are closed operators on a Hilbert space R.
13. SelfAdjointness: Part 2. The KatoRellich Theorem
132
Definition 13.1. An operator B is called Abounded if D(B) D D(A).
It is obvious, but important to note, that any bounded operator B E G(H) is Abounded for any linear operator A.
Proposition 13.2. 4'p(A) ¢ 0 and B is Abounded, then there exist nonnegative constants a and b such that
IlBull < all Aull + bllull,
(13.1)
for all u E D(A). Proof. We equip D(A) with the graph norm defined by
IIulLA =(Ilul12+IIAull')i.
(13.2)
It is easy to see that A is a closed operator if and only if D(A) is closed in the graph norm (see Appendix 3 for a discussion of this). Furthermore, this norm is induced by an inner product, (u, v)A = (u, v) + (Au, Av).
If A is closed, the linear vector space D(A) is a Hilbert space with this inner product, which we call RA
Problem 13.1. Verify that (A, D(A)) is closed if and only if D(A) is closed in the
graph norm. Then show that if z E p(A), RA(z) is a bounded operator from N onto RA 
Since 7(A = D(A) as sets, and B is relatively Abounded, we have that BRA(z) is everywhere defined on N.
Problem 13.2. Prove that BRA(Z) is closed on N. By the closed graph theorem, Theorem A3.23, an everywhere defined closed operator is bounded. This means that there exists an a > 0 (depending on z E p(A)) such that (13.3) IIBRA(z)fIIHA :5allfllx,
for all f E H. For any f E N, we showed that u = RA(z)f E D(A) (and each u E D(A) can be written in this way), so f = (A  z)u. Using this representation of f in (13.3), we get IlBull
=
IIBRA(z)f II
aII(A  z)ull
allAull + b Ilull, where b = aIz1. This proves the theorem.
0
Definition 13.3. The smallest nonnegative constant a such that (13.1) holds for all u E D(A) is called the bound of B relative to A, or the Abound of B. We remark that B may be unbounded in general. If B is relatively Abounded, it follows from (13.1) that for all u E D(A), there exists a constant c > 0 such that IlBull < cIIuIIA,
13.1 Relatively Bounded Operators
133
which is to say that B : HA ) 7I is bounded. Problem 13.3. Consider the Laplacian on L2(R" ). Prove that any partial derivative 0j, j = 1, ..., n, as a linear operator with domain H 1 (W1 ), is relatively Laplacian bounded. Compute the relative Laplacian bound of these operators.
Example 13.4. Let us consider a potential V E L2(1183) + L°°(R3), as in Example 8.2. Then V is Abounded with relative bound zero. Proof. Let V = V1 + V2, with V1 E L2(1183) and V2 E LC°(]R3). The multiplication operator V2 is a bounded operator, IIV2ull
IIV2IIOOIlull,
_
0 such that JIG;, 112
:5 Ell V 112
(13.7)
Thus, from (13.4), (13.6) and (13.7), it follows that for all A large enough, IIVRA(iA)f11 00 _< EIIf112
From (13.3) and (13.1), this implies that IIVuII
for any e > 0 and all u E D(A).
EIIAUII+ E >.Ilull,
(13.8)
134
13. SelfAdjointness: Part 2. The KatoRellich Theorem
Notice from this last inequality that the constant in front of the Auterm can be made small only by increasing the coefficient of the last term. This phenomenon commonly occurs: The constant a in (13.1) can be made small only at the expense of the second constant b.
The reason that relatively bounded operators are so important is due to the following theorem.
Theorem 13.5 (KatoRellich theorem). Let A be selfadjoins, and let B he a closed, symmetric, and A bounded operator with relative A bound less than one. Then A + B is selfadjoint on D(A). The proof of the theorem relies on the following lemma.
Lemma 13.6. Let A be selfadjoint and B be A bounded with relative hound a. Then
IIB(A00'IJ 0,
II Wull < all(o)3ull + bllull,
(13.11)
for all u E D((A)1/2). Next, we show that (0)1/2 is Hobounded with relative bound zero. For any u E D(Ho) C D(0),
II(A)IuII2 = (Du, u) 5 (Hou, u) < IlHoull (lull,
(13.12)
13.2 Schrodinger Operators with Relatively Bounded Potentials
137
using the positivity of U. Now for any a. h E R, and for any c > 0,
ab < e a2 +(4E2)l b2. Applying this to the right side of (13.12), we get (13.13)
II(O)'ull < EIIHoull + (2E)l IIuII.
Substituting (13.13) into the right side of (13.11), we obtain a relative bound for W, for all it E D(Ho), Wit 11
0 is arbitrary, this proves that W is Hobounded with relative bound zero. By the KatoRellich theorem. A + U + W = H is selfadjoint on D(Ho). Problem 3.6. Under the hypotheses of Corollary 13.8, prove that Ct(R") is a core
for H. Finally, we prove that  A determines the essential spectrum of H =  A + V for a certain class of relatively bounded potentials V. This result should be compared with Theorem 10.7.
Theorem 13.9. Assume that V is real and Abounded with relative Abound < 1, and that V(x) a 0 as Ilxll + oc. Then H = A + V is .selfadjoint on D(H) = H2(R") and aess(H) = a(A) = [0, 00).
Proof.
(1) The Schrodinger operator H = A + V is selfadjoint on H2(]f8"), by Theorem 13.5, since the relative Abound of V is less than one. We first verify the hypotheses of Theorem 10.6 in order to be able to conclude that aess(H) = Z(H). By the Aboundedness of V, the local compactness of A (see Example 10.2), and the second resolvent identity, Proposition 1.9, it follows that H is locally compact. We must verify (10.7). Choose 0 E C0°(R") as in Theorem 10.7, and set 0n(x) = 0(x/n). We compute the commutator,
[H, on](H  i)' = ( p2/n +2(pOn) P )(H 
i)I.
(13.14)
0(x/n), we have the estimates Ilpionll = 0(n1) and IIp2o,Il = O(n2). Moreover, by Problem 13.3 and the relative boundedness of V, one easily shows that pt(H  i)is bounded. Hence, there is a constant c > 0 such that Since
II[H, on)(H  i)' II < cn1, which verifies (10.7), and hence Z(H) = Qess(H)
138
13. SelfAdjointness: Part 2. The KatoRellich Theorem
(2) We have already shown in Theorem 7.6 that cress(A) = cr (A) = [0. 00).
We first show that aes,(H) C [0, oo). Let n. E aeSY(H) and {u } be a Zhislin sequence for H and L Since supp uF. C R" \ Bk(0),
lim IIVunll = 0. nx Since D(H) = D(A), by the relative boundedness of V, we have
II(,+A)unll
oo, X E R. Note that
since A is selfadjoint, 11(A  iX)v!12
=
IIAv112 + X211v112
IIAv112 + IIvI12
11(A  i)v112.
Letting v > (A  i),)1 v in this inequality, we have
IIvI12 ? II(A  i)(A  iX)1v112, so (A  i)(A  iX)1 is uniformly bounded in X. Next, for U E D(A), we obtain from Corollary 5.7,
11 (A  i)(A  i),)lull
=
II(A  i),)1(A  i)uII IXI1 II(A  i)uII + 0,
14.2 Weyl's Theorem: Stability of the Essential Spectrum
141
asIXI > oo.Since (Ai)(Ai),)' converges strongly to zero aslxl + 00 on a dense set and is uniformly bounded, it converges strongly to zero.
(2) To prove the lemma, we first write
B(A  0)' = B(A  i)'(A  i)(A  0.)'. Since B(A  i)' is compact and (A  i)(A  iA)l  0, the operator B(A  i),)' converges uniformly to zero by Theorem 9.19. Proof of Theorem 14.2. For any E > 0, choose % such that
IIB(A  ia.)' ll 2 if n = 4, and p > n/2 if n > 5. This class is quite natural with regard to the KatoRellich theorem. If the real potential V is in the KatoRellich class, then the Schrodinger operator H = A + V is selfadjoint on D(H) = D(A) = H2(IR"). It is easy to see that not all KatoRellich class potentials are relatively compact. The potential function V = I is not relatively compact, as follows from Problem 10.2. Note that such a perturbation of the Laplacian shifts the essential spectrum. Indeed, from Theorem 13.9, one sees that an additional condition, like the vanishing of the potential at infinity, is sufficient for the invariance of the essential spectrum. We refine the KatoRellich class to capture this characteristic.
Definition 14.7. A potential function V (x) is called a Kato potential if V is real and V E L2(R") + L"(W )E, where the a indicates that for any E > 0, we can decompose V = V1 + VZ with V1 E L2(R') and V2 E L°°(R"), with II V211,,, < E.
Example 14.8. The Coulomb potential: V (x) = c Ilx II 1, c constant. For any E > 0, let xE (Ilx II) be the function that is 1 on {x I Ilx II < (cc)') and vanishes outside {xI lixll < 2(ce)' }. Then, we decompose the potential as V(x) = cxE(llxll)Ilxll1 +c(l  xE(IIxiD)Ilxll1 = VI(x) + V2(x). We have V1 E L2(R' ) and
sup Ic(1  xE(llxll))Ilxll11 _< E. XER^
Problem 14.1. Let V be a real, continuous function with lim11 Show that V is in the Kato class.
11
V(x) = 0.
Theorem 14.9. If V is a real Kato potential, then V is relatively Acompact. Proof. We give the proof for the threedimensional case. We know from Theorem 13.7 that V is Abounded, that is,D(V)D D(A). We must check that V( A+i) I is compact. For any c > 0, we decompose V as V = VI + V2, where VI E L2(R3) and VZ E L°O(1183)E. Then, for any f E L2(R3), IIVZ(
+i)'fll
IIV2hochi(A+1) Ifll <Elifll,
(14.3)
14. Relatively Compact Operators and the Weyl Theorem
144
and it follows that 11 V2( z +i)111 < e. Next, with the help of (4.18), we observe that V1(A  z)1 , Im z 0, is an integral operator with kernel given by I
j 1111.
ylille
K(x, Y) = V1(x)[47r 2Ilx 
Indeed, for any f E L2,
(VI(A  z)I f)(x)
V1 (x) f [4n`Ilx 
=
ylille11.V,l f(y)dy
83
f K(x,y)f(y)dy.
(14.4)
The square root is defined as the principal branch: Re z1/2 > 0. This kernel is a HilbertSchmidt kernel for [4nzl ul 12e2julz' du 0, we define VE(x) = max(V(x)E, 0) and WE(x) = min(V(x)E, 0), so that V(x)  E = WE(x) + VE(x). We have sketched V, WE, and VE in Figure 14.1. We then have
HE=A+VE+ WE. By Theorem 10.3, the operator A + VE is locally compact. Since supp (WE) is compact, it follows that WE is (A + VE)compact. Hence, by Weyl's theorem, Qess(H  E) = cress(A + VE).
(14.6)
14.4 Persson's Theorem: The Bottom of the Essential Spectrum
145
FIGURE 14.1, The potentials VE and WE for the potential V.
On the other hand, as VE > 0, the operator A + VF > 0, and so a(A + VE) C [0, oo). Thus aess(H  E) C [0, oo),
so aes,(H) C [E, oo).
Since this is true for any E > 0, ae..(H)
and a(H) = ad(H) is purely
discrete.
14.4
C
Persson's Theorem: The Bottom of the Essential Spectrum
Persson [Per] discovered a beautiful geometric description for the bottom of the essential spectrum of a semibounded Schrodinger operator H =  A + V. It is based on the same principle as the modification of the Weyl criteria in the presence of local compactness: The behavior of the potential V at infinity determines aesS(H). We state and prove a simplified version of Persson's theorem (see also [Ag 11). This theorem can be applied to other elliptic partial differential operators on unbounded domains in R" or on noncompact manifolds (see, for example, [FrHi]).
Theorem 14.11. Let V be a realvalued potential in the KatoRellich class, and let H = A + V be the corresponding selfadjoint, semibounded Schrodinger operator with domain H2(R"). Then, the bottom of the essential spectrum is given by
infae..(H) = sup [inf
KCR° oho
(0, Hr)
110 11 2
I 0 E Cp (II8"\K)}J ,
(14.7)
where the supremum is over all compact subsets K C R".
As a preliminary, we need an expression for the bottom of the spectrum. This is easily obtained from a modification of Proposition 12.1, equation (12.1), which
expresses the lowest eigenvalue of a selfadjoint operator as the infimum of a quadratic form.
Proposition 14.12. Let H be a Schrodinger operator as in Theorem 14.11. We
146
14. Relatively Compact Operators and the Weyl Theorem
have the formula,
inf or (H) = inf { (0, HO) IIpII2 I p c Co(ff ")}
.
(14.8)
Proof. If infa(H) is in the discrete spectrum, the result follows from Proposition 12.1. Otherwise, we assume that ko = inf a(H) E aeS,(H). Let k denote the right side of (14.8), where we take the infimum over ¢ E H2(R"). We leave it as an exercise to pass from this result to the proposition, which is possible since Co(II8'1) is a core for H. Let (u, } be a Weyl sequence for H and ,lo. For any e > 0, there exists NE such that n > NE implies that (un, Hun) > A0  e,
(14.9)
so that A > A0. As for the reverse inequality, suppose that % < A0. We can find a sequence of vectors vn, with 1Ivn II = 1, such that lim (v,,, (H  a.)vn) = 0.
(14.10)
n. cc
Since we have II(H  A)v,1 = inf [I
(
, (H  A)vn)
I
II
II
'
,
(14.11)
we see that the right side of (14.11) can be made arbitrarily small. By the Weyl criterion for the essential spectrum, this implies that either % E aess(H) or ,1 is an eigenvalue. In either case, A E a(H), which contradicts the assumption that X
(¢, (H + X))O) II0II2 (14.17)
E(H, 1C).
Results (14.16) and (14.17) together imply that
inf cress(H) > E(H),
(14.18)
which proves one part of the theorem.
(2) Conversely, let Eo  inf Qess(H). Since the essential spectrum of H is closed, we have that Eo E 6ess(H). Then by Theorem 10.6, there exists a Zhislin sequence {u"} for H and E0. For any compact 1C C W?, we write E(H,1C) =
inf
mECo (R",W)
(0' (H  E0W + Eo 110112
We claim that for any E > 0, there exists aE E Co
.
j
(14.19)
such that
IIYEII=land 1(1E, (H  Eo)1E)I < E.
(14.20)
This proves that for any c > 0,
E(H,/C)> Eo  E,
(14.21)
which, together with (14.18), proves the theorem. To prove the claim, it follows from the definition of the Zhislin sequence that there exists an index n 1 such that (14.22) II(H  Eo)u",11 0 such that Fr = {zl Iz  AI = r} C p(TK) for all small IKI; (ii) we let PK be the Riesz projection for TK corresponding to the contour Fr, then PK  P0 as K + 0 in norm. Proposition 15.3. Suppose k is a stable eigenvalue of To. Then forall I K I sufficiently
small, any operator TK has discrete eigenvalues tii(K) near X of total multiplicity equal to the multiplicity of A.
Proof. By part (ii) of Definition 15.2, PK 0, for K small, so TK has spectrum inside the contour Fr. The discreteness of the spectrum follows from the fact that the multiplicity of A is finite and from Lemma 15.4. Hence, since PK is a continuous, projectionvalued function of K. dim(Ran PK) is a finite constant and equal to dim(Ran Po), for all small IKI.
Lemma 15.4. Let P and Q be projections. If dim(Ran P)
dim(Ran Q), then
IIP  QII > I. Problem 15.2. Let M and N be two subspaces of 71 with dim M > dim N. Then 3x E M such that x is orthogonal to N. (Hint: By considering a subspace of M, if necessary, we can assume that dim M and dim N are finite. Take orthonormal bases {x, Jand { yj m > n, for M and N, respectively. Let x = Y"_1 aixi, and solve (x, yi) _ Ym 1 ai (xj, yi) = 0 for i = 1, ... , m. The matrix A [(y;, xi)] is an n x m matrix. Then 3x E M, x ¢ 0, and (x, yi) = 0, i = 1, ... , n if and only if Aa = 0 has a nontrivial solution. But ker A {0}, and so we can solve the equations.)
Proof of Lemma 15.4. Let dim(Ran P) < dim(Ran Q). Let U E Ran Q n [Ran P]1, lull = 1. Such a u exists by Problem 15.2. Then
(PQ)u=Qu=u, which implies that 11 (P  Q)uII = 1, so 11P  Q11 > 1.
15.3
Criteria for Eigenvalue Stability: A Simple Case
We begin by considering a special case, which we will generalize in the next section. Consider a selfadjoint operator H on a Hilbert space 7( and a family of operators HK defined by
152
15. Perturbation Theory: Relatively Bounded Perturbations HK = H + K W,
(15.1)
where we assume
Condition A. The perturbation W is Hhounded. Note that we do not assume that W is symmetric. By employing the techniques used in the roof of the KatoRellich theorem, we have (a) for any K E C, the operator HK is closed and D(HK) = D(H), independent of K;
(b) for any it E D(H), HKu is an entire analytic function of K. Problem 15.3. Prove properties (a) and (b) of the family HK. Suppose, in addition, that W is symmetric. Obtain an explicit bound on the size of IK I in (a) so that HK is selfadjoint for K E R.
We have had occasion to discuss analytic families of operators in Chapter 1, Theorem 1.2. We now study them more carefully. We begin with a simple case. Definition 15.5. A family of bounded operators BK is called analytic at Ko if the map K  BK is analytic as an operatorvalued function on a small disk BE(Ko). That is, for K near KU, BK has a uniformly convergent power series expansion: BK = E, 0(K  KO)" b,,, where, for each n, bn is a bounded operator.
Problem 15.4. (1) Show that BK is an analytic family about K = Ko if and only if K + BKu is analytic for any u c W. This is the notion of strong analyticity. (Hint: Use the principle of uniform boundedness (see [RS 1 ]) to show that II BK II < Co. for all K E BE (KO). Then use the Cauchy formula for the nth derivative of an
analytic function.)
(2) Reformulate the notion of analyticity in terms of Morera's theorem [R]. (3) Show that BK is an analytic family about K = Ko if and only if K * (BK U, U)
is analytic for any u, v E H. This is the notion of weak analyticity. We first study families of operators HK as in (15.1) which satisfy condition A. Lemma 15.6. Assume condition A and then let HK be as in (15.1).
(i) Let G be any bounded, open set such that d C p(H). Then G C p(HK) for I K I sufficiently small.
(ii) F o r any A E C as in (i), the map K  ( , l  H K ) is analytic in K f o r I K I sufficiently small.
15.4 TypeA Families of Operators and Eigenvalue Stability: General Results
153
Proof. (1) Let ;, cz G. Then
(E p(H) and
KW(AH)1)(A H).
H  KW = By condition A, W(h 
is hounded. Hence for IKI small enough, H)1 is invertible, and hence (n.  HK) is invertible. Thus, X E p(HK) for IKI sufficiently small, that is, G C p(HK) provided IKI < (maxAE6 II W(a.  H)1 II)1 IIK W(%  H)'11
(2) To show that K equation:
H)1
< 1, so I  KW(% 
(A 
(A  FIK)1 = (A
HK)_

1
is analytic, we use the second resolvent
H)1 +K(A
 H)1 W(;,  HK)
1.
Upon iterating this equation, we obtain
(AHK)1
x [(KW(A_H)_)1].
=(AH)1 [5
(15.2)
1=o
Since II(W(A  H)1)'11 < II W(,l 
H)1II'
< M', for all A E G. the
series converges absolutely for IKI < M1. Hence, K analytic family of bounded operators for IKI < M.
HK)1 is an
Theorem 15.7. Let HK be as in (15.1), and assume condition A. Then all discrete eigenvalues of Ho = H are stable. Moreover, if Ai(K) are the eigenvalues of HK near the eigenvalue a. of Ho, then the total multiplicity of the ),.;(K)'s equals the multiplicity of A.
Proof. Let A be a discrete eigenvalue of Ho. Due to part (i) of Lemma 15.6, if r > 0 is such that rr  {zl Iz  XI = r} C p(Ho) (such an r exists as A E Qd(Ho)), then r r C flK p(HK), for IKI sufficiently small. Let PK  (2iri)1 A,, R(K, z)dz, where R(K, z)  (HK  z)1. Then due to part (ii) of Lemma 15.6, K > PK is analytic about Ko = 0 (see Problem 15.5). Hence A is stable. The remaining part of the theorem follows from Proposition 15.3. Problem 15.5. Prove that K * PK is analytic about K = 0. (Hint: Use the expansion (15.2).)
15.4
TypeA Families of Operators and Eigenvalue Stability: General Results
In this section, we discuss general analytic families of type A. Not only can we establish a stability result, but we will be able to describe the manner in which
15. Perturbation Theory: Relatively Bounded Perturbations
154
the eigenvalues )t (K) depend upon K. It follows from the proof of Theorem 15.7 that a.; (K) depends continuously on K as K > 0. In many cases, we will be able to show that this dependence is, in fact, analytic. We now introduce a family of operators generalizing (15.2) and condition A.
Definition 15.8. Let R be a nonenipty, open subset of C and suppose that for each K E R, TK is a closed operator with p(TK) 0. Then the family of closed operators TK. K E R, is said to be typeA if
(i) D(TK) is independent of K (which we call D);
(ii) for each u E D, T, u is strongly analytic in K on R.
Problem 15.6. Prove that a vectorvalued function ua on an open, connected subset R of C is weakly analytic (i.e., a E R > (ua, v) analytic) if and only if it is strongly analytic (i.e., a E R + ua analytic in the usual topology on N). In analogy with Lemma 15.6, we have the following lemma, which describes the main technical aspects of typeA analytic families.
Lemma 15.9. Let TK, K E R, be a typeA analytic family of operators. Then for any Ko E R, we have the following two statements. (i) Let G bean y bounded, open set such that G C
Then G C p(TK) for
IK  KoI .Sufficiently small.
(ii) For any ;, E G as in (i), K
Th )
is analytic in K for IK  KoI
sufficiently small.
Proof. (i) Let ,l E G and write TK = TKO + VK, where VK = TK  TK,,. Then by the second resolvent formula,
(ATK)~1 =(I
(15.3)
Now D = (,X  T,O1)7(, and so VK(a. is a closed, everywhere defined operator. Hence, by the closed graph theorem, Theorem A3.23, it is bounded. Furthermore, it is analytic in K for IK  KoI sufficiently small (see TKo)l
Problem 15.4). Consequently, because VK (),.  TK(,)1  0 as J K  KoI > 0, it follows that II VK(A 1, for all K sufficiently close to Ko. Then 1  Vk (X is invertible, and so ),. E p(TK) by (15.3).
(ii) For any A E G as in (i) and for IK  KoI small, we can iterate equation (15.3) to obtain 00
(J
TK)1
=(ATKo)Y' [VK(A(15.4) 1i=0
Now, for any e > 0 3 3 > 0 such that IK  KoI < S implies that II VK(; < E. Choose e < 1. Then as each term in the series on the TKO)111
15.4 TypeA Families of Operators and Eigenvalue Stability: General Results
155
right side of (15.4) is analytic on (KIIK  Kol < 8}, bounded by F", and as E" = (I  c) 1, it follows by the Weierstrass Mtest that the series is analytic in K On (K IK  Kol < 8}. Ell
I
The stability of isolated eigenvalues with respect to perturbations given by typeA families is easy to prove. If, in addition, the eigenvalue is nondegenerate, then we can also prove that the corresponding eigenvalue of TK is analytic in K.
Theorem 15.10. Let TK be an analytic family' of type A about Ko = 0. Let n. be a discrete, nondegenerate eigenvalue of To. Then there exists an analytic family) 0 such that Fr C nK P(TI) for IKI sufficiently small. Moreover, PK (the Riesz projection for TK and 1'r) is analytic in K. Hence, by the stability Theorem 15.2, TK has a nondegenerate, discrete eigenvalue for IKI small.
Problem 15.7. Prove that PK is analytic in K for small IKI.
(2) To prove the analyticity of ,(K), let t/r,
IIt/i1I
= 1, be an eigenfunction of
T o for A so that To* = AVf. Then, PKt/r + Vr as K ' 0, so I I P K ; 1. Moreover, TK PKt/I = A(K)PKi/r, so (z  TK)1 PK tG = (z  ,(K))1 PKY
Taking the inner product of this equation with *, we get (z
)L(K))1
= (V' PK*)1 (*, (z 
TK)1 PK`Y).
(15.5)
Now (V, PK>(r) is analytic for IKI small and (V, PoV) = 1, so (tf, PKVi)1
is analytic for lid small. Moreover, the second factor on the right side of (15.5) is clearly analytic in K. Hence, (z  X(K))1 is analytic in K, and as z E p(To), this implies that ),.(K) is analytic in K for IKI small enough. We now turn to the general theorem on stability for a discrete, degenerate eigenvalue. In general, a degenerate, discrete eigenvalue Ao splits into several eigenvalue
branches ,t(K) under a perturbation. These are the eigenvalues of TK which converge to as K  0. In the case for which TK, K E R is selfadjoint, each branch is an analytic function of K. In the nonselfadjoint case, the eigenvalues are branches of a multivalued function, that is, a function defined on a Riemannian surface. Theorem 15.11. Let TK be an analytic family of typeA about Ko = 0. Let )`o be a discrete eigenvalue o f To. Then there exist families At(K ), I = 1, ... , r, of discrete eigenvalues of TK such that (i) X, (0) = Ao and the total multiplicity o f the eigenvalues X/ (K), 1 = 1, is equal to the multiplicity of A.o;
...
, r,
(ii) each family A1(K) is analytic in KUP for some integer p (i.e., A/(K) has a Puiseux expansion); if TK is selfadjoint for K real, then the eigenvalues are analytic in K.
15. Perturbation Theory: Relatively Bounded Perturbations
156
Proof.
(i) Suppose that 1l is finitedimensional and dim 7{ = n < oo. Then TK is an analytic typeA family of matrixvalued functions, and the eigenvalues %./(K) are solutions of
det(J, TK)=0. Let F(ti, K)  det(A  TK ). Then F is a polynomial of degree n in % with leading coefficient 1:
F(A,K)=A"+a,,(K)A"
+...+a2(K)A+al(K),
where ai(K) are analytic in K. We know that F(Xo. 0) = 0. It follows from a classical result (see, for example, !Kn]) that for JKJ small, F(', K) has roots %/(K) near Ao, and these roots are given by the branches of one or more multivalued analytic functions having at worst algebraic singularities at K = 0. Consequently, these roots have a Puiseux expansion in K I /p, p E 7G+, and satisfy n.,(0) = ,o. If T, is selfadjoint for K real, then %./(K) is real and
p must be 1.
(ii) Suppose To has a discrete, degenerate eigenvalue ).o. Then dim Ran Po is finite. Since PK is analytic in K for JKI small, and PK + Po, we know that dim Ran PK = dim Ran Po, for all IKI small. Let MK = Ran PK. Then TK PK
restricted to MK is a linear transformation on a finitedimensional space. We want to apply the method of part (i), but the subspaces MK depend on K. Hence we use a simple device to map all the subspaces MK to Mo. We prove in Lemma 15.12 in the appendix to this chapter that for small IK I there exists an invertible map SK : MK > Mo that, together with its inverse, is analytic in K. Define EK = SK PK TK S, 1; then EK : Mo + Mo is analytic in K by construction. Moreover, as a (PK TK I MK) = a (EK ), the results of part (1) applied to EK establish the theorem for PKTK, and hence for TK.
Let us note that the idea behind the proof of this theorem is simple. The Riesz projection PK allows us to consider the problem for the finiterank operator PKTK.
The only technical problem concerns the construction of an analytic, invertible map SK : MK + Mo with analytic inverse. Once such a map is constructed, the spectral problem reduces to the study of the roots of the equation F(J', K) = 0 for K near Ko = 0. This is done using the implicit function theorem.
15.5
Remarks on Perturbation Expansions
We briefly consider the Taylor expansion of the eigenvalues k(K) about K = 0 for the case when Ao  A(0) is a nondegenerate, discrete eigenvalue of a typeA analytic family HK = H + KW, where W is relatively Hbounded. The Taylor series for A(K) is called the RayleighSchrddinger series. This case was studied
15.6 Appendix: A Technical Lemma
157
in Section 3 of this chapter. For K near 0, the operator HK has a nondegenerate eigenvalue that is analytic in K. The projection PK onto the eigenspace of HK with eigenvalue >`(K) was shown to be analytic in K (Problem 15.5), and PK * PO. To obtain a perturbation expansion, note that if S20 is an eigenfunction of Ho H0S2o = A0S20, then II PK Q011 + II S2o II as K > 0, whence it follows that for IK I
small: (Qo, HK
(K) _
Xo +K
(Qo, W PKQo)
II PK c2O II 2
II
(15.6)
PK QO II2
Now PK is expressible in terms of the resolvent: _ 27ri
PK
rr
(z 
HK)1dz,
for a contour Fr C fK p(HK ), and it follows from the Neumann expansion (15.2) that °C PK=EK' dz(zH)1[W(zH)1]1. 27r i 1
(15.7)
rr
1o
Substituting (15.7) into (15.6), we find the expansion oc cc
J,(K) = I,.o + K
anKn
\
_m =o bmKm where
an = and
bm =
1 27ri
1J ,
27r'
rr
00
) _ Ao+Ya/K1
,
(15.8)
I=1
J dz(Qo, [W(z rr
dz(S2p, (z  H)1 [W(z 
H)1]m
2o)
Although the terms are complicated at higher orders, in principle they can all be written out. The lowestorder term is a1 = (0o, WS20), which is a wellknown expression for the eigenvalue shift to first order. By Theorem 15.10, series (15.8) has a nonzero radius of convergence.
15.6
Appendix: A Technical Lemma
In order to treat the case of degenerate eigenvalues, we use the following result of Kato (see [K]). Lemma 15.12. Let MK = Ran PK be as in the proof of Theorem 15.11. Then there exists, for IKI sufficiently small, an invertible map SK : MK MO such that SK and its inverse are analytic in K about K = 0.
15. Perturbation Theory: Relatively Bounded Perturbations
158
Proof. (1)
We first construct two matrixvalued functions U and V that are holomorphic in a neighborhood of K = 0. Since pK2 = P, we have PK PK + PK PK = P,
(15.9)
and upon multiplying the left and right sides of this equation by PK, we obtain PK PK PI = 0.
(15.10)
[ P,, PK J. This is holomorphic in K about Ko = 0. By (15.9), we have (suppressing the K) Let QK
PQ = PP'P  PP' = PP', (15.11a)
QP = P'P, so that
[Q, P] = P',
(15.1 lb)
by (15.9)(15.11). We consider the firstorder linear system X'(K)= QKX(K).
(15.12)
By the standard theory of ordinary differential equations (e.g., the method of successive approximation; see [HiSm]), the initialvalue problem for (15.12) has a unique solution given X(0). Let U(K) be the solution corresponding to X(O) = 1. Similarly, let V(K) be the unique solution of
Y'(K) = Y(K)QK,
(15.13)
corresponding to initial condition Y(0) = I. Both U(K) and V(K) are analytic in K about K = 0.
Problem 15.8. Use the method of successive approximations to prove that the initialvalue problems for (15.12) and (15.13) have unique holomorphic solutions about K = 0.
(2) The matrixvalued functions U(K) and V(K) are inverses. To see this, we write the derivative
(VU)'=VQU+VQU=0, by (15.12) and (15.13). Thus, the function (VU)(K) is a constant, and by the initial conditions, (VU)(K) = (VU)(0) = I. Since the spaces are finitedimensional, this implies that V(K) = U(K) 1 . Finally, we have to show that U(K) : Mo > MK. This is equivalent to showing that
PK = U(K)P0U(K)',
15.6 Appendix: A Technical Lemma
159
or U(K) PO= PKU(K).
(15.14)
Consider the derivative of PK U:
(PU)'
=
_
P'U + PU' = P'U + PQU [Q, P]U + PQU = QPU,
(15.15)
where we used (15.1lb) and (15.12). Equation (15.15) says that PU is a solution of (15.12) with initial condition Po. By uniqueness, this means that
PKU = U PO, which is (15.14). Hence, we take SK  U(K) = V(K) in Theorem 15.11.
16
Theory of Quantum Resonances I: The AguilarBalslevCombesSimon Theorem
16.1
Introduction to Quantum Resonance Theory
We have studied the discrete and essential spectrum of a selfadjoint operator H. In particular, if H = A + V is a Schrodinger operator with V in the KatoRellich class, then H is selfadjoint on D(0) = H2(R" ). If, in addition, limlxI_,Cc V(x) _ 0, then we know that aess(H) = [0, oc). Under these conditions on V, it is physically reasonable that there are no bound states, that is to say, eigenvalues, of H with positive energy. This is because, as follows from the principles of quantum tunneling discussed in Chapter 3, the wave function for such an eigenvalue will tunnel through the potential barrier and eventually reach the region where E > V. In this unbounded region, the particle will behave like a free particle and escape to infinity. To be more specific, suppose V is such that a classical particle with positive energy E > 0 is trapped by potential barriers (see Figure 16.1). In contrast to the situation of Chapter 3, we now assume that the barriers decay to zero at infinity. That is, we assume that the set, which we call the classically forbidden region at energy E as in Chapter 3, defined by {x V(x) > E} = CFR(E), separates R" I
into two disjoint connected sets: the potential well W(E) at energy E, and the exterior region 6(E). We assume that the potential well W (E) is bounded and that the complement of W(E)U CFR(E) in R", which is E(E), is unbounded. Any classical trajectory with energy E beginning in W (E) will remain in W (E) for all times. We now consider a quantum state *t with initial condition *0 localized in W (E) and such that the energy of >/ic is approximately E. If the potential barrier CFR(E) is very large, we expect that i/r, remains localized in W(E) for a long time. However, the quantum tunneling effect will eventually cause the wave packet to
162
16. Theory of Quantum Resonances 1.
xo
FIGURE 16.1. A typical resonance situation: the potential well W(E) and the classically forbidden region CFR(E).
decay away from W(E). In fact, if BR(O) is a ball of radius R centered at 0 and XR is the characteristic function on BR(0), one can show that lim T ,x, T
f o
T
II XR
,
ll'dt = 0.
(16.1)
This is part of the RAGE theorem; see [RS3]. Note that if H had an eigenvalue at E > 0 and if we take *o to be the corresponding eigenvector, then IIXRVJrII = II XR V'o II, and the integral in (16.1) is a nonzero constant.
Thus, although t/i, does not behave like a bound state, if W(E) is deep and the potential barriers are high, it is suggested by the WKBform of the wave function in the CFR(E) given in (3.6), that t/r, remains concentrated for a long time in W(E). Such an almostbound state, characterized by the fact that it has a finite lifetime, is called a quantum resonance. We will give a precise definition in Section 16.2. In theoretical physics, resonances are used to describe states with finite lifetimes, such as unstable particles. They are usually associated with poles of the meromorphic continuation of the Smatrix. The famous BrietWigner formula associates a resonance at energy E with a large increase or bump in the scattering
crosssection at energy E. However, it is quite difficult to give a mathematical description of resonances in this way. Rather, we will work with the resolvent of H and its meromorphic continuation to define quantum resonances. In the remainder of this chapter, we will define quantum resonances as poles of the meromorphic continuations of certain matrix elements of the resolvent. These poles will be identified as the eigenvalues of nonselfadjoint operators constructed
from H. We give a complete exposition of the theory developed by Aguilar and Combes [AC], Balslev and Combes [BC], and Simon [Sim9]. In the first two papers, dilation analytic techniques were used to prove the absence of singular continuous spectra for 2 and Nbody Schrodinger operators, respectively. Simon [Sim9] applied these methods to define quantum resonances. In the following chapters, we will study the existence of resonances in the generalized semiclassical regime for various twobody Schrodinger operators. The notion of "semiclassical regime" makes precise the earlier statement "for W(E) large enough." The term "generalized" refers to the fact that the parameter that we vary is not necessarily h (as in Chapters 1112) but may be the electric or magnetic field strength, or so on, depending on the problem. The main idea, however, is that in varying a parameter
16.1 Introduction to Quantum Resonance Theory
163
the quantum system becomes close, in a prescribed way, to a system in which quantum tunneling is suppressed. Such a system is similar to the classical system. For such a "quasiclassical system," the resonances appear as actual bound states of an approximate Hamiltonian. We want to emphasize that the resonances of H do not correspond directly to any spectral data for the selfadjoint operator H. Resonances can be detected by examining the spectral concentration of the continuous spectrum of H (see [RS41), but this does not provide a convenient criterion. Since resonance energies are complex (with negative imaginary part), there are no L2eigenfunctions of H at these energies. (We will not discuss zero energy resonances.) In the cases for which the Smatrix for H and Ho = A exists and has a meromorphic continuation, the quantum resonances of H appear as the poles of this continuation. Therefore, they are intrinsic for this class of Schrodinger operators. The study of resonances is closely connected with the absence of positive eigenvalues for certain families of Schrodinger operators, as indicated previously. We will, in fact, use the absence of positive eigenvalues to establish that resonances have nonzero imaginary parts. Although we will not prove the absence of positive eigenvalues in this book, we give a theorem that will suffice for the examples we study in the following chapters. This theorem can be proved using the method of Froese and Herbst [FrHe]. A textbook discussion can be found in [CFKS].
Theorem 16.1. Suppose V is in the KatoRellich class and satisfies the following two properties: (1) The potential vanishes at infinity, lim lIx U.00 V W = 0.
(2) The operator x VV(x) is relatively Laplacian compact. Then H = A + V has no strictly positive eigenvalues. Before turning to the technical machinery necessary to make these ideas concerning resonances precise, let us sketch the general scheme further. Suppose we are given a Schrodinger operator H(A) depending on a parameter A. We suppose that for A large, quantum tunneling effects are strongly suppressed (,l = I /h is the typical case). This will lead us to an approximate operator Ho(A), which will have only eigenvalues in some positive energy interval I that interests us. By contrast, H(A) will typically have essential spectrum in I. We want to compare and H0(A) in the interval I. This seems difficult at first because of the different spectral properties of H0(A) and H(A) in I. The first step is to replace H(A) by a family H(A, 0), B E D C C, of nonselfadjoint operators. This family has the property that the essential spectrum near I has been moved off the real axis. As the essential spectrum has been deformed, we call H(A, 6) a spectral deformation family for H. We develop the general theory of spectral deformation in Chapter 17 and study its application to Schrodinger operators in Chapter 18. We can now apply perturbation theory to compare H(X, 6) with Ho(X). These two operators will not be close in any usual sense of perturbation theory discussed
164
16. Theory of Quantum Resonances 1.
so far. However, the theory we now develop requires only that the difference
V(n, 0)  H(%., 0)  Ho(n) be relatively small when localized to the region (corresponding to CFR(F_ ), E E 1)
where V(n) is large. The perturbation theory works because, as we have seen in Chapter 3, quantum mechanical quantities. for example, wave functions, are small
in such a region. It is reasonable to expect that the difference of the resolvents localized to such a region will also be small. This type of perturbation theory, which
differs from the analytic perturbation theory of Chapter 15, is called geometric perturbation theory. We discuss this, along with other general aspects of eigenvalue stability, in Chapter 19. This will allow us to conclude the existence of complex eigenvalues of H(? , 0) near the (real) eigenvalues of Ho(n). Finally, these complex eigenvalues are related to the poles of the meromorphic continuation of the matrix
elements of the resolvent, and hence to the resonances of H, by the AguilarBalslevCombesSimon argument. This establishes the existence of resonances of H near the eigenvalues of Ho(n) in I for large n. We now turn to the definition of resonances and the AguilarBalslevCombesSimon argument.
16.2
AguilarBalslevCombesSimon Theory of Resonances
The AguilarBalslevCombesSimon theory gives a precise meaning to the notion of quantum resonance. Let us consider a Schrodinger operator H with essential
spectrum [0, oc) and bound states below zero. The resolvent of H, RH(z), is analytic in C \ a(H). In Corollary 5.7, we proved that the operator norm of the resolvent RH(Z) is bounded above by 1/1 Im zl. This is a reflection of the fact that
for z E a(H), the resolvent is an unbounded operator on L2(R"). If we relax the condition that the resolvent be considered on the space L2(R" ), it is a classical result
that the resolvent of a Schrodinger operator (for suitable potentials V) remains a bounded operator, as Im z + 0, between other, weighted, Hilbert spaces. This result, called the limiting absorption principle, is discussed in Reed and Simon, Volume IV [RS4] and in Cycon, Froese, Kirsch, and Simon [CFKS]. Another way in which the boundary values of the resolvent can be controlled is to consider matrix
elements of the resolvent between vectors in L2(W) with certain nice properties. In general, there is a discontinuity in the matrix elements (f, RH(z)g), for suitable f, g E L2(1I' ), as we approach II8+ from above and below. This can be seen for the free Schrodinger operator Ho = A by studying the explicit kernel of (A z)1.

Problem 16.1. By methods of Fourier transform (see Appendix 4 and Section 4.3), construct Green's function Go(x, y; z) = (A  z)1(x, y) in even and odd dimensions. These can be expressed in terms of the Bessel function HO). Using these representations, study the discontinuity along R+ and study the meromorphic continuation of Ro(z). Your computations should lead to the following formula for the kernel:
16.2 AguilarBalslevCombesSimon Theory of Resonances
Ro(z`)(x, y) _
(L) 4
(27r
x}
165
H. ,(zlx  yD
Given that there is a discontinuity across the essential spectrum, one is tempted to construct a meromorphic continuation. This is one of the accomplishments of the AguilarBalslevCombes theorem. Hence, we are interested in studying the meromorphic continuation of matrix elements of RH(Z) through the discontinuity along IR+.
Definition 16.2. The quantum resonances of a Schrodinger operator H associated with a dense set of vectors A in the Hilbert space 71 are the poles of the meromorphic continuations of all matrix elements (f, RH(z)g), f, g E A, from
{zECIImz>0)to(zECIImz 0, the function Un/.. E L2(R ). It is easy to verify, then, that for
D  {z I IIm zI < 7r/4}, the map (6.i/i)c DxA >UoVf is an L2analytic map. This verifies (AO) and (Al) (except for the density of UAA, which is proven in Proposition 17.10). As for (A2), for 6 E I a simple computation shows that H0(0) = e29 A = e 20 Ho This formula defines Ho(O) for 6 E D. It is clear that (A2) is satisfied. Furthermore, the formula shows that Qess(Ho(6)) =
e2i Im B
The effect of the spectral deformation is to rotate the essential spectrum of Ho about
the origin through an angle of 2Im 0. We define 0 =_ C' U (z I arg z > n/2}. Hence, if we define DE as in (A3), we can take c2 = {z arg z > 2e}. Of course, we can take a as close to n/4 as we like. This increases the size of the sector Q . The spectrum of this family of operators is sketched in Figure 16.3. We will discuss in detail the technical aspects of Example 16.3 in Chapters 17 I
and 18, in which we present the theory of spectral deformation and its application to Schrodinger operators. However, this example contains all of the basic ideas concerning spectral deformation and resonances, and should always be kept in mind.
Given assumptions (A0)(A3), we can now formulate the AguilarBalslevCombes theorem concerning meromorphic continuations.
Theorem 16.4. Let H be a .selfadjoint Schrodinger operator with spectral deformation family U and analytic vectors A such that (AO)(A3) are satisfied.
(1) For f, g E A, the function Ffx(z)
RH(z)g),
(16.2)
defined for Im z > 0, has a meromorphic continuation across ress(H) = II8+
into c , for any s > 0.
168
16. Theory of Quantum Resonances I.
(2) The poles of the continuation of Ffg(z) into S21 are eigenvalues Qf till the operators H(O), B E D+, such that aess(H(6)) fl Q _ 0.
(3) These poles are independent of U in the following sense. If V is another spectral deformation family for H with a set of analytic vectors AV such that (A I)(A3) are satisfied and A n AV is dense, then the eigenvalues of
H(9) = VeHVA ', B E D+, in 0 are the same as those of H(8) in this region.
The AguilarBalslevCombesSimon theory identifies quantum resonances, as defined in Definition 16.2, as the eigenvalues of the spectrally deformed Hamiltonians H(8) in the lower halfplane. In the next two chapters, we will discuss how to construct such families of operators and identify their spectrum. The price we pay for such a description is the fact that we must work with nonselfadjoint operators. Despite this, the theory is a powerful tool for investigation of resonances in quantum mechanical systems (as we will study in Chapters 20 and 23) and for the numerical computation of resonances in atomic and molecular systems [Sim101. We also mention that the technique of spectral deformation allows one to study the perturbation of embedded eigenvalues. In particular, if H has an embedded eigenvalue X, then ; remains in the spectrum of H(6) and will be an isolated eigenvalue. We can then apply the methods of analytic perturbation theory to H(6) and X.
16.3
Proof of the AguilarBalslevCombes Theorem
We now give the proof of the AguilarBalslevCombes theorem. We invite the reader to follow through the proof with the simple example of the Laplacian using the representation of the resolvent given in Problem 16.1. Proof. (1) With Ffg(z) defined in (16.2), assumption (A0) implies that Ffg is analytic on C \ R. Fix Z E C. For B E II8 fl D, Ue is invertible with U;1 = UB , and so we can write
Ffg(z) = (Uef, (UORH(z)UA')Ueg).
(16.3)
By (Al), 0 E D > Uef, Ue$ are analytic maps. Furthermore, we have UoRH(z)UB ' = RH(e)(Z)
(16.4)
Condition (A2) guarantees that 0 E D > RH(e)(z) is an analytic map provided z that
a(H(B)). Since we can write U6 f on the left in (16.3), we see
169
16.3 Proof of the AguilarBalslevCombes Theorem
H E D + Ffg(z;9) = (Uhf. RH(e)(z)Uog)
(16.5)
is an analytic map provided z stays away from a(H(9)). We now choose
F > 0 and fix z E Q'. The function Fig(z, 0), defined for 9 E D n R, can now he extended in 6 into DE by (A2) and (A3). We fix 9 E DF according to (A3) so that ae (H(9)) n 0t, = 0. It follows that Ffg(z, 0) Q, n C. Now can be meromorphically continued in z from Q into Q recalling (16.3), which says that Ffg(z. 0) = Ffg(Z), Z E S2f , the identity principle for meromorphic functions IT1] says that there exists a function meromorphic on c2 which equals F fg(z) on Q'. This function provides the meromorphic continuation into QE
(2) The meromorphic continuation of Ffg(z) into Q. is given by the matrix elements of RH(O)(Z) in the states fe and go, which denote the continuation
of Uo f and Ue$, respectively. Condition (Al) states that such vectors in
U(9)A, 9 e D, are dense. Consequently, if H(9) has an eigenvalue at X(9) E QT, Ffg(z) will have a pole there. Conversely, if the continuation of Ffg(z) has a pole at a.(9) E QE , then it must be an eigenvalue of H(9).
Problem 16.2. Verify this last statement by showing that the corresponding Riesz projection for H(9) is nonzero.
Problem 16.3. By the above analysis, conclude that the poles of the continuation of Ffg() into Q are independent of 9. (Hint: Use uniqueness.) Conclude that if
E Qe is an eigenvalue of H(9), 9 E DF, then it is also an eigenvalue of H(9'), E DF, provided it remains away from oes,(H(9')). Problem 16.4. Prove the third part of Theorem 16.4. (Hint: Again use uniqueness.) With these problems, the proof of Theorem 16.4 is completed.
We would like to call attention to certain points of Theorem 16.4.
Corollary 16.5. Under the hypothesis of Theorem 16.4,
(i) ad(H(9)) n Q = 0, 0 E D+; (ii) any A E ad(H(9)), 0 E
D+,
is independent of 9 provided X V or .. (H(O));
(iii) if A is an eigenvalue of H in Q, n IR, then A E ad(H(9)) provided A aess(H(9)) for 0 E D+.
16. Theory of Quantum Resonances I.
170
Proof.
(i) Since Ffg(z;0) = Ffg(z) for 0 E D+ and Z E S2F, and H is selfadjoint. a pole of the left side would imply that H had a complex eigenvalue in SZE .
(ii) The continuation of Ffg(z) is unique and independent of 9. Hence as A E ad(H(9)) is a pole of this continuation (for some f, g c A), it is independent of 9.
(iii) If )A E ad(H), the corresponding projection is
P = (27ri)I
RH(z)dz,
(16.6)
where r is a simple, closed contour about ),. If A remains isolated from aeSS(H(9)), 9 E D+, then we proceed as follows. Let f, g E A. and consider the matrix element (16.7)
for 9 E ll n D, where Po  U9PU©'
=(2iri)i
RH(o)(z)dz.
(16.8)
The right side of (16.7) has a continuation in 9 into D. By uniqueness of the continuation and the identity (16.7), the Riesz projection PO, 0 E D+, exists and is nonzero. Hence, ), E ad(H(9)). Now suppose that A is an embedded eigenvalue of H in S2F n R. In Section 6.4, we proved that the projection for A is given by
P = s  lim(iE)(His)' 0 F
Again, we conjugate P with Uo, 9 E R n D, and take matrix elements for f, g E A to obtain
(f, Pg) = lim(iE)(U©f, RH(o)(A+iE)Uog). Fo
(16.9)
The right side can be continued into D. Hence, the weak limit exists and is nonzero:
Py = w  lim (i E)RH(o)(;, + iE). E+0
However, as P2 = P, an application of the first resolvent formula shows that Pot = Po. Hence, PO is a Riesz projection for H(9) and, as PO =/ 0, this implies that )< E a(H(9)). (Note that this argument applies as long as A remains isolated from aeSs(H(9)) or, at most, A remains on the boundary of El
Problem 16.5. Prove that PH = Po as in the proof of Corollary 16.5.
16.4 Examples of the Generalized Semiclassical Regime
171
Problem 16.6. Suppose A is an isolated eigenvalue of H of finite multiplicity. Assuming (AO)(A3), use typeA analyticity to give another proof that.. E (r(H(H)), provided A is away from cre,s(H(0)), H E D+. (Hint: Use analyticity and the fact that U0 is unitary for 0 e R fl D.) It follows from Theorem 16.4 and its corollary, that the resonances of H, R(H ),
in the sector Q. C C` can be given as
R(H) n c
= U 6d(H(6)).
(16.10)
HED,!
Of course. we take e as large as possible, but it may be that by the spectral deformation method we do not get all resonances of H.
16.4
Examples of the Generalized Semiclassical Regime
We want to mention some physical situations where resonances have been shown to exist. We will only consider twobody, timeindependent Hamiltonians. There are
two basic types of resonances whose existence and properties can be established using the techniques described later in this section: (1) shape resonances and (2) Stark effect resonances. In addition, resonances play an important role in the study of the Zeeman effect. We will treat shape resonances in detail. The Hamiltonian
H(h) _ h2 + V is considered in the semiclassical regime of small h. The potential V vanishes at infinity, V > 0 (for simplicity), and forms a positive barrier that traps classical particles with energies in an interval (0, E0). By rescaling
H(h), we consider H(X) = A+A2V as A > oo. In the large a, regime, quantum tunneling through the barrier is suppressed. This is seen by noting that the distance across the barrier in the Agmon metric diverges as A + oo. We can see the same phenomena in many Stark Hamiltonians. A Stark Hamiltonian describes a particle moving in a static electric field F and potential V. It has the form H(F) = A + V + F x. Suppose V is an attractive Coulomb potential so H(0) has bound states. When BFI 0, the spectrum of H(F) is R and there are no longer any eigenvalues. We think of the eigenvalues of H(0) as becoming resonances since, due to F, the quantum particle can tunnel through the Coulomb barrier in the direction of F. Again, when IFI > 0, the Agmon distance across this barrier diverges. Hence, the electric field F for which IFI is very small is the semiclassical regime for this problem. A similar situation occurs in one dimension when V is a periodic potential. This is known as the Stark ladder problem. Each potential minimum of V contributes a resonance when F 0, and due to the periodicity of V, we get a sequence or ladder of resonances. Finally, we can consider a hydrogen atom in 1183 in an external, constant magnetic field in the direction. This is the Zeeman effect. The Hamiltonian for a particle in the magnetic field (0, 0, B) with vector potential A = B(y. x. 0) is
;
2B2
H(B)=A+(x2+yz)aBL,+V, 01
172
16, Theory of Quantum Resonances I.
where a is a constant involving the electric charge and L=  i (x
 y x) is
the third component of angular momentum. Neglecting the Lterm. let us analyze Ho(B)+ V, where + HO (B)
of
B 4
(x,
(+ v) +
; 1
dz
on L2(R3) = L 2(LR2)OL2(R). The twodimensional operator 110 2 (B) A.,,, + a' a' (x` + y2) is simply a twodimensional harmonic oscillator Hamiltonian. The a spectrum is discrete (Theorem 10.7), and the eigenvalues {Ef(B)}, are called the Landau levels. Since a(d ) = 10, o o) on L2(R), it is not hard to check that x;
o(Ho(B)) = U[Ei
x)
by constructing approximate eigenfunctions 4i(x, y)e' , where Ei L (B )O; . We now consider the perturbation V, which we take to be
V(x) _  Ilxll  i.
(16.11)
that is, Coulombic. If V had the special form  Izl ', the effect would be to add a sequence of eigenvalues {ei }, with e, * 0, accumulating at each E. In the case with V Coulombic, (16.11), we find eigenvalues accumulating at EL . Even though these eigenvalues are not analytic in B at B = 0, the RayleighSchrodinger expansion predicts their behavior in an asymptotic sense. We will see that this behavior is related to the resonances of H(i B), that is, to the tunneling situation 0, the lifetime of these resonance states grows, portrayed in Figure 16.4. As B and so small B is the semiclassical regime. In terms of tunneling phenomena, if we apply dilation analyticity to
22 V(x,y,z)=a B (x2+y2)Irl,
FIGURE 16.4. The potential for the atomic Zeeman effect with a purely imaginary magnetic field.
16.5 Notes
we obtain V0(x, 1, Z) =
e20
a2 B2
(x2
4
+V 2)
aB  Irl
173
.
the sign of the oscillator potential becomes negative. At this value, Setting 0 the eigenfunctions of the Coulomb potential can tunnel through the barrier formed by the inverted oscillator potential and escape to infinity. The width of the barrier
becomes infinite as B * 0.
16.5
Notes
The theory of resonances in quantum systems is almost as old as quantum mechanics. In Chapter 20, we will discuss the shape resonance model, which dates from 1928. Resonances occur in virtually all areas of quantum physics: the theory of atoms and molecules, nuclear and elementary particle physics, and the theory of solids. As mentioned in the Introduction to this chapter, the physical picture of a resonance in these systems is a state that decays very slowly. Resonances are often described as the poles in the meromorphic continuation of the Smatrix or of the resolvent. This relationship between poles and time decay can be seen very easily if one formally writes the inverse Laplace transform representation of the timeevolution group for the Schrodinger operator H:
eirH
(2ni)i
F o Im J
etE(, (H E 
ie))dE,
where we have neglected any negative eigenvalues. Suppose that the matrix element of the resolvent can be continued in the complex energy plane so that the contour can be deformed into the fourth quadrant. If the continuation of the resolvent has
a pole at Eo  i r, with r > 0, then by the residue theorem, the coefficient of this term in the integral is
eitEptr
which decays exponentially in time. This is called the singlepole approximation
and suggests that the phenomena of resonances can be described by poles of the meromorphic continuation of matrix elements of the resolvent. Mathematical justification of this singlepole approximation is quite difficult. We will discuss some recent results in Section 23.4 (see Gerard and Sigal [GS], Hunziker [Hu3], and Skibsted [Skl], [Sk2], [Sk3]). The mathematical theory of resonances seems first to have been established for onedimensional and spherically symmetric systems. One can look for solutions to the ordinary differential equation with complex energy and certain asymptotic behavior at infinity. The existence of resonances can be reduced to the existence of complex zeros of a Fredholm determinant. For a discussion of the physics of resonances and the onedimensional and spherically symmetric cases, we refer to the book of R. G. Newton [Ne]. We also refer the reader to the paper of Harrell and Simon [HaSi], which, among other things, discusses resonances for onedimensional
174
16. Theory of Quantum Resonances I.
and spherically symmetric systems and gives a good historical overview. The reader can find there a discussion of the Zeeman problem and the Stark effect (see Chapter
23). The situation is more difficult in arbitrary dimensions. Since resonances do not belong to the spectrum of the selfadjoint Schrodinger operator (which is to say, the resonance states are not eigenfunctions of H in the Hilbert space), it is difficult to study them by looking at H alone. One way in which one can infer the existence of resonance states by examining H alone is through the notion of spectral concentration mentioned in the Introduction. The notes to Chapter XII of Reed and Simon, Volume IV [RS4] contain references to this idea and some earlier investigations. Resonances defined in the AguilarBalslevCombesSimon theory are intrinsic to the Schrodinger operator H = A + V and the set of analytic vectors. Aspects of the relationship between H and the set of analytic vectors have been explored by Howland [Hol I. In many situations, resonances can be identified with the poles of the meromorphic continuation of the scattering matrix, which depends only on the pair [Ho = A, H). Balslev established this result for shortrange, dilation analytic potentials in [BI]. Gerard and Martinez [GM2] proved that a suitably defined Smatrix for a family of analytic, twobody, longrange potentials has a meromorphic continuation and that the poles of this continuation coincide with the resonances. Other results on the connection between scattering theory, the poles of the scattering matrix, and resonances can be found in Hagedorn [Hal and in Jensen [J2]. There are other approaches to spectral deformation. Local distortion methods in momentum space were developed by Babbitt and Balslev [BB I and by Balslev [B2]. Using this method, Jensen [JI] showed that resonances, defined as poles of the meromorphic continuation of the resolvent, coincide with the poles of the meromorphic continuation of the scattering matrix for a class of twobody potentials.
Resonances in quantum mechanics can also be defined without the analyticity assumptions of the AguilarBalslevCombesSimon theory. They can be defined directly in terms of their physical properties, as done by Lavine [La]. If the twobody potential is exponentially decaying, one can prove that the Green's function has a meromorphic continuation as a bounded operator between exponentially weighted Hilbert spaces. A theory of resonances for Schrodinger operators with potentials that are the sum of an exponentially decaying potential plus a dilation analytic potential was developed by Balslev [B3] and extended by Balslev and Skibsted [BS I]. These authors also explore the relation between the poles of the scattering matrix and resonances for these potentials. Orth [Or] developed a theory of resonances for families of perturbations based on the limiting absorption principal and the Livsic matrix. Gerard and Sigal [GS 1 developed a theory of resonances in the semiclassical regime directly in terms of the propagation properties
of a resonance. They introduce the notion of a quasiresonance as a solution to the Schrodinger equation with certain outgoing propagation properties. They also show that such quasiresonances exhibit the expected exponential timedecay behavior (see Section 23.4). We mention the approaches of Davies [DaJ, the wave
16.5 Notes
175
equation approach of Lax and Phillips [LP], and the microlocal approach of Helffer and Sjostrand [HSj4] (we will comment more on the microlocal approach in Chapter 23).
One can also ask if the resonances defined by the AguilarBalslevCombesSimon theory give rise to the characteristic scattering signatures that physicists associate with resonances. The scattering theory for shape resonance models was studied by Nakamura [N2, N31. The famous WignerBriet formula was proved in the semiclassical regime by Gerard, Martinez, and Robert [GMR].
17
Spectral Deformation Theory
17.1
Introduction to Spectral Deformation
The AguilarBalslevCombesSimon theory of resonances identifies the resonances of a selfadjoint operator H with the complex eigenvalues of a closed operator H(O), which is obtained from H by the method of spectral deformation. In this chapter, we present the general theory of spectral deformation. This technique is applicable to many situations in mathematical physics, such as Schrodinger operator theory, quantum field theory, plasma stability theory, and the stability of solutions to certain nonlinear partial differential equations [Si4]. We will discuss the application to Schrodinger operators in Chapter 18. Originally, spectral deformation theory was formulated for the dilation group. The Schrodinger operators to which it applied had to have dilation analytic potentials. Through the contributions of various researchers, these constraints were relaxed (see the Notes at the end of this chapter). The theory that we discuss here was presented by Hunziker in [Hu2]. The basic idea behind the spectral deformation method is as follows. We consider oneparameter families of diffeomorphisms on ]I8" generated by smooth vector fields. The families we choose will admit an extension to smooth maps on a neighborhood of ' in C" as the parameter B becomes complex. For real B, any
such family induces a family of unitary operators UA on L2(W'). In this chapter, we concentrate on the construction of spectral deformation families, given smooth vector fields on R" and a corresponding set of dense, analytic vectors. In Chapter 18, we study how the spectrum of the conjugated Schrodinger operator H (B)  UH H UB ', t9 E R, deforms as 0 becomes complex. We remind the reader
17. Spectral Deformation Theory
178
that the basic strategy is illustrated in Example 16.3, where the dilation analyticity of the Laplacian is studied.
Vector Fields and Diffeomorphisms
17.2
R" be a smooth mapping (actually, only C' is necessary). Consider, for H E JR, the related family of maps OH: R" > J8" defined by Let g: W1
0H(x)=x+Hg(x).
(17.1)
Since 0tt(x) = x, we expect Op to be invertible for 0 sufficiently small.
Examples 17.1.
(1) If g(x) = x, then O)(x) _ (I +H)x. Comparison with the map x > eox, for 101 small, shows that g is the infinitesimal generator of dilations on R". This comparison can be done by taking the Taylor expansion of e&x in H about 0 = 0. This map plays a particularly important role in the theory.
(2) For a fixed vector e c JR", we define g(x) = e. We then have 00(x) _ x + Be. This is simply a oneparameter family of translations on R" in the direction e. Such maps 00 are invertible and form a oneparameter group of diffeomorphisms on R". This family of maps plays a role in the theory of the Stark effect, as we will see in Chapter 23.
To study the maps 09, H E R, we need to consider the derivative. We let Df denote the derivative of a map f : W1 > J8". The derivative Df is a linear map on R". Associated with this map Df is the n x n real Jacobian matrix relative to some basis of J". We will write
Df(xo)= \2axi (xo)f .
(17.2)
The determinant of this matrix is the Jacobian determinant, which we denote by Jf(x) = detl
"" (x) axi
.
(17.3)
When we consider a family 00, we will write Jo for J0,,. Returning to the family Oo, note that DOe(x) = 1 +6(Dg)(x), (17.4) where I is the n x n identity matrix. We denote by Mi the inverse of the supnorm of the derivative of g, 1 rsup IIDg(x)IIJ Ml = I
L Eli'
where II
I
(17.5)
II denotes the operator norm on the set of linear transformations on ft8".
17.2 Vector Fields and Diffeomorphisms
179
Lemma 17.2. Let g: W' > R1 be a smooth vector field. Then the map ¢y defined in (17.1) is a diffeomorphism of R" for 101 < Mi.
Proof. It follows from (17.4) that if J01 MI ' < 1, then D06 is invertible. The inverse is given explicitly by
(D4 )' _
(1)"9"(Dg)n,
(17.6)
n
which, for any x E R", is an absolutely convergent series provided 101 < Mi. Hence, by the inverse function theorem (see, for example, [HiSm]), 4o is invertible on I[8" provided 101 < Mi. 0
Remark 17.3. Some insight can be gained from the theory of ordinary differential equations (ODE) (see, for example, [HiSm]). Suppose v: R" * R" is a vector field with II DvlIx < oo. We consider the ODE d 4,o
dO
=vo0e,
(17.7)
with initial condition
Standard theory then implies that there exists a solution 08, 6 E R, to this ODE (17.7) such that for any x E IR", 4)0(x) = x and 00 o 00,(x) _ 4)o+o'(x), 0, 6' E R. We say that 0e is the global flow generated by v. If we consider the map 0 E IR ¢o(x) E R", for fixed x and for 101 small, the Taylor expansion of this map about
9=0is 00(x) = x + 0 V(x) +
d2 Wt2
4'(x)
02,
(17.8)
S
for some s E (0, 6). We see that (17.1) is the infinitesimal version of the global flow generated by g (if such a global flow exists). Problem 17.1. Consider the two vector fields given in Examples 17.1. Verify that the global flows exist, and check the assertions in Examples 17.1.
Example 17.4. The theory of exterior dilations relative to the ball BR(0) is based on a smooth map g: R " * R" which satisfies
g(x) = 0,
jxI < R,
g(x) = x,
jxl > 2R.
Problem 17.2. Complete the construction of this vector field, and verify that it generates a global flow.
180
17. Spectral Deformation Theory
17.3
Induced Unitary Operators
We now consider the behavior of functions under the action of the maps 00 generated by a smooth vector field g. For any f E S(R" ), we define a map UN on S(IIR" ) by
(17.9) (Uef)(x)= Ja(x)'f(Oo(x)), B E R. Proposition 17.5. The map Uf) defined in (17.9) maps S(w") into S(W). For I jBj < M1 and real, Ue extends to a unitary operator on L2(R") and Ue strong/v as 0  0.
Proof. We leave it as a problem to check the first part of the proposition. As for the second, we note that for f c S(R"), IIUof!j2
=
f Jo(x)If(Oo(x))l2dx
f
Jo(4tie 1(v))I.f(v)12det[DOO 1(y)1dv,
(17.10)
where we took v  ¢o(x). Consider the identity, valid for all y E R", 00 0,9
y.
1
(17.11)
Differentiating both sides of (17.11) with the aid of the chain rule, we obtain DOo(.00 1(y)) (DOf7 1)(y) = 1.
(17.12)
Taking the determinant of both sides of (17.12), we obtain
Jo(09 1(y)) det[DO, 1(y)] = 1. We can use this formula in the right side of (17.10) to obtain
lIUofll = Ifll,
0 E R.
(17.13)
and so Uo is an isometry. We define a map Vo on S(R" ), which is similar to U0, as
(Vef)(x) = Joe I(x)f(0B 1(x)).
(17.14)
We leave it as an exercise to check that Ve is an isometry and that on S(R"), V9 U9 = U0 Vo = 1.
(17.15)
for 161 < M1. Hence, by the density of S(R") in L2(R") and facts (17.13)(17.15), we see that Up is unitary and that U; 1 = V0r for jBj < M1.
Problem 17.3. Fill in the details, and complete the proof, of Proposition 17.5. For the last part, the Lebesgue dominated convergence theorem, Theorem A2.7 of Appendix 2, will be useful,
17.4 Complex Extensions and Analytic Vectors
181
Problem 17.4. Suppose 6 E R  0e is a global flow on R" as described in Example 17.4. Defining UI) as in (17.9), show that (Ue 19 E R) forms a strongly continuous, oneparameter unitary group.
Examples 17.6. For the vector fields in Examples 17.1, we have the following unitary groups. The dilation group is given in IRS" by
(Uef)(x) = e 'T f (ex).
(17.16)
The oneparameter translation group in IRS" with direction e is given by
(Uef)(x)= f(x+9e).
(17.17)
Note that in both examples, UB 1 = U_s.
17.4
Complex Extensions and Analytic Vectors
As discussed earlier in this chapter, we are interested in deforming R' into C" by allowing the parameter 9 to become complex. We now investigate the manner in which we can extend the operators Ue, implementing ¢y, from 6 E R to 8 E C, at least for small 101. Of course, for O E C, the operators UB will no longer be unitary. Formula (17.9) indicates that such an extension will be possible provided J9(x)112
and f have extensions into some complex neighborhood of R'. Formula (17.4) shows that Jp(x)1 /2 extends analytically to complex 8, provided 101 < M1, since the determinant map preserves the analyticity in 6 of the matrix on the right side of (17.4). The condition 161 < M1 guarantees that the argument of the function stays away from the branch point.
As to the second problem, the analyticity of f, we need to find a dense set of functions in L2(R") that are the restrictions to R" of functions analytic on a small complex neighborhood of R" in C", and such that f o 00 remains in L2(IR") as a function of x for 161 < M1. For such a function f, the map 6 E C, 181 < MI, to the vector UA f E L2(R") will be strongly analytic. We introduce a large class of analytic functions as follows. Definition 17.7. Let A be the linear space of all entire functions f (z) having the property that in any conical region CE,
C.  {z E C" I Ilm zI 0, we have for any k E N, lim
WIzI'If()I=0.
Iz
ZEC,
We note that A is not empty since any entire function of the form
f (z) = ea`2 p(z),
182
17. Spectral Deformation Theory
for a > 0 and any polynomial p, belongs to A. We now use the linear space A to define our set of functions in L2(W'). Definition 17.8. The set of analytic vectors in L'(W?) is the set of i/i E L2(W1) such that 3f E A and fi(x) = f (x), x E JR".
This definition is a generalization of one coming from the theory of unitary group representations. We recall Stone's theorem (see [RS21), which establishes a onetoone correspondence between strongly continuous, oneparameter unitary groups {UH 8 E If8} and selfadjoint operators. Each such group is generated by a selfadjoint operator A, in the sense of Problem 17.5, and conversely. In the finitedimensional case when A is a symmetric matrix, we have 1
UH = exp(i8A),
where we define the exponential function of a matrix by its power series. For a matrix A, this power series converges absolutely. In the infinitedimensional case, if A is a bounded, selfadjoint operator, then the series again converges in norm. In the general unbounded case, however, the series does not converge in norm and one has to define the right side through the functional calculus. However, the series
might converge strongly, at least for certain vectors. A vector * E L2(R') is said to be analytic for A if the power series oc
on
Eni1 n
n=0
has a nonzero radius of convergence. On such vectors *, the map 8 E If8 4 U0 V1
can be continued to a small complex neighborhood of the origin. Problem 17.5. Consider the unitary groups of Examples 17.1. Construct the selfadjoint operators A that generate these groups. Here, the formula Hm
(Ue 8
1)f = iAf
(17.18)
will be useful (think of the matrix case). Prove that A is selfadjoint. Construct the set of analytic vectors for both generators (i.e., give an explicit description of the sets).
There is a theorem of Nelson [RS2] that says that a closed symmetric operator A is selfadjoint if and only if its domain contains a dense set of analytic vectors. Combined with Stone's theorem, we see that in the case that our vector field g generates a global flow, so that the corresponding family U. 0 E R, is a unitary group, then we have immediately a dense set of analytic vectors. We can use these vectors to extend UHg, 8 E JR, into a small complex neighborhood of zero such that UBg remains in L2(lf1") (see [HS2] for this construction). However, in the general
17.4 Complex Extensions and Analytic Vectors
183
case that g induces only a local flow, we cannot appeal to these general theorems. Hence, we have to construct an appropriate set of vectors, as in Definition 17.8. We now explore the properties of A. Lemma 17.9. The set of functions in A restricted to R".form a dense, linear subset
of L2(l'!" ). Furthermore, for any f E A, f (z) E L' (R") for z E C, any e > 0. Problem 17.6. Prove Lemma 17.9. The denseness of the Hermite functions in L2(1R") will be useful.
We now consider the action of a spectral deformation family U on A. To do this, we must make an additional assumption on the vector field g that generates the family: (L) 31? > 0 and a constant
0, such that Ilg(x)II < Ilx11 +c1 `dx E II8" with
IIxII>R. We will always assume this in what follows. Consistent with condition L, we now normalize g so that MI = 1. Note that the vector fields of Examples 17.1 satisfy both of these conditions. We will see in the next section that the linear growth of g outside BR(0) greatly facilitates the calculation of the essential spectrum of H(0) for 9 E C. We define a domain Do in C about the origin by
D0= j6 EC 1101
R}, V0 E Do. Writing 0 = 01 +i62 E Do, we have
g(x)+0, Ilg(x)II'
(17.20)
IIRe zll`
=
11x 11`+201x
IIIm z11`
=
02I1g(x)II2.
(17.21)
(/21011 Ilgll)  4 Ilxll' +0rIlg(x)II'
(17.22)
Using the simple inequality I01x
g(x)I :
184
17. Spectral Deformation Theory
and (17.20) and (17.21), we obtain IIRe 2112  IIIm 2112 > 1 IIXII2
 101211g(X)III.
(17.23)
Next, restricting to Ilx 11 > R, the linearity assumption on g implies that for
18120, 11 Re Z1122  IIIm 2112
> >
2IIXII2
 IBI2(IIXI12 + C2)
EIIXI12C3 Ellg(x)112  C4 eltIm z112  C4,
(17.24)
where C4 is a finite constant. In the last inequality, we used (17.21). From (17.24), we obtain IIIm
z112
< (1 +E)' IIRe 2112 +c.
Since we have 11 Re z 11 > R, for any e > 0, we obtain IIIm zll < (1  E)IIRe z11, showing that Ran 0 , 9( x ), 0 E Do and llx 11 > R, lies in a cone C. Hence,
by the decay assumptions on f, f o 0e E L2(lR') (the Jacobian factors are uniformly bounded). Finally, to prove strong analyticity of Ue f , we note that the map
0EDo(i,Uef), for any i/r E L2(R"), is analytic on Do. Because weak analyticity implies strong analyticity, the result follows. Problem 17.7. Prove that weak analyticity implies strong analyticity in a Hilbert space.
(2) To prove that UBA is dense in L2(RI ), for any 8 E Do, we first note, as above, that the Jacobian factors are uniformly bounded for B E Do. Consequently, it suffices to show that the functions
f(0e(x))= f(x+Bg(x))
(17.25)
are dense for f E A. For any h E CO'W), define a sequence of functions hk by hk(X) _
(k) n
f
dy.1e(y)h(y)ek(xY09(Y))2.
(17.26)
17.4 Complex Extensions and Analytic Vectors
185
Since supp h is compact, the functions hk are entire and decay like akz2 for any 8 E Do (see Problem 17.8). Hence, for each k E N, hk E A. Fixing 0 E Do, we consider hk o ¢o. We prove that s  limk._,x hk o 00 = h, which proves the density of A in L2(R" ). Let the exponent in (17.26) for hk o 0,9(X) be designated by kVfo(x, y), where
Vo(x, Y) = [(x  Y) + 8(g(x)  g(y))]2 In the same manner as in part (1), we can show that IIRe*01122 II1m*01I2> EIIxY112,
for some e > 0, so that Il ek*o 11 < eekllxyll'`.
We leave it as a problem to show that
( ) fdYJo(Y)e_ko
= 1.
(17.27)
It then follows that
f f
Ih(x)  hk o 000)1
dyl JJ(y)I I h(x) 
h(y)lekl*el2
dyl J8(y)l lh(x) 
h(y)le"klla.y112
(17.28)
Problem 17.8. First, prove in detail that hk E A. Then, verify the identity (17.27). It is useful to compute the integral for 0 E Do fl R and then to use analyticity.
(3) To finish the proof, we take the L2norm of the expression in (17.28) and obtain an upper bound
f
2
Changing variables from (x, y) to (u cok;
J
du
f dvlh(u) 
< cok if du < clk1,
f
h(y)leekllxyll2
dylh(x) 
k1/2y), we obtain
k112x, v 2
h(v)leeluv112
dvlh(u)le6'11uu112
+ f dvlh(v)leelluUII
f
I
186
17. Spectral Deformation Theory
since h has compact support. This proves that lim 1Ih  hk o ¢y II = 0,
kroc
and so UBA is dense in L2(R"), 9 E Do.
17.5
Notes
The theory of dilation analytic potentials is presented in Reed and Simon, Volume IV [RS4]. This theory works well for atoms since the Coulomb potential centered at the origin is dilation analytic. Indeed, one of the first applications by Simon [Sim9] of the resonance theory discussed here was to the helium and other, multielectron, atoms. It was soon realized, however, that the potential for multicentered problems like molecules, which consist of several atoms, is not dilation analytic. Motivated by this and related problems, there were successive modifications of the dilation analytic theory. Simon introduced exterior dilation analyticity in [Siml 11. This paper shows that only analyticity near infinity is necessary. The problem of spectral deformation for molecules was also one of the motivating factors of Hunziker's work [Hu2]. The use of flows generated by general vector fields in configuration space was developed by Hislop and Sigal in [HS2]. Hunziker [Hu2j realized that the global flow is not necessary and worked only with the firstorder approximation to the flow. This form of the theory is presented in this chapter. Another problem that could not be treated by dilation analyticity is a Schrodinger operator with a nonsmooth, exponentially decaying potential; for example, a potential of compact support. It was realized that this could be treated by spectral deformation methods using flows generated by vector fields in the Fourier transform variable. Since the potential operator V becomes a convolution operator in this representation, there are new technical difficulties. This was solved by Sigal in [Si2], by Cycon [Cy],
and, from a pseudodifferential operator perspective, by Nakamura in [N4] and [N5].
18
Spectral Deformation of Schrodinger Operators
The method of spectral deformation in configuration space, developed in the last chapter, is quite general. It has been applied to a variety of problems. Our main application is to the semiclassical theory of shape resonances. For this, we need to study the behavior of Schrodinger operators under spectral deformations. In this chapter, we first study the effect of local deformations on the Laplacian and its spectrum. We then show that the effect of adding a relatively compact potential, which is the restriction of a function analytic in some neighborhood of W1, does not change the essential spectrum. This shows that the hypotheses of the Aguilar
BalslevCombesSimon theory are satisfied and opens the way for a study of resonances.
18.1
The Deformed Family of Schrodinger Operators
We apply the theory of spectral deformation to study Schrodinger operators H = A + V. We assume that U  J U0 1101 < 1 / h} is a spectral deformation family associated with a vector field g normalized so that M, = 1. We also assume that g satisfies condition L of Chapter 17. As for H, we assume that it is selfadjoint with domain D(H) = H2 (R"). We will impose more conditions on V later. Let De be the disk Do  {B E C 110 1 < I /.}. Consider, for 0 E Do f1 R, the family of unitarily equivalent operators
H(6)=UeHUB'=Pe+Ve,
(18.1)
188
18. Spectral Deformation of Schrodinger Operators
where pe = UAp'UH 1,
pi = i ,
(18.2)
and
VH=UoVW.
(18.3)
(Vo f)(x) = V (x + Bg(x))f W,
(18.4)
The operator Vo is given by
for suitable f E LZ(W'). The operator pe can be computed explicitly. We first compute, for any f E S(R' ), (U0
ax,i
UB 1f) (x)= JoW()z
(Uf)((x)).
(18.5)
The derivative of the map 0,9 is given by
40(x)'
Jin(x) = [Ddo(x)lij =
axJ
We denote the inverse of the matrix J by J1 and write
(JI (x))ei = J(x)`e We will use the standard convention that repeated indices are always summed over their full range of values. Using the chain rule, we find
(aai UB'f) (00(x)) = J(x)re
aae
Je(00(x))2f(x).
(18.6)
Here, we write .J (x) for the Jacobian determinant of B 1(x). From equation (17.12), we see that
70(00(x)) = e(x) and so from (18.5) and (18.6), we get U08xiU0J8J1i(x)aaeJe
(18.7)
The adjoint of this operator (0 is real) is given by (UB
ax,
UB 1)* = J,9
axe
(X)
(18.8)
Multiplying results (18.7) and (18.8) together and suppressing the xdependence, we obtain, from (18.2),
pe °
U; I
= Jg
7 Pe Jti J0 Jmi pmJJ 2
(18.9)
18.1 The Deformed Family of Schrodinger Operators
189
A particularly useful form of (18.9) is
pt`s _ VaoV +go,
(18.10)
where the matrixvalued function ao is n
[ao(x)lij = E 'm(x)Jim(x), m=1
and go is the function
go(x)
4
[oo.bo)] Jkm Jem r _x (log Jo) I + 2
Ij kmjkm .. (log Je)] xe a
.
(18.12)
Problem 18.1. Verify these calculations.
The functions [ao(x)]ij and go(x) are analytic in 9 on Do. They satisfy the following estimates. Since I axj I is bounded, it follows that J9(x) is uniformly bounded in x and 9 E Do. By expanding the determinant, we see that IJ6(x)l < I+ C1 101,
for 101 sufficiently small. Similarly, from the definitions of ao and go, we find
[ael = I + [bel, where [bo]ij < c101 and
Ige(x)I < C101,
for all 191 sufficiently small.
Proposition 18.1. The family of operators pa, 9 E Do, defined in (18.2) is a typeA analytic family of operators with domain D(p2) = H2(R")
Proof. We first verify the constancy of domain. Suppose u E H2(W). Since ao is smooth, we have pi aij pj u = aij pi pj u + [pi, ai j ]pj u. It follows from this and the boundedness of ao and go that IlpeuII < c111p2u11 +c211u11,
and so D(p2) = H2(Rn) C D(p22). The proof of the opposite inclusion is more involved. We begin with some matrix estimates. Recall that ao = (J1)(J`1)T
190
18. Spectral Deformation of Schrodinger Operators
Let bo = aH' = J'J. Since J = 1 +9(Dg), where Dg is the Jacobian matrix of g, we have, for any i; E C", 112 II
+ 20 Re(i;', (Dg)4) + 02 11(Dg) 112.
(18.13)
Recall that by our normalization, 1l Dg ll < 1. It then follows that 3co > 0 such that ll
(18.14)
Since b0 is invertible, if we replace i by aoE in (18.14), we obtain I(
. ae ) I > col (
,
(18.15)
c is strictly positive. For any u E H2(R"), we can write IIP2u112
1: (Pku, PZPku) k
< cE(I(Pku, P,29 Pku)I +I(Pku.g0Pku)l),
(18.16)
k
where we used (18.10), (18.15), and the definition of pot. Commuting Pk to the left in each term of (18.16), we obtain for any 8 > 0, I(P2U,PHU)I
I
(1 + co) > Mo,
for some 0 < Mo < oc, and
(,b0) E So
z
f
arg(z+Mo)I < arg 1 +i 2 (l +co)
(18.19)
n
R}. We impose two conditions on V so that (1) the transformed potential V o 0e admits a continuation to 0 E Do as a relatively p2bounded operator, and (2) we can compute aCSS(H(6)), given the result of the previous section on aess(pe ). Definition 18.3. A realvalued function V on R" is an admissible potential for a spectral deformation family U if
(VI) V is relatively p2compact; (V2) V is the restriction to R" of a function, which we also denote by V, that is analytic on the truncated cone CR, forany e > 0 and some R > O sufficiently large. Lemma 18.4. Let V satisfy (V 1) and (V2). Then V o 0e extends to 0 E Do as an analytic, relatively pycompact operator.
18.4 The Spectrum of Deformed Schrodinger Operators
193
Problem 18.5. Prove Lemma 18.4 using condition L on the vector field g and Proposition 18.1. Theorem 9.8 will be useful in proving the relative compactness. Corollary 18.5. Let V satisfy (V 1) and (V2). Then the selfadjoint operator H(9) _ P192 + Vq, defined for 9 E Do fl R, extends to an analytic typeA family of operators on Do with domain H2(R").
18.4
The Spectrum of Deformed Schrodinger Operators
The last results we need concern the spectrum of H(9) = pH +Ve, for an admissible
potential V, and the resonances. Recall from Section 16.2 that we expect the essential spectrum of H(9) to deform off of the real axis into the lower halfplane for Im 9 > 0. In the case of dilation analyticity discussed in Example 16.3, we saw that cress(e20A) = e2' Im9R+. It follows from Proposition 18.2 that this is typically the case for p. when the spectral deformation family U is generated by a vector field g that approaches x at infinity, in the sense of condition L. We expect that the addition of an admissible potential will not change the essential spectrum. As for the resonances, we can apply the AguilarBalslevCombes theorem of Section 16.2 to our family of deformed Hamiltonians H(9) and the set of analytic
vectors A. Note that A is independent of the specific vector field used, so the resonances of H, obtained as in Section 16.2, depend only on H and the set A. Theorem 18.6. Let U be a spectral deformation family for the Schrodinger operator H = A + V, with V satisfying (V 1) and (V2). Then for any 0 E Do, I
E
.
(18.25)
Let R(H) be the set of resonances as defined in the AguilarBalslevCombesSimon theory, and let So be the open region in C bounded by I1 and aess(H(0)), for 9 E Do fl C*. Then,
ad (H(9)) fl Sy C R(H).
(18.26)
In particular, the resonances in this sector depend only on H and A.
Remarks 18.7. We will not prove (18.25) here completely, but only a weaker version. The proof of (18.25) uses a version of Weyl's theorem, Theorem 14.6, for closed operators. We state this theorem without proof. The proof can be found, for example, in [K].
Theorem 18.8. Let T be a closed operator on a Hilbert space 7l and A be a relatively Tcompact operator. Then aess(T) = aess(T + A). Problem 18.6. Suppose T and A satisfy the hypotheses of Theorem 18.8 and that aess(T) = W(T). Prove that aess(T + A) = aess(T)
194
18. Spectral Deformation of Schrodinger Operators
Proof of Theorem 18.6. Fix H e Do, let So = {z arg z = 2 arg(1 +9)}, and I
define Co = C \ Se. We will show that aeSs(H(9)) fl Co = 0, and so aess(H(9)) C S. This is a weaker version of (18.25) but sufficient for our purposes. The statement (18.25) follows from Proposition 18.2 and Theorem 18.8. By Proposition 18.1, (pH  z)I is analytic in z on Co. Lemma 18.4 and this fact imply that i
Fe(z) = V9(pe  z)
is an analytic, compact, operatorvalued function on Co. We apply the Fredholm alternative, Theorem 9.12, to study Fe(z).
The operator 1 + Fe(z), z E Co, is invertible if and only if I V a(Fe(z)), which is discrete, except possibly at 0. Suppose 1 E a(F9(zo)), zo E Co. By analyticity, 3r > 0 such that for z e Dr = {z Ilz  zoll < r} implies that II Fo(z)  Fo(zo)II < 1/2. Similarly, by Theorem 9.15, there exists a finiterank operator F such that II F9(zo)  FII < 1/2. Consequently, by Theorem A3.30, the operator 1 + Fe(z) + F is invertible and analytic on Dr. Let us write I + Fe(z) as I
1 + Fe (z) = (I  Go(z))(I + F9(z)+ F),
where the finiterank, analytic, operatorvalued function G9(z) = F(1 + Fe(z) + F)1. Because of the canonical form of a finiterank operator, we can consider
1  Go(z) as an analytic matrixvalued function. Let D(z) = det(1  Ge(z)) be the determinant of this matrix. It follows from the preceding equation that I + Fe(z) is invertible on Dr if and only if D(z) 0. By the assumption that 1 E a(F9(zo)), we know that D(zo) = 0. Since D(z) is an analytic function on Dr, either D(z) = 0, VZ E Dr, or there exists a discrete set of points, with no limit point in Dr, where D(z) = 0. We now show that the first alternative cannot hold. As a consequence, the set of points in Co where 1 + Fe(z) is not invertible is discrete. Since R_ C Co, it follows from (18.23) that
II(pe +,k) III < c;,1, Since Ve is relatively
p2_
A > 0.
compact, for any a > 0, 3 fi(a) > 0 such that
II Veull 5 allpull +8(a)Ilull, for any u E H2(W ). Consequently, for any e > 0,
IIVe(pe+A)'II <S, for all;, sufficiently large. This shows that 1 + Fe(z) is invertible for all z E Co with Re z sufficiently negative. Using the connectivity of Co, if the first alternative
mentioned above held, that 1 + Fe(z) is not invertible on Dr, then by repeating the above argument, we would arrive at the region where we know I + Fe(z) is invertible, and hence obtain a contradiction.
18.5 Notes
195
Therefore, the value  I cannot be in a discrete, countable set P of zi in CH. This means that (I + F(4(z))1 exists on Co \ P. Writing
Ho  z _ (I + F0(z))(p2  z), we see that H0  z is invertible on CH \ P since it is the product of two invertible operators. This also shows that (HO  z)1 is meromorphic on Co \ P.
Problem 18.7. Complete the proof of the theorem using the AguilarBalslevCombes theorem, Theorem 16.4, and fill in the details of the argument for the invertibility of 1 + This finishes the proof of Theorem 18.6. The combination of analyticity and compactness used in the proof of the theorem
yields a result of interest in its own right. The following theorem is called the analytic Fredholm theorem in [RS 11.
Theorem 18.9. Let F(z) be a compact, analytic, operatorvalued function on a connected, open subset D in C. Then, exactly one of the following holds: (1) the operator 1 + F(z) is not invertible for any z E D;
(2) the operator 1 + F(z) is invertible on D\S, where S is a discrete subset of D with no limit points in D. Problem 18.8. Prove that the poles of (H(6)  z)1 in Ce are of finite order. Problem 18.9. The purpose of this problem is to study translation analyticity. Let g(x) be a smooth vector field that is bounded with bounded derivatives and such that g + 1 as llxll + oo (see Examples 17.1). Suppose that V is the restriction to II8 of a function V(z) analytic on a strip {z IIm zl < a}. Assume that V is relatively p2compact. Prove the analogues of Proposition 18.2 in this situation. In particular, aess(H(0)) C {z I Re Z E R and Im z = 0} for I$I < a. I
18.5
Notes
References to the spectral deformation of Schrodinger operators are given in the Notes for Chapter 16. The calculations of this chapter follow the paper of Hunziker [Hu2]. Translation analyticity was introduced by Avron and Herbst [AvHe] to study Stark Hamiltonians. These Schrodinger operators have the form H = Hp+V , where Hp
is the free Stark Hamiltonian. The constant vector F E R" is the electric field. Let us fix coordinates so that F = IIFlle,. Let Ue be the unitary implementation
196
1 8. Spectral Deformation of Schrodinger operators
of the translation group x > x + 6. It is easy to see that, formally, the free Stark Hamiltonian transforms as
=o+IlFI!xi+IIF116. It can be proved, as outlined in Problem 18.9, that QeSS(Ho(6)) _ {z IIm z = II F II Im 01. The main problem with translation analyticity is that the Coulomb potential is not translation analytic. The free Stark Hamiltonian, on the other hand,
does not transform well under dilations. In fact, the spectrum of the free Stark Hamiltonian with a complex electric field is empty! Herbst showed in [He2] how to resolve these difficulties and use dilation analyticity to define resonances for the free Stark Hamiltonian with a Coulomb potential.
19
The General Theory of Spectral Stability
In Chapter 15, we studied the question of the stability of a discrete eigenvalue Ao of an operator To under an analytic perturbation TK. The main result, Theorem 15.11, states that the family TK, for K in a small complex neighborhood of 0, will have eigenvalues near Ao of total algebraic multiplicity equal to that of A0. This is what we mean by the stability of Ao with respect to the family TK. With regard to the perturbation theory of discrete eigenvalues, this theorem is quite satisfactory. There are, however, other interesting situations that we want to study. We describe two main cases here: (1) Nonanalytic perturbations. Let S be a sector in cC with vertex at 0. Consider a family TK, K E S, of closed operators depending in a controlled manner on the complex parameter K E S. Suppose that there exists a closed operator To such that TK * To in some sense. Furthermore, let us suppose that .lo is a discrete eigenvalue of To. What can we say about ad(TK) in a neighborhood of a.p for K E S near 0?
(2) Embedded eigenvalues. Suppose that Ao is an eigenvalue of a selfadjoint operator To lying in the essential spectrum of To . How can we study the behavior of such an eigenvalue under perturbations of To?
As we will see, both of these situations occur quite naturally in examples in quantum mechanics. The examples involve both eigenvalues and resonances. We will study aspects of nonanalytic perturbation theory in this chapter that follow
the works of Vock and Hunziker [VH], Hunziker [Hu4], and Simon [Siml2]. Basic material on asymptotic perturbation theory is given in Kato [K] and in Reed and Simon, Volume IV [RS4]. We study this problem because of its importance
19. The General Theory of Spectral Stability
198
for several problems in quantum mechanics, and because we need to generalize the notion of stability of the discrete spectrum in order to treat problems in the semiclassical regime. Indeed, as the examples in the next section will show, we must broaden the notion of stability to include not only eigenvalues but also resonances. By considering the stability of eigenvalues in nonanalytic situations and for nonselfadjoint operators obtained, for example, by spectral deformation, we will develop a theory in which eigenvalues and resonances are treated equally. We typically encounter two situations for which we wish to study the effects of a perturbation on an eigenvalue: an embedded eigenvalue of the unperturbed operator To, described in case (2), and nonanalytic perturbations of isolated eigenvalues for which the eigenvalue disappears as soon as the perturbation is turned on (examples (2) and (3) of the next section). Perturbation theory of embedded eigenvalues is most commonly associated with spectral resonances, as discussed in Chapter 16. In these examples, the key idea is that one first moves away the essential spectrum and then applies perturbation theory to study the stability of the eigenvalue. In general, the perturbed eigenvalue of the spectrally deformed operator will be a resonance of the perturbed, selfadjoint operator. We will illustrate this approach with a simple example in the last section. We will return to the problem of embedded eigenvalues in our discussion of resonances for the Helmholtz resonator in Chapter 23. Nonanalytic perturbations are associated with the failure of the RayleighSchrodinger perturbation series to converge about K = 0. Yet, it is found that the truncated series gives remarkably good information about the perturbed eigenvalue or resonance. This is the topic of asymptotic expansions and Borel summability. We will
not discuss these topics here, but readers may refer to [RS4], [Siml2], and [Hu4]. The main result of this chapter is the stability criteria for families of Schrodinger operators, Theorem 19.12. It should be compared to the estimates in Theorems 5.6
and 5.9, which give conditions for determining the spectrum of a selfadjoint operator.
19.1
Examples of Nonanalytic Perturbations
We give three classic examples of nonanalytic perturbations in quantum mechanics.
(1) The anharmonic oscillator. We consider a onedimensional Schrodinger operator of the form HK = P2+X2 +KX4
(19.1)
acting on L2(R). For K = 0, the Hamiltonian Ho in (19.1) is just the harmonic oscillator Hamiltonian studied in Chapter 11. The spectrum is purely discrete, as follows from Theorem 10.7, and is given in (11.5). Note that the perturbation x4 is not relatively Hobounded. The properties of the perturbed operator depend dramatically on the sign of the coupling constant K. When
19.1 Examples of Nonanalytic Perturbations
199
K > 0, the potential VK (x)  x22 + K x4 is strictly positive and Theorem 10.7 again indicates that the spectrum is purely discrete. When K < 0, the poten
tial satisfies VK(x)  oo as IxI > oo and the Hamiltonian is unbounded from below.
Problem 19.1. Prove that the essential spectrum of HK, for K < 0, is R, by constructing appropriate Zhislin sequences.
Because of this change in spectral type, from discrete spectrum to only essential spectrum as K passes through 0, it is impossible that the eigenvalues
are analytic in K about K = 0. What is true, however, is that the operator family HK is analytic in a sector S containing the positive real axis and with vertex at K = 0. In fact, the family HK is analytic of typeA about any Ko E S. One of the amazing aspects of this example is that if one formally computes the RayleighSchrodinger series for the eigenvalues given in Section 15.5, the values obtained are remarkably close to the eigenvalues of HK for K > 0.
(2) The Stark effect. We consider the Schrodinger operator for a hydrogen atom in a uniform, external, electric field in the x1direction. The unperturbed operator is Ho _ A  1/IIx11, the hydrogen atom Hamiltonian, acting on L2(R3). The perturbation is Fx1, where F is the electric field strength and plays the role of the coupling constant. For F 0, the FarisLavine theorem (see [RS21) can be used to prove that the operator is essentially selfadjoint on Co00(R" ). What are the spectral properties of
H(F) = Ho  Fx1 = A  1/11x11  Fx1,
(19.2)
for various values of F? When F = 0, the spectrum is again well known: It consists of a sequence of infinitely many negative eigenvalues of the form (1/m2 1 m E Z), accumulating at 0, and essential spectrum [0, oc). Are the eigenvalues stable with respect to the perturbation? The perturbation is not relatively Hobounded, so some strange behavior might be anticipated. For F 0, the essential spectrum of H(F) is the entire real line and there are no eigenvalues. Thus, the negative eigenvalues of Ho cannot be stable with respect to the perturbation in the sense of Chapter 15. We will see in Chapter 23 that they become resonances of H(F). The formal RayleighSchrodinger series gives remarkably good estimates of the shift of the real part of the eigenvalues. The imaginary part of the resonance, however, is exponentially small in F and cannot be computed in perturbation theory.
(3) A shape resonance model. This simple model is discussed by Hunziker in [Hu4). One again considers the hydrogen atom Hamiltonian Ho . O 1/11x11, acting on L2(R3). The perturbation is given by VK(x) _
KIIxII
I+KIIXII'
(19.3)
200
19. The General Theory of Spectral Stability V. (X) }
K=0 ltx 11
0
K>0
FIGURE 19.1. The radial section of the potential VK, for K > 0.
for K real, and we define
HK  Ho  VK.
(19.4)
Since 1imil,11, VK(x) = 1, it is easy to check that inf aess(HK) _ 1, for K =/ 0. The potential is sketched in Figure 19.1. It is similar to Figure 16.1.
We expect that the eigenvalues of Ho in the interval (1, 0) will become resonances of HK. Once again, the perturbation is nonanalytic, and the notion of stability given in Chapter 15 is inadequate to explain this phenomenon. We will devote the next few sections to developing a notion of stability applicable to the eigenvalues of the unperturbed operators given in the above examples. In the last section, we will comment on the case of perturbation of eigenvalues embedded in the essential spectrum.
19.2
Strong Resolvent Convergence
In dealing with the general question of eigenvalue stability, we must weaken our notion of the convergence of the perturbed family to the unperturbed operator. In the case of analytic perturbations studied in Chapter 15, we encountered the situation of normresolvent convergence: For any z in the common resolvent set, IIRK(z)  Ro(z)II > 0
(19.5)
as K > 0. This has as a consequence the norm convergence of the projectors, which is the first criterion for stability as given in Definition 15.1. Normresolvent convergence is not a necessary condition for the norm convergence of the projections. Indeed, in many situations of nonanalytic perturbations, the perturbation is not relatively bounded, so one cannot expect normresolvent convergence. We will replace this with strong resolvent convergence. We now consider a family {TK } of closed operators depending on a complex parameter K E S C C. We assume that S is a sector of the form
S = {K E C 100 < arg(K) < 91 },
19.2 Strong Resolvent Convergence
201
where, for some e > 0, 7r E < 9o < 01 < 7r +E. We are actually only concerned with the behavior of the family for K E S with IK I small. By K , 0. we mean the limit along some sequence in S approaching 0. Definition 19.1. A family of closed operators {TK, K E S} converges strongly in the generalized sense to a closed operator To if
(i) 3 a nonempty, connected subset A C C such that A C p(TK) tl p(To) for all K E S and small;
(ii) for some z E A, we have strong resolvent convergence, that is, for any u E N. (19.6) s  lim RK(z)u = Ro(z)u. K0 Problem 19.2. Prove that if {TK } is an analytic typeA family of operators on a neighborhood D about 0, then the family satisfies the conditions of generalized strong convergence.
An immediate question concerns the extension of the strong resolvent convergence from some z E A to other points of A. In the case of normresolvent convergence, (19.5), this can always be done. Problem 19.3. Prove that normresolvent convergence at one point implies it at any other point in the corresponding connected component of the common resolvent sets.
It is clear that the question can be answered in the affirmative whenever there are uniform bounds on the resolvents in some neighborhood of z in A. Definition 19.2. Let { TK } be a family of closed operators as in Definition 19.1. The region of boundedness for the family {TK } is a subset Aj, C C defined by
Ab = {z
I
IIRK(z)II
M, VK E S small},
for some positive constant M.
Problem 19.4. Prove that the region of boundedness is an open subset of A, and use this to extend the strong resolvent convergence condition (19.6) to this set. Given that a family {TK } converges to an operator To in the strong resolvent sense, what can be said about the relationship between the spectral properties of the family and those of To? Under normresolvent convergence, an isolated eigenvalue of To
cannot spread into an interval of essential spectrum because of the constancy of the dimension of the spectral projections following from (19.5). In general, we say that the spectrum cannot suddenly contract as the parameter approaches 0 (the dimension of the projection cannot go from infinite to finite as K > 0). This shows that there is some stability of the spectrum. In the examples of Section 19.1, we have a change of spectral type which indicates that we cannot have normresolvent convergence. Strong resolvent convergence, however, is much weaker. The only
202
19. The General Theory of Spectral Stability
general statement that can be made applies to the case when the family {TK } and the limit operator To are selfadjoint [K]. Then, the spectrum of TK does not suddenly
expand at K = 0. One way to say this is that dim PK > dim P0. In the following sections, we will study the stability of a discrete eigenvalue of To in the general case of a perturbation under the conditions of strong resolvent convergence. This will necessitate additional assumptions, for it is possible that a discrete eigenvalue of TO can become embedded in or dissolve into the essential spectrum of TK, for any K ' 0, under the conditions of strong resolvent convergence.
19.3
The General Notion of Stability
The notion of stability for a discrete eigenvalue of an operator To with respect to a perturbation TK is given by Kato [K]. We extend the notion of core introduced in Chapter 8. For a closed operator T with domain D(T), a dense subset Do C 7l is a core for T if T I Do is closable with closure T. We first list our assumptions of the family TK for which we can define the notion of a stable eigenvalue.
Definition 19.3. Let TK be an analytic family of closed operators for K E S, a sector in C defined by
S°{KEC 10o<arg(K) 0, 7r  E < Bo < 01 < Tr + e. Such a family {TK, S} is called a continuous family of operators if there exists a closed operator To, having a common core Do with the family TK, such that for any u E Do,
s  lim
K O
TKu = Tou.
Proposition 19.4. Let TK and To be a family of operators as in Definition 19.3. If p(T0) n Ob 0, then TK converges strongly to To in the generalized sense. Proof. Let zo E p(To) n Ob. By the second resolvent formula, we have RK(zo)  Ro(zo) = RK(zo)[TK  To]Ro(zo).
(19.7)
Applying this equation to a vector v = (To  zo)u, with u E Do so that Ro(zo)v E Do, we obtain strong convergence on a dense set. By Problem 19.4, we can extend this to any z E p(T0) n Ob. For the situation described in Definition 19.3, let us suppose that Xo is an isolated
eigenvalue of To with finite algebraic multiplicity, so X0 E ad(To). Let us recall the definition of the Riesz projections from Chapter 6, equation (6.1). Let I, be a simple closed curve about Xo in the resolvent set of To. Then we define i
P"0
2Iri
rAo
Ro(z)dz.
(19.8)
19.4 A Criterion for Stability
203
Recall that, in general, the projection PP, is not orthogonal. We define PK similarly with RK(z) in place of Ro(z) in (19.8), when it makes sense.
Definition 19.5. An eigenvalue A.o E a (To) is stable with respect to the family of perturbations TK, as in Definition 19.4, if the following two conditions hold:
(1) d E > 0 such that the annular region AS(Ao)  {z 10 < Iz  Aol < 8} is in the boundedness set Oh of the family TK, for K E S and small, and in the resolvent set of To.
(2) Let PK be the operator constructed as in (19.8) with To replaced by TK. Then we require that
lim IIPK  PAall =0. K0
(19.9)
It follows from this definition that for all K small, the operators HK have discrete spectra inside AS(),.o) with total algebraic multiplicity equal to that of ko.
Problem 19.5. Suppose that the continuous family {TKS} with limit operator To satisfies the first condition of Definition 19.5. Prove that the Riesz projector PK converges strongly to Po.
19.4 A Criterion for Stability We will now concentrate on a special class of families of operators TK which arise from Schrodinger operators. We do this in order to utilize local compactness results and the techniques of geometric spectral analysis introduced in Chapter 10. Much
of what is presented here can be extended to families of operators that are local perturbations of elliptic differential operators. The main difficulties in proving a stability result arise in locating the resolvent set and estimating the resolvent of TK. The resolvent set might be empty or might be the entire complex plane. Also, we noticed earlier that the region of boundedness plays a crucial role in the theory. As Problems 19.4 and 19.5 show, it is important to have information about this region to prove strong convergence of the projections and in order to prove the first condition of stability. Recall that our previous estimates on the resolvent of an operator, Corollary 5.7 and Theorem 5.8, required that the operator be selfadjoint. There are no such estimates for a general, closed operator. Thus, we have to put conditions on the family TK so that we can control the resolvent set and the region of boundedness Ob. Finally, as Problem 19.5 shows, projections for a continuous family of operators will converge strongly. More information is needed for the second condition of stability. In order to motivate the additional conditions on a continuous family of operators
sufficient for stability, we need another concept. One of the basic tools for the spectral analysis of closed operators is the numerical range.
204
19. The General Theory of Spectral Stability
Definition 19.6. Let T be an operator with domain D(T). The numerical range of T, denoted by O(T ), is the subset of C defined by
O(T) = {(u. Tu) I U E D(T) and Ilull = 1}.
(19.10)
An important fact about the numerical range of T is that it is a convex subset of C. The proof is not relevant to the discussion here, so we refer to the book by Stone [St). It is easier to check the following properties of the numerical range.
Problem 19.6. Let T be a closed operator, let U(T) be its numerical range, and
define an open set 0(T)  C \()(T). For any z E A(T), the operator T  z is injective and has closed range. The following proposition is an immediate consequence of Problem 19.6.
Proposition 19.7. Let T be a closed operator. If Ran(T  z) is dense in the Hilbert space for any z E A(T ), then A(T) C p(T) and
II(T  z)'11 < 1/{dist(z. O(T) )}. Having introduced the concept of the numerical range, the following conditions on a potential are more natural.
Definition 19.8. A family of complexvalued functions VK = V, + VI E Liac(1P") K is called a continuous family of potentials with sector ,5 if (i) V 1 is relatively p2 = Abounded in the sense that for any a > 0, there
exists a constant P (a) > 0, such that for all u E Co (R n) II VI(U)II
a IIp2U11 +R(a) Ilull
(19.11)
uniformly for K E S; (ii) V,,2 is multiplication by a complex function in Llo (R") such that the range
of the function lies in a halfplane bounded away from the negative real axis,
0 < Re(e" VZ),
(19.12)
for some real yK with I yK I < n/2  E, for some a independent of K and for
allKES;
(iii) the potential VK converges strongly to V0 as K > 0 within S on Co (118").
We use continuous families of potentials to construct families of Schrodinger operators on S. Let HK = p2 + VK be initially defined on Co (III") Lemma 19.9. Let {HK, S) be the densely defined family of operators introduced above for a continuous family of potentials VK.
19.4 A Criterion for Stability
(1) For any u E Co
205
") with (lull = 1, we have
(u, p2u) < aRe {e'Y" (u, HKU) } +b,
(19.13)
for finite, positive constants a and b independent Oft. (2) The numerical range of HK lies in the halfplane 11:
O(HK)Cll  zE(C i  E < arg zQ 0, it follows that (cos yK 5) > sin c 6. Hence, result (19.13) follows from both (19.18) and this observation with suitable a and b independent of K.
206
19. The General Theory of Spectral Stability
The statements about the numerical range are proved by observing that for any u E Co ()R") with (lull = 1, condition (19.13) implies that 0 < a Re((e`Y^ HK )) + b.
(19.19)
In particular, the numerical range lies in a halfplane.
Let us recall that for selfadjoint Schrodinger operator H with Co (R") as a core, the condition
II(H  z)ull ? Ellull, for all u E Co(R"), implies that z E p(H). For nonselfadjoint operators. this is no longer true in general, and we need additional information on the adjoint H* of H. This is the motivation for the following definition. Definition 19.10. A family of Schrodinger operators of the form (19.20)
where V a continuous family of potentials with sector S, is called a continuous family of Schrodinger operators with sector S if (i) the adjoint of HK, denoted H,*, is the complex conjugate of HK, that is, if J is the complex conjugation operator, then (19.21)
H,* = JHKJ;
(ii) the family HK is an analytic family of operators on any open region in the sector S.
Examples 19.11. (1) Consider an admissible potential V as in Definition 18.3. Let g be the vector field for the corresponding spectral deformation family. The Schrodinger operator is
Ho=Pe+V8,
(19.22)
on the sector of analyticity of V. Recall that Ve(x) = V(Oo(x)).
(19.23)
Ho = p2+VA,
(19.24)
From (18.10), we can write
with the potential Ve satisfying condition (19.11). It is easy to check that this is a continuous family of Schrodinger operators.
(2) The anharmonic oscillator HK, given in (19.1), is a continuous family of Schrodinger operators on the sector S = {z larg zl < 7r/2  E}. In this case, VB = 0 and one can check condition (19.12). I
19.5 Proof of the Stability Criteria
207
Problem 19.7. Show that a continuous family of Schrodinger operators is a continuous family of operators in the sense of Definition 19.3 and hence converges in the generalized strong sense to Ho, as in Definition 19.1.
We are now in position to state the main result on continuous families of Schrodinger operators due to Vock and Hunziker [VHJ (see also Hunziker [Hu4]).
Theorem 19.12. Let HK, K E S, be a continuous family of Schrodinger operators as in Definition 19.10. Suppose that ),0 is a discrete eigenvalue of Ho. The eigenvalue A0 is stable if for all u E Co (IIx II > n), for some n > 0, there is an E > 0 such that II(HK  Ao)uII >_ EIIUII > 0.
19.5
(19.25)
Proof of the Stability Criteria
We now proceed with the proof of the stability criteria, Theorem 19.12, for a continuous family of Schrodinger operators. Let HK be a continuous family of Schrodinger operators on a sector S satisfying the conditions of Definition 19.10. We suppose that Ao is an isolated eigenvalue of Ho with multiplicity mo. We need two preliminary lemmas. The first concerns local compactness of a continuous
family, and the second gives some important statements that are equivalent to inequality (19.25).
Lemma 19.13. Let F E C' (IR" ). Let K + 0 be a sequence in S, and choose a sequence uK E D(HK) satisfying
II1KII+IIHKuKII n) and for all z E As, we have (19.40) II(HK  z)ull > (2) Ilull > 0. This condition also holds for the pair { HK , z J. By the local compactness lemma, Lemma 19.13, and the fact that Co (1R") is a common core for the family and the adjoint family, we conclude that Ran(HK  z) is dense and closed, that is, equal to L2(R"), for any z E As. Inequality (19.40) therefore implies that IIRK(z)II
0.
(19.46)
Extending this to u E D(H,) and using the invertibility of (HK  z) for z E As, it follows from (19.46) that
(f) II(I 2
 F)RK(z)uKII < II(HK  z)(1  F)RK(z)uKll
< I1(1F)uKII+II[RK(z),flu, 11.
(19.47)
(19.48)
Let I C As be a simple closed contour of radius r < S. Integrating both sides of inequality (19.48) about F, we obtain _
(2) II(I  F)uK11 < r11(1  F)uK11+2nCo
fr Idzl
II[HK,
F]RK(z)uK11
(19.49)
where we used the uniform boundedness of IIRK(z)Il on i' and for all K small. Since [HK, FJRK(z) is a compact operator that is uniformly bounded on P and w  IimKi()uK = 0, it follows that the integrand on the right side of (19.49) converges to zero. Since we have chosen r < 3 < E/2, we obtain a false inequality from (19.49) unless II(1  F)uK 11 > 0. This, however, contradicts (19.45).
19.6
Geometric Techniques and Applications to Stability
Theorem 19.12 reduces the question of eigenvalue stability to the calculation of a lower bound on the norm of the operator HK  Ao acting on smooth states supported in the complement of an arbitrarily large region, say BR(0). Quite often, we will have local information about the potential on subsets of such a region. We can utilize this by using a partition of unity in the complement of BR(0) and estimating each piece separately. The use of partitions of unity for such estimations plays an important role in geometric spectral analysis. We will give some general formulas in this section. These will be used in our discussion of resonances in Chapters 2023.
19.6 Geometric Techniques and Applications to Stability
211
For a function j E C2(P" ), we define a firstorder differential operator WW by
W(j)W jl=V Vj 
(19.50)
Note that this operator is localized to the support of V j. We will use two types of partitions of unity { jj }: a standard partition for which yNi j; = 1, and sometimes it will be convenient to choose the functions such that yNI j;2 = 1. In this section, we will work formally and assume that Co (R") is a core for the
Schrodinger operator H = A + V . (1) Lowerbound inequality. Let { j; }^'1 be a partition of unity with END jz =
1. Then, for H = A + V, we have 11 (H  Z)11
112
(II(H
?

z)j+u112
 II Wtu112)
,
(19.51)
for U E D(H) and any z E C. We have written W; = W (j; ). If this identity is to be applied to (19.18), for example, the partition of unity is chosen to be one on the exterior of some ball. Problem 19.9. Prove this inequality. We can improve this estimate for families of continuous Schrodinger operators (or a family depending on some parameter) using local compactness. Let us consider a finite partition of unity { j;,K }, for the exterior of B" (0), where n is as in Theorem 19.12. Suppose that, in addition, the functions satisfy (19.52) 8, j;,, > 0, as Ilx II
f oo and K + 0, for multiindices a, with la I = I and 2. The
proof of the following lemma can be found in [Hu4]; one direction follows immediately from (19.25), and the other direction uses the local compactness lemma, Lemma 19.13.
Lemma 19.15. Inequality (19.18) holds if and only if
II(HK  Xo)if,Kull > ellij.Kull > 0,
(19.53)
for some e > 0 and for all jl.K.
(2) IMS localization formula. Suppose that {j}1 is a set of C2functions such that yNi j 2 = 1. The IMS localization formula (see [Si 11) for a Hamiltonian H is
N
H = Y(j1 Hjr  I V j, I2). Problem 19.10. Prove the IMS formula by computing [H, [H, j;ll.
(19.54)
212
19. The General Theory of Spectral Stability
(3) Geometric resolvent formulas. Geometric resolventformulas, that are similar to the second resolvent formula, provide a powerful tool for comparing the resolvents of operators that are the same when acting on functions localized to certain regions of R", but differ in other regions where the resolvents can be controlled. We will see many examples in the next chapters. The heart of geometric perturbation theory method is to estimate H = A+V by simpler Hamiltonians H, , i = I_ , N, Hi _ A + Vi, with Vi differing from V in a region of R" that is classically forbidden for the interval of energies I we are considering. For an interval 1, we define CFR(1) = nEet CFR(E). Typically, the Vi's are obtained from V as follows. Let {j i } N t be a partition of unity for 11g" such that j, E C2, j, > 0, and N t ji = 1. The operators {H,}^', are selfadjoint Schrodinger operators on L'(RI) with potentials V, having the property that Vi
I supp ji = V,
(19.55)
so that supp(V  Vi) C CFR(I). Each Vi is extended to 18" in a suitable manner. Each H, is simple in the sense that the resolvent Ri(z) _ (Hi z)t can be analyzed. We relate R1(z) to R(z) = (H  z)t by the geometric resolvent equation. As above, let Wi be the firstorder differential operator defined by
Wi  [0, A.
(19.56)
We assume that Wi is relatively Hibounded. Lemma 19.16. Let H and { Hi } be constructed as above using a partition of unity { ji }N t. Then, for all z in the intersection of'the resolvent sets of H and each Hi, N
R(z) 
N
R(z)Wi Ri(z).
ji R, (z) t=t
(19.57)
i=t
Problem 19.11. Prove Lemma 19.15.
19.7
Example: A Simple Shape Resonance Model
We illustrate the ideas of this chapter with the shape resonance model of Section 6.2. This model is due to Hunziker [Hu4]. Our discussion will also serve as an introduction to the theory of shape resonances in the semiclassical regime, which will be discussed in Chapter 20. We are concerned with a family of Schrodinger operators, HK, K E R, acting on L2(1R3), of the form
HK = A 
+IKIIIx11.
Ilxll
I
(19.58)
19.7 Example: A Simple Shape Resonance Model
213
The potential is sketched in Figure 19.1. When K = 0, this is the hydrogen atom Hamiltonian. There is discrete spectrum in the interval (1, 0), and the essential spectrum is the halfline [0, oo). For K ¢ 0, the essential spectrum shifts to the halfline [1, oo). Theorem 16.1 can be used to show that the Schrodinger operator HK  HK + 1,
has no positive eigenvalues so that HK has no eigenvalues greater than 1. We prove that the eigenvalues of Ho in the interval (1, 0) become resonances of HK , K E R.
The first step is to deform spectrally the family of operators HK. Because of the simple form of the potential, we can use dilation analyticity. For 9 E R, the dilation transformation x H eox generates the spectral deformation family. The unitarily equivalent family of Schrodinger operators, for 9 E R, is HK __ e_20
0 2
IIx1I
0
1 +e6KIIXII
(19.59)
For fixed K, this is an analytic typeA family of Schrodinger operators with respect
to 9 for larg 91 < 7r/2. Moreover, using the results of Chapter 18, it is easy to verify that
creSs(HD)= {z E C largz=2i Im0)
(19.60)
creSs(HK)={1+zIzECandargz=2iIm0}.
(19.61)
and, for K E R,
It also follows from the AguilarBalslevCombes theorem that the discrete spectrum of Ho in the interval (1, 0) remains the discrete spectrum of Ho . We want to show that these eigenvalues are stable with respect to the perturbation H.
We now fix 0 < Im 9 < 'r/2 and study the Schrodinger operators He as functions of K. The family HK , up to a factor of
e_26,
is a continuous family of Schrodinger operators, in the sense of Definition 19.10. with sector S = (K E C I E < arg K + Im 0 < Jr  e }, for any E > 0. The form of the sector can be verified by examining the denominator of the perturbing potential. Because of this, we can apply Theorem 19.12 to establish the stability of a discrete eigenvalue of
Ho in the interval (1, 0). In order to do this, we use the geometric method and Lemma 19.15. It is important to note that the stability estimate (19.25) cannot be obtained simply by a numerical range argument. This is because the numerical range, being a convex set, includes the entire sector {1 + z I Z E C and arg z = 2i Im 0). Consequently, the resolvent estimate of Proposition 19.7 gives no useful information in a neighborhood of any eigenvalue in the interval (1, 0). This is the typical situation one encounters in dealing with resonances. Instead, we have the following result. Lemma 19.17. The set of ),for which the stability estimate (19.25) fails to hold is the range of the function cI defined by
214
19. The General Theory of Spectral Stability eeKllxll
(K, p,x)r>
onSxII83x1R3
I +e'Kllh II
.
As the reader can check, the complement of this set in C includes a neighborhood
of the interval (1 + e, E), so that the stability estimate (19.25) holds for the eigenvalues in this region. Consequently, by Theorem 19.12, these eigenvalues are stable.
Sketch of the proof of Lemma 19.17. Since the Coulomb potential contribution to V9 is a Kato potential, we do not expect it to contribute to the determination of the stability set. Now, if ? is in the range of the map (P, we can use the previous
comment to construct a sequence K,,  0 in the sector S and corresponding functions UK,, E CO (11x11 > n) such that
lim (HA,
ty o0
 )uK  0.
Conversely, we construct a finite partition of unity Ja,K of the range of the map (h, and corresponding operators HK a, such that (le°nl'
II(HK  HK.a)Ja.Kull
n) and 8 > 0 small. The operators HK'a have numerical range contained in the range of (D. If ? is not in the range of (P, it then follows from Lemma 19.15 and (19.62) that the stability criterion (19.53) holds for each a, for suitable n and 8. To construct the partition of unity, let 0  Im 0 + arg K, and define a function e`'s
V(O,s)=
l+e'0s
for 0 < s and E < 0 < rr  E, as above. We first choose a finite partition of unity Ja of II8+ as follows. For any 8 > 0, choose finitely many points sa so that I V(O, S)  V(0, sa)I < S,
for all c < 0 < rr  e and for all s in supp Ja. The partition (Ja.K. is defined by Jc.K(x)  Ja(IKe°I 11x11) We then define the operators/.f
The reader can verify the claims made above.
= e'°p'V(0,sa).
20
Theory of Quantum Resonances II: The Shape Resonance Model
20.1
Introduction: The Gamow Model of Alpha Decay
The shape resonance model was developed by Gamow [Ga] and by Gurney and Condon [GC] to describe the decay of an unstable atomic nucleus by alphaparticle emission. The idea is very simple. The atomic nucleus is modeled by a potential barrier of finite width which traps the alpha particle. A typical situation is shown in Figure 16.1. According to quantum theory, the wave function for the alpha particle, initially localized in the potential well between the barriers, will oscillate between the barriers. However, because the barriers have finite thickness, as measured by the Agmon metric, the wave function penetrates the barriers, and hence the particle has a nonzero probability of escaping to infinity. In fact, the probability that the particle will escape to infinity in infinite time is 1. To say this another way, there are typically no bound states for this model, and consequently a decay condition such
as (16.1) holds: The wave function will eventually leave every bounded region. The classical limit of this model is clear: The barriers are infinitely high, so the distance in the Agmon metric across the barrier is infinite. This forces the wave function to vanish in the classically forbidden region, and the alpha particle is in a bound state. As we will show by a rescaling of the Schrodinger operator, this is equivalent to taking Planck's constant to zero. The semiclassical regime, therefore, is described by very large potential barriers relative to the energy of the alpha particle. We describe the alpha particle as a quantum resonance of a Hamiltonian H(A) _ A +,k2 V, where V has the "shape" of a confining potential barrier of finite thickness determined by X. We will work in the large A regime.
216
20. Theory of Quantum Resonances II: The Shape Resonance Model
The lifetime of the resonance is controlled by the characteristics of this barrier given by V and by the semiclassical parameter A. The lifetime is the inverse of the imaginary part of the resonance E  i I', I' > 0. In this and the next two chapters, we will prove the existence of resonances in the large n. regime and give an upper bound on the resonance width I'(A). Let us point out some differences between this problem and the shape resonance model of Section 19.1. The perturbation in the model we study is formally around A = oo. The lowlying eigenvalues of the problem of a potential well with infinitely high potential barriers will play the role of the unperturbed eigenvalues. We will show that these are stable under the perturbation given by large, but finite, X. This Hamiltonian has no bound states, so one must first apply the method of spectral deformation to it to move away the essential spectrum. The goal is now to show that this spectrally deformed Hamiltonian has an eigenvalue close to a lowlying eigenvalue of the unperturbed problem. For this, we need a stability estimate similar to the one appearing in Theorem 19.12. This will be obtained using localization methods as in Section 19.6. In the potential well region, the spectrally deformed Hamiltonian is close to the unperturbed model. In the classically forbidden region, we will use tunneling estimates to control the perturbation. In the exterior region,
where the energy is above the potential, we need to use quantum nontrapping methods to control the resolvent. These estimates, of interest in their own right, are discussed in the next chapter.
20.2
The Shape Resonance Model
The model we study is described by a potential V on R' having the following geometric and complex analytic properties. We first describe the geometric properties.
(V I) The potential V E C2(lR") is a realvalued, bounded function such that for some E > 0 and for any multiindex a = (ai, ... , Cl,), a; E N U {0}, with a a,i ial _ E"_i ai and 8' = aX dtKa , the potential V satisfies
la,V(x)l 0.
(V2) The potential V has a single, nondegenerate minimum at x0 such that V(x0) > 0.
The nondegeneracy of V at x0 means that the n x it matrix of second partial derivatives of V at x0,
A= r8
z
V
L 8xi dXj
1
(xo)J ,
(20.2)
is a positive definite matrix. By a simple translation, we can assume that x0 = 0. The results derived here can be extended to the case of several nondegenerate minima. One can also allow the potential V to have local singularities (see [HS2]).
20.2 The Shape Resonance Model
217
Conditions (V 1) and (V2) have the following geometric consequences. For any E > V (O), we define the classically forbidden region for V at energy E, denoted by CFR(E) as in Section 16.1, as the subset of R" given by
CFR(E)  {x I V(x) > E}.
(20.3)
Condition (VI) guarantees that this set has compact closure, for, if not, there exists a sequence (x,), with Iix,II + co, such that lim"y,,, V (x,,) > E, which contradicts (V 1). If E > sup,,.ER V(x), then CFR(E) = 0. In the following text, we will always assume that E > V(0) is chosen such that CFR(E) 0 and that Int CFR(E) 0. This is actually no additional restriction, since we will work with lowlying eigenvalues of the potential well.
For such an E > V(0), the complement of CFR(E) in 1[8" consists of two, disjoint, closed regions. One region is compact and contains the origin:
W(E)  {x I V (x) < E and x is path connected to 0}.
(20.4)
We call W (E) the potential well for V and E. The orbits x(t) of a classical particle
with energy E' < E and initial conditions (xo, p0) such that xo E W(E) and po + V(xo) = E' remain in W(E) for all time. Problem 20.1. Consider the classical equations of motion: d
dtx(t) = p(t),
d
dt p(t) =
VV(x(t)).
Given initial conditions (x0, po) so that xo E W(E) and E' = po + V(xo), as z p(t) vanishes above, prove that x(t) E W(E) for all time. (Hint: Use the fact that when V(x(t)) = E'.) The other region, which we call the exterior region, E(E), is unbounded. A classical particle with energy E and initial conditions (xo, po), such that x0 E E(E) and po+V (xo) = E, will move out to infinity (see the discussion in Section 21.2). ; We define classical turning surfaces relative to E as follows:
S(E) S+(E)
=
W(E),
(20.5)
8£(E).
(20.6)
These surfaces are disjoint. Each is compact and connected with
BCFR(E) = S(E) U S+(E).
(20.7)
These sets and surfaces are illustrated in Figure 20.1. The name "shape resonance" comes from the fact that it is the geometry of the potential well W(E) that generates the resonances. As we will illustrate below,
the resonances are associated with the quantization of the closed orbits of the
218
20. Theory of Quantum Resonances 11: The Shape Resonance Model
CFR (E)
S_(E) Xo S (E)
S'(E)
S'(E)
FIGURE 20.1. A shape resonance potential: the potential well W(E), the classically forbidden region CFR(E), the exterior region e(E), and the classical turning surfaces
Sf(E).
well. Although it is believed that resonances should exist for H = A + Va Schrodinger operator with potential V satisfying (V 1) and (V2) and some form of (V3) and (V4) aheadthis has not yet been proven except in the semiclassical regime, which we now describe.
20.3
The Semiclassical Regime and Scaling
The semiclassical Hamiltonian is
H(h) = h2A + V,
(20.8)
where h is a small, nonnegative parameter (see Section 1 1.1 for a discussion of the semiclassical regime). Factoring h2 from H(h) and defining A = h1, we obtain
H(h) = h2(A+A2V) = h2H(;').
(20.9)
We now consider Intuitively, as ;, + oo, the potential barrier becomes very large, decoupling the well W(E) from the exterior £(E). Since, however, the bottom of the well A2 V (O) also increases with A, this situation is difficult to work with unless V(0) = 0. We do another rescaling to achieve this. Let U;,, A E R+, be the transformation on L2(R") defined by
(UAf)(x)=tif () 0, we set S  (Si, SZ) and define a truncated cone 1'fi with respect to S+(Eo) by F6 = {z E C" I IIm zI 0. Our third condition on V is
(V3) The potential V I Ext S+(E0) is the restriction of a function, also denoted by V, which is analytic in 1s, for some Si, 62 > 0. To formulate the final condition on V, we note that the geometric conditions (VI) and (V2) and the analyticity condition (V3) enable us to construct a vector field relative to V in the region Ext S+(Eo) and satisfying condition L. We will use this vector field to construct a spectral deformation family for H(A) as in Chapters 1618. Let S C R" be a Ck, k > 1, compact, connected surface with 0 E Int S so that lib" = Int S U Ext S. Definition 20.1. A C2 vector field g on R' is said to be exterior to the surface S if
(i) g I /nt S= 0 and g I Ext S is nonvanishing; NO supxEExt S 11Dg(x)II = M1 < oo;
(iii) 3R > 0 and cR > 0, finite, such that Iig(x)II
for all x, IIx II > R.
cRIIxII.
222
20. Theory of Quantum Resonances 11: The Shape Resonance Model
Proposition 20.2. Let V satisfy (V 1)(V3). Then there exists a vector field r exterior to S+(e'; n.), with MI = I and satisfying condition L, so that the potential V is an admissible potential for the spectral delorniation family U generated hr v. The spectrally deformed ftunily of Schrodinger operators,
H(;,H)=Pry+V(0e(X);A).
(20.26)
is an analytic typeA family on some disk D,,, 0 < a < 1 //, where a is determined by 8i, 8> > 0. Before proving this proposition. let us state the final condition on V.
(V4) Let H(ti, H), 4 E D,,, be any spectral deformation family for H(A) constructed as in (20.26). Then 3 b > 0 such that for 0 < Im 0 < a Im 0 < a. and for any function U E C' (Ext S+(e; ti)), II(H(A. 9)  eo)ull > bllm 0l where i
(111th +
Il gi3lu11
)
(20.27)
0 is any function supported in Ext S+(e;.k) \ Ext S+(e'; A).
In Chapter 21, we will formulate the socalled nontrapping condition on a potential V and verify that this condition implies the lower bound (20.27). It will then be clear that there are many potentials satisfying (Vl)(V3) for which (V4) holds.
Proof of Proposition 20.2. (1) Construction of the vector field. For any e E R+, let C e E COO(R) be such that oe(s) = 0 fors > e and ce is a bounded, nonnegative function for s < e with limn,_oc &e(s) = 1. We can take, for example, ye(s) _
e
s(eS)i
s
0. We define a function Qre(x) by =.Oe(V(X)),
cpe(X)
0
x E Ext S`(e), otherwise,
(20.29)
and a vector field Ve(X)
(20.30)
Problem 20.4. Show that v,.(x) is a vector field exterior to the surface S+(e).
20.5 Spectral Stability for Shape Resonances: The Main Results
223
(2) By Lemma 17.2, ue defines a diffeomorphism on R" by oN(x) = x +0ue(x),
(20.31)
for 101 < MI, where MI is defined in (17.5) and M, I appears in condition (ii) of Definition 20.1. By simple computation, it is clear that we can assume that MI = I by choosing the constant CR < 1. With this choice, condition L is also satisfied. This family of diffeomorphisms (20.31) has two properties that are easily verified for all A sufficiently large and e < E0:
(1) 0e maps Ext S+(e) onto itself; (2) for 0 E D, Ran 0()(x) C ps, s, (S+(e)), for some 31, 62 > 0, depending on CR and M, of conditions (ii) and (iii) in Definition 20.1. The proposition now follows from these results, Theorem 18.6 and the discussion in that chapter, and the following problem. It is important to note that the potential Vti is not relatively p2compact, but the shifted potential V, +,AV(0) = is relatively p22compact. By Theorem 18.6, C aess(H(n.;0))={7AV(0)EClargz=2arg(I+9)}.
Problem 20.5. Verify that (V3) implies that, for all A sufficiently large, VV,IExt S+(e'; A) has an analytic continuation to the truncated cone
rs.a = {z E C" I Ilm zI 0 be the constant appearing in (V4). We choose a I so that
0 < a, < I (Im 6)b,
(20.46)
for 0 E Du , and set ao = ; a I . An immediate corollary of Theorem 20.5 is an estimate on Ro(z) __ (Ho(A)  z)1, for z c A(eo).
228
20. Theory of Quantum Resonances II: The Shape Resonance Model
Corollary 20.6. For all A sufficiently large, and for all z E A(eo), 4aI
IiRo(z)II
0 and any function q with supp rl C Int S+(e'; A) \ Int S+(e; ),).
Problem 20.7. Prove Corollary 20.6.
We want to apply formula (20.40) to study R(z)  (H(A, 9)  z)1, z E A(eo), for ,k large. We will show that a consequence of (V4) and Corollary 20.6 is that R(z) is analytic in A(eo). We will prove the following a priori estimate on R(z). Theorem 20.7. Assume conditions (V 1)(V4), and fix Im 9 E Du . Then, for all A sufficiently large 3 0 < cl < oo such that b'z E A(eo),
II(H(A, 0)  z)' II < c,.
(20.47)
We defer the proof of these results until the next section. As an intermediate step in the proof, we need to know that RI(z) = (HI (X, 0)  z)1 satisfies an a priori estimate similar to (20.47). The proof of the following theorem is similar to that of Theorem 20.7 and will also be given in the next section.
Theorem 20.8. Assume conditions (V I)(V4), and fix 9 E D. Then for all A sufficiently large, R1(z) is analytic inside ( z Iz  eoI = ai) and satisfies the I
estimates IIR1(z)II
IlvaiRi(z)II

1/2, given in (20.21 ) and (20.22).
For E E R. let i; E be a smooth function as defined in (20.33). Setting E = e, defined above, we introduce a cutoff function (20.56)
j0.e(x) _ e(V] (x; with VI (x; X) given in (20.37). We note that
(20.57)
VOJO.e = V1O.e
In order to use the geometric resolvent formula (19.51), we need a partition of unity satisfying
Tz I
(20.58)
Ji.e = 1. i=O
We obtain this by setting (20.59)
jl.e(x) _ 1  JO,e(X ))'
It is left as Problem 20.9 to show that there exist smooth functions E(s) so that ji.e E C2(R"). As in (20.35), we have for I < i, j < n,
IdadJk.e(x)I = 00,
?b),
(20.60)
where a and B are nonnegative integers satisfying a+fi = 2 and k = 0, 1. Analogous to (20.57), we have (20.61) Vl jl.e = VJI,e
Problem 20.9. Show how to construct a smooth function iE(s), as in (20.56), so that (1  (i;E(s))2)I/2 is twice continuously differentiable. Using such a function, verify claims (20.57), (20.60), and (20.61). In what follows, we consider e fixed as above and write ji for ji,e.
Proof of Theorem 20.7. We first prove that for all ,l large enough, 3 co > 0 such that for any u E Co (W) and for all z E A(eo), II(H(A, 0)  z)ull > collull, 0 E Dp D. From (19.51) and Problem 19.7, we have the geometric resolvent equation I
II (H(,k, 0) 
2
57 II(H(A, 0)  z)jiul1 Z  7 (u),
(20.62)
i __O
where the remainder R(u) is given by
R(u) _ t 11 [A, i=0
jiluli2.
(20.63)
20.7 Resolvent Estimates for HI (X, B) and H(A. B)
231
We first estimate the main terms of (20.62). Because of the choice of jo, we have (20.64)
(H(,.. 0)  z)jou = (Ho(A)  z)jou.
It follows from Corollary 20.6 that 3c1 > 0 such that for all ,, sufficiently large, (20.65)
II(H(,l, 0)  z)joull ? cl lljoull. Next, condition (V4) and the definition of A(eo), (20.41), imply that n
II(H(A, 0)  z)jiull > (Im 9)b I IIjlUII +
Ilndiull)
,
(20.66)
for some rl with supp rl C Int S+(e'; ;,) \ Int S+(e, ),.) and all z e A(eo). Estimates (20.65) and (20.66) give a lower bound for the main term on the right in (20.62). As for the remainder term R(u), (20.63), we expand the commutator
[ A, ji ]u = (2(V ji) V + (Oji ))u. Estimate (20.60) on IVji I leads to c2),.2(
i=0
id)
8)
(20.67) k=1
where rr 1 is a function supported on a set slightly larger than suppi V ji I. Combining
estimates (20.65), (20.66), and (20.67), we find that the right side of (20.62) is bounded from below as II(H(X, 9)  z)uI12
>
(CO  c1A125)IIuII2
+(c2lIm 012b2

n
c3X125) T
11ql aku112,
k=l
(20.68)
where co depends on b and IIm 01, and ct, c2, and c3 are finite, positive constants. Since 8 > 1/2, we find from (20.68) that for all large A, II(H(),, 0)  z)ull > co11u11.
(20.69)
Recall that Co (R") is a core for H(A, 0). This follows from the fact that (VI) and (V2) imply that Co (R") is a core for H(X) and the typeA analyticity. Hence,
estimate (20.69) shows that ker(H(,k, 0)  z) has no nontrivial elements. The estimate also shows that Ran(H(A, 0)  z) is closed as (H(A, 0)  z) is a closed operator. If g E [Ran(H(A, 0)  z)]1, then g E ker(H(X, 9)*  z). We leave it to Problem 20.10 to show that an estimate similar to (20.69) holds for H(X, 0)*. Consequently, (H(A, 0)  z) is invertible with a bounded inverse. This proves the theorem.
232
20. Theory of Quantum Resonances II: The Shape Resonance Model
Problem 20.10. Prove that for u E D(H) and z E A(eo), 11(H(%, 9)*  z)u11 > co11ull.
by replacing the argument used in the proof of Theorem 20.7. Note that it is necessary to consider Theorem 18.6 for aeti,(HO .6)*). It remains to prove Theorem 20.8 on R, (z). We outline the necessary modifications to the proof of Theorem 20.7 in Problem 20.11.
Problem 20.11. Prove Theorem 20.8. (Hints: The main modification occurs in the estimate of (HI(),, 0)  z) on supp jo. For H = A + V, use the Schwarz inequality to establish
II(H  E)jou11 > I1jou11' (jou. (H  E)jou). Apply this to H, (A 6), taking into account the simplifications of H, (),., 0) on supp jo. Finally, to estimate IInakR1(z)II, retain the kinetic energy term on the right side of (20.62), and use the simple inequality IIf112+11g112 > 211.11 IIxll
to obtain 11akjou11'11ioul1 ' > a11akjoull a211joull.
Apply this inequality by making a judicious ,1dependent choice of a.)
20.8
Notes
Recent papers on the semiclassical theory of shape resonances are by Ashbaugh and
Harrell [AH], Combes, Duclos, Klein, and Seiler [CDKS], Heifer and Sjostrand ]HSj4], Hislop and Sigal [HSI, HS2], and Sigal [Si5]. Ashbaugh and Harrell consider onedimensional and spherically symmetric shape resonance potentials of compact support in the large ,l regime. They prove the existence of resonances and compute upper bounds on the width. The other papers consider the multidimensional case and the methods are similar to those presented here. Another approach to the semiclassical theory of resonances was developed by Helffer and Sjostrand in [HSj4]. We will comment more on this in Chapter 23. For the relation between the HelfferSjostrand definition of resonances and the AguilarBalslevCombesSimon theory, we refer to the paper of Helffer and Martinez [HeM]. Using techniques similar to those discussed here, the existence of shape resonances in the semiclassical regime was proved by Combes, Duclos, Klein, and Seiler in [CDKS]. Shape resonances potentials of compact support were studied by Nakamura [N4, N5]. The question of the order of the pole at a resonance in the semiclassical regime was studied by Kaidi and Rouleux [KRJ.
20.8 Notes
233
The theory of resonances for generalized semiclassical regimes (see Section 16.4) has been applied to a variety of problems. Among the problems in the quantum mechanics of twobody systems, resonances in the Stark effect were studied by Graffi and Grecchi [GrGr], Harrell and Simon [HaSi], Herbst [He3], Sigal [Si5], and Titchmarsh [T21. The Zeeman effect and resonances in magnetic fields are studied in Briet [Br], Helffer and Sjostrand [HSj4] and Wang [W2]. Resonances in the BornOppenheimer model for molecules were studied by Martinez [M3, M41 and in a model of molecular dissociation by Klein [K12]. We mention some other applications of the theory. Resonances in quantum mechanical systems often correspond to bounded orbits in a corresponding classical system. This idea has been applied to various geometrical situations. Resonances
for the LaplaceBeltrami operator on an R" with certain spherically symmetric Riemannian metrics, such that there exist families of trapped geodesics, were studied by DeBievre and Hislop [DeBH]. Barrier top resonances were studied by Briet, Combes, and Duclos in [BCD2]. This is a case of resonances generated by closed, hyperbolic trajectories, which were analyzed by Gerard and Sjostrand in [GSj]. The resonances for a Schrodinger operator with a periodic potential were studied by Gerard [G2] and Klopp [Kip). Gerard also considered resonances in the scattering of atoms off surfaces of semiinfinite periodic structures in Gerard [G 11. We refer to Reed and Simon, Volume IV [RS4] for a discussion of resonances in the scattering of an electron off a helium atom.
21
Quantum Nontrapping Estimates
Let H,K be a family of Schrodinger operators for K E S, some sector in C, having 0 as a limit point. Suppose that Ao is an unperturbed eigenvalue of HK. Lowerbound
estimates on (H,K  Ao)u, for functions u supported in the complement of some compact region, play an essential role in proving the stability of the eigenvalue with respect to the perturbation HK. We have explored some numerical range and localization methods for establishing such estimates in Chapter 19. In our discussion of shape resonances, we assumed a condition, (V4), which would be rather difficult to verify for a given potential. In this chapter, we formulate a nontrapping condition directly on the potential, relative to a given vector field used for the spectral deformation, and show that the stability estimate (V4) can be derived from this condition on V. The nontrapping condition implies that the potential outside a compact region does not produce any resonances in a neighborhood of the unperturbed eigenvalue Xo.
21.1
Introduction to Quantum Nontrapping
In order to motivate the quantum nontrapping condition associated with Hunziker's method of spectral deformation, let us recall a very simple form of the virial theorem (see [RS4]). Consider a potential V E CO0(W) satisfying for a = ( a I , . . ,a,,), IaI =0, 1,
18aV(x)I l),
(21.12)
(t; y, rl)
_
(V V)(x(t;y, i])),
(21.13)
with the initial conditions x(0; y, r)) = y and (0; y, r)) = r). If we fix the energy E > 0, then for given initial conditions (y, rl) with h(y, r)) = E, the flow (x(t; y, ri), i (t; y, rl)) remains on the energy surface {(x, i') l h(x, = E) __ SE for all time. We are interested in the situation when the surface SE is unbounded. The bounded case was dealt with in Problem 20.1.
Definition 21.2. An energy E > 0 is nontrapping for the Hamiltonian system (21.12) and (21.13) if for any R >> 1 large enough, 3 T =_ T(R) such that Ilx(t; y, ?7) 11 > R,
for allltl > T when lyl < Rand E =
11 T7112 +V(y).
2
This timedependent condition indicates that V will not produce bounded orbits from initial conditions on the energy surface SE and with II v II < R. We would like a timeindependent condition directly on V which guarantees that V is nontrapping in the sense of Definition 21.2.
21.2 The Classical Nontrapping Condition
239
To formulate such a condition, we first note that a repulsive potential is nontrapping, for if the force F = x  VV > 0, then we immediately have 11X(t)II > 0, which shows that IIx(t)II > cot +c1. On the other hand, if E > V, then IIEII = 12(E  V(x))1""2 > 0, and so by Hamilton's equations,
I1i(t)II =12(E  V(x(t)))]', provided the right side is strictly positive. It follows that in this case the potential is nontrapping at energy E. If, however, there is a surface on which E = V (x), then we may find a bounded orbit for some initial conditions.
Problem 21.4. Consider a potential on R1, VW=e
_r'
and show that every positive energy E ' I is nontrapping. Explore the Hamiltonian
flow for E = 1. Is E = I NT? We can now write a condition on V that allows a local repulsive character of V to compensate for the vanishing of E  V and, conversely, that allows the local positivity of E  V to dominate the possible local attractivity of V. The quantity we study is
SE(x; V)  2(E  V (x))  x VV(x),
(21.14)
which is a generalization of the virial (21.2). Note also the relationship between this quantity and the commutator (21.4). If we define a classical analogue of the operator A by a(x, i;) = x , then the Poisson bracket between the Hamiltonian h in (21.11) and a is {h(x, ), a(x, l; )}
h
= F=]
=
as
ah
as
ax;  ax; aF;
SE(x;V),
(21.15)
provided we restrict to the energy surface SE. We have the following simple proposition.
Proposition 21.3. Suppose V E C 1 and an energy E > 0 is such that SE(x, V) > co > 0, for all x E R" and some co > 0. Then for initial conditions (y, rl) such that h(y, r7) = E, the flow satisfies
Ilx(t; y, 0II > cit +c2,
for some cl > 0, some c2, and all t E R.
Proof. Let a = x . The Poisson bracket of a with h is {h.
a}
_
(ag, h
2
a.,, a 
a a,r, h)
V)  2V  x VV.
(21.16)
240
21. Quantum Nontrapping Estimates
If we evaluate (21.16) along a trajectory on the energy surface SE, we find
{h, a}=2(EV)
(21.17)
By Hamilton's equations and the NT condition, we obtain da
dt
(X (t;
v,
{h, a} > co > 0,
q),
(21.18)
so that
Again, by Hamilton's equation, i = z, and so I
d
2 dt
,
Ilx(t;Y, n)II > cot +c1,
C
which allows us to verify the condition of Definition 21.2.
The classical NT condition shows that a trajectory on a nontrapping energy surface SE will eventually escape to infinity. No bounded orbits can be created by the potential with initial conditions lying on SE. To compare this with the quantum NT condition of Definition 21.1, let us examine the classical case in more detail. For this, it is convenient to work in one dimension. We can generalize the virial (21.14) by replacing x by a general vector field v. We then obtain the NT condition
 vV' + 2(E  V )v' > eo.
(21.19)
This is close to the quantum condition (21.7) and suffices in the quantum onedimensional situation (see (21.9)). In analogy with the proof of Proposition 21.3, we can define an observable a by
a, = v(x) ,
(21.20)
with v the vector field appearing in (21.19). The Poisson bracket of a and h is
da
_
{h,a,;}=i`v'V'v
=
2(h(x, z)  V(x))v'(x)  V'(x)v(x).
dt
Restricting to an energy surface h(x, ) = E, we see that
da dt
= {h, a
= SE (x, V) >> eo.
}
(21.21)
SE
Hence, a is a classical observable that increases along the trajectories on the surfaces SE. The existence of such an observable implies that there are no bounded trajectories on SE.
21.3 The Nontrapping Resolvent Estimate
21.3
241
The Nontrapping Resolvent Estimate
In this section, we prove that a nontrapping condition (21.7) on a potential V implies an a priori lower bound on the spectrally deformed Hamiltonian HO'. 0) in the semiclassical regime. This reduces the verification of condition (V4) for shape resonance potentials to showing that they are nontrapping. We discuss this in the next section. Let eo c o,(K) as in Chapter 20. We defined in (20.21) and (20.22) the energies e' = eo + )`a and e  eo + 2A8, for 8 > ; . We assume that V (x; ),) is nontrapping on Ext S+(e; k) with vector field v constructed in Proposition 20.2, Let H(;., 0) be the corresponding spectral deformation family. We can take 0 = i,8, 0 < 0 < a, and write Hp().)  H(X, ia), for simplicity. We prove the following nontrapping estimate. Theorem 21.4. Let V (x; A) be nontrapping at energy e in Ext S+(e; ),.) for all large. Then 3 co > 0 such that Vu E C'(Ext S+(e; X)), Vl large and sufficiently
small 0
0, all 0 > 0 small, and A sufficiently large. (Note that (21.43) implies cos 0 > co > 0 for large A.) (3) Interior estimate. We now treat the jo. Fterm in (21.34). The kinetic energy condition is Re(u, jo,FP?/3jo.FU) = Re(jo,FU, PPai/3,tmPrJo,FU)
+Re(u, JO,Fgiju).
(21.46)
From the estimates obtained after Problem 18.1, we can bound (21.46) from below by n
Y(1
 COT)IIJo,FaeuII2 
CZp.121 IIJO,FUI12,
(21.47)
e=I
for co, cI, c2 > 0 and 8 > 1/2. As for the potential energy term, we note that V(x; A) I [Int S+(F'; A) \ Int S+(e; A)] > F', which holds for A sufficiently large. Using this fact and the expansion (21.39) about
0= 0, we find Re(u, (Vip(x;),)  eo)jo,FU) >
(2Coo) IIjO,FUI12.
(21.48)
Combining (21.47) and (21.48), we find Re (jo,FU, (HP(A)  eo)jo,FU) >
GI
xs
 cod  Ci
Al2s
IIjo,FU112
n
cof3)IIJo,F8PU112, t=1
9
> CofIIjO,FU112+
for all A large and )3 > 0 small.
10
n
IIjo,FaeU112 cjAZ8I1U112, P=1
(21.49)
246
21. Quantum Nontrapping Estimates
(4) Error terms. Finally, we consider the k = 0 term in the sum (21.34). Writing p 2 as in (18.10), we define At, = ajo.fnr(pill Jo.f,) and
At = ajp.me(PmjO,F)
The commutator in (21.341) for k = 0 is [Peo, JO.F]JO.F = 2Af(Pejo,F)+(Ae + Ae)jo,FEpt,
from which we obtain n
1(u,[p ,jO,F]Jo,FU)I
R, for some
R > 0,
V(x)=f(x)Ilxl[
,
(21.55)
where p > I and the positive, bounded function f is smooth with bounded derivatives. In addition, f must satisfy some local conditions given below. The behavior of V inside BR(0) is rather arbitrary, subject to the conditions mentioned above, and there can be local singularities. Let us consider a positive energy level E > 0. We can choose E so that in a neighborhood of the classical turning surface S+(E), the potential V has the form (21.55). We will denote by Si various strictly positive constants whose precise value we will adjust below. Let vE be a vector field exterior to the surface S+(E + 281 ), as in Definition 20.1, VE (X) = rhE+23, (V (x))x,
(21.56)
where the form of the function 4E+28, will be chosen below. We indicate the proof that such a potential V is nontrapping on Ext S+(E + 81) with respect to the vector field VE.
We can find 82 such that E  V > 83 > 0 for x E Ext S+(E  282) because of the smoothness of V. Let us introduce a partition of unity j + j; = 1 for Ext S+(E + 281) so that
ji
I ExtS+(E  282) = 1
and
jl I IntS+(E  82) = 0; see Figure 21.2.
248
21. Quantum Nontrapping Estimates
E+262 Ext S'(E+ 2 fit)
83
12
1
I
i
0
S'(ES1) S (E251) S (E+ 2 S2)
FIGURE 21.2. Geometry and partition of unity used for the verification of the nontrapping condition.
Computing the first term in (21.7), we obtain
V1 = VE VV =PE+2S1(V(X))[Pf  x' Vfl 11X 11
(21.57)
and for the second term, we obtain
V2 = 2(E  V(x))(Pi[OE+2s,8iVxj +OE+2S,Sijlpj)(P2)
1.
(21.58)
We now estimate from below Vi on supp jk. We choose OE+26, as in the proof of Proposition 20.2 and satisfying 4E+26, > S4 on s < E + 31, and OE+26, < 1165, on s < E + S1. It then follows that for any u E Co (Ext S+(E + 311)), we have
(u, PA[OE+2b,8iVxj +OE+2Si8ij]Pju)(P2)u 1 > (84_
co)
(21.59)
SS
/
since 8; Vx j is uniformly bounded on this exterior region.
The function f' must satisfy the condition that (p  (x  V f) f 1) > K > 0 on the region where the gradients of ji are supported. This condition is not too restrictive, for if we recall that near S+(E) we have that llx 11
[ f/E 11I P, and thus
the condition is implied by (p  [ f/E]1 /P[ Ilof IIf1 I) > K > 0. On the support of j1 fl Ext S+(E + S1), it follows from (21.57) that V1
=
QE+25,(V(X))V(X)[P  x  (of)f1l
>
[E  2821K,
(21.60)
and from (21.58) and (21.59) that
V2(x) > 251[C, +S4],
(21.61)
where C1 is independent of 81. It follows from these expressions that by taking 31 small, if necessary, there exists an E1 > 0 such that (u, jj [V, +V2Ju) > E,11j,u112.
(21.62)
21.5 Notes
249
The term involving E  V should dominate on the support of j,. From (21.57), we find V2(x) > 263
[
4
CO
(21.63) 65
and
V1(x) > 64lif ll
Rp+1Ko,
(21.64)
where Ko > I pf  x V f I IIxiL1, for Ilx11 > R. By adjusting b4, if necessary, we see that there exists E2 > 0 such that
(u, j;[V, +V2]u) > (2I1.i2ull'.
(21.65)
This equation, together with (21.62), establishes the nontrapping condition for potentials of the form V (x) = f (x) llx II P for II x 11 > R. It follows that any potential
that is a sum of such functions outside of some compact region will satisfy the nontrapping condition. From this discussion, we see that there are nontrivial shape resonance potentials that satisfy conditions (VI)(V4) of Chapter 20 (we simply have to take f to be the restriction of a function analytic in some truncated slab inC" ). We will discuss the nontrapping condition for the Stark effect and for Stark ladder resonances in Chapter 23.
21.5
Notes
Nontrapping conditions play a significant role in controlling the resonances arising from the exterior of the potential. Nontrapping conditions also enter in the theory of resonances for the scattering of waves by obstacles (see Section 23.3 for a discussion of the Helmholtz resonator, the basic paper of Morawetz [Mo I], and the book [Mo2], and the book by Lax and Phillips ILP]). As in Theorem 21.4, nontrapping conditions in quantum mechanics enter into stability estimates. Nontrapping is closely related to the existence of resonance free domains. In the shape resonance case, the nontrapping condition is used to prove that the exterior Hamiltonian, H, (;., 0), has no resonances in certain regions of the lower half of the complex plane. These ideas were studied in the papers of Briet, Combes, and Dulcos [BCD1], Combes, Dulcos, Klein, and Seiler ICDKS], Combes and Hislop [CH], DeBievre and Hislop [DeBH], Helffer and Sjostrand [HSj4] and Klein [KI1]. They were further generalized by Gerard and Sjostrand [GSj]. Nontrapping conditions also enter into semiclassical resolvent estimates. These estimates are a form of the limiting absorption principle for which the dependence of the boundary value of the resolvent on the semiclassical parameter is explicit. Let H(h)  h2i + V be a selfadjoint Schrodinger operator with a C2 decaying potential. Suppose that the potential V satisfies a nontrapping condition at energy E with respect to a vector field f on the complement of the classically forbidden
250
21. Quantum Nontrapping Estimates
region. An example of such a semiclassical resolvent estimate is
Jim sup 11(1+11x112)2(H(h)EiF)'(1+IIX112) 211 < Ch*, E*0
for any a > 1/2 and h sufficiently small. Such estimates are used in the study of scattering in the semiclassical regime. We also mention the relation between the Mourre estimate (see [CFKS]), which in some sense generalizes the nontrapping condition, and semiclassical resolvent estimates. We refer the reader to some papers discussing these issues: Robert and Tamura [RTI ], [RT2]; Yafaev [Y]; Jensen [J3]; Gerard and Martinez [GM 11; Gerard [G3]; Graf [Gr]; Hislop and Nakamura [HNI; and X. P. Wang [W I].
22
Theory of Quantum Resonances III: Resonance Width
Introduction and Geometric Preliminaries
22.1
The goal of this chapter is to prove exponentially small upper bounds on the imaginary part of resonances in the semiclassical regime. In Chapters 20 and 21, we proved the stability of a lowlying eigenvalue e; E a(K) of finite multiplicity mi > 1, with respect to the perturbation H(A, 0), Im 0 > 0, for all k sufficiently large. We first showed in Theorem 20.5 that there exist mi not necessarily distinct functions {ej(i)(A)}T' I such that ej(i)(X) E Q(Ho(A))
(22.1)
lim ej(i)(A) = ei.
(22.2)
and
Next, we considered an annular region A(ei ), described in (20.41) and (20.46). For a i > 0 satisfying 0 < a 1 < (Im 6)b, we defined
;
A(ei)zIIai<Jeizl 0 such that A > Ao implies that lej(i)(A)  e; I < !a, and that a(Ho(A)) n {z I Iz  e; I < a,} = {el(; )(a,)}. We then proved in Theorem 2 exist eigenvalues {zi,k(A)} of H(A, 0) lying 20.4 that for a. sufficiently large, there in {z Iz  ei I < 1ai) n (C of total algebraic multiplicity mi. From (20.55), we conclude that I
IRe zi.k(A)  e; 1 0 and co depending on e and (22.5)
IIm :r.k(A)I < cb;
Our aim is to improve this second result on Im z,.k(A). For simplicity of notation, let us fix Pj E cr(K) and call it eo as in Chapter 21. Correspondingly. we write epo)(A) and zo,t(n.). We consider the Agmon metric po, as in Definition 3.2, for V (x; n.) and eo:
po(x..v) = inf )ET,
I
f
i
[V(y(t );^)  eol+jY(t )Idt
(22.6)
0
where y E P,, satisfies y E AC[0. 11. y(0) = x, and y(l) = Y. Recall that for two subsets T. S C R", the podistance between them is defined as po(S. T) = inf po(.v. Y).
(22.7)
VES
VET
Our improvement of (22.5) is the following. Theorem 22.1. Let So(A.) be the distance in the Agmon metric po (22.6) between
S(eo:A)and S+(eo;n.).Thenforanye > 0andforallAsufficienth'large, 3cE > 0 such that IIo.k()  eJA(o)(A)I < c,e
(I
(22.8)
for some jk and, in particular, IIm :o.k(A)I < cFe
`(' F'S°(k'
(22.9)
Furthermore, we have
S0(A) = AdA 0(0,S'(0))+0 (.l)
(22.10)
as,k + oe, where
d (x,v)=
inf
{fi V(y(t))V(0)1(t)Idt
(22.11)
is independent of X.
Let us note that the corresponding real part (22.8) does not really improve (22.4) and that (22.9) is the main result. The exponentially small imaginary part is characteristic of resonances that have their origin in tunneling phenomena. By contrast, resonances arising from a perturbation of the form Ho +,k V will generally have a nonvanishing, imaginary, part secondorder in A. The width is then predicted by Fermi's golden rule (see [RS41). As we have discussed, the exponentially small shift (22.9) cannot be predicted in any finiteorder perturbation theory in X.
22.2 Exponential Decay of Eigenfunctions of Ho(X)
253
To prove (22.8), we construct approximate eigenfunctions of HO', 0) and ZOAX) from the eigenfunctions of Ho(X.) for the cluster of eigenvalues (ej(o)O.)). This is a good approximation because the remainder term is localized in the CFR(e0) for
the potential V0. Thus, we can use the exponential decay of the corresponding eigenfunctions of Ho(A) to control the remainder. Let us recall some relevant geometry of V discussed in Chapters 20 and 21. For e0 E a (K ), we define e'
=
eo + As,
e
=
eo + 2X8.
for 3 > 1/2. The potential Vo(x; X), defined in (20.36), satisfies Vo = V on Int S+(e'; Al). We used a partition of unity { j; } '_0, constructed in Section 20.7,
satisfying E j7 = I and supp(Vj,) C Int S+(e'; A.) \ Int S+(e; A).
(22.12)
jo I Int S+(e; A.) = 1,
(22.13)
j,
(22.14)
so that
I Ext S+(e', A) = 1.
Given our construction of the spectral deformation family of H(k), if u E Co (Int S+(e'; A)), then
H(A, 6)u = Ho(A)u.
(22.15)
This observation is essential for the construction of approximate eigenfunctions.
22.2
Exponential Decay of Eigenfunctions of Ho(A)
Let (ea(;)(,k)}nj'' i be the set of eigenvalues of Ho(A) satisfying (22.2). For simplicity, we write ej (A) for these eigenvalues, eo for the unperturbed eigenvalue, and mo for
the total multiplicity. Let {uj } , be an orthonormal set of corresponding eigenfunctions for Ho(A): Ho(A)uj = ej(A)uj. Because of the choice of the partition { j; }i_0 in (22.12)(22.14), and the convergence (22.2), we know that suppI V j; I C CFR(ej(A)) for the potential V (x; A) for all large A. We can then apply the methods
of Chapter 3 on the decay of eigenfunctions for eigenvalues below the bottom of the essential spectrum. It is important to note the following difference. Here, we want the decay in A of the eigenfunctions uj restricted to a subset of the classically forbidden region rather than decay in x. This results in some simplifications. In particular, as the CFR(eo) for VV is bounded in the Agmon metric, the technicalities of Lemma 3.7 are not needed. Recall, finally, from Proposition 3.3 that po as given in (22.6) satisfies
254
22. Theory of Quantum Resonances III: Resonance Width
IV4po(x, y)I'` < (V(x;,l)  e0)+ almost everywhere. We need the following variation of Theorem 3.4.
(22.16)
Theorem 22.2. Let x be supported in le  Int S+(e';.k) \ Int S+(e; Al), and recall that So(A) is defined in Theorem 22.1 as the Agmon distance from S(eo; n.) to S+(eo; ),). Then for any E > 0 2 a.F and constants c, ct. > 0 such that for all A>%F, IIXu1II < ca.e (1F)So(A)
(22.17)
and
IlxVujII _ IIVOI12 +coeA 110112.
(22.20)
We next define a smooth cutoff function Xo by Xo(x) =
x E Ext S(e; A), x E Int S (e'; a.).
j 1' l 0,
(22.21)
We see from (20.50) that IVXol = O(X;S). Setting 0 = of Xou j in the left side of (22.20), we obtain an upper bound: Re(e2f Xou j, (Ho(X)  e j(X))Xouj) ([IVXOI2 +2(VXo Of)XoJe2 uj, u1) .
o)e2(1e)po(o,s(e:x)),
< coil oxoll.(IIVXollx + where obtain
ll
(22.22)
1 on supp(VXo). Combining this with the lower bound (22.20), we
Il
V (ef Xou j )112 + C
EcoAa 11 'k23'
of Xou j 112
2(1E)po(O.S(e;;0)
(22.23)
Let us define o(x) = po(0, x)  po(0, S(e, A)). Then (22.23) immediately implies that
f
i.
(22.24)
22.2 Exponential Decay of Eigenfunctions of Ho(A)
255
Furthermore, since we have
Ile1xoVujII < IIV(efXouj)II+EeoXlllefXoujll, where Xo is supported on a slightly bigger set then Xo, we obtain, from (22.23) and (22.24), fe
2(1F)PO(a)xo(x)Iouj(x)12
0. Since we can take 1/2 < 3 < 1, the result now follows from
0
(22.26) and (22.29) by taking % sufficiently large.
Problem 22.1. Beginning with estimate (22.25), derive estimate (22.18) for Vu j. Problem 22.2. Prove estimates (22.10) and (22.11) on the asymptotic behavior of
Problem 22.3. The goal of this problem is to obtain decay estimates on the resonance eigenfunctions localized in CFR(e'; V;,). Let (0j) m', be an orthonormal basis of generalized eigenfunctions (see Section 6.2) for H(?, 6) in Ran P(X. 6). Let {uk)k °i be an orthonormal basis for Ran Po(%), as used above.
f )Oj. (Hint: The condition H(ti, 9)* (1) Show that P(A. O) f = Ejn'0, H(?, 6) (complex conjugate) implies that P(n., 9)* = 9).)
(2) Let M!j =
u j); show that the mo x mo matrix M is invertible and that IIM111 < co, for all large k.
(3)
Let jo be as in (22.12) and (22.13). Prove the following geometric resolvent formula (a variant of Lemma 19.16):
R(z)jo  joRo(z) = R(z)WoRo(z), where W0 =
(22.30)
jo].
(4) Employing formula (22.30), derive the relation P(A, 6)jOUk = jOuk + rx,k
(22.31)
by integrating over FO in A(eo). The remainder is given by rx.k =
(27rt)i
fro R( z)Wouk(ek(A)  z)l dz.
Estimate (1  jo)uk and rx,k using Theorem 22.2 (in particular, (22.24) and (22.25)), the stability theorem, Theorem 20.7, and (20.50). (5) Make use of parts (1) and (2) and formula (22.31) to obtain the identity mo
T(MT )ij (JOU1 + rx.j ), j=1
and conclude that for
f
= 1 on CFR(e; Vx), 10,(x)12 < cbk
where po is defined after (22.23). Extend this result to ai aj0k using (22.15) and condition (V 1).
22.3 The Proof of Estimates on Resonance Positions
22.3
257
The Proof of Estimates on Resonance Positions
There are at least two techniques available for estimating resonance positions. We will prove Theorem 22.1 by constructing an approximate basis for the resonance subspace Ran 9). We outline in Problem 22.7 another approach, used in [HSj3] and [Si5], based on estimates of the resonance eigenfunctions on the surface S+(e: A). We also refer to the paper of Howland [Ho2l, which discusses the resonance width in some simple onedimensional models. Lemma 22.3. Let {uj }j"') he an orthonormal family of eigenfunctions of Ho(1') for eigenvalues (ei(A))'7" i as in Section 22.2 with lim;, . ej(A) = eo. Let (ji }t)=0 be the partition of unity constructed in (22.12)(22.14). We define functions i/rj jouj, j = 1.... , m(). Then for any 1 /2 < S < I 3A.o such that A > A.p implies
HP., 0)i/ij = ei(a.)ij +rj(A),
(22.32)
where (22.33)
Ilrj(A)II < and
(i/i. Vrj) = Sij +
O(e2(IF)So'.)).
(22.34)
Proof. By the choice of jo, (22.15) and (20.36), we have jouj E D(H(A. 9)) and
H(A. H)fj = Ho(A)jouj = e1(A) V' + [A, joluj. As Ilojollx = O(A'  s) and supply jol C le, Theorem 22.2 implies that
Il[A, Jo]ujII
0 independent of ;, and /j. To conclude the estimate, we need a case of the Sobolev trace theorem (see [Ag2] for the general theorem and the proof): For S2 c R" sufficiently regular, the map H t (12) + L2(8S2) is continuous. Conclude from this and (22.56) that for any regular region We containing Ste, JIM zo,j(A)I
0 we have S(s, 8) > d(C. E)  25 > 0,
we see that the Euclidean distance along the tube Z(e) controls the width of the resonances. This is consistent with the earlier comments indicating that the tube behaves as a classically forbidden region. Thus, there is an effective tunneling through the tube. The origin of this tunneling is the Poincare inequality, (23.32). Applying this inequality to the tube Z(e), and noting that the right side of (23.32) is just the square root of the energy, we see that an eigenfunction of fixed energy cannot be supported in the tube for all small E. An eigenfunction corresponding to a fixed eigenvalue is squeezed out of the tube as it narrows.
278
23. Other Topics in the Theory of Quantum Resonances
We now sketch some aspects of the proof of this theorem. We have introduced the analogues of K and H0(A). As for the exterior region, let E(s) = Int(E U Z(s)), and denote by Hi (e) the Dirichlet Laplacian on E(s). This operator is the analogue of Hi (A). Note that both Ho(s) and Hi (s) are defined on the forbidden region Z(s) (as in the Stark ladder problem where the two approximate operators agreed on a subset of the forbidden region). A spectral deformation family for H and HI (s) is easily constructed on the exterior of BR(0), where R is chosen large enough so that S2(e) C BR(0). We choose a vector field v on I[g" so that
(1) v = 0 on BR(0);
(2) v(x) = x  Rx  ; z
on W1 \ BR+i (0), where z
 x IIx II 1;
(3) 11v(x)  v(y)II < llx  vll, x, Y e R". Such a vector field approaches the generator of dilations as IlxII * oo. All the results of Chapter 17 apply in this case. We denote the spectrally deformed families
by HF(p)and HI (s; t), p E {z E C I Im z c i(1, 1), IRe zI small}. The essential spectrum of both families is {z E C I arg z = 2 arg(1 + µ)}. The first step of the stability proof is to demonstrate that A0 E (7(AC) is stable under the perturbation Ho(e). Proposition 23.14. Let A0 E a(AC) with multiplicity No. Then, for n > 3, 3 so, c > 0 such that `ds < so, Ho(e) has No eigenvalues XI (e), ... , (counting multiplicity) satisfying for all j = 1, ... , No, IAo  >~)(s)I < CE 2'.
For n = 2, the estimate is
(23.34)
for any 0 < fl < 1/2.
The proof of this proposition relies on a comparison of Ho(e) and the direct sum
operator AC ® hE = Ho(s), where hF is the Dirichlet Laplacian on Z(e). The effect of he is small because the Poincare inequality implies that inf o(hr,) > CE2.
Hence, (h  Zr' is analytic in any neighborhood of 1,.o for all s sufficiently small. The resolvents Ro(z) and Ro(z) of H0(s) and Ho(e), respectively, can be compared using Green's theorem since the two operators differ only by the addition of a Dirichlet boundary condition on the surface Dr joining Z(e) and C. For any U. V E L2(C(s)), we obtain (u, (Ro(z)  Ro(z))v) = f (T RO(z)*u)(BRo(z)v).
(23.35)
Op
where T is the restriction or trace map for De,
BRo(z)v = (n O)c(Ac  z)'v ® (n O)z(hr  z)I v. (n O)C is the outward normal for C, and (n V)z is the outward normal for Z(e). The trace map is controlled by the Sobolev trace theorem as mentioned in Problem
23.3 Resonances of the Helmholtz Resonator
279
22.7. Iterating (23.35), we again obtain an analogue of the geometric resolvent formula for which the interaction term is localized on the small disk D6. The proposition and bound (23.34) follow from a careful estimate of these operators. The second step of the stability proof consists in proving the stability of the family of eigenvalues {AS(E)IN(, of HO(E) near .u under the perturbation H8. We use
a different form of geometric perturbation theory. Let 1to = L2(C(E)) e L2(£(E)) and WE, µ)  Ho(E) ® Hl (E. ti). We must compare Ho(E, M) acting on Ho with H8(µ) acting on h = L2(S2(e)). This is a twoHilbertspace perturbation theory similar to the one encountered in the discussion of the Stark ladder resonances; see (23.20). Let (j,)2?_, he a partition of unity of W1 satisfying _ _i J? = I and
Ji
{x
I
I d(x, £) > 28} = I
and J2
1
{x I d(x, £) < 8} = 1,
and so supp VJi C {x 18 < d(x, £) < 28} and IVJ; I is independent of e. We define a map J : 7l > Ro by
Ju=J1u®J2u, and so J*J = Ix. Let R(z) = (H8(µ)  z)] and Ro(z) _ (Ho(e, µ)  z)'. The geometric resolvent equation we need is
R(z) = J*Ro(z)J + R(z)MRo(z)J
(23.36)
on h, where the interaction M: to .> R is given by
M(ul ® U2) = {A, Jl }ul + (0, J2]u2. (This should be compared with (23.20).) Note that M is supported in the tube Z(e). In analogy with (22.36)(22.38), we define
K(z) = JMRo(z): Ho > ho. The main technical lemma is the following. Let ,Ei be the exponent of E appearing in Proposition 23.14 and c be the constant. Let I'E be a contour around Xo of radius 2ceP.
Lemma 23.15. For any 8 > 0, 3 cb > 0, Eo > 0 such that fore < Eo and uniformly
on r, 11K(z)II < cbE22b
(23.37)
Let X be a function localized near supplVJ; I. The proof of this lemma follows from estimates on the localized resolvents X (Ho(E)  z)', analogous to Corollary
20.7, and X (Hi (E, µ)  z)1, analogous to Theorem 20.8. The former estimate follows from the Poincare inequality. The latter estimate is much easier than in the shape resonance case. It is not hard to check that H, (E, µ) has no spectrum on and
inside re.
280
23. Other Topics in the Theory of Quantum Resonances
Given this lemma and (23.36), the proof of stability is easy. We solve (23.36) for R(z) on 1'f to obtain (recall (23.37))
R(z) = J*Ro(z)J + J*Ro(z)(1  K(z))' JMRo(z)J.
(23.38)
This formula is integrated along I'f. Denoting by P(E, µ) and Po(E) the resulting projections, we conclude that II P(E, µ)  J* Po(E)J II
0 3 ca > 0 such that for all e small enough l i e ((
a)d, (.C)/EuF
(23.39)
II L2(Z(F)) < ca,
and similarly for Vu, (dE is defined in Theorem 23.13). This proposition can be used in the arguments of Section 22.3 to obtain the upper bound (23.33) on the resonance width. The proof of Proposition 23.16 relies on the Poincare inequality (23.32) and Agmon's weighted inequalities, as in Section 22.3. The form of the Poincare inequality applicable here is
f
1012 < e2(l +cE) f
IVo12,
(23.40)
Z(F)
(F)
for appropriate 0 on C(E). Substituting f 0 for 0 in (23.40), where f is a real weight function, we obtain
E2(1+cE)' f (F)
If012
0 such that
Im zo(X) > cA2e d0(O.S+(0))
Other related results, including asymptotic expansions, can be found in [HSj4]. Lower bounds for resonances for onedimensional Schrodinger operators were obtained by Harrell [H2] using the methods of differential equations. Fernandez and Lavine IFL] obtained lower bounds for resonances widths for Schrodinger operators with potentials having compact support and for the Dirichlet Laplacian
in the exterior of a starshaped region in three dimensions. Lower bounds for resonances of onedimensional Stark Hamiltonians with negative potentials of compact support were computed by Ahia [Ah].
Appendix 1. Introduction to Banach Spaces
Al.! Linear Vector Spaces and Norms Definition A1.1 A linear vector space (LVS) X over a field F is a set with two binary operations: addition (a map of X x X + X) and numerical (or scalar) multiplication (a map of F x X > X), which satisfy the commutative, associative, and distributive properties. We are primarily interested in spaces of functions and will take F to be either R or C. Examples A1.2. (1) X = Ill" with the operations (+, ), the usual componentwise operations, and with F = R.
(2) X = C(L0, 1]), all continuous, complexvalued functions on the interval [0, 1 J. Addition is pointwise, that is, if f, g E X then (f +g)(x) = f (x)+g(x ), and scalar multiplication for F = C is also pointwise.
(3) X = lP, where 1P consists of all infinite sequences of complex numbers: (xi , ..., x,,, ...), x; E C such that x Ixi I P < oo, for 1 < p < oo. Again, addition and scalar multiplication are defined componentwise. Problem ALL Show that lP is an LVS. (Hint: Use the Minkowski inequality:
286
Appendix 1. Introduction to Banach Spaces
Definition A 1.3. Let X be an LVS over F, F = ll or C. A norm Il II on X is a map II
(i)
II
Ilx 11
:
X
R+ U {0} such that
> 0 and if Ilx II = 0 then x = 0 (positive definiteness),,
(ii) lIAxll = lxl Ilxll. % E F (homogeneity);
(iii) lix + I'll < IIx 11 +
11N. 11, x, V' E X (triangle inequality).
Examples A1.4. ( 1 ) X = 8" and define llx 1 1 = (j:;'=1 lx, I2) I/ , x e E. This is a norm, called
the Euclidean norm, and has the interpretation of the length of the vector x.
(2) X = C({0, 1]) can be equipped with many norms, for example, the L"norms: If f ii
° (d> If(x)IV'dx)
, f E X. 1 < p < oc, and the supnorm
(P=00): Il.f Ilx ° sup lf(x)l, f E X. xE(o. I
I
(3) X = lP has a norm given by Ilx II = (_Y;_`I Ix;1'')
Problem A1.2. Show that the maps X + ll
I/",
x E X.
claimed to be norms in Examples
A 1.4 are norms.
Definition A1.5. An LVS X with a norm 11 is a normed linear vector space (NLVS), that is, an NLVS X is a pair (X, II 11) where X is an LVS and 11 is a norm on X. II
11
A 1.2 Elementary Topology in Normed Vector Spaces Elementary topology studies relations between certain families of subsets of a set X. A topology for a set X is built out of a distinguished family of subsets, called the open sets. In an NLVS X, there is a standard construction of these open sets. Definition A1.6. Let X be an NLVS, and let a E X. An open ball about a of radius r, denoted Br(a), is defined by
Br(a)Ix EX111xall 0
i=)
as m, n * oc. We must show (1) that {x(n) } is convergent to some x, and (2) that
x E l°. (1) Condition (A 1.1) implies that each sequence of real (or complex) numbers {x(n)}n, (fixed i) is Cauchy, and from the completeness of R (or C) it is
convergent. Hence there is an x; E R (or C) such that xi = limn.mxi"). (xi, x2, ...). We show that xt") * x in the 1Pnorm. From the Let x definition of (xtn)}, for each e > 0 there is an NE E N such that m, n > NE )/n implies [Fk E. Now we take the limit of the left xim)Ip])lp
side as n > oo (fixed k) and get Sm(k) = [Ek=) Ixi < E. As a sequence in k, Sm(k) is monotonic (i.e., Sm(k) < S,(k') if k < k) and bounded, and hence limk_,w Sm(k) exists and is less than e. This is the 1pnorm of (x  x(m)), that is, lix  x(m)IIp < E. Hence, x(m) X.
(2) To prove that x E I P, we use Minkowski's inequality (Problem Al. 1) and begin by considering a finite sum as above. We write k iL=.1r
)'PL
(iX1i")
k iL=.n
Ix xi i") 1
n
+ r lxIP) r k
iL=.1
1
.
290
Appendix 1. Introduction to Banach Spaces
Since x(") E lt' and such that
Ilx(n)
 xllp f 0, given E > 0, we can choose n > NE k
lim
\\,
Ixilp)
< IIx`"'IIp+E,
i=1
and hence IIx1Ip < oo (i.e., x E 1P).
We now introduce some other topological ideas associated with Banach spaces.
Definition A1.19. Let X be an NLVS. A subset A C X is dense in X if each point x E X can be approximated arbitrarily closely by points in A, namely, for each
x E X there is a sequence {an} C A such that limn« a = x. Equivalently, for any x c X and any E> 0 there exists a E A such that IIx  a I I< E. Examples A 1.20.
(1) The rational numbers Q are dense in the real numbers. (2) The Weierstrass approximation theorem states that the set of all polynomials in x on [0, 1] (i.e., functions of the form ykI cix' +co) is dense in C([0, 1 ]) with the supnorm.
(3) Let X be a Banach space, and let A C X be a subset that is not closed. Then A is dense in A, its closure. Definition A1.21. A Banach space X is separable if it contains a countable, dense subset. (A set is countable if its elements can be put in onetoone correspondence with the set of integers 7L.)
Examples A1.22. (1) The set of polynomials on [0, 1] with rational coefficients is a dense, countable set in C([0, 1]) with the supnorm, so this Banach space is separable.
(2) The set of all elements x E lp with xi E Q is a dense, countable set in U', so lp is separable. (3) Simply for reference, the space L°O(R), the LVS of all (essentially) bounded functions on R with the (ess) supnorm, is nonseparable.
Remark A1.23. All the spaces used in the chapters are separable, unless specifically mentioned.
A 1.4. Compactness
291
A 1.4 Compactness We introduce another important topological notion associated with Banach spaces (which is, of course, an important notion in its own right).
Definition A1.24. Let X be a (separable) Banach space. A subset Y C X is called compact (respectively, relatively compact) if any infinite subset of Y contains a convergent sequence and the limit of this sequence belongs to Y (respectively, to X).
Note that if Y C X is relatively compact, then Y, the closure of Y in X, is compact. Moreover, if Y is compact, then Y is a closed subset of X. This follows since if {y } is a Cauchy sequence in Y, it is an infinite subset of Y. Since X is
complete, {y,) is convergent and the limit y =limn.x yn E Y by definition of compactness. Hence, we have the next result.
Proposition A1.25. Let X be a Banach space. If Y C X is compact, then it is closed and bounded (i.e., 3M > 0 such that Ilyll < M Vy E Y). Proof. It remains to show that Y is bounded. Suppose not; then for each n E 7L, there exists yn E Y such that Ilyn ll > n. Then {yn) is an infinite subset of Y. By of {y, , where or : 7L+ > Z+ compactness, we can choose a subsequence is an orderpreserving map, which converges. Then, as Y is compact, 3y E Y such that I1y,(n1ll converges to Ilyll. But, IIya(n)II > a(n) > n > cc, and so we get a E3 contradiction. Hence Y is bounded. It is a wellknown result that the converse of Proposition A 1.25 is not true: There are many closed and bounded subsets of certain Banach spaces which are not compact. Examples A 1.26.
(1) The problem just described cannot occur in RN or CN: the HeineBorel theorem states that a set is compact if and only if it is closed and bounded. (2) Let X = C([0, fl), and let Y C X be the subset of all functions in X which, together with their derivative, are uniformly bounded, that is, Y = If E X 111 f II oo < M and II f 'II oo < M). The Ascoli Arzela theorem states that Y is compact in X.
We now want to relate the definition in A1.24, which depends on the metric structure of X, to the general topological notion of compactness.
Definition A1.27. Let A C X, X any topological space. A collection of open subsets (Vi) of X is called an open covering of A if A C Ui Vi.
Theorem A1.28. A subset K C X, X a Banach space, is compact if and only if any (countable) open covering of K contains a finite subcovering of K. Proof. (1) Suppose K C X is compact, and let {Vi } be an open covering of K. Suppose on the contrary that there exists no finite subcovering. Then K \ Un 1 Vi is
292
Appendix 1. Introduction to Banach Spaces
nonempty for any n. Let X,, E K \ U" V; . By the compactness of K, there x,1 E K. Since { V; } exists a convergent subsequence Ix,,,) and x = covers K, X E Vj, for some j. Since Vj is open, x E Vj for all large n', say n' > No. Then { Vi , ... , VN0,, Vj) (where E is a finite covering of K, giving a contradiction. I
(2) Suppose K has the finite covering condition but is not compact. Let {x } be an infinite (countable) sequence in K with no limit points. The sets U
{x ,
are closed (U,, has no limit points) and nn U = n U =
Then Un is open and {U`} forms an open cover of K. Then there exists a finite subcovering { Un, } of K with which is certainly not true by construction.
Appendix 2. The Banach Spaces LP(IR71), 1 < p < oo
We present the basic theory of this family of Banach spaces. The reader can find any of the proofs not presented here in any standard text on real or functional analysis; see, for example, Reed and Simon, Volume I [RSI], Royden [Ro], or Rudin [R].
A2.1 The Definition of LP(IRn), 1 < p < oo The most important Banach spaces for us are the LPspaces, I < p < oo. The value p = 2 plays an especially important role, and we will describe its properties in greater detail later. There are two basic ways to obtain the LPspaces, and both are useful. The first, which we will sketch here, uses the idea of the completion of an NLVS. The second involves Lebesgue integration theory, so we will merely mention the results.
Definition A2.1. The support of a function f : n c R" > C, denoted by supp f, is the closed set that is the complement of the union of all open sets on which f vanishes.
Problem A2.1. Prove that supp f = {x E n I f (x) V 0}. Definition A2.2. The set of infinitely differentiable functions on R" with bounded (and hence compact) supports is denoted by Co (R" ). If l C R" and is open, then Co (S2) consists of all infinitely differentiable functions with compact supports in
294
Appendix 2. The Banach Spaces L"'(IR"), I < p < oc
We note that it is easy to see that C0 (R) is an LVS over C or R. We will always
take LVS over C in this section. For each p, 1 < p < oc, and any f E C (S2), define the nonnegative quantity
Ilfllp = C If(x)IV' dr]
71
(A2. I )
S2
which exists as a Riemann integral because supp f is compact and because f is bounded on its support. We see that (C((Q). II 11p) is an NLVS. Problem A2.2. Prove that for any open 2 C R". (C((Q). II II p) is an NLVS for 1 < p < oo. (Hint: Use the simple identity, 1./'+ glp < 21' (I fI" + glp).) II,) a Banach space that contains Our goal is to associate with X1, _ (Ci (S2), XP as a dense set. Recall that we have already seen in Remark AI.14 that XP is not complete. There are Cauchy sequences in X p whose limits do not exist in X p. Hence, we want to enlarge Xt, by adding to it "the limits of all Cauchy sequences in Xt," This will result in a "larger" space, called the completion of X. It is a Banach space that contains a copy of Xp as a dense subset. We will sketch the idea of the completion of an NLVS. In this manner, we obtain the Banach spaces in which we are interested. II
Definition A2.3. For any open 2 C R' and 1 < p < oo, the Banach space Lp(S2) is the completion of (Co°(S2), II
Ilt,).
Let (X, II II) be an NLVS that is not complete. We show how to obtain a Banach
space X with a dense subset X, such that X is isomorphic to X. That is, the sets X and X are in onetoone correspondence, and if i E X corresponds to x E X, then 11i 11 = Ilxll (the norm on the left is the induced norm on X). {x" } and Step 1. Consider the collection of all Cauchy sequences in X. Let a denote any two Cauchy sequences. We say that two Cauchy sequences # are equivalent if lim,,,x 11X  y" II = 0.
Problem A2.3. Show that (i) this defines an equivalence relation on the set of Cauchy sequences, and (ii) if {x I is Cauchy, then limcc llx,, II exists. Let [a] denote the equivalence class of a. We define a set X to be the set of all equivalence classes of Cauchy sequences in X. The set k has a distinguished subset X, which consists of all equivalence classes of constant sequences (i.e., if x E X, then let ax = {x" = x}). We see that [ax] consists of all Cauchy sequences converging to x. Hence, is isomorphic with X. Problem A2.4. Show that f( is an LVS, and verify that X is isomorphic to X.
A2. 1. The Definition of LP(OR"), l < p < ac
295
Step 2. We define a norm on X as follows. Let [a l E X, and take Ix,) E [a]. Then define IIlee ll1X = "lim llx"II
.
Problem A2.5. Verify that II  IIX is a norm on X. (Be sure to check that it is "well defined," that is, independent of the representative chosen.)
Note that if lax] E X. then naturally II[a.;IIIX = IIXII so that the isomorphism between X and X is isometric. We conclude that (X .
II
II z
is an NLVS with a subset k isometric with X.
Step 3. We sketch the proof that (X, II iix) is complete. Suppose {[a'']} is a Cauchy sequence in X. Let {xk " } be a representative. Then for e > 0 there exists AE such that for N, M > A r, we have [a
N}
 [aM]
E
X
;l xIIx"x411 0 such that IIxN x411 < E/2 for i > i0. For N > M, choose an index iN > ip for which IIxN  x411 < E/2. We can choose the map ip > iQ). Since {x7} is Cauchy, N p iN to be orderpreserving (i.e., P > Q there exists i i > 0 such that j, k > i i implies
IIxX"11 max(io, i 1). We claim {x N } C X is a Cauchy sequence. This follows since
IIXN X411 1.
f(x)= Then, f E L1(IR), but f2 V L1(IR). We do, however, have the possibility that fg belongs to some LP'space if f and g belong to some other V and L'spaces. Theorem A2.4 (The Holder inequality). Let p. q, r E R+ be such that 1 / p+ 1 /q =
11r. If f E L" and g E Lq, then fg E Lr and IIfgiir < IIflip IIgiIq Proof. (1) We first prove the following inequality for any a, b E IR+ and any p, q E ll+ such that I/ p + 1 /q = I :
ab < plap +q'I bq .
(A2.2)
Consider the function
a = F(b) = bq1 or, equivalently,
b= Fl(a)=ap1, where we use the fact that (p  1)(q  1) = I. We consider the rectangle [0, bo] x [0, aol in the baplane, where ao, bo > 0 are arbitrary. Then the curve a = F(b) divides the rectangle into two regions, I and II. The area of region I, which lies below the curve a = F(b), is r bo
Al=J
F(b)db=qlbo,
0
whereas for the complementary region,
All =
Fl(a)da = plao J0 ao
It is easy to see that we always have aob0
which is just (A2.2).
f in L P (R" )
Theorem A2.7 (Lebesgue dominated convergence). Suppose fn E LP(R") and f" + f almost everywhere. If there exists g c LP(ll ) such that If"IP < ISIP almost everywhere, then f E LP and f" + f in LP. We need one other result about sequences of functions that converge in the LPnorm. Theorem A2.8. Suppose that { fn) is a sequence of functions in L' (I[8") that converges to f in the L 1 norm. Then, there is a subsequence { fn(k) } that converges pointwise almost everywhere to f. We refer the reader to [RS 1 ] or [Ro] for the proof.
Appendix 3. Linear Operators on Banach Spaces
A3.1 Linear Operators Definition A3.1. Let X and Y be LVS. A linear operator A from X to Y is a linear map defined on a linear subspace D(A) C X, called the domain of A, into Y. We write A : D(A) Y. A linear map has the property that for A E C (or
and f, g E D(A), A(Af + g) = AAf + Ag. Note that D(A) is a linear space by definition, so f, g E D(A) implies that f +;lg E D(A).
Examples A3.2. (1)
Let X = Y = 111;". Any linear transformation A :
Ilk"
11 ®" (i.e., D(A) = R")
can be written in the form (Ax)i = Fj=i Aijxj, for some n2 quantities Aid E R. This matrix representation of A depends on the choice of basis for ". The numbers xi are the coefficients of x relative to this basis.
(2) Let X = Y = C([0, 1]), and let a E X. Define A : X + X by (Af)(x) = a(x) f (x), f E X. Then A is a linear operator with D(A) = X. The operator A is "multiplication by the function a" and is called a multiplication operator.
(3) Let X = Y = C([0, 1]) and let D(A) = C1([0, 1]), the LVS of continuous functions that are differentiable and whose first derivatives are continuous. Note that D(A) ¢ X. Define an operator A, for f E D(A), by (Af)(x) _ (df/dx)(x). Then A: C'([0, 1]) + C([0, 1]) is a linear operator.
302
Appendix 3. Linear Operators on Banach Spaces
(4) Let X = Y = C([0, 1]) and let K(s, t) be continuous in .s and ton [0, 11 x [0, 11. Let f EX. and define (A f)(t) = f, K(t. s) f (s )ds. It is easy to check that this integral defines an element of X for each f E X. Hence, we may take D(A) = X, and A is a linear operator.
(5) Let X = C([0, I]) and Y = FR,. For each f E X. define Af =
f f(s)ds.
Then A: X * Y with D(A) = X is a linear operator.
(6) Let X = Y = C([0, I]). The map (Af)(x) = f2(x). defined on all of X, is not linear. This is an example of a nonlinear operator.
Problem A3.1. Verify all the statements in Examples A3.2.
Definition A3.3. A linear operator A : D(A) C X + Y is densely defined if D(A) is dense in X.
Remark A3.4. Note that the definition of a linear operator includes both the domain and the action of the operator on elements in the domain. If the domain is changed, the operator is also changed. All the operators in Examples A3.2 are densely defined. A linear operator that is not densely defined has many undesirable properties (for example, we shall see that it does not have a uniquely defined adjoint). For this reason, we will work only with densely defined operators. Moreover, we will assume that X is an NLVS. If D(A) C X is not dense, we will replace X by D(A) __ X1, the closure of D(A) in X, which is a Banach space. We then consider the operator A with dense domain D(A) in X1. It is possible to define algebraic operations on the family of linear operators from X to Y. Assume X and Y have the same field of scalars F. In particular, we have
(1) Numerical multiplication: If A E F, and X E D(A), we define ),A by (;,A)(x) = A(Ax). Thus, if >
Y even when A is defined only on D(A) C X. Moreover, by operator we will always mean linear operator.
Definition A3.5. An operator A : X  Y is continuous if for each convergent Ax,, _ x E D(A), we have sequence C D(A) with x,,) = Ax. Problem A3.2. Show that A is continuous if for each sequence {xn } C D(A) with Ax,, = 0. x x = 0, we have
Example A3.6. With reference to part (2) of Example A3.2, the operator A, multiplication by a E C({0, 1]), is continuous on X = C({0, 1 ]) with any norm on X as given in Example A1.4 (2). This follows from the fact that a is bounded: Let supti.Elo.il la(s)I = Ilallx Then, it follows that
IIAfII. < IIaIla1IflI c and
IlAfll,, < Ilallcll.fllp.
1 Y be a linear operator. Then the following two statements are equivalent: (1) A is continuous; (2) A is bounded.
Proof. (2)
(1) If A is bounded, we observed that A is continuous; in fact, we can extend A to a bounded operator on X and the extension (which is bounded) is continuous.
(1)
(2) Suppose A is continuous but unbounded. This means the exists a sequence {xn } in D(A), with Ilxn II < M, such that II Axn II > n Ilxn II . Let
yn = xnIIAxnll'. Then {yn} C D(A), IIynII < n1, so yn ;, 0, but 0 but yn > 0. This contradicts the IIAynil = 1; that is, limnx Ay,, continuity of A, so A must be bounded.
Definition A3.11. Let A : X  Y be bounded. Then II All, the norm of A, is defined by IIAII
=_ inf{K > 0 IIIAx1Iy < Kllxllx, for all x E X},
which is equivalent to
IIAII = sup Ilxllx' IlAxlly 1EX
iXiIO
Problem A3.4. Prove the equivalence of these two definitions. Note that we have assumed that if A is bounded, then D(A) = X, as follows from Proposition A3.9. Moreover, we have
IIAxhl < IIAII llxll, x E X. Examples A3.12 (with reference to Examples A3.2).
A3.2. Continuity and Boundedness of Linear Operators
305
FIGURE A3.1 The functions
(1) All linear operators A : II8"  R" are bounded. Note that there are actually many norms on the set of n x n matrices. The most common are II A II x = sup;
1
la;1 I, and Il A lltr = Tr(A*A)11'2. The norm defined in Definition A3.11
is the operator norm. This norm is distinguished by the fact that it satisfies II A* A II
= II A 112, where A* = AT and A 7 is the transpose of A.
(2) Let Af = af, with a, f E X  C([0, 1]). The multiplication operator A is bounded in all norms on C([0, 1]) (although this space is a Banach space only in the supnorm). In the Banach space (C([0, 1]), II llx). we = Ilallo. To see this, note that for any f E X, 11Afl1 «; = Ilalloo.Uponchoosing f = I E X, Ilaflloo < llalloollflloc,andsohIAII
have 11 All
we have 11Af11oo = Ilalloc, so Ila1loo < 11A11.
(3) Let Af = f', f E C'([0, 1]), and X = C([0, 1]). Then A is unbounded. To see this, we can, for example, note by Theorem A3.9 that if A was bounded we could extend it to all X, but there are plenty of functions in X for which
f does not exist! Alternately, consider a sequence of functions f whose graphs are given in Figure A3.1. Then each f E C' ([0, 11), 11 f" II = 1, but 11 f" 1100 + oo as n > oc. Consequently, there exists no finite K > 0 such that IIAf,11
K 11 f,, 11oc = K, for all n.
(4) Let Af (t) = fo" K(t, s) f(s)ds, where X = C([0, 1]) and K is continuous on [0, 1] x [0, 1]. In the supnorm on X, IIA11 = sup$E[0,11 fo I K(s, t)ldt.
Problem A3.5. Compute the bound on A given in Example A3.12 (4). We now return to the algebraic structure on operators introduced in Section A3. 1. There, this structure was complicated by the fact that the domains depended on the operators. In light of Proposition A3.9, we do not encounter this problem if we restrict ourselves to bounded operators since we can then take the domain to be all of X. Definition A3.13. Let X and Y be two Banach spaces. We denote by £(X, Y) the family of all bounded linear operators from X to Y.
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Appendix 3. Linear Operators on Banach Spaces
Let us make a few remarks. First, in this definition, the space X need not be a Banach space; an NLVS will do. Second, we will always assume that a bounded operator on X has been extended so that its domain is all of X. Finally, if Y = X, we write C(X) for C(X, X)).
Proposition A3.14. If A, B E £(X, Y) and c E C (or IR), then cA and A + B E L(X, Y) and
IIcAll = cIllAll,
IIA+BII < IAII+IIBII. Moreover, if A E C(X, Y) and B E 1(Y, Z), then BA E C(X, Y) and
IIABII < IIAII IIBII Problem A3.6. Prove Proposition A3.14. Corollary A3.15. £(X, Y) is an LVS over C (or R). II, as defined in Definition A3.11, is a It follows from the last two results that norm on L(X, Y). This follows from Proposition A3.14 and the facts that II A II > 0 and 11 All = 0 if and only if A = 0. This shows that C(X, Y) is an NLVS. Hence we can ask if £(X, Y) is complete in this norm. I
Theorem A3.16. Let X be an NLVS and Y be a Banach space. Then C(X, Y) is a Banach space with the norm II II given in Definition A3. 11. Proof. Let {An} be a Cauchy sequence in C(X, Y). We must show that there exist
A E £(X, Y) such that A = limn,a, An. Take any x E X and consider the sequence {Anx}. Then we have IIAnx  Amx1I
:
11 An  Am II IIxII,
by linearity and boundedness. Hence, {Anx} is a Cauchy sequence in Y, and as Y
is a Banach space, it converges. Let y = limnw Anx, and define A : X  Y by Ax = y. Problem A3.7. Continue the proof of Theorem A3.16. Show that A is well defined on X, linear, and bounded. (Hint: Use the fact that IIIAnll  IIAm!II II An  Am II, so { II A, II is Cauchy and hence converges. For boundedness of A, use }
IIAxI! < nlimollAnxlI < nl IlAnll Completion of the proof We show that limn_,oc An = A: IIA  An II = supll(An  A)xll IIxII.Ko
A3.3. The Graph of an Operator and Closure
307
and
II(An  A)x(l = IIAnx  Axll
F(A)  { (x, Ax) I x E D(A) }. Problem A3.8. Show that I' C X x Y is a linear subspace (we assume that X and Y have the same field of scalars). Note that X x Y is an LVS under componentwise addition and scalar multiplication. We denote by {x} x Y =_ {(x, y)Iy E Y} C X x Y. One may ask when a subset r' C X x Y is the graph of some linear operator A : X + Y. Of course, I' must be a subspace of X x Y. A convenient condition on r is the following.
Proposition A3.18. A linear subspace I' C X x Y is the graph of some linear operator A if and only if
({0} x Y) n r = {(0, o)).
(A3.1)
Proof.
(1) If there exists a linear operator A such that r = r(A) and (0, y) E r, then y = A(0) = 0, so condition (A3.1) holds. (2) Assume (A3.1) holds. Let D(A) = {x I (x, y) E F, for some y E Y }. For each x E D(A), define Ax = y, where (x, y) E F. We want to show that A is well defined and linear. Suppose there exist y, y' E Y such that (x, y) and (x, y')
are in F. Since F is linear, (x, y)  (x, y') = (0. y  y') E F and by (A3.1), y = y'. Hence, A is well defined. Second, if xi, x2 e D(A) and X E C, then (XI +x2. Ax, + Ax2) E r. This implies that (x, +x2, A(x, +x2)) E r, so A(x, + x2) = Axe + Axe. Since X(x,, Ax,) _ (?.x,, AAx, ), A(?`.x,) = ),.Ax,. 0 Hence, A is linear.
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Appendix 3. Linear Operators on Banach Spaces
Definition A3.19. Let X. Y he NLVS. For each (x. Y) E X x Y, define 1(X' Y)11 = Ilxllx + IIyIIY.
Lemma A3.20. (X x Y, II II) is an NLVS. 
Proof. We have seen that X x Y is an LVS. For the norm, we must check the conditions of Definition A 1.3: (i)
II(x, r)II > 0 since Ilxllx > 0, Ilylly > 0 and it follows that II(x y)ll = 0 if and only if x = 0 = Y.
(ii) 11%(x. y)II = II(Xx. )'v)11 = I%I(Ilxllx + Ilylly) = IAIII(x. y)II.
(iii) II(x, y)+(x', v')II = II(x +x'. y +y')II
= Ilx+x'llx+Ily+y'lIY < (Ilxllx + llylly)+(llx Ilx + Ily'lly) = ll(x, Y)II + ll(x',
Y'
)11.
11
Problem A3.9. Prove that the map X x Y * R defined by II(x, y)IIE
Ilxllx + Ilyl(Y ]'
is a norm on X x Y. Show that the topology on X x Y determined by II HE is equivalent to the topology induced by the norm in Definition A3.19. This means that an open set in one topology contains an open set in the other.
Problem A3.10. Show that if A E £(X, Y), then 1(A) C X x Y is closed. (We will see later that the converse is not true). (Hint: r (A) is closed if for any (x, y) E T(A), in T(A), lim,,.+ (x,,, convergent sequence {(x,,, that is, x E D(A) and Ax = y.)
Definition A3.21. Let A be an operator on X with D(A). An operator A with domain D(A) is called the closure of A if 1(A) = r (A). An operator A is said to be closable if it has a closure.
Remark A3.22. Suppose A is a linear operator with graph 1(A). It may happen (and does!) that 1(A) is not the graph of any operator (precisely how this may happen is given in Proposition A3.18). In this case A is not closable. Suppose, however, that A is closable. Then the closure A is unique and satisfies Ax = Ax, X E D(A). This follows directly from Definition A3.21. An important connection between everywhere defined operators (i.e., D(A) = X) and boundedness is given by the next result. Theorem A3.23. (Closed graph theorem). Let X, Y be Banach spaces. Let A be defined on X (i.e., D(A) = X). If A is closed (i.e., A = A), then A is bounded.
The proof of this is given in Reed and Simon, Volume I [RS I]. As the proof requires more machinery, the Baire category theorem, we will not give it here.
A3.4. Inverses of Linear Operators
309
Note that there are everywhere defined, unbounded operators. By Theorem A3.23, these will not be closed!
Examples A3.24. (1) Let X = (C([0, 11), 11 11,,) and A = d/dx with D(A) = C' ([0, 11), as in Example A3.2 (3). We claim that (A, D(A)) is closed. To see this, let (fn, Afn)
be a convergent sequence in r(A). Then f  f in X and Afn = f,,  g in X. By a standard result on uniform convergence, f E C 1 ([0, 11) and f' = g. Consequently, f E D(A) and (f, Af) E I'(A). We can modify this example slightly to obtain a closable operator whose closure is (A, D(A)). Simply take AI = d/dx and D(A 1) = C2([0, 1]). Problem A3.11. Prove that (Ai, D(A )) is closable and that its closure is (A, D(A)).
(2) Here is an example of an operator that is not closable. Let X = L2(1[8), and choose some fo E X with 11fo1l = 1. Define B, on D(B) = Co (R),
by Bf = f(0)fo. Then (B, D(B)) is a densely defined operator and is not closable. Consider, for example, a function h e C'([1, ]), h > 0, h(0) = 1, and f h = 1. Let hn(x) h(nx). Then Ilhnll = 1/n * 0 as n * oc. However, Bh,t = hn(0).fo = ,fo, so 11Bhn11 = 1. This shows that (0, fo) E P (B), and so it cannot be the graph of an operator, by Proposition A3.18.
A3.4 Inverses of Linear Operators Let X and Y be Banach spaces (over C for simplicity), and let A be a linear operator from X to Y. We associate with A three linear subspaces:
D(A) C X, domain of A; Ran(A) C Y, range of A, Ran(A) __ {y E Y I y = Ax, some x E D(A)};
ker A C X, kernel of ker A  {x E D(A) I Ax = 0). Definition A3.25. An operator A1 : Ran(A) * D(A) is called the inverse of A if A1 A = idD(A) (the identity map on D(A)) and AA1 = idRan(A) Theorem A3.26. An operator A has an inverse if and only if ker A = {0).
Proof. (1) G If ker A = (0), then for any y E Ran(A), there exists a unique x E D(A) such that Ax = y, for suppose xi, x2 E D(A) and Ax1 = y = Axe. Then,
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Appendix 3. Linear Operators on Banach Spaces
by linearity, Axe  Ax, = A(xj  x2) = 0, so xi  x2 E ker A. But ker A = {0}, whence x] = x2. Because of this we can define an operator A1 : Ran(A) > D(A) by A  ' v = x, where A.x = y. The operator A' is well defined and linear. Moreover, A v = A (Ax) = x, so A ' A = idD(A), and for y E Ran(A), AA1y = Ax =Y, so AA"' = idRan(A) (2) = Suppose A1 : Ran(A)
D(A) exists. If X E ker A, then x = A 1 Ax =
0, so ker A = {0}.
As this theorem shows, the condition for the existence of A' is simply that ker A = {0}. This guarantees that the inverse map is a function. However, we have In particular, we would like A to be bounded on little information about A all of Y.
Definition A3.27. A linear operator A is invertible if A has a bounded inverse defined on all of Y.
Problem A3.12. Prove that if A is invertible, then A
is unique.
Example A3.28. Let X = C([0, 1]), and define A by
(Af)(t) = I f(s)ds, f E X. 0`
This operator A has an inverse by the fundamental theorem of calculus: A d/dt. But A is not invertible, since A1 is not bounded! Warning: According to our definition, A may have an inverse but not be invertible.
Theorem A3.29. Let A and B be bounded, invertible operators. Then AB is invertible and (AB)1 = B'1A'. Problem A3.13. Prove Theorem A3.29.
Theorem A3.30. Let T be a bounded operator with II T II < 1. Then I  T is invertible, and the inverse is given by an absolutely convergent Neumann series:
(1T)'T" n=0
that is, limN,,x, y0k=Tk converges in norm to (1
 T)
.
Proof. Since y,=o II Tk II < Eko II T Ilk, and the series Ek II T Ilk converges (as II T II < 1, it is simply a geometric series), the sequence of bounded operators _k p Tk is norm Cauchy. Since Y is a Banach space, this Cauchy sequence converges to a bounded operator. Now we compute 0C
00
(1  T) E Tk = E(Tk  Tk+i) = 1, k=0
k=0
A3.4. Inverses of Linear Operators
311
where the manipulations are justified by the norm convergence of the power series. Similarly,
(Tk)(1 T)= 1, k.0
so the series defines (1  T)1.
Theorem. A3.31. Let A be invertible, and let B be bounded with
IIBII
 0. For I