Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday: Quantum Field Theory, Statistical Mechanics, and ... (Proceedings of Symposia in Pure Mathematics)

Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday Quantum Field Theory, Statistical Mechanics, and Nonrelativistic Quantum Systems

Proceedings of Symposia in

PURE MATHEMATICS Volume 76, Part 1

Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon's 60th Birthday Quantum Field Theory, Statistical Mechanics, and Nonrelativistic Quantum Systems A Conference on Spectral Theory and Mathematical Physics in Honor of Barry Simon's 60th Birthday March 27-31, 2006 California Institute of Technology Pasadena, California

Fritz Gesztesy (Managing Editor) Percy Delft Cherie Galvez Peter Perry Wilhelm Sehlag Editors

American Mathematical Society N,

Providence Rhode Island

2000 Mathematics Subject Classification. Primary 35J10, 35P05, 47A55, 47A75, 47D08, 81Q15, 81T10, 81Uxx, 82B05, 82B10.

Library of Congress Cataloging-in-Publication Data Spectral theory and mathematical physics : a festschrift in honor of Barry Simon's 60th birthday : Quantum field theory, statistical mechanics, and nonrelativistic quantum systems / Fritz Gesztesy... [et al.], editors. p. cm. - (Proceedings of symposia in pure mathematics ; v. 76, pt. 1) Includes bibliographical references. ISBN-13: 978-0-8218-4248-5 (alk. paper) (Part 1) ISBN-13: 978-0-8218-3783-2 (alk. paper) (Set)

1. Spectral theory (Mathematics)-Congresses. I. Simon, Barry, 1946-

II. Gesztesy, Fritz,

1953QC20.7.S646S64

2006

515'.7222-dc22

2006047073

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Contents Part 1. Quantum Field Theory, Statistical Mechanics, and Nonrelativistic Quantum Systems Preface

ix

A Mathematical Genealogy A Selection of Barry Stories

xvii

Quantum Field Theory, Statistical Mechanics Perspectives in Statistical Mechanics MICHAEL AIZENMAN

3

Coherent Infrared Representations in Non-Relativistic QED THOMAS CHEN and JURG FROHLICH

25

Quantum Spin Systems After DLS 1978 BRUNO NACHTERGAELE

47

Barry Simon's Contributions to Quantum Field Theory LON ROSEN

69

Nonrelativistic Two-Body and N-Body Quantum Systems, Resonances Isoperimetric Inequalities for Eigenvalues of the Laplacian MARK S. ASHBAUGH and RAFAEL D. BENGURIA

105

Non-Self-Adjoint Operators and Pseudospectra E. B. DAVIES

141

Barry Simon's Contributions to Non-Relativistic Quantum Mechanics: Two-Body and N-Body Schrodinger Operators and Resonances RICHARD FROESE

153

N-Body Quantum Scattering and Quantum Resonances: An Overview 169

CHRISTIAN GERARD

Mathematical Analysis of Born-Oppenheimer Approximations GEORGE A. HAGEDORN and ALAIN JOYE V

203

CONTENTS

vi

Perturbation Theory and Atomic Resonances Since Schrodinger's Time EVANS M. HARRELL II

227

On a Theorem for Quantum Mechanical Scattering Theory ANDREW LENARD

249

Analytic Criteria in the Qualitative Spectral Analysis of the Schrodinger Operator VLADIMIR MAZ'YA

257

The Spectral Geometry of Geometrically Finite Hyperbolic Manifolds PETER PERRY

289

Topics in the Theory of Positive Solutions of Second-Order Elliptic and Parabolic Partial Differential Equations YEHUDA PINCHOVER

329

Complex Scaling in Atomic Physics: A Staging Ground for Experimental Mathematics and for Extracting Physics from Otherwise Impossible Computations WILLIAM P. REINHARDT

357

Recent Results on the Bethe-Sommerfeld Conjecture ALEXANDER V. SOBOLEV

383

Electric and Magnetic Fields, Semiclassical Limit Recent Developments in Quantum Mechanics with Magnetic Fields LASZL6 ERDOS

401

Aspects of the Integer Quantum Hall Effect GIAN MICHELE GRAF

429

Barry Simon's Work on Electric and Magnetic Fields and the Semi-Classical Limit IRA W. HERBST

443

Some Bound State Problems in Quantum Mechanics DIRK HUNDERTMARK

463

Part 2. Ergodic Schrodinger Operators, Singular Spectrum, Orthogonal Polynomials, and Inverse Spectral Theory Preface

ix

Random and Ergodic Schrodinger Operators, Singular Continuous Spectrum A New Approach to Spectral Gap Problems JEAN BOURGAIN

499

CONTENTS

vii

Strictly Ergodic Subshifts and Associated Operators DAVID DAMANIK

505

Lyapunov Exponents and Spectral Analysis of Ergodic Schrodinger Operators: A Survey of Kotani Theory and Its Applications DAVID DAMANIK

539

Spectral Properties of Schrodinger Operators with Decaying Potentials SERGEY A. DENISOV and ALEXANDER KISELEV

565

On the Formation of Gaps in the Spectrum of Schrodinger Operators with Quasi-Periodic Potentials MICHAEL GOLDSTEIN and WILHELM SCHLAG

591

Ergodic Schrodinger Operators (on one foot) SVETLANA JITOMIRSKAYA

613

The Integrated Density of States for Random Schrodinger Operators WERNER KIRSCH and BERND METZGER

649

Exotic Spectra: A Review of Barry Simon's Central Contributions YORAM LAST

697

Orthogonal Polynomials, Inverse Spectral Theory Riemann-Hilbert Methods in the Theory of Orthogonal Polynomials PERCY DEIFT

715

Inverse Spectral Theory as Influenced by Barry Simon FRITZ GESZTESY

741

Orthogonal Polynomials: From Jacobi to Simon LEONID GOLINSKII and VILMOS TOTIK

821

Orthogonal Polynomials: The First Minutes SERGEY KHRUSHCHEV

875

Spectral Theory via Sum Rules ROWAN KILLIP

Barry Simon's List of Publications

907

931

Preface This Festschrift is dedicated to Barry Simon on the occasion of his sixtieth birthday.

Barry Simon was born on April 16, 1946, to parents Minnie and Hy Simon. Together with his older brother Rick, he grew up in a one-bedroom apartment on Kings Highway in Brooklyn, New York. Barry attended James Madison High School in Brooklyn. He obtained a perfect score on the American High School Mathematics Examination (AHSME) in 1962. Barry's decision to go into physics was influenced by Sam Marantz, an inspired physics teacher he had in high school. Mr. Marantz was also the one who suggested he apply to Harvard; before that Barry's first choice for undergraduate studies was Caltech.

Barry received his undergraduate education at Harvard. He was elected to Phi Beta Kappa in his junior year and graduated with a B.A. summa cum laude from Harvard College in 1966. He was a Putnam Competition Winner (top six) in 1965. He then entered Princeton University and received his Ph.D. in Physics in 1970 under the direction of Arthur Wightman. Barry would not have thought of Princeton for graduate school except that when he asked George Mackey if there was anyone who combined physics with mathematical proof, Mackey told him about Wightman.

In 1968, Martha Katzin entered Princeton Graduate School in Mathematics where she eventually got her Ph.D. under the direction of Robert Gunning. In January 1971, shortly after Barry became an Assistant Professor at Princeton, they were married. Martha has taught in a variety of schools in the New York/New

Jersey area and then California and is currently a Lecturer in Mathematics at California State University at Northridge. Barry and Martha have five children (Rivka, Benjamin Pesach, Zvi, Aryeh and Chana) and, so far, seven grandchildren.

Barry's subsequent career at the Departments of Mathematics and Physics at Princeton University, after being an instructor in Mathematics there in 1969/70, reads as follows: Assistant Professor (1970-72), Associate Professor (1972-76), and Professor (1976-81). He was on leave from Princeton as a Visiting Professor at IHES, Bures-sur-Yvette, France; the CNRS, Marseille, France; and the ETH, Zurich, Switzerland, in 1972/73; and at Yeshiva University, New York City, in 1976/77.

During the academic year 1980/81, he was again on leave as a Sherman Fairchild Distinguished Visiting Scholar at the California Institute of Technology, Pasadena, ix

PREFACE

x

California. He joined Caltech as Professor of Mathematics and Theoretical Physics in 1981. Since 1984 he has held the position of IBM Professor of Mathematics and Theoretical Physics at Caltech. He was a Distinguished Visitor at the University of California at Irvine in 2002 and the Lady Davis Visiting Professor at Hebrew University, Jerusalem, in 2004/05. Barry has received numerous honors, among which we mention the following: Sloan Foundation Fellowship, 1972-76; Invited Speaker, International Congress of Mathematicians (ICM), 1974 (Vancouver); Invited Speaker, American Physical Society (APS), 1975 (Anaheim) and 1980 (Los Angeles); Invited Speaker, American Mathematical Society (AMS), 1977 (New York) and 1990 (Louisville); Invited Speaker, International Association of Mathematical Physics (IAMP), 1981 (Berlin); the Medal of the International Academy of Atomic and Molecular Science, 1981; the Stampacchia Prize, 1982; a Guggenheim Fellowship, 1988/89; Corresponding Member of the Austrian Academy of Sciences, 1990; D.Sc.h.c., Technion-Israel Institute of Technology, Haifa, Israel, 1999; Fellow of the American Academy of Arts & Sciences, 2005; Honorary Fellowship of the University of Wales Swansea, 2006. From Barry's vast administrative experience we mention, in particular, his position as Chair of the Scientific Organizing Committee for the International Congress of Mathematical Physics at Swansea, 1988; Vice President of the AMS, 1988/89; Executive Officer (Department Chair) for Mathematics, Caltech, August 1997-July 2003.

Barry has served as an Associate Editor or Editor of a variety of journals and book series. Currently, he serves on the editorial board of the Journal of Operator Theory (since 1979), Communications in Mathematical Physics (since 1979), Cambridge University Press (since 2002), Journal of Approximation Theory (since 2005), and Inverse Problems and Imaging (since 2006).

Barry supervised eleven Ph.D. students at Princeton University and fifteen Ph.D. students at Caltech so far. Throughout his career, Barry has mentored many scientists. His webpage of "mentees" lists 43 names. According to the Mathematics Genealogy Project, he has 73 mathematical descendants. At this point, Barry has about 108 coauthors and he lists over 350 papers and proceedings contributions. Thus far he has published fourteen books in science (plus four computer books) and co-edited one Festschrift volume. With so many co-authors, Barry has had a tremendous impact on people and their careers, but also on science itself, both through his vast research accomplishments and through his highly influential books. In particular, his four-volume book series, Methods of Modern Mathematical Physics, I-IV, written jointly with Mike Reed in the period 1972-79, played the role that Courant-Hilbert I, II did for the preceding generation. It has educated and profoundly influenced a whole generation of mathematical physicists, and will continue to play that distinguished role in the foreseeable future. His other books, most notably Functional Integration and Quantum Physics and Trace Ideals and Their Applications, are standard references that have influenced research in many areas.

PREFACE

xi

As a sign of Barry's impact, we note that the last time we checked, the author citation database of MathSciNet (based on citations in bibliographies of papers reviewed in the last five years or so) said that Barry is cited 5,217 times by 2,430 authors.

A gifted and lucid expositor of science, encyclopedic in his knowledge, and a grand master of mathematical structure and abstract analysis, Barry has been a phenomenal force in mathematical physics and applied mathematics in the broadest sense. As the acknowledged and undisputed authority on spectral theory for Schrodinger operators, he created a school, and much of what we know today about spectral phenomena, including exotic aspects in connection with singular contin-

uous spectra, is due to Barry and coworkers around him. While in recent years Barry's focus has been primarily on problems in nonrelativistic quantum mechanics and orthogonal polynomials, his research has had great breadth, with significant contributions to quantum field theory, statistical mechanics, and abstract functional analysis. This Festschrift had its origins in a conference called SimonFest held at Caltech,

March 27-31, 2006, but it is not a Proceedings volume in the usual sense. Barry requested that both his birthday conference and this Festschrift concentrate not so much on what the speaker or writer has done recently, but instead on reviews of the state of the art, with a focus on recent developments and open problems. While the number of speakers at Simonfest was originally limited by the number of hour slots in a full week, the contributions to this Festschrift contain a few additionally selected reviews. In the end, the bulk of the articles in this Festschrift are of this state of the art survey form with a few that instead review Barry's contributions to a particular area. Here in Part 1, the focus is on the areas of Quantum Field Theory, Statistical Mechanics, Nonrelativistic Two-Body and N-Body Quantum Systems, Resonances, Electric and Magnetic Fields, and Semiclassical Limit. In Part 2, the focus is on the areas of Random and Ergodic Schrodinger Operators, Singular Continuous Spectrum, Orthogonal Polynomials, and Inverse Spectral Theory.

Those who know Barry are aware of the fact that one of his secret weapons to be so productive is Cherie Galvez, his superb secretary (aka Administrative Assistant). We borrowed her services for this Festschrift. She not only was her usual efficient self, but was a wonderful editor and TeXpert. She also did a great job of dealing with author corrections and making sure that the styles of the different

manuscripts meshed. We appreciate her hard work and thank her for her efforts. Since we wanted to be able to thank her, Cherie is not a signer of this Preface, although she is rightly listed as one of this Festschrift's editors. At SimonFest, Barry told the story that one day, shortly after Dirk Hundertmark (who is German) started as a postdoc, Barry overheard him saying on the phone "Cherie is her weight in gold worth." He then said to Dirk: "You made two mistakes. First, you used German grammar. Second, Cherie is not worth her weight in gold; she is worth my weight in gold."

PREFACE

xii

We are grateful to Sergei Gelfand, Christine Thivierge, and the staff at AMS for their support throughout the preparations of this volume. We also thank all the authors for their contributions and the referees for their invaluable assistance.

We sincerely thank the following sponsors of SimonFest for their financial support: Caltech's Center for the Mathematics of Information (CMI); Caltech's Division of Physics, Mathematics, and Astronomy; International Association of Mathematical Physics (IAMP); International Union of Pure and Applied Physics (IUPAP); U.S. National Science Foundation (NSF). Happy Birthday, Barry! Percy Deift Ritz Gesztesy Peter Perry Wilhelm Schlag October 2006

A Mathematical Genealogy The following pages include the mathematical ancestries of Barry Simon and Martha Katzin-Simon as well as the mathematical descendants, to date, of Barry Simon.* As the formal advisor system is about 150 years old, relationships older than this must be regarded with caution. Doctorates were awarded in years shown in parentheses. Information for this genealogy was obtained from the Mathematics Genealogy Project: http : //genealogy. math. ndsu. nodak. edu/.

*The editors are grateful to Stephen L. Clark for compiling and formatting these mathematical family trees. xiii

A MATHEMATICAL GENEALOGY

xtv

Ehrhard Weigel (1650)

Otto Mencken (1668)

Johann C. Wichmannshausen (1685)

Jacob Bernoulli

Christian A. Hausen (1713)

Johann Bernoulli (1694)

Abraham G. Kaestner (1739)

Leonhard Euler (1726)

Johann F. Pfaff (1786)

Joseph L. Lagrange

Carl F. Gauss (1799)

Simeon D. Poisson, Jean-Baptiste J. Fourier

Christian L. Gerling (1812)

Gustav P.L. Dirichlet (1827)

Julius Pliicker (1823)

C. Felix Klein (1868)

C.L. Ferdinand Lindemann (1873)

Arnold J.W. Sommerfeld (1891)

David Hilbert (1885)

Karl H. Herzfeld (1914)

Erhard Schmidt (1905)

John A. Wheeler (1933)

Salomon Bochner (1921)

Arthur S. Wightman (1949)

Robert Clifford Gunning

Barry M. Simon (1970)

Martha Katzin-Simon (1974)

(1955)

A MATHEMATICAL GENEALOGY

Robert Greene (1979) Tara Nanda (1982) Carlos Tomei (1982) Anthony O'Connor (1972) Jay Rosen (1974) Robert Israel (1975) Percy Deift (1976)

xv

Licio Bezerra (1990) Suzana de Moraes (1993) Luiz Duczmal (1997)

Maria de Fatima de Almeida (1998) Humberto Bortolossi (1999) Hamilton Bueno (1999) Ricardo Leite (2000)

- Serge Parmentier (1989) Luen Li (1983) Stanley Alama (1988) Roger Oba (1988) Spyridon Kamvissis (1991) Thomas Kriecherbauer (1993) Kenneth McLaughlin (1994) - Jeffrey DiFranco (2004)

H Evans Harrell II (1976)

H George Hagedorn (1978) -

Chern Lu (1998) Jinho Baik (1999) Konstantin Aslanidi (2001) Jeffrey Beh (2003) Roman Svirsky (1985) Glenn James (1990) Dale Smith (1990) Edward Green (1991) Patricia Michel (1994) Walter O'Connell, Jr. (1998) Xue-Feng Yang (1998) Jan Derezinski (1985) Sam Robinson (1986)

Armin Kargol (1994) Steven Jilcott, Jr. (2000)

Barry Simon

Mark Ashbaugh (1980)

Antti Kupiainen (1980) Steven Levin (1980)

E

Peter Perry (1981) Keith Miller (1982) Byron Siu (1984) Nestor Caticha Alfonso (1985) Barton Huxtable (1987) Kristiana Odencrantz (1987) Clemens Glaffig (1988) Askell Hardarson (1988) John Lindner (1989) Vojkan Jaksic (1992) Yunfeng Zhu (1996) Alexander Kiselev (1997) Andrei Khodakovsky (1999) Rowan Killip (2000) Andrej Zlatos (2003) Irina Nenciu (2005) Mihai Stoiciu (2005)

Craig Haile (1997) Kimberly McHale (1997) Andres Avila (1999) Lotfi Hermi (1999) Guotian Lin (1994) Jani Lukkarinen (2001) Ville Hakulinen (2002) Timo Korvola (2003) Mikko Stenlund (2006) Emiliano De Simone (2006) Albert Schueller (1996) Zhiqiang Wu (1997) Douglas Riley (1999)

Azida Hosein (2003)

A Selection of Barry Stories The organizers of SimonFest invited Barry's friends, colleagues, and potential conference attendees to submit "Barry Stories" to a webpage

http://www.math.caltech.edu/SimonFest/stories.html. The response was so extensive that we can't include them all here (but they will remain posted at the website for some time) so we asked Barry to pick his favorites. - The Editors

In the early eighties, Murph Goldberger was the president of Caltech. Murph was a great fan of Barry since their Princeton days and concocted a scheme to try and lure Barry from Princeton to Caltech. The scheme started innocently enough by inviting Barry to spend a year in sunny California as a Fairchild Scholar. Murph's plan indeed worked like a charm and a year of leave turned out to be a permanent move.

Princeton, at the time, was the Mecca of mathematical physics. It probably had the largest math physics group. Caltech, at the time, and as far as mathematical physics was concerned, was like the rest of Saudi Arabia: a desert (with a lot of oil). As you probably know, Barry thrives in company. So he looked for someone to accompany him to the desert. After a lot of fruitless searching, Barry must have remembered that I had some practical experience living in a real desert and asked me to join him. The arrangement was curious. I must have been the only Princeton Assistant Professor who had his paycheck sent to California. And, I remember that the bank teller once asked me if I could swing such a trick for her. She preferred the Bahamas, however. Anyway, since I had no official standing at Caltech, I decided to decorate myself as the Fairbaby Scholar.

The Fairkid year we spent together was a remarkable year, for I had Barry almost all to myself. In Princeton, where I first met Barry as his postdoc, I would

get as an appointment the walks to the parking lot, and occasionally a ride to Edison spending a pleasant Shabat with Barry and Martha. How much leisure we had in this Fairkid year is evidenced by the fact that, hold on to your chairs, Barry and I used to go swimming regularly at lunch time at the Caltech swimming pool! Can you imagine how many papers did not get written because of my bad influence on Barry! Swimming at the Caltech pool was very good for me: Here was a place where I could beat Barry fair and square. Barry, on the other hand, felt very guilty about this waste of time. To comfort

himself he told me that the lost hours at the pool would be paid up by gained xvii

xviii

A SELECTION OF BARRY STORIES

years in better health. Well, what could be more appropriate than to celebrate this wisdom twenty years later, at his 60th anniversary party. Let me now tell you a story involving Ira Herbst and Barry. I was an overwhelmed Wigner fellow at Princeton, working with Ira and Barry and we were quite close. Ira came back startled from one job interview, I forgot where. The chairman had asked him if he ever wrote a wrong paper. Ira was shocked and said "Of course not!" The reaction of the chairman was unexpected. He said: "Then you do not write enough." Now, nobody could ever accuse Barry of not writing enough. But I take the credit for writing a wrong paper with Barry. Here is how this came to be: There was a lot of excitement in Caltech when Voyager first sent those spectacular pictures of the rings of Saturn. I had learnt that Saturn appears to have infinitely

many rings from a math grad student at Caltech. Barry and I were working at the time on almost-periodic Schrodinger equations and were fascinated with fractal spectra. I knew about Hill lunar theory and Barry had just written a review of Arnold's book on Mechanics. This led us to make a theory where the near incommensuration of the periods of the moon of Saturn (and also involved the Sun) gave rise to rings with fractal structure. Peter Goldreich, the czar of planetary physics in Caltech, did not like the theory because it was linear stability. But Feynman did because it was simple. The theory could not account for the order of magnitude for the observed gaps and so turned out to be wrong. Barry's time axis is divided into BC and AC: Before Caltech and After Caltech. In the BC era Barry had difficulties deciding if his heart lay with constructive field theory, statistical mechanics or quantum mechanics. In the AC period the die was cast in favor of quantum mechanics and spectral theory. Barry also contracted a chronic strain of PC flu. Since I am an old-timer let me reminisce about the prehistoric BC era. At Princeton there were the lunch time seminars and the Math-Phys seminars. In the lunch time seminar, Barry was the prima ballerina. (Can you imagine Barry on his tiptoes?) He would normally either tell a new result of his or a new result

of someone else. With Dyson, Lieb and Wightman in the audience, most grad students and postdocs were too terrified to expose their slowness if they were to ask an innocent question. Most of the time, nobody dared open his mouth. The notable exception was the fresh grad student from Harvard, Alan Sokal, who never had a fear of authority and was sufficiently smart and self-confident to argue with Barry. The math phys seminars were a different business. There was an outside speaker most of the time. Wigner would usually show up and ask his typical Wignerian questions. Barry would sit in the audience and write a paper. From time to time he would look up from his notes and ask a question that would unsettle most speakers: Someone in the audience seemed to know more about what he was talking about than himself. Sometimes, at the end of the talk, Barry would go to the board and give his version of the proof, which was always slick. Barry, you are now 60. Most of us probably do not enjoy being reminded about our advanced age, but I think that one of the nice things about you, Barry, is your optimism. You probably enjoy being 60! I wish you fun with math and good health in the next sixty years. Yosi Avron

A SELECTION OF BARRY STORIES

xix

In 1983 Derek Robinson invited me to visit him at the Mathematical Institute in Canberra for a month. As one of the inducements, he mentioned that he had also invited Barry Simon for the same period. The prospect of seeing Barry, Derek and kangaroos was enough to make my decision, and in July 1983 I set off on the gruelling journey. I was very surprised shortly before my departure to hear from Derek that he was unable to avoid a commitment to go to a conference in Japan, and would not return until ten days after my arrival. Barry arrived about the same time as myself, and I asked him a problem about heat kernels of Schrodinger operators which I had solved in one dimension by a method that could not be extended to higher dimensions. At that time I had read several of the papers on hypercontractivity, a concept that was invented specifically to solve problems in quantum field theory, in which there are an infinite number of degrees of freedom. Barry listened to my question carefully and agreed that some progress should be possible. The very next time I saw him he told me that he had solved the problem by an improvement of the standard hypercontractive estimates that made use of the finite dimensionality. He then proceeded, without notes, to give me a lecture on the subject, explaining every step in detail, including the infinite summation LP bounds to L2 - i L°° bounds by procedure that allowed one to pass from L2 controlling the constants involved. Although very impressed, I had the temerity to suggest at the end of his lecture

that although he had clearly done what was needed, I did not like his solution aesthetically, and would prefer an account that depended on differential inequalities. I had the feeling that this would yield better constants and be in some sense more

natural. On the next occasion Barry rewrote the entire account in this language, and we realized that this was going to have enough ramifications to occupy the entire month. By the time Derek got back we were fully committed to the project, and I now feel embarrassed that I did not spend nearly enough time talking to him. Derek was, however, the person who coined the term ultra-contractivity, which was the focus of almost all of my research over the next ten years. A research relationship is, of course, not symmetrical, and for Barry it was just one of many different projects he pursued in that decade. Indeed I wonder whether

his main memory of the month was quite different. He had gone to Canberra with Martha and his four children; together they might have comprised the entire Orthodox Jewish community there. With typical thoroughness he arranged for kosher meat to be delivered to him from Sydney every few weeks. The Jewish butcher had agreed to prepare a parcel and then put it in a deep freeze before sending it by rail. On one occasion it did not arrive, and I was involved in a very strange mathematical discussion, which was punctuated every hour or so by long conversations between Barry and the delivery service, in which he explained that if his parcel did not arrive within a few hours he would have to throw it away. I believe that eventually everything was okay. Surviving in Canberra for a month with his family while observing Orthodox rules was just another challenge of the type that Barry always seemed to relish. Perhaps the absence of such problems in London, which has a very large kosher Jewish community, was one of the reasons why he has visited me on a number of occasions there. - E. B. Davies

A SELECTION OF BARRY STORIES

xx

Close interactions with Barry have their great rewards

and perils!

While Barry's lightning mental agility, his extraordinary talent to strip the unnecessary clutter surrounding an argument, getting straight to its core, and his o

remarkable ability to see connections to related topics other than the obvious ones in question, are legendary, the following observations will sound familiar to many of us:

Scenario #1: You joyously walk into Barry's office to present him with a new idea, just to exit a few minutes later, your idea having been shred to pieces. "Back to the drawing board" is sometimes his comment, with a broad grin on his face. All this sounds more cruel than it is: After all, you have just been saved from going down a cul-de-sac and you can start regrouping! Scenario #2: You proudly walk into Barry's office to show him something new. Barry thinks for a second, then jumps up to the blackboard and explains to you in no uncertain terms what you "really had in mind." That's great, because at this point you realize a joint paper will eventually be written. Scenario #3: You march into Barry's office and this time you're convinced you have a blockbuster at your fingertips. You start to explain to Barry, and then he says "time out" and silence fills the room. After a bit of eerie silence you realize this time you're going to write a very nice joint paper with Barry. Of course, after a few more moments scenario #2 will be repeated, but that's quite alright! Scenario #4: Barry asks you into his office and explains what he was struggling with lately. (He likes to put it this way, though: "I was banging my head against the wall about this...") After he suggests jointly working on this you return to your office with a big puzzle in your hands. Those rare instances in which you can actually do the job asked of you and complete the argument are priceless.

Barry likes to pick on me since I'm usually not afraid of computing anything, well, almost anything (while he doesn't have the patience to do so!). So once in July of 1997, he confessed to me that he had a terrible contradiction in his long manuscript on "The classical moment problem as a self-adjoint finite difference operator" (it later appeared in Advances in Math. 137, 82-203 (1998)), but he just couldn't find the error. So I was supposed to look at this. It was an intricate o

puzzle! I spent a day on it and well after midnight was sure I had found the error. So I e-mailed him what I thought was the culprit and slumped home to the apartment. Next morning I opened my e-mail and there was Barry pointing out that I was dead

wrong. It was "back to the drawing board" as he still grinned during our brown bag lunch meeting that day. I was dejected! Well, I had another day before going away with my wife to Hawaii, and I was not about to let this ruin our vacation! So I frantically computed like a dog and finally saw the light: This was it! I decided to treat myself to dinner and left after e-mailing him my second attempt to find the culprit. After returning from dinner late that night, I had received quite a different message from Barry. It started out: THANK YOU, THANK YOU, ... and went on like this for half a page.

- Ritz Gesztesy

A SELECTION OF BARRY STORIES

xxi

Below is an amusing little story that shows Barry's good sense of humor and how quickly he could think of a clever comment.

Barry had many graduate students at Princeton. Although he treated us well, he used to kid around about abusing us or treating us as some sort of subhumans. It was all in good fun, and I always found it amusing. In November 1976, the night that Jimmy Carter was elected president, Barry

invited some of us to to have dinner and watch the election results on the television at the Simons' house in Edison, New Jersey. Shortly after I arrived, I was hanging my coat up in the closet when Barry's daughter Rivka pointed

at me and said, "Daddy, what's that man's name?" Without the slightest hesitation, Barry replied, "Rivka, that's not a man, that's a graduate student!" - George Hagedorn

One of the rare categories in which I can compete pretty well with Barry is in the execrableness of our handwriting. I was at MIT as a postdoc when we wrote our long paper on the Stark effect. Back in those days, one actually wrote articles by hand and a secretary, using a device called a "typewriter," turned them into manuscripts, leaving blanks for formulae to be inserted. As a lowly postdoc I didn't get first pick of the secretarial staff, and the manuscript ended up in the hands of a well-intentioned but struggling secretary who would produce about one page per day, which was usually sent back multiple times with corrections, often amusing. One day a favorite adjective of Barry's, "operator-theoretic," came back as "operator neurotic," and I knew the manuscript was taking its toll on her. With lots of encouragement and little gifts she finished the manuscript after months of work, as the term ended. But she didn't return the next term-it was doubtless the last mathematical manuscript she ever typed. o

Barry may or may not have said "This is my cross to bear" according to Ed Nelson, but might he ever have been a cross bear? The first time Barry's o

bearlikeness crossed my mind was at a party at his home in New Jersey, where even

graduate students were invited, and little Rivka appeared looking for one of her toys. I think it may have been a stuffed animal, a "big horse" or something like that. Finding the toy was way beyond the powers of a graduate student, but Barry walked by in the next room so I sent Rivka in that direction, telling her "Just follow the big bear, and he'll find your toy." Bears have acute hearing, and I remember

him growling, "I heard that!" but Rivka didn't hesitate at my description of her father.

Many years later Barry gave a seminar at Georgia Tech. In the audience was a colleague from another university, whose rather nice early work had been discovered and promoted by Barry, but who had been inactive for some time while struggling with mental illness. At the end of the seminar the visitor asked some rather peculiar questions, which went on and on. Barry handled them with his usual aplomb, but clearly something was amiss, so after everyone left I took my friend off to find his

medication. When he was again rational, he told me that half way through the seminar, Barry had turned into an enormous grizzly bear before his eyes, and that

A SELECTION OF BARRY STORIES

xxii

he needed to distract the bear with mathematical questions, or the audience would be devoured! Barry has always been remarkable for his vast knowledge of mathematics, so it was many years before I can recall ever telling him a published theorem he didn't already know. One day I saw Barry in Princeton shortly after a meeting and told him about an old inequality for PDEs, which, as I could tell from his intent look, was new to him. I said, "It seems to be useful. Do you want to see the proof?" His response was "No, that's OK." Then he went to the board and wrote down a flawless proof on the spot. o

- Evans Harrell

Soon after I arrived in Princeton as a postdoc, Barry suggested that we work on proving Borel summability of the perturbation series for the Schwinger functions in two-dimensional 04 quantum field theory. He had an idea which he showed me and after a couple of days I had another idea. But I soon realized that these two ideas o

were not enough. Up to this time I was a very hard worker but having thought about how ideas percolate up through the subconscious, I decided to relax and let them do just that. WRONG! A couple of weeks later a preprint arrived from Geneva with a proof.

Barry, Yosi Avron and I were working on magnetic fields. As everyone knows Barry is a very fast worker and he writes up his work even faster. Barry and Yosi felt we should write something and as usual I wanted to get more done first. One day the two of them arrived in my office and began trying to convince me again that we should write something up. I protested, at which point Barry took his hand from behind his back and with a smile produced a manuscript which he had presumably written the night before. o

- Ira Herbst

Many of you know Barry from his academic work and community achievements.

I have a rather, uh, different perspective. I had the distinct honor and privilege of co-writing a handful of computer books with Barry, including several Mother of All Windows books, and The Mother of All PC Books. I'll never forget Barry's squeals of delight when he found foolish inconsistencies in Windows, the way his voice would drop low-and he'd talk fast-when he was working through a particularly snarly problem, and the way he'd rub his hands with glee when a solution suddenly appeared. Barry wrote about PCs with extraordinary clarity and wit. The Mother books became (in)famous for their casts of characters-no dry technical mumbo-jumbo here. My favorite character from the early Mother books was the eight-legged cockroach (and bug expert) known as Erwin. We gave Erwin the enviable assignment of pinpointing and explaining bugs in Windows, a task for which he was eminently qualified.

A SELECTION OF BARRY STORIES

xxiii

Barry describes Erwin's birth this way in The Mother of All Windows 95 book, from the perspective of uber-iconic Mom, the Mother of the Mother books, as it were:

Erwin has been with me since my first book, pointing out bugs and warning folks about the unthinkable. He's a dashing eight-legged refugee from the 1930s. The physicists in the audience will no doubt recall Erwin Schrodinger, one of the founders of quantum mechanics, who invented a famous "thought experiment" in which a vial of poison gas might (or might not) kill a cat. Schrodinger's cat became justly famous among the psi-squared crowd. A few years ago, a computer book writer had the temerity to refer to Schrodinger's cat in a book submitted to IBM. The IBM Thought Police wouldn't put up with such an offensive allusion to a cuddly animal, so they changed the manuscript, exorcising Schrodinger's cat and introducing in his stead Schrodinger's cockroach, an animal that could be (presumably) sometimes dead without offending the more delicate readers of IBM manuals.

Barry is one of the most intensely intelligent people I've ever met-and delightful, in every sense of the term. Except for the puns. The puns were really, really bad.

Hey, Barry! Wanna write another Windows book? NO! Put DOWN that brickbat! - Woody Leonhard

In the late 1960s, Barry was a graduate student in physics at Princeton and attended some courses I taught. I soon learned that I did not need to prepare with great thoroughness; it was enough to get things approximately right and Barry from where he was sitting would tell us how to get them precisely right. I miss o

Barry. o Once Barry was engaged in an acerbic priority dispute with someone at another

institution. I offered to intervene, but Barry said, verbatim, "This is my cross to bear." (I told this story at a meeting in June 2004 and Barry said mildly, "I would never have said such a thing!") o Once Barry wrote a paper on hypercontractive semigroups and when he got it back from the typist, every instance of "hypercontractive" was rendered as "hypercontraceptive." Barry received an ugly, uncivil letter from a mathematician complaining that

o

he had not been given sufficient credit in a volume of Reed-Simon for his work on a certain topic. Barry responded with a dispassionate two-page letter calmly reviewing the entire history of the topic and the contribution of each person to it. He concluded the letter with a one-sentence paragraph: "I hope that you will receive this letter in the same spirit in which you sent yours." o Shortly after moving to Caltech, Barry came east for a visit. He said that someone had stolen his attache case. When we asked whether he had lost anything of importance, he replied, "Only the paper I wrote on the flight." o One Saturday afternoon I saw Barry and Martha strolling with the Cappells. It was a different Barry. Gone were the intense energy and concentration, and in

xxiv

A SELECTION OF BARRY STORIES

their place were peace, calm, and repose. I had a strong and somewhat wistful sense of the Sabbath as a gift. - Ed Nelson

In his third year at Caltech, Barry taught a graduate class in group representations. There were two undergraduates in the class, Zinovy Reichstein and me. I couldn't get up at 9 am for lectures and Zinovy took beautiful notes. I typically photocopied Zinovy's notes every week prior to doing the homework. One week Zinovy had to be away. Forewarned I showed up in class. Barry walked into class, did a double-take when he spotted me and announced "It's undergraduate number conservation!" It never bothered Barry that I did not come to lectures, even though I was one of his undergraduate advisees. When I went to graduate school and expressed some dismay at the less than supportive or encouraging attitudes of the faculty, he wrote me long encouraging letters. I don't think I ever understood how someone as busy as he was, with every minute of his time scheduled, could find the time to write such letters. Many years later, I was an Assistant Professor at Princeton when Barry came to give a talk. I saw him in the corridor as I was on my way to an undergraduate Senior Thesis oral examination. I greeted him and asked him if he recognized me. He remembered me after a little while, and then was amused (I think) when I told him that I was on my way to an undergraduate oral examination, and reminded him of my first oral examination at Caltech in rigorous statistical mechanics, when

he asked me to outline Onsager's solution. He started that particular exam by informing me that he had wanted to put a sophomore through a really rigorous oral ever since his own sophomore year.

- Vipul Periwal

As an undergraduate I took Quantum Mechanics from Barry, little knowing that I would later wind up a student. At the same time, I was taking Functional Analysis from Ira Herbst. The Quantum Mechanics course met on Tuesdays and Thursdays and there was a break in the middle of the lecture. One day I had my copy of Methods of Modern Mathematical Physics, Vol. I, Functional Analysis with me. Barry walked up to me during the break and broke into a big smile followed by mock indignation when he saw that I had a copy of Reed-Simon. "Don't read that stuff!" he admonished me. "It'll pollute your mind! It's worse than comic books!" Long before Barry became a department chair he was already a master recruiter.

Several years later, I had the good fortune to begin thesis work with Barry as a graduate student.

- Peter Perry

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xxv

In the mid-1970s, Barry visited Moscow. One day he went into a store to buy some eggs. He handed over a 10-ruble bill to the storekeeper and said "Eggs" in Russian; it was the only word in Russian which he knew. She asked him whether he wanted to spend all 10 rubles (a considerable amount in those days) on eggs. But this was a different phrase which Barry didn't understand, and in reply he just smiled his charming smile. She then gave him a check for a hundred eggs. The following day, Barry gave a seminar at the university. It was his last day since he was leaving Moscow the next day. After his talk, he distributed the eggs among the participants. Following the American tradition, undergrads and graduate students received the largest number of eggs and professors received almost nothing. - Yakov Sinai

Quantum Field Theory, Statistical Mechanics

Proceedings of Symposia in Pure Mathematics Volume 76.1, 2007

Perspectives in Statistical Mechanics Michael Aizenman Dedicated to Barry Simon on the occasion of his sixtieth birthday ABSTRACT. Without attempting to summarize the vast field of statistical mechanics, we briefly mention some of the progress that was made in areas which have enjoyed Barry Simon's interests. In particular, we focus on rigorous non-perturbative results which provide insight on the spread of correlations

in Gibbs equilibrium states and yield information on phase transitions and critical phenomena. Briefly mentioned also are certain spinoffs, where ideas which have been fruitful within the context of statistical mechanics proved to be of use in other areas, and some recent results which relate to previously open questions and conjectures.

CONTENTS

An Appreciation Statistical Mechanics in Relation to Field Theory Aspects of Equilibrium Statistical Mechanics Some Cherries from the Pie: Essential Results Derived Through Non-Perturbative Methods 5. Related Recent Developments Epilogue 1.

2. 3. 4.

References

1. An Appreciation In this chapter of the Festschrift celebrating Barry Simon's contributions to the fields which have enjoyed his attention, we focus on statistical mechanics. I shall list here some of the subject's core topics, and mention a selection of results related to issues which have attracted Barry's interests. These represent only a small part 2000 Mathematics Subject Classification. 82-02.

Key words and phrases. phase transitions, critical phenomena, statistical mechanics, field theory, rigorous results. Supported in part by NSF grant DMS 0602360. ©2007 by the author. Faithful reproduction for non-commercial purposes is permitted. 3

M. AIZENMAN

4

of the results which were derived in this vibrant field during the period addressed here.

The results presented below came from a community of people who throughout

their work have stimulated and informed each other, activities in which Barry Simon has excelled. He has done that with flair and in his unique style: with remarkable energy, mathematical skill, eye for the essence of the argument, and intellectual generosity towards students, colleagues, and predecessors. Thinking of a way to convey Barry Simon's impact, I am reminded of a question which one is occasionally asked to comment upon: In what way would the world have been different without the contribution of this individual? I usually find this to be a rather humbling and somewhat troubling question. However, if this is in regard to Barry Simon, my answer is: Most likely the world would have been poorer, my ignorance much greater, and our accomplishments fewer.

And I believe that this could also be said by all of Barry's generation in our field, some of his elders, and the many students and postdocs he has generously informed and inspired. Thank you - Barry!

2. Statistical Mechanics in Relation to Field Theory Statistical mechanics is a subject originating in some profound observations tying the orderly behavior which is the subject of thermodynamics with an underlying chaos at the lower scales. A similar claim can be made about the origins of the order seen in most of physics, as we now see in hindsight, of quantum mechanics and quantum field theory. Looking more closely at statistical mechanics, one finds a rich collection of interesting phenomena, challenging questions, and lessons for other disciplines. While Barry Simon has embraced the subject close enough to eventually write a book about it [110], his perspective has, at least initially, been driven by the relation of statistical mechanics with constructive field theory. The challenges of the latter ([54]) have carried a sense of urgency, a glimpse of which can be found in the introduction to Simon's "The P(0)2 Euclidean (Quantum) Field Theory" ( [102]). The two subjects are related in a number of ways:

i. Similarity The Gell-Mann low formula,

r(x1

...

xn) _

(OJT [e(zf R(x)d4X)0(xl)...O(xn)] 10) (OJT [e(ifH(x)d4x)] 10)

(2.1)

displays a formal similarity with the Gibbs state expression for the correlation function of local (spin) variables in a thermal equilibrium state: O1...0` e-QH(°) 01

---

Un

PERSPECTIVES IN STATISTICAL MECHANICS

5

As discussed in [102], the formal similarity is made even more compelling by the observation that, in Bosonic field theory, under analytic continuation into imaginary time vacuum, expectation values of products of field operators are transformed into amplitudes associated with functional integrals over non-negative measures. The integrals are rendered convergent through suitable ultraviolet and infrared cutoffs, and the corresponding Schwinger functions then fall within the realm of classical statistical mechanics. In that situation, the general perspective of this field and some of its specific tools become applicable. Some of the dramatic consequences which this reduction has had on the program of constructive field theory can be seen in [60, 62, 102], and are described in the contribution of Rosen in this volume. ii. Criticality underlying the FT path integrals One of the lessons of statistical mechanics is that in an extensive system of variables, with short range interactions, at generic choices of parameters the correlations decay exponentially. The correlation length is typically not much greater

than the range of the interaction, and it is only in the vicinity of critical points that correlations exhibit structures of much greater scales. For a field theory based on local interactions, the interaction range vanishes on the scale of the continuum limit. Hence, in a constructive approximation, for which convergence is accomplished through ultraviolet cutoffs, the underlying system of the local variables needs to be very near a critical point. Some familiarity with critical phenomena is, therefore, essential for the understanding of a continuum field theory.

iii. Statisitcal mechanics as a constructive tool for field theory Scaling limits of the fluctuating component of the local order parameter in statistical mechanics are described by Euclidean fields. To some extent, this relation has fueled the interest in statistical mechanics within the community of constructive field theorists, in particular through the hope to construct the Euclidean Od field theory through limits of critical Ising-type models. This can indeed be done, but to the chagrin of some, only below the upper critical dimension, which in the above case is d, = 4. The "no-go" theorems related to the phenomenon of the upper-critical dimension ([1, 42, 51]) may have somewhat diminished the interest in the field on the part of those who came to it looking for help with the tasks of constuctive FT. Nevertheless, statistical mechanics has continued to serve as a rich source of interesting challenges and insights which enrich other fields. Among the new challenges one could find the effects of frustration and disorder in the parameters (Imry-Ma effect and spin glass phenomena). Ideas which were developed in the context of statistical mechanics have affected developments in various other areas. Such "spinoffs" have included topics of discrete mathematics with relation to computer science, as well as techniques for addressing the spectral and dynamical properties of random Schrodinger operators.

3. Aspects of Equilibrium Statistical Mechanics Without trying to provide a proper summary of the essential results in this subject, and as a prelude to a selected few presented next, let me mention a nonexhaustive list of themes which have drawn the attention of mathematical physicists working in this area.

6

M. AIZENMAN

1. Derivation of Thermodynamics Convergence of the free energy density Entropy and its properties Free energy for the long range Coulomb interaction Finite size effects

One of the first accomplishments of Boltzmann's and Gibbs' statistical mechanics has been in presenting an intellectually coherent basis for the laws of thermodynamics and, in particular expanding our understanding of entropy-the concept at the root of thermodynamics. The book by Ruelle on the subject [98] was an invitation extended to mathematically-minded researchers and, indeed, in short order a slew of interesting results have followed. Among the general and foundational results of the subject is convergence of the free energy density for extensive systems with short range interactions. Key contributors have included van Hove [69], Ruelle [96], Fisher [38], and Griffiths [55].

Further studies were needed to address the question for long range forces, such as the Coulomb interaction ([78]), which of course is of deep interest. The proof of convergence for quantum Coulomb systems was accomplished in a fundamental paper of Lieb and Lebowitz [82]. The work benefitted from the input of Simon, who contributed a technical appendix [100]. Further studies were needed to extend our understanding of entropy to the quantum domain. A major accomplishment was the proof of subadditivity of entropy by Lieb and Ruskai [83], in another article featuring an appendix by Simon. It was also noted that the von Neumann notion of entropy does not offer a fully satisfactory quantum counterpart to the Kolmogorov-Sinai entropy of classical dynamical systems, and neither is it fully satisfactory in the setting of statistical mechanics, where the dynamics correspond to translations. After some search, other proposals were developed [27].

II. Mapping the Phase Diagrams High temperature and low temperature regimes Phase transitions for long range interactions Conditions for the existence of symmetry breaking Establishing the criticality of a phase transition

One of the next tasks for statistical mechanics is to produce the tools for mapping the phase diagram and description of the distinct phase regimes in the thermodynamic space, whose parameters include the temperature T = (k/3)-1, the overall strength of the coupling, and control parameters such as the magnetic field. Of particular interest is understanding the conditions under which there will be phase transitions associated with discrete or continuous symmetry breaking ([90, 56, 86, 30, 48, 35, 49, 110, 521). The structure of the Gibbs equilibrium states at high temperatures can be approached through fairly general and robust methods, at least for systems with rapidly decaying interactions. The tools include cluster expansions, such as the improved Meyer expansion [91, 97, 118], Dobrushin's uniqueness-of-state technique (bounds for a generally defined "influence kernel") [28, 104], and the generalized polymer expansion of Kotecky and Preiss [75]. For low temperatures, the basic tools include the Peierls argument [90] and the corresponding bounds and expansion, the more general Pirogov-Sinai theory [94], and occasionally a duality map converting low to high temperature regime.

PERSPECTIVES IN STATISTICAL MECHANICS

7

Naturally, the analysis gets to be somewhat trickier near the boundaries of the distinct phases, which is where critical phenomena are found. In particular, a non-perturbative argument is needed for the important issue of establishing that a phase transition is critical, which means here that the correlation length diverges as the transition point is approached. More is said on this topic below.

III. Structure of the Gibbs States Correlation functions: bounds, inequalities, and relations Extremal state decomposition; applications for issues of uniqueness Effects of long range interactions, and in particular 1x-yI2 interactions in 1D

Some could find it disappointing that in dimensions d > 2 most systems of interest are not solvable. In lieu of solutions, for some systems useful information can be obtained through correlation inequalities through which much can be learned about the phase structure, properties of the Gibbs states, and the critical behavior. Among the general results of statistical mechanics is the statement that, in one dimension, short range interactions do not yield phase transitions, but phase transitions and symmetry breaking are possible if the interactions decay slowly enough. The threshold case, which is of interactions falling as 1/Ix-y12, has attracted attention for a number of reasons. The question of the exact condition for the threshold decay rate has attracted a number of different ideas, leading to the conclusion that it is JJ,y .:: J/Jx - y2 with /iJ = 1 [117, 33, 1081. Through an intriguing energy vs.-entropy argument, Thouless has suggested that in the borderline case of Ising spins with the 1/1x - y12 ferromagnetic interaction, there should be found an unusual discontinuity of the spontaneous magnetization at the transition temperature. Anderson, Yuval, and Hamman [15] have pointed out that this particular model is quite significant for the Kondo problem, in which context the one dimensional parameter is time. They also introduced a very insightful renormalization group analysis which explains the system's essential features. The curious prediction of Thouless was eventually established rigorously [9, 7]. However, related analysis also showed that the phenomenon is not accounted for by the original argument [71].

IV. Critical Phenomena Critical exponents Universality The "renormalization group" perspective The phenomenon of upper critical dimension(s) Among the major notions to emerge in the late sixties and early seventies of the twentieth century have been the realization of "universality" in critical phenom-

ena, and the "renormalization group" perspective on the subject [39, 119]. The translation of the latter into proper mathematical statement is not easy [36], and to large extent still remains to be done. In this situation, it was deemed of value to have even partial rigorous results, to either correct or to offer supportive evidence to the validity of the sometimes far-reaching claims and, by implication, build confidence in the heuristic methods. Various results have been obtained, often with a characteristic time delay of about a decade and usually by means which have been very different from the earlier physicists' heuristic arguments.

M. AIZENMAN

8

V. Structures Emerging from Local Interactions Random path / random cluster representations Random surface models, and their phase transitions The spin correlations which exist in the equilibrium states of ferromagnetic models can often be presented as the result of correlated excitations which are associated with random paths, as in (4.7) below, or with random clusters-as seen in the Fortuin and Kasteleyn representation of the Q-state Potts models [40]. Subtle correlations may then be robustly expressed in terms of properties of the associated stochastic geometric models. This has led to a rather fruitful line of research, with progress made through both analogies and exact relations of spin systems with percolation type models (see [1, 8, 7, 58] and references therein). In gauge models one finds relevant excitations associated with random surfaces. In this context, an interesting extension of the percolation transition is found in a random plaquette model in three dimensions [6]. The standard percolation transition is related there by duality to a confinement-deconfinement transition, which has a natural definition for a system of randomly occupied plaquettes. This is but one example of interesting issues associated with random surface models [14].

VI. Scaling Limits The emergence of stochastic geometry Insights and challenges of the quantum gravity method Conformal invariance and SLE It is generally understood that the scaling limit of the fluctuations of the local order parameter would naturally be described by fields. More could be said about

the limiting distribution of the fractal stochastic geometric structures which are associated with the long range correlations exhibited in these models at criticality. The theory of the critical fluctuations has been developed most effectively in two dimensions where conformal symmetry applies and has particularly strong consequences [17]. An intriguing tool has been the "quantum gravity" method, which

in effect means the study of statistical mechanics on fluctuating surfaces. In that setup, various characteristic exponents can be calculated through the asymptotics of random matrices. Curiously, the resulting scaling laws are predicted to bear an explicit relation to the corresponding ones on a rigid plane [32]. Another approach has recently been enabled through the introduction, by Schramm [99], of the SLEK, family of processes which are endowed with the "conformal Markov property" [76, 95]. There have been many interesting mathematical results in this area, including the description of some of the scaling limits [113, 25], as well as various results on critical exponents [76]. In general, the latter fit very well with the quantum gravity predictions, although a broad statement still remains to be established.

VII. Disorder Effects Effects of randomness in the coupling strengths; the Imry-Ma phenomenon Spin glass phenomena

In addition to its intrinsic interest, the Imry-Ma phenomenon is remarkable in providing a rare example where a rigorous result was derived before a consensus has emerged in the physics community concerning two conflicting predictions ([70, 23, 13]).

PERSPECTIVES IN STATISTICAL MECHANICS

9

On the last listed topic-spin glass models significant progress [65, 116] was recently enabled through a surprisingly effective interpolation argument which was introduced by Guerra and Tonineli. Curiously, Barry Simon's interest in statistical mechanics was stimulated though a different, yet similarly surprising and stimulating, contribution which Guerra made at the beginning of the time frame which we bear in mind writing these notes [59].

4. Some Cherries from the Pie: Essential Results Derived Through Non-Perturbative Methods Rather than review the body of results derived on the topics listed above, which may take the talents of Barry Simon to write, I shall present here only some examples. These are selected by few common themes: i. They demonstrate that often "soft arguments," such as inequalities which by their nature can be viewed as an imprecise tool, can address questions which are beyond the reach of hard analytical methods; ii. The examples are related to some of the basic models which have driven the interests of the intellectual community in which Barry Simon was active; and iii. For many of these results there is an underlying random walk perspective. Various insights have sprung from this perspective on a variety of topics.

4.1. Absence of Phase Transitions for the Ising Model and the 04 Euclidean Field Theory at h $ 0. The determination of the phase diagram of a model is of course of fundamental importance. For ferromagnetic Ising models, the task can be greatly simplified through correlation inequalities. In particular, the combination of the GHS and FKG inequalities (Griffiths-Hurst-Sherman [57] and Fortuin-Kasteleyn-Ginibre [41]) allows one to conclude that a first order phase transition can occur only along the line of symmetry h = 0. The GHS inequality states that in spin models, with o = ±1 and the Hamiltonian

H(a) _

Jx,vo-xoy + > hxcx {x,y}

(4.1)

x

with J.,. > 0, the cluster functions (u for Ursel) e

ue(xI,..., x1) :=

ahxlaahxf In Z(A; h J) = (ax1; ...; QXe)

satisfy

sgn h u3(xi, x2i x3) < 0 and, at h = 0: u4(xi, ..., x4) < 0Given that the mean magnetization, M(I, h) = (QO),3,h, obeys a2 M(,3, h)

W

=

2 E u3 (0 , x, Y) ,

(4.2)

(4 .3)

x,y

the inequality (4.2) allows one to conclude that M(/3, h) is concave as function of

h at h > 0, and convex for h < 0. The important-and otherwise difficult to reach-implication is that for h 0 the magnetization is continuous in h. The FKG inequality allows one in such case to rule out first order phase transitions (which require the existence of more than one Gibbs state, in the infinite volume

M. AIZENMAN

10

limit). Related results for the line h = 0,

oo) were derived by yet different

inequalities of Lebowitz [77].

A yet stronger statement is allowed by the Lee-Yang theorem [80, 84] which implies that in this class of models the infinite volume free energy density is analytic in h for h 0. An important implication is that at h 0 the correlation functions decay exponentially in the distance [79].

The results described next have allowed to extend the above results also to the Euclidean 04 field theory, for which exponential decay of correlations translates into a mass gap [103, 63].

4.2. Construction of 04 Variables out of Ising Spins. The analyticity methods and the correlation inequalities which were initially derived for systems of Ising spins (on arbitrary graphs) admit natural extensions to broader families of systems of ferromagnetically coupled variables. Following up on the observation of Griffiths that these methods extend to systems whose spin variables can be presented as linear combinations of ferromagnetically coupled Ising spins and that various examples of interest can be obtained by taking limits of such variables, Simon and Griffiths [111] showed that the class also includes the important case, historically and conceptually, of the continuous 04 measure

p(dO) =

e-,,,,4+b,,2

dq ,

(4.4)

atA>0,b>0. Thus, the observations made above about Ising models also extend to functional integrals of the form f ... f F(O) e E Jy,(0y-0y)2 + r_. hxOye- Ey any+bO. ll dq,,, . (4.5) (F Z(A;h,J,A,b)

Of particular interest has been the scaling (continuum) limit, for which the lattice

spacing is taken as a -> 0, and one considers the limiting probability measure for T(x)a :_ ((a)-lqx/a which is to be interpreted in the distributional sense. For the construction of a meaningful limit, both the coupling constants and the field strength renormalization ((a) are adjusted while a -p 0, so as to guarantee convergence, in a weak sense, and regular values (neither

Wn(xl ..., xn) = l 0(IF (xl)

...

0, nor oo) for

IF (x,,,))a .

4.3. Insights from Lee-Yang Theory for the Scaling Limits. An important consequence, noted by Newman [87], is that any scaling limit of the lattice 04 model will be non-gaussian if and only if the suitably scaled limit of u4(xl, ..., x4) - (0x1; ...; 0x4) does not vanish. The inequalities presented in [1] and [42] yield such a conclusion through direct bounds, yet the insight from the Lee-Yang theory was instructive.

4.4. A Random Path Perspective. A new range of insights and tools become available through random path representations, and/or random cluster representations, of the correlation functions in some of the essential models. For Ising ferromagnetic spin systems, after suitable expansion of the Gibbs factor and summation of the spin variables, one finds [57, 1], for h = 0: (Ux1...Uxk) -

>an={x1,...,xk} w(m) L.am=0 w(m)

(4.6)

PERSPECTIVES IN STATISTICAL MECHANICS

11

where m = {mb} ranges over "random current" configurations, each described by a collection of (integer) fluxes defined over the lattice bonds. The weight function w(m) is a product of local terms. The sum is restricted by a constraint on the set of sites at which m has odd flux, which is denoted here by 8m. Configurations with a prescribed set of such "sources" consist of an assortment of many current loops and few "current lines" linking pairs of sources, as is indicated in Figure 1. In other words, in this representation the spin-spin correlations are expressed through the amplitudes associated with source-insertion operators in a loop-soup integral. Under partial summation over the background loops, one obtains a random-path representation: (Uyl...O'X2 )

E

pairings of

I

P('Y1, ...,''

)

"Yl :xil'""xjl'

{21,...,x25} 7n.:2in.~Xjn

where y, ranges over paths connecting the designated sources. The weight for a collection of paths p(ryl, ..., -yn,) factorizes, approximately, if the paths are remote from each other.

M

FIGURE 1. A schematic depiction of a current configuration (m), and the corresponding pair of source-linking paths ('yl and rye)

This representation captures the fact that without the interactions, the spin variables are decoupled and the correlations are built through pair interactions. The representation is made particularly effective through some convenient identities, which allow one to reduce various truncated correlations to path intersection amplitudes [1, 5]. As will be indicated below, such geometrization of correlations has far reaching consequences.

The path representation for the correlation functions is not unique, though some have particular technical advantages. Related representations have also been worked out for 04 correlators in terms which are native to the FT functional integrals (4.5), [42]. Their implications are extensively discussed in the monograph [37].

4.5. Proofs of the Criticality of (Certain) Phase Transitions. Phase transitions at which the correlation length diverges are referred to as critical-a condition which is not met at the usual first order transitions. Criticality plays an important role for the emergence of universality in critical phenomena, and for the existence of scaling limits. It is therefore of value to be able to establish the criticality of phase transitions of interest. The question seems to fall beyond the reach of the standard perturbative methods, as these tend to diverge at phase transitions. It was therefore gratifying to find a useful tool for this purpose in the form of the

M. AIZENMAN

12

family of inequalities which has evolved from the "Simon inequality" [106] for ferromagnetic Ising spin systems. Particularly effective is the Lieb improved version [81] which, after further improvement, states that for any pair of sites {x, y} within the

set on which the model is defined and any domain D which includes x but not y, the correlations of the Ising ferromagnetic model, G(x, y) - (Qxoy), satisfy:

(0 <) G(x, y) <

G(x, u)D 13Ju,v G(v, y)

(4.8)

UED, vEZd\D

where G(x, u)D is the correlation function for the system restricted to D. Applying the inequality with D chosen as finite boxes of increasing size, (4.8) allows one to conclude the correlation length has to diverge as the temperature approaches the transition point.

It may be added that the spark which led to (4.8) originated in a discussion of the work by Dobrushin and Pechersky [29], where it was shown that, quite generally, fast enough power law decay of generalized correlation functions implies exponential decay. The extraction of a simple, pedagogical, and useful principle, at least for a subclass of models, is a good example of Barry at work. It is also an example of a robust idea: a number of inequalities of this genre, valid for different spin models, have then appeared in close succession-close enough to be published in the same issue of CMP. The inequality (4.8) can be understood through the random walk perspective on the spread of correlations in the model. (Although the original derivations did not make use of the representation (4.7), equation (4.8), which was not in the original batch of the similar inequalities, has an easy derivation using the technique of [1]). Useful variants of this relation occur also in various other models: a very similar

statement is valid for the connectivity function in percolation models [8], and a related inequality is satisfied by the (fractional) moments of the Green function of the discrete Schrodinger operators with random potentials [11].

4.6. Systematic Criteria for Mapping the High Temperature Regime. Another reason for interest in inequalities of the Simon-Lieb type, is that (4.8) yields a sequence of finite volume criteria for the high temperature phase. For each L < oo, let NL be defined by the condition

13L = max 3:

Gp(0, u)D ,(3Ju,v < 1

.

(4.9)

uE[-L,L]d, V eZd\[-L,L]d

The inverse temperatures ,(3L can in principle be evaluated to the desired accuracy through finite computations. The inequality (4.8) allows one to conclude that any 13L provides a rigorous bound on the actual transition temperature 33, and it can also be shown that these bounds actually converge: for each L < oo:

and

QL

<

lira NL =

L-.oo

lic,

i3c

(4.10)

.

A more general method for such bounds which, in principle, permits one to successively map the entire high temperature regime, is provided by the work of Dobrushin and Shlosman [31]. While it was only very rarely used as a tool for practical computations, the finite volume principle has interesting implications for the nature of the high temperature regime.

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13

The method has also inspired similar statements for the study of the regime of Anderson localization for the aforementioned random Schrodinger operators [50, 11, 53].

4.7. Continuous Symmetry Breaking (in d > 2 Dimensions). A serious challenge, whose robust resolution could provide a tool for addressing a number of issues, is to establish that under suitable conditions there is continuous symmetry

breaking. An outstanding development was the series of results which started with the work of Frohlich, Simon, and Spencer [48]. Their argument proceeds through the gaussian domination bound, which says that for reflection positive spin models with a pair interaction and periodic boundary conditions, for dual momenta p c A*\0: (IS(p)I21) <

d

2/dE(p)

(4.11)

Here, S(p) = A-1/2 lxEA and for p :,k 0, E(p) _ - >XEA ezp'JX. It helps to note that the quantity on the left side in (4.11) plays a dual role: the Fourier transform of the two point function G(x,y) = (SXSy) is given by (IS(p)12I), and E(p) (1 S(p)121) gives the mean value of energy in the p mode. By the latter obser-

vation, (4.11) says that the "equipartition law" provides a rigorous bound. The former observation is used to show that in dimensions d > 2, at low temperatures there is symmetry breaking, which can be attributed to Bose-Einstein like macroscopic occupation of the p = 0 mode, proven by an estimate which resembles the BE calculation [37]. The gaussian domination bound was proven using the "Chessboard Inequality,"

which is based on the reflection positivity. The inequality is a remarkable tool. Among its other applications are bounds on expectation values of products of local observable through thermodynamic quantities [43]. In particular, it permits one to establish the existence of phase transitions though arguments of thermodynamic flavor.

Reflection positivity (RP) is a tool with contradictory aspects, and limitations which are not always intuitive. For instance, there is still no rigorous proof of the existence of symmetry breaking in the quantum Heisenberg ferromagnet, although such a result was established for the antiferromagnetic model for the suitable dimensions [34]. When RP applies, its results are spectacular and physically well motivated, but when its exacting condition is not satisfied even to a small degree, it provides no information. One could say it is a gem of an argument.

4.8. The Mermin-Wagner Phenomenon and the Kosterlitz-Thouless Phase for Plane Rotors in Two Dimensions. A celebrated general statement, known as the Mermin-Wagner theorem [86], is that in two dimension there is no continuous symmetry breaking. The rigorous proof of that for the compact symmetry group of rotations was initially provided by Dobrushin and Shlosman [30]. The phenomenon derives from the fact that in d < 2 dimensions, the ground state configuration of a large system, with rotation-invariant interactions of short-range, can be rotated against the ordered boundary conditions with only small energy cost. The energy penalty vanishes in the infinite volume limit, just as the related

M. AIZENMAN

14

variational quantity, for d < 2:

inf{

(4.12)

(

which for two dimensions vanishes as 1/ log L. However, the argument outlined above is incomplete, as what is needed for the proof is an estimate of the free energy associated with the rotation of a state with thermal disorder. Surprisingly, the harder part to deal with is the fist order term, which vanishes for the totally aligned configurations and thus does not show up in the ground state calculation. A particularly simple and non-perturbative argument for the case of compact symmetry group was devised by Pfister [93] (see also [52]). Altogether, the so-called Mermin-Wagner Theorem is one of the essential results of statistical mechanics. It is of interest that at the thresholds for the feasibility of symmetry breaking one finds borderline models with unusual behavior: low temperature phases at which there is neither long range order nor rapid decay of correlations, where the correlation functions decay by temperature-dependent power laws. For continuous symmetry such behavior is encountered in the two component X-Y (0(2)) model in two dimensions, in what is known as the Kosterlitz-Thouless (KT) phase [74] (long range order is ruled out there by the Mermin-Wagner Theorem). The rigorous proof of the existence of this phase, by Frohlich and Spencer [49] (see also [85]), was a notable accomplishment. The techniques which were developed for this purpose have included elements of the multiscale analysis which has found many other applications since, in particular, in the theory of Anderson localization [50, 45, 53]. At the threshold for discrete symmetry breaking is the one dimensional Ising model with Jx,y = 1/1x - y12. This system also exhibits temperature-dependent power law decay of correlations, which here occurs before the onset of long range order [71]. The existence of this unusual low temperature behavior shows that the Thouless phenomenon, discontinuity of the spontaneous magnetization, which is exhibited there [9, 7] is not accounted for by the argument which were initially used for its prediction [117]. Related energy-entropy arguments, and their limitations, were discussed by Simon and Sokal [112]. It may also be added here that KT-like phases are not expected to occur for O(N) models with N > 2, in two dimensions. The reasoning is interesting: a calculation shows that under a renormalization group scheme at both very low and very high temperatures, the temperature flows upward, and it was surmised that this flow connects the two regimes. However, the analysis gets to be more complicated at intermediate temperatures. The renormalization group map is somewhat ill defined there, and that leaves a serious gap in the argument. It was argued in [89] that the question which is left open is an interesting one, and that its resolution may be significant also for clarifying other assertions of "asymptotic freedom," which is an issue of great significance.

As a prelude to the next topic, it may be added that an intermediate phase is expected to occur in two dimensions for discrete two-component "clock" models with discrete rotational symmetry. Its essential characteristic is the existence of power law decay of correlations at power which varies with the temperature, and

without long range order. Such a phase does not occur for translation invariant Ising spin systems, in any dimension.

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15

4.9. The Coincidence /3T = OH for Ising and Percolation Models. For ferromagnetic Ising spin models in d > 1 dimensions, the edge of the high temperature regime, which is characterized by the exponential-decay of correlations, coincides with the threshold for the non-vanishing of the long range order parameter. That is, such a system does not exhibit an intermediate phase which shares the characteristics of the KT phase. (The two transition temperatures were initially designated /T1 and /H1, for Temperly and Hammersley, correspondingly). Such statements are of basic interest, as they reflect on both the phase diagram and on the critical behavior, yet the proofs are beyond the reach of the available expansions. Nevertheless, it was found that the above assertion can be addressed at certain generality, of the translation invariant Ising spin models. An analogous statement also holds for percolation models on transitive graphs (with application also to the contact process). As it turned out, one understands each of the models better by considering the two simultaneously. It is somewhat remarkable that the above statement, and other information about the behavior at the critical point, can be proven by means of soft looking partial differential inequalities which bear no specific reference to the critical temperature. Of course, the non-linearity of the expressions contains the seeds for the information about possible critical behavior, as the analysis shows ([8, 26, 4] ). The proof is enabled by a pair of partial differential inequalities which are valid for any homogeneous (i.e., invariant under a transitive group of lattice symmetries) system of Ising spins. Similar relations are valid for percolation models, for which h is an auxiliary parameter which requires some explanation, a controls the bond

density (px,y = 1 - e--O), and M(a, h = 0+) is the percolation probability. These are:

yaM<jJl MhM

("Burgers inequality") (4.13)

M < huh M +

a1JIMn-2aM

+ Mn-1

("fin bound")

(4.14)

with n = 4 for Ising (04) and n = 3 for percolation (03). The observation that the combination of the two inequalities yields OT = OH,

or PT = PH, was made first in the context of percolation models ([4, 5]). The inequality (4.13) appeared earlier in the context of Ising systems, for which it follows

from the GHS inequality. Newman [88] has noted that the relation resembles the Burgers equation for the evolution of the profile of a velocity field on R, under the correspondence: {M, h, a} H {V, x, t}. The Burgers equation provides a simple example of evolution yielding shocks, which in our case correspond to first order phase transitions. An interesting aspect of the relation (4.14) is that it contains hints of the 0n structure of the relevant diagrams in the two models. The point, which is better understood by looking at the proof, also suggests other insights, including that the upper critical dimensions may be d, = 4 for short range Ising models, and d, = 6 for percolation, which is indeed the case, as is mentioned below. It may be added that the equality PT = PH has a further implication in two dimensions, for which the model is self dual under a transformation which exchanges

the high and low temperature regimes. There, the knowledge that there is no intermediate phase permits one to determine the critical temperature, or density,

M. AIZENMAN

16

as the unique self-dual point. Before the availability of the general result, the case of 2D percolation was addressed by Kesten [73] through other arguments.

4.10. Mean Field Critical Exponent Bounds. The differential inequalities presented above also permit one to conclude bounds on the critical exponents which are expected to be associated with the singular behavior in the vicinity of the critical point, as in:

x=

hM(O,h=0) M(O,, h)

M(,3,h=0)

-)3cI-ry

10

phi-116 1

,

3

(4.15)

3IQ.

The interest in such exponents is greatly enhanced by their universality. The latter is the statement of the independence of critical behavior from many of the local details of the model. The conceptual explanation of this experimentally observed fact was provided by the renormalization group picture. A particular implication

is that the exponents measured in real phase tansitions may agree exactly with those associated with the simpler mathematical models. Neverthelss, even seemingly simple models are not solvable in dimensions d > 2, and the mathematical expression of the renormalization group ideas is still an incomplete task. In this situation, even partial results are of value. While the proof of strict power law behavior is still incomplete, partial differential inequalities allow one to conclude meaningful bounds. For instance, the inequality (4.16)

ax lh=O < iJix2

which for Ising systems follows from GHS, allows one to conclude that y_ > 1, where the subscript indicates that the exponent pertains to the approach of 0, from below. The full argument requires that attention be paid to suitable cutoffs, as is explained in [8]. With somewhat more involved integration and interpolation [5] (see also [115, 37]), the inequalities (4.13) and (4.14) permit one to complete the list to:

,3 < (n-2)

ry_>1

6> n-1

(4.17)

(with n as explained below (4.14)) A notable feature here is that the exponents are shown to be bounded by the values they assume in the corresponding mean-field models. The reader may find a detailed discussion, and a far more comprehensive list of references, in the monograph [37].

4.11. Upper Critical Dimensions. A high point of the random walk methods was reached when it was realized that they permit one to establish the existence of the phenomenon of the upper critical dimension. By that we mean the existence of d,, such that in dimensions d > do the critical exponents assume their mean-field values, and in particular the relations (4.17) hold as equalities [1, 42, 5, 37] (this is usually also true at d = d, apart from logarithmic corrections [51]).

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The value of the upper critical dimension depends on the model, and the interaction range. For short-range models:

d_

4

4 6 8

weakly self repelling walks (presumably also strongly SAW) [24, 66] ferromagnetic systems of Ising spins, or 04 variables [1, 42], percolation, at least with spread enough connections [8, 16, 66], "lattice animals" Q66]) (4.18)

The phenomenon is driven by the fact that in high dimensions, loop effects are of diminishing significance. That is exemplified through the asymptotic vanishing, for d > 4, of the probability of mutual intersection for the paths of random walks whose end points are sampled randomly within a region of increasing diameter. The same turns out to be true for the paths with the weights p(y, y') in (4.7). For the proofs of the above statements, it has been very helpful to first establish that a sufficiency criterion for the mean-field behavior of the critical exponents is

the finiteness at the critical point of a certain model dependent diagram. The relevant diagrams tend to be in the form Dk

G*

(0, 0) =

G13(0, u1) G1(ul, u2) ...G1(uk-1) 0) U1,....Uk_1

J_ir/2,,r/2]d G(p) ddp where

(4.19)

is the Fourier transform of G(o, ). The diagrammatic condition is:

lim sup Dk < oo

(4.20)

with k = 2 for Ising/04 systems [114, 1, 42], k = 3 for percolation [8, 16, 20], and k = 4 for "lattice animals" [66] (which are mentioned here mainly for variety). The proof that the diagrammatic condition is satisfied under suitable conditions

usually needs to be provided by other methods. That task can be accomplished through either the infrared bound (4.11)-in the few cases where it is applicable, or through the technique of the "lace expansion" which was initiated by Brydges and Spencer [24]. Through a number of developments, this method has grown into a robust and versatile tool [66].

4.12. Scaling Limits of Critical Models. Our brief tour now returns to the topic which was mentioned at the beginning of this account. A striking feature of the upper critical dimension is that for the Ising/c4 systems, in the scaling (continuum) limit of the infinite system, the fluctuations of the local order parameter have the distribution of a gaussian field [1, 42, 51]. For an explicit statement, one may take the model as formulated on the lattice (aZ)d with periodic boundary conditions at distance L, and consider the distribution of the block spin variables

SR =

17

Na

1:

v,,,

(4.21)

.f(x)cr

(4.22)

XE[0,R]n(aZ)d

or more generally

S(.f) = Na xE[0,1]n(aZ)d

18

M. AIZENMAN

with some positive, compactly supported, F E C0(Rd). The normalizing constant Na is to be determined through the second moment condition: (S2

1)L,,, = 1.

(4.23)

If the distributional limits for variables of the form S(f) exist, one may define a random field T(-), in the sense of a random distribution for which variables of the form f f (x)4(x)dx have the limiting distribution of S(f). The question is then what are the limiting distributions of the variables S1 and S(f), in the continuum limit, for which a -- 0 and the temperature is either fixed at TT or taken to approach that value (at a rate for which the localization length does not vanish on the continuum scale). The results of [1, 42] state that in d > 4 dimensions, if T is fixed at TT, and the limit L -+ oo taken first, then one gets the normal gaussian distribution:

D- lim lim S1 = N(0,1) a-+0 L-+oo

(4.24)

with suitably adjusted normal law, N(0, b1), for limits of S(f ). Related statements are available for the cases where T is allowed to be adjusted with a without leaving residual long-range order. This phenomenon is nicely explained by the tendency, in high dimensions, of the random paths to stay out of each other's way. The resulting factorization of

the joint amplitude (4.7) reduces to Wick's formula, i.e., gaussian structure for the correlation function. A more complete discussion of this observation, and its consequences, can be found in [1, 42, 37].

5. Related Recent Developments There have recently been a number of developments which tie in with the topics discussed above. Of these, let me mention certain results which address a seeming contradiction concerning the critical behavior in high dimensions, d > d,. From experience we know that some of it may appear surprising. Once one is informed that above the upper critical dimension the critical exponents no longer vary with the dimension, two different paradigms suggest themselves for their calculation. One approach is to carry the calculation on the tree graph version of the model. Calculations on tree graphs are usually more approachable. As graphs, trees can be viewed as infinite dimensional, and it is natural to expect that a homogeneous graph with no loops provides a good approximation for a situation in which loop effects are no longer relevant. The other path towards the exponent calculation is to employ the mean field approximation. The two paradigms lead to common values for many of the exponents, including those listed in (4.17) (where the inequalities are then saturated). However, they also differ on some of the predictions. In particular, for Ising/04 spin systems there are drastic differences in the suggested behavior of the scaling limits. The following results shed some light on these differences. The mean-field ferromagnetic interaction for a finite system is 1

2IAL'aI

.

YEAL,a.

A calculation, which in effect was already mentioned-being at the basis of the Griffiths-Simon construction (4.4)-implies that at the critical point the limiting

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19

distribution of the full block spin variable SL is not gaussian. From this perspective there is room to wonder about consistency with (4.24), which asserts normal limit for the distribution of the block spin variable when that is sampled from an infinite system, for d > 4. In a joint work with Papathanakos ([10], in progress) we prove that if one takes the local (non-mean field) Hamiltonian with the periodic boundary conditions, the

scaling limit of the total block spin variables is also not gaussian. Furthermore, even the "local" block spin S1 will have non-gaussian scaling limits, when L is taken to infinity while a -* 0, provided the increase of L is not too fast. (The limit is, nevertheless, gaussian if L -- oo is taken first, as in (4.24).) A familiar and convenient measure of the deviation from the gaussian distribution is provided by the renormalized coupling constant, which can be formulated with S - S1 as:

g-

(S4) - 3(S2)(S2) (S2)2

(5.2)

and thus the renormalized coupling constant depends on {L, a}. The new result is that for periodic boundary conditions and T = TT: lim inf gL,a > 0 (5.3) Here

where the limit is taken at fixed L, or also at L = L(a) H oo provided the increase is limited to a sufficiently low power of 1/a. The explanation of this phenomenon requires the discussion of the winding paths which appear in the random path expansion (4.7) under the periodic boundary conditions. (One can find related effects of the periodic boundary conditions in the rather simpler context of loop erased walks, for which interesting results were presented recently in [18, 92].) A similar phenomenon was conjectured to occur also for percolation models, for which one may define an effective 03 coupling constant along the lines which were explored in [8, 3]. In a recent work, Heydenreich and van der Hofstad [68] have proven, up to logarithmic terms, the percolation version of the conjecture which was formulated by this author concerning that case. The boundary conditions affect there both the effective coupling constant and the size of the maximal connected cluster in a large finite system at p = pc (thus reconciling the results and predictions of [3, 20]). Although the technique is quite different, the coupling algorithm which was introduced in [68] has been of value for the above results for the Ising case ([10]).

Epilogue In concluding, I hope that this brief article has served to remind Barry Simon of some good times of constructive and joyful jostling and camaraderie. I would like to thank my collaborators for the pleasure of learning through joint works, and to express my regret at being able to mention here only a small part of the many interesting works on the subject.

M. AIZENMAN

20

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of finite graphs. I. The scaling window under the triangle condition, Random Structures Algorithms 27 (2005), 137-184. [21] C. Borgs, J. T. Chayes, H. Kesten, and J. Spencer, The birth of the infinite cluster: finitesize scaling in percolation, Comm. Math. Phys. 224 (2001), 153-204, Dedicated to Joel L. Lebowitz. [22] C. Borgs and J. Z. Imbrie, A unified approach to phase diagrams in field theory and statistical mechanics, Comm. Math. Phys. 123 (1989), 305-328. [23] J. Bricmont and A. Kupiainen, Lower critical dimension for the random-field Ising model, Phys. Rev. Lett. 59 (1987), 1829-1832.

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DEPARTMENTS OF PHYSICS AND MATHEMATICS, PRINCETON UNIVERSITY, PRINCETON, NJ 08544, U.S.A.

E-mail address: aizenman®princeton.edu

Proceedings of Symposia in Pure Mathematics Volume 76.1, 2007

Coherent Infrared Representations in Non-Relativistic QED Thomas Chen and Jiirg Frohlich Dedicated to Barry Simon on the occasion of his 60th birthday, in admiration and friendship ABSTRACT. We consider dressed one-electron states in a translation-invariant model of non-relativistic QED. To start with a well-defined model, the interaction Hamiltonian is cut off at very large photon energies (ultraviolet cutoff)

and regularized at very small photon energies (infrared regularization). The infrared regularization is then removed, and the representations of the canonical commutation relations of the electromagnetic field operators determined by the dressed one-electron states are studied using operator-algebra methods. A key ingredient in our analysis is a bound on the renormalized electron mass uniform in the infrared regularization. Our results have important applications in the scattering theory for infraparticles.

CONTENTS

Introduction Definition of the Model Statement of the Main Theorems 4. Infraparticle Scattering 5. Proofs of Theorems 3.1 and 3.2 6. Proof of Proposition 5.1 1.

2. 3.

References

1. Introduction In this note, we consider a translation-invariant model of non-relativistic Quantum Electrodynamics (QED) describing a non-relativistic Pauli (spin 2) electron interacting with the quantized electromagnetic field. An infrared regularization 2000 Mathematics Subject Classification. 81V10, 81Q10.

Key words and phrases. non-relativistic QED, infraparticle states, infraparticle scattering theory. ©2007 American Mathematical Society 25

T. CHEN AND J. FROHLICH

26

(parametrized by a number a << 1) and a fixed ultraviolet cutoff are imposed on the interaction Hamiltonian. Let H(p, a) denote the cutoff fiber Hamiltonian corresponding to the conserved momentum p on the fiber Hilbert space Hp. This space is isomorphic to C2 ®a, where a denotes the photon Fock space, and C2 accounts for the spin of the electron. It is proved in [2] and [4] that, for sufficiently small values of the finestructure constant, H(p, a) possesses a ground state eigenvalue E(p, or) (of multiplicity two for spin 2, see [12, 5, 16]) at the bottom of its essential spectrum. Let 9p,o denote the corresponding ground state eigenspace. The unit rays determined by the eigenvectors T.,, (p, o,) E Sp,a, 11 %F,, (p, a) 11 = 1, can be parametrized by u E S2 C R3, with (IF. (p, a) , rrTu(p, a)) = u (rr is the vector of Pauli matrices, see (2.18)). For an introduction to the mathematics of non-relativistic QED, see, e.g., [16]. Let Kp := {k E JR3 I kj > p} be the set of photon momenta corresponding to

photon energies > p. (We choose units such that h = c = 1. The finestructure constant is a = e2.) By 3p we denote the symmetric Fock space over the onephoton Hilbert space L2 (Kp, d3k) ®C2 of wave functions describing the pure states of a photon of energy > p; the factor C2 accounts for the two possible polarizations

of a photon. Let S(,ap) denote the algebra of all bounded operators on 'gyp. We define a C*-algebra, 2(, by setting II

-

II

2[:= V 13(,3p) p>O

where the closure is taken in the operator norm. We are interested in the representations of 21 determined by dressed one-electron states via the GNS construction. We define the infrared-regularized states wp,a(A)

(&.(p,a),AWu(p,a))

,

AE21,

for a fixed choice of u c S2. We prove that, for momenta p with 0 < Ii < and any sequence a,, \ 0 (n ---+ oc), there exists a state wp on 2( given by wp(A) = limb-. wp,o,, (A), for all A E 21, for some subsequence (o,,,,). By the GNS construction, the state wp determines a representation of 2t. For p # 0, this representation turns out to be quasi-equivalent to a coherent state representation of 21 unitarily inequivalent to the Fock representation. It will be determined explicitly. For Nelson's model, similar results were proven in [9, 10]. However, the more complicated coupling structure of the Hamilton operator of non-relativistic QED makes a key argument in [9] inapplicable. The difficulty arises from the fact that the interaction term in QED is of minimal substitution type and hence quadratic in creation and annihilation operators, while, in Nelson's model, it is linear. We arrive at our main result by making use of the uniform bounds on the renormalized electron mass recently derived in [4] and [2]. An important application of our results concerns infraparticle scattering theory, in particular Compton scattering. Recently, some significant progress in scattering theory was made by Pizzo in [14], where infraparticle scattering states are constructed for Nelson's model after a complete removal of the infrared regularization. The proof uses, and significantly extends, ideas proposed in [9, 10]. A bound on the renormalized particle mass uniform in the infrared regularization a > 0 is assumed in [14] without proof.

INFRARED REPRESENTATIONS IN NON-RELATIVISTIC QED

27

The construction of an infraparticle scattering state in [9, 14] crucially involves

a dressing transformation. To construct the latter, it is necessary to identify a coherent state representation that is quasi-equivalent to the GNS representation determined by wp. This was achieved in [9] for Nelson's model, but has not been accomplished for non-relativistic QED, due to the difficulties noted above. This is the main reason why attempts to construct an infraparticle scattering theory for non-relativistic QED have been unsuccessful, so far, even after the appearance of Pizzo's work. With Theorem 3.2 of the present paper, we provide this important missing ingredient. Further modifications necessary to adapt Pizzo's analysis to non-relativistic QED are outlined, but a detailed discussion of these matters is beyond the scope of the present paper.

2. Definition of the Model We consider an electron of spin 2 coupled to the quantized electromagnetic field, with a fixed ultraviolet cutoff imposed on the interaction Hamiltonian. The Hilbert space of one-electron states is given by 'Hed = L2(R3) ® cC2

(2.1)

.

The Fock space of the quantized electromagnetic field in the Coulomb gauge is given by ®,j(n)

_

,

3(0) = C

(2.2)

,

n>O

where the fully symmetrized n-fold tensor product space ,j(n) = Symn(L2(TR3) ® C2)®n

(2.3)

denotes the n-photon Hilbert space. The factor C2 accounts for the two transverse polarization modes of a photon, and Symn symmetrizes the n factors in the tensor product, in accordance with the fact that photons are bosons. A vector

E j corresponds to a sequence

= (4) (°), (D (1),

where 4)n) = .1)(n) (k1, AI, .

. .

,

4)(n

... , (D (n), ...)

(, j(n)

kn, An), kj E JR3 is the momentum, and Aj E {+, -}

labels the two possible helicities of the j-th photon. The scalar product on j is given by "D1

,

2) =

1n1 ,

2n)

n>O

Let f denote the Fourier transform of f. For A E {+, -} and f E L2(1R3), we introduce annihilation operators a,\ (f)

Z(n

->

2(n-1

(2.4)

defined by (aa (f) .1))(n-l) (k1

A1,

... , k,1, An_1)

_ v/n J dkn J * (kn) ,D(n) (k1, A1, ... , kn-I, An-1, kn) A)

(2.5)

and creation operators

aa(f)

z(n)

z(n+1)

with a*(f) _ (a), (f))*

.

(2.6)

T. CHEN AND J. FROHLICH

28

These operators satisfy the canonical commutation relations [aa (f ), a*, (9)]

= (f, 9)L2ba,a,

[aa (f ), aa, (9)]

=0,

(2.7)

for all f,g E L2(II83), where ap denotes either as or a*. The Fock vacuum is the unique unit vector Q f = (1, 0, 0, ...)

(2.8)

aa(f)Slf = 0,

(2.9)

in a with the property that for all f E L2(II83).

Since as (f) is linear and as (f) is antilinear in f, one can write

aa(f) =

f

3dkaa(k)f(k)

,

a.\(f) =

f 3dkf(k)aa(k)

(2.10)

where ao(k) are operator-valued distributions also referred to as creation and annihilation operators. They satisfy the commutation relations

= ba a, 6(k - k')

[a,\, (k') , as(k)]

=

[aa,(k'), aa(k)]

0

(2.11)

for all k, k' E R3 and A, A' E {+, -}, and as (k) Q f = 0

(2.12)

for all k, A.

The Hilbert space of the system consisting of a single Pauli electron and the quantized radiation field is given by the tensor product space ?-l = 'H 1 ®;

.

(2.13)

The Hamiltonian is given by

H(a) = 1 (iVy ® 1 f - V Ao(x))2 + /T . B, (x) + lea ® Hf ,

(2.14)

where

Hf =

Jdkka(k)aA(k)

(2.15)

a=1,2

is the free-field Hamiltonian, and Aa (x )

=

B , (x) =

f

dk Ik

dk

r,, (Ikl) [ ca (k )e -Zkx ® aa( k ) + h . c. k

ik A Ba (k) e -2kx ® a,\ (k ) + h .c.

( 2 . 16)

denote the (ultraviolet cutoff) quantized electromagnetic vector potential in the Coulomb gauge, and the magnetic field operator, respectively. The function acQ

INFRARED REPRESENTATIONS IN NON-RELATIVISTIC QED

29

imposes an ultraviolet cutoff and an infrared regularization parametrized by a << 1. One may choose it to satisfy

x/a for x E [0, a], for x E [a, 2], C°° and positive, for x E (2, 1), for x > 1 . (ri, r2i r3), with

dcQ(x) =

1

(2.17)

0

Moreover, r

rI _

CO

11

_

i

0

r3

_

1

0

r2 -. 0 0 -1 ) 0J , denotes the vector of Pauli matrices. We remark that H(a) defines a selfadjoint ,

1

(2.18)

operator on 7-1 bounded from below; see, e.g., [15].

For a charged particle of spin 0, the Zeeman term proportional to B, (x) is absent. The momentum operator of the system is given by

Ptot = iV. ® 1 f + led ® Pf ,

(2.19)

where

Pf =

J

dk k as (k) as (k)

(2.20)

is the momentum operator of the electromagnetic field. The model under consideration is translation invariant in the sense that

[H(a), Ptot] = 0

(2.21)

.

We decompose the Hilbert space into a direct integral, 71 =

/®

J

dp7-lp

(2.22)

,

]1$3

where Hr, the fiber Hilbert space corresponding to a total momentum p, is isomoris invariant under exp[-itH(a)], we may consider the phic to (C2 ®a. Since restriction of H(a) to Hp,

H(p,a) = H(a)

t

= a (p - Pf - VGAo)2 + r BQ + Hf

,

(2.23)

P

where, henceforth, AQ - A,(0) and BQ - BQ(0). We will use results established in [4] and [2] on the nature of the infimum of the spectrum of H(p, a), for p sufficiently small. We define E(p, a) := infspec{ H(p, a) } (2.24) The following theorem is proved in [4, 5]. .

THEOREM 2.1. Assume that p < 3. There exists a small positive constant ao independent of a such that, for all a < ao, the following holds: For every a > 0, E(p, a) is an eigenvalue at the bottom of the essential spectrum, and, by The corresponding eigenspace p,Q has rotation symmetry, is a function only of I dimension 2 for spin 2. The functions E(p, a), 3Ip1 E(p, a) and 8 E(p, a) are uniformly bounded in 1

a > 0. There is a constant co > 0 independent of a and a such that the second derivative aIPl E(p, a) =

1 - 2 (VpTu(p, a) , (H(p, a) - E(p, a)) VpWu(p, a)) ,

(2.25)

T. CHEN AND J. FROHLICH

30

'u (p, a) I I= 1, u H S2 (see Section 1), satisfies

where W u (p, a) H Fp,,,

1 - coa < 02 E(p, a) < 1

(2.26)

,

and

E(p, a) -

a

22

- 2 (Qf ,

AQ Qf

1V E(p,a) - p

co

<

p2 2

< coapI.

(2.27)

The renormalized electron mass, m ren ('n 0)

-_

1

(2.28)

02 E(p, a)

is bounded by

1 < mren(p, a) < 1 + COa ,

(2.29)

uniformly in o, > 0, i.e., the radiative corrections increase the mass of the electron by an amount of 0(a). In [2], a convergent, finite algorithm is devised to determine mren(0, 0) to any given precision, with rigorous error bounds.

We remark that (2.27) implies that VpE(p, a) = 0 if and only if p = 0, for all momenta p, with IpI < 3, all a > 0, and a sufficiently small.

3. Statement of the Main Theorems In this paper, we prove accurate upper and lower bounds on the expected photon number in the dressed one-electron state Wu (p, o-) and study the GNS representation determined by Wu(p, o-) in the limit o, \ 0, for momenta p with IpI < 3

3.1. Estimates on the Expected Photon Number. Our first main result is the following theorem:

THEOREM 3.1. Assume that IpI < 3f, and let

Nf=> X

J

dk a* (k) a , \

denote the photon number operator. Then, for all a < a0 (where a0 is the same constant as in Theorem 2.1), and independently of u H S2, the following hold: For p 0, so that V E(p, a,) 0,

(-ca +

c'alVpE(p,o')I2log 1)+

('u(p,a), NfTu(p,a)

< Ca + C'aIVpE(p,a)I2log 1

Q

(3.1)

for non-negative constants c, C, and 0 < c' < C' independent of p, a and a > 0; (here r+ := max{0, r}). For p = 0 (with V E(0, a) = 0),

Wu(0,a), NfWu(0,a)) < Ca, uniformly in or > 0.

(3.2)

INFRARED REPRESENTATIONS IN NON-RELATIVISTIC QED

31

3.2. Infrared Representations. For p > 0, let 2tp

(3.3)

B(,Jp)

denote the algebra of bounded operators on Up := ®Upn)

(3.4)

n>O

with (n)

Symn(L2(Kp, dk) ®C2)®n ,

(3.5)

where

Kp :_ {kE1R31Iki>p}.

(3.6)

As indicated above, we define a C* algebra 21 as the direct limit

21 := V 2(p

11-11

(3.7)

p>0

where (.)

II

denotes the closure with respect to the operator norm. We define a state wp,a on 2( by II

wp,a(A) = (Wu(p,a), AT,, (p, o,))

(3.8)

,

for A E 2l, corresponding to a vector W,,,(p, a) E l;p,a (the space of dressed oneelectron states, i.e., ground states of the fiber Hamiltonian H(p, a)). The choice of u E S2 is arbitrary but fixed; our results will not depend on u. We prove that, in the limit a \ 0, wp,a tends to a well-defined state, wp, on 2l which determines a GNS representation that is quasi-equivalent to a coherent state representation.

THEOREM 3.2. Assume that 0 < IpI < 3, and let wp,a be as defined above.

Then, for all a < a0 (where ao is the same constant as in Theorem 2.1), the following hold:

1. Let fail denote an arbitrary sequence with limi .

ai = 0. Then there

exists a subsequence {aid } and a state wp on 2[ such that 3l rn wp,a, (A) = wp (A) ,

(3.9)

for all A E 21. The state wp is normal on the subalgebras 2(p, for p > 0. 2. The state wp,a satisfies wp,a(aa(k)*aa(k))

-

Iwp,a(aa(k))I2

ca

r,IIkl)

(3.10)

uniformly in a > 0, where ICa is the cutoff function (2.17), and f dk wp(aa(k)*aa(k)) - Iwp(aa(k))12

< Ca,

in the limit a \ 0, for some finite constants c, C.

(3.11)

T. CHEN AND J. FRDHLICH

32

3. Let 7rp denote the representation of 2t, H the Hilbert space, and ft E H,p the cyclic vector corresponding to (wp, 2t) by the GNS construction, for all A E 2t). Moreover, let (with wp(A) = (SZp , np(A)1 vp,u,a(k)

,/_a_E (k) - V E(p, ,)

(3.12)

k opE(p, o,)

kj

and

vp,A(k) := liM vp,Q,A(k)

(3.13)

.

Then, irp is quasi-equivalent to 7rFook o ap (where 7rFo,k is the Fock repre-

sentation of 2t), and ap is the *-automorphism of 2t determined by

ap(a' (k)) = aa(k) + vp A(k)

.

(3.14)

4. The Fock representation and 7rp are related to each other as follows. (i) If p = 0

lim wo,a(Nf< ca ,

(3.15)

and iro is (quasi-) equivalent to 7rFock.

(ii) If p 54 0, irp is unitarily inequivalent to the Fock representation, and

lim wp,o(Nf) = oc .

cr\O

(3.16)

However, wp has the following "local Fock property":

(a) For every p > 0, the restriction of wp to 91, determines a GNS representation which is quasi-equivalent to the Fock representation. (b) For every bounded region B in physical x-space, the restriction of wp to the local algebra 2t(B) determines a GNS representation which is quasi-equivalent to the Fock representation of 2t(B). Similar results also hold for a charged particle with spin 0.

4. Infraparticle Scattering In this section, we comment on the significance and implications of our results for the scattering theory of infraparticles, more precisely Compton scattering, in view of recent work of Pizzo. A framework for an infraparticle scattering theory in Nelson's model was outlined in [9], and the existence of one-electron scattering states for a > 0 was estab-

lished. The existence of scattering states in the limit a \ 0 has only recently been proven by Pizzo for Nelson's model in [14], using results in [13]. The only unproven hypothesis in [14] is that the renormalized electron mass satisfies mreS1(p,a) < c, uniformly in a > 0, for lpI < MO (in our units). Uniform bounds on the renormalized electron mass in non-relativistic QED are proven in [4, 5] and [2], and also hold for Nelson's model; (but it has to be assumed there that the infrared regularization Ko (Ik l) is non-zero in an open neighborhood of kj = 0). The infrared regularization in [14] is implemented by a sharp cutoff x(Ikl > a), because [14] uses results of [9, 13] (where this choice is technically convenient). Replacing x(Ikl > a) by r,,(Ikl) in [14] can be implemented with

INFRARED REPRESENTATIONS IN NON-RELATIVISTIC QED

33

minor modifications. The methods of [4, 5] then yield the bound aIPE(p, a) > c, uniformly in a > 0. An inequality similar to (3.10) for Nelson's model plays a central role in [9] and [14], since it explicitly identifies a coherent state representation which is quasiequivalent to the GNS representation defined by wP. This coherent state representation determines the correct choice of a "dressing transformation" for the asymptotic (free) comparison dynamics, which is an essential ingredient for the construction of infraparticle scattering states. With (3.10), we provide such a dressing transformation for non-relativistic QED. However, due to the more complicated structure of the interaction Hamiltonian in non-relativistic QED, as compared to Nelson's model, there are some additional modifications which we sketch without detailed proofs.

We start by recalling some basic results in [14], but formulated for QED. Let

E :_ {p E R3 p1 < s }

(4.1)

denote the ball of admissible infraparticle momenta (in [14], the bound pJ < -L 20 is used). Let WP,a(t) :_

e-itxf exp

(4.2)

IIA(vp,a,,)] eitHf

where t denotes time. The function vp,a,A(k) is defined in (3.12), and 11A (f) := i(aa (f) - a , (f )). As proposed in [9], a natural candidate for the asymptotic, freely moving comparison state is given by

f dp WP,a(t)

e-2(P-Pf)x ez7o(VpE(p,o,),t) h(p) e itE(p,a)

Wu(p, a)

E

E xel 0 (4.3)

Here, h(p) e-itE(P,o) Tu(p a) describes a freely moving electron with wave function h (in Schwartz space, and supported in E). The operator WP,a(t) describes a freely time-evolving cloud of physical soft photons surrounding the electron. The integral

over p and the factor e-i(P-Pf)x implement the inverse Fourier transform. The purpose of adding a scalar phase factor -ya(V E(p, a), t) (which we do not specify in detail here) is similar as in Dollard's classical construction of modified wave operators for Coulomb scattering, [7]. While the limit a \ 0 of the one-electron states T u (p, o,) does not define vectors in the Fock spaces HP, the limit a \ 0 of the vectors (4.3) defines vectors in the physical Hilbert space Hei 0 Next, we sketch the main construction in [14]. In [14], a discretized (Riemann sum) version of (4.3) is used as the free comparison state, where the resolution of the discretization becomes arbitrarily fine, as

t -+oo. Let TnE) := 2'/', for some 0 < e << 1 (to be fixed appropriately). Let N(t)

E = U rj (t)

(4.4)

j=1

be a decomposition of E into disjoint cubic cells, where the number of cells N(t) is time-dependent, and given by

N(t) = (2- )3

,

for

T(e) < t < T,+1

.

(4.5)

T. CHEN AND J. FROHLICH

34

The analysis in [14] assumes that (4.6) > c > 0, aPE(p, a) = (mren(p, uniformly in a > 0. A key element of the construction in [14] is to render the infrared cutoff at time-dependent, with at converging to 0 at a prescribed rate, as t - oo. a))-1

Accordingly, let N(t)

E 'Oh,at,j(t,X)

h,at(t,x)

j=1

with

h,, j (t, X)

eitHt)

dp W(V7,

t)e-a-Pf)x

(4.8)

fF3(t) ex'Yot(Vi,VpE(p,at) t)

h(p)

e-itE(p,at) Qiu(p, at)

,

and

V :_ VPE(pj, at)

,

where pj is the center of the cell F3 (t). Here, Wt (Vj, t) is defined as the operator obtained after replacing VPE(p, at) by Vj in Wp,at (t), and yat (Vj, VpE(p, at), t) is a scalar phase factor. The main result of [14], formulated for the model of non-relativistic QED studied here, can be stated as follows. Let

at -

t-0

(4.10)

,

for ,Q > 1 sufficiently large, and N(t), T(,E) as in (4.5), with e sufficiently small. Then the limit (out)

z/>h

I' ,,Ph,at (t) = S - lim t-00

(4.11)

exists in the one-particle Hilbert space l = ?{et ®1i, for the model of non-relativistic QED defined in Section 2. A similar result holds for t -. -oo, yielding a state phi"`) The vectors 0(in/out) are infraparticle scattering states.

The strategy of the proof of [14] comprises two main steps, which can be sketched as follows.

4.1. Step 1: Control of the Norm. This step consists in proving that the norm IIVGh,at(t)IIN is uniformly bounded in t, and that, in fact, lim

t-*oo

4'h,at (t)

x = IIhh2 .

(4.12)

Introducing the matrix elements Mi,7 (t)

\

//11

(Wh,at,i(t) , Wh,at,i (t)1

,

(4.13)

one easily sees that the sum over diagonal terms, i = j, yields the right hand side of (4.12), in the limit t -+ oc. The off-diagonal matrix elements are shown to satisfy I Mi,; (t) I

< c(t)

,

i#jl

(4.14)

INFRARED REPRESENTATIONS IN NON-RELATIVISTIC QED

35

where

c(t) N(t)2 \, 0

(4.15)

,

as t -> oo, so that lim t-i00

Mi,j (t)

(4.16)

0

i0i Since the centers of the cells Fj(t) label distinct asymptotic velocities of infraparticle states, this result implies that, asymptotically, the latter become mutually orthogonal, for i # j. One uses here dispersive estimates for the free infraparticle propagation, which are derived from the uniform bounds on 8P2E(p, a), for o > 0. For further details, see [14].

4.2. Step 2: Strong Convergence. In this step, one proves that {VVh,at (t)}t defines a Cauchy sequence in the one-particle Hilbert space 7-l. To this end, let t2 > tj >> 1. The main result of [14] is an estimate of the form 11 Y h,at2 (t2) - Y'h,,,, (tI)

7{

< tl 6

,

6>0.

(4.17)

The proof in [14] is organized as follows. Let 4'h,at,r(t') (s) denote the vector obtained from z/'h,Qt, (t') by first replacing at, ---> at and then t' -> s, while keeping the cell decomposition

F(t') := { Fj(t') }N(i)

(4.18)

corresponding to time t' fixed. Assuming t2 > tI >> 1, the left hand side of (4.17) is estimated by II Oh,at2,r(t2)(t2) - 0h,at1,r(tl)(t1) 1 1x < (I)

+ (II) + (III)

(4.19)

with the following definit ions. The term

(I)

I

Y' h,a12,r(t2)(t2) - Oh,at2,r(tl)(t2) I

l

rt

(4.20)

is the error made by replacing r(t2) by the coarser cell decomposition r(ti) in Oh,at2,r(t2)(t2), while keeping the infrared cutoff and the argument t2 fixed. One can control (I) similarly as the off-diagonal terms in (4.14). The term (II)

(4.21)

Oh ,at2,r(t1)(t2) - 4'h,atl,F(t1)(t2) rt

is the error made by subsequently changing the infrared cutoff from ate to at1 inOh,at2,r(t1)(t2). It admits a bound that involves a positive power

of at, = ti The term (III) :_

Wh,atl,r(ti)(t2) - 'Nh,atl,r(ti)(ti)

(4.22)

R (t2) replaced by Oh,atl ,r(tl) (t2) . is the left hand side of (4.17) with Oh"t2 ,r(t2) I

To bound (III), one applies Cook's argument to "/'

f t2

4Oh,at1,.71 (t2) - Y'h,at1,.71 (tl) = J

tl

CAS 19.Oh,atl,j1 (S) .

(4.23)

T. CHEN AND J. FROHLICH

36

This is the most involved part of the analysis, and the integrand on the right hand side of (4.23) must be subdivided into many different terms for which one can either prove rapid decay in s or (asymptotically) precise cancellations.

4.3. Modifications of [14] for QED. Most of the constructions in [14] can be adopted directly to yield the corresponding ones in non-relativistic QED. The following minor modifications are necessary.

The infrared regularization is implemented by a sharp cutoff x(jkj > a) in [14]. It must be replaced by an infrared regularization k(k) which

is zero at kj = 0, but non-zero in an open neighborhood of Jkl = 0. Implementing this modification in [14] (invoking results of [4, 5], instead of [9, 13]) is straightforward. The dressing transformations in [14] are slightly different from the ones used in non-relativistic QED. In [14], the integral kernel corresponding to vp,,,a(k) has the form 1 Vr- x(a < kj < 1) (4 . 24) kH - k VpE(p, a) kjz while, here, there is an additional factor VPE(p, a) EA(k); see (3.12). This does not lead to any non-trivial changes of the considerations in [14]. However, some other modifications are less straightforward, due to the more complicated interaction term of non-relativistic QED. In the application of Cook's method, there is a derivative (eisx(at)Wat (Vj s)e-iSH(at))

a',

(4.25)

/l

which contains a term of the form iei5H(at) [H(at)

- Hf Wat (Vi

(4.26)

s)]e-isH(ot)

(we recall that the interaction term in H(a) depends on x). Due to the linear coupling in Nelson's model, the above commutator is given by

[H(at) - Hf Wat (V j, s)] = W, (Vj, s)0at,v, (x, s) ,

,

(4.27)

where Oat,v, (x, s) is a scalar function that has rapid decay in s. For QED, O,t,vj (x, s) is replaced by an operator linear in VPH(p, at) (for total momentum p). The modifications arising here are technically somewhat demanding and involve an application of the uniform bounds on the renormalized electron mass. A more detailed analysis of scattering theory along the lines of [14] would be appropriate.

5. Proofs of Theorems 3.1 and 3.2 Our proofs closely follow [9], where the statements of Theorems 3.1 and 3.2 were established for Nelson's model.

In our proofs of Theorem 3.1 and part 2 of Theorem 3.2, the first step is to employ the usual "pull-through formula," which yields an explicit expression for aa(k)qlv,(p, a) in terms of q/,, (p, a). However, this is not the end of the story, in contrast to [9], where the result corresponding to Theorem 3.2 for Nelson's model was established. In non-relativistic QED, the different coupling structure

INFRARED REPRESENTATIONS IN NON-RELATIVISTIC QED

37

in the Hamiltonian H(p, a) poses considerable difficulties. Our method involves application of the uniform bounds (2.29) on the renormalized electron mass, which has only recently become available. Our main technical result is formulated in the following proposition. PROPOSITION 5.1. Under the hypotheses of Theorem 2.1, the vector a,\ (k)1P.(p,o,) can be decomposed into

aa(k)W,(p,a) = Ii(p,a;k,A) + c2(p,a;k,A),

(5.1)

where

and

11"D 2(Ao,;k,A)II C

cZ-a- K° Ii

I)

for a constant c that is independent of or and a. The uniform bound on the renormalized electron mass (2.29) enters the estimate for the vector'D2(p, a; k) . (We recall that icQ denotes the cutoff function in (2.16).)

5.1. Proof of Theorem 3.1. The statement of Theorem 3.1 is an immediate consequence of Proposition 5.1.

5.2. Proof of Theorem 3.2, Part 1. For the existence of a convergent subsequence, we refer to [9]. The proof comprises the following main steps.

Let K,:= {k E R3 I Jkl > p} for 0 < p < 1, and let 3p denote the Fock space over the one-photon Hilbert space L2(Kp) ® C2. Let 2t, denote the C*-algebra of bounded operators on 3p. One first establishes the existence of an operator Cp affiliated with 2tP which has a compact resolvent on 3p, and which satisfies

wp,a(CP) < M(P) < 00

(5.4)

uniformly in or > 0. For instance, the operator CP

a-+,-

kI?p

dkaa(k) [ - Ak + Ikl2] a,\ (k)

(5.5)

has these properties in the present case (see also [9, 10] and [11]). It follows that {wp,, }1>0 C 2tP is norm compact; see [11]. The dual 2LP of 2tp P

is a Banach space, because 2tp is a von Neumann algebra. Hence, for any sequence {aj converging to zero, there exists a subsequence {aj,0 converging to zero such that {wp,vj 0 converges to a normal state w(P) on 2tP.

Choosing p,,, = n for n c N, we get, by Cantor's diagonal procedure, a subsequence {aj, }-0 converging to 0 such that {wp,aj }1_0 converges on 2t , for all n < oo. Hence, fop,,,, }i_0 converges on Vn 2t , and thus on 2t, to a state wp on is a normal state. 2t*. wp I %P

T. CHEN AND J. FROHLICH

38

5.3. Proof of Theorem 3.2, Part 2. This is an immediate consequence of Proposition 5.1. Indeed, we have that 2

(,DI(p,a;k,A), 4,i(p,a;k,A)),

(T (p,a), 411(p,a;k,A))

(5.6)

since 4)1(p, a; k, A) is a scalar multiple of Wz,(p, a), and IWv,(p, a)II = 1. Therefore,

T,l(p,a), a* (k) as(k)W (p, a)) (-,b I(p,a;k,A), 41I(p,a;k,A))+Pi(p,a;k,A) 2

(Wv,(p,a), 4)1(p,a;k,A))

+PI(p,a;k,A) z

(T,, (p, a) , as (k)W ti, (p, a))

+ pi (p, a; k, A) - P2 (p, a; k, A)

(5.7)

where

PI (p,a;k A)

,DI(p,a;k,A))

= ,D1(p,a;k,A),

(5.8)

((D2(p,a;k, A), 4)2(p,a;k,A))

+ and

(p, a) , ' i (p, a; k, A)) (4)2 (p, a; k, A)

P2 (p, a; k, A)

T,, (p, o,))

,

+ (`Wu(p,a), 4' 2(p,a;k,A))(4i(p,a;k,A), 'Pu(p'a)) 2

(5.9)

Clearly, PI (p, a; k, A) I

,

I P2 (p, o k, A) l C 2II (D1(p, a; k, A) II II -1)2 (p, a; k, A) II

+ IID2(p,a;k,A)II2

+ c,

< calVpE(p,a)I

c

(Ikl) Ik12

(5.10)

IkI

This proves the claim.

5.4. Proof of Theorem 3.2, Part 3. We sketch the proof, and refer to Lemma 3.1 in [9] for details (see also [3, 6, 8]). We consider the coherent * automorphisms a,,, (A) = Wp,a AWP,a , A E 2l

,

(5.11)

where

Wp,a = exp [i

HA(vp,a,A)]

(5.12)

see (3.14), and IIA (f) = i (aa (f) - a (f )). In the limit or \ 0, the states

pp,

Wp(ap,a('))

(5.13)

converge to

Np = w (ap(')),

(5.14)

where ap(A) = n - limaNo ap a(A), for A E 2(; see [9].

Next, one proves that the representation ir1 = 7rp o ap admits a positive, selfadjoint number operator. This implies that 7rµp is quasi-equivalent to the Fock

INFRARED REPRESENTATIONS IN NON-RELATIVISTIC QED

representation, for 0 < IpI < 3,

39

To this end, we define the local number

[6].

operators

NP := E f A=+,-

dk a(k) aa(k)

for p > 0

,

(5.15)

,

kl>P

where exp[itNP] E 9A p. Let HPP denote the Hilbert space and 1,P E 'H,,,, the cyclic

vector corresponding to yp by GNS construction. One can show that 7rN,P(exp[itNP])7rP,,,(A)52N,P converges strongly, as p

0,

for all A E VP>02(P, and all t E R. The limit of 7r, (exp[itNP]), as p', 0, t E ][8, defines a strongly continuous unitary group on fµ,. Its generator defines a positive, selfadjoint number operator on 7-IP,P. Since A E VP>0 21P7 there is some let

f

NP"P

> 0 such that A E 2tP. Let p' < p < p, and dk a- (k) a, (k)

(5.16)

.

'
Then, II7rµP

- eitNP, )A)

((eitNP

QOP 112 µp(A*eitNP,,PA)

= 2µp(A*A) -

-

YP(A*e-itNP,,PA)

.

(5.17)

Using that 7µP (1 -

eitNP, P)

_-i

/pt

J0

7r" (eisNP,

P NP',P)

(5.18)

a straightforward calculation shows that !lp(A*e'tNP,,PA)I

Iµp(A*A) -

< tI11A1121J P'
ItI IIAII2 Ip - P I z

dk[wp(aa(k)aa(k)) - Iwp(aa(k))12 J (5.19)

,

which tends to zero as p \, 0. In the last step, we used (3.10). Our results imply that 7rp is quasi-equivalent to the coherent representation corresponding to (3.14) by the GNS construction.

5.5. Proof of Theorem 3.2, Part 4. Theorem 2.1 implies that VPE(p, a) : 0, for IpI < 3. Thus if p

0 if and only if p

0, (3.1) in Theorem 3.1 implies that

wp,,(Nf) _ (` .(p,a), Nf`I'u(p,a)) >

(- ca + c'aI vpE(p, a) I 2 log 1) Q +

(5.20)

which diverges to oo as or \ 0. Hence, (3.16) follows. However, if p = 0, one obtains (3.15) from VpE(p, a) = 0. The local Fock properties of wp are derived from the following considerations. Using Proposition 5.1, it is easy to see that WP' 01 (NP)

<

dk [ 4,I (p, a; k, a) 112 + I4b2 (p, a; k, \) 11 2

2 ,

< C(p) ,

IkI>P

(5.21)

T. CHEN AND J. FROHLICH

40

uniformly in a > 0. By similar considerations as in the proof of part 3 of Theorem 3.2, one concludes that the representation of 21, corresponding to wp by GNS construction is quasi-equivalent to the Fock representation for every p > 0. Let B c R3 denote a bounded region in physical x-space, and let 2t(B) denote the corresponding local algebra. Then, the restriction of wp to 2t(B) defines a GNS representation of 2t(B) which is quasi-equivalent to the Fock representation. This can be shown by a straightforward adaptation of results in [11] to the present model.

6. Proof of Proposition 5.1 It remains to prove Proposition 5.1, our key analytical result in this paper. To this end, we first derive the following representation of a,\ (k) IV "(p, a).

LEMMA 6.1. Assume that 0 < Jkl < 1, pj < 3, and a < a0 (where a0 denotes the same constant as in Theorem 2.1). Let E(p, a) denote the ground state eigenvalue of H(p, a), and let T,,(p, o,) be an eigenvector in the corresponding twodimensional eigenspace.

Then the operator H(p - k) + jk - E(p, o,) is invertible, and 1

aA(k)W v,(p, a) =

H(p-k,o-)+kj -E(p,v) [,/-a-

ikl) EA(k) V H(p, a)

+ V/0i

i

T (k AEA(k))]T"(p,a) 12

In the scalar case, E(p, o-) is a simple eigenvalue, and the magnetic term (proportional to T) is absent.

PROOF. We recall the definition of the fiber Hamiltonian

H(p, a) = I (p - Pf - /A)2 + VG-T Ba + Hf .

(6.2)

The "pull-through formula" says that

a,\ (k)H(p,a) _ (2(p - Pf - k - /A)2 + /T - B, + Hf + Ik1)aa(k)

- vla + Va I

EA(k) . (p - Pf - FAQ) I

I)

T (ik A E,\ (k)) ,

(6.3)

where T = (TI, T2, T3) is the vector of Pauli matrices. We observe that

V H(p, a) = p - Pf - iAQ ,

(6.4)

EA(k) - VpH(p - k, a) = EA(k) . VpH(p, a)

(6.5)

and that

INFRARED REPRESENTATIONS IN NON-RELATIVISTIC QED

since EA(k) k

41

0, by the Coulomb gauge condition. Hence

aA(k)E(p, a)I'ti,(p, a)

aA(k)H(p, a)xP,,,(p, a)

[(H(p-k,a)

+ Ik1)a,\ (k)

kl) EA(k).V H(p,a)

+V"a-

+V"a

T

1

(6.6)

(ik A EA(k))]'1'. (p, a)

,

I

so that [H(p - k, Q) + kl - E(p, a)] aa(k)'1',,(p, a) [V/a- r

(6.7)

ikl) T (ik A Ea(k))]W"(p, a)

ikl) EA(k) - V H(p, a) +

2

Furthermore, the bounds

H(p - k, o,) + kl - E(p, a) > E(p - k, a) + kl - E(p, o,) >

101

(6.8)

follow from (6.18) below. Hence, H(p - k, a) + l k - E(p, o,) (see left side of (6.7)) is invertible, for any 0 < k j < 1 and pl < 3 We conclude that 1

aa(k)'1',,(p,a) =

H(p - k, a) + kl - E(p, or) [vr,-

kik EA(k) . V H(p, or)

+ 12

(6.9)

T (ik A EA(k))i ]1'. (p, a)

,

as claimed. PROOF OF PROPOSITION 5.1. We note that

(V H)(p, a)q1u(p, a) = V (H(p, a)q,u(p, a)) - H(p, a)Vpq1,,(p, a) V (E(p, a)q1.(p, a)) (V E)(p, a)'1'.(p, a) - (H(p, a) - E(p, a))VpT.(p, a) From (6.1), we get aa(k)T (p, a)

=

(6.10)

(6.11)

(I) + (II) ,

where

_Vla_

i kl)

(I)

(II) _

(Ikl) kl2

(EA(k) . V E(p, a))

H(p - k, Q) + k - E(p, a)

W" (p, a)

1

H(p-k,a)+lkl -E(p,a) [(H(p, a) - E(p, a))EA(k) . Vpql.(p, a)

- T (ik A e (k))IV,,,(p, g)]

.

(6.12)

T. CHEN AND J. FROHLICH

42

Let us first bound (II). To this end, we first prove that for 0 < IkI < 1 and lpI < 3 <3.

(H(p, a) - E(p, a)) H(p - k, a) +I IkI - E(p, a)

(6.13)

op

We note that

H(p - k, a) = H(p,a)+2 - k.VPH(p,a)

(6.14)

so that

H(p-k,a) - E(p,a) = H(p,a) - E(p,a) -

H(p,a) +

2

z

>

H(p, a)

>

(1

- E(p, a) -

-

2

(V pH(p, a))2 + 2

2S

3IkD)(H(p, a) - E(p,

2 - 4I kI (6.15)

a)) +

E(p,a)),

+21kI(Hf using the Schwarz inequality with S =

3IkI

< 3 From

IBQI < c 1 + Hf the operator on the last line is bounded by

3IkHf - VG-c 1+Hf - E(p,a)] using E(p, a)

2

(6.16)

,

3IkI [2

>

(3)z + c'v] ,

(6.17)

2z + ca for Ipl < 1. Therefore, 2

H(p - k, a) - E(p, a) + I k I

1 - 31 k1)(H(p, a) - E(p, a)) + 2

>

z

+Ikl(1 >

4

2

(3) -

IkI

1

3(H(p,a)-E(p,a)) +

(6.18)

10 '

for Ikl < 1. This implies (6.13). It is then easy to see that IV I)11

<

c ra- (I

l) [ (H (p, a) - E(p, a)) H(p - k, a) + l kl - E(p, a) 1

z op

2

H(p- k, o,) + 1kl - E(p,a) op

(H(p,a)-E(p,a))zVPIP. (p,a)) 1

n

+ I I H(p - k, a) + I k I - E(p, a)

C a(Ikl) [ Iklz

cVG '.a(Ik IkI

1

1

IkII

mren(p,a)

Iik A eA(k)1] op

-1] 2

1

+

(6.19)

INFRARED REPRESENTATIONS IN NON-RELATIVISTIC QED

43

where

mren(p) a) _ [1 - 2(VpWu(p,a), (H(p,a) - E(p,a))V

(6.20)

Wu(p,o-))]

is the renormalized electron mass. We recall from (2.29) that mren(p, a) -1l < CO, for pl < 3, uniformly in a > 0. Next, we discuss the term (I). We use the resolvent identity and (6.14) for

H(p-k,a)+lkl -E(p,a)

(6.21)

H(p,a) -E(p,a)+lkl+ z2 H(p - k, a) + kl - E(p, a)

k VpH(p, a)

1

H(p,a)-E(p,a)+lkl+

22

Accordingly,

(I)

_ (Il) + (12)

(6.22)

,

where (II)

(E,\ (k) .V E(p,a))

VI-a-

=

lklz

Ifi-I

na(I kl)

1

lkl2

lkl+ 2

1

22

H(p,a)-E(p, a) +lkl+-

T.(p,a) (6.23)

2 (ea(k) . V E(p, a))IF.(p, a) .

We note that the L2-norm of this term diverges logarithmically in the limit a \, 0. Moreover, (12)

=

ikl)(ea(k)-V (k - V pH(p, a)) na(Ik1) lkl z

E(p,a))H(p-k, a)+kl -E(p,a) 1

H(p,a)-E(p,a)+kj+ 22

W (p, a)

1

lkl

+

(--A(k) -VpE(p,a)) 2

21

H(p - k, a) + l kl - E(p, a) (k pH(p, a))`yu(p, a) (121) + (122)

(6.24)

with (12I)

klz

1

lkl +k2

(ea(k)-V E(p,a))

1

H(p-k,a)+lkl -E(p,a)(k VpE(p>a))u(p,a) k . V E(p, a) . (1)

lkl + z

(6.25)

T. CHEN AND J. FROHLICH

44

and

kl + k2 (EA(k) - VPE(p, 5))

Va

(122)

(6.26)

T

H(p-k,a)+Ikl -E(p,a) (H (n a) - E(p a))k . VPT U(p a) ,

,

We find that 11(122)11

<_

3,la-'O'

IkI

+ k2 IkllopE(p, a)I 2 1

1

2

H(p-k,o)+IkI -E(p,a)Ilop (H (p, a) - E(p, a)) H(p-k,o)+Ikl

(H (p, )

-E

2

-E(p,a)

op

T. -12

1

mren(p, a)

IkI

< ca K,(Ikl) IVpE(p,a)I

(6.27)

IkI

using (6.13).

Hence, solving for (I) (recalling that (6.25) is a multiple of (I)),

(1) _ 1 -

k - VPE(p, a) i

IkI+ 2

[(II) + (122)]

(6.28)

,

where k - VpE(p, a) I < IkI Ipl (1 + ca) <

(6.29)

121

for Ipl < s and a sufficiently small, see (2.27). Noting that l_

k.VPE(p,a)+ IkI+22

ia(Ikl) EA(k).V E(p,a) VrC-e

k12

Ikl

cvIa-

ii

l)

41..(p,0')

II IF. (p, a) II

,

(6.30)

we find that

a,\ (k)` 0(p,a) = 4)I(p,a;k,A) + 4)2(p,a;k,A),

(6.31)

where

,DI(p,a;k,A) = -\EA(k).V E(p,U) iki k l)

IkI

-k

1V

E(p

(p,a) (6.32)

and

II-'D2(p,a;k,A)II

This establishes Proposition 5.1.

<-

cv/a-K' (Ikl) IkI

(6.33)

INFRARED REPRESENTATIONS IN NON-RELATIVISTIC QED

45

Acknowledgements. We are grateful to A. Pizzo for some useful discussions of the work in [14]. T. C. was supported by NSF Grant DMS-0524909.

References [I] V. Bach, T. Chen, J. Frohlich, and I. M. Sigal, Smooth Feshbach map and operator-theoretic

renormalization group methods, J. Funct. Anal. 203 (1), 44-92 (2003). [2]

V. Bach, T. Chen, J. Frohlich, and I. M. Sigal, The renormalized electron mass in nonrelativistic Quantum Electrodynamics, to appear in J. Funct. Anal.

http://arxiv.org/abs/math-ph/0507043 [3]

[4]

0. Bratteli and D. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vols. I and II, Springer-Verlag, New York (1981). T. Chen, Operator-theoretic infrared renormalization and construction of dressed 1-particle states in non-relativistic QED, ETH Dissertation 14203 (2001).

http://xxx.arxiv.org/abs/math-ph/0108021 T. Chen, Infrared renormalization in non-relativistic QED and scaling criticality, submitted.

http://xxx.arxiv.org/abs/math-ph/0601010 G.-F. Dell'Antonio, S. Doplicher, and D. Ruelle, A theorem on canonical commutation and anticommutation relations, Commun. Math. Phys. 2, 223-230 (1966). J. D. Dollard, Asymptotic convergence and the Coulomb interaction, J. Math. Phys. 5, 729738 (1964).

J.-P. Eckmann and J. Frohlich, Unitary equivalence of local algebras in the quasifree representation, Ann. Inst. Henri Poincare 20, 201-209 (1974).

J. Frohlich, On the infrared problem in a model of scalar electrons and massless, scalar bosons, Ann. Inst. Henri Poincare, Section Physique Theorique 19, 1-103 (1973). J. Frohlich, Existence of dressed one-electron states in a class of persistent models,

Fortschritte der Physik 22, 159-198 (1974). J. Glimm and A. Jaffe, The .\(04)2 quantum field theory without cutoffs, Acta Math. 125, 203-267 (1970).

F. Hiroshima and H. Spohn, Ground state degeneracy of the Pauli-Fierz model with spin, Adv. Theor. Math. Phys. 5, 1091-1104 (2001). A. Pizzo, One-particle (improper) states in Nelson's massless model, Ann. Henri Poincare 4, 439-486 (2003).

A. Pizzo, Scattering of an infraparticle: The one particle sector in Nelson's massless model, Ann. Henri Poincare 6, 553-606 (2005).

M. Reed and B. Simon, Methods of Mathematical Physics. II. Fourier Analysis, SelfAdjointness, Academic Press, New York (1975). H. Spohn, Dynamics of Charged Particles and Their Radiation Field, Cambridge University Press, Cambridge (2004).

(Thomas Chen) DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY, 807 FINE HALL, WASHINGTON ROAD, PRINCETON, NJ 08544, U.S.A.

E-mail address: [email protected] (Jurg Frohlich) INSTITUTE FOR THEORETICAL PHYSICS, ETH HONGGERBERG, 8093 ZURICH, SWITZERLAND

E-mail address: [email protected]

Proceedings of Symposia in Pure Mathematics Volume 76.1, 2007

Quantum Spin Systems After DLS 1978 Bruno Nachtergaele Dedicated to Barry Simon on the occasion of his sixtieth birthday ABSTRACT. In their 1978 paper, Dyson, Lieb, and Simon (DLS) proved the existence of Neel order at positive temperature for the spin-S Heisenberg antiferromagnet on the d-dimensional hypercubic lattice when either S > 1 and d _> 3

or S =

and d is sufficiently large. This was the first proof of spontaneous breakingz of a continuous symmetry in a quantum model at finite temperature. Since then the ideas of DLS have been extended and adapted to a variety of other problems. In this paper I will present an overview of the most important developments in the study of the Heisenberg model and related quantum lattice systems since 1978, including but not restricted to those directly related to the paper by DLS.

CONTENTS 1.

2. 3. 4. 5. 6. 7. 8. 9.

Introduction General Properties Neel Order and Other Applications of Reflection Positivity The Classical Limit and Beyond Spin Chains and Haldane's Conjecture Perturbation Theory Estimates of the Spectral Gap Ordering of Energy Levels Other Topics

References

2000 Mathematics Subject Classification. 82B10, 82B20, 82B26. Key words and phrases. Heisenberg model, reflection positivity, symmetry breaking, quantum spin systems, Neel order, spectral gap, Haldane's conjecture. ©2007 Bruno Nachtergaele. This article may be reproduced in its entirety for non-commercial purposes. 47

48

B. NACHTERGAELE

1. Introduction The purpose of this article is to review a number of important developments in the mathematical physics of quantum spin systems that have occurred since Barry Simon first got involved with the subject. A comprehensive account of such developments clearly would not fit in the space allotted here. Therefore, I have selected seven topics that either have a relation to the seminal paper by Dyson, Lieb, and Simon [19] or that for some other reason I believe interesting and important enough to deserve to be discussed in some detail here. In Section 9, I list a number of works I regretfully had to leave out. There is one development that stands out by its absence: a proof of long-range order in the Heisenberg ferromagnet at positive temperature in dimensions > 3 has eluded the valiant attempts of many. One can trace some of these attempts in the literature [14, 85, 22], and the authors of [19] also comment on an erroneous proof they thought they had, but certainly most failed approaches have not left a written record. It is not a hopeless problem, in my opinion. It is much like other open phase transition problems such as hard-core bosons at finite density and temperature in I[83, or the XY-model in a uniform external field. Proving long-range order in the Heisenberg ferromagnet should be simpler than the problem of proving ferromagnetic long-range order in an itinerant electron model such as the Hubbard model [83, 31]. There are of course many other open problems in quantum spin systems. I will mention some of them in the following sections. And many interesting problems have been solved in the almost three decades since DLS1978. The ferromagnetic phase transition is a nagging reminder that we lack the mathematical arguments to tackle some of the most basic and clearly formulated problems of equilibrium statistical mechanics. Other problems in mathematical physics suffer from being physically not well-understood, from being mathematically ill-posed, from being too complex, but not the Heisenberg ferromagnet. In the years following DLS, Simon made several other contributions to statistical mechanics that will stand the test of time: elegant estimates, useful inequalities, and a great book [5, 76, 78, 80, 77, 79]. Is there a chance Barry will once more turn his attention to statistical mechanics? The few years in the late seventies and the early eighties he worked in statistical mechanics were a spectacular success. Maybe I can entice him with the following spectral problem for the Heisenberg ferromagnet which lies somewhere in the middle between his more recent interests and full-fledged statistical mechanics. The problem is to prove that the Hamiltonian of the spina isotropic Heisenberg ferromagnet on Zd in the GNS representation of one of its translation invariant

pure ground states (which are the fully polarized states) has purely absolutely continuous spectrum except for the ground state eigenvalue 0. In one dimension this result follows from the work of Babbitt and Thomas [8]. So the open problem is for d > 2. To formulate the problem precisely, we introduce the relevant Hilbert space as 12(Fd), where 1d is the set of finite subsets of Zd . Let {S1X I X E Fd} be the canonical orthonormal basis of R. One should think of 11x as the state vector with all spins pointing up except for those finitely many that are located at x c X, which are down. The Hamiltonian H is then the unique selfadjoint operator with

QUANTUM SPIN SYSTEMS AFTER DLS 1978

49

domain containing the canonical orthormal basis and such that H110 = 0,

HQ x =

E

(9x - 1XU{y}\{x}), for all 0:/' X E Td.

xEX,yVX,1x-y1=1

Note that, up to an overall sign, H is also the generator of the symmetric simple exclusion process on Zd. Another expression for H is given in (2.1). It is easy to see that H is non-negative definite, and from the definition it is obvious that 0 is an eigenvalue. It is not hard to prove that the eigenvalue 0 is simple. We conjecture that the positive real line consists of purely absolutely continuous spectrum in all dimensions. In particular, there are no strictly positive eigenvalues.

2. General Properties The mathematical setting for studying quantum spin systems rigorously, as described in textbooks such as [75], [12], and [79], was developed in the late sixties starting with Robinson [74]. This is not to say that there are no rigorous results about quantum spin models before that date. In particular, Lieb and coworkers wrote several landmark papers that continue to be important today (see, e.g., [52, 49]). In this general formalism it was possible to obtain several fundamental theorems for large classes of quantum spin models. An early example is the uniqueness of the equilibrium state at any finite temperature for one-dimensional models with short-range interactions by Araki [7]. In this section we first briefly review the main elements of the general setup and then discuss some recent results in this context. For the purpose of our discussion, a quantum spin is any quantum system with a finite-dimensional Hilbert space of states. A quantum spin system consists of a finite or infinite number of spins, which we will label by the elements of a set V. When V is an infinite set, typically corresponding to the vertices of a lattice or a graph, one often considers families of quantum spin systems, labeled by the finite subsets X C V. Certain properties are easily stated for infinite sets V directly, but to define the dynamics and specific models we will first assume that V is finite. In this case, the Hilbert space of states is

xV = ®Cnx xEV

where the dimensions nx > 2 are related to the magnitude of the spins by nx = 2sx+1, and sx E { 2,1, 2, . J. For each spin, the basic observables are the complex n x n matrices, which we will denote by Mn. The algebra of observables for the system is then AV = ® Mn. = X3(7-ly) .

xEV

Given a Hamiltonian, a selfadjoint observable Hy = Hv E AV, the Heisenberg dynamics of the system is defined as follows: for any t E R, an automorphism at is defined on A by the formula

at (A) = U7(t)AUv(t)

where Uv(t) = e-Ztx One of the most important examples of a quantum spin system is the Heisenberg model. In general, this model is defined on a graph (V, E) which consists of a set

of vertices V and a set of edges E. E is a set of pairs of vertices denoted by

B. NACHTERGAELE

50

e = (xy) for x, y c V. Let S., i = 1, 2, 3, denote the standard spin sx matrices associated with the vertex x, and for each edge e = (xy), let Jxy E R be a coupling constant corresponding to e. The Heisenberg Hamiltonian (also called the XXX Hamiltonian) is then given by

HV= - L (xy)EE

where Sx denotes the vector with components Sx, SS, Sx3. The most commonly studied models are those defined on a lattice, such as Zd, with translation invariance. For such Hamiltonians, the magnitude of the spins is usually constant, i.e., sx = s,

the edges are the pairs (xy) such that Ix - yj = 1, and Jxy = J. Depending of the sign of J, the Heisenberg model is said to be ferromagnetic (J > 0) or the antiferromagnetic (J < 0). For extended systems, i.e., those corresponding to sets V of infinite cardinality, more care is needed in defining the quantities mentioned above. Hamiltonians are introduced as a sum of local terms described by an interaction, a map 1 from the

set of finite subsets of V to AV, with the property that for each finite X C V, 4)(X) E Ax and 4)(X) = -D(X)*. Given an interaction 4), the Hamiltonian is defined by

HV=E'D (X) xcv Infinite systems are most often analyzed by considering families of finite systems, indexed by the subsets of V, and taking the appropriate limits. For example, the C*-algebra of observables, A, is defined to be the norm completion of the union of the local observable algebras Uxcv Ax. To express the decay of interactions and correlation functions, we need a distance function on V. Let V be equipped with a metric d. For typical examples, V will be a graph and d will be chosen as the graph distance: d(x, y) is the length

of the shortest path (least number of edges) connecting x and y. The diameter, D(X), of a finite subset X C V is D(X) = max{d(x, y) I x, y c X}

.

In order for the finite-volume dynamics to converge to a strongly continuous one-parameter group of automorphisms on A, one needs to impose a decay condition

on the interaction. For the sake of brevity, we will merely introduce the norm on

the interactions that will later appear in the statement of the results described in detail here. For weaker conditions which ensure existence of the dynamics, see [12, 75, 79, 57]. We will assume that the dimensions nx are bounded:

N = sup nx < oo, xEV

and that there exists a A > 0 such that the following quantity is finite: II

IIA := sup > IXI II-D(X)II N2,Xi

eaD(X) < oo .

xEV X3x

For translation invariant systems defined on a lattice with an interaction 4 such that II a < oo for some A > 0, Lieb and Robinson [51] proved a quasi-locality property of the dynamics, in the sense that, up to exponentially small corrections, II

QUANTUM SPIN SYSTEMS AFTER DLS 1978

51

there is a finite speed of propagation. More precisely, they obtained an estimate for commutators of the form

[at(A), B] where t E R, A E Ax, B E Ay, and X, Y C V. Clearly, such commutators vanish if t = 0 and X fl Y = 0. Quasi-locality, or finite group-velocity, as the property is also called, means that the commutator remains small up to a time proportional to the distance between X and Y. We will now formulate an extension of the LiebRobinson result to systems without translation invariance or even an underlying lattice structure. It will be useful to consider the following quantity

CB(x,t)

Sup II [at(A), B] II

AEA

IIAII

for x E V, t c R, B E AV. The basic result is the following theorem [63]. THEOREM 2.1. For x c V, t E IR, and B E AV, we have the bound CB(x t) < e21t1 II4'IIACB(x, 0) + E

e-;kd(x,y) (e2Itl IlIDlla

_ 1)

CB (Y, 0)

yEV:y#x

Our proof avoids the use of the Fourier transform which seemed essential in the work by Lieb and Robinson and appeared to be the main obstacle to generalizing the result to non-lattice (V, d). If the supports of A and B overlap, then the trivial bound II [Tt (A), B] II < 211 All II B II is better. Observe that for B E Ay, one has that CB (y, 0) < 211 B 11 XY (y),

where xY is the characteristic function of Y, and therefore if x V Y, then one obtains for any A E Ax a bound of the form II [Tt(A),B]II <_ 2IYI IIAIIIIBII

(e2ltl 11 4111,\

_ 1) e-Ad(x,Y)

where

d(x, Y) = min{d(x, y) I y E Y} . Moreover, for general local observables A E Ax, one may estmate II[Tt(A),BI II <_ N21XIIIAII E CB(x,t), xEX

in which case, Theorem 2.1 provides a related bound. Theorem 2.1 has been further extended to provide bounds in the case of longrange interactions by Hastings and Koma [35]. The next general result relies on the Lieb-Robinson bound to prove that a nonvanishing spectral gap above the ground state implies exponential decay of spatial correlations in the ground state. This result can be regarded as a non-relativistic analogue of the exponential clustering theorem in relativistic quantum field theory [23]. The idea that a Lieb-Robinson bound can be used as a replacement for strict

locality in the relativistic context is natural but, as far as I know, it was used in the literature for the first time only recently by Hastings in [34]. In the physics literature, the term massive ground state implies two properties:

a spectral gap above the ground state energy and exponential decay of spatial correlations. It has long been believed that the first implies the second, and the next theorem proves that this is indeed the case. The converse, that exponential decay must be necessarily accompanied by a gap is not true in general. Exceptions

B. NACHTERGAELE

52

to the latter have been known for some time [61], and it is not hard to imagine that a spectral gap can close without affecting the ground state. For simplicity of the presentation, we will restrict ourselves to the case where we have a representation of the system (e.g., the GNS representation) in which the model has a unique ground state. This includes most cases with a spontaneously broken discrete symmetry. Specifically, we will assume that our system is represented on a Hilbert space 7-l, with a corresponding Hamiltonian H > 0, and that 1 E 7-l is, up to a phase, the unique normalized vector state for which HQ = 0. We say that the system has a spectral gap if there exists 8 > 0 such that spec(H) f1 (0, 8) = 0, and in this case, the spectral gap, y, is defined by y = sup{8 > 0 I spec(H) fl (0, 8) = 0} . Our theorem on exponential clustering derives a bound for ground state correlations which take the form (2.2)

(c2, ATzb(B)f)

where b > 0 and A and B are local observables. The case b = 0 is the standard (equal-time) correlation function. It is convenient to also assume a minimum site spacing among the vertices: inf d(x,y) =: a > 0. x,yEV

(2.3)

x:A y

We proved the following theorem in [63].

THEOREM 2.2 (Exponential Clustering). There exists p > 0 such that for any y E V and all A E Ate, B E Ay for which (1k, B1) = 0, and b sufficiently small, there is a constant c(A, B) such that x

(1, Aaib(B)1) < c(A, B)e

(1+4µ d y)

One can choose

yA

4II"DIIa+y

and the bound is valid for 0 < yb < 2pd(x, y).

() 25

The constant c(A, B), which can also be made explicit, depends only on the norms of A and B, (in its more general form) the size of their supports, and the system's minimum vertex spacing a. For b = 0, Theorem 2.2 may be restated as I (52, ABSZ) - (1, A1) (1, B1 I < c(A, B)e-µd(x'y).

(2.6)

Note that there is a trivial bound for large b > 0 (2.7) I(Il,Aaib(B)f)I 0 regime, the estimate (2.4) can be viewed as a perturbation of

(2.6). Often, the important observation is that the decay estimate (2.4) is uniform in the imaginary time ib, for b in some interval whose length, however, depends on d(x, y).

In a recent work, Hastings and Koma obtained an analogous result for models with long-range interactions [35]. It has been known for some time that the spontaneous breaking of a continuous

symmetry, such as the SU(2) rotation symmetry of the XXX Heisenberg model (2.1), precludes the existence of a spectral gap in the infinite volume ground state.

QUANTUM SPIN SYSTEMS AFTER DLS 1978

53

This is often called the Goldstone Theorem of statistical mechanics [44, 86]. Matsui [56] adapted the proof of the Goldstone theorem to show the absence of a spectral gap above the 111-interface ground states of the XXZ ferromagnet defined in (3.8)

in all dimensions d > 2, a result anticipated by Koma and Nachtergaele who had previously proved it for d = 2 [41]. In dimensions < 2, spontaneous continuous symmetry breaking cannot occur at positive temperatures. This is called the Mermin-Wagner-Hohenberg theorem, a general version of which was proved by Frohlich and Pfister [28], and for systems of fractal dimension d < 2 by Koma and Tasaki [43]. In the next section we review some results on existence of spontaneous symmetry breaking in quantum spin models at T = 0 and T > 0.

3. Neel Order and Other Applications of Reflection Positivity One of the main results in the 1978 paper by Dyson, Lieb, and Simon (DLS), and an extension by Neves and Perez [71] to include ground states, and refinements

by Kennedy, Lieb, and Shastry [38, 39], is the existence of long-range order in the isotropic spin-S antiferromagnet on hypercubic lattice Zd at sufficiently small positive temperatures in dimensions d > 3, and any spin magnitude S > 2, and also in the ground state (T = 0), if d > 2 and S > 1, or d = 3 and S = 2 DLS was a breakthrough based on another breakthrough, namely, the method of infrared bounds to prove continuous symmetry breaking developed for classical systems by Frohlich, Simon, and Spencer [30, 29]. The method was soon generalized to cover a wide variety of models [27, 26] that have a property called reflection positivity. There is no a priori reason why the infrared bound philosophy should require reflection positivity, a property not shared by the Heisenberg ferromagnet [81]. But in spite of serious attempts to find ways to prove infrared bounds for quantum spin systems not relying on reflection positivity, no such method has

been found to date, and we still do not have a proof of long-range order in the ferromagnetic Heisenberg model at positive temperature. For the XXZ model with Hamiltonian

Hv = -

1

(SlSy + S2Sy) + SISy V C 7Gd

(3.8)

an Ising-type phase transition was proved by Frohlich and Lieb [27], for d > 2 and sufficiently large A (since their proof assumed reflection positivity). Kennedy [36] showed how to prove the phase transition for all A > 1 without using reflection positivity but instead by relying on a version of the Peierls argument using a clever cluster expansion method. Thus it may appear that to make further progress, one should circumvent the reflection positivity property. This is certainly true in part. It is worth mentioning, however, that the use of reflection positivity in quantum lattice models pioneered in DLS has been successfully adapted to solve a number of other problems. Let us just mention a few interesting examples: dimerization in the Hubbard-Peierls model on the ring and the spin-Peierls problem [50], the flux-phase problem [48, 47], and singlet ground states of quantum lattice systems [46, 24, 53, 54]

54

B. NACHTERGAELE

4. The Classical Limit and Beyond In a very elegant paper, [45], Lieb proved the existence of the classical limit for a broad class of quantum spin systems for which the Hamiltonians can be considered

as a multilinear function, H, of the standard spin operators. The main result of Lieb's is then upper and lower bounds for the quantum partition function with the spin operators taken to be of magnitude S. To state the result, we define ZQ(0,S) = Tre-pH({Sx})

where the spin operators in the Hamiltonian are taken to be of magnitude S and the trace of the corresponding Hilbert space of dimension (2S + 1)IVI, and where V is fixed. The corresponding classical partition function is defined by Zc(,3) = f e-AH({sti}) TT d1x xEV

where d1 is the normalized invariant measure on the 2-sphere (unit vectors in R3). Then Lieb proved

Zc(0)CZQ(/3,S)
(4.9)

with c a universal constant. This result suffices to prove convergence of the mean free energy (assuming it exists for the classical system), as well as convergence of the expectation of averaged quantities (intensive observables). In [76] Simon generalizes Lieb's result by proving the same relations for systems of G-spins, where G is an arbitrary compact Lie group. The Hamiltonian is assumed to be multilinear function operators X', which form the basis for the Lie algebra of G in a fundamental representation of weight A, copied at each x E V. He then considers the sequence of quantum systems where each Xx' is replaced by 7r(Xi), with 7r a representation of weight nA, n = 1, 2, 3, .... Simon's main result is then Zc(i3) C ZQ(13,n)

Zc()3

\1+ cn

(4.10)

where Zc is again a classical partition function for a suitable classical system which he defines. The bounds (4.9) and (4.10) by themselves are not good enough to deduce a phase transition for the quantum model at sufficiently large S or n, assuming that the classical model has a phase transition, although one may expect such a result. Very recently, Biskup, Chayes, and Starr [9] refined the Berezin-Lieb inequalities, which are at the core of the results of Lieb and Simon, to include estimates of matrix

elements e-pH in coherent state vectors, not just for the partition function. This enables them to prove the existence of phase transitions in the quantum model for sufficiently large spin, assuming that both the classical and the quantum models are reflection positive and that chessboard estimates prove a phase transition for the the classical model. Another refinement of the classical limit, very much akin to studying the central limit theorem as a refinement of the law of large numbers, was undertaken in [59, 60].

QUANTUM SPIN SYSTEMS AFTER DLS 1978

55

5. Spin Chains and Haldane's Conjecture Quantum spin chains, i.e., systems defined on the one-dimensional lattice Z, and their ground states are a fascinating subject by themselves. They have received a lot of attention in recent years due to their relevance for quantum information theory. But even before quantum information theory became fashionable again,

the subtleties and great variety in behavior attracted the attention of condensed matter physicists and mathematical physicists alike. Haldane gave the subject a big impetus when he predicted that the Heisenberg antiferromagnetic chain has qualitatively different ground states depending on whether we have S E Z or S E Z + 2, where S is the magnitude of the spins in the chain [32]. The Bethe Ansatz solution of the spin-! chain shows a unique ground state with polynomial decay of correlations and no spectral gap. Haldane conjectured that the spin-1 chain would also have a unique ground state (in the thermodynamic limit), but with a spectral gap above it and exponentially decaying correlations (this is sometimes called the Haldane phase). Affleck and Lieb [3] proved that for half-integer spin chains a unique ground state indeed implies the existence of low-lying excitations. A related implication was proved by Aizenman and Nachtergaele [4]: a unique ground state must necessarily have no faster than polynomial decay of correlations. That the Haldane phase exists was rigorously demonstrated by Affleck, Kennedy, Lieb, and Tasaki, who proposed the spin-1 chain with Hamiltonian (now called the AKLT model)

2+

H[Q,b] _

2Sx

' 5x+1 +

s(S.

5.+1)2]

(5.11)

X

and proved that it has a unique ground state, which they explicitly constructed, with exponential decay and a spectral gap above the ground state [2]. Inspired by the AKLT model and a construction by Accardi [1], Fannes, Nachter-

gaele, and Werner introduced finitely correlated states [20] and proved that they provide the exact ground state of a large family of spin chains, including the AKLT model, with similar properties. All correlations of finitely correlated states (also called quantum Markov states or matrix product states) can be explicitly computed by a transfer matrix-type formula. Let Az be the algebra of local observables of the spin chain with each spin represented by a copy of A = M,,,, and let B be another finite-dimensional C*-algebra, say B = Mk. Let E : A 0 B -. B be a completely positive map that is unit preserving, i.e., IE(1A 0 118) = Il8. It can be shown that for such IE there exists

at least one state p on B such that p(IE(1A 0 B)) = p(B),

for all B E B.

Suppose p is such a state. It is useful to introduce the maps IEA : B - B, EA (B) _ IE(A®B). It is then straightforward to verify that, with these objects and properties, the following formula defines a translation invariant state w on Az: w(A1 ®A2 ®... ®AN) = TrpIEA, 0 IEA2 0 ... 0 LEAN (ll)

We say that w is the finitely correlated state generated by the triple (B, IE, p). A particularly important class of maps IE are the so-called pure maps. Let V : Ck _, Cd®Ck be linear and isometric, i.e., V*V = ll. Then, E(AOB) = V*A®BV, defines a pure completely positive unital map in the sense that there is no non-trivial decomposition E = El +E2, with B1 and IE2 completely positive maps. Such finitely

B. NACHTERGAELE

56

correlated states are called purely generated. Pure finitely correlated states can be generated by a pure map E [21]. We conclude this section by specifying the triple (B, E, p) that generates the ground state of the AKLT model with Hamiltonian (5.11). Since it is a spin-1 chain, A = M3. It turns out that B = M2 and E(X) = V*XV, with V : C2 -p C3 ® C2, given by

V I1/2) =

V I-1/2) =

1/311) ®I-1/2) - V/"2-/310) o 1/2)

1/3 I-1) ®I1/2) - V'2-/310) ®1-1/2)

where Im) denotes an eigenbasis of S3. Up to a phase factor, V is also the unique isometry satisfying the following intertwining relation VD (1/2) = DM ® D(112)V where D(S) is the spin-S representation of SU(2). The state p is given by

p(B) = 2 Tr B .

6. Perturbation Theory A major goal in the perturbation theory of quantum spin systems is to show that the set of interactions for which the model has a unique ground state with a nonvanishing spectral gap above it (in the thermodynamic limit) is open in a suitable topology on the space of interactions. Significant steps toward this goal have been

made by a number of authors and the results they obtained have a scope and generality that justifies calling them quantum Pirogov-Sinai theory [16, 11, 10]. It should be noted, however, that the applicability of these results is restricted to models that can be regarded as quantum perturbations of classical models. In some examples, suitable perturbation methods have also been used to show the absence of gaps or eigenvalues [17, 37]. The remarkable paper by Kennedy and Tasaki [40] was the first to make a serious attempt to get away from perturbing classical models. A new result by Yarotsky, which we discuss below, can be seen as taking that line of approach one step further. Yarotsky's result makes it possible to prove stability of the massive phase provided that there is a nearby generalized valence bond solid model [2, 20, 61] that can be used as a reference point for the perturbation in the space of interactions. In particular, Yarotsky [87] proves that the AKLT chain of (5.11) is contained in an open set of interactions with this property. The general theorem proved by Yarotsky can be stated for the following class of models defined on Zd, d > 1. For these models the Hilbert space fx, at x E 7Ld, is allowed to be infinite-dimensional. Let 7-lv = ®xEV 7-lx, for any finite V C Zd, be the Hilbert space associated with V. The unperturbed model has finite-volume Hamiltonians of the form HV0 _

hx

(6.12)

x,Vo+xCV

where hx is a selfadjoint operator acting non-trivially only on 7ivo+x, for some finite Vo. The main assumption is then that there exists 0 j4 S2x E 7-L, such that with a spectral gap Q°v = ®xEV 1 is the unique zero-energy ground state of of magnitude at least JVol above the ground state. Explicitly,

Hv>IVol(ll -I1°v)(1°v1).

(6.13)

QUANTUM SPIN SYSTEMS AFTER DLS 1978

57

The perturbed Hamiltonians are assumed to be of the form 0(r) + 0(e)

Hv = Hlo, +

(6.14)

x,ua+xCV

where Oa

and

are selfadjoint operators on 7ly,,+x satisfying I(0, 0(r)V))I < allh1,20II2

II0(b)II <<< 0,

(6.15)

and suitable constants a and 0. One can call ON a "purely for all 0 E relatively bounded" perturbation, while 0(6) is simply a bounded perturbation. THEOREM 6.1 (Yarotsky [87]). Let Hv be of the form (6.14), and such that the three relations in (6.13) are satisfied. For all is > 1, there exists 8 = 8(ic, d, Vo) > 0 such that if condition (6.15) is satisfied with some a E (0,1), and,3 = 6(1-a)" (d+1) then

(1) Hv has a non-degenerate gapped ground state 1 v : HvIv = Ev1v, and for some -y > 0, independent of V, we have

Hv > Evlsw>(IlvI + (Ev +-r)(11 - Ilv)(IlvI) .

(6.16)

(2) There exists a thermodynamic weak*-limit of the ground states Stv, V/Zd (AQv, Qv)

w(A)

(6.17)

where A is a bounded local observable. (3) There is an exponential decay of correlations in the infinite-volume ground state w: for some positive c and µ, and Ai E B(7-ly. ),

Iw(A1A2) -w(A1)w(A2)I < cIVhI+1v2le-µd(V1,v2)IIAIIIIIA2II .

(6.18)

(4) If, within the allowed range of perturbations, the terms Ox (or the resolvents (h , +Ox-z)-1 in the case of unbounded perturbations) depend analytically on some parameters, then the ground state w is also weak* -analytic in these parameters (i.e., for any local observable A its expectation w(A) is analytic).

Application of this result to the AKLT model (5.11) yields the following theorem.

THEOREM 6.2. Let = V E A[o,r]. Then there exists A0 > 0, such that for all A, JAI < A0, the spin chain with Hamiltonian

H=HAKLT+A

4141

has a unique infinite-volume ground state with a spectral gap and exponential decay of correlations. Here E A[x,x+r] is 1D translated by x.

To prove this theorem, Yarotsky shows that the AKLT model itself can be regarded as a perturbation of a particular model, one he explicitly constructs, to which Theorem 6.1 can be applied.

B. NACHTERGAELE

58

7. Estimates of the Spectral Gap The most essential result in the AKLT paper [2] is the proof of a non-vanishing

spectral gap in the thermodynamic limit. Also in [20] one of the core results is an explicit lower bound for the spectral gap of the valence bond solid models, i.e., the generalizations of the AKLT model introduced in that paper. A further generalization to models with discrete symmetry breaking and hence a finite number of distinct infinite-volume ground states, is studied in [61]. In that paper lower bounds for the spectral gap are obtained by an adaptation of the so-called martingale method of Lu and Yau [55] to the quantum context. In the following we will prove a simple version of this method for obtaining lower bounds for the spectral gap in extended systems. Then we will discuss some generalizations and improvements. Consider a quantum spin model on AZ, with local Hamiltonians of the form b-1

H[a,b] = E hx,x+1 x=a

where hx,x+1 is the translation of a fixed nearest neighbor interaction h0,1i acting non-trivially only at the pair of sites {x, x + 1}, with h0,1 a non-negative definite element of A[0,1]. Suppose that w is a zero-energy ground state of this model in the sense that w(hx,x+1) = 0, for all x E Z. This is the situation encountered for VBS models and there are other interesting examples [61, 15]. Our aim is to show that, under some rather general conditions on w, this model has a non-vanishing spectral gap above the ground state w, i.e., there exists -y > 0 such that spec(H.) fl (0, -y) = 0 (7.19)

where H, is the GNS Hamiltonian of the model in the state w, which satisfies H,SZw=0. The martingale method provides lower bounds for the smallest non-zero eigenvalue of the finite-volume Hamiltonians H[a,b], uniform in the volume. We start with a lemma that implies that this is enough to establish (7.19). LEMMA 7.1. Define

ry = sup{8 > 0 I spec(HH) fl (0, S) = 01,

assuming that the set on the RHS is non-empty and put ry = 0 otherwise. Then ry > lim inf A, (n) n>2

(7.20)

where A, (n) is the smallest non-zero eigenvalue of H[1,n] .

PROOF. Let 1'o be defined by the RHS of (7.20). Then, we need to show that (L,7r(X*)HH1r(X)1w) >-'Yo(Qw,7r(X*)H,2,7r(X)f2w)

for all X E Aloe We may assume that X E A[a,b], for some a < b E Z. By using the fact that Hw7r(X)Qw 1 kerHw, we see that (1w,1r(X*)HH7r(X)1lw)

= lim l mZ limy(Q,, 7r(X*)ir ([HA [HA2, [HAS, X]]]) Qw) (1W 7r(X*)H[a-3,b+3]7r(X)fw).

QUANTUM SPIN SYSTEMS AFTER DLS 1978

59

The last expression is the expectation of an element of A[a_3,b+3] in W. Let P[a-3,b+3]

be the density matrix of the restriction of w to A[a_3,b+3]. Then, (LL,7r(X*)Hw r(X)1w) ? TrP[a-3,b+3]X*H[a-3,b+31 X Tr (H[a-3,b+3]P[a-3,b+3]X*H[a-3,b+3]) H[a-3,b+3]

> A, (b - a + 5) TrP[a-3,b+3]X*H[a-3,b+31 X

A, (b - a + 5)(Q,,,,7r(X*)HH7r(X)1L)

.

As X E AA for all A that contain [a, b], we can take the lim inf of the last inequality to conclude the proof.

With this lemma, the problem of finding a lower bound for the spectral gap of the GNS Hamiltonian is reduced to estimating the smallest non-zero eigenvalue of the local Hamiltonians. Due to the translation invariance, it is sufficient to consider intervals of the form [1, L]. As before, L-1 (7.21)

H[1,L] _ E hx,x+1 X=1

where hx,x+1 is a translation of h1,2, acting non-trivially only at the nearest neighbor

pair {x, x + 1}, and we assume that h1,2 > 0 and that ker H[1,L] : {0}. We will denote by rye the smallest non-zero eigenvalue of h1,2, i.e., the gap of H[1,2]. It is obvious that L-1

ker H[1,L] = n ker hx,x+1 .

(7.22)

X=1

For an arbitrary subset A, let GA be the orthogonal projection onto

E

ker

(7.23)

hx,x+1

x,{x,x+1}CA

For intervals [a, b], 1 < a < b < L, G[a,b] is the orthogonal projection onto the zero eigenvectors of Ebx=a hx,x+1, and G{x} = 11 for all x. From these definitions it immediately follows that the orthogonal projections GA satisfy the following properties: GA2 GA1 = GA1 GA2 = GA2 if Al C A2 GA1 GA2 = GA2GA1 if Al n A2 = 0 hx,x+1 > ry2(1l - G[x,x+1]) Define operators E, 1 < n < L, on ?[1,L] by

E=

(7.24-b) (7.24-c)

1l-G[1,21

ifn=1,

G[1,n] - G[i,n+i]

if 2 < n < L - 1, ifn = L.

G[1,L]

(7.24-a)

(7.25)

One can then easily verify, using the properties (7.24-a)-(7.24-c), that {En 1 1 < n < L} is a family of mutually orthogonal projections summing up to 11, i.e., L

En* = En,

EnEm = am,nEn,

E En = 11 . n=1

Next, we make a non-trivial assumption.

(7.26)

B. NACHTERGAELE

60

ASSUMPTION 7.2. There exists a constant e, 0 < c < 1//, such that for all

1
EnG[n,n+l]& < E2En

(7.27)

IIG[n,n+l]EnII < c.

(7.28)

or, equivalently,

Note that, due to (7.24-a), G[n,n+l]En = G[n,n+l]G[l,n] -G[i,n+l]. This relates Assumption 7.2 with Lemma 6.2 in [20], where an estimate for IIG[n,n+l]G[l,n] G[l,n+l] II is given for general valence bond solid chains with a unique infinitevolume ground state. The same observation also implies that [G[n,n+l], G[l,n]] _ [G[n,n+l], En] which, if (7.28) holds, is bounded above in norm by 2E. The next theorem is a special case of Theorem 2.1 in [61]. Just like Theorem 6.4 in [20], it provides a lower bound on the gap of the finite-volume Hamiltonians, but it achieves this in a slightly more efficient way. We will repeat the proof for the particular case stated here because it is simple, short, and instructive. THEOREM 7.3. With the definitions above and under Assumption 7.2, the following estimate holds for the smallest non-zero eigenvalue of H[1,L]: (7.29)

A1(L) ? `Y2(1 - v2c)2

i.e., the spectrum of H[1,L] has a gap of at least -y2(1 - v"2-c)2 above the lowest eigenvalue, which is 0.

PROOF. From the properties (7.26) of the En and the assumption that G[1,L]V) _ 0, it immediately follows that L-1

(7.30)

II0II2 = E IIEnVII2. n=1

One can estimate IIEnOII2 in terms of (0 I hn,n+i'') as follows. First, insert G[n,n+l] and the resolution {Em}, L-1 IIEn0II2 = (W I (fl - G[n,n+l])En'V) + (4'

EmG[n,n+1]En0) .

I

(7.31)

M=1

Using (7.24-a) and (7.24-b), one easily verifies that Em commutes with Gn,n+i if either m < n- 2 or m > n+ 1. In these cases, EmG[n,n+l]En = G[n,n+l]EmEn = 0

because the En form an orthogonal family. By this observation, we obtain the following estimate. / For any choice of constants c1, c2 > 0, /1

I (11- G[n,n+l])Ef b) + ((En-1 + En)W I G[n,n+l]EnW)

IIEn4'II2 = ( 2c1

+

(V)

I

2C2

(Il - G[n,n+l])P) + 2 (0 I En's) (4'

I EnG[n,n+1]En4) + 2

I (En-1 +En)2W)

where we have applied the inequality I(01I02)I<2 110,1I2+2C II02II2

(7.32)

QUANTUM SPIN SYSTEMS AFTER DLS 1978

61

for any c > 0, to both terms of (7.31). The first term in the right side of inequality (7.32) can be estimated with the interaction using (7.24-c). The third term can be estimated with (7.27). It then follows that

2 - cl - c2) I E.V 2 -

c211(En-1 + En) 112 <-

61')'2

( hn,n+1

>

The term containing E,,,-1 is absent for n = 1, and ETV) = 0 if n = L. We now sum over n and use (7.30) to obtain 2

\

(2_cl- c2 - 2c2 l 11112

C1^/2

Finally, put cl = 1- ev and c2 = e/v and one obtains the estimate (7.29) stated in the theorem. The theorem above, which is stated in the form given in [61], can be generalized in different directions. First, one can formulate the conditions for local Hamiltonians on an increasing sequence of finite volumes that are not necessarily intervals, or that grow to infinity in a more restricted way. It is also not necessary that the model is translation invariant, as long as the inequality of Assumption 7.2 holds and y2 is replaced with the minimum of the gaps of H[n,11+1], over all n = 1, . . , L - 1. An application where this is relevant was given in [15]. A nice improvement of the lower bounds provided by Theorem 7.3 was made by Spitzer and Starr in [82]. We will state their main result without proof. It is useful to generalize slightly, as done in [82], by introducing a family 6,,n, for m, n > 1. To this end, we consider finite systems on intervals of the form [a, b], with a < -m, b > n, and define e(m, n) by .

e(m, n) = sup{(V), G[o,1]V))

E H[a,b], G[-n,,,o]bi = b, G[-,n,n]b = 0, 11011 = 11.

I

Then, em,n, for m, n > 1, is defined by

e,n,n = sup e(m, n). In' >m,

THEOREM 7.4 ([82]). With the same notation as in Theorem 7.3, we have the bound

A1(L) >- A, (n + m) I 1 - 2

(1 - e,,,, ))

.

Even in the case m = n = 1, this `theorem gives a strictly better lower bound than Theorem 7.3. To see this, note that e1,1 = e2, A1(2) = rye and, for all 0 < e <

1//, one has 1<

1-2 e2(1-e2) <2. (1 - V/-2e)2

So, depending on the value of e, the bound of Theorem 7.4 is always better than the one provided by Theorem 7.3 by up to a factor of 2. By somewhat different but not unrelated methods, excellent estimates were obtained by Caputo and Martinelli in [13] for the higher spin XXZ chains. They proved that the spectral gap is bounded above and below by quantities of the form constant times S, supporting the conjecture that the limit lims_._.,In ry(S)/S exists and is strictly positive [42].

B. NACHTERGAELE

62

8. Ordering of Energy Levels One easily checks that the Heisenberg Hamiltonian HV, defined in (2.1), commutes with both the total spin matrices and the Casimir operator given by ST

=

Sx, i = 1, 2, 3,

and

C = Sv SV

.

xEV

The eigenvalues of C are S(S+1) where the parameter S E {Smin, Smin+l, ... , Smax},

with Sm,, = Ex sx. S is called the total spin, and it labels the irreducible representations of SU(2). Let n(S) be the eigenspace corresponding to those vectors of total spin S. One can show that f(S) is an invariant subspace for the Hamiltonian HV and therefore, the number E(HV, S) := min spec HV J

(s)

is well-defined.

Supported by partial results and some numerical calculations, we made the following conjecture in [66]. CONJECTURE 8.1 ([66]). All ferromagnetic Heisenberg models have the ferromagnetic ordering of energy levels (FOEL) property, meaning

E(Hv, S) < E(HV, S'),

if S' < S.

In [49], Lieb and Mattis proved ordering of energy levels for a class of Heisen-

berg models on bipartite graphs, which includes the standard antiferromagnetic Heisenberg model. The FOEL property mentioned above can be considered as the ferromagnetic counterpart. To compare, a bipartite graph G = (V, E) is a graph such that its set of vertices V has a partition V = A U B where A fl B = 0 and any edge (xy) E E satisfies either x E A and y E B, or x E B and y E A. For such a graph, one considers Hamiltonians of the form

H=-Hv+HA+HB where HV, HA, and HB are ferromagnetic Heisenberg Hamiltonians on the graph G, and arbirary graphs A and B, respectively. Let Sx = LXEX sx' for X = A, B, V. The Lieb-Mattis theorem [49, 46] then states that (i) the ground state energy of H is E(H, SSA - SBA); (ii) if ISA - SBI < S < S', then E(H, S) < E(H, S'). One can see this property illustrated in Figure 1. We first obtained a proof of FOEL for the spin-! chain in [66]. In that paper we also prove the same result for the ferromagnetic XXZ chain with SUq(2) symmetry. Later, in [67], we generalized the result to chains with arbitrary values of the spin magnitudes sx and coupling constants Jx,x+i > 0. In short, we have the following theorem. THEOREM 8.2. FOEL holds for all ferromagnetic chains.

The main tool in the proof is a special basis of SU(2) highest weight vectors introduced by Temperley-Lieb [84] in the spin- 2 case and by Frenkel and Khovanov [25] in the case of arbitrary spin. Some generalizations beyond the standard Heisenberg model have been announced [67, 68].

QUANTUM SPIN SYSTEMS AFTER DLS 1978

63

Spectrum of SU(2) - symmetric Heisenberg model : j = 1, L = 5

10 r

1

0

2

3

2)

5

m

WC 4

0

5 I

-5

I

-4

I

-3

I

-2

I

I

I

I

-1 1 0 2 SI3>-eigenvalue / M-sector

I

3

1

4

5

The spectrum of a ferromagnetic Heisenberg chain consisting of five spin-1 spins with constant couplings. On the horizontal axis we have plotted the eigenvalue of the third component FIGURE 1.

of the total spin. The spectrum is offset so that the ground state energy vanishes. The arrows on the right, with label S, indicate the multiplets of eigenvalues E(H, S), i.e., the smallest eigenvalue in the subspace of total spin S. The monotone ordering of the spin labels is the FOEL property. On the left, we have indicated the largest eigenvalues for each value of the total spin. The monotone ordering of their labels in the range 1, ... , 5 is the content of the Lieb-Mattis theorem [49] applied to this system. The FOEL property has a number of interesting consequences. The first immediate implication of FOEL is that the ground state energy of H is E(H, Smax), corresponding to the well-known fact that the ground state space coincides with the subspace of maximal total spin. Since there is only one multiplet of total spin Sm, FOEL also implies that the gap above the ground state is E(H, Smax 1) - E(H, Smax). In the case of translation invariant models, this is the physically expected property that the lowest excitations are simple spin-waves. Another application of the FOEL property arises from the unitary equivalence of the Heisenberg Hamiltonian and the generator of the symmetric simple exclusion process (SSEP). To be precise, let G = (V, E) be any finite graph, and define Q

to be the configuration space of n particles, for n = 0,1, ... , V J, consisting of 77 : V -+ {0, 1}, with ExEV 71(x) = n. For any (xy) E E, let r1y > 0. The SSEP

B. NACHTERGAELE

64

is the continuous time Markov process on SZ,,, which exchanges the states (whether

there is a particle or not) at x and y with rate rxy, independently for each edge (xy). The case n = 1 is the random walk on G with the given rates. Alternatively, this process is defined by its generator on l2(SZn)

Lf (iJ) =

rxy(f (i1) - f (7lxy)) (xy)EE

where qxy is the configuration i with the values at x and y interchanged. One verifies L > 0, L1 = 0, and therefore L generates a Markov semigroup {e-tL}t>o, such that f f(ii)it(d) =

f(etLf)((d)

where µo is the initial probability distribution on the particle configurations. It is easy to show that for each n there is a unique stationary measure given by the uniform distribution on Sln. The relaxation time, which determines the exponential rate of convergence to the stationary state, is given by 1/A(n), where .\(n) > 0 is the spectral gap (smallest eigenvalue > 0) of L as an operator on 12(1 ). Aldous, based on discussions with Diaconis [6], made the following remarkable conjecture concerning \(n):

CONJECTURE 8.3 (Aldous). \(n) =.\(1) for all 1 < n < IVI - 1. Assuming the conjecture, one may determine the gap by solving the one-particle problem. To make the connection with FOEL, observe Iv'

l2(in) - (C2)®IVI

7-lV.

n=0

This is the Hilbert space of a spin- a model on V. An explicit isomorphism is given by

f

1: A77) In) n

where 1rl) E 7-l is the tensor product basis vector defined by S.' 171) _ (71x-1/2)1,1)

Under this isomorphism, the generator, L, of the SSEP becomes the XXX Hamiltonian H. To see this, it suffices to calculate the action of L:

Lf(rl)- E(Lf)(ri) 177) 77

_ E E rxy (f (xy)

f

I)

77

= EErxy(1 -txy)f(,1) I?1) (xy)

17

=Hi1 where txy is the unitary operator that interchanges the states at x and y, and the last step uses 1

and Jxy = 2rxy.

QUANTUM SPIN SYSTEMS AFTER DLS 1978

65

The number of particles, n, is a conserved quantity for SSEP. Since, St t = -V/2 + n, the corresponding conserved quantity for the Heisenberg model is the third component of the total spin. Under the isomorphism, the invariant subspace of all functions f supported on n-particle configurations is identified with the set of vectors with S3 = Smax n, which we will denote by 7-1,x. The unique invariant measure of SSEP for n particles on V, which is the uniform measure, corresponds

to the ferromagnetic ground state with magnetization n - V J/2. The eigenvalue A(n) is the spectral gap of H,, JR.. It is then c. "_y to see that the FOEL property implies that )(n) = A(1). Since we proved FOEL for chains, we also provided a new proof of Aldous' Conjecture in that case [33].

9. Other Topics To conclude, I would like to give a sample of the many topics in quantum spin systems that I have omitted in the preceding sections. First, I have not discussed developments in exact solutions of quantum spin chains by the Bethe ansatz and related techniques, which is a subject all by itself. I also did not elaborate on the many recent results on the XXZ model related to interface and droplet states. For a short review on this topic, see [62]. There are also some interesting connections between quantum spin chains and problems in combinatorics related to counting alternating sign matrices and packed loop configurations on a finite square lattice with matchings specified at the boundary [72, 73, 18]. Another topic of recent activity is the behavior of the dynamics and non-equilibrium properties of quantum spin chains [58, 65, 69, 70]. Quantum spin systems are an inexhaustible source of mathematical problems, many of which are of physical interest. Somebody should write a book about them.

Acknowledgements. This article is based on work supported by the U.S. National Science Foundation under Grant No. DMS-0303316. I have freely borrowed and modified some passages from my recent preprints [68] (with Shannon Starr) and [64] (with Robert Sims), and my unpublished lecture notes of the VolterraCIRM International School on Quantum Markov Chains and Their Applications in Physics and Quantum Information, 'Lento, Italy, 14-20 December 2001. I also thank Robert Sims and an anonymous referee for pointing out several errors in a previous draft.

References [1] L. Accardi, Topics in quantum probability, Phys. Rep. 77 (1981), 169-192. [2] I. Affleck and F. D. M. Haldane, Critical theory of quantum spin chains, Phys. Rev. B 36 (1987), 5291-5300.

I. Affleck and E.H. Lieb, Proof of part of Haldane's conjecture on spin chains, Lett. Math. Phys. 12 (1986), 57-69. [4] M. Aizenman and B. Nachtergaele, Geometric aspects of quantum spin states, Commun. Math. Phys. 164 (1994), 17-63. [5] M. Aizenman and B. Simon, A comparison of plane rotator and Ising models, Phys. Lett. 76A (1980), 281-282. [3]

[6] D. Aldous,http://www.stat.berkeley.edu/-aldous/Research/OP/sgap.html.

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J. Frohlich and E. H. Lieb, Phase transitions in anisotropic lattice spin systems, Commun. Math. Phys. 60 (1978), 233-267. J. Frohlich and C. Pfister, On the absence of spontaneous symmetry breaking and of crystalline ordering in two-dimensional systems, Commun. Math. Phys. 81 (1981), 277-298. J. Frohlich, B. Simon, and T. Spencer, Infrared bounds, phase transitions and continuous symmetry breaking, Commun. Math. Phys. 50 (1976), 79-95. , Phase transitions and continuous symmetry breaking, Phys. Rev. Lett. 36 (1976), 804-806.

[31]

J. Frohlich and D. Ueltschi, Hund's rule and metallic ferromagnetism, J. Statist. Phys. 118 (2005), 973-996.

[32] F. D. M. Haldane, Continuum dynamics of the 1-d Heisenberg antiferromagnet: identification with o(3) nonlinear sigma model, Phys. Lett. 93A (1983), 464-468. [33] S. Handjani and D. Jungreis, Rate of convergence for shuffling cards by transpositions, J. Theor. Prob. 9 (1996), 983-993. [34] M. B. Hastings, Lieb-Schultz-Mattis in higher dimensions, Phys. Rev. B 69 (2004), 104431. [35] M. B. Hastings and T. Koma, Spectral gap and exponential decay of correlations, Commun. Math. Phys. 265 (2006), 781-804.

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[36] T. Kennedy, Long range order an the anasotropac quantum ferromagnetic Heasenberg model, Commun. Math. Phys. 100 (1985), 447-462. [37] , Expansions for droplet states in the ferromagnetic XXZ Heasenberg chain, Markov Processes and Rel. Fields 11 (2005), 223-236. [38] T. Kennedy, E. H. Lieb, and B. S. Shastry, Existence of Neel order in some span-1 Heasenberg antiferromagnets, J. Statist. Phys. 53 (1988), 1019-1030. [39] , The XY model has long-range order for all spins and all dimensions greater than one, Phys. Rev. Lett. 61 (1988), 2582-2584.

[40] T. Kennedy and H. Tasaki, Hidden symmetry breaking and the Haldane phase in S = 1 quantum spin chains, Commun. Math. Phys. 147 (1992), 431-484. [41] T. Koma and B. Nachtergaele, Low-lying spectrum of quantum interfaces, Abstracts of the AMS 17 (1996), 146; and unpublished notes. [42] T. Koma, B. Nachtergaele, and S. Starr, The spectral gap for the ferromagnetic spin-J XXZ chain, Adv. Theor. Math. Phys. 5 (2001), 1047-1090. [43] T. Koma and H. Tasaki, Classical XY model in 1.99 dimensions, Phys. Rev. Lett. 74 (1995), 3916-3919.

[44] L. Landau, J. F. Perez, and W. F. Wreszinski, Energy gap, clustering, and the Goldstone theorem in statistical mechanics, J. Statist. Phys. 26 (1981), 755-766. [45] E. H. Lieb, The classical limit of quantum spin systems, Commun. Math. Phys. 31 (1973), 327-340.

, Two theorems on the Hubbard model, Phys. Rev. Lett. 62 (1989), 1201-1204. , Flux phase of the half-filled band, Phys. Rev. Lett. 73 (1994), 2158-2161. [48] E. H. Lieb and M. Loss, Fluxes, Laplacians, and Kasteleyn's theorem, Duke Math. J. 71 [46] [47]

(1993), 337-363. [49] E. H. Lieb and D. Mattis, Ordering energy levels of interacting spin systems, J. Math. Phys. 3 (1962), 749-751.

[50] E. H. Lieb and B. Nachtergaele, The stability of the Peierls instability for ring-shaped molecules, Phys. Rev. B 51 (1995), 4777-4791. [51] E. H. Lieb and D. W. Robinson, The finite group velocity of quantum spin systems, Commun. Math. Phys. 28 (1972), 251-257.

[52] E. H. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain, Ann. Phys. (N.Y.) 16 (1961), 407-466. [53] E. H. Lieb and P. Schupp, Ground state properties of a fully frustrated quantum spin system, Phys. Rev. Lett. 83 (1999), 5362-5365. [54] , Singlets and reflection symmetric spin systems, Physica A 279 (2000), 378-385. [55] S. L. Lu and H. T. Yau, Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics, Commun. Math. Phys. 156 (1993), 399-433. [56] T. Matsui, On the spectra of the kink for ferromagnetic XXZ models, Lett. Math. Phys. 42 (1997), 229-239.

, Quantum statistical mechanics and Feller semigroup, Quantum Probability Communications, QP-PQ, X, pp. 101-123, World Scientific Publishing, River Edge, NJ, 1998. [58] T. Matsui and Y. Ogata, Variational principle for non-equilibrium steady states of the XX model, Rev. Math. Phys. 15 (2003), 905-923. [59] T. Michoel and B. Nachtergaele, Central limit theorems for the large-spin asymptotics of quantum spins, Prob. Theory Rel. Fields 130 (2004), 493-517. [60] , The large-spin asymptotics of the ferromagnetic XXZ chain, Markov Processes Rel. Fields 11 (2005), 237-266. [61] B. Nachtergaele, The spectral gap for some spin chains with discrete symmetry breaking, Commun. Math. Phys. 175 (1996), 565-606. [62] , Interfaces and droplets in quantum lattice models, XIIIth International Congress of Mathematical Physics (London, 2000), pp. 243-249, International Press, Boston, 2001. [63] B. Nachtergaele and R. Sims, Lieb-Robinson bounds and the exponential clustering theorem, Commun. Math. Phys. 265 (2006), 119-130. [64] , Recent progress in quantum spin systems, arXiv:math-ph/0512020, to appear in Markov Processes Rel. Fields (J. T. Lewis special volume). [57]

[65] B. Nachtergaele, W. Spitzer, and S. Starr, On the dynamics of interfaces in the ferromagnetic heisenberg XXZ chain under weak perturbations, Contemp. Math. 327 (2003), 251-270.

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, Ferromagnetic ordering of energy levels, J. Statist. Phys. 116 (2004), 719-738. [67] B. Nachtergaele and S. Starr, Ferromagnetic Laeb-Mattis theorem, Phys. Rev. Lett. 94 [66]

(2005), 057206. [68]

, Ordering of energy levels in Heisenberg models and applications, Mathematical Physics of Quantum Mechanics, Selected and Refereed Lectures from QMath9 (J. Asch and

A. Joye, eds.), pp. 149-170, Lecture Notes in Physics 690, Springer Verlag, Berlin-New York, 2006.

[69] Y. Ogata, The diffusion of the magnetization profile in the XX-model, Phys. Rev. E 66 (2002), 066123. [70]

,

The stability of the non-equilibrium steady states, Commun. Math. Phys. 245

(2004), 577-609.

[71] J. F. Perez and E. J. Neves, Long range order in the ground state of two-dimensional antiferromagnets, Phys. Lett. 114A (1986), 331-333. [72] A. V. Razumov and Yu. G. Stroganov, Spin chains and combinatorics, J. Phys. A: Math. Gen. 34 (2001), 3185-3190. , Spin chains and combinatorics: twisted boundary conditions, J. Phys. A: Math. [73] Gen. 34 (2001), 5335-5340. [74] D. W. Robinson, Statistical mechanics of quantum spin systems, Commun. Math. Phys. 6 (1967), 151-160. [75] D. Ruelle, Statistical Mechanics: Rigorous Results, W. A. Benjamin, New York, 1969. [76] B. Simon, Mean field upper bound on the transition temperature in multicomponent ferromagnets, J. Statist. Phys. 22 (1980), 491-493. , Absence of continuous symmetry breaking in a one-dimensional n-2 model, J. Statist. Phys. 26 (1981), 307-311. [78] , The rate of falloff of Ising model correlations at large temperature, J. Statist. Phys. 26 (1981), 53-58. [79] , The Statistical Mechanics of Lattice Gases, Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1993. [80] B. Simon and A. D. Sokal, Rigorous entropy-energy arguments, J. Statist. Phys. 25 (1981),

[77]

679-694.

[81] E. R. Speer, Failure of reflection positivity in the quantum Heisenberg ferromagnet, Lett. Math. Phys. 10 (1985), 41-47. [82] W. Spitzer and S. Starr, Improved bounds on the spectral gap above frustration free ground states of quantum spin chains, Lett. Math. Phys. 63 (2003), 165-177. [83] H. Tasaki, Ferromagnetism in the Hubbard model: a constructive approach, Commun. Math. Phys. 242 (2003), 445-472. [84] H. N. V. Temperley and E. H. Lieb, Relations between the `percolation' and `colouring' problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the `percolation' problem, Proc. Roy. Soc. London Ser. A 322 (1971), 252280.

[85] B. Toth, Improved lower bound on the thermodynamic pressure of the spin-2 Heisenberg ferromagnet, Lett. Math. Phys. 28 (1993), 75-84. [86] W. F. Wreszinski, Charges and symmetries in quantum theories without locality, Fortschr. Phys. 35 (1987), 379-413. [87] D. A. Yarotsky, Ground states in relatively bounded quantum perturbations of classical lattice systems, Commun. Math. Phys. 261 (2006), 799-819.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, DAVIS, ONE SHIELDS AVENUE,

DAVIS, CA 95616-8366, U.S.A.

E-mail address: [email protected]

Proceedings of Symposia in Pure Mathematics Volume 76.1, 2007

Barry Simon's Contributions to Quantum Field Theory Lon Rosen To Barry Simon on the occasion of his 60th birthday, with fond memories of our collaboration together ABSTRACT. Beginning with a brief history of constructive quantum field theory, we present a review of Barry Simon's contributions to the field. We emphasize his role in advocating Euclidean methods, his introducing ideas and techniques from statistical mechanics such as correlation inequalities and the Lee-Yang Theorem, and his developing functional analytic methods such as hypercontractivity.

CONTENTS 1.

2. 3. 4. 5. 6. 7.

Introduction Barry Simon's Publications in Quantum Field Theory A Brief History of Constructive Quantum Field Theory Early Work (1969-1972) The Euclidean Revolution (1972-1975) Later Work (1975-1977)

Conclusion References

1. Introduction Barry Simon: mathematical omnivore, super synthesizer, lightning quick thinker, speed writer, consummate simplifier, magical trickster, abstract artist, .... Rather than dwelling on these celebrated characteristics, I'd instead like to say something here of a personal nature. I had the joy of collaborating with Barry for about six years at the beginning of my research career. I cherish that experience. I found 2000 Mathematics Subject Classification. 81T08, 82B05. Key words and phrases. constructive quantum field theory, classical statistical mechanics, Euclidean methods, correlation inequalities, hypercontractivity.

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Barry to be a most generous and enthusiastic collaborator. Thus it was with a deep sense of gratitude that I accepted this assignment. Mind you, when we were collaborating, I was also grateful that Barry had so many other research projects going (in quantum mechanics, functional analysis,

statistical mechanics, ...) and was also writing several textbooks. This enabled me, working full-time on quantum field theory (QFT), to keep up with him-more or less-and to make a reasonable contribution to our joint work. In preparing this review, I had a sense of deja vu: thank goodness he published in so many different areas and that my portfolio consisted of "only" 30 rather than 300 items. Doubtless there are other contributors here who share this sense of relief. Yes, a consummate simplifier and magical trickster. How many of you have had the experience of presenting a complicated argument to Barry and having him fire back a much simpler one, usually involving some clever tricks? This happened to me when I first went to Princeton to give a seminar on higher order estimates, and it kept on happening. Although not all workers in the field appreciated these

characteristics,' Barry remained faithful to his mantra (due to Hilbert?): "One doesn't really understand a theorem until one regards its proof as trivial." His ability to cut to the "simple" idea at the heart of an argument and his expository skills in reaching a wider audience have made a real contribution to QFT, and no doubt to all his fields of interest.

To give some structure to my narrative, I plan to proceed chronologically through Barry's QFT publications, as listed in the following bibliography ([B1][B29]). There is a lot of material and a plethora of technical details! Accordingly, I shall relegate some of these technicalities to footnotes and suppress all the rest. Other references may be found in a separate bibliography at the end of the paper.

2. Barry Simon's Publications in Quantum Field Theory [B1] Convergence of regularized, renormalized perturbation series for superrenormalizable field theories, Nuovo Cimento 59A (1969) 199-214 [B2] Borel summability of the ground state energy in spatially cutoff (0')2, Phys. Rev. Lett. 25 (1970) 1583-1586 [B3] (with R. Hoegh-Krohn) Hypercontractive semigroups and two-dimensional self-coupled Bose fields, J. Funct. Anal. 9 (1972) 121-180 [B4] (with L. Rosen) The (02")2 Hamiltonian for complex coupling constant, Trans. Amer. Math. Soc. 165 (1972) 365-379

[B5] On the Glimm-Jaffe linear lower bound in P(0)2 field theories, J. Funct. Anal. 10 (1972) 251-258

[B6] Continuum embedded eigenvalues in P(0)2 field theory, Proc. Amer. Math. Soc. 35 (1972) 223-226

[B7] Summability methods, the strong asymptotic condition, and unitarity in quantum field theory, Phys. Rev. Lett. 28 (1972) 1145-1146 [B8] (with F. Guerra and L. Rosen) Nelson's symmetry and the infinite volume behavior of the vacuum in P(0)2, Commun. math. Phys. 27 (1972) 10-22

[B9] (with F. Guerra and L. Rosen) The vacuum energy for P(0)2: Infinite volume limit and coupling constant dependence, Commun. math. Phys. 29 (1973) 233-247

[B10] (with F. Guerra and L. Rosen) Statistical mechanical results in the P(0)2 quantum field theory, Phys. Lett. 44B (1973) 102-104 [B11] Correlation inequalities and the mass gap in P(0)2, I. Domination by the two point function, Commun. math. Phys. 31 (1973) 127-136

'I quote without attribution: "a premature emphasis on the simplicity and elegance of the details can divert energy away from central issues, and thereby delay or obstruct progress."

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[B12] (with R. Griffiths) Griffiths-Hurst-Sherman inequalities and a Lee-Yang theorem for the (04)2 field theory, Phys. Rev. Lett. 30 (1973) 931-933 [B13] Positivity of the Hamiltonian semigroup and the construction of Euclidean region fields, Helv. Phys. Acta 46 (1973) 686-696 [B14] (with R. Griffiths) The (04)2 field theory as a classical Ising model, Common. math. Phys. 33 (1973) 145-164 [B15] The Glimm-Jaffe 0 bound: A Markov proof, in Constructive Quantum Field Theory (Erice), G. Velo and A. S. Wightman, eds., Springer-Verlag, New York, 1973, pp. 125-131

(with F. Guerra and L. Rosen) The pressure is independent of the boundary conditions for P(0)2 field theories, Bull. Amer. Math. Soc. 80 (1974) 1205-1209 [B17] The P(0)2 Euclidean (Quantum) Field Theory, Princeton University Press, Princeton, 1974 [B18] (with F. Guerra and L. Rosen) The P(0)2 Euclidean quantum field theory as classical statistical mechanics, Ann. of Math. 101 (1975) 111-259 [B19] Correlation inequalities and the mass gap in P(0)2, II. Uniqueness of the vacuum for a class of strongly coupled theories, Ann. of Math. 101 (1975) 260-267 [B20] (with F. Guerra and L. Rosen) Correlation inequalities and the mass gap in P(0)2, III. Mass

[B16]

gap for a class of strongly coupled theories with nonzero external field, Commun. math. Phys. 41 (1975) 19-32 [B21] (with E. Seiler) On finite mass renormalizations in the two-dimensional Yukawa model, J. Math. Phys. 16 (1975) 2289-2293 [B22] (with E. Seiler) Bounds in the Yukawa2 quantum field theory: Upper bound on the pressure, Hamiltonian bound and linear lower bound, Commun. math. Phys. 45 (1975) 99-114 [B23] (with J. Rosen) Fluctuations in P(O)1 processes, Ann. Prob. 4 (1976) 155-174 [B24] (with E. Seiler) Nelson's symmetry and all that in the Yukawa2 and (cp4)3 field theories, Ann. Phys. 97 (1976) 470-518

[B25] (with F. Guerra and L. Rosen) Boundary conditions for the P(¢)2 Euclidean field theory, Ann. Inst. H. Poincare 25A (1976) 231-334 [B26] (with J. Frohlich) Pure states for general P(0)2 theories: Construction, regularity and variational equality, Ann. of Math. 105 (1977) 493-526 [B27] (with J. Frohlich and T. Spencer) Phase transitions and continuous symmetry breaking, Phys. Rev. Lett. 36 (1976) 804-806 [B28] (with J. Frohlich and T. Spencer) Infrared bounds, phase transitions and continuous symmetry breaking, Commun. math. Phys. 50 (1976) 79-85 [B29] (with L. Yaffe) Rigorous perimeter law upper bound on Wilson loops, Phys. Lett. 115B (1982) 145-147

3. A Brief History of Constructive Quantum Field Theory Before describing Barry's contributions to QFT, I would first like to review a little of the history of an exciting period in mathematical QFT, albeit from a barrycentric point of view. By the late '50s, although QFT had achieved many striking numerical successes and was regarded as the fundamental theory of elementary particles, it was plagued by mathematical improprieties. Calculations involved the suspicious cancellations of infinities and were based on perturbation series which almost surely diverged. In order to put the field on a firm mathematical

foundation, Wightman in 1956 [49] and then later with Garding [50] formulated sets of axioms which captured the essential properties of a QFT. For the next decade axiomatic QFT flourished as researchers explored the structure and consequences of the Wightman axioms and established many deep and beautiful theorems. For an account of the axioms, these results, and the many references, see the book by Streater and Wightman [47].2 Faced, however, with the growing embarrassment

that there was no known example of a QFT model satisfying the axioms other than the "trivial" free field theories, Wightman (Barry's thesis advisor) strongly 2See also Jost [20]. An alternate set of axioms which stressed the algebraic structure of a field theory was proposed by Haag and Kastler [17].

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advocated rigorous model construction, especially in lower space-time dimensions where the divergences were less severe. The era of constructive QFT (CQFT) can be viewed as beginning with the Ph.D. theses of two of Wightman's students, Jaffe (1965) and Lanford (1966) (see [19]). Here is a (very incomplete) description of the goals of CQFT in the Hamiltonian framework. Suppose we wish to construct a QFT model in d space-time dimensions built from a number of fields, each corresponding to a constituent input particle3and we do! In this review I shall consider only models for a single scalar boson of mass m > 0, whose field will be denoted O(x), and the Yukawa model for a boson interacting with a fermion-antifermion pair, whose fields will be denoted fi(x), fi(x). When I wish to emphasize that I am dealing with time-zero fields, I shall attach a subscript o. Formally, the Hamiltonian or total energy operator of the model has the form H = Ho + AV + 6H(A), where A is the coupling constant, the free Hamiltonian Ho is given as an integral over space of a quadratic expression in the time-zero fields, and the interaction is given as the integral of a polynomial of degree > 2 in the fields. Typically, V is a highly singular perturbation of Ho and the role of the counterterms in 6H is to cancel the resulting divergences. It is well known

how to define the time-zero fields as operator-valued distributions and the free Hamiltonian as a positive self-adjoint operator on Fock space F, the free particle

Hilbert space.4 As for V, we must use Wick powers :(x):, :io(x) o(x):.... in order to eliminate the divergences due to self-contractions. Wick powers can be introduced in several ways, e.g., in the traditional way by splitting the field into the sum of an annihilation and a creation term, and moving the creation terms in the Wick power to the left, or by means of generating functions.5 For details, see [B17] or [12]. 31t is possible that the model may yield different output particles, which are not in 1-1 correspondence with the input fields.

4For example, in the case of a scalar boson, we can take the single particle space to be R = L2 (Rd-1), the n-particle space to be the symmetric tensor product Fn = R ® . OR (n factors), and Fock space .F to be the (complete) direct sum .F = En'=0 where Fo = C. Given a linear operator A on 7-1 there are two ways of lifting A to F, (and hence to .7=): F(A) = A®...®A (n factors), and the "second quantization" of A,

dF(A) =A®I®... ®I+I ®A®...®I+...+I ®...®I ®A. Thus, if A generates a contraction semigroup on 7{, then dF(A) generates the contraction semigroup F(e t 4) on 7 . The free boson Hamiltonian Ho = dF(w) where w = (-0+m2) 2 , where A is the Laplacian operator on R. The corresponding structures for fermions are similar except that the vectors in R carry spinor indices and symmetric tensor products are replaced by antisymmetric ones.

5We sketch the generating function definition for a time-zero boson field 0 = Oo in one dimension whose two-point VEV is

MxMy)) = 2w-1(x - y) = a f

A,

w(k) = (k2 + m2) z .

For any reasonable approximation h(x) to the delta function 1(x), we introduce the ultraviolet cutoff field Oh(x) = h * O(x) = f h(x - y)o(y)dy. Wick powers are defined by s

:et0h(x): _

ni

to = etOh(x)-2

(Oh(X)2)

=

n=0

(so :0h: is actually a Hermite polynomial in 00. In the special case of one dimension, if we smear with a real L2 function g then f :0n(x):g(x)dx - limp-b f : (x):g(x)dx is a well-defined self-adjoint operator.

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The spatially cutoff P(O)d model for a self-interacting scalar boson in d spacetime dimensions has interaction V(g) = f :P(0o(x)):g(x)dd-lx where the polynomial P is bounded below, and g > 0 is the spatial cutoff function. For d = 2 and g E L' f1 L2, V (g) is a well-defined self-adjoint operator on T, and the spatially cutoff Hamiltonian

H(g) = Ho + AV(g) = Ho + A I :P(Oo(x)):g(x)dx

(1)

is densely defined. The (scalar) Yukawa model Yd in d dimensions has interaction

V (g) = f Formal perturbation theory suggests the form of the counterterms bH (as a power series in the coupling constant A). Does such a choice of 6H work to cancel Oo(x)g(x)dd-ix.

the divergences to every order in A? This is a notoriously difficult and much studied question. Of the many contributions to perturbative renormalization theory since 1950, I recommend the beautiful renormalization group approach of Gallavotti and co-workers [5].6 Perturbation theory provides the following classification: A model is renormalizable if only a finite number of types of counterterms are required in off (so that renormalization can be interpreted as an adjustment of physical parameters such as mass and charge). A model is superrenormalizable if counterterms in 6H are required only to some finite order N in perturbation theory. The following models

are superrenormalizable: P(q)2 (for which only a vacuum energy counterterm is required), Y2 (for which N = 2), 03 (N = 3), Y3 (N = 6). (The value of N is an indicator of the degree of difficulty of a rigorous construction.) 03, 04 and Y4 are renormalizable. 04 is non-renormalizable. A rigorous construction follows this strategy: 1) impose enough cutoffs (volume,

high momentum, ...) so that all terms in H are well-defined; 2) choose 6H wisely so as to cancel divergences (which become infinite when the cutoffs are removed); 3) remove the cutoffs via well-defined but possibly delicate limits. Perturbation theory has proved to be a useful guide (particularly in the choice of OH) but cannot be used

as a proof.? In the Hamiltonian era, model construction began with these basic questions: Is H bounded below? Is H self-adjoint? These are difficult operator questions. For instance, for P(0)2i formal semiboundedness is clearly ruined by Wick ordering. The key to unlocking these problems was the use of "Q-space" where the boson fields are diagonalized (so that Q-space gives the Schrodinger representation of a QFT). There are many ways of realizing the Q-space representation (for details see

[B17]), but for our purposes all that matters is that there is a probability measure space (Qo, dµo) with gaussian measure dµo such that under the unitary equivalence of F with L2 (Qo, dµo) the field Oo (f) is a gaussian random variable on Q. In the

case d = 2, for f E S(R), the field qo(f) has covariance given by the two-point

6Their approach is based on a discretization of the renormalization group by momentum scales. For a continuous renormalization group approach based on Polchinski's differential equation, see the text by Salmhofer [40] and references therein. 7Indeed, extreme caution must be exercised with perturbation theory: even if the renormalization procedures succeed in curing the divergences at every finite order of the formal perturbation series, the series almost certainly diverges and may not be asymptotic to anything (as is now believed to be the case for 04 and QED4)!

L. ROSEN

74

vacuum expectation value (VEV)5 'Qo

Oo(f)Oo(g)dµo = (Qo, Oo(f)Oo(g)Qo)T = z(f,w 19)> !(R),

(2)

where SZ° is the Fock vacuum. It's fairly straightforward to show that the P(0)2 interaction V(g) is in LP(Qo, dµ0) for all p < oo. In 1966 Nelson [26] made the critical observation that although the P(q)2 interaction V (g) is not bounded below, it is large negative on a set of very small measure8 with the result that e-v(9) E LP(Qo,dfto) for all p < oo. Glimm [8] extended this result into a proof that the P(0)2 Hamiltonian H(g) was bounded below. Self-adjointness soon followed [9, 36, 43]. Under the leadership of James Glimm and Arthur Jaffe (who were joined in 1973 by Tom Spencer), results for superrenormalizable models poured forth over the next decade. For references to the work of all the contributors to this endeavour, many of whom were students or postdocs of Glimm and Jaffe, see the extensive bibliography in their book [12]. Euclidean methods, based on replacing time t by it, have a long history in QFT, starting with the desire of physicists to rid themselves of the indefinite relativistic metric x2 = t2 Kac , trying to make sense of Feynman's mysterious "measure" on paths in his path integral representation of the Schrodinger unitary group eitx was similarly motivated when he considered instead the semigroup e-1H for which "Feynman measure" is replaced by bona fide Wiener measure [21]. In a boson field theory, fields O(x) and 0(y) at space-like separated points ((x - y)2 < 0) commute. Since any two Euclidean (= Schwinger) points (x", is) and (y, it) are "spacelike" separated, Euclidean field theory (EFT) apparently replaces non-commuting operator-valued distributions with commuting random distributions. Schwinger [42] considered VEV's at Euclidean points in the late '50s. But it was Symanzik [48] who undertook a serious study of EFT with the Euclidean fields enjoying a central status as generalized random variables. He studied the properties of the VEV's at Schwinger points, which he called Schwinger functions Sn(xl) ... , xn), and he emphasized the similarities between EFT and classical statistical mechanics (CSM). However, what was unclear in Symanzik's program was how to recover a relativistic QFT from its corresponding EFT. In the early '70s, Nelson, inspired by Symanzik's work, developed an axiomatic framework for (bosonic) EFT which imposed more structure than the Wightman axioms, namely, the existence of a probability measure space9 on which the fields are defined as generalized random variables [28]. Emphasizing the roles of Euclidean covariance and the Markov property,10 Nelson showed how to analytically continue

a Markov EFT back to recover a relativistic QFT, thereby rescuing the Symanzik 8For large n, µo{gIV(g) < -(ln_)'n} < e-' where n = 2 degP, and a and c are positive constants. 9As a consequence of this additional structure, a Nelson field theory satisfies Nelson-Symanzik positivity, which asserts the positivity of the integral of a positive function of the fields with respect to the underlying probability measure. See the discussion of [B13] below. 10For

a closed set A C Rd let A' be the closure of Rd\A and 8A = A fl A', let EA be the a-algebra generated by {0(f) I supp f C Al and let EA denote the conditional expectation with respect to EA. The Markov property states that for any closed A C Rd and for any EA measurable function F, EA (F) = ESA(F). That is, for questions about the exterior (the "future"), knowledge of the boundary (the "present") is as good as knowledge of the closed interior (the "past"). The Markov property seems to be difficult to verify in practice, and for more singular models OS positivity8 has become the preferred tool.

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75

program from no man's land (one man's land?). Just like the time-zero relativistic field 00, the free Euclidean boson field O(x) can be represented as a gaussian random distribution on a probability measure space (Q, dµ) except that now the covariance is given by the two-point Schwinger function S2(x, y) = (-0+m2)-1(x, y). Taking d = 2, we have as in (2)

fQ 0(.f)0(9)dµ = (f, (-A+ m') 19)L2(R.2).

(3)

In the mid-'70s Osterwalder and Schrader [32, 33] proposed an alternate set of Euclidean axioms for the Schwinger functions S,,,(xl,... , x,,,) (with no explicit Euclidean fields) which featured the important "OS positivity condition," 11 and they analyzed the equivalence of their axioms with the Wightman axioms.12 For an account of the relations among these various axiom schemes, see Barry's book [B17].

I conclude this historical introduction with the observation that at the same time as these developments in axiomatic/ constructive QFT were occurring, there were parallel developments in rigorous statistical mechanics, which you can read about in Ruelle's book [39]. The two fields involved many similar issues (e.g., the infinite volume = thermodynamic limit) and there were many close analogies (e.g., the Gibbs formula for correlation functhe Gell-Mann-Low formula for VEV's tions), but for the most part quantum field theorists and statistical mechanicians seemed to exist in parallel but distinct universes.

4. Early Work (1969-1972) [B1] is Barry's first published paper. The problem was to investigate the convergence of the Cell-MannLow perturbation series for the time-ordered VEV's of the Y2 model. The only divergences in perturbation theory are the second order vacuum energy and the boson mass diagrams. Because of the cancellations due to fermi statistics, one might hope for convergence, at least for the model regularized with momentum and volume cutoffs. 110S positivity amounts to the statement that a vector in the Wightman Hilbert space (albeit with its arguments continued to the Euclidean region) has a non-negative norm. This positivity is distinct from Nelson-Symanzik positivity in Nelson's framework. Although the concept of OS positivity or "reflection positivity" originated in an axiomatic setting, it developed into a powerful constructive tool. Here's why: Let F+ consist of functions of the fields supported in the half-space R+ _ {(a, t) t > 0}. .F+ is generated by the functions F = C40(f) where I

f E Co (R+). Let O be the reflection operator in the hyperplane t = 0 which acts on test functions and by extension on functions of the field. For example, for F = ez0M, OF = ez0(©f), where O f (X, t) = f (X, -t). OS positivity states that f 6FF dµ > 0 for F E T+. Consequently, the bilinear form b(F, G) = f 6FG dµ satisfies the Schwartz inequality b(F, G) < b(F, F) a b(G, G) I for F, G E Y. This innocent-looking "reflection inequality," iterated and applied to different hyperplanes, has led to many significant estimates, notably in Barry's papers [B24] and [B26]. For a comprehensive account of "multiple reflection" bounds, see Chapter 10 of [12]. 12The temperedness assumption on the Schwinger functions in the OS axioms is problematic. In [32] the temperedness axiom (EO) turned out to be inadequate to reconstruct the relativistic theory. In their second paper [33] Osterwalder and Schrader replaced (E0) with an axiom (EO) which restored the equivalence of the OS and Wightman axioms but which is probably impossible to verify in practice. So they proposed a more practical temperedness assumption (E0') involving an n! growth condition on S. which also allows a recovery of the Wightman functions, albeit with a superfluous growth condition.

76

L. ROSEN

Barry based his approach on Caianiello's analysis of the unrenormalized model.

Caianiello controlled the large n growth of the VEV (Vn) _ (J:

:

O)n) as

follows. With all the cutoffs in place, the bose factor (cpn) - bnn!1/2, and the fermi

factor ((: :)n) can be evaluated as a determinant and estimated by Hadamard's inequality to give a number growth f nn!l/2 Thus n, ((AV)n) (cA)n where the constants b, f and c depend on the cutoffs. This yields convergence of the sum over n for small A. Renormalization complicates matters. In particular, the mass renormalization upsets the lovely determinantal structure (and possibly Caianiello's bounds). It

was necessary to analyze the resulting objects, namely determinant expansions without two-cycles (i.e., less mass bubbles) as well as one-cycles. One can already see Barry's penchant for inventing colourful terminology as he coined the name "bubblessian" for these objects.13 Barry succeeded in carrying out Caianiello's strategy for the expansion involving bubblessians and thus obtained convergence of the renormalized, regularized expansion for small A. However, the method obviously could not sustain a removal of the momentum cutoff and Barry sadly concluded that "the question of convergence of non-regularized fermion theories is about as unsettled as it was before this discussion." I have decided not to suppress this disappointing outcome because it is interesting to see how a master struggled and developed in the apprentice stage of his career. With hindsight we can see that, riding the horse-and-buggy technology of the late '60s, Barry was up against an intractable problem whose solution had to wait for the Euclidean revolution and the seminal work of Erhard Seiler on Y2 in 1975 [44]. Seiler's starting point was the Matthews-Salam formula for the Euclidean VEV's where the fermi fields have been "integrated out," leaving the function 0A(0) = det(1 - ASOXA) of the boson field 0 (here S is the fermi propagator and XA the volume cutoff function). Working with the Euclidean theory, Seiler could exploit the machinery of LP estimates, and, working with OA (O) rather than the individual terms in its perturbation series, he could delete the mass bubbles with the expeditious replacement of 0A(c) by PA(O) = det3(1 Thereby protecting the fermi cancellations internal to pA(q), Seiler was able to establish its critical L' properties. ASOXn)13.

One could excuse Barry if he refused to consider the Y2 model ever again. However, he soon joined forces with Seiler and contributed significant results to Y2 [B21, B22, B24] which I discuss below. Further Y2 contributions were made by McBryan [24, 25], Cooper and Rosen [1], and Magnen and Seneor [23], culminating ten years after [B1] in Renouard's proof of Borel summability15 of the infinite volume Y2 model [35]. The number growth obtained by Renouard for the renormalized VEV's (ZS)A was n, -d4 (ZS)A _ Cnn!1/6+E where e > 0 and c is independent of the momentum cutoffs. Intuitively, I believe that the best one could do is cnn!E. In any case, it seems that the (Euclidean version of the) perturbation series which Barry studied diverges, but to my knowledge this has not been verified. 13In the more standard terminology of regularized determinants [45] the bubblessians Bn(xl,...,xn) with matrix elements S(x;,x,) are generated by det3(1+SO) where 0 is, say, a Schwartz space function and det3(1 + A) = det(1 + A) exp (-TrA + 2TrA2). That is, det3(1 + SO) = En' o n, f ... f dxl ... dxnB, (xl,... , xn)O(xl) ... ¢(xn). The relations between bubblessians and determinants may be read off from the

of these formulae.

QUANTUM FIELD THEORY

77

In [B2] Barry established the Borel summability of the ground state energy for

the spatially cutoff 02 Hamiltonian H(g). This paper carried over to QFT ideas which Barry had been working on in connection with the anharmonic oscillator, and it applied general results which he was simultaneously working on with Raphael Hoegh-Krohn [B3]. Barry considered HA = H(Ag) with a complex coupling constant A in a domain

of the form Da,,b = {A jarg A < a, jAj< b} where < a < 7r, b > 0. The main technical result of the paper was the norm resolvent convergence (HA - z)-1 (Ho - z)-1 for z V a(Ho) as A _# 0 in Da,,b. From this it follows [B3] that for sufficiently small b and A E Da,b, HA has a unique eigenvalue E(A) near the eigenvalue 0 of Ho. Moreover, E(A) is an analytic function of A and its perturbation series >o a,,,A' is asymptotic. Obtaining strong control14 over the remainder of the series via number operator estimates, Barry then applied Watson's Theorem to conclude that the series T,o a? An is Borel summable and thus completely determines E(A).15 In their paper on hypercontractivity [B3], Barry and Hoegh-Krohn developed

an abstract perturbation theory for hypercontractive semigroups with a minimal use of cutoffs and they applied this general theory to the P(0)2 model. Their work can be regarded as an abstraction of results of Nelson [26], Glimm [8], Glimm and Jaffe [9], Rosen [36], and Segal [43]. The following statement from the introduction of [B3] amounts almost to a personal credo of Barry, the "abstract artist" : "Finally, by developing the theory in an abstract setting, we gain

all the advantages of abstraction. Our proofs have been pared down to the essentials; the abstract setting suggests certain questions which weren't obvious in the concrete situation but whose

answer would be illuminating; and it is possible to extend the theory to other cases of interest." The term "hypercontractive" was invented in this paper: the contraction semigroup U(t) on the probability measure space L2(M, dµ) is called hypercontractive if a) U(t) is a contraction on L1 for all t > 0, and b) U(T) is bounded from L2 to L4 for some T > 0.16 This concept played a central role in the early days of CQFT (19661972) until the Euclidean revolution pushed the Hamiltonian and hypercontractive semigroups off centre stage. However, hypercontractivity continued to flourish as a tool in analysis. For applications to results for logarithmic Sobolev inequalities,

elliptic PDE's, best constants in classical inequalities, Kato's inequality, ... , see the recent review article by Gross [15]. Hregh-Krohn and Simon considered perturbations H = Ho + V of a hypercontraction generator H0, where V was a (real) function in LP for some p > 0 with e-v E LP for all p < oo. Relying on the Duhamel formula to control the perturbation, they established general results for H including self-adjointness, continuity in V (of the resolvent and semigroup of H), and new analyticity results. In particular, for the P(0)2 model, they exploited the fact that the negative part V_ of 14This means that JE(a) _EN a,A-1 < cdN+l(N+1)!IA "+1 where c and d are constants. 15More precisely, the Borel transform E(A) = >o 1 a,A'n is defined for Jai < d-1, E(A) has

an analytic continuation to {A I I arg A < a - 2 }, and E(A) = f '° e-tE(At)dt for A E Da_ z b 16A general and very elegant form of hypercontractivity, due once again to Nelson [30] is

this: Theorem. Let A be a contraction on the one-particle Hilbert space f. Let 1 < p < q < oo. Then r (A) is a contraction from LT' to L9 if IJAII < 9-1. Example. For e-tH0, where H0 is the free hose Hamiltonian with mass m, the T in b) is explicitly given by T = In 3/2m.

L. ROSEN

78

V was a tiny form perturbation17 of Ho to prove that various objects associated with HA = Ho + AV, namely, the resolvent, vacuum energy and vacuum vector had analytic continuations off the positive real axis. From this they deduced that the perturbation series for the ground state energy and equal-time VEV's were asymptotic.18

In [B4], my first collaboration with Barry, we extended Barry's results on analyticity [B2] to the P(0)2 model, using my technique of higher order estimates [37]. In particular, for the spatially cutoff PM2 Hamiltonian Ha with A in the cut plane C\(-oo, 0) we established LP smoothing properties 19 of e-, H>' for z in

the right half plane. Let N be the number operator, defined by N = n on F,,. For I arg Al < 4 , we established higher order estimates for H), in which powers of N, as well as one power of H0, can be dominated by powers of Ha.20 Using these +c)-1 properties we obtained norm resolvent convergence as in [B2], i.e., (Ha

-

(Ho + c)-1 as A - 0 in the sector Da,b. As in [B3] this enabled us to deduce analyticity of the ground state energy E(A), the vacuum vector Q(A), and the equal-time VEV's W(A) = (11(A), Oo(fi) ... for A E Da,b with a < it and b sufficiently small. We further showed that the perturbation expansions for

these objects are asymptotic, and, for the special case of the 04(g) model, we proved Borel summability of W(A). By 1976 these results had been extended to the infinite volume theory by a number of authors including Dimock, Eckmann, Magnen, Seneor, Epstein, Frohlich, and Osterwalder. In [B5] Barry gave a new short proof of the Glimm-Jaffe linear lower bound (LLB) for P(0)2 [11], which I first explain. Let HI be the P(0)2 Hamiltonian with a sharp cutoff function g(x) = Xi (x), the characteristic function of [-1/2,1/2], let

El = inf o,(Hi), and let H1 = Hl - Ei. If P is normalized so that P(0) = 0, then El < 0.21 The LLB states that El > -c l where c is a constant independent of 1. The generalized LLB (GLLB) controls the vacuum energy for perturbations of Hl: Let 6P be a polynomial such that P ± SP is bounded below (e.g., a lower degree polynomial), and let Hi = Hi+6V where 6V = f g(x) :SP(ql(x)): dx with Jg(x)J < 1

and D = diam supp g < oo. Let E1 = inf u(Hi) and 6E1 = E1 - El. Glimm and Jaffe proved that there is a constant c independent of l and g such that 1bEl < c D.

(4)

This yields the bilinear form inequality ±8V < Hl + c D, which survives the limit 1 -> oo. The particular example 6P(q) = 0 yields the so-called "q-bounds" which imply that in the infinite volume theory the fields are well-defined bilinear forms and the Wightman functions exist as tempered distributions (Theorems A and

B of [11]). The Glimm-Jaffe proof was a tour de force. They bounded F(t) = (11° e-t''11o) by a complicated inductive expansion which paved the way for the even more difficult proofs for the 03 model.22 17i.e., for all e > 0 there is a b such that V_ < cHo + b as quadratic forms 18uniformly on a domain Dab as in [B2] but with a < it/2. 19For 0 = arg z E (-ir/2, 7r/2) with R(zA) > 0, and p, q E ( '`101 , there is a T > 0 such

that if z > T, then e-,H,\ is a bounded operator from LP(Qo,dµo) to L9(Q0idpo). 20For example, for r a positive integer and c > 0 sufficiently large, 21since (520, H100) = 0 and 52o is not an eigenvector for HI

N'''(H)+c)-r

is bounded.

22That the exponential decay of F(t) determines E1 is based on the important principle of "vacuum overlap." The operator e-HI is positivity improving and so, by a Perron-Froebenius

QUANTUM FIELD THEORY

79

Barry's simplified proof of the LLB in [B5] was based on using local number operators NJ defined over the one-particle space 7-t = 7-t_ 21 where the inner product

is given by (1), rather than over L2(R).23 He concluded that any method for proving a LB for N + V (g) automatically yields a LLB. Barry submitted his paper in May of 1971. Later that year, Francesco Guerra showed us how the LLB follows magically from Nelson's symmetry (see below) and so, like many other advances in the Hamiltonian era, [B5] was buried in a Euclidean avalanche. In the short paper [B6] Barry observed that there are spatially cutoff P(0)2 models whose Hamiltonians H(g) have eigenvalues embedded in the continuous spectrum. His "super synthesizer" reasoning was based on an analogous question in quantum mechanics for the helium atom Hamiltonian H(O) = Ho + /3V

Al -- 02 -

2

2

XI

x2

/3

+ xi - xz

on 7-l = L2(R6). The spectrum of Ho can be explicitly computed as the sum of two hydrogen spectra: there are eigenvalues Em n = - m2 - n2 , m, n = 1, 2, ... , and continuous spectrum on [-1, oc). Thus, for m, n > 1, E.,,,,,n is in the continuum. If one turns on the Coulomb repulsion (/3 > 0), one expects these eigenvalues to dissolve. However, some of the eigenvalues embedded in (-1, -1-4) are protected (for small /0) by the superselection rule associated with the symmetry S = (-1)JP, where J is the total angular momentum and P is the total parity.24 Barry considered the P(0)2 Hamiltonian HA,a(g) = Ho - AV2(g) + /3V(g) EA,Q(g) where V2(g) = f g(x) :O(x)2: dx; V(g) = f g(x) :P(O(x)): dx where P is an even polynomial of degree > 4 which is bounded below; 0 < A < m2 and /3 > 0; g 0 0 is a smooth function of compact support satisfying 0 < g(x) < 1; and EA,p(g) is chosen so that inf o-(Ha,p(g)) = 0. The spectrum of the quadratic Hamiltonian HA,o is completely known [38]: it has continuous spectrum [m, oc) and eigenvalues

< ej are in nle1 + + njej where j > 1, n1,... , nj = 1, 2, ... , and el < (0, m).25 Thus Ha,o has eigenvalues embedded in the continuum, most of which are expected to dissolve when /3 > 0 is turned on. The symmetry analogous to S is (-1)lv where N is the number operator, and its superselection rule protects the embedded eigenvalue 2e1.26 argument, its largest eigenvalue a-El is simple with a unique positive eigenvector Q, (see Theorem

V.17 in [B17]). Therefore (Oo,O1) > 0, i.e., the vacua overlap, and e-tE, 23For

(Oo,S21)2e-tEt < F(t) <

a bounded interval J C R, the local number operator N,I = dh(Pj) where Pj is

the projection in 7-l onto 7-(j, the closure in 71 of smooth functions with support in J. The disadvantage of these local number operators is that disjoint intervals have an overlapping count, i.e., NINj # 0 if In J = 0. This is not a major problem because the overlap is exponentially small for distant intervals due to the exponential decay of the kernel w-1 (x - y) in the inner product. 247-1 decomposes as a sum of subspaces 7-l, ®7-1, left invariant by H(Q), where 7i, (resp. 7-La) is the subspace of vectors of natural (resp. unnatural) parity, corresponding to the eigenvalue

+1 (resp. -1) of S. For small ,Q, H(,3)R. has continuous spectrum only on [-11, oo) so that the eigenvalues in (-1, - 11) are not embedded in the continuous spectrum of "unnatural parity." They do not dissolve. 25This follows from the fact that H,\ ,o is unitarily equivalent to dI'(µ9) where µg = [-0 + m2 - Ag(x)] 112 has at least one eigenvalue e1 in (0, m) [38]. 26T decomposes as a sum of invariant subspaces F0 ® .Fe, where the odd subspace .F0 (resp. the even subspace Fe) is the eigenspace for the eigenvalue -1 (resp. +1) of (-1)N. Restricted to .Fe, Ha o has a "protected" eigenvalue 2e1 below the continuous spectrum [m+e1, co). By taking

L. ROSEN

80

The letter [B7] is advertising copy for the strong asymptotic condition or SAC.14 With the physics community as audience, Barry pointed out that SAC makes it possible to validate formal statements involving divergelA perturbation series, such as unitarity of the S-matrix.

5. The Euclidean Revolution (1972-1975) In 1971 Francesco Guerra opened the door into another world for Barry and me: the Euclidean world. I can still remember Francesco quietly stepping up to the chalkboard in Princeton, writing down a list of incredible claims for the P(0)2 model, and claiming he could show us elementary proofs. Given our knowledge of Hamiltonian technology and our sense of what problems were within our incremental reach, Barry and I simply didn't believe him. One magical hour later, we believed!

Guerra's results were for the P(0)2 Hamiltonian H1, its vacuum energy El, vacuum vector Il1i and vacuum energy density al = -El/1.27 His main claims were:

1) A simple proof of the LLB: El > -c l + d;

2) a, = liml_,

0 al exists28 and equals sup, a1;

3) (van Hove phenomenon) Sll - 0 as l -- oc. Guerra's arguments were based on Nelson's theory of Euclidean Markov fields.

Indeed, in his now famous-but then almost totally ignored-Berkeley lecture of 1971 [27], Nelson had remarked on a simple proof of the LLB. The irony was not lost on Barry and me that we needed a visiting Neapolitan to make us aware of a fellow Princetonian's ideas. The main tool that Guerra borrowed from [27] was "Nelson's Symmetry": 29 (ho,e-tH,ho)

= (00,e-1H`ho)

(5)

When recast in terms of Euclidean fields, this mysterious symmetry becomes transparent as a corollary of Euclidean invariance. If you have not already had the pleasure of doing so, I recommend that you look at Guerra's two-page paper [16] to see how easily he deduced 1)-3) from (5). After Barry and I recovered from our shock and awe, the three of us launched an exciting four-year collaboration in which Barry was the driving force. Our first paper [B8] basically sharpened Guerra's results. For example, we showed that 1) al is strictly increasing, and, if P is normalized, El is strictly decreasing; 2) a1 is Lipschitz continuous; 3) /31 - -El - la,, is convex, decreasing and hence convergent to /3m = inf1 /3130; 4) there are positive constants c and d such that for large 1, e-o1 < IQIII < e-dl A small, Barry arranged that 2e1 > m so that 2e1 is in the continuum for HA,0. By the norm resolvent convergence of HA,,3 to H>,,0 as Q --> 0, this embedding survives for small ,Q (by a result of [18]).

27The Hamiltonian HI with a sharp spatial cutoff and with P(O) = 0 is described in the discussion of [B5] above. 28The convergence of the energy density in the infinite volume limit had been conjectured and its proof pursued. Osterwalder and Schrader [31] had obtained convergence for a simplified model in which the "Hamiltonian" was defined using the number operator N in place of Ho. 29Actually, all that Guerra needed was the single consequence of (5): a, > at it ln(llo, Sb). 30This gave the beginning of an asymptotic series for ai: cal = a +,31-1 +0(1-1).

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81

Probably the most convincing illustration of the power of the Euclidean methods in [B8] was an elementary proof of the Glimm-Jaffe GLLB (4), as I now demonstrate. By a limiting argument it's sufficient to prove (4) for g a step function of the
: SP(c(x)) : dx. By rotating the roles of l and t we obtain a generalized Nelson's Symmetry (Q0, e-tH1Qo) _ (Q0, e-(ao+2)Hte-(ai-ao)Ht(91)

... (6)

where on the right side the Hamiltonian semigroup which acts on the "time" interval (ai-1, ai) has "space" cutoff Xt and interaction polynomial P + g1SP. Therefore, (Q0, e tH' o) < (Q0, e (2ao+l)HtQo) y

le

i)Ht(9,) II(Q0, e

(az

(l

2a )Ht

o) 2

.

i=1 (7)

Reapplying Nelson's Symmetry, we bound the first factor by (1l

e tH1}2ao 1) z

<e

2 Ez+2ao

and similarly the last factor bye z Ei-2-- . By the LLB, the product of norms in e-(a,-a,_1)ct = eDct Hence by (7), (7) is bounded by fl' l tlim

ln(1lo e tH` Qo)

a (Ei+zao +

Dc < -El + Dc

by 1). We conclude that SEl = El - El > -cD. The reverse inequality follows from considering Hl = Hl - SV. I can still remember Barry's excitement on discovering this remarkably simple proof and then on showing it to Jim Glimm. When we wrote [B8], I must confess that we still hadn't fully grasped the power of the Euclidean Markov theory. We stated: "... we take [Nelson's Symmetry] as given and otherwise do not use Markov field techniques. We do so because we find the probabilistic techniques less familiar than the LP methods of Fock space and Q-space and we suspect the same is true for other quantum field theorists." 31 The above GRS proof of the GLLB does not actually produce the best norm If 11 in the q-bound: +Oo(f) < 11f 11(H, + 1). In one of his contributions [B15] to the 1973 Erice Lectures, Barry showed that by utilizing the Markov property more effectively it was possible to give a fairly short proof of the 0-bound with improved control on the norm, e.g., If 11 could be taken as jjf jjL1 or 1jf jj H_1. In [B9] we generalized the results of [B8] to spatial cutoff functions g more general than a characteristic function Xi, and we explored the minimal conditions possible on g that guarantee that the P(0)2 model is well-defined. These generalizations are mostly of academic interest but involved some ingenious bootstrapping between finite and infinite volume. Let E(Ag) be the vacuum energy for the

Hamiltonian H()g) defined in (1) with degP = 2n; let al(A) = -E(AXl)/l and a,, (A) = lima ,m al(A). Some of the results we obtained in [B9] were: 31It is worth noting that although the focus and methods of the Hamiltonian era were about to be swept away by the Euclidean tsunami, some results could not be readily proved in the Euclidean

formalism without using the Hamiltonian semigroup = transfer matrix. The monotonicity 1) of ca, is a case in point.

L. ROSEN

82

1. There are constants a and b such that aA(ln.))n

for large \

bA

for small A.

2. Using Nelson's Symmetry we proved the elegant bound

-E(g) < f ')o00

a.(g(x))dx.

This provided a way of transferring bounds on a,, back to E(g), e.g.: 3. For any e > 0, there are constants c and d such that32

-E(g) C c J g2dx + dJ g52

g1+Edx. >2

4. We could also show that H(g) was semibounded and essentially self-adjoint on C°°(Ho) with "almost minimal" hypotheses on g. As suggested by 3), instead of g E Ll rl L2 we showed that it was sufficient to assume that g E L2 + L'+E E > 0.33

The 1973 letter [B10] was an announcement for [B18] which didn't appear until two years later. Since the intervening publications [B11]-[B17] all depend on material in [B18], I shall now violate chronological order and review [B10] and [B18] together. The magnum opus [B 18] 34 investigated the analogy between CSM and EQFTor E(Q)FT, as Barry likes to write because of the commuting Euclidean boson fields. This was a project dear to Barry's heart. As I've already referred to under the label

"super synthesizer," one of Barry's great strengths is to notice parallels between different problems and fields, and to deploy ideas from one subject as a powerful tool in the other subject. It's impossible to do justice here to all the ideas and results in [B18]. I'll merely summarize the contents of the paper, and select a couple of topics for a more detailed discussion. The starting point for [B18] was what we called the Feynman-Kac-Nelson (FKN) formula. Similar in structure to the FK formula [21], the FKN formula identifies the Euclidean Markov theory as playing the role of a Euclidean covariant path space for the Hamiltonian theory. (Nelson's Symmetry is an immediate 32This control over E(g) allowed a more general convergence result for the vacuum energy

density: Let {gj } be a sequence of functions such that 0 < gj < cXj for some constant c and

II9j - Xjlli/7 - 0 as j --> on. Then E(gj)/II9jii1 -> -a.. (We were unable to remove the conditions that the gj's be uniformly bounded or have compact support in [B9], but we did eliminate the latter condition in [B25].) 33The notation g E L2 + L1+' means that g = f + h where f E L2 and h E L1+'. This Ig(k)I2(lnw(k))"-1w(k)-1dk < on, which is certainly a necessary condition implies that gj2n = f condition on g for H(g) even to be defined on C°° (Ho) since II:02n (g):Oo 11 > cI912n We conjectured

that this condition was also sufficient for semiboundedness and self-adjointness of H(g). 34We submitted the 147-page manuscript [B18] to the Annals of Mathematics. The editor was unwilling to have an entire issue monopolized by one paper, no matter how important the authors thought it was. So he came up with a Solomonic solution, namely, to cut the paper in half (well, into parts of 79 and 68 pages) and to publish the "halves" in each of two consecutive issues. Curiously, however, he put Barry's eight-page paper [B19] immediately after Part II, perhaps as an apology or perhaps to redress the imbalance in the lengths of the halves.

QUANTUM FIELD THEORY

83

corollary.) A special case of the FKN formula is tn_

(tn

(Qo, e-(,'

)H'...e-(t2-t1)H`0o(x1)e-(t1+21)H`Qo) (8)

A(, .l

where t > 0 and - < tl < t2 <

Ilo) O(xn) tn) ... O(xl, tl) e-u`tdy

f

(9)

e-U`t dµ

< to < 2 , (8) lives in Fock space with 00 the

time zero field and H1 the P(0)2 Hamiltonian of [B5], and (9) lives in the Euclidean world with the integrals taken over Euclidean Q-space, d,u the gaussian measure of t 2 (3), 0 the Euclidean field, and Ult - f z, f z :P(O(x, s)): dads. The Wick ordering L

here is with respect to the Euclidean measure dµ.35 The limit t -+ oo is easy in (8). By a Perron-Froebenius argument22 the "transfer matrix" Ti = e-1 ' has a simple (largest) eigenvalue 1 with positive eigenvector Q1. Thus, as t oo, TI' 0 -* (521, QO)c1 and we obtain from (9): 0o(xn)e-(tn-to-1)H,

(521,

...

e-(t2-t')H`00(x1)521)

lb(x1, t1) e-Uitdµ = t_00 lim f Oxn, tn) f...e-vet dµ

This "half solves" the infinite volume limit. In [B18] more general regions A C R2 than rectangles were considered, with associated probability measure dvA = e-U"dp/ f e-U"dµ where UA = f fA:P(O(x,t)): dxdt. dvA defines a Gibbs expectation on Q-space whose correlation functions are the Schwinger functions. Taking a more abstract point of view, we defined a state p to be a family {PA} of non-negative, normalized functions on Q-space, labelled by bounded sets A C R2 and satisfying certain consistency and regularity conditions.36 With each state p, the entropy associated to A was defined to be S(p, A) = - f PA In PA dµ.

The structure of the Euclidean P(0)2 model as a model of CSM was then apparent:

P(0)2 E(Q)FT

CSM Gibbs expectation in A Partition function in A Pressure in A Correlation function in A State p Entropy in A of a state p

f dvA ZA = f e-UAdp. PA = TA-'j In ZA

Schwinger function SA(xl,... , xn) = f q5(x1) ... Family {PA} of consistent densities S(p, A)

dvA

f pA In pA dµ

35To make Wick subtractions with respect to a gaussian measure dµ with covariance C on Euclidean Q-space, we follow the same procedure as with the time zero field4; i.e., we place an ultraviolet cutoff on the Euclidean field (to be removed later) by Oh = h * 0 and define Wick powers via the generating function ±h (x) t o = etOh(-)-2

:et0h(X): = n=O

361f A

f Oh(x)2dA = etOhW-yt2 f f h(x-y)h(x-y')c(y,y')dydy`

n.

C A', EA(PA) = PA; PA E LP(dp) for some p > 1.

L. ROSEN

84

While this analogy opens up the possibility of analyzing boson field theories with the ideas and techniques of CSM [39], there is one significant difference. In CSM the free measure is local and it is the interaction which couples different regions, whereas in EQFT it is the reverse.37 Here is a brief summary of (some of) the contents of [B18]: 1. An exposition of the free Euclidean Markov field theory following Nelson [29]. 2. A study of the spatially cutoff P(cb)2 model based on the FKN formula. 3. Introduction of conditioned theories, in particular, models with Dirichlet boundary conditions (B.C.). 4. Hypercontractive estimates for the free measure dtt including "Sandwich" and "Checkerboard" Estimates. 5. Approximation of the P(0)2 model on R2 by Markov field theories on the lattice 6Z2, 6 > 0. 6. Using 5), a proof of correlation inequalities for P(0)2 of Griffiths [14] and FKG [3] type.

7. Applications of the correlation inequalities such as monotonicity of the mass gap with respect to various parameters (the coefficient of the quadratic term in P, the bare mass, the spatial cutoff parameter 1). 8. Existence of the infinite volume limit of the pressure PA as A -+ oo in the sense of van Hove38 for both free and Dirichlet B.C. (see the discussion of [B26] below).

9. Properties of the entropy39 and convergence of the entropy per unit volume,40 i.e., s(p) = limn-,m S(p, An)/IAnl exists if An -+ oo in the sense of Fisher. 10. Formulation of the DLR equilibrium equations for P(0)2.41 11. Formulation of the Gibbs Variational Principle for P(0)2 states.42 37The measure dp is non-local because its covariance (-0 +

m2)-1 is.

38The notion of convergence of An C Rd to oc in the sense of van Hove or in the stronger sense of Fisher are familiar from CSM (see [39] or Appendix C of [B18]). Roughly speaking, van Hove convergence controls the regularity of the An's and Fisher convergence ensures that they don't get too bent out of a basic square shape. 39For example, S(A) = S(p, A) is monotone increasing and "weakly subadditive" in A, i.e., if Al and A2 are disjoint regions, then S(A1 U A2) < S(A1) + S(A2) + r(A1, A2), where r(A1, A2) vanishes exponentially in the distance between Al and A2. We called this weak subadditivity because of the last term. It is absent in CSM but present in EQFT due to the non-independence of disjoint regions. 40The proof of convergence relied on the Checkerboard Estimate and required additional

assumptions on the state p, namely, translation invariance and a growth condition such as )2

where d(An) is the diameter of An. 41The DLR = {Dobrushin, Lanford, Ruelle} equations of CSM give meaning, independently

IIPA, IILP < ead(A,

of limit procedures, to the statement that a state is an equilibrium state for a given interaction at a given temperature. We proposed, as the DLR equations for a P(t)2 state p, that PA =

where OaA E L1(dµ) is a function of the fields on OA, i.e., 08A is EA-

measurable for any open set A 3 aA. We called such a state Gibbsian. The factor '5,A takes into account the part of the interaction outside A, and the 08A's satisfy consistency equations corresponding to those of the pA's. We gave a simple 1D example in [B 18] showing that for these consistency equations to have a unique solution they must be supplemented by B.C. as A - oo. 42The mean interaction associated to a state p is given by u(p) = f UAPAdµ. It's easy to see that u(p) is independent of A because of the locality of the interaction. We established the Gibbs

variational inequality s(p) - u(p) < a for a suitable class C of states p, and we conjectured, on the basis of CSM, that the Gibbs variational equality held: supp,c[s(p) - u(p)] = a. with equality if p is Gibbsian. In [B25] GRS established equality in the sense that the sup = ate,

QUANTUM FIELD THEORY

85

12. A list of fifteen open problems that arose naturally from our interpretation of the P(0)2 EQFT as a model of CSM.43 Our results on the DLR equations and the Gibbs Variational Principle were only of a preliminary nature. We introduced these notions from CSM in the hope that ultimately they would provide a characterization of infinite volume states in a EQFT, independently of cutoff procedures. A few years later, in collaboration with Jiirg Frohlich, Barry succeeded in carrying these ideas much further [B26]. From all this material, I shall now examine topics 4)-6) a little more carefully. An important element in the control of the infinite volume limit in CQFT is the ability to take advantage of the exponential decoupling of distant regions [10]. We found an elegant way to express this decoupling on Euclidean Q-space, which I now describe in d dimensions. If ul and u2 are independent random variables, then Ilulu2MM1 = Ilu1IIiIMu2MM1. For general random variables one can do no better than IIu1IIp1 Ilu2llp2 where pi and p2 are conjugate Holder indices (pi l+pZ 1 =

Ilulu2111

1). However, we showed that distant regions decouple exponentially in the sense that if uJ is EA3-measurable then pl and p2 may be taken exponentially close to 1 for large r = dist(A,, A2); more precisely, Ilulu2Ill <- IlulIlpi IIu2IIp2 if (P1 - 1)(p2 - 1) = e(r),

(10)

where e(r) = O(rd-le-2mr) for large r.44 When the geometry is simple, one can be more specific about e(r). For instance, if Al and A2 can be separated by two parallel hyperplanes a distance r apart, then e(r) = e-2mr in (10). This estimate can be generalized by induction to cover a "checkerboard" situation. We state this generalization in the simple case of a square checkerboard in 2D: Let {Aij}' _1 be an array of 1 x 1 squares separated by strips of width r. Let uij be EAij -measurable. Then 11 fz j uijll1 < Flz,j Il uijIlp, where p = [ 1 1+e' ] 2.45 The usefulness of the Checkerboard Estimate is that it allows a factorization over an arbitrary number of disjoint regions even though such regions are not strictly independent. Turning to 5), we consider the lattice approximation in d dimensions in which Euclidean space Rd is replaced by the lattice 6Zd = {n6 I n E Zd} with spacing 6 > 0 and the Laplacian A is replaced by the finite difference operator (Abf) (n6) = 6-2 [_2df(n) +

n1-n

f (n'b)]

(11)

where the sum takes place over the 2d nearest neighbours n' of n, which we denote m2)-1 by n''-n. Now the free field measure dp with covariance (-A + can be formally written as d,u = const. e 2 (0, (-o+m2)0)do.

(12)

and in [B26] Frohlich and Simon proved the stronger result that the actual infinite volume states constructed for P(c)2 gave equality. 43A number of these problems were solved in the next three years by a variety of people, Barry in particular. 440ur proof consisted of controlling the norm IeA1 eA211 of the product of single-particle projectors and lifting this bound to Q-space by hypercontractivity.16 45The vacant strips were needed in the proof of convergence of the entropy per unit volume. As r becomes large, p -> 1 and the regions &3 become stochastically independent.

L. ROSEN

86

(11) suggests that the quadratic expression in the exponent in (12) is negative "on-diagonal" and non-negative "infinitesimally off-diagonal." To realize this "ferromagnetic" property in a rigorous way, we introduced a new type of ultraviolet cutoff field, the lattice cutoff field 06(n), n E Zd. We defined 06(n) by smearing with a function46f6,,,,(x) concentrated around x = nd and specially chosen so that the variables q,,, = 06(n), n E Zd, have covariance matrix C = 6-d(-06 + i.e., C-1 - A is given by (see (11)) m2)-1

I m2Sd + 2d8d-2 if n = n' A,nn, _

if n-n'

-6d-2 10

(13)

otherwise.

Given a bounded region A C Rd, let A6 = An SZd be the N lattice points in A, and let UA,6 = Sd En6EA6 :P(Cp6(n)):, where Wick powers are defined with respect

to the covariance C. Given h E Co (Rd), let 06(h) = 6d EnEZd g6(n)h(n6). For the P(q5)d theory the (smeared) Schwinger functions with space cutoff A and lattice cutoff S are defined by

SA,6(hl,...,h,.) =

f g6(hl) ...

06(h,)e-UA,sdp

fe-UA,6dlt

(14)

These definitions have accomplished several things: 1) The integrands in (14) involve only a finite number of field variables, q,,,, n = 1,.. . , N (where we assume for convenience that supp hj C A). Thus the

numerator in (14) reduces to a sum of terms of the form constj q,,1 ... qn,, e-6dv:P(4,):dµA,6(q) with gaussian measure ditA,6(q) = const. e 2gciA1q dNq, where the N x N matrix CIA is the covariance matrix C restricted to the indices in A. Similarly for the denominator in (14).

2) The approximation is locality preserving in the sense that the Gibbs factor e-6dE:P(4_): involves a sum of terms each depending only on a single qn,. 3) But what is most critical for our purposes is the structure of the matrix CJA1 in the gaussian density. Although 54 (C-')IA = AAA, equality almost holds: CIA1 = AIA - BaA, where BaA is a positive semi-definite matrix with non-negative elements that is concentrated on the "boundary" 9A6 of A6.47 Thus, -CIA1 is ferromagnetic, i.e., it has non-negative off diagonal elements. Moreover, except for the boundary variables, CIA1, like A, links only nearest neighbours. This is the Markov property on the lattice. What is the significance of the boundary matrix BaA? Different choices of Ban correspond to different choices of B.C. The above BaA which arises from "integrating out" the field variables outside A6 gives free B.C. On the other hand, if we took BaA = 0, we would obtain Dirichlet B.C. (for more on B.C., see the discussion of (CAA)-1

[B25] below). Let me summarize. Regardless of our choice of B.C., the lattice approximation

realizes the P(¢)2 theory as a lattice spin system with ferromagnetic interactions that act between nearest neighbours, except possibly for the boundary variables. 46 f6,,

(.) =

(2n)-a f d T5

eik(x-ns) w(k) dk, where T6 = [-'r/a, 7r/a], w(k) = (k2 -F,n2) 2 , and w6 (k)

w6(k) = [26-2(d - Ed 1 cosSki) F n2] 1 . 47 OA6 = {n8 E A6 1 3n'-n with n'S

A6}.

QUANTUM FIELD THEORY

87

The Gibbs factor arising from the polynomial P is local and does not couple spins but only mediates the interaction of spins by perturbing the distribution of each uncoupled spin. Thus we defined a ferromagnetic measure on RN to be one of the form dv(q) = Fi(g1) ... FN(gN)e-gMgdNq, where M is an N x N positive definite matrix with non-positive off-diagonal elements, and the Fi are positive, bounded, continuous functions on R. If each Fi is even, we say that v is even. We write (f), = f f (q)dv(q)/ f dv(q) Our proof of Griffiths inequalities for even ferromagnetic measures was adapted

directly from Ginibre's elegant analysis [6]. For functions f and g in the Ginibre class g,48 the first and second Griffiths inequalities (G-I and G-II) read (f), > 0 and

Y; g)' = (fg)v - (f)v(g)v > 0.

(15)

A ferromagnetic measure on RN satisfies condition (A) of [3], and so we also obtained the FKG inequality, i.e., (15) holds for f and g increasing functions on RN.49 There is no need for v to be even. The G-I, G-II and FKG correlation inequalities transferred immediately to the continuum theory once we controlled the limit S - 0. For P(q5)2 the convergence of the lattice approximation with free B.C. was just a variation of the standard proofs of semiboundedness and removal of the ultra-violet cutoff. In particular, objects like 06(h) and e-UA,a converge in LP(Q, dµ) for any p < oo so that, by Holder's inequality, the Schwinger functions (14) converge. For the case of Dirichlet B.C. some mild regularity conditions on A were required. Let SAX be the P(ql)2 Schwinger function with Gibbs measure dvA where X denotes the B.C., e.g., X = F (free) or D (Dirichlet). We established that, for even P, SAX satisfied G-I: SS (xi, ... xr) > 0, and G-II: )

SA (xl, ... , xr+s) > SA (xi, ... , xr)SA (xr+i) ... , xr+s).

(16)

Moreover, the FKG inequality held for any semibounded P, i.e., for F, G in the cone I of increasing functions of the field50

(F; G)

FGdvA

- f Fdv' f Gdvv > 0.

(17)

It should be noted that, while the G-II and FKG inequalities are both statements of positive correlation, the classes of allowed functions are different and the class of measures is more restricted for G-II. Do Griffiths inequalities hold without

the evenness condition on P? For lattice spin systems the answer is yes: using Ginibre's method, we showed that G-I and G-II hold if P is a semibounded polynomial of the form P = Pe - Q where Pe is even and Q is an odd polynomial that is positive and increasing on (0, oo). The roadblock to extending this result to P(0)2 is that the Wick subtractions upset the hypotheses51 on Q, and so we could only allow Q(x) = ax, a > 0. We also tried to establish Griffiths inequalities involving 48g is defined as the set of functions on Rr' that are sums of products fi (qi) ... fN (qN) where each fi(gi) is of the form fi(gi) = e(gi)gi(Igi1) where gi is a positive, increasing, polynomially bounded function on (0, oo) and e(qi) = 1 or sgngi. For example, the monomial qi1 ... qjN E

where i1 , ... , iN are non-negative integers. Ginibre's key step was establishing his condition (Q3): for gl , ... , g.. E 9 and any choice of the ± signs, f f i l 1 [gi (p) ± gi (q)] dv(p)dv(q) > 0.

49i.e., f(-) > f (y) if xi > yi for each i = i, ... , N 501 consists of functions of the form f (¢(h1), ... , 4(hr)) where hl,... , hr > 0 are in Co and f : R' - R is increasing. 51For r > 1 the rth Hermite polynomial is not positive and increasing on (0, oo).

L. ROSEN

88

Wick powers, but the only possibility seemed to be G-II involving :02: since for it the Wick subtraction is a constant. We explored uses of the correlation inequalities modelled after the traditional applications in CSM, e.g., an increase in correlation lengths or persistence of a phase transition if the interaction is made more ferromagnetic. For P(cb)2 "more

ferromagnetic" means that the coefficient of 02 has been decreased or that the nearest neighbour interactions have been increased. For example, F B.C. are more ferromagnetic than D B.C. because there are additional couplings across OA. The analogue of [correlation length]-1 is the physical mass or the gap in the spectrum of the Hamiltonian above the vacuum.52 However, what eluded us was reproducing one of the most important applications of correlation inequalities in CSM, namely, using G-II to prove monotone convergence of the P(0)2 Schwinger functions as A increases. Our inability to do so was most perplexing. Suppose A C A' and we wish to compare SF with S S. Lacking G-II for Wick powers, we couldn't simply turn on UA'\A monotonically. Nor could we start with SK in the lattice theory and replace BaA with B&A' because this does not give a consistent change in ferromagnetic couplings. As for D B.C., starting with SD and turning on the couplings across OA does give an increase in the strength of the ferromagnetic couplings and hence an increase of the Schwinger functions except for one snag: it's necessary to revise the Wick ordering to match the new D B.C. on aA'?!? Ed Nelson saved us by pointing out in [30] that the solution was to use hybrid

B.C., namely, D B.C. on the gaussian measure and F B.C. for the Wick ordering. The correlation inequalities obviously held for these B.C. which we called "half-Dirichlet" (HD).53 The Griffiths inequalities thus implied that for even P the Schwinger functions S D (x1, ... , x,) with distinct arguments were monotonically increasing in A and converged to an infinite volume limit.54 This convergence

provided the first construction of the PM2 model without cutoffs in the strong coupling regime, where the questions of dynamical instability and phase transitions

become interesting. A happy ending! Not quite-there's more to the story (see [B26] below).

The paper [B11] was inspired by work of Lebowitz [22] on the classical Ising

model with spins si = ±1. Lebowitz proved that the exponential decay of the truncated 2-spin correlation (si; sj) as I i - jI - oo represents the correct decay of all truncated spin correlations. His main technical tool was the FKG inequality which he used to deduce that 0 < (nA; nB) C 4

(si, sj),

(18)

iEA,jEB 52See pages 224-228 of [B18].

53The convergence of the lattice approximation for HD B.C. involved controlling the dif-

ference in Wick ordering (which gives rise to terms which diverge at 8A like a power of L(x) = j In dist(x, 8A) I) and placing another mild condition on the region A (fn L(x)3 dx < 00 for any positive integer j). 54The required upper bound on the Sn D is supplied by correlation inequalities (Sn D < Sn )

and the q-bounds for F B.C. In the special case of P(x) = ax4 + bx2 + cx (a > 0, c < 0), we obtained convergence of SAD with full D B.C. because G-II for :02: gave us control over the change in Wick ordering.

QUANTUM FIELD THEORY

89

where ni = '(I + si) is the occupation number (0 or 1) at site i and nA = 1 1iEA ni for a set of sites A. Since GRS had just established the FKG inequality for P(0)2i Barry realized that it should be possible to prove an analogous result for P(ql)2. However, unlike

the si, the fields 0(f) are unbounded. Barry's solution to this technical nuisance was to chop off the fields to make them more Ising-like: Let B(x) = x if x < 1 and B(x) = sgnx if xJ > 1. Define the "spin" o(f) = B(O(f )) and the "occupation number" v(f) = 2 (1 + o (f )) (which takes values in [0, 1]). For non-negative f, fi E C0 (R2), let EA = EiEA v(fi) and HA = UiEA v(fi) where A is a finite index set. These constructs provide useful, non-trivial observables in the class Z of increasing functions of the field. It's easy to see that the following are all in Z: 0(f), 0'(f), 0(f)-Q(f ), EA, HA, EA-HA. This observation allowed Barry to deduce55 the analogue of (18): 0 < (HA; HB) <_ (EA; EB) <

(19) iEA jEB

where the truncated expectations are as in (17) (with the indices A and X suppressed). By appealing to the FKN formula (9), Barry finessed (19) into the conclusion

that in the spatially cutoff P(0)2 theory the field 0o(f) couples the vacuum 52 = S2(g) to the first excited state of H = H(g) -E(g).56 As corollaries of this coupling, Barry drew these conclusions about the infinite volume P(0)2 model: 1) If P is even, then the first excited state is odd, i.e., it is taken into its negative

by the 0 - -0 symmetry. 2) The vacuum is unique if (0(0); O(x)) -- 0 as x --> oc.

3) There is a mass gap > m if for each f c Co (R2) and c > 0, (0(f t); 0(f)) O(e-('-E)t)

as t --> oo, where ft(x, s) = f (x, s - t). The importance of 2) and 3) is that to answer questions about the uniqueness of the vacuum and the size of the mass gap one need only look at the two-point function.

Barry's paper [B12] with Griffiths is an announcement for their paper [B14], and so I shall review these two papers together. What Simon and Griffiths accomplished was to approximate the 02 lattice spin model (and hence the continuum 02 model) by a classical Ising model, where "classical" means that the spins take the values +1 and are coupled by ferromagnetic pair interactions. The purpose of the approximation was to carry over to 02 special results known for the classical Ising model, such as the Lee-Yang Theorem and the GHS correlation inequality of Griffiths, Hurst and Sherman. 55The first inequality in (19) is just FKG applied to the increasing functions HA and IIB; the second inequality follows from adding the FKG inequalities (EB; EA-IIA) > 0 and (EB-1IB; HA) > 0.

56We sketch the ideas. For vectors P, 1 in Fock space F, let (,P, 1)1 = (T, 4)) Q) (Q, 4)). We wish to show that for any e > 0, there is an f E Co (R) so that for large ce-('v-1)t where ry > 0 is the gap in the spectrum of H above 0. t (co(f)Q,e-tIco(f)11)L > For 9i E C o (R), 9i > 0, i = 1, ... , n, let fi(x, s) = 9i(x)o(s), and .fi (x, s) = 9i(x)a(s - t). Taking the functions indexed by A to be fi, ... , f f and those indexed by B to be . . f L,, we deduce from (19) and the FKN formula that (v(fl) ... v(fn)Qo, e-tF-(.fi) ... v(.fn)Oo)1 < a Ei j-1(0o(gi)Q e-ta0o(9j)Q)1 Since the vectors v(fi) ... v(fn)Qo are total in F, the gi's . . .

can be chosen so that the left side is > ce-('v-')t, and we obtain the desired conclusion.

L. ROSEN

90

The basic idea is so clever that I can't resist showing it to you. The problem is -aq4-bq2 of the 04 model to approximate the single spin probability density F(q) = by the suitably resealed probability density for the sum s = s1+ +sn of n classical Ising spins. The 02 ferromagnetic measure on RN can then be approximated by a classical Ising system of nN spins. Let do > 0 be the strength of the ferromagnetic coupling (to be chosen below). Now s takes values in Zn = {-n, -n+2,..., n-2, n} with probability edn8z edn82 = n ed" F_#, s,s, = e 2 C2

2

81 .. .,8n

g1+. ..+Sn.=s

s1

s1,...,sn

\(n + s)/2

(20)

where cj is an n-dependent normalization constant. For large n and s = o(n), a

i

straightforward asymptotic analysis of the binomial coefficient yields57 n

((n +

s)/2,,,

1

4

c3e-ZnS2_ 12nT8

(21)

Thus the quartic term will emerge from the binomial coefficient provided we 8ns where 8n = can control the quadratic term. We rescale s by so that 12 s4 = as4. Then we extend the discrete variables with values in BnZn to a continuous variable q whose values lie in In = [-(n + 1)8n, (n + 1)8n) and whose probability density Fn(q) agrees with the distribution of s in the sense that Fn is piecewise constant with Fn(q) = c4w(9j) if q is within 8n of the value sj of s. Explicitly, Fn(q) = c4w([q]n), where [q]n is the unique element of Zn in (q/8n - 1, q/8n + 1]. Note that In - (-oo, oo) as n -- oo. By (20) and (21), (12an3)-1/4

Fn(q) - c4e

2

12

[qln

[e]m+dn[eln,.,

c5

aa4+(dn 2;,)bn 2g2.

Choosing do = -L-b8n2 (which is positive for large n), we obtain Fn(q) Simon and Griffiths, of course, provided a more careful error analysis.

c5-aq4-bq2

With their approximation in hand, Simon and Griffiths obtained the GHS Inequality58 and the Lee-Yang Theorem59 for the (a04 + b02 - µO)2 model 60 and the expected consequences analogous to those in CSM; for example:

1) concavity of the "magnetization" M(µ) = (0(0)), 2) monotonicity of the mass gap m(µ), 3) analyticity of the "pressure" = energy per unit volume ca,, (µ) for Re i > 0, 4) M(µ) = c X (p) for u 54 0. As explained in §8 and §9 of [B14], these analogies with CSM clarified the meaning of a phase transition for the 0z model and confirmed the validity of the "conventional wisdom" or Goldstone picture. 57This is the famous De Moivre-Laplace result with an s4 correction.

58Let (-) denote the expectation value for the spatially cutoff (a¢4 + b02 - µo)2 model, where a > 0 and µ > 0, with F, D, or HD B.C. (or for the corresponding infinite volume theory). Then for any non-negative Co' (R2) functions f,g,h: (O(f)ct(g)q5(h)) +2(q(f))(q(g))(O(h)) (0(f))(o(g)o(h)) - (O(g))(O(f)O(h)) - (O(h))(O(f)4)(g)) < 0. 59Let denote the expectation value for the spatially cutoff (a04 +b02)2 model with F, D, or HD B.C. where a > 0 (or for the corresponding infinite volume theory). Let F(z) = (ezcblf)) where f is a non-negative function in Co (R2). Then F(z) # 0 if Rez > 0. 60Simon and Griffiths observed that the Lee-Yang Theorem and the GHS Inequality fail for some 6th degree polynomials, and Spencer has pointed out that, for n > 2 and A small, the A 02n model cannot be approximated by classical Ising models because of the sign of the 4th Ursell function.

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In [B13] Barry examined the relationship between the Wightman-Garding axioms (WGA) and those of Nelson (NA), by addressing the question: What must be added to WGA to guarantee the existence of an EQFT satisfying NA? Now there are obviously technical assumptions in NA which go beyond WGA, e.g., the existence of and regularity bound61 on the time-zero field operators 00(f). Aside from these technicalities, Barry emphasized that the additional ingredients required in WGA were that the vacuum vector SZ be cyclic with respect to the time zero fields, and that the Hamiltonian semigroup be positivity preserving.62 With these extra conditions and some minor technical assumptions, Barry showed how, starting with a Wightman field theory, one could construct a Euclidean Markov field theory related to it via the FKN formula. His proof was based on the result from probability theory that a positivity preserving semigroup generates a Markov process. Barry also gave an example of a WGA model in one space-time dimension for which there was no NA counterpart. In this example, with Hamiltonian H =

p4 + p2 + q2 and "field" q(t) = eztxge-ztx the failure of e tH to be positivity preserving was an expression of the fact from probability theory that only second order differential operators are suitable as generators of Markov processes. The title of [B16], "The pressure is independent of the boundary conditions for P(0)2 field theories," announced the central result of [B25]. Postponing the details and significance of the result until the discussion of [B25], I just want to explain here what the title means. Given a bounded, open region A C R2, we can consider several different P(0)2 theories in A depending on what B.C. we choose.

In [B16] we considered the four classical choices X = F (free), D (Dirichlet), N (Neumann), and, for A a rectangle, P (periodic). When X B.C. are placed on the covariance or Green's function C = (-0X + m2)-1 the corresponding gaussian measure is denoted dpA A. There is the additional complication that the Wick ordering in the interaction UX = fA : P(O(x)) :y dx can be defined with respect to any of the CX's. Out of the sixteen possible B.C. B = (X,Y), we considered only those which were matched, B = (X, X), which we called X B.C., and, from the unmatched cases, we considered B = (X, F), which we called "half X" (HX) B.C. The main result announced in [B16] was that for the P(0)2 model and B = F, D, N, P, HD, HN, HP, the limits limA-0 aA all exist and are equal. The principal use of this result is to provide flexibility in the choice of B.C. Barry's book [B17] about the P(0)2 model was based on lectures he gave in Zurich in 1973 and is a good example of how his skill at "speed-writing" granted a wide audience access to current research. In writing his book, Barry emphasized the probabilistic approach-to such an extent that this confirmed quantum field theorist often felt as though he were reading a book written by a born-again probabilist. The author's new outlook is signalled right off the top with the dedication: "To Ed Nelson, who taught me how unnatural it is to view probability theory as unnatural." Since much of the content of [B17] overlaps with material in Barry's papers [B9]-[B20] and since I have already made frequent reference to [B17] in my review, 61100(f) I < If II (H+ 1)' for some n and Schwartz space norm If II Such a bound is expected to hold in theories with a finite field strength renormalization. 621f F : R' - [0, oo) and G : R" -+ [0, oo) are bounded and measurable, and 91, ... , g E S(Rd-1), then (F(0o(f1)..... Oo(fm))Q, e 1HG(go(91), ... , 00(9n)A > 0 fort > 0.

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I shall content myself with picking out just a few of the features of the book not covered by the research papers. In Chapters II and IV, Barry gave a detailed account of the various possible axiom schemes (Wightman-Garding, Nelson, OsterwalderSchrader) and of the relations among them. As a matter of fact, his contribution to our understanding of the axioms goes beyond dissemination. Barry's questioning of the proof of a technical lemma in Osterwalder and Schrader's first paper [32] led to their modification12 of the axioms and the overhaul of their reconstruction theorem

in [33]. I also note the acknowledgement that Konrad Osterwalder paid Barry in his 1973 Erice lectures on the OS axioms: "I would also like to thank Prof. B. Simon for sending me prior to publication a copy of Chapter II of his Zurich lecture notes

[B17]. It was very helpful during the preparation of these notes." In his analysis of the Nelson axioms, Barry found it necessary (with the inventor's approval) to strengthen the regularity axiom61 on the field to be a bound on loo(f) l instead of ±00(f). Maybe I should add the label "gap detector" to my list. Finally, I mention that in Chapter IX there is a brief discussion of Lebowitz' inequality for the (a04 + b02)2 model63 with some interesting applications to the energy specrum of the model.64

In [B19] Barry considered the P(0)2 model with P(0) = a04+b02-µ0, where a > 0 and tt:yk for which he and Griffiths had established the Lee-Yang Theorem. Nelson's observation that the HD Schwinger functions converge monotonically as

A/R 2 leads to an infinite volume theory with expectation ( ),, whose relativistic QFT satisfies all the Wightman axioms with the possible exception of uniqueness of the vacuum. In [B19] Barry established the missing axiom.65 He reasoned as follows. By the Lee Yang Theorem aAD(µ) _ JAI In f e-fA'P'd,a is analytic for p in the right half complex plane. By [B16] we know that aA D (µ) converges on the positive real axis as A - oo. It follows that aAD(µ) converges uniformly on compact subsets of the right half plane, and, by the Cauchy integral formula, so do its derivatives. In particular, dµ2 aA D = T' (O(xA); O(xA))A D converges, yielding the bound (q5(XA); O(xA)). D < dJA1 for A a square. By a geometric argument and the use of G-II, Barry extended this bound to the infinite volume theory: (O(xA); O(XA))m <_ d1AJ for A a square.

(22)

If the vacuum were not unique, the two-point criterion 2) of [B11] implies that there is a c > 0 such that66 (O(x); O(y)), _> c, and so (O(xA); O(xA))ac > cIA12. This contradicts (22). Barry's arguments established the following hierarchy among the various crite-

ria for dynamical instability for the PM2 model with P(O) = a04 + b02, a> 0: the infinite volume HD theory for P does not have a unique vacuum the magnetization M(µ) for P(0) - p4 is discontinuous at p = 0 the pressure am(p) for P(0) - p is non-differentiable at µ = 0 63Lebowitz' inequality asserts that the fourth Ursell function u4 < 0 at zero magnetic field. 64Probably the most striking application of these ideas is to the spectrum of the anharmonic

oscillator. For example, if E1, E2, E3 are the first three eigenvalues of the Hamiltonian H = p2 + aq4 + bq2 (a > 0), then E3 - E2 > E2 - El . 65Uniqueness of the vacuum was already known for the AP(0)2 theory for arbitrary semibounded P and small A > 0 [13]. 66This conclusion uses the fact that, by the FKN formula, (fi(x); 0(y)) - is a monotone decreasing function of Ix - yl.

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the infinite volume HD theory for P does not have a mass gap.

[B20] is the third paper in the series [B11, B19, B20] on correlation inequalities and the mass gap for P(cb)2. Let ml(p) be the mass gap for the Hamiltonian Hl (µ) on [-1/2,1/2] with interaction polynomial P(0) = a04 +b02 -µ0 and periodic B.C., i.e., mi (µ) = E2 (µ) - El (µ) where E1(µ) < E2 (µ) are the first two eigenvalues of H1(µ). According to the Goldstone picture, in the critical region b << 0, P has two equal minima if p = 0 and so one expects two equilibrium states and no exponential clustering, i.e., ml (µ) = 0. However, if p 4 0, P has a unique minimum and one expects clustering and ml (µ) > 0. In his cluster expansion analysis of the P(0)2 model with a large external field [46], Spencer confirmed the validity of this picture by proving that indeed ml (p) > 0 for lj >> 1. The goal of [B20] was to extend this result to all µ 0. That such a "continuation" was possible was suggested by Penrose and Lebowitz' beautiful use of superharmonic functions to prove the clustering of Ising-type models [34]. That the application of their methods to P(0)2 was not entirely trivial was due to a mismatch in B.C.: Spencer's analysis required P B.C., whereas the existence of the infinite volume P(q)2 theory was known only for D or HD B.C. The resulting technical difficulties over B.C. involved questions on which Barry, Francesco and I were working at the time. So, on Barry's invitation, we put the GR back into GRS in [B20]. What is the role of superharmonicity? What is superharmonicity? A function

f from an open set S1 C C to R U oo is called superharmonic if i) f is lower semicontinuous on 52, and ii) if {z z - aI < r} C Q then f (a) > z- ff" f (a + reia)dO. The analyticity provided by the Lee-Yang Theorem enabled us to prove that the function ml (µ) on (0, oo) has an extension to a superharmonic function Ml (µ) > 0 on the right half plane.67 Now it follows from Green's formula that if the superharmonic function f is non-negative on an open set 1 and strictly positive on an arc in S2, then it is strictly positive on all of Q. Thus Spencer's positivity result m1(µ) > c(µ) > 0 for large y continues superharmonically to all µ > 0 (with a similar result for all p < 0)! What about the mismatch in B.C.? By Barry's two-point criterion, we needed to show that for non-negative f E Co (R2) and p 0, (O(ft)i0(f))o00 = (0(ft)0(f))oo - ((O(.f))00)2 =

0(e-,I).

(23)

Courtesy of Spencer and superharmonicity, there is such decay for the (.)p theory. While it follows from G-II that (q(f t); 0(f )) D < (O(ft); 0(f))', the same inequality on the one-point function in (23) goes the wrong way. What saved the situation

was the equality of pressures ap = aD which implies that liml_0 (O(f ))P = ((f)). Thus, by a rather circuitous route winding between different B.C. and

67Modulo technicalities concerning compact operators and analyticity, the general idea is this. If the eigenvalue b(µ) of an analytic family of compact operators like B(µ) = e-HI({i) does not cross another eigenvalue then it has an extension to an analytic function and the corresponding eigenvalue of H1(µ), E(µ) = -In b(µ), has an extension to a harmonic function. Crossings produce "kinks" and the extension is only superharmonic. In general, a difference of superharmonic functions is not superharmonic. But the Lee-Yang Theorem rules out crossings for the lowest eigenvalue E1(µ) and so it has a harmonic extension and therefore ml (µ) = E2 (µ) - E1(µ) has a superharmonic extension.

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finite and infinite volume, we succeeded in proving that the infinite volume D or HD (a04 + b02 - t0)2 theory had a mass gap for any y 0 (and a > 0).68

6. Later Work (1975-1977) As I mentioned in the discussion of [B1], after Seiler brought the Y2 model into the Euclidean world [44], Barry made a productive return to EY2 in a series of three joint papers with Seiler. In their first paper [B21], S2 = {Seiler, Simon} removed an apparently unnecessary condition which Seiler had imposed in his pioneering paper.

In order to serve you their results, I must first set the table with a fair amount of notation. For simplicity, I shall put the coupling constant A equal to 1 and both the boson and fermion masses equal to m. The (Euclidean) boson covariance is C = (-A + m2)-1 with associated gaussian measure dµ, as in the P(cb)2 model, and the fermi propagator is S = (ip+ m)-1 where JC7= ry V = y0 -2 + yl aal and y0 and yl are the 2 x 2 Euclidean Dirac matrices satisfying yµry + y yµ = -2Sµ,,. The fermion single particle space is R = R12 ® ?-l1 where h12 is the Sobolev space 2 L2((p2+m2) 2' d2p). If A is an operator on 1-l we let A 'A be the n-fold antisymmetric

tensor product of A with itself on A N, the n-fold antisymmetric tensor product of 1-l with itself. For A a bounded, open set in R2, let K = KA be the operator on 7-l with kernel K(x, y) = S(x, y)O(y)XA(y).69 Following Seiler, the analysis of EY2 starts with the Matthews-Salam formula for the (smeared) Schwinger functions SA with volume cutoff A, after the fermi fields have been (formally) integrated out. In this representation, SA = Z, 1 f FA(q)dlc

where FA(O) is a complicated function of the boson field on which the fermions have left their antisymmetric imprint. In particular, FA contains products A R of the operator R = (1 - K)-1.70 The advantage of eliminating the fermi fields is that it allows one to bound SA using LP estimates on the various factors of FA. For our purposes, we can suppress the harmless part of FA which involves the smearing functions, and focus on the operator norm of the problematic factor

Um =

ARdt1 - K)

e-BA-b 1Af b2

3

is the operator norm on Anf, BA = fn : Tr (K2 + KtK) : where Kt is the adjoint of K as an operator on L2(R2) ® 2L2(R2), b is a finite boson mass Here

).7' The renormalization parameter, and det3(1 - K) = det((1 - K)J,K+2 Tr K term arises from the Wick ordering :W:, and the illegal 2Tr K2 terms in 680ur bounds on superharmonic functions showed that Ml (µ) > cp for c > 0 as µ - + 0+. Letting y be the critical exponent measuring the rate at which m1(µ) --> 0 as µ -. 0+, i.e., m1 (µ) - µI for small µ, we had thus proved the (not best possible) bound y < 1. 69Since 0(y) is a distribution, this definition is only formal. Indeed, K, and all expressions involving K, must be interpreted in the standard way by introducing cutoffs, performing cancellations as required, and then removing the cutoffs via well-defined limits. We ignore these technicalities. 70We can understand the appearance of R by considering the formal fermi "gaussian" integral S-1'P)6'j6'b = RS. In effect, 1XA plays the role of an external f e-fA'''k'I e(' T = (S-1 field for the Dirac operator S-1 = i7V+ m. 43'3 is well-defined if 71The regularized determinant detn(1 - A) = det((1 -'XA)-1

A)eFi=1TI

A E C., the trace class of compact operators with JJAJJn = [Tr(A*A) 2 ] showed that K E C3 a.e. with respect to dµ (in fact, that K E C2+, a.e.).

n1

< eo (see [45]). S2

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det3 and BA cancel each other. The :Tr KtK: term in BA has the form c fA :02: which is the allowed boson mass counterterm predicted by perturbation theory. Observe that at those O's where R = (1 - K)-1 is singular, det3(1 - K) has a compensating zero. Seiler and subsequent authors have developed a number of "determinant bounds" to control the operator norm of the product An Rdet3(1 - K). So long as one lets the fermi statistics collect all the dangerous factors of R into a single factor An R which is then tamed by the det3, one avoids the combinatorial nightmares which tormented Barry in [B1]. The procedure is automatic and bubblessianless! In [44] Seiler was obliged to take the parameter b sufficiently positive to ensure

that u(0) E LP. However, this restriction seemed unnnecessary because, in his analysis of the Y2 Hamiltonian [7], Glimm avoided such a restriction by separating out the contributions from the low fermi momenta which didn't require countert-

erms and estimating them in a way different from the contributions from high momenta. In [B21] S2 carried out an analogous strategy: they split K = L + H as a sum of a low momentum part L and a high momentum part H, and derived ^ n R det3(1 - K) which separated the L and H cona new determinant bound on sp/l\itting tributions. By choosing the point sufficiently large they were able to free up an arbitrarily large part of the boson mass counterterm which was not needed for the H bounds but which could be used instead to dominate the term e-6 fA:02:

for any b E R. In this way they proved that u(q) E LP (dµ) for all p < oo.72 Continuing their analysis of EY2 in [B22], S2 proved the LLB which had been so important in the P(0)2 theory: ZA =

- KA) e-BAdp < ecjAj.

They obtained similar bounds on the (unnormalized) Schwinger functions (ZS)A which gave both the correct volume dependence and the correct local number dependence, as I now describe. Cover R2 with an array of unit squares A. Suppose the smearing functions f = (fl, . . . , fn) for the boson fields are in 7-1_1 and are each localized in a unit square, with a total of nA localized in the unit square 0, and that the smearing functions g = (g1, ... , gj) and h = (h1i ... , hk) for the fermi fields 0 and are in 7-1_2 1 . Then there are constants c1 and c2 such that73 (ZS)A(f,g,h)I
(24)

(24) improves on the n!2' number dependence in [44]. Note that the external fermi fields don't contribute to the number growth because of the cancellations due to fermi statistics.74 Finally, S2 observed that their Euclidean LLB yields the 72This result, as well as the main result of [B22], was obtained independently by McBryan [24, 25] by different methods. 73Some of the techniques deployed by S2 were a) an improved determinant bound with a more refined splitting of K so as to control the linking of distinct squares; b) the Checkerboard Estimate; and c) detailed CP class estimates on various single particle operators involving D = (-A+ m2) and the characteristic functions of squares. 740n the basis of (24) one might think that the number growth from ((fA :x V):0)1) might only be cnn! 2 and lead to a summable perturbation series. However, the renormalization procedures needed to deal with the fact that K C1 seem to produce a non-summable (but Borel summable) number growth, as I mentioned in the discussion of [B1]. a1

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Hamiltonian LLB of Glimm [7] and Schrader [41], obtained by much more arduous methods.

The joint paper [B23] with Jay Rosen on fluctuations in P(o)1 processes is not really a QFT paper, but I decided to include it in my review because it was inspired by work in CQFT and it includes an application to P(q)2. In one dimension, the P(q)1 "field" or path q(t) is a Brownian motion with drift .7,5 Rosen and Simon obtained sharp upper and lower bounds on the fluctuations of the paths q(t), and they determined the dependence of the bounds on the coefficients of the poly-

nomial P(q). In the application to P(0)2 only upper bounds on the fluctuations were possible.76 These bounds showed that the higher the degree of P the more restrained the fluctuations are, and distinguished the interacting measure from the free measure.

Unfortunately, space restrictions prevent me from presenting a detailed report

on [B24, B25, B26], all of which are substantial. I shall simply summarize the major contributions in these papers. S2's goal in [B24] was to develop for Y2 and 03 those techniques which had proved so successful for P(0)2i for instance: Nelson's Symmetry, Guerra's Theorem on the (monotone) convergence of the vacuum energy density al, the overlap of the free and interacting vacuum vectors, the GLLB, .... They faced several obstacles. The first was that Nelson's Symmetry failed for Y2 and 04 1 The S2 diagnosis was

that the vacuum energy counterterms lacked symmetry in l and t (for Y2 with volume cutoff an 1 x t rectangle) or in 11, 12, and t (for 03 with volume cutoff an 11 x l2 x t box). Their cure was to modify the definitions of the counterterms in the Matthews-Salam formula and in the Hamiltonian so as to be symmetric. In this way they restored Nelson's Symmetry for Y2 and in a formal sense77 for 03. The second obstacle, the absence of a Markov property, was overcome by replacing uses of the Markov property by that of OS positivity.1' OS positivity was critical in their proof of volume independent "multiple reflection" bounds (e.g., see 5) and 6) below). Establishing the vacuum overlap was also problematic, but S2 succeeded in doing so for Y2 by using OS positivity in the spatial direction and for 03 (in a formal sense) by using Nelson-Symanzik positivity. We now summarize the main results of [B24]: 1) direct proof of the Matthews-Salam formula for Y2 using an iterated Duhamel (- Phillips) expansion; 2) Euclidean construction of the Y2 and 03 Hamiltonians; 3) Guerra's Theorem78 for Y2 and 03; 4) for the 03 and Y2 + P(q)2 models;

751f q is the (unique) ground state of the Hamiltonian H = p2+P(q), then dq(t) = 2 z dW (t)+ r(q(t)dt, where W is Brownian motion and r = 20'/Q. 76Let Oh be the regularized field h * where h E Co (R2) and let {xj} be a sequence of

points in R2 with xj - xk I > CIj - kI for some C > 0. Then limj- qh(xj)/(ln j) 2- < oo. The proof used the GLLB of [B9]. 77Their identity expressing Nelson's Symmetry for 03 was the formal limit of a legitimate momentum cutoff identity. 78For Y2, -E1/l and 1nZl,t/lt are monotone increasing in l and t and converge to the same limit a00 as 1, t on. For 03, lim11 12 (-E1112)/1112 and limj1 i2 t-.,, In Zi1t2t/1112t both exist

and equal ate.

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5) volume independent bounds generalizing (24) for the normalized Y2 Schwinger functions:79

Ig;ll-Ilh,ll ?

lim

fi fill-1 l,t-oo I Sz,t(f,g,h)I
z

(25)

6) volume independent bounds on the normalized 03 Schwinger functions: n

lim 1

1, 1 2,

t-a 00

I SI1,12,t (.f1, ... , fn) I < n! fl Ilfi ll i=1

is a Schwartz space norm. The motivation for GRS's long, technical paper [B25] was the lesson from CSM where I If II

that the freedom to choose from a variety of B.C.'s is of great strategic value in taking the infinite volume limit. I have already described the main result of [B25], namely, that the P(0)2 pressure a,, is independent of B.C., and I illustrated the usefulness of this result for the proof that the (ao4 + b02 - P0)2 model has a mass gap for p # 0. Each of the four classical B.C.'s has its own particular merits: the F B.C. covariance, being diagonal in momentum space, is the simplest to work with;80 D B.C. provide the simplest way to erect barriers between regions and play the role

of the "free" B.C. of CSM; P B.C. are closest to the infinite volume theory in the sense that (0(x))P is a constant; and N B.C. provide a "repulsion" between regions (ZA is submultiplicative in A) and this leads immediately to an infinite volume estimate given a finite volume one. Here is a summary of some of the main results of [B25] on B.C. for the P(0)2 model:

1) a general analysis of gaussian81 B.C. (including those with non-zero mean); 2) the theory of conditioning and resulting conditioning inequalities; 3) bounds on covariances; e.g., CA < cCx as operators on L2(A); 4) limA-.0o aA = a00 for X = F, D, N, P, HD, HN, HP; 5) van Hove convergence of aA to a, as A -+ 00;82 6) convergence of E(gn)/IIgfIIL1 to -ate as gn -i 1, for fairly general sequences of cutoff functions; 7) equality in the Gibbs Variational Principle;42 8) transfer matrix formulae involving traces for periodic states; 9) asymptotic bounds on a, as the coupling constant or coefficients of P become large;83

10) convergence of the lattice approximation (and the consequent correlation inequalities) for P, N, HP, HN B.C.; 79The bound (25) implies a bound which is suitable as input for the OS reconstruction theorem: For f E S(R2), define the Schwartz space norm If I« = From (25), limi,t_oo IS1,t(f,g,h)I < n!2 rjz IfiH-1Uj Ig;I_2Ih;I_2 800n the other hand, when restricted to a finite volume A, the differential operator inverse to CFIL2(A) has a complicated non-local B.C.

81By this we meant that the associated measure dµ13 has the form duB = F13d(u I EA) where FB > 0 is a gaussian concentrated of (9A 82The applications 5)-7) all relied on our control of the Dirichlet pressure an and the conditioning inequalities of 2). 83Such bounds are useful in obtaining best constants in :O j: bounds [4].

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11) inequalities among the Schwinger functions for different B.C.;84 12) Checkerboard Estimates for D, N, P B.C. We elaborate a little on 2). The basic idea of conditioning is that if we have two covariances C < C, then we can realize the gaussian process associated with C as the independent sum = 0+0' where 0' is the gaussian process associated with C' = C. This situation applies to the classical B.C.'s since CD < C or P < Cn .

If ZA and ZA are the partition functions associated with the measures dµc and dµc, then we immediately deduce that ZA < ZA by integrating out the difference field 0' and applying Jensen's inequality. In this way, ZD _< Z° r P <_ ZD By similar reasoning, if A = Al U A2 where Al fl A2

we proved the submultiplicity

ZN < ZN Z and supermultiplicity Zf > ZD ZD Z. Barry's paper [B26] with Jiirg Frohlich is in some sense the last word on the P(ql)2 model, and my commentary on it will be my last word on Barry's contributions to QFT. I will leave the papers [B27] and [B28], which cross over into statistical mechanics, to other writers, and I shall not review the short paper with Yaffe [B29] on bounding Wilson loops in gauge field theories. The FS = {Frohlich, Simon} results in [B26] are truly breathtaking and settled a number of important questions which had remained open during the decade-long

study of the P(ci)2 model. I (essentially) quote from their abstract: "Among our results are: (1) For any polynomial P bounded from below, we construct two Euclidean states (expectations) (- )P,+ not necessarily distinct, which satisfy all OsterwalderSchrader axioms including clustering and obey the DLR equations for P. (2) Equality (.)p,+ = (.)p,- holds if and only if the pressure c (µ) corresponding

to the polynomial P(x) - ,ax is differentiable at µ = 0, and in this case the state ( )P,± is independent of a large class of different (in particular, classical) B.C.

(3) All P(ci)2 expectations thus far constructed are locally absolutely continuous with respect to the free field gaussian expectations with LP Radon-Nikodym derivatives, for all p < oo. (4) The strong Gibbs variational equality holds, for all states constructed so far for a given P." I'd like to make a few admiring comments about these results. Recall from my discussion of [B18] that the existence results for P(ci)2 in the strong coupling regime had relied on the Griffiths inequality G-II and thus held only for polynomials of the form P(x) = Pe(x) - px where Pe is even. By using the FKG inequality instead of G-II in their construction, FS could handle an arbitrary semibounded polynomial. Here's what they did. Choose h sufficiently large that measures dv , for infinite volume states can be constructed for the polynomials Q±(x) = P(x)Thx by Spencer's large external field method [46]. Letting f be a non-negative function in Co (R2), we consider expectations of eOM (which generate the Schwinger functions). Choosing the upper sign, we wish to take A -4 oo for the expectation EA = f f e-hO(X^)dvj, i.e., once we have exploited the large 840ur results here were more limited than we anticipated on the basis of the conditioning SB' for B = HD and B' = F, HN, HP (if P(x) = Pe (x) - µx where Pe is even and µ > 0) and for B = D and B' = F, N, P (if in addition deg P < 4). inequalities of 2). We were able to establish only that SB <

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99

external field h, we turn it off. The limit exists by monotonicity: EA > EA' if A C A'. To see this, we interpolate between EA and EA' by

E(s) _ f CO(f)e-sh0(xA'vA)-hO(XA)dv+/f e 00 Now E'(s) _ h(eP(f)O(XA'\A)) +h(e"(f))((p(XA'\A)), where the expectation e-3hcb(XA'\A)-h'(XA)dv+ satisfies the FKG inequality. Hence E'(s) < 0. This cf yields the state (- )p,+ and choosing the lower sign yields the state (- )p,_. FS developed a set of elegant estimates based on multiple reflectionsll which generalized the estimates of [4] and [B24] and which they called Chessboard Estimates. The point of these estimates was to use the pressure to bound interacting expectations. By means of these estimates, FS established very strong regularity properties for the boundary factor OaA in the DLR equations4l (e.g., that 'L'aA E LP(dy) for all p < oo). The FS proof of equality in the Gibbs Variational Principle42 improved on the GRS result in [B25]: better than just showing that there were states that came arbitrarily close to achieving equality, FS established actual equality for every infinite volume P(0)2 state that had been constructed (hence the adjective "strong" in (4) above).

7. Conclusion If one looks back at Barry's many results in QFT and tries to extract a short list of his most significant contributions, one might come up with something like this:

1) the introduction of many ideas and results from CSM into CQFT, in particular, the invention of the lattice approximation to prove correlation inequalities [B18];

2) his strong advocacy of Euclidean methods in CQFT (both as a researcher and as a propagandist); 85

3) the classical Ising approximation of [B14] and its consequences; 4) the abstraction of the notion of hypercontractivity in [B3] which led to a major mathematical industry (as described in [15]); 5) the determinant inequalities of [B21, B22, B24]. We have now reached the end of the tour of Barry Simon's life in CQFT (19681977). As your overworked tour guide, I'm relieved that he left the field, but, as a quantum field theorist, I consider his departure to have been a serious blow to progress in constructive quantum field theory.

References [1] Cooper, A. and Rosen, L., The weakly coupled Yukawa2 field theory: cluster expansion and Wightman axioms, Trans. Amer. Math. Soc. 234 (1977) 1-88 [2] Feldman, J., The A0s field theory, Commun. math. Phys. 98 (1974) 93-120 [3] Fortuin, C. M., Kasteleyn, P. W. and Ginibre, J., Correlation inequalities on some partially ordered sets, Commun. math. Phys. 22 (1971) 89-103 [4] Frohlich, J., Schwinger functions and their generating functionals. I, Hely. Phys. Acta 47 (1974) 265-306; II, Adv. Math. 23 (1977) 119-180 [5] Gallavotti, G., Renormalization theory and ultraviolet stability for scalar fields via renormalization group methods, Rev. Modern Phys. 57 (1985) 471-561

85Thus I add yet another label to my initial list-Barry Simon: Euclidean revolutionary!

L. ROSEN

100

[6] Ginibre, J., General formulation of Griffiths' inequalities, Commun. math. Phys. 16 (1970) 310-328

[7] Glimm, J., The Yukawa coupling of quantum fields in two dimensions. I, Commun. math. Phys. 5 343-386; II, Commun. math. Phys. 6 (1967) 61-76 [8] Glimm, J., Boson fields with non-linear self-interaction in two dimensions, Commun. math. Phys. 8 (1968) 12-25 [9] Glimm, J. and Jaffe, A., A A(04)2 quantum field theory without cutoffs. I, Phys. Rev. 176 (1968) 1945-1951

[10] Glimm, J. and Jaffe, A., A A(04)2 quantum field theory without cutoffs. III, The physical vacuum, Acta Math. 125 (1970) 203-261 [11] Glimm, J. and Jaffe, A., A x(04)2 quantum field theory without cutoffs. IV, Perturbations of the Hamiltonian, J. Math. Phys. 13 (1972) 1568-1584 [12] Glimm, J. and Jaffe, A., Quantum Physics, A Functional Integral Point of View, SpringerVerlag, New York, 1987

[13] Glimm, J., Jaffe, A. and Spencer, T., The Wightman axioms and particle structure in the P(¢)2 quantum field model, Ann. of Math. 100 (1974) 585-632 [14] Griffiths, R. B., Correlation in Ising ferromagnets. I, II, III, J. Math. Phys. 8 (1967) 478-484; 484-489; Commun. math. Phys. 6 (1967) 121-127 [15] Gross, L., Hypercontractivity and Logarithmic Sobolev Inequalities, in Diffusion, Quantum Theory, and Radically Elementary Mathematics: A Celebration of Edward Nelson's Contributions to Science, ed. W. G. Faris, Princeton University Press, 2006

[16] Guerra, F., Uniqueness of the vacuum energy density and van Hove phenomenon in the infinite volume limit for two-dimensional self-coupled Bose fields, Phys. Rev. Lett. 28 (1972) 1213-1215

[17] Haag, R. and Kastler, D., An algebraic approach to quantum field theory, J. Math. Phys. 5 (1964) 848-861

[18] Hgegh-Krohn, R., On the spectrum of the space cutoff P(ql) Hamiltonian in two space-time dimensions, Commun. math. Phys. 21 (1971) 256-260 [19] Jaffe, A., Lanford, O. and Wightman, A. S., A general class of cutoff model field theories, Commun. math. Phys. 15 (1969) 47-68 [20] Jost, R., The General Theory of Quantized Fields, American Mathematical Society, Providence, 1965 [21] Kac, M., Probability and Related Topics in Physical Sciences, Interscience, New York, 1959 [22] Lebowitz, L., Bounds on the correlations and analyticity properties of ferromagnetic Ising spin systems, Commun. math. Phys. 28 (1972) 313-321 [23] Magnen, J. and Seneor, R., The Wightman axioms for the weakly coupled Yukawa2 model in two dimensions, Commun. math. Phys. 51 (1976) 297-313 [24] McBryan, 0., Finite mass renormalizations in the Yukawa2 quantum field theory, Commun. math. Phys. 44 (1975) 237-243 [25] McBryan, 0., Volume dependence of Schwinger functions in the Yukawa2 quantum field theory, Commun. math. Phys. 45 (1975) 279-294 [26] Nelson, E., A quartic interaction in two dimensions, in Mathematical Theory of Elementary Particles, ed. R. Goodman and I. Segal, MIT Press, Cambridge, 1966 [27] Nelson, E., Quantum fields and Markoff fields, in Partial Differential Equations, ed. D. C. Spencer, American Mathematical Society, Providence, 1973 [28] Nelson, E., The construction of quantum fields from Markoff fields, J. Funct. Anal. 12 (1973) 97-112

[29] Nelson, E., The free Markoff field, J. Funct. Anal. 12 (1973) 211-227 [30] Nelson, E., Probability theory and Euclidean field theory, in Constructive Quantum Field Theory, ed. G. Velo and A. S. Wightman, Springer-Verlag, New York, 1973 [31] Osterwalder, K. and Schrader, R., On the uniqueness of the energy density in the infinite volume limit for quantum field models, Helv. Phys. Acta 46 (1972) 746-754 [32] Osterwalder, K. and Schrader, R., Axioms for Euclidean Green's functions. I, Commun. math. Phys. 31 (1973) 83-112 [33] Osterwalder, K. and Schrader, R., Axioms for Euclidean Green's functions. II, Commun. math. Phys. 42 (1975) 281-305 [34] Penrose, O. and Lebowitz, J. L., On the exponential decay of correlation functions, Commun. math. Phys. 39 (1974) 165-184

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101

[35] Renouard, P., Analyticite et sommabilite "de Borel" des fonctions de Schwinger du modele de Yukawa en dimension d = 2, I. Approximation "a volume fini", Ann. Inst. H. Poincard 27 (1977) 237-277; II, La "limite adiabatique", Ann. Inst. H. Poincare 31 (1979) 253-318 [36] Rosen, L., A a02, field theory without cutoffs, Commun. math. Phys. 16 (1970) 157-183 [37] Rosen, L., The (cb2n)2 quantum field theory: higher order estimates, Commun. math. Phys. 24 (1971) 417-457 [38] Rosen, L., Renormalization of the Hilbert space in the mass shift model, J. Math. Phys. 13 (1972) 918-927 [39] Ruelle, D., Statistical Mechanics, Benjamin, New York, 1969 [40] Salmhofer, M., Renormalization, An Introduction, Springer, Berlin, 1999 [41] Schrader, R., Yukawa quantum field theory in two space-time dimensions without cutoffs, Ann. Phys. 70 (1972) 412-457 [42] Schwinger, J., On the Euclidean structure of relativistic field theory, Proc. Nat. Acad. Sci. U.S.A. 44 (1958) 956-965 [43] Segal, I., Construction of nonlinear local quantum processes: I, Ann. of Math. 92 (1970) 462-481

[44] Seiler, E., Schwinger functions for the Yukawa model in two dimensions with space-time cutoff, Commun. math. Phys. 42 (1975) 163-182 [45] Simon, B., Trace Ideals and Their Applications, Cambridge University Press, Cambridge, 1979; second edition, Mathematical Surveys and Monographs, 120, American Mathematical Society, Providence, 2005

[46] Spencer, T., The mass gap for the P(ct)2 quantum field model with a strong external field, Commun. math. Phys. 39 (1974) 63-76 [47] Streater, R. F. and Wightman, A. S., PCT, Spin B Statistics, and All That, Benjamin, New York, 1964

[48] Symanzik, K., Euclidean quantum field theory, in Local Quantum Theory, ed. R. Jost, Academic Press, New York, 1969 [49] Wightman, A. S., Quantum field theory in terms of vacuum expectation values, Phys. Rev. 101 (1956) 860-866 [50] Wightman, A. S. and Girding, L., Fields as operator valued distributions in quantum field theory, Ark. fur Physik 28 (1965) 129-184

DEPARTMENT OF MATHEMATICS, THE UNIVERSITY OF BRITISH COLUMBIA, 1984 MATHEMATICS

ROAD, VANCOUVER, B.C., CANADA V6T 1Z2

E-mail address: [email protected]

Nonrelativistic Two-Body

and N-Body Quantum Systems, Resonances

Proceedings of Symposia in Pure Mathematics Volume 76.1, 2007

Isoperimetric Inequalities for Eigenvalues of the Laplacian Mark S. Ashbaugh and Rafael D. Benguria* Dedicated to Professor Barry Simon on his sixtieth birthday ABSTRACT. We give a survey of universal inequalities for low-lying eigenvalues of the laplacian.

CONTENTS 1.

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Introduction Rayleigh-Faber-Krahn Inequality Szego-Weinberger Inequalities The Payne-Polya-Weinberger Inequality in R Monotonicity of g and B The PPW Bound for Domains in S' The Eigenvalue Gap A2 - Al Steklov Eigenvalues Annular Domains Inequalities Between Dirichlet and Neumann Eigenvalues Polya's Conjectures

The Bass Tone of a Drum and Its Inradius The Eigenvalue Ratio A3/A1 An Isoperimetric Inequality for Ovals in the Plane Open Problems

References

1. Introduction The purpose of this article is to give an overview of some isoperimetric inequalities for eigenvalues of the laplacian. Since the literature on the subject is extensive (see, e.g., the reviews [32, 138, 171]), we will restrict ourselves primarily 2000 Mathematics Subject Classification. Primary 35P15; Secondary 35J05, 49R50. Key words and phrases. eigenvalues of the laplacian, isoperimetric inequalities. * Supported by Fondecyt, project 102-0844. ©2007 American Mathematical Society 105

106

M. S. ASHBAUGH AND R. D. BENGURIA

to the consideration of isoperimetric results for low-lying eigenvalues with special attention to results connected to eigenvalue ratios. Isoperimetric inequalities have a long history in mathematics dating back to the Greeks and Dido's problem, i.e., the classical isoperimetric inequality in Euclidean geometry (see, e.g., [136, 170] for reviews of the subject). With the introduction of the Calculus of Variations in the seventeenth century, isoperimetric inequalities found their way into mechanics and physics (see, e.g., [153]). The theme of this manuscript, isoperimetric inequalities for eigenvalues of the laplacian, has its roots in the work of Lord Rayleigh on the Theory of Sound [157]. It was determined in the nineteenth century that the basic equation that describes the small vibrations of an elastic medium is the wave equation. The normal modes and proper frequencies that characterize the vibrations of a fixed, homogeneous membrane correspond to particular solutions of the wave equation. They are determined by the solution of the eigenvalue problem for the Dirichlet laplacian on a bounded domain in 182. Consider a bounded domain 12 C R' with a piecewise smooth boundary aQ. Then .\ is a Dirichlet eigenvalue of 12 if there exists a function u E C2 (Q) n CO (12) (Dirichlet eigenfunction) satisfying the boundary value problem

-Du = Au

in 12,

(1.1)

with u = 0 in au, where 0 is the Laplace operator. Dirichlet eigenvalues (for n = 2) were introduced in the study of the vibrations of the clamped membrane in the nineteenth century. In fact, they are proportional to the squares of the eigenfrequencies of the membrane with fixed boundary. See [106] for a review and historical remarks. Provided 12 is bounded and the boundary 1312 is sufficiently regular, the Dirichlet laplacian has a discrete spectrum of infinitely many positive eigenvalues with no finite accumulation points [148], 0 < A1(1l) < A2 (12) < A3 (12) < ...

(1.2)

(with Ak(12) -4 oo as k -+ oo). The Dirichlet eigenvalues are characterized by the max-min principle [53, 158] dx .\k = sup inf fn IVu12 u2 dx fn

(1.3)

where the inf is taken over all u E H01(9) \ {0} orthogonal to cpl, 02, , ,Pk-1 E H01(9), and the sup is taken over all choices of IcOi}k i . For simply connected domains it follows from the max-min principle (1.3) that the lowest eigenvalue A1(12) is nondegenerate and the corresponding eigenfunction ul can be taken to be positive in the interior of Q. For future reference, we let {uk}k 1 denote an orthonormal basis of real eigenfunctions corresponding to the Dirichlet eigenvalues {.\k}' 1 (listed with multiplicity), i.e., ffl ui uj = Si.9 for all i, j and -Auk = Ak Uk for each k. For higher values of k the nodal lines of the k-th eigenfunction Uk divide 12 into no more than k nodal domains [53]. Dirichlet eigenvalues are completely characterized by the geometry of the domain Q. The inverse problem, i.e., to what extent the geometry of 12 can be recovered from knowledge of {Ak}k 1, was posed by M. Kac in [96]. If n = 2, for example, and 1312 is smooth (in particular 1312 does not have corners) the distribution

ISOPERIMETRIC INEQUALITIES FOR EIGENVALUES OF THE LAPLACIAN

107

function (trace of the heat kernel)

E e akt

4At

-

L

+

-(1- r) +

o(tli2),

(1.4)

k=1

as t -- 0+, where A is the area, L the perimeter, and r the number of holes of 52, so at least these features of the domain can be recovered from knowledge of all the eigenvalues. However, complete recovery of the geometry is impossible, as was later shown by Gordon, Webb, and Wolpert [70], who constructed two isospectral

domains in I[82 with different geometries (the same result for dimension n > 4 had been accomplished earlier by Urakawa in [176]). Since the area A and the perimeter L of the membrane are determined from the asymptotic expansion of the distribution function (1.4), it follows from the classical isoperimetric inequality that one can hear whether the membrane is circular. One can make similar remarks about the Neumann eigenvalues of the laplacian

for a bounded domain 1 C R' with smooth boundary. For further elaboration of their background and notation, see the first paragraph of Section 3 below. For the Euclidean case, both the Dirichlet and Neumann eigenvalues scale as one over length squared, and therefore one can compare them with geometric properties of the underlying domain, and try to find universal inequalities. The simplest such inequality to be considered is a universal (hopefully sharp) bound on A .X1. This universal inequality is precisely the Rayleigh-Faber-Krahn inequality (see Section 2, below). Other universal inequalities involve bounding the first nontrivial Neumann eigenvalue by an expression involving A, the so-called Szego-Weinberger inequality (see Section 3, below). In Section 4 we consider universal inequalities for ratios of Dirichlet eigenvalues, in particular the Payne-Polya-Weinberger (henceforth PPW) isoperimetric inequality for )2/)1. Section 5 is devoted to discussing monotonicity properties of special functions involved in the proof of the PPW in-

equality. In Section 6 we discuss the analog of the PPW inequality on spaces of constant sectional curvature. In Section 7 we discuss universal inequalities for the gap between the first two Dirichlet eigenvalues of the laplacian for convex domains. Inequalities for the Steklov eigenvalues are discussed in Section 8. In Section 9, we discuss universal inequalities for eigenvalues of the laplacian in annular domains. In Section 10, we briefly survey inequalities between Dirichlet and Neumann eigenvalues. P61ya's conjectures are discussed in Section 11. Connections between the lowest eigenvalue of the Dirichlet laplacian and the inradius of the underlying domain are discussed in Section 12. In Section 13 we discuss universal inequalities for A3/A1. In Section 14 we discuss an isoperimetric inequality for ovals in the plane. Finally, in Section 15 we mention some of the most intriguing open problems concerning universal inequalities for low-lying eigenvalues.

2. Rayleigh-Faber-Krahn Inequality The Rayleigh-Faber-Krahn inequality for the fixed membrane (i.e., n = 2) states that Al >

2

7 jo,1

A

where jo,1 = 2.4048... is the first zero of the Bessel function Jo(t), and A is the area of the membrane (Bessel functions are used throughout this manuscript, here we follow the notation of [1]). Equality is attained in (2.1) if and only if the membrane

M. S. ASHBAUGH AND R. D. BENGURIA

108

is circular. In simple words, among all membranes of a given area, the circular shape gives the lowest fundamental frequency. This inequality was conjectured by

Lord Rayleigh (see [157], pp. 339-340), based on exact calculations for simple domains, and a variational argument for nearly circular domains. In 1918, Courant [52] proved the weaker isoperimetric result that among all membranes of the same perimeter L, the circular one yields the least lowest eigenvalue, i.e.,

Al ?

4fL2o'1

(2.2)

with equality if and only if the membrane is circular. Rayleigh's conjecture was proven independently in the 1920s by Faber [64] and Krahn [101]. The corresponding isoperimetric inequality in dimension n, Al

(Cn 1 2/n

> \ FQI /

2

gn/2-1'1'

(2.3)

was proven by Krahn [102] (an English translation of this article of Krahn with commentary can be found in [124]). In (2.3), denotes the first positive zero of the Bessel function J,,,,(t), IQI is the volume of the domain, and Cn = 7rn/2/h(n/2+ 1) is the volume of the unit ball in n dimensions. Equality is attained in (2.3) if and only if SZ is a ball.

The proof of the Rayleigh-Faber-Krahn inequality rests upon two facts: a variational characterization of the lowest Dirichlet eigenvalue and the properties of symmetric decreasing rearrangements of functions. The variational characterization of the lowest eigenvalue is given by [53, 158]

f IVul2dx A, (Q) =

inf

uEH0 (n)\{0}

f0 u2 dx

(2.4)

Concerning decreasing rearrangements, let S2 be a measurable subset of R , then the symmetrized domain Q* is a ball with the same measure as Q. If u is a real-valued measurable function defined on a bounded domain 52 C R , its spherical decreasing rearrangement u* is a function defined on the ball Q* centered at the origin, such

that u* depends only on distance from the origin, is decreasing away from the origin, and is equimeasurable with u. The spherical increasing rearrangement of u, denoted u, is defined in the same way but with "nondecreasing" replacing "nonincreasing" in the definition above. We refer to [25, 97, 72, 118, 153, 169, 171, 172] for properties of rearrangements of functions. Since the function u and its spherical decreasing rearrangement are equimeasurable, their L2-norms are

the same. What Faber and Krahn actually proved is that the L2-norm of the gradient of a function vanishing on a52 is decreased under spherical decreasing rearrangement (the usual proof of this fact uses the rearrangements of the level sets of u, Federer's co-area formula, and the classical isoperimetric inequality; see [169, 172] for details, or [114] for a different approach). The fact that the L2-norm of the gradient of a function decreases under rearrangement, combined with the variational characterization (2.4) and the fact that u E Ho (1) implies u* E Ho (S2*), immediately gives the Rayleigh-Faber-Krahn inequality. In S', the analog of the classical isoperimetric inequality was proven by Schmidt [162]. The analog of the Rayleigh-Faber-Krahn inequality for domains on Sn was

ISOPERIMETRIC INEQUALITIES FOR EIGENVALUES OF THE LAPLACIAN

109

proven by Sperner [167] (see also [68]). For a discussion of Rayleigh-Faber-Krahn-

type inequalities for domains in manifolds, we refer to the book by Chavel [44] (pp. 86-94) and references therein. Rayleigh-Faber-Krahn-type results have also been obtained for Schrodinger operators (see, e.g., [125]), and more recently for Schrodinger operators with magnetic fields [62]. A Rayleigh-Faber-Krahn result for wedge-like membranes was obtained by Payne and Weinberger [146]. For the classical isoperimetric inequality there are well known stability results, that control the isoperimetric defect L2 - 47rA from below in terms of the radii of the incircle and the circumcircle [36] (see also [39, 137]). In the same spirit, Melas proved a stability result for the Rayleigh-Faber-Krahn inequality for convex domains [131]. On the other hand, Avila [23] proved a stability result for the Rayleigh-Faber-Krahn inequality for domains in S'. There is also a much older result of Payne and Weinberger [147] which shows that A Al is close to the Rayleigh-Faber-Krahn lower bound if the isoperimetric defect is small. Rayleigh-Faber-Krahn-type results have also been obtained for more general operators, in particular, for the p-laplacian: Let S2 C ll.' be a bounded domain, and suppose that 1 < p < oo. Let Al = A1(p, S2) = inf

f

I puJP dx (2.5)

fn JuI P dx

where the inf is taken over all functions u E W0'P, u t_ 0. It is well-known that A1(p, 1) > 0 and a nonzero minimizer u = u(p, 1) exists and satisfies the Euler equation,

div(lDulP-2Du) + AIuIP-2u = 0,

(2.6)

in SZ with u E W0 'P. The operator div(lDulP-2Du) is the p-laplacian (for p = 2 it is just the ordinary laplacian). Then, Al is the first eigenvalue and u the first eigenfunction of the p-laplacian on Q. Al is simple and isolated, and u has one sign (see, e.g., [34, 122]. For the p-laplacian we also have the following RayleighFaber-Krahn inequality [34, 129] (see also [67]): Al (p, 11) > A,(p,1*).

(2.7)

Equality is obtained in (2.6) if and only if S2 is a ball. More recently, a Rayleigh-Faber-Krahn result has been proven for the lowest eigenvalue of the laplacian for smooth (not necessarily bounded) domains on the Euclidean space R' with a gaussian weight [33]. Finally, we mention that the Rayleigh-Faber-Krahn inequality extends almost immediately to a result for the second eigenvalue, A2 (for this discussion we confine

our attention to the case of Euclidean domains, though certainly extensions to domains in Riemannian manifolds are possible and even straightforward). This follows from the fact that any eigenfunction for A2 has exactly two nodal domains [53], and that A2 is the first eigenvalue of either nodal domain. From this fact, the lower bound A2

(2)

2/n to/2-1,1

(2.8)

> follows immediately. In words, this just says that to minimize A2 among domains of equal volume one should take the limiting case of two equal balls. This inequality

M. S. ASHBAUGH AND R. D. BENGURIA

110

was first observed by Krahn [102] in his second, longer paper proving the RayleighFaber-Krahn inequality in n dimensions. There he also suggested that perhaps Ak is minimized in the (limiting) case of k identical balls in Rn. While this is generally not true (in 12 a single disk has a lower A3 than do three equal disks of the same total area), it may be true for certain choices of k and n. For more information in this direction, the reader might consult [182] or [82], among others.

Krahn's result for )'z was little-noticed and hence has been rediscovered by others several times since its original publication in 1926. For example, it was later found by Hong [92] and, independently, by Peter Szego (see p. 336 of Polya's paper [151]). For recent discussions on minimizing A2 see, e.g., [84, 85].

3. Szego-Weinberger Inequalities The lowest nontrivial eigenvalue for the free membrane also satisfies an isoperi-

metric inequality. Let Il be a a bounded domain in 1I with smooth boundary. Then we can list the Neumann eigenvalues of the laplacian on Il as 0 = po (1) < ,u1(I) < P2 (Q) < ..., i.e., this is a listing, with multiplicities, of the eigenvalues of the problem

-Du = pu with

8u

an

=0

in Q,

(3.1) (3.2)

on 011,

where n denotes the outward normal to OQ.

For the free membrane problem (i.e., for n = 2), Szego [168] proved, for a simply-connected domain 11, z

pl(y) <

'1 = pl(Q*),

(3.3)

where p1,1 = 1.8412... is the first positive zero of the derivative of the Bessel function J1(t). This isoperimetric inequality had been conjectured by Kornhauser and Stakgold [100]. The corresponding result for dimension n,

(Cn

2/n

pl (y)

(3.4)

pn/z,1 = 11(Q*),

91 / was proven by Weinberger [178] for arbitrary bounded domains. Here, Cn is the

volume of the unit ball in dimension n, as above, and p,,,,,1 denotes the first positive zero of the derivative of the "Bessel" function t1-n''J,,,,(t). In dimension n = 2,

Weinberger [178] also noted (based on a conversation with Szego) that Szego's proof yields 1

Al

+

1

pz (Q)

2A 7rp1

_

1

1 (3.5)

pl (11*) + pz(S2*)

for a simply-connected domain 0. In (3.3) and (3.5) equality is attained if and only if SZ is a disk, and in (3.4) if 0 is a ball. Although the Szego-Weinberger inequality appears to be the analog for Neumann eigenvalues of the Rayleigh-Faber-Krahn inequality, its proof is significantly different. In fact, the proof of the Szego-Weinberger isoperimetric inequality is much closer in spirit to that of the PPW inequality that we discuss in the next section. We recall that the Rayleigh-Faber-Krahn inequality was done in two steps: (i) the Rayleigh-Ritz characterization of A1, and (ii) the use of spherically decreasing rearrangement, in particular, the fact that the L2-norm of an Ho function remains

ISOPERIMETRIC INEQUALITIES FOR EIGENVALUES OF THE LAPLACIAN

111

the same while the L2-norm of its gradient decreases under spherically decreasing rearrangement. In the Neumann problem, the eigenfunction corresponding to the first nontrivial eigenvalue of the ball is not spherically symmetric, and thus the method we just described cannot be used in exactly the same way. On the other hand one can exploit the fact that the first nontrivial Neumann eigenvalue for the ball in n dimensions is n-fold degenerate. The main steps in the proof of the Szego-Weinberger result are thus the following: (i) The Rayleigh-Ritz inequality for the first nontrivial Neumann eigenvalue

()

µ1 < foVPJ2dx

3.6

f0P2dx

provided fo P dx = 0, and P 0 0. Here P is a trial function for µ1, orthogonal to the constants. (ii) Center of Mass result. Use (3.6) written as

jP2dx < j VP2 dx

(3.7)

n

n times with the n different trial functions x, Pi = g (r)

(3.8)

where the xi's denote the usual Cartesian coordinates for I[8n and r denotes radial distance from the origin. Weinberger [178] (see also [25], pp. 153-154) used a topological argument based on the Brouwer fixed point theorem to show that one

can always choose the origin so that the trial functions Pi defined by (3.8) are orthogonal to the constants. We call this a center of mass result since it generalizes

away from the case with g(r) = r, where one would choose the origin to be the center of mass of 1 when S2 is viewed as a uniform mass density in R'. (iii) Choice of the variational function g. Substituting each of the Pi's defined by (3.8) into (3.7) and summing, one arrives at the basic estimate fsz [(g)2 + (n - 1)(g/r)2] dx.

(3.9)

f0g2dx The choice of g is dictated by the fact that we want equality in (3.6) when the Pi's are of the form (3.8) and SZ is a ball. Based on the knowledge of the eigenfunctions for the first nontrivial Neumann eigenvalue for a ball, one thus takes

g(r) =

(3.10)

.1-n/2

Jn/2(pn/2,lr/R), for 0 < r < R, and g(r) = g(R) for r > R, where R is the radius of the ball 11* and p,,,,,1 denotes the first positive zero of the derivative of tl-mJm(t). (iv) Monotonicity properties. From the definition of g and properties of Bessel functions, one can prove that g(r) is a nondecreasing function of r and B (r) _ (g')2 + (n - 1)(g/r)2 (3.11) is a nonincreasing function of r (see [178] or [25] for details). (v) Symmetric rearrangement. Using the basic estimate (3.9), the monotonicity properties of g and B, and properties of rearrangements, one has µ1(i) <_

fn B(r) dx < fsz B(r) dx _ P 12 1 fo g2 dx

f0, g2 dx

R2

(3.12)

112

M. S. ASHBAUGH AND R. D. BENGURIA

which is precisely the Szego-Weinberger inequality. The analog of the Szego-Weinberger inequality in spaces of constant sectional

curvature was considered by Ashbaugh and Benguria in [17]. In particular, we proved the analog of (3.4) when S2 is a domain contained in a hemisphere of S. Earlier variants of this result are due to Bandle (see her book [25]) and Chavel [43] (see also [44]). In [17] we also gave a proof of the analog of (3.4) for any bounded, smooth domain 1 in IH[', a result which had been noted earlier by Chavel [43].

4. The Payne-Polya-Weinberger Inequality in R' The next simplest isoperimetric inequality concerns the quotient of the two lowest Dirichlet eigenvalues .\2/.\1. In 1955, Payne, Polya, and Weinberger (henceforth PPW) proved the universal bound, A2 (Q)

A10 _< 3

(4 . 1)

7

for the quotient between the first two Dirichlet eigenvalues of a bounded smooth domain SZ C R2 (using twice the Rayleigh-Ritz characterization of .\2 with trial functions xu1 and yul, and combining the two inequalities appropriately) [143, 144]. This was extended to domains in ]l in [174], where the corresponding bound reads A2(Q)/Al (Q) < 1+4/n. Based on exact calculations for simple domains, PPW conjectured the isoperimetric inequality _2(_) < Al

A2

j111

2.539...

for

Q C Il82

(4.2)

01

The bound (4.1) was successively improved in [37, 57, 49]. Finally, the conjectured inequality (4.2) and the corresponding inequalities in n dimensions for all n were proven by Ashbaugh and Benguria [7, 9] (see also [10]). Notice that while Dirichlet eigenvalues satisfy stringent constraints (like the one embodied by (4.2) for any bounded domain in ]R2), no such constraints exist for Neumann eigenvalues,

other than the fact that they are nonnegative. In fact, given any finite sequence {0 = /10 < µ1 < µ2 < < µk_1}, there is an open bounded, smooth, simply connected domain of R2 having this sequence as the first k Neumann eigenvalues of the laplacian on that domain [51]. The proof of the PPW inequality (4.2) has several features in common with the proof of the Szeg6-Weinberger inequality. However, the necessary facts about special functions are much harder to establish for the PPW inequality. A key fact which goes some ways towards explaining this difference is that in the SzegoWeinberger case, the first (Neumann) eigenfunction is explicitly known (it is just a constant), while in the proof of PPW, neither the first (Dirichlet) eigenfunction ul nor its symmetric decreasing rearrangement is explicitly known. Since in the course of the proof integrals occur which involve ul and its symmetric decreasing rearrangement ui explicitly, a comparison technique, due to Chiti, is needed to estimate these integrals in terms of integrals of known functions. The proof of (4.2) is performed in the following six steps: (i) The Rayleigh-Ritz characterization of the second Dirichlet eigenvalue, through the use of the gap formula [158, 88]: IVPI2u2 dx , A2 - al < J

f P2ui

dx

(4.3)

ISOPERIMETRIC INEQUALITIES FOR EIGENVALUES OF THE LAPLACIAN

113

where fu P ui dx = 0, and Pt- 0. To get the isoperimetric result out of (4.3), we must make very special choices of the function P, in particular, choices for which (4.3) is an equality if S2 is a ball. (ii) A center of mass result: In n dimensions one uses (4.3) for n trial functions xi

Pi = g(r)

-r

i = 1,2,...,n,

(4.4)

where the xi's represent the usual Cartesian coordinates for R n and r denotes radial distance from the origin. To insure orthogonality of Piu1 to ul for 1 < i < n, one must make a special choice of origin, using a topological argument due to Weinberger [178] based on the Brouwer fixed point theorem. We call this a center of mass result since it generalizes away from the case with g(r) = r, where one must choose the origin to be precisely the center of mass of S1 thought of as a mass distribution with density function ui. (iii) Averaging over the Pi's and choosing g: Using (4.3) with the special choices given by (4.4) (following similar ideas as in the proof of the Szeg6-Weinberger inequality), we arrive at the inequality fn [(gl)2 + (n - 1)(g/r)2] ui dx A2 - A1 <

(4 5) .

fsz g2 ui

ax Once we have arrived at (4.5) it is convenient to use results on rearrangements. For that, it is crucial to understand the behavior of the two functions g(r) and

(g')2 + (n - 1)(g/r)2. The choice of g is dictated by the fact that we need to have equality in (4.3) when SZ is a ball (but precisely which ball has yet to be determined). This determines g and (g')2 + (n - 1)(g/r)2 up to a scaling of the independent variable and their behavior for large r (which does not matter if S2 is a ball but is necessary for the general situation). Given the knowledge of the eigenfunctions corresponding to Al and A2 for the ball, one takes

g(r) = w(ryr)

(4.6)

where w is defined by

w(t) =

Jn/2(jn/2,1t)

for 0 < t < 1, (4.7) Jn/2-1(jn/2-1,1t) and w(t) = limt-1- w(t), for t > 1. That is, g needs to be the ratio of the "radial parts" of the eigenfunctions ul, u2 for some n-ball (so a scaling of that same ratio w for the unit ball). Since g(r) = w(ryr), we have (g')2 + (n - 1)(g/r)2 = 2B(ryr), with

B(t) = w'(t)2 + (n - 1)(w(t)/t)2.

(4.8)

(iv) Monotonicity properties of special functions: From the definition of w and the properties of Bessel functions one can prove that w(t) is a nondecreasing function of t and B(t) is a nonincreasing function of t (see Section 5, below). (v) Use of symmetric rearrangement: Using rearrangements and the monotonicity properties of B and w, one has B(ryr)u1 dx < n

- fn

I

- fn

B(ryr)*u12 dx <

B(ryr)u12 dx

(4.9)

w(ryr)ZU12 dx.

(4.10)

and

f

- f . w(ryr)*u12 dx -> f

w(ryr)ZU1 dx >

1*

M. S. ASHBAUGH AND R. D. BENGURIA

114

(vi) Chiti comparison argument: Let B1 be the ball with radius chosen so that the first Dirichlet eigenvalue of B1 is also A1. It is clear that this radius is jn/2-1,1/ A1. Let z be the first Dirichlet eigenfunction of B1 normalized such that fBl z2 dx = ui. The final step, essential to proving (4.2), is a result of Chiti [47, 48, 49] comparing the spherical decreasing rearrangement ui of ul with z. Chiti's comparison theorem implies that f(r)ui2 dx > f f(r)z2 dx, ft 2*

1

if f is increasing, and the reverse inequality if f is decreasing. The result of Chiti builds upon the work of Talenti [169], which makes use of rearrangements results, Federer's co-area formula [65], and the classical isoperimetric inequality. The Rayleigh-Faber-Krahn inequality implies that B1 C SZ* (in fact B1 is a proper subset of 11* unless Il itself starts out as a ball). It follows from (4.11) and the monotonicity properties of B and w that

J B(yr)u12 dx < *

B(yr)z2 dx

(4.12)

JBI

and

fo w (yr)2u12 dx >_ f w(ryr)2z2 dx. *

(4.13)

1

Combining these inequalities with (4.9), (4.10), (4.5), (4.7), (4.8), and the definition of z, and taking -y = Xi/jn/2-1,1, we finally get

A2 - Al <

2

fB, B(yr

z2

a fB, w(yr)

2z

dx

dx

= a22 (02

- a2)

(4.14)

where a = in/2-1,1 and ,Q = j,,,/2,1. From here the inequality A2 A1

jn2 < j2 /2,1 = A2(1l*) A1(0*)

(4.15)

follows immediately.

Remarks. (i) Note that inequality (4.2) and the fact that equality holds if and only if SZ is a ball means that one can "hear" the shape of a circular drum from just its first two frequencies. Moreover, if all one seeks is to know if the drum is circular or not, one needs only the ratio of the first two frequencies and not their individual values. And, as Melas [131] tells us, we can "hear" that a drum is close to circular

if the ratio of its first two tones is close to its value for a circular drum. That is, using an equivalent characterization in terms of eigenvalues, if A2/A1 is very close to its value for a disk, then in a certain rigorous sense the boundary of the domain is very close to being circular, at least for convex domains. (ii) There are by now several classes of (not necessarily sharp) universal inequalities between Dirichlet eigenvalues. For example, one has the general bound of Payne, Polya, and Weinberger, Am,,,+1 - A. C

4

n

(A1 + A2 + .... f A,,,,),

(4.16)

ISOPERIMETRIC INEQUALITIES FOR EIGENVALUES OF THE LAPLACIAN

which holds for m = 1, 2, 3,

115

... and for any bounded domain in Il [143, 144]. One

also has the stronger Hile-Protter bound [91], Ai

i=1

)m+1

>mn

- Ii -

(4.17)

4

and, in addition, an even stronger result due to Yang [183]. We will not review the many generalizations of these bounds that have been obtained (instead, we refer

the reader to the review articles [15, 2], and also to the recent articles [4, 21]). See also [183, 45], and also [12, 13]. (iii) A very useful approach to proving universal inequalities for eigenvalues of the laplacian (such as those given in (ii) above) makes use of commutators (see,

e.g., [73, 93, 74, 78, 79, 80, 111, 21]). These take as their starting points the works of PPW, Hile and Protter, Yang, and others, as mentioned in the preceding remark. Interestingly, a connection can also be made to the sum rules studied in quantum mechanics from its early days [111]. (iv) There are a number of other interesting isoperimetric results which are corollaries of (4.2). Some of the most interesting involve the geometric object known as the maximal inner conformal radius r, or conformal radius for short. Polya and Szego (see [153]) had established that for a simply-connected domain r2A1 is maximized at the disk (or, equivalently, that among all domains of a given conformal radius r the one giving the largest Al is the disk). Combining this with (4.2) now shows that A2 enjoys the same property: A2 is maximized by the disk among all simply-connected domains of a given conformal radius. Beyond that,

Baiiuelos has remarked that these results also combine to show that under the same conditions, the fundamental gap A2 - Al is maximized at a disk. Indeed, the same is true of A2 - cAl for any constant c less than or equal to A2/A1 of a ball. Thus, for any c < ji,l/j2,1, r2 (A2 - cA1) < .7i,1 - cjo,1, which is isoperimetric, with equality if and only if Il is a disk.

5. Monotonicity of g and B In our original proof of the PPW inequality (4.15) [7, 9], we used the product representation for Bessel functions and a certain inequality between ratios of zeros

of Bessel functions to prove the monotonicity properties of the functions w (or g) and B introduced above. The proof of these properties can be simplified by analyzing a Riccati differential equation satisfied by the function tw(()) (5.1) q(t) = The main advantage of this proof is that it can be generalized to other situations (e.g., to a proof of the analog of the Szego-Weinberger bound for domains in spaces

of constant sectional curvature, or to a proof of the PPW inequality for domains in S'''). In this section we will sketch the proof of the monotonicity properties of w and B in this approach. In terms of q, we can write B = [q2 + (n - 1)] t2 and

(5.2)

T22

B' = 2[q q' + (q - 1)(q2 + (n - 1))/t]

.

(5.3)

M. S. ASHBAUGH AND R. D. BENGURIA

116

From (5.1) and (5.3) it is clear that the monotonicity properties of w and B will follow if we can show that

0
(5.4)

q' < 0

(5.5)

and

hold for 0 < t < 1. Our strategy for proving these inequalities is as follows. By Taylor expansion at 0 one finds that q(0) = 1 and that q < 1 on an interval just to the right of 0. Similarly, by Taylor expanding at t = 1, one finds q(1) = 0 and q > 0

just to the left of t = 1. The expansions here are straightforward though the key comparisons depend on additional facts concerning eigenvalues for balls. In fact,

2j2/z-11 q(0) = 1, q(0) = 0, q(0) =

n

2.2

1

(5.6)

n + 2

and one must show that j2

in/2,1 2

n

< n+2'

for n > 2, a result that follows using reasonably simple trial functions in a Rayleigh quotient for jn/2-1,1 (see [10], Lemma 2.2). Similarly, q(1) = 0,

q'(1) = ((n - 1) - (.7n/2,1 - jn/2-1,1))/3

(5.7)

and one can show that q'(1) < 0 via another Rayleigh quotient argument. Again, see [10] for details. The next step is to show that q cannot cross below 0, for 0 < t < 1. If it did, one could find values 0 < a < b < 1 such that q'(a) < 0, and q'(b) > 0. On the other hand, the differential (i.e., the Riccati) equation for q is given by 1

q = -(l3z - a2)t + t (1 - q)(q + n - 1) + 2a JJn 2((at) q,

(5.8)

with a = in/2- 1,1 and /3 = jn/2,1 Thus, whenever q = 0 for t E (0, 1) we have

q' = -(/32 - a2)t + n t

(5.9)

The right side of (5.9) is strictly decreasing with increasing t, so it is not possible to have points a, b, with a < b, q(a) = q(b) = 0 and q'(a) < 0, and q'(b) > 0. Therefore q > 0 in (0, 1). In an analogous way one can prove that q < 1 (see [10]). Finally, to prove that q is decreasing in (0, 1), we argue by contradiction. Assume that q' is somewhere positive on (0, 1). Then, there are three points, t1, t2, t3, say, with

0 < t1 < t2 < t3 < 1, such that q(tl) = q(t2) = q(t3), and q'(tl) < 0, q'(t2) > 0, q'(t3) < 0. Think of the right side of (5.8) as a function of q and t. For fixed q (with 0 < q < 1) the right side of (5.8) is a convex function of t, which follows from the convexity of Jn/2(at)/Jn/2_1(at), a fact which is not entirely trivial to establish (see [10], Lemma 2.3). Since t2 above is a convex combination of tl and 6, this immediately yields a contradiction. Thus q' < 0 in (0, 1).

ISOPERIMETRIC INEQUALITIES FOR EIGENVALUES OF THE LAPLACIAN

117

6. The PPW Bound for Domains in S' The analog of the PPW bound for domains contained in a hemisphere of Sn also holds [19]. This follows from Sperner's isoperimetric inequality (i.e., the analog of the Rayleigh-Faber-Krahn inequality for domains in5n) and the following theorem of [19] (a fixed Al result): Let SZ be contained in a hemisphere and let B1 denote the geodesic ball in 5n having the same value of Al as I (i.e., A,(1) = A, (BI)). Then, A2(Q) < )2(B1),

(6.1)

with equality if and only if 1 is itself a geodesic ball in 5n. The proof of (6.1) follows a similar sequence of steps as that sketched in the proof of the PPW bound in Euclidean space. The form of the trial functions Pi must be modified slightly, again with optimality for the case that SZ is itself a geodesic ball as the motivating factor. Thinking of S' as the unit sphere in IR''+1 and with the center of mass point for SZ fixed at the north pole, one considers Pi

g(0)

0,

i = 1,2,...,n,

(6.2)

= where 0 is the azimuthal angle of a point from the axis through the north pole (in Sn, 0 plays the role of a geodesic radial variable). We saw that in the Euclidean case, the special functions that enter into the solution of the Dirichlet problem for a ball were Bessel functions. In the case at hand, they are Legendre or associated Legendre functions. Then one defines as before the function g(O) as the quotient between the radial parts of the first two Dirichlet eigenfunctions of the LaplaceBeltrami operator for the geodesic ball. Following the same scheme as we did earlier, one must prove monotonicity properties of g(O) and the function z

B(0) = g'(0)2 + (n - 1) s9(O)0)

(6.3)

which appears naturally in the problem when using the averaged version of the gap

formula (the analog of (4.5)) with trial functions Pi given by (6.2). In order to prove the monotonicity properties one proceeds as in Section 5, above, but now using the function q(0) - sin 0 g'(0) g(B)

(6.4)

This time the monotonicity properties of g and B follow from these two properties of q,

0
andq'<0,

(6.5)

for 0 out to an appropriate radius 01. The proof of (6.5) is obtained by a careful analysis of the properties of a Riccati equation satisfied by q, analysis that involves certain key convexity and monotonicity properties of some combinations of associated Legendre functions (see [19] for details). The rest of the proof proceeds as before, using symmetric decreasing rearrangements (with respect to the north pole), and the analog of the Chiti comparison argument. Our proof only holds for domains contained in a hemisphere of Sn because the proof of the properties (6.5) only holds if 01 < it/2. Remarks. (i) In fact, in addition to (6.1), we prove that for a domain SZ contained in a hemisphere, .X2/A1 is maximized by the spherical cap of the same area (volume).

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M. S. ASHBAUGH AND R. D. BENGURIA

This follows from (6.1) and the fact that for spherical caps A2/A1 is an increasing function of radius (see [19]), at least up to the hemisphere. (ii) To prove the analog of the sharp PPW bound for A2 when Al is fixed for domains in Hn is still open. That is, no one has yet proved the 1F' analog of (6.1). We also note that an isoperimetric ratio result for 1 C 1FF' saying that A2/A1 is maximized at the ball of the same volume Q* is impossible, for reasons detailed in the introductory section of [19]. (iii) The question of finding sharp estimates for the ratio between the first two Dirichlet eigenvalues in other more general settings can also be addressed. For example, while a sharp PPW bound has been conjectured for the p-laplacian on

bounded domains in R' (for 2 < p < oo), it has only been proven for p = 2 [9], and recently for the limiting case p = oo [95]. For some particular Schrodinger operators, a sharp bound of this type has also been proven [71].

7. The Eigenvalue Gap A2 - Ai For an arbitrary domain 1 in

', the gap between the first two eigenvalues,

known as the fundamental gap, can be very small. (Think, for example, of a domain

made out of two equal balls joined by a thin tube.) However, if we restrict the domain St to be convex, this gap can be bounded from below in terms of the diameter of the domain. In 1983, van den Berg [177] (see also [184], Problem 44, and [5]), made the following conjecture: Consider a bounded, convex subset 1 of W', and let A1(Sl) and A2(1) be the first two Dirichlet eigenvalues of the laplacian on Q. Then 2

(7.1) A2 (Q) - A1(1) ? i2 , where d is the diameter of the set SZ (one can get as close as desired to the conjectured lower bound by choosing cigar-like domains; the lower bound is attained for an interval of length d in one dimension). Since 1983, many authors have ob-

tained lower bounds on this eigenvalue gap [165, 187, 123, 166]. The best general results to date give (7.1) but without the factor 3 on the right. Recently, a number of authors have obtained results for the multidimensional case which are sharp

but require certain symmetry conditions on the domain. These authors include [30, 56, 29, 185] (see also [59]). We note, however, that the general conjecture remains open for an arbitrary, bounded, convex domain Q. Related to this inequality is the Payne-Weinberger inequality [145]: The first nonzero eigenvalue p, for the free membrane problem for a convex region 51 satisfies the inequality

µl ?

Ir d2 ,

(7.2)

where d is the diameter of Q. This inequality is sharp in the sense that µ1d2 tends to 7r2 for a parallelepiped all but one of whose dimensions shrink to zero (or in the limit of an infinite strip). Remarks. (i) The original conjectures for the sharp lower bound to the fundamental gap in terms of the diameter were actually for Schrodinger operators on a convex domain with a convex potential, i.e., for the operator H = -0 + V (x) on a convex domain S2 C R' with potential V a convex function on Q. This more general conjecture also remains open, though the results of many of the authors cited above also apply to this case, giving partial results.

ISOPERIMETRIC INEQUALITIES FOR EIGENVALUES OF THE LAPLACIAN

119

(ii) Lower bounds for the gap between the first two Dirichlet eigenvalues have also been considered for convex domains in S'l (see, e.g., [108] and [123]). For spectral gap estimates on compact manifolds, see, e.g., [134]. 8. Steklov Eigenvalues

The Steklov eigenvalue problem on a bounded smooth domain SZ C TR' consists of finding nontrivial solutions u to

an = v u on 81.

Au = 0 in S1,

(8.1)

Here a/an denotes the outward normal derivative. One seeks values of the constant v for which a nontrivial solution u exists. Such v's are called the Steklov eigenvalues of 0. This problem is also the eigenvalue problem for the Dirichlet-to-Neumann map for the laplacian on Q. Further general information on this subject can be found in [25], pp. 95-96, or [161], pp. 37 (Ex. 3.11) and 94 (Ex. 11.2), for example. The reader should be aware that "Steklov" is also often transliterated as "Stekloff." The setting for this problem may be usefully generalized in various ways. For one, we could consider other operators replacing the laplacian A in the first part of (8.1). For example, the eigenvalue problem for the Dirichlet-to-Neumann map for the operator A + A has been employed to very good effect by Friedlander [69] in his comparison of Dirichlet and Neumann eigenvalues (his results are recounted

in Section 10, below). However, in the sequel we consider only the case of the laplacian. We can also consider modifying the second part of the definition by introducing a nonnegative (and nontrivial) weight function g defined on as1 and asking that

au _

On -

v g u on 00.

(8.2)

A reasonable class of functions from which to take g is L°°(8 1). The Steklov problem for the laplacian, with or without a weight function, has a countable sequence of eigenvalues {vi}°_1 (counted according to their multiplicity) with corresponding eigenfunctions {ui}°_1. These eigenvalues can be listed in -p oo (see, for example, [25]). We can increasing order as 0 = v1 < v2 < v3 < (and will) assume that the corresponding eigenfunctions are real and orthonormal in the L2 sense on a1. Thus, {ui}°O1 forms a real orthonormal basis for L2(a51). For g a positive weight function, the Steklov eigenvalues v can be conveniently studied variationally via the Rayleigh quotient

ffz y 2

(8.3)

where cp represents a real trial function (one can allow complex-valued trial functions cp if one interprets I V 2 in the numerator as V7. V p and if one replaces cp2 in the denominator by Icpl2 = pcp, where - denotes complex conjugation).

For the Steklov problem, vl = 0 (with ul = const.; note that the variational formulation precludes v1 < 0), and the first eigenvalue of real interest is v2. A number of authors have given estimates of v2 for more or less arbitrary bounded domains, or for domains in suitably restricted classes (convex, star-shaped, simplyconnected, ...). We recount here some of the highlights in these development. We focus, in particular, on upper bounds, since these can be obtained using RayleighRitz estimates much as the inequalities for the other eigenvalue problems already

120

M. S. ASHBAUGH AND R. D. BENGURIA

discussed. For lower bounds, one might consult various works of Kuttler and/or Sigillito, for example. For convex planar domains with analytic boundary, there is also the nice isoperimetric result of Payne [141] (or see [25], pp. 161-163), that v2 is greater than or equal to the minimum curvature 'min of the boundary, i.e., v2 > min with equality if and only if the domain is a disk. Weinstock's inequality ([179]): Consider the Steklov eigenvalue problem for a bounded simply-connected smooth domain SZ C R2 with weight g, and total mass M - fag g. The eigenvalue v2 satisfies the inequality v2 < MI

(8.4)

with equality if and only if SZ is a disk and g is constant on 952. This inequality was subsequently strengthened in several stages. What follows will all be in the setting of bounded, simply-connected planar domains with sufficiently smooth boundary. First, it was observed by Hersch and Payne [89] that Weinstock's proof actually yields the stronger inequality, including both v2 and v3i > M 1+ 1 v3

(8.5)

7r

V2

Later, by an improved argument, Hersch, Payne, and Schiffer [90] were able to strengthen this inequality even further to 47x2

M2'

V2 v3

(8.6)

Again, there is equality if and only if 1 is a disk and g is constant. That this inequality is stronger than the one before is a simple consequence of the elementary inequality between the geometric and harmonic means of two positive numbers. Various other inequalities for Steklov eigenvalues in two dimensions are proved in

the papers already cited. For a good summary including the discussion of some further developments, see [25]. Finally, for a generalization to n dimensions, we have the recent result of Brock [38] n+1 2(

i=2 v

)

> nRg,

(8.7)

where R is the radius of the n-dimensional ball BR (= 1 *) having the same measure as 52 and g is defined by

fa0(1/9) 9

(8.8)

JOBRI

In this result, too, equality obtains if and only if SZ is a ball and g is constant on 952. Using the notation C. = 7rn/2/I'(n/2 + 1), we have, by the well-known formulas for volume and surface area of an n-dimensional ball, IBR I = CnRn = 52j and I9BRI = nCnRn-1(< I852J, this last inequality by virtue of the classical isoperimetric inequality). Brock's result holds for arbitrary bounded smooth domains in ][8n. Note that while on the one hand its two-dimensional specialization is somewhat weaker than

the two-dimensional result observed by Hersch and Payne in that its right-hand side is easily shown to be less than or equal to that of the other inequality (by a straightforward application of the Cauchy-Schwarz inequality and use of the classical isoperimetric inequality), it is more general since it applies to all bounded

ISOPERIMETRIC INEQUALITIES FOR EIGENVALUES OF THE LAPLACIAN

121

smooth domains and not just simply-connected ones. And, of course, the bound itself applies in all dimensions. The reason the two-dimensional results are restricted to simply-connected domains is that their proofs rely on the existence of conformal maps taking arbitrary simply-connected domains into a disk. That this is essentially a two-dimensional phenomenon explains both the merits and the drawbacks of the method. Another recent result, which is an isoperimetric lower bound to the sum of the squared reciprocals of all the Steklov eigenvalues save the first, can be found in Dittmar [58]. This result holds for bounded simply-connected plane domains with sufficiently smooth boundary and equality obtains if and only if the domain is a disk. There is also an interesting isoperimetric result due to Edward [61] which involves all the Steklov eigenvalues.

9. Annular Domains In this section we discuss certain problems for the eigenvalues of the laplacian

for annular domains. We start with domains in '. A problem that made the rounds in the 1990s (and perhaps even earlier) is,

Suppose you have a Dirichlet eigenvalue problem inside a disk (or ball) and you have a smaller disk (ball) that you may place as an "obstacle" anywhere inside the larger disk (ball). Where should you place the obstacle so as to maximize (or minimize) the first eigenvalue?

To fix notation, let D1 denote the larger disk and D2 the smaller. Then the question is, among all placements D2 C D1 to find that which maximizes A1(D1

D2) and that which minimizes it (here our disks are taken as open sets, and the notation A denotes the closure of the set A). This question is answered rather easily by the Hadamard variational formula, which tells how a simple eigenvalue can be expected to change at first order due to a specified change in the underlying domain, and a simple reflection and domain monotonicity argument (followed by use of the boundary maximum principle). The answer is that to maximize .A1 one should place the smaller disk in the center of the larger one, while to minimize one should do the opposite, i.e., one should push the small disk to the boundary, so that D2 remains in D1 but D2 just touches the boundary of D1 (so, in fact, the two boundary circles will be tangent). Also, the analogous statement is true in higher dimensions. The proof proceeds as follows. One considers a given placement of the small

disk inside the larger one, then one reflects about the center of the smaller disk (or ball) in the hyperplane perpendicular to the line connecting the centers, and then one uses domain monotonicity of Dirichlet eigenvalues, applies the boundary maximum principle, and finishes off by applying the Hadamard variational formula.

Assuming that the small disk is in "general position," with neither its center coinciding with that of the larger disk or with its boundary touching that of the larger disk, then the reflection hyperplane cuts our domain D1 \ D2 into a smaller piece and a larger piece. We choose to view the smaller piece (and the part of the eigenfunction that lives on it) as reflected into the larger piece and the two eigenfunctions combined by subtraction (the one from the smaller piece is subtracted). By a domain monotonicity argument (and the maximum principle), this difference

122

M. S. ASHBAUGH AND R. D. BENGURIA

must be positive on the reflection of the smaller piece of the domain (we assume here, and throughout, that the first eigenfunction ul has been taken to be positive on f2, as is always possible by a choice of sign). Since the difference is 0 along the reflection hyperplane, and, most crucially, along the part of the boundary of D2 which bounds the reflected piece, one sees that the inward normal derivative of ul on the "larger side" of the reflection hyperplane is pointwise larger than the inward normal derivative of ul at the corresponding point on the "smaller side." One then has only to invoke the Hadamard variational formula (for A a simple eigenvalue of -A with u a normalized real eigenfunction corresponding to A), as at t=o

an

where n denotes the outward normal to the boundary and v is a vector field (ostensibly defined on 0, but ultimately needed only on all) detailing how, at first order in t at t = 0, our domain variation occurs. Thus, behind the scenes we have a mapping, Mt, defined for t small, that sends SZ to a new domain, Qt. In detail, each x E 1 gets sent to y = Mt (x) E Ot and, under the assumption that Mt is C' in t, say, the vector field v is gotten via the identification Mt (x) = x + t v"(x) + o(t). Thus to leading order Mt leaves points alone (i.e., Mo is the identity mapping and,

in particular, S2o = 1) and then f tells us how points are instantaneously pushed at first order by the mapping. In the present case, of course, the vector field v is just a constant on aD2 (and vanishes on aD1) since all we are considering are rigid motions of the obstacle D2. Sorting out the signs in the Hadamard variational formula and using the pointwise comparison of normal derivatives at the points on aD2 identified by reflection (these pointwise comparisons being the result of the reflection argument given above), and noting that rigid motions of D2 parallel to the line connecting the centers lead to "equal and opposite" dot products n v, we see that movement of D2 towards the center (so moving D2 so that its center moves directly towards that of D1) causes Al to increase, while movement of D2 away from the center causes Al to decrease. Since movement along a radius in this fashion covers all possible positions of D2 relative to D, (up to symmetries, i.e., rigid movements of the entire configuration D, \ D2), we see that Al is maximized when D2 is centered and minimized when D2 is pushed to its most extreme eccentric position, i.e., when it just touches the boundary of D1. These arguments have been written up, and applied in greater generality, in the paper of Harrell, Kroger, and Kurata [76] (for some further discussion, see [63]), in the paper [156] of Ramm and Shivakumar, which briefly recounts the argument above, citing a personal communication between one of the present authors (MSA) and Professor Ramm (see also the unpublished manuscript [20]), and in the paper of Kesavan [99]. Some further discussion occurs in Henrot's paper [82], pp. 448-449. Remarks. 1. These investigations were prompted by queries of Davies ([76]) and, independently, Ramm ([20]). We do not know where Kesavan learned of the problem, though it may have been through the paper of Ramm and Shivakumar (quite possibly in an early version), as this paper is referenced in [99]. It is also noted there, but as an afterthought suggested by the referee, that a web-based version of Ramm and Shivakumar's paper cited as "www.math.ksu.edu/ramm/r.html, publication 383," does sketch a proof of their conjecture, i.e., the eigenvalue monotonicity result proved above.

ISOPERIMETRIC INEQUALITIES FOR EIGENVALUES OF THE LAPLACIAN

123

2. Note that the result presented above can be viewed as a (rather special) equal areas (volumes) rearrangement result. There are many other results in the literature that address eigenvalue problems for the laplacian for annular domains (and their higher dimensional analogs). We briefly discuss several of these. An early result of Payne and Weinberger [147] says that among all annular domains in 1R2 of given area A and with outer boundary 91 o of fixed length Lo, if we impose Dirichlet boundary conditions along 9 o and leave the rest of the boundary (D1l \ DSZo) "free" (or we could impose Neumann boundary conditions on the rest of the boundary), then Al is maximized when the outer boundary is a circle of length Lo and the inner boundary is the concentric circle such that the area between the circles is A (in fact, for this result to hold there could be many "inner boundaries" to the original domain). Also, see [25], pp. 144-146, for a generalization of this result. A counterpart to the Payne-Weinberger result is the result of Hersch [87] that for a bounded multiply-connected domain in ll 2 of fixed area A and with Dirichlet boundary condition imposed along one "inner boundary" F0 of fixed length Lo and all other boundaries free, the first eigenvalue .A1 is maximized when the inner boundary is a circle (of length L0), there are no other inner boundaries, and the outer boundary is a circle concentric with the inner boundary of such radius that the area between the two boundaries is A. The outer boundary is free, i.e., one should impose Neumann conditions along it. Again, Bandle [25] discusses this result (pp. 147-149) and gives some extensions. Note that both of the theorems above allow one to obtain computable upper bounds to the first eigenvalue in appropriate situations, as both of the maximizing problems are amenable to exact solution (via separation of variables and the use of Bessel functions).

Some more recent results with a similar flavor are those of Exner, Harrell, and Loss [63]. These authors consider what we shall call "thickened" curves and surfaces in W''. For example, in JR2 one could consider a closed, nonintersecting curve and thicken it by including all points within a distance d of the curve to get a two-dimensional domain Q. If d is small and the curve is nice (C2, say), then SZ will also be nice and we can hope to understand its eigenvalues in terms of the underlying curve. (That all the curves and surfaces we consider are nice enough in this sense, and d small enough, will be an implicit assumption in our entire discussion of the results of Exner, Harrell, and Loss.) In particular, within certain restricted families of curves we may be able to determine the curve or curves that maximize (or minimize) a given eigenvalue of the laplacian on 51, especially the first. In variants of this overall setting, one may decide to include only those points within distance d of the curve and lying either inside or outside. If one moves up in dimension (to ]R', n > 2, say), but stays with curves, then the thickening process leads to tubes, for which we again might hope to be able to say something about eigenvalues based on the underlying geometric structure (and, indeed, there are results in this direction). But, in keeping with our focus on domains of "annular type," the generalization to which we shall adhere in the present discussion is that where a bounded closed hypersurface is considered in ][8n with the hypersurface homeomorphic to Sn-1. We can then thicken this hypersurface into a shell by including all points of IIR" within distance d of it. Again, in this case, as for a

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M. S. ASHBAUGH AND R. D. BENGURIA

closed curve in 1[82, we can choose to include only the points within distance d of the surface and inside (resp., outside) it, should we so choose. We are now in a position to state some of the results of Exner, Harrell, and Loss. First, in the setting of closed curves in R2, among all curves of a given length and for a fixed choice of d (suitably small), the first eigenvalue of the thickened curve is uniquely maximized when the curve is a circle. (This holds whether the thickening is done to both sides, or in either one-sided sense. But one should make a choice and adhere to it.) Second, if we move up to closed surfaces in 1183 homeomorphic to S2 and we fix the area of the surface and the volume of Il, and insist that the surface be convex and that the thickening be done only towards the inside of the surface, then again the first eigenvalue Al is uniquely maximized when the surface is a (Euclidean) sphere.

Remarks. 1. Note that in the case of closed curves in R2 the constraint on the length and specification of d (and how the thickening is done) completely determine the area of Q. Thus the result for closed curves is actually an "equal areas" result and as such could be compared or contrasted with the Rayleigh-Faber-Krahn inequality and other such results. Obviously the second result is an "equal volumes" result as stated, and as such bears similar comparisons. Note that equal volume in this case entails changing the value of d when one changes from one surface to another. 2. There is a sense in which the results of Exner, Harrell, and Loss discussed above relate directly to eigenvalue problems on the underlying curves or surfaces

themselves. This can be seen if one considers the limiting situation when d --> 0 and renormalizes the eigenvalues (this works because with d going to infinity the eigenvalues also go to infinity, but in a controlled and uniform way). The basic operator then becomes the laplacian on the curve or surface as induced from the Euclidean metric of 1[8', possibly modified by potential terms coming from curvature effects. Thus, one arrives at the study of a sort of geometric Schrodinger operator,

some cases of which are discussed in Section 14 of this paper. These also have physical relevance in the setting of electrical properties of nanoscale structures (the consideration of quantum wires, quantum optics, and the like).

Finally, we mention that some interesting results for the first (resp., second) Dirichlet eigenvalue of concentric annular domains on the sphere S2 have been obtained by Shen and Shieh [163] (resp., Shieh [164]). By "concentric annular domains" in this setting we mean domains defined as the region between two concentric circles on the sphere. For the purposes of our descriptions, we shall speak in terms of the north and south poles, the equator, and lines of latitude and longitude. Furthermore, we shall always consider the circles that bound our annular domains

to have their centers at the poles so that these circles become lines of latitude. Obviously there is no loss of generality in this restriction, and once the results are stated the reader will have no difficulty applying them to annular domains in arbitrary position. For A1, the result is that among spherical bands (belts) of fixed area on S2, the one which maximizes Al is the symmetric band centered on the equator (an equatorial belt). Indeed, Shen and Shieh show that Al increases monotonically

as one pushes the belt from its extreme position as a polar cap to the centered equatorial position. This fits nicely with the Rayleigh-Faber-Krahn result on S2, as the polar cap is the minimizer of Al among all domains of a given area, and thus initially as one slides the belt away from the pole Al must go up. Shen and Shieh add to that by showing that it just keeps increasing until its other extreme

ISOPERIMETRIC INEQUALITIES FOR EIGENVALUES OF THE LAPLACIAN

125

position is reached. (Note that if one removes the restriction that the boundary circles be centered at the poles, but insists that they remain concentric, then the main result is that Al is maximized by any belt of the prescribed area centered on a great circle.) The second paper shows that the same main statement can be made for A2 if

one restricts the total area of the band to be less than or equal to 27r (the area of a hemisphere). That is, Shen shows that among concentric annuli (i.e., bands) on S2 of a given area A < 27, the band centered on the equator maximizes A2. Though monotonicity is not shown, it is conjectured, and monotonicity is shown for the analogous problem in R2. Moreover, it is shown that .\2 for any band is doubly degenerate (i.e., A2 has multiplicity 2) and that the nodal line of any second eigenfunction is given by two pieces of longitudes (parts of the same great circle through the poles, where again we are restricting our description to the case of latitudinal bands).

10. Inequalities Between Dirichlet and Neumann Eigenvalues A different type of universal inequality that one can consider involves a comparison between Dirichlet and Neumann eigenvalues. Several inequalities between Dirichlet and Neumann eigenvalues have been established. Let 0 < Al < A2 < ... denote the Dirichlet eigenvalues and 0 = µo < µl < µ2 < ... denote the Neumann eigenvalues for a bounded domain S2 C R', with a smooth boundary. The min-max characterization of Dirichlet and Neumann eigenvalues gives the easy comparison, for all k > 1. (10.1) /1k-1 < Ak, It was suggested by Payne (see [139] or, for example, [142], p. 155) that for all k > 1, (10.2) /1k < Ak, and perhaps more, especially for convex domains. The first such inequality, (10.3) µl < A1, was proven by Polya [149] (motivated by the work of Kornhauser and Stakgold [100]). This is an immediate consequence of the Szeg6-Weinberger inequality, the Rayleigh-Faber-Krahn inequality, and an explicit calculation for balls, through the chain of inequalities,

pi (52) < µ1(S2*) < A1(S2*) < A1(S2),

(10.4)

as first suggested in [100]. The first proof of (10.3), however, as given by Polya [149], used a different intermediate result, which circumvented any direct need for the Szego-Weinberger inequality. (In connection with (10.4), we note that µ1(S2*)/Al (Q*) = p,2n/2/jn/2-1 < 1, for all n.) A little later Payne [139] proved that if in two dimensions S2 is convex, /2k+1 < Ak,

k > 1.

(10.5)

Then, Aviles [24] showed the strict inequality, (10.6) k=1,2, ... for domains in Rn with a C2+" boundary having nonnegative mean curvature.

µk+1 < Ak,

Concurrently and independently, Levine and Weinberger [110] proved this same result as part of a family of results for domains in R based on hypotheses on the principal curvatures of 852. Their first result was (10.6) under the same nonnegative

M. S. ASHBAUGH AND R. D. BENGURIA

126

mean curvature condition that Aviles used, but then they took progressively more restrictive hypotheses on the principal curvatures, obtaining stronger and stronger inequalities, until reaching the convex case (and the result, (10.8), given explicitly below). Finally, in 1991, Friedlander [69] proved Payne's suggestion, µk < ,\k,

for k > 1,

(10.7)

for domains SZ C R" with a C' boundary. (For an alternative proof see [130], and more recently [66].) Levine and Weinberger [110] had shown earlier, among their other results, that for 1 a bounded, smooth, convex domain in R' (n > 2), for all k > 1.

µk+._1 < Ak,

(10.8)

Inequalities between Dirichlet and Neumann eigenvalues have also been considered for domains in S [22, 94]. In particular, one has (10.9)

{t1(Q) < AI (1),

for smooth domains S2 contained in a hemisphere of S'2, which follows from Sperner's inequality [167], our analog of the Szeg&-Weinberger inequality (valid for domains

contained in a hemisphere) [17], and exact calculations for geodesic balls. Notice

that (10.9) cannot hold for general domains in S', since ,a = Al = n at the hemisphere in 5', and µ1 > Al for geodesic balls larger than the hemisphere. In [22], the analog of the Aviles bound is proven for domains in S'ti, more precisely, that µk (Il) < Ak (Si) for a domain Il C 5" whose boundary is everywhere of nonnegative mean curvature (with strict inequality if the mean curvature is ever positive). This

and further results, based on a more differential-geometric approach, are to be found in [94].

11. P61ya's Conjectures For large values of k, if SZ C R', Weyl [180] (see also [181, 155]) proved ~_

47r2k2/"

Ak

(11.1) (CCIQ1)2/n'

where S2 and Cn = 7r"/2/I'(n/2 + 1) are, respectively, the volumes of 1 and of the unit ball in '. Relation (11.1) is usually referred to as Weyl asymptotics or as Weyl's law. For any plane-covering domain (i.e., a domain that can be used to tile the plane without gaps or overlaps, allowing rotations, translations, and reflections of itself), Pdlya [152] proved that

Ak>

4Ak

forallk=1,2,...

(11.2)

and conjectured the same bound for any bounded domain in R2 (here A denotes the area of the domain). Polya's conjecture in n dimensions is equivalent to saying that the Weyl asymptotics of Ak (11.1) is in fact a lower bound for Ak, i.e., Ak >

47r2k2/ri

(C"

)2/'

'

(11.3)

for k = 1, 2. .... This can be shown for tiling domains in I[8n, following Polya's proof of the two-dimensional result. The best result to date towards a proof of the

ISOPERIMETRIC INEQUALITIES FOR EIGENVALUES OF THE LAPLACIAN

127

Polya conjecture (11.3) is the bound of Li and Yau [113] n

k

41r2k1+2/n

EAa> n+2(CCI1 )2/n' i=1

(11.4)

for all k = 1 , 2, ... , proven using the asymptotic behavior of the heat kernel of 52 and the connection between the heat kernel and the Dirichlet eigenvalues of a domain (see, e.g., [55] for a review and related results; see also [132] for a recent improvement on (11.4) that includes a correction term). From (11.4) it follows easily that the individual eigenvalues Ak satisfy Ak >

n 47r2k2/n n+2 (CCI52I)2/n

(11.5)

for all k > 1. In a sense (11.4) is an "integrated" form of the conjectured bound (11.3).

There are related results also for Neumann eigenvalues. For large values of k, Weyl [180] proved µk N

47r2k2/n

(11.6)

(C'nI1I)2/n

For any plane-covering domain, Polya [152] and, in full generality, Kellner [98] proved 47rk

µk C A

for all k = 1, 2....

(11.7)

and conjectured the same bound for any bounded domain in R2. The analogous conjecture in n dimensions is µk <

47r2k2/n

(11.8)

(C.IQI)2/n'

for k = 1, 2, .... Thus far, Kroger [103, 104, 105] has obtained the most significant

results towards a proof of Polya's conjecture for Neumann eigenvalues, (11.8). In particular, he has proven the following integrated version of (11.8) n

k

pi < n+2 i=1

47r2k1+2/n

(11.9)

(CnIQI)2/n'

for all k = 1, 2, .... He has also proven that µk

-

n+2 \

2/n 47r2 k2/n

2

k = 0,1,2,...,

(C'nlgl)2/n'

(11.10)

for the individual µk's. We note that a proof of Polya's conjectures for both Dirichlet and Neumann eigenvalues would imply Friedlander's result (10.7). Polya's conjectures for 1 C ll 2 were first published in [150] (in a weaker and preliminary form for the Neumann case), and later, in definitive form, in [152].

Remark. Inequalities of the form 47r2 k2/n Ak

(11.11)

> an for some constants an < n/(n + 2) (cf. (11.5)) were proven for bounded domains in [35, 50] and later for arbitrary domains in [159, 160, 133, 116]. The result of (CnI1I)2/n'

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M. S. ASHBAUGH AND R. D. BENGURIA

Li and Yau gives (11.11) with a,= n/(n+2). Polya's conjecture says that (11.11) should hold with a,,, = 1 for an arbitrary bounded domain Q.

12. The Bass Tone of a Drum and Its Inradius The inradius rO of a domain Il C R2 is the radius of the largest disk that can be inscribed in Q. It follows from domain monotonicity that among all planar regions of fixed inradius, the disk has the highest bass tone (throughout this section we will call the square root of the lowest Dirichlet eigenvalue of ft ai (Q), its bass tone; this represents the actual lowest vibrational frequency of a uniform drum of shape S2 up to an overall constant factor determined by physical parameters). In 1951, P61ya and Szego [153] found the first lower bounds for r' A1(1), for convex domains, and raised the question of finding the best lower bound

A - inf r2 ai(l)

(12.1)

for general simply connected domains. In 1965, Makai [127], proved the lower bound

A> 1

(12.2)

(see also [126, 81, 135, 173, 154, 54]). If 1 is an infinite strip of width 2, one obtains r Al (1l) _ 7x2/4, hence,

A < 4 = 2.467...

(12.3)

(7r2/4 is, in fact, a sharp lower bound for r2 A1(1) among all convex domains [86]). Using probabilistic methods, Baiiuelos and Carroll [26, 27, 28, 42], improved the lower bound on A and obtained

A>0.6197...

(12.4)

On the other hand, by considering an explicit, nonconvex domain, they also showed

that A < 2.1292... (12.5) While (12.4) and (12.5) are the best estimates on A to date, determining its exact value, and characterizing the optimizing domain(s), are still open problems. It was long ago observed by Polya that the inradius rD is a much better geometric indicator of the bass tone (or A1) of a membrane (drum) than its area A, the Rayleigh-Faber-Krahn inequality notwithstanding, in the following sense. Specifically, even though one has the Rayleigh-Faber-Krahn lower bound, there can be no upper bound in terms of A since a membrane of given area can have arbitrarily large first eigenvalue (consider a x b rectangles with fixed area ab). On the other hand, the results above show that a simply-connected domain has its bass tone bounded

both above and below by constant multiples of 1/ro, and, in fact, the constants are not that far apart. This was quite true already in Polya's time, and he is said to have remarked that rO determines the bass tone of a drum to within an octave. Indeed, with the improved constant due to Banuelos and Carroll there is only a factor of approximately 3.05 between the upper and lower bounds (/0.6197 0.7872 vs. jo,i 2.4048). For convex domains, of course, the agreement is even better: The ratio between jo,1 and 7r/2 is approximately 1.53. Thus if one estimates the bass tone by taking the geometric mean of the two bounds for a convex drum, one cannot be off by more than about 25% either way.

ISOPERIMETRIC INEQUALITIES FOR EIGENVALUES OF THE LAPLACIAN

129

It might also be observed that the fundamental gap A2 - Al of a simplyconnected domain has an isoperimetric upper bound in terms of inradius, viz., A2 - Al < (72,1 - jo,1)/r2 . This follows at once from the result (due to Baiiuelos) quoted at the end of Section 4, using c = 1 and rO < r (see [153]). Unfortunately, a rectangle with one dimension arbitrarily large shows that no corresponding lower bound is possible (i.e., there can be no "inradius lower bound" for the fundamental gap). Moreover, this remains true even if we restrict consideration to convex domains.

13. The Eigenvalue Ratio A3/A1 The problem of maximizing the ratio A3/A1 in two dimensions, has been considered by several authors. The maximum is certainly not achieved by the disk (for the

disk, A3/A1 .:: 2.539, while for the f by f rectangle, A3/A1 = 35/11 :: 3.182). Marcellini [128], found the upper bound 3.917, which was improved in 1994 by Ashbaugh and Benguria [14] to 3.905 and then in 1996 to 3.831 [18], so the optimal bound is between 3.182 and 3.831. Levitin and Yagudin [112], via a numerical search, found a certain dumbbell shape for which A3/A1 is approximately 3.202. Recently, Trefethen and Betcke [175] have computed numerically this ratio for a domain consisting of the union of the disks of radius 2.002323 centered at +1.191322. For this particular domain, A3/A1 falls short of the value of Levitin rectangle, the and Yagudin by approximately 0.35%. For each of the by domain considered by Trefethen and Betcke, and several domains found by Levitin and Yagudin (maximizing A3/A1 within one class or another), one has the property that A3 = A4. It has been conjectured by Ashbaugh and Benguria [15] that the domain that maximizes A3/A1 should satisfy A3 = A4 (notice also that the optimal upper bound for A2/A1 is attained for the ball, at which A2 = A3). This conjecture is still open. Some indications in its support may be found in [112] (see, in particular, Theorem 4.2).

14. An Isoperimetric Inequality for Ovals in the Plane The next problem we will consider is a conjectured isoperimetric inequality for closed, smooth curves in the plane. It has attracted considerable attention in the literature during the last decade (see, e.g., [60, 77, 63, 75, 40]), and it has many interesting connections in geometry and physics. In particular, Benguria and Loss [31] have shown a connection between this problem and a special case of the Lieb-Thirring inequalities [120, 117], inequalities which play a fundamental role in Lieb and Thirring's proof of the stability of matter (see, in particular, [119] and the review article [115]). Denote by C a closed curve in the plane, of length 27r, with positive curvature ic, and let

H(C)

S2

+r2

(14.1)

acting on L2 (C) with periodic boundary conditions. Let A1(C) denote the lowest eigenvalue of H(C). Certainly, A1(C) depends on the geometry of the curve C. It has been conjectured that A1(C) > 1, (14.2) with equality if and only if C belongs to a one-parameter family of ovals which include the circle (in fact, the one-parameter family of curves is characterized by a

130

M. S. ASHBAUGH AND R. D. BENGURIA

curvature given by ic(s) = 1/(a2 cos2(s)+a-2 sin2(s))[31]). It is a simple matter to see that if C is a circle of length 27, the lowest eigenvalue of H(C) is precisely 1. The fact that there is degeneracy of the conjectured minimizers makes the problem much harder. The conjecture (14.2) is still open. Concerning (nonoptimal) lower bounds, Benguria and Loss proved [31]

)1(C) > 2,

(14.3)

and more recently Linde [121] found the best lower bound to date,

\1(C) > I 1+ + 8

)

(14.4)

0.60847.

Even though this is the best bound to date, it is still some distance from the conjectured optimal value 1. Very recently Burchard and Thomas [41] have shown that the ovals that give A1(C) = 1, alluded to before, minimize A, (C) at least locally, i.e., there is no small variation around these curves that reduces )u (C). This is an important indication

that the conjecture (14.2) is true, though, of course, it is not enough to prove it and thus it remains open. In recent years several authors have obtained isoperimetric inequalities for the lowest eigenvalues of a variant of H(C), and we give a short summary of the main results in the sequel. Consider the Schrodinger operator H9(C)

2

(14.5)

+grc2

defined on L2 (C) with periodic boundary conditions. As before, C denotes a closed curve in R2 with positive curvature ic, and length 27r. Here, s denotes arclength. If g < 0, the lowest eigenvalue of H9 (C), say A1(g, C), is uniquely maximized when

C is a circle [60]. When g = -1, the second eigenvalue, A2(-1, C), is uniquely maximized when C is a circle [77]. If 0 < g < 1, A, (g, C) is uniquely minimized when C is a circle [63]. It is an open problem to determine the curve C that minimizes .\ (g, C) in the cases, 4 < g _< 1, and g < 0, g -1. If g > 1 the circle is not a minimizer for .Al (g, C) (see, e.g., [63, 75] for more details on the subject).

15. Open Problems As we have seen throughout the paper, there are many open problems concerning universal and isoperimetric bounds for low-lying eigenvalues of the laplacian. To finish, we will list a few more of the most interesting conjectures in the field (see [140, 142, 184, 15, 3] and references therein for additional open problems). Other interesting open problems (and much else of interest) can be found in the forthcoming book by Henrot [83]. Perhaps the most significant outstanding problems concerning eigenvalue ratios are the following conjectures of Payne, Polya, and Weinberger: (i)ForSlC1182 <

A2 + A3

Al

(Q)

A2 + A3(Q*)

5.077

and its n-dimensional analog (i.e., for S2 C R n) + An+l A2 + A3 + A2 + A3 + A1

l

(15.1)

Al

Al

. + A.+1

(S2*).

(15.2)

ISOPERIMETRIC INEQUALITIES FOR EIGENVALUES OF THE LAPLACIAN

131

For nonoptimal bounds on these quantities, see [144, 37, 128, 13, 14, 181. In particular, the best bound proved to date for (A2 + A3)/A1 in two dimensions is approximately 5.507 [18], while the conjectured best bound is given by (15.1). (ii) For SZ C IR', and m = 1 , 2, 3, ... ,

A-+, (Q) < A..

(c*).

(15.3)

Al

This inequality has been proved for the cases m = 1, 2, and 3 [8, 11]. (iii) Indeed, it might well be true that for integers m > 1,

2, (S) <

A1(SZ*).

(15.4)

This was not conjectured by Payne, Polya, and Weinberger, although it would represent a substantial strengthening of (ii) above. The inequality has been proved for m = 2 (see [11]). For further discussion and indications in this direction, see [11, 16]. A similar conjecture could be made for Ak,,,,/Am(1), which would be that Akm/a,,, (1) < supfiAk/A1(1), but for this to be truly useful one would need to know or have good estimates for supfiAk/A1(SZ), and the present state of our knowledge for k > 2 is not nearly as good as it is for k = 2 (the case addressed by (15.4)).

(iv) For 1 C I[82, show that A4

Al

(Q) <

A4

(S *)

4.5606.

(15.5)

Al

The best bound to date is (A4/A1)(S2) < (A2/A1(S2*))2 (v) For Il C 1182, show that

6.4452 [11].

A3(SZ) > \3(11*),

(15.6)

(A2 + A3)(1) > (A2 + A3)(SI*).

(15.7)

or even that Corresponding inequalities can be formulated for SI C ll ; see [3] for details. (vi) Friedlander's conjecture: It has been observed by Friedlander that possibly

µk+._1 < Ak,

for all k > 1.

(15.8)

This is (10.8) from Section 10, so it is known for all convex domains in WL. The novelty here is that this inequality, with index shift n - 1 in dimension n, might indeed be true for all domains (one should disregard a claimed example to the contrary mentioned near the end of [110] and in [109], p. 126). This conjecture, if true, would represent a substantial improvement upon (10.7). (vii) For a convex domain SI C 1[8', show that µ2

< 4.

(15.9)

14

This bound is the analog of van den Berg's conjecture for the case of Neumann eigenvalues. It would be saturated by any rectangular parallelepiped having all but one of its dimensions tiny.

M. S. ASHBAUGH AND R. D. BENGURIA

132

References [1] M. Abramowitz and I. A. Stegun, editors, Handbook of Mathematical Functions, National Bureau of Standards Applied Mathematics Series, vol. 55, U.S. Government Printing Office, Washington, D.C., 1964. [2] M. S. Ashbaugh, Isoperimetric and universal inequalities for eigenvalues, Spectral Theory and Geometry (Edinburgh, 1998), E. B. Davies and Yu. Safarov, editors, London Math. Soc. Lecture Notes, vol. 273, Cambridge University Press, Cambridge, 1999, pp. 95-139.

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VII: Spectral Theory of Differential Operators, Springer, Berlin, 1994 (Series Editor, M. A. Shubin). [162] E. Schmidt, Beweis der isoperimetrischen Eigenschaft der Kugel im hyperbolischen and sphdrischen Raum jeder Dimensionenzahl, Math. Z. 49 (1943), 1-109. [163] C. L. Shen and C.-T. Shieh, Some properties of the first eigenvalue of the Laplace operator on the spherical bands in S2, SIAM J. Math. Anal. 23 (1992), 1305-1308. [164] C.-T. Shieh, On the second eigenvalue of the Laplace operator on a spherical band, Proc. Amer. Math. Soc. 132 (2004), 157-164. [165] I. M. Singer, B. Wong, S.-T. Yau, and S. S.-T. Yau, An estimate of the gap of the first two eigenvalues in the Schrodinger operator, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), 319-333. [166] R. G. Smits, Spectral gaps and rates to equilibrium for diffusions in convex domains, Michigan Math. J. 43 (1996), 141-157. [167] E. Sperner, Zur Symmetrisierung von Funktionen auf Sphdren, Math. Z. 134 (1973), 317327.

[168] G. Szego, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal. 3 (1954), 343-356. [169] G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa (4) 3 (1976), 697-718.

[170] G. Talenti, The standard isoperimetric theorem, Handbook of Convex Geometry, vol. A, P. M. Gruber and J. M. Wills, editors, North-Holland, Amsterdam, 1993, Chap. 1.3, pp. 73-123.

[171] G. Talenti, On isoperimetric theorems of mathematical physics, Handbook of Convex Geometry, vol. B, P. M. Gruber and J. M. Wills, editors, North-Holland, Amsterdam, 1993, Chap. 4.4, pp. 1131-1147.

[172] G. Talenti, Inequalities in rearrangement invariant function spaces, Nonlinear Analysis, Function Spaces and Applications, vol. 5 (Prague, 1994), M. Krbec, A. Kufner, B. Opic, and J. Rakosnik, editors, Prometheus Publishing House, Prague, 1994, pp. 177-230. [173] M. E. Taylor, Estimate on the fundamental frequency of a drum, Duke Math. J. 46 (1979), 447-453.

[174] C. J. Thompson, On the ratio of consecutive eigenvalues in N-dimensions, Studies in Appl. Math. 48 (1969), 281-283. [175] L. N. Trefethen and T. Betcke, Computed eigenmodes of planar regions, Contemp. Math., to appear (2005). [176] H. Urakawa, Bounded domains which are isospectral but not congruent, Ann. Sci. Ecole Norm. Sup. (4) 15 (1982), 441-456. [177] M. van den Berg, On condensation in the free-boson gas and the spectrum of the Laplacian, J. Statist. Phys. 31 (1983), 623-637.

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[178] H. F. Weinberger, An isoperimetric inequality for the n-dimensional free membrane problem, J. Rational Mech. Anal. 5 (1956), 633-636.

[179] R. Weinstock, Inequalities for a classical eigenvalue problem, J. Rational Mech. Anal. 3 (1954), 745-753.

[180] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen, Math. Ann. 71 (1911), 441-479. [181] H. Weyl, Ramifications, old and new, of the eigenvalue problem, Bull. Amer. Math. Soc. 56 (1950), 115-139.

[182] S. A. Wolf and J. B. Keller, Range of the first two eigenvalues of the Laplacian, Proc. Roy. Soc. London Ser. A 447 (1994), 397-412.

[183] H.-C. Yang, An estimate of the difference between consecutive eigenvalues, preprint IC/91/60 of the Intl. Centre for Theoretical Physics, Trieste, 1991 (revised preprint, Academia Sinica, 1995). [184] S.-T. Yau, Open problems in geometry, Proc. Sympos. Pure Math., vol. 54, Part 1, Differ-

ential Geometry: Partial Differential Equations on Manifolds (Los Angeles, 1990), R. Greene and S.-T. Yau, editors, Amer. Math. Soc., Providence, R.I., 1993, pp. 1-28. [185] D. You, Sharp inequalities for ratios of partition functions of Schrodinger operators, Potential Analysis 18 (2003), 219-250. [186] Q.-H. Yu, On the first and second eigenvalues of Schrodinger operator, Chinese Ann. of Math. 14B (1993), 85-92. [187] Q.-H. Yu and J.-Q. Zhong, Lower bounds for the gap between the first and second eigenvalues for the Schrodinger operator, Trans. Amer. Math. Soc. 294 (1986), 341-349. (Mark S. Ashbaugh) DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MISSOURI, COLUMBIA,

MISSOURI 65211-4100, U.S.A.

E-mail address: [email protected] (Rafael D. Benguria) DEPARTAMENTO DE FISICA, P. UNIVERSIDAD CATOLICA DE CHILE, CASILLA 306, SANTIAGO 22, CHILE

E-mail address: [email protected]

Proceedings of Symposia in Pure Mathematics Volume 76.1, 2007

Non-Self-Adjoint Operators and Pseudospectra E. B. Davies Dedicated to Barry, in recognition of the constant flow of ideas that he has provided over the years

ABSTRACT. The theory of pseudospectra has grown rapidly since its emergence from within numerical analysis around 1990. We describe some of its applications to the stability theory of differential operators, to WKB analysis and even to orthogonal polynomials. Although currently more a way of looking at non-self-adjoint operators than a list of theorems, its future seems to be assured by the growing number of problems in which the ideas are clearly of relevance.

CONTENTS 1.

2. 3. 4. 5. 6.

Introduction The Basic Ideas Generalized Pseudospectra Truncation Numerical Range and Functional Calculus Orthogonal Polynomials

References

1. Introduction The readers of this Festschrift volume will hardly need persuading that interesting problems and techniques can enter mathematics from physics or probability, but it is less usual for a major new idea in spectral theory to originate in numerical

analysis. This paper is an introduction to the theory of pseudospectra, a way of imposing some order on the chaos of non-self-adjoint (NSA) phenomena.

Since 1990 this subject has developed rapidly, and has been shown to give substantial insights into the properties of NSA matrices and operators, including 2000 Mathematics Subject Classification. 47A10, 47A12, 34L05, 65F15, 33C45, 33Dxx.

Key words and phrases. non-self-adjoint operators, spectrum, pseudospectra, numerical range, orthogonal polynomials. ©2007 American Mathematical Society 141

E. B. DAVIES

142

second order differential operators. It is relevant to the study of resonances if one uses the method of complex scaling, and also models involving `optical potentials' which replace certain channels by dissipative terms in the Schrodinger operator, but it has many other applications. Michael Berry has a series of papers on the importance of NSA 2 x 2 matrices in physics. I am happy to be able to announce on this occasion that as the result of my collaboration with Barry Simon, it has recently proved useful in resolving an outstanding question about orthogonal polynomials on the unit circle. However, most of this review will be a description of the basic ideas, to be found in four recent books on the subject. The most important is by Trefethen and Embree, the first of whom is responsible for the explosion of activity in the field; the authors illustrate it with a wealth of applications and numerical examples which demonstrate its relevance; [20]. The second, by Bottcher and Silbermann, is about Toeplitz operators, but makes substantial use of pseudospectral ideas; [1]. The third volume, by this author, covers many topics, some related to pseudospectra, and some not; [5]. Finally Hinrichsen and Pritchard [11] contains a lot of related material. Everything written here may be found in one or more of these books, except where explicit references are given.

2. The Basic Ideas If one wishes to compute the spectrum of a NSA linear operator, or even a large NSA matrix, one quickly comes up against problems which make one suspect that one has mistakes in the code. Although some of the eigenvalues appear where one expects, others may be in completely wrong positions. The reason is spectral

instability: the fact that IAf-Af1I <EIIfII for E = 10-16 (say) does not imply that A is close to the spectrum of A. It is best to describe the problem in terms of the resolvent norms. It is easy to prove that IIR(z, A)II > dist(z, Spec(A))-1.

This becomes an equality if A is a normal operator (i.e., AA* = A*A), but nothing which bears any resemblance to a reverse inequality holds in general. Since z is an approximate eigenvalue in the sense described above if and only if IIR(z, A) 11 > e-1, this implies that there might be large regions of the complex plane far from the spectrum which are associated with approximate eigenvalues. The graphics package EigTool enables one to compute the pseudospectral regions Spec,(A) := {z : IIR(z, A)II > e-11 very efficiently; see [21]. The alternative definition

Spec,(A) := U{Spec(A+B) : IIBII < e} provides the very helpful insight that one cannot hope to distinguish numerically between Spec(A) and Spec,(A) if e is the rounding error of one's computer. This is not just theory. The NSA harmonic oscillator H := P2 + c2Q2 acting in L2 (R) has compact resolvent and its eigenvalues are A,,, := c(2n + 1) where n = 0, 1, .... For every complex c the norms of the spectral projections Pn increase at an exponential rate as n --> oo, and the precise constant is given in [7]. Its pseudospectral diagram is given for c := i1/2 in Figure 1, and started my interest in the subject. The resolvent norm is equal to 105/2 for s := 0, 1, 2, ... on the contours drawn, the outermost contour corresponding to s := 0. The more or less

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143

uniform spacing of the contours confirms the exponential growth of 1R(reie, A) 11

as r --> +oo, provided 0 < 0 < ir/2. Note that the norms of the resolvents and of the spectral projections are so large in the top right-hand part of the figure that one needs to be careful to avoid errors due to an inaccurate discretization of the differential operator or rounding errors in the computation. 30

25

20

15

10

5

0

-5 -5

0

5

10

15

20

25

30

FIGURE 1. Pseudospectra of the NSA harmonic oscillator

Although e-Ht is compact for all t > 0 and its kernel is given by Mehler's formula, the expansion

e- Ht = n=O

where PP,, are the spectral projections, is only norm convergent for t greater than a certain positive critical constant; [7]. Understanding the pseudospectra of this example depended on the development of NSA WKB analysis; [2, 3]. The basic idea is to use WKB methods to produce explicit approximate eigenfunctions associated with points in the (real) phase space which satisfied a certain inequality expressed in terms of the classical Hamiltonian. The approximate eigenvalue is the (complex) value of the classical Hamiltonian at the relevant point, and the error vanishes as h - 0. It was realized by Zworski that

E. B. DAVIES

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the idea was applicable to general pseudodifferential operators and was simply a new way of looking at an old theorem of Hormander; [22]. It has inspired a number of new theorems on the subject; [10, 4, 15, 19, 18]. EXAMPLE 1. The standard n x n Jordan matrix Jn is defined by (Jn)r,s

1

ifs = r + 1,

0

otherwise.

1

The resolvent norm is most easily computed if one uses the 11 norm on Cn. Starting from the formula

((zI - J.)-1)"',

zr-s-1

if r < s, otherwise.

0

one immediately obtains IzI-n

II (zI -

J)-1 n 1I 1 =

-1

1-

This diverges at an exponential rate as n - oo for every z satisfying Izi < 1. The same phenomenon occurs if one uses the standard 12 (Euclidean) norm on Cn

and the pseudospectra are again rotationally invariant in C. One says that the pseudospectra fill up the unit circle at an exponential rate even though Spec(Jn) _ {0} for every n. The following theorem may be used two ways. If one only knows an operator

A to within an error S > 0 then its pseudospectra Spec,(A) do not have any significance for e < S, although they are numerically stable for substantially larger e. Conversely, if one is only interested in the shape of the pseudospectra of A for e > S, one may add any perturbation of norm significantly less than 6 to A before carrying out the computation. The following theorem is equivalent to Theorem 52.4 of [20].

THEOREM 2. Let A1i A2 be two bounded operators on 13 satisfying 11A1-A21$ <

6. If we put

ar(z)

.

_

0

if z E Spec(Ar),

IIR(z,Ar)II-1

otherwise,

for r = 1, 2 then Spec,(Ar) = {z : 0,(Z) < E} and 1a1(z) - a2(z)J < 6

for all z c C.

3. Generalized Pseudospectra The standard theory of pseudospectra is a study of properties of the operator

family z -> (zI - A) where A is a bounded or unbounded linear operator; see [20]. However, the key ideas and theorems extend to other operator families. The parameter space need not be C, and in [4] it is the set of all points in the phase space associated with the differential operator. Even if we restrict to z E C as the parameter one may consider polynomial operator pencils such as

A(z) :=

n

E Arzr r=1

NON-SELF-ADJOINT OPERATORS AND PSEUDOSPECTRA

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where A, are operators on some Banach space B. Whole books have been devoted to this subject, for example [12], but we mention just one example. EXAMPLE 3. Given bounded operators A, B on some space 13 one seeks solutions of an abstract telegraph equation 2

at + Af =0

t +B e that are of the form ft := eztg. Assuming that suitable technical conditions are satisfied this leads one directly to Ag + zBg + z 2 g = 0.

This can be rewritten as a conventional eigenvalue equation for the 2 x 2 block matrix

X:= (-AB I0

The eigenvalue problem for the general 2 x 2 block matrix Y'_

AB CD

is often reformulated as a generalized eigenvalue problem for the analytic family

M(z) := C(zI - A)-1B + D - zI where one assumes that z V Spec(A).

For simplicity of exposition we assume below that the parameter is a complex number and that the operators concerned are bounded, although many of the theorems hold in more generality. The theorems below are taken from [5], but their proofs are obvious adaptations of the same results for the standard notion of pseudospectra. By definition the spectrum of an operator family is the set {z : A(z) is not invertible}.

The pseudospectra are defined to be the sets Spec, ((A(.)) := S U Spec(A(.))

where S is the set of z for which there exists an `approximate eigenvector' f E 13 satisfying IIA(z)f1I < rMIfII.

Note that if A and B are bounded invertible operators the equations A f = zB f ,

B-lAf = zf and f = zA-1Bf have the same eigenvalues z, but the pseudospectra may look very different, depending on the norms of the inverse operators (among other things).

THEOREM 4. The following three conditions on an operator family A(.) are equivalent.

(i) z E (ii) There exists a bounded operator D : 131 -> B2 for which IID II < E and A(z) + D is not invertible; (iii) Either z E Spec(A(.)) or IIA(z)-111 > e-1.

Some information about the shape of the pseudospectra is provided by the next theorem.

146

E. B. DAVIES

acting on a Hilbert THEOREM 5. Given an analytic operator-valued family space put JJA(z)-111-1 if A(z) is invertible, or (Z) otherwise. 0

Then a is continuous and its local minima all lie in

4. Truncation Pseudospectra are relevant not only in the study of highly non-normal operators, but also when truncating an operator to a subspace. The phenomena described below have escaped the attention of many people because they have no analogue for self-adjoint operators: the truncation of a self-adjoint operator is always selfadjoint. If L is the standard convection-diffusion operator

(Lf)(u) := -f"(u) + f'(u) acting in L2(R) then L is normal and its spectrum is the parabola

{x+iy:x=y2}. If one truncates the operator to L2 (0, a) and imposes Dirichlet boundary conditions,

one obtains a highly non-normal operator Lo, which has compact resolvent and spectrum {1/4 + ir2n2/a2 : n = 1, 2, ...}.

It follows that the spectrum of Lo, converges as a -* oo to [1/4, oc), and has no connection with the spectrum of L. In this particular case one can get the right limit by using periodic boundary conditions rather than Dirichlet boundary conditions.

However, there is no theorem which states that this always works for operators with variable coefficients in one dimension, and if one replaces the intervals by an increasing sequence of regions in R' periodic boundary conditions may not make sense. To add to the complexity of this problem the spectrum of L restricted to L2(0, co) and subject to Dirichlet boundary conditions at 0 is equal to the boundary plus interior of the parabola. See [14] and [20, Chap. 12].

If one adopts the pseudospectral point of view the situation is immediately clarified. Although the spectrum of Lo, remains within [1/4, oo) as a increases, the resolvent norms I I R(z, A,,,) I I diverge rapidly provided z is inside the parabola, but they remain uniformly bounded for z outside the parabola. In fact the pseudospectra of Lo, converge to the pseudospectra of the half-line operator as a -i cc. See [14] and [20, Chap. 12]. Truncation can also lead to problems for a self-adjoint operator if there is a gap between two parts of its essential spectrum. If one uses standard finite element packages to compute the spectrum, one routinely finds a large number of eigenvalues in the spectral gap. This does not happen for eigenvalues below the bottom of the essential spectrum, because of variational theorems. The phenomenon is called spectral pollution, and there are various ad hoc procedures for avoiding it. There is also a rigorous way of doing so, but at substantially greater computational cost. See [8] for a `geometrical' account of the rigorous numerical procedures involved.

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5. Numerical Range and Functional Calculus The material in this section is to appear in [6]. Let A be a bounded operator on a Banach space B and let y be a closed curve surrounding the spectrum of A. If f is an analytic function in the interior of y which extends continuously to y then one may define

f (A) = -1 1 f (z)R(z, A) dz, 7

and deduce the estimate

If(A)II < ' Iyl If 11. provided II R(z, A) II < c for all z E y, Iy I is the length of y and IIf I

I

is the maximum

value of If I on y. Unfortunately the resolvent norm may be very large on y even if y remains at a considerable distance from the spectrum of A. If A is an n x n matrix with n distinct eigenvalues Ar then the alternative formula n

1: f(Ar)Pr

f(A)

r=1

where Pr are the rank one spectral projections of A may run into related problemsthe norms of spectral projections can be arbitrarily large even for 2 x 2 matrices, and for larger matrices this problem is commonplace. Dismissing non-diagonalizable Jordan matrices because they form a set of zero Lebesgue measure is a serious mistake: in many problems one may indeed write A = SDS-1 where D is diagonal, but the condition number k(S) = IISII IIS-1II of the diagonalization is so large that the procedure is computationally infeasible. It is argued in [6] that the usefulness of diagonalization depends on how small r,(S)e + IBII can be made, where A = SDS-1 + B, e is the rounding error of the computations, B is a small perturbation and D is diagonal. Numerical experiments suggest that the answer is frequently of order e1/2, and this has been proved for certain types of Jordan matrix A which cannot themselves be diagonalized. The problem of computing f (A) can sometimes be resolved by using the concept of numerical range. The set

Num(A):={(Af,f):IfIi=1} is always convex, and satisfies

Spec(A) C Num(A) c {z :

IzI

IIAII}.

It is easy to compute, but not by evaluating (Af, f) for a large collection of unit vectors! There is a theorem stating that IIR(z, A)II < dist(z, Num(A))-1

so that if y encloses Num(A) rather than just Spec(A), and dist(y, Num(A)) is not too small, then one obtains a useful estimate on I I f (A) I . I

Whether or not this is possible depends on the domain of analyticity of f .

Consider the calculation of At for the highly NSA matrix

I r/n ifs=r+1, Ar,3

c

10

if r = s, otherwise.

E. B. DAVIES

148

Figure 2 plots IIAt11 against t for n = 100 and c = 0.6. Computing the square root of a NSA matrix may be a highly unstable procedure. The computations were carried out by diagonalizing A+B, where B is a random matrix of norm 10-8. This number was chosen because it is the square root of the rounding error in the numerical computations; the reason for this choice has already been mentioned. The stability of the computations for different choices of B was checked.

500 450

400 350 300

250 200 150 100

50

0L 0

0.5

1

1.5

2

FIGURE 2. Graph of IIAtjj for 0 < t < 2

This figure illustrates another point, that the short time behaviour of a oneparameter semigroup may be very different from the long time asymptotics. The latter is controlled by the spectrum of the operator, but the former is not. In particular one cannot prove that the solution of some dynamical equation arising in applied mathematics is stable in any useful sense simply by showing numerically, or even rigorously, that all of its eigenvalues have negative real parts. See [20, Chap. 20] for examples that demonstrate that hydrodynamic stability, in particular, is not always implied by such `eigenvalue stability'.

Figure 3 provides a contour plot of the norms of the complex powers of the same matrix. The computation was carried out by the same random approximate diagonalization method. The contours are placed where the norm equals 103/5 for s = 1, ..., 25, the smallest closed loops being for s = 1. Note that the norms are of order 1 when x + iy is an integer. The use of a random perturbation does not affect the accuracy of the figure, but it enables one to compute all the quantities concerned with a single diagonalization.

NON-SELF-ADJOINT OPERATORS AND PSEUDOSPECTRA

149

0.8

0.6

0.4

0.2

00

-0.2 -0.4 -0.6

-0.8

-0.5

0

0.5

1

1.5

2

2.5

FIGURE 3. Contour plot of IIA'+zyll for -0.5 < x < 2.5 and -1 <

y<1

6. Orthogonal Polynomials If y is a probability measure on the unit circle S, one can apply Gram-Schmidt to construct a sequence of monic polynomials 4% on S that are orthogonal within L2(S, dy). A certain sequence of contractions A on L2(S) lie at the centre of the theory of such orthogonal polynomials. They are defined by

A. where M,, is the unitary operator of multiplication by z and P is the orthogonal projection with range

L,,:=lin{4D,.:0
prove that if A has an approximate eigenvalue z near S then z is close to a true eigenvalue, with precise estimates. This was a key ingredient in the study of the

E. B. DAVIES

150

zeros of certain random orthogonal polynomials 4),, (z, w), which also relies heavily on earlier work of Stoiciu; [16, 17]. The main theorem is

THEOREM 6 ([9]). Let A be an n x n matrix and suppose that z E C satisfies Spec(A). Then

zj > IJAII and z

dist(z,Spec(A)) IR(z,A)U < cot (4n) and the constant on the RHS is optimal.

The paper also contains a much more elementary proof in which the RHS is replaced by the weaker bound 2n. A related but weaker result has been also proved by Nikolski in [13]. The theorem is applied as follows. If there exists f E Cn and E > 0 such that jfAf -zf11 <,511f 11 then

IIR(z, A) II > E-1 so 71

dist(z, Spec (A)) < Ecot (4n

)

The proof that the constant is optimal depends on identifying the `worst possible' matrix, which is

/1

2

0

1

0

0

0

0

.

. .

2

2

.

2

2

1

2

0

1

Mn =

... ...

This matrix has a number of special properties and is a discretization of the standard Volterra integral operator

(Vf)(x) := fox f (s) ds acting on L2(0, 1). The proof of the theorem relies upon the fact that z is close to the boundary of the numerical range of A. Indeed there is a more general version. THEOREM 7 ([9]). Let A be an n x n matrix and suppose that z c C is not in the spectrum or the interior of the numerical range of A. Then dist(z, Spec(A)) IIR(z, A) 11 < cot (f)

and the constant on the RHS is optimal.

I hope that I have persuaded you that NSA spectral theory has features quite foreign to those brought up in the SA tradition. It is unrealistic to guess what the theorems are without being prepared to do some numerical experimentation, but the starting point must be to forget everything you ever learned from the SA theory.

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References

[1] Bottcher A, Silbermann B: Introduction to Large Truncated Toeplitz Matrices. Springer, New York, 1999.

[2] Davies E B: Pseudospectra, the harmonic oscillator and complex resonances. Proc. R. Soc. London A 455 (1999) 585-599. [3] Davies E B: Semi-classical states for non-self-adjoint Schrodinger operators. Commun. Math. Phys. 200 (1999) 35-41. [4] Davies E B: Semi-classical analysis and pseudospectra. J. Diff. Eqns. 216 (2005) 153-187. [5] Davies E B: Linear Operators and Their Spectra. Cambridge University Press, expected 2007. [6] Davies E B: Approximate diagonalization. In preparation, 2006. [7] Davies E B, Kuijlaars A B J: Spectral asymptotics of the non-self-adjoint harmonic oscillator. J. London Math. Soc. (2) 70 (2004) 420-426. [8] Davies E B, Plum M: Spectral pollution. IMA J. Numer. Anal. 24 (2004) 417-438. [9] Davies E B, Simon B: Eigenvalue estimates for non-normal matrices and the zeros of random orthogonal polynomials on the unit circle. J. Approx. Theory, to appear. [10] Dencker N, Sjostrand J, Zworski M: Pseudospectra of semi-classical (pseudo) differential operators. Comm. Pure Appl. Math. 57 (2004) 384-415. [11] Hinrichsen D, Pritchard A J: Mathematical Systems Theory I. Modelling, State Space Analysis, Stability and Robustness. Texts in Appl. Math. 48, Springer, Berlin, 2005. [12] Markus A S: Introduction to the Spectral Theory of Polynomial Operator Pencils, Transl. Math. Monographs, Amer. Math. Soc., Providence R.I., 1988. [13] Nikolski N K: Condition numbers of large matrices, and analytic capacities. St. Petersburg Math. J. 17 (2006) 641-682. [14] Reddy S C, Trefethen L N: Pseudospectra of the convection-diffusion operator. SIAM J. Appl. Math. 54 (1994) 1634-1649. [15] Redparth P: Spectral properties of non-self-adjoint operators in the semi-classical regime. J. Diff. Eqns. 177 (2001) 307-330. [16] Stoiciu M: Zeros of Random Orthogonal Polynomials on the Unit Circle, Ph.D. dissertation, California Institute of Technology, 2005.

http://etd.caltech.edu/etd/available/etd-05272005-110242/ [17] Stoiciu M: The statistical distribution of the zeros of random paraorthogonal polynomials on the unit circle. J. Approx. Theory 139 (2006) 29-64. [18] Trefethen L N: Wave packet pseudomodes of variable coefficient differential operators. Proc. R. Soc. London A 461 (2005) 3099-3122. [19] Trefethen L N, Chapman S J: Wave packet pseudomodes of twisted Toeplitz matrices. Comm. Pure Appl. Math. 57 (2004) 1233-1264. [20] Trefethen L N, Embree M: Spectra and Pseudospectra, The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton, 2005. [21] Wright T G: EigTool sofware available at

http://www.comlab.ox.ac.uk/pseudospectra/eigtool [22] Zworski M: A remark on a paper of E B Davies. Proc. Amer. Math. Soc. 129 (2001) 29552957.

DEPARTMENT OF MATHEMATICS, KING'S COLLEGE, STRAND, LONDON, WC2R 2LS, ENGLAND

E-mail address: [email protected]

Proceedings of Symposia in Pure Mathematics Volume 78.1, 2007

Barry Simon's Contributions to Non-Relativistic Quantum Mechanics: Two-Body and N-Body Schrodinger Operators and Resonances Richard Froese

Dedicated with respect and admiration to Barry Simon ABSTRACT. This article reviews Barry Simon's contributions to the theory of two-body and N-body Schrodinger operators and to the theory of resonances.

CONTENTS 1.

2. 3. 4.

Introduction Two-Body and General Schrodinger Operators N-Body Schrodinger Operators Resonances

1. Introduction Non-relativistic quantum mechanics and Schrodinger operators occupy a central place in Barry Simon's prodigious output. This article is a review of Simon's work on two-body and N-body Schrodinger operators, and on resonances. A few early papers reviewed here do include results about the general theory of Schrodinger

operators (that is, without any decay assumptions on the potential), but I have excluded some more recent papers on one-dimensional Schrodinger operators that seemed more closely related to work on inverse problems and random potentials. Also notable among the excluded results are papers dealing with electric and magnetic fields, the semi-classical limit and Thomas-Fermi theory. These will be covered in other articles. Imposing these severe restrictions on the subject matter still leaves a formidable collection of papers. I have grouped them into three sections: two-body and general Schrodinger operators ([4], [5], [13], [24], [35], [59], [70], [71], [80], [99], [103], [106], [114], [115], [135], [137], [179], [181], [218], [226], [227]), N-body Schrodinger 2000 Mathematics Subject Classification. 81Q10, 35P05, 35P25. Key words and phrases. Barry Simon, Schrodinger operators. ©2007 American Mathematical Society 153

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Barry Simon's papers on two and N-body Schrodinger operators and resonances

operators ([9], [25], [43], [44], [45], [46], [51], [55], [83], [84], [85], [95], [96], [97], [119], [122], [124], [125], [131], [132], [133], [134], [160], [175], [199], [304]), and resonances ([19], [20], [105], [107], [121], [156], [276]). Here and throughout this article, references are to Simon's publication list at

http://www.math.math.caltech.edu/people/biblio.html and on page 931 of this Festschrift. Note that upper case (resp. lower case) roman numerals that appear below refer to books (resp. review articles and conference proceedings.) The following graph displays the distribution of these papers among all of Simon's research articles. (The horizontal axis is the publication date. The graph shows research articles only, not review papers or books.) As the graph shows, this review mainly considers topics that occupied Simon during the early and middle parts of his career.

I should emphasize that this article is about Barry Simon's contributions, and not a review of the field. Barry Simon himself is a masterly reviewer, and in addition to his research articles, he has authored books as well as several booklength reviews, dealing with the topics discussed here. These books and reviews have been (and continue to be) profoundly influential in my own education, and in shaping the entire field. I was fortunate to participate in the writing of [X], together with Cycon, Kirsch and, of course, Simon. This book started out as a week long lecture series by Simon, delivered at breathtaking pace in a small castle in Thurnau, Germany. It aims to survey the field of Schrodinger operators with an emphasis on results that were

recent at the time of writing in the early 1980's. Other reviews that are relevant to this article are the 1982 Bulletin article Schrodinger semigroups [xxi] and the 2000 review Schrodinger operators in the twentieth century [xxxix]. For this article

I have often had to compress the contents of entire long research papers into a sentence or two. So I am fortunate to be able to refer the reader to these articles for more information, in particular, to references to works by other authors. Each of Simon's reviews contains a reference list of several hundred articles. Finally, it is impossible to discuss Simon's pedagogical work on Schrodinger operators without mentioning the series Reed and Simon ([II], [III], [V], [VII]) from which several generations of mathematical physicists have learned the subject.

Almost all the papers under review here concern the theory of Schrodinger operators of the form

H=-A+V

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acting in L2(W1). In the context of non-relativistic quantum mechanics the most important questions about such an operator for a given potential V are (i) self-adjointness: To begin with, the expression for H is only a formal sum. How can H be defined as a self-adjoint unbounded operator? (ii) spectral theory: What is the nature of the spectrum and where is it located? What are the smoothness and decay properties of the eigenfunctions? (iii) scattering theory: What is the long time behaviour of the Schrodinger time evolution e-itH? The definition of resonances requires some extra structure and is discussed below.

2. Two-Body and General Schrodinger Operators This section deals with work on two-body potentials, that is, potentials V (x) that tend to zero in some sense as jxj -+ oo, as well as some results (mostly about self-adjointness) that continue to hold under the addition of an L°° term to the potential. Note that two-body operators become one-body operators after the removal of the center of mass. This is briefly discussed in the N-body section below.

2.1. Quadratic Forms, Singular and Positive Potentials. [5] On positive eigenvalues of one-body Schrodinger operators, Commun. Pure Appl. Math. 12 (1969), 531-538 [13] Hamiltonians defined as quadratic forms, Commun. Math. Phys. 21 (1971), 192-210 [24] Essential self-adjointness of Schrodinger operators with positive potentials, Math. Ann. 201 (1973), 211-220 [35] Essential self-adjointness for Schrodinger operators with singular potentials, Arch. Rat. Math. Anal. 52 (1973), 44-48 [41] Quadratic forms and Klauder's phenomenon: A remark on very singular perturbations, J. Funct. Anal. 14 (1973), 295-298 [59] (with W. Faris) Degenerate and non-degenerate ground states for Schrodinger operators, Duke Math. J. 42 (1975), 559-567 [71] (with P. Deift) On the decoupling of the finite singularities from the question of asymptotic completeness in two body quantum systems, J. Funct. Anal. 23 (1976), 218-238 [76] An abstract Kato's inequality for generators of positivity preserving semigroups, Ind. Math. J. 26 (1977), 1067-1073 [86] Kato's inequality and the comparison of semi-groups, J. Funct. Anal. 32 (1979), 97-101 [135] (with M. Aizenman) Brownian motion and Harnack's inequality for Schrodinger operators, Commun. Pure Appl. Math. 35 (1982), 209-273

In the early part of his career, Simon contributed much to the understanding of how to handle singularities of the potential in basic questions about Schrodinger operators. Singularities are important because the most important potentials in physics, like the Coulomb potentials Clxl-1, are singular. Moreover, the kinetic energy (Laplacian) term in a Schrodinger operator dominates a potential as singular as -aIxI-2 for small enough a so one would like to be able to handle potentials up

to this borderline. It is a fact of life, though, that for most properties there is no simple best possible class of potentials. Simon's thesis ([13] and the book [I]) is a systematic exploration of the Rollnik

class of potentials. A potential V defined on R3 is in the Rollnik class R if the

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norm given by IIVIIR = fR6

V (x)IIV(y)I d3xd3y

x - yI is finite. Simon proposes the class R as a replacement for L2(R3) in a collection of theorems about two-body Schrodinger operators. For example, considering the question of self-adjointness, the basic Kato-Rellich theorem allows H = -A + V to be defined as a self-adjoint operator sum on the usual domain D(0) of the Laplacian provided V E L2(R3) + L0(1R3). This definition allows singularities as bad as IXI-3/2. To allow for potentials as singular as JxI-2 (the physically relevant borderline), Simon proposes the condition V E R + L°°(1R3). With this condition, H can no longer be defined as an operator sum, but must be defined as the operator associated to the sum of quadratic forms, via the KLMN theorem. For these Hamiltonians defined as quadratic forms, Simon then considers bound state problems, existence and completeness of wave operators under an Ll condition, and existence of eigenfunction expansions. The bound state results rely on the fact that for Rollnik potentials V V 1/2 (-0 - E) -1 IV 11/2 is Hilbert-Schmidt. This allows Birman-Schwinger type arguments to bound the number of negative energy bound states and restrict the number and position of positive energy bound states. In particular, for small enough Rollnik norm there are no positive energy bound states. The question of the existence of positive energy bound states is a subtle one, since a tunneling argument predicts that they do not exist for a two-body potential. Yet, there is a well-known example due to Wigner and von Neumann of a potential decreasing like IxI-1 that does have a positive eigenvalue. Before his thesis, Simon already published a result on positive eigenvalues in [5], generalizing work of Kato

and Odeh. In this paper he rules out eigenvalues in [E0, oo) for potentials V E L2 (1183) + L°° (1[83) that are smooth near infinity and can be written as V = V1 + V2 where as IxI -> oc, JxJ Vi(x) ---> 0, V2 (x) - 0 and limsupx VV2(x) = Eo < oo. The

proof treats the equation for an eigenfunction as an ODE in the radial variable for an L2(S2) valued function, and proceeds via differential inequalities. Simon returns to the question of self-adjointness in [24], [35], this time essential self-adjointness on Co for Schrodinger operators in higher dimensions. The natural condition given by the Kato-Rellich theorem for dimension n > 4 is V E LP(R') + L0(R') for p > n/2 (but p = n/2 for n > 5 is also known). For positive potentials one expects to be able to handle more singular potentials. In [24] Simon proves essential self-adjointness on Co for positive V in L2(R,) + L°°(IR' ). Clearly, a local L2 condition is required to define H on Co. In this paper Simon conjectures that V = V+ - V_ with V+ EL o, and V_ as above should suffice for essential selfadjointness on Co. This was proved by Kato, and again by Simon in [35]. A crucial ingredient in Kato's proof is the distributional inequality Dlul > Re(sgnuDu). (For complex u, sgnu equals u/Iul at points where u# 0 and vanishes at points where u does.) Simon explores the connection between this type of inequality and positivity preserving semigroups in [76] and [86]. The paper [35] also proves self-adjointness for positive potentials V E L2 (1l \0) provided V > Cm,r-2 for an explicit constant

C.

Singularities on a distinguished set also play a role in [59], where Simon and Faris examine the question of degeneracy of the ground state. A theorem of Faris insures the non-degeneracy of the ground state for H = -A+V defined via quadratic

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forms if V = V1 +V2 with V1 > 0, V1 E Li c (I(gn) and V2 E L°° (R'1) +L''V2 (I[8n) (here

n > 3, there are similar conditions for n = 1, 2). On the other hand, Klauder noted that the singularity in H = -d2/dx2 + x2 + I xI -a for a > 1 effectively decouples the positive and negative axis leading to a doubly degenerate ground state. (The phenomenon was analyzed by Simon in [41] in terms of form sums.) Looking to higher dimensions, Simon and Faris consider potentials in L' c(Rn\K), where K is a closed set of measure zero. They prove that if W'\K is connected, then -A + V has a non-degenerate ground state, even though V can clearly be very singular on K. Two proofs are given: one using path integrals and one using differential operator methods. On the other hand if Rn\K has more than one component, there exists V E L o,,(Rn\K) such that the ground state of -A + V is degenerate. In [71] Deift and Simon also consider potentials V that can be very singular on closed sets of measure zero, this time in the context of scattering theory. The

main result asserts that if wave operators for Ho = -A and H = -A + V are complete then so are the wave operators for Ho and H' = -A + xV, where x is the characteristic function of the exterior of a ball. A mild regularity condition on the negative part of V is required. By the beginning of the 1980's, in the book [X] and review article [xxi], Simon was advocating the class K as a natural class of potentials for self-adjointness and other properties, based on his work with Aizenman in [135]. A potential on W' for v > 3 is in this class if

Ix-yl2-"IV(y)Id"y=0

lim sup J

x-yl
x

lim I (-A + E)-1 V II , = 0.

E Here II

.

I

lo.,,,,, denotes the L°° --+ L' operator norm.

2.2. Weak Coupling and Threshold Behaviour. [70] The bound state of weakly coupled Schrodinger operators in one and two dimensions, Ann. Phys. 97 (1976), 279-288 [80] (with R. Blankenbecler and M.L. Goldberger) The bound states of weakly coupled long-range one-dimensional quantum Hamiltonians, Ann. Phys. 108 (1977), 69-78 [115] (with M. Klaus) Coupling constant thresholds in nonrelativistic quan-

tum mechanics, I. Short range two-body case, Ann. Phys. 130 (1980), 251-281 [226] (with F. Gesztesy and G.M. Graf) The ground state energy of Schrodinger operators, Commun. Math. Phys. 150 (1992), 375-384

[227] (with G.M. Graf) Asymptotic series for the ground state energy of Schrodinger operators, J. Funct. Anal. 112 (1993), 442-446

The A dependence of an eigenvalue of a Schrodinger operator -A + AV can be understood using regular perturbation theory, as long as the eigenvalue stays isolated from the rest of the spectrum. However, as A tends to zero, the eigenvalue

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will eventually be absorbed in the continuous spectrum, at the latest when A becomes zero. Simon has written a number of papers about what happens near the point where the eigenvalue is absorbed. The papers [70] and [80] deal with values of A close to zero. They address the questions of whether there are eigenvalues for small but non-zero A, and, if so, how they behave as A tends to zero. Simon's paper [70] begins with the observation that a negative square well potential in one or two dimensions, no matter how shallow, always has a bound state. In three or higher dimensions, shallow square well potentials have no eigenvalues. This observation is generalized as follows. In one dimension, if f (1 + lxl2)IV(x)ldx < oc then -d2/dx2 + AV has a bound state for all small positive A if and only if f V(x)dx < 0. A similar result holds in two dimensions. This paper also considers the analyticity of the lowest eigenvalue e(A) at A = 0 (true in one dimension and false in two) and the threshold behaviour. For example, in one dimension e(A) - cA2 if f Vdx < 0 and e(A) - cA4 if f Vdx = 0. In [80] Blankenbecler, Goldberger and Simon revisit the one-dimensional situation, but for longer range potentials for which f (I + Jx12)IV(x)ldx = oc. Roughly speaking,

this paper considers V(x) - -ajxj-0 for 1 < Q < 3. New phenomena appearing include the existence of infinitely many bound states for small coupling (in the range 1 <,3 < 2). For short range potentials V, if an eigenvalue e(A) is absorbed in the continuous

spectrum, that is, e(A) --> 0, as A

.

A0, what is the leading order of e(A)? For

example, is e(A) - (A-A0)' ? This question is answered in detail in Klaus and Simon [115]. The answer is highly dimension dependent, with different results depending on whether the dimension is in {1}, {2}, {3}, {4}, {5, 7, 9, ...} or {6, 8, 10,. ..}. In the early 1990's, Simon again considered Schrodinger operators with weak

coupling, that is, -A + AV for A small. But now V is a potential that does not go to zero at infinity. Let e(A) denote the infimum of the spectrum. Of course, in this setting e(A) need not be an eigenvalue. In [226] Gesztesy, Graf and Simon address the question of when e(A) < 0 (notice the strict inequality) for A 0.

They prove for a large class of potentials that do not decrease at infinity that -cA2 < e(A) < -dA2 for positive constants c and d. In a follow up, Graf and Simon [227] give a class of potentials, including some almost periodic potentials, for which e(A) has a complete asymptotic series at A = 0. 2.3. Some Scattering Results. [99] (with E.B. Davies) Scattering theory for systems with different spatial

asymptotics on the left and right, Commun. Math. Phys. 63 (1978), 277-301

[103] (with C. Radin) Invariant domains for the time-dependent Schrodinger equation, J. Diff. Eqn. 29 (1978), 289-296 [106] Phase space analysis of simple scattering systems. Extensions of some work of Enss, Duke Math. J. 46 (1979), 119-168 [218] Absence of ballistic motion, Commun. Math. Phys. 134 (1990), 209212

Here I will briefly describe some papers on scattering that have not been mentioned above, and are not discussed with N-body results below. In [99] Davies and Simon study scattering theory for systems with different spatial asymptotics on the left and the right. An example is a potential in one dimension that approaches different limiting values as x - ±oo. A physically important example is a half solid, that is, a potential in R' that vanishes in a half space and is periodic in the other

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half space. This paper contains results about existence and completeness of scattering, where the comparison dynamics may be different on each side. In the case of a half solid, there is a proof of the decomposition of the absolutely continuous spectral subspace into scattering states and surface states. Motivated by a result in classical mechanics, Radin and Simon in [103] show that there is a dense set of vectors such that 1xe-itx f11 is bounded by C(1 + Itl) for f in this set, provided V is Laplacian bounded with bound less than one (in either the form or operator sense). Here x is the vector valued position operator. This bound cannot be improved in general, since it describes ballistic motion, that is, the time behaviour for a freely moving particle. However, if H has only point spectrum one might expect there to be no ballistic motion. This is proved by Simon in [218], where he shows that for such Hamiltonians limt_.m t-2lixe-itx fl12 = 0 for f with compact support. The paper [106] is a long exploration of the method introduced by Enss in twobody, and later in three-body, scattering. Enss's method created a sensation when it appeared, since it was able to reproduce the strongest known results at the time in a transparent manner, with beautiful connections to classical mechanics. Simon extends the method to a large number of scattering systems, including pseudodifferential operators, Stark and Zeeman effects, optical and acoustical scattering, complex potentials, magnons and half solids (in some sense making obsolete the methods just introduced in [99]). 2.4. Multiple Wells, Continuum Eigenfunctions and Eigenvalue Splittings. [114] (with M. Klaus) Binding of Schrodinger particles through conspiracy of potential wells, Ann. Inst. H. Poincare A30 (1979), 83-87 [137] Spectrum and continuum eigenfunctions of Schrodinger operators, J. Funct. Anal. 42 (1981), 347-355 [179] (with W. Kirsch) Universal lower bounds on eigenvalue splittings for one-dimensional Schrodinger operators, Commun. Math. Phys. 97 (1985), 453-460 [181] (with W. Kirsch) Comparison theorems for the gap of Schrodinger operators, J. Funct. Anal. 75 (1987), 396-410

The paper [114] with Klaus was motivated in part by Sigal's work on the Efimov effect. The Efimov effect is the phenomenon occurring for certain N-body Hamiltonians where short range (even compactly supported) pair potentials can produce infinitely many bound states. The main result in [114] concerns two compactly supported critical potentials V1 and V2. Here critical means that -A + V > 0 but -A + AV has a negative eigenvalue for any A > 1. In this situation Simon and Klaus prove that the potential Wr = Vl (x) + V2 (x - r), where r is large enough so that the supports are disjoint, produces a bound state. In three dimensions the bound state energy E(r) ti /31rl-2, where /3 is a universal constant. (The remark in this paper about the behaviour of E(r) in higher dimensions is inaccurate and was corrected in papers of Y. Pinchover.) In [137] Simon shows that the spectrum of a fixed Schrodinger operator on a class of polynomially weighted spaces is independent of the weight. This holds for a large class of potentials V = V+ + V_ with V. > 0 and V+ E K,l,°O V_ E K,,. A consequence of this theorem is the fact (first proved by Sch'nol) that if Ho = E0 has a polynomially bounded solution, then E is in the spectrum of H. In [179] and [181] Kirsch and Simon consider the question of estimating the splitting between two eigenvalues of a Schrodinger operator. The paper [179] treats

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the one-dimensional case, where eigenvalues are simple. For V E C0 supported in [a, b] a bound of the form E,,, - En,-1 > irA2 exp(-A(b - a)) is obtained, where A = SUPEE(E, 1E --V(x) 1. In higher dimensions, since eigenvalues other than the ground state can be degenerate, it only will be possible to estimate E1 - E0 in general. The paper [181] gives variational lower bounds on this number that follow from realizing a Schrodinger operator as a Dirichlet form. These bounds lead to a quadratic lower bound E0(k) - go (0) > Ck2 for the lowest band function E0(k) for a periodic Schrodinger operator.

3. N-Body Schrodinger Operators A Schrodinger operator describing N particles with masses mi moving in IRn and interacting via pair potentials V j has the form N 1 i=1

2mi

Vj(xi-xj). 1
This operator, acting in L2(InN), has purely absolutely continuous spectrum arising from the uniform motion of the centre of mass. To define the N-body operator H, we must remove the centre of mass. I will give a brief description of this, as well as some of the terminology used below. To remove the centre of mass (according to the Sigal-Sigalov prescription), consider the inner product on RnN defined by the masses via the formula N

((xlr ... I XN), (Y1, ... YN))

Mi(Xi, yi), i=1

where the bracket on the right side denotes the usual inner product on R. Decompose RnN into an orthogonal direct sum X ® Y, where

Y={(x,...,x):xEW} is the diagonal subspace and N 01-

i=1

The T h e subspace Y can be thought of as co-ordinates for the displacement of the centre of mass, while the subspace X contains internal co-ordinates. From the direct sum decomposition, we obtain L2(RnN) = L2(X) ® L2(Y). The first (kinetic energy) term in H, being - times the Laplace operator for this metric defined by this inner

product, can be written -Ox ® I - 10 Ay, where Ax and Ay are the Laplace operators for the respective subspaces. Since the potential is constant along the diagonal, it can be written V ® I, where V is given by the formula above, but thought of as the restriction of the original potential to X. Thus

H=H®I+I®(-XAy). Ax + V acting in L2 (X). This defines the N-body Hamiltonian H = -!Ax For any partition C = {c1, ... , ck} of {1, . . . , N} into disjoint subsets, or clusters, there is an associated cluster Hamiltonian H(C) obtained from H by dropping

potentials V,j when i and j lie in different subsets. Thus H(C) is the Hamiltonian describing the particles within each cluster interacting, but clusters moving independently of the others.

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There is an associated direct sum decomposition of X into Xc1 ® ® Xck ®Yc giving rise to the tensor product decomposition L2 (X) = L2 (Xc1) ®. . ® L2 (Xck) ®

L2(Yc). The subspaces Xc contain the internal co-ordinates for the cluster ci while Yc contains co-ordinates for relative motion of the clusters. The cluster Hamiltonian has the associated decomposition

H(C)=hcl

®...®I®I+...+I®...®hck

®I+I®...®I®(-2Ay,)

where hck is the Nk-body Hamiltonian for the particles in the cluster ck. Here Nk is the number of particles in the cluster ck. This decomposition therefore sets the stage for an inductive treatment of N-body Hamiltonians. 3.1. The Pointwise Bounds Series. [43] Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, I, Proc. Amer. Math. Soc. 42 (1974), 395-401 [46] Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, II, Proc. Amer. Math. Soc. 45 (1974), 454-456 [51] Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, III, Trans. Amer. Math. Soc. 208 (1975), 317-329 [95] (with P. Deift, W. Hunziker and E. Vock) Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, IV, Commun. Math. Phys. 64 (1978), 1-34

[133] (with R. Carmona) Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, V. Lower bounds and path integrals, Commun. Math. Phys. 80 (1981), 59-98 [134] (with E. Lieb) Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, VI. Asymptotics in the two cluster region, Adv. Appl. Math. 1 (1980), 324-343

This series of papers considers the question of exponential decay of eigenfunc-

tions. Let T(x) be an eigenfunction of an N-body Hamiltonian with eigenvalue E < E = inf Qess(H). One way of measuring the decay of T(x) is by a so-called L2 bound, that is, a statement like ea*xIW(x) E L2 indicating that in some average sense, T(x) decays faster than e-aIxl. A stronger statement is a pointwise bound of the form I,p(x)I < Cae-aixi

which is required to hold for every x. This kind of bound (and the corresponding lower bound) is the subject of these papers. In [43], the first paper of this series, Simon proves that such a pointwise estimate

holds for all a < E - E. In addition, Holder continuity of and VT is proved. Prior to this paper, O'Connor had proved L2 bounds by showing that 4 has an L2 analytic continuation to a tube around R3N-3 and applying a Payley-Wiener argument. Simon shows that this continuation can be made in L1, which leads to the desired bound. Vii are required to satisfy The papers [46] and [51] are actually not about N-body Hamiltonians, but consider potentials V (x) that grow at infinity. In this case, one expects faster than exponential decay, and this is what Simon proves. The exponential bounds described so far are isotropic, that is, they depend only on I x I and not on the direction of x. However, the behaviour of an N-body potential

at infinity depends very much on the direction. Therefore, it is not unreasonable to expect that in the upper bound, e-aIxI could be replaced by e-f (x) where f is homogeneous of degree one. Such non-isotropic bounds are the subject of the milestone paper [95] with Deift, Hunziker and Vock. In this paper, the Combes-Thomas

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technique is used to prove non-isotropic bounds JkF(x)I < Ce-f(-) for homogeneous functions f whose gradient satisfies an inequality involving the spectrum of cluster Hamiltonians, that is, Hamiltonians defined by keeping only potentials that do not

go to zero in a certain direction. Between this paper and the next in the series, the optimal f satisfying the gradient inequality was found by Agmon, and is now known as the Agmon metric. The next paper [133] with Carmona, is another landmark paper. In this paper, probabilistic methods are used to prove that the Agmon metric provides an exponential lower bound for the ground state 41. Combining this with the upper bound, we have W(x) <

CEe(1+E)f(-).

This bound shows that, in some sense, the Agmon metric captures exactly the behaviour of the ground state. Actually, the paper [133] considers not only N-body Hamiltonians, but also potentials that grow at infinity. For these potentials there is also a corresponding Agmon metric and upper bound, and Carmona and Simon provide the corresponding optimal lower bound for the ground state. In the final paper [134] of this series, Lieb and Simon go beyond upper and lower bounds above, to compute the first term in an asymptotic expansion. They consider the ground state T in a region of configuration space where the particles form two widely separated clusters. They are able to prove asymptotics of the form `L((1, rr S2, R) =

O(e ryR))

where S1 and S2, the internal co-ordinates for the two clusters, stay bounded while the intercluster distance R tends to infinity. Here 01 and 02 are the ground states of the corresponding cluster Hamiltonians. 3.2. Geometric Methods. [83] (with P. Deift) A time-dependent approach to the completeness of multiparticle quantum systems, Commun. Pure Appl. Math. 30 (1977), 573583

[84] Geometric methods in multiparticle quantum systems, Commun. Math. Phys. 55 (1977), 259-274 [85] N-body scattering in the two-cluster region, Commun. Math. Phys. 58 (1978), 205-210 [124] (with V. Enss) Bounds on total cross-sections in atom-atom and atomion collisions by geometric methods, Phys. Rev. Lett. 44 (1980), 319-321 [125] (with V. Enss) Finite total cross-sections in non-relativistic quantum mechanics, Commun. Math. Phys. 76 (1980), 177-210

[304] (with Y. Last) The essential spectrum of Schrodinger, Jacobi, and CMV operators, J. d'Analyse Math. 98 (2006), 183-220

Geometric methods for N-body Hamiltonians aim to exploit the special form of an N-body potential near infinity in configuration space by using partitions of unity. In regions of configuration space where two particles are widely separated so that the corresponding potential is small, we should be able to ignore that potential. Such localizations are now pervasive in the subject, so it is interesting to sense the excitement they caused in the late 1970's. This is from the introduction to Simon's paper [84]:

"... there is some point in seeing the sweep of results presented here obtained by geometric methods without the pernicious influence of resolvent equations. Not that these equations do not have their usefulness but it seems to me that they have overly

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dominated our thinking. I will regard this paper as a success if it helps strike a balance in the reader's mind; even better, I would hope to convey the excitement of the geometric ideas." In the first paper [83] in this series, Deift and Simon take the first steps in a geometric time-dependent proof of asymptotic completeness for multiparticle systems. They consider wave operators for a three-body system of the form

W = st-Too -

limeitx- Joe-itHPa'.

Here Ha is a cluster Hamiltonian obtained by dropping two of the three potentials in the original H, and Jc, is a cutoff function that localizes to the region where the dropped potentials are small. For example, if the remaining potential in Ha is V1,2, then J0 would localize to the region where IX1 - x21 is small, but Ix1 - x31 and Ix2 - x31 are large. Deift and Simon prove that if asymptotic completeness holds, then these wave operators exist. They also show that if the wave operators exist, and scattering is complete for two body subsystems, then asymptotic completeness holds.

In the paper [84], Simon uses geometric methods to provide simple proofs of a number of basic results about N-body Schrodinger operators. The first of these is the HVZ theorem, first proved separately by Hunziker, Van Winter and Zhislin. Let E denote the bottom of the essential spectrum of H E = inf Qess(H)

and denote by E2 the infimum over the spectra of the cluster Hamiltonians H(C) as C ranges over all two cluster partitions E2

C:#nf

inf Qess(H(C)).

The quantity E2 is the smallest energy the system of particles can have if we divide

them into two groups and move the groups far apart. The HVZ theorem asserts that E= E2. The second result is a theorem first proved by Combes. It concerns the situation where it always requires energy to break a two cluster partition into three groups. In other words, E2 < E3, where E3 = inf inf Qss(H(C)). C:#(C)=3

Given the HVZ theorem, we see that scattering is possible in the energy range [E, E3]. Combes' theorem asserts that scattering is complete in this energy range. The final part of this paper deals with the question of when there are finitely many and when there are infinitely many bound states. When proving Combes' scattering result Simon required the interparticle potentials VVj(x) to decay like xjlxl-1_E in the . In [85] Simon improves this to case of central potentials, that is, potentials that only depend on Ixl. The papers [124] and [125] address the problem of estimating the total scattering cross section. Actually, most of this paper is not about N-body scattering, but the geometric time dependent approach makes it possible to consider the N-body problem too, and there is a result about the cross section for a two cluster channel in this paper. The total cross section in the two body case is defined as follows.

Fix an incident direction 6 and suppose that g(x) is a function of x e, that is,

R. FROESE

164

g(x) = G(x

e). Define g(k) to be the Fourier transform of G. Then utot(k) is

defined via 11(S

-

1)gMI2

= fatot(k)(k)2dk.

Here S denotes the scattering operator. Since g is not square integrable, the left hand side must be defined as a limit limR-goo IJ(S - 1)ghR1I2 for a sequence of cutoff functions hR. The paper contains both upper and lower bounds on Qtot. In particular, the leading behaviour of the cross section for large coupling constant is obtained.

The recent preprint [304] with Last is proof that the geometric method has not run out of steam. This paper presents a unified framework for identifying the essential spectrum of a broad class of operators that includes the HVZ theorem as a special case. This method is able to handle potentials that do not have a limit at infinity. The proof is based on localization formulae. But in contrast to the earlier work, the support of the cutoff function is on balls of a fixed size, and only single commutators are used. 3.3. Absence of Singular Continuous Spectrum. [131] (with P. Perry and I. Sigal) Absence of singular continuous spectrum in N-body quantum systems, Bull. Amer. Math. Soc. 3 (1980), 1019-1024 [132] (with P. Perry and I. Sigal) Spectral analysis of N-body Schrodinger operators, Ann. of Math. 114 (1981), 519-567

Perry, Sigal and Simon's paper [132] was a landmark in the history of the subject. It contained the first proof, for general N-body systems, of the absence of singular continuous spectrum and the fact that eigenvalues can accumulate only at thresholds. The immediate antecedent of this work was a paper of Mourre who considered abstract positive commutator estimates, now known as Mourre estimates, of the form EAi[H, A] EA > aEA + K where EA is the spectral projection of H for the interval A, A is a self-adjoint operator which in applications to N-body systems is the generator of dilations, and K is compact. Mourre showed how this sort of estimate could lead to the desired spectral information for H in the interval A. The hypotheses in Mourre's work allowed him to apply his theorem to two- and three-body systems. Perry, Sigal and Simon's major technical achievement was the proof of the Mourre estimate for N-body systems for a large class of potentials. 3.4. Coulomb Systems. [45] (with E. Lieb) On solutions of the Hartree-Fock problem for atoms and molecules, J. Chem. Phys. 61 (1974), 735-736 [55] (with E. Lieb) The Hartree-Fock theory for Coulomb systems, Commun. Math. Phys. 53 (1977), 185-193 [97] (with E. Lieb) Monotonicity of the electronic contribution of the BornOppenheimer energy, J. Phys. B11 (1978), L537-L542 [119] (with M. Hoffmann-Ostenhof and T. Hoffmann-Ostenhof) On the nodal structure of atomic eigenfunctions, J. Phys. A13 (1980), 1131-1133 [160] (with M. Hoffmann-Ostenhof and T. Hoffmann-Ostenhof) A multiparticle Coulomb system with bound state at threshold, J. Phys. A16 (1983), 1125-1131

[175] (with E. Lieb, I. Sigal and W. Thirring) Asymptotic neutrality of large Z ions, Phys. Rev. Lett. 52 (1984), 994-996

TWO-BODY AND N-BODY SCHRODINGER OPERATORS AND RESONANCES

165

[1991 (with E. Lieb, I. Sigal and W. Thirring) Approximate neutrality of large-Z ions, Commun. Math. Phys. 116 (1988), 635-644

The most important multiparticle Hamiltonians from the physical point of view are Coulomb systems describing the interaction of N electrons with k nuclei via Coulomb potentials. A common approximation is to consider the limit of infinite nuclear mass. In this case the nuclei are static, fixed at positions R,.... Rk. The corresponding Hamiltonian is N

N

k

H=-J:Di-J:J:zjlxi-Rjl-1+ 1 xi-xjl-1. i=1

i=1 j=1

1
The zj are positive numbers giving the charges of the nuclei. In the case of static nuclei, no removal of the centre of mass is needed and H acts in L2(IR3N). However,

to take account of the effect of spin, we must let H act in LQ(R3N; C2N)

A

function in this space can be thought of as T (x1, 01; ... ; XN, QN) with xi E R3 and Qi E { , -11 with the requirement that IQ is antisymmetric under exchanges of 2 xj, Qj. xi, Qi and

The Hartree-Fock approximation to the ground state energy is obtained by minimizing (T, HT) over a restricted set of T given by Slater determinants T (x1, 011, ... , xN, o-N) = (N!) -112 det(ui(xj, oj))

where ui (x, a) for i = 1, ... , N is an orthonormal set of trial functions. The EulerLagrange equations for this minimization problem are called the Hartree-Fock equations. They are a complicated set of non-linear equations, and it is not clear that they have a solution. In the paper [55] (and the announcement [45]), Lieb and Simon give a positive solution to this fundamental problem. Their proof is variational, that is, they prove the existence of a minimizer of the energy functional, rather than directly proving the existence of a solution to the Hartree-Fock equations. In addition, they obtain properties of the minimizers, ui, such as exponential decay. The proof is related to another approximation, the Thomas-Fermi approximation, which however is not covered in this review. Lieb and Simon in [97] study the true ground state energy of H as a function of the centres Ri. They consider first the case when there are two nuclei and one electron. In this case they prove that the ground state energy is an increasing function of IR1 - R2 I. Secondly, they consider the case of one electron and k nuclei whose positions are scaled to AR1 i . . . , ARk. In this case the ground state energy is non-decreasing as A increases. The proof uses correlation inequalities. The paper [119] with M. Hoffmann-Ostenhof and T. Hoffmann-Ostenhof is an

early work about nodal properties of eigenfunctions. The result is that near a zero of an eigenfunction the eigenfunction takes on both signs. Two proofs are sketched. Actually, the Coulomb nature of the potentials is not important in the proof. In [160] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and Simon consider H(A) = -O1 - A2 - x21-1 + A1x1 - x21-1 at the critical value of the coupling constant A where the ground state is absorbed in the continuum. They show that at the critical value the bound state persists. The final paper in this group [199] (with an announcement in [175]) joint with Lieb, Sigal and Thirring concerns the maximum number N(Z) of electrons that an atomic Hamiltonian with charge Z can bind. This number is defined as follows. Ix11-1

R. FROESE

166

Consider the Hamiltonian H above with N electrons, one nucleus, that is, k = 1 and

charge z1 = Z. Call this Hamiltonian H(N, Z). Let E(N, Z) be the ground state energy, that is, the infimum of (W, H(N, Z) T) taken over antisymmetric T. The Ruskai-Sigal theorem states that there is an N such that E(N + 1, Z) = E(N, Z). The smallest such N for a fixed Z is N(Z). Sigal had shown that lim sup N(Z)/Z < 2 and Lieb had improved this to N(Z) < 2Z + 1. This paper contains a proof of the fact that

N(Z)/Z , 1 as Z , oo. The proof begins by proving a classical analog of this theorem. This is then used in a proof modeled on Sigal's. 3.5. Dilation Analyticity. [25] Quadratic form techniques and the Balslev-Combes theorem, Commun. Math. Phys. 27 (1972), 1-9

[44] Absence of positive eigenvalues in a class of multiparticle quantum systems, Math. Ann. 207 (1974), 133-138

These papers develop ideas introduced by Balslev, Combes and Thomas for studying the spectrum of N-body Hamiltonians and the decay of eigenfunctions. In [25] the original dilation analyticity argument of Balslev and Combes on the absence of singular continuous spectrum is extended to a wider class of potentials using quadratic form techniques. Paper [44] proves the absence of positive eigenvalues for a class of N-body Hamiltonians with dilation analytic potentials. In addition to dilation analyticity, the proof uses the Combes-Thomas method of proving exponential decay.

3.6. Other N-Body Papers. [9] On the infinitude or finiteness of the number of bound states of an Nbody quantum system, I, Helv. Phys. Acta 43 (1970), 607-630 [96] Scattering theory and quadratic forms: On a theorem of Schechter, Commun. Math. Phys. 53 (1977), 151-153 [122] (with M. Klaus) Coupling constant threshold in non-relativistic quantum mechanics, II. Two-cluster thresholds in N-body systems, Commun. Math. Phys. 78 (1980), 153-168

The paper [9] is an early work on the question of whether an N-body Hamiltonian has finitely many or infinitely many bound states. The mechanism for infinitely many bound states discussed here is the presence of longer range intercluster potentials (as opposed to the Efimov effect which may occur for potentials of compact support) and the proof uses trial functions and minimax. In [96], Simon shows how Cook's method can be adapted to prove existence of wave operators where the difference is only defined as a quadratic form.

The paper [122] with Klaus is a continuation to the N-body setting of the study in [115] of the behaviour of an eigenvalue e(A) as a function of the coupling constant A at the point where the eigenvalue is absorbed into the continuum. The results in [115] are highly dimension dependent. So, in the N-body case, it is important to determine the effective dimension of the bottom of the continuum, since this will give the right asymptotic behaviour of e(A). For example, if the

bottom of the continuous spectrum is determined by a two-cluster breakup, the effective dimension is the dimension of the Laplacian giving the intercluster kinetic energy. This is the case treated in this paper.

TWO-BODY AND N-BODY SCHRODINGER OPERATORS AND RESONANCES

167

4. Resonances [19] Convergence of tame-dependent perturbation theory for autoionazing states of atoms, Phys. Lett. A36 (1971), 23-25 [20] Resonances in n-body quantum systems with dilatation analytic potentials and the foundations of time-dependent perturbation theory, Ann. of Math. 97 (1973), 247-274 [105] (with E. Harrell) The mathematical theory of resonances whose widths are exponentially small, Duke Math. J. 47 (1980), 845-902 [107] The definition of molecular resonance curves by the method of exterior complex scaling, Phys. Lett. 71A (1979), 211-214 [121] (with J. Morgan) The calculation of molecular resonances by complex scaling, J. Phys. B14 (1981), L167-L171 [156] (with N. Corngold and E. Harrell) The mathematical theory of resonances whose widths are exponentially small, II, J. Math. Appl. 99 (1984),447 457 [276] Resonances in one dimension and Fredholm determinants, J. Funct. Anal. 178 (2000), 396-420

A resonant state 0 of a quantum system described by a Schrodinger operator H is a state that behaves almost like a bound state, or eigenvector, but is ultimately unstable. Heuristically, 0 should be an eigenvector with complex eigenvalue E-iF/2 12 decays like e-rt. But, of course, a self-adjoint operator so that I (0, e-i,Ho) a4 H cannot have complex eigenvector, and giving a mathematical definition of resonances was a challenge from the first days of quantum mechanics. "No satisfactory definition of a resonance can depend only on the structure of a single operator in an abstract Hilbert space." This remark of Howland was dubbed Howland's Razor, in reference to Ockham, by Simon in his review article [xv] for quantum chemists. The reason is that it is possible to display two unitarily equivalent Hamiltonians (Stark Hamiltonians with differing field strengths) that, on physical grounds, should have different resonances. Simon goes on to say that the only "satisfactory" definition of a resonance is as a scattering pole, that is, a pole in the analytic continuation of the scattering amplitude. The extra structure involved is then a reference operator Ho (for a time dependent definition of the scattering operator) or the geometric structure of space. However, other definitions of resonances are easier to work with, and Simon's papers on resonances deal mostly with the dilation analyticity definition. In this definition, one starts with the complex dilated Hamiltonian

H(8) = -e-200 + V (eex) defined for 0 in a strip in the complex plane for suitable V. Aguilar and Combes (for two-body Hamiltonians) and Balslev and Combes (for N-body Hamiltonians) proved that as 0 moves from the real axis into the complex plane the essential spectrum pivots into the complex plane around the thresholds. Along the way complex eigenvalues are uncovered which, once exposed, do not depend on 0. These complex eigenvalues are the (dilation analytic) resonances. Simon's papers [19] and [20] use the dilation analytic definition of resonance to give a mathematical meaning to perturbation theory for embedded eigenvalues,

and to the Fermi Golden rule. Important physical examples of this phenomenon are the so-called autoionizing states for helium. Here one starts with the N-body Hamiltonian describing a helium atom, but with the electron-electron interaction

R. FROESE

168

turned off. Explicitly,

Ho = -01 - A2 - 2/1xi

- 2/Ix21.

Since this Hamiltonian has the form h ®1 + 10 h it is easy to compute the spectrum

and one finds there are eigenvalues at -n-2 - m-2, n, m E Z embedded in the continuous spectrum, which starts at -1. When the electron-electron interaction, 01jx1 - x21 is turned on (that is, 0 is increased from zero to some small value) one expects that the embedded eigenvalues dissolve. The basic idea of these papers is to first perform a complex dilation. This turns the embedded eigenvalues into isolated discrete eigenvalues, albeit of a non-self-adjoint operator. Nevertheless, regular perturbation theory can be applied to these, giving a series expansion for the perturbed eigenvalue. The leading order for the imaginary part of this series yields Fermi's Golden rule for the lifetime of the resonance. It can be used to show that the eigenvalue really does move into the complex plane to become a resonance. For the autoionizing states there are additional issues stemming from the high multiplicity of the eigenvalues, and Simon also addresses these. Another physically important resonance problem is the hydrogen atom in a constant electric field, with Hamiltonian given by

H(F) = -A - 1/jxj + Fx3. When F = 0 this is the Hamiltonian for the hydrogen atom with negative eigenvalues accumulating at zero. However, for all non-zero real F the spectrum is purely absolutely continuous and covers the entire real line. Clearly, this is a very singular perturbation problem. Nevertheless, Herbst succeeded in extending the Balslev-Combes framework to this problem and obtained an expansion E(F) - Eo + E A2,,,F2n for the resonances. The coefficients A2n in this expansion are all real, which implies that the imaginary part RE(F) of E(F), which he shows are non-zero, must be very small as F tends to zero. In fact there is a formula proposed by physicists, called the Oppenheimer formula, for the width of the resonance associated with the lowest eigenvalue of H(0). It is given by

s(E(F))

exp 6F

2F The papers [105] (with Harrell) and [156] (with Corngold and Harrell) deal with this and related situations (including the anharmonic oscillator and the Bender-Wu formula). One of their results is a proof of the Oppenheimer formula. The proofs proceed by reducing the problem to a problem in ODE's. Simon's most recent paper on resonances [276] deals with the one-dimensional

case. There are two main results in this paper. One is an expansion formula for a Fredholm determinant appearing in my own work on the counting function for resonances. This formula allows Simon to connect my approach with the original approach of Zworski. The second result is a counting result for antibound states. Acknowledgements. It is a pleasure to thank Fritz Gesztesy, Peter Hislop, Peter Perry and Wolfgang Spitzer for reading an early version of this manuscript and offering suggestions for improvements. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF BRITISH COLUMBIA, VANCOUVER, B.C., CANADA V6T 1Z2

E-mail address: rfroesec math.ubc. ca

Proceedings of Symposia in Pure Mathematics Volume 76.1, 2007

N-Body Quantum Scattering and Quantum Resonances: An Overview Christian Gerard To Barry Simon, with admiration, on the occasion of his 60th birthday ABSTRACT. We survey the progress in the last twenty years in N-body quantum scattering and resonance theory.

CONTENTS 1.

2. 3. 4. 5. 6. 7.

Introduction N-Particle Quantum Systems N-Particle Systems in Constant Electromagnetic Fields Stationary Scattering Theory Resonances in Quantum Mechanics Time-Dependent Picture of Resonances Resonances in the Semi-Classical Limit

References

1. Introduction The aim of this article is to review two domains of mathematical physics which underwent tremendous progress in the last twenty years, namely, scattering theory and resonance theory for Schrodinger operators. Scattering theory deals with the study of the long time behavior of the evolution e-itHu of a quantum state u. The evolution of bound states of H is, of course, trivial. Concerning states from the continuous spectral subspace of H, one expects that the evolution a-itxu becomes asymptotic to a superposition of simpler evolutions. If H models a system of N interacting particles, these simpler evolutions describe bound clusters of particles moving freely away from each other. Cast in 2000 Mathematics Subject Classification. 81U05, 81U10, 81U20, 81Q.

Key words and phrases. N-body quantum scattering, S-matrix theory, resonance theory, semiclassical analysis.

169

C. GERARD

170

precise mathematical terms, this property is the asymptotic completeness, to which Sections 2 and 3 are devoted. Comparing the asymptotic evolutions of a given state when t -+ -00 with those

when t -j +oc leads to the scattering matrix and scattering amplitudes. From the scattering amplitudes, one obtains the scattering cross-sections, which have a direct physical meaning, since they describe the probabilities of various scattering processes. Some properties of the scattering matrix will be described in Section 4. If one looks at the evolution e-UHu for intermediate times (i.e., neither too small nor too large), one encounters some states u exhibiting a metastable behavior, i.e., e-'tHu = e-st(E-ir)u + remainder, for r > 0. The complex number E - iF is called a resonance of the Hamiltonian H. The appearance of complex frequencies in a self-adjoint setting is quite a fascinating fact. To give a mathematically precise definition of what a resonance is and to study the properties of these resonances is the subject of resonance theory. Eventually the definition of resonances of H as poles of a meromorphic extension of (z - H)-1 across the continuous spectrum proved the most useful. We will discuss this question in Section 5. Nevertheless, physicists prefer to view resonances as complex poles of the scattering matrix which, when it can be rigorously established, shows the deep link between resonances and scattering theory. Some results in this direction will be described in Subsection 5.2. To study more precisely the location of resonances or their influence on spectral quantities, it is necessary to work in some asymptotic regime. A first regime is the small coupling limit, where one studies how an embedded eigenvalue Eo of a Hamiltonian Ho turns into a resonance E(g) - iF(g) for the perturbed Hamiltonian Ho+gV by coupling to the continuous spectrum. This will be discussed in Section 6. Another regime is the semi-classical limit where, for example, resonances for the

Schrodinger operator -h20 + V (x) are studied in the limit h - 0. The presence of resonances near the real axis is then connected with the existence of trapped trajectories for the corresponding classical problem. We will survey the rather impressive body of results in this area in Section 7.

It is now time to add a few comments about the content and scope of this review. I had to make some choices to keep the review within the page limit. As a rule, I have tried to focus on results for models of non-relativistic quantum mechanics, which usually means the Schrodinger equation. I have deliberately restricted my discussion of scattering theory to N-particle Hamiltonians. This was the domain which motivated the advances in time-dependent scattering theory. Nevertheless, the time-dependent method proved flexible enough to be applied to many other problems that are not reviewed here. Similarly, I have omitted several important results in resonance theory, for example, those concerned with Breit-Wigner formulas, refined estimates on the number of resonances, or resonances for other equations of mathematical physics. I strongly suggest that the reader read the contribution by Evans Harrell in this Festschrift, which is specifically devoted to quantum resonances and contains many

results, in particular, on the important topic of perturbation theory that are not covered in this review.

To conclude this introduction, I hope that this review will convey my own feeling of admiration for the collective effort that led to a much better understanding of scattering and resonance theory that we have today.

N-BODY QUANTUM SCATTERING AND QUANTUM RESONANCES

171

Barry Simon was one of the leading contributors to this effort: for scattering theory it suffices to mention his extension of Enss' time-dependent method [105], his

papers on the geometric method in scattering theory [29, 101], and the landmark paper with Perry and Sigal [89] on the Mourre estimate for N-particle Schrodinger operators. For resonance theory, there were, among many others, his seminal paper [100] on perturbation theory for resonances and the Fermi golden rule and the paper [104] on exterior complex scaling. In my younger days, when after a brief excursion in microlocal analysis I moved to mathematical physics, reading Barry Simon's books and articles was always a stimulating and enlightening experience. As a testimonial of my admiration, I dedicate this review to him on his 60th birthday.

Acknowledgments. During the preparation of this survey, the author benefited from very useful discussions with Volker Bach, Jan Derezinski, Jakob Yngvason, Thierry Ramond and Erik Skibsted. My thanks go also to Pedro Filiot for his careful reading of the manuscript, and to the referees for their helpful suggestions. I am still solely only responsible for any remaining errors or omissions.

2. N-Particle Quantum Systems It is not so exaggerated to say that N-particle Schrodinger Hamiltonians describe all of non-relativistic quantum mechanics (at least as long as interaction of atoms with radiation is disregarded). While for questions related to bound states problems, it is often necessary to replace the Schrodinger equation with simpler effective theories, for the basic questions of scattering theory, the true model can be completely analyzed. We will try to describe the main results in this domain in the next two sections. There are several books and review articles devoted to scattering theory of N-particle systems: for the time-dependent approach, let me mention the book by Derezinski and Gerard [33], which contains a thorough exposition of scattering theory of N-particle systems both in classical and quantum mechanics, and the review articles by Hunziker and Sigal [67, 68]. For the stationary approach, I recommend the books by Yafaev [133, 137]. The microlocal approach to stationary scattering theory is explained in a nice book by Melrose [84] and a review article by Vasy [128]. I have focused in this section on results and methods that have appeared since the mid-eighties, starting with the time-dependent results of Enss, Sigal and Soffer. Prior to these, the results achieved by time-dependent methods were more limited: existence of channel wave operators was proved long ago by Hack [58], and extended

later by Hunziker to potentials with local singularities [66]. There were also some completeness results by Iorio and O'Carroll [72] for small coupling constant, and by Lavine [78] for repulsive potentials. In both cases only the free channel is open, which simplifies the analysis.

2.1. N-Particle Hamiltonians. Consider a system of N non-relativistic particles in Euclidean space '. It turns out that for the scattering theory, the statistics, i.e., the bosonic or fermionic character of the particles, and their spin do not play any significant role. Particle spin is usually conserved for the models described in this section, and the inclusion of statistics presents only combinatorial problems.

Therefore, one usually assumes that the N particles are spinless and distinguishable, and the system is described with the Hilbert space L2(RN"). Its time

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172

evolution is described by a Hamiltonian of the form N

H

Dj + > Vij (xi - xj), -1 j=1 2m3 1
(2.1)

mj being the mass of particle j and Vij the interaction potential between particles i and j, and Dj = i-1 Vx, . The most important case for physics is, of course, the Coulomb interaction, where v = 3 and Vii (X) =

iii

qj being the electric charge of particle i. A related Hamiltonian arises when certain particle masses (like nuclei) are much larger than others and can be considered as infinite. The heavy particles are hence treated as fixed classical sources. The Hamiltonian is then N

1

j=1 2mj

(2.2)

1
j=1 k=1

the points Xk E 1', 1 < k < P being the positions of the heavy particles. Note that the Hamiltonian in (2.1) is invariant under space translations, while the Hamiltonian in (2.2) is not.

2.2. Agmon Hamiltonians. A nice framework, the Agmon Hamiltonians, has been introduced by Agmon [6]. Besides its mathematical elegance, it allows one to treat in a unified way the two cases considered above. It also allows one, for example, to easily incorporate many particle interactions, such as potentials vi,j,k(x) = v(xi - xj, xi - xk), where v : R2i --+ 1l tends to zero at infinity.

Let X be a Euclidean space. The scalar product on X will be denoted by x.x = x2. We also get a scalar product on the momentum space X', denoted by p.p = p2. Using the Lebesgue measure associated to the scalar product, we can consider the Hilbert space L2(X), on which the Laplacian

Ho =1D2 2 for D = -iVy is well defined. Assume that there exists a finite family {Xa}aEA of subspaces of X, closed under intersections and containing X. The set A is equipped with an order relation by saying that a < b if Xa D Xb. Since the family {Xa} is closed under intersection, we see that A equipped with < becomes a semi-lattice; in other words, for each a, b E A, there exists a smaller element c such that a < c, b < c, denoted by a V b. One has, of course, XaVb = Xa fl Xb.

Let also amen, amax denote the smallest and largest elements of A, so that Xamin

_

X, Xamax = naEA Xa-

For each a E A, we choose a real potential Va : X -* 1(8 invariant under translations in Xa: Va(X) = Va(X - y), V Y E Xa.

Then an Agmon Hamiltonian is an operator on L2 (X) of the form

H = 2 D2 + E Va(x). aEA

N-BODY QUANTUM SCATTERING AND QUANTUM RESONANCES

173

Note that H commutes with the translations along the space Xamax, so if Xamax {0}, one can separate the motion of the center of mass. Using the scalar product on X, we can introduce the subspaces

Xa:=Xa, so that a < b if Xa C Xb. If we denote by 7ra, Ira the orthogonal projections on

Xa and X' , we set x = xa + Xa, for xa := 7rax, Xa := 7raX,

which allows us to write L2(X) = L2(Xa) ® L2(Xa).

Note that Va can be considered as a real function on Xa. Clearly, to hope for a reasonable scattering theory, one has to assume that Va

(Xa)

--_ 0, when I xal - 00,

(2.3)

at least in some averaged sense. Similarly, if we set

Xa:={pEX, I (p,x)=0,VxEXa},Xa' := {pEX, I (p,x)=0,`dxEXa}, we can set

p = pa + Pa, so that p2 = (pa)2 + (pa)2. For a E A, one can introduce the cluster Hamiltonians:

Ha := 2 D2 +Va(x) for

Va(x) := I: Vb(X). b
Note that Va commutes with translation by vectors in Xa, so that Va can be considered as a real function on Xa. Then we get Ha = 2 (Da)2 + 2 (Da)2 + Va(xa) _: z (Da)2 +H a,

where Ha is a Hamiltonian on L2(Xa). Note that Ha has the same Agmon Hamil-

tonian structure on Xa, with the set of indices Aa = {b E A I b < a}. This is a first indication of the lattice structure of the N-body problem, which plays a fundamental role. Another notation that will be needed is the inter-cluster potentials: Ia (X)

E vb(x), bl(a

so that

H=Ha+Ia. We can now easily discuss the separation of the center of mass. If Xamax {0}, the Hamiltonian H commutes with the translations by vectors in Xamax. We write

H=

(Damax)2 + Hamax z

where Hamax acts on L2 (Xamax ), and it is clearly sufficient to study In the sequel we will always assume that Xamax = {0}, i.e., that the motion of the center of mass has been separated out. Let us now explain this formalism for the two Hamiltonians introduced above. Ha---.

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For the Hamiltonian (2.1), one sets X = IAN", equipped with the quadratic form

(x, x) :_ i=1

Note that with the above notation we have 2

2mz(Di)

2

D2

We choose for A the set of cluster decompositions, i.e., the set of partitions a = {C1,. .. , Ck} of {1, ... , N}. For a pair {i, j}, one says that {i, j} < a if {i, j} C C1 for some C1 E a. To a cluster decomposition a = {C1, ... , Ck} E A, one associates the collision subspace

Xa := {x E RN" I xi = xj if {i, j} < a}, corresponding to a particle configuration where all particles in a cluster of a are at the same point of R", which describes the configuration space for the center of masses of the clusters of a((. The space X' of internal cluster motion is

Xa={XERN" Emixi=0, 1
iEC,

The order relation introduced above is the familiar order relation on partitions, meaning that a < b if a is finer than b. We see then that amin = {{1}, .. . , {N}}, Xan,in = X, and amax = {{1, ... , N}}, Xan,ax = {x (E ISBN" I xi = xj, V i, j}. For the Hamiltonian in (2.2), we choose for A the set of partitions of {0, 1, ... , N}. To a cluster decomposition a = {Co, C1,.. , Ck} where 0 E Co we associate the .

subspace

N"I xi=0ifiECO,xi=xjif{i,j}
EINU

Hence, particle indices belonging to Co label the particles close to the origin in R'. In fact, for the scattering theory, all the heavy particles can be viewed as sitting at the origin of configuration space. The space Xa is again

Xa={ ((

E

IR

"

mixi=0, 1 <1
2.3. Asymptotic Velocity. The use of asymptotic observables in scattering theory has a long history (see, for example, [9]). In the time-dependent approach, it appeared first in the approach of Enss [36, 37]. The introduction of the asymptotic velocity observable by Derezinski in [30, 31] allows for a very elegant formulation of scattering theory. The proofs of Theorems 2.1 and 2.2 can be found in [31, 32] or in [33, Ch. 6].

Let us formulate a simple set of assumptions under which the results below hold. We assume va(xa) = Va,s (Xa) + va,l(xa), lva,s(xa)I µs > 1, C(xa)-µg

l ayava 1(xa)I < C(xa)-I-I-I`', 0 < I01

(2.5)

2, /2 > 0.

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175

THEOREM 2.1. Assume (2.5). Then there exist an X-valued self-adjoint observable P+, called the asymptotic velocity, such that: s-

t

e-htH = F(P+), VF E C. (X ).

li n eitHF

(t)

Here C,,(X) is the space of continuous functions on X vanishing at infinity. Note, for example, that the fact that P+ has a dense domain is reflected physically by the fact that any state propagates with finite speed. The proof of Theorem 2.1 relies on propagation estimates first proved by Graf [54].

The use of the asymptotic velocity allows one, for example, to naturally split an initial state into parts which belong to various scattering channels. In fact, if we set

Z.

(2.6)

Xa\ U Xb, bra

we see that {Za}aEA is a partition of X. Hence,

1=

1{za}(P+). aEA

For u E L2(X), 1{zo,}(P+)u is the part of u that asymptotically separates into the clusters of a. Let us now collect some properties of P+, which are deduced from a Mourre estimate [88, 89, 44].

i) P+ commutes with the Hamiltonian H. THEOREM 2.2. ii) 1{o}(P+) = 1PP(H). iii) One has a(P+, H) = U {(Sa,T + 2Sa) I {{

{{

tt

Sa E

Xa, T E

aPP(Ha)}.

aEA

Property iii) can be viewed as a poor version of asymptotic completeness. One should compare it with the description of the joint energy-momentum spectrum in relativistic quantum field theory. Here the Galilee invariance of the non-relativistic N-body problem is manifest.

2.4. Short-Range Wave Operators. Let us now describe the short-range wave operators for Agmon Hamiltonians. We consider in this subsection an Agmon Hamiltonian with Xama.x = {0}, i.e., without trivial translation invariance. For simplicity, we will assume the following form of the short-range condition:

IVa(xa)I < C(xa)-µ, It > 1.

(2.7)

The need to take care of the precise decay rate of the pair potentials at infinity, instead of the very weak assumption (2.3), is well known from the study of the one-body problem. For a Schrodinger operator H = aD2 + V(x) for V decaying at infinity like I xI -µ for p > 1, one can use as comparison dynamics the free dynamics

e-'tHO, Ho = 2D2, while if V decays like I xI -µ for 0 < µ < 1, one has to use a modified dynamics. The standard example is Coulomb scattering, where one uses the Dollard dynamics to define modified wave operators.

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The wave operators should label the various possible scattering channels, describing bound clusters of particles, whose center of masses has an asymptotically free motion. Bound states of a set of clusters a are labeled by states in 7-IPP(Ha), the space of bound states of H. This leads to the following definition of wave operators: DEFINITION 2.3. The short-range wave operators for an Agmon Hamiltonian H are defined by: e'txe-itHalPP(Ha),

QT a := s- lim

t-++ o0

aEA

provided the above strong limit exists.

Note that with our definition, we have StST amax = 1PP(H). Clearly, the wave operators Sts a, if they exist, are isometries and satisfy the intertwining relations s sr,aHa = HSZsr

a

This definition leads to the two familiar questions in time-dependent scattering theory: Existence of wave operators. Find conditions on the potentials va such that the limits e'tHe-itHalPP(Ha), s- lim a E A exist. t-+±oo

Asymptotic completeness. Find conditions on the potentials Va such that

® Ran Sts ,a = L2(X), aEA

or equivalently,

f f: _

sr,aQsra = 1 aEA

Note that in the literature, one sometimes finds the following weaker definition, called the weak asymptotic completeness: Ran SZs,a = aEA

Ran Qsr,a aEA

Weak asymptotic completeness implies that the scattering operator is unitary, and is sufficient to get a complete physical description of a scattering experiment. However, I don't know of examples where weak asymptotic completeness has been shown without proving at the same time (strong) asymptotic completeness. Let us now explain how the existence of asymptotic velocity easily implies the existence and completeness of short-range wave operators. THEOREM 2.4. Assume (2.7). Then: i) The short-range wave operators exist. ii) The system is asymptotically complete: (@ Ran (Sts ,a) = L2(X). aEA

To prove ii), it clearly suffices to show that Ran(Sls a) = 1{za}(P+)7t. The c part is easy since it amounts to considering the dynamics generated by Ha. The D part is also easy: for a state in 1{za}(P+)7-l, one has Ixbj > ct for all b 0 a

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177

and hence the inter-cluster potential Ia decays like t-µ. Since µ > 1, one can use the Cook argument to show that u E Ran(SZS ,a). The first success of time-dependent methods for N-body Hamiltonians was due to Enss, who proved the asymptotic completeness for short-range [34, 35] and then - 1) three-body systems. Enss relied heavily on the RAGE long-range (for µl > theorem and on compactness arguments. Later Sigal and Soffer [97] were able to prove asymptotic completeness for general N-body short-range Hamiltonians.

Their method was based on Kato's H-smoothness argument. In the Sigal-Soffer proof, an essential role was played by the Mourre estimate due to Mourre [88], later extended and simplified by Perry, Sigal and Simon [89] and Froese and Herbst [44]. Later a very elegant, fully time-dependent proof was given by Graf [54].

2.5. Long-Range Wave Operators. The case of long-range pair potentialsroughly speaking, decaying at infinity like xl -µ for 0 < µ < 1-is much more difficult for general N than for N = 1. A scattering state corresponding to a cluster decomposition a is described by a state in Ran 1z (P+). Because of the long-range nature of the pair potentials, the inter-cluster potential la cannot be entirely neglected on such states. Because the clusters of a move away from each other, it is possible to replace Ia by an effective time-dependent potential la (t, x), which can, for example, be defined as

F(ln(t)alt > 1)Ia(x),

Ia(t,x)

where the inter-cluster distance Ixla is Ixla := inf Ixbl.

If the pair potentials Va satisfy (2.5), then

8t Ia(t, x) E 0(t-u'-I'l), Ial < 2, for each 0 <

< Then it follows directly from the very definition of the asymptotic velocity P+ that the limits s-

lim Ua(0,t)e-'thhi jP+)

t-- +oo

exist, where Ua(t, s) is the dynamics generated by the time-dependent Hamiltonian Ha + I. (t, x)

The problem is that the dynamics Ua(t, 0) couples the internal and external dynamics. In order to prove asymptotic completeness, the essential step is to prove that these two dynamics decouple, i.e., to prove that one can replace Ia (t, x) by Ia(t, 0, xa) After this step, the internal and external dynamics are completely decoupled, and it is a simple matter of one-particle long-range scattering to introduce a modified evolution for the time-dependent Hamiltonian aDa Ia (t, 0, xa).

The simplest way of proving this fact is to show a bound on the size of the clusters. Clearly, for a state in 1Za (P+), the size of the clusters is o(t). If one can show

that the clusters have a size 0(t13), for some 0 depending on µ, then one gets

Ia(t,x)-Ia(t,0,xa) =O(IxaI)IVxla(t,x)I

=0(t1-1-N,).

The crucial progress was made by Derezinski [32], who showed that for a state in 1Zo. (P+), the size of the clusters is bounded by O(t 2T ). This exponent has again a

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classical interpretation: for the one-body problem, there exist zero energy classical 2 trajectories which go to infinity like t22+ . For the Kepler problems, these trajectories are the well-known parabolic trajectories. Note that parabolic trajectories are known to exist also for the classical N-body problem of celestial mechanics. The condition that 2+µ < µ yields the condition

p> v3 -1-0.732. Hence, one obtains the following result:

THEOREM 2.5. Assume (2.5) for pi > i) The long-range wave operators

S2+ lim lr,a = s- t-++oo

- 1. Then

eitxe-iso,(t,D..)e-itxalpp(Ha) exist.

ii) The system is asymptotically complete

®Ran(Ql a) = L2(X). aE.A

Here Sa(t, pa) is a convenient modifier for the time-dependent potential Ia(t, 0, Xa) (since E.ti > 2 one can take, for example, the Dollard modifier). This result is due to Derezinski [32] for general N-body systems. As mentioned above, for N = 3, the result was obtained earlier by Enss [39] for N = 3 with the same decay condition but by different arguments. Independently, Sigal and Soffer proved asymptotic completeness for N = 4, and µ = 1 (i.e., for Coulomb decay) in [98] and later for arbitrary N and µ > 1- 2-iV-2 in [99]. Let us also mention other

papers on asymptotic completeness for short- or long-range N-particle systems [77, 122, 139].

2.6. Further Results. The condition p > J-1 in Derezinski's result suffices to cover the physical case of Coulomb interactions. Nevertheless, it is interesting mathematically to study the case of slower decay. So far, positive results are essentially limited to three-particle problems. One can focus attention on the so-called exceptional states, i.e., possible states not in the range of the wave operators. One is faced with the slightly depressing task of working very hard to study states which ultimately are shown not to exist! For three-body problems, these states consist of a pair, with internal energy asymptotically 0, and a third particle moving away from it. One can focus attention on the motion of the pair, and describe the interaction with the third particle by an effective time-dependent potential. Hence, one is faced with the study of the dynamics generated by the time-dependent Hamiltonian

H(t) = 2 D2 +V(x) + W(t, x), where V is a long-range potential and ax W (t, x) E 0(t-HaH-µ). Asymptotic completeness for N = 3 was shown by Wang [131] for pi > under the condition that the negative part va of the potentials decays like xaJ-ry for some -/ > 2(1-µ0µl 1 For potentials which are negative near infinity, asymptotic completeness for N = 3 was shown by Gerard [45], again for pi > 2, assuming the pair potentials satisfy a virial condition 2Va(xa) + xawa(xa) < -C(xa)-µ1 . Later Skibsted [114] was able to eliminate the virial condition, replacing it by the weaker condition that va(xa) < -C(xa)-µ" near infinity.

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Again, these positive results rely on a detailed study of classical one-body problems in an external time-dependent potential. There also exist counterexamples to asymptotic completeness, due to Yafaev [136], for the three-body problem in one space dimension with pair potentials decaying slower than xJ-2. In these counterexamples, quantum mechanics comes back in the picture: they are essentially based on Born-Oppenheimer approximation. Counterexamples similar in spirit were known for time-dependent [132] and time-independent Schrodinger operators [138].

3. N-Particle Systems in Constant Electromagnetic Fields A system of N quantum particles can be put in an external classical field. The simplest case is that of a constant and uniform electromagnetic field.

3.1. Constant Electric Fields. A system of N non-relativistic particles of masses mi and charges qi in a constant external electric field E E IR" is described by the Stark Hamiltonian N

H

1

D2

+gjE.xj+

j=1

11

Vij(xi-xj).

(3.1)

1
There are several aspects of Stark Hamiltonians not present for usual N-particle Hamiltonians. These aspects were first understood for two-body Stark Hamiltonians by Avron and Herbst [10]. First, Stark Hamiltonians are unbounded below. This causes some problems if one wants to include singular potentials. Also the proof of the Mourre estimate, an essential tool in the study of the spectral properties of H, relies on an induction argument over subsystems. Because of the electric field, the internal energy of subsystems is neither bounded from above nor from below if the total energy is bounded, which complicates this inductive step. Second, the notion of a short-range potential is different for Stark Hamiltonians: because the position xt along the electric field of a particle grows like t2, potentials decaying as xJ-µ for p > 1 in the direction of the electric field are now short-range potentials. Finally, because of tunneling, bound states typically disappear in a constant electric field. Again it is easy and convenient to pass to the framework of Agmon Hamiltonians: a constant electric field corresponds clearly to a vector F E X'. However, to describe trajectories, it is convenient to identify F with a vector E in X, using the scalar product on X. One can then define Agmon-Stark Hamiltonians:

H=

2 D2

+ E.x + E V" (X). aEA

For a E A, one can write E = Ea + Ea, where Ea E Xa, Ea E Xa. For the physical case (3.1), we see that Ea = 0 if EiEC, qj = 0 for each 1 and Ea = 0 iff m = m, for each pair {i, j} C C1, C1 being a cluster of a. The cluster Hamiltonians are then

H. =H - Ia = 1 (Da)2+Ea.xa+Ha, where the internal Hamiltonian Ha = 1(Da)2 +Ea.xa +Va(xa) acting on L2(Xa) is again an Agmon-Stark Hamiltonian. Clearly, if Ea = 0 then the electric field

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only acts on the center of mass motion of the clusters of a, and the Hamiltonian Ha is a usual N-particle Hamiltonian. This can be easily formulated by using the partition X = UaEA Za, where we recall that Za := Xa\ U Xb. bra

Clearly, there exists a unique a E A such that E E Za, and one has

a=max{aEAIEa=O}. Let us give a set of assumptions (see [61, 62, 4, 5] for much weaker conditions) under which the results below hold. One says that a potential Va is Stark short-range if

wa(xa)l = O(lxal-"`a),

where ha > a if Ea 0, µa > 1 if Ea = 0. One says that Va is Stark long-range if for 0 < Ial < 1, axll Va(xa) =

where µa>0ifE'

IO(Ixal-lia-Jaj),

0,Pa >/-1if Ea=0.

The conditions above are the natural analogs of the conditions (2.7). Both have a purely semi-classical interpretation. Let us mention, however, that results on the nature of the spectrum for one-dimensional Stark Hamiltonians can be proved under much weaker hypotheses (see, for example, the recent paper [25]). An important intermediate result (see [61] and references therein), is: THEOREM 3.1. Assume Ea

0. Then one has

o'pp(Ha) = 0, asc(Ha) = 0.

This implies that only the clusters a < a can contribute to the scattering channels of H. The key result, proved in [62] (see also [4, 5]) is that the motion of the system is asymptotically localized in a neighborhood of the field direction:

i) Let q e C°°(X) be a function homogeneous of degree 0 near infinity, and supported in a conical neighborhood of iE- not intersecting the subspaces Xb for b X a. Then one has

THEOREM 3.2.

e-9tHu = q(x)e-'txu + o(1), t -p +oc, V u E L2(X).

ii) One has e- 9tH u

=

F( ia 2I

> JEI - c le-stHu+o(1), t

+oo, Vu c L2(X).

In the short-range case this allows for replacing the full evolution e-itH by e-'tH6. Since Ha is a usual N-particle Hamiltonian, one can complete the proof by invoking the known results in Subsection 2.4. In the long-range case one has to introduce a modified dynamics for the motion of the center of mass. Due to the fact that in a constant electric field lxl grows like t2, it suffices to add a pure phase factor Sa (t) = This gives the following results:

ft(s2)

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181

THEOREM 3.3. Assume the potentials va are Stark short-range. Then the wave operators e'txe-itxa lPP(H' exist for all a < a. SZ r,a := s- lim ) t

foo

The wave operators are complete: Ran1l ,a = L2(X). a
THEOREM 3.4. Assume the potentials Va are Stark long-range. Then the wave operators SZ: lr,a

e'txe-sa(t)e-itHa1PP(Ha) exist for all a < a.

lim := s- t_±00

The wave operators are complete:

®RanQIr, a = L2(X). a
Asymptotic observables do not play a central role for Stark Hamiltonians because the motion is essentially localized in the direction of the electric field. Nevertheless, it is possible to construct and study asymptotic observables in this case also (see [1]).

3.2. Constant Magnetic Fields. A system of N non-relativistic particles of masses mi and charges qj coupled to a constant magnetic field is described by the Hamiltonian: N

H=

j=1

2m ,

uij (xi - xj ).

(D - giJxi)2 +

(3.2)

1
7

Here Jx is the vector potential associated with the magnetic field. It is convenient to work in the transversal gauge. In this case, if the space dimension v is 3, we have

J where the magnetic field is

0

-b

0

b

0

0

0

0

0

,

(0, 0, 2b), b > 0, and if v = 2, 0 b

b

-

0

)

-

If v = 3, we write the coordinates of the particles as

xi = (yi, zi), where yj E R2, zi E 11, and if v = 2, we simply omit the z variable in the formulas. Again it is convenient to pass the formalism of Agmon Hamiltonians. We set X = IRN", equipped with the Euclidean structure x.x defined in (2.4), and write (with obvious notations):

X=Y®--z. The N-particle vector potential is Ax = (g1Jx1, ... , gNJxN), and should be considered as an antisymmetric operator A : X -> X', or equivalently, as an antisymmetric bilinear form on X.

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If we assume that all particles are charged, then Z = KerA.

The main new feature of magnetic N-particle Hamiltonians is that while invariant under translations of the center of mass, the generators of these translations do not commute with one another. Therefore, care has to be taken with the separation of the center of mass (see [11, 46]). Actually, it turns out that separation of the center of mass is not really necessary to study the scattering theory of H-it can be replaced by a correct definition of bound states and scattering states (see [46]). The generator of translations of the center of mass is the pseudo-momentum of the center of mass: Kama.x

Damax + (Ax)amax,

which satisfies the following commutation relations: [(Z> Kamax), 1(x') Kamax)] = -2(X Aamax,amaxx ), x, x' E Xamax,

where Aamax,amax is the restriction of A to Xamax X Xamax The component of Kamax

along Z is the usual momentum Dzamax . The other two components commute if the total charge N

Q = j qi i=1

vanishes. A similar analysis can be applied to the channel Hamiltonian

Ha=2(D-Ax)2+EVa, b
which now commute with

Ka := Da + (AX)a. This leads to the distinction between charged clusters (whose total charge is nonzero) and neutral clusters. For the analysis of Ha, it is convenient to construct unitary operators Ua where the non-commuting components of Ka are transformed into (multiples of) positions and momentum variables. We will not give their definition here but refer to [46, 49] for details. From these transformations, one sees quite easily that: A charged pair of two particles has a bounded motion in the y variable, and a free motion in the z variable. A neutral pair of two particles has a ballistic motion in the y variable and a free motion in the z variable. An example of this fact for classical particles can easily be constructed using the properties of the Lorentz force (see [49]). Using these transformations and proving a Mourre estimate for the motion in the z direction, one can show the following result [46]: Because of the special role played by the z direction in the three-dimensional case, one has first to separate out the motion of the center of mass in the z direction. A convenient way to do this is to define a subspace abound C L2 (X) corresponding to states where the N particles stay together but can move in the y and z directions (if the total charge is 0) or only along z (if the total charge is not 0). The space of scattering states is then

ascatt

abound-

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183

THEOREM 3.5. Assume that the system has no neutral subsystems and that the pair potentials are short-range. Then the wave operators 52S a :

s- lim

eitHe-itH°Ha

t-,+00

exist for a J ama,.

The system is asymptotically complete: Ranhl r,a = 7lscatt aEA,a#amax

Here Ha is a projection on states describing bound clusters of a (its precise definition uses the reducing transformation Ua). Note that in this theorem the total charge of the system can be 0. The size of the clusters in the y variables is asymptotically bounded. The analog situation in two dimensions is much simpler: there are no scatter-

ing states, and if the total charge is non-zero, o,(H) is a closed countable set of eigenvalues with infinite multiplicity (see [49]). One can next consider the case of long-range interactions. As in the standard case, the main problem is to decouple internal and external motion, and to this end, to prove a bound on the size of the charged clusters. A bound in the z direction can be obtained by the same arguments as in the standard case. The proof of a bound in the y direction is very different and relies on the center of orbit observable:

C:= 2 (y + A1Dy) Classically, the two components of C are the position of the center of the circle followed by the classical particle. It is easy to see that a bound on C yields a bound on y because C-y=2A-1(Dy-Ay),

and Dy - Ay is controlled by the total energy. Using then a construction inspired by the Graf vector field, but using now the indefinite charge metric N

4(y, y) := i=1

it is possible to show that along the evolution, C (and hence y) is bounded by O(t«) for each a > 2. This leads to the following result:

THEOREM 3.6. Assume that the system has no neutral subsystems and that - 1). Then the wave the pair potentials are long-range (with decay rate pi > operators Qsr,a

S

hm eitHe-iSa(t,Dza)e-itHaHa exist.

t-++00

The system is asymptotically complete:

® Rant a = xscatt

aE.A,a am x

Here the modification Sa(t, Dza) is, for example, a standard Dollard modifier in the z direction. With some considerable effort, it is also possible to prove asymptotic completeness for three-particle systems having one or two neutral pairs. This result holds in two or three space dimensions ([48, 49]).

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Note finally that in the results above, it was necessary to assume that all particles are charged. Adachi [2, 3] has studied the case where all particles except one are charged.

3.3. Concluding Remarks. Time-dependent scattering theory, essentially based on the Mourre estimate and the phase space propagation estimates introduced

by Sigal and Soffer, has proved to be an extremely efficient and flexible tool to answer the basic questions of scattering theory. The N-body scattering theory in the eighties-which, to the outsider I was at the time, looked like a jungle-has turned into a clean and orderly "jardin a la francaise," a sure sign of a mature and well-understood theory. However, open ground still lies near the boundaries of this garden. For example,

scattering theory in constant magnetic fields is not completely understood in the presence of neutral clusters of particles. Similarly, there are very few results about dispersive N-particle systems, where the non-relativistic kinetic energy N E2

p i-1 2mi is replaced by a general kinetic energy w(p), like the relativistic one,

pi +m?. i=1

Also, systems in time-periodic electromagnetic fields are not completely understood (see, for example, [87]).

When one thinks hard about the time-dependent method, one realizes its defects:

The first obvious one is that it is of little use in studying scattering amplitudes. Here stationary methods seem to rule. A second defect is that this method does not allow for understanding mean motion. Since it is based only on differential inequalities, it cannot control a motion which is a superposition of a mean ballistic motion and a bounded periodic one. This is the very reason why dispersive systems or neutral clusters in a magnetic field are not well understood (except for N = 3 in the latter case). A new method to control mean motion would certainly allow one to handle many interesting problems still open. I think this is a challenging and important question.

4. Stationary Scattering Theory The stationary approach to scattering theory, relying on the study of the resolvent (H - z)-1 when Imz -+ 0, is much older than the time-dependent approach. It seems, in general, less efficient in proving the basic results of scattering theory like asymptotic completeness, but it is the only way to get results on scattering quantities like properties of the scattering matrix. For N-body scattering theory, it started with the approach of Faddeev [40]. He used clever resolvent identities (the celebrated Faddeev equations) and the stationary method to study asymptotic completeness for a certain class of short-range potentials in dimension 3 or bigger for three-body systems [40]. Unfortunately, his method required imposing certain implicit assumptions on the potentials.

N-BODY QUANTUM SCATTERING AND QUANTUM RESONANCES

185

After Faddeev, the stationary method was developed by Ginibre and Moulin [53] and Thomas [124] for three-body systems, by Hagedorn [56] for four-body systems, and by Sigal [93] for N-body systems. All of these papers had the same drawback, namely, implicit assumptions. The only exception was the work of Loss and Sigal [81], which contained a stationary proof of the asymptotic completeness of a certain (rather small) class of three-body systems without implicit assumptions. Another approach is the one of Kato [76], based on the concept of H-smoothness of an operator. For a long time, its application was restricted to two-body problems. However, progress in the time-dependent method, coming from the consideration of propagation estimates in phase space, led to similar progress in Kato's method. Yafaev [134] was able to prove asymptotic completeness for short-range N-body

Hamiltonians by proving that a certain differential operator is H-smooth. Note that a certain function used in Yafaev's construction, related to Graf's function, played a crucial role in the estimate on the size of clusters of Derezinski. Later Yafaev [135] was also able to derive a stationary formula for the scattering matrix S(A), which allowed him to show that S(A) is strongly continuous in A. The

scattering matrix is defined using the two spaces approach to scattering theory. Clearly, if one labels the scattering channels by indices a = (a, Aa), where a is a cluster decomposition and .Xa an eigenvalue of Ha for an eigenstate (repeated as usual with its multiplicity), one can introduce the auxiliary Hilbert space := ®a, a0amaxL2(Xa),

auxiliary Hamiltonian

H

®a, a#ama.xA + 2 Da,

and identification operator

J := E Ja, Jaua

.

a ® Ua,

a, a54amax

in such a way that a whole set of channel wave operators can be described with just 11+ := s- lim e'tH Je-'tH t_±00

The scattering operator is then S = SZ+*S2-.

It commutes with H and hence can be decomposed as a direct integral

S=

S(A)dA, o (H)

where S(A) defined A a.e. is the scattering matrix. Yafaev's proof relies on two ingredients: First, one can replace the identification operator J by a better one, J, localizing in the region where propagation takes place. Second, one has to use resolvent estimates of the type II GaR(A ± ie)Gbll E 0(1),

where Ga is a first-order differential operator of the form Ga = x(x)I xI a Va , where x is a cutoff function in a conical neighborhood of Za and Da is the projection

of the gradient V

on the direction orthogonal to xa.

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Related but more detailed results were obtained by Isozaki [73, 74] for threeparticle Hamiltonians. In [73] Isozaki considers three-body Hamiltonians for particles in R3 and looks at scattering matrices with two- to three-particle scattering processes. The scattering matrix has kernels So,a (A; 0, w), 0 E S5, W E

He shows first that So,,, is continuous in all variables when 0 is outside the collision planes. He also derives pointwise asymptotics of S0, a(A; 0, w) when 0 approaches a collision plane Xa,. The singular terms in these asymptotics is related to zero energy eigenfunctions and zero energy resonances for the Hamiltonian H. In [74] related results are obtained for the spatial asymptotics of generalized

eigenfunctions.

Note that Isozaki's results are related to those of Vasy that we will briefly describe below. The difference is in the way one describes the singularity of a given distribution. In Isozaki's approach, asymptotics of the distribution near the points where it is not continuous are derived. Vasy uses the microlocal approach where singularities of a distribution are described by studying its wavefront set.

4.1. Microlocal Approach to Stationary Scattering Theory. In this subsection we will briefly describe the results of Vasy [125, 126, 127] on singularities of generalized eigenfunctions and scattering matrices for N-particle Hamiltonians. Vasy follows the decidedly microlocal approach initiated by Melrose for two-body scattering (see, for example, the book [84]): properties of a distribution are described in phase space and PDE methods like pseudodifferential calculus and propagation of singularities theorems are used. The goal is, of course, to study asymptotic properties for jxj --+ 00 of generalized

eigenfunctions of H, which can be considered as particular cases of distributions u E S'(X) solving

(H-.X)u=0. In discussing asymptotic properties of u when xj -+ oo, it is convenient to introduce a radial compactification of X = Wn to the upper half-sphere Sn. We follow Vasy's notation and denote the points in Euclidean space R' by w: RC : IEBn D W H (1/(1 + w12) 2 , w/(1 + Iw12)') E S.

The boundary at infinity of S+ is Sn-1 = BS+. The images of collision planes Xa, in S+ are denoted by Xa, (extended up to the boundary), and one sets Ca, := Xa, n Sn-1.

Clearly, {Ca}aE.A inherits the semi-lattice structure from {Xa}. The following analog of the sets Za in (2.6) will also be needed: Ca = Ca.\ Ubga Cb.

(4.1)

Extending jwj in lwl < 1 to make it smooth, one gets the boundary defining function x := jwj-1. Various spaces of functions and distributions have natural definitions in this setup. For example, the space of polyhomogeneous symbols of order m E ll S 5(W')

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becomes x-mC°°(S+) (in particular, the space S(W') becomes C°°(S+), the space of C°° functions on S+ vanishing to infinite order on S'-1). Similarly, the image of the weighted Sobolev space (w)-PHm(I[8') is denoted by H-,P(S+). With this notation, the limiting absorption principle takes the form R(A ± i0) : Hk,t (S+) -, Hk+2,1' (S+), 'dl > z , 1' < _I

.

To define a microlocal calculus, one needs essentially three objects: 1) a "garbage space" S, i.e., a space of functions considered as being "smooth." For the N-body problem, this would be the space S(Rn) (note that the notion of smoothness here is decay at spatial infinity). 2) a phase space X (typically the cotangent space of some manifold).

3) a class of functions on X with a quantization rule m H Op(m) sending a symbol m onto a (pseudo-differential) operator Op(m) with a good asymptotic calculus. One requires that Op(m) sends S into itself for all m, and that there is a good notion of "essential support," i.e., that one can make sense of the property that a symbol m "vanishes" at a point x E X. Using these three objects, one can define a notion of wavefront set of a distribution u by saying that a point x E X does not belong to the wavefront set of u if there exist m with m(x) 0 such that Op(m)u E S. The main result one often proves in this setting is the so-called propagation of singularities theorem, which says that if u satisfies Op(m)u E S, for m of "real principal type," then the wavefront set of u is a union of Hamiltonian curves of m in X. It is out of the scope of this review to give a precise definition of these objects,

for which we refer the reader to [125, 126, 127], and we will only indicate the definition of the phase space. The scattering vector fields is the space V. (S+) = xV[b](S+)

where Vb(S+) is the space of vector fields on S+ tangent to the boundary S'-1. If one introduces local coordinates (x, Yi) ... , yn_ 1) on S+ near aS+, then a basis of VSc(S+) is given by the vector fields

x28,x8yj,1<j
scT*s+ = Ua TG, _k" The scattering wavefront set ScWFu of a distribution u is now a closed subset of SC t* S7

On this phase space, Vasy introduces a rather intricate notion of broken Hamiltonian curves, roughly defined as follows: First, Hamiltonian vector fields are scaled

by a factor of lwl. In this way a free trajectory which took an infinite time to go to one point on Sn-1 to its antipodal now does this journey in a time equal to ir.

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In the free region, a broken Hamiltonian curve is simply a Hamiltonian trajectory for this rescaled Hamiltonian. When it hits a collision plane Ca, it is either reflected or can continue along Ca respecting the energy conservation (at that point, of course, eigenvalues of subsystems must be taken into account). The main result of [127] is now that if u c S'(R') is a solution of (H-A)u = 0, then SCWFu is a union of broken Hamiltonian curves.

Vasy also proved a similar result about the wavefront set of the scattering amplitudes. These being distributions on products of spheres, the wavefront set is now the usual wavefront set for distributions on a compact manifold.

5. Resonances in Quantum Mechanics 5.1. Definition of Resonances. Resonance theory is the way to describe mathematically metastable states. The first to rigorously define resonances for rather simple models was Friedrichs [42], using Fredholm theory. This approach was further developed by Howland [63] to include Schrodinger Hamiltonians. A different approach, based on the consideration of a non-self-adjoint operator was used by Lifsic [80] and Grossman [54]. For the wave equation in exterior domains (the so-called acoustic scattering), resonances were defined directly by Lax and Phillips [79] as eigenvalues of the generator of the Lax-Phillips semi-group, or equivalently, as complex poles of the scattering matrix. The Lax-Phillips theory is really adapted to the wave equation: the Hamiltonian has to have 1[8 as its spectrum, and moreover, the Huygens principle

plays an important role. Note that there have been recent attempts [121, 19, 120] to extend the Lax-Phillips approach to Hamiltonians which are bounded from below, like Schrodinger Hamiltonians. However, it seems that meromorphic continuation of the scattering matrix has to be assumed. The class of potentials for which resonances can be defined was greatly enlarged

by the complex scaling method due to Aguilar and Combes [8], and extended by Balslev and Combes [16] to N-particle Hamiltonians. To apply the complex scaling method to a Hamiltonian H, one starts from a unitary group eRA (typically the

group of dilations), such that Ht = eitAHe-i1A can be easily computed. One assumes that for fixed z E C, Imz > 0, the resolvent Rt(z) = (Ht - z)-' can be analytically extended in t to a sector {Argz < Bo} in the upper half plane. If A is the generator of dilations and H = - 2 A+V (x), this requires that V is dilation analytic, i.e., can be analytically extended in such a sector. Using Fredholm arguments, one

can then show that for 0 < Oo < 0, the operator Ro(z) can be meromorphically extended in {-O < Argz}, with poles having finite rank residues. The poles of this extension are, by definition, the resonances of H. The complex scaling method was subsequently generalized to allow for more general potentials, giving rise to the exterior complex scaling (see, for example, [96]), where the generator of dilations A is replaced by a vector field vanishing near the origin, and to the complex distortion method (see, for example, [64, 94]), where the family of deformed Hamiltonians Ht = e'tAHe-i1A is replaced by its first-order approximation Ht = H + it [A, H].

5.2. Resonances as Poles of the S-Matrix. Physicists detect resonances (corresponding to unstable particles) by looking for specific "resonance shapes" in scattering cross-sections. These shapes seen for real energies are thought to be the

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indication of a complex pole of the scattering matrix S(q). To identify poles of S(A) with poles of the resolvent (H - A)-1 is hence an important problem. Note first that in acoustic scattering, where one can apply the Lax-Phillips theory, resonances can be directly defined as poles of the scattering matrix. For two-particle Schrodinger operators, this question was solved in increasing degree of generality by Shenk and Thoe [91], Simon [94], Babbitt and Balslev [12], Jensen [75], Balslev [13], Agmon [6] and Gerard and Martinez [50]. For N-particle Schrodinger operators, only partial results are available. Balslev

[14, 15] proved the meromorphic extension of the scattering matrix below the three clusters threshold and Hagedorn [57] considered the 2-clusters to 2-clusters scattering matrix for four-particle systems. There are also papers by Sigal [92, 95] on the N-particle case. Finally, for long-range potentials, the meromorphic extension of the 2-clusters to k-clusters scattering matrices was shown by Bommier [20].

6. Time-Dependent Picture of Resonances To connect resonances (defined, for example, as poles of the meromorphic con-

tinuation of the resolvent) with metastable states is an important problem which is still the subject of active research. A related problem is the problem of spectral concentration, i.e., to show that a resonance A = E - iF manifests itself by the fact that the spectral density of some state has a peak near the real energy E. It is clear that besides the time parameter (required to tend to oc), some other parameters are needed. In fact, metastable states associated to resonances very far from the real axis would decay so fast as to be unnoticeable. A first choice is to take as extra parameter the coupling constant: here one looks at how eigenvalues embedded in the continuum turn into resonances under a perturbation. This was first studied by Simon [100] where the first rigorous justification of the Fermi golden rule was given: under complex scaling, the continuous spectrum moves out of the way and one can then apply regular perturbation theory to the dilated Hamiltonian Ho. The time behavior of a resonant state was first studied by Skibsted [115, 116], who considered two-body potentials which are sums of a dilation analytic and an exponentially decreasing part. If 0 is a resonance eigenfunction for the resonance A, then uniform bounds are given on the difference e-itH )R0 4'

-

Rt,

where OR = 1{IxI
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where F(g) > 0 is a lifetime, obtained from the Lifsic matrix. A related result was obtained by Hunziker [65], where now dilation analyticity was assumed, and a stronger result: e-itH9,bo)

o,

(

= a(g)e-*)a(9) + O(g2)(1 + itl)-m

was proved, where A(g) is the resonance starting from )o. A purely time-dependent approach to resonances generated by an embedded eigenvalue was developed by Soffer and Weinstein [117], later improved in [28].

They start from the same form Hg = Ho + gV as above, acting on L2([8'). As usual, one assumes that A0 is a simple eigenvalue of Ho embedded in the continuous spectrum with eigenvector ')o. The essential hypotheses are a local decay estimate for the dynamics e-itH0 at energies close to A0: II(x)

-itHO(1

-

Po)(x)-a II E 0(t-r+1), r > 2,

where Po = J'bo)(?Poj, and the non-vanishing of the Fermi golden rule:

F := ir(Voo, b(Ho - X0)(1 - Po) Voo) > 0. From these assumptions, it is possible to deduce several results for g small enough: First, the spectrum of H9 is absolutely continuous near A0 and a local decay estimate holds also for Hg. Moreover, the evolution of any state 0 spectrally localized near Ao can be described as follows: e-itHgo

+ R(t),

= (1 + 0(g)) 1 e-sw(9)t(4'0, 4') Wo + e-itH0

where the complex frequency w(g) is of the form

Wl

w(g) = A0 + g(')o, V O) - g2 (V')o, vp(Ho - A0)-1 VOo) + D(g3),

(here vp denotes the principal value) and the remainder term R(t) satisfies the bounds JJ(x)-°R(t)JJ E O(g), JI (x)-aR(t)JI E

for t > g-2-a.

Related results were also obtained by Merkli and Sigal [86]. The new idea here was to formulate the local decay estimate in terms of the generator of dilations

A=

(x.Dx + Dx.x). 2

This extension allows one, for example, to apply the abstract results to N-particle Hamiltonians. Recently the problem was again considered by Cattaneo, Graf and Hunziker [24]. This time the assumptions are a Mourre positive commutator estimate and some smoothness estimates of H w.r.t. the conjugate operator A. The familiar metastable behavior of a sum of an exponentially decaying term and a polynomially decaying one is obtained for the expectation values (,'o e-itH9X(Hg)Jo),

where X is an energy cutoff supported near A0.

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7. Resonances in the Semi-Classical Limit We will now describe another asymptotic regime where a lot of results have been obtained about resonances, namely, the semi-classical regime. The typical problems studied here are resonances for semi-classical Schrodinger operators:

H = -1 h2 + V (x), on L2

(7.1)

in the limit when h -+ 0, or also resonances in acoustic scattering, where

H=

on LZ(W1\1l),

where Il is a bounded open set such that R'\ul is simply connected, and -AD is the Laplacian with Dirichlet boundary condition on 9Q. For acoustic scattering, the semi-classical limit clearly corresponds to the high energy limit, and this is still so for perturbations of the Laplacian which persist at high energies, such as, for example, perturbation of the free metric in Rn in a compact region. Due to lack of space, we will not discuss several related problems, such as resonances for Laplacians on manifolds, or resonances for other equations of mathematical physics, like, for example, elasticity. We refer the reader to the nice survey article by Zworski [142] for further results in this direction.

7.1. Asymptotic Expansions for Resonances. Let us consider the Hamiltonian in (7.1) assuming, for example, that V is analytic in a domain {z E Cn I RezI > R, IImzI < 61Rez1}, and tends to 0 at infinity in this domain of Cn. Assume

one wants to compute the resonances of H near a real energy level Ao > 0. One expects that these resonances are related to the properties of the classical flow for the Hamiltonian p(x, S) = 2

Z

+ V (X)'

near the energy surface p-1({Ao}). It is surprising that in several non-trivial cases, one can get complete asymptotic expansions of the resonances in some domains of C, typically of the form [A0 - c,.\o + c] + i[-C, 0] or [A0 - e, Ao + E] + i[-Ch, 0], especially if one compares the situation with the corresponding problem for the eigenvalues of a semi-classical Schrodinger operator near a compact energy surface where, in general, one cannot get very precise results. This is due to the fact that the classical flow can be much simpler if the energy surface is not compact. The first rather easy result is the following: if the energy level A0 is nontrapping, then there are no resonances in a fixed (h-independent) complex neighborhood of A0. This was proved by Briet-Combes-Duclos [21] for a more restricted notion of non-trapping, and a proof using the natural non-trapping condition can be given using the machinery of Helffer and Sjostrand [60]. The first case where asymptotic expansions for resonances were computed was the case of shape resonances, where p-1({Ao}) has a compact connected component because of the presence of a potential barrier. This was considered by Combes, Duclos, Klein and Seiler [26], by Helffer and Sjostrand [60] and by Sigal [96]. One obtains resonances exponentially close to eigenvalues of a model problem with a modified potential obtained by filling the outside of the potential well. The constant

So in the e-So/h error term is the Agmon distance between the two connected components of the energy surface. Related to these works is the paper [59], devoted to resonances in a small electric field and the famous Bender-Wu formula.

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Later, complete asymptotic expansions of resonances near A0 were obtained when the set of trapped trajectories in p-1({A0}) is rather simple, namely, either a non-degenerate critical point of p(x, ) (see [22] and [107]) or a closed hyperbolic trajectory [51]. This latter result was extended recently when the space dimension is equal to 2 [110]. For acoustic scattering, the situation corresponding to the one in [51] is the case of two strictly convex obstacles, with a single ray trapped between the two obstacles. This was studied by Ikawa [69] and Gerard [45].

7.2. Resonance Free Domains. A question related to the one considered in the previous subsection is to determine, starting from assumptions on the classical flow on p-1({A0}), some domains free of resonances when h -> 0. The first result of this type is, of course, the result for non-trapping energy levels mentioned above. The corresponding result for acoustic scattering follows from the propagation of singularities theorem of Melrose and Sjostrand [85], and yields absence of resonances in domains {Imz > -Cln(JRezJ), RezJ > R}. For non-trapping obstacles with analytic boundaries, the resonance free domain was extended to

{Imz > -CJRezJ1/3 JRezJ > R} by Bardos, Lebeau and Rauch [18], using the propagation theorem of Lebeau for Gevrey 3 singularities. For several convex obstacles, a strip free of resonances was obtained by Ikawa [70]. A corresponding result for semi-classical Schrodinger operators, assuming the set of trapped trajectories has a hyperbolic structure, was given in [52]. Finally, there is a general result due to Burq [23], valid without any assumption on the classical dynamics. Let

H=

a,(x)(hD,,)a a1<2

be a second-order differential operator with dilation analytic coefficients, tending

to -h2A at infinity (boundary conditions on an obstacle can also be included). Then for any compact interval I C]0, +oo[, there exist e, C, h0 such that H has no resonances in

{z E C I Rez E I, JImzJ < ee-Clh}, for h < h0. The proof of Burq relies on clever Carleman estimates (weighted L2 estimates with exponential weights). For compactly supported potentials in dimension 3, a similar result was obtained earlier by Fernandez and Lavine [41], using a different method.

7.3. Counting Resonances in Balls. The problem of counting the number of resonances in balls B(0, r) when r - oo has attracted a lot of interest since the eighties. To formulate the various results, it is convenient to use in the Schrodinger case the same convention as in the acoustic case: namely, resonances are poles of the meromorphic continuation of (H - A2)-1 rather than of (H - A)-1. This convention is particularly convenient for operators in R1 for n odd, since the resolvent (H - .A2)-1 then extends meromorphically to C. If n is even, the resolvent extends meromorphically to the Riemann surface of log Z.

One then denotes by N(r) the number of resonances (counted with algebraic multiplicity) of H in the ball {IzI < r}.

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The problem started with the result of Melrose [82], who for acoustic scattering in odd dimension n, showed the bound

N(r) < Cr', exhibiting the same behavior as the Weyl estimate for eigenvalues. This was later extended by Zworski [141] to the Schrodinger case with a compactly supported potential, who showed the same bound. In both cases the proof was done by finding an entire function h(z) whose zeros contain the resonances, satisfying a bound lh(z)l <

An application of Jensen's inequality then gives the desired upper bound on the number of resonances N(r). Note that in the one-dimensional case for compactly supported potentials, Zworski [140] obtained an asymptotic formula: N(r) ,,, 2(b - a)

I

r

where [a, b] is the support of the potential V (see also [43, 106, 27] for related results on one-dimensional resonances). A rather different approach was introduced by Sjostrand in [108]. Resonances are considered as eigenvalues of a non-self-adjoint operator P acting on some microlocally weighted spaces (the famous H(AtG) spaces from [60]). To bound the number of eigenvalues of P in a domain, one uses Weyl's inequalities N N

111zzI >_ fµj, i=1

i=1

between eigenvalues zj and characteristic values pj of P (the eigenvalues of P*P2 ). Clearly, a bound from below on the smallest eigenvalue of P*P and a bound from above on the number of eigenvalues of P*P will lead to a bound on the number of resonances. The required bounds are obtained using the Helffer-Sjostrand machinery from [60] and a delicate construction of an escape function, a function G(x, ) such that {p, G} > CI VG12, in W, where W is a neighborhood of the set K of trapped trajectories in p-1({Xo}).

The final result can be formulated as follows: Let M(r, 6) be the number of resonances in {z I Iz - Ao - irI < r + 6}, for Coh < 6 < r << 1. Then one has

M(r, 6) < CV (r, C6)h-n, where V (r, 6) is the volume in phase space of the set


{(x,l;) EW I

In some cases where the flow near the set of trapped trajectories in p-1 [A0 -E, Ao+E]

has a hyperbolic structure, it is possible to improve this bound to

M(r,6) <

CS--12(6/h)-n,

where m E N is a kind of Hausdorff dimension of the set of trapped trajectories.

Later the counting problem was studied in a more abstract framework by Sjostrand and Zworski [111]. In this paper the "black box" setting made its first appearance. The black box framework can be seen as a generalization of the Friedrichs

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model: one assumes that the Hilbert space can be written as R = HR,) ® L2(W'\B(0, Ro)),

where HR,, is an auxiliary Hilbert space. It is traditional to think of 7-1Ro as L2(B(0, Ro)), so that, for example, the operator 1B(o,Ro) is actually the orthogonal projection on 7-1Ro. We will use this slightly disturbing convention also. An abstract operator H acting on 7-l is considered, assuming the following:

1) Hu = -Du if u = 0 near B(0, Ro). 2) 1B(o,R,,) (H + i)-1 is compact. 3) If X E Co (Tl') and X = 1 near B(0, Ro), then for g E Co (R), one has ITr(g(h2H))XI <

where 4)(t) is an increasing function greater than tn. In applications, one gets, for example, 4)(t) = t'16 if H satisfies the hypo-elliptic estimate IkIIs+2e

<- CS(IIHull8 + Ilulis),

where 11.113 is a Sobolev-type norm. Then Sjostrand and Zworski show that

N(r) < CO(Cr), for some constant C and r > 1. The method relies on exterior complex scaling, again defining resonances as complex eigenvalues of a dilated Hamiltonian H0. To be able to count resonances not only

inside cones, it is necessary to also consider a fractional power HH'3 of H9. The bounds eventually rely again on Weyl inequalities. The results in [111] apply to odd or even space dimensions (in the case of even dimension, only resonances with IArgzl < C are estimated). The black box framework was generalized and the proofs simplified by Vodev [129]. The Hamiltonian H does not need to be self-adjoint; it suffices to assume that its resolvent set is not empty. The new idea is to define the resonances as poles of the cutoff resolvent:

Rx(z) = X(H - z2)-1X, where X E Co (Rn) is equal to 1 near B(0, Ro). (The fact that Rx(z) has a meromorphic extension is easy, but one has to show that the poles do not depend on the choice of X.) One has then to make a clever choice of an entire family K(z) of compact operators such that the poles of Rx(z) are among the poles of (1 - K(z))-1. The estimate on N(r) is again deduced from a Jensen estimate on the determinant

det(1 - K(z)). The case of n even was treated in [130], extending earlier results of Intissar [71]. Since the resolvent extends to the Riemann surface of log z, the estimates are done on the counting function N(r, a) = 0{a, I IA, I < R, IArg.j I < a}, when r, a - oo.

7.4. Lower Bounds on the Number of Resonances. The methods used to count resonances in balls, while giving the correct asymptotics in some simple situations (for example, for radial potentials), are still quite crude. Therefore, it was important to also get lower bounds. The first method to obtain lower bounds originates with the trace formula of Bardos-Guillot-Ralston [17] and Melrose [83], valid for obstacle or metric perturbations in odd space dimensions:

Tr(cos(vHt) - cos( Ho)) = Ee'AiItI, in t a;

0,

(7.2)

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where ) are the resonances of H, and Ho is the natural comparison operator (typically the free Laplacian). The proof of this trace formula relies heavily on the Lax-Phillips formalism. For self-adjoint operators with compact resolvent, the similar Poisson formula allows one to get estimates on the counting function by studying the singularity of

the left hand side at t = 0. Here, because the formula is valid only outside t = 0, one has to rely on a possible singularity of the left hand side at a time to > 0 to get a hold on the resonances. This was done by Sjostrand and Zworski in [112]. Their starting point is to start from a locally finite set {.Xj} in C with Im.Xj > 0 and O{Aj

I

lAj I < rl C- 0(r7),

and to consider a distribution u(t) equal to E e'Aj Itl in t

0.

Aj

The microlocal singularities of u near some to > 0 are related to a lower bound on the counting function: Np(r) := O{a; I Imaj < p log jAjI, IA; I < r},

in logarithmic neighborhoods of R. In applications, the distribution u(t) is of course

the left hand side of (7.2). In specific cases, such as, for example, the acoustic scattering with two strictly convex obstacles with Dirichlet boundary condition, or several strictly convex obstacles with Neumann boundary condition, it is possible to compute the singularity of u near some point to and to deduce from it lower bounds on the number of resonances in logarithmic regions. In [113], the Lax-Phillips formalism was combined with the black box setup to extend the applications of the trace formula. Later, great progress was made by Sjostrand in [109], who extended the trace formula to a wide abstract framework containing, for example, dilation analytic long-range potentials. The setup for Sjostrand's trace formula starts with two black box-type Hilbert spaces xi = Hi,Ro ® L2(1Pn\B(0, Ro)) and Hamiltonians Hi i = 0, 1. If A2 is a quantity depending on the index i = 0, 1, then one sets I

[AE] :=AI-Ao. 0

One assumes that in jxj > R0, one has Hi=

lal<2

ai,a(x)(hDx)a, with [a,,a(x) ] E

O((x)-'-')

E > 0.

(7.3)

0

One also considers for R >> Ro two reference operators H , acting on Hi,Ro L2(T\B(0, Ro)), where T is the torus ll /RZT, so that Hi coincide with Hi near B(0, R0) (it is easy to guess how to give a precise meaning to this notion using cutoff

functions). The reference operators H have discrete spectrum and one assumes that they satisfy a Weyl-type law: a(Ha) f1 [-A2, A2] = O(A/h)"

for n2 > n.

It is easy to see using the decay hypothesis (7.3) that for f c Co (]ER), the difference 1

[f (HE) ] 0

is trace class.

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If, in addition, one assumes that the coefficients in (7.3) are dilation analytic in a wedge of angle Bo > 0, one can define the resonances of Hi in the wedge {-20o < argz < eo}. If W is a bounded open set in {-20o < argz < eo} and if one sets 1± = W n l , W_ = W n {Imz < 0}, then for any functions f holomorphic near W and X E Co (1[8) equal to 1 near 1+, one has T!r[(Xf)(HE) ] = [ 0

E

-

AEResHEf(A)nW_

f (h') ] +O(h pEo(HE)nI_

max(no,ni))

0

As an application of this trace formula, Sjostrand shows, for example, in [109] how to deduce existence of resonances from the existence of analytic singularities of some

measure on the real line related to phase-space volume. Let H = -h20 + V(x), where V is a dilation analytic potential, and let v+(E) :=

dx. fV(x)>E}

Assuming that E0 E ll belongs to the analytic singular support of v+, one obtains that for every complex neighborhood W of Eo:

{Res(H) n W} > C(W)h-n, for 0 < h < h(W). Noting that v+ certainly has a singularity at E0 if E0 is a local maximum of V, we see that in this case one obtains a lower bound for the counting function which is of the same order as the upper bound. To conclude this subsection, let us discuss another method to obtain lower bounds on the number of resonances, based on the construction of quasi-modes. Stefanov and Vodev [118, 119] and later Tang and Zworski [123] showed that the existence of quasi-modes implies the existence of resonances close to the quasieigenvalues.

References Adachi, T.: Asymptotic observables for N-body Stark Hamiltonians. Ann. Inst. H. Poincare Phys. Theor. 68 (1998), 247-283. Adachi, T.: Scattering theory for a two-body quantum system in a constant magnetic field. J. Math. Sci. Univ. Tokyo 8 (2001), 243-274. Adachi, T.: On spectral and scattering theory for N-body Schrodinger operators in a constant magnetic field. Rev. Math. Phys. 14 (2002), 199-240. Adachi, T., Tamura, H.: Asymptotic completeness for long-range many-particle systems with Stark effect. J. Math. Sci. Univ. Tokyo 2 (1995), 76-116. Adachi, T., Tamura, H.: Asymptotic completeness for long-range many particle systems with Stark effect, II. Comm. Math. Phys. 174 (1996), 537-559. Agmon, S.: Lectures on Exponential Decay of Solutions of Second Order Elliptic Equations. Lecture Notes in Math. 29, Princeton University Press, Princeton, 1982. Agmon, S.: Spectral theory of Schrodinger operators on euclidean and non-euclidean spaces. Comm. Pure Appl. Math. 39 (1986), 453-516.

Aguilar, J., Combes, J. M.: A class of analytic perturbations for one-body Schrodinger Hamiltonians. Comm. Math. Phys. 22 (1971), 269-279. Amrein, W. 0., Georgescu, V., Jauch, J. M.: Stationary state scattering theory. Helv. Phys. Acta 44 (1971), 407-434. Avron, J., Herbst, I.: Spectral and scattering theory of Schrodinger operators related to the Stark effect. Comm. Math. Phys. 52 (1977), 239-254. Avron, J., Herbst I., Simon, B.: Schrodinger operators with magnetic fields, II. Separation of the center of mass in homogeneous magnetic fields. Ann. Phys. 114 (1978), 431-451.

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LABORATOIRE DE MATHEMATIQUES, UNIVERSITE DE PARIS-SUD, 91405 ORSAY CEDEX, FRANCE

E-mail address: christian.gerardmmath.u-psud.fr

Proceedings of Symposia in Pure Mathematics Volume 78.1, 2007

Mathematical Analysis of Born-Oppenheimer Approximations George A. Hagedorn and Alain Joye Dedicated to Barry Simon in celebration of his 60th birthday ABSTRACT. We review mathematical results concerning Born-Oppenheimer approximations in molecular quantum mechanics.

CONTENTS

Introduction 1. A Historical Introduction 2. The Standard Time-Dependent Approximation 3. The Time-Independent Approximation 4. Propagation Through Level Crossings 5. Propagation Through Avoided Crossings with Small Gaps References

Introduction The goal of this paper is to review rigorous mathematical results concerning Born-Oppenheimer approximations. We make no attempt to cover the enormous physics and chemistry literature on the subject. We begin with a description of the development of the subject in Section 1 without any precise mathematical statements. We present more precise results concerning the standard time-dependent Born-Oppenheimer approximation in Section 2. We describe the time-independent approximation in Section 3. In Section 4, we discuss extensions of the time-dependent approximation to accommodate electron energy level crossings. Finally, in Section 5, we discuss the analogous extensions to accommodate avoided crossings with small gaps. 2000 Mathematics Subject Classification. Primary 81Q05, 81V55; Secondary 92E10, 92E20. Key words and phrases. molecular quantum mechanics, Born-Oppenheimer approximation. The first author is supported in part by National Science Foundation Grant DMS-0303586. ©2007 American Mathematical Society 203

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1. A Historical Introduction In 1927, just one year after the publication of the Schrodinger equation, Max Born and J. Robert Oppenheimer published a truly remarkable paper [5] on molecular quantum mechanics. A year later, Fritz London published a little-known paper [69] that contained the idea that a related approximation should apply to the dynamics of molecules involved in a chemical reaction. The paper [5] is famous, and essentially everything that is known about the structure of molecular bound states relies on the approximation developed in it. By contrast, paper [69] is almost unknown and hardly ever cited, but its ideas are the basis for almost everything that is known about molecular dynamics. Both of these papers exploit the disparity between the masses of nuclei and the masses of electrons. The protons and neutrons that make up nuclei have masses that are 1,836 times the mass of an electron. Using this large mass ratio, the two papers develop the following physical intuition: Because of their much smaller masses, electrons typically move very rapidly compared to the nuclei. So, the electrons can

quickly adjust their state in response to the slow nuclear motion. If they start in their nth bound state, they stay in their nth bound state, determined as though the nuclei were not moving. These electronic bound states and electronic energies depend on the nuclear positions, and the electronic energy levels play the roles of effective potentials for the nuclear motion. Furthermore, because of their large masses, the nuclei obey a semiclassical approximation. From this intuition, Born and Oppenheimer developed a quantitative model for molecular bound states (in which the nuclei could vibrate and the whole molecule could rotate), and London presented a qualitative model for understanding chemical reactions. These approximations agreed well with experimental results, but rigorous verification of their validity was not established for half a century. Mathematicians had missed an enormous opportunity to make a contribution to chemical physics much earlier. Born and Oppenheimer recognized the role of the expansion parameter e, where E4

was the electron mass divided by the mean nuclear mass. They expanded the energy of a molecule formally through fourth order in e and found a zeroth order approximation for the full molecular wave function. Their expansion for the energy was particularly beautiful. The zeroth order term was the electronic energy with the nuclei at an optimal configuration. The first order term was zero. The second order term was the energy of the harmonic approximation to the vibrational motion of the nuclei. The third order term was zero. The fourth order term contained the energies of the rotational motion, anharmonic corrections to the nuclear vibrations, and non-adiabatic corrections to the electron energy. The spectra of small molecules reflect this structure. Electronic transitions are typically in the ultraviolet or visible ranges. The much lower energy vibrational transitions are in the infrared. Even lower still are the rotational transitions that are in the microwave range. Without going into technicalities, we can describe the zeroth and second order terms more precisely. We let the vector X denote the positions of the nuclei in the center of mass frame of reference. The electron Hamiltonian h(X) is then defined as the sum of the quantum mechanical kinetic energy operator for the electrons

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plus the total potential energy of the electrons and the nuclei. It is a self-adjoint operator on the Hilbert space for the electrons that depends parametrically on X. Discrete eigenvalues of h(X) (that are chosen to depend continuously on X) are called electron energy levels.

Born and Oppenheimer assumed that some electronic level E(X) had a local minimum at some point X0. The zeroth order term in their expansion was E(Xo). They then approximated E(X) by its second order Taylor series near X0. This led to an exactly solvable Harmonic oscillator problem that gave the vibrational levels at second order in E. The first rigorous paper in this subject was published in 1973 by Seiler [88]. He verified the formal results of Born and Oppenheimer for an exactly solvable, but unphysical, harmonic oscillator model. A rigorous proof of the validity of the conclusions of Born and Oppenheimer for a physically realistic model was announced [10] in 1975, and the proof was published by Combes, Duclos, and Seiler [11, 12] in 1980. The ideas of London's work were made precise to leading order in e and proved in 1980, under the assumption that the potentials between the particles were smooth [26].

The next developments came roughly five years later. In both the time-independent [28] and time-dependent [27] situations, rigorous expansions were devel-

oped to arbitrarily high order in e under the assumption of smooth potentials. These papers relied on the powerful bookkeeping tool called the "Method of Multiple Scales" from the applied mathematics literature. The technique gave a systematic way of separating adiabatic correction terms from semiclassical correction terms. In retrospect, Born and Oppenheimer had used their brilliant physical intuition to do exactly this separation of terms, one by one, through the fourth order calculation. The high order results led to the question of whether or not one could do the expansions to arbitrary order in e for Coulomb potentials. This was a delicate technical problem because the singularities of the Coulomb interactions between the electrons and the nuclei gave rise to cusps in the electronic wave functions. As a result, many expressions from the smooth potential case did not make sense in the Coulomb case because of a lack of differentiablity. In 1986, Hunziker [48] proved that electron energy levels were analytic as functions of the nuclear positions, even in the Coulomb case. By exploiting his techniques, Born-Oppenheimer expansions to all orders in e were developed for the time-dependent case in [30]. The time-independent expansion was developed to arbitrary order for diatomic molecules in [29], and for general molecules in [66]. These papers were technically very complicated, but the underlying idea was quite simple: Hunziker's results showed that the electron wave function had directional derivatives of all orders in certain directions in the nuclear configuration space. By setting up the expansion in just the right way, one needed only those directional derivatives.

Similar questions have arisen in scattering theory. For situations with smooth potentials, see [62, 83, 64, 84]. For Coulomb potentials, see [65, 54]. Fairly recently, Nenciu, Martinez, Sordoni, Panati, Spohn, and Teufel [75, 81, 82, 91, 92, 94] have developed new approaches to the subject of the timedependent approximation. These papers consider isolated subsets of the spectrum

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of h(X) that may be more general than a simple isolated eigenvalue. They do a more general adiabatic approximation for the electrons, and take the semiclassical limit for the nuclei as an independent step, only when necessary or desired. When the chosen subset of the spectrum of h(X) contains many states, the associated nuclear evolution can be complicated. The underlying innovation of these papers is related to what is done in some practical computations in chemical physics. The semiclassical approximation for the nuclei can lead to larger errors than the adiabatic approximation for the electrons in real systems. Thus, even when considering only one electronic level E(X), some chemical computations handle the nuclei quantum mechanically using the effective

potential E(X). Since there are fewer approximations, this should be expected to yield better results. From a technical point of view, pseudodifferential methods tend to be the most effective tools regarding these issues for providing mathematical justification of these approximations.

The next development in this subject was the study of the effects of electron energy level crossings for the time-dependent approximation. A basic assumption of the standard theory was that if the electrons were in their nth energy level, then this level was isolated from all other electronic levels. The first step in addressing what would happen when this assumption was not satisfied was to classify level crossings. Since the early days of quantum mechanics [95], it has been known that crossings generically occur on submanifolds of various codimensions, depending on symmetry considerations. Generic crossings of electronic levels with the minimal multiplicity allowed by the symmetry group were classified in [32] and [33]. As expected, the most striking distinction between these crossings was the codimension of the set of nuclear positions where the levels crossed. These were codimensions 1, 2, 3, and 5. The second step was to propagate molecular wave packets through each of these crossings. This was first done in [33]. For codimension 1 crossings, the leading order result is that the system simply follows the smooth energy level as though no crossing had occurred. However, a new component of order e can be created that propagates according to the dynamics of the other smooth electronic level as the wave packet moves through the crossing. In the higher codimension cases, the leading order wave packet splits into two components as the system moves through the crossing. One piece follows one electronic level and the other piece follows the other level. In contrast to the codimension 1 cases, the final probabilities for ending up on one surface or the other depend strongly on the detailed structure of the nuclear wave packet just before it hits the crossing. Propagation through crossings can play a significant role in chemical reactions because it provides a mechanism for the electrons to move efficiently from one level to another. For example, a key reaction in human eyesight relies on propagation through a codimension 2 level crossing in the chromophore retinal in the protein rhodopsin. See, e.g., [46] and the references cited there. In the last several years, other approaches to the study of level crossings have been developed. Fermanian-Kammerer and Gerard [16, 17, 18, 19, 20, 21] introduced objects called two-scale measures that resolve the coupling of levels in an 6dependent neighborhood of the crossing submanifold. In subsequent works, Lasser

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and Teufel [67, 68] lifted these results to Wigner functions. These approaches provide a very nice phase space description of propagation through level crossings. In particular, they can accommodate quite general initial conditions at the price of losing information on the phases of the wave function. For example, FermanianKammerer and Lasser [22] describe the time behavior of the wave function's Wigner measure for codimension 2 crossings, and Lasser and Teufel [68] have developed an elegant branching process that describes the splitting of the wave function in the codimension 2, 3, and 5 cases. For an application to the specific molecule Pyrazine, see [23].

Colin de Verdiere [7, 8] has developed another approach that can handle phase

space level crossings that are more general than those that occur in the BornOppenheimer context. This approach is based on the construction of microlocal normal forms for matrix valued symbols in a neighborhood of phase space level crossings. It provides a description of the propagation of WKB-type states as well as semiclassical wave packets through the crossings. A related subject is the motion through "avoided crossings." These are nuclear configurations where two electronic levels approach very close to one another, but do not actually cross. A classification of generic avoided crossings is given in [34]. The authors have studied propagation though avoided crossings with small gaps between levels in [36, 37]. In these papers, two electronic levels are assumed to have an avoided crossing in which the gap between the two levels is 0(e). Under these circumstances, the leading order wave function splits into two pieces, one that propagates on the upper level and one that propagates on the lower level. In case the levels involved in an avoided crossing are known only approximately through perturbation theory in e, Exner and Joye [14] show that the same conclusions hold as well.

Rousse [85, 86] has generalized these results to study the case where the gaps are O(ep) for p in an interval around 1. For p < 1 the leading order term ignores the avoided crossing. For p > 1, the system behaves as though there were an actual crossing. The critical case p = 1 is that studied in [36, 37].

There are recent results about pushing the standard time-dependent approximation even further when the electronic Hamiltonian is analytic in the nuclear configuration variable. In the absence of crossings, the authors proved [40] that by optimal truncation of the time-dependent expansion, one obtains approximations in which the errors behave like CI exp { - C2/e2 }, where C2 > 0. The optimal truncation involves the following: When expanding to order c , the error terms have norms CN fN. By very careful estimation of these errors, one can prove that CN < A B' N!. The exponential estimates are obtained when N is chosen to behave like N - g/e2 for an appropriate choice of g. Descriptions of exponentially accurate results that use pseudodifferential operator techniques can be found in [75, 76, 81, 91]. Under appropriate hypotheses, results with exponentially accurate error bounds have been proved in the context of stationary scattering theory. See, e.g., [70, 60, 58, 2]. For some very special systems, the large time asymptotics of the leading order exponentially small non-adiabatic correction terms for solutions to the timedependent Schrodinger equation have been determined in [42]. These transitions

208

G. A. HAGEDORN AND A. JOYE

are associated with avoided crossings with a fixed gap as E -> 0. The transition amplitudes and the momenta of the nuclei after the transitions are larger than what one might naively guess. Also the form for the nuclear wave function is not what one might naively guess in some cases. In [42], the time development is hidden. A very nice open problem is to understand how these non-adiabatic terms arise as a function of time. Although this has not been solved, preliminary steps towards the solution can be found in [41, 3, 4]. In the last decade, there have been more results concerning the time-independent

approximation. Sordoni [89] has examined highly excited rotational states of diatomic molecules. Rousse [86, 87] has studied highly excited vibrational states when the nuclei have one degree of freedom. In this case, a Bohr-Sommerfeld condition determines the correct vibrational levels for the nuclei. Rousse holds the energy fixed as c --> 0, so the nuclei are in vibrational state n, where n = O(E-2). Very recently, Hagedorn and Toloza have applied optimal truncation techniques to the time-independent Born-Oppenheimer approximation. They did this first [44] for a toy model with just two electronic levels and the nuclei with one degree of freedom. They generalized this to fairly realistic electron Hamiltonians [45], but with the nuclei restricted to one degree of freedom.

In the last twenty-five years, many other interesting developments have occurred in closely related areas.

Resonances. In traditional two-body scattering, tunnelling effects often give rise to resonances. In molecular systems there are more such mechanisms. For example,

non-adiabatic transitions by the electrons can allow a system to escape from a well. This leads to the theory of predissociation. Klein [63] studied predissociation

in 1987. Martinez studied resonances and proved that in many instances, they had exponentially long lifetimes [71, 72, 73, 74]. Resonances associated with codimension 2 level crossings have also been studied by Nedelec [77, 78, 79, 80] and Fujiie, Lasser and Nedelec [24].

Resolvent Estimates. Many detailed results on scattering matrices and cross sections depend on (weighted) estimates for the resolvent of the Hamiltonian near

its spectrum. For molecular systems, Jecko [49, 50, 51, 52] has studied these estimates in a variety of circumstances, including situations related to level crossings. These papers construct conjugate operators in the sense of Mourre for the Hamiltonian, which is a challenge when the levels may cross.

Molecules Interacting with Laser Pulses. With the development of ultrashort laser pulses, enormous amounts of information about molecular dynamics can be obtained experimentally. Much of the theory related to these experiments relies on the idea that electrons absorb photons on such a short time scale that the nuclei do not have time to move. Jilcott [55, 56] has made this idea precise and has proven its validity. Hagedorn, Rousse, and Jilcott [43] have corrected a calculational error in [55, 56] and have examined transition amplitudes to different vibrational levels for the nuclei when a laser pulse causes an electronic transition. Electronic transitions typically excite 0(E-1) different vibrational states. The paper

MATHEMATICAL ANALYSIS OF BORN-OPPENHEIMER APPROXIMATIONS

209

[43] rigorously calculates the Franck-Condon factors that govern the strengths of the associated spectral lines.

Results related to Berry Phases.

Many papers on Born-Oppenheimer approximations assume the chosen electronic level E(X) has multiplicity 1, and that the phases of the electronic eigenstates can be chosen in accordance with the "adiabatic connection" on the bundle of eigenstates over the nuclear configuration manifold. When this cannot be done, the system displays Berry phases. Herrin and Howland [47] examine a diatomic model that has a Berry phase associated with rotations of the whole molecule. They observe that the rotational energy levels are altered in this case. In related work, Faure and Zhilinskii [15] demonstrate a precise link between geometrical properties of electronic eigenspaces and the structure of the spectrum of the full molecular Hamiltonian.

Time-Independent Information about Crossings and Avoided Crossings. Time-independent techniques can yield information about transitions between levels associated with avoided crossings. For example, in models where the nuclei have only one degree of freedom and the electronic Hilbert space is finite dimensional,

the time-independent Schrodinger equation is equivalent to a system of ODE's. In this representation, the e - 0 limit looks formally like an adiabatic problem where X has become the time variable. The situation is more complicated than the standard adiabatic problem because the resulting Hamiltonian is not self-adjoint. However, in some situations, exponentially small scattering amplitudes associated with avoided crossings can be calculated quite explicitly. See, e.g., [70, 60, 57, 58, 9, 61]. Also, solutions to the time-independent Schrodinger equation near codimension 2 crossings have been found in [25].

Chemistry without the Born-Oppenheimer Approximation? One can ask whether or not certain fundamental concepts used in chemistry can be formulated without the Born-Oppenheimer approximation. For example, when there are isomers and/or identical nuclei in the system, do chemical structure diagrams make sense if the nuclei and electrons are simply treated as quantum particles? For a recent discussion of this topic, see the review article [93].

2. The Standard Time-Dependent Approximation In this section we describe the coherent state approach to the time-dependent Born-Oppenheimer approximation in some detail. This approach yields a rather simple, complete description of the molecular wave function, although it requires special initial conditions and the semiclassical approximation for the nuclei. The more sophisticated approaches described in Section 1 can allow more general initial nuclear states and/or avoid the semiclassical treatment of the nuclei. However, many of them study Wigner distributions on phase space rather than wave functions for the nuclei. To make precise statements, we need some preliminary notation and results concering semiclassical wave packets for the nuclei. We have tried to minimize the technicalities, but some are unavoidable.

G. A. HAGEDORN AND A. JOYE

210

2.1. Semiclassical Wave Packets. The "coherent state" approach to semiclassical quantum mechanics makes use of generalizations of the usual harmonic oscillator eigenfunctions. These wave packets are described in detail in [35]. Although the notation is different, they coincide with "generalized squeezed states" [13].

In d dimensions, these states are cps (A, B, h, a, 77, x), where j is a d-dimensional multi-index, A and B are complex d x d matrices, a and 77 are d-dimensional vectors, and x is the d-dimensional variable. We always assume A and B satisfy Bt A

-

At B =

0,

and

B* A + A* B = 21, where the superscript t denotes the transpose. The first of these conditions is equivalent to B A-1 being symmetric, i.e., (real symmetric) + i (real symmetric). The second condition is equivalent to (Re BA-1)-1 = AA*. We let JAI denote the positive square root of A A*. When the multi-index j equals zero, these states are the complex Gaussians 7r_ d/4 hd/4 [det A] -1/2 Po(A,B,h,a,rl,x) x

(x - a)

ex

a represents a classical position, and q represents a classical momentum. The position uncertainty is given by the covariance matrix h1/2 JAI and the momentum uncertainty is given by h1/2 JBI. The choice of square root for [det A] -1/2 is arbitrary, but is chosen to depend continuously on time when A depends on time.

For non-zero values of j, the easiest way to define the semiclassical wave packets is to introduce raising operators [35]. These act on Schwartz functions by

( A. (A, B, h, a,q) * 0)(x)

-

,(

d

d

B(nm(xn_an)

-iE

Anm(-iht

n

n=1

- 77n)

(x)

For any non-zero multi-index j, we define

wj (A,B,h,a,17, ) _

(Al(A,B,h,a,rl)*)j1

x

(A2(A,B,h,a,77)*)j2

... (Ad(A,B,h,a,q)*)Jd coo (A,B,h,a,rl, - )

For fixed a, r7, A, and B, these functions form an orthonormal basis of L2 (lRd),

indexed by j. Numerous properties of these functions are discussed in [35]. The main property that we need is that modulo an 0(hl/2) error, eis(t)/l p (A(t), B(t), h, a(t), r7(t), x) satisfies the time-dependent Schrodinger equation ifit

00

2

_

-2AO+V(x)O

(2.1)

MATHEMATICAL ANALYSIS OF BORN-OPPENHEIMER APPROXIMATIONS

if V E C3(Rd) is bounded below, a(t)

V (X)

=

211

< C eMX2 , and

ra(t)

(2.2)

- (VV)(a(t))

(2.3)

77(2 )2

S(t)

_

A(t)

= i B(t)

-

V (a( t ))

B(t) = i V(2) (a (t)) A (t).

( 2 . 4)

(2.5) (2.6)

z 0V

Here V(2) denotes the Hessian matrix . The solution to (2.2)-(2.6) is axi axk completely determined by the classical phase space flow generated by the potential V.

By taking time-dependent superpositions of the eiS(t)/h oi (A(t), B(t), h, a(t), rl (t), x)

(with finitely many different j) this result can be improved to give O(h/2) accurate approximations for any m if V E Cm+3(1[84). When V is analytic, these results can be improved by using optimal truncation techniques to produce approximations with O(e-r/h) errors [38, 39]. Similar results on the propagation of observables can also be found [6].

2.2. Electron Energy Levels and Phases of Eigenstates. A basic assumption of the standard Born-Oppenheimer approximation is that the electron Hamiltonian h(X), acting as a self-adjoint operator on the electronic Hilbert space flel, depends smoothly (in the strong resolvent sense) on X for all X in some connected open set X C IIBd. We further assume that h(X) has an isolated nondegenerate eigenvalue E(X) that depends smoothly on X E X. The function E(X) is called an electron energy level. For simplicity in this discussion, we assume that the eigenvector 4D(X) corresponding to E(X) can be chosen so that

(4b(X), v VX -D(X)) = 0

(2.7)

for any vector v. This condition is satisfied if X has trivial homology groups and h(X) commutes with a conjugation for all X E X. In this case (2.7) is satisfied if 4)(X) is chosen to equal its conjugate. In more complicated situations with Longuet-Higgins phases or Berry phases, we can use a time-dependent choice of phase of 4)(X) to satisfy the appropriate analog of (2.7).

2.3. The Multiple Scales Technique for the Time-Dependent Approximation. One way or another, all the mathematical results on the time-dependent Born-Oppenheimer approximation make use of two length scales in the nuclear variables. The authors' favorite way of doing this is to use the method of multiple scales.

The choice of time variable for the time-dependent Schrodinger equation can be made so that the nuclear motion has a non-trivial limit. After this and some trivial

G. A. HAGEDORN AND A. JOYE

212

resealing of the nuclear variables, the equation we wish to solve can be written as i E2

_ - 2 OX IF + h(X) IY,

at

(2.8)

where T E L2 (Rd, H'1) Since E2 appears in (2.8) precisely where h appears in (2.1), we make use of the semiclassical wave packets. The chosen electron energy level E(X) plays the role

of the potential in the equations (2.2)-(2.6) in the Born-Oppenheimer situation. The method of multiple scales replaces the nuclear configuration variable X by two variables

x = X and y = (X - a(t))/E, where a(t) describes the classical configuration for the nuclei. Rather than attempting to solve (2.8) directly, we first study solutions W(x, y, t) to

2 0x

i E2

+

-

E3 Vx Vy W

- 2 AY W

(2.9)

[h(x) - E(x)] W + E(a(t) + Ey) W.

Then we obtain a solution q i(X, t) to (2.8) as T (X, t) = ih (X, X-a(t), t) Anticipating the same dynamical phases as in the semiclassical wave packets, we look for solutions to (2.9), of the form

W (x y t) =

eiS(t)/e2

X(x y t).

Then X(x, y, t) must satisfy 4

i E2

at

2 Ox - E3 Vx . Vy

2

- 2 AY - i E277(t) Dy

+ [E(a(t) + Ey) - E(a(t)) - E (VE)(a(t))

+ [h(x) - E(x)],

y]

X

We substitute a formal expansion

X(x, y, t) = 00(x, y, t) + E01(x, y, t) + E2 02 (X, y, t) + ... into this equation, expand E(a(t)+cy) in its Taylor series in powers of c, and equate terms of equal powers of E on the two sides of the resulting expression. Order 0. The O(Eo) terms require

[h(x) - E(x)] 'o (x, y, t) = 0. Thus,

00 (X, y, t) = go (X, y, t) 4) (x), where go is, as of yet, not determined. Order 1. The O(E1) terms require

[h(x) - E(x)] &1(x, y, t) = 0. Thus,

'b1(x, y, t) = g1(x, y, t) 4(x), where gl is, as of yet, not determined.

MATHEMATICAL ANALYSIS OF BORN-OPPENHEIMER APPROXIMATIONS

213

Order 2. The O(c2) terms are much more complicated. They require i'

o = - 2 Ay )o +

00 - i77(t)

E(2) (a(t))

-V

2

where the dot 2,

o + [h (x) - E(x)] 02, 22

denotes the time derivative, and E(2) (a) denotes ax,, (a)yzyj. The utility of the multiple scales technique now becomes evident because we can separate this equation into parts. The first part consists of all components that are in the (D(x) direction in the electronic Hilbert space. The second part consists of all components that are orthogonal to (D(x) in the electronic Hilbert space. Equality must be satisfied for both of these components. The two equations we obtain are

zt

i 90 + i q(t) - vxgo = - Ay 9o + E(2) (a (t)) 2

go

(2.10)

2

and

[h(x) - E(x)] 02(x, y, t) = i9o(x, y, t) rl(t) . (VA) W .

(2.11)

For any j, equation (2.10) is solved by

go(x, y, t) =

E-d/2 e2s(t)IE2

Oj (A(t), B(t),1, a(t),77(t), y),

where E(X) plays the role of the potential in determining the solution to (2.2)-(2.6). Note that these solutions have no x dependence, and if we substitute (X - a(t))/e for y, they are normalized in L2(lRd). We henceforth assume that each such function g,,, is independent of x. This removes some of the ambiguity of expressing functions of X in terms of either x or y. To solve (2.11), we note that [h(x) - E(x)] is invertible as an operator on the orthogonal complement of the span of 4)(x). We denote this inverse by r(x), and then,

W2(x, y, t) = 4'2 + wz, where

0z = igo(y, t)r(x)rl(t). (Vxlb)(x) and

= 92(x, y, t) 'D (x), with 92(X, y, t) not yet determined. Order n > 3. The higher order terms are dealt with in the same way, although the technical details are more complicated. We project the O(en) equation into the 4b(x) direction and into the direction perpendicular to D(x) in the electronic Hilbert 0121

space. In the 4D(x) direction, we determine gn_2 and hence Onl _2 by using techniques

associated with the semiclassical wave packets. In the orthogonal direction we employ the reduced resolvent operator r(x) to determine O,- . Thus, the multiple scales bookkeeping allows us to separate terms that we treat by semiclassical techniques from those that we handle by adiabatic techniques. For the details, see [27, 31, 33].

After doing these formal calculations, a rigorous proof of the validity of the time-dependent approximation is a consequence of the following lemma. We apply

G. A. HAGEDORN AND A. JOYE

214

this lemma to the function N

t)

E En n=0

Wn (X' X Ea(t)

EN+1

i

N+1

t

X - a(t)

(X

t)

1 + EN+z N+2 (XI

X - a(t) E

t\ /I

to obtain the result with an 0(EN+1) error. LEMMA 2.1. Given a molecular wave function WE (X, t), define

(E(x, t) = ie2

8tE(X,

t) +

2

(AxWE)(X, t)

h(X)`I'6(X, t).

If III(, t) 11 < tz (E, t), then WE(X, t) differs in norm from the exact solution to (2.8) with initial condition WE(X, 0) by at most fo µ(E, s) ds/E2.

When h(.) satisfies an analyticity condition we can improve these results further by employing an optimal truncation technique. We obtain a time-dependent approximation with an error bound of the form 0 (exp(-F/E2)) by using the above construction and choosing N - g/E2 for an appropriate choice of g. The technical details can be found in [40].

3. The Time-Independent Approximation We can use the Method of Multiple Scales to construct quasimodes and quasienergies for the time-independent Born-Oppenheimer approximation also. The results of [11] show that under appropriate hypotheses, these quasienergies correspond exactly to the low-lying eigenvalues of the full Hamiltonian. For simplicity, we present the formal calculations near an isolated minimum of an electron energy level. In most cases of interest, after the removal of the center of mass motion of the molecule, there is still a two or three dimensional rotational symmetry that causes the minimum to be attained on a non-trivial manifold (isomorphic to the two-sphere S2 or the full rotation group SO(3)). This physical situation is technically more complicated because degenerate perturbation theory is required.

In our example, we assume that an electron energy level E(X) has a local minimum at some point X = a with the second derivative matrix E(2) (a) strictly positive. Again, to avoid degenerate perturbation theory, we assume the square roots Wk of the eigenvalues of this matrix are not rational multiples of one another. We then introduce the two variables

x=X

and

y = X-a E

We then look for approximate solutions to the equation

-

4

2

0x

-

E3 VX vv

-

2

2

A YT + [h(x) - E(x)]xF

00

+ E(a + Ey) W + E E' (Tn(x) - Tn(a + Ey))

_ E(E)

n=4

The operators Tn(.) will be chosen during the perturbation calculations so that certain functions will be independent of x.

MATHEMATICAL ANALYSIS OF BORN-OPPENHEIMER APPROXIMATIONS

215

We now substitute formal series expansions

=

£(E)

co

+ E £1 + E2 £2 + .. .

= ')o + E 7p1 + E2 02 + .. . into (3.1). We then expand E(a+Ey) and T, (a+Ey) in their Taylor series in powers of E and equate terms of the same orders on the two sides of the resulting equation.

Order 0. The order co terms require [h(x) -E(x)] i,bo(x, y) + E(a)'zlio(x, y) _ £o'bo(x, y) This equation has solutions

£o = E(a) and

00(x, y) = go (X, y) -D (x),

where go is not yet determined. We later choose T4(x) so that go has no x dependence.

Order 1. After some cancellations, the order el terms require [h(x) - E(x)]V)1(x, y) + (VE)(a) - y'bo(x, y) = £1' o(x, y). Since E(x) has a local minimum at a, (VE)(a) = 0. Thus, we learn that and

'b1 (x, y) = 91(x, y) 4) (x),

where 91 is not yet determined. We later choose T5 (x) so that gl has no x dependence.

Order 2. After some simplification, the order E2 terms require [h(x) - E(x)] 'b2(X, y) -

2

(AvOo)(x, y) + E(2)(a)

'bo(x, y) = £20o(x, y)

2 the direction orthogonal We project this equation into the span of 4)(x) and into to 4)(x) in the electronic Hilbert space to obtain two conditions:

- 2 (A'go) (y) + E(2) (a)

90 (y) = £2 90

2 [h(x) - E(x)] V)2(X, y) = 0. The first of these equations is a harmonic oscillator problem. We obtain go(y) as product of Hermite polynomials times a Gaussian, and and

d

£2 =

k1

2 (flk+)

wk,

where each vibrational quantum number nk is a non-negative integer. The second equation simply tells us that

'b2(x, y) = 92(x, y) 4)(x). We will choose T6(x) so that 92 has no x dependence.

G. A. HAGEDORN AND A. JOYE

216

Order 3. The order d terms require 2

1

[h(x) - E(x)] V)3 + (- Dy + E(2,(a)2

£2)

O1

- (Vy90) (Vx41)(x) + E(3)(a) 3, *0 =

£3 4'0

We again project into the 4)(x) direction and the perpendicular direction in the electron Hilbert space. For the terms in the (D(x) direction, we also project into the go (y) direction and the perpendicular direction in L2 (Rd, dy). Thus, we get three conditions. From the terms orthogonal to (D(x), we have

[h(x) - E(x)] b3 = (Vygo) . (Vx"b)(x) By applying the reduced resolvent operator r(x) for [h(x) - E(x)], we obtain Y)3(x, y) = 93(x, y) 4 (x) + r(x) (Vygo) . (Vx(D)(x) We will choose Ta(x) so that 93 has no x dependence. Since any homogeneous cubic

polynomial in y times go(y) is orthogonal to go(y), the terms in the direction of go (y) (D (x), require

£3=0. The terms that are multiples of D(x) but orthogonal to go(y) require C- 2 Ay

+ E(2) (a) 2 - £2 91 = - E(3) (a) 3,

go.

We do not precisely normalize our quasimodes, so we can take gl orthogonal to go. We then solve the equation for g1 by applying the reduced resolvent operator for 1A y + E(2) (a) 2 £2) .

-

Order 4. The order e4 terms require

/

2

1

[h(x) - E(x)] I4 + I -20y + E(2)(a)2 - £2 I

'b2

- (Vy91) (Vx'))(x) - 2 go (y) (Ax 4)) (x) + E(3)(a) 3, V)1 + E(4)(a) 4i Vo + T4 (x) V)o - T4 (a) Oo = £4')0.

We break this into three equations by taking the same projections as for the order E3 terms. The multiples of go (y) (x) yield - 2 (,D(x), Ox

(9o(y), E(3)(a) 3, 9i(y))y + (go (y), E(4)(a) 44go(y))y

+ T4(x) - T4(a)

We can solve this equation by taking

T4(x) =

2

(,D(x), Ox(b(x))R.

£4.

MATHEMATICAL ANALYSIS OF BORN-OPPENHEIMER APPROXIMATIONS

217

and E4 = - T4(a) + (go (y), E(') (a) 3,91(y))y + (go(y), E(') (a) 4,9o(y))y

We solve the other two equations by applying the reduced resolvents of [h(x) -E(x)] and (- 2 Ay + E(2) (a) E2) as we did for the order E3 terms. 22

-

Order n > 5. At higher orders, we simply mimic what we did for the order E4. The order 0 terms determine T,,,(x), 9, the component of Eli,, orthogonal to fi(x), and g.-2(Y).

We do these formal calculations to any order n > 2. We then drop the terms in the wave functions and energy that _have not yet been determined. Next, we replace x by X and y by (X - a)/E in to obtain the quasimode T,. We easily check that O(En'+1), II H(E)', - E(E)1E where H(E) is the full Hamiltonian. The spacing between levels is O(E2) for our simple model, so this yields non-trivial spectral information. Hagedorn and Toloza attempted to apply optimal truncation techniques to this expansion, but could not get the required estimates because of difficulties with the T",(x). By abandoning the multiple scales technique and working just in the y variable, they succeeded in applying optimal truncation techniques to a different expansion. By doing so, they obtained quasimodes with errors of order exp(-r/E2) for some models [44, 45].

4. Propagation Through Level Crossings We present fewer details in this section because the subject is more technical.

Again, we present a summary of the authors' work on the subject. The other approaches deal with Wigner distributions on phase space instead of the wave functions and rely heavily on pseudodifferential operator techniques and two-scale measures. Before studying propagation through level crossings it is necessary to study the structure of different kinds of level crossings. This subject is studied in [32] and Chapter 2 of [33], which we now briefly summarize without any proofs. Symmetry considerations are central to the classification of different types of crossings. In quantum mechanics, symmetry operations may or may not involve time reversing operators, which are anti-unitary (i.e., they have all the properties of unitary operators, except that they have U(AO) = .X U(O) instead of A U(b)). Thus, standard group representation theory is not sufficient, and one must allow "corepresentations" which involve linear and anti-linear operators.

The relevant symmetry group G is the set of all X-independent unitary and anti-unitary operators on the electron Hilbert space that commute with the electron Hamiltonian h(X) for all X. Each electron energy level is associated with an irreducible representation (or corepresentation) of G. Minimal multiplicity irreducible representations are one dimensional, and minimal multiplicity irreducible corepresentations are either one or two dimensional. In standard situations, the two dimensional irreducible corepresentations occur when electron spin is taken into account and the molecule has an odd number of electrons. We henceforth consider

G. A. HAGEDORN AND A. JOYE

218

only electronic levels of the minimal multiplicity allowed by the representations or corepresentations. Generic crossings of levels associated with inequivalent irreducible (co)representations are the simplest. The two eigenvalues are simply two different functions of X, and they generically take equal values on a codimension 1 submanifold. Near the crossing, the electronic eigenfunctions assoicated with the two levels can be chosen

to depend smoothly on X. So, in a basis that depends smoothly on X, the two levels are described by a diagonal matrix. In [32, 33], these are crossings of types A, C, D, E, F, G, and H. Crossings of type A involve eigenvalues corresponding to inequivalent representations. The others come from eigenvalues that correspond to inequivalent corepresentions, and irreducible corepresentations, themselves come in three different types. The situation is much more interesting when two levels associated with equivalent (co) representations cross. In [32, 33], these are crossings of types B, I, J, and K, and the manifolds where the levels are equal generically have codimensions 3, 2, 3, and 5, respectively. The type B situation occurs when there are no time-reversing operators in the symmetry group. Near the crossing, the behavior of the two levels is similar to that of the general traceless Hermitian matrix C

X1

X2 + iX3

X2 -ZX3

-X1

/

'

whose eigenvalues are ± Xi + X 22+ X3 . The eigenvectors cannot be chosen to depend smoothly on X and have non-trivial Berry phases. The type I situation comes from a one dimensional corepresentation. The behavior of the levels near the crossing is similar to that of the general traceless real symmetric matrix X1 C

X2

X2 _X1

'

whose eigenvalues are ± X1 + X2 X. The eigenvectors cannot be chosen to depend smoothly on X, and they exhibit a degenerate Berry phase that is frequently called a Longuet-Higgins phase. If the eigenvectors are locally chosen to be real, then they pick up a minus sign as the parameter X is moved around the crossing manifold. The chemists use the term "conical intersection" for these crossings. For our specific example, the plot of the eigenvalues with (X1, X2) horizontally and energy vertically yields the cone E2 = X1 + X2 with the crossing at the vertex. The type K crossings come from a two dimensional irreducible corepresentation.

Their local behavior is like the direct sum of a type B matrix and its conjugate. The canonical example is X1

X2 + 2X3

X2 - 2X3

-X1

0

0

0

0

0 0

0

X1 X2 + ZX3

X2 - 2X3

0

-X1

whose eigenvalues are + Xi + X2 + X3 and are doubly degenerate.

MATHEMATICAL ANALYSIS OF BORN-OPPENHEIMER APPROXIMATIONS

219

Type J crossings occur on codimension 5 submanifolds. The canonical example

is the matrix X1

0

X2 + 2X3

X4 + 2X5

0

X1

-X4 + iX5

X2 - iX3

0

-X1

X2 - 2X3 -X4 - iX5 X4 - iX5

X2 + 2X3

-XI

0

whose eigenvalues are Xi + X2 + X3 + X4 + X5 and are doubly degenerate. The bundles of eigenvectors have a "non-Abelian Berry phase" that is given by a

2 x 2 unitary matrix instead of a complex number. The complicated geometrical structure of these bundles is described in [1]. The main results of [33] describe what happens when a standard time-dependent Born-Oppenheimer wave packet encounters any of the types of generic, minimal multiplicity crossings. At codimension 1 crossings, nothing interesting happens to leading order in E. The zeroth order wave packet follows the smooth electron energy level as though the second level were not there. Depending on the behavior of the derivatives of the electron states with respect to the nuclear coordinates, a term of order e associated with the second level can be created as the zeroth order term passes through the crossing. All other levels of h(X) are involved only at second order in e, so they are irrelevant through first order. The higher codimension crossings cause much more significant effects because the electronic states are singular near the crossings. In [33], propagation through these crossings is studied by using matched asymptotic expansions. The analysis is very technical, but the underlying idea is quite simple. One proves that there exist two numbers e and ' that satisfy 3 < ' < < 1, such that all of the following results can be proven. An incoming standard Born-Oppenheimer state away from the crossing is chosen so that its associated classical position a(t) hits the crossing submanifold at some time that we take to be t = 0. The leading order standard approximation

agrees with an exact solution up to errors of order Pi for some p1 > 0 for all t E [-T, -e"), for all n < e. Next, an "inner expansion" is derived that agrees with an exact solution up to errors of order eP2 for some p2 > 0 when Itl < X. This wave packet is arranged so that it agrees with the incoming wave packet up to errors of order eP3 with p3 > 0

for times t = -e" with ' < /, < 6 where both expansions are valid.

For small positive times e" with ' < is < , a similar matching is done to outgoing standard time-dependent Born-Oppenheimer states that have errors of order P' for t E (e", T], when is < . So, for any time t E [-T, T], at least one of the three expansions is valid to leading order in E. The inner expansion is obtained by replacing the original time variable t by a rescaled time s = t/e and replacing the original variable X by y = (X - a(t))/E. In the new variables, a formal asymptotic expansion is substituted into the Schrodinger

equation. The leading order terms satisfy a hyperbolic PDE that has very simple characteristics. Along each characteristic, it can be solved explicitly in terms of parabolic cylinder functions of complex order and complex argument. The behavior of these special functions is known well enough to yield a proof that the matching can be done rigorously.

G. A. HAGEDORN AND A. JOYE

220

When a wave packet propagates through these crossings, it splits at leading order, e0, into two separate wave packets. One piece propagates according to the dynamics of one electronic level involved in the crossing. The other piece has dynamics governed by the second level. These separate wave packets have complicated

phases, but their absolute values can be found in a relatively simple way because the leading order equation for the inner solution is hyperbolic. Along each characteristic one finds that a Landau-Zener formula dictates the probability to end up on the upper or lower of the two levels. However, there is a different minimal gap between the electronic levels felt along different characteristics. The parts of the wave function that go right through the crossing manifold feel no gap, and they follow the smooth electronic level. If they start on the lower level, they go to the upper level, and vice versa. The pieces of the wave function that stay far from the crossing (on a distance scale of order e) obey the adiabatic theorem and stay on the same level because they feel a relatively large gap. The Landau-Zener formula gives the precise result for intermediate gaps. The techniques underlying the crossing results of Fermanian-Kammerer, Gerard, Teufel, and Lasser [16, 17, 18, 19, 20, 21, 22, 23, 67, 68] rely on two-scale measures to resolve the behavior of solutions when they are a distance of order e from the crossing submanifold. By concentrating on Wigner functions instead of wave functions, these techniques yield somewhat cleaner results because they do not yield phase information that is contained in the wave packets. Lasser and Teufel have developed a very nice algorithm for the propagation of Wigner functions through crossings.

5. Propagation Through Avoided Crossings with Small Gaps The authors have also studied what happens when wave packets go through avoided crossings with small gaps. An avoided crossing is a situation where two levels get close to one another, but do not actually cross. The expression "small gaps" refers to levels that come an O(e) distance from one another. For this small gap situation, the electron Hamiltonian must depend on E. That may appear to be unphysical, but for real molecules, there is a fixed value of e, and levels can have avoided crossings that have gaps that are roughly equal to C. Gaps on the order of e are the most interesting. As commented in Section 1, Rousse [85, 86] has shown that avoided crossings with gaps of order eP with p < 1 yield give rise to trivial phenomena to leading order. When p > 1, they behave like actual crossings. As in the case of crossings, there are several different types of generic, minimal multiplicity avoided crossings. They are classified, and normal forms are found for

each type in [34]. In that paper, avoided crossings are defined as crossings that have been perturbed in a generic way to produce avoided crossings. Canonical examples of electron Hamiltonians of each type with gaps of size 2j are given by the following matrices:

Type 1

E(X1) = f Xi

62

(multiplicity 1). 6

CXl 6

-X1

MATHEMATICAL ANALYSIS OF BORN-OPPENHEIMER APPROXIMATIONS

221

Type 2 E(X1) = + Xi + 62 (multiplicity 2). X1

8

8 0 0

-X1 0 0

0 0

0 0 8

X1

-X1 1

8

Type 3 E(X1i X2) = ±,/X? + X2 + S2 (multiplicity 1). C

X1

X2 + i8

X2 - i8

-X1

Type 4 E(X1 i X2) = ± X1 + X2 + 62 (multiplicity 2). X1

X2 + i8

0

0

X2 - i8

-X1

0

0

0 0

0 0

X2 - i8 X2 + i8 -X1 X1

Type 5 E(X1i X2, X3) = f Xi + X2 + X3 + S2

(multiplicity 2).

0 X1

X2 + 2X3

8

X2 -ZX3

X2 - 2X3

-6

-8 -X1

0

8

X2 + ZX3

0

-X1

X1 0

Type 6 E(X1, X2, X3, X4) _ ± VI'X2 + X2 + X3 + X4 + 82 (multiplicity 2). X1

0

0

X1

X2 - iX3 -X4 - i8 X4 - i8 X2 + 2X3

X2 + 2X3

X4 + i6

-X4 + i8 X2 - iX3 -X1 0 -X1 0

Papers [36, 37] study the propagation of wave packets through each type of The underlying basic idea is again avoided crossings when 8 is taken to be to use matched asymptotic expansions as in the case of actual crossings. Although the analysis is quite similar, there is the additional technical complication that the classical mechanics for the nuclei now depends on c in a complicated way as e -+ 0. The results for each type of avoided crossing are somewhat similar to those for codimension 2, 3, and 5 crossings. As the wave packet propagates through the avoided crossing, it splits at leading order into two pieces. One piece propagates according to the dynamics of the upper level and the other piece propagates according to the dynamics of the lower level. The leading order inner solutions again involve hyperbolic PDE's whose solutions can be written in terms of parabolic cylinder functions. A Landau-Zener formula describes the transition probability along each characteristic. Similar difficulties occur when the molecular Hamiltonian of [36] is known only by means of perturbation series. However, the perturbation theory of BornOppenheimer wave packets in presence of avoided crossings can still be performed. See [14].

G. A. HAGEDORN AND A. JOYE

222

Although no one seems to have used them to do so, the techniques of Colin de Verdiere, Fermanian-Kammerer, Gerard, Lasser, and Teufel can almost certainly produce analogous results. If the gap is small and fixed, instead of being O(E), then transitions associated with avoided crossings are O(exp(-F/E2)). Under rather restrictive hypotheses, the large time behavior of the leading order non-adiabatic component is found in [42]. The nuclear variable X is restricted to one dimension and the electron Hamiltonian is an n x n matrix that depends analytically on X, with appropriate limiting behavior as X -> ±oo. In the remote past, the allowed incoming states include the product of a semiclassical wave packet cj in X times an electronic eigenfunction. If the initial electronic level and a second level have a single avoided crossing with a sufficiently small gap, then the leading order transition wave function on the second level is Gaussian, with norm of order E-i exp(-a/c2) for a precise a > 0.

Both its amplitude and its average momentum are larger than what one would obtain by treating the nuclei as classical point particles and applying the LandauZener formula for the electrons. The proof in [42] uses generalized eigenfunction expansions associated with the full Hamiltonian and a WKB technique [57, 58] that is valid for complex X. Results of a similar nature, but for more general systems of PDE's are obtained in [59]. Further time-independent generalized eigenfunction results are obtained in [60, 70, 58]. The results of [42] describe only the large time behavior. It is a beautiful open problem to analyze the dynamics up to this exponential accuracy when the gap is fixed and small.

Acknowledgements: We wish to thank B. Sutcliffe, S. Teufel, R. G. Woolley, and the referee for their constructive comments and several references.

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(George A. Hagedorn) DEPARTMENT OF MATHEMATICS AND CENTER FOR STATISTICAL MECHANICS AND MATHEMATICAL PHYSICS, VIRGINIA TECH, BLACKSBURG, VA 24060-0123, U.S.A.

E-mail address: hagedornmmath.vt.edu (Alain Joye) INSTITUT FOURIER, UNITE MIXTE DE RECHERCHE CNRS-UJF 5582, UNIVERSITE DE GRENOBLE I, BP 74, F-38402 SAINT MARTIN D'HERES CEDEX, FRANCE

E-mail address: [email protected]

Proceedings of Symposia in Pure Mathematics Volume 76.1, 2007

Perturbation Theory and Atomic Resonances Since Schrodinger's Time Evans M. Harrell II This review is dedicated to Barry Simon, whose vision in [160] laid out a program of more than thirty years of rigorous mathematical research on resonances and perturbation theory. ABSTRACT. Quantum theory makes a sharp distinction between bound states and scattering states, the former associated with point spectrum and the lat-

ter with continuous spectrum. Resonances associated with quasi-stationary states bridge this distinction, and have posed mathematical challenges since the beginning of the Schrodinger theory. Here the development of the mathematical underpinnings of resonance theory in atomic physics is reviewed, with particular reference to the role of the (DC) Stark effect, and time-independent perturbations of bound states in the two-body problem in atomic and molecular physics.

CONTENTS 1.

2.

3.

Resonances in the Early Days of the Schrodinger Theory Mathematical Foundations of Quantum Resonance Theory and the Analysis of One-Dimensional and Separable Models Quantum Resonance Without Symmetries

References

1. Resonances in the Early Days of the Schrodinger Theory It was a perturbation calculation that convinced physicists that Schrodinger's theory was superior to the earlier quantum theory of Bohr and Sommerfeld based on phase integrals. The old quantum theory provided an explanation for the spectral lines of isolated hydrogen, but there was no easy or systematic procedure to handle perturbations of exactly integrable models. Schrodinger considered the Stark 2000 Mathematics Subject Classification. 81-02, 81U05, 34L25, 35P25, 47A40. Key words and phrases. quantum resonance, scattering, complex scaling, semiclassical, tunneling, Stark effect. This work was supported by NSF grant DMS-0204059. ©2007 by the author

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effect [175, 116, 108], the shifts caused to hydrogen's emission spectrum by the application of a constant electric field, in 1926 [153]. In Schrodinger's model the energies are eigenvalues of the hydrogen Stark Hamiltonian, which in scaled units reads

-V2 + 1 + KX1.

r

(1)

Schrodinger adapted Rayleigh's [146] procedure for generating the Taylor series in powers of ,c for the eigenvalues and eigenfunctions of a family of linear operators of the form Ho + nW. At first order he recaptured the formulae of Epstein [45, 46] for the shifts in the spectral lines, and with the more systematic procedure he was able to obtain second-order corrections and better agreement with experiment. Although the Stark Hamiltonian was known to be separable in parabolic coordinates, and to correspond to an integrable classical system [45, 46], it was clear from Schrodinger's procedure that separability and integrability played no essential role in the calculation, and that perturbative corrections could be calculated to arbitrary order. Due to this first major success of the Schrodinger theory, both Schrodinger's equation and his perturbation scheme were swiftly adopted, and the old quantum theory was soon scrapped [33, 127]. (Notoriously reluctant to accept the new theory was Stark [176] himself.) To this day the perturbation expansions of Rayleigh and Schrodinger are easier to calculate and more successful than the perturbation methods used in classical mechanics, on which the Bohr-Sommerfeld theory was cobbled together. Given the impact of Schrodinger's analysis on the development of modern physics, it is ironic to note much later proofs that: The operator (1) has no eigenvalues at all as soon as is > 0 [181, 182, 15]; and The Taylor coefficients according to Schrodinger's prescription follow a precise asymptotic law [58, 68], and the radius of convergence of the series is 0. The quantum states observed in the Stark effect are not truly bound, but are instead resonance phenomena, although Schrodinger did not recognize this in [153]. While resonances of classical oscillators formed part of the tool-kit of every physicist at the dawn of quantum mechanics, it took some time for physicists to address or even articulate the basic questions about quantum resonances: 1. What is the definition of a resonance energy? 2. Is there a "resonance state," and how is it defined? 3. How can the resonance energy be calculated? 4. How can the time-decay of a resonance be quantified? The first step was taken by Oppenheimer, in one of his earliest works [137]. Oppenheimer stated that the Stark Hamiltonian (1) had no eigenvalues, and rather

cavalierly cited an article of Weyl [187] with a loose connection to this claim. He then attempted to calculate the rate of decay of the wave function associated with the ground state of unperturbed hydrogen by approximating a solution to the Schrodinger equation including the electric field and estimating a matrix element connecting this state to the unperturbed hydrogen state. Although Oppenheimer did not use the term "resonance," he addressed the fourth question for the first time within quantum theory, and implicitly considered something one could regard as a resonance state, a special non-normalized solution of Schrodinger's equation. Not long thereafter Weisskopf and Wigner [186] defined a resonance as a bump in the graph of the scattering amplitude and explicitly connected it to the presence

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of a pole in the analytic continuation of the scattering amplitude, located close to the real frequency axis. The analogy they made was with a system of classical damped oscillators, in which the Fourier transform of the displacement has a pole in the complex plane at position wo - i , contributing a Lorentzian line shape a proportional to 1

(2)

(W -Wp)2+ 4 in the response function. From the shape of the graph, F is referred to as the width of the resonance. With this analogy and the observed exponential decay in time of a radioactive

species, early resonance theorists attempted to fit the probability of decay of a resonance state, perhaps as coupled to a radiation field, to an exponential function, assuming that (To,exp(-itH)Wo) 12 ,,, Cexp(-Ft),

(3)

where'TO is some sort of resonance state. (In contrast, for generic scattering states, polynomial decay in time is the most that can be expected [96, 92].) If the ansatz of exponential decay in time is accepted, then formal arguments lead one to expect

that the r in (3) should be comparable to the r in (2) and given by a version of the Fermi Golden Rule [47], according to which to leading order r is proportional to the square of a matrix element coupling the resonance state to the continuum. A criticism of the usual formal arguments for (3) is to be found in [161, 148]. Although early arguments for the exponential-decay ansatz were rather vague about what constitutes the resonance state To, in practice it was often chosen as an eigenstate of an unperturbed H0, or as such an eigenstate corrected to some finite order in perturbation theory. Alternatively, the left side of (3) might be replaced by the projection of exp(-itH)WYo onto a set of continuum states localized near the energy of To. A difficulty with (3) is that it is one of those deep truths that are clearly false! For small t the expression in (3) should behave as 1-0(t2), which is inconsistent with exponential decay. On the other hand, it can be shown that if (3) holds for some '1'o E 'H as t --f oo, then a(H) = R: Following Herbst [75] (see also [161]), observe that (3) would imply that the spectral measure dµpo := (To, P(-.,a]'ho) is absolutely continuous and has an analytic continuation to the strip {Im < }. If 2 the support of dµq,o were not the whole real axis, then analytic continuation would imply that dµwo = 0 identically on the strip. Although exponential decay in time is possible for the Stark effect Hamiltonian (1), for which the spectrum is R, it is consequently excluded in typical situations where the Hamiltonian is bounded from below, and the most that can be hoped for is transient exponential decay according to (3). A more realistic expectation for the decay law is

(To, exp(-itH)qlo) = exp(-i(E - iF/2)t) + b(t),

(4)

where the remainder b(t) is small for intermediate times but relatively significant for small t and dominant for large t. The physical mechanism of most resonances has generally been understood to be the existence of a quasi-stationary state [24, 142] caused by perturbation of a true bound state. Two variants of this mechanism are encountered: 1. Shape resonances, which correspond to confinement of a particle by a barrier, through which tunneling occurs, as in Gamow's model of alpha emission [54], in which the potential energy was a radial function proportional to X{a
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The Stark effect may also be viewed as a kind of shape resonance: Although the strength of the electric field may be small, the perturbation interaction remains large somewhere far from the origin. Conceptually, a shape resonance arises as a classical bound state liberated by tunneling. Its signature is a very slow decay, with a characteristic time identifiable with the reciprocal of F in (2).

2. Embedded bound states [134, 43] that "dissolve into the continuum" under a slight perturbation. The most important model is that of the Auger effect, in helium. If the nucleus is treated as fixed in space and the electron-electron interaction is neglected, then the helium spectrum is the set sum of two scaled Balmer (hydrogen) spectra, consisting of negative bound states at { - 4 } and a continuum covering the non-negative real values. The continuous spectrum of this operator begins at - 4 and overlaps many eigenvalues and thresholds. The introduction of the electron-electron interaction destroys the symmetry of the operator and the embedded eigenvalues are expected to couple to the continuum and disappear. The decay of resonances associated with embedded eigenvalues is not as slow as that of shape resonances. Other mechanisms can arise if the interaction depends on time (e.g., [190]), but will not be pursued here.

There is a unified way to understand resonances and true eigenvalues of a quantum Hamiltonian, namely, as poles of the resolvent operator (H - A)-1 or its integral kernel, the Green function. If the pole occurs on the real energy axis, it corresponds to an eigenvalue in the usual way, but additional poles may occur when the resolvent is analytically continued onto an "unphysical sheet." Schrodinger Hamiltonians are supposed to be self-adjoint and therefore have only real spectra, but the non-real poles in the continued resolvent might be located at the eigenvalues of some related non-self-adjoint operator. Non-real eigenvalues had been introduced into physics as early as 1884 by Thomson [180], as a way to understand resonances in electromagnetic theory, and they appeared in quantum scattering theory since the 1950's [115, 49], somewhat artificially. The idea of finding poles in the resolvent as continued analytically across a branch cut on the real energy axis was introduced by Schwinger in [155]. Although quite plausible, the connection between resonances defined as poles of a continued resolvent and those defined as poles in the scattering

amplitude or zeroes in the Jost function was initially murky. (Theorems equating them in some generality have been proved in, e.g., [66, 18].) The next section of this review will recount some rigorous mathematical approaches to atomic resonances that were elaborated in one-dimensional and highly symmetric models in the 1970's and 80's. Section 3 will describe later developments in which assumptions of symmetry were abandoned as more sophisticated methods of scattering theory and PDEs became available. Finally, some alternative approaches to resonance theory that have been developed will be mentioned.

2. Mathematical Foundations of Quantum Resonance Theory and the Analysis of One-Dimensional and Separable Models In the 1930's and 40's, the work of Rellich [151] and Kato [101] put timeindependent perturbation theory for non-relativistic quantum theory on a firm mathematical footing, particularly for Hamiltonians with discrete spectra. Families of operators depending on a coupling constant n were understood as analytic

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functions of a complex variable, taking values in an operator algebra. Most often the family of operators has the form H,. := Ho + KW, and if W is relatively bounded, in the sense that D(W) 3 D(Ho) and there are constants a and b such that for all f E D(Ho), IWf1I < allHofll +blIf1I, (5) then the spectral projectors of H,,, for isolated subsets of the spectrum depend analytically on r,, and consequently the eigenvalues of H, can be analytically continued, provided that they do not meet other parts of the spectrum. Wave operators were introduced by Moller in [126], which enabled Kato [102] and Birman [22] to lay the mathematical foundations of quantum scattering theory

over the following decades (see [41, 147, 9, 135, 143, 82, 189]). Moller's work was also the starting point for the development of acoustic scattering theory in the hands of Lax and Phillips [113]. In the early period rigorous results about resonance scattering were rare. One of the first topics connected with resonances to receive attention was the mechanism whereby an eigenvalue could dissolve into the continuum. In [50] Friedrichs discussed some simple models in which an eigenvalue embedded in the continuum dissolves under a perturbation, and made connections with quantum scattering theory. Soon afterwards Titchmarsh attempted to make sense of the perturbation series that Schrodinger had developed for the Stark ef-

fect. In [181] he showed for the first time rigorously that a version of the Stark Hamiltonian does not have any eigenvalues, and introduced the notion of spectral concentration.

DEFINITION 1. Let Hn be a sequence of self-adjoint operators with spectral projectors En(S). Let T and {Sn} be subsets of R. Then the part of the spectrum of Hn in T is concentrated on Sn provided that En(T - Sn) _# 0

in the strong sense. (The notation of [103] is followed here.)

Titchmarsh showed that the asymptotic series for Stark "eigenvalues," as derived by Schrodinger, defines nested intervals on which the spectrum of (1) is concentrated [182]. Spectral concentration gave some insight into the third basic question of how the resonance energy can be calculated, at least in an asymptotic sense.

Titchmarsh's analysis was extended by Conley and Rejto [34] and by Riddell [152]. Howland also analyzed spectral concentration in a variety of contexts [85, 86, 87, 88]. In [133], Nenciu showed under wide circumstances that if the unperturbed operator has finitely degenerate eigenvalues in a given interval, then there is spectral concentration near those eigenvalues to order p if and only if there is an asymptotic basis of pseudo-eigenvectors to order p. Nenciu's analysis included general Stark Hamiltonians. Nenciu also showed that if there are pseudo-eigenvectors to order p, then they remain quasi-stationary for a correspondingly long time. A perturbative treatment of resonances seemed, however, to offer little hope of a precise estimate of the rate of decay of a shape resonance, for which F is not merely "several million times" smaller than wo, as Weisskopf and Wigner marveled more than once in [186], but, as an exponentially small function of n, identically 0 in perturbation theory. In addition to the limitations of spectral concentration for quantitative calculations, it was not even clear that spectral concentration could adequately be used

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to define quantum resonance. Indeed, in 1975 Howland [89] showed that although resonances imply spectral concentration, it is possible for spectral concentration to occur in the absence of a corresponding singularity in a continued resolvent. Howland is also credited with a fundamental insight about possible definitions of a resonance. In contrast to Occam's razor, which cuts away complications in scientific theory, Howland's razor cuts away simplicity. As formulated in [161], this principle states: No satisfactory definition of a resonance can depend only on the structure of a single operator on an abstract Hilbert space.

For example, the family of Stark Hamiltonians (1) is unitarily equivalent for all non-zero real c [15]. How could the energy or lifetime of the resonance depend on ,c, if the definition were a unitary invariant? Two key techniques opened the way to making sense of the perturbation series for resonances and precisely calculating resonance widths in the 1960's and 70's: complex scaling and summability methods. The inspiration of Aguilar, Balslev, and Combes [6, 17] was to complexify Euclidean symmetry groups and to use analytic continuation to define resonance energies as non-real eigenvalues of the family of operators produced by conjugating the Hamiltonian with the complexified groups. In the original version of complex scaling, Aguilar, Balslev, and Combes exploited symmetry under dilatations. The unitary group of dilatations depends on a real parameter 9 such that x c R" --> eex, hence [U(O)fl (x) := evO/2 f (eex). For suitable potentials one can treat 0 as a complex variable and regard HO := U*(0)HU(O) as an analytic family of operators. An easy calculation shows that if 9 is continued off the real axis, the essential spectrum of the complex-scaled one-particle Laplacian is simply rotated in the complex plane. Correspondingly, for a many-particle free Hamiltonian, the essential spectrum becomes the union of rays in the complex plane emanating from threshold energies. Assuming relative compactness for the potential energy function, the same will be true for o-ess(H,9). Meanwhile, by KatoRellich perturbation theory, isolated eigenvalues of HO are analytic in 9 and can be continued as long as they remain isolated. The kicker is that variations of 0 in real directions correspond to unitary transformations of H9, so the spectrum is constant with respect to such variations. An eigenvalue is therefore constant as a function of 0 on any connected component of its domain of regularity. It follows that for "dilatation-analytic" potentials, eigenvalues are independent of 0 except that they may appear or disappear when the essential spectrum passes over them. Resonances correspond to non-real eigenvalues of the complex-scaled operator. The complexscaling definition of a resonance thus complies with Howland's razor, but without special dependence on the introduced parameter. Resonance eigenvalues may well

depend on perturbation parameters ,c occurring in H, but in the usual situations they are isolated and non-degenerate, and hence accessible to well-controlled perturbation expansions in powers of rc. Complex scaling also offers a definition of a resonance state as the eigenfunction corresponding to a non-real eigenvalue and its analytic continuation in 9. In an influential article [160], Simon synthesized and clarified the theory of resonances and dilatation analyticity, showing that it applies to important physical models, including Coulomb and Yukawa interactions. This article cemented the importance of complex scaling as a tool in quantum physics [31], and it has become a standard reference in the physical literature for the mathematical underpinnings of

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the technique (e.g., [150, 99, 23, 84, 27, 125, 40]). For mathematical physicists, Simon articulated the challenge of a better understanding of the time-decay of resonance and related matters: The goals of the time-dependent theory are much more ambitious than merely proving certain eigenvalues dissolve .... The time-dependent theory is supposed to compute a characteristic lifetime T, for the decay of a state ... It turns out to be a very hard problem to define the lifetime directly. Simon identified a suitable interpretation of the Fermi Golden Rule for F in (3) and (4) as an estimate of the imaginary part of the Taylor coefficient a2 of a resonance eigenvalue associated with a bound state at r, = 0. (The coefficient a1 is real by first-

order perturbation theory.) Using Stone's formula and second-order perturbation theory, one can express the resonance width for a perturbed non-degenerate discrete bound state as 27rd (4)o,WP(A)W.Do/ (6)

2=

-ao

where 4)o is the unperturbed eigenfunction and P(A) is the spectral projector for (-oo,A) \ {Ao}, cf. [148]. In the period around and following [160], analytic function theory helped to clarify the issues raised in [160] and to unify the understanding of high-order perturbation theory, resonances, and complex scaling. Inspiration was drawn from the detailed understanding of the eigenvalues Ak(rc) of the one-dimensional anharmonic oscillator (7) p2 + x2 + icx4

and their analytic continuations that had been achieved beginning with the work of Bender and Wu [19]. Bender and Wu had observed that the perturbation coefficients for an eigenvalue A(rc) as functions of rc can be efficiently calculated with a difference equation, and the perturbation series diverges. They also tied the analysis of the functions Ak(g) to the WKB approximation of semiclassical quantum mechanics (more accurately called the Liouville-Green approximation [114, 64, 136]). At that time a popular method for interpreting divergent series was Pade approximation, and in [119] it was proved that the diagonal Pade approximants for the anharmonic oscillator perturbation series converge to the eigenvalues. Afterwards, Graffi, Grecchi, and Simon [61] noted that an understanding of the domain of analyticity of the eigenvalues [159, 118] and control on the growth in n of an allow the use of Borel summation to define a unique analytic function f (rc) to which

the series is asymptotic in a sector. (The Borel sum works by summing the series with an replaced by (m -1)!' then analytically continuing onto the positive real axis, and recovering f (n) with a Laplace transform.) The Borel sum became a standard method that in due course was adapted for many other classic models where the perturbation series diverges, including the Zeeman effect [14, 13, 16], and the hydrogen molecular ion [38, 60], which resemble the anharmonic oscillator in that discrete eigenvalues remain discrete under singular perturbation. The idea of a "Bender-Wu theory" relating high-order perturbation theory, summation methods, and tunneling for the complex eigenvalues in resonance models [19, 20, 21] was made plausible by considering (7) when rc is continued to negative values, in which case, at least at the formal level, the model exhibits non-real eigenvalues and quasi-stationary states. In [159] Simon showed that Taylor coefficients an are

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proportional to the moments on the negative ic axis of the imaginary part of the continued anharmonic oscillator eigenvalues. In resonance models, given the ability to displace the essential spectrum and analytically continue the resolvent to a second sheet, where resonances can be defined as eigenvalues, an analogous formula can often be derived with Cauchy's theorem. For the Stark effect it reads [77, 68]

a2n = -

R

I 7r

t-2n-1F(ic)dic + O(R-2n).

o

In the remainder of the 1970's, the vision of a perturbation theory of resonances advanced rapidly, with Avron and Herbst's analysis of Stark Hamiltonians using translation and dilatation analyticity [15, 74]; A Bender-Wu analysis of high-order perturbation theory and summability for

Stark resonances [77, 58, 161, 158, 59, 78]; and Precise tunneling asymptotics for the resonance widths for the Stark effect, i.e., the (corrected) Oppenheimer formula [191, 68], among other mileposts. By the end of the decade, most of the standard models of quantum resonance brought about by perturbations of eigenvalues were mathematically well-understood. The general theory, however, left much to be desired. Dilatation analycity was an oddly strict condition to put on the potential energy for a number of reasons. First, from the point of view of operator theory, the resolvent (H - A)-' is already an analytic function of A on the resolvent set of H, without regard to the details of H, such as whether its coefficients depend analytically on auxiliary parameters in H that have nothing to do with A. It does not help much that some such parameter, like 0 in the dilatation argument, is introduced ad hoc, even if resonance eigenvalues are independent of 0, but for whether or not they exist. Second, the special assumptions needed to achieve the magical results of complex scaling excluded some of the simplest models, like those where the potential is radial and of compact support, in which case the original version of dilatation-analyticity is not able to encompass the familiar use of the Sommerfeld outgoing radiation condition [174] to define a resonance state. In [162] Simon proposed the method of exterior complex scaling to respond to this shortcoming. Many other variants of complex scaling, relying on other symmetries or localizing in other ways, were introduced in due course (e.g., [81, 71]). The method reached its culmination in the 1980's in the hands of Hunziker [90, 82]. The really sharp results attained by 1980 on high-order perturbation theory and estimates of exponentially small quantities like F were restricted to one-dimensional, separable, or otherwise highly symmetric Hamiltonians, usually for one particle. This restriction was not so much due to a shortcoming of the theory of resonances

as it was due to the method of calculation. In a circumstance where r is exponentially small, the difficulty is to disentangle it from the perturbation expansion for the energy, despite r being smaller than the perturbative error at all orders. A successful strategy for accomplishing this is to use integral identities for r in terms of an identifiable resonance eigenfunction, and then estimate the exponential growth of the latter with sufficient accuracy. For example, suppose that one has identified a resonance with a non-real eigen-

and, with the aid of complex scaling, has picked out a canonical

value A - i 2

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"resonance solution" Pr solving (-A + V (x))-D, = (A - i 2 )fir, where V is a realvalued potential function. Of course I r is not normalizable in L2, but one can multiply by Kr and integrate over a finite region S. With Green's identity, it is easily calculated that F

Js

1 'r 2d"x = 2

LIm

d"x.

(8)

s Thus with a good choice of S and uniform control on the resonance solution (Pr, an estimate of F can be obtained, which is accurate in proportion to the pointwise accuracy with which 4)r is known. In that period, excellent techniques were available for uniform approximation of solutions to ordinary differential equations (e.g., [136]), including ones that grew or decayed exponentially, but not for partial differential equations. Not only are geometric complications minimal for ordinary differential equations, but, most usefully, the solution space of the time-independent Schrodinger equation is two-dimensional. With the Wronski identity, variation of parameters, etc., a plethora of explicit relations can be found for special solutions, whether for their dependence on the position

x or on a spectral parameter. In the case of the Stark effect and other separable problems, uniform approximation of solutions of ordinary differential equations enabled the proof of "tunneling" formulae like the corrected Oppenheimer formula [68].

One of the earliest canonical models of the subject, shape resonance, was relatively late to receive rigorous mathematical analysis. This model was introduced in the 1920's by Gamow [54] and by Gurney and Condon [65] to understand alpha emission. A particle is supposed to be confined in a potential well surrounded by a high potential barrier, outside of which the potential is zero, or at least tends to zero. In the radially symmetric case one can separate variables and study the operator 2

-d22 + U(r) +/3V(r)

(9)

on L2(R+) with a Dirichlet boundary condition at the origin. The barrier V may be supposed non-negative and strictly positive a.e. on its support, a finite interval such as [1, 2]. The operator (9) is considered in the strong-coupling limit, and as /3 -+ oo it tends formally to 2

dr2 +U(r)

(10)

for r < 1, with a Dirichlet condition at r = 1. Outside the finite region, the timeindependent Schrodinger equation is identical to the Helmholtz equation obtained by separating the wave equation, and Gamow imposed an outgoing radiation condition on (9) equivalent to that used earlier by Sommerfeld [174] in his analysis of the wave equation. (Thomson [180] had an even earlier version of an outgoing radiation condition.) The outgoing condition was specifically connected with the notion of a quantum resonance by Kapur and Peierls [100, 141] and Siegert [156], and is perhaps the most common way to define resonance states in the physical literature (e.g., [192, 108]). The outgoing radiation condition is precisely recovered in the shape-resonance problem for compactly supported radial potentials when the method of exterior complex scaling is brought to bear. Ashbaugh and Harrell [11], using uniform approximation of solutions of ODEs and the implicit function theorem, established in this situation that the eigenvalues

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of (10) turn into eigenvalues or resonances of (9). A systematic perturbation theory was developed, in fractional powers of 0, and exponentially small resonance widths were calculated, using a version of (8). Around this time another model that became popular was known as the StarkWannier Hamiltonian, in which a periodic background potential is combined with that of a constant electric field. Wannier introduced this model in 1960 [185], and mathematical analysis was initiated by Herbst and Howland in [76]. Resonances

in this model occur in infinite evenly spaced "ladders," and were studied with perturbative methods analogous to those used for the more venerable canonical models [2, 94, 62, 106, 10, 63, 29] Estimates of the density and distribution of resonances go back in the physical literature to Regge [149]. In one dimension, theorems of this form were obtained by Melrose [122] and Zworski [193, 194, 195], who chose to define resonances as poles of the analytic continuation of the S-matrix and obtained a Weyl-type distribution result; and by Froese [51, 52], who preferred the Birman-Schwinger kernel K(n) of IV (x) I1/2sgn (V (x)) (Ho + ic2) -1 I V (x)11/2. Actually, due to the special features of ordinary differential equations, there are close connections among

Birman-Schwinger kernels, S-matrices, phase shifts, Jost functions, m-functions, and Green functions, and consequently in one dimension resonances can be defined more or less equivalently in terms of singularities or zeroes of complex functions concocted from any of these. This point was brought out, for example, in an article of Simon [164], which unified the results of Froese, Melrose, and Zworski by focusing

on the properties of the Fredholm determinant d(r,) := det (1 + K(ic)),

the zeroes of which correspond to resonances [122, 7, 56, 57, 194]. In addition to estimates of the distribution of resonances of several one-dimensional models, Simon obtained distribution estimates for antibound states.

3. Quantum Resonance Without Symmetries In the last two to three decades the mathematical theory of quantum resonance has broken free of the symmetries and special circumstances needed to make the canonical models accessible. Advances have depended on three important developments: Better estimates of the growth properties of solutions of elliptic PDEs; an application of the Schur complement of matrix theory to Schrodinger operators, known in the trade as the Livsic-Feshbach matrix; and Mourre theory in scattering. The understanding of solutions of the multidimensional Schrodinger equation advanced in the 1980's. The notion of an Agmon metric, as expounded in [3], allows estimates of the exponential decay in space of Schrodinger wave functions without separation of variables or other uses of symmetry. This construction is quite general; for example, for a Schrodinger operator H, if one can find a function A(x) > 0 such that

H-p>A(x)>0

(11)

in the sense of quadratic forms, then solutions are governed by a metric PA defined by ds,24 = A(x)IdxI2. Agmon's essential result was that if (11) holds and a solution 0 of Hz/' = po has controlled growth in the sense that on an exterior domain S2, sz IV) I2 exp (-2(1 - J)PA(x)) dx < oo,

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then on average it decays exponentially in the sense that V)12 I

exp (2(1 - E)pA(x)) dx < oc.

fz

With epsilonic loss, Agmon's theorem shows that when integrated, solutions of the time-independent Schrodinger equation satisfy exponential-growth estimates much like the familiar tunneling formulae in one dimension. Moreover, the function A(x) provides flexibility to adapt the method to particular circumstances. Similar estimates have been proved on adapted regions, such as exterior sectors [3], applied to Green functions [39], etc.

Agmon's techniques made estimates like (8) tractable in the absence of reparability. In the hands of Helffer and Sjostrand [72], Simon [163], and others, systematic asymptotic expressions for solutions of multidimensional Schrodinger equations were developed in semiclassical limits, whether as a parameter h -+ 0 or

in other circumstances with a large or small parameter. Quantum resonances figured among the semiclassical phenomena on which these microlocal techniques were brought to bear (e.g., [73]). Many articles of Helffer, Sjostrand, and collaborators

are synopsized in [70, 165, 107]. In the late 1980's and 90's, a number of the results on canonical models that had been obtained with one-dimensional methods were re-obtained without symmetry assumptions. Among these results were perturbation theory and width estimates for shape and Stark resonances [73, 157, 95, 165, 71], and stability, resolvent estimates, and perturbation theory for shape resonances without radial symmetry [32, 81, 82]. In the same period, the notion of a resonance state as satisfying an outgoing condition, or a truncated version of such a state, was further developed in the multidimensional setting, and conditions were found under which exponential decay in time (4) could be proved for such states [170, 171, 172, 91, 105, 112]. One of the earliest to appreciate the value of the Livsic-Feshbach matrix in the theory of quantum resonances was Howland, who, in a somewhat overlooked article [89], used it to develop a perturbation theory for resonances of self-adjoint Schrodinger operators, arising from embedded eigenvalues. (Liv"sic had introduced the matrix in relation to non-real eigenvalues of explicitly non-self-adjoint operators [115]. Feshbach independently developed similar ideas in an article on resonance theory for nuclear physics [49].) The power of the Liv"sic-Feshbach technique is that the analysis of the spectrum of an infinite-dimensional operator is replaced by that of a finite-dimensional operator, albeit one that depends on the spectral parameter. DEFINITION 2. Let H be self-adjoint and let P be a finite-dimensional projector (normally the projector onto the unperturbed eigenvector 00 of a reference operator HO). The Livsic-Feshbach matrix is the finite-dimensional operator B(z) acting on

K := Ran P such that

(B(z) - zl)-I = P(H - zl)-1P,

(12)

when the right side is "compressed" to K.

The matrix-valued function B(z) is meromorphic on the complement of the essential spectrum of H and has only real singularities [89]. Provided that IC C D(H), the Hamiltonian H can be written as a block-partitioned operator with

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respect to K and K', and B(z) - z is, in the terminology of matrix theory, the Schur complement of (1 - P)(H - z)(1 - P) in H - z1 [154, 36, 42]. It follows by a calculation from (12) that, with P := (1 - P) and H := PHP,

B(z) = PHP - PHP(H - z)-'PHP.

(13)

(Again, this formula is to be interpreted as compressed to K.) Replacing H by Ho + KcW and taking P as the orthogonal projector corresponding to an eigenvalue )o E up(Ho) with normalized eigenvector 4Po yields the Feshbach formula

B(z, ic) = \ol + icPWP - ic2F(z,,c), where F(z, rc) = PWP(H - z)-'PWP [49]. When P is one-dimensional, B(z, ic) and F reduce to scalar functions that satisfy B(z, rc) = )o + rc (4)o, Who) - rc2F(z, rc).

(14)

Observe that the first-order term is identical to that of Rayleigh-Schrodinger perturbation theory for the first-order correction to a non-degenerate eigenvalue, and that -ic2F(z, rt) resembles the second-order correction. From (14) the leading-order expression for the resonance width is found to be

r 2

= -ImF(.1o + ic, 0).

(15)

In [89] Howland observed that the Livsic-Feshbach matrix allows an analogue of Kato's regular perturbation theory for non-isolated eigenvalues of a family of operators H(,c), and he demonstrated that this theory embraces not only the situation of an isolated eigenvalue, but also perturbations of certain embedded eigenvalues and resonances of complex-scaled Hamiltonians as in [160]. Howland's work on the Livsic-Feshbach matrix was extended in the 1985 dissertation of Orth and a related article [138]. There, a resonance is defined in the model case of a non-degenerate unperturbed eigenvalue ao as follows:

DEFINITION 3. Suppose that there exists a dense subspace H+ containing K, W(K), and all possible eigenvectors of H(ic). If for A in some neighborhood of Ao

and is near 0, ((1 - P)H(ic)(1- P) -1)-' can be continued analytically in z to the real axis as a bounded operator from x+ onto its dual 7-l_, and the continuation is Lipschitz continuous with Lipschitz constant O(tc-2), then B(z, ic) can likewise be continued to the real axis, and the resonance eigenvalue near Ao is the fixed point of the equation A (n) = B(A(ic), .\).

With this definition, Orth showed that resonances are associated with spectral concentration, as in [89], and proved the existence of resonances for a large class of many-body potentials. Orth required no ad hoc analyticity of the potential with respect to dilatations. Instead, the dilatation group makes its appearance in an assumption that products of its generator and the one-body potentials are bounded maps between appropriate Sobolev spaces. According to Mourre theory [128, 97], this allows control on resolvents, localized in energy and space. Hence the subspace x+ can be taken as D(A2), where A is the symmetrized generator

of dilatations, z (x V + V x). The Howland-Orth definition of a resonance for H(ic) has become widely accepted.

Mourre theory refers to the use of local commutator estimates to prove that quantum scattering is well-behaved. The role of commutators in scattering theory

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dates from the 1960's, when Putnam [145] showed that if there is a bounded selfadjoint operator A such that B := i[H, A] > 0, and 0 is not an eigenvalue of B, then the spectrum of H is purely absolutely continuous. Putnam's theorem led to many further developments, notably the theory of smooth perturbations in the hands of Kato [104], Lavine [109], and others. Mourre's innovation in [128] was to localize Putnam's condition in energy: Suppose that the commutator B satisfies certain technical conditions, and that on some real interval A there exists a positive number a and a compact operator K, such that

PABPA>aPA+K.

(16)

Mourre discovered that under this condition H has no singular continuous spectrum in A, and only finitely many eigenvalues in any compact subset of A. Moreover, if K = 0, then the expression II(IAI + 1)-1(H - A - i6)-1(1A1 + 1)-111

(17)

remains bounded for A E A as 6 - 0+ (see also [53, 144]). Mourre's method was extended to higher-order commutators and higher powers of resolvents in [97]. An exposition of Mourre theory is to be found in [37]. Connections between Mourre theory and quantum resonances were first made by Orth [138] in his many-body analysis, and by Herbst and Skibsted in the context of the Stark effect [80]. Later, Soffer and Weinstein [173] posited an outright hypothesis on the time-decay of exp(-iHot), when localized in space and in energy, and under that assumption proved that embedded eigenvalues away from thresholds dissolve into the continuum as resonances and decay according to a version of (4), provided that F, as given by (6), satisfies a certain bound from below. (Agmon, Herbst, and Skibsted had earlier shown in [5] that embedded eigenvalues are generically unstable against perturbations.) The decay hypothesis of Soffer and Weinstein is a familiar consequence related to (17) of a Mourre inequality, and therefore holds under wide circumstances away from thresholds. Soffer and Weinstein focused more on the notion of a resonance state than did Howland and Orth. In particular, with the decay hypothesis on the propagator, the decay law (4) is shown to apply to all initial functions in the range of a smoothed version of the spectral projector onto an interval A containing an unperturbed eigenvalue but no other part of the point spectrum of Ho. Most earlier analyses of resonance states and their the time-evolution had defined them with outgoing radiation conditions, and were consequently more restrictive. Some technical assumptions of Soffer and Weinstein were relaxed in [35]. In particular, given sufficient regularity of the unperturbed resolvent, P was required in [35] merely to be positive rather than bounded away from zero in a specific way. Making similar assumptions, Merkli and Sigal [124] considered the nature of a metastable state, showing that any state suitably localized in energy can be written as the sum of a resonant part and a dispersive part. The resonant part will stay near the unperturbed state for a long time, and then decay according to (4). In [98], Jensen and Nenciu recently considered how an eigenvalue E at a (nondegenerate) threshold of Ho becomes a resonance of Ho+,W, in the Howland-Orth sense, when the first-order correction to E is positive. Jensen and Nenciu carry out an asymptotic analysis of the Livsic-Feshbach matrix, and with the aid of an expansion of the resolvent at zero energy, they derive a modified Fermi Golden Rule

for the width r, proportional to cm/2 with m odd.

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In another recent article [30], Cattaneo, Graf, and Hunziker synthesize the current state of quantum resonance theory under rather basic assumptions, viz.: That Mourre's inequality (16) holds for H(ic) and an auxiliary operator A; that the multiple commutators of A with H(n) and W exist up to some finite order N; and that D(H) is mapped by exp(isA) into itself. Their main theorem is that if N is sufficiently large, and if F > 0, then for some smoothed spectral projector g onto an interval A containing A0,

(0, exp(-iH(r)t)g(H(r))O)

+ O(k2)) + b(r, t), (18) where b is an error term vanishing suitably as r, --+ 0 and decaying polynomially in time. Boundary values of the resolvent are also shown to be regular in A. Concurrently with the evolution of resonance theory as described above, with the focus on perturbations and the exponential-decay law (4), important mathematical work was done on resonances from some independent points of view. Although Lax and Phillips stated in the introduction to their monograph [113] that their subject was "classical-in contrast to quantum mechanical scattering theory," a section on the Schrodinger equation with compactly supported potential was included. Observing the simple relationship between the Schrodinger S-matrix for compactly supported potential energy and an "acoustic" scattering matrix for the wave equation with a compact obstacle, Lax and Phillips showed in [113] that the Schrodinger scattering matrix has a meromorphic continuation onto the nonphysical sheet, poles of which constitute resonances in their set-up. The counterpart for the wave equation of semiclassical analysis for the time-dependent Schrodinger equation is the propagation of wave fronts along rays. Correspondingly, the LaxPhillips approach to scattering has brought out connections between resonances and rays that are trapped. An obstacle is said to be trapping if there exist rays reflecting according to Snell's law that are either closed or of arbitrary long duration. For problems with a potential, trapping requires trajectories of the classical flow at some energy E, that do not tend to infinity in time, around which one can construct quasi-modes [167, 178]. Physical intuition argues that if rays can be trapped, then there will be quasi-stationary wave packets, whereas in the absence of trapping, scattering should take place without delays, and consequently the poles in the analytic continuation of the S-matrix ought to be relatively far from the real axis.

With or without trapping, there are universal, non-asymptotic lower bounds to shape-resonance widths, which show exponential dependence if parameters like Planck's constant are introduced. Such resonance-free regions were first found for one-dimensional shape resonances by Harrell in [67], using ODE comparison theorems and integral identities. A similar result was obtained in higher dimensions by Fernandez and Lavine [48] for states satisfying a strict outgoing radiation condition. Related semiclassical bounds were then studied by Helffer and Sjostrand [73], Briet et al. [25], and Burq [28], who allowed long-range potentials. Other restrictions on the location of individual resonances have been obtained with various methods

in [69, 131, 117, 12, 1]. If the potential is assumed non-trapping, the expectation is that resonances should be few and weak. Under non-trapping assumptions, Melrose, Zworski, and others have delineated resonance-free regions and estimates of the numbers and distribution of resonances [122, 193, 130, 194, 123, 83, 195, 121]. In contrast, it can be proved that if the potential is trapping, or if quasi-modes are assumed,

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then there are resonances exponentially close to the real axis; for results of this sort, see [32, 26, 55, 166, 167, 177]. Theorems about the distribution of resonances in many dimensions generally lack the detail of the results in one dimension

[122, 193, 194, 51, 52, 164], and often they distinguish between even and odd dimensions, a familiar circumstance for the wave equation that is less compelling for the Schrodinger equation. For example, Sjostrand and Zworski [168] proved a distribution theorem for resonances only in odd dimensions, which was also valid in even dimensions, although that was proved only much later [167]. Broadly viewed, the acoustic resonance theory of Lax and Phillips follows the same strategy as quantum resonance theory, of reducing the subject to analytic Fredholm theory for families of compact operator. Variants of this strategy have been developed by a number of authors (e.g., [56, 139, 57, 140]), and most currently by Agmon [4], who regarded the resolvent (H - A)-1 as an operator from H to some larger space. Assuming that the resolvent can be continued as a meromorphic function of A, Agmon extracts in a canonical way an operator with discrete spectrum corresponding to the poles of the resolvent. A conceptually distinct approach to resonance theory takes quasi-stationary time-evolution of the wave function as the starting point. A persistent advocate of this point of view has been Lavine [110], who stressed the notion of the sojourn time of a quantum state, closely related to the Eisenbud-Wigner time-delay [44, 188, 120, 93, 135, 132, 8]. Lavine prefigured Mourre by using commutators with the generator A of dilatations, and in particular derived a formula for the sojourn

time in terms of B := i[H, A]. (See [111, 93, 8, 129, 183, 184] for further developments connected with sojourn times and time-delays.) The program laid out out by Simon in [160] has largely come to fruition, as the questions laid out in that article about resonances are now answered not only in special models but in wide generality. The mathematical methods developed in the interim are suitable for making numerical calculations of resonance energies and decay rates, and are in the process of joining Rayleigh-Schrodinger theory and complex scaling in the arsenal of physicists for everyday calculation. In the mature phase of the theory of quantum resonances, there will continue

to be challenging problems to analyze. For example: Can the restriction to odd dimensions in articles such as [98] be relaxed? Can the asymptotics of resonances in many dimensions be estimated with an exactness approaching that of the onedimensional case [122, 193, 194, 51, 52, 164]? The most interesting possibilities on the horizon, however, may be a full merger of the different conceptual approaches to resonances, whether under an abstract umbrella like Agmon's theory [4], or one

beginning with a time-delay and proceeding from that to information about the analysis of continued resolvents.

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SCHOOL OF MATHEMATICS, GEORGIA TECH, ATLANTA, GA 30332-0160, U.S.A.

E-mail address: harrell®math.gatech.edu

Proceedings of Symposia in Pure Mathematics Volume 76.1, 2007

On a Theorem for Quantum Mechanical Scattering Theory Andrew Lenard For the 60th birthday of my friend Barry who advanced the mathematics of quantum mechanics by more than anyone ABSTRACT. It is shown that from a phase independent definition of the quantum mechanical scattering condition, it follows that suitable choice of phases can be made for the conventional, phase dependent, scattering limit to hold.

In quantum mechanics one ordinarily describes states of a system by vectors I' of unit norm in a separable complex Hilbert space (space of wave functions) H. Actually, though, a multiplicative arbitrary phase factor makes no difference, as evidenced by the formula

(A) A, = (', A') (1) for the expected or average value in the state 0 of an observable (quantum random variable) to which-according to the canons of quantum theory-a self-adjoint operator A corresponds. To be precise, therefore, one should say that states are mathematically specified as rays of 7-l, where by a ray one means some totality of unit vectors differing from each other only by a phase (a multiplicative complex number factor with absolute value 1). Nonetheless, one is often justified in dealing with actual vectors 0 instead of the rays they generate; but in principle, at least, this requires justification in any particular case. It was perhaps the late Eugene Wigner who first explicitly recognized and stressed this by proving the following theorem,' fundamental for instance for the study of symmetries in quantum theory: Given a one-to-one transformation T of the set of rays in 'H itself with the property that it preserves the absolute value of the inner product between any two,2 then there exists either an essentially unique unitary operator or an essentially unique conjugate-unitary operator U acting in 71 that induces in a natural manner the transformation T. It is not my purpose to elaborate on this and spell out 2000 Mathematics Subject Classification. Primary: 47A40; Secondary: 81U05, 81U10. Key words and phrases. quantum scattering theory, wave operators, rays in Hilbert space. 'A particularly perspicuous and wholly elementary proof was given by V. Bargmann in J. Math. Phys. 5, p. 862 (1964), where other references too may be found. 2Well defined, since it is plainly independent of which vectors in the respective rays one uses in the inner product. ©2007 American Mathematical Society 249

A. LENARD

250

with mathematical precision what it means, only to mention it as a reminder, in order to discuss another theorem that-in the spirit of Wigner's-reduces a phase independent description to one that involves definite choices of phases. Its context is quantum mechanical scattering theory. Let us briefly recall what is involved, at least as it is conventionally formulated

in quantum mechanical scattering theory textbooks. This mathematical setup is as follows. In a separable infinite-dimensional complex Hilbert space Ii, there are given two continuous one-parameter unitary operator families U(t) and U0(t) (-oo < t < oo) called the propagator and the free propagator, with self-adjoint generators H and Ho (the Hamiltonian and the free Hamiltonian), respectively. The former specifies the natural time evolution in the space of states of some quantum system by means of the formula

1(t) = U(t)0(0),

(2)

whereas the latter specifies, by an analogous formula, a time evolution "without interaction," the quantum analogue of straight-line uniform motion of particles in classical mechanics. These two are assumed to be so related that, in the limit t --> oo, U(t) is "well approximated" by U0(t), or somewhat otherwise stated, one is asymptotic to the other in the limit. Quite precisely expressed, this means the existence of the limit

lim U(-t)Uo(t)' = 0.

t-.00

(3)

for every vector l of 7-l, in the usual sense of (strong) convergence in Hilbert space. This limit defines an isometric operator S2c, (the "Moller operator") by the equation

10. _ 0.0

(4)

that plays an important role in the development of scattering theory.

Such is-in a nutshell-the beginning and foundation of quantum scattering theory. But it is clear that this formulation cannot be taken literally, inasmuch as two unit vectors 0 and 0 in 7-l, differing only by a phase factor t;, denote-as said above-the same quantum mechanical state from the point of view of physical interpretation. Thus, if in scattering theory one multiplies with any variable phase e(t), that is, if one replaces U(t)/ by e(t)U(t)O, the physical meaning of evolution in the space of states remains the same; yet (3) plainly cannot remain valid under such change if one chooses a sufficiently oscillatory phase function e(t), for then the limit will cease to exist.

This forces one to formulate quantum scattering theory ab initio in a phase independent manner, that is, instead of unit vectors one should use rays in Hilbert space. A ray generated by some unit vector 0 determines, and is uniquely determined by, a projection operator Ep onto it. In terms of it the basic formula (1) can be written (A)AV = Tr(E pA) (5) Or even better, instead of merely projection operators onto vectors, why not allow

convex combinations ("mixtures") of these? Such operators W ("von Neumann operators" or "density matrices") can be abstractly characterized as being positive definite and having the finite trace 1. The natural time evolution of some von Neumann operator W = W(O) under the propagator U(t) is given by the formula

W(t) = U(t)WU(t)*

(6)

ON A THEOREM FOR QUANTUM MECHANICAL SCATTERING THEORY

251

(* denotes "adjoint"), plainly independent of modifying U(t) by any variable phase factor. Scattering theory from this vantage point-should be the requirement that to each von Neumann operator W, there should correspond another one WW,

such that the natural evolution of the latter should, as t - oo, approach the free evolution of the former. Mathematically, "to approach" always means the existence of some limit. But for operators "limit" may mean any number of different things. Perhaps the most

natural way of attributing a precise meaning to the word is to introduce a distance function ("metric") d(W, W') in the space of von Neumann operators by the following reasonable definition

d(W, W') = sup{TrA(W - W')I: IIAII < 1, A* = Al,

(7)

where IIAII denotes the usual operator norm of A. This turns the space of states, now mathematically represented as the space of von Neumann operators, into a metric space, allowing one to make mathematically precise statements regarding the limiting behavior of states that depend on one or several discrete or continuous parameters. With this definition, the precise-and phase independent-formulation of the scattering condition becomes lt

d(U(t)W,, U(t)*, Uo(t)WUo(t)*) = 0.

(8)

It is not hard to see that this is equivalent to lt

d(Uo(t)*U(t)W,,, U(t)*Uo(t),W) = 0,

(9)

and when one defines the unitary operator-valued function

Q(t) = U(-t)Uo(t) =

U(t)-1Uo(t)

= U(t)*Uo(t)

(10)

this can be abbreviated to lt

00

1l(t)*WC,Q(t) = W,

(11)

provided one remembers that "limit" is meant here in the sense of the metric (7) introduced above in the space of von Neumann operators. The question then naturally arises: Is this formulation of quantum scattering theory related to the conventional textbook formulation, as expressed by (3) above, and if so how? The short answer is: Yes, it is related; and, in fact, in the simplest possible manner. The situation is quite analogous to the theorem of Wigner mentioned at the beginning. The precise statement is the following THEOREM. Let S2 (t) be an operator-valued function of the real variable t, acting

in a complex separable Hilbert space 7{, subject to no other conditions than, first, that it is isometric for all t, and second, that given any von Neumann operator W, there exists a von Neumann operator W,,, such that (11) holds. Then there exists a phase function ea(t) such that for all 0 in 7-1, the vector-valued function o(t)ct(t)o has a limit in 7-{ as t -* oo. Note that the theorem does not presuppose the continuity of S2(t) or any other degree of regularity as a function of t; nor is it assumed that the S2(t) are necessarily unitary (in distinction to being merely isometric); and, in particular, 1(t) need not arise in the manner (10).

A. LENARD

252

It is convenient to precede the proof of the theorem with two preliminary propositions. LEMMA 1. Under the hypotheses of the theorem, for every vector x of unit norm in the Hilbert space 7-t, there exists a vector z of unit norm and a phase function t; (t)

such that z = limt-,,, (t)ul(t)x. Moreover, if for a given x, two pairs z, fi(t) and z', '(t) have the stated property, then there exists a complex number 77 of absolute value 1 such that z' = riz and limt_,0 £(t) = 71-

PROOF. Let x be given, 11xII = 1, and let W be the projection operator onto the subspace generated by x. Plainly tli

m 00

d(cl(t)*W0052(t), W)

= lim sup{TrA(5l(t)*W0SZ(t) - W): IIAII c 1, A* = Al t

Do

> lim sup{I(1 (t)y, W011(t)y) - I (y, x)121: IIyII = 1}

- tC50

(12)

and therefore lim sup{I (1l(t)y, W0052(t)y) - I(y, x)121: IIyII = 1} =0

t

(13)

00

whence, in particular,

tl 00 (1l(t)x, W0051(t)x) = 1.

(14)

Consider now the spectral decomposition of W00 in terms of its normalized eigenvectors wn with corresponding eigenvalues An > 0, E An = 1. In terms of these we have, by the Cauchy inequality, I (1(t)x, W0Q(t)x)I =

i

A.I(1(t)x, wn)I2

n

a l 1/2 (l((t)xw)l4)1/2. n

/

( 15)

n

But since I (IZ(t)x,wn)12 < 1I1(t)xh12 < 11Q(t) 11 11x112 < 1

(16)

n

it follows that 1,

(Q(t)x,wn)14 < n

(17)

n

and therefore (S2(t)x,WC"0(t)x)I <

(A)

1/2

< 1.

(18)

n

Here we may let t --j oo, so with (14) it follows that (19)

n

But 0 < An < 1 implies A < An, so

ON A THEOREM FOR QUANTUM MECHANICAL SCATTERING THEORY

253

This shows that, for each n, An can only be either 0 or 1; and evidently, the latter may happen only for one case, say n = 0. (14) then becomes th

-00

I (S2(t)x,wo)12 = 1.

(21)

But

At)x,woW < 110(t)xII < Pt)11 < I.

(22)

This shows that

tli

II1(t)xMI = 1.

(23)

Write the complex number (S2(t)x, wo) in terms of its polar coordinates as r(t)t; (t) with r(t) > 0 and lt;(t) I = 1. Then (21) shows that limt_,00 r(t) = 1. But then 0 < 116(t)1 (t)x - woMM2 = 1 + j1 (t)xI12 - 2r(t) < 2 - 2r(t),

and as the rightmost function tends to zero as t ---> oo, we have tli m 6(t)S2(t)x = wo.

(24)

This completes the proof of the existence statement in the lemma, with z = wo. As regards the second part, note that from (24), it follows that (z, z') = tlim 6(t) e' (t) Pt)xII2,

(25)

so that with (23), limt-,0 l;(t) e'(t) = limt_,0 fi) exists, say 77, and must lie on the unit circle of the complex plane. Thus z and z' are proportional and z' = riz. This completes the proof of Lemma 1.

The correspondence that assigns to any unit vector x a pair z, e(t) is extended by homogeneity in a natural way from unit vectors x to vectors of arbitrary norm, keeping the phase function the same (of course, when x = 0 then z = 0 and the phase function is irrelevant). LEMMA 2. If x' and x" are two vectors in the Hilbert space and z', l;'(t) (resp. Z", "(t)) are corresponding pairs as in Lemma 1, then limt.0 £ (t) exists and is of absolute value 1.

PROOF. Let x' and x" be two vectors in the Hilbert space and z', 1='(t) (resp.

z", "(t)) the corresponding pairs, as assumed. Let A' and A" be two arbitrary complex numbers, and furthermore, z, l= (t) be the pair that corresponds to the vector x = A'x' +.X"x" via Lemma 1. Consider the following identity

z-

(t) (A'x' + A"x")

(26)

+ A'e(t) (S2(t)x' - t; (t) z')

+ a" t; (t) (1(t)x" - "(t) z") The expressions on each of the last three lines tend to zero. Hence we have also z

tli m e(t) (A'

e' (t) z' + A'/ err (t) z")

(27)

But from this and t Z= lim e(t)1l(t)(.\'x' +."x")

(28)

A. LENARD

254

follows JA'x' + A"x"112 = jizll2 = tli

'(t) z' + A"

IIA'

(t) z"II2

(29)

Viewing this as the equality of two quadratic forms in the arbitrary variable complex numbers A', A", we extract the equality of the coefficients of )' A" on the two sides, obtaining

tl

(x x(30)

m'(t)C(t)(z,

Plainly (z', z") = 0 if and only if (x', x") = 0. If this is not the case, then the existence of the limit

V/ (t)

lim

t--.oo

= (x. x")

(31)

'(t) - (zt z")

is established. In the exceptional case of those inner products vanishing, let a third vector x"' in Hilbert space be such that neither (x', x"') nor (x", x"') vanish. Then the limit required still exists, namely,

= lim

'(t)

oo

t

lim

f"(t) V/ (t)

t--.oo "'(t) V (t)

(x"r, x") (x. X///)

_ (z"' z")

(32)

1

Note that the ratios of inner products on the right hand sides of (31) and (32), being limits on the unit circle, are necessarily themselves of absolute value 1 as required. This completes the proof of Lemma 2. PROOF OF THE THEOREM. Let x0 be a non-zero vector in the Hilbert space,

arbitrary but to be regarded as fixed, and let zo, l;o (t) be a pair corresponding to it via Lemma 1. Let z, fi(t) be a pair that corresponds to an arbitrary vector x. Another pair, equivalent by the lemma, is z' = Oz, '(t) = fo(t), where 0 = a limit that exists by Lemma 2 and has absolute value 1. Then limt-,", lim

t-+oo

o(t)1l(t)x = z',

(33)

a formula that defines the (obviously linear) transformation: x -> z' in the Hilbert space such that W11 = 11x1j, that is to say, the transformation fZo,, so defined is isometric. (28) may then be written as the existence of a strong limit lim 0(t)f2(t) = 000

(34)

i

as required. This completes the proof of the theorem.

Note that our theorem does not require any smoothness assumptions on the operator family Q(t), other than the existence of the limits indicated in (3). It is somewhat surprising perhaps that so seemingly weak an assumption still suffices for such a strong conclusion, namely, existence of the limit (34). In scattering theory Q (t) = U(-t)Uo(t), where U(t) and Uo(t) are the "natural" (resp. "free") strongly continuous unitary propagators for the scattering system considered. The continuity hypothesis in this situation allows one to show that in this case the variable phase factor fi(t) that enters the statement of the theorem may be chosen to be a differentiable function of t. Let us show why this is the case. With 11(t) = U(-t)Uo(t), we have lim 1;(t)U(-t)Uo(t) = SZ00.

i

(35)

ON A THEOREM FOR QUANTUM MECHANICAL SCATTERING THEORY

255

Setting t + tl for t and making use of the group property of the unitary operator functions, we get immediately

s-li l;(t+tl)U(-t)U0(t) =

(36)

The last two limits imply that also

t-li -(tfi(+t)ti) 9 =

(37)

But this is possible only if the complex number limit

slim

+ ti) = w(tl)

(38)

exists. Thus (replacing tI by t) we have

w(t)SZ = U(t)Q,,U0(-t).

(39)

Because U(t) and U0(t), by hypothesis, are strongly continuous functions of t, the function w(t) with values on the unit circle is necessarily continuous. Moreover, (38) implies that w(t)w(ti) = w(t + t1) and therefore there exists a real number at such that w(t) = e"t Vt (40) If x be any vector, we have from (37) that t li (41) ,, x , U(-t)Uo(t)x)] = 1.

m 00

On account of I(1l x, U(-t)U0(t)x)j < 1, (41) implies tli

(t)(1 x, U(-t)Uo(t)x) = 1.

(42)

Let x be in the domain of the self-adjoint generator Ho of the unitary group Uo(t); then the vector-valued function Uo(t)x depends (strongly) differentiably on t. We

now consider that the inner product (52x, U(-t)Uo(t)x) may be written in the equivalent forms (U(t)52,,,x, Uo(t)x) = w(t) (52 0U0(t)x, Uo(t)x), and the last way

of writing makes it clear that it is a differentiable function of the parameter t. This shows that f (t) may be replaced, if needed, by an asymptotically equivalent differentiable function.

A Historical Note The theorem proved here originated from conversations with the late Professor

Josef M. Jauch in the early 1970s when he visited Indiana University. Jauch, a distinguished theoretical physicist (and my Ph.D. advisor at the State University of Iowa during 1949-53) was at the time working on the mathematical foundations

of scattering theory, a work that he was unfortunately not allowed to finish on account of his premature death. He was aware that the usual asymptotic approach of quantum evolution to a "free" evolution (the "scattering limit") is not valid for the case of Coulomb scattering, where a slowly oscillating phase function is an ob-

stacle to the existence of the conventional limit. Therefore-but also for reasons of principle-he sought a presentation of scattering theory that would obviate reference to any phase at all. This led him to formulate the asymptotic condition of scattering theory in a manner indicated by (7) and (8) above. In conversations with him, I learned of this topic, and of Jauch's challenge to dig out of his scattering condition (in the present essay, the hypotheses of the

256

A. LENARD

theorem) a description closer to the conventional one. After some thought about.

the matter, I concluded that the customary asymptotic limit did in fact exist, depending merely on the existence of a certain phase function that could be proved

rather than just assumed. Some notes of this were shown to Jauch (at the time still handwritten, as word processors were not yet in common use), who was very pleased by this mathematical development. The present write-up is basically from these notes, improved only in minor ways. In the summer of 2002 a conference took place in the Hungarian university town of Pecs, celebrating the many contributions to physics and mathematics of the late Eugene Wigner who was born 100 years before. I intended to speak at this conference on the topic of this paper; and I looked forward eagerly to the trip, topped by the anticipated pleasure of talking in my native tongue with the many Hungarian colleagues attending. But, unfortunately, I could in the end not go in good conscience because at the time my sister, mortally ill, was approaching the end of her life. But a friend at the conference, John Klauder, volunteered to read the manuscript for me, for which I remain much indebted to him.

A quirk of this story is that when it came to publication of papers read at the Pecs conference, my contribution was disregarded as "not sufficiently topical." Be that as it may, it remained unpublished until now. But perhaps that makes it especially appropriate to send it to my friend Barry Simon, whose 60th birthday was celebrated recently by a conference in Pasadena, in the hope that it will cause him a bit of pleasurable reading, and as a small token of my affection, as well as my respect, for the rich literature he produced over the years on the mathematics of our favorite subject: Quantum Mechanics.

DEPARTMENT OF MATHEMATICS, INDIANA UNIVERSITY BLOOMINGTON, IN 47405, U.S.A.

E-mail address: lenard@indiana. edu

Proceedings of Symposia in Pure Mathematics Volume 76.1, 2007

Analytic Criteria in the Qualitative Spectral Analysis of the Schrodinger Operator Vladimir Maz'ya Dedicated to Barry Simon on the occasion of his 60th birthday ABSTRACT. We survey a number of topics in the qualitative spectral analysis of the Schrodinger operator -A + V. In particular, some old and new results concerning the positivity and semiboundedness of this operator as well as the structure of different parts of its spectrum are considered. Attention is focused on conditions both necessary and sufficient, as well as on their sharp corollaries.

CONTENTS 1.

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Introduction Wiener Capacity Equality of the Minimal and Maximal Dirichlet Schrodinger Forms Closability of Quadratic Forms Positivity of the Schrodinger Operator with Negative Potential Trace Inequality for 52 = R' Positive Solutions of (0 + V)w = 0

Semiboundedness of the Operator -0 - V Negative Spectrum of -h0 - V Rellich-Kato Theorem

Sobolev Regularity for Solutions of (-0 + V)u = f Relative Form Boundedness and Form Compactness Infinitesimal Form Boundedness Kato's Condition K.,, Trudinger's Subordination for the Schrodinger Operator Discreteness of the Spectrum of -0 + V with Nonnegative Potential

Strict Positivity of the Spectrum of -0 + V Two-Sided Estimates for the Bottom of the Spectrum and Essential Spectrum

2000 Mathematics Subject Classification. 35P, 47F05. Key words and phrases. Schrodinger operator, qualitative spectral analysis, negative spectrum, discreteness of spectrum, essential spectrum, Wiener's capacity, pointwise multipliers. Supported in part by NSF Grant DMS-0500029. ©2007 American Mathematical Society 257

V. MAZ'YA

258

19. 20.

Structure of the Essential Spectrum of Hv Two Measure Boundedness and Compactness Criteria

References

1. Introduction The purpose of this article is to survey various analytic results concerning the Schrodinger operator -A + V (x) obtained during the last half-century, including several quite recently. In particular, the positivity and semiboundedness of this operator, as well as the structure of different parts of its spectrum, are among the topics touched upon.

In the choice of material I aim mostly at results of final character, i.e., at simultaneously necessary and sufficient conditions, and their consequences, the best possible in a sense. Another motivation for the inclusion of any particular problem in this survey is my own involvement in its solution. Naturally, the selection of topics is far from exhaustive. For instance, conditions for the essential selfadjointness of the Schrodinger operator and for the absence of eigenvalues at positive energies, as well as properties of eigenfunctions and bounds

on the number of eigenvalues, are not considered here. These and many other themes of qualitative spectral analysis have been discussed in the comprehensive surveys [91, 12, 22], and, of course, in the classical Reed and Simon treatise [75][78]. However, the information collected in what follows has been quoted rarely-if ever-in the quantum mechanics literature. Due to space limitation, I write only about the linear Schrodinger operator with scalar potential, although some of the results below can be modified and applied to nonlinear, magnetic, relativistic Schrodinger operators and even general elliptic operators with variable coefficients. For the same reason, proofs are supplied just in a few cases, mostly when a source does not seem readily available.

2. Wiener Capacity The capacity of a set in R' will come up frequently in the present article. This notion appeared first in electrostatics and was introduced to mathematics by Wiener in the 1920's. Since then several generalizations and modifications of Wiener's

capacity have appeared: Riesz, Bessel, polyharmonic capacities, p-capacity and others. They are of use in potential theory, probability, function theory and partial differential equations. The capacities provide adequate terms to describe sets of discontinuities of Sobolev functions, removable singularities of solutions to partial differential equations, sets of uniqueness for analytic functions, regular boundary points in the Wiener sense, divergence sets for trigonometric series, etc. (see, e.g., [1, 14, 43, 56]). Some applications of Wiener's and other capacities to the theory of the Schrodinger operator are presented in this survey. Let Il be an open set in R and let F be a compact subset of Q. The Wiener (harmonic) capacity of F with respect to 1 is defined as the number

Q IVu12dx

capiF = inf{J

:

u E Co (S2), u > 1 on F}.

(1)

ANALYTIC CRITERIA FOR THE SCHRODINGER OPERATOR

259

We shall use the simplified notation capF if 1 = R'. The capacity cap0F can be defined equivalently as the least upper bound of v(F) over the set of all measures v supported by F and satisfying the condition < 1,

where G is the Green function of the domain Q. If Si! = R3 then it is just the electrostatic capacity of F. It follows from the definition (1) that the capacity is a nondecreasing function of F and a nonincreasing one of Q. We have Choquet's inequality cap0(F1 fl F2) + cap0(FI U F2) < cap0 Fl + cap0 F2 for arbitrary compact sets Fl and F2 in SZ [17]. It is easy to check that the Wiener capacity is continuous from the right. This means that for each 6 > 0, there exists

a neighborhood G, F C G C G C SZ such that for each compact set Fl with F C Fl C G, the inequality cap0 FI < cape F + e holds.

Let E be an arbitrary subset of ft The inner and outer capacities are defined as numbers

cape E = sup cap0 F,

F compact in Q,

cap0 E = inf cap G, GDE -Q

G open in Q.

FCE

It follows from the general Choquet theory that for each Borel set both capacities coincide [17]. Their common value is called the Wiener (harmonic) capacity and will be denoted by cap0 E. By v.,, we denote the volume of the unit ball in Wn and let mes,,,F stand for the n-dimensional Lebesgue measure of F. By the classical isoperimetric inequality, the following isocapacitary inequalities hold (see [53, Sect. 2.2.3])

cap0 F > nv2/' (n - 2) I(mesn

S)(2-n)/n

F)(2-n)/nI-1

- (mesa

if n > 2

(2)

and

cap0 F > 47r (log

mes2 1 - I ) mes2 F)

if n = 2.

(3)

F)(n-2)/n.

(4)

In particular, if n > 2 then capF > nv2/n(n - 2) (mesn

If S2 and F are concentric balls, then the three preceding estimates come as identities. Using Wiener's capacity, one can obtain two-sided estimates for the best constant in the Friedrichs inequality IIuIIL2(B1)

C II"UIIL2(B1),

(5)

where BI is a unit open ball and u is an arbitrary function in C°° (BI) vanishing on a compact subset F of BI.

V. MAZ'YA

260

PROPOSITION 2.1. [48] The best constant C in (5) satisfies

C < c(n) (cap F)

(6)

112,

where c(n) depends only on n.

It is shown in [61] that (6) holds with c2(n) = 4vn,n-1(n2 - 2). Proposition 2.1 has the following partial converse. PROPOSITION 2.2. Let

cap F < 7 cap B1, where y E (0, 1). Then any constant C in (5) satisfies

C > c(n,ry)(cap F)

(7)

(8)

1/2.

This assertion was proved in [48] (see also [53, Ch. 10]) with a sufficiently small y = y(n). The present stronger version is essentially contained in [60]. Both propositions were used in [42] to derive a discreteness of spectrum criterion for the magnetic Schrodinger operator.

3. Equality of the Minimal and Maximal Dirichlet Schrodinger Forms Let V be a nonnegative Radon measure in Q. Consider the quadratic form

Q[u, u] = f IV UI2 dx + f IuI2 V(dx).

(9)

The closure of Q defined on the set

{u E C' (R') : suppu C 11, Q[u, u] < oo}, which may contain functions with noncompact support, will be denoted by Qmax. Another quadratic form Qmin is introduced as the closure of Q defined on Co (a). The question as to when the equality Qmax = Qmin holds has been raised by Kato [39]. Simon studied a similar question concerning the magnetic Schrodinger operator in 1Rn and proved Theorem 3.1 below in the case where V is an absolutely continuous measure [89]. The following necessary and sufficient condition for this equality was obtained in [15].

THEOREM 3.1. (i) If either n < 2 or n > 2 and cap(Rn\1) = oo, then Qma _ Qmin

(ii) Suppose that n > 2 and cap(Rn\1) < oo. Then Qmax = Qmin if and only if

V(1 \F) = oo

for every closed set F C S2 with capF < oo.

(10)

Example. Consider the domain SZ complementing the infinite funnel {x = (x', xn) : xn > 0, Ix'I < f (xn)}, where f is a continuous and decreasing function on [0, oo) subject to f (t) < c f (2t).

One can show that cap (R \cl) < oo if and only if the function f (t)n-2 for n > 3 and the function (log(t/ f (t)))-1 for n = 3 are integrable on (1, oo). REMARK 3.1. Note that the equality Qmax = Qmin is equivalent to (10) in the particular case SZ = R. Assume that V(dx) = V (x) dx, where V is a positive function in L o°(Rn) locally bounded away from zero. According to [15], condition (10) is necessary for the essential selfadjointness of the operator V-1i. on L2(V dx) with domain Co (W )

ANALYTIC CRITERIA FOR THE SCHRODINGER OPERATOR

261

4. Closability of Quadratic Forms a nonnegative Radon measure in ft Let 1 be an open set in By L2(1) we denote the completion of C000 (1) in the norm V uL2(s1) One says that the quadratic form

f Iu12V(dx)

kzi L2(o,V(dX))

(11)

which is defined on Co (1k) is closable in L2'(52) if any Cauchy sequence in the space

L2(1,V(dx)) converging to zero in L2(1) has zero limit in L2(Q, V(dx)). We call the measure V absolutely continuous with respect to the harmonic capacity if the equality cap E = 0, where E is a Borel subset of S2, implies V(E) = 0. For example, let V be the cp-Hausdorff measure in R', i.e.,

inf E p(ri),

V(E) = lim

E->+O (B(i)}

where cp is a nondecreasing positive continuous function on (0, oo) and {C3U)} is any covering of the set E by open balls 13M with radii ri < e. It is well known that this measure is absolutely continuous with respect to the Wiener capacity if

"o W(t)t1-' dt < no, J0

and that this condition is sharp in a sense (see [14]). Needless to say, any measure V absolutely continuous with respect to the ndimensional Lebesgue measure is absolutely continuous with respect to the capacity.

The following result was obtained in [49].

THEOREM 4.1. The quadratic form (11) is closable in L2'(1l) if and only if V is absolutely continuous with respect to the Wiener capacity. The closability of (11) in L2(52) is necessary for the Schrodinger operator -0V, formally associated with the form S[u, u] :=

Jn

I V u12 dx -

f

lul2 V(dx),

u E Co (SZ),

(12)

Z

to be well-defined. Dealing with the Schrodinger operator formally given by the expression -0+V and acting in L2(1) we need the following notion of a form closable in L2(1l).

By definition, the quadratic form Q defined on Co (52) by (9) is closable in L2(1) if any Cauchy sequence in the norm Q[u, u]1/2 which tends to zero in L2(S2) has zero limit in the norm Q[u, u]1/2. This notion is equivalent to the lower semicontinuity of Q in L2(1l) [40, 87, 88].

THEOREM 4.2 (see [53], Sect. 12.4, 12.5). The form Q defined on Ca (1) is closable in L2 (9) if and only if V is absolutely continuous with respect to the Wiener capacity.

We assume in what follows that the measure V is absolutely continuous with respect to the Wiener capacity.

V. MAZ'YA

262

5. Positivity of the Schrodinger Operator with Negative Potential The next result was obtained in [47, 49] (see also [53, Thm. 2.5.2]). THEOREM 5.1. Let 1 be an open set in ]Rn, n > 1, and let V be a nonnegative Radon measure in Q. The inequality

L.

luI2V(dx) <

V uj2 dx

(13)

holds for every u E Co (52) provided

V(F) cap0F

1

4

(14)

for all compact sets F C Q. A necessary condition for (13) is V(F) < 1,

cap0F -

(15)

where F is an arbitrary compact subset of Q. REMARK 5.1. In inequalities (14), (15) and elsewhere in similar cases, we tacitly assume that vanishing of denominator implies vanishing of numerator, and we may choose any appropriate value of the ratio.

REMARK 5.2. The necessity of (15) is trivial. The proof of sufficiency of (14) in [49, 53] shows that (14) implies (13) even without the requirement V > 0, i.e., for an arbitrary locally finite real-valued charge in Q. REMARK 5.3. The formulation and proof of Theorem 5.1 do not change if we assume that 52 is an open subset of an arbitrary Riemannian manifold. Theorem 5.1 immediately gives the following criterion. COROLLARY 5.2. The trace inequality

L holds for every u E C0 (Sl) if and only if sup

V(F)

FCC ap0F The bounds

< oo.

and 1 in (14) and (15) are sharp. The gap between these sufficient

4 and necessary conditions is the same as in Hille's nonoscillation criteria for the operator

-u"-Vu,

V>0,

on the positive semi-axis R': xV((x, oo))

4

and xV((x, oo)) < 1

(17)

for all x > 0 [36]. By the way, conditions (17) are particular cases of (14) and (15)

withn=1 and 52=ll

.

O

ANALYTIC CRITERIA FOR THE SCHRODINGER OPERATOR

263

Combining Theorem 5.1 with isocapacitary inequalities (2)-(4), we arrive at sufficient conditions for (13) whose formulations involve no capacity. For example, in the two-dimensional case, (13) is guaranteed by the inequality < 47r V(F) mes2 ci log

mes2 F The sharpness of this condition can be easily checked by analyzing the well known Hardy-type inequality

f

dx < 4 fn IV u(x)I2 dx IxI2(1 g Ix1)2

where u c Co (SZ) and 52 is the unit disc. For some history of Hardy's original inequality see, e.g., [23], [97, Sect. 16]. For more references on best constants for extended Hardy-type inequalities, see, for instance, [24, 35], and the references therein. The sufficiency of (14) for the inequality (13) can be directly obtained from the following more precise result. THEOREM 5.3 ([54]). Let n > 1 and let v be a nondecreasing function on (0, oo) such that

V(F) < v(cap0F) for all compact subsets F of Q. If

j

00

00 v(T)d(T1)Iv'(T)I2dT

0

for all absolutely continuous v with v' E L2(0, oo) and v(0) = 0, then (13) holds. One example of the application of this assertion is the following improvement of the Hardy inequality which cannot be deduced from (14):

f

dx

1

/

'\Bl u21 1 + (n - 2)2(log IzI)2) Ix I2

4

(n - 2)2

J.

\B1

Du 2 dx,

where u E C000 (R n\B1) and n > 2.

Here is a dual assertion to Theorem 5.3 which is stated in terms of the Green function G of S2 and does not depend on the notion of capacity.

THEOREM 5.4. Let VF be the restriction of the measure V to a compact set F C Q. Inequality (13) holds for every u E Co (S2) provided

G(x,y)VF(dx)VF(dy) < 4V(F)

(18)

jo 10

for all F. Conversely, inequality (13) implies

ffG(x,Y)VF(dx)VF(dY) < V(F).

(19)

SKETCH OF THE PROOF. Let u be a nonnegative function in Co (S2) such that

u> 1 on F. Then V(F) <

(J

1/2

/

G(x,y)VF(dx)VF(dy))

IIVuIIL2(o)

V. MAZ'YA

264

which in combination with (18) gives (14). The reference to Theorem 5.3 gives the sufficiency of (18). Let (13) hold. Then 2

< V(F)

L

IIVu1I2

L2(o).

Omitting a standard approximation argument, we put

u(x) = fG(xY)VF(dY) and the necessity of (19) results. The next assertion follows directly from Theorem 5.1.

COROLLARY 5.5. The trace inequality (16) holds if and only if there exists a constant C > 0 such that

JjG(x,y)VF(dx)VF(dY) <_ CV(F)

(20)

for all compact sets F in Q. REMARK 5.4. Note that Theorem 5.1 and Corollary 5.5 include the case n = 2 when the existence of the Green function is equivalent to cap(1R2\11) > 0 (see [43]). REMARK 5.5. Obviously, the pointwise estimate

i

G(x, y) V(dx) < 4

(21)

implies (18) and hence it is sufficient for (13) to hold.

It is well-known that the operator S obtained by the closure of the quadratic form S[u, u] defined by (12) generates a contractive semigroup on Lp(1), p E (1, oo) if and only if

c,

gulp-2uSudx > 0

(22)

for all u E Co (1) ([45, 62], and [76, Thm. X.48]). The following analytic conditions related to (22) can be deduced from Theorem 5.1. COROLLARY 5.6. Let p c (1, oo) and p' = p/(p - 1). The operator S generates a contractive semigroup on Lp(1) if V(F)

1

< FcQ caPOF - pp' sup

and only if

V(F)

4

sup < FCO capOF pp

ANALYTIC CRITERIA FOR THE SCHRODINGER OPERATOR

265

6. Trace Inequality for 11 = Rn The inequality, u

2V(dx) < c f IVu12 dx, `du E Ca (Rn),

(23)

R

deserves to be discussed in more detail. First, (23) for n = 2 implies V = 0 . Let n > 2. Needless to say, by Theorem 5.1 the condition sup V(F) < o0 F capF

(24)

where the supremum is taken over all compact sets F in Rn, is necessary and sufficient for (23). Restricting ourselves to arbitrary balls B in Rn, we have by (24) the obvious necessary condition V(B) (25) < oo. sup (mesnB)1-2/n

On the other hand, using the isocapacitary inequality (4), we obtain the sufficient condition sup (me

< oo,

F

(26)

VF))-2/n

where the supremum is taken over all compact sets F in Rn. Moreover, the best value of C in (23) satisfies 4vn2/n

C

V(F)

- n(n - 2) --F'

(meSnF)1-2/n

and the constant factor in front of the supremum is sharp [49]. Although (25) and (26) look similar, they are not equivalent in general. In other words, one cannot replace arbitrary sets F in (24) by balls. Paradoxically, the situation with the criterion (20) in the case 52 = Rn is different. In fact, Kerman and Sawyer [41] showed that the trace inequality (23) holds if and only if for all balls B in Rn / V(dx) V(dy) < CV(B). (27) JB Ix - y1n-2 Maz'ya and Verbitsky [63] gave another necessary and sufficient condition for

I

(23): SuxP

Il(I1V)2(x)
(28)

where IS is the Riesz potential of order s, i.e., ISV(x) :=

J

xV (yi n) s

The following complete characterization of (23) is due to Verbitsky [99]. For every dyadic cube Po in Rn, sup

V(P)2 00 E (mesnP)1-2/n < V (Po) 1

(29)

where the sum is taken over all dyadic cubes P contained in P0 and C does not depend on P0.

V. MAZ'YA

266

Let us assume that V is absolutely continuous with respect to mesa, i.e.,

V(F) =

V(x) dx. F IF

Fefferman and Phong [28] proved that (23) follows from sup

JB

Vt dx (mesaB)1-zt/a

B

w ith some t E (1 , n /2)

< oo

.

(30)

This can be readily deduced from Sawyer's inequality [82]

f (M tf )

v( dx) < C sup

( 31 ) (mesaB) ) -2t/7_" f f 2 d x, where v is a measure, f > 0 and Mt is the fractional maximal operator defined by 2

lB f (y) dy

.Mt f (x) = sup

B (mesnB) 1-t/a where B is an arbitrary ball centered at x. In fact, for any 5 > 0

Iif (x) = (n-1)

f 0

r-n

f

(x)+(t-1)-1b1-tMtf

f (y) dy dr < (n-1) (S.M f

Br(x)

(x)),

where M is the Hardy-Littlewood maximal operator. Minimizing the right-hand side in 5, we arrive at the inequality

Iif (x) <- (t - 1)t (Mtf (x))11t(M f (x))1-1/t

(32)

([34], see also [1, Sect. 3.1]). Now, (32) implies III1 f II L2(Vdx) < C IIMtfIIL2(Vtdx) IIM fiIL2(R-).

Therefore, by (31) and by the boundedness of M in L2(Rn) we arrive at the inequality

J Vt dx III1 f IILZ(Vdx) < C suB ((me n

1/zt

IIf

BB)1-2t/a

which is equivalent to the Fefferman-Phong result. Their result was improved by Chang, Wilson and Wolff [16] who showed that if co is an increasing function [0, oo) -+ [1, oo) subject to

j(r(r))_1dr < 00,

(33)

then the condition fB, V(x) cp(V(x)(diamB)2) dx < 00

su B11

(mesa B)1-2/n is sufficient for (23) to hold. The assumption (33) is sharp. By Remark 5.5, the condition V(dy) sup < 00 x Ix - yIn-2 "

f

(34)

(35)

ANALYTIC CRITERIA FOR THE SCHRODINGER OPERATOR

267

is sufficient for (23). It is related to Kato's condition frequently used in the spectral

theory of the Schrodinger operator (see Section 14 below for more information about Kato's condition). Note that (35) is rather far from being necessary since it excludes (23) with V(dx) _ xH-2dx, i.e., the classical Hardy inequality, unlike other conditions just mentioned. We conclude this section with the observation that the multiplicative inequality

u2 V(dx) < C (J

fRis equivalent to

IV u12dx)T (f

sup

u2 dx)1 T,

V(B)

0 < T < 1,

< 00

(36)

(37)

(mes,,,B)1-2T/n

([53, Thm. 1.4.7]; see Section 15 below for further development).

7. Positive Solutions of (A + V)w = 0 Inequality (23) is intimately related to the problem of existence of positive solutions for the Schrodinger equation -A w = V W

in R'.

(38)

Allegretto [5] and Piepenbrink [72] showed that the existence of w > 0 is equivalent

to the positivity of the operator -A - V for relatively nice V. Since then there has been an enormous literature on this subject. In addition to the detailed list of references in the contribution to this Festschrift by Pinchover [74], we refer to Agmon [2], Allegretto [6], Metafune and Pallara [66], Moss and Piepenbrink [68], Persson [71], Piepenbrink [73], and Schechter [83] for additional results of this type. The following result due to Hansson, Maz'ya and Verbitsky [33] shows that the condition 11(I1V)2 < CIIV (39) is equivalent (up to values of C) to the existence of w > 0. THEOREM 7.1. (i) If -A w = V w has a nonnegative (weak) solution w, then I1V < oc almost everywhere and there exists a constant C1 = C(n) such that

I,((I1V)2)(x)
(40)

(ii) Conversely, there exists a constant C2 = C2 (n) such that if (40) holds with C2 in place of C1, then there exists a positive solution w to (38) which satisfies

IV log w(x)I < CI, w(x) and

w(x) > C 0xI + 1)-' for some positive constants C and c.

In addition, if I2V(x) < oo a.e., then there is a solution w such that e12 v(X) < W (x) < ec l2 v(X).

All the constants depend only on n.

V. MAZ'YA

268

8. Semiboundedness of the Operator -A - V We formulate some consequences of Theorem 5.1 concerning the inequality

in IV

u12

dx - f Iu12V(dx) > -C f Iu12 dx,

u E Co (Q).

THEOREM 8.1 ([49]; see also [53], Sect. 2.2). (i) If

limsup{ 6-.o

V(F) :FCSZ, diamF<S}< cap0F

-

(41)

then the quadratic form S[u, u] defined by (12) is semibounded from below and closable in L2(1). (ii) If the form S[u, u] is semibounded from below in L2(f ), then lim sup{ 6-.o

M(F)

cap0F

: F C S2, diamF < b} < 1.

(42)

COROLLARY 8.2. The condition

lim sup { s_ o

V(F) F C SZ, diamF < S} = 0 cap0F

(43)

is necessary and sufficient for the semiboundedness of the form

Sh[u,u] := h

f

IV u12dx -

f IuI2 V(dx)

(44)

in L2 (1) for all h > 0. COROLLARY 8.3. The inequality

In Iu12V(dx) < C

f

(IV

u12

+ Iui2) dx,

(45)

where u is an arbitrary function in Co (SZ) and C is a constant independent of u, holds if and only if there exists 8 > 0 such that sup{

V(F) cap0F

F C 52, diamF < b } < oo.

9. Negative Spectrum of -h0 - V Investigation of the negative spectrum of the Schrodinger operator with negative potential can be based upon the following two classical general results. LEMMA 9.1 ([29]). Let A[u, u] be a closed symmetric quadratic form in a Hilbert space H with domain D[A] and let y(A) be its positive greatest lower bound. Further, let B[u, u] be a real-valued quadratic form, compact in D[A]. Then the form A - B is semibounded below in H, closed in D[A], and its spectrum is discrete to the left of y(A).

LEMMA 9.2 ([30]). For the negative spectrum of a selfadjoint operator A to be infinite, it is necessary and sufficient that there exists a linear manifold of infinite dimension on which (Au, u) < 0.

ANALYTIC CRITERIA FOR THE SCHRODINGER OPERATOR

269

Note also that by Birman's general theorem [9], the discreteness of the negative spectrum of the operator

Sh= -h/ -V

forallh>0

generated by the closure of the quadratic form Sh [u, u] defined by (44) is equivalent to the compactness of the quadratic form (46) z

with respect to the norm u12

+ Iu12)

dx)

Analogously, the finiteness of the negative spectrum of the operator Sh is equivalent to the compactness of the quadratic form (46) with respect to the norm II V uIIL2(sl) (We note that the idea that discreteness of the negative essential spectrum is equivalent to some kind of relative compactness is originally due to Weyl, and we refer, e.g., to Reed and Simon [78, Sect. XIII.4] for a detailed discussion.) Historically, the first paper on finiteness of the negative spectrum is due to Bargmann [7]. Moreover, by [8] (see also [9] and [86]), the number of negative eigenvalues of Sh coincides with the number of eigenvalues Ak, Ak < h-1 of the Dirichlet problem

-Du-AVu=0,

uEL2I(f2).

The next theorem contains analytic conditions both necessary and sufficient for the Schrodinger operators Sh to have discrete, infinite or finite negative spectra for all h > 0. These characterizations were obtained by the author from either sufficient or necessary conditions, analogous to (41) and (42), for the operator S

independent of the parameter h to have a negative spectrum with the properties just listed ([47, 49]; see also [53, Sect. 2.2]). THEOREM 9.3. (i) A necessary and sufficient condition for the discreteness of

the negative spectrum of Sh for all h > 0 is

V(F) cap0F FCO\BP, diamF<1 sup

0

asp

oc.

(47)

Here and elsewhere B. = {x c Rn : 1xI < p}. (ii) A necessary and sufficient condition for the discreteness of the negative spectrum of Sh for all h > 0 is sup FCQ\BP

V(F) -+ 0 as p -+ oo. cap0F

(48)

(ii) A necessary and sufficient condition for the negative spectrum of Sh to be infinite for all h > 0 is V(F) sup = oo. FCO cap0F

(49)

Needless to say, simpler sufficient conditions with cap0F replaced by a function of mesnF follow directly from (47)-(49) and the isocapacitary inequalities (2)-(4).

V. MAZ'YA

270

10. Rellich-Kato Theorem By the basic Rellich-Kato theorem [79, 37], the selfadjointness of -A + V in L2(Rn) is guaranteed by the inequality IVuIIL2(R)
V(y)12 dy < c(n).

(51)

Here and elsewhere Br(x) = {y E ll8n Iy - x1 < r}. This class of potentials is known as Stummel's class Sn [94] which is also defined for higher dimensions by :

=0

forn>5,

- y1)IV(y)12dy) =0

forn=4.

Ix-y14-nIV(y)12dy)

1lm(sup J B,.(x)

lm(supJ

log(Ix B,.(x)

Although the condition V E Sn is sufficient for (50) for every n, it does not seem

quite natural for n > 4. As a matter of fact, it excludes the simple potential V(x) = clxl-2 obviously satisfying (50), if the factor c is small enough.

If n > 5, a characterization of (50) (modulo best constants) results directly from a necessary and sufficient condition for the inequality IIVuIIL2(R")
uEC0 (W),

found in [50]. We claim that the Rellich-Kato condition (50) holds in the case n > 4 if and only if there is a sufficiently small c(n) subject to

f IV(y)2dy F

sup

c(n)

(52)

cap2F (the values of c(n) in the sufficiency and necessity parts are different). Here and elsewhere cap,,, is the polyharmonic capacity of a compact set F defined in the case diamP<1

2m
IVm ul2 dx :

u c C, (Rn), u > 1 onF},

For 2m = n one should add IIutti2(R) to the

where V n = {am/axil last integral. The condition

IF I V (y)12 dy

sup diamF<S

F

caP2F

,0 as b-+0

(53)

is necessary and sufficient for (50) to hold with an arbitrary a and b = b(a). An obvious necessary condition for (50) is r4-n

for n > 5 Supr<1,xEIl8" (log r) 1 fB,.(x) IV(y)12 dy < c(n) for n = 4,

f Supr<1,xER"

fB,(x) I V (y)12 dy < c(n)

(54)

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271

where c(n) is sufficiently small. Standard lower estimates of cap2 by mesa combined with the criterion (52) give the sufficient condition

for n > 5 for n = 4.

JSupdiamF
fF jV (y) I2 dy < c(n)

(55)

Though sharp and similar looking, (54) and (55) are not equivalent. We omit a discussion of other sufficient and both necessary and sufficient conditions for (52) to hold parallel to that in Section 5 (analogs of the Fefferman-Phong, KermannSawyer, Maz'ya-Verbitsky, and Verbitsky conditions mentioned in Section 5). Finally, we observe that the inequality IIVuMIL2(Rn) <_ CIAU11L2(R-)IIMIIL2(R

)

holds for a certain T E (0,1) and every u E C0 (II8) if and only if for all r E (0,1)

sup J

I V (y) I2 dy < C rn-4T B,.(x)

(see [53, Thm. 1.4.7]).

11. Sobolev Regularity for Solutions of (-A + V)u = f Let m be an integer > 2 and let W2(ll8n) denote the Sobolev space of functions u E L2(Rn) such that V,,,,u E L2(Rn). Obviously, the operator

-0 + V : W2 (Rn) -* W2,-2 (Rn) is bounded if and only if V is a pointwise multiplier acting from W2 (Rn) into W2,-2 (R-). (We use the notation V E M(W2(lRn) - W2 -2(lRn)). According to [58, Ch. 1], necessary and sufficient conditions for V E M(W2(lRn) -* W2,-2 (R n)) have the form (56) Sup f (IVm-2V(y)I2 + IV(y)I2) dy < oo xEIP

i(x)

for n < 2m and J IDm-2V(y)12dy sup

F

diamF<1

capmF

+ sup J

V (y) 2 dy < oo

(57)

Bl(x)

xE1R'

for n > 2m.

Two-sided estimates for the essential norm of a multiplier, W(1R7) -+

W2,-2 (R n) found in [58, Ch. 4], lead to the following regularity result:

THEOREM 11.1. Let cp e W2 (lRn, loc) be a solution of

(-A+V)cp = f, where f E

(R

If n < 2m and V E W2 -2 (R n' loc) then cp E W2 m (I[8n, Joe).

Let n > 2m. Suppose there exists a sufficiently small constant c(n) such that for all sufficiently small S > 0,

IFdy

sup

diamF
Then cp E W2 m(R n' 10C).

capmF

JIV(y)I2dy

+ sup x

d (x)

6n-2m

< c(n).

(58)

272

V. MAZ'YA

Equivalent forms of the conditions (57) and (58) of Kerman-Sawyer, Maz'ya. Verbitsky, and Verbitsky type, as well as various separately necessary or sufficient conditions involving no capacities, are available (compare with Section 5). Similar higher regularity LP-results can be obtained from estimates of the essential norms of multipliers: Wp (Rn) -> WP -2(Rn) obtained in [58, Ch. 4].

12. Relative Form Boundedness and Form Compactness Maz'ya and Verbitsky [64] gave necessary and sufficient conditions for the inequality

f Iu(x)12V(x)dx
u E C, (Rn)

(59)

to hold. Here the "indefinite weight" V may change sign, or even be a complexvalued distribution on 118n, n > 3. (In the latter case, the left-hand side of (59) is understood as I (Vu, u) I, where (V , - ) is the quadratic form associated with the corresponding multiplication operator V.) An analogous inequality for the Sobolev space W21 (IR"), n > 1 was also characterized in [64]:

JR

lu(x)I2V(x) dx < c f (IVu(x)I2 + lu(x)I2) dx,

u E Co (Rn).

(60)

Such inequalities are used extensively in spectral and scattering theory of the Schrodinger operator Hv = -A+V and its higher-order analogs, especially in questions of selfadjointness, resolvent convergence, estimates for the number of bound states, Schrodinger semigroups, etc. (See [9, 10, 11, 18, 22, 27, 28, 76, 85, 90], and the literature cited there.) In particular, (60) is equivalent to the fundamental concept of the relative boundedness of the potential energy operator V with respect to -A in the sense of quadratic forms. Its abstract version appears in the so-called KLMN Theorem, which is discussed in detail together with applications to quantum-mechanical Hamiltonian operators, in [76, Sect. X.2]. It follows from the polarization identity that (59) can be restated equivalently in terms of the corresponding sesquilinear form:

I I 3. (61) Here the energy space L2 '(R ) is defined as the completion of CO' (1Rn) with respect to the Dirichlet norm IIVUJIL2, and LZ 1(IRn) is the dual of L2(ll ). Similarly, (60) means that H is a bounded operator which maps W21(W) to WZ 1(ll ), n > 1.

Previously, the case of nonnegative V in (59) and (60) has been studied in a comprehensive way (see Section 5). For general V, only sufficient conditions have been known. It is worthwhile to observe that the usual "naive" approach is to decompose V into its positive and negative parts: V = V+ - V_, and to apply the just mentioned results to both V+ and V_. However, this procedure drastically diminishes the class of admissible weights V by ignoring a possible cancellation between V+ and V_ . This cancellation phenomenon is evident for strongly oscillating weights considered below. Examples of this type are known mostly in relation to quantum mechanics problems [4, 19, 69, 95].

ANALYTIC CRITERIA FOR THE SCHRODINGER OPERATOR

273

The following result obtained in [64] reflects a general principle which has a much wider range of applications. THEOREM 12.1. Let V be a complex-valued distribution on Rn, n > 3. Then (59) holds if and only if V is the divergence of a vector-field f : Rn -> Cn such that

J

lu(x) I2 If (x) I2 dx < const f IVu(x)12 dx,

(62)

where the constant is independent of u E C0 (Rn). The vector-field I' E L2(Rn, loc)

can be chosen as f = VA-1V. Equivalently, the Schrodinger operator Hv acting from L2'(R n) to L2 -'(R n) is bounded if and only if (62) holds. Furthermore, the corresponding multiplication operator V : L2(Rn) -i L2 1(Rn) is compact if and only if the embedding LZ(Rn) C L2(Rn, If I2 dx) is compact.

Once V is written as V = div the implication (62)- (59) becomes trivial. In fact, it follows using integration by parts and the Schwarz inequality. This idea has been known for a long time in mathematical physics (see, e.g., [19]) and the theory of multipliers in Sobolev spaces [58]. On the other hand, the proof of the converse

statement (59)- (62) where f = VA-'V is rather delicate. We note that the notion of form compactness has its origin in work by Rellich and Riesz (see, e.g., Reed-Simon [78, Sect. XIII.14 ]). Theorem 12.1 makes it possible to reduce the problems of boundedness and compactness for general "indefinite" V to the case of nonnegative weights If 12, which is by now well understood. In particular, combining Theorem 12.1 and the criteria in Section 5 for If I2, one arrives at analytic necessary and sufficient conditions for (59) to hold. As a corollary, one obtains a necessary condition for (59) in terms of Morrey spaces of negative order. COROLLARY 12.2. If (59) holds, then, for every ball Br(x0) of radius r, IVA-1V(x)I2 dx <

Crn-2,

where the constant does not depend on x0 E Rn and r > 0. COROLLARY 12.3. In the statements of Theorem 12.1 and Corollary 12.2, one

can put the scalar function (-A)-2'V in place of r = VA-'V. In particular, (62) is equivalent to the inequality:

f u(x)2I(-0) 2V(x)2dx
(63)

for all u E C0 (Rn). The proof of Corollary 12.3 uses the boundedness of standard singular integral operators in the space of functions f E L2,1oc(Rn) such that L l u(x)121 f (x)12 dx < c

f

for all u E Co (Rn); this fact was established earlier in [63].

V. MAZ'YA

274

Corollary 12.3 indicates that a decomposition into a positive and negative part, appropriate for (59), should involve expressions like (-A)- 2 V rather than V itself.

Another important consequence is that the class of weights V satisfying (59) is invariant under standard singular integral and maximal operators. REMARK 12.1. Similar results are valid for inequality (60); one only has to replace the operator (_A)-1/2 with (1 - A)-1/2, and the Wiener capacity capF with the corresponding Bessel capacity (see [1]). It suffices to restrict oneself, in the corresponding statements, to cubes or balls whose volumes are less than 1. We now discuss some related results in terms of more conventional classes of admissible weights V. The following corollary, which is an immediate consequence of Theorem 12.1 and Corollary 12.3, gives a simpler sufficient condition for (59) with Lorentz-Sobolev spaces of negative order in its formulation.

COROLLARY 12.4. Let n _> 3, and let V be a distribution on R' such that (-A)-2V E L,,,, (Rn), where LP,,, denotes the usual Lorentz (weak LP) space. Then (59) holds.

For the definition and basic properties of Lorentz spaces Lp,q(Rn), we refer to -. )-- 2 V E Ln, oo is equivalent to the estimate [93]. Note that, in particular, (-A)

I (-A)-ZV(x)I2 dx < C (mesnF)1 2/n.

(64)

The following corollary of Theorem 12.1 is applicable to distributions V, and encompasses a class of weights which is broader than the Fefferman-Phong class (30) even in the case where V is a nonnegative measurable function. COROLLARY 12.5. Let V be a distribution on Rn which satisfies, for some t > 1, the inequality

f

I(-0)-2 V(x)I2t dx <

Crn-2t

(65)

r (xo )

for every ball Br(xo) in Rn. Then (59) holds.

Note that by Corollary 12.2 the preceding inequality with t = 1 is necessary for (59) to hold. REMARK 12.2. A refinement of (65) in terms of the condition (34) established by Chang, Wilson, and Wolff [16] is readily available by combining (34) with Theorem 12.1.

To clarify the multidimensional characterizations for "indefinite weights" V presented above, we state an elementary analog of Theorem 12.1 for the SturmLiouville operator Hv = -d2/dx2 + V on the half-line. THEOREM 12.6. The inequality

J holds for all u E Co (ll

Iu(x)I2 V(x) dx < C

f

Iu'(x) 12 dx,

(66)

) if and only if 2

00

sup a J a>0

a

JV(t)dt

dx < oo,

(67)

ANALYTIC CRITERIA FOR THE SCHRODINGER OPERATOR

275

00

where r(x) =

fV(t) dt is understood in terms of distributions.

Equivalently, Hv : L2(I[8+) - L21(R+) is bounded if and only if (67) holds. Moreover, the corresponding multiplication operator V is compact if and only if a

Ja

jF(x)12 dx = o (1),

where a -> 0+

and a -' +oo.

(68)

For nonnegative V, condition (67) is easily seen to be equivalent to the standard Hille condition [36]: 00

sup a a>o

Ja

V(x) dx < oo.

(69)

A similar statement is true for the compactness criterion (68).

13. Infinitesimal Form Boundedness Maz'ya and Verbitsky [65] characterized the class of potentials V E D'(Rn) which are -A-form bounded with relative bound zero, i.e., for every e > 0, there exists C(e) > 0 such that I (Vu u)I < e I Vu11L2(JR_) +C(e) I Vu E Co (I[8"`). (70) I

I

In other words, they found necessary and sufficient conditions for the infinitesimal form boundedness of the potential energy operator V with respect to the kinetic energy operator Ho = -A on L2(IRn). Here V is an arbitrary real- or complexvalued potential (possibly a distribution). This notion appeared in relation to the KLMN theorem and has become an indispensable tool in mathematical quantum mechanics and PDE theory. The preceding inequality ensures that, in case V is real-valued, a semibounded selfadjoint Schrodinger operator Hv = Ho + V can be defined on L2(Rn) so that the domain of Q[u, u] coincides with W2 (Rn). For complex-valued V, it follows that Hv is an m-sectorial operator on L2(Rn) with Dom(Hv) C W 2 (W') ([76, Sect. X.2], [26, Sect. IV.4]). The characterization of (70) found in [65] uses only the functions IV(1-A)-1 V1

and I(1 - A)-' V1, and is based on the representation:

V=divI'+'y,

f(X)=-V(1-A)-'V'

ry=(1-0) 1V

In particular, it is shown that, necessarily, P E L2 (Rn, loc)n, and, when n > 3, lim

sup 62-n

a-'+0 xoEIIB"

J

(71)

'y E L1(IIBn, loc),

(IF(x)I2 + 17(x) 1) dx = 0,

(72)

6(xo)

once (70) holds.

In the opposite direction, it follows from the results in [65] that (70) holds whenever lim

sup 52r-n

a-'+o xo EIIFn

(lF(x)I2 + 17

x)1)r

dx = 0,

(73)

B,5 (xo)

where r > 1. Such admissible potentials form a natural analog of the FeffermanPhong class (30) for the infinitesimal form boundedness problem where cancellations between the positive and negative parts of V come into play. It includes functions with highly oscillatory behavior as well as singular measures, and properly contains the class of potentials based on the original Fefferman-Phong condition where V

276

V. MAZ'YA

is used in (73) in place of If I' + 1-yl. Moreover, one can expand this class further using the sharper condition (34) applied to f12 + y1 A complete characterization of (70) obtained in [65] is given in the following theorem which provides for deducing explicit criteria of the infinitesimal form boundedness in terms of the nonnegative locally integrable functions IrI2 and I-y1. THEOREM 13.1. Let V E D'(1Rn), n > 2. The following statements are equivalent:

(i) V is infinitesimally form bounded with respect to -A. (ii) V has the form (71) where P = -V(1 - A)-' V, y = (1 - A)-' V, and the measure p E M+(Rn) defined by

dp = (Ir(x)I2 + ly(x)I) dx

(74)

has the property that, for every e > 0, there exists C(e) > 0 such that

f

I IVuH

I u(x)12 dp < e

Vu E Co ( z).

2(R-),

,.

(75)

(iii) For /t defined by (74), (P)z

E

lim sup 1 p = a +0 Po:diamPo<6 µ(PO) PCP (mesa P) i-zn 0

0,

(76)

where P, Po are dyadic cubes in l[8n, i.e., sets of the form 2i(k + [0, 1)n), where i e Z, k E Zn. (iv) For /c defined by (74), 6

µ(F) = 0, lim sup +0 F: diam F<6 cap F

(77)

where F denotes a compact set of positive capacity in Rn. (v) For /2 defined by (74), z

lim

1(n) - 0,

su

6-'+0 xoEIl$

(78)

/2(B6(x0))

where /B5(xo) is the restriction of /c to the ball B6(xo).

(vi) For /2 defined by (74), lim sup 6 +0 x, xoEi'

Gl

*

(G1

z

* AB,(xo)) (x) = 0, Gl * /AB,(xo) (x)

(79)

where Gl * µ = (1 - A)-2'[t is the Bessel potential of order 1. In the one-dimensional case, the infinitesimal form boundedness of the SturmLiouville operator HV = -d2ldx2 + V on L2(ll8') is actually a consequence of the form boundedness. THEOREM 13.2. Let V E D'(IR'). Then the following statements are equivalent. (i) V is infinitesimally form bounded with respect to -d2/dx2.

(ii) V is form bounded with respect to -d2/dx2, i.e.,

(Vu,u)I
(n1),

VU ECo (l[8').

ANALYTIC CRITERIA FOR THE SCHRODINGER OPERATOR

277

(iii) V can be represented in the form V = dF/dx + -y, where x+1

sup xER1

(IF(x)I2 + I-y(x)I) dx < +oo.

(80)

x

(iv) Condition (80) holds where

I' (X) _ f sign (x - t) e--tl V(t) dt,

f

y(x)

1

e--tI V(t) dt 1

are understood in the distributional sense. (v) V belongs to the space WZ 1(1[81, unif), with the norm where i c Co (]E81), 71 (0) = 1.

sup II7I(x - .) VIIw21(R1),

xEll'

The statement in Theorem 13.2 can be found in [85, Thm. 11.2.1], whereas (ii)=(iv) follows from results in [64, Thm. 4.2], and [65, Thm 2.5]. The equivalence of (ii) and (v) is proved in [59, Cor. 9].

14. Kato's Condition Kn Among well-known sufficient conditions for (70) which ignore possible cancellations, we mention: Q E L'2 (Rn) + L°° (][8n) (n > 3) and Q E Lr (][R2) + L°° (]82),

r > 1 (n = 2) (see [13]), as well as Kato's condition Kn introduced in [38]: lim

Q(

sup

dx = 0,

n > 3,

-

(81)

IQ(x) I dx = 0,

n = 2.

(82)

6-.+0 xoER" JBb(xo) Ix - X01 n-2

lim

sup J 5(xo) log

6-'+o xoER"

x - xp

Kato's class proved to be especially important in studies of Schrodinger semigroups, Dirichlet forms, and Harnack inequalities [2, 4, 90]. Theorem 13.1 yields that (70) actually holds for a substantially broader class of potentials for which IrI2 + I'yI E Kn. We emphasize that no a priori assumptions were imposed on C(c) in this theorem. An observation of Aizenman and Simon states that, under the hypothesis C(e) < a eb

P

for some a, b > 0 and 0 < p < 1,

all potentials V which obey (70) with V1 in place of V are contained in Kato's class. This was first proved in [4] using the Feynman-Kac formalism. In [65], a sharp result of this kind is obtained with a simple analytic proof. It is shown that if (70) holds with IVI in place of V E Li°°(][8n), then for any C(e) > 0, V (x) I

sup f b(xo) Ix - xp n2 dx xoEIIE"

6-2

V (x) I dx < c J

sup fB,6(xo) log x0ER2

c

Ix - x01

6-2

S2

dsn > 3,

C(s) s2 logs

ds,

n

2,

(83) (84)

where c is a constant which depends only on n, and 6 is sufficiently small. Here C(s) = inf,>o {C(e) + s e} is the Legendre transform of -C(c). In particular, it follows that the condition C(e) < a eb E-P for any p > 0 is enough to ensure that V E K2 in the more subtle two-dimensional case.

V. MAZ'YA

278

15. Trudinger's Subordination for the Schrodinger Operator In [65] inequality (70) is studied also under the assumption that C(e) has power growth, i.e., there exists co > 0 such that (V u, u)I < e IVuIIL2(Rn) +CC -11 uIIL2(Rn),

Vu E Coo" (Rn),

(85)

for every e E (0, eo), where ,(3 > 0. Such inequalities appear in studies of elliptic PDE with measurable coefficients [98], and have been used extensively in spectral theory of the Schrodinger operator [4, 38, 76, 81, 85, 22, 23, 44, 90]. As it turns out, it is still possible to characterize (85) using only Irk and I'yI defined by (71), provided 13 > 1. It is shown in [65] that in this case (85) holds if and only if both of the following conditions hold: sup

P-1 62 p++l

xoER"

(x) 1 2 dx < +oo,

(86)

I y(x)I dx < +oo,

(87)

415 (X0)

0<6<60

sup x0E1R"

60+1--f 6(x0)

0<6<60

for some 6o > 0. However, in the case /3 < 1 this is no longer true. For 0 = 1, (86) has to be replaced with the condition that r is in the local BMO space, or respectively is Holder-continuous of order (1 - /3)/(1 + /3) if 0 < /3 < 1.

In the homogeneous case eo = +oo, (85) is equivalent to the multiplicative inequality (36) where T = /3/(1 + /3) E (0, 1). In spectral theory, (36) is referred to as the form -r-subordination property (see, e.g., [3, 31, 32, 46], and [81, Sect. 20.4]). For nonnegative potentials V, where V coincides with a locally finite measure p on ]R n, inequality (36) is equivalent to (37) [53, Sect. 1.4.7]. For general V, the following result is obtained in [65]. If r > 2, then (36) holds if and only if VA-1V lies in the Morrey space £2, A (]jn), where A = n + 2 - 4T. The Morrey space Gr' A (I[8n) (r > 0, A > 0) consists of f E Lr (]8n, loc) such that xoE1 p6>0 B6(o)Ik1n

fBF,5(xo) f (x)

r dx < +00.

We refer to [70] for an overview of Morrey spaces. For T = 2(36) holds if and only if VA-1V E BMO(W0), and for 0 < T < 1, whenever VA-1V is in the Holder class CI-2, (Wn). These different characterizations are equivalent to (37) if V is a nonnegative measure.

16. Discreteness of the Spectrum of -0+V with Nonnegative Potential Friedrichs proved in 1934 [29] that to the left of the point lim inflxl.,, V(x),

the spectrum of the Schrodinger operator Hv = -A + V in L2(Rn) with a locally integrable potential V is discrete, and hence all spectrum is discrete provided V(x) -> +oo as xl -- oo. By Rellich's criterion [80], the spectrum of Hv is purely discrete if and only if the ball (Hvu, u) + (u, u) < 1 is a compact subset of L2 (]l8n). This and the discreteness of spectrum easily imply that for every d > 0 V(x)dx

+oo

as

Qd --+ 00,

(88)

Qd

where Qd is a closed cube with the edge length d and with the edges parallel to coordinate axes, Qd -* oo means that the cube Qd goes to infinity (with fixed d).

ANALYTIC CRITERIA FOR THE SCHRODINGER OPERATOR

279

In 1953 Molchanov [67] proved that this condition is in fact necessary and sufficient in case n = 1 but not sufficient for n > 2. He also discovered a modification of (88) which is equivalent to the discreteness of spectrum in the case n > 2: inf F

JQd\F

V(x)dx - +oo as Qd -> oo,

(89)

where the infimum is taken over all compact subsets F of Qd which are called negligible. The negligibility of F in the sense of Molchanov means that capF < y cap(Qd), where cap is the Wiener capacity and -y > 0 is a sufficiently small constant. More precisely, Molchanov proved that one can take y = cn, where cn, = (4n)-4n(cap(Q1))-1 for n > 3. As early as in 1953, Gelfand raised the question about the best possible constant

cn (personal communication). We describe results from [60], where a complete answer to this question is given. Let V be a nonnegative Radon measure in an open set SZ C Rn, absolutely continuous with respect to the Wiener capacity. We will consider the Schrodinger operator Hv which is formally given by the expression -A + V. It is defined in L2(cl) by the closure of the quadratic form Q[u,u] with the domain C0 (Il). Instead of the cubes Qd, a more general family of test bodies will be used. Let us start with a standard open set 9 c f8n. We assume that g satisfies the following conditions:

(a) 9 is bounded and star-shaped with respect to a ball BP; (b) diam(Q) = 1. For any positive d > 0 denote by Ccd(0) the body {x : d-lx E 9} which is

homothetic to g with coefficient d and with the center of homothety at 0. We will denote by 9d a body which is obtained from 9d(0) by a parallel translation: 9d(Y) = Y + 9d(0) where y is an arbitrary vector in 1W'. The notation Jcd - oo means that the distance from Qd to 0 goes to infinity. Definition. Let y E (0,1). The negligibility class Nry (gd; 1) consists of all compact sets F C Od satisfying the following conditions:

9d\c cFC9d

(90)

and

cap F < y cap gd. (91) Now we formulate the main result of the work in [60] about the discreteness of spectrum. THEOREM 16.1. (i) (Necessity) Let the spectrum of Hv be discrete. Then for every function ry : (0, +oo) -* (0, 1) and every d > 0

inf

FEJV,(d) (gd,O)

V(9d \ F) -> +oo

(ii) (Sufficiency) Let a function d neighborhood of 0, and satisfy

as

c!d --4 00.

(92)

y(d) E (0, 1) be defined for d > 0 in a

lim sup d-2 y(d) = +oo.

(93)

djO

Assume that there exists do > 0 such that (92) holds for every d E (0, do). Then the spectrum of Hv in L2(1) is discrete. We make some comments about this theorem.

V. MAZ'YA

280

REMARK 16.1. It suffices for the discreteness of spectrum of Hv that the con-

dition (92) holds only for a sequence of d's, i.e., d c {dl, d2i ... }, dk - 0 and z

REMARK 16.2. The condition (92) in the sufficiency part can be replaced by a weaker requirement: there exist c > 0 and de > 0 such that for every d E (0, do). there exists R > 0 such that d-n inf V(gd \ F) > cd-2y(d), (94) FcAf (d) (Gd,O)

whenever cd n (11 \ BR) zA 0 (i.e., for distant bodies cd having nonempty intersection

with Il). Moreover, it suffices that the condition (94) is satisfied for a sequence d = dk satisfying the condition formulated in Remark 16.1. Note that unlike (92), the condition (94) does not require that the left-hand side goes to +oo as GJd -j oo. What is actually needed is that the left-hand side has a certain lower bound, depending on d for arbitrarily small d > 0 and distant test bodies cd. Nevertheless, the conditions (92) and (94) are equivalent because each is equivalent to the discreteness of spectrum. REMARK 16.3.

If we take y = const E (0, 1), then Theorem 16.1 gives

Molchanov's result, but with the constant y = cn replaced by an arbitrary constant y c (0, 1). So Theorem 16.1 contains an answer to the abovementioned Gelfand's question.

REMARK 16.4. For any two functions y1i y2 : (0, +oo) - (0, 1) satisfying the requirement (93), the conditions (92) are equivalent, and so are the conditions (94), because any of these conditions is equivalent to the discreteness of spectrum. It follows that the conditions (92) for different constants y E (0, 1) are equivalent. In the particular case when the measure V is absolutely continuous with

respect to the Lebesgue measure, we see that the conditions (89) with different constants y E (0, 1) are equivalent.

REMARK 16.5. The results above are new even for the operator -0 in L2 (S2) (for an arbitrary open set SZ C R' with the Dirichlet boundary conditions on aQ). In this case the discreteness of spectrum is completely determined by the geometry of Q. More precisely, for the discreteness of spectrum of Ho in L2(ll) it is necessary and sufficient that there exists do > 0 such that for every d E (0, do) lim inf cap(Cd \ Il) > -y (d) cap ggd,

cd-oo

(95)

where d - y(d) E (0, 1) is a function, which is defined in a neighborhood of 0 and satisfies (93). The condition (95) with different functions y, satisfying the conditions above, are equivalent. This is a nontrivial property of capacity. It is necessary for the discreteness of spectrum that (95) holds for every function y : (0, +oo) --> (0, 1) and every d > 0, but this condition may not be sufficient if y does not satisfy (93) (see Theorem 16.2 below).

The following result demonstrates that the condition (93) is precise.

THEOREM 16.2. Assume that y(d) = 0(d2) as d - 0. Then there exists an open set 11 C Rn and do > 0 such that for every d E (0, do) the condition (95) is satisfied but the spectrum of -0 in L2(SZ) with the Dirichlet boundary conditions is not discrete.

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REMARK 16.6. In the recent work [96] Taylor found another criterion of discreteness of the spectrum of Hv stated in terms of the so-called scattering length of V.

17. Strict Positivity of the Spectrum of -A+ V We say that the operator Hv, the same as in Section 11, is strictly positive if its spectrum does not contain 0. Equivalently, we can say that the spectrum is separated from 0. The strict positivity is equivalent to the existence of A > 0 such that Q[u,u] > AMMuMML2(O),

u E Co (52).

(96)

The characterization of positivity of the spectrum in the next theorem with Molchanov's negligible sets in the formulation was found in [51] (see also [53, Sect. 12.5]). The present stronger version was obtained by Maz'ya and Shubin in [60].

THEOREM 17.1. (i) (Necessity) Let us assume that Hv is strictly positive, so that (96) is satisfied with a constant A > 0. Let us take an arbitrary 7 E (0, 1). Then there exist do > 0 and x > 0 such that d_n

inf

FEN/ (gd,O)

V (cd \ F) >

(97)

for every d > do and every CJd.

(ii) (Sufficiency) Assume that there exist d > 0, x > 0 and 7 E (0, 1), such that (97) is satisfied for every CJd. Then the operator Hv is strictly positive. Instead of all bodies Jcd it is sufficient to take only the ones from a finite multiplicity covering (or tiling) of W2. REMARK 17.1. Considering the Dirichlet Laplacian in L2 (52), we see from The-

orem 17.1 that for any choice of a constant ry E (0, 1) and a standard body g, the strict positivity of -0 is equivalent to the following condition I d > 0, such that cap(Cgd n (R''\1)) > 'y cap(Qd) for all CJd.

(98)

In particular, it follows that for two different 7's these conditions are equivalent. Noting that Rn\1 can be an arbitrary closed subset in R', we get a property of the Wiener capacity, which is obtained as a byproduct of our spectral theory arguments.

18. Two-Sided Estimates for the Bottom of the Spectrum and Essential Spectrum Let A = A(f, Hv) denote the greatest lower bound of the spectrum of the Schrodinger operator Hv handled in the two preceding sections. By Qs we denote the set of all cubes Qd having a negligible intersection with the complement of I (in Molchanov's sense), i.e., QQ = {Qd : cap(Qd \ 1) < 'y cap Qd}, where ry is sufficiently small.

(99)

For V = 0 Maz'ya [52] and for the general case Maz'ya and Otelbaev [57] (see also [53, Ch. 12]) obtained the two-sided estimate c1D_2


2

,

(100)

V. MAZ'YA

282

with D = D(S2, V) given by

D = sup

d: do-2 > inf V(Qd\F) } .

(101)

F

QdEQ1

Here cl and c2 are positive constants which depend only upon n and the infimum is taken over all sets F, satisfying

and capF < -y cap Qd.

Qd \ Tl C F C Qd

Clearly, the strict positivity of Hv is equivalent to the condition D < 00. In particular, if V = 0, then D becomes the capacitary interior diameter of Q: the maximal size d of cubes Qd with the negligible intersection with 1[8n \ 1 (i.e., cubes Qd E Qq). The notion of the capacitary interior diameter was introduced in [52]. Explicit estimates for cl and c2 in the case V = 0 are given in [61], where also the smallness condition of ry is replaced with -y E (0, 1). The paper [57] (see also [53, Ch. 12]) also contains a two-sided estimate for the bottom of the essential spectrum of Hv in L2(1). We will denote this bottom by A = A(1; Hv). The estimate has the form c1D,2 < A < (102) where

D,,, = lim D(Sl \ BR, V, y).

(103)

R-oo

Note that the discreteness of spectrum of Hv is equivalent to the equality A = +oo. Therefore, (102) implies that D,),) = 0 is necessary and sufficient for the pure discreteness of the spectrum of Hv.

19. Structure of the Essential Spectrum of Hv As in Sections 16-18, let Hv = -A + V, V >_ 0. The following result is due to Glazman [30]. LEMMA 19.1. If the spectrum of Hv is not purely discrete, the essential spectrum

of Hv extends to infinity. Moreover, if 0 belongs to the essential spectrum of Hv, then this spectrum coincides with [0, oc).

PROOF. Let A be the bottom of the essential spectrum. Then there exists a in Co (S2) subject to the conditions

sequence of real-valued functions Ilcpv

1,

p-0

weakly in L2(SZ),

(104) (105)

(Hv -A) VvIIL2(Q) --> 0.

We set

u _ p,exp(i(a - A) 1/2

n

xk), k=1

where a > A. We see that u satisfies (104) and that n 2

2

II(Hv - a) uV JjL2(cz) = II(Hv - A) coVIIL2(n) +4(a - A)Il J:a1Pv/axkIIL2(Q). k=1

Since the right-hand side does not exceed II (Hv - A) ' IIL2(Q) + 4n(a - A) Q[W,, wv],

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283

we have - a) u11j1L2(Q) < 11(Hv

-A)

4n(a - A) J (Hv - A)

r 11 IIL2 (0)

+ 4nA(a - A).

By (105) lira sup II(Hv - a) u1111L2(0) <_ p(a), 11 -+00

where p(a) = 2(nA(a - A))1/2. It follows that any segment [a - p(a), a + p(a)] contains points of the essential spectrum. If, in particular, A = 0 then every positive a belongs to the essential spectrum.

In concert with this lemma, the pairs (S2, V) can be divided into three nonover-lapping classes. The first class includes (1k, V) such that the spectrum of Hv is discrete. The pair (1, V) belongs to the second class if the essential spectrum of Hv is unbounded and strictly positive. Finally, (1, V) is of the third class if the essential spectrum of Hv coincides with [0, oo). By (101) and (102) the three classes can be described as follows: (i) (S2, V) belongs to the first class if and only if D00 = 0. (ii) (Q, V) belongs to the second class if and only if D00 > 0. (iii) (Q, V) belongs to the third class if and only if D = oo (or, equivalently, D00 = oo). By Theorem 16.1, the condition (i) is equivalent to (92), and (ii) holds if and only if (97) is valid. Finally, (iii) is equivalent to the failure of (95) by Theorem 17.1 and Lemma 19.1. In other words, (iii) holds if and only if lim inf inf V(Qd \ F) = 0 Gd-'o0 FE.M (d)(4d,H)

for every d > 0.

There are still open problems in the study of conditions insuring the absolute continuity of the spectrum. Simon conjectured in [92] that the absolutely continuous spectrum of HV with V subject to (V(x))2(1 + Ixl)1-" dx < 00 fRis the positive real axis. An interesting sufficient condition supporting this conjecture: V = divI' and V(x)I + If(x)l < c (1 + e > 0, has recently been obtained by Denisov in [25]. IxI)-112-E

20. Two Measure Boundedness and Compactness Criteria The following lemma is a particular case of a more general result from [50] (see also [53, Sect. 2.3.7]).

LEMMA 20.1. Let p and v be two nonnegative Radon measures in Il C R', n > 1. The inequality f f (106) Iu12 dp < C(J IVu12 dx + j Ju12 dv)

J

sz

h olds

for all u c Co (1) if and only if there exists a constant K > 0 such that for all open bounded sets g and G subject tog C G, G C ft the inequality p(g) < K(capG g + v(G)) (107)

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284

holds.

This criterion becomes more transparent in the one-dimensional case because the capacity admits an explicit representation. THEOREM 20.2 (see [55]). Let n = 1 and let o-d denote the open interval (x d, x + d). Inequality

f lul2 dµ < C(J u'12 dx + J Iu12 dv)

(108)

holds for all u c CO '(Q) if and only if µ(ad(x))

const(r-1 + v(ad+T(x))),

(109)

where x, d, and r are such that o-d+T(x) C Il, is valid without complementary assumptions about µ and v. The sharp constant C in (108) is equivalent to

su x,d,7

µ(ad(x))

r 1 + v(ad+T(x))

where x, d, and r are the same as in (109). We define the space W2 1(v) as the closure of Co (Il) with respect to the norm II f I I W2

(v) = (f If, (X)12 dx +

J

I f (x) I2 dv)

The condition (109) is a criterion of boundedness for the embedding operator I Wti (v) - L2(µ) The next theorem contains a two-sided estimate for the essential norm of I. We recall that the essential norm of a bounded linear operator A acting from X into Y, where X and Y are linear normed spaces, is defined by ess IIAll = inf IIA - TII T

with the infimum taken over all compact operators T : X --+ Y. THEOREM 20.3. [55] Let

E(µ, v) :i

µ(ad(x)\ [- M, M)

sup x,d,T r

+ v(o'd+T(x))

There exist positive constants c1 and c2 such that cl E(µ, v)1/2 < essIIIII C C2 E(i, v)1/2.

In particular, the operator I is compact if and only if

lim sup A(ad(x)\[-M, M]) = 0

M-oo x,d,,r r_1 + v(ad+T(x))

where x, d, and r are the same as in (109).

(110)

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[84] M. Schechter: Spectra of Partial Differential Operators, 2nd ed., North-Holland Series in Applied Mathematics and Mechanics, 14, North-Holland, Amsterdam, 1986 [85] M. Schechter: Operator Methods in Quantum Mechanics, Dover Publications, Mineola, NY, 2002.

[86] J. Schwinger: On the bound states of a given potential, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 122-129.

[87] B. Simon: Lower semicontinuity of positive quadratic forms, Proc. Roy. Soc. Edinburgh, Sect. A 79 (1977), 267-273. [88] B. Simon: A canonical decomposition for quadratic forms with applications to monotone convergence theorems, J. Funct. Anal. 28 (1978), 377-385. [89] B. Simon: Maximal and minimal Schrodinger forms, J. Operator Theory 1 (1979), 37-47. [90] B. Simon: Schrodinger semirgoups, Bull. Amer. Math. Soc. 7 (1982), 447-526. [91] B. Simon: Schrodinger operators in the twentieth century, J. Math. Physics 41 (2000), 3523-3555.

[92] B. Simon: Schrodinger operators in the twenty-first century, Mathematical Physics 2000, pp. 283-288, Imperial College Press, London, 2000. [93] E. M. Stein, G. Weiss: Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, NJ, 1971. [94] F. Stummel: Singuldre elliptische Differentialoperatoren in Hilbertschen Raumen, Math. Ann. 132 (1956), 150-178. [95] K. T. Sturm: Schrodinger operators with highly singular, oscillating potentials, Manuscr. Math. 76 (1992), 367-395. [96] M. Taylor: Scattering length of positive potentials, Canad. Math. Bull. 49 (2006), 144-151. [97] H. Triebel: The Structure of Functions, Birkhauser, Basel, 2001. [98] N.S. Trudinger: Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa 27 (1973), 265-308. [99] I. Verbitsky: Nonlinear potentials and trace inequalities, The Maz'ya Anniversary Collection, Vol. 2 (Rostock, 1998), pp. 323-343, Oper. Theory Adv. Appl., 110, Birkhauser, Basel, 1999.

OHIO STATE UNIVERSITY, COLUMBUS, OH 43210, U.S.A.; UNIVERSITY OF LIVERPOOL, LIVERPOOL, L69 3BX, U.K.; LINKOPING UNIVERSITY, SE-581 83 LINKOPING, SWEDEN E-mail address: vlmaz®math.ohio-state.edu; vlmaz®mai.1iu.se

Proceedings of Symposia in Pure Mathematics Volume 76.1, 2007

The Spectral Geometry of Geometrically Finite Hyperbolic Manifolds Peter Perry Dedicated to Barry Simon on his sixtieth birthday ABSTRACT. Selberg's trace formula for a compact Riemann surface gives a pre-

cise, quantitative relationship between periods of closed geodesics and eigenvalues of the Laplacian. In this paper we discuss analogues of Selberg's trace formula for hyperbolic manifolds of infinite volume and the relationships they reveal between periods of closed geodesics and scattering resonances. We also discuss results for scattering theory on asymptotically hyperbolic manifolds which shed further light on this relationship and reveal interesting connections with conformal geometry.

CONTENTS

Introduction Geometrically Finite Hyperbolic Manifolds Spectral Theory for the Laplacian Spectral Geometry of Non-Compact Riemann Surfaces Spectral Geometry of Convex Co-Compact Manifolds Spectral Geometry of Asymptotically Hyperbolic Manifolds

1.

2. 3. 4. 5. 6.

References

1. Introduction For those of us who came of age in the 1980's, "geometric scattering theory" evokes the geometric approach to quantum-mechanical scattering which began with the pioneering work of Deift-Simon [34], Enss [40, 42, 41], Mourre [115], Sigal-

Soffer [154], and others, reviewed in this Festschrift in the survey of Christian Gerard [54]. Here I would like to survey some recent developments in "geometric scattering theory" in Melrose's [112] sense: the application of ideas and techniques 2000 Mathematics Subject Classification. Primary 58J50, Secondary 35P25. Key words and phrases. Selberg zeta function, scattering resonances, scattering operator. Supported in part by NSF Grant DMS-0408419. ©2007 American Mathematical Society 289

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of scattering theory to the study of the Laplacian on a complete, non-compact Riemannian manifold. For complete Riemannian manifolds with "simple geometry at infinity," the study of the Laplace operator naturally leads to questions of spectrum, analytic continuation of the resolvent, and scattering resonances familiar to workers in Schrodinger operators

Manifolds of constant negative curvature and infinite volume are a class of manifolds with "simple geometry at infinity" which are particularly appealing in Mathematical Physics. On the one hand, the geodesic flow is ergodic, and there are infinitely many closed orbits. On the other, the Laplacian has at most finitely many eigenvalues and infinitely many resonances, and so provides a model of chaotic

quantum scattering where the resonances carry dynamical and geometrical information. The relationship between geodesic flow and the resonances can be analyzed in depth using ideas originally developed by Selberg [151] in the context of analytic number theory. The simplest example of the class of manifolds to be considered here is the hyperbolic cylinder X which arises as the quotient of the Poincare upper half-plane by an abelian discrete group of isometries. Recall that the Poincare upper-half plane is1

H2={(y,x):yER, x>0} with the hyperbolic metric ds2

= x-2 (dx2 + dy2)

For any f > 0, the dilation (y, x) --* (efy.etx)

is an isometry of H2 and generates an abelian discrete group F. The quotient X = F\IH12 is a topological cylinder Rt x S1 0 with the hyperbolic metric ds2 = dt2 + f2 cosh2 t d02

This manifold has closed geodesic of length f at t = 0, and the "boundary at infinity" is the union of two circles. Let OX be the positive Laplacian on X. By Fourier analysis in the 0 variable, it is easy to see that the operator OX - 4 is a direct sum over m E Z of one-dimensional Schrodinger operators 2

2

L,,

- dt2 + C 2 Q ) + 4

sech2 t

The spectral resolution of this operator can be explicitly calculated in terms of Legendre functions (see, for example, [75] and [30]). From this calculation it is clear that OX has purely continuous spectrum in [4, oo). To study the resolvent (OX - z)-1, we write the spectral parameter z as s(1 - s) for Re(s) > 2, corresponding to cut plane C\[4, oo). The resolvent has a meromorphic continuation to the complex s-plane with an infinite lattice of scattering resonances at the points

s= -k+

27rim

'The notation (y, x) for an ordered pair with x > 0 is motivated by a notational convention in geometric scattering theory in which x is a smooth positive function which vanishes to first order at the boundary at infinity; see Section 6 below!

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for k = 0, 1, 2.... and m c Z. These should be thought of as quasi-bound states associated to the closed geodesic f whose exponential rate of decay is determined by Re(s). They are exactly the zeros of the trivial "zeta function" 00

Zx(s)=]J(1 -

e-(s+k)2)

k=o

The resonances and the resonance eigenfunctions determine a "resonance wave expansion" for solutions of the non-Euclidean wave equation utt + (Ox - DU = 0; see [30].

We will see that Selberg's zeta function can be defined for a large class of hyperbolic manifolds, and connects the set of periods of closed orbits with the resonances of the Laplacian in much the same way as in the simple example above. Scattering theory plays an essential role in developing the trace formula and showing that the zeta function is a quotient of entire functions of bounded order. The theory of entire functions may then be used to derive the trace formula. Many of the wellknown consequences of Selberg's trace formula for compact surfaces, such as the asymptotics of the counting function for closed geodesics and the equidistribution of geodesics in homology classes, also hold in the infinite-volume setting. On the other hand, the distribution of resonances reflects features of the classical orbits such as the Hausdorff dimension of the "trapped set." In what follows, I will first review hyperbolic geometry, give some important examples of hyperbolic manifolds, and recall basic results about the spectrum of the Laplace operator. I will then discuss spectral and scattering theory for hyperbolic surfaces of infinite area, where results are most complete. In higher dimensions, I restrict attention to the class of convex co-compact hyperbolic manifolds where many of the same connections between quantum resonances and classical orbits have been established. Finally, I discuss recent work on scattering for asymptotically hyperbolic manifolds-a geometrically natural generalization of hyperbolic manifolds-where analogues of the trace formula hold and surprising connections with conformal geometry come to light.

This article follows the trajectory of my own work and the related research with which I am most familiar. For the relationship of questions discussed here to larger questions in the theory of scattering resonances, see, for example, the surveys of Zworski [162, 163, 170]. For a guide to the extensive literature on "quantum chaos" for compact and finite-volume hyperbolic surfaces which we have not touched on in this paper, see, for example, Sarnak's lectures [149] and [150]. Another important line of research not discussed in depth here is the work of Bunke and Olbrich on scattering theory for geometrically finite hyperbolic manifolds. Their approach to scattering theory yielded a proof of Patterson's conjecture [128] characterizing the singularities of the zeta function in cohomological terms; see the monograph of Juhl [88], Bunke and Olbrich's paper [23], and references contained therein. Melrose's Stanford lectures [112] provide a broader perspective on "geometric scattering theory" as discussed here. For a survey of "counterexamples" consisting of non-isometric pairs and families of non-compact manifolds with the same resonances, see [57]. Finally, Chang's lectures [28] provide the context in conformal geometry and geometric analysis in which Graham and Zworski's results [61], discussed in Section 6, may be more fully appreciated.

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Acknowledgements. This article is tendered as a small expression of my appreciation for the wonderful job Barry Simon did as a thesis advisor, for the productive years that I spent at Caltech under his sponsorship, and for the enjoyable and rewarding career that has followed. On this occasion it should be duly noted that, although I moved outside the mainstream of Schrodinger operator theory, I could not escape the reach of Barry's scholarship; see [33]! I am also grateful to the mathematicians with whom I have had the opportunity to collaborate on some of the work discussed here, including David Borthwick, the late Robert Brooks, Richard Froese, Carolyn Gordon, Peter Hislop, Chris Judge, Jeffrey McGowan, Ruth Gornet, S. J. Patterson, Dorothee Schueth, and Floyd Williams. I thank David Borthwick for sharing with me a first draft of his forthcoming book on scattering theory for Riemann surfaces. Finally, I am grateful to the referee and to Laurent Guillope for a careful reading of the manuscript and many suggestions that improved the finished product.

2. Geometrically Finite Hyperbolic Manifolds Hyperbolic manifolds arise as quotients of real hyperbolic space by a discrete group F of isometries. Let us first recall some basic notions of hyperbolic geometry and discrete groups of hyperbolic isometries. For a complete and systematic development of hyperbolic geometry, we refer the reader to the monograph of Ratcliffe [144].

Real (n + 1)-dimensional hyperbolic space 1H!n+1 may be realized either as the open unit ball {z E R-+1 : Izl < 1} with metric 4 dz2

ds = 1-Iz12

or as the upper half-space R'+1={(y,x):yEll

,

x>0}

and metric

g = x-2(dx2 + dy2) Isometries of H'+' are Mobius transformations leaving the unit ball (resp. the upper half-space) invariant. The isometries of lH['+1 act transitively, i.e., any point

in H'+1 can be mapped to any other point of H'+1 by some isometry. These isometrics may be divided into three types. The first type is an elliptic motion or non-Euclidean rotation conjugate in the ball model to a Euclidean rotation about the origin. The second type is a parabolic motion conjugate to a mapping of the form

(y, x) - (Ay + b, x) in the upper half-space model (where A is an orthogonal matrix and b a fixed vector in l[8'l). The third type is a loxodromic motion conjugate in the upper half-space model to a mapping of the form (2.1) (y, x) H et(y) (A-,y, x) where fly > 0 and Ay E O(n). We will denote by al (-Y), ... , an(y) the eigenvalues of A. The group of isometries of Hn+1 is isomorphic to the group of conformal transformations on the "boundary at infinity."

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Geodesics in the upper half-space model are straight lines parallel to the x-axis, and semicircles intersecting the boundary x = 0 normally. Any geodesic is determined by its endpoints at the boundary. Hemispheres intersecting the boundary normally are geodesic hyperplanes which divide H's+1 into two isometric half-spaces. Let d( , - ) denote hyperbolic distance. We say that two geodesics 'y1(t) and -Y2(t)

are equivalent if d(yl (t), y2 (t)) remains bounded as t - oo. It is not difficult to see that equivalence classes of geodesics are labelled by boundary points in Sn (in the ball model), so that Sn is the "geometric infinity" of Hn+1 In dynamical terms, Sn parameterizes the possible set of directions for classical paths moving to infinity. If F is an orientation-preserving discrete group of isometries of Hn+l, we say that F is geometrically finite if there is a finite-sided fundamental domain for F in Hn+1. We will always assume that F is torsion-free (i.e., has no elements of finite

order), so that X = F\Hn+1 is a Riemannian manifold with the Riemannian metric induced from the Poincare metric on JH[n+l. The limit set of F, denoted A(F), is most easily described in the unit ball model: it is the subset of Sn consisting of accumulation points of F-orbits in the Euclidean topology of Bn+1. It is not difficult to see that if zo is any fixed point in 1H[n+1, the set of accumulation points of the orbit {y(zo) : y c F} is independent of the choice of zo and so is equal to A(F). The limit set is a closed subset of Sn, and its complement, denoted S2(F), is called the

domain of discontinuity of F acting on S. The (hyperbolic) convex hull of the limit set is the union of all geodesics in jHIn+1 with endpoints in the limit set.

The set of closed geodesics of F are in one-to-one correspondence with the hyperbolic conjugacy classes of F. A hyperbolic element in F is called primitive if it is not the power of another element in F; analogously, a closed geodesic is called primitive if it is not the power (in the sense of composition of paths) of another closed geodesic. The primitive closed geodesics of X are in one-to-one correspondence with the primitive hyperbolic conjugacy classes of F. The group F is called convex co-compact if X has infinite volume but any fundamental domain for the action of F on TH[n+1 intersects the convex hull of the limit set in a compact convex subset of 1HIn+l. If F is convex co-compact, it contains

no parabolic elements and the quotient X = F\Hn+1 is the interior of a compact manifold with boundary (see Example 2.2 below). The exponent of convergence of F is the infimum over those real s for which the Poincare series

E exp [-sd(x, 'y(x))] ryEF

converges; here d( , ) is hyperbolic distance. The exponent b is independent of x E Hn+1 by the triangle inequality and the fact that y is an isometry. We now consider several important examples of hyperbolic manifolds. EXAMPLE 2.1 (Hyperbolic surfaces). Suppose that F is a geometrically finite discrete group of hyperbolic isometries acting on H2. The quotient X = F\1HI2 has the decomposition X = X LJ (LJZ °1C2) LJ (LJ FlFj)

(2.2)

where X is a compact manifold with boundary, and the Ci and Fj are "cusp" and "funnel" ends described as follows. The Ci are cuspidal ends isometric to S10 x [0, oo)t with the metric

ds2 = dt2 + h2 e-2td02

(2.3)

294

P. PERRY

(so that, metrically, the "infinity" of the cusp is a single point), where hi is the length of the horocycle joining Ci to the compact manifold with boundary X. The Fj are funnel ends isometric to S1 x [0, oo)t with the metric ds2 = dt2 + £3 cosh2(t)d92

(2.4)

(so that, metrically, the "infinity" of the cusp is a circle). Here fj is the hyperbolic length of the geodesic circle where Fj is joined to X. All of the closed geodesics of X lie in X. The integers nc and nF count the respective number of cusp and funnel ends of X, and we assume that at least one of nF and nc is nonzero. The boundary of X consists of nF simple closed geodesics where the funnels are attached, and nc circles where the cusps are attached. The boundary at infinity of X thus consists of nc ideal points and nF circles. If nF = 0, then X has finite hyperbolic volume but if nF > 0, then X has infinite hyperbolic volume. In what follows, we will always suppose that nF > 0. Topologically, X is a surface of genus h (the number of holes in X) with nc punctures and nF discs removed, and the Euler characteristic of X is given by

X(X) = 2 - 2h - 2nF - 2nc

(2.5)

EXAMPLE 2.2 (Convex co-compact hyperbolic manifolds). Suppose that r is a convex co-compact discrete group of hyperbolic isometries acting on Hn+1 By definition, the hyperbolic convex hull of the limit set intersects any fundamental domain for the action of p on Hn+1 in a convex compact set, the Nielsen region for F. This descends to X = F\lH[n+1 as a compact convex set, X, called the convex core of X (in two dimensions, X has smooth boundary, but this need not be the case in higher dimensions). The Euler characteristic of X is that of the compact manifold with boundary X. Thus X is the union of X and finitely many "ends" diffeomorphic to (0, 1) x S for a compact manifold S without boundary. These ends are analogous to the funnels of Example 2.1 but need not carry a simple warped product metric if dim(X) > 2. Thus, in general, there is no simple model Laplacian for the ends of a convex co-compact hyperbolic manifold in dimension three or higher. There is a natural geometric compactification of X to a manifold with boundary. Let M = F\1(F) (recall that S2(F) is the domain of discontinuity of F acting on the boundary at infinity of 1<-I[n+l); this manifold carries a natural conformal structure since the transformations of F act conformally on 51(F). Then

X = X U M is a compact manifold with boundary. The smooth manifold M parameterizes the directions in which a geodesic may escape to infinity just as the

manifold Sn parameterizes these directions for geodesics in the space H'+'. In example 2.1, a surface with nc = 0 is convex co-compact, and M is a disjoint union of nF circles. EXAMPLE 2.3 (Classical Schottky groups). For simplicity we will describe classical Schottky groups acting on 1H13, following [21]; see also Section 2 of [73] for an

introduction to Schottky groups and further references. Suppose that

{Cj :1<j<2g} is a collection of disjoint circles on the Riemann sphere S2 (viewed as the boundary

of the unit ball in R3). For each i, 1 < i < g, let yi be a Mobius transformation with ryi(C2i_1) = C2i and taking the interior of C2i_1 to the exterior of C2i. Each circle Ci bounds a hyperbolic half space Hi, and each -yi extends to a hyperbolic

SPECTRAL GEOMETRY OF HYPERBOLIC MANIFOLDS

295

isometry with %i(aH2i-i) = aH2i and'yz(H2i_1) = IHI3\H2i. If F is the free discrete group generated by the -yi, the group F is called a Schottky group. A fundamental domain for its action on 1H13 is H3 - UZ91Hj. It is not difficult to see that the quotient manifold X = F\1H13 is handlebody of genus g whose boundary at infinity is a Riemann surface of genus g. Schottky groups have the property that geodesic

flow on the quotient X can be coded as symbolic dynamics, and the dynamical zeta function for geodesic flow can be studied using the Ruelle theory of transfer operators; see [73] and see our discussion in Section 5.3 below.

3. Spectral Theory for the Laplacian Lax and Phillips [94, 95, 96] studied the spectral theory of the Laplace operator on geometrically finite hyperbolic manifolds in the framework of Lax-Phillips scattering theory [97]. THEOREM 3.1 ([94, 95]). Suppose that F is a geometrically finite discrete group of isometrics of 1H1n+1 so that X = F\IH1n+1 has infinite volume. Then AX has at most finitely many eigenvalues in the interval [0, n2/4) and purely absolutely continuous spectrum with no embedded eigenvalues in [n2/4, oo).

The absolute continuity follows from the translation representation established

in [95], which provides much more detailed information about the behavior of solutions to the non-Euclidean wave equation than is discussed here. It follows that the resolvent (Ax - z)-1 is meromorphic in the cut plane [n2/4, oo) with at most finitely many poles due to eigenvalues. It is convenient to set z = s(n - s) and define

Rx(s) = (Ax - s(n -

s))-1

initially in the half-plane Re(s) > n/2, corresponding to the cut plane [n2/4, oo). The continuous spectrum corresponds to the "critical line" Re(s) = n/2. In the following sections we will consider meromorphic continuation of the resolvent and the geometry of the resolvent resonances. The Patterson-Sullivan theorem [126, 160] relates the exponent of convergence 6 for the discrete group r to the spectrum of the Laplace operator.

THEOREM 3.2 ([126, 160]). Suppose that r is a geometrically finite discrete group of hyperbolic isometries acting on 111Tn+1 with exponent of convergence 6. Then

the limit set A(F) has Hausdorff dimension S. Moreover, if 6 > n/2, the lowest eigenvalue of the Laplacian is 6(n - 6), and the positive eigenfunction can be recovered from a Poisson integral over the Hausdorff measure on the limit set. The theorem is due to Patterson if dim(X) = 2 and to Sullivan for general n. It implies that the resolvent is pole-free for Re(s) > 6. If 6 < n/2, it is still the case that the first pole of the meromorphically continued resolvent is given by 6 unless 6 - n/2 is an integer; see [127]. We close this section with a brief survey of related work on the spectral and scattering theory of hyperbolic manifolds of infinite volume. The spectral theory of the Laplacian on infinite volume, hyperbolic surfaces (i.e., nF > 0 in Example 2.1) was first studied by Elstrodt [38], Fay [45], Guillope [70], and Patterson [124]; these surfaces, as quotients of 11112 by a geometrically finite discrete group, are included in the class of geometrically finite hyperbolic manifolds (in any dimension) studied by

Lax and Phillips in [95, 96]. If nc = 0 but nF > 0 then X is a convex co-compact

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hyperbolic surface; we discuss this case and its higher-dimensional analogues in greater detail in Section 5. The case nF = 0 but nc > 0 includes the quotient of IH12 by arithmetic subgroups of PSL(2, III) and has been the subject of intensive investigation by many mathematicians; among the many studies of spectral and scattering theory in this setting, we mention only the pioneering work of Faddeev [44] and the 1976 study of Lax and Phillips [93]. Colin de Verdiere [29] studied a counting problem for geodesics on hyperbolic surfaces related to the counting problems discussed in Section 4.3; his analysis uses the spectral theory of the Laplacian. Mandouvalos [100] studied three-dimensional hyperbolic manifolds using techniques of analytic number theory, while Agmon [1, 2] and Perry [131, 132] applied Schrodinger-operator techniques to study spectral and scattering theory for convex co-compact hyperbolic manifolds. Mazzeo and Melrose [104] used very different techniques to study the Laplace

operator on a large class of manifolds, the asymptotically hyperbolic manifolds, that include convex co-compact hyperbolic manifolds. Their techniques have been applied to a number of problems on asymptotically hyperbolic manifolds, and we will return to their work later. Froese, Hislop, and Perry [51, 52], building on earlier work of Froese and Hislop [49], used Mourre's commutator method and Schrodinger-operator techniques to study the spectral and scattering theory for general geometrically finite hyperbolic manifolds in three dimensions, including those with so-called non-maximal rank cusps. These authors prove that the scattering operator, and hence the resolvent, admit a meromorphic continuation to the complex plane. Recently, Mourre's commutator technique has been used by Bouclet [18] to prove high-energy resolvent estimates for asymptotically hyperbolic manifolds. Cordoso and Vodev [31] proved optimal high-energy estimates on the resolvent

for a class of manifolds including hyperbolic manifolds with cusps. Bunke and Olbrich [24] studied spectral theory of the Laplacian and analytic continuation of the scattering operator for general geometrically finite discrete groups (including those with irrational cusps) using techniques of representation theory. They also refined and proved Patterson's conjecture [128] which relates the singularities of the zeta function to group cohomology of the group F and provides a unified approach to the "spectral" and "topological" zeros of the zeta function (see Theorem 5.3 below).

Recently, Guillarmou [63] has proven analytic continuation of the resolvent for a class of geometrically finite discrete groups acting on ]H[n'+i, n > 1, which includes the class studied by Froese, Hislop, and Perry but excludes manifolds with irrational cusps. The reader may consult, for example, the survey of Hislop [81] for further discussion and references to the literature up to 1993.

4. Spectral Geometry of Non-Compact Riemann Surfaces In this section, we consider the spectral geometry of geometrically finite, noncompact hyperbolic surfaces of infinite volume, as described in Example 2.1 above. Because these surfaces have ends isometric to explicit model spaces, particularly sharp results have been proven on the distribution of resonances, the asymptotics of the length spectrum, and the inverse resonance problem. For a more comprehensive

introduction to the spectral geometry of these surfaces, we refer the reader to Borthwick's forthcoming book [13].

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There is a natural renormalized integral on X, the 0-integral of a smooth function on X with suitable asymptotic behavior at infinity, which will play an important role in some of the results that follow. To define it, recall the cusp and funnel coordinates (t, 0) from Example 2.1 and set, for E E (0, 1): X (E) = X LJ (Uz 1CC(E)) LJ (H

1Fj(E))

where

C, (E) = {m E CZ(E) : t(m) < -log E} and

Fj (E) = {m E Fj (E) : t(m) < - log E}

The compact submanifolds X (E) exhaust X. If f E C°° (X) and dg denotes Riemannian volume measure with respect to the hyperbolic metric, we set

J Off dg = FP Ef°

f dg

(4.1)

X(E)

where FP denotes the Hadamard finite part (see Definition 1 of [77] and see Hormander [84], Section 111.3.2, page 70 for discussion). The existence of the 0-integral depends on the asymptotic behavior of f in the cusp and funnel ends. To given sufficient conditions, put p = et where t is the appropriate cusp or funnel coordinate, and say that f is polyhomogeneous if in each end f admits an asymptotic expansion of the form N3

°o

f (P, 0) - E E fj,k (e)P'

logk p

j=-N k=0

as p 1 0. Observe that, in each cusp or funnel, p2dg is a smooth function of p and 0. It follows that fX (E) f dg has an asymptotic expansion of the form Ej k Cj,kEj(log (1/E))k (where the j may be negative); the Hadamard finite part is the coefficient c0,0. The zero-trace need not be invariantly defined (the definition given here depends on an explicit choice of coordinates in the cusp and funnel ends) but under some circumstances it leads to geometric invariants. A simple example is the 0-volume 0- vol(X) =

°

J

dg = -21rX(X)

(4.2)

as may easily be seen from the Gauss-Bonnet theorem for X and the explicit form of the cusp and funnel metrics. The 0-trace of an operator with smooth kernel is the 0-integral of the kernel of the diagonal.

4.1. Resolvent and Scattering Operator. For surfaces with nF > 0, the Laplacian AX has purely continuous spectrum of infinite multiplicity in [1, oo) and at most finitely many eigenvalues in [0, 4) (see Theorem 3.1). It follows that the resolvent (AX - z)-1 is meromorphic in the cut plane (C\[4, oo). It will be convenient to make the quadratic transformation z = s(1-s) of the spectral parameter so that the continuous spectrum lies on the line Re(s) = a and the "physical region" corresponds to the half-plane Re(s) > 2. We define

Rx(s) = (OX - s(1 - s))-1

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298

as an operator on L2(X) for Re(s) > a and s(1-s) not an eigenvalue of Ox. Using the explicit form of model cusp and funnel resolvents, Guillope and Zworski [751 prove:

THEOREM 4.1 ([75]). The resolvent Rx(s) extends to a meromorphic operator valued function from Co (X) to C°° (X) with poles of finite rank.

More precisely, Rx (s) has isolated singularities and the Laurent expansion of Rx (s) at each such singularity has a finite polar part whose coefficients are finiterank, smoothing operators. Resolvent resonances are poles of the meromorphically continued resolvent. We denote by Rx the collection of poles of the resolvent (including those real s with s > a and s(1- s) an eigenvalue of the Laplacian), and define the multiplicity of a singularity C E Rx, denoted m{, to be the rank of the residue 1

2,7ri

,(2s - 1)Rx(s) ds y

where ry is a contour enclosing S and no other singularity of Rx(s); note that if > z and ((1 - (;) is an eigenvalue, then m( is the usual multiplicity. Let


Px(s) = 11 (1

- s/()" emc(s/C+s2/2(2)

(4.3)

cE7

converges to an entire function of s having order at most two. We will use the Hadamard product below to describe analytic properties of Selberg's zeta function on X. In [76], Guillope and Zworski show that Nx(r) also obeys a lower bound Nx(r) > C2r2 (where C2 is geometrically defined and may vanish for some X). Scattering eigenfunctions for Ox are smooth solutions of the eigenvalue equa-

tion (Ox - s(1 - s))u = 0 for Re(s) = 2. In the cusp ends, these can be chosen to take the asymptotic form u(t, 0) + bips + O(P°°) where p = e-t (referring to the coordinates in (2.3)) while in the funnel ends aipl_s

u(t, 0) ,,,

pl_s

f j (0) + Psgj (0) + 0 (P)

where again p = e-t (referring to the coordinates in (2.4)). It turns out that 2, a scattering eigenfunction is uniquely for fixed s with Re(s) = z and s determined by the numbers {ai}°1 and the functions { fj} Fl, i.e., by data in C"c ®((DnF1C°°(Sl)). Thus the map Sx(s) : Cnc ® ({a }i C1

,

(®,F C°°(sl)) c ® (®nF C°°(S1)) 7=1 7=1 {fj}h=l) ({bi}n al {gj}F

is well-defined. From its definition it is clear that

Sx(s)Sx(n - s) = I This map is called the absolute scattering operator for X and relates "incoming" to "outgoing" asymptotic data for scattering eigenfunctions. The operator Sx(s) is

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a pseudodifferential operator (see Section 5.1 below where the singularities of the scattering operator for convex co-compact manifolds are described). The geometry of X suggests a natural "comparison dynamics" for scattering. Each funnel end (imposing Dirichlet boundary conditions at t = 0) has an absolute scattering operator SF, (s) acting on C°°(Sl) which may be explicitly calculated by separation of variables in the eigenvalue equation. If we define a "comparison" scattering operator acting on (C'° ® ((D F1C°°(S1)) as So(s) = 1 ® (@3 c1SF, (s))

(here 1 is the identity operator on (Cnc), the relative scattering operator is given by

Sx(s) = Sx(s)So(s)-1

In [76] it is shown that, for Re(s) = a and s = the relative scattering operator takes the form Sx (s) = I + Tx (s) where Tx (s) is a trace-class operator on the natural Hilbert space C' ® Thus the determinant Dx(s) = det (I +Tx(s))) is well-defined The associated scattering phase is ax (s)

2-.

log det (Sx(s))

Guillope and Zworski [76] prove that the scattering phase has Weyl-type asymptotics. THEOREM 4.2 ([76]). The relative scattering phase obeys the asymptotic formula o"x(s)

41

0-vol(X) 1812 - nQ Is, log Is, + 0

(I81)

4.2. Resonances and Selberg's Zeta Function. The set Rx (including the multiplicities) may be regarded as the analogue of the eigenvalues of the Laplacian on a compact surface, and it is natural to ask how the resonances reflect the underlying geometry of X as encoded in the closed geodesics of X. For a compact manifold X, Selberg's zeta function gives a precise relationship between the eigen-

values of X, the length spectrum of X, and the Euler characteristic x(X) for X. The zeta function is defined as the Euler product 00

Zx(s) = 11 11 (1 - e-(3+k)t(7)) yEPxk=O

where the outer product ranges over primitive closed geodesics of X, and £(y) denotes the hyperbolic length of y. This product converges for Re(s) > 1 as a simple geometric argument2 shows that the number of geodesics of length r grows at most like exp(r). It follows from Selberg's trace formula (Selberg [151]; see also, 2Fix an origin 0 in hyperbolic space and consider the lattice {-y(0) : -y E F}. For each y the points 0 and -y(O) are separated by a geodesic hyperplane, so 0 is contained in an open half-space H. An open fundamental domain for I' is given by J7 = fl.yErH.y. Since closed geodesics correspond to conjugacy classes of hyperbolic elements, counting lattice points in a metric ball of radius r provides an upper estimate on the counting function for closed geodesics. The volume of a geodesic ball about 0 grows as the exponential of the geodesic distance. If X (and hence.) has finite volume, there can be at most O(exp(r)) lattice points within a metric ball of radius r, and if X has infinite volume, the same is true.

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e.g., McKean [107] for an exposition, but see Hejhal [80], Theorem 4.11, page 72 and comments following for a precise formulation and a correction to [107]) that the zeta function has an analytic continuation to an entire function of order two having spectral zeros at those s with Re(s) > a and s(1 - s) is an eigenvalue of the Laplacian, a zero at s = 0 of multiplicity 1 - X(X), and topological zeros at the integers s = -k, k = 1, 2, ... , with multiplicity -(2k + 1)X(X). The zeta function remains well-defined when X is a hyperbolic surface of infinite

volume, and it is natural to ask whether an analogous relationship holds. Guillope [71, 72] showed that, for such X, the zeta function has a meromorphic continuation to the complex plane and zeros in the resonance Rx. Borthwick, Judge, and Perry [15] determined the order and divisor of the zeta function. THEOREM 4.3 ([15]). Let X = F\ffH2 where r is finitely generated, and suppose

that nF > 0. Selberg's zeta function extends to a quotient of entire function of order at most two with the following singularities: (1) Spectral zeros (with multiplicity) at points S E Rx

(2) Topological zeros at s = -k, k = 0, 1, 2,..., of order (2k + 1)(-X(X)) (3) Topological poles at s = 2 - k, k = 0, 1, 2, ... , of order mc.

For finite area surfaces (nF = 0) the result is a well-known consequence of Selberg's trace formula. The proof of Theorem 4.3 relies heavily on the scattering theory developed by Guillope and Zworski in [76] and an explicit formula relating the resolvent of the Laplacian Ox to the zeta function (see Section 5.2 below for a formula valid when nc = 0). The factorization Zx(s) = e4(3)F(s - 2)n`Z.(s)pX(s) holds, where

Z. (s) _

[

(27)SF2(s)21-x(X)

F(s)

J

(F2(s) is Barnes' double F-function) and q is a polynomial of order at most two. There is also a natural renormalized "determinant of the Laplacian" (see [14]

for the case nc = 0 and [15] for the general case; compare [148] and [35] for compact surfaces, and [37] for non-compact surfaces of finite volume)

Dx(s) = det(Ax - s(1 - s)) which obeys the formula Dx(s) = eFs(s-1)+GZX (S) 1Z. (S)

if nc = 0 (here F and G are constants which depend on the renormalization; see [14]), and a similar formula with additional singular factors if nc > 0 (see [15], Theorem 5.1). Here the determinant is defined, not by the usual (-function regularization, but rather by the formula 1

(2s

d f)\2

-1ds

log Dx (s) = 0- Tr (Rx (s)2)

which is motivated by the identity

d ()21ogdet(A_AI)=Ir[(A_AIy2] true for symmetric matrices.

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301

Using the analytic properties and asymptotics of the zeta function, one can prove ([15], Corollary 1.2):

THEOREM 4.4 ([15]). The resonance set Rx determines the length spectrum

of X, the Euler characteristic x(X), and the number of cusps nc. The length spectrum determines x(X) and nc up to finitely many possibilities. The length spectrum, x(X), and nc together determine the resonance set. The discrete group F has a realization as a finitely generated, torsion-free subgroup of SL(2, ][8), and the lengths of closed geodesics of X correspond to traces of elements y E F. By combining Theorem 4.4 with Teichmuller theory, we can prove a rigidity result analogous to those of McKean [107] for compact Riemann surfaces and Muller [116] for Riemann surfaces of finite area. THEOREM 4.5 ([15]). Let X be a complete, geometrically finite hyperbolic sur-

face of infinite area. Then the length spectrum of X determines X up to finitely many possibilities. In particular, the resonance set determines X up to finitely many possibilities.

There are examples of non-isometric Riemann surfaces of infinite area with the same scattering resonances (see Remark 2.15 in [76] and [20], both of which exploit Berard's formulation [11] of Sunada's method [161]), so the conclusion of Theorem 4.5 is not trivial It is natural to ask what happens when one considers perturbations of the hy-

perbolic metric. For compactly supported perturbations of the metric, all of the essential features of spectral theory the meromorphic continuation of the resolvent, basic estimates on the counting function for closed geodesics, etc.-remain unchanged, as is proved by Guillope and Zworski in [76]. Borthwick, Judge, and Perry [14] considered relative determinants for metrics on infinite area surfaces X with nc = 0 of the form g = e2`0T where r is a hyperbolic metric and cp is a smooth, compactly supported function. An important role is played in this work by the uniformization theorem of Mazzeo and Taylor [105] for asymptotically hyperbolic metrics on surfaces. Using the relative determinant, they show first of all that the resonance set of the perturbed metric determines the Euler characteristic of X, and hence the diffeomorphism type of the surface, up to finitely many possibilities. They also show that the determinant obeys a Polyakov-type formula for conformal variations of the metric, and obtain an analogue of Osgood, Phillips, and Sarnak's celebrated result [121] that the set of metrics on a compact surface with given spectrum is precompact in a natural C°° topology on isometry classes. A complementary formulation of the duality between closed geodesics and resonances has been given by Guillope and Zworski [77], who prove a wave-trace formula analogous to the Duistermaat-Guillemin trace formula [36] for elliptic operators on compact manifolds. The set Px of primitive closed geodesics of least period £(y) is in one-to-one correspondence with the conjugacy classes of primitive elements in F. Let P. denote the Poincare once-return map for geodesics viewed as closed orbits in the cotangent bundle T*M. Guillope and Zworski prove:

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THEOREM 4.6 ([77]). Let X = F\1H12 , where F is finitely generated, having nc cusps. The following formula holds in distribution sense for t E IR:

0-Tr[cosltJ

,11

_ 0-vol(X) cosh(t/2)

,

sinh2 (t/2)

Sir

+ 4 coth (JtJ /2)

+ [nc ('Yo - log 2) -

hz

6(t)

i=1

p

(yYk 1/2

+12 .y

k=1 I I -

6 (I tl - k (7'))

Here the sum runs over primitive closed geodesics and yo is Euler's constant.

This is the "geometric" side of the trace formula. The "spectral" side of the trace formula is the Poisson formula 0-Tr [cos

(t/Lx

-

mse

2 4

(112-3)Itl

sEIZx

obtained by the authors in [76]. Guillope and Zworski use their trace formula to obtain a lower bound on the distribution of resonances in a strip. To state it, say that f (r) = 1l(g(r)) if and only if there does not exist a constant C so that f (r) < Cg(r). They prove: THEOREM 4.7 ([77]). For any hyperbolic surface X of finite geometry and any E E (0, z), the estimate

#ERx holds.

They conjecture that the right-hand side should be replaceable by a lower bound of the form rl+b where 6 is the exponent of convergence for F; see Section 5.3 for results in this direction.

4.3. Counting Closed Geodesics. Selberg's zeta function encodes information on the lengths of closed geodesics of X, so it is natural to ask what information about the length spectrum (i.e., the set of lengths counted with multiplicity) may

be read off from the resonance set Rx. It is expected that the first zero of the zeta function (which is spectrally determined) determines the asymptotics of the counting function. This is indeed the case and can be stated in a precise way. Let

N(T) _ # {y E Px : $(y) < T} and to describe the asymptotics of N(T) let li(x) = 12X

dt log(t)

Note that

li(x) = logx (1 +0 (logX))

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303

For a compact or finite-volume non-compact Riemann surface, it is known that

N(T) = li(ef) +

li(eakT) + 0 (e(3/4)T) k=1

as T - oo, where the ak are determined by "exceptional" eigenvalues A = 4 -a' of the Laplacian with A < 4 (see, e.g., Hejhal [80] and references therein for compact surfaces, and Sarnak [147] for the finite-volume case). The proof of (4.5) relies on Selberg's trace formula [151]. Independently, Guillope [69] (nc > 0) and Lalley [92] (nc = 0) showed that the counting function for closed geodesics on a non-compact hyperbolic surface (or equivalently on the compact manifold with boundary X; see (2.2)) has leading asymptotics IT

N(T) ^

ST

where 6 is the exponent of convergence of the group F. Naud [119] gave a sharp result on N(T) for Riemann surfaces with finite geometry and infinite volume (e.g., an infinite-volume hyperbolic surface with cusps, which is not convex co-compact). His analysis is based on Guillope and Zworski's work on scattering asymptotics [76] and their wave trace formula [77] (see Theorem 4.6). If nc > 0 it follows that 6 > (see Beardon [10]). 2

THEOREM 4.8 ([119]). Let X = F\1H12 be geometrically finite with nF >_ 1 and

6>1. Then, as T-* +oo, P

N(T) = li(eIT) + E li(eaiT) + 0 (e(6/2+1/4)T) k=1

where 1 < ap < ap_1 <

< as = 6 and ak(1 - ak) lies in the point spectrum of

OX

Naud also obtains a sharper version of the asymptotic formula if 6 < 1 (so that nc = 0). First, he uses transfer operator techniques to show that the first resonance of the resolvent, occurring at s = 6, is isolated in the sense that all other resolvent resonances lie in a half-plane Re(s) < 6 - e for some e > 0 [118]. (Stoyanov [159] has recently proven a similar result for zeta functions of convex co-compact discrete groups in any dimension.) Let RX denote the set of poles of the resolvent RX(s) and let

0 = sup {Re(s) : s E RX, s

6}

Using this result, he is able to prove: THEOREM 4.9 ([119]). Let X = F\1H12 with nc = 0, and 6 < 2, and let 3+ = max(,3, 0) Then

N(T) = li(eST) + 0 (e(6+0+ )T/2)

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5. Spectral Geometry of Convex Co-Compact Manifolds We now consider a class of hyperbolic manifolds in higher dimension which are roughly analogous to the hyperbolic surfaces of infinite volume without cusps: the convex co-compact hyperbolic manifolds considered in Example 2.2. Recall that these manifolds admit a geometrically natural compactification to a manifold with boundary: X = X U M where M is a smooth manifold with a natural conformal structure. A defining function for the boundary M is a non-negative smooth function x on X which is strictly positive on X and vanishes exactly to first order on M. There are two reasons why the geometry of resonances for convex co-compact hyperbolic manifolds can be thoroughly analyzed. First, the hyperbolic metric g takes the form g = x-2g where x is a defining function for M = aX and g extends

to a smooth nondegenerate metric on X. In this situation the Mazzeo-Melrose construction of the resolvent [104] is available (see also [74] where a somewhat more explicit construction is given for a class of manifolds that includes the convex co-compact hyperbolic manifolds). Second, Selberg's zeta function (whose precise definition if n > 2 we will come to in a moment) may be studied using Ruelle's thermodynamic formalism (see Ruelle [145] and Fried [48]). These techniques together lead to a connection between resonances and classical orbits precisely analogous to the connection already discussed for hyperbolic surfaces. This connection can be exploited to study the distribution of classical orbits.

In what follows, we will need a renormalized integral analogous to (4.1). If X is convex co-compact, x is a defining function for aX, and f E C°° (X) has a polyhomogeneous asymptotic expansion in x near x = 0, one can define the 0-integral of f and the 0-volume of X in analogy to (4.1) and (4.2), now taking X(,-) _ {m E X : x(m) > E} In general, the 0-integral of a smooth function with polyhomogeneous expansion at

x = 0 depends on the defining function and its derivatives at the boundary, but under some circumstances it yields interesting invariants. In an appendix to [130], Epstein proves that if dim(X) = 2m is even, the formula

0-vol(X) = (-1) (2m)!

X(X)

(5.1)

holds, where X(X) is the Euler characteristic of X.

5.1. Resolvent and Scattering Operator. If X is convex co-compact, it follows from Theorem 3.1 that the resolvent

Rx(s) = (OX - s(n - s))-' is a meromorphic operator-valued function for Re(s) > n/2 (corresponding to the cut plane C\[n2/4, co)) viewed as a map from L2(X) to itself. As before, it is important to study the analytic continuation of Rx (s) in order to define and discuss resonances.

We view the convex co-compact manifold X as the interior of a manifold with

boundary, X, and we let x be a defining function for M. Denote by C°°(X) the space of smooth functions in X and by C°°(X) the space smooth functions which vanish to all orders at x = 0. Functions in C°° (X) have Taylor expansions in x to all orders at x = 0. Finally, we denote by C°°(X) the space of smooth

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305

functions on X with no growth restriction as x 10. It follows from a more general result of Mazzeo and Melrose [104] (see Theorem 6.1 in what follows) that the resolvent operator Rx (s) : C°° (X) -, C°° (X) admits a meromorphic continuation to the complex s-plane whose poles are finite-rank smoothing operators. More precisely, the singularities of Rx(s) are isolated and the Laurent expansion at each such singularity has a finite polar part whose coefficients are finite-rank smoothing operators. The resonance set R.x is the set of all singularities of Rx (s). If E R.x we define the multiplicity of (, m( to be the rank of the residue of Rx (s) at s = (, i.e., m( = dim Ran PP

(5.2)

where

f(2s - n)Rx (s) ds (5.3) Zi and -y is a simple closed contour enclosing ( and no other singularity of R(s). If Re(() > n/2 then ( is real and m( is the dimension of the L2-eigenspace of the PC =

Laplacian with eigenvalue ((n - (). The scattering operator for X will play an important role in what follows, and is perhaps best understood in the framework of geometric scattering theory as formulated by Melrose [112] (especially Chapter 8); see also Borthwick [12] and Joshi-Sa Barreto [86] where the resolvent and scattering operator for asymptotically hyperbolic manifolds are analyzed in detail. We consider the following "Dirichlet problem" for generalized eigenfunctions u E C°° (X) with boundary data f E COO (M):

(Ox-s(n-s))u=0

(5.4)

u = xn-SF + xSG

Flax =f where F and G belong to C°°(X) and Re(s) = n/2 with s

n/2. The form of

the solution is dictated by the form of the Laplace operator on X: in any regular neighborhood of infinity, we can choose coordinates (x, y) E (0, 1) x R so that Ox = - (xax)2 + n (xOx) - x202

Indeed, the indicial operator for Ax is the operator

I(Ox) = - (x5x)2 + n (xax) which satisfies

I(Ox)xs = s(n - s)xs Using this fact, it is easy to see that for 2s - n not an integer, solutions having the form above (considered as power series in x) are formally determined given the boundary values of F and G at x = 0. In fact, for Re(s) = n/2 with s n/2. There is a unique solution to the Dirichlet problem (5.4): this follows from the absence of solutions u c L2(X) and the "boundary pairing formula" (see, e.g., [61] where this formula is discussed in the context of asymptotically hyperbolic manifolds). It follows from the uniqueness that the mapping Sx (s) : C°° (M) -> C°° (M)

f'- GIM

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is well-defined. This map is called the absolute scattering operator. From its definition, it is not difficult to see that

Sx(s)Sx(n - s) = I where I is the identity operator on C°°(M). The scattering operator has a meromorphic continuation to s E C, as was shown for example by Agmon [1, 2] and Perry [131, 132]. Its formal logarithmic derivative Sx(s)-1S'x(s) was studied in [130]. In the next section, we discuss analogous but more general results for the scattering operator on asymptotically hyperbolic manifolds. THEOREM 5.1. The scattering operator Sx (s) has a meromorphic continuation to C with the following properties: (i) For each s where it is defined, Sx (s) is a pseudodifferential operator acting in suitably chosen local coordinates on M as convolution with the singular kernel

cnF(n/2 - s)

Ixl_2s

F(s)

modulo a smoothing operator.

(ii) At s = n/2 + k, k E N, Sx(s) has a first-order pole of infinite rank. (iii) At each singularity ( of s H Sx(s) in C\ 2 (n + N), the Laurent expansion of Sx (s) near s = ( has a finite polar part whose coefficients are smoothing operators. (iv) The operator Sx(s)-1SX(s) has first-order poles with finite-rank residues at each singularity ( and

vC _ -Tr (Res [Sx(s) 1Sx(s) S=C is an integer.

The infinite rank poles of Sx (s) turn out to be of great interest in conformal geometry; see Theorem 6.14.

It is important to note that, even in the model case X = Hn+l, the scattering operator as we have defined it has poles of infinite rank in the "physical" halfplane Re(s) > n/2 at each of the points s = n/2 + k, k = 1, 2, ... ; see Section 2 of [61] for an illuminating explanation of this phenomenon via an example from scattering theory on the half-line. The residues of Sx(s) at the poles s = n/2 + k, k = 1, 2, ..., are actually differential operators Pk of order 2k associated with the conformal structure on M. We will set dk = dim ker(Pk)

(5.5)

These operators will play a very important role in the analysis of scattering and in connection with conformal geometry.

5.2. Selberg's Zeta Function and Its Divisor. If X is convex co-compact, the discrete group F consists of loxodromic elements (see (2.1)), and closed geodes-

ics of X are in one-to-one correspondence with conjugacy classes of F. For each conjugacy class, the length £(y) and the eigenvalues a1(y), ... , an(y) of the rotation matrix A, are independent of the choice of representative y. If we view the closed geodesic as a closed orbit in the cotangent bundle of X under geodesic flow, then £(y) is its period and the Poincare once-return map P., may be computed in terms of the invariants £('y), a1(y), ... , an(y).

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307

Selberg's zeta function is defined as follows. Let {-y} be a listing of representatives of conjugacy classes of primitive elements of P. Then

Zr(s) = 11

e-(s+ki+...+k,)P(7)

(1 - 0,1(_J')ki ... an('y)kn)

JJ

{-y} kl, ,k,,,>0

This infinite product can be shown to converge absolutely for Re(s) > n. It is important to note that, as s - oo through real values, Zr(s) approaches 1 exponentially fast. Selberg's zeta function can also be identified with the dynamical zeta function for geodesic flow on X. If -y is a primitive element of F, the Poincare once-return map P.y for the associated geodesic satisfies the identity I det (I - Pr)11/2 = e(n/2)E(7)G,y

where n

G.y = 11 (1 -

c,j(-y)e-e(7))

I

k=1

Elementary manipulations with (5.6) give the formula

Zr(s) = exp

- 1: E

1 exp(-(s - n/2))Q(ryn') I det (I - Py-) 1/2

m m=1-yEPX

which exhibits the zeta function as a dynamical zeta function for geodesic flow on X. Using Ruelle's thermodynamic formalism [145] and results of Fried [48], one can prove: THEOREM 5.2 ([130]). Let F be an orientation-preserving, torsion-free, convex co-compact discrete group. Then Selberg's zeta function Zr(s) is a quotient of entire functions of order at most n + 1.

The singularities of the zeta function are determined by the poles of the scattering operator together with the Euler characteristic of the manifold X. For dim(X) even, this result is due to Patterson and Perry [130]; for dim(X) odd, it is due to Bunke and Olbrich [23]. In what follows, x(X) denotes the Euler characteristic of X and may be positive, negative, or zero.

THEOREM 5.3 ([23, 130]). Let F be an orientation-preserving, torsion-free, convex co-compact discrete group, and let X = F\Hn+I Selberg's zeta function Zr(s) has the following singularities: (1) A zero of order mC if ( > n/2 and ((n - O is an eigenvalue of OX (2) A zero of order dim ker (I + Sx (n/2)) at s = n/2 (3) A zero of order v( if Re (() < n/2 ( is a singularity of SX(s)S'X(s), and ((n - () is not an eigenvalue of AX, and of order vC - mn_C otherwise (4) A zero (or pole) of order hn(k)x(X) at s = -k where

hn(k) _ (2k + n)

(k + 1)

(k + n

1)

n!

Note that singularities of types (3) and (4) may coincide.

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308

Let us explain the main ideas of the proof for dim(X) even. The key element in the proof is the identity Zr(s) + Zr(n - s) = 0-Tr(RX(s)-RX(n-s)) (5.7) Zr(s) Zr(n - s)

+ (2k

(-1)F(s)P(n - s) n!

F(s - n/2)F(n/2 -s) X(X)

This follows from the identity, due to Patterson [128] (and familiar in calculations with the Selberg trace formula ula in other settings)

Zr'(s) = f. [Gr(z, w, s) - Go (z, w, s)] iz=w dvol(z)

(5.8)

for a fundamental domain F in Hn+l, where Gr(z, w, s) is the lift of the resolvent kernel to ]H[n+1 and Go(z, w, s) is the resolvent kernel for the Laplacian on IH[n+1

Patterson's identity exploits the fact that both the logarithm of the zeta function and the resolvent can be expressed as a sum over the group F. To obtain (5.7) from (5.8), one uses the identity I'( Go(z, w, s) - Go(z, w, n - s) _ n-n/2 F(s /2)F(n/2 - s) rr(/) F(n) (which follows from the explicit formula for Go(z, w; s)) and Epstein's formula (5.1). The first right-hand term in (5.7) gives rise to zeros of the zeta function associated to eigenvalues and resolvent resonances; see (6.13), and note that the multiplicity of these zeros is determined jointly by the resolvent resonances and the numbers dk = dim ker Pk, as has been emphasized by Guillarmou [62] (see Theorem 6.9 in what follows). The second right-hand term gives rise to poles whose multiplicity is determined by the Euler characteristic. Brooks, Gornet, and Perry [21] constructed pairs of non-isometric Schottky manifolds in three dimensions with the same eigenvalues and resonances, so the resonances do not uniquely determine the isometry class of a metric on a convex co-compact hyperbolic manifold.

5.3. Distribution of Resonances. Selberg's zeta function can be used to study the distribution of scattering resonances by exploiting its connection with classical dynamics. We let RX be the set of all resolvent resonances, counting multiplicity and RX,s be the set of poles of the scattering operator. Let

NX(R)=#{sERX:Isl
NX,S(R)_#{sERX,,s: Isl
Nx (R) < CRn+2 holds. This bound is not optimal in the sense that there are hyperbolic manifolds for which NX(r) ^_ CRn+1 (see, e.g., the cylindrical manifolds considered in Appendix

B of [130] or, for n = 2, the solid torus studied in [142]). Perry [140] used the results of Patterson-Perry [130] and Bunke-Olbrich [23] to prove the lower bound NX,s (R) > CRn+1

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309

for convex co-compact hyperbolic manifolds in any dimension. Note that the difference between the counting functions N,8 and Nr depends on the numbers dim ker Pk;

one expects these numbers to be zero "generically" (they are known to vanish if dim(X) = 2) but on the other hand, there are examples when they actually saturate the counting function (see the comments in Section 2 of [68] and especially the explicit computation in Proposition 2.2 there). A finer question is the relationship between the distribution of resonances and the "trapped set" of geodesics. Zworski [166] proved THEOREM 5.4 ([166]). Suppose that X is a convex co-compact hyperbolic sur-

face. For any a c [0, 1] and a, b > 0, the estimate

# Is E RX : a - Re(s) < a Im sla + b, Is! < r} <

Crl+a+b(I-a)

holds, where 6 is the Hausdorff dimension of the limit set.

Note that for a = 0, it follows that # Is E RX : 2 - Re(s) < C1, Isl

< r} < C2rl+b

This gives a "Weyl-type" upper bound for the counting function for resolvent resonances in strips. The dimension of the recurrent set for geodesic flow in T*X

is 2(1 + 5) so that the exponent can be taken to represent the dimension of the trapped set. Guillope, Lin, and Zworski [73] studied Selberg's zeta function for convex co-

compact Schottky groups using Ruelle transfer operators. Among other results they show that Selberg's zeta function for such groups is entire and not simply a quotient of entire functions as shown by Patterson and Perry. They proved that, in strips parallel to the imaginary axis, the zeta function obeys the bound IZr(s)I < exp (C Islb) where S is the exponent of convergence of the Schottky group P. This implies the following estimate on zeros of the zeta function in strips: THEOREM 5.5 ([73]). Suppose that F is a convex co-compact Schottky group and let Z be the set of zeros of Selberg's zeta function, counted with multiplicity. Then for any Co > 0 there is a Cl so that the estimate

# {( E 2 : Re(s) > -Co, r < IIm(s)I < r + 1} < Clrb holds.

Any convex co-compact hyperbolic surface is Schottky (see Section 6 of [73] and references given there). In this case, the estimate on zeros of the zeta function, combined with Theorem 5.3 imply an estimate on the distribution of resonances for convex co-compact surfaces. THEOREM 5.6. [73] Suppose that X is a convex co-compact hyperbolic surface.

For any Co > 0 there is a constant Cl with the property that

# Is E RX : Re(s) > -Co, r < IIm(s) I < r + 1} < Clrb Numerical evidence is presented suggesting that these estimates are optimal.

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5.4. Counting Closed Geodesics. The counting function for closed geodesics may be studied using the zeta function much as in the surface case described in Section 4.3. As before, we denote by Px the set of primitive closed geodesics of X and set

N(T) = # {yEPx:f('y)
N(T) ,,, exp(6T) dT

asT ->oo. The key to the proof is to show that the first zero of Selberg's zeta function is, in all cases, the exponent of convergence b, and occurs with multiplicity one. For d > n/2 this is an immediate consequence of the Patterson-Sullivan theorem [124, 160] and the characterization of the divisor of Z(s). Ford < n/2 one makes a careful study of the relationship between resolvent poles and scattering poles in the region 0 < s < n/2. A similar result for certain discrete groups with parabolic elements was obtained by Dal'bo and Peigne [32] by very different methods. Guillarmou and Naud [68] proved a more refined asymptotic formula in case X is a convex co-compact hyperbolic manifold with d > n/2. In what follows, we write eigenvalues Ai of Ax as ai(n - ai) where ai > n/2. THEOREM 5.8 ([68]). Let X = 1,\lHlni+1 be a convex co-compact hyperbolic manifold with b > n/2. Then

li (eajT) + C

N(T) = li(eST) + 0n(b)
( e/3"(b)T T

I

where 3n(6) _ (2 + d) n/(n + 1). The proof of this theorem rests on a formula for the 0-trace of the wave group on a convex co-compact hyperbolic manifold together with the results of PattersonPerry [130] and Bunke-Olbrich [23] on the divisor of Selberg's zeta function. The trace formula is of interest in its own right; recall the numbers dk defined in (5.5).

THEOREM 5.9 ([68]). Let X = 1'\Hn+1 be a convex co-compact hyperbolic manifold. Then, as distributions on l[8\ {0}, 0-tr (cos (t 2ix

- n2/4))

mse-(V/2-s)Itl

SER.x

dke-IkIt

+1 kEN

-

cosh(t/2)

2-n -'A(X)

sinh(Itj /2)n+1

where

A(X)

0

x(X)

if (n + 1) is even if (n + 1) is odd

(5.9)

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We also have

0-tr (cos (t

f (Y) Ox - n2/4)) _ E E 21-r7 S(itl - ke(y)) kl

(5.10)

ryEpxk=1

+ B(X)

cosh(t/2) (sinh (Itl /2))n+1

where

if (n + 1) is even if (n + 1) is odd

cn 0-vol(X)

B(X) _

0

This trace formula should be compared with Theorem 4.6. In that case, it is known that the integers dk are all zero. The identities (5.9) and (5.10) are the "spectral" and "geometric" sides of the trace formula. One can also ask how geodesics are distributed between homology classes, a question which was already studied for compact manifolds of negative curvature by Katsuda-Sunada [89] and Phillips-Sarnak [143]. Epstein [43] studied the distribution of geodesics in homology classes for hyperbolic manifolds of finite volume and any dimension. More recently, Sharp [153] (see [152] for related results) has obtained precise asymptotics for the counting function of geodesics in a homology class for finite area hyperbolic surfaces For hyperbolic manifolds, the homology group H1 (X, Z) is the abelianization of the fundamental group 7r1(X) . We assume that H1 (X, Z) is non-trivial and let cp be a homomorphism from H1(X, Z) onto ZT. Since H1(X, Z) = r/ [r, r] and the geodesics of X are in one-to-one with conjugacy classes of r, we may regard cp as a mapping from the closed geodesics to Z and partition the closed geodesics of X into equivalence classes labelled by elements a E zr. Let correspondence

r.(T)=#{yEPx:£(y)
where ( ) denotes the usual inner product on W. As in the work of previous ,

authors, the spectral problem AU = Au u('Y(z)) = X0(y)u(z)

plays a role. We denote by A (O) the lowest eigenvalue of the spectral problem, which is unique for small 0. McGowan and Perry [106] prove: THEOREM 5.10 ([106]). Let F be a convex co-compact discrete group of isome-

tries of ]H[n+1 with 6 > n/2. Let X = r\]H[n+' and suppose there is an onto homomorphism cp : H1 (X, Z) -> Z r. Then 7ra(T)

exp(6T)

tr/2+1 (co + c1T-1 +

c2T-2

... )

where co is strictly positive and depends only on r and the Hessian of A(O) at 0 = 0.

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The proof of Theorem 5.10 uses a Selberg trace formula for the Laplacian twisted by a character, following the approach of Phillips and Sarnak. Using very different methods, Babillot and Peigne [7, 8] studied the distribution of geodesics in homology classes for hyperbolic manifolds of Schottky type with cusps; for these groups F\H'+1 has infinite hyperbolic volume but may have both regular and cusp neighborhoods at infinity. Schottky groups admit a coding of the geodesic flow by methods of symbolic dynamics; the authors use this symbolic coding and Ruelle's theory of transfer operators to obtain asymptotic formulas for Ira (T) of the form exp(ST)

it (T)

Cr P(T)

where the form of P(T) depends on the Hausdorff dimension 6 of the limit set for F (see also the comments in [43] concerning equidistribution for hyperbolic manifolds of finite volume with cusps).

6. Spectral Geometry of Asymptotically Hyperbolic Manifolds Many of the results of the preceding sections are illuminated by considering the case of asymptotically hyperbolic manifolds. To define this class, we first recall the class of conformally compact manifolds. Suppose that X is a smooth manifold

with boundary whose interior we will denote by X. We set M = OX. A smooth Riemannian metric g on X is called conformally compact if there is a defining function x for M and a smooth metric g on X with the property that x2g = g on X. In this case we set h = g1M. If, in addition, the differential dx satisfies the 19 denotes the norm on one-forms condition ldx(p)19 = 1 for each p c M (here induced by the metric g), the sectional curvatures of g approach -1 as x 10, and I

the metric g is called asymptotically hyperbolic. If x is a defining function for M, then so is

_

e2c'x for any cp E C°° (X). For

this reason, the metric h = x2gl M is determined only up to a conformal factor. Thus an asymptotically hyperbolic metric g on X determines an equivalence class [h] of conformally equivalent metrics on M. The pair (M, [h]) is called the conformal infinity of (X, g). Suppose that (X, g) is an asymptotically hyperbolic manifold with conformal infinity (M, [h]). There is a collar neighborhood (0, e) x M of the boundary and we

denote by H(x) a smooth function on (0, r) taking values in the symmetric twoforms on M. Graham and Lee [60] (see also [58] and Section 2 of [65]) showed that for each boundary metric h E [h], there is a unique boundary defining function x so that the metric g assumes the model form

g=

dx2 X2

H(x)

+ x2

(6.1)

where H(e) is a metric on the hypersurface x = e, and H(0) = h. We will call such a function a special defining function. We will call a metric g even modulo x2k+1 if k

H(x)

hjx2' +O(x2k+1)

(6.2)

j=0

for smooth symmetric two-tensors hj on M. Guillarmou shows that this condition is independent of the choice of special defining function (see [65], Lemma 2.1). Hyperbolic metrics are even modulo O(x°°).

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A special class of asymptotically hyperbolic metrics will play an important role in what follows. Fefferman and Graham [46] showed that, given a conformal manifold (M, [h]) where M bounds a compact manifold X, there is an asymptotically hyperbolic metric g with conformal infinity [h] which satisfies

n odd n even with an additional condition if n is even. Such metrics are called asymptotically Einstein. Hyperbolic metrics are exact solutions of the Einstein equation Ric(g) +

Ric(g) + ng =

O(xn )2)

ng=0. As in the case of convex co-compact hyperbolic manifolds, we can define the notion of 0-integral for a smooth function on X as

J Off dg = FP Ef0

f dg

x>E

if the finite part exists. This will be the case if f admits an asymptotic expansion near x = 0 of the form Mi

f (x, y) ,., E

aj,k (y)x (- log x)k

j>-N k=0

for smooth functions ajk on M. The 0-trace of an operator T with smooth kernel K is the 0-integral of its diagonal, i.e.,

0- Tr T =0 fK(m,m) dg(m) In general, the 0-integral and 0-trace will depend on the choice of defining function but for certain operators and appropriate classes of metrics, it leads to an invariantly defined quantity; see the papers of Albin [5, 6] and the work of Guillarmou [67] discussed below. The 0-trace is a natural trace on the algebra of Vo-pseudodifferential operators developed in [104].

6.1. Resolvent and Scattering Operator. To discuss the spectral theory of the Laplacian, we need to describe its form in a neighborhood of infinity. Suppose that (X, g) is an asymptotically hyperbolic manifold and x2g I M = h. Choose a special defining function, as described above, so that g takes the form

g = x-2(dx2 + H(x)) where H(0) = h. The Laplace operator then takes the form OX = I(xax)

+x2Ah(x) +xTr(h-1(x)axh)(xax)

(6.3)

(6.4)

where

I(xax) _ -(xax)2 +n(xax)

(6.5)

is the indicial operator that describes the degeneracy of the Laplacian near M. The Laplacian on an asymptotically hyperbolic manifold belongs to an algebra of pseudodifferential operators, the 0-pseudodifferential operators, studied in [104]. The development of the 0-calculus is part of a larger program of analysis on manifolds with corners which has many applications in geometric scattering theory; see [112] for an exposition and references. The importance of the 0-calculus in the present context is that it permits the inversion of elliptic operators in the calculus, including the Laplacian.

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If (X, g) is an asymptotically hyperbolic manifold, the Laplacian has at most finitely many eigenvalues in the interval [0, n2/4) and no eigenvalues A with A > n2/4 [102]. Let

Rx(s) = (OX - s(n - s))-1 initially defined for Re(s) > n/2. To describe the meromorphic extension of the resolvent, we define the function spaces C°° (X ), C°° (X ), and C°° (X) in analogy to those defined on convex co-compact hyperbolic manifolds. Mazzeo and Melrose [104] proved: THEOREM 6.1 ([104]). Suppose that (X, g) is an asymptotically hyperbolic man-

ifold. The resolvent RX(s), viewed as an operator from C°°(X) to C°°(X), admits a meromorphic continuation to the complex plane with the points s E 2 (n removed.

The resonance set and the multiplicity of a resonance are defined as in Section 5.3. The Mazzeo-Melrose construction gives the following important mapping property of the resolvent: THEOREM 6.2 ([104]). The resolvent Rx(s) is a continuous map from C°°(X) to x8C°°(X).

If the metric g has constant sectional curvatures -1 in a neighborhood of infinity, Guillope and Zworski [74] proved a sharper result on the meromorphic continuation. THEOREM 6.3 ([74]). Suppose that (X, g) is asymptotically hyperbolic and that Then the resolvent RX(s), viewed as a mapping from C°°(X) to C°°(X), admits a meromorphic con-

g has constant curvature -1 in a neighborhood at infinity.

tinuation to the complex plane with isolated singularities. At each singularity , the resolvent has a Laurent expansion with finite polar parts whose coefficients are finite-rank smoothing operators.

These authors also prove the estimate

Nx (r) < CX(I +r)n+2 on the counting function for resonances; this estimate is not optimal in the dimension.

In his doctoral thesis, Guillarmou ([62] and [65]) illuminated the relationship between essential singularities of the resolvent and asymptotic behavior of the metric We will say that RX (s) is finitely meromorphic in a region Q C C if s -+ RX (s) has no essential singularities in Q, the poles of Rx (s) form a discrete set, and the

polar part of the Laurent expansion at each singularity consists of a finite series whose coefficients are finite-rank smoothing operators. Guillarmou proves:

THEOREM 6.4 ([62, 65]). Suppose that (X, g) is an asymptotically hyperbolic manifold and that g is even modulo O(x21,+1). Then the resolvent extends to a finitely meromorphic function in the half-plane Re(s) > n/2 - k - 2

In particular, if g is even modulo O(x°°) if and only if the meromorphically continued resolvent has no essential singularities. On the other hand, he shows that there are metrics which are even modulo O(x21c+1) for which the resolvent has an

essential singularity at s = (n - 1)/2 - k (see [65], Corollary 4.3). Indeed, if one topologizes the asymptotically hyperbolic metrics by the topology inherited from

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315

x-2C°°(X,T*X ®T*X), essential singularities of the meromorphically continued resolvent occur generically (see [65], Theorem 1.5). To define the scattering operator for Ax, we proceed exactly as in Section 5.1. Consider solutions u E C°° (X) to the generalized eigenvalue problem

(Ox-s(n-s))u=0 Note that solutions with Re(s) = n/2 correspond to the continuous spectrum of Ax. A formal power series analysis based on the indicial operator (6.5) suggests that the general solution should take the form u = xn-sF + x8G

where F and G belong to C°°(X). The Taylor series of F and G at x = 0 are formally determined by the functions f = FIM and g = GIM. Moreover, it can be shown that the Dirichlet problem

(Ax - s(n - s)) u = 0 u = xn-sF + xSG FPM

(6.6)

=f

has a unique solution for given f E C°°(M) and fixed s with Re(s) = n/2, s $ n/2. Thus the mapping Sx (s) : C°° (M) -> C°° (M)

(6.7)

f - GIM is a well-defined linear operator from C°° (M) to itself. From its definition,

Sx(s)Sx(n - s) = I The operator Sx (s) depends on the choice of defining function x,: if t = e`'x is another defining function for cp E C°° (X) with VIM = T, it is not difficult to see

that

Sx(s) = e-sTSx(s)e(n-s)T (6.8) Note that h = x2g = e2T h. In this sense the scattering operator is conformally covariant.

Although initially defined for Re(s) = n/2 and s $ n/2, the scattering operator extends to a meromorphic family of pseudodifferential operators. We denote by a(A) the principal symbol of the pseudodifferential operator A. Joshi and Sa Barreto [86] prove THEOREM 6.5 ([86]). Suppose that (X, g) is an asymptotically hyperbolic mani-

fold, fix a defining function x for M = 8X, and let h = x2gl.=o. Then the mapping

s H Sx(s), initially defined on the line Re(s) = n/2 and s

n/2, extends to a

meromorphic function on (C\ (n - N) taking values in the pseudodifferential operators on C°° (M). The operator2 Sx (s) is a pseudodifferential operator of order 2s - n with principal symbol 2n_2s

I'(n/2 - s) (As-n/2 ) f(s - n/2) or h

where Ah is the Laplacian on (M, h). At each singularity in C\ (n - N) the scata tering operator has a Laurent expansion with finite polar point whose coefficients are smoothing operators.

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Joshi and Si Barretto use the Mazzeo-Melrose construction to obtain the meromorphic continuation for general, asymptotically hyperbolic metrics. The paper [86] also proves the following interesting result on inverse scattering.

THEOREM 6.6 ([86]). Suppose that gl and 92 are asymptotically hyperbolic metrics on a fixed manifold X, and that the associated scattering operators Si(s) and S2(s) have the property that Si(s) - S2(s) is a pseudodifferential operator of order 2 Re(s) - n - k. Then the metrics gl and 92 have the property that x2 (gl - 92) vanishes to order k at x = 0. A stronger inverse result was recently proven by Sa Barreto by very different methods. Using the boundary control method of Belishev [9] and the author's study of radiation fields for scattering on asymptotically hyperbolic manifolds, Sa Barreto [146] proves: THEOREM 6.7 ([146]). Let (Xi, gi) and (X2ig2) be two asymptotically hyperbolic manifolds with the same boundary Ml = M2 = M. Suppose that Sl (s) and S2(s) are scattering matrices on Xl and X2 corresponding to special defining functions xl and x2i and suppose that Sl(n/2+iA) = S2(n/2+i.\) for every A E R\ {0}. Then, there is a diffeomorphism : Xl --p X2, smooth up to M, so that = I on

M andT*92=gi Graham and Zworski [61] give an elegant proof that Sx (s) admits a meromorphic continuation using Theorem 6.2; let us summarize the main ideas here. First, it is easy to construct a formal power series solution of (Ax - s(n - s)) u = 0 in a neighborhood of the boundary (where X has the structure of a product manifold M x (0, e)) having the form 00

ul - r aj

n-s+j

j=0

Here the aj are smooth functions on M, ao = f, and the remaining aj are determined by recursion, using the fact that the indicial operator (6.5) satisfies

[I(xax) - s(n - s)]

xn-s+j

= j(2s - n -

j)xn-s+j

This gives, on the formal level at least, a mapping 1(s) : C°°(M) - xn'-sC°°(X) with the property that (Ax - s(n - s)) ,,P(s) f E C°°(X). Moreover, it is easy to see that this map is analytic in s away from the points s = n/2 + k, k = 1, 2, . Since the resolvent Rx (s) maps d- (X) to x3C°° (X), we can solve the initial value problem by setting

u = Ib(s) f - [Rx(s) o (Ax - s(n - s)) o 4)(s)] (f)

(6.9)

Comparing (6.9) with the definition of the scattering operator (6.7) via the Dirichlet problem (6.6), we obtain the formula

Sx(s)f =

-x_s [Rx(s)

o (Ax - s(n - s)) o ((s)] (f)lx=(,

Given the analytic continuation of the resolvent, this formula yields the analytic continuation of the scattering operator. It also connects the resolvent resonanc and scattering resonances explicitly. The scattering operator has poles at the resolvent resonances but also has poles at the points s = n/2 + k where k is an integer. They arise in the factor -D (s) above

SPECTRAL GEOMETRY OF HYPERBOLIC MANIFOLDS

317

owing to the crossing of indicial roots in the formal power series solution that defines

'(s). The residues Pk = ,-n/2+k Res Sx (s)

(6.10)

are actually elliptic differential operators of order 2k acting on C°° (M) whose coefficients depend on the the first 2k derivatives of the metric at the boundary. In all cases, these operators are conformally covariant in the sense that Pk = e-(n/2+k)TPke(n/2-k)T

if h = e2T h. In the special case that g is an asymptotically Poincare-Einstein metric, these operators are natural operators associated to the conformal structure on M, that is, they depend only on the metric h = x2g restricted to M. We will discuss them in greater depth below. Recently, Perry and Schueth [141] showed that, given any metric on a complete Riemannian manifold (X, g) of dimension n > 9 admitting an isometric action by 0(n), there are continuous families gt of non-isometric perturbed metrics which are identical to g outside a compact set and have the same resolvent resonances and scattering resonances. In particular, there are continuous families of hyperbolic or asymptotically hyperbolic metrics on simply connected spaces of dimension n > 9 which have this property.

6.2. Trace Formulas. In this section we describe progress toward trace formulas recently obtained by Joshi-S6 Barreto [87] and Guillarmou [67]. Joshi and Sa Barreto [87] have studied the trace of the wave group for asymptotically hyperbolic manifolds. They study the solution operator U(t) for the wave equation

utt + (Ox - n2/4) u = 0 u(x,0) = f(x) ut(x, 0) = 0

and show that U(t) is a Fourier integral operator. They show that the 0-trace of U(t) exists and defines a distribution on R. Note that this distribution need not be independent of the choice of defining function. Using their representation of U(t) as a Fourier integral operator, they are able to prove the following analogue of Duistermaat-Guillemin's results for the trace of the wave group on a compact manifold.

THEOREM 6.8 ([87]). Let (X, g) be an asymptotically hyperbolic manifold. The singular support of 0-tr (U(t)) is contained in the set of periods of closed geodesics of (X,g).

More recently, Guillarmou [67] has obtained Krein-type formulas relating the 0-trace of the spectral family for the Laplacian to a renormalized determinant of the scattering operator. The interesting feature of these formulas is that there is no "comparison operator" as in the Krein theory of the spectral shift, but instead the renormalizations are "intrinsic" to the geometric setting. It will sometimes be convenient to work with a renormalized scattering operator Sr (S) =

22s-n 1'(n/2 - s) S,x (s)

I'(s - n/2)

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318

for which infinite-rank poles are absent. It can be shown that the operator

T(s) = -Sr(s)-'S,(s)

(6.11)

has at most first-order poles whose singularities are finite-rank operators. Moreover, the traces

v( = Tr

Is=(

T(s)

(6.12)

are integral and count the multiplicity of scattering poles (see [130] for the case of convex co-compact hyperbolic manifolds and Guillarmou [67] for the case of asymptotically hyperbolic manifolds). The poles of the scattering operator "essentially" correspond to the poles of the resolvent. The precise relationship was analyzed by Guillope and Zworski [76] for hyperbolic surfaces and by Borthwick-Perry [16] (using Agmon's [4] perturbation theory of resonances) and Guillarmou [62, 66] for asymptotically hyperbolic manifolds. Borthwick and Perry also show that the resonances are generically simple,

i.e., singularity of the resolvent at the resonance is a first-order pole with rankone residue (compare [90] where an analogous result for Schrodinger operators is proved). Recall the definitions (6.12) and (6.10), and recall that mS is the multiplicity of the resolvent resonance at . Borthwick-Perry [16] and Guillarmou [66] proved:

THEOREM 6.9 ([16, 65]). Let (X, g) be an asymptotically hyperbolic manifold, and suppose that ( is a resolvent resonance. Then VC

Vn/2-N

m( - mn_( = vC-dimkerPk

=n/2-k

For S V n/2 - N, this result was proved by Borthwick-Perry [16]; the precise description of the case ( E n/2 - N is due to Guillarmou [66]. If ( < n/2, the term mn_C is zero unless c(n-() is one of the finitely many eigenvalues of the Laplacian. For a geometrically natural class of asymptotically hyperbolic manifolds, Guillarmou [67] has defined a renormalized determinant of the scattering operator and used it to construct an analogue of Krein's spectral shift. He considers asymptotically hyperbolic manifolds with dim(X) is even and metrics g on X which are even modulo 0(x°). In this setting, Guillarmou exploits the work of Kontsevich-Vishik [91] and Lesch [99] to define a trace for pseudodifferential operators such as T(s) above which are very far from being trace class. Although we cannot fully describe

the ideas here, the starting point is to consider, for an elliptic pseudodifferential operator A, the analytic trace-valued function Z(s) = Tr (BA-s). If B is a fixed pseudodifferential operator, A is elliptic and Re(s) is sufficiently large, then Z(s) is well-defined and one wants to define TR(B) = Z(0) by analytic continuation. This can be carried out for suitable classes of pseudodifferential operators B. There is also a determinant of Sr(s) which extends meromorphically to the complex plane, formally defined as follows. Noting that Sr(s) is actually self-adjoint for s real and sufficiently large, one can define a Ray-Singer-type determinant for such s by det(Sr(s)) = exp (-
((s, (s, t) = Tr (Sr(S)_t)

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319

is well-defined (with the usual trace) for t large by the ellipticity of Sr(s). The determinant can be analytically continued, and it can be shown that det(Sr(s)) has TR [ST(s)Sr(s)-1] as its logarithmic derivative. Let Gx (x, y, s) be the Schwarz kernel of the resolvent operator Rx (s). Then Gx (x, y, s) - Gx (x, y, n - s) is actually a smooth kernel since the diagonal singularities cancel. Guillarmou [67] proves: THEOREM 6.10 ([67]). Suppose that (X, g) is an asymptotically hyperbolic man-

ifold, that dim(X) is even, and that g is even modulo O(x°°). Then the formula

0-Tr ((n - 2s) [Rx(s) - Rx(n - s)]) = TR [S'r(s)Sr(s)-1]

(6.13)

where TR is the Kontsevich-Vishik trace, holds meromorphically in s E C.

The left-hand side of (6.13) can naturally be regarded as the derivative of a renormalized spectral density (via Stone's formula). This formula is analogous to formulas appearing in the Krein theory of the spectral shift, but the renormalization here is purely geometric. The determinant has a meromorphic continuation whose singularities are determined by the resonances of the resolvent together with the kernels of the operators Pk. More precisely: THEOREM 6.11 ([67]). Under the hypotheses of Theorem 6.10, the function det(Sr(s)) extends meromorphically to C. If C E C then the divisor of det(Sr(s)) at s = is given by -m( + Mn-( - 1n/2_Ny(() dim ker Sr (n - () + 1,,/2+N(() dim kerS,(()

Here is denotes the characteristic function of the set S and a pole is assigned negative sign. For the case of convex co-compact hyperbolic manifolds, Guillarmou proves a

beautiful formula relating the determinant of the scattering operator to the zeta function.

THEOREM 6.12 ([67]). Let X = F\Hn+1 be a convex co-compact hyperbolic manifold having even dimension, let X(X) be the Euler characteristic of X, and let Zx(s) be Selberg's zeta function for X. Finally, let mn/2 be the rank of the residue

of Rx (s) at s = n/2. Then det Sr (n + iA) 2

Zx (n2 + iA )

Zx(n/2 -

exp (-iX(X)G(z))

where

G(z) _

(-2)(n+1)/27r1/2F (2) fZ F(n/2 + it)F(n/2

- it)

dt

F(it)F(-it) 6.3. Scattering Theory and Conformal Geometry. In this subsection, n!!F(n)

we describe recent work of Graham and Zworski [61] which connects the scattering

operator on (X, g) with invariants of the conformal structure on the conformal manifold (M, [h]). First, let us briefly describe the conformal invariants studied in [61]. Graham, Jenne, Mason, and Sparling [59] proved the existence of certain natural, conformally covariant differential operators on a conformal manifold. A differential operator on

(M, h) is natural if it can be expressed in terms of the metric h, its curvature,

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and covariant derivatives of h. A family of operators Ph defined for h E [h] is conformally invariant of weight (w, w') if PT = ew'TPhe-wT

for h = e2T h. Note that, in this sense, the scattering operator Sx(s) for an asymptotically hyperbolic manifold is a conformally covariant pseudodifferential operator

of weight (s - n, -s) THEOREM 6.13 ([59]). Let (Mn, [h]) be a conformal manifold. For k E N (or

1 < k < n/2 if n is even), there exist natural, conformally covariant differential operators Pk of weight (k - n/2, -n/2 - k) so that the leading symbol o,(Pk) is the principal symbol of Oh.

These "conformally invariant powers of the Laplacian" (often referred to in the literature as the "GJMS operators") play a fundamental role in conformal geometry. The operator n-2 R

Pl=Ah+ 4(n - 1)

where R is the scalar curvature, is the Yamabe operator (see, e.g., Lee and Parker [98]). It occurs in the study of conformal deformations of metrics to a metric of constant scalar curvature. In even dimensions, Branson's Q-curvature [19] is an analogue in conformal geometry of the scalar curvature (indeed, in two dimensions, Q = R/2); it transforms under conformal deformations h -- e2Th as enTQ

= Q + Pn/2T

and its integral is a conformal invariant. Graham and Zworski [61] proved: THEOREM 6.14 ([61]). Let (Mn, [h]) be a conformal manifold and let (X,g) be

a Poincare metric associated to [h]. Suppose that k c N if n is odd, or k < n/2 if n is even. Finally, suppose that (n/2)2 - k2 is not an eigenvalue of the Laplacian Ax. Then the scattering operator SX (s) has a simple pole at s = n/2 + k and

ckPk = -

Res SX(s) s=n/2+k

where the Pk are the GJMS operators and ck

(_1)k - 22kk!(k - 1)!

Moreover, if n is even, Pn/21 = 0 and cn/2Q = SX(n)1

The authors deduce the self-adjointness of the Pk from the self-adjointness of SX(s) for real s. A different analysis, using formal power series arguments in the place of scattering theory, was later given by Graham and Fefferman [47].

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[154] Sigal, I. M.; Soffer, A.: The N-particle scattering problem: asymptotic completeness for short-range systems. Ann. of Math. (2) 126 (1987), no. 1, 35-108. [155] Simon, Barry: Phase space analysis of simple scattering systems: extensions of some work of Enss. Duke Math. J. 46 (1979), no. 1, 119-168. [156] Simon, Barry: Trace Ideals and Their Applications. London Mathematical Society Lecture Note Series, 35. Cambridge University Press, Cambridge-New York, 1979. [157] Sjostrand, Johannes: Geometric bounds on the density of resonances for semiclassical problems. Duke Math. J. 60 (1990), no. 1, 1-57. [158] Sjostrand, Johannes; Zworski, Maciej: Asymptotic distribution of resonances for convex obstacles. Acta Math. 183 (1999), no. 2, 191-253. [159] Stoyanov, Luchezar: Ruelle zeta functions and spectra of transfer operators for some Axiom A flows. Preprint, 2006. See http://www.maths.uwa.edu.au/-stoyanov/geodAA.pdf. [160] Sullivan. D.: Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. I.H.E.S. Publ. Math. 50 (1979), 259-277. [161] Sunada, T.: Riemannian coverings and isospectral manifolds. Ann. of Math. (2) 121 (1985), no. 1, 169-186. [162] Zworski, Maciej: Counting scattering poles. Spectral and Scattering Theory (Sanda, 1992), pp. 301-331, Lecture Notes in Pure and Appl. Math., 161, Dekker, New York, 1994 [163] Zworski, Maciej: Distribution of resonances for convex co-compact hyperbolic surfaces. Journees "Equations aux Derivees Partielles" (Saint-Jean-de-Monts, 1997), Exp. No. XVIII, 9 pp., Ecole Polytech., Palaiseau, 1997. [164] Zworski, Maciej: Poisson formulae for resonances. Seminaire sur les Equations aux Derivees Partielles, 1996-1997, Exp. No. XIII, 14 pp., Ecole Polytech., Palaiseau, 1997. [165] Zworski, Maciej: Poisson formula for resonances in even dimensions. Asian J. Math. 2 (1998), no. 3, 609-617. [166] Zworski, Maciej: Dimension of the limit set and density of resonances for convex co-compact hyperbolic quotients. Invent. Math. 136 (1999), 353-409. [167] Zworski, Maciej: Resonances in physics and geometry. Notices Amer. Math. Soc. 46 (1999), no. 3, 319-328.

[168] Zworski, Maciej: Resonance expansions in wave propagation. Seminaire: Equations aux Derivees Partielles, 1999-2000, Exp. No. XXII, 11 pp., Semin. Equ. Deriv. Partielles, Ecole Polytech., Palaiseau, 2000. [169] Zworski, Maciej: A remark on isopolar potentials. SIAM J. Math. Anal. 32 (2001), no. 6, 1324-1326.

[170] Zworski, Maciej: Quantum resonances and partial differential equations. Proc. International Congress of Mathematicians, Vol. III (Beijing, 2002), pp. 243-252, Higher Ed. Press, Beijing, 2002.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF KENTUCKY, LEXINGTON, KY 40506-0027,

U.S.A. E-mail address: [email protected]

Proceedings of Symposia in Pure Mathematics Volume 76.1, 2007

Topics in the Theory of Positive Solutions of Second-Order Elliptic and Parabolic Partial Differential Equations Yehuda Pinchover

Dedicated to Barry Simon on the occasion of his 60th birthday ABSTRACT. The purpose of the paper is to review a variety of recent developments in the theory of positive solutions of general linear elliptic and parabolic equations of second-order on noncompact Riemannian manifolds, and to point out a number of their consequences.

CONTENTS 1. 2. 3.

4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Introduction Principal Eigenvalue, Minimal Growth, and Classification Perturbations Indefinite Weight Large Time Behavior of the Heat Kernel Nonuniqueness of the Positive Cauchy Problem and Intrinsic Ultracontractivity Asymptotic Behavior of Eigenfunctions Localization of Binding The Shuttle Operator Periodic Operators Liouville Theorems for Schrodinger Operators and Criticality Polynomially Growing Solutions and Liouville Theorems Criticality Theory for the p-Laplacian with Potential Term

References

2000 Mathematics Subject Classification. Primary 35J15; Secondary 35B05, 35C15, 35K10. Key words and phrases. Green function, ground state, heat kernel, Liouville theorem, Martin boundary, positive solution, p-Laplacian. ©2007 American Mathematical Society 329

330

Y. PINCHOVER

1. Introduction Positivity properties of general linear second-order elliptic and parabolic equations have been extensively studied over the recent decades (see for example [47, 68] and the references therein). The purpose of the present paper is to review a variety of recent developments in the theory of positive solutions of such equations and to

point out a number of their (sometimes unexpected) consequences. The attention is focused on generalizations of positivity properties which were studied by Barry Simon in the special case of Schrodinger operators. Still, the selection of topics in this survey is incomplete, and is according to the author's working experience and taste. The reference list is far from being complete and serves only this expose. The outline of the paper is as follows. In Section 2, we introduce some fundamental notions that will be studied throughout the paper. In particular, we bring up the notions of the generalized principal eigenvalue, criticality and subcriticality of elliptic operators, and the Martin boundary. Section 3 is devoted to different types of perturbations and their properties. In Section 4, we study the behavior of critical operators under indefinite perturbations. In Sections 5 and 6, we discuss some relationships between criticality theory and the theory of nonnegative solutions of the corresponding parabolic equations. More precisely, in Section 5, we deal with the large time behavior of the heat kernel, while in Section 6, we discuss sufficient conditions for the nonuniqueness of the positive Cauchy problem, and study intrinsic ultracontractivity. In Section 7, we study the asymptotic behavior at infinity of eigenfunctions of Schrodinger operators. The phenomenon known in the mathematical physics literature as `localization of binding', and the properties of the shuttle operator are discussed in Sections 8 and 9, respectively. The exact asymptotics of the positive minimal Green function, and the explicit Martin integral representation theorem for positive solutions of general Zd-periodic elliptic operators on Rd are reviewed in Section 10. We devote Section 11 to some relationships between criticality theory and Liouville theorems. In particular, we reveal that an old open problem of Simon (Problem 9.1) is completely solved (see Theorem 11.2). In Section 12, we study polynomially growing solutions of Z' -periodic equations on Rd. We conclude the paper in Section 13 with criticality theory for the p-Laplacian with a potential term.

2. Principal Eigenvalue, Minimal Growth, and Classification Consider a noncompact, connected, smooth Riemannian manifold X of dimension d. For any subdomain S2 C_ X, we write D Cc S2 if D is a compact subset of Q.

The ball of radius r > 0 and center at xo is denoted by B(xo, r). Let f, g c C(1l), we use the notation f g on D C S2 if there exists a positive constant C such that C-lg(x) < f (x) < Cg (x) for all x E D. By 1, we denote the constant function taking at any point the value 1. We associate to any subdomain I C_ X an exhaustion of S2, i.e., a sequence of smooth, relatively compact domains {1 },'?__I such that I 54 0, Qy C Slj+I and Uj__152j = Q. For every j > 1, we denote SZ = I \ Qj. We say that a function f E C(S2) vanishes at infinity of 1 if for every E > 0 there exists N E N such that If (x) I< E for all x E S2n,. We associate to any such exhaustion {S2j}j__1 a sequence

of smooth

cutoff functions in Q such that Xj(X) - 1 in Q j, xj (x) - 0 in S2 \ S1j+1, and 0 <

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Xj (x) < 1 in Q. Let 0 < a < 1. For W E Ca (52), we denote Wj (x) = Xj (x) W (x) and Wj* (x) = W (x) - Wj (x). We consider a linear, second-order, elliptic operator P defined in a subdomain 52 C X. Here P is an operator with real Holder continuous coefficients which in any coordinate system (U; x1i ... , Xd) has the form d

P(x, ax)

d

aij (x)aiaj +

bi(x)ai + c(x),

(2.1)

where ai = alaxi. We assume that for each x c 52 the real quadratic form Edj=1 aij (x)t;'ilj is positive definite on Rd.

We denote the cone of all positive (classical) solutions of the elliptic equation Pu = 0 in 52 by CP(c). We fix a reference point x0 E 521. From time to time, we consider the convex set 1CP(c) :_ {u E CP(S2) I u(xo) = 1}

of all normalized positive solutions. In case that the coefficients of P are smooth enough, we denote by P* the formal adjoint of P. DEFINITION 2.1. For a (real valued) function V E Ca(c), let A0(P, 52, V) := sup{A E R I CP-av(52) # 01

be the generalized principal eigenvalue of the operator P with respect to the (indefinite) weight V in Q. We also denote

a,, (P,c,V) := sup A0(P,c\K,V). KC=H

For a fixed P and 52, and V = 1, we simply write A0

A0(P, 52,1) and A,,',

(P, Q, 1).

DEFINITION 2.2. Let P be an elliptic operator of the form (2.1) which is defined

on a smooth domain D C= X. we say that the generalized maximum principle for the operator P holds in D if for any u E C2 (D) n C(D), the inequalities Pu > 0 in D and u > 0 on aD imply that u > 0 in D. It is well known that ao(P, 52,1) > 0 if and only if the generalized maximum principle for the operator P holds true in any smooth subdomain D C= Q. The following theorem is known as the Allegretto-Piepenbrink theory, it relates A0 and a,,, in the symmetric case, with fundamental spectral quantities (see for example [1, 17, 76] and the references therein). THEOREM 2.3. Suppose that P is symmetric on Co (52), and that A0 > -oo. Then A0 (resp. Ate) is equal to the infimum of the spectrum (resp. essential spectrum)

of the Friedrich's extension of P.

Therefore, in the selfadjoint case, A0 can be characterized via the classical Rayleigh-Ritz variational formula. In the general case, a variational principle for A0 is given by the Donsker-Varadhan variational formula (which is a generalization of the Rayleigh-Ritz formula) and by some other variational formulas (see for example [51, 68]).

DEFINITION 2.4. Let P be an elliptic operator defined in a domain 52 C_ X. A function u is said to be a positive solution of the operator P of minimal growth in

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a neighborhood of infinity in S2 if u E Cp(ci) for some j > 1, and for any 1 > j, and v c C(ii U rill) rl Cp(ci ), if u < v on acj, then u < v on S2i . THEOREM 2.5 ([1]). Suppose that Cp(SZ)

0. Then for anyxo E S2 the equation

Pu = 0 has (up to a multiple constant) a unique positive solution v in i \ {xo} of minimal growth in a neighborhood of infinity in Q. By the well known theorem on the removability of isolated singularity [29], we have:

DEFINITION 2.6. Suppose that Cp(c) 0. If the solution v of Theorem 2.5 has a nonremovable singularity at xo, then P is said to be a subcritical operator in Q. If v can be (uniquely) continued to a positive solution v of the equation Pu = 0 in 52, then P is said to be a critical operator in SZ, and the positive global solution v is called a ground state of the equation Pu = 0 in Q. The operator P is said to be supercritical in S2 if Cp(c) _ 0. REMARKS 2.7. 1. In [74], Simon coined the terms '(sub)- (super)-critical operators' for Schrodinger operators with short-range potentials which are defined on Rd, where d > 3. The definition given in [74] is in terms of the exact (and particular) large time behavior of the heat kernel of such operators (see [75, p. 71] for the root of this terminology). In [43], Murata generalized the above classification for Schrodinger operators which are defined in any subdomain of Rd, d > 1. The definition of subcriticality given here is due to [52]. 2. The notions of minimal growth and ground state were introduced by Agmon in [1].

3. For modified and stronger notions of subcriticality, see [24, 52]. Outline of the proof of Theorem 2.5. Assume that Cp(S1) 54 0 and fix xo E Q.

Then for every j > 1, the Dirichlet Green function GP' (x, y) for the operator P exists in ij. It is the integral kernel such that for any f E C0 (1l), the function uj (x) := f1 GP' (x, y) f (y) dy solves the Dirichlet boundary value problem

Pu= f inu3,

u=0 on act.

It follows that GPj (., xo) E Cp(ci3 \ {xo}). By the generalized maximum principle, {GP' (x, xo)}jo1 is an increasing sequence which, by the Harnack inequality, converges uniformly in any compact subdomain of 52 \ {x0} either to Go (x, x0), the positive minimal Green function of P in S2 with a pole at x0 (and in this case P is subcritical in S2) or to infinity. In the latter case, fix x1 E 52, such that x1 x0. It follows that the sequence {GP' xo)/GP' (x1, x0)} converges uniformly in any compact subdomain of S2\{xo}

to a ground state of the equation Pu = 0 in 11, and in this case P is critical in Q.

COROLLARY 2.8. (i) If P is subcritical in S2, then for each y E S2 the Green function Gn y) with a pole at y exists, and is a positive solution of the equation Pu = 0 of minimal growth in a neighborhood of infinity in Q. Moreover, P is subcritical in S2 if and only if the equation Pu = 0 in S2 admits a positive supersolution which is not a solution. (ii) The operator P is critical in S2 if and only if the equation Pu = 0 in S2 admits (up to a multiplicative constant) a unique positive supersolution. In particular, dimCp(1l) = 1.

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(iii) Suppose that P is symmetric on Co (S2) with respect to a smooth positive density V, and let P be the (Dirichlet) selfadjoint realization of P on L2(Q, V(x)dx). Assume that A E apoint(P) admits a nonnegative eigenfunction cp, then A= A0 and P-AOV is critical in Il (see for example [43]). (iv) The operator P is critical (resp. subcritical) in 52 if and only if P* is critical (resp. subcritical) in Q.

As was mentioned, (sub) criticality is related to the large time behavior of the heat kernel. Indeed, (sub) criticality can be also defined in terms of the corresponding parabolic equation. Suppose that A0 > 0. For every j > 1, consider the Dirichlet heat kernel kP' (x, y, t) of the parabolic operator L at + P on SZj x (0, oo).

So, for any f E Co (1), the function uj (x, t) = f0 kP (x, y, t) f (y) dy solves the initial-Dirichlet boundary value problem

Lu = 0 in S2j x (0, oo),

u = 0 on 852 x (0, oo),

u = f on SZj x {0}.

By the (parabolic) generalized maximum principle, {ko (x, y, is an increasing sequence which converges to kP(x, y, t), the minimal heat kernel of the parabolic operator L in Q. LEMMA 2.9. Suppose that A0 > 0. Let x, y E Q, x

fkOp(x,y,t)dt
resp.

y. Then

fk0p(x,y,t)dt=oo

00

if and only if P is a subcritical (resp. critical) operator in Q. Moreover, if P is subcritical operator in 11, then GP' (x, y) =

kP(x, y, t) dt.

J0

(2.2)

For the proof of Lemma 2.9, see for example [68]. Note that if A
Subcriticality (criticality) can be defined also through a probabilistic approach. If the zero-order coefficient c of the operator P is equal to zero in S2, then P is called

a diffusion operator. In this case, P1 = 0, and therefore, P is not supercritical in Q. Moreover, for such an operator P, one can associate a diffusion process corresponding to a solution of the generalized martingale problem for P in Q. This diffusion process is either transient or recurrent in Q. It turns out that a diffusion

operator P is subcritical in S2 if and only if the associated diffusion process is transient in SZ (for more details see [68]). A Riemannian manifold X is called parabolic (resp. non-parabolic) if the Brownian motion, the diffusion process with respect to the Laplace-Beltrami operator on X, is recurrent (resp. transient) [32]. Suppose now that P is of the form (2.1), and P is not supercritical in Q. Let co E Cp(SZ). Then the operator Pw acting on functions u by 1

P'u :_ -P(cpu) is a diffusion operator, and

kP (x, y, t) _

p(x)

kp (x, y, t)P(y)

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Therefore, P is subcritical in Sl if and only if P`' is transient in Q. We have the following general convexity results. THEOREM 2.10 ([54]). (i) Let V E Ca(SZ), V # 0 and set

S+ = S+ (P, SZ, V) _ {A E 1l I P - AV is subcritical in SZ}, So = So(P, S2, V) _ {A E 1l I P - AV is critical in 1l}. Then S := S+ U So C R is a closed interval and So C 8S. Moreover, if S S is bounded if and only if V changes its sign in Q.

(ii) Let W,V E Ca(c), then the function .o (p)

(2.3) (2.4)

0, then

Ao(P - MW,c,V) is a

concave function on the interval {µ E JR I IAo(µ)I < oo}.

PROOF. For 0 < s < 1, and V0, Vi E Ca (S2), let

P3:=P+sV1+(1-s)Vo. Assume that uj are positive supersolutions of the equations Pju = 0 in SZ, where j = 0, 1 (so, Pjuj > 0). It can be verified that for 0 < s < 1, the function [uo(x)]1_9

u9(x)

[u1(x)]9

is a positive supersolution of the equation Psu = 0 in Q. Moreover, for any 0 < s < 1, us E Cp (S2) if and only if Vo = V1, and uo, ul E Cp0 (SZ) are linearly dependent. The lemma follows easily from this observation. COROLLARY 2.11 ([54] and [76]). Suppose that Ps := P + sV1 + (1 - s)Vo is subcritical in c for s = 0, 1. Then for 0 < s < 1 we have y)]1_9

GPs (x, y) 5 [GP0 (x,

8

[GP, (x, y)]

.

(2.5)

REMARK 2.12. The dependence of .\o on the higher order coefficients of P is more involved. In [12] it was proved that in the class of uniformly elliptic operators with bounded coefficients which are defined on a bounded domain in IRd, \o is locally

Lipschitz continuous as a function of the first-order coefficients of the operator P. Ancona [7] proved that under some assumptions, \o is Lipschitz continuous with respect to a metric dist(P1, P2) measuring the distance between two elliptic operators P1 and P2 in a certain class. Ancona's metric depends on the difference between all the coefficients of the operators P1 and P2. If P is subcritical in S2, then Cp(c) is in general not a one-dimensional cone. Nevertheless, one can construct the Martin compactification QM of c with respect to the operator P (with a base point x0), and obtain an integral representation of any solution in CP(c). More precisely, the Martin compactification is the compactification of SZ such that the function

G (x' y) sz

KP (x, y)

on S2 X St \ {(xo, x0)}

GP (xo, y)

has a continuous extension KP(x,rl) to S2 x (QM \ {xo}), and such that the set P of functions rl)j,,cq separates the points of QM. . The boundary of QM is denoted by ap S2 and is called the Martin boundary of St with respect to the operator is called the Martin function of the pair P. For each t; E aP S2, the function KP

(P, Q) with a pole at l;. Note that for t; E 8P S2, we have

E ACp(S2). The

POSITIVE SOLUTIONS OF ELLIPTIC AND PARABOLIC PDE

335

aP 1 such that is an extreme point of the convex set 1CP(S2) is called the minimal Martin boundary (for more details, see [43, 47, 68, 81, and the references therein]). The Martin representation theorem asserts that for any u E 1CP(Sl) there exists a unique probability measure p on aP S1 which is supported on am pSl such that set 0,,MP1l of all

u(x) = f M KP(x, ) dlu() P

There has been a great deal of work on explicit description of the Martin compactification and representation in many concrete examples (see for example [41, 47, 50, 68, 81, and the references therein]). We present below two elementary examples of Martin compactifications. In Section 10, we discuss a recent result on the Martin compactification of a general periodic operator on Rd. EXAMPLE 2.13. Let Sl be a smooth bounded domain in Rd, and assume that the coefficients of P are (up to the boundary) smooth. Then aP Q is homeomorphic to 81 , the Euclidean boundary of Sl, and for any y E asl,

avCP(x

K'sz(x, y)

,

y)

(2.6)

y) ' where av denotes the inner normal derivative with respect to the second variable. Note that y) is the Poisson kernel at y E 811.

EXAMPLE 2.14. Consider the equation HAu := (-0 + ,\)u = 0 in Rd. Then CH,, (Rd) $ 0 if and only if A > 0. It is well known that Ho = -0 is critical in Rd if and only if d < 2. Moreover,

r(I/ d

I

A = 0, and d > 3,

d/2

41r

HA (x, y) _

V (27r)-d/2

Cx

- yl

K,(v'A-lx - yI)

A > 0,

where v = (d - 2)/2, and K is the modified Bessel function of order v. Clearly,

lim G A (x, y) = 1. yI-oo Therefore, the Martin compactification of Rd with respect to the Laplacian is the one-point compactification of Rd, and we obtained the positive Liouville theorem: /C-A(Rd) = {1}. Suppose now that A > 0. Then for any l; E Sd-1, GHa (x, y)

lim

-c l yl --goo GHd (0, y)

and therefore, the Martin boundary of Rd with respect to HA is the sphere at infinity. Clearly, all Martin functions are minimal. Furthermore, u E CHA (Rd) if

and only if there exists a positive finite measure p on Sd-1 such that u(x) =

f

1

d

e"\"' dµ(t;)

Y. PINCHOVER

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REMARK 2.15. We would like to point out that criticality theory and Martin boundary theory are also valid for the class of weak solutions of elliptic equations in divergence form as well as for the class of strong solutions of strongly elliptic equations with locally bounded coefficients. For the sake of clarity, we prefer to concentrate on the class of classical solutions.

3. Perturbations An operator P is critical in 0 if and only if any positive supersolution of the equation Pu = 0 in S2 is a solution (Corollary 2.8). Therefore, if P is critical in 1 and V E Cc (Q) is a nonzero, nonnegative function, then for any A > 0 the operator

P + AV is subcritical and P - AV is supercritical in Q. On the other hand, it can be shown that subcriticality is a stable property in the following sense: if P is subcritical in 1 and V E Cc (Q) has a compact support, then there exists e > 0 such that P - AV is subcritical for all JAI < c, and the Martin compactifications QM and QM av are homeomorphic for all J.\ < e (for a more general result, see Theorem 3.6). Therefore, a perturbation by a compactly supported potential (at least with a definite sign) is well understood. In this section, we introduce and study a few general notions of perturbations related to positive solutions of an operator P of the form (2.1) by a (real valued) potential V. In particular, we discuss the behavior of the generalized principal eigenvalue, (sub) criticality, the Green function, and the Martin boundary under such perturbations. Further aspects of perturbation theory will be discussed in the following sections.

One facet of this study is the equivalence (or comparability) of the corresponding Green functions.

DEFINITION 3.1. Let Pj, j = 1, 2, be two subcritical operators in Q. We say that the Green functions GP, and Go are equivalent (resp. semi-equivalent) if P2

Go GP2 on Q x Q \ { (x, x) l x E Qj (resp. GP', (-,yo) some fixed yo E Q).

GP2 (-,yo) on Q \ {yo} for

LEMMA 3.2 ([52]). Suppose that the Green functions GP, and GP2 are equivalent. Then there exists a homeomorphism -b : B,MP1 S2 am Q such that for each minimal point E a Pl11, we have K ° 1 ( . , ) KP2 -b (t )) on Q. Moreover, the cones Cp, (S2) and Cp2 (S2) are homeomorphic. REMARKS 3.3. 1. It is not known whether the equivalence of G° and GP2

implies that the cones Cp, (S2) and Cp2 (Il) are affine homeomorphic. 2. Many papers deal with sufficient conditions, in terms of proximity near infinity in S2 between two given subcritical operators PI and P2, which imply that Go and GP2 are equivalent, or even that the cones Cp, (1) and Cp2 (Q) are affine homeomorphic, see Theorem 3.6 and [4, 7, 43, 46, 52, 53, 69, and the references therein].

We use the notation E+ = E+(V, P, Q)

sE+ = sE+(V, P,1)

{a E I[8 I GP_av and GP are equivalent},

{A E R I GP_av and GP are semi-equivalent} . The following notion was introduced in [53] and is closely related to the stability of Cp(11) under perturbation by a potential V.

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337

DEFINITION 3.4. Let P be a subcritical operator in S2, and let V E Ca(c). We say that V is a small perturbation of P in SZ if lim

j°°

sup 'YCQ

f

G° (x, z) IV(z) IGP(z, y) dz1ll

GP(x, y)

= 0.

(3.1)

The following notions of perturbations were introduced by Murata [46]. DEFINITION 3.5. Let P be a subcritical operator in SZ, and let V E Ca(c). (i) We say that Vris a semismall perturbation of P in S2 if GP(xo, z) I V(z)I GPO (z, y)

lim { sup j00 YCQ

dz

= 0.

(3.2)

GOP (X0' y)

(ii) We say that V is a G-bounded perturbation (resp. G-semibounded perturbation) of P in S2 if there exists a positive constant C such that GO (x, z) I V (z) I GP(z,y)

fn

GPO (x, y)

dz < C

(3.3)

for all x, y E SZ (resp. for some fixed x E SZ and all y E i \ {x}). (iii) We say that V is an H-bounded perturbation (resp. H-semibounded perturbation) of P in Il if there exists a positive constant C such that dz < C f GP (x, z)IV(z)lu(z) u(x)

(3.4)

st

for all x c S2 (resp. for some fixed x E i) and all u c CP(c). (iv) We say that V is an H-integrable perturbation of P in Il if < oo

(3.5)

for all x c Il and all u c CP(S2). THEOREM 3.6 ([46, 53, 54]). Suppose that P is subcritical in Q. Assume that

V is a small (resp. semismall) perturbation of P* in Q. Then E+ = S+ (resp. sE+ = S+), and 8S = So. In particular, S+ is an open interval. Suppose that V is a semismall perturbation of P* in S2, and A E 5o. Let cpo be the corresponding ground state. Then cpo GP x0) in ii. Suppose that V is a semismall perturbation of P* in S2, and A E S+. Then the mapping

GP_av (x, z)V(z)u(z) dz f an affine homeomorphism of CP(c) onto CP_av(SZ), which induces a homeoT (u) := u(x) + A

(3.6)

morphism between the corresponding Martin boundaries. Moreover, in the small perturbation case, we have T(u) u in fZ for all u E CP(SZ). REMARKS 3.7. 1. Small perturbations are semismall [46], G-(resp. H-) bounded

perturbations are G- (resp. H-) semibounded, and H-semibounded perturbations are H-integrable. On the other hand, if V is H-integrable and dimCP(1l) < oo, then V is H-semibounded [46, 52]. There are potentials which are H-semibounded perturbations but are neither H-bounded nor G-semibounded. We do not know of any example of a semismall

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(resp. G-semibounded) perturbation which is not a small (resp. G-bounded) perturbation. We are also not aware of any example of an H-bounded (resp. H-integrable) perturbation which is not G-bounded (resp. H-semibounded) [61]. 2. Any small (resp. semismall) perturbation is G-bounded (resp. G-semibounded), and any G-(resp. semi) bounded perturbation is H-(resp. semi) bounded perturbation. 3.

If V is a G-bounded (resp. G-semibounded) perturbation of P (resp. P*)

in S2, then GP and GOP \V are equivalent (resp. semi-equivalent) provided that I.X(

is small enough [46, 52, 53]. On the other hand, if Go and GP+v are equivalent (resp. semi-equivalent) and V has a definite sign, then V is a G-bounded (resp. G-semibounded) perturbation of P (resp. P*) in Q. In this case, by (2.5), the set E+ (resp. sE+) is an open half line which is contained in S+ [54, Corollary 3.6]. There are sign-definite G-bounded (resp. G-semibounded) perturbations such that E+ C S+ (resp. sE+ C S+) [61, Example 8.6], [47, Theorem 6.5]. Note that, if V is a G-(resp. semi-) bounded perturbation of P (resp. P*) in S2 and O E C" (1) is any function which vanishes at infinity of Sl, then clearly the function O(x)V(x) is a (resp. semi-) small perturbation of the operator P (resp. P*) in Q. 4. Suppose that GP and G°v1 are equivalent (resp. semi-equivalent). Using the resolvent equation it follows that the best equivalence (resp. semi-equivalence) constants of GP and GP±Iv,.l tend to 1 as j ---+ oo if and only if V is a (resp. semi-)

small perturbation of P (resp. P*) in Q. Therefore, zero-order perturbations of the type studied by Ancona in [7] provide us with a huge and almost optimal class of examples of small perturbations. (see also [4, 43, 46, 53, and the references therein] ).

Grigor'yan and Hansen [33] have introduced the following notions of perturbations.

DEFINITION 3.8. Let P be a subcritical operator in S2, and fix h E Cp(SZ). A nonnegative function V is called h-big on Q if any solution v of the equation (P + V)v = 0 in S1 satisfying 0 < v < h is identically zero. V is non-h-big on 1 if V is not h-big on Q. REMARK 3.9. If V is an H-integrable perturbation of P, then it is non-h-big for any h E Cp(1) (see Proposition 11.1). The following notion of perturbation does not involve Green functions.

DEFINITION 3.10. Let P be a subcritical operator in 1 C X. A function V E C" (Q) is said to be a weak perturbation of the operator P in I if the following condition holds true. (*): For every A E 1l there exists N E N such that the operator P - .Vn(x)

is subcritical in 1 for any n > N. A function V E C"(11) is said to be a weak perturbation of a critical operator P in S2 if there exists a nonzero, nonnegative function W E Co (Q) such that the function V is a weak perturbation of the subcritical operator P + W in Q. REMARKS 3.11.

1. If V is a weak perturbation of P in Q, then aS = So and

A,, (P, S2, ±V) = oc ([60], see also Theorem 7.1).

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2. If V is a semismall perturbation of P in 52, then V1 is a weak perturbation of P in 52, but G-bounded perturbations are not necessarily weak. 3. Let d > 3. By the Cwikel-Lieb-Rozenblum estimate, if V E Ld/2 (Rd), then

IVI is a weak perturbation of -A in R W. On the other hand, (1 + xj)-2 is not a weak perturbation of -A in Jd, while for any e > 0 the function (1 + IxI)-(2+6) is a small perturbation of -A in IRd, d > 3 [43, 52].

4. Indefinite Weight Consider the Schrodinger operator HA := -A - AW in IRd, where A E R is a spectral parameter and W E Co (Rd), W 0 0. Since -A is subcritical in Rd if and only if d > 3, it follows that for d > 3 the Schrodinger operator HA has no bound states provided that JAI is sufficiently small. On the other hand, for d = 1, 2, Simon proved the following sharp result. THEOREM 4.1 ([73]). Suppose that d = 1, 2, and let W E Co (IRd), W 0 0. Then HA = -A - AW has a negative eigenvalue for all negative A if and only if fad W(x)dx < 0. The following result extends Theorem 4.1 to the case of a weak perturbation of a general critical operator in Q. THEOREM 4.2 ([60]). Let P be a critical operator in 52, and W E Ca(52) a weak

perturbation of the operator P in Q. Denote by cpo (resp. cpo) the ground state of the operator P (resp. P*) in SZ such that cpo(xo) = 1 (resp. cp*(xo) = 1). Assume that Wcpocpo E LI(1).

(i) If there exists A < 0 such that P - AW(x) is subcritical in 52, then

l,.

W(x)cpo(x)cpo(x) dx > 0.

(4.1)

(ii) Assume that for some nonnegative, nonzero function V E C0(52) there exists A < 0 and a positive constant C such that GP+v-Aw(x, xo) < Ccpo(x)

and GP+v-aw(xo, x) < Ccco(x)

(4.2)

for all x E 52 \ 521 and A < A < 0. If the integral condition (4.1) holds true, then there exists A < 0 such that P - AW(x) is subcritical in 0. (iii) Suppose that W is a semismall perturbation of the operators P + V and P* + V in 52, where V > 0, V E Co (52) . Then there exists A < 0 such that P - AW(x) is subcritical in 52 if and only if (4.1) holds true.

5. Large Time Behavior of the Heat Kernel As was already mentioned in Section 2, the large time behavior of the heat kernel is closely related to criticality (see for example Lemma 2.9). In the present section, we elaborate this relation further more. Suppose that A0(P, 52,1) > 0. We consider the parabolic operator L

Lu = ut + Pu

on 52 x (0, oo).

(5.1)

We denote by l-lp(52 x (a, b)) the cone of all nonnegative solutions of the equation Lu = 0 in 52 x (a, b). Let kf (x, y, t) be the heat kernel of the parabolic operator L in Q.

If P is critical in Q, we denote by 'po the ground state of P in 52 satisfying cpo(xo) = 1. The corresponding ground state of P* is denoted by cpo.

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DEFINITION 5.1. A critical operator P is said to be positive-critical in 1 if cpoco

E LI (Sl), and null-critical in Sl if cpocpo V L' (Il)

THEOREM 5.2 ([55, 62]). Suppose that A0 > 0. Then for each x, y e S2

I lim eAOtkP(x, y, t) = t-+oo

fst oo(z) P0(z) dz

otherwise.

0

Moreover, we have the following Abelian-Tauberian type relation

lim eAOtkP(x, y, t) = lim (ao - A)GP_A(x, y).

a/a0

t_00

(5.2)

REMARK 5.3. The first part of Theorem 5.2 has been proved by Chavel and Karp [13] in the selfadjoint case. Later, Simon gave a shorter proof for the selfadjoint case using the spectral theorem and elliptic regularity [77]. We next ask how fast limt_,, eAOtkP(x, y, t) is approached. It is natural to conjecture that the limit is approached equally fast for different points x, y E Q. Note that in the context of Markov chains, such an (individual) strong ratio limit property is in general not true [14]. The following conjecture was raised by Davies [20] in the selfadjoint case. CONJECTURE 5.4. Let Lu = ut + P(x, aax)u be a parabolic operator which is defined on S2 C X. Fix a reference point xo E Q. Then lim

kP (x' y, t) = a(x, y)

t-.oo k' (xo, xo, t) exists and is positive for all x, y E Q.

(5.3)

If Conjecture 5.4 holds true, then for any fixed y E 1 the limit function a(., y) is a positive solution of the equation (P - Ao)u = 0 which is (up to a multiplicative function) a parabolic Martin function in 7-(p(I x ][8_) associated with any Martin

sequence of the form (y, tn) where to - -oo (see [20, 63, and the references therein] for further partial results).

6. Nonuniqueness of the Positive Cauchy Problem and Intrinsic Ultracontractivity In this section, we discuss the uniqueness of the Cauchy problem

fLu:=ut+Pu=0 on1 x(0,T), lu(x, 0) = uo (x)

on SZ,

in the class of nonnegative continuous solutions. So, we always assume that u° E

C(X),andu0>0. DEFINITION 6.1. A solution of the positive Cauchy problem in QT := Q x [0, T) with initial data uo is a nonnegative continuous function in QT satisfying u(x, 0) _ uo (x), and Lu = 0 in S2 x (0, T) in the classical sense. We say that the uniqueness of the positive Cauchy problem (UP) for the operator L in QT holds, when any two solutions of the positive Cauchy problem satisfying the same initial condition are identically equal in S1T.

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341

Letfk(xY,t)u(Y)dY=u(x) u c Cp(1). By the parabolic generalized maximum principle, either for some (and hence for all) x E

,

t > 0,

(6.2)

Jk(xYt)u(y)du
,

t > 0,

(6.3)

z

or

see for example [19]. Note that both sides of (6.3) are solutions of the positive Cauchy problem (6.1) with the same initial data uo = u. Therefore, in order to show that UP does not hold for the operator L in S2, it is sufficient to show that (6.3) holds true for some u E Cp(1 ). It is easy to show [19] that (6.3) holds true if and only if there exists A < 0 such that

-Af GP_\(x, y)u(y) dy < u(x)

(6.4)

for some (and hence for all) x E Q. Furthermore, it follows from [45] that (6.4) is satisfied if

J G' (x, y)u(y) dy < oo

(6.5)

for some (and hence for all) x E Q. Thus, we have: COROLLARY 6.2. If 1 is an H-integrable perturbation of a subcritical operator P in Il, then the positive Cauchy problem is not uniquely solvable. REMARKS 6.3. 1. A positive solution u c Cp (1) which satisfies (6.2) is called a positive invariant solution. If P1 = 0 and (6.2) holds for u = 1 one says that L

conserves probability in SZ (see [32]). We note that if P is critical, then the ground state cpo is a positive invariant solution. It turns out that there exists a complete Riemannian manifold X which does not admit any positive invariant harmonic function, while A0(-A, X, 1) = 0 [57]. 2. For necessary and sufficient conditions for UP, see [36, 48] and the references therein.

The following important notion was introduced by Davies and Simon for Schrodinger operators [21, 22, 23].

DEFINITION 6.4. Suppose that P is symmetric. The Schrodinger semigroup et' associated with the heat kernel ko (x, y, t) is called intrinsic ultracontractive (IU) if P - A0 is positive-critical in f2 with a ground state cpo, and for each t > 0 there exists a positive constant Ct such that Ctcoo(x)coo(y) Ct 1Wo(x)(Po(y) k' (x, y, t) REMARKS 6.5. 1. If e-tW is IU, then

lim eAOtko(x, y7 t) _ Wo(x)wo(y)

t°°

Vx, y E f2.

(6.6)

f0 [Wo(z)]2 dz

uniformly in 1 x SZ (see for example [8], cf. Theorem 5.2). 2. If SZ is a bounded uniformly Holder domain of order 0 < ce < 2, then e-t(_ is IU on Q [8]. 3. Let a > 0. Then e-t(-°+Ixl') is IU on Rd if and only if a > 2.

)

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Intrinsic ultracontractivity is closely related to perturbation theory of positive solutions and hence to UP, as the following recent result of Murata and Tomisaki demonstrates. THEOREM 6.6 ([46, 49]). Suppose that P is a subcritical symmetric operator, and that the Schrodinger semigroup e-'p is IU on Q. Then 1 is a small perturbation of P on Q. In particular, UP does not hold in Q. On the other hand, there are planar domains such that 1 is a small perturbation of the Laplacian, but the semigroup e-t(-A> is not IU (see [9] and [61]).

7. Asymptotic Behavior of Eigenfunctions In this section, we assume that P is symmetric and we discuss relationships between perturbation theory, Martin boundary, and the asymptotic behavior of weighted eigenfunctions in some general cases (for other relationships between positivity and decay of Schrodinger eigenfunctions, see [2, 76, 78]). THEOREM 7.1 ([61]). (i) Let V E C"(S2) be a positive function. Suppose that P is a symmetric, nonnegative operator on L2(SZ, V(x)dx) with a domain Co (i).

Assume that V is a weak perturbation of the operator P in Q. Suppose that P admits a (Dirichlet) selfadjoint realization P on L2(c,V(x)dx). Then P has a purely discrete nonnegative spectrum (that is, Qess(P) = 0). Moreover,

c(P) = Qdiscrete(P) = Qpoint(P) = {An}n 0) where limn--goo An = 00. In particular, if NO :_ N0(P, S2, V) > 0, then the natural embedding E : R ---> L2 (SZ, V(x)dx) is compact, where 7-l is the completion of Co (SZ) with respect to the inner product induced by the corresponding quadratic form. (ii) Assume further that P is subcritical and V is a semismall perturbation of the operator P in Q. Let {con},° o be the set of the corresponding eigenfunctions (Pcon = AnVcon). Then for every n > 1 there exists a positive constant Cn such that (7.1)

1cPn(x)I < CncPo(x).

(iii) For every n > 1, the function con/coo has a continuous extension 'On up to the Martin boundary aP i, and 'On satisfies 4n(S)=(Wo(S))

IN

(z,

sz

)V(z)cPn(z)dz

Ao 0KP(z,V(z)coo(z)dz

for every l E aP S2, where 00 is the continuous extension of

xo) to the

Martin boundary aP S2.

REMARKS 7.2. 1. By [21], the semigroup a-'p is IU if and only if the pointwise eigenfunction estimate (7.1) holds true with Cn = ct exp(tAn) Ikon 112, for every t > 0

and n > 1. Here ct is a positive function of t which may be taken as the function such that ko(x, y,t) < ctcoo(x)coo(y), where kP is the corresponding heat kernel. It follows that if e-'p is IU, then the pointwise eigenfunction estimate (7.1) holds true with Cn = inft>o{ctexp(tAn)}jjconjj2. We note that in general {Cn} is unbounded [30].

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343

Recall that if e-'' is IU, then 1 is a small perturbation of P (see Theorem 6.6). In particular, part (iii) of Theorem 7.1 implies that if e-tF' is IU, then for any n > 1, the quotient cp,i/cpo has a continuous extension zb,,, up to the Martin boundary aP S2. 2. Murata [44] proved part (ii) of Theorem 7.1 for the special case of bounded Lipschitz domains. See also [35] for related results on the asymptotic behavior of eigenfunctions of Schrodinger operators in Rd.

8. Localization of Binding Let V E Ca(lRd) and R E Rd. Throughout this section, we use the notation VR(x) := V(x - R). For j = 1, 2, let Vj be small perturbations of the Laplacian in Rd, d > 3, and assume that the operators Pi -A + Vj (x) are nonnegative on Co (I ). We consider the Schrodinger operator

PR := -A+VI(x) +VR(x)

(8.1)

defined on TRd, and its ground state energy E(R) := Ao (PR, Rd, 1). In this section

we discuss the asymptotic behavior of E(R) as JRI -> oo, a problem which was studied by Klaus and Simon in [38, 74] (see also [56, 68]). The motivation for studying the asymptotic behavior of E(R) comes from a remarkable phenomenon known as the Efimov effect for a three-body Schrodinger operator (for more details, see for example [80]). DEFINITION 8.1. Let d > 3. The space of functions

Kd := S V E C'(Rd) 1 lim sup

f

I V W1

M °°xEEd z>M Ix - zId2

dz = OJ

(8.2)

is called the Kato class at infinity. REMARK 8.2. Let d > 3. If V E Kd, then V is a small perturbation of the Laplacian in Rd. THEOREM 8.3 ([56]). Let d > 3. For j = 1, 2, let Vj (x) E Kd be two functions such that the operators P. = -A + Vj (x) are subcritical in lRd. Then there exists ro > 0 such that the operator PR is subcritical for any R E Rd \ B(0, ro). In particular, E(R) = 0 for all RI > ro.

Assume now that the operators P. = -A + V. (x), j = 1, 2, are critical in Rd. It turns out that in this case, there exists ro > 0 such that E(R) < 0 for JRI > ro, but the asymptotic behavior of E(R) depends on the dimension d, as the following theorems demonstrate (cf. [38, the remarks in pp. 84 and 87]). THEOREM 8.4 ([80]). Let d = 3. Assume that the potentials Vj, j = 1, 2 satisfy IVj(x)I < C(x)-Q on R3, where (x) :_ (1 + Jx12)I/2 ,3 > 2, and C > 0. Suppose that Pj = -A + Vj(x) is critical in R3 for j = 1, 2.

Then there exists r0 > 0 such that the operator PR is supercritical for any R E R3 \ B(0, ro). Moreover, E(R) satisfies lim I R12E(R) = -/32 < -1/4,

IRI->oo

where /3 is the unique root of the equation s = e-s.

(8.3)

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THEOREM 8.5 ([58]). Let d = 4. Assume that for j = 1, 2 the operators P, = -A + V. (x) are critical in IR' , where V E Co (R4) Then there exists r0 > 0 such that the operator PR is supercritical for any R E R4 \ B(0, ro). Moreover, there exists a positive constant C such that E(R) satisfies -CIRI-2

< E(R) < -C IIRI-2(log Rj)-1 for all

J R1 > ro.

(8.4)

THEOREM 8.6 ([56]). Let d > 5. Suppose that V, j = 1, 2 satisfy IV (x) I < in Rd, where 0 > d - 2, and C > 0. Assume that the operators Pj _ -A + V (x), j = 1, 2, are critical in Rd. Then there exists r0 > 0 such that the operator PR is supercritical for any R E Rd \ B(0, ro). Moreover, there exists a positive constant C such that E(R) C(x)-p

satisfies

-CIRI2-d < E(R) < -C-IIRI2-d for all R1 > r0.

(8.5)

What distinguishes d > 5 from d = 3, 4, is that for a short-range potential V, the ground state of a critical operator -A + V (x) in Rd is in L2 (Rd) if and only if d > 5 (see [75] and Theorem 3.6).

9. The Shuttle Operator In this section we present an intrinsic criterion which distinguishes between subcriticality, criticality and supercriticality of the operator P in Q. This criterion depends only on the norm of a certain linear operator S, called the shuttle operator which is defined on C(DD), where D C Q. The shuttle operator was introduced for Schrodinger operators on Rd in [15, 16, 83, 84]. Using Feynman-Kac-type formulas [79], Gesztesy and Zhao [28, 84] have studied the shuttle operator for Schrodinger operators in Rd with short-range potentials (see also [27]), and its relation to the following problem posed by Simon.

PROBLEM 9.1 ([75, 76]). Let V E Li (R2). Show that if the equation (-O+ V)u=0 on R2 admits a positive L°°-solution, then -A+V is critical. Gesztesy and Zhao used the shuttle operator and proved that for short-range potentials on 1R2, the above condition is a necessary and sufficient condition for criticality (see also [42] and Theorem 3.6 for similar results, and Theorem 11.2 for the complete solution). On the other hand, Gesztesy and Zhao showed in [27, Example 4.6] that there is a critical Schrodinger operator on R with `almost' shortrange potential such that its ground state behaves logarithmically.

Let P be an elliptic operator of the form (2.1) which is defined on Q. We assume that the following assumption (A) holds: (A): There exist four smooth, relatively compact subdomains Q j, 0 < j <

3, such that ffj C 52j+,, j = 0, 1, 2, and such that CP(113) Cp(Qo)=

0 and

REMARKS 9.2. 1. If assumption (A) is not satisfied, then we shall say that the spectral radius of the shuttle operator is infinity. In this case, it is clear that P is supercritical in Q. 2. Assumption (A) does not imply that Cp(1) 0.

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345

Fix an exhaustion {52j}000 of Q, such that S2j satisfy assumption (A) for 0 < j < 3. By assumption (A) the Dirichlet problem

Pu=0 in112i U= f

ona12

(9.1)

is uniquely solved in 112 for any f c C(4912), and we denote the corresponding operator from C(a12) into C(S12) by TH2. Moreover, for every f c C(a111), one can uniquely solve the exterior Dirichlet problem in the outer domain 1i, with `zero' boundary condition at infinity of Q. So, we have an operator TQi : C(a11) -> C(Q1) defined by THl f (x) := lim u f,j (x), j-'00 where u f,j is the solution of the Dirichlet boundary value problem:

Pu = 0 in 11 n11j, u = f onOQ*, u=0 ona(1l n11j)\911. For any open set D and F C D, we denote by RD the restriction map f f IF from C(D) into C(F). The shuttle operator S : C(0111) -- C(a11) is defined as follows:

S := We denote the spectral radius of the operator S by r(S). We have

(9.2)

THEOREM 9.3 ([59]). The operator P is subcritical, critical, or supercritical in 11 according to whether r(S) < 1, r(S) = 1, or r(S) > 1. The proof of Theorem 9.3 in [59] is purely analytic and relies on the observation that (in the nontrivial case) S is a positive compact operator defined on the Banach space C(3111). Therefore, the Krein-Rutman theorem implies that there exists a

simple principal eigenvalue vo > 0, which is equal to the norm (and also to the spectral radius) of S, and that the corresponding principal eigenfunction is strictly positive. It turns out that the generalized maximum principle holds in any smooth subdomain D C= SZ if and only if vo < 1, and that vo < 1 if and only if P admits a positive minimal Green function in Q. The shuttle operator can be used to prove localization of binding for certain nonselfadjoint critical operators (see [59]).

10. Periodic Operators In this section, we restrict the form of the operator. Namely, we assume that P is defined on 1R and that the coefficients of P are Zd-periodic. For such operators, we introduce a function A that plays a crucial role in our considerations.

Its properties were studied in detail in [3, 37, 41, 50, 67]. Consider the function A : Rd -' R defined by the condition that the equation Pu = A(1;)u on Rd has a positive Bloch solution of the form

uc(x) = where

(10.1)

E Rd, and cp E is a positive 7Gd-periodic function.

I[8 d. THEOREM 10.1. 1. The value is uniquely determined for any 2. The function A is bounded from above, strictly concave, analytic, and has a nonzero gradient for any i; E TRd except at its maximum point. e-C*xPeC" on the torus Td. Then 3. For E IRd, consider the operator is the principal eigenvalue of P(.) with a positive eigenfunction cpc. Moreover, is algebraically simple.

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4. The Hessian of A(l;) is nondegenerate at all points

C Rd.

Let us denote (10.2)

A0 = maRx

It follows from [3, 41, 67] that A0 = A0, and that P - A0 is critical if and only if d = 1, 2 (see also Corollary 11.5). Thus, in the selfadjoint case, A0 coincides with the bottom of the spectrum of the operator P. Assume that A0 > 0. Then Theorem 10.1 implies that the zero level set

8= JZ E R d I

(10.3)

is either a strictly convex compact analytic surface in Rd of dimension d -1 (this is the case if and only if A0 > 0), or a singleton (this is the case if and only if Ao = 0). In a recent paper [50], Murata and Tsuchida have studied the exact asymptotic behavior at infinity of the positive minimal Green function and the Martin boundary of such periodic elliptic operators on Rd. Suppose that A0 = A(do) > 0. Then P is subcritical, and for each s in the unit sphere Sd-1 there exists a unique l;s E 7- such that CE=

For s E Sd-1 take an orthonormal basis of Rd of the form {es,i, ... , es,d-1, s}. For E Rd, let cps and co* be periodic positive solutions of the equation P(1;)u = A(1;)u and P*(e)u = A(1;)u on Td, respectively, such that lTd pE(x)W*(x) dx = 1.

Then the minimal Green function GPd of P on Rd has the following asymptotics as ix - yl -> oo: THEOREM 10.2 ([50]). 1. Suppose that A0 > 0. (d-3)/2 e-(x-y)'l;s

(x)) o*s (y) 1 VA(Ss) l ,d (x,y)(21rlx-yl)(d-1)/2[det(-es,j-HessA(Ss)es,k)]1/2 GP [1+00x-y1-)1'

where s

(x - y)/lx - yl.

2. Suppose that A0 = A(!;o) = 0 and d > 3. Then the minimal Green function GPd of P on Rd has the following asymptotics as ix - yl --> oo: II$d

Gp (x, y)= {det

)]} 1/2 1[_HessA(!;o)]

r

1/2 (x

- ) 1d

2 2L 1

+ O (lx-yl -1 )].

Combining the results in [3, 50], we have the following Martin representation theorem. THEOREM 10.3 ([3, 50]). Let ° be the set of all t; E Rd such that the equation

Pu = 0 admits a positive Bloch solution u&) = e(x) with W£ (0) = 1. Then u is a positive Bloch solution if and only if u is a minimal Martin function of the equation Pu = 0 in Rd. Moreover, all Martin functions are minimal. Furthermore, u E Cp (W') if and only if there exists a positive finite measure µ on "E such that u(x) = JE u(x)

dµ().

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347

Theorem 10.3 (except the result that all Martin functions are minimal) was extended by Lin and the author to a manifold with a group action [41]. It is assumed that X is a noncompact manifold equipped with an action of a group G such that GV = X for a compact subset V C= X, and that the operator P is a G-invariant operator on X of the form (2.1). If G is finitely generated, then the set of all normalized positive solutions of the equation Pu = 0 in X which are also eigenfunctions of the G-action is a real analytic submanifold E in an appropriate finite-dimensional vector space R. Moreover, if ° is not a singleton, then it is the

boundary of a strictly convex body in R. If the group G is nilpotent, then any positive solution in Cp(X) can be uniquely represented as an integral of solutions In particular, u c Cp (X) is a positive minimal solution if and only if it is over a positive solution which is also an eigenfunction of the G-action.

11. Liouville Theorems for Schrodinger Operators and Criticality The existence and nonexistence of nontrivial bounded solutions of the equation Pu = 0 are closely related to criticality theory as the following results demonstrate (see also Section 12). PROPOSITION 11.1 ([31], [61, Lemma 3.4]). Suppose that V is a nonzero, nonnegative function such that V is an H-integrable perturbation of a subcritical operator P in SZ and let u E Cp(1). Then for any e > 0 there exists uE E Cp+Ev(f ) which satisfies 0 < uE < u and the resolvent equation

uE(x) = u(x) - e J GP+Ev(x, z)V(z)u(z) dz.

(11.1)

In particular, if P1 = 0, then for any e > 0 the operator P + eV admits a nonzero bounded solution.

In [18, Theorem 5], Damanik, Killip, and Simon proved a result which, formulated in the following new way, reveals a complete answer to Problem 9.1 posed by Simon in [75, 76] (see also [28, 42] and Theorem 3.6). An alternative proof based on criticality theory is presented below. THEOREM 11.2 ([18]). Let d = 1 or 2, and q E Li ,(I[8d). Suppose that Hq -A + q has a bounded positive solution in CH, (]Rd). If V E Li ,(I[8d) and both Hq±v > 0, then V = 0. In other words, Hq is critical. PROOF. Theorem 2.10 implies that we should indeed show that Hq is critical.

Assume that Hq is subcritical. Take a nonzero nonnegative W with a compact support. Then by Theorem 3.6, there exists e > 0 such that HQ-,w > 0. Let M < N. For d = 1 take the cutoff function aM,N(x) :=

10

Ixj>N,

1

1xI < M,

1 - Nx1-M M < xj < N, and for d = 2 aM,N(x) :=

0

xI > N,

1

1xI < M,

logN-logjxI M < x < N. log N-log M

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Let V) be a positive bounded solution of the equation Hqu = 0 in W1. Then for appropriate N, M with M, N -/-1-> oc (see [[18]), we have 0
W(aM,N b)2 dx < J d [IV(aM,NY')I2 + q(aM,NY')2] dx

= f IVaM,NI2V)2 dx

0,

Rd

and this is a contradiction. REMARKS 11.3. 1. Theorem 11.2 is related to Theorem 1.7 in [11] which claims

that for d = 1, 2, if Hq admits a bounded solution that changes its sign, then .o <0. This claim and Theorem 11.2 do not hold for d > 3 [10]. 2. For other relationships between perturbation theory of positive solutions and Liouville theorem, see [32, 33]. After submitting the first version of the present article to the editors, we proved the following result which generalized Theorem 11.2 and the Liouville-type theorems in [11].

THEOREM 11.4 ([64]). Let S2 C X be a domain. Consider two Schrodinger operators defined on S2 of the form

Pi := -V (ASV) + V

= 0, 1,

(11.2)

such that Vj E L o_ (Q- R) for some p > d/2, and Aj : Q - I[8 d2 are measurable matrix valued functions such that for any K C c there exists µK > 1 such that PK1Id < A, (x) < lKId bx E K, (11.3) where Id is the d-dimensional identity matrix. Assume that the following assumptions hold true. (i) The operator Pl is critical in Q. Denote by cp E Cp1(S2) its ground state. (ii) .o(Po, Q, 1) > 0, and there exists a real function 0 E Ho,(c) such that 0+ 0, and Po'b < 0 in 9, where u+(x) := max{0, u(x)}. (iii) The following matrix inequality holds

02(x)Ao(x) < Cc02(x)A,(x)

a.e. in f2,

(11.4)

where C > 0 is a positive constant. Then the operator Po is critical in SZ, and 0 is its ground state. In particular, dimCp0(c) = 1 and.o(Po, SZ, 1) = 0. The proof of Theorem 11.4 relies on Theorem 13.6.

COROLLARY 11.5 ([67]). Assume that the coefficients of the elliptic operator

P := -V. (AV) + V are Zd-periodic on R . Then the operator P - .o is critical in Rd if and only if d < 2. REMARK 11.6. One can use [41] to extend Corollary 11.5 to the case of equivariant Schrodinger operators on cocompact coverings. Let X be a noncompact nilpotent covering of a compact Riemannian manifold. Suppose that P := -A+V is an equivariant operator on X with respect to its nilpotent deck group G. Then P - A0 is critical in X if and only if G has a normal subgroup of finite index isomorphic to Zd for d < 2.

POSITIVE SOLUTIONS OF ELLIPTIC AND PARABOLIC PDE

349

12. Polynomially Growing Solutions and Liouville Theorems Let H = -A+ V be a Schrodinger operator on Rd. Then Snol's theorem asserts

that, under some assumptions on the potential V, if H admits a polynomially growing solution of the equation Hu = 0 in Rd, then 0 E v(H). Snol's theorem was generalized by many authors including Simon; see for example [17, 76] and [72]. In [39, 40] the structure of the space of all polynomially growing solutions of a periodic elliptic operator (or a system) of order m on an abelian cover of a compact Riemannian manifold was studied. An important particular case of the general results in [39, 40] is a real, second-order Z'-periodic elliptic operator P of the form (2.1) which is defined on Rd. In this case, we can use the information about positive solutions of such equations described in Section 10 and the results of [39] to obtain the precise structure and dimension of the space of polynomially growing solutions. DEFINITION 12.1. 1. Let N > 0. We say that the Liouville theorem of order N for the equation Pu = 0 holds true in Rd, if the space VN(P) of solutions of the equation Pu = 0 in Rd that satisfy lu(x) I < C(1 + xj)I\' for all x c Rd is of finite

dimension.

2. The Fermi surface Fp of the operator P consists of all vectors ( E Cd such that the equation Pu = 0 has a nonzero Bloch solution of the form u(x) = e'S'xp(x), where p is a Z'-periodic function. For a general 7Gd-periodic elliptic operator P of any order, we have: THEOREM 12.2 ([39]). 1. If the Liouville theorem of an order N > 0 for the equation Pu = 0 holds true, then it holds for any order. 2. The Liouville theorem holds true if and only if the number of points in the real Fermi surface Fp n R is finite.

For second-order operators with real coefficients, we have: THEOREM 12.3 ([39]). Let P be a Zd-periodic operator on W' of the form (2.1)

such that A0 > 0. Then 1. The Liouville theorem holds vacuously if A(0) > 0, i.e., the equation Lu = 0 does not admit any nontrivial polynomially growing solution. 2. If A(0) = 0 and Ao > 0, then the Liouville theorem holds for P, and dim VN (P)

-

d+NN

1

1

3. If A(0) = 0 and A0 = 0, then the Liouville theorem holds for P, and

dimVN(P)=I dNN I-I dNN22 which is the dimension of the space of all harmonic polynomials of degree at most N in d variables. 4. Any solution u E VN(P) of the equation Pu = 0 can be represented as

x'pj(x)

u(x) = i N with 7Zd-periodic functions pj.

Y. PINCHOVER

350

13. Criticality Theory for the p-Laplacian with Potential Term Positivity properties of quasilinear elliptic equations defined on a domain S2 C Rd, and in particular, those with the p-Laplacian term in the principal part, have been extensively studied over the recent decades (see for example [5, 6, 25, 26, 34, 82] and the references therein). Let p E (1, co), and let S2 be a general domain in TRd. Denote by Ap(u) V (IVulp-2Vu) the p-Laplacian operator, and let V E L0 (S2) be a given (real) potential. Throughout this section we always assume that

Q(u) := j (jVujp +Vjujp) dx > 0

Vu E Co (SZ),

(13.1)

z

that is, the functional Q is nonnegative on C0 (S2). In [66], Tintarev and the author studied (sub)criticality properties for positive weak solutions of the corresponding Euler-Lagrange equation 1

1Q'(v) :_ -Ap(v) +VIvIp-2v = 0

in S2,

(13.2)

along the lines of criticality theory for second-order linear elliptic operators that was discussed in Sections 2-4. DEFINITION 13.1. We say that the functional Q is subcritical in S2 (or Q is strictly positive in 52) if there is a strictly positive continuous function W in S2 such

that

Q(u) > fWu'dx

du E Co (S2).

(13.3)

z

DEFINITION 13.2. We say that a sequence {un} C Co (S2) is a null sequence, if un > 0 for all n E N, and there exists an open set B C= S2 such that fB l un l p dx = 1, and

lim Q(un) = lira f (IVunlp + VlunI') dx = 0.

(13.4)

n-oo n-+oo u We say that a positive function cp E Cl , (S2) is a ground state of the functional Q in S2 if cp is an L ,,P,_(Q) limit of a null sequence. If Q > 0, and Q admits a ground state in S2, we say that the functional Q is critical in Q. The functional Q is supercritical in S2if C0 (S2). The following is a generalization of the Allegretto-Piepenbrink theorem. THEOREM 13.3 (see [66]). Let Q be a functional of the form (13.1). Then the following assertions are equivalent (i) The functional Q is nonnegative on Co (S2). (ii) Equation (13.2) admits a global positive solution. (iii) Equation (13.2) admits a global positive supersolution.

The definition of positive solutions of minimal growth in a neighborhood of infinity in S2 in the linear case (Definition 2.4) is naturally extended to solutions of

the equation Q'(u) = 0. DEFINITION 13.4. A positive solution u of the equation Q'(u) = 0 in ci is said to be a positive solution of the equation Q'(u) = 0 of minimal growth in a neighborhood of infinity in S2 if for any v E C(it U 33521) with l > j, which is a positive solution of the equation Q'(u) = 0 in Q1' the inequality u < v on 8S21i implies that u < v on f2i .

POSITIVE SOLUTIONS OF ELLIPTIC AND PARABOLIC PDE

351

If 1 < p < d, then for each x0 E Il, any positive solution v of the equation Q'(u) = 0 in a punctured neighborhood of x0 has either a removable singularity at x0i or Ix x01-(d,p) p < d, v(x)

-

as x - xo,

-logjx-x0I p=d,

(13.5)

where cti(d, p) := (p - d)/(p - 1), and f - g means that lim,-xo [f (x)/g(x)] = C for some C > 0 (see [29] for p = 2, and [70, 71, 82, 66] for 1 < p < d). The following result is an extension to the p-Laplacian of Theorem 2.5. THEOREM 13.5 ([66]). Suppose that 1 < p < d, and Q is nonnegative on Co (S2). Then for any xo E SZ the equation Q'(u) = 0 has (up to a multiple constant) a unique positive solution v in SZ\{xo} of minimal growth in a neighborhood of infinity in Q. Moreover, v is either a global minimal solution of the equation Q'(u) = 0 in S2, or v has a nonremovable singularity at xo.

The main result of this section is as follows. THEOREM 13.6 ([66]). Let S2 C_ Rd be a domain, V E L o (Q), and p E (1, oo). Suppose that the functional Q is nonnegative on Co (SZ). Then (a) The functional Q is either subcritical or critical in Q. (b) If the functional Q admits a ground state v, then v satisfies (13.2).

(c) The functional Q is critical in SZ if and only if (13.2) admits a unique positive supersolution. (d) Suppose that 1 < p < d. Then the functional Q is critical (resp. subcritical) in 52 if and only if there is a unique (up to a multiplicative constant) positive solution

of the equation Q'(u) = 0 in S2 \ {xo} which has minimal

cpo (resp.

growth in a neighborhood of infinity in SZ and has a removable (resp. nonremovable) singularity at xo. (e) Suppose that Q has a ground state cpo. Then there exists a positive continuous function W in S2, such that for every 0 E Co (SZ) satisfying f0 zbcpo dx # 0 there exists a constant C > 0 such that the following Poincare-type inequality holds: p

Q(u) + C

fn

Ou dx

> C-1

W Iujp dx

J

du E Co (SZ).

(13.6)

REMARKS 13.7. 1. Theorem 13.6 extends [65, Theorem 1.5] that deals with the linear case p = 2. The proof of Theorem 13.6 relies on the (generalized) Picone identity [5, 6]. 2. We call GQ(.,x0) (after an appropriate normalization) the positive minimal p-Green function of the functional Q in S2 with a pole at x0.

3. Suppose that p = 2, and that there exists a function such that

E L2 (S2) and C E R

2

Q(u) + C

J

z/'u dx

>0

Vu E Co (SZ),

(13.7)

sz

then the negative L2-spectrum of Q' is either empty or consists of a single simple eigenvalue.

We state now several positivity properties of the functional Q in parallel to the criticality theory presented in Sections 2-4. For V E L0 (11), we use the notation

Qv(u) := j(lVujp + V jujp) dx

(13.8)

Y. PINCHOVER

352

to emphasize the dependence of Q on the potential V. PROPOSITION 13.8. Let V. E Lo" (Q), j = 1, 2. If V2 > V1 and Qv1 > 0 in S2, then Qv2 is subcritical in Q. PROPOSITION 13.9. Let S21 C SZ2 be domains in Rd such that S22 \ SI1

0. Let

Qv be defined on Co(SZ2).

1. If Qv > 0 on Co (12), then Qv is subcritical in Ii1. 2. If Qv is critical in Q1, then Qv is supercritical in 112. V1. For s c R we denote Q3(u) := sQvl(u) + (1 - s)Qvo(u), (13.9) and suppose that Qv, > 0 on Co(1l) for j = 0, 1. Then the functional QS > 0 on Co' (I) for all s E [0, 1]. Moreover, if Vo # V1, then QS is subcritical in S2 for all s E (0, 1). PROPOSITION 13.10. Let Vo, V1 E LOO (S2), Vo

PROPOSITION 13.11. Let Qv be subcritical in Q. Consider Vo E L°°(11) such that Vo

0 and supp Vo C Q. Then there exist 0 < T+ < oo, and -oo < T_ < 0

such that Qv+svo is subcritical in SZ for s c (T_, r+), and Qv+r+vo is critical in Q. Moreover, r_ = -oo if and only if Vo < 0.

PROPOSITION 13.12. Let Qv be a critical functional in Il, and let coo be the corresponding ground state. Consider Vo E L°° (1l) such that supp Vo C Q. Then there exists 0 < r+ < oo such that Qv+svo is subcritical in SZ for s E (0, T+) if and only if

I. Vo(x)cpo(x)P dx > 0.

(13.10)

Acknowledgments. The author expresses his gratitude to F. Gesztesy and M. Murata for their valuable remarks. The author is also grateful to the anonymous referee for his careful reading and useful comments. This work was partially supported by the Israel Science Foundation founded by the Israeli Academy of Sciences and Humanities, and the Fund for the Promotion of Research at the Technion.

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DEPARTMENT OF MATHEMATICS, TECHNION-ISRAEL INSTITUTE OF TECHNOLOGY, HAIFA 32000, ISRAEL E-mail address: pincho®techunix.technion.ac.il

Proceedings of Symposia in Pure Mathematics Volume 76.1, 2007

Complex Scaling in Atomic Physics: A Staging Ground for Experimental Mathematics and for Extracting Physics from Otherwise Impossible Computations William P. Reinhardt Barry: may your analyticity continue!

ABSTRACT. In the early 1970s problems related to the spectral theory of Schrodinger operators, especially those relating to the properties of the N-body Coulomb problem, gave rise to a new class of operators where rigorous spectral analysis could be carried out. These dilatation analytic operators almost immediately suggested computational implementation of methods now known as complex scaling or coordinate rotation. At first these operator generalizations of Cauchy contour distortion suggested a new, and greatly simplified, approach to the problem of locating the energies, and widths, of N-electron atomic "resonances." It was then quickly realized that for a whole series of problems of atomic physics in external fields that the computational simplifications re-

sulting from extensions of the theory to include non-compact perturbations would be even greater. In such cases, the imaginary parts of the resonance eigenvalues give rise to predictions of field induced ionization rates, rates of atomic ionization near a metallic surface, single and multi-photon ionization rates. These developments, which resulted from a highly unusual convolution of parallel developments in computation and rigorous mathematics, are overviewed. Fully detailed applications of complex scaling to scattering problems themselves required another two decades to mature. Here the notion of many-body exterior complex scaling has recently allowed progress on even the most challenging of all the "simple" problems in atomic collision theory: electron impact ionization of neutral atoms, with its resulting three-body charged particle continuum final state. It will not be overlooked that Barry Simon and his many collaborators and colleagues were intertwined in all of these developments: they initiated the theory, stimulated and suggested computations in many areas, and quickly caught up with the sometimes surprising results of "experimental" computations in others.

2000 Mathematics Subject Classification. 35P05,25; 47A10,12,20; 47B15; 47F05. Key words and phrases. Schrodinger operators, dilatation analyticity, resonances, Stark effect, Zeeman effect, operator analyticity, complex eigenvalues, scattering theory.

This work is supported, in part, by the US National Science Foundation through Grant PHY01-40091. ©2007 American Mathematical Society 357

W. P. REINHARDT

358

CONTENTS 1. 2. 3. 4. 5. 6.

Introduction Dilatation Analyticity for the N-Body Atomic Coulomb Problem Atomic Structure in Time-Independent External Fields The AC Stark Problem Scattering The Literature and an Overview of Recent Developments

References

1. Introduction "Physics" is on the upper lip of a branch cut. Roger G. Newton, in Scattering Theory of Waves and Particles, 1966. Branch cuts are wherever you choose them to be. Bernard Budiansky, Applied Mathematics 201, Harvard, Fall 1964.

Contour distortions are everywhere, being Cauchy. Wick [92] rotated time, t, down the axis, t --i -it; the time-dependent Schrodinger equation 0,0(y, t) i

= HV) (T, t),

(1)

with non-relativistic Hamiltonian operator H = -a V2 + V (.t), where we take h = e = m = 1, these being Planck's constant, the magnitude of the charge on an electron or proton, m the mass of an electron, as we concentrate on atomic and electron physics, here and in the hereafter, becomes a real time diffusion equation: a7p (x, t)

at

allowing "diffusion quantum Monte Carlo" to have become a thriving computational cottage industry. Boltzmann's canonical e_ H, 3 = 1/kT, morphs into Schrodinger's propagator e-iHt as temperature 3 is replaced by it, nominally for any Hamiltonian, H, as Wiener integrals become path integrals [73, Chapter 9].

In the early 1970s it was realized that [1, 5, 81, 76, 89] rather than making contour distortions in specific integral (spectral) representations of resolutions of the identity or propagator, for very special classes of many-body Schrodinger operators-those possessing dilatation analyticity-messy details could be avoided by analytic continuation of the scalar inter-particle distances in the operators themselves. Thus, for example, the radial Coulomb Schrodinger Hamiltonian for the Hydrogen atom _ 1 d2 l(l + 1) Z

H(r)

2 dr2 +

r

2r2

(3)

whose discrete, bound-state, eigenvalues are the celebrated Bohr energy levels for Hydrogen-like (a fixed nucleus of charge +Z, plus one electron of charge -1) atoms or ions , z2 Z2me4 (4) n = 1 2 , 3 , ... ), 2n2

(

- 2n2h2

,

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359

DILATION TRANSFORMATION

H-310- H (6)

a(H)

a (H(6))

A

B

FIGURE 1. Effect of a dilatation transformation on the spectrum of the Hamiltonian of a one-body dilatation analytic potential: the bound states are invariant, and the continuum rotates into the lower half-plane. becomes

H (rexe ) _ _

ie

e 2

+e

dr2

22e

under the complex rotation r --+ rexe.

l ( 1) _

e

r

Z

( 5)

(6)

Why is this of interest? The complex scale transformation r -- reie leaves the discrete eigenvalues of (3) invariant, while "rotating" the positive energy continuous

spectrum, which normally runs from [0, oo), into the lower half complex energy plane by the angle 20, i.e., [0, ooe-2x°) as shown in Figure 1.

The discrete spectrum corresponds to the various bound quantum levels of the electron moving in the potential field of the nucleus, and these eigenvalues actually accumulate at E = 0 in the case of the long-range -Z/r Coulomb potential, although this is not shown in the figure. The discrete spectrum is observed in many

ways in the laboratory, one being absorption of photons, leading to transitions between the various energy levels. This is the spectrum of the physicist, rather than the mathematician, and such spectra will actually appear in Section 4. The continuous positive energy spectrum corresponds to the quantum motion of an electron free to escape to "infinity" in coordinate space; such states describe various scattering processes, including those which begin and end with separated particles, and those during which a bound charged particle become free and vice-versa. The value of this transformation becomes evident if we consider L2 matrix elements of the resolvent (7) ((z - H)-I) as a function of the single complex variable z. Figure 1 now admits a second

interpretation: the discrete spectrum of H now appears in the form of poles of the resolvent, the onset of the continuous spectrum at E = 0 is now a branch point (it is actually also an essential singularity in the Coulomb case), and the continuous spectrum is now a branch cut characteristic of a particular representation of ((z -

H)-') in the cut Argand plane.

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W. P. REINHARDT

The difference between 1A and 113 is then of some consequence: Following Newton's motto [53], above, physics often appears as the z --> E + ie limit of a matrix element like that of (7), a good example being the observation that in the E + ie limit of an appropriate matrix element Im[((z - H)-')] is proportional to the total photo-absorption cross-section of an atomic or molecular system. If the particular representation of said matrix element has a cut along the positive real E axis this complicates computation, as "physics is on the upper lip of this cut" [53]. Things are even worse if (z - H)-1 is the kernel of an integral equation: this is precisely the case for the famous Lippmann-Schwinger equation [53, 86] of nonrelativistic scattering theory, where the limit E+ir corresponds to "outgoing wave" boundary conditions. The matrix element, and integral kernel corresponding to the spectrum of Figure 1A is then "singular" right where we wish to find the physics.

Distortion of the continuous spectrum, as in Figure 1B, clears the way for nonsingular computations, allowing us to take Budiansky's motto to heart: put the branch cuts where you want them, not where you find them! The cuts are usually "found" on the real axis simply because the operators are Hermitian. As elementary

quantum theory assumes Hermitian operators from the start, one might naively think one has no choice in this matter; but the dilatation transform shows that this is not the case. Even further, the sector which is exposed when comparing Figures 1A,B is actually the analytic continuation of ((z - H)-1]) onto the higher Riemann sheet obtained by continuation from the upper half-plane, and should there be non-analyticities of interest there (i.e., resonance poles of ((z - H(6))-1), H(O) being the now complex and non-Hermitian Schrodinger operator), they might be directly estimated via direct computation of a discrete (complex) eigenvalue. The above remarks belong to the theory of potential scattering from a spherically symmetric potential and the theory of the spectrum of the Hydrogen atom. While the atomic physics of the Hydrogen atom is still of interest in precision measurements of physical constants, and to the extension to the full QED and QCD treatment of this simple system, theoretical and computational atomic (and molecular) theorists, and even chemists, want to understand and compute energy levels, lifetimes of resonances, and scattering cross-sections for many-electron atoms and molecules or single and many electron systems in external fields. The world of such people, mostly unused to dabbling in pure mathematics, underwent a first order phase transition with the publication of the papers [1, 5, 81, 76, 89] which extended the concept of Cauchy analyticity and contour distortion to the far more powerful theory of dilatation analytic operators. We still have not fully recovered, and are still finding new ways to make use of the theory of dilatation analytic Schrodinger operators. Rather than choose from discussions of the use of the theory for complicated examples, where the methods of dilatation analyticity are now part of the standard toolbox of computational theorists, often appearing under the nicknames complex scaling, or complex coordinates, we focus in the present overview on applications to some simple one- and two-body problems, which turn out to be complicated enough, are by no means yet exhausted, and where one might actually hope that rigorous mathematical progress might be made. Perhaps this latter is too overly optimistic, as all of the systems to be described, with the exception of the DC Stark effect in atomic hydrogen which is separable, have Hamiltonian classical dynamics which is chaotic in some parts of the appropriate phase space, but not in others. That is, they are generic non-linear

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361

classical Hamiltonian systems, not possessing a full set of integrals of the motion [6]. One, then, certainly does not expect any of them to have tractable closed form solutions in quantum theory, either. Nonetheless, the general nature of the theory of dilatation analyticity of Coulomb Schrodinger operators has allowed considerable theoretical progress to be made. The organization of the paper is thus: Section 2 states, simply and only semirigorously, the extension of the analytic continuation connecting Figures 1A and 1B above to the general N-electron atomic system, and includes some examples of "rigorous" statements concerning the spectra of atoms which basically fell, mannalike, directly from the new mathematical developments. Section 2 also overviews some simple computational applications of the power of analytic continuation by indicating how "resonances" may be directly calculated. Namely, the energies and lifetimes of unstable atomic states determined directly from finite dimensional matrix representations of dilatation-analytic Schrodinger operators without explicit enforcement of any asymptotic boundary conditions whatsoever, other than the simple requirement that the coordinate space wave-function vanish at large particle separations. The original statements of the dilatation analyticity of many-body Coulomb Hamiltonians excluded the presence of static electric and magnetic fields, namely, the Stark and Zeeman effects, these being non-compact with respect to the Coulomb Hamiltonian. The presence of such fields are typical textbook examples of the use of perturbation theory as they affect atomic energy levels in elementary quantum mechanics. Paradoxically, naive application of perturbation theory gives rise to strongly divergent expansions in both cases; see [79] for an overview, a topic mainly left for discussion elsewhere, and [45, 30, 65, 23] for the Stark case, and [3, 79] for the Zeeman problem. Here application of the theory of dilatation analyticity was seen, empirically [63] to provide a reliable basis for computational exploration of atomic structure in time-independent external fields, as overviewed in Section 3, where in the Stark case successful computations preceded the appropriate theoretical developments, which are now in place [29]. In the case of applications to atomic structure in external fields the ability to compute without specific enforcement of boundary conditions enabled computations which would still be difficult, or might not be computationally feasible at all, if normal, traditional, asymptotic boundary conditions needed to be imposed. This is especially the case in time-dependent external fields as discussed in Section 4. In Section 5 we give a single example of the use of a generalization of dilatation analyticity (exterior complex scaling [78]) to a famously difficult problem in computational scattering theory: that of electron impact ionization of a neutral atom, which results in a final state with three asymptotically free but strongly interacting charged particles, with each pair having an interaction energy falling off as 1/rij, rig being the pair separation for the (i, j) pair, a hideously slow decay, implying very long-ranged correlations among the three charged particles. Finally, in Section 6, a brief overview of the now quite large literature of applications is given, with emphasis on reviews; and, with mention of some of the newest work, so that mathematicians and mathematical physicists interested in "what's out there," much of which undoubtedly needs formal shoring up, will have a lot to chew on.

W. P. REINHARDT

362

2. Dilatation Analyticity for the N-Body Atomic Coulomb Problem Elucidation of the properties of atomic bound states and resonance is greatly enhanced by use of dilatation analyticity. A simple dilation transformation (Im(0) = 0, is always assumed here for simplicity of presentation, although it need not be [50]) scales all inter-particle distances, ri3 = jri - r"jj, by a single complex "rotation" ri3 -p ezerij. Assuming an infinite mass, and thus fixed, nucleus (the BornOppenheimer approximation) of charge +Z, and a distance Ri from the ith electron, the non-relativistic Schrodinger Hamiltonian for an N-electron atom becomes N

HAtomic

(0) i=1

(e--2iB 02 + Ze -iB) + Ri J 2

N

e-iB

rid

(8)

i<j=l

where the first sum includes the kinetic energy of each electron and its attraction to the stationary nucleus, and the second sum consists of the repulsive electronelectron Coulombic pair interactions, which make the problem non-separable. Even for N = 2 with a fixed nucleus, and so already a restricted three-body problem, analytic solutions of the Schrodinger equation are not available, and the corresponding classical dynamics is characterized as "mixed" in phase space, as regions of both regular and chaotic motion co-exist. Thus, there are neither "good" classical actions, nor "good" quantum numbers, except those relating to global energy conservation and rotational and particle interchange symmetries. Much of the physics of two-electron atoms is simpler in quantum theory, than classically, as low lying electronic states are simpler and more stable than might be expected. This is largely due to the predominance of the nuclear attractions, and also due to the quantum spectrum of each electron being bounded from below. This simplicity does not carry over for the higher lying states which have all the characteristics of statistical theory of nuclear structure, and show overlapping resonances with the corresponding Ericson fluctuations, a.k.a. quantum chaos [6]. A recent review of "two-electron atoms and ions" is [84]. We will not, except in passing, make mention of molecules. The spectral properties of HAtomic (0) follow from the theory of dilatation analyticity, and the numerical properties, useful for applications, follow from taking finite matrix representations of HAtomic (0) in appropriate L2 basis sets, giving the HAtomic (0). corresponding matrix representation As is often the case, if the basis set consists of real functions of the distance coordinates the resulting matrix representation will be complex-symmetric. It is sometimes useful to use intrinsically complex basis functions, and an alternate version of the theory may be developed using appropriately complex scaled basis functions, with the 0 = 0 version of the Hamiltonian above, as in [41], an option not further discussed here.

2.1. Theory. Under the dilatation transformation, the spectrum of the Nelectron atom is shifted as shown in Figure 2. The physics and spectral theory are now far more complex: there are bound states of the N-electron system, these are the discrete spectrum; there are states where one, or more than one, electron escapes from the atom, forming the various parts, or segments, of the continuous spectra; there are complex eigenvalues, seen for appropriate ranges of the rotation angle, 0, wherein differing parts of different Riemann sheets of the function are exposed, these correspond to possibly long lived resonance, or quasi-bound, ((z-HAtomic)-I)

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363

DILATION TRANSFORMATION

EFFECT ON THE SPECTRUM

H --H (0) a(H(0))

a(H)

Bound state in continuum Thresholds

Y

X X X &I v°XH 11

Bound

Complex threshold E = 0 (final threshold)

-30

x. 1

states

Resonance -'

/

`- Resonance

(Hidden)

(Exposed)

Complex threshold (Hidden)

FIGURE 2. Effect of a dilatation transformation on the spectrum of the Hamiltonian of an N-body system with pairwise dilatation analytic potentials: the bound states are invariant, and the various continua rotate into the lower half-plane. Parts of the spectrum previously hidden on higher Riemann sheets are now exposed. These include complex resonance eigenvalues and complex thresholds, with their own rotated continua.

states of the system. These complex discrete eigenvalues,

and -y are both real, are not eigenvalues of the original self-adjoint operator, Such complex eigenvalues describe quantum states of the system which have lifetime exp(-yet); the imaginary part of the eigenvalue thus determines the E

HAtomic

lifetime of a decaying state. It is a key observation, and of great importance in almost all of the applications to be discussed below, Sections 3-5, that such lifetimes emerge directly as the imaginary parts of the discrete complex eigenvalues of HAt°mic(0). These are typically far more difficult to compute directly, as such direct computation requires solution of a "scattering" problem, requiring connection of physical regions of coordinate

space, large enough to contain the bound decay product, to proper asymptotic states which correctly describe the physics as the particles separate to "infinity." These latter boundary conditions are often computationally highly tedious, and very expensive to enforce, especially if large numbers of channels are open: thus computationally oriented atomic theorists immediately grasped the importance of the theory of dilatation analyticity. A complicated scattering calculation is converted to a far simpler diagonalizing of a finite dimensional matrix in an L2 basis, wherein the actual nature of the continuum is simply ignored; although some thought and cleverness in choice of basis is required. This will be particularly the case when the resonance lifetimes desired are the rates of particle detachment or ionization due to static and/or oscillatory external fields (or both!), as discussed in Sections 4 and 5. In these cases even proper formulation of the boundary conditions for "free" charged particles is problematic. Continua can begin on the real axis, corresponding to an asymptotically free electron and a ground or excited but bound state of the remaining atomic ion.

364

W. P. REINHARDT

Continua may also originate at complex branch points, which correspond to the interaction of an electron with an already decaying state of the residual ion. Finally at E = 0, all particles are unbound, and the final scattering threshold is attained: the continuum beginning at E = 0 is an N-body continuum. To illustrate the power of the mathematical theory of complex scale transformations in actually answering "physically motivated" questions in terms the theory, brief reference is given to the cases of "bound states in the continuum" and the problem of bound sates and resonances "at energies above E = 0."

2.2. Hints at Two Applications of the Theory. May there be "bound states" in the continuum for a typical atomic system, that is, are there parts of the discrete spectrum of HAtomic which are degenerate with the continuous spectrum? Figure 2 illustrates such a state, which would correspond to an L2 eigenfunction of both HAtomic and HAtomic (0), for Re[O] > 0. In particular, should such an eigenvalue

exist, would it be an analytic function of parameters in the Hamiltonian? Standard Kato perturbation theory [42] tells us that isolated (i.e., non-degenerate) members of the discrete spectrum of an atomic Hamiltonian, HAtomic are, in fact, locally analytic functions of the electron-electron coupling, and the nuclear charge Z. This clearly does not allow such discussion of a discrete eigenvalue "embedded" in an overlying part of the continuous spectrum, and so not isolated. However, under a dilatation transformation the continuous spectrum is rotated into the lower halfplane, while the discrete eigenvalue remains invariant. Kato perturbation theory may thus be applied, and analyticity follows: one

outcome of this is the realization that if an embedded eigenvalue of HAtomic is real over any range of (real) parameters in HAtomic, then it remains real unless it becomes degenerate with a threshold [62]. This observation began the process of understanding the radius of convergence of the so-called 1/Z perturbation expansion for two-electron atoms: early computations suggested [83] that for small Z, where Z is now thought of as a continuous real parameter and not the physically realistic (quantized) integral value of the nuclear charge, the ground state of HAtomic moved past the first scattering threshold, and then for still smaller nuclear charge Z, (even weaker binding) became a resonance.

Reinhardt [62], using dilatation analyticity, pointed out that this would lead to a contradiction, and suggested that the radius of convergence corresponded to the degeneracy of the ground state with the first threshold, Z = Zcrit Simon et al. [34] showed that exactly one L2 discrete eigenstate may co-exist, at Z = Zcrit degenerate with the threshold energy, but not for smaller values of Z. (See also Simon [80].) Morgan et al. [4] finally succeeded in proving that the radius of convergence was exactly Zcrit a physically and mathematically satisfying result. In a related observation, Simon [77] provided a simple proof that ((z - H)-1), including its analytic continuation from the upper half-plane, is an analytic function

of z, in the restricted half-plane Re(z) > 0. Thus there are no bound states, or resonances (these being, perhaps naively, being defined as parts of the point spectrum of HAtomic(O), for Re[O] > 0) at energies above the final threshold E = 0,

as in Figure 2. Does this imply that there are no "features" in the scattering cross-section above this final threshold? There is not yet a clear answer: in an experiment involving the scattering of electron from the hydrogen negative ion at a total energy, E > 0, a (very broad) resonance-like feature was observed [61] in

APPLICATIONS OF COMPLEX SCALING IN ATOMIC PHYSICS

365

DILATION TRANSFORMATION EFFECT ON THE NUMERICAL SPECTRUM

H-00-1-4 (8)

a (H(8))

a(H)

Bound state in continuum

Thresholds

.x x xX -'. and B states Resonance---J (Hidden)

l

0 Bound

states

X

-h \

++

.

-r

"-- Resonance (Exposed)

FIGURE 3. Effect of a dilatation transformation on the spectrum of the matrix representation of the Hamiltonian of an N-body dilatation analytic system: the bound states are invariant, and the now discretized continua rotate into the lower half-plane. Compare with the actual spectrum shown in Figure 2. If diagonalized in a basis defined by "M" basis functions, the numerical spectrum will typically have M complex eigenvalues. the 1970s. This difficult experiment has not been repeated. Theorists were puzzled [85, 18, 60], and their results contradictory.

2.3. Computational Applications to Atomic Helium, and a Comparison to a "Stabilization" Method. The eigenfunctions of HAt°mic(B), for 0 > 0, corresponding to discrete bound or resonance eigenvalues, are L2 as functions of the rotated coordinates ei0rij for values of 0 for which they appear as isolated parts of the point spectrum of HAtomic (0). The discrete eigenvalues may thus be computed via standard linear variational theory, i.e., they are approximated as eigenvalues of Atomic the matrix representation H (0) wherein H Atomic (0) is projected onto a dis-

crete finite basis of basis functions, {0i, i = 1, ... , M}. This discretization results in what we call here the numerical spectrum of discrete eigenvalues, which attempts to approximate the actual spectrum of HAtomic (0). A typical numerical spectrum corresponding to the normal and rotated spectra of Figure 2, might look something like that of Figure 3. Evidently, the N approximate complex eigenvalues fall into several classes: some seem to approximate the discrete spectrum, below the first threshold; some approximate branch cuts as "rows of poles" of ((z-H(0))-'); and some appear to lie between the rotated continua (branch cuts of ((z-H(0))-')), and these last are the approximations to the desired resonances. A bound state in the continuum is now simply an isolated real eigenvalue. In short, there is a perhaps surprising ability of the discrete approximation ((z -K(0))-1) to actually approximate ((z - H(0))-1).

In fact, in the upper half-plane, ((z - )7(0))-1) often gives rapidly convergent results, away from the real axis, and such data may be useful in constructing actual "real energy' scattering cross-sections (see Section 5.1), as well as providing the

W. P. REINHARDT

366

-0.001

t

im (z)

-0.002

-0.003

Re (z) -PFIGURE 4. Convergence of the numerical approximation to a single "resonance" eigenvalue of a metastable state of Helium. Shown for three different basis sets is the value of the resonance eigenvalue as a function of the rotation angle 0. There is evidently an approximate stationarity of the numerical approximation to this complex eigenvalue. (Figure redrawn from [19] with permission.) Atomic

discrete approximations to resonance eigenvalues of H (0). Figure 4 shows the properties of a "resonance" eigenvalue, this being one of the first applications to determination of a resonance energy and width for a decaying (auto-ionizing) state of atomic Helium near the n = 2 excited state of the Helium positive ion, as a function of rotation angle 0.

It is easily noted [19] that this 0 - trajectory pauses near a converged value of the desired complex eigenvalue: eigenvalues corresponding to the rows of poles mimicking the rotated continuous spectrum show no such convergence properties. Complex scaling has been used to investigate swarms of such resonances in far more highly excited states of Helium, where the widths actually become larger than the mean spacing between neighboring resonance energies, a criterion for the onset of "Ericson fluctuations" which are a symptom of quantum chaos, in an attempt to answer the classical-quantum correspondence principle question: Where is the chaos in the spectrum of Helium? as discussed by Blumel and Reinhardt [6].

What would the eigenvalue of Figure 4 look like at 0 = 0? It would be superimposed on the discretized numerical continuum as illustrated in Figure 3 without the coordinate rotation. Figure 5 shows the discrete eigenvalues of the finite matrix representation of a scattering problem with a resonance whose real part, e, is near energy "0.4", all

APPLICATIONS OF COMPLEX SCALING IN ATOMIC PHYSICS

367

1.2

1.0

.6

.4

.2

10

20

30

40

50

N

FIGURE 5. Stabilization of a real approximation to a resonance eigenvalue, from diagonalization of an N-dimensional matrix approximation to a model resonance Hamiltonian, as a function of the

scale length of the basis set (see Hazi and Taylor [28]), redrawn and used with permission. As both the resonance and discretized continuum eigenvalues are real, there are avoided crossings, but the real part of the resonance is quite stable with respect to variation of the scale length, as defined by the basis set size, N. Were a complex rotation applied, the approximate continuum would separate from the complex resonance eigenvalue, for appropriate values of 0, and the resonance eigenvalue would stabilize in the complex plane as in Figure 4.

as a function of a parameter defining the coordinate span of the basis [28]. For no coordinate rotation, all eigenvalues of the self-adjoint Schrodinger operator are real, and superimposed, so the approximate continuum overlies an approximate resonance eigenvalue. That is "one" of the discrete eigenvalues is attempting to be stationary, while also showing the expected Wigner-von Neumann avoided crossings, while the others depend strongly on the scaling of the basis-it is precisely this "stabilization" that would correspond to the convergence of the complex eigenvalue of Figure 4 were the eigenvalues separated, as they automatically are for more suitable choices of 0, where the basis set discretized continuum is then nicely out of harm's way. The stabilization methods (see [28] and references therein) were developed, and successfully applied by Taylor et al., well before the dawn of dilatation analyticity.

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W. P. REINHARDT

3. Atomic Structure in Time-Independent External Fields 3.1. The Stark Problem. Extending the atomic Coulomb Hamiltonian,

HAtomi°

to include a static external electric field, f, magnitude F, is easy: N

H Atomic,Stark

-

Z+ i=1

+ Ri

N

N

ri. 1

i<j=1

(9)

i=1

Real physicists decry this simplicity: if the Hamiltonian is known so easily, it can't be "physics," so following Dirac, it must have been reduced to "chemistry." This rather hides the complexity of this seemingly small change. Even for the oneelectron problem of the Hydrogen atom in an external electric field (the Stark effect, or the revisionist LoSurdo-Stark effect [23, 2]), where it is now assumed that the field vector is along the z-axis (F. F = Fz = Fr cos(O), where F = I Fl, and where now 0 an angle in 3D configuration space, an ambiguity we leave unresolved as it is always context-clear), H(r)Stark

1

d2

+ l(l + 1)

- Zee + Fz,

(10)

2 dr2 2r2 r rather severe, and for decades unresolved, problems arise. Laboratory, although not

astrophysical, external electric fields, with magnitude F, are typically very small on the scale of the internal electric fields which determine the structure of an atom. Nonetheless, a standard Rayleigh-Schrodinger perturbation expansion of the Bohr energy levels of (10), as a power series in F, has zero radius of convergence [79]. Titchmarsh [88], likely the first mathematician to take problems arising in atomic physics seriously, noted that the Hamiltonian HStark has no discrete spectrum: all of the bound states at F = 0 become resonances, with perhaps long (growing as Fexp(a/IFI), for small F), but finite lifetimes. The fact that the lifetime has an essential singularity in F, of course, coincides with the failure of perturbation theory to converge. Said another way, addition of the non-compact perturbation, Fz, dramatically alters the spectral properties of the atomic Schrodinger Hamiltonian operator. The continuous spectrum is now the whole real axis, (-oo, oc).

What has happened to the formerly discrete spectrum of quantum bound states? We all exist in small but real fields, and so for all practical purposes, our physicians are correct to neglect them, except when using them purposefully to take an X-ray or MRI, as few of us die, or have our lifetimes determined by the rate of our electrons leaking off to infinity, or even down the hall. Titchmarsh [88] noted that instead of being on the real axis, the poles of ((z - HStark)-1) are now to be found by suitable analytic continuation from the upper half z-plane, across the cut, which now runs from -oc to +oo along the whole real energy axis, onto a higher Riemann sheet. He had the correct answer, but in those days no one knew

what to do with that explicit and tempting statement in a manner leading to a computational algorithm.

Reinhardt, as a numerical experiment [63] (and in full knowledge that Fz spoiled traditional dilatation analyticity) posited that the matrix representation of 1 H(r, 0)Stark = e- i20 [_ 2

dd

+ l( 2r 1) ]

-

Z

e

r

+ e2BFz

(11)

APPLICATIONS OF COMPLEX SCALING IN ATOMIC PHYSICS

369

Re (E) -0.525 -0.005 f

-0.526

-0.527

-0.528

-0.529

F = 0.1 a.u.

-0.006

0 = 0.2

(3)

w_

E

\

-0.007 (8)

f

I

(9)\ -0.008

(4) \

(i0)

/ / (5)

1

I

/

(7)

FIGURE 6. Convergence of a complex Stark resonance eigenvalue for the ground state of atomic Hydrogen, as a function of basis set dimension, rather than rotation angle. Here there are eight values of the angular momentum in a partial wave expansion centered at the nucleus, so the actual dimensionality of the matrix representation of the discretized, rotated, Stark Hamiltonian is 8M where the value of M runs from 3 to 12.

might have interesting properties. It was quickly found that well converged complex Stark eigenvalues of H(r, O)Stark allowed essentially exact estimates for the Hydrogen atom Stark problem, without any enforcement of asymptotic boundary conditions, as it was simply assumed that the discrete eigenvalues of (11) would be L2. Figure 6 shows numerical convergence of a complex eigenvalue, as a function of basis set size, a useful augmentation to the variational 0 - trajectory methods of Figure 4. Worries about "where did the Stark continuum actually go" were partially allayed in collaborative work with Avron, started at the Aspen Center for Physics [10]. Reinhardt then presented a physics seminar entitled "Is the Atomic Stark Hamiltonian Dilatation Analytic?" at the University of Virginia: Ira Herbst was in the audience! The mathematical analysis then began in earnest. Herbst concluded [29] that the continuum actually vanished for any finite 0 54 0, puzzling at first to those grounded in simple physical models, but later "explained" in terms of such a model by Cerjan et al. [9]. However, numerical diagonalization of the N-dimensional complex symmetric matrices that typically arise in numerical applications of the complex scaling method, just as typically, give N complex eigenvalues.

Figure 7 shows such a set of complex eigenvalues: some of these are approximations to decaying Stark resonances, but others must somehow represent the basis set

W. P. REINHARDT

370

Im z

X

X X

X X

X

X

X

X

X

X

X

Re z

X X

FIGURE 7. Numerical spectrum of the complex rotated Stark Hamiltonian, discretized in a a finite basis. An approximation to an actual resonance eigenvalue of the Stark broadened and shifted Hydrogenic eigenvalues is near the real axis at the far left of the figure. What are we, then, to make of the "other" eigenvalues, which seem tilted at angle 0 with respect to the real axis? This is the edge of the numerical range [10] of the dilatated Stark Hamiltonian, and remnants of it seem clear. Paradoxically, the actual continuum has run off to infinity, and so should not appear anywhere in the figure. discretized continuum, which, Herbst-Simon-wise, actually isn't there! One possible interpretation of these spectral remnants would be to develop the appropriate analysis of the pseudo-spectrum, of the highly ill-conditioned matrix representations of H(r, O)stark following the work of Davies [15]. More recently, Nicolaides and Themelis have discussed the Stark problem and complex scaling, using traditional Airy function boundary conditions [54], validating earlier assumptions. Extension of the one-electron Hydrogenic Stark problem to the case of the already resonant (decaying) two-electron excited states of the Hydrogen negative ion (H-) in an additional applied external electric field were begun shortly thereafter in a complex scaling approach [91]. At the time, many channel scattering computations for a two-electron system in the presence of an external field were not imaginable. The problem of the effect of a static electric field on two resonances, near the n = 2 excited state of H itself, in H- (a Feshbach resonance just below the n = 2 first excited state of atomic H, and an unusual and broad shape resonance, lying, in energy, just above that threshold) continue to be problematic thirty years later. Ho describes complex scaling work, following [91], on the effect of electric fields on these two states, which are already decaying resonances in the absence of external fields, [32, 33], respectively. Bryant [7] overviews the status of theory and experiment, again with a long history, and a related astrophysical puzzle, namely, that the shape resonance is not observed in astrophysical spectra. One might think that this is simply because H- is a rare and unusual species. Such is not the case, the passage of visible light through the atmosphere of our own sun is greatly affected by the frequency dependent photo-absorption of H-, which dominates the

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371

outer parts of the solar atmosphere and which determines the stellar opacity in the visible part of the spectrum: were H- absent we, as a vision-driven species, would perhaps have evolved differently, or not at all.

3.2. Ionization of Atoms/Ions Near a Metallic Surface. It should be mentioned that quite similar numerical techniques, using complex scaling, have allowed surface physicists and chemists to determine the lifetime of atomic Hydrogen as it loses its electron as it approaches a metallic surface [56]. This same technique should be applicable to the surface induced ionization of any atomic or molecular species. Within the Born-Oppenheimer approximation (namely, a fixed atomic nucleus, and fixed "edge" of a metallic surface), it is still not an easy problem to write down an adequate but simple "one-body" effective potential for a single electron interacting with both the nucleus and the surface: image potentials, polarization of the surface, and a correct description of the potential as the ionizing electron actually passes through the surface and into the conductor must be included. However, once this is accomplished to a suitable degree of precision (i.e., "just good enough"), the actual computations (and theoretical issues) are essentially the same as for the problem of complex scaling for the DC Stark effect computations discussed above. Physically and chemically important results have been obtained for a series of atomic species by Nordlander and Tully [55].

3.3. The Zeeman Problem. From an experimental and theoretical point of view the problem of magnetic fields is seemingly quite different from that of the Stark problem, where all states become finite lived resonances. In an external magnetic field, atomic bound states remain bound for small fields: perturbation theory nonetheless is divergent, and Simon et al. [3, 79] reignited interest in solution of that problem. Astrophysically, the problem of atomic structure and spectra (i.e., the pattern of light absorption as a function of frequency, rather than the operator spectrum of mathematics) in intense electric and magnetic fields is of great practical importance: the understanding and modeling of such effects allow measurement of such fields in far away and unusual stellar objects. Here on earth, experiments of Garton et al. [47] revealed a very different challenge, as illustrated in Figure 8. Here the experiment shows, at its original fairly low resolution, the accumulation of the atomic eigenvalues (of (4)) near the ionization threshold (IP, for ionization potential, in the figure) via absorption of light exciting a Ba (Barium) atom to higher and higher quantum levels, and eventually ionizing the atom, forming Ba+,

all of this now in the presence of a strong external DC magnetic field. Complex "new" structures are also seen above the IP. These field induced oscillations are not themselves resonances, but are interference patterns as orbiting and ionizing electrons leave and return to the atomic center [66, 21]. However, at much higher resolution one finds that the experimental spectra, at energies below and above the ionization limit, consists only of myriads of broad and narrow field ionizing resonances. Figure 9 shows a comparison of a high resolution laboratory experiment for the photo-ionization of atomic Lithium atoms in a strong magnetic field [38] compared with a complex scaling computation [16] at the same ionization threshold, and the same magnitude of the external field, for atomic Hy-

drogen. Here, fortuitously, the Li+ core interacts weakly with the highly excited p-wave outer electron, so that this comparison makes sense. The experiment would

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FIGURE 8. Photo-absorption of gaseous atomic Barium in a strong magnetic field, near the ionization threshold, where the Bohr-like eigenvalues actually accumulate and merge with the ionization continuum [47]. Magnetic field induced oscillations (which are not resonances) are clearly seen [66, 21] in this low resolution spectrum.

Figure 9 shows a related set of spectra at far higher resolution. (Redrawn from [66] with permission.)

be far more difficult for atomic H, and a full, first principles, computation for Li correspondingly harder. So there is some luck here, but complex scaling for computational atomic physics in an external magnetic field certainly seems to get the job done.

4. The AC Stark Problem The simplest representation of the modification of an atomic (exemplified here by Hydrogen again) Hamiltonian in a classical Maxwellian oscillatory field results from ignoring the smaller magnetic interactions and positing

H(r) AC-stark =

1

d2

+

1(1 + 1) 2r2

Ze2

+ Fz cos(wt), (12) 2 dr2 r where w is now the (circular) frequency of the external electromagnetic field. Addition of this time-oscillatory interaction with the external field is referred to as the AC (as opposed to DC, as we are now in the realm of engineers!) Stark problem, and formerly bound energy levels of the atomic system both shift, and broaden due to the presence of the field. Often it is only the shifting we need to consider, as the

APPLICATIONS OF COMPLEX SCALING IN ATOMIC PHYSICS EXPERIMENTAL SPECTRUM

373

(Li)

at 6.1 T

I

2-

I -10

i

COMPUTED SPECTRUM at 6.1 T 30

10

Energy [cxn-']

FIGURE 9. Comparison of experiment [38] and complex scaling computation [16] of the spectrum of an atom in an intense (6.7T) static magnetic field, shown as approximate reflections of one another. At this high resolution the broad interference structures of Figure 8 are hard to see, but show up clearly in the powerspectrum. field induced ionization is exponentially small. Fantastical physical effects, such as that of "slow light" [25], follow from such considerations wherein several atomic levels are coupled via several different pumping frequencies, w. However, from a mathematical point of view, the Hamiltonian of (12) has no discrete spectrum, and the continuum, as in the DC Stark case, runs along the real axis from -00 to +00.

In intense laser fields it is the resulting ionization rates that are of interest. Chu and Reinhardt, and collaborators (see [12, 48, 35]), again in initial numerical experiments, took the ansatz H(r, 0)AC-stark = e-z20 [_

e

1(1

2 dr2 +

1) ]

- r

Z

+ eZBFz cos(wt),

(13)

and applied the ancient methods of Floquet theory [75] to obtain a time-independent matrix representation, suited to the presence of a single oscillatory field. The imaginary parts of the complex quasi-energies, obtained from numerical diagonalizing of this complex scaled Floquet matrix, contain non-perturbative information about single or multi-photon ionization rates. This has turned into, especially in the hands of Chu and co-workers [13], a highly flexible method which can now handle multiple frequencies in addition to the presence of static external electric and magnetic fields, or the single AC frequency of usual Floquet theory. Chu and co-workers [8] have also extended the notion of "density functionals" to appropriately reproduce the Bohr (4) accumulation of bound atomic eigenvalues

near ionization thresholds, allowing dilatation analyticity to be applied the the spectra of complex atoms in strong external fields. Maquet and Moiseyev and their respective co-workers [90, 44, 87] have used these methods in studies of atoms as non-linear devices which in strong laser fields yield long series of "harmonics" of the original laser frequencies. No doubt these extensions of the computational theory

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go well beyond the applicability of rigorous results, although Yajima [94], early on, made considerable progress for the single frequency, single electron problem.

5. Scattering 5.1. Early Efforts. A brief historical comment: It would be an error to think that the generalization that dilatation analyticity represents came de novo out of vacuum fluctuations-many atomic and mathematically oriented physicists were thinking in the complex plane. Dispersion relations were the rage of the day in particle physics [11, 52, 22]. Titchmarsh [88] talked about Stark complex eigenvalues as poles of an analytically continued resolvent; Schlessinger [72] was computing scattering amplitudes via conformal mappings from (numerical) data collected on the real axis at energies below the first scattering threshold; Oxtoby et al. [69] extended that work via the Fredholm determinant, (or Jost Function), which allowed fully off shell analytic continuations; and, especially, Nuttall and co-workers [57, 60, 49, 58, 20] were early explorers of the complex plane for computational

e- + H, and nuclear scattering amplitudes, difficult or impossible to compute by the standard methods of the time. These "numerical analytic continuation" methods achieved some measure of success: elastic scattering amplitudes, and thus, via the optical theorem, total inelasticities were computed for energies above the three-body break up threshold for electron [71] and positron [93] Hydrogen atom scattering; photo-ionization amplitudes, rather than just total cross-sections, were computed, Johnson et al. [40] using a Coulomb wave operator representation due to Chandler and Gibson [24]. However, full advantage of dilatation analyticity proved difficult to realize: the powerful theories of the analyticity of L2 matrix elements of the resolvent ((z - H(O))-') do not apply to similar matrix elements computed with respect to the usual unperturbed scattering functions e+ikT. For short-ranged (exponentially bounded) potentials U(x), the product U(r)e±i/CT can withstand some measure of continuation to complex r, but these simplicities fail for the long-ranged interactions of atomic physics. T-matrix and Lippmann-Schwinger approaches thus proved to be far more limited than the great progress made when all desired information could be obtained from the relatively simple computation of a single complex eigenvalue of a dilatated Schrodinger operator. What was needed was a combination of very large scale computation plus a return to a direct analysis of the Schrodinger wave-function, first implemented in 1999 in a direct attack on the long outstanding problem of the three-body charged particle continuum.

5.2. The Three-Body Charged Particle Continuum. The related problems of "double photo-detachment" (where a single photon knocks two electrons out of an atom or ion, leaving behind a center of charge) and "electron impact ionization" of atomic systems, result in essentially identical and almost computationally intractable final states, even for the fundamental (but still restricted) fixed nucleus two electron problems

fiiw+H-

p++e-+e-

e-+Hp++e +e

(14) .

(15)

Here H denotes a neutral Hydrogen atom (p+ + e-), H- two electrons, e-, bound to a single proton, p+, and 11w a photon of sufficient energy to eject two

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FIGURE 10. Real part of a full scattered wave radial expansion amplitude including the two-electron continuum for the impact ionization problem. In the case illustrated, the total angular momentum is L = 2, and the state is a singlet. (Redrawn from [70] with permission, and where complete details may be found.)

electrons from H-. Attempts at direct computations relating to these processes are made profoundly difficult by the three free charged particle continuum, where each pair of particles interacts via a potential which decays only as +l/rid, for all three independent (i, j) pairs. This is closely related to the gravitational three-body problem, except for some sign changes, where an also restricted three-body problem was known to Poincare, and likely guessed by Newton, to be non-integrable, that is, not reducible to quadratures. The most recent computational progress on this formidable problem has been made, using the Simon method of exterior complex scaling [78], wherein the transformation rig - eiorij is only made for values of the inter-particle distance greater than a pre-set distance r-. In potential scattering, of a single particle from a fixed potential V(r), this is an old method, e.g., Taylor [86], updated to many-body problems in the Simon paper. Rescigno et al., in producing a cover article for Science [70], and approaching the problem of low energy electron impact ionization of Hydrogen, used exterior scaling for the two radial coordinates of both outgoing electrons. That is, the distance of each electron from the nucleus, p+, was independently so scaled. This avoids, as in the many examples of computations of resonance properties described in earlier sections, the problem of imposing the correct three-body asymptotic form of the wave function when both electrons are a large distance from the nucleus. It was the avoidance of this complex, and difficult to impose boundary condition, which is actually analytically known only for distances rather larger than handled within the computations carried out, that allowed computations to be completed. Various partial cross-sections, where the electrons emerge at specific angles with respect to one another and with respect to the incoming electron beam, were then computed via direct computation of fluxes from the Schrodinger wave-function. A slice through a two electron radial wave-function is shown in Figure 10.

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FIGURE 11. Schematic illustration of an electron-atom impact ionization experiment. The initial scattering state is a stationary atom and a single incoming electron, as on the left. The final state (see (14) and (15)) has two outgoing electrons, shown here as detected in a co-planar setting of the detector, with two detection angles as indicated. (Redrawn from [70] with permission.)

FIGURE 12. A sample of data from the co-planar, equal energy outgoing electrons, triple differential cross-section, comparing theory and experiment for the electron-impact ionization of atomic H at an incident energy of 17.6eV. Further details, and a much more thorough comparison of theory and experiment appear in [70]. (Data set courtesy of C. W. McCurdy, 2005, redrawn and published with permission.)

Comparison [70] with angle resolved final state distributions for equal energy final state electrons (see Figure 11) emerging in a co-planar manner are shown as a function of their absolute and relative angles with respect to the incoming beam of electrons are shown in Figure 12. Needless to say, this a triumph of mathematical development and ultra-large scale computation.

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6. The Literature and an Overview of Recent Developments The results in Sections 3-5 all refer to the determination of locations and widths of decaying resonance states, a now well-developed field which had already reached the first conference proceeding stage in 1978 [46], with a conference summary and

critique by Simon [82]; reviews, intended for the atomic and molecular physics and theoretical chemistry communities appeared in the 1980s by Reinhardt [64, 67], Ho [31], and Junker [41] who focuses on the use of complex scaled "basis functions" rather than operators. In 1998 Moiseyev [50], in tutorial fashion, brings the literature up to date to include scattering as well as resonance applications, overviewing both analytically soluble systems and numerical applications. A more recent review, Chu and Telnov [13], overviews applications of complex scaling to atomic physics in intense single and multi-frequency laser fields, a very rapidly expanding area for both theory and experiment, appeared in 2004. Recent examples of the use of complex scaling, and generalizations, in atomic, molecular, and many-body physics include: Applications to the chemical dynamics following electron capture by a system as complex as a water molecule [26, 27]. These authors actually used the complex Kohn variational principle [95], a seemingly quite different method, but one clearly influenced by the mindset of the complex scaling community, and which neatly solved the problem of the original Kohn principle, where poles of an implicitly discretized continuum showed up on the pos-

itive real axis and made extraction of scattering information a true art from; see Schwartz [74]. Applications to electron-atom scattering, in the spirit of [70] continue [37, 36].

Properties of complex atoms in strong external fields [90, 43, 8] which include further developments in this area, some of which [44] employ a competitive technique, that of complex absorbing potentials on a finite boundary; and, finally, the tunneling of Bose-Einstein condensates through external barriers, in studies of the possibility of formation of coherent "atomic laser-like beams" of ultra-cold atoms [51].

Having mentioned Bose-Einstein condensates, it should be noted that two-body atomic collisional Feshbach resonances [14], in ultra-cold "atom-atom" scattering, have allowed magnetic tuning of the effective "pseudo-potential" pair interaction well suited to describing the near zero energy collisions of atomics in highly dilute quantum gases. This tunability combined with density control of both Bose and Fermi condensates of cold atoms have placed gas phase atomic physics in a position to explore zero temperature quantum phase transitions, and the mechanistics of super-conductivity and superfluity, central themes of traditional condensed matter physics.

N-atom resonances in these coherent many-body quantum systems also include non-linear collective excitations such as solitons [68, 17], which for appropriate transverse trapping have no known decay mechanism, although it is highly likely that they are long lived resonances, and thus have finite lifetimes due to correlations

of the quantum particles. So, we end with a challenge to the mathematics and physics community: What is the analog of dilatation analyticity which will make simple the understanding and computation of the decay of such non-linear emergent excitations?

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Acknowledgments. The author's earliest memory of being intrigued by utilization of methods depending on analytic continuation dates back to the academic year 1964-65 where, in Applied Mathematics 201, Harvard Professor Bernard Budiansky (1925-1999), in despair and frustration over the fact that, at the time of the first hour exam, no one in this graduate class of about 150 students understood how to take advantage of the ambiguity in placing branch cuts to solve a simple problem, announced: "This is not some rarefied and useless figment of the imagination; there are still whole careers to be made by those who understand the principle of analytic continuation, and its implications!" As this was also clearly the pre-image of the Barry Simon yet to be, and who at the time was but a (rather unusual!) Harvard junior, this paper is dedicated to his memory. As for Barry, himself: the years 1977-84 were quite a rush: Thank you, Barry!

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[84] G. Tannor, K. Richter, and J.-M. Rost: The theory of two-electron atoms: Between ground state and complete fragmentation, Rev. Mod. Phys. 72 (2000), 497-544. [85] H. S. Taylor and L. D. Thomas: Short lived resonance state of H-, Phys. Rev. Letts. 28 (1972), 1091-1092.

[86] J. R. Taylor: Scattering Theory: The Quantum Theory of Non-Relativistic Collisions, John Wiley, New York, 1972.

[87] D. A. Telnov and S.-I. Chu: Ab initio study of high-order harmonic generation of H2+ in intense laser fields: Time-dependent non-Hermitian Floquet approach, Phys. Rev A 71 (2005), 013408-1-10.

[88] E. C. Titchmarsh: Eigenfunction Expansions Associated with Second-Order Differential Equations, Part 77, Oxford University Press, Oxford, 1958. [89] C. J. van Winter: Complex dynamical variables for multi-particle systems with analytic interactions. I, J. Math. Anal. Appl. 47 (1974), 633-670. [90] J. Wassaf, V. Veniard, R. Taieb, and A. Maquet: Roles of resonances and recollisions in strong-field atomic phenomena: Above-threshold ionization, Phys. Rev. A 67 (2003), 053405-1-12.

J. J. Wendoloski and W. P. Reinhardt: Effects of an external electric field on 1P° resonances of H-, Phys. Rev. A 17 (1978), 195-200. [92] G. C. Wick: Properties of Bethe-Salpeter wave functions, Phys. Rev. 96 (1954), 1124-1134. [91]

[93] J. R. Winick and W. P. Reinhardt: Moment T-matrix approach to e+-H scattering. II. Elastic scattering and total cross section at intermediate energies, Phys. Rev. A 18 (1978), 925-934.

[94] K. Yajima: Resonances for the AC-Stark effect, Commun. Math. Phys. 87 (1982), 331[95] J. Z. H. Zhang, S.-I. Chu, and W. H. Miller: Quantum scattering via the S-matrix version of the Kohn variational principle, J. Chem. Phys. 88 (1988), 6233-6239.

DEPARTMENT OF CHEMISTRY, CAMPUS Box 351700, UNIVERSITY OF WASHINGTON, SEATTLE,

WA 98105-1700, USA E-mail address: [email protected]

Proceedings of Symposia in Pure Mathematics Volume 76.1, 2007

Recent Results on the Bethe-Sommerfeld Conjecture Alexander V. Sobolev Dedicated to Barry Simon on his 60th birthday ABSTRACT. The Bethe-Sommerfeld conjecture for self-adjoint differential op-

erators with periodic coefficients states that the number of spectral gaps is finite. The objective of this paper is to give a short survey of recent results on the conjecture with the emphasis on the method initiated by Skriganov, which involves arithmetic properties of lattices.

CONTENTS 1. 2.

3. 4. 5.

Introduction Results and Discussion Lattice Points, Perturbation Theory Proof of Theorems 2.1, 2.2, 2.4, and 2.5 Results for Other Operators

References

1. Introduction Under very broad conditions, spectra of elliptic differential operators with periodic coefficients have a band structure, i.e., they represent a union of closed intervals (bands), possibly separated by spectrum-free intervals (gaps) (see [35] and [27]). It was conjectured by Bethe and Sommerfeld in the 1930's that the number of gaps

in the spectrum of the Schrodinger operator H = -A + V with a periodic electric potential V in dimension three must be finite. It is relatively straightforward to see that this conjecture holds for potentials which admit a partial separation of variables, as shown by Eastham (see [5, p. 121]). For general potentials this problem turned out to be quite difficult, and the first rigorous results appeared only in the 2000 Mathematics Subject Classification. 35P15, 11H06. Key words and phrases. periodic Schrodinger operator, lattice point counting, band-gap spectral structure. ©2007 American Mathematical Society 383

384

A. V. SOBOLEV

beginning of the 1980's. It is known that the number of gaps is generically infinite

if d = 1 (see Reed and Simon [35]). For d > 2 there are at least three different approaches which lead in one way or another to the justification of the conjecture. This first approach was put forth by Skriganov, and it presents a combination of number-theoretic ideas and analytic tools. This approach was used in [34] by Popov and Skriganov to give the first rigorous proof of the conjecture for the case d = 2. Then Skriganov [38] obtained a proof for all dimensions d > 3 under the assumption that the r, lattice of periodicity, is rational. For d = 3 the result was

extended to arbitrary r in [39]. A short elegant proof for the case d = 2 was given by Dahlberg and Trubowitz in [4]. We describe its main technical point after formula (3.6). The case of rational lattices was revisited in paper [41] by Skriganov and Sobolev. In [41] the original Skriganov's proof [38] was simplified by separating the number-theoretic part from the spectral one.

The relevance of the lattice type for the Bethe-Sommerfeld conjecture will become clearer in Section 3, where we discuss counting properties for rational and irrational lattices. At this point we only note that for rational lattices, the spectral bands at high energies A remain of size of order one, whereas for the irrational lattices, their length tends to zero as A - oc. Moreover, as the dimension increases, the bands may shrink faster as A - oo (see the remark after Theorem 2.5), which makes it easier for a perturbation to open new gaps in the spectrum. This explains why the proof of the conjecture becomes more problematic in higher dimensions if one does not require r to be rational.

The other two methods are indirect in the sense that the finiteness of the number of gaps in both of them is inferred from other spectral properties of the operator in question, the study of which presents a separate difficult problem. The first of these "indirect" approaches is based on the high energy asymptotics of the Bloch eigenvalues and eigenfunctions. It was first applied by Veliev in [43] to prove

the validity of the Bethe-Sommerfeld conjecture for d = 3. Another proof can be found in the paper [17] by Karpeshina, see also the book [19], to which we refer for a comprehensive account of this approach and further bibliography. We point out that in [17, 19] the conjecture was proved for a wide class of singular potentials, including Coulomb potentials. The third method was developed by Helffer and Mohamed in [13]. Using the microlocal techniques, they derived a suitable two-term asymptotic formula for the generalized integrated density of states of the operator H at large energies, which implied the validity of the Bethe-Sommerfeld conjecture for d = 2, 3, 4. Note that this is the paper where the result for d = 4 is obtained without the assumption of rationality of the lattice for the first time. Recently Parnovski [30] announced a new proof of the conjecture without any restrictions on the dimension or the lattice. However, we are not in the position to comment on the details-the proof needs to be better understood before it finds its place among the existing results. The Bethe-Sommerfeld conjecture for the polyharmonic operator in L2 (RI)

H = Ho + V, Ho = (-A)l,

1 > 0,

(1.1)

with a periodic real-valued function V, was studied in less detail. The first result is due to Skriganov (see [37, 38]), who showed that the number of gaps is finite if 21 > d, d > 3. Later the polyharmonic operator was studied by Karpeshina in [18] (see also [19] and references therein) in the framework of the analytic perturbation

BETHE-SOMMERFELD CONJECTURE

385

theory. The high energy asymptotics of the Bloch eigenvalues and eigenfunctions found in [18] implied the Bethe-Sommerfeld conjecture for 41 > d+ 1, d > 2. Under the same restriction on 1, d, the number of gaps was announced to be finite in the note [47] by Yakovlev (see also [48]) for operators of the form Po+V with an elliptic pseudo-differential operator P0 with constant coefficient and a convex homogeneous symbol of order 21. However, the full version of the Yakovlev's proof does not seem to be available. In [47] and [31, 32] under the condition 4l > d + 1, the perturbation V is allowed to be an arbitrary self-adjoint periodic operator. Later, Parnovski and Sobolev [31, 32] extended the result for potential perturbations to the case 81 > d + 3, d > 2. For the Schrodinger case l = 1, the latter condition is equivalent to the requirement that d = 2, 3 or 4. These are exactly the dimensions for which the conjecture was justified in [13]. At the same time, the approach of [31, 32] is based on the elementary perturbation-theoretic argument, and it is simpler. Moreover, it treats all admissible dimensions d and orders l in a unified fashion. One should say that studying the case of arbitrary 1 (rather than 1 = 1 only) is useful and instructive, as it allows one to understand better the mechanisms responsible for the quantitative characteristics of the spectrum, and to explore how far one can push the perturbation-theoretic argument in its investigation. The aim of the present paper is to give a short survey of the existing results on the Bethe-Sommerfeld conjecture. We concentrate mainly on the approach (or a variation thereof), initiated by Skriganov. The method has two ingredients. First, one observes that the spectrum of the unperturbed operator Ho can be found explicitly and its study reduces to the classical number-theoretic problem of counting lattice points in balls of a large radius. The second step is to develop an appropriate perturbation theory which would allow to include the perturbation V. The finiteness of the number of gaps is a legitimate question for other operators as well, but the list of relevant results is very short. At the end of this paper we make a few comments about the periodic magnetic Schrodinger operator and the limit-periodic operators. As far as stronger periodic perturbations are concerned, Green (see [12]) constructed an example of the Laplace-Beltrami operator with a periodic metric having arbitrarily many gaps in the spectrum. This survey does not pretend to be complete. For instance, we do not discuss the unpublished very technical work by Veliev [45, 46] (see also the preceding paper [44]), where he extends his methods from [43] to treat the operator H in arbitrary dimensions d > 3.

2. Results and Discussion Let H be as defined in (1.1), with a real-valued function V, which is periodic with respect to the lattice r C 1[8d. Let 0 = Or = 1Rd/F and Ot = Or = Rd /Ft be the fundamental domains of the lattice r and of the dual lattice Ft respectively. Denote p = Ot1, the Lebesgue measure of Ot. Introduce in Sj = L2(0) the operators

H(k)=Ho(k)+V, Ho(k) = (-iV + k)2, depending on the quasi-momentum k E Ot, with the natural domain H21(0). For each H(k), the spectrum consists of discrete eigenvalues .i(k), that we arrange in ascending order counting multiplicity. By a simple perturbation theory, these

A. V. SOBOLEV

386

are continuous functions of k. The spectrum of the initial operator H can be now represented as the union

(H) = Utj, £j = U Aj(k), j

kEOt

where the closed intervals Qj are called spectral bands. Define two quantitative characteristics of overlapping of the bands: multiplicity of overlap, which measures the number of bands covering given point A:

m(A) = #{j: A E Ij}, and overlap length, which shows how far the bands penetrate into each other:

Jmaxj max{t [A - t, A + t] C Pj },

A E Q(H),

l0,

A V a(H).

:

Both these functions were first introduced by Skriganov (see [38]). The quantities m(A) and ((A) can be linked with the counting functions N(A; k) = #{j : Aj (k) < Al, n(A; k) = #{j : Aj (k) < Al, of the operator H(k): Im(A) = maxk N(A; k) - mink n(A; k) > maxk N(A; k) - mink N(A; k), (2.1)

((A) = sup{t : mink N(A + t; k) < maxk N(A - t; k)}.

The object which puts both functions together is the 6-variation, defined for all

A,8Ell: W (A, 6) = max N(A - S; k) - min N(A + b; k).

(2.2)

For instance, if W(A; b) > 0 for some A E Il8 and 6 = 8(A) > 0, then m(A) > W(A; 0) and ((A) > S.

If V = 0, we attach the subscript 0 to all objects introduced above: e.g., No(A; k), WO (A; S), etc. In order to state the main results, we need the function O(P) _ Od(P) =

1,

d j4 1 (mod 4),

e-A(log log p)4

d = 1 (mod 4),

defined for all p > 0 with some positive constant A.

THEOREM 2.1. Let V be a bounded self-adjoint r-periodic operator, i.e., V commutes with shifts by vectors of the lattice F. If 41 > d + 1, d > 2, then there exist numbers p1 = pi (V, d) and A, c > 0 such that m(p21) > cAd210(P), c(P21) > cA21-d2l0 (P),

(2.4)

for all p > p1, where A, c may depend on the lattice, but do not depend on V. Clearly, either of the estimates (2.4) implies that there are no gaps in the spectrum starting from the point Al = pit, i.e., that the Bethe-Sommerfeld conjecture 1 (mod 4), and V II is holds. Also, the theorem remains true if 41 = d + 1, d small enough. If the perturbation V is a multiplication by a real-valued smooth function, then the condition 41 > d + 1 can be generalized to 81 > d + 3:

BETHE-SOMMERFELD CONJECTURE

387

THEOREM 2.2. Let V E C°° (0) be a real-valued function, and let 81 > d + 3, d > 2. Then the assertion of Theorem 2.1 remains true. Theorems 2.1 and 2.2 were proved in [31] and [32] respectively, with a function for which the exponential in (2.3) is replaced by p-6 with an arbitrary 8 > 0. Correspondingly, the constant c in (2.4) becomes 8-dependent. The conditions on the smoothness of V in Theorem 2.2 can be relaxed, but we do not go into the details here. As mentioned earlier, for the Schrodinger operator (i.e., with 1 = 1) the requirement 81 > d + 3 implies that d = 2, 3 or 4. The case of arbitrary dimensions d > 5 can be dealt with if one assumes that the lattice r is rational.

DEFINITION 2.3. A lattice r C Rd is said to be rational if for any two vectors _Y1, _f2 E r, the inner product satisfies the relation 'Y172 = /drr12,

0 is a real-valued constant independent of ryl, -Y2 i and r12 = r21 is an integer number. Otherwise the lattice is called irrational. where Or

Although the two theorems below are stated for 1 = 1, they can be generalized

to arbitrary 1, but to keep things simple, we state them in the form they were proved in [41]:

THEOREM 2.4. Let d > 4, 1 = 1, and the lattice r be rational. Let V be a continuous r-periodic real-valued function. Then there exist positive constants Ao = Ao(r, V), 80 = 8o(1, V) > \\0 and c, C` such that for all A > Ao

c^d2z < m(A) < CA'Z2,

(2.5)

if d > 5 and m(A) > c(loglogA)-1A,

if d = 4. If d = 4 and r contains a cubic sublattice, then m(A) > cA.

Moreover, for A > A0,

((A) ? 80. The constants C, c depend on the lattice r and do not depend on V. For d > 5, the estimates (2.5) cannot hold for irrational lattices:

(2.6)

THEOREM 2.5. Let d > 5, 1 = 1, and let r be an irrational lattice. Suppose that V satisfies the conditions of Theorem 2.4. Then m(A) = o(A d22 ),

A --* oo.

(2.7)

The lower bounds of the type (2.5) were first established by Skriganov in [38] and were based on properties specific for rational lattices. Later, Skriganov and Sobolev [41] significantly simplified this proof by separating the number-theoretic and spectral parts. Comparing the estimates (2.4) and (2.6) for ((A), we see that in contrast to the irrational lattices, the overlap length ((A) for rational ones is always guaranteed to remain well separated from zero as A ---> oo. This explains why the rational lattices are easier to study than the irrational ones. In the next section we present the components of the proof of Theorems 2.1, 2.2, 2.4, and 2.5.

A. V. SOBOLEV

388

3. Lattice Points, Perturbation Theory 3.1. Lattice Points in the Ball of a Large Radius. Let C C 118d be a set, and let N[C] be the number of points -yt c Ft in C. Denote e(k) = { E ]18d

+ k E C},

:

k e Ot.

Using the notation (f) _ f f(k)dk

t

for the average value of a function f E L'(Ot), one easily shows that

(NO -

(3.1)

)]) = ICI.

We are interested in the number of lattice points in the ball

0. Notice that the eigenvalues of the unperturbed operator Ho(k) coincide with yt +

E rt, so that

k121, ryt

(3.2) No(P21; k) = N[B(p, k)]. Formula (3.1) shows that (N[B(p, )]) = wd pd, where wd is the volume of the unit ball in 118d. We need lower bounds on the variation of the number N[B(p, k)] when the parameter k varies. To describe this variation, we first introduce the following integral norms describing the deviation of the above number from its average value: -Wdt1-1PdIIp' PC

ap(p) = IIN[B(P)]

[1,00),

a. (P) = sup I N[B(P, k)] - wd µ-1 pd I. k

IIp denotes the norm in LP(Ot). Another convenient quantity is the 6variation WO (A; b); see (2.2). By (3.2) for all A > 0 and arbitrary 6 : 181 < A, we Here II

-

have

Wo(A, S) = max N[B((A - 6)2'1 , k)] - min N[B((A + 6)2'1 , k)].

This 6-variation was first introduced in [40] for l = 1. The question of estimating the quantities ap(p) and Wo(A, 6) falls in the same category with the famous circle problem (see, e.g., [7, 25]). The circle problem is usually associated with estimating the number a,, from above and has been investigated quite well. We start with the classical bound initially proved by Sierpinski [36] for d = 2 and generalized to d > 3 by Landau [28]:

(P) = 0(p2,,) xl =

l

2

+

1=1,2,....

(3.3)

l + 1, Today more precise results are known. For relevant results in dimensions d = 2 and 3, 4, we refer to papers by Huxley [15] and Kratzel and Nowak [26] respectively. For dimensions d > 5, Gotze [11] (see also the preceding paper [3]) established the universal bound 2

aoo(P) = Q(Pd-2)

(3.4)

If one aims to obtain more precise estimates, one is led to differentiate between the rational and irrational lattices. It was shown in [11] (see also [3]) that for d > 5, o(Pd-2)

aoo(P) =

(3.5)

BETHE-SOMMERFELD CONJECTURE

389

if and only if the lattice r is irrational. Furthermore, in [16], examples of irrational latices were constructed, for which

N[B(P, 0)] =

Pd+C(P2

µ for any c > 0. For our purposes we need lower bounds for a1(p) and Wo(A, S). For a sufficiently large po > 0, the quantity al (p) satisfies the bound d-1 al (P) >- csp 2 -b, VP ? Po,

(3.6)

where

S=

r0, arbitrary positive,

Sl

if d 1 (mod4); if d = 1 (mod 4).

Here and below we denote by C, c (with or without indices) various positive conCp(d-1)/2 stants whose value is of no importance. As was shown in [23], a2(p) < for all d, so that (3.6) is sharp if d 1 (mod 4). For these values of d, the estimate (3.6) can be easily derived using an argument due to Dahlberg and Trubowitz; see [4] and also [13] (A similar idea was also used by Skriganov; see [39, Lemma 12.2].) This argument is so elementary that we repeat it here. Denote N(p, k) = N[B(P; k)].

Just as in [4] and [23], we easily conclude that for any y E r N(P; -Y)

f N(p, k)eZrykdk

=f

e

dkOkIP

and, in particular, N(P; 0) = (N(P)) = wd pd

Observe that 2a1(P) = 2

f

IN(P, k)-N(P; 0) Idk > (IN(P; y)I+IN(P; 2'Y)I),

dy E r\{0}. (3.7)

Computing the Fourier coefficient, we obtain that (27r)d/2y-d/2Pd/2Jd/2(P`Y),

`Y = I'YI > 0. N(p;'y) = We point out the following elementary property of Bessel functions:

2v 0 1 (mod4)

(3.8)

for all sufficiently big z > 0. Indeed, the Bessel function has the asymptotics (see [1])

Jv(z)

z g(z) + O(z-3/2),

with

g(z) = sin (z + a7r), a

= - 2v4- 1

The required estimate will be proved if we show that g(z) I + Ig(2z) I > c,

z > zo

A. V. SOBOLEV

390

for some zo > 0. The roots of g(z) and g(2z) are -a7r + 7rn and -a7r/2 + 7rm/2, m, n E Z respectively. Since a is not an integer, these roots never coincide. This proves (3.9) and (3.8). Now (3.8) implies that N(P; y) + IN(P; 2 Y) ? cp d21,

(3.10)

so that (3.7) immediately yields the required lower bound (3.6) for d 54 1 (mod 4).

For the dimensions d = 1 (mod4), (3.6) was established for the first time in [31]. In [31] it was also proved that there exists a sequence p, , oo such that d-1

Cpn2 (log

0,1(Pn)

Pn)_Q

with a > 0. This shows that (3.6) cannot be improved in the power scale. However, it can be made more precise in the log-scale: THEOREM 3.1. Let d > 2. Then for some positive constants A, M > 0, one has

a1(P) > MAPd20(P), for all sufficiently large p > 0, where 0 is defined in (2.3).

(3.11)

The above estimate for d = 1 (mod 4) was proved by Konyagin, Skriganov, and Sobolev in [24], and for d 1 (mod 4), it simply repeats (3.6). Remembering that for any bounded function f on (fit with the property (f) = 0, one has 2µmin f(k) < -11f Ill, 2pmkax f(k) > 11f ill,

we obtain from (3.11) and (3.2) that Imaxk No (P21; k) mink N o (P21; k)

>

µ-1 wd Pd + MP d 21 0(P)

(3.12)

µ-1 wd Pd - MP d z l 0(P)

From these we can easily derive the following lower bounds for the 8-variation:

COROLLARY 3.2. Let F C Rd, d > 2 be an arbitrary lattice.

Then for all

sufficiently large p > po(r) and all 6 E [0, bo(p)], where So(p) = MdWI

p21

d

d210(P),

we have the bound Wo(P21, S) > MP d 21 O(P).

Using the formulas (2.1), one immediately obtains (2.4) for the functions m(A) and ((A),.\ = p21 for the unperturbed operator Ho. For rational lattices, the lower bounds for the 6-variation can be made more precise. The next three theorems are proved in [40]. From now on we consider the case 1 = 1 only. The positive constants Ao, So, c featured in the theorems below depend only on F.

THEOREM 3.3. Let F C Rd be a rational lattice, and let d > 5,1 = 1. Then there are three positive constants So, )to and c, such that for all S E [0, So] and all A > Ao, we have Wo(A, 6) >

cAd22.

(3.13)

BETHE-SOMMERFELD CONJECTURE

391

According to (3.4), for all 6 E R, W,, (A; 6) = O(A T'),

d > 5,

(3.14)

so that (3.13) is sharp. The next theorem deals with the four-dimensional case: THEOREM 3.4. Let r C 1184 be a rational lattice, and let 1 = 1. Then there are three positive constants So, A0 and c, such that for all 6 E [0, 60] and all A > A0, we have

W0(A, 8) > cA(log log A)-1.

(3.15)

It is not yet clear whether one can get rid of the log log-factor in (3.15) for general rational lattices. However, for the case of a cubic lattice r, this can be done:

THEOREM 3.5. Let I = (2irZ)4, 1 = 1. Then for each 8 E [0, 2-15], all sufficiently large A > A0 > 0 and some c > 0, one has the bound W0(A, 8) > cA.

(3.16)

The main tool in the proof of the above theorems are asymptotic formulas for the number of representations of integers by positive definite quadratic forms with integer coefficients.

In [38] Skriganov obtained a number of conditional results which relate the behaviour of ,,(p) as p - oc, with the number of gaps in the spectrum of the

Schrodinger operator H = -A + V with a periodic perturbation V, which is not supposed to be local. In particular, Theorem 15.3 from [38] combined with the estimates (3.4), (3.5), gives the following proposition.

PROPOSITION 3.6. Let d > 5. Then there exists a number t > 0 and a (nonlocal) periodic perturbation V with the norm 11VII = t such that the number of gaps in the spectrum of H is infinite. If, in addition, the lattice is irrational, then for any t > 0, there exists a (nonlocal) periodic perturbation V such that II V II = t and the number of gaps in the spectrum of H is infinite. This shows that the lattice properties are relevant for the band-gap structure of the operator -A + V, although one should remember that the perturbation V in the above proposition is not local.

3.2. Perturbation Theory. The results on lattice points counting in the previous section lead to lower bounds for the overlap multiplicity and overlap length for the unperturbed operator. In this subsection we provide necessary perturbation-

theoretic results which allow one to extend the mentioned lower bounds to the perturbed operator. The following estimate is crucial for the proof of Theorem 2.2: PROPOSITION 3.7. Let d > 2, 21 > 1. Suppose that V E C°°(O) and that

10 V(x)dx = 0. Then II N(P21) - No(P21)1

for sufficiently large p.

Cpd+l-41 log

(3.17)

P,

(3.18)

A. V. SOBOLEV

392

The bound (3.18) was derived by Karpeshina in [20] as an intermediate result for obtaining the corresponding estimate for the integrated density of states D(p21) = (N(p21)). Indeed, the unperturbed density of states Do(p21) coincides with wd pd, so that (3.18) leads to D(p21) = wd pd + pd+1-410(log P),

P - oo. Actually, the conditions in [20] require only a finite smoothness of the potential, but we do not go into these details here and refer to the original source instead. Note also that for 1 = 1, a similar asymptotics for D(p2) was established in [13] with the remainder estimate O(pd-3+E) with arbitrary E > 0. For rational lattices, the reduction to the unperturbed operator is implemented in the following theorem:

THEOREM 3.8 ([41]). Let d > 2, 1 = 1, and let V be a continuous real-valued 1-periodic function satisfying (3.17). Then for any 6 E R, e > 0, 7o (A, 8 +

E) - O()xd-1) < W(A; 8) < WO (A; 8 - E) + O()"d-1)

(3.19)

for large A, where W is defined in (2.2) and x1 in (3.3). Note that the condition (3.17) in Proposition 3.7 and Theorem 3.8 is imposed for convenience only. For the general V, the results can be recovered in the obvious way.

To be precise, the result obtained in [41] gives a two-sided bound for m(A) of the form (3.19) with 8 = 0. However, a minor modification of the proof in [41] immediately leads to (3.19). We do not give the full proof of this theorem, but make a few comments on its pivotal points. The fact that the presence of the potential V has a relatively small effect on the 8-variation W admits a perturbation-theoretic explanation. Assume for simplicity

that

V(x)

veez0X BEO

where O C Ft is a finite subset, and ve = v=e. The eigenvalues of the unperturbed Hamiltonian Ho(k) are given by A(') (k) _ (w + k)2, w E It. The analysis of these eigenvalues under the perturbation V is dramatically different for d = 1 and d > 2. If d = 1, then the standard perturbation theory yields a complete asymptotic expansion of the eigenvalues. On the contrary, for d > 2, the unperturbed eigenvalues split in two groups behaving differently under the perturbation V, depending on whether or not the value w is close to the set A, which is the union of the hyper-

planes { E Rd : O(2 + 0) = 0}, 0 E O. The eigenvalues A(') (k) with w away from A are well separated and can be more or less completely described by the perturbation theory; see Feldman, Knorrer, and Trubowitz [8] and Karpeshina's book [19]. The eigenvalues with w close to A move by a quantity of order JIV II under the perturbation V. This effect is due to the small divisors arising when the eigenvalues A(") (k) get close to one another. In the relevant literature these exceptional eigenvalues are sometimes called resonant, unstable or singular, see [9, 19]. It was shown in [9, 19] that their behaviour can be described by means of some effective one-dimensional Schrodinger operators. When proving Theorem 3.8, one needs to estimate the number of those points in the resonant set that can affect the counting function N(A; k). Since A is a union

BETHE-SOMMERFELD CONJECTURE

393

of hyperplanes, this problem amounts to counting lattice points inside balls of the reduced dimension d - 1. This leads to the error term O(A'"d-1) by virtue of the classical bound (3.3). In view of (3.13), (3.15), and (3.16), for rational lattices and d > 4, this error is dominated by the 8-variation Wo (A; 8), and hence the bound (3.19) leads to Theorem 2.4. For the sake of comparison, note that the estimate (3.18) also requires the study of the resonant set, but it involves only the average number of lattice points, and hence one can use the formula (3.1) without appealing to number-theoretic results.

4. Proof of Theorems 2.1, 2.2, 2.4, and 2.5 4.1. Proof of Theorems 2.1 and 2.2. We prove only lower bounds for the multiplicity m(A). Introduce the notation Ti(P) = II N(P21) - No(P21)II i.

Si(P) = IIN(P2d) - (N(P21))II1,

Before proceeding with the proof, we establish a few elementary estimates. First of all, observe that S1(P) ? al (P) - Tl (P) - I (N(P21) - (No (P21)) I

> ai (P) - 2T1(P)

As in the proof of (3.12), we get for A = p21, 2p max N(A; k) > (N(A)) + Si (p) W d Pd + Sl (P) - Tl (P)

Wd Pd + al (P) - 3T1(P)

Similarly,

2p min N(A; k)

Wd Pd - al (P) + 3T1(P).

k

The two previous bounds, together with (2.1), imply that [al (P) - 3Ti (p)].

m(P21) >

(4.2)

It

We emphasize that the above bound contains only L1-norms. Assume that the conditions of Theorem 2.1 are satisfied, i.e., V is a bounded periodic perturbation. Denote v = II V II . By a straightforward perturbation argument, N(A; k) - No (A; k)

No(A+ v; k) - No (A; k),

No (A; k) - N (A; k)
IN(A;k) -No(A;k)I < No(A+v;k) -No(A-v;k), so that, by (3.2), Ti(p)

(N[B(P+)]) - (N[B(P-)]) = Wd(P+ - P ),

p± = (A ± v) z

Consequently,

Ti(P) =

O(vpd-21).

(4.3)

A. V. SOBOLEV

394

Thus the bound (4.2) in combination with (4.3) and (3.11) implies that m(p21) > cpd2ll o(P)

-

Cvpd-21

Under the condition 41 > d+ 1, the first term dominates, which immediately proves the proclaimed lower bound for Suppose now that the conditions of Theorem 2.2 are satisfied. Now we need to use the bound (3.18), which, after substitution into (4.2), leads to m(p21).

m(p21) >

cPd

1

-Cpd+1-41 loge

0(P)

The condition 81 > d + 3 ensures that the first term dominates again, which leads to the bound (2.4) for m(p21).

REMARK 4.1. The proof of the Bethe-Sommerfeld conjecture for l = 1 given in [13] uses a variation of the method described above. Namely, instead of estimating Sl (p) from below by a1(p), as in (4.1), Helffer and Mohamed find the asymptotics of the generalized density of states, i.e., of Fourier coefficients N(A; -y)

= fo N( A; k)ei7kdk

for arbitrary y c r as A -+ oo:

N(p2; ^t) = No(P2; Y) + O(pd which, in view of (3.10) implies

3+E),

IN(P 2;.y)I+IN(p2;2^t)I>cpdzl

,

VC

> 0,

d=2,3,4.

(4.4)

Then using the same argument as in the proof of (3.6) for d # 1 (mod 4) (see above), they obtain for S1 (p) the bound S, (p) > cpd21,

d= 2,3,4

(4.5)

for sufficiently large p.

4.2. Proof of Theorems 2.4 and 2.5. Since xd_1 < (d - 2)/2, the lower bounds in Theorem 2.4 follow immediately from Theorem 3.8 and estimates (3.13), (3.15), (3.16). The upper bound in (2.5) is a consequence of (3.19) and the upper bound (3.14). To prove (2.7), note that in view of (3.5), lim lim sup A- d22 VV Wo (\, c) = 0. C-0 A- cc Now, using (3.19) with 6 = 0, we obtain (3.14).

5. Results for Other Operators 5.1. Periodic Magnetic Schrodinger Operator. Consider now the operator with the magnetic potential a = (al, a2, ... , ad):

H = (-iV - a)2 + V, where a1, a2i ... , ad, V are r-periodic real-valued C°°-functions. As shown by Herbst

and Hempel in [14], under certain assumptions on the magnetic potential, there are gaps in the the spectrum of the operator H. Mohamed proved in [29] that there can be only finitely many of them if d = 2. The method is a generalization

BETHE-SOMMERFELD CONJECTURE

395

of the approach put forward in [13], and it is based on the asymptotics of the Fourier coefficients of the counting function N(A; k); see Remark 4.1. Applying microlocal techniques developed in [13], Mohamed proved for d = 2 the estimate of the form (4.4), which allowed him to deduce (4.5), leading to the justification of the Bethe-Sommerfeld conjecture. Another proof of the conjecture for d = 2 was given by Karpeshina in [21], relying on a modification of the methods described in [19] for the operator (1.1). The approach involves subtle asymptotic formulas for the Bloch eigenfunctions and eigenvalues of the perturbed operator. A different spectral picture emerges when, instead of the periodicity of the magnetic potential, one assumes that the magnetic field B = curl a is periodic. Assume

that B = const

0. Remembering that for d = 2 the spectrum of the operator

Ao = (-iV - a)2 consists of isolated equidistant eigenvalues of infinite multiplicity (Landau levels), it is clear that at least for sufficiently small perturbation V, the spectrum of A = A0 + V will have infinitely many gaps. If d = 3, the spectrum of A0 fills the half-line [IBI, oo) and it makes sense to ask whether the perturbed operator A has finitely many gaps in the spectrum. Under the assumption that the flux of the magnetic field is rational, and the potential V is sufficiently small, the number of gaps was shown to be finite by Geiler, Margulis, and Chuchaev in [10].

Later, Elton in [6] removed the restriction on the size of the perturbation. The idea follows the proof of Theorems 2.1 and 2.2 above, but the technical difficulties are more substantial. In particular, one needs a new bound of the form (3.6) for the counting function of the Floquet fibres A0(k) of the unperturbed operator A0. Another ingredient is the following estimate similar to (3.18): T1(P) = O(Pl-E),E > 0.

The estimate of the type (3.6) was found in [10], and it is sufficient to justify the Bethe-Sommerfeld conjecture for small perturbations. This is the above upper bound for T1 that allows one to extend the result to perturbations of arbitrary size, and its proof is the main focus of [6].

5.2. The Limit-Periodic Operators. We say that a sublattice A C r has

index p = 1, 2, ... , if I OA I = pI Or I. Denote the set of all sublattices of index p by 91p. We say that the potential V is limit-periodic if it has the form 00

V = E > VA, P=1 AE91p

where all VA's are A-periodic, and the series converges absolutely in the sup-norm.

We are interested in the spectrum of the perturbed polyharmonic operator (1.1) with a real-valued limit-periodic potential V. For d = 1 and 1 = 1, this problem has been studied relatively well; see, e.g., Avron and Simon [2] or Pastur and Tkachenko

[33]. It was shown that generically the spectrum is a Cantor set. In the multidimensional case, it is natural to ask whether the spectrum of the operator (1.1) contains a half-line, i.e., there is a number Ao E 11 such that [)o, oo) C a(H). (5.1) With some mild conditions on the decay of VA asp -- oc, supposing that 81 > d+3, d # 1 (mod 4) (cf. Theorem 2.2) and that V is periodic in one direction, Skriganov and Sobolev [42] showed that the spectrum indeed contains a half-line. The proof is

A. V. SOBOLEV

396

based on the study of the operators (1.1) with periodic potentials Vs, s = 1, 2, ... , approximating V in the sup-norm. As the lattice of periods for Vg grows together

with s, in general, the spectral bands of the approximating periodic operators shrink, which makes it harder to control their overlap. However, since one chooses all potentials VS to be periodic in one direction with a period independent of s, one

finds that the spectral bands and their overlap lengths stay well separated from zero as s - oc. This guarantees (5.1) for some Ao. For the operator (1.1) with l > 6 and d = 2, the inclusion (5.1) was proved by Karpeshina and Lee in [22] under the assumptions that the potential V is composed of periodic potentials with doubling periods, and that the size of VA decays superexponentially. Using the methods developed for periodic operators in [19], in combination with the ideas of the KAM method, the authors proved that for every sufficiently large A, there is a family of generalized eigenfunctions that are close to plain waves, thereby proving that the spectrum contains a semi-axis.

References [1] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, Dover, 1972. [2] J. Avron, B. Simon, Almost periodic Schrodinger operators. I. Limit periodic potentials, Comm. Math. Phys. 82 (1981), 101-120. [3] V. Bentkus, F. Gotze, On the lattice point problem for ellipsoids, Acta Arithm. 80 (1997), 101-125.

[4] B. E. J. Dahlberg, E. Trubowitz, A remark on two dimensional periodic potentials, Comment. Math. Helvetici 57 (1982), 130-134. [5] M. S. P. Eastham, The spectral theory of periodic differential equations, Scottish Academic Press, 1973.

[6] D. Elton, The Bethe-Sommerfeld conjecture for the 3-dimensional periodic Landau operator, Rev. Math. Phys. 16 (2004), 1259-1290. [7] P. Erdos, P. M. Gruber, J. Hammer, Lattice Points, Longman, 1987. [8] J. Feldman, H. Knhrrer, E. Trubowitz, The perturbatively stable spectrum of a periodic Schrodinger operator, Invent. Math. 100 (1990), 259-300. [9] , Perturbatively unstable eigenvalues of a periodic Schrodinger operator, Comment. Math. Helvetici 66 (1991), 557-579. [10] V. A. Geller, V. A. Margulis, I. I. Chuchaev, Spectrum structure for the three-dimensional periodic Landau operator, Algebra i Analiz 8(3) (1996); English transl. in St. Petersburg Math. J. 8 (1997), 447-461. [11] F. Gotze, Lattice point problems and values of quadratic forms, Invent. Math. 157 (2004), 195-226.

[12] E. L. Green, Spectral theory of Laplace-Beltrami operators with periodic metrics, J. Diff. Eq. 133 (1997), 15-29. [13] B. Helffer, A. Mohamed, Asymptotics of the density of states for the Schrodinger operator with periodic electric potential, Duke Math. J. 92 (1998), 1-60. [14] R. Hempel, I. Herbst, Strong magnetic fields, Dirichlet boundaries, and spectral gaps, Comm. Math. Phys. 164 (1995), 237-259. [15] M. N. Huxley, Exponential sums and lattice points, III, Proc. London Math. Soc. III 87 (2003), 591-609.

[16] V. Jarnik, Uber Gitterpunkte in mehrdimensionalen Ellipsoiden, Math. Ann. 100 (1928), 699-721.

[17] Yu. E. Karpeshina, Perturbation theory for the Schrodinger operator with a periodic potential, Trudy Mat. Inst. Steklov 188 (1990), 88-116; Engl transl. in Proc. Steklov Math. Inst., Issue 3 (1991), 109-145. [18] , Analytic perturbation theory for a periodic potential, Izv. Akad. Nauk SSSR Ser. Mat., 53 (1989), 45-65; English transl. in Math. USSR Izv. 34 (1990), 43-63.

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, Perturbation Theory for the Schrodinger operator with a Periodic Potential, Lecture Notes in Math. 1663, Springer, 1997. , On the density of states for the periodic Schrodinger operator, Ark. Mat. 38 (2000), 111-137.

[21]

,

Spectral properties of periodic magnetic Schrodinger operator in the high-energy

region. Two-dimensional case, Comm. Math. Phys. 251 (2004), 473-514. [22] Yu. Karpeshina, Y.-R. Lee, Spectral properties of polyharmonic operators with limit-periodic

potential in dimension two, preprint arXiv:math-ph/0601008.

[23] D. G. Kendall, R. A. Rankin, On the number of points of a given lattice in a random hypersphere, Quart. J. Math. Oxford 2 (1953), 178-189. [24] S. Konyagin, M. M. Skriganov, A. V. Sobolev, On a lattice point problem arising in the spectral analysis of periodic operators, Mathematika 50 (2003), 87-98. [25] E. Kratzel, Lattice Points, Kluwer, 1988. [26] E. Kratzel, W. G. Nowak, Lattice points in large convex bodies, II, Acta Arithmetica 62 (1992), 285-295.

[27] P. Kuchment, Floquet Theory for Partial Differential Equations, Birkhauser, 1993. [28] E. Landau, Zur analytischen Zahlentheorie der definiten quadratischen Formen (Uber die

Gitterpunkte in einem mehrdimensionalen Ellipsoid), Sitzber. Preuss. Akad. Wiss. 31 (1915), 458-476. [29] A. Mohamed, Asymptotics of the density of states for the Schrodinger operator with periodic

electromagnetic potential, J. Math. Phys. 38 (1997), 4023-4051. [30] L. Parnovski, On the Bethe-Sommerfeld conjecture, preprint 2005. [31] L. Parnovski, A. V. Sobolev, On the Bethe-Sommerfeld conjecture for the polyharmonic operator, Duke Math. J. 107 (2001), 209-238. , Lattice points, perturbation theory and the periodic polyharmonic operator, Ann. [32] H. Poincare 2 (2001), 573-581. [33] L. A. Pastur, V. A. Tkachenko, Spectral theory of a class of one-dimensional Schrodinger operators with limit-periodic potentials, Trans. Moscow Math. Soc. 51 (1989), 115-166.

[34] V. N. Popov, M. Skriganov, A remark on the spectral structure of the two dimensional Schrodinger operator with a periodic potential, Zap. Nauchn. Sem. LOMI AN SSSR 109 (1981), 131-133 (Russian). [35] M. Reed, B. Simon, Methods of Modern Mathematical Physics, IV. Analysis of Operators, Academic Press, 1975. [36] W. Sierpinski, 0 pwenem zagadnieniu z rachunku funkcyj asymptotychnych, Prace Mat.Fiz. 17 (1906), 77-118; in French: Sur un probleme du calcul des fonctions asymptotiques, Oeuvres Choisies, (S. Hartman, A. Schinzel, eds.), Warszawa: Editions Scientifiques de Pologne, 1974.

[37] M. Skriganov, Finiteness of the number of lacunae in the spectrum of the multidimensional polyharmonic operator with a periodic potential, Mat. Sb. 113 (155) (1980), 131-145; English transl. in Math. USSR Sb. 41 (1982), 115-125. [38]

,

Geometrical and arithmetical methods in the spectral theory of the multi-

dimensional periodic operators, Proc. Steklov Math. Inst. 171 (1984), 121 pp. [39] , The spectrum band structure of the three-dimensional Schrodinger operator with periodic potential, Invent. Math. 80 (1985), 107-121. [40] M. M. Skriganov, A. V. Sobolev, Variation of the number of lattice points in large balls, Acta Arithmetica, 120 (2005), 245-267. , Asymptotic bounds for the spectral bands of periodic Schrodinger operators, Algebra [41] i Analiz 17 (2005), 276-288; English transl. in St. Petersburg Math. J. 17 (2006), 207-216. [42] , On the spectrum of a limit-periodic Schrodinger operator, Algebra i Analiz 17 (2005), 164-189; English transl. in St. Petersburg Math. J. 17 (2006), 815-833. [43] O. A. Veliev, Asymptotic formulas for the eigenvalues of a periodic Schrodinger operator and the Bethe-Sommerfeld conjecture, Funkt. Anal. i Prilozhen. 21 (1987), 1-15; English transl. in Functional Anal. Appl. 21 (1987), 87-99. [44] , On the Spectrum of Manydimensional Periodic Operator, Functions Theory, Functional Analysis and Their Applications, Kharkov University, 49 (1988), 17-34 (Russian). , Asymptotic analysis of the periodic Schrodinger operator, preprint mp-arch 05-90. [45] [46] , On the polyharmonic operator with periodic potential, preprint mp-arch 04-204.

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[47] N. N. Yakovlev, Asymptotic estimates of the densities of lattice k-packings and k-coverings, and the structure of the spectrum of the Schrodinger operator with a periodic potential, Dokl. Akad. Nauk SSSR, 276 (1984); English transl. in Soviet Math. Dokl. 29 (1984), 457-460. [48]

, On spectra of multi-dimensional pseudo-differential periodic operators, Vestn. Mosk. Univ. Ser. 1 Mat, Mekh, No 3 (1985), 80-81 (Russian).

SCHOOL OF MATHEMATICS, UNIVERSITY OF BIRMINGHAM, EDGBASTON, BIRMINGHAM B15 2TT, U.K.

E-mail address: [email protected]

Electric and Magnetic Fields, Semiclassical Limit

Proceedings of Symposia in Pure Mathematics Volume 76.1, 2007

Recent Developments in Quantum Mechanics with Magnetic Fields LAszlo Erdos Dedicated to Barry Simon on his sixtieth birthday ABSTRACT. We present a review of the recent developments concerning rigorous mathematical results on Schrodinger operators with magnetic fields.

CONTENTS 1.

2.

3.

4. 5.

Introduction Basic Qualitative Properties Quantitative Properties of One-Body Operators Many-Body Magnetic Systems Random Schrodinger Operators with Magnetic Fields

References

1. Introduction The mathematical formulation of quantum mechanics, given by Schrodinger, Pauli and Dirac, has posed an enormous challenge: can mathematics, with its own tools and standards, rigorously justify or even predict physical phenomena of the quantum world? Similarly to the development of the differential and integral calculus, strongly motivated by Newton's classical mechanics, new mathematical tools

have been created (most notably by von Neumann, Weyl, Wigner and later by Kato). Functional analysis, representation theory and partial differential equations would have been much poorer mathematical disciplines without quantum mechanics.

Electromagnetic fields play a central role in quantum physics; their rigorous inclusion in the theory is certainly one of the key goals of mathematical physics. 2000 Mathematics Subject Classification. 81Q10, 81Q70. Key words and phrases. Schrodinger operator, Pauli operator, magnetic fields. Partially supported by EU-IHP Network "Analysis and Quantum" HPRN-CT-2002-0027 and by Harvard University. ©2007 American Mathematical Society 401

402

L. ERDOS

Quantum electrodynamics (QED) postulates that electric and magnetic fields are to be described within a unified relativistic theory. Although the framework for QED has been clear since the 1930's, the mathematical difficulties to even formulate the theory rigorously have not yet been resolved. In the low energy regime, however, massive quantum particles can be described non-relativistically. Electric and magnetic fields, with a good approximation, can be considered decoupled. Since typical magnetic fields in laboratory are relatively weak, as a first approximation one can completely neglect magnetic fields and concentrate only on quantum point particles interacting via electric potentials. The rigorous mathematical theory of Schrodinger operators has therefore started with studying the operator H = --1- p2 + V (x) on L2 (Rd) and its multi-particle analogues. Here x E Rd is the location of the particle in the d-dimensional config-

uration space, p = -iVy is the momentum operator and m is the mass, that can be set m = with convenient units. The Laplace operator describes the kinetic energy of the2 particle and the real-valued function V (x) is the electric potential. Although both the kinetic and potential energy operators are very simple to understand separately, their sum exhibits a rich variety of complex phenomena which differ from their classical counterparts in many aspects. The mathematical theory of this operator is the most developed and most extensive in mathematical physics: the best recent review is by Simon [151]. As a next approximation, classical magnetic fields are included in the theory, but spins are neglected. The kinetic energy operator is modified from p2 to (p + A)2 by the minimal substitution rule: p r-4 p - eA and we set the charge to be e = -1.

Here A : Rd -* Rd is the magnetic vector potential that generates the magnetic field B according to classical electrodynamics. In d = 2 or d = 3 dimensions B = V x A is a scalar or a vector field, respectively. In d = 1 dimension the vector ei'P(p+A)2e-ip = p2, potential can be removed by a unitary gauge transformation, cp = f A, therefore magnetic phenomena in R1 are absent (they are present in the case of Sl).

We will call the operator (p + A)2 + V the magnetic Schrodinger operator. In general, even the kinetic energy part contains non-commuting operators, [(p + A)k, (p + A)e] # 0, and the theory of (p + A)2 itself is more complicated than that of p2 + V. The simplest case of constant magnetic field, B = const, is explicitly solvable. The resulting Landau-spectrum consists, in two dimensions, of infinitely degenerate eigenvalues at energies (2n + 1) 1 B1, n = 0, 1, .... Notice that the magnetic spectrum is characteristically different from that of the free Laplacian. The eigenfunctions are localized on a scale JBJ-1/2; this corresponds to the cyclotronic radius in classical mechanics (Landau orbits). The interaction of the spin with a magnetic field is proportional to the field strength. In the low energy regime this effect is comparable with the energy shift due to inclusion of A into p2. Since electrons are spin-! particles, the spin, in principle, should not be neglected whenever magnetic fields are considered. Nevertheless, magnetic Schrodinger operators constitute an important intermediate step to understand magnetic phenomena. The state space of a spin-! particle is L2 (Rd, C2) (in d = 2, 3) and the momentum operator is the Dirac operator, VA := o (p+A), where o = (al, 92, Q3) is the vector of the Pauli matrices. The kinetic energy is given by the Pauli operator, B,

(1.1)

RECENT DEVELOPMENTS IN MAGNETIC FIELDS

403

and external potential may be added as before. The last identity is a special case of the Lichnerowicz formula known from spinor-geometry. In most of this review we restrict ourselves to these operators and their multi-

particle generalizations. However, we briefly mention that in dimensions higher than 3 or on configuration spaces with a non-flat Riemannian metric, the vector potential is canonically defined as a one-form, a, and the magnetic 2-form, 0 = da, is its exterior derivative. In the conceptually most general setup for the spinless case, the Hilbert space of states consists of the L2-sections of a U(1)-bundle over an orientable Riemannian manifold, M, representing the configuration space, and the momentum operator is the covariant derivative, V, on this bundle. In this formulation, the vector potential does not appear directly but the magnetic field is (i-times) the curvature 2-form of V. Proper description of the spin involves covariant derivatives on sections of a Spin-bundle with Pauli matrices replaced by Clifford multiplication [46]. In relativistic theories, electron-positron pair-creations cannot be neglected and one studies the full relativistic Dirac operator, a (p + A) + (3m, where (a, 0) is the vector of the Dirac matrices and m is the mass. Due to the lack of semiboundedness

of the Dirac operator, its definition, even without a magnetic field, is a complex issue that is not yet satisfactorily resolved in the many-body situation ("filling the Dirac sea"). We will not pursue this direction here since the current research focuses more on the non-magnetic aspects of the Dirac operator.

A consistent quantum theory requires to quantize the electromagnetic field as well. Ideally, this should be done within the framework of the Dirac operator (relativistic QED) but this problem is beyond the reach of the current techniques. A more tractable model is the non-relativistic QED, where quantized electromagnetic field is introduced in the Pauli operator, i.e., pair-creations are neglected. This overview gives an admittedly biased summary of a few recent key results involving magnetic Hamiltonians. Many people have contributed to these questions and a selection was unavoidable; the author apologizes to everyone whose work has been left out. The choices reflect the author's taste and the pressure of the editors to keep to the page limit. In Section 2 we present results related to the proper definitions of these operators. In Section 3 we discuss one-particle spectral theory, including Lieb-Thirring type bounds and semiclassical methods. In Section 4 we consider multi-particle problems, including stability of matter, large atoms and scattering. Finally, Section 5 is devoted to random Schrodinger operators with magnetic fields. Barry Simon was undoubtedly one of the initiators and most important contributors to the endeavor to put Schrodinger operators on solid mathematical ground. His work was especially pioneering in the theory of magnetic fields. Among many of his achievements in this area, I would just mention here those two that had the biggest impact on my own work. Barry was the first who systematically exploited path integral methods for magnetic fields upon an initial suggestion of Nelson (see e.g., [152]). Secondly, his seminal papers with Avron and Herbst [5] have become the classical reference "handbook" about magnetic fields. This overview is dedicated to his 60th birthday.

404

L. ERDOS

2. Basic Qualitative Properties 2.1. Definitions. Along the development of the rigorous theory of Schrodinger operators without magnetic fields, it was apparently Kato who first initiated the natural program to extend this theory to the most general magnetic fields. The unique self-adjoint extension of the operator (p+A)2+V without any growth condition on A was shown in 1962 by Ikeda and Kato [88]. This result indicated that magnetic operators should not simply be viewed as second order differential operators with variable coefficients. For most mathematical purposes it is misleading to look at (p + A)2 as p2 + A p + p A + A2. The A-field plays a special role in magnetic problems: it balances the derivative of the phase of the wave function. This effect is inherently present in the form (p + A)2. Kato proved his celebrated distributional inequality, Al-%I > Re [sgn 0V], for any 0 E L', 0I E Li , in 1973 [93]. Simon realized its connection to the semigroup inequality, leto'I < e"l'bl in 1977 [146]. A more general abstract setup was considered in [147], and independently in [80], leading to the magnetic versions of these inequalities. For regular vector potential, a simple proof of the semigroup diamagnetic inequality, le-t(P+A2,)I < l

,

(2.2)

via the Feynman-Kac formula was given in Simon's paper [146] quoting an argument of Nelson in a private communication. It was apparently Nelson who pointed out the probabilistic approach to Simon (see the history in Simon's book [152]), but the real power behind the rigorous path integral method for magnetic fields was realized in a series of papers of Simon and collaborators [5] (see also [25]). More singular vector potentials were considered with analytic methods in [94]. Finally, in his seminal paper [148], Simon gave a simple proof of the diamagnetic inequality (2.2) where the operator H = (p+A)2 was defined under the most general conditions, namely for A E Li, and for H defined as the operator associated with the maximal quadratic form. The domain of the maximal form contains all z/) E L2 with (p + A)V) E L2 in distributional sense. Using (2.2) and semigroup smoothing, Simon showed that the CO' is a form core for H. A non-negative potential V E L'10i can be added to H without any difficulty. The optimal conditions for CO' being the operator core for H are A E L I., and divA E Ll . This was conjectured by Simon and proved by Leinfelder and Simader, [102]. Leinfelder [101] has also showed the unitarity equivalence under any gauge transformation, A - A+V gyp, that stays within the above classes. Again, a non-negative potential V E L2,,i , can be added to H without any difficulty and the Leinfelder-Simader theorem extends to a certain class of negative potentials as well (V = V1 + V2 < 0, V1, V2 E Li C, Vi (x) _> -(const) 1x12, V2 bounded relative to

-A with a bound smaller than one). The most general conditions on potentials with non-trivial negative part, V_ 0, are hard to use directly. The typical argument uses the KLMN theorem (Theorem X.17 [139]) that defines self-adjoint operators by adding a relative form bounded perturbation (with bound less than 1) to a semi-bounded closable quadratic form. The boundedness of V_ relative to (p + A)2 + V+ is, however, hard to check. With the help of the diamagnetic inequality, the boundedness of V_ relative to p2 + V+

is sufficient. We recall that V_ being in the Kato class, V_ E 1C, implies even

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infinitesimal boundedness. Using Stratonovich stochastic integrals, the FeynmanKac formula can be extended to A E Li , vector potentials if V_ is relative bounded by p2 + V+ ([84]). If one prefers to use Ito stochastic integrals, then, additionally, divA E L2 is also necessary for the Feynman-Kac formula. The definition of the magnetic operator with Neumann boundary conditions was carefully worked out recently in [87] and the proof of the diamagnetic inequality and Kato's inequality were extended to this case using the method of Simon [148]. The most general diamagnetic result for the Neumann case is obtained in [86] that uses no regularity assumptions on the domain and on V+. Similarly to the non-magnetic case worked out in the fundamental paper by Simon [150], with the help of the (magnetic) Feynman-Kac formula one can prove smoothing and continuity properties of the semigroup and its kernel. This work has been carried out in [17] with great care and with many fine details. To summarize the results, one assumes that the vector potential A belongs to the so-called (local) magnetic Kato class, i.e., A2, divA E and the potential is Kato decomposable (V+ E 1Clo,, V_ E 1C). Then the LP-semigroup is continuous in time and if A and V are approximated locally in the Kato-norm, then the approximating semigroups converge. Moreover, the Feynman-Kac formula defines a continuous representation of the semigroup kernel. 10,

The definition of the Pauli operator can be directly reduced to that of the magnetic Schrodinger operator using (1.1) and treating or - B as a (matrix-valued) potential term. However, the supersymmetric structure of the Pauli operator (at least in even dimensions) allows one to define the Pauli operator directly and for more general magnetic fields. On topologically trivial domains only the magnetic field has physical relevance. The weakest necessary condition on a B, if considered as a potential, is B E Ll . However, not every L1 field can be generated by an L oC vector potential, hence (p + A)2 might not be defined even as a quadratic form. Therefore it is desirable to define the Pauli operator directly, by circumventing the vector potential. This idea has been worked out in d = 2 dimensions in [49], where A was replaced by a scalar potential, h, satisfying Ah = B, and the Pauli quadratic form was given by

s

q('W),'W)

4J

la=(e-h,b+)I2e2h+4 f

la=(eh

-)12e-2h

V=

("J

(2.3)

This definition is applicable for any measure-valued magnetic field and it is consistent with the standard one for fields that can be generated by Li, vector potential. However, for singular fields the form core is not Co' any more. Strangely enough, a similar construction does not seem to apply for the magnetic Schrodinger operator and the higher-dimensional generalizations are also open.

2.2. Compact Resolvent, Essential Spectrum, Absolute Continuous Spectrum. A basic qualitative fact about magnetic fields is that their inclusion into the free spinless Laplacian, very roughly speaking, increases the bottom of the local spectrum by 1 B(x)1. This intuitive statement makes sense only if the spectrum of the localized operator can be defined and if B(x) is sufficiently regular. The key mathematical reason is the Lichnerowicz identity (1.1) that shows that (p+A)2 on

spinors is a non-negative operator plus -a B. Viewing this identity restricted to spinors with a spin direction opposite to the field, one obtains a useful lower bound on the magnetic Schrodinger operator. One can also see this by using the fact that

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the components of (p + A)2 do not commute: the commutator is the magnetic field (up to a factor i). This trivial but crucial observation is the core of many results

throughout the next sections. We emphasize that this effect holds only for the spinless magnetic Schrodinger operator and not for the Pauli operator. This idea has been elaborated by several authors to investigate the location of the essential spectrum and resolvent compactness of the magnetic operators. For the Schrodinger operator it has been shown that for sufficiently regular fields, the condition that the strength of the magnetic field goes to infinity is equivalent to the compactness of the resolvent (see [72] and references to previous results therein, in particular [5]). The regularity assumptions were weakened in [144] by using functions belonging to the so-called reverse Holder class. For the Pauli operator without external potential it is conjectured that its resolvent is never compact. This has been shown for sufficiently "well-behaving" magnetic fields in [75]. Under stronger conditions about the magnetic field at infinity, the essential spectrum of the Pauli operator was also identified in [75]. In a recent work of Last and Simon [100] a different characterization of the essential spectrum was given in terms of the union of the spectra of certain limit operators at infinity. The localized eigenfunctions of the Landau spectrum in case of a constant magnetic field indicate that in d = 2 dimensions the magnetic field has a strong localization effect, while in d = 3 dimensions the free motion parallel with the field guarantees absolutely continuous spectrum. It is somewhat surprising that by a small change of the constant magnetic field, the spectrum can become purely absolutely continuous even in d = 2. This was first observed and proved by Iwatsuka [92] for magnetic fields that are translation invariant in one direction and tend to two different values at plus and minus infinity in the other direction. The classical analogue of this model is actually a very simple geometric picture. Since the cyclotronic radius depends on the field strength, the closed Landau orbits become spirals whose average velocity is non-zero and perpendicular to the gradient of the field.

A similar phenomenon can be created by an external potential in constant magnetic field or by Dirichlet boundary conditions along an edge of the sample that extends to infinity. Under suitable conditions the states can be classified as edge states and bulk states. The edge states are localized along the boundary and they give rise to pure absolutely continuous spectrum inside the Landau gaps. They carry non-vanishing chiral edge currents. This picture persists even under perturbations

with a small (possibly random) potential [57]. The edge states exhibit a level repulsion that is even stronger than that of the Gaussian ensembles expected for the usual Anderson model in the extended states regime [125].

2.3. Zero Modes and Multiplicity. The supersymmetric structure of the Pauli operator is responsible for the spectrally rigid and typically large kernel of Hp in d = 2. The Aharonov-Casher theorem [3, 27, 128] states that dim KerHp is given (essentially) by the total flux, divided by 27r, z f B. As a special case of the Index Theorem, for smooth data and on a compact manifold, it basically relies on algebraic identities. Still, in its most general form on R2 it was only recently proved in [49] (for finite total flux) and [141] (for non-negative field) using (2.3). For arbitrary field it is false [49]. In the strong field limit, under some regularity assumptions, the local density of Aharonov-Casher zero modes converge to B [34].

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The elements of the kernel the Pauli operator (the so-called zero modes) in d = 2 are conceptually much easier to understand than in d = 3. Naive extensions of the two-dimensional constructions to three dimensions fail, and they even seem to indicate that there are no zero modes in d = 3. A fundamental observation of Loss and Yau [124] is that the equation

DA'b=0,

A,B,0EL2

does have a solution on R3, albeit quite complicated. This seemingly innocent fact implies, among others, that non-relativistic matter with a magnetic field cannot be stable unless the fine structure constant is sufficiently small [60]. The explicit construction of [124] does not shed much light on the conceptual origin of the zero modes. It turns out that two-dimensional Aharonov-Casher zero modes on S2 can be lifted to R3 using the Hopf map and spinor geometry [46] (see also [2, 32] for other examples). In particular, magnetic fields with arbitrary number of zero modes can be constructed. Although many zero modes are obtained in this conceptual way, still not all explicit zero modes of [124] are covered. On the other side, it is known that magnetic fields with zero modes form a slim set in the space of all magnetic fields [10], [33]. It is an interesting open question to connect the existence, or even the number of the zero modes with the geometry of the magnetic field. Currently we do not even have a conjecture for a general characterization of magnetic fields with zero mode. For the spinless magnetic Schrodinger operator no supersymmetric structure is available to analyze the ground states and even to compute the bottom of the spectrum is complicated, apart from the strong field regime (Section 3.2.1). Since the Perron-Frobenius theorem does not apply to the magnetic Laplacian, the ground state can be degenerate, although for generic field it is simple. Still, the strength of magnetic field restricts the possible multiplicity. Based upon similar observations by Colin de Verdiere on graphs, it was conjectured in [23] that on a two-dimensional manifold M, the total curvature of the line bundle, i.e., the total flux, fm IBI, gives an upper bound on the multiplicity of the magnetic ground state. This was proved in [42] modulo constants depending on the geometry of the base manifold. The same bound with constants depending only on the genus of M is still an intriguing open question. The proof in [42] relies on an upper bound on the ground state energy in terms of the total flux and this intermediate result necessarily depends on the geometry of M. The construction of an appropriate trial state uses the scalar

potential h (with Oh = B) instead of the vector potential in order to control the energy solely by the LI norm of B.

2.4. Magnetic Operators on the Lattice. The magnetic Schrodinger (and Pauli) operator can also be defined on the lattice. The magnetic field is defined on the plaquets, while the magnetic vector potential, Ab, is a function on the bonds. The magnetic translation operator along the bond b amounts to a multiplication by a complex phase eiA, in addition to the usual hopping. The field on each plaquet is the oriented sum of the vector potentials along the boundary. The field and the vector potential are defined only modulo 27r. Although this definition is very natural, the spectral properties of the lattice magnetic Schrodinger operator differ vastly from the continuous version. Even for a constant magnetic field B on a regular two-dimensional square lattice (Harper

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operator), the spectrum exhibits a complex fractal behavior ("Hofstadter butterfly") depending sensitively on the Diophantine properties of a = B/2'ir. With a simple transformation, this operator can be reduced to the almost Mathieu operator; a simple prototype of a one-dimensional discrete Schrodinger operator with an almost periodic potential: H,,,A = .\ cos(27raD,,) + cos x .

(2.4)

The continuous Schrodinger operator with a constant magnetic field and periodic potential leads to a similar equation. The Cantor-like spectrum of Ha,a was first proven in [14] for a dense set of parameter values and later in a series of papers, Helffer and Sjostrand performed a detailed quantitative semiclassical analysis [76] initiated by Wilkinson [166] to identify a large set of parameter values a with Cantor spectrum if A = 1. With quite different techniques, Last obtained a similar result and he also computed the Lebesgue measure of the spectrum [99] for all A. Finally, the Cantor spectrum has recently been proven by Puig [137] (A 0, a is Diophantine) and by Avila and Jitomirskaya [4] for all conjectured values of the parameters: A 54 0, a irrational ("Ten martini problem," as it was named and popularized by Barry Simon).

2.5. Diamagnetism and Paramagnetism. Diamagnetism plays a crucial role in the analysis of the magnetic Schrodinger operators since it gives an easy a priori comparison of magnetic and non-magnetic operators, like (2.2). However, the apparent strength of the basic diamagnetic inequality is somewhat misleading when it comes to quantitative results. On one hand, it completely neglects magnetic effects; operators with two different but non-zero magnetic fields are not comparable with this method. In particular, diamagnetism in a strong sense, i.e., monotonicity of the energy in the magnetic field strength, does not hold in general because of the de Haas-van Alphen oscillation effect (see [77] for a rigorous proof in the weak field regime with a periodic external potential). On the other hand, diamagnetism is applicable only for the exponential statistics, tr e-13H = >i of the eigenvalues, Aj, in particular for the ground state (,3 -- oo). Going beyond these constraints is notoriously difficult and there are only a few results and many open questions. Loss and Thaller proved [123] that the heat kernel of a two-dimensional Schrodinger operator H = (p+A)2 with an arbitrary magnetic field B(x) can be estimated by e

_tH

B _ e - 47r sink Bt

(x, y) <

(=_v)2 4t

if B(x) > B(> 0). The right hand side is smaller than the free heat kernel and its exponential behavior Pz e- Bt correctly reflects a spectral shift at the ground state energy by at least B. However, it does not retain the full Gaussian off-diagonal decay of the magnetic heat kernel with a constant field. With the help of this inequality, sharp LP - L4 bounds were shown in [123]. The proof heavily uses the Gaussian character of the heat kernel of the constant field operator. Several counterexamples [39] show that this result is basically the best one could hope for: there is no strict diamagnetic comparison between two non-homogeneous magnetic fields or even between two homogeneous magnetic fields with a potential. The Gaussian off-diagonal decay cannot be fully recovered. Such type of decay apparently

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requires real analyticity of the magnetic field and potential in the angular direction [37, 132, 159]. In the large field limit, the diamagnetic effect is so strong that the magnetic Schrodinger operator converges (in resolvent sense) to the free Laplacian with Dirichlet boundary conditions on the regime where the magnetic field vanishes [79]. In other words, strong magnetic fields act like Dirichlet walls, confining the electron motion to regimes where the field is zero. In contrast to the diamagnetism of the (spinless) magnetic Schrodinger operator, the Pauli operator tends to be paramagnetic. This issue was apparently raised first in [82] and Lieb proved (appendix of [5]) that the ground state energy of the Pauli operator with potential cannot increase as a constant magnetic field is turned on. However, paramagnetism fails for non-homogeneous fields [7, 71, 39]. For many-fermion systems, one studies the sum of the low lying eigenvalues of the one-body operator. This statistics is more singular, it is beyond the exponential statistics offered by the heat kernel and surprising phenomena occur. The magnetic Schrodinger operator on a square lattice turns out to be paramagnetic at half-filling. It is the maximal flux (7r on each plaquet) that minimizes the sum of the first A/2 magnetic eigenvalues on a torus of volume A. The result actually holds on any bipartite graph that has a periodicity at least in one direction. After some special cases presented in [104] and proved in [108], the general result was proven by Lieb [105]. The proof uses reflection positivity and seemingly it cannot be extended to other filling factors or to graphs without periodicity, leaving the general case as an intriguing open question. Diamagnetism for sums of the Schrodinger eigenvalues fails in the continuum as well. For a compact domain in Rd and for a constant magnetic field B, let A3 (B) be the j-th magnetic eigenvalue. The sum of the first N eigenvalues, E 1 Aj (B), may decrease by turning on a non-zero magnetic field, but it can never drop below the semiclassical bound [43]. The proof heavily relies on the homogeneity of the magnetic field. For this case a stronger diamagnetic inequality was proven: tr[Xf ((p + A)2)] < tr[Xf (p2 )] for an arbitrary non-negative, convex function f decaying to zero at infinity. Here X is the characteristic function of an open set, the operators are defined in the whole R'. This stronger diamagnetic inequality fails for non-homogeneous fields [43] but still the semiclassical bound for the eigenvalue sum is conjectured to hold.

2.6. One-Body Scattering. A short range magnetic field, IB(x)I does not substantially influence the non-magnetic scattering theory, in particular asymptotic completeness holds. Long range potentials can also be included. The most general result is due to Robert [140]; previously Loss and Thaller treated C(x)-3/2-E the IB(x)l < case in [121] and they also considered the Dirac operator
[122].

The borderline case, when B(x) decays as xJ-1 at large distances, is substantially more involved. In this case, there is no decay on the vector potential. In the simplest d = 2-dimensional, axially symmetric situation, the lack of decay leads to a dense point spectrum in the low energy region while the spectrum is absolutely continuous above an energy threshold [129]. To study the scattering, for simplicity one considers d = 2 and assumes that the field is homogeneous of degree -1, i.e., in

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polar coordinates (r, 0) it is given by B = b(B)/r. The analogous problem for nonmagnetic scattering is the case of a homogeneous of degree zero electric potential, V(x) = U(x/lxl), where the generic classical trajectories are asymptotically straight and they select the directions of local extrema of U. The classical trajectory in the borderline magnetic case turns out to be a logarithmic spiral if the magnetic field has a definite sign. If the total flux is zero, J b(O)d9 = 0, then the trajectories are approximately straight lines (in the direction of the zeroes of b(9)); if the total flux is non-vanishing but b has no definite sign, then both types of behavior may occur. The corresponding quantum scattering follows these trajectories. Part of this picture has already been proven in the recent work [26]; the rest is work under preparation. Scattering in constant magnetic field was first studied in [5] where the asymptotic completeness of the one-particle scattering for short range and Coulomb potential was shown (a different proof given in [149]). The general long range potential was treated independently by Laba [97, 98] and Iwashita [91]. 2.7. Miscellaneous. In this section we mention two results whose non-magnetic counterparts are classics but their standard proofs are quite rigid and their extensions to magnetic fields were considerably more involved. The Rayleigh-Faber-Krahn isoperimetric inequality on the lowest Dirichlet eigenvalue of a domain of given area predates quantum mechanics. Its magnetic analogue asserts [38] that the lowest magnetic Dirichlet eigenvalue, A(S2, B), of a planar domain 1 C RZ with a fixed area and with a fixed homogeneous magnetic field B = const is attained exactly for the disk .\(1l, B) > ,A (D, B),

Area(1l) = Area(D),

D = disk.

(2.5)

The constant magnetic field plays the role of the homogeneity of the membrane in Rayleigh's original formulation of the problem. In the non-magnetic case, the minimal eigenvalue in any dimension is attained for the ball. The minimizing domain for constant magnetic fields in dimensions d > 3, however, is unknown. Isoperimetric results for the magnetic Neumann Laplacians are also unknown. Note that (2.5) does not hold for the Neumann case since the ground state has a tendency to favor non-spherically symmetric geometry (Section

3.2.1), but the disk geometry should be extremal for other spectral variational problems in this case as well. The standard proof of the original Faber-Krahn inequality uses rearrangement methods that are applicable for positive functions. The magnetic ground state of a general domain is genuinely complex and its amplitude, its phase, and the vector potential must be rearranged separately. A Schrodinger operator with a periodic external potential has purely absolutely continuous (AC) spectrum by a classical theorem of Thomas [161]. The periodicity of the magnetic field itself does not guarantee AC spectrum (e.g., B = const zA 0 in d = 2), but a periodic vector potential does. Note that this latter implies not only the periodicity of the magnetic field but also that the flux is zero in the unit cells.

The absolute continuity of the magnetic Schrodinger spectrum with a small periodic vector potential was first proven in [79]. The proof was reduced by perturbation to the original analyticity argument of Thomas and it could not be extended beyond the perturbative regime. In [15] a representation similar to (2.3) was used

to transform a periodic vector potential into a periodic external potential and a

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modification of Thomas' argument applied. A periodic metric can also be included [131]. This approach, however, works only in d = 2 dimensions. The general case was obtained by Sobolev [158] who proved that the spectrum is purely a.c. for the magnetic Schrodinger operator with a sufficiently smooth periodic vector potential in any dimension. The proof combined Thomas' argument with a pseudodifferential technique.

3. Quantitative Properties of One-Body Operators 3.1. Lieb-Thirring Inequalities. One of the fundamental results about the standard Schrodinger operator -A + V is the Lieb-Thirring bound [119] on the moments of negative eigenvalues, Ej, in terms of integral norms of the negative parts of the potential, V_

tr [-A + V]" =

J[V]d+-y/2

jEj 1-1 < Ld,7

with a finite constant Ld,,y for d > 3, -y > 0; d=2, -y> 0 or d = 1, -Y> 1/2. This bound plays a crucial role in the proof of the stability of matter and it provides a basic a priori estimate for the semiclassical formulas and for justification of the Thomas-Fermi theory for the ground state energy of atoms and molecules. By the diamagnetic inequality, the usual proof of the Lieb-Thirring (LT) bound for the non-magnetic operator, -A+V, applies directly to the magnetic Schrodinger operator, (p + A)2 + V as well. The same holds for the Cwikel-Lieb-Rozenblum (CLR) bound on the number of eigenvalues (-y = 0). The presence of a magnetic field should, in principle, improve these estimates, but no such non-trivial result is available.

The systematic study of Lieb-Thirring bounds and semiclassics for the Pauli operator started with a series of seminal papers by Lieb, Solovej and Yngvason [116, 117, 118]. For the d = 3-dimensional Pauli operator, H = [o (p + A)]2 + V, with a constant magnetic field, B, and external potential, the following bound was proven for the sum of negative eigenvalues of H [116]

E I < (const) f[v]5/2 + (const)

f

JBI [V]3/2

(3.6)

where [V]_ = -min{0, V}. A similar bound holds in d = 2 dimensions as well [118]. The first term in (3.6) is the corresponding Lieb-Thirring estimate for -A + V. Due to the paramagnetism, the Pauli energy may be below the nonmagnetic energy and the additional term f JBI [V]3/2 is indeed necessary. The number of eigenvalues can be infinite in d = 2, 3 dimensions, so there is no CLRbound for the Pauli operator. For non-homogeneous magnetic fields, the bound (3.6) does not hold. Most importantly, the existence of the Loss-Yau zero modes shows that, in the perturbative regime, the lowest eigenvalue itself may scale linearly in [V]_. Moreover, the pointwise density of the Loss-Yau zero mode scales as max (x) I2 ,-, B3/2 Therefore, a general LT estimate in the strong field regime must contain a term that grows as the 3/2 power of B. To prove an LT estimate with the B3/2 scaling, the spin-coupling term a B in (1.1) is treated as a potential and the diamagnetic inequality is used for (p + A)2. Several papers [36, 156, 145, 19] used this idea with different assumptions on the

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magnetic field. The most general result in this direction is due to Lieb, Loss and Solovej [113] showing that

E; I < (const)

J

[V]5/2 + (const)IIBII2/2 II[V]4.

(3.7)

The proof introduces the so-called running energy scale method. It consists of artificially scaling down the Pauli kinetic energy in an energy-dependent way to reduce the negative effect of o, B. The main advantage of this method is that it uses no other assumptions on B apart from the finiteness of its L2-norm. Note that f B2 is the field energy. Although a term growing as B3!2 (in the large field regime) is necessary for a general LT bound, a smaller power is sufficient if some control on VB is allowed. Especially, the linearity in B of the bound (3.6) reflects the basic fact that the space with a magnetic field cannot be considered isotropic: the magnetic field affects only the quantum motion in the transversal directions. Under a control on the H1 norm of B, the LT bound in [20] scales as the 17/12 power of the field. With more regularity on B and V the lower power 5/4 was obtained in [44]. Finally, the correct linear behavior in the field strength under a stronger regularity assumption was proved in [47] and [48]. The proof in [47] is shorter, but the estimate is not local: a large irregular magnetic field far away from the support of [V]_ should not influence the eigenvalue sum too much, but the estimate in [47] does not reflect this. A conceptually different proof was given in [48] that relies on a much stronger localization and approximation technique. The main difficulty behind these proofs is to control the density of Loss Yau zero modes. It is amusing to note that it was a substantial endeavor to show that zero modes may exist at all (Section 2.3). On the other hand, it is quite difficult to prove an upper bound on their number in terms of the expected first power of the magnetic field [48].

3.2. Semiclassics and Strong Fields. We have seen that magnetic fields can cause surprising effects when the magnetic lengthscale is comparable with other lengthscales in the problem. However, in the semiclassical and/or in the strong field regimes, lengthscales are typically separated, rendering simpler formulas available

in the limit. One studies the magnetic Schrodinger or Pauli operators with two parameters:

H(h, b) :_ (hp + bA)2 + V or [o,. (hp + bA)]2 + V

,

where h << 1 is the semiclassical parameter and b is the field strength; in most cases b >> 1 (assuming that A and V are fixed). The magnetic field is bB(x) = b curl A(x). Under these scalings, the magnetic field can typically be approximated by a (locally) homogeneous one, since the magnetic lengthscale (b/h)-1/2 is short. If, in addition, hb << 1, then the gap between (local) Landau levels shrinks to zero and magnetic effects usually do not contribute to the main term in the asymptotic regime. This is especially the case for the standard semiclassics when h -* 0 and b is fixed [25]. If, on the other hand, hb 74 0, then the main term is typically obtained by replacing the magnetic field by a locally constant one.

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The key technical step is, therefore, to localize the problem to a sufficiently small scale, where the data (especially the magnetic field) can be considered homogeneous. Similarly to the non-magnetic theories, two basic techniques have been developed: pseudodifferential calculus and coherent states. Note that the presence of a constant magnetic field changes the structure of the phase space. For example, in three dimensions, the phase space is U' 0 R4, where v labels the Landau levels. The phase space for each level is four-dimensional; it consists of three position coordinates and only one momentum coordinate which represents the free motion parallel with the magnetic field. The momenta transversal to the field are not present due to the localization effect of the field. Accordingly, one either has to develop a pseudodifferential calculus that treats the harmonic oscillators on each Landau level exactly or one has to construct magnetic coherent states. 3.2.1. Results on individual eigenvalues. Semiclassical estimates on individual eigenvalues and eigenfunctions have mainly been carried out for the ground states.

For the magnetic Schrodinger operator without potential the ground state is localized near the minimum of the field strength ("magnetic bottles"; see [5] and [22]). The basic observation is that due to the positivity of the Pauli operator and the Lichnerowicz formula (1.1), the magnetic Schrodinger operator can always be estimated from below by B(x), at least in two dimensions and if B > 0. A similar result holds in higher dimensions, at least locally. This concentration of ground state is especially visible in the large field regime. Note that if an external potential is not present, the semiclassical limit is formally identical to the large magnetic field limit, h = 1/B. Precise analysis of this phenomenon was initiated by several groups with different methods. With the help of the Feynman-Kac formula, the magnetic ground

state energy can be turned into a question about the rate of decay of an oscillatory Wiener integral, see [126] and for a more precise bound [35], [162]. Ueki has also explored the connection with the hypoellipticity of the 8b problem [163]. Montgomery [130] has analyzed the case when the two-dimensional magnetic field

vanished along a curve. In this case min JBI = 0, hence the leading term in the large field asymptotics vanishes, and Montgomery obtained the subleading term that involved the curvature of the zero locus. This approach was later generalized in [73].

On a domain with boundary, however, the Lichnerowicz formula (1.1) does not hold, unless Dirichlet boundary conditions are imposed. In particular, the ground state energy of the magnetic Schrodinger operator with Neumann boundary condition is smaller than B even for a constant magnetic field. In this latter case the ground state is localized near the boundary, more precisely near the point with largest curvature of the boundary. The second term in the semiclassical expansion of the ground state energy is determined by the curvature, similarly to Montgomery's

result. A similar phenomenon occurs in three dimensions as well with a proper definition of an effective curvature [74]. Recently a complete expansion for the energy of the low lying eigenvalues in d = 2 dimensions was carried out in [55]. We remark that the Neumann boundary problem naturally arises at the minimization of the Ginzburg Landau energy functional describing superconducting states. Several people have contributed to these results; see [74, 55] and references therein.

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3.2.2. Results on Cumulative Spectral Quantities. The first result on spectral statistics, where the magnetic field played a non-trivial role, is probably due to Colin de Verdiere [22] and Tamura [160] (independently) who proved a Weyl-type asymptotics for the number of eigenvalues with a non-homogeneous magnetic field and a confining potential. The magnetic field increases at infinity, ensuring that magnetic effects contribute to the large energy asymptotics. Colin de Verdiere used the magnetic extension of the classical Dirichlet-Neumann bracketing, while Tamura estimated the short time asymptotics of the magnetic heat kernel. Several authors extended these results; see [127] for references. A technically somewhat similar problem is the rate of the eigenvalue accumulation near the Landau levels due to perturbation by a decaying potential. The basic idea is that the magnetic field strongly localizes the particle and its interaction with the potential can be computed fairly precisely. The accumulation rate is explicitly given by the decay rate of the external field; see [138] and references therein. The next semiclassical question concerns the moments of negative eigenvalues in the spirit of [115]. Here we consider only the more interesting case of the Pauli operator, H(h, b) := [o . (hp+bA)]2+V. For simplicity, we work in d = 3 dimensions and we will approximate only the eigenvalue sum E(h, b) := tr [H(h, b)] The corresponding semiclassical expression is

ESc(h, b) := -h-3 fR P (hbB(x), [V (x)])dx 3

with

P(B, W) :=

82

(W3/2 + >[2vB - W]3/2)

.

(3.8)

V=1

This formula can be simply deduced from the structure of the phase space outlined above.

For a homogeneous magnetic field, the semiclassical limit lim

E(h, b)

h-o E,,(h, b)

=1

(3.9)

was proved uniformly in the field strength b [117] (the two-dimensional result was obtained in [118]). The main ingredients were the magnetic Lieb-Thirring inequality (3.6) and new magnetic coherent states. For the non-homogeneous case, a Lieb-Thirring inequality that scales as B3/2 (see Section 3.1) allows one to prove

the semiclassical formula only up to hb = 0(1) [157]. With the improved LiebThirring inequality [44] and a new construction of coherent states, the proof can be extended to b << h-3. This result is already sufficient to cover the full semiclassical regime of the large atoms [45]; see Section 4.2. Uniform semiclassics can be obtained with the help of the uniform Lieb-Thirring inequalities [47, 48]. The development using pseudodifferential calculus has focused on obtaining precise spectral asymptotics for the local traces of the form tr [ xcp(H) ], where x is a spatial cutoff function. These efforts have culminated in the book of Ivrii [89] where precise remainder estimates were proven in great generality. His recent work

investigates the same questions with irregular data [90]. A more concise result using these ideas is the proof of a certain local version of (3.9) for homogeneous

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fields in [154], improved later in [155] to include Coulomb singularities. The microlocal technique also gives higher order corrections to the leading term. However, these methods require, in general, strong regularity assumptions on the data. Moreover, some non-asymptotic a priori estimate (Lieb-Thirring bound) is necessary to remove the cutoff X. In addition to the energy, other physical quantities are also of interest. Fournais

has studied the quantum current in a magnetic field and proved the corresponding semiclassical formula. Note that the current is a second order effect in the semiclassical expansion and it becomes a leading term only after non-trivial cancellations. Both microlocal techniques similar to [154] and coherent states methods similar to [117] have been tested, e.g., in [52] and [53].

3.3. Peierls Substitution and Corrections to the Semiclassics. The Bloch decomposition for a single particle Schrodinger operator with a periodic external potential can be extended to include weak electromagnetic fields. The basic idea due to Peierls is to substitute the minimally coupled magnetic momentum p+A into the band functions, E,,,(p), obtained from the non-magnetic Bloch decomposition. If the electromagnetic field varies on a much larger scale than the periodic background, then the problem is effectively semiclassical with a scale-separation parameter e (for the general theory, see [135]). The resulting pseudodifferential operator can be analyzed with well-developed mathematical tools. Algebraic methods were applied in [13]; for a systematic presentation of the pseudodifferential approach, see [77]. For example, it was shown in [65] that near a fixed energy level the original Hamiltonian is isospectral to a pseudodifferential operator with the same principal symbol as the Peierls Hamiltonian has. The detailed behavior of the density of states, in particular, the de Haas-van Alphen effect for the oscillation of the magnetization, was shown in [78]. The de Haas-van Alphen oscillation is due to a subleading effect in a semiclassical type expansion. However, it determines the current to leading order. The electromagnetic field is weak, but on the long time scale, t - e-1, it yields an order one change in the dynamics. To describe these effects on the dynamics correctly, Panati, Spohn and Teufel [136] developed a time dependent version of the Peierls substitution. The classical equations are corrected by an effective magnetic moment (Rammal-Wilkinson term) and an "anomalous velocity" term due to the curvature of the Berry connection. The latter, in particular, provides a simple semiclassical explanation of the quantum Hall current.

We remark that an oscillation similar to the de Haas-van Alphen effect is exhibited for the Harper operator (2.4) for magnetic fluxes (per unit cells) that are near a rational number; see [62] and references therein. The Harper operator itself can also be viewed as a Peierls substitution by quantizing the classical symbol

cost + cos ij with [,

21rica.

In this case the distance of a to a

nearby rational number with small denominator plays the role of the semiclassical parameter [76].

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4. Many-Body Magnetic Systems 4.1. Magnetic Stability of Matter. In a system of charged fermionic quantum particles subject to a Coulomb interaction the ground state energy per particle is uniformly bounded, independently on the number of particles. This fundamental fact is called the stability of matter. For an excellent review of the progress in the last 35 years, see [106]. The first proof of the stability of matter with the non-relativistic kinetic energy, -A = p2, is due to Dyson and Lenard. A simpler proof was given later by Lieb and Thirring using the Lieb-Thirring inequality. Stability of matter also holds if the non-relativistic kinetic energy operator, p2, is replaced by JpJ = v"--0 ("relativistic" kinetic energy) and the fine structure constant, a, is sufficiently small. The first proof was given by Conlon, improved by Fefferman-de la Llave and finally the optimal bound was obtained by Lieb and Yau; see also a recent improvement and references in [112]. In the relativistic case, the kinetic energy and the potential energy both scale as [length]-1, therefore the energy per particle can be bounded from below only if the Hamiltonian is non-negative, i.e., N

1: Ipjl+aVc>0

(4.10)

j=1

where V. stands for the Coulomb potential of N electrons and K nuclei with charges Z. This inequality was proven by Lieb and Yau in [120] (Theorem 2) for a < 1/94

and Za < 2/7r, the second condition being optimal. Theorem 1 of [120] has a weaker result (a < 0.016, Za < 1/7r) but its proof can easily be generalized to the magnetic case as well, where the kinetic energy JpI can be replaced by its magnetic counterpart lp + Al. This follows from a simple application of the diamagnetic inequality, as pointed out in [113]. With the Pauli kinetic energy, however, even the hydrogen atom is unstable because in a strong magnetic field the electron can be strongly localized around the nucleus without a penalty in the kinetic energy. The ground state energy of the hydrogen in a constant magnetic field B diverges as (log B)2, if B is large [5]. If the energy of the magnetic field is added to the total energy, then stability is restored: N

(p+A)j]2+V+ j=1

1

87ra2

JB2 > -C(Z)(K + N)

(4.11)

where the constant depends only on the charges of the nuclei. The parameter a (fine structure constant) must be sufficiently small in order for (4.11) to hold. The existence of a Loss-Yau zero mode [124] shows that the total energy can be negative if a is large. Actually, an absolute upper bound on a and an upper bound on (maxZj)a2 are both necessary, where Zj are the nuclear charges. The bound (4.11) is called the stability of matter interacting with a classical magnetic field. It was first proven for atoms [60] and single electron molecules [107]. The general case was settled by Fefferman [50] for a very small a. Lieb, Loss and Solovej [113] gave a much shorter proof that also holds for the physical value of the fine structure constant. The backbone of this proof is the magnetic Lieb-Thirring inequality (3.7).

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Using the Birman-Koplienko-Solomyak trace inequality and the magnetic version of the Lieb-Yau bound (4.10), Lieb, Siedentop and Solovej [114] also proved

the stability of matter for the Dirac operator. The particles must be restricted to the positive energy subspace of the one-particle Dirac operator a (p + A) +,3 (filling the "Dirac sea"). It is important to note that the Dirac sea must be defined via the gauge invariant Dirac operator. Restriction onto the positive energy subspace of the free Dirac operator (as it is often done in perturbation theory) leads to instability of matter for any a. Ultimately, the electromagnetic field must also be quantized. Imposing an ultraviolet cutoff and using results from [113], Bugliaro, Frohlich and Graf proved stability of matter for the Pauli operator with a quantized electromagnetic field [21]. The essential observation is that to restore the magnetic stability for the Pauli operator with a classical field, it is sufficient to add the field energy only near the nuclei. On a finite volume and with an ultraviolet cutoff, the classical and quantized field energies can be compared. Lieb and Loss [109] showed stability of matter for the Dirac operator with a quantized electromagnetic field and with a suitable one-body spectral projection, similar to [114]. The quantization of the electromagnetic field poses several complications, such as ultraviolet cutoff and mass renormalization, and little is known about how to rigorously include them into a fully consistent theory. However, one problem has been settled satisfactorily: the existence of atoms in non-relativistic QED. Because of the quantized field, the ground state of the total system (atom and photons) is at the bottom of a continuous spectrum, and it is not at all obvious that it is an eigenvalue. Moreover, the so-called binding condition also has to be satisfied, i.e., the energy of a system of N electrons is actually lower than that of a system with fewer electrons, otherwise the ground state may contain no electron at all. For small values of the parameters (ultraviolet cutoff parameter and fine structure constant) the existence of atoms was shown in [8] and for arbitrary parameter values in [70, 110]. No infrared cutoff was needed, unlike for the scattering problem (Section 4.3). More recently, the thermodynamic limit for non-relativistic Coulomb matter with quantized electromagnetic field was investigated and the lower bound was proved [111].

4.2. Large Atoms. One of the main motivations to study semiclassical spectral asymptotics for -h2A+V originates in the seminal paper of Lieb and Simon, [115], where the exactness of the Thomas-Fermi theory for the ground state energy of atoms with large nuclear charge, Z >> 1, was proven with semiclassical methods. In the presence of magnetic fields the Thomas-Fermi theory is more complex.

Depending on the strength of the magnetic field compared with Z, there are five different regimes.

Within the classical Thomas-Fermi theory, the kinetic energy as a functional of the density is always given by the Legendre transform of the pressure (3.8). This classical magnetic Thomas-Fermi theory, however, holds only for weak and moderate magnetic fields (B << Z3). More precisely, for B << Z4/3 the magnetic effects are absent in the leading term of the large Z asymptotics. For B _ Z4/3 the full magnetic Thomas-Fermi theory is needed. If Z4/3 << B << Z3, the usual Thomas-Fermi theory still applies, but only the first summand in the pressure function (3.8) is needed. The atom is spherical in all these cases, but the energy is

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affected by the magnetic field. These results were proven in [117] with the help of the magnetic Lieb-Thirring inequality and the semiclassical result (3.9). For a strong magnetic field, B > (const)Z3 with a positive constant, the electrons are confined to the lowest Landau band and the shape of the atom is a long

cylinder along the field. For B - Z3 the atoms in the transversal direction have a non-trivial structure that can be described by a new density functional theory relying on minimizing density matrices instead of density functions [116]. Finally, if B >> Z3, the atom becomes effectively one-dimensional and a one-dimensional

Thomas-Fermi caricature applies. Analogous results in d = 2 were obtained in [118].

If the magnetic field goes to infinity, but Z is fixed, then the ground state energy

diverges as -4Z2(logB/2)2. For one-electron atoms this has been established in [5]. The energy of the many-body system, after factoring out the divergent (log B)2 term, is given by the ground state energy of an effective one-dimensional bosonic Hamiltonian with Dirac delta interactions [11]. This idea has been extended to prove resolvent convergence and explore other effective Hamiltonians in [18] and references therein. The correctness of the Thomas-Fermi theory in the semiclassical regime (B Z3) for non-homogeneous magnetic fields was proven in [45] after extending the Lieb-Thirring inequality [44] and constructing appropriate coherent states. The uniform magnetic Lieb-Thirring inequality [47] should also allow the extension of the strong field regime from [116] to non-homogeneous magnetic fields. The asymptotic behavior of the total magnetization and of the current for large atoms in homogeneous fields was obtained in [53, 54]. A bound on the maximal ionization is proven in [143].

4.3. Multiparticle Scattering in a Magnetic Field. A detailed presentation of scattering theory and asymptotic completeness is given in the contribution of C. Gerard in this Festschrift; here we just briefly mention the most important results involving magnetic fields. N-body asymptotic completeness is well understood for non-relativistic particles without magnetic field. Scattering in the presence of a constant magnetic field is a much more delicate question since a classical charged particle moves on circles. Therefore, charged subsystems can scatter only parallel with the field, while neutral systems may move out to infinity in all directions. The general theory has been developed and asymptotic completeness has been proved by Gerard and Laba for the case when all possible subsystems are charged (the best reference is their book [64]). The zero charge case in general is still open. The special case of three particles with Coulomb forces was solved in [63] and one charged particle was considered in [1]. A constant electric field can also be included. Skibsted [153] reduced this prob-

lem to scattering of non-interacting subsystems all having the same charge/mass ratio. In the presence of a quantized electromagnetic field, the scattering of photons on a bound electron (Rayleigh scattering) and the scattering of electrons dressed with photons (Compton scattering) have been studied. The mathematical framework is non-relativistic quantum mechanics with a quantized field with ultraviolet cutoff to ensure that the Hamiltonian is well defined. For total energies below the electron-

positron pair creation threshold, this non-relativistic caricature of QED is well

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justified. One also has to cope with the infrared (IR) problem ("soft bosons"): there could be infinitely many photons with a finite total energy. For technical simplicity, scalar photons are considered. This set of problems was initiated by Frohlich who first investigated the infrared problem and constructed wave operators (without completeness) [56]. Rayleigh scattering was first tackled in [28] where the photons were massive to avoid the IR problem and the potential was confining. Later it was extended to the physically more realistic massless photons, but with an IR-cutoff, and instead of a confining potential, the total energy was set below the ionization threshold which also guarantees spatial localization of the electrons [58]. During scattering, photons are absorbed by the bound electrons, lifting them to higher excited states,

but no electron can escape. Thus, after some time, the electron cloud relaxes to the ground state and emits photons that propagate essentially freely to infinity in space. Note that the analysis of Bach et al. guarantees that all excited states are unstable [8, 9]. For Compton scattering [59], the total energy is assumed to be sufficiently small so that the speed of the massive electron is less than 1/3 of the speed of light. This technical assumption safely separates free photons from the dressed electron after long time evolution. The asymptotic completeness in this model means that

the long time evolution of the state is a linear combination of asymptotic states consisting of a freely moving electron dressed by a photon cloud plus freely moving excess photons.

5. Random Schrodinger Operators with Magnetic Fields Since the proof of the Anderson localization with the powerful multiscale method of Frohlich and Spencer [61], random Schrodinger operators have become one of the main research directions in mathematical physics. The typical problems concern the self-averaging properties (deterministic spectrum, existence of density of states), the asymptotics of the density of states (Lifshitz tail), and establishing the dense point spectrum in the localization regime. These questions have been recently studied with magnetic field as well; most results are in the most relevant d = 2 dimensions.

5.1. Constant Magnetic Field. First we consider the problems where the magnetic field is constant and only the external potential is random, i.e., the random Landau Hamiltonian (for a recent survey, see [103] and references therein). In

this case, the standard self-averaging techniques work to establish deterministic spectrum, the translation must simply be replaced by the magnetic translation. The Lifshitz tail in the low energy regime for Gaussian random potential is very universal and it is insensitive to the magnetic field (it even holds for certain random magnetic

fields). For Gaussian randomness even the density of states is bounded and the Wegner estimate holds [87]. Localization with algebraically decaying eigenfunctions was proved in [51], exponential localization in [163]. These proofs are valid only at very low energies, using the deep wells of the Gaussian randomness; in particular, this regime is far from the analogue of the band-edge localization.

For repulsive Poissonian obstacles, the precise Lifshitz tail is more delicate, similar to the classical vs. quantum dichotomy in the non-magnetic setup. If the single-site potential has a slow decay, then classical effects dominate and the Lifshitz tail can be computed from a simple mean-field argument. Otherwise the quantum

420

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localization energy competes with the entropy of the large domains free of impurities. For magnetic fields, the threshold decay is not algebraic but Gaussian. The classical regime was investigated in [16] (in three dimensions [85]), the main result in the quantum regime was obtained in [40] using Sznitman's coarse-graining probabilistic method (see also [83, 41]). Since in the Poisson model the energy is not monotone in the random variable so Wegner estimates are much harder to obtain, localization for a constant magnetic field was investigated for Anderson-type i.i.d. random potentials. Wang has given a full asymptotic expansion of the density of states away from the Landau bands [165]. More explicit quantitative results are known on the Lifshitz tails, including the double logarithmic asymptotics at each band edge [96]. The pure point spectrum at a certain distance away from the Landau levels was proven independently by Wang [164], Combes and Hislop [24] and in a single-band approximation by Dorlas, Macris and Pule [29]. Since d = 2 is the borderline dimension for the necessary large distance decay of the Green's function of the non-magnetic Lapla-

cian, an additional decay must be extracted from the presence of the magnetic field. This eventually required basic results from two-dimensional bond percolation theory. Precise estimates on the localization length and dynamical localization near the Landau band edges was obtained by Germinet and Klein [67] using their extension of the Frohlich-Spencer multiscale analysis [66]. In contrast to the localization regime, the presumed regime of delocalization in Anderson-type models is poorly understood and there are almost no mathematical results. Apart from the Bethe lattice, there is only one model, where the existence of the mobility edge has been rigorously proven: the random Landau Hamiltonian [68]. More precisely, it is shown that there exists at least one energy E near each

Landau band, so that the the local transport exponent, O(E), is positive. The local transport exponent measures the extension of the wave packet in a suitable averaged sense for large times. (Dynamical) localization is characterized by O(E) = 0. Moreover, there is an important dichotomy for /3(E): it is either zero or at least 1/2d.

The key quantity in the proof in [68] is the Hall conductance. In the regime of dynamical localization, the Hall conductance is constant in the mobility gap (see [12], strengthened later in [31]). On the other hand, the Hall conductivity jumps by one at each Landau level for the free Landau Hamiltonian and it is also constant, as a function of the disorder parameter, in the gaps. Therefore, it must jump somewhere inside the bands, at least for sufficiently small disorder, but then the complete band cannot belong to the dynamically localized regime, completing the argument of delocalization in [68]. The Hall conductance for quantum Hall systems (two-dimensional disordered samples subject to a constant perpendicular magnetic field) at energies falling into

the mobility gap, A, can actually be defined in two different ways. The bulk conductance is defined on R2 by the Kubo-Streda formula (see [6]) QB (A) = -itrPP [ [PA, A,], [Pa, A2] ]

where PA is the spectral projection onto (-oo, A] and Ai is the characteristic function of {xi < 0}, i = 1, 2. The edge conductance is defined in a half-plane sample (x2 > -a, eventually

a -ioo)by UE = -itrO (H) [H, Al]

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(modulo non-trivial technicalities), where p is a smooth spectral cutoff function that is one below A and zero above A. This quantity gives the derivative of the current

flowing through the line x1 = 0 with respect to lowering the chemical potential along the edge. These two conductances are the same. For intervals A falling into the spectral gap, this was proven in [142] with K-theoretical methods and later in [30] by basic functional analysis. The proof given recently in [31] is also valid for intervals that contain strongly localized spectrum only (such intervals are often called mobility gaps).

5.2. Random Magnetic Field. Since a magnetic field itself enhances localization, one expects that a random magnetic field is localizing even stronger than a random potential. Technically, however, random magnetic fields are harder to fit into the multiscale analysis, mainly because the vector potential is non-local. There is a substantial difference between the zero and non-zero flux cases, the former being easier. In particular, stationary random vector potentials always generate magnetic fields that have zero flux on average.

The existence of the integrated density of states (IDS) and its independence of the boundary conditions in the thermodynamic limit (uniqueness) has first been shown by Nakamura for both the discrete and continuous Schrodinger operator with a random magnetic field with non-zero flux [133, 134]. In both cases, Lifshitz tails were also obtained. Recently, Hundertmark and Simon gave a short proof for the existence and uniqueness of the IDS [86]. Anderson localization has been shown for a Gaussian vector potential by Ueki in [163]. The Germinet-Klein multiscale analysis has been extended to include very general random magnetic fields, but the Wegner estimate requires a random vector potential (which implies zero average flux) in addition to other technical conditions. For the discrete random magnetic Schrodinger operator, the method of Nakamura [133] has been extended to obtain the Wegner estimate and localization [95]. However, the zero flux condition is enforced in a strong sense: neighboring cells are paired and the magnetic flux is opposite in these pairs. A deterministic background magnetic field atop of the local random vector potential is allowed in [81] where the Wegner estimate was proved in the continuous model. A small stationary random vector potential was included in a Schrodinger operator with a periodic background potential in [69], where Lifshitz tails were proven under a special non-resonance condition. Band-edge localization for Schrodinger operators with random magnetic field is widely believed to hold in the most general case. The additional assumptions

on the zero flux (which, in one form or another, is present in all papers so far) seems to be only technical. However, there is no agreement in the physics literature about the possible existence of the continuous spectrum for such operators, unlike in the non-magnetic case, where the existence of the extended states is universally accepted by physicists and "only" the mathematical proof is missing. The

Landau orbits and their quantum counterparts, the strongly localized magnetic eigenfunctions, are characteristic only to the constant magnetic field and they are not universal. There is a competition between a possible weaker form of the Landau localization, that may still hold for random fields, and the resonance effects that enhance delocalization.

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Acknowledgements. The author is indebted to Jan Philip Solovej for his help in preparing the manuscript. Special thanks to Christian Gerard, Gian Michele Graf, Dirk Hundertmark, Frederic Klopp, Michael Loss, Benjamin Schlein, Stefan Teufel,

Jakob Yngvason, Simone Warzel and to the referee for their critical reading and corrections. Part of this work was done at Harvard University. References

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MATHEMATISCHES INSTITUT, LMU, THERESIENSTRASSE 39, D-80333 MUNICH, GERMANY E-mail address: [email protected]

Proceedings of Symposia in Pure Mathematics Volume 76.1, 2007

Aspects of the Integer Quantum Hall Effect Gian Michele Graf To Barry, with admiration ABSTRACT. We review some of the concepts which shaped the understanding

of the integer quantum Hall effect, as well as the mathematical results they led to. Also described are the underlying physical motivations. Emphasis is placed on the equivalence of different approaches.

CONTENTS 1.

2. 3.

4.

Introduction Results Motivations Some Snapshots from the Proofs

References

1. Introduction The integer quantum Hall effect (IQHE) has been at the crossroads of several developments in mathematical physics, such as non-commutative geometry, index theory, localization, and the adiabatic theorem. Some of the most illuminating contributions to these topics are coauthored by Barry Simon. The fractional quantum Hall effect, which we shall not review here, further ties to conformal and topological field theory [21] and to the classification of certain integral lattices [19]. The IQHE appears in some two-dimensional samples at temperatures close to zero and in a strong, transverse magnetic field. The current density j and the stationary electric field E, both lying in the plane of the sample, are empirically related by the Hall-Ohm law

j=crE,

(1)

2000 Mathematics Subject Classification. 81V70 (primary); 82D30 (secondary). Key words and phrases. quantum Hall effect, index, conductance, localization. ©2007 American Mathematical Society 429

G. M. GRAF

430

which introduces the conductivity tensor a. The phenomenon is that, under appropriate conditions, transport is dissipationless, j E = 0, which means that the tensor is antisymmetric, or = I

0 CH

O)

(2)

Even more remarkably, the Hall conductance OH is quantized in multiples of e2/h, where e is the electron charge and h is Planck's constant (i.e., of 1/27r if natural

units are used); and the quantization of o H is accurate to within 10-8 when the magnetic field or the density of electrons are varied over a sizeable range, a fact called a plateau. Its width may be comparable to the separation between plateaus. Three basically distinct explanations have been proposed for the IQHE. Common to all, at least in their original formulation, is that they treat the electrons as independent particles, except for the Pauli principle. (i) The Hall conductance may be identified with the charged pumped across a closed ribbon when the magnetic flux threading it is increased by one flux quantum [26]. The Hamiltonians before and after that change are unitarily conjugate and the state has evolved adiabatically in between because the transport is dissipationless. The occupation of states follows the spectral flow as a function of the flux and that charge is the number of occupied eigenvalues having crossed the Fermi energy in the process. Clearly, that number is an integer. (ii) Linear response theory computes the current j induced by a weak electric field E in the bulk of the sample, which yields the Kubo formula for the Hall conductance. That expression can be related to a Chern number [33], which is an integer.

(iii) The Hall current is ascribed to states flowing at the edge of the sample. In special cases it may be identified with the number of edge channels [23, 10], which is an integer. The Hall conductance as defined on the basis of (i) has been linked [26, 23] to the current flowing along the ribbon when an electric potential is applied across it. That current may be interpreted as flowing either in the bulk [26] or at the edges [23] of the ribbon. In real experiments it is a combination of both possibilities [23]. Following their original formulation, approaches (i)-(iii) have not only gained in mathematical precision, but the physical concepts involved have been clarified as well. For instance, the argument [26] for (i) depended on the fact that eigenvalues moving down (resp. up) under the spectral flow are associated with the inner (resp. outer) edge. If the Hamiltonian is symmetric against rotations along the ribbon, as it was first assumed, then the eigenvalues are monotone in the flux throughout its variation, permitting the conclusion on charge transport stated above. If it is not, as for example implied by the presence of weak disorder, eigenvalue crossings formerly protected by symmetry may now be avoided, destroying monotonicity. This issue, which reflects the possibility of eigenstates tunnelling between edges, admittedly

remained to be investigated in [26]. It can be avoided altogether if the outer edge is pushed to infinity [4], which turns the ribbon into a punctured plane. In that geometry, the Hall conductance can be identified with the relative index [4] of a pair of projections. As another example, the argument [33] for (ii) assumes a periodic Hamiltonian and hence rational magnetic flux per unit cell and no disorder. This is however not needed, since the unit cell may be replaced by a torus of boundary

ASPECTS OF THE INTEGER QUANTUM HALL EFFECT

431

conditions [29] or fluxes [5] (in the latter case the approach extends to interacting particles). Another generalization [6] of the Chern character, more appropriate to the thermodynamic limit, is by means of non-commutative geometry. Disorder is crucial for the formation of plateaus. There, the Fermi energy varies within an interval where bulk states are localized, a so-called mobility gap. Such a variation changes the occupation of states (and hence the electron density), but not that of those participating in transport. By contrast, a variation over the extended states spectrum, or over a spectral gap, changes both (resp. neither). Disorder thus affects the analysis in two main ways: First, it destroys a symmetry, like the above mentioned rotations or periodicity. This applies even if the simplifying assumption is made that (a) the Fermi energy falls in a spectral gap. Second (b), the Fermi energy actually falls in a mobility gap. It it therefore of utmost importance that definitions of Hall conductance and their equivalence be compatible with disorder. For (i, ii) this was achieved in [6, 4]; for (iii, a) in [31, 24] and for (iii, b) in [16].

2. Results We shall momentarily present three definitions of Hall conductance related to the above pictures (i)-(iii). A mathematical setting, in which they are conveniently placed, is that of discrete Schrodinger operators [14]. The bulk is represented by the lattice Z2 E) x = (X1,X2) with Hamiltonian HB = HB on £2(Z2). Its matrix elements HB(x, x'), (x, x' E Z2), are of short-range in the sense that

IHB(XIx')I (eµl"I - 1) =: C1 < oo

sup

(3)

xEZ2 x'EZ2

for some It > 0, where xj = Ix11 + Ix21. A bounded, open interval A C IR, which shall contain the Fermi energy, is assumed to lie (a) in a spectral gap or, more generally, (b) in a mobility gap: (a)

Ano,(HB)=o;

(4)

(b) For some v > 0,

sup E jg(HB)(x, x')I(1 +

xj)-veWlx-x'1

< oc ,

gEBi(A) x x'EZ2

(5)

where B1(0) denotes the set of Borel measurable functions g which are constant in {AIA < Al and in JA IA > Al with jg(A)l < 1 for all A E R. In particular, the spectrum is pure-point in A [25]. Denoting by EM the characteristic function of M C IR, the assumption is completed by dimE{a}(HB) < oo

,

(A E A)

(6)

,

i.e., no eigenvalue in A is infinitely degenerate. Condition (5) is basically a statement about dynamical localization. It has been established in [1] and more explicitely in [30], where the above property is related to the SULE property, as well as in [2], where g is allowed to be constant, rather than zero, outside of A. The condition holds true almost surely for ergodic Schrodinger operators whose Green's function G(x, x'; z) = (HB -z)-1(x, x') satisfies a moment condition [3] of the form

limsup E(IG(x,x';E+irl)I')

EEO,,q-0


G. M. GRAF

432

for some 0 < s < 1, but can also be proved [22] using multi-scale analysis [20]. Condition (6) appears to be essential for a plateau, in view of the fact that for the Landau Hamiltonian (though defined on the continuum rather than on the lattice) QH jumps as the Fermi energy crosses an infinitely degenerate Landau level. The condition, in fact, simple spectrum, follows almost surely from the arguments in [32]. We stress however that the hypotheses (a) and (b) themselves are deterministic. Translation covariance or ergodicity of HB are not assumed here. The three definitions of Hall conductance shall now be associated with a Flux, the Bulk and the Edge. The physical motivations relating them to approaches (i)-(iii) mentioned before will be given in the next section. (i) The definition is based on the index of a pair of projections and depends on some unitaries associated with gauge transformations. Let P and Q be two orthogonal projections on a Hilbert space, so that P - Q is compact. Then

Ind(P, Q) = dim{o I Po = b, Q' = 0} - dim{ '% 1 Pz,b = 0, QO = '}

.

(7)

Let U(x), (x e Z2), satisfy IU(x)l = 1 and JU(x) - U(y)I < Cl Il

-x

,

(Ix - YI <-

(8)

for some C1, C2. Along a large loop encircling the origin counterclockwise,

the phases of U(y)/U(x), which are small for single bonds (x, y), add up to a multiple, N(U) E Z, of 27r. We assume that the winding number N(U) equals 1. Then aF(A)

where PA =

21Ind(PA, UPAU*)

(9)

,

(HB).

(ii) The definition makes use of switch functions: Let A(n), (n E Z), be a function which equals 0 for large negative n (resp. 1) for large positive n. Then 5B (A) = i tr PA [[PA, Al], [PA, A2]]

(10)

,

where Al and A2 are switch functions of x1 (resp. X2). (iii) The sample with an edge is modeled as a half-plane Z x 7Gq,, where Za _

{n E Z I n < a}, with the height a of the edge eventually tending to oo. The Hamiltonian Ha, = Ha on £2(Z X Za) is obtained by restriction of HB under some largely arbitrary local boundary condition, described as follows. Denoting by Ja : £2(Z X Za) --f £2(Z2) the extension by 0, we assume that

E

IEa(x,x')JeN'(Jx2+aI+Ix1-xiI)

< 00 .

xEZ aEZ2 x'EZxZ

We then set

vE = i lim lim trp'(Ha)[Ha,Al]A,?,a(A2) , ,7-0 a-+oc

(12)

ASPECTS OF THE INTEGER QUANTUM HALL EFFECT

433

where p E C°° (I[8) satisfies

P(A) =

1

(A
0,

(A > A)

(13)

and

A" .(X) = r f0

e-77te'x°tXe- Hatdt

is the average over a time - 17-1 of a bounded operator X with respect to the Heisenberg evolution generated by Ha. The main results are as follows: First of all, these quantities are well-defined for A E A. Second, they are independent of various auxiliary objects, such as U, A1, A2 and Ea. In particular, they do not change when U, A1, or A2 are replaced by some translates. For concreteness only, the reader may think of Up(x) = U(x - p) with U(x) = IX

(14)

X1

and p = (pl,p2) C Z2* = Z2 +

as well as Ai,P = O(xi - pi); for Dirichlet boundary conditions, Ho, = Ja HB Ja, the locality condition (11) holds true by (3). Third, the conductances are independent of A C 0 (resp. p) with (13), which is the manifestation of a plateau. Fourth, QF =QB =QE Finally, 27rO'F is manifestly an integer.

Some of these definitions were first formulated under slightly different assumptions. For instance, under assumption (a), the definition (12) of the edge Hall conductance can be replaced by the simpler QE = i tr p'(Ha) [Ha, Ai]

(15)

for any a c Z. In an ergodic setting, Al and A2 in (10) may be replaced by xl (resp. X2) and the trace by the trace per unit area, UB = i try Pa [[PA, xl], [Pa, x2]]

,

(16)

which takes an almost sure value [7]. The same replacement can be made in (15), with the trace now becoming trace per unit length [31]. It is now appropriate to review the genesis of these results. The definition (16) was proposed in [6] and identified as a 2-cocycle. It was shown to be an integer by relating it to a Fredholm index, through a formula of Connes [13]. In [4] that index was formulated as the index (7) of a pair of projections and endowed with independent physical motivation. It was proved to be equal to the expectation of (10), again in the ergodic setting and using Connes' formula, but without explicit reference to non-commutative geometry. Translation invariance of (9) was shown and used in [6, 4]. It was noted in [16] that the same is true for (10), and not just for its expectation, which allowed to turn QF = QB into a deterministic statement. In [31], which is placed in the ergodic setting and under assumption (a), QE was displayed as a winding number, which provided an independent reason for its quantization. It was also shown to be a 1-cocycle, from which the conclusion vE = vB was drawn using K-theory. The equality OE = QF, again for (a) but in a deterministic setting, was shown in [15, 27]. In the more general case (b), QE = trB

G. M. GRAF

434

is due to [16], with a special case due to [12]. In all this, the pioneering role of non-commutative geometry [6, 31] is manifest.

3. Motivations We provide the physical pictures underlying the definitions (i)-(iii) of Hall conductances. The discussion is largely heuristic. (i) In one of its guises the Laughlin argument relates the Hall conductance with the charge pulled from infinity when a flux quantum is slowly added near the origin. By the continuity equation, the charge Q inside a loop C changes at the rate dQ dt

_

J - v s,

where v is the outward normal. By Faraday's law, a change in the flux 4) is accompanied by an electric field,

i E Tds=where T is the tangent vector. If the change is slow, the field is nearly stationary, so that (1, 2) apply, resulting in AQ = aHO4). This is to be compared with the quantum mechanical computation of OQ. The state of the system is U(t, 0)PAU(t, 0)*,

where U(t, 0) is the evolution generated by the Hamiltonian, now depending on time through the flux. By the time, t = to, the flux has increased by 04) = 2ir the Hamiltonian becomes unitarily conjugate to H through U(x) with winding number 1. Its Fermi projection is UPaU*. In the limit of a large loop and of a slow process, the change OQ equals the excess number of electrons in the evolved many-body state, as compared with the ground state for the same flux: OQ = Ind(U(to, O)PAU(to, 0)*, UPAU*) = Ind(PA, UPAU*)

The second equation follows because of the additivity (Ind(P, R) = Ind(P, Q) + Ind(Q, R)) and the continuity (11P - QJJ < 1 Ind(P, Q) = 0) of the index, implying

Ind(U(t, 0)PaU(t, 0)*, PA) = 0 (ii) The Kubo formula for conductance is derived by adiabatically switching .

an electric field and considering the linear response of the system. The function -A2 can be seen as an electric potential of unit drop for a field pointing in the positive x2-direction, while the operator i[H, A1] stands for the total current in the x1-direction. The expectation of the latter in the state perturbed by the former yields o12i i.e., aH in eq. (2). The time-dependent Hamiltonian is

H(t) = H - A2f(t)

,

(t<0),

where f (t) slowly interpolates between 0 and 1, e.g., f (t) = e7t for some small 77 > 0. As in (i) the unperturbed density matrix is PA. The perturbed density matrix p(t) satisfies the initial value problem

dtp(t) = -i[H(t), p(t)]

,

t

li

e,Htp(t)e-'Ht = Pa

To first order, in the electric field the solution is

P(0) - Pa = i

f0 1-00

dt e'lte'Ht [A2 PA]e-'Ht

ASPECTS OF THE INTEGER QUANTUM HALL EFFECT

435

and we obtain, after an integration by parts, aB = lim ni tr[H, AI] (p(0)

PA) = li o irl tr

/0

J-00

dt e0t (e-,xtAie'xt - A1) [A2, PA]

(17)

Since PA is a projection, we have [A2, PA] = PA[A2, PA](1-PA)+(1-PA)[A2, PAPA.

The substitution of this into (17) amounts, by cyclicity, to the substitution of AI there by the expression (1 - PA)A1PA + PAAI (1 - PA) _ [[A,, PA], Pa]

The l.h.s. contributes two terms to (17) containing ef;Ht similar to one another, one of which is /0

irltr J

dte'tte-Wt(1

-00

- PA)AlPAe'Ht[A2, PA]

As we now sketch, it vanishes for rl -4 0. Representing the propagators as e-'xt = f e-'t`tdPN,, we are led to irl J

dt

ente-'(tt+-`)t

-00

=-

77

µ + irl

(18)

with µ+ > A, p_ < A. Since this quantity vanishes pointwise as rl -4 0 for (p+, p_) in the stated region, one is tempted to conclude that QB = -i tr [[AI, Pa], Pa] [A2, PA] = i tr PA [[AI, Pa], [A2, Pa]]

which is (10). The passage from (17) to (10), or rather its analogue in the ergodic setting, cf. (16), has been put on a firm basis; see [7], but also [2, 9]. In the present setting the result can be obtained using methods of [17]. In both cases it is crucial that A lies in a mobility gap, which allows one to control the small denominator in (18).

(iii) For a simpler start let us first discuss the definition (15) for QE valid in the case of a spectral gap. We interpret p(Ha,) as the 1-particle density matrix of a stationary quantum state. Though some current is flowing near the edge, we should discard it, as it is supposed to be canceled by current flowing at an opposite edge

located at x2 = -oc. The idea is that the two opposite edges of a macroscopic sample are infinitely separated from a microscopic perspective, and we focus on one of them. The drop in electrical potential used in (ii) is now given a different physical realization: The Fermi energy is lowered by 6 at the first edge, but not at

the second. Then a net current I = itr((p(Ha + 6) - p(Ha))[Ha,AI]) is flowing, resulting in QE = lira

I

i tr p'(Ha) [Ha, All

The current operator i[Ha, Al] is relevant only on states along a strip near xI = 0, and p'(Ha) only near the edge x2 = a, because p'(HB) = 0 due to (4). The intersection of the two strips is compact, which is basically why the trace exists. In the presence of a mobility gap, however, this property of p'(Ha) fails. In search of a proper definition of QE for this case, we consider only the current flowing across the line xI = 0 within a finite window 0 5 x2 < a next to the edge. This

G. M. GRAF

436

amounts to modifying the current operator to be i[Ha, A1]A2 (or a symmetrized version thereof), with which one may tentatively use lim itrp'(Ha)[Ha,A1]A2 a-oo

(19)

as a definition for QE. Though this limit exists, it is not the physically correct choice. States in the range of p'(Ha) supported far away from the edge are close to bound states of the bulk Hamiltonian, HB'OA = Ana, or linear combinations thereof. Such states may carry persistent currents (whence the operator in (15) is no longer trace class), but no current across the line xl = 0, since (OA, [HB, A1] 5A) = 0.

This cancellation is the rationale for ignoring the part x2 < 0 of the line x1 = 0 by means of the cutoff A2 in (19). However, the cancellation is not achieved on states located near the end point x = (0, 0). The contribution missed by (19) is (0A, i[HB, A1] (1 - A2)6) = -(z/'A, i[HB, A1]A2ba) from each bound state. By weighting them with p'(A), we amend the definition (19) of the edge conductance:

QE= lim itrP'(Ha)[Ha,A1]A2-i E p'(A)trE{a}[HB,A1]A2E{a} a-.oo

(20)

.

AEE&

The sum, which is over the eigenvalues in A, happens to be absolutely convergent,

but there is no general reason for it to vanish. In fact, it can be shown to be non-zero for the Harper Hamiltonian with a Cauchy distributed random potential. Alternatively, one may use in (19) and instead of A2 a cutoff which commutes with the dynamics generated by Ha, at least in some limit. Its use will not create spurious contributions which call for compensation. Such a possibility is realized by the time average of A2 and leads to definition (12). Unlike (20), it is stated purely in terms of the edge Hamiltonian Ha. Nevertheless, the two definitions agree [16]. We also remark that the time averages in eqs. (12) and (17), though of different physical origin, are mathematically related, which is instrumental to the proof of QE =QB.

4. Some Snapshots from the Proofs 4.1. The Equality QF = 0B in Case (b). In the case of finite-dimensional projections P and Q, the index can be computed as Ind(P, Q) = tr(P - Q). The generalization [4] to the infinite-dimensional case is

Ind(P, Q) = tr(P - Q)2-+1 if P - Q E .72,+1 for some odd integer 2n + 1, where JP, (1 < p < oo), are the trace ideals. In particular, tr(P - Q)3 = tr(P - Q) if P - Q E J1, which can be seen from the identity

(P - Q) - (P - Q)3 = [PQ, QP] = [PQ, [Q, P - Q]]

(21)

In the application to the QHE, where P = PA, Q = UP,\ U*, the difference luckily is not trace class, since the contrary would imply tr(PA - UPAU*) = 0 by evaluating the trace in the position basis. However, the third power of this difference is trace class, which yields QF =

tr(PA - UPAU*)3 = 2,7r

P, (x, y)PA(y, z)PA(z, x)S(x, y, z) x,y,zEZ2

S(x y, z) = - 2'(1- U(y)) (1 - U(z)) (1 - U(-))

,

(22)

ASPECTS OF THE INTEGER QUANTUM HALL EFFECT

437

To show its equality with

vB=i

Pa(x,y)P (y,z)Pa(z,x)[(Ai(y)

Ai(x))(A2(z)-A2(y))

(1 H 2)]

,

x,y,zEZ2

(23)

the authors of [4] assumed that the projection is ergodic (or covariant) with respect to magnetic translations. Translation invariance of QF then implies that (22) is a translation invariant function of the randomness and almost surely equal to its expectation. After taking the expectation of both equations (22), (23), the expressions IE(PA(x, y)PP11(y, z)PA(z, x)) are constant under a common shift a of the

summation variables x, y, z. By trading one them against a, the sums over the latter involve only U (resp. Ai). Moreover, for U = xl lxl, they are related through a formula of Connes, to be discussed below. The result is UF = IE(UB). A slightly different use of translation invariance, which does not depend on ergodicity, was made in [16]. Let U = Up and Ai = Bi,p as in (14). Since in fact both OF and aB are independent of p, averaging of (22), (23) over AL = {p E Z2* pJ < L} results I

in OT = 7rL 2

>

Px(x, y)Pa(y, z)Px(z, x)S(p, x, y, z)

,

(24)

,

PEAL

x,y,zEZ2

S(p, x, y, z) = sin L(x, p, y) + sin L(y, p, z) + sin L(z, p, x)

(25)

,

respectively in QB =

L2

/

.

P (x 71)P1(91 z)P" (z x). ,

,

,

PEAL

x,y,zEZ2

.[(B(yl -PI) - 9(x1 - pi))(0(z2 - P2) - O(y2 -P2)) - (1 H 2)]

.

(26)

These two expressions, which do not depend on L by derivation, will be shown to be equal in the limit L -p oc. Because the decay (5) applies to PA, the summation ranges x E Z2, P E AL can then be replaced by x E AL, P E Z2*. At this point, Connes' formula [13] S(p, x, y, z) = 2 Area(x, y, z) 11

(27)

pEZ2.

may be used in (24), where Area(x, y, z) is the triangle's oriented area, namely a (x - y) A (y - z). On the other hand, the corresponding sum in (26) also yields 2 Area(x, y, z), since E (O(yi

- pi) - 0(xi - pi)) = xi - yi

p,EZ*

The proof of QF = aB is completed by P,\L (y, z) = 6yz - PA(y, z).

4.2. Connes' Formula. The role of the sine function in (27), (25) is less special than one might think, as noted by [11]: For a fixed triplet u(I), u(2), u(3) E Z2, let cxi(p) = L(u(i+I), p, u(i+2)) E (-7r, 7r) be the angle of view from p c 7G2* of u(i+2) relative to 0+I) (with cxi(p) = 0 if p lies between them). Let g(a) be a bounded function satisfying g(-a) = g(a) and

g(a) = a + 0(03)

(28)

G. M. GRAF

438

near a = 0. Then, g(ai(p)) = 27rArea(u(') u(2) u(3))

.

(29)

PE7Z2* i=1

The proof is as follows. We may assume the triangle to be positively oriented. The statement (29) is true for g(a) = a. Indeed, for each p E 7G2*, 3

i=1

inside

1

ai(p) = 27r

1/2

on the boundary of outside

for p

0

the triangle.

(30)

Thus, for g(a) = a the l.h.s. of (29) is 27rx the number of dual lattice sites within the triangle (counting a boundary site with weight 1/2). This number equals the triangle's area. The above observation reduces (29) to the statement that for f (a) = g(a) - a 3

E f(ai(p)) = 0 . PEZ2 i=1

(31)

A significant difference between f and g is that the individual terms f (ai(p)) are summable in p E Z2, since by (28) f (ai(p)) = O(IpJ-3) for JpJ -i oo. However, each of the three individual sums changes sign under the reflection with respect to the midpoint of the corresponding edge, (u(z+1) + u(i+2))/2 E (Z/2)2 (which is a symmetry of the lattice Z2). Thus, even the individual sums (at given i) vanish.

4.3. The Equality QF = QE in Case (a). In this setting QE is given by (15), without the need to pass to the limit a -f oo. We may thus take Ho,=o on $2(Z x Z0) as the edge Hamiltonian. Unlike for HB, the interval A is not a spectral gap for HO. Its spectrum in A may actually be absolutely continuous [28, 8, 18], which is a manifestation of states extending along the edge. As a result, the matrix elements of a spectral projection (Ho), (A E A), will no longer decay rapidly away from the diagonal, which is the property that in the bulk case ensured

P -UPU* E .73

(32)

for P = PA = E(_,,,A) (HB) and U as in (8). By contrast, we also have PA = p(HB) with p as in (13) due to assumption (a), and the property (32) extends to P = p(Ho). The price to pay is that P is no longer a projection, but that does not seem to be a crucial aspect of the Laughlin argument. Morally, we may identify OF with (27r)-1 tr(P - UPU*), while eq. (21) is cautioning us that the correct computation of the trace is, in the case of projections, by discarding the trace of the commutator on the r.h.s. which, though not defined, is formally zero. That identity reads [(1- P)(1- Q), (1 - Q)(1 - P)] (P - Q) - (P - Q)3 = 2 [PQ, QP] 2 + (1 - 2P)(P - P2) - (1 - 2Q)(Q - Q2)

+2{P-Q,P-P2+Q-Q2} if it is not restricted to projections, where

denotes the anticommutator. It

suggests considering the expression

K(U) := tr(2 {P - Q, (P - P2) + (Q - Q2)} + (P - Q)3)

(33)

ASPECTS OF THE INTEGER QUANTUM HALL EFFECT

439

as a replacement for tr(P - Q) when P and Q = UPU* are unitarily conjugated. More precisely, we consider unitaries U which are multiplication operators with respect to some fixed basis, like the position basis of P2(Z x Zo), and operators P P* such that (P - Q)3, (P - Q) (P - P2), p(P) - p(Q) E 3 for p(A) = A - A2 and p(A) = (1 - 2A) (A - A2). We remark that these properties are satisfied for P = p(Ho) and for U as in (8), if U = 1 on all but a finite piece of the edge. For

instance, (P - Q)(P - P2) is then associated with such a piece, because p - P2 vanishes away from the edge. Hence it is trace class. The important property is that K(U) is unaffected by changes of U which are trace class. This is used as follows: Let U - 1 be supported in a cone whose rays point into the lower half-plane Z x Z0, and let its curl, i.e., the magnetic flux, be concentrated near the vertex. Moving the vertex without changing the fan of the cone is an example of such a change of U. If the vertex too is placed well inside the lower half-plane, the cone does not intersect the edge and the first term in (33) is negligible. In this limit K(U) reduces to 27rOF. If, on the other hand, the vertex is pulled across the edge and well into the upper half-plane, then the second term in (33) is associated with the intersection of the cone with the lower half-plane, which in the limit is a negligible (though infinite) tail. Moreover, and still inside the lower half-plane, U may be represented as an exponential, which near the edge is of the form e2'riA' W. The remaining first term in (33) can then be computed as

3 tr(P - Q) ((P - P2) + (Q - Q2))

6 tr(P - Q) (p - P2) f27r lzz 6

J

dcp- tr(P - ei`°A1 Pe i"oA1) (P - P2)

= 6 27ri tr[P(Ho), A1] (p(Ho)

- P(Ho)2)

= 6 2iri tr[Ho, Al]P'(Ho) (P(Ho) - P(Ho)2) = 27ritr[Ho, AI]p'(Ho) = 27raE , where p = 3p2 - 2p3 satisfies the same assumption (13) as p does.

4.4. The Equality QE = 0B in Case (b). We sketch some of the steps towards this identity when OE is defined by (20). If the other definition (12) is chosen, the argument runs along similar lines. The statement may be rephrased as

lim itrp'(Ha)[Ha,A1]A2=UB+i > p'(A)trE{A}[HB,AI]A2E{a}

a-.oo

(34)

AEE&

The operator on the l.h.s. is geometrically associated with the finite but growing portion 0 < x2 < a of the the line x1 = 0. It therefore has no chance to converge in trace class norm as a oo. To see that its trace nevertheless does, we look for an operator Z(a) E J with tr Z(a) = 0, and replace the operator with iP (Ha)[Ha,A1]A2 - Z(a)

,

(35)

hoping that convergence in that norm now holds true. A first attempt is Z(a) = i[p(Ha), AI]A2i which satisfies the two requirements; in particular, its trace is seen to vanish by computing it in the position basis. A partial cancellation between the

G. M. GRAF

440

two terms is made manifest using the Helffer-Sjostrand representations

P(Ha) =

2' f d2zazp(z)Ra(Z) P (Ha) = i fd2Z3p(Z)Ra(Z)2, where Ra(z) = (Ha -

z)-1

,

and p(z) on the r.h.s. is a quasi-analytic extension of

p(x). It yields in fact [P(Ha),A1]A2 = -27r f d2zazp(z)Ra(z)[Ha,A,]Ra(z)A2

,

(36)

P (Ha)[Ha,A1]A2 = -27r fd2zap(z)Ra(z)2[Ha,Ai]A2.

The two expressions would look even more similar if, in the second line, one power of the resolvent could be moved to the end of the expression. This however is just a commutator which may be absorbed into a redefinition Z(a). Then (35) reads

- i f d2za2P(z)Ra(z)[Ha,A1](A2Ra(z)-Ra(z)A2) _ --L f d2zazp(z)Ra(z)[Ha,A1]Ra(z)[Ha,A2]Ra(Z) 27r

27r

(37)

This expression is geometrically associated with the intersection of the lines x1 = 0 and x2 = 0, which is independent of a. It is therefore reasonable that it has a limit as a ---> oo, which is indeed obtained by replacing the subscript a with B. It remains to show that the trace of the bulk quantity T so obtained equals the r.h.s. of (34). To this end we use (37), (36) in reverse, but now with a - B, and obtain T = -i[p(HB), A1]A2

- 27r1 fd 2 zazP(z)R(z)[Hs,A1]A2R(z)

.

Unlike for a < oc, the two terms are not separately trace class. We next compare the expression with (17): While the first term, e-;xtAleiH1 is necessary to ensure that the whole expression is trace class, it is formally only the second one which contributes to the trace, as explained there: QB = -i tr Al [A2i Pa], or O'B = i tr[A1 i PA]A2. These expressions are not well defined, but the following is a correct representation for QB:

aB(Ao) = -itrE_[Paa,A1]A2E_ -itrE+[Paa,A1]A2E+ - E i tr E{a} [Pao, A1]A2E{a} ,

(38)

AEE&

where E_, E+ are the spectral projections for HB onto {A A < Al (resp. {A A > 0}). Since QB(Ao) is independent of A0 E 0, we may replace PAO by p(HB) in (38). We then frame T similarly with Ej_, E{A}, without changing its trace. The first term is then just 0B; the contributions with E± from the second vanish because E+R(z) and R(z)E+ are analytic on the support of p(z) or of p(z) - 1. The remaining contribution is I

-1 27r

f d2za2P(z)(A - z)-2 trE{A}[HB, A1]A2E{A} , AEE&

which equals the last term in (34).

ASPECTS OF THE INTEGER QUANTUM HALL EFFECT

441

Acknowledgements. This contribution would not have been possible without all I learned from collaborations with M. Aizenman, Y. Avron, P. Elbau, A. Elgart, J. Frohlich, and J. Schenker, to all of whom I am indebted. I thank A. Elgart for a critical reading of the manuscript.

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[30] R. del Rio, S. Jitomirskaya, Y. Last, and B. Simon. Operators with singular continuous spectrum. IV. Hausdorff dimensions, rank one perturbations, and localization. J. Anal. Math., 69:153-200, 1996.

[31] H. Schulz-Baldes, J. Kellendonk, and T. Richter. Simultaneous quantization of edge and bulk Hall conductivity. J. Phys. A, 33(2):L27-L32, 2000. [32] B. Simon. Cyclic vectors in the Anderson model. Rev. Math. Phys., 6:1183-1185, 1994. [33] D. J. Thouless, M. Kohomoto, M. P. Nightingale, and M. den Nijs. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett., 49:405-408, 1982.

THEORETISCHE PHYSIK, ETH-HONGGERBERG, CH-8093 ZURICH, SWITZERLAND

E-mail address: [email protected]

Proceedings of Symposia in Pure Mathematics Volume 76.1, 2007

Barry Simon's Work on Electric and Magnetic Fields and the Semi-Classical Limit Ira W. Herbst Dedicated to Barry Simon on his 60th birthday ABSTRACT. We review some highlights of Barry Simon's work on Schr6dinger operators with external electric and magnetic fields. In addition, we summarize Barry's work on the spectrum of -A+A2 V (x) as A -> oo where V has multiple

wells. In particular, we outline one of his arguments giving a formula for the asymptotics of the splitting of the ground state eigenvalue in the case of a double well.

CONTENTS

Magnetic Bottles The Zeeman Effect in Hydrogen Enhanced Binding Translation Invariance Universal Diamagnetism Schrodinger Operators with Constant Electric Field The Rayleigh-Schrodinger Perturbation Series One-Dimensional Electric Fields Semi-Classical Analysis References 1.

2. 3. 4. 5. 6. 7. 8. 9.

It was an exciting time back in the seventies when I was a postdoc at Princeton, and perhaps the most exciting and rewarding experience for me was my interaction with Barry Simon and Yosi Avron. This began with a three-person collaboration on Schrodinger operators with magnetic fields. We wrote a series of six ([5] -p [10]) papers, and I will first try to give you some highlights of these works with some additional comments about Barry's work on magnetic fields in which I was not involved. 2000 Mathematics Subject Classification. 46N50, 35P05. Key words and phrases. Schrodinger operator, external field, double well. ©2007 American Mathematical Society 443

L W. HERBST

444

I will then present some of Barry's work on Schrodinger operators with electric fields and finally turn to highlights of his work on the semi-classical limit. I apologize in advance for not including an up-to-date list of references for the material considered here. For a more complete set of references on magnetic fields, please refer to the contribution of Laszlo Erdos in this Festschrift.

1. Magnetic Bottles The usual intuitive notion of compactness for bounded operators in L2(Rn) goes back to the idea that if the symbol of a pseudo-differential operator, p(x, ), tends to zero as jxj + oo, then the operator should be compact. Thus, it may have surprised some to learn that although for no magnetic field B = da does

(( - a(x))' + i)_1 , 0 xj -- oo, there are many magnetic fields for which the resolvent of H = (-iV -a(x))2 is compact. In [5] it is shown that if B(x)j - oc as xj -- oo in such

as

a way that B(x)/IB(x)l does not twist too wildly, then H has a compact resolvent. This is based on the simple commutator estimate (,P, H 2Im((-i5j - aj (x)) V, (-iOk - ak(x)co) = i(,p, [-i5j - aj (x), -iOk - ak(x)]Q) = (V, BjkP) with Bik = D ak - DDkaj. Localizing, we obtain

THEOREM 1 ([5]). Suppose B is continuous and {oa} is a covering of R by cubes of fixed size with A, centered at a c 1R' . Suppose for each a there are two unit vectors ea and e', such that inf B(x)(e,,, ea) --* oc as jal -+ oo. xEO Then H = (-iV - a(x) )2 with B = da has a compact resolvent. Theorem 1 is a precise expression of the consequence of a localized zero point energy becoming large as jx - 3 oo. This zero point energy is absent with the Pauli operator (o- (p - a(x) ))2, p = -iV, but bound states can occur there for other reasons [37, 21].

Magnetic fields such that H has a compact resolvent are dubbed magnetic bottles of the third kind in [5]. A simple example in three dimensions is B (x = (x 1i x2, -2x3). See [29] for further developments in understanding under what conditions H = (-iV - a(x) )2 + V (x) has a compact resolvent. There is an example of Miller and Simon [38] in two dimensions which shows

that 0(,) behavior at infinity of the magnetic field B is borderline for certain spectral behavior: They consider magnetic fields essentially given by b/jxjry with b a non-zero constant and prove THEOREM 2 ([38]).

(a) If 0 < y < 1, H has dense point spectrum in [0, oo). (b) If 1 < 'y, H has purely absolute continuous spectrum. (c) If ry = 1, the spectrum of H is dense point in [0, b2] and absolutely continuous in (b2, oo).

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In arbitrary dimension n > 2, if the 1-form B(x)(x, ) is o(1) at infinity, then H has no eigenvalues [20]; thus indeed, behavior is borderline. The latter result is independent of even or odd dimension. There is a conjecture in [5] which is still open as far as I know:

In R3, if B(x) is continuous and o(1) at infinity, then H has some continuous spectrum. Intuitively, one expects the particle to be able to follow some field lines out to infinity.

Let us indicate how Miller and Simon prove (c) above because there are two interesting ideas there: We ignore the singularity at x 0 and write

H=(p-a)z=-0+b2_2L, r=xJ, r L = -i8/89 in polar coordinates. The first step is to write H as the direct sum of operators

H=®°°-_ (-A + b 2

2rm

where 'H n is the kernel of (L - m). General considerations thus show the spectrum of H is absolutely continuous in (b 2, oc) and pure point in [0, bz). The second ingredient is the general statement that no matter how a(x) behaves at infinity, aegg((p -a) 2) = [0, oo) if lim IB(x)l = 0. IxHoo

The proof of this statement uses a Weyl sequence fin, and the ability to build a gauge transformation a -- a + V fn = an, into 0,,, where a,,, is small on the support of 0n. In the case at hand, this shows the spectrum is dense pure point in [0, b2].

2. The Zeeman Effect in Hydrogen The Zeeman effect refers to the effect of a constant magnetic field on the structure of the spectrum of an atom. Consider the Hamiltonian for the hydrogen atom in a constant magnetic field:

H=2

(-i0

ea(x))z

- e2

B=V x a= Be3i

with B a non-zero constant. If a(x) = a B x x, then z

z

H=-Z O- BL3+e8B (x12 +x2)1

e2

Here L3 is the third component of the angular momentum. THEOREM 3 ([7, 9]).

(1) The ground state of H has L3 = 0(2) The binding energy for the ground state is monotone increasing in B. (3) The Rayleigh-Schrodinger series for the eigenvalues of H (a power series in Bz) is Borel-summable to the correct eigenvalue.

I. W. HERBST

446

The first two results were proved using FKG [22] inequalities and the third using dilation analyticity at a fixed L3. The FKG inequalities imply the following: Suppose V and W are functions of xI j, ... , Ixnl and in the region {x : xj > 0 for all j}. aia,V(x) < 0 if i j and

aiW > 0 all i. Suppose 1 C Rn is such that 1p depends only on ixI I,-, Ixnj and is decreasing in each xjj. Then the normalized ground states, 'v and "v+w, of -A + V and -A + V + W satisfy

f f IOv+w(x)12dnx >

J

IOv(x)I2dnx.

(1)

For (1), see [7] and [32]. Here is an outline of the proof of (1) and (2) of Theorem 2: We add cz2, z = x3i to H, go to a subspace of fixed L3 = £, and make a simple unitary transformation to give the operator he,, = (2m)

1

( 1\

a2

aa2z2

t2

ape

41

e2 B2

eB P - e 2 (p2 +z2)-12 +Ez - pt

p2

+ 8m

p2

2

(2)

in L2 (R+ x ]R, dp dz). Here p = xl -+x22. Assume B > 0. The ground state ')e of he,, will have energy E(2, e) satisfying

at

\

(mp

B

The ground state of H + cz 2 is infinitely degenerate when the Coulomb potential

is removed so it is no surprise that '9E = 0 for 2 > 0 with no Coulomb term. According to the inequality (1), if we add back in the Coulomb term p-2 t) increases. Thus, taking e 10, we obtain the result. Similarly, to prove monotonicity of the binding energy with JBI, differentiate es - E(0, c) with respect to CBI. This derivative is zero without the Coulomb term '2-M and increases when the Coulomb term is added back in. For additional material on the ordering of the energy levels in this system, see [12] and [26]. A formal series (usually not convergent for x

0)

00

E anxn n=0

is said to be Borel-summable to f (x) for small positive x if the Borel transform

B(x) _

anxn/n!

has an analytic continuation to a neighborhood of [0, oc) and

f (x) = J

B(tx)e-tdt

0 00

for small positive x. Sufficient conditions for this to be true (see [27] for Watson's

theorem and [39] or [51] for a more refined result quoted here) require f to be analytic in {z : Iz - R/21 < R/2} for some R > 0 and a bound

f(z) - Eanznl < ICzIN+I(N+ 1)! n=0

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in this region. In [7] these conditions are proved using dilation analyticity and a simple formula for the eigenvalue involving contour integrals of the resolvent of the Hamiltonian. For additional work on the perturbation series for the eigenvalues in the Zeeman effect in hydrogen (in particular on Bender-Wu formulas), see [2] and [3].

3. Enhanced Binding Note that in two dimensions H = (-iV-a(x))2 with V x a = B = constant 0 has pure point spectrum with each eigenvalue infinitely degenerate. (The infinite degeneracy is because any Landau orbit can be translated anywhere in the plane without changing its energy. See below.) It seems then that by adding a scalar potential, binding in three dimensions should be easier in constant B field since it is only in the direction along B that a particle can escape to infinity. In one dimension an arbitrarily small attractive V will produce a discrete bound state ([42, 14]). This is not true in three dimensions if B = 0 but is true with B r 0 ([7, 10]). Consider first the one-body problem in three dimensions. Take

H = (-iV - 2 B x x)2 + V (x),

B = be3i

where b > 0 is constant. THEOREM 4 ([7, 10]). Suppose V is 0-compact and for large enough R, V (x)

0 for I xH > R. In addition, suppose V is not zero a.e. outside the cylinder x2 + x2 < R2. Then H has at least one eigenvalue below the continuum, and if V is azimuthally symmetric, H has infinitely many eigenvalues below the continuum.

We sketch a proof in the case V is also bounded with limb,,I-,, V(x) = 0. Choose a trial state 0(x) = f (x3)cpe(x1i x2), where L3W = fWe, f > 0 and cpe is a normalized ground state of the corresponding two-dimensional purely magnetic Hamiltonian which thus has energy b. One easily sees that Qess(H) = [b,oc). By the min-max principle, we can increase V, if necessary, to make V integrable. Note

that 2

(

,

(H - b) b) = Cf, (- dx3 + V )

f)

where Ve(x3)

=

I Ve(xl, x2)12V (XI, x2, x3)dxldx2.

J It is easy to see that f Vi (X3) dX3 < 0 for all large 2 (since Ve I2 has most of its weight far from the origin for these $). If we take f (X3) = fo(ax3) with fo E Co (R), and fo (0) = 1, we find

1li o(O, (H - b)O) = f Ve(x3)dx3 < 0.

This completes our outline of the proof. More physically relevant perhaps is a result in [7] about negative ions of multielectron atoms. We take units where the Hamiltonian of an ion of nuclear charge n - 1 and n > 2 electrons is n

H(b) j=1

i<j

I. W. HERBST

448

where B = be3, S is the sum of the electron spins, and H(b) acts in H, the antisymmetric subspace of ®;1 1L2(II83; C2).

THEOREM 5 ([7, 10]). Fix b > 0. Then the operator H(b) has infinitely many bound states in 7-l below the physical continuum. For small b, the binding energy of each such state is at least as large as cb3, (c > 0 depends on the eigenvalue of L3).

We give some ideas about the proof of the first statement of the theorem. Using an inductive proof, we first show that the neutral atom has a ground state rl of definite L3. It is important to have an estimate for the decay of rl but this is not difficult. The idea is to construct a trial function on the space of fixed large L3, where the nth electron is far from the neutral atom. We put this electron in a state yn)((an), rn = (xn, yn, zn), for large £ and small a and ( (1) = 1, ((-1)z = 0. But if we leave the other n - 1 electrons in state rl, the potential due to the shielded nucleus is not enough to give binding because of exchange terms whose signs are not known. We take into account an induced dipole moment of the neutral atom, due to the far away electron. The state we choose is b = Pcp, P = antisymmetrization, and n-1

cP=rl (1+Y(r

-al Z''I cc (xn, yn)S (an), j=1

where

g(r) = -j3 sgn(z)7(z)/(1 + Ir12),

and ry is a smooth even function, zero in a neighborhood of zero, but one in a neighborhood of infinity. We take $ large, a _ f 2 , C3 a positive constant and estimate

(0, (H(b) - E)0) Here E arises from moving the nth electron off to infinity in the z-direction with zero z-kinetic energy and putting it in a ground state Landau orbit with spin state ((a). For large £, this expectation is negative.

4. Translation Invariance Consider an N-body problem in constant B for example, with two-body potentials:

0 where the particles interact,

1( <j

The Hamiltonian is not invariant under xj - xj + c all j, but since V x a(x + c) = V x a(x), we can follow the usual translation by a gauge transformation to obtain invariance of H. The resulting unitary group {U(c)}cEa3 satisfies U(ci)U(c2) = ezQ"U(e2)U(el),

where Q = >j ej is the total charge and (P is the flux of B through the parallelogram with sides c1 and c2i 4 = B B. (c1 x c2). Thus, if Q = 0,

H = f H(k)d3k,

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where k is a quasimomentum. In fact, H(k) = H(0) + 2M - l x B R [7], R = E eixi, M = E mi. (It turns out that in a quantized radiation field it is the quasimomentum + photon momentum which is conserved.) If Q 54 0, then as with a single particle in two dimensions, the generators of translations perpendicular to B satisfy the CCR. And thus, because of von Neumann's theorem on the canonical commutation relations, there is an infinite degeneracy in the strongest sense:

H=h®I, where magnetic translations perpendicular to B act in the second factor. If

0,

U(c) = 10 u(c). Note that the Hilbert space associated with the second factor must be infinitedimensional. This material can be found in [6].

5. Universal Diamagnetism Barry was the originator of some far-reaching inequalities involving magnetic Schrodinger operators. The first leads to universal diamagnetism for spinless particles:

THEOREM 6. Suppose a E Li c(Rn) and V-, the negative part of V, is bounded.

Let E(a) = inf a((-iV - a)2 + V). Then E(a) > E(0). PROOF. Suppose u E Co (JPn) (a form core for our Hamiltonian by definition). Then VIUI2

= 2 Re(u(V - ia)u),

so

V IuF + E2 _

Re(u(V - ia) u) u2 + E2

< I (V - ia)ul. Taking c 10 and integrating, we obtain 11 (V - ia)u112 + (u, Vu).

II V IU1112 + (u, Vu)

The result follows from the variational principle.

In fact, Theorem 6 follows from the more general result (here p = -iV) e-t((p-a)2+V)

f

<

e-t(-o+V)

If

(3)

which Barry first conjectured in [48]. Upon hearing Barry's conjecture, Ed Nelson supplied the path integral proof which appears in the same paper. Barry gave another very elegant proof [50] which simply uses the Trotter product formula [41] e-t(p-a)2

= s - hm M-00

(e-t- (pi-ai)2 ... e- -ML (per.-a. )2

and the fact that for j = 1, .... n, 2

for some (real) fj with ajfj = aj.

, /1

2

I. W. HERBST

450

There is much additional work on comparison of semigroups which Barry has been involved in (see [36, 49]). In [49], Barry proves the equivalence of a semigroup comparison such as (3) with a generalized Kato inequality for the generators of the two semigroups. In [36], Barry and Dirk Hundertmark prove the pointwise bound Ct((P a)N,,+V)

-e

t((p a)e

D,A+V) f <

(e-t(-Div,n+V)

- e-t(-OD,A+V) (4)

where the subscripts N, A indicate Neumann boundary conditions on the boundary of the open set A and similarly D, A indicate Dirichlet boundary conditions on M.

Here A is an arbitrary open set and a and V are very general. The inequality (4) is very useful in proving uniqueness of the integrated density of states under rather general conditions (see [36]). In contrast to (3), the proof [36] is not trivial, basically because there does not (yet?) exist a sufficiently useful path integral representation for the semigroup involving Neumann boundary conditions. An interesting conjecture of universal paramagnetism for fermions [35] which for the Pauli operator reads

inf spec((p - a)2 + o, B + V) < inf spec(-A + V)

(5)

was later shown to be incorrect by Avron and Simon in [11]. However, as is shown in [35], (5) is true if B = constant.

6. Schrodinger Operators with Constant Electric Field After some initial work on resonances for the two-body Stark problem ([24, 4, 30]), Barry and I followed this up with the more involved N-body problem, in particular, the Stark effect in atoms and molecules. For definiteness, consider Oj

H(f)=E j

2m7 .

+fe Eejxj+EVj(xi-xj) i<j

and H(f), the same Hamiltonian with center of mass motion removed. We assume the Vij are dilation analytic so that resonances can be defined as poles of matrix elements of the resolvent of H(f), continued to the second sheet. These matrix elements are, of course, between dilation analytic vectors. We proved the following stability result: THEOREM 7 ([33, 34]). Suppose E0 is a discrete eigenvalue of H(O) of multi-

plicity m. Then, given e > 0, there is an fE > 0 so that if 0 < f < fE, H(f) has m resonances (counting multiplicity) in the disk {A : JA - E0 < e}, and they all converge to E0 as f 10.

Unexplored problem. Are resonances of H(0) stable under perturbation to H(f )? This seems to be a very difficult problem in the framework of dilation analyticity.

7. The Rayleigh-Schrodinger Perturbation Series To deal with degeneracy, define an eigenvalue of H(0) as "normal" if on all symmetry subspaces the degeneracy, if any, is removed to first order in f (see below for a precise definition). Then

ELECTRIC AND MAGNETIC FIELDS AND THE SEMI-CLASSICAL LIMIT

451

THEOREM 8 ([33, 34]). Suppose Eo is a normal discrete eigenvalue of H(O)

Then for any b > 0, there is an R5 > 0 and t functions E1(f),...,EQ(f) analytic for f in with multiplicity Q. {f

:

If I < Ra, -Min (p, 2) + 6 < arg f < 7 + Min cp,

2

where the V3 are dilation analytic in {B Im01 < (p}, which for real f > 0 are the resonances of H(f). The Rayleigh-Schrodinger series for the putative perturbed eigenvalues of H(f) are Borel-summable to the resonances of H(f).

The region of analyticity of the resonances has its origins in the region of analyticity in the two complex variables (f, 0) of the operator

-e-290 + f6 xee, which is

{(f,0): f 40, 0<argf+3Im0
FIG. 1

In contrast to the resonances of H(f), the coefficients in the Rayleigh-Schrodinger series are real; thus the recovery of the resonances from the Borel transform for real f has a slightly different form from that given in the discussion after Theorem 3 (see [33] and [51]).

I. W. HERBST

452

We also show a relationship between the asymptotics of the width of the resonance and the asymptotic behavior of the Rayleigh-Schrodinger coefficients an. For simplicity, for a non-degenerate eigenvalue of H(O) in a state of definite parity (assuming V3 (u) = Vj(-u))

E(f) ,,,

a2nf2n

with

F(f) + O(R-2n). a2= - II IR f2.ndf In joint work ([28, 17]), Barry and his co-authors derived rigorously the Oppenheimer formula, r(f)=4f-'e_ 3f

1

(I+0(f)))

for the width F(f) = -2 Im E(f) of the ground state resonance of hydrogen, but as far as we know, for multielectron atoms the leading behavior of r(f) is not rigorously known.

In most cases of interest the spectrum of the complex dilated operator H(f, 0) in the region of analyticity, f 0, 0 < 3 Im 0 + arg f < ir, I Im 0 1 < co, is purely discrete, consisting of resonances only. This is true unless for some non-trivial decomposition of the particles into clusters, the total charge to total mass ratios of all the clusters are all equal. Definition of normal eigenvalue of H(0): An eigenvalue E0 of H(0) is normal if there exist mutually orthogonal projections Qj commuting with the dilation gen-

erator, H(0), and X = e >j ej (xj - R) where R = Ej mj xj /(Y:j mj) such that with P = orthogonal projection onto the eigenstates of H(0) with eigenvalue Eo:

(i) Range >j Qj > Range P; (ii) P(QjXQj)P has non-degenerate eigenvalues on Range P. In practice the Qj are orthogonal projections onto eigenspaces of a set of generators of symmetry groups which commute with X, H(0), and the dilations. The proofs of Theorems 7 and 8 depend on being able to analyze the behavior of the resolvent (z - H(f, B))-1 for z near a discrete eigenvalue of H(0, 0) and for appropriate (f, 0) with If small. This presents interesting difficulties even in determining the spectrum of operators of the form HC1(f, 0) ® I + I ® H 2 (f, 0), where C1 and C2 are non-interacting subsystems. We used Weinberg-van Winter equations for appropriate semigroups, and a theorem of Gearhart [23] which allows one to determine the spectrum of a contraction semigroup e-tA on a Hilbert space from the behavior of the resolvent of A. See [31] for a review of this work where the analysis is somewhat simplified in that only the usual Weinberg-van Winter equations for resolvents are used. / In [25], Graffi and Grecchi treated Stark resonances in atoms obtaining similar results using different methods.

8. One-Dimensional Electric Fields There are three very interesting papers which Barry co-authored involving operators of the form z

H = - dx2 - fx+V(x), where V could be random. The first shows:

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THEOREM 9 ([13]). Suppose f > 0, V C C2(II8) with V, V', and V" bounded. Then a(H) = Qa,c(H).

For me, the interesting thing about this result is that there is no restriction on the absolute size of V'(x) so that the force due to V could overwhelm the force due to the electric field, at least pointwise. Of course, on the average V'(x) is small in the sense that sup

1

a+r

aE2r I-,

V'(x)dx

0,

as r

oc,

and this is what is used in [13]. A similar result was proved previously in the half-line case in [52]. But if V is not smooth, the situation can be quite different as the other two papers co-authored by Barry show: THEOREM 10 ([18, 19]). Suppose 00

V(x,w) _

an(w)6(x - n), n=-oo

where {an} is a sequence of iid random variables with bounded probability density and zero mean. Then, for small f , a(H) = aPP (H) while for large f , app (H) = 0, almost surely.

In their proof of the above theorem, the authors note that the problem reduces

to a discrete one which has a lot in common with the model described by the Hamiltonian

-A + Ann-2',

A, random and i.i.d., A = discrete Laplacian in 1 dimension, which they also analyze. It is unknown in their electric field model whether for large f the continuous spectrum corresponds to diffusion. Although rather singular, the potential of Theorem 10 is still Laplacian form bounded with form bound zero. Thus it is interesting to ask how smooth a bounded potential needs to be to get a result such as Theorem 9. This question is answered in [16] (see also [40]).

9. Semi-Classical Analysis Beginning in 1983, Barry began a series of papers ([43] --- [47]) studying the low lying eigenvalues of the operator H(A) = - z A + A2V(X)

in the limit A T oc, where V is a smooth non-negative function with isolated zeros, and H(A) acts in L2(W'). He proved some beautiful results.

THEOREM 11 ([43]). Suppose the zeros of V are finite in number and nondegenerate in the sense that at each zero, a, [c5z3 V (a)] is positive definite. Suppose

in addition that V(x) > 6 > 0 for large jxj. Let Eo(A) < EI(A) < E2(A)...

be the eigenvalues of H(A) counting multiplicity (the number of such eigenvalues oc as A T oo). Order the eigenvalues eo < el < e2...

I. W. HERBST

454

(counting multiplicity) in the union of the spectra of the operators

+

2(aiajV(a))Xix31

as a varies over the zeros of V. Then

A 'Ej(A) = ej. This result is not far from intuition since if V(O) = 0, A-1H(.X) is unitarily equivalent to - a A + AV (x/ f) and the Taylor series of V at 0 gives

AV (x/V) _ EaaV(0)xixj +O(A-2). i,j

But Barry goes much further to give asymptotic expansions to all orders of the Ej (A) and their corresponding eigenfunctions. A much finer result is THEOREM 12 ([44, 45]). With the assumptions of Theorem 11 and in addition that V has only two zeros, a and b, let Sto (A) be the normalized ground state of H(A) and suppose that for any small e > 0,

liminf

f

x-aIGE

Aloo

IQo(A)(x)12dx > 0.

IQo(A)(x)12dx fx

(6)

Then

lim -A-' log(El(A) - Eo(A)) = p(a, b),

ATCO

where

f

T

p(x, y) = inf

f

T

a

T

V(ry(s))ds : -1(-T) = x, 'y(T) = y, T > 0 T (7)

Thus, p(a, b) is the infimum of the action for an orbit going from a to b with the potential -V. There is an orbit minimizing the action if we allow T = oo. This orbit is called an instanton. It is a solution of Newton's equation with potential -V, which goes from a to b in infinite time. Actually p(x, y) is better known to some as the Agmon metric p(x, y) = inf { f 1

2V('y(s)) l'Y(s)ids : 'Y(0) = x, 'y(1) = y1

(8)

The equality of (7) and (8) was shown by Carmona and Simon [15]. The assumption (1) is satisfied, for example, if the double well arises because of a Euclidean symmetry R of V such that Ra = b. Then for small E > 0, lim ATE

f

I

= lim f

p(, ()`)(x)I2dx = 2

ATE

po(A)(x)I2dx.

x-b)GE

Part of the proof of Theorem 12 uses large deviation results (a la DonskerVaradhan), but Barry also shows how to obtain the result using "standard" PDE methods. (We give a sketch below.) Another theorem of the same genre concerns periodic V.

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THEOREM 13 ([46]). Suppose V is non-negative, smooth, and periodic with only one (non-degenerate) zero per unit cell. Let A(A) be the width of the ground state band. Then urn -A-' log 0(A) = min{p(a, b)

:

a and b are distinct zeros of V J.

Finally, Barry examines the effect of a seemingly negligible potential W in the situation of Theorem 11, but where V has only two zeros, a and b, connected by a Euclidean symmetry of order 2. W is assumed smooth, non-negative, and zero in a neighborhood of a and b but does not necessarily respect the symmetry of V. He is interested in differences between four different energies:

E0(A), El (A), the first two eigenvalues of - !A+ A2V (x), and

Eo (A), El (A), the first two eigenvalues of - a O

) 2 (V (x) + W (x) ).

Of course, because of the tunneling from a to b, W will have an effect, but a less complicated one than one might expect. THEOREM 14 ([47]). Let d = p(a, b),

dl = min{2p(a, supp W), 2p(b, supp W)}, d2 = max{2p(a, supp W), 2p(b, supp W)},

where p(x, S) = inf{p(x, y) : y c S}. Let eo and el be the first two eigenvalues of e-ad 0

( e- ad

0

/

and eo and e"1 the first two eigenvalues of e-adl C

e- Ad

e- ad e-ad2

Then the leading behavior of the differences of any of the E's and E's is given by the leading behavior of the differences of the corresponding e's and Vs in any of the three cases: (i)

d < dl < d2;

(ii)

dl < d < d2;

(iii)

dI < d2 < d.

Thus one can compute lim -A-I log(E1(.X) - Eo(A)) = d = lim

-A-I log(E1(.X)

aToo

AToo

- Eo(.\))

if d < dl, but if dl < d < d2,

lim -) I log(El(A) - Eo(A)) = dl.

XToo

Notice that in cases (i) through (iii) the size of W is not relevant, just the Agmon distance of its support from the zeros of V. This shows why certain cases (iii). of equality have been omitted in (i)

I. W. HERBST

456

Let us give a very brief sketch of the beautiful result lim A-' log(Ei(A) ATOO

Eo(A)) _ -p(a, b),

under the assumptions of Theorem 12. For simplicity we also assume the existence of a Euclidean symmetry R of the potential such that Ra = b. We first need the behavior of SZo (A, x) for large A:

PROPOSITION 15 ([44]). Let S2o(A,x) be the positive normalized ground state of H(A) and d(x) = min{p(x, a), p(x, b)}. Then with the hypotheses above

lim -A I log Qo (A, x) = d(x),

ATOO

uniformly on compact sets. Barry gives two proofs of this result, one relying on large deviations estimates of path integrals and the other based on PDE methods of Agmon. We sketch Barry's PDE proof. First we note that Barry's work in [43] shows that as A T oo, + (b)

SZ0

(9)

where (a is the ground state for -

1

A + 2

1:

aiajV(a)(x - a)i(x - a)j, 2

and similarly, bb.

This is a very rough result but a bit more work gives pointwise estimates in any neighborhood of the form I x - al < CAA, , which show, for example, that with Pa (x) =

A-,./4

SZo (A, A- 2 x +a);

ka (x) = A-v'4S}a (A-2' x + a);

lim O (x)/1a(x) = 1,

(10)

uniformly on compact sets and similarly with a replaced by b. We will use this later. We now sketch a proof of the upper bound half of Proposition 15: First note

that Id(x) - d(y)j

p(x, y) : Ix - yI

J0

1

2V-(1 - t)x + ty) dt,

where the second inequality comes from taking a straight line path from x to y. If follows that Iod(x)I2 < 2V(x), a.e. Given real f : ][8v - ll , let Hf(A) = f(x)H(A)e-f(x)

= H(A) -

2IVf(x)I2+(V.Vf+Vf.V)/2,

ELECTRIC AND MAGNETIC FIELDS AND THE SEMI-CLASSICAL LIMIT

457

and note that lief (H(A) - Eo(A))V)l = II (Hf(A) - Eo(A))ef*ll

> Re(ef0, (Hf(A) >- (ef0, (A2V

- Eo(A))efV))/ll efVGll

-1

- Eo(A))ef*) /Ilef

Vfl2

If we take f (x) = A(1 - c)2' d(x), the usual Agmon analysis [1] shows efgo E L2 if A is large enough since

A2V-2lofl2-Eo(A)>6>0,

(12)

for large x. If we take zb _ j o with q(x) = 0 if Ix - al < coA-2 or Ix - bI < coA-2, then, in fact, for large A (12) is true on the support of ri and thus (11) implies Ilef (H(A) - Eo(A))nQoll >- 6 Ief,gQoll,

which (with (10) and a similar result with a -+ b) leads to eX[(I-E)d-E]S2o

< const.

(13)

Now 12o is subharmonic outside a neighborhood of {a, b} of the form min{Ix al, Ix - bI} < c1A-2. Thus the L2 estimate (13) along with (10) gives a similar pointwise estimate and the upper bound lim sup A-' log 12o (A, x) < -d(x). Atoo

The lower bound of Proposition 15 relies on the following comparison result based on unpublished work of Agmon: LEMMA 16. Suppose 2 A11 = W12 where W and 12 are non-negative in the cylinder Do = {(xi, x1) : 0 < xl < all + 6), Ix1 < R}. Here x1 E I1 Y

.

Define a > 0 by

2 = sup { R + W (x) : 2

x E Do } ,

where eo is the lowest eigenvalue of -20 in L2 of the ball of radius 1 in Rv-1 with Dirichlet boundary conditions. Then, if xjj < 2R,

12(a, x1) > 0e-,a(1 - e-26aa) min{12(0, y1) : Iy1I < R}, = min{rl(y) : lyl < 2} and q > 0 is the eigenfunction corresponding to eo normalized with IHII,,, = 1. where

The proof of Lemma 16 is based on subharmonic comparison which compares 12 to

4' = '1'rl (x1/R) l e-axl - e-2aa(1+5)eaxil where

ry = min{12(0,x1)

Note that 12 > 0 on 8Do which implies 12 >

Ix 1I < R}.

in Do. We do not give details.

I. W. HERBST

458

The idea behind the lower bound for SZo is to approximate a geodesic from a to x by a piecewise linear path connecting the points xo = a, x1i X2.... , xn = x with segments so small that n

i=I

where vi is sup{V(x) : x c Di} and Di is a cylinder of radius along xi - xi-1 and length (1 + 2S)Ixi - xi_11. If we take

2-'A-2

with axis

zaj(A)2 = and skip a few of the initial cylinders until A2vj > Eo(A), we can apply Lemma 16 to obtain n SZo(.\ x) > e- ra,Ix. -x3 _11on fl(1 e26a.(A)) (14)

-

i=1

where we use (10) to infer that in any region Ix - aI < CA-2, SZo(A,x) > 1 for large A. From (14), we obtain lim i f A-1 log S2o (A, x) > -p(x, a). ATOO

A similar argument with a - b gives the lower bound. We now sketch the proof of the upper bound on E1(A) - Eo(A): Clearly, if (f Q0, 0o) = 0, then E1(A) _ Eo(A) < (.f1o, (H(A) - Eo(A))f1o) 1

(15)

1 f Qo 112

To make sure (f Q0, Qo) = 0, define

f =g-

2

dog dx,

where P(x,a) - P(x,b)

g(x) = h

p(a, b)

)

'

and h is a smooth, odd function with Ih(t) I = 1 outside a small neighborhood of 0. Actually, we should smooth out the argument of g, but for clarity we refrain from doing so in this sketch. Because of our assumption of symmetry, actually f Slog dx = 0. Note that since Q0 concentrates near a and b, 11gS2o II - 1 The numerator of (15) is z f Qo(Vg)2dx, and is supported in a small neighborhood of {x : p(x, a) = p(x, b) } = B. But min{d(x) : x c B} = a p(a, b),

which leads to lim sup A-' (El (A) - Eo (A)) < -P(a, b) ATc

Finally, we sketch the proof of the lower bound on El (A) - Eo(A): Let

Il(A,x)

9a (x) = S20 (A x)

ELECTRIC AND MAGNETIC FIELDS AND THE SEMI-CLASSICAL LIMIT

459

where S21 is the first excited state of H(A) normalized so that IIQ1II2 = 1 and S21(A, b) > 0. Let -y be a geodesic from a to b and notice that p(a,

Max {Min{p(a, y(t)), p(b, y(t))} : t c [0, 1]} =

b)

2

since, of course, p(a, -y(t)) + p(b, -y(t)) = p(a, b). Find a tubular neighborhood Tb of -y small enough so that for large A and x E T5, 1o (A, x)2 >

e-A(p(a,b)+E).

Again from Barry's work in proving Theorem 11 [43], he obtains (since Sb - (a is orthogonal to Sb + (a) lim

=0

1 - 72= ((b - Sa)

,TOO

(see (9)), and for the same reason that (10) is true,

ga(x) = T1,

(16)

uniformly in a neighborhood of the form Ix - al < A- 2 (respectively, Ix - bI < a- 2 ).

Choose coordinates (t, u), u E Ri-1 in Tb, so that -y(t) has u = 0. Shrink Tb, if necessary, so that

{x: t=O,Iul
Tb={x: 0
El(i) -Eo(n) = z fIVg2dx f IVgal21odx

a/T6

(2 J

\

\

vgdx I/

(17)

a

To estimate fTb I Vga l2 from below in Tb, notice that 9,\(t = 1, u) - 9a (t = O,u)IZ <

f0 l a at gA

2

dt.

Integrating over u and using (16) gives

(v

a 2 dt du, at gA

1)/2 <

changing variables from (t, u) to x introduces only another constant so that for some c" > 0, c"A-(v-1)12 <

IT5 IV ga2dx.

This and (17) give the desired lower bound.

I. W. HERBST

460

References [1] Agmon, S., Lectures on exponential decay of solutions of second-order elliptic equations: Bounds on eigenfunctions of N-body Schrodinger operators, Princeton University Press and University of Tokyo Press, 1982. [2] Avron, J. E., Bender-Wu formulas for the Zeeman effect in hydrogen, Ann. Phys. 131, no. 1 (1981), 73-94.

[3] Avron, J. E., Adams, B. G., Cizek, J., Clay, M., Glasser, M. L., Otto, P., Paldus, J., and Vrscay, E., Bender-Wu formula, the SO(4, 2) dynamical group, and the Zeeman effect in hydrogen, Phys. Rev. Lett. 43 (1979), 691-693. [4] Avron, J. E. and Herbst, I., Spectral and scattering theory of Schrodinger operators related to the Stark effect, Commun. Math. Phys. 52 (1977), 239-254. [5] Avron, J., Herbst, I., and Simon, B., Schrodinger operators with magnetic fields, I. General interactions, Duke Math. J. 45 (1978), 847-883. [6] Avron, J., Herbst, I., and Simon, B., Schrodinger operators with magnetic fields, II. Separation of center of mass in homogeneous magnetic fields, Ann. Phys. 114 (1978), 431-451. [7] Avron, J., Herbst, I., and Simon, B., Schrodinger operators with magnetic fields, III. Atoms in homogeneous magnetic field, Commun. Math. Phys. 79 (1981), 529-572. [8] Avron, J., Herbst, I., and Simon, B., Schrodinger operators with magnetic fields, IV. Strongly bound states of hydrogen in intense magnetic field, Phys. Rev. A20 (1979), 2287-2296. [9] Avron, J., Herbst, I., and Simon, B., The Zeeman effect revisited, Phys. Lett. 62A (1977), 214-216.

[10] Avron, J., Herbst, I., and Simon, B., Formation of negative ions in magnetic fields, Phys. Rev. Lett. 39 (1977), 1068-1070.

[11] Avron, J. and Simon, B., A counterexample to the paramagnetic conjecture, Phys. Lett. A 79 (1979/80), no. 1-2, 41-42. [12] Baumgartner, B., and Seiringer, R., On the ordering of energy levels in homogeneous magnetic fields, Lett. Math. Phys. 54, no. 3 (2000), 213-226. [13] Bentosela, F., Carmona, R., Duclos, P., Simon, B., Souillard, B., and Weder, R., Schrodinger

operators with electric field and random or deterministic potential, Commun. Math, Phys. 88 (1983), 387-397.

[14] Blankenbecker, R., Goldberger, M. L., and Simon, B., The bound states of weakly coupled long-range one dimensional quantum Hamiltonians, Ann. Phys. 108 (1977), 69-78. [15] Carmona, R. and Simon, B., Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, V. Lower bounds and path integrals, Commun. Math. Phys. 80 (1981), 59-98.

[16] Christ, M. and Kiselev, A., Absolutely continuous spectrum of Stark operators, Ark. Mat. 41 (2003), 1-33. [17] Corngold, M., Harrell, E., and Simon, B., The mathematical theory of resonances whose widths are exponentially small, II, J. Math. Appl. 99 (1984), 447-457. [18] Delyon, F., Simon, B., and Souillard, B., From power localized to extended states in a class of one dimensional disordered systems, Phys. Rev. Lett. 52 (1984), 2187-2189. [19] Delyon, F., Simon, B., and Souillard, B., From power pure point to continuous spectrum in disordered systems, Ann. Inst. H. Poincare 42 (1985), 285-309. [20] Eastham, M. S. P. and Kalf, H., Schrodinger Operators with Continuous Spectrum, Pitman, Boston, 1982. See Theorem 6.3.1. [21] Erdos, L. and Solovej, J. P., The kernel of Dirac operators on S3 and R3, Rev. Math. Phys. 13, no. 10 (2001), 1247-1280. [22] Fortuin, C., Kasteleyn, P., and Ginibre, J., Correlation inequalities on some partially ordered sets, Commun. Math. Phys. 22 (1971), 89-103.

[23] Gearhart, L., Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc. 236 (1978), 385-394.

[24] Graffi, S. and Grecchi, V., Resonances in the Stark effect and perturbation theory, Commun. Math. Phys. 62 (1978), 83-96. [25] Graffi, S. and Grecchi, V., Resonances in the Stark effect of atomic systems, Com-

mun. Math. Phys. 79 (1981), 91-109. [26] Grosse, H. and Stubbe, J., Splitting of Landau levels in the presence of external potentials, Lett. Math. Phys. 34, no. 1 (1995), 59-68.

ELECTRIC AND MAGNETIC FIELDS AND THE SEMI-CLASSICAL LIMIT

461

[27] Hardy, G. H., Divergent Series, Oxford Univ. Press, Oxford, 1949, page 191. [28] Harrell, E. and Simon, B., The mathematical theory of resonances whose widths are exponentially small, Duke Math. J. 47 (1980), 845-902. [29] Helffer, B. and Mohamed, A., Caracterisation du spectre essentiel de l'operator de Schrodinger avec un champ magnetique, Ann. Inst. Fourier (Grenoble) 38 (1988), 95-112.

[30] Herbst, I., Dilation analyticity in constant electric field I: The two-body problem, Commun. Math. Phys. 64 (1979), 279-298. [31] Herbst, I., Schrodinger operators with external homogeneous electric and magnetic fields, in Rigorous Atomic and Molecular Physics (G. Velo and A. Wightman, eds.), pp. 131-183, Plenum Press, NY, 1981. [32] Herbst, I. and Pitt, L., Diffusion equation techniques in stochastic monotonicity and positive correlations, Prob. Theory Related Fields 87, no. 3 (1991), 275-312. [33] Herbst, I. and Simon, B., Dilation analyticity in constant electric field, II. The N-body problem, Borel summability, Commun. Math. Phys. 80 (1981), 181-216. [34] Herbst, I. and Simon, B., The Stark effect revisited, Phys. Rev. Lett 41 (1978), 67-69. [35] Hogreve, H., Schrader, R., and Seiler, R., A conjecture on the spinor functional determinant, Nuclear Phys. B 142, no. 4 (1978), 525-534. [36] Hundertmark, D. and Simon, B., A diamagnetic inequality for semigroup differences, J. Reine Angew. Math. 571 (2004), 107-130. [37] Loss, M. and Yau, H.-T., Stability of Coulomb systems with magnetic fields, III. Zero energy bound states of the Pauli operator, Commun. Math. Phys. 104 (1986), 283-290. [38] Miller, K. and Simon, B., Quantum magnetic Hamiltonians with remarkable spectral properties, Phys. Rev. Lett. 44 (1986), 1706-1707.

[39] Nevanlinna, F., Zur Theorie der asymptotischen Potenzreihen. (Finnish) Ann. Acad. Sc. Fennicae (A) 12, Nr. 3, 81 S. (1916). [40] Perelman, G., On the absolutely continuous spectrum of Stark operators, Commun. Math. Phys. 234 (2003), 359-381. [41] Reed, M. and Simon, B., Methods of Modern Mathematical Physics I: Functional Analysis, Revised and Enlarged Edition, Academic Press, San Diego, 1980, page 275. [42] Simon, B., The bound state of weakly coupled Schrodinger operators in one and two dimensions, Ann. Phys. 97 (1976), 279-288. [43] Simon, B., Semiclassical analysis of low lying eigenvalues, I. Non-degenerate minima: Asymptotic expansions, Ann. Inst. H. Poincare 38 (1983), 295-307. [44] Simon, B., Semiclassical analysis of low lying eigenvalues, II. Tunneling, Ann. of Math. 120 (1984), 89-118.

[45] Simon, B., Instantons, double wells, and large deviations, Bull. Amer. Math. Soc. 8 (1983), 323-326. [46] Simon, B., Semiclassical analysis of low lying eigenvalues, III. Width of the ground state band in strongly coupled solids, J. Funct. Anal. 63 (1985), 123-136.

[47] Simon, B., Semiclassical analysis of low lying eigenvalues, IV. The flea on the elephant, J. Funct. Anal. 63 (1985), 123-136. [48] Simon, B., An abstract Kato's inequality for generators of positivity improving semigroups, Indiana Univ. Math. J. 26, no. 6 (1977), 1067-1073. [49] Simon, B., Kato's inequality and the comparison of semigroups, J. Funct. Anal. 32, no. 1 (1979), 97-101.

[50] Simon, B., Maximal and minimal Schrodinger forms, J. Operator Theory 1, no. 1 (1979), 37-47.

[51] Sokal, A., An improvement of Watson's theorem on Borel summability, J. Math. Phys. 21 (1980), 261-263.

[52] Walter, J., Absolute continuity of the essential spectrum of -d2/dx2+q(x) without monotony of q, Math. Z. 129 (1972), 83-94.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF VIRGINIA, CHARLOTTESVILLE, VA 22903,

U.S.A. E-mail address: iwhmvirginia. edu

Proceedings of Symposia in Pure Mathematics Volume 76.1, 2007

Some Bound State Problems in Quantum Mechanics Dirk Hundertmark Dedicated to Barry Simon on the occasion of his 60th birthday ABSTRACT. We give a review of semi-classical estimates for bound states and their eigenvalues for Schrodinger operators. Motivated by the classical results, we discuss their recent improvements for single particle Schrodinger operators as well as some applications of these semi-classical bounds to multi-particle systems, in particular, large atoms and the stability of matter.

CONTENTS

Semi-Classical Bounds for Single-Particle Schrodinger Operators Multi-Particle Coulomb Schrodinger Operators More on Bound States for Atoms References 1.

2. 3.

In this survey, we focus on results for bound states of Schrodinger operators related to the semi-classical limit and Coulomb potentials. We will not discuss a large part of the existing literature on the general theory of bound states for Schrodinger operators. With no attempt at completeness, we nevertheless would like to mention at least some part of this literature: For one-particle Schrodinger operators, see, for example, [19, 151]. Two-body cluster results are discussed in [4, 84, 145, 153, 154, 175]; finiteness results of the discrete spectrum for Nparticle systems can be found in [1, 41, 166]; and for results on the Efimov effect, see, for example, [3, 37, 125, 157, 162, 172].

1. Semi-Classical Bounds for Single-Particle Schrodinger Operators The origin of semi-classical estimates can be traced back to the dawn of quantum mechanics in the beginning of the last century. Around 1912, Hermann Weyl 2000 Mathematics Subject Classification. 35J10, 81Q10. Key words and phrases. bound states, semi-classical inequalities, stability of matter, Coulomb Hamiltonian. Supported in part by the National Science Foundation grant DMS-0400940. ©2007 by the author. Faithful reproduction of this article, in its entirety, by any means is permitted for non-commercial purposes. 463

464

D. HUNDERTMARK

published a series of papers [168, 169, 170] (see also [171]) on the frequencies of an oscillating membrane and the radiation inside a cavity, verifying a conjecture of Jeans and Lorenz on the connection between the asymptotic behavior of these frequencies and the volume of the cavity. While in itself being a classical problem, this work was the starting point of a substantial branch of analysis and mathematical physics, especially in quantum mechanics. Consider a bounded domain A C Rd and the eigenvalue problem for the Dirichlet Laplacian -DA cp = Ecp,

that is, the partial differential equation d

a2

j

andcplOA=0.

(1)

Furthermore, let E1 (A) < E2 (A) _< E3 (A) < ... be an ordering of the eigenvalues of (1) and define the counting function

ND(E):=

1

E3 (A)<E

which counts the number of eigenvalues of -AD below E. Weyl showed that

ND(E) =

CAI Ed/2 + o(Ed/2)

as E

oc.

(2)

d/2

Here wd = r(1+(d/2)) is the volume of the unit ball in Rd. Weyl's formula holds for all bounded domains A C Rd.

The origin of Weyl's equality is easy to see: If A is a centered cube of side length a, then the eigenfunctions and eigenvalues of the Dirichlet Laplacian are known explicitly and given by d

u(x) = 11 sin("Q"

n , E N` and

=1

E=

2

a2

2

In2

d

nv,

a2 =1

that is, ND (E) is precisely the number of points n E N d within the ball of radius E'/2, which behaves asymptotically as wd (27r)d adEd/2 + o(Ed/2) for E ---> oc.

(3)

Similarly, the counting function NN (E) of the Neumann Laplacian has the same asymptotics as the Dirichlet Laplacian. They differ by a surface term which is of lower order in the high energy asymptotics, at least for domains with a nice boundary A. Weyl's crucial idea was to approximate a general domain A C Rd by cubes and to use (3). Using variational arguments, he showed wd(a )d#{disjoint

lim E-d/2ND(E) = lim cubes of side length a in A}. a-.0 E-oo 27r Since #{disjoint cubes of side length a in Al = ad (JAI + o(1)), one obtains (2).

BOUND STATES IN QUANTUM MECHANICS

465

1.1. P61ya's Conjecture. In 1961, Polya [127] conjectured that the asymptotic result (2) holds as a uniform bound on ND(E) for all E > 0 with some constant P(d), that is, ND (E) < P(d) JAI Ed/2 for all E > 0,

(4)

or, equivalently,' En > (P(d) I AI)-2/d n2/d for all n E N.

Polya also conjectured that the sharp constant in (4) is given by Weyl's asymptotic result, P(d) = (2 dr)d . He was able to prove this conjecture for the special class of tiling domains A, that is, disjoint congruents of A are assumed to cover 1(8d. The argument is rather simple. Scaling AT := rA, one gets another tiling domain with ND(E) = ND (r-2E) (by scaling of the kinetic energy). So with B = ball of unit volume in 1[8d, we get D( ND (r 2E) < NB (r

2

E) T

where AT = #{disjoint congruents of rA which are subsets of B}. Fix E > 0 and let r -* 0. By Weyl's asymptotic we know

NB (r 2E) = r -d

((27rWd)d

Ed/2

and, obviously, A, = r-d(IAI-1 + 0(1)) as r ---> 0. Together these estimates give the bound D( r-2

ND (E) < lim NB

AT

CJd

E)

T2--j-)d Ink

Ed/2

for a tiling domain A. Unfortunately, the sharp result is not known for general bounded domains. The best result is due to Li and Yau [93], who showed

()d/2

N (E)

(27r)d

IAI Ed/2

(5)

More precisely, they proved the sharp bound n

> j=1

d

d+2

2/d 10d

(27r)JAI

n'+2 forallnENd

and deduced (5) from this simply by observing En > n E; I E. Laptev [88] gave a much simpler argument than the original one by Li and Yau in 1996. Moreover, he showed that if Polya's conjecture holds for a domain A, C 1[8d1, then it holds for all domains A = Al x A2 for all d2 E N and domains A2 C 1[842. This paper was the first instance where the idea of "stripping off' dimensions appeared, which later turned out to be the key for a refined study of semi-classical inequalities for moments of eigenvalues of Schrodinger operators, see Section 1.7.

'Indeed, if N(E) < CE', then putting E = E one sees n = N(En) < CE,', that is, E, > C-l/-nl/a. Conversely, N(E) = FEj<E 1 1 < CEa.

D. HUNDERTMARK

466

1.2. Weyl Asymptotics for Schrodinger Operators. In the early 1970's, Birman-Borzov, Martin, and Tamura [17, 122, 161] proved semi-classical asymptotics for the number of the negative eigenvalues2 of Schrodinger operators with a

Holder continuous and compactly supported potential V. Let El < E2 < E3 < < 0 be a counting of the negative eigenvalues of -A + V on L2 (R) and set N(V) := #{negative eigenvalues of - A + V}. Assume that V is non-positive for the moment and introduce a coupling constant A. We want to study the large A asymptotics of N(AV). Assuming the DirichletNeumann bracketing technique developed by Courant and Hilbert, a short argument

can be given as follows; for details, see [128]. We let Aa = {-a < xj < 2, j = 1, ... , d} and define Aa,j = j + A. for j E aZd. Assume that V is constant on the cubes Aa,j, that is, V(x) = Va,j for x E Aa,j. Using Dirichlet-Neumann bracketing, one gets

Nn ,, (AVa,j).

I NA , (AVaj) < N(AV) < jEaZd

jEaZd

Applying Weyl's asymptotic result (2) with E

N(AV) =

1:

Wd

(21r)d Adl2ad

IVa,j

_

AIVa,j

,

I d/2 + o(Ad/2) as A - oo.

jEaZd

So if V is continuous, non-positive, and of compact support, N(AV) = (21d

Ad/2

fd

V(x)dl2 dx + o(Ad/2)

(6)

This formula is, in fact, a semi-classical asymptotic. Let be the so-called classical symbol associated with the operator -A + AV. Using the Fubini-Tonelli

theorem and scaling, the volume of the negative energy region in phase space is given by

ffI2+av(x)
wdAd/2

f V(x)d]2 dx. d

Thus, formula (6) says that the number of negative eigenvalues of -A+AV asymptotically, for large A, behaves like the classical allowed phase-space volume divided by (27r)d,

lira A-d/2N(AV) = (21 ^-00

d

ff(2 +V(x))° d dx,

where we set ro = 1 for r < 0 and 0 for r _> 0. So each eigenfunction corresponding to a negative eigenvalue occupies a volume (2rr)d in phase space. This is in perfect agreement with the Heisenberg uncertainty principle, according to which an electron occupies a volume of at least (27r)d in phase space; see also [45]. One should also note that Weyl's original result fits very well into this more general framework. Indeed, and again somewhat formally, for a bounded open set

A, one can recover the Weyl asymptotic by setting V = 001Ac - 1A (with the 2For a probabilistic proof, see [78] or [152].

BOUND STATES IN QUANTUM MECHANICS

467

convention oo 0 = 0). Then (6) gives

NA (E) = N(EV) 1

(27r)d fL2+EV(x)<0 Wd

(27r)

d JAI

E

d dx + o(Ed/2)

+ o(Ed/2)

which is Weyl's asymptotic.

1.3. The Birman-Schwinger Principle. Around 1961, Birman [16] and Schwinger [139] independently found a way to reduce an estimate on the number of negative bound states to the study of a bounded integral operator. The idea of their argument is as follows: First assume that V = -U is non-positive. Let cp be

an eigenvector of -0 - U for the negative energy E. That is, (-0 - U)cp = Ep, with E < 0. Rearranging gives (-A - E)cp = Ucy, which in turn is equivalent to cp = (-A-E)-1Ucp, since for E < 0, (-O-E) is boundedly invertible. Multiplying this with VU and setting V) = v"U-W, we see -0 - U has a negative eigenvalue E if and only if the Birman-Schwinger operator

)-I U KE _ /U(-A - E)-'V-U-

(7)

has eigenvalue 1. Moreover, since the map E -> nth eigenvalue of KE is (strictly) monotone increasing, a careful analysis shows that the corresponding eigenspaces

have the same dimension. In short, denoting Al(KE) > A2(KE) > ... > 0 the <_ 0 eigenvalues of the Birman-Schwinger operator KE and by El < E2 < E3 < the negative eigenvalues of -A - U, the Birman-Schwinger principle is the statement that 1 = An (KE.) for all n. (8)

If V is not non-positive, one uses the min-max principle to see that NE(V) < NE(-V_). Here NE(V) = #{eigenvalues of - 0 + V < E}. Thus, setting U = -V-, using the Birman-Schwinger principle, and the monotonicity of A (KE), one sees

NE(V) < #{eigenvalues of KE > 1}. In particular, NE(V) < tr[KE] for all n E N. On 1883, KE is a Hilbert-Schmidt operator with integral kernel e-./=EIx yl KE(x,y) = V_(x) V/ _(y)

-yJ

implying the bound

V-(x)V (y) x - yl As E -* 0, we get the Birman-Schwinger [16, 139] bound NE(V) < tr[KE] =

1

(47r)2

N(V) = No(V) C

ff

(y) (47r)2

X- yl 2 ff V (x)

dxdy.

dxdy.

(9)

Note that (the negative part of) a potential V_ is in the Rollnik class [148] if and only if the right-hand side of (9) is finite. On the other hand, this simple bound does not have the right large coupling behavior since it only shows that N(AV) < (const))2 which grows much faster than

D. HUNDERTMARK

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the asymptotic growth A3/2 shown by large coupling asymptotic. Nevertheless, the Birman-Schwinger principle is at the heart of all proofs of semi-classical eigenvalue bounds.

1.4. The CLR and Lieb-Thirring Bounds. The semi-classical Weyl-type asymptotics for the number of bounds states leads naturally to the question whether there is a robust bound of the form

f N(V) < (2ir)d Co'd JJ 1

1 d dx 12+v
10 )

=Lpdf V-(x)d/2dx d

for arbitrary potentials V for which the right-hand side is finite.3 Here Lo,d = Co,dwd/(27r)d, where wd is the volume of ball of radius one in d dimensions.

More generally, one can ask a similar question for higher moments4 of the negative eigenvalues. That is, whether there is a semi-classical bound for

S"(V) = tr[(-A + V)'-] of the form Sry(V) <

C1, d (21r)d

f

+V(x))' dd x.

(11) (12)

Or, equivalently, whether S-'(V) < Lry d with

L ry,d = C.y,d L i d, w here L..,d =

V(x)ry+dl2 dx

J 2 - 1)y d = (a i )d f (

(13) 2d7rdi 2r(ry+1 +d/2)

is

the so-called classical Lieb-Thirring constant. In particular, Lc d = wd/(21r)d and

Lild = d+a

Id)d

Of course, one has to assume that the negative part of the

potential V_ E L'Y+d/2 (1Rd) One recovers N(V) from Sry (V) by N(V) = S° (V) _ lim.y_o S7(V). Of course, the physically most interesting cases are y = 0, the counting function for the number of bound states, and y = 1, which gives a bound for the total energy of a system of non-interacting fermions in an external potential given by V. Also a simple approximation argument (see, e.g., [149]) shows that a bound of the form

(10) allows one to extend the semi-classical asymptotic (6) to all potentials in Ld/2 (Rd).

It is easy to see that bounds of the type (12) can only hold for y > 0 in d < 2 and y > 0 in d > 3. In fact, since any non-trivial attractive potential has at least one bound state [150, 83] and [87, pp. 156-157], there can be no semi-classical bound of the form (13) for y = 0 in d = 2; and in one dimension, this bound is even impossible5 for 0 < y < z 3More precisely, V+ should be locally integrable and V_ E Ld/2(Rd) 4Often called Riesz' moments.

5Let c > 0, b be the Dirac measure at 0, and note that, by a one-dimensional Sobolev embedding, the operator -82 - c6 is well-defined as a sum of quadratic forms. Take a sequence of approximate delta-functions 6 converging weakly to 6. Then -82 - c6 converges to -82 - c6 in strong resolvent sense. If -y < Z, the right-hand side of (13) goes to zero, but the ground state

of -82 - c6 stays bounded away from zero (it converges, in fact, to -c2/4 = single negative eigenvalue of -82 - c6).

BOUND STATES IN QUANTUM MECHANICS

469

The inequalities (12) and (13) were proven by Lieb and Thirring [117, 118] in 1975-76 in the cases ry > 2 in d = 1 and y > 0 for d > 2. The proof of (10), the famous Cwikel-Lieb-Rosenblum bound, is considerably more complicated than the proof for ry > 0. It also has an interesting history: Rozenblum announced his proof of (10), which is based on an extension of a machinery developed by Birman and Solomyak [18], in 1972 in [130]. This announcement went unnoticed in the West. Independently of Rozenblum, Simon established

in [149] a link between the bound (10) and the then conjectured fact that the Birman-Schwinger operator K° given in (7) for E = 0 is a certain weak trace ideal6

for d > 3. This conjecture by Simon on the asymptotic behavior of the singular values of Ko motivated Cwikel and Lieb for their proofs of the CLR bound. In [31], Cwikel proved Simon's conjecture and Lieb, [94], used semigroup methods to bound tr[F(Ko)] for suitable functions F. Rozenblum's proof appeared in 1976 in [131], Lieb's was announced in 1976 in [94], and Cwikel's proof was published in 1977. Of the three methods, Lieb's gives by far the best estimates for the constants Co,d. A very nice and readable discussion of Lieb's method can be found in Chapter 9 of [152] and Chapter 3.4 of Ropstorff's book [129]. In particular, Ropstorff discusses the fact that an extension of Lieb's method to higher moments y > 0 gives the upper bound Cy,d < 27(y+d/2)(1+O((y+d/2)-1)) as ry+d/2 -> oc. Later proofs of the CLR bound were given by Li and Yau [93] and Conlon [27]; see also [92, 1321. The Lieb-Thirring inequalities (12) fit beautifully into the large coupling asymptotics. At least on a formal level, it is easy to lift the asymptotics for N(AV) _ S°(AV) to moments y > 0 by the following observation: For any -Y > 0, )ry = y fo,3 t7-1 dt

(S

=y

100

Jo

(s + t)°

dt

for all real s (here s° = 1 if s < 0 and zero if s > 0). Freely interchanging integrals and traces gives f S'Y (AV) = tr(-A + AV)" = y tr(-A + AV + t)°

J

=ryL°°(

0

(27r)d

1

0 0,0

ff(2 + V(x) + t)° d dx +o(Ad/2) C-ldt

ff(2 + AV (x))ry
Thus urn

A-(7+d/2)S-y(AV)

_

(2x)d

o(A'r+d/2).

ff(2 + V(x))I (14)

Ly,d f V

(x)y-+dl2

dx.

Of course, to make this sketch rigorous, one needs to handle the error terms more carefully, which we skip. This large coupling asymptotics shows that the best possible constants L y,d in the Lieb-Thirring inequality have the natural lower bound L,y,d > L' Id, or, equivalently, C.y,d > 1. 6A compact operator is in a trace ideal SP if its singular values are in the space 1P(N) and it is in the weak trace ideal Sw if its singular values are in the weak-lP space l,P (N); see, for example, [156].

D. HUNDERTMARK

470

1.5. A Sobolev Inequality for Fermions. Besides being mathematically very appealing, the y = 1 version of the Lieb-Thirring bound gives a Sobolev inequality for fermions whose d = 3 version has a nice application to the stabilityof-matter problem. For notational simplicity, we will not take the spin of the particles into account. The following gives a duality between a Lieb-Thirring type bound and a lower bound for the kinetic energy of an N-particle fermion system. Thus the kinetic energy inequality for fermions is an immediate corollary of the Lieb-Thirring bound for y = 1. THEOREM 1. The following two bounds are equivalent for non-negative convex functions G and F : 1E8+ -> R+ which are Legendre transforms of each other: The Lieb-Thirring bound,

EjJ = trL2(Ed)(Ho +V)_ < f G((V(x))_) dx,

(15)

(usually with Ho = -0, but this does not matter in the following) and the ThomasFermi bound, Ho

\

? fF(p(x))dx,

L2(Rd) / for all antisymmetric states 0C AN,L12(Rd) with norm one. Here

(16)

n=1

P,G(x):=N f

aN-1d

IZ/J(x,x2,...,xN)I2dx2...dxN

is the so-called one particle density associated with the antisymmetric N-particle

state.

For an explicit relation between F and G, see (18) and (19). Using the traditional Lieb-Thirring bound with y = 1, one immediately gets the following COROLLARY 2. For normalized

E

AN L2 (R d),

N

, - E Ojv j=1

L2(1RNd)

Kd F

with TF

Kd

d

d+

2(d+2 2

cl

Ld)

f P, (x) d2dx d+

C1,d2/d

>

2/d

-

ad

d

47r2

d+ 2 U)2/d'

One should note that the right-hand side of this bound is exactly the ThomasFermi prediction for the kinetic energy of N fermions and Kd F is the Thomas-Fermi

constant; see section 2.2. In particular, if C1,d is equal to one, then the ThomasFermi ansatz for the kinetic energy, a priori only supposed to be asymptotically correct for large N, should be a true lower bound for all N. This is a situation very much similar in spirit to Polya's conjecture. REMARK 3. Taking the spin of electrons into account, that is, assuming that b E A N (L2 (Rd, (Cq)) is normalized (and q = 2 for real electrons), one has the lower bound

N

E-Ojo) j=1

> (qCl

d)-2/dKdTF f

(x)(d+2)/ddx.

(17)

BOUND STATES IN QUANTUM MECHANICS

471

PROOF OF THEOREM 1: This proof is certainly known to the specialist, but we include it for completeness. In fact, the reverse implication is the easy one, (15) = (16): Fix N E N and let El < E2 < ... < EN < 0 be the first N negative eigenvalues of the one-particle Schrodinger operator H = Ho+V. Usually,

Ho = -0, but this does not really matter. By the min-max principle, we can assume without loss of generality that V is non-positive, V = -V_ = -U. If H has only J < N negative eigenvalues, then we put Ej = 0 for j = J + 1, . , N. Consider HN = EN1(Hoj - U(xj)) on AN L2(W1) be the sum of N independent copies of H. More precisely, one should write HN = N1 Hj with . .

®H®10 ... 01.

H3 =1

N-j-1 times

j times

Let cpl, ... , cON be the normalized eigenvectors corresponding to the eigenvalues

Ej (if J < N, pick any orthonormal functions for j > J) and put N

cpl A ... AWN GA L2 (R d),

the normalized antisymmetric tensor product of the cpj's, that is, a Slater determinant. Then N

N

J:IEnI n=1

N

-EEn -V),EHnV n=1 n=1 (0, N

N

\\

//,,//''

n=1

\

n=1

Since En 1 Un is a sum of one-body (multiplication) operators, we have (V), E'1 UnV)) f U(x)p,0(x) dx, by the definition of the one-particle density. Thus, taking (16) into account, one gets N

dx -

IEnj < f n=1

r

J

F(pV, (x)) dx

=

f(u(x)p(x) dx - F(p.,(x))) dx

<

f sup(U(x)t - F(t)) dx t>o

G(U(x)) dx,

where we were forced to put

G(s) := sup(st - F(t))

(18)

t>o

since we only know that pp (X) > 0. (16): This is certainly standard-the argument in the original case goes (15) through nearly without change. By min-max and the Lieb-Thirring inequality (15), we know that for any non-negative function U and any normalized antisymmetric N-particle V),

\

n=1

(Ho - U)z/i) > - tr(Ho - U) > /

f G(U(x)) dx.

D. HUNDERTMARK

472

Thus N n=1

HoV)) > K

_

UnV))

- fc U(x)) dx (

n=1 f

U(x)p,,(x) dx -

J

G(U(x)) dx

= J [U(x)p, (x) - G(U(x))] dx again by the definition of the one-particle density. Hence N

n=1

Ho>

>

U>p

[U(x)P,G (x) - G(U(x))] dx

J

G(U(x))] dx

sup U(x)>o

= f sup [sp(x) - G(s)] dx = fF(P(x)) dx, s>O

where, of course, we put

F(t) := sup(st - G(s)). s>o

REMARK 4. Since F and G are Legendre transforms of each other and since the double Legendre transform of a convex function reproduces the function (under suitable semi-continuity and convexity assumptions), we see that the Lieb-Thirringtype inequality (15) and the Thomas-Fermi-type kinetic energy bound (16) are dual to each other. In particular, one implies the other with the corresponding optimal

constants. This could be interesting in the hunt for sharp constants, since Eden and Foias gave in [36] a direct and rather simple proof of the kinetic energy lower bound in one dimension which, via the duality result of Theorem 1, still gives the best estimate for C1,1.

Following Lieb and Thirring, the bound in Theorem 1 has a beautiful application to the stability-of-matter problem which we will discuss a little bit in Section 2.3.

The Lieb-Thirring inequalities also found other applications, especially in the theory of non-linear evolution equations, as a tool to bound the dimension of attractors [29, 55, 59, 100, 133, 164].

1.6. Classical Results for the Lieb-Thirring Constants. The moment inequalities due to Lieb and Thirring are an important tool in the theory of Schrodinger operators since they connect a purely quantum mechanical quantity with its classical counterpart. Moreover, as we already saw, a dual version of it, the Sobolev inequality for fermions, is related to the theory of bulk matter. So a good understanding of the Lieb-Thirring coefficients is of some importance for our understanding of quantum mechanics. In general dimensions d c N, one now knows the following properties of C,y,d: Cy,d > 1, which follows from the Weyl-asymptotics. Monotonicity in -y: C.y,d < C7,,,d for all ry > yo (Aizenman and Lieb [2]).

BOUND STATES IN QUANTUM MECHANICS

473

Cy,d > 1 as soon as 'y < 1 (Helffer and Robert [66]7). The best bounds on C,y,d are due to Lieb [100], but they are explicitly dimension dependent and grow like Cy,d = 27ra as a = y + d/2 -> oo; see [129, Chapter 3.4]. Some special bounds in small dimensions: The bounds CI,1 < 27r

CI,2 < 6.03388 C1,3 < 5.96677

are due to Lieb [100] and, after twelve years of additional work, were slightly improved by Blanchard and Stubbe [20] in 1996 C1,1 < 5.81029

C,2 < 5.17690 C1,3 < 5.21809.

Unnoticed by the Schrodinger operator community, Eden and Foias [36] gave a simple and direct proof for the Thomas-Fermi-type kinetic energy inequality in one dimension. Using the duality between the ThomasFermi-type bound and the Lieb-Thirring inequality, their approach shows

that C1,1 < 7r/./ < 1.82, which is still the best available bound in this case.

There is a natural lower bound on Co,d using the fact that the CLR bound implies a Sobolev inequality8 [58], [118, eq. (4.24)], see also the discussion in [152, pp. 96-97], In three dimensions, it gives 4.6189 < Co,3 < 6.869

where the upper bound is from Lieb [94]. In particular, this shows that in dimension 3, Lieb's result is at most 49% off the best possible. In fact, the above lower bound is conjectured to be the correct value [58, 118, 152]. The monotonicity in -y is probably easiest to understand in the phase-space picture (12) of the Lieb-Thirring bounds: Let s- = (s)_ = (JsJ - s) be the negative part a of s. For any 0 < yo < y, there exists a positive (!) measure y on ll + such that9 (s)ry =

f(s + t) o

(dt).

71n particular, this disproved part of a conjecture of Lieb and Thirring made in [118]. 8More surprisingly, the Sobolev inequality together with the fact that -0 generates a Markov semigroup implies the CLR bound, see [92]. 91n fact, p.(dt) = ctry-YO-1 dt for an explicit constant c; just do the integral on the right-hand side by scaling.

D. HUNDERTMARK

474

With this we have

tr(-A + V)-' = <

j(tr(_+ V + t)°

2d(ISI2+V(x)+t)-'°d dxp(dt)

-,.,dd f f fa dd

(2 (21r)d

(dt)

dx

ff fLd(

12+V(x))ry d dx,

by freely interchanging the integrations and the trace. In particular, this shows C.y d <

Cy0,d.

Thus monotonicity in y is just a simple consequence of the Fubini-Tonelli theorem and scaling. In one dimension, much more is known: C3/2,1 = 1 (Gardner, Greene, Kruskal, and Miura [57]; Lieb and Thirring [118]). By monotonicity, this implies, C.y,1 = 1 for y > 2.

For 1 < y

2, an explicit solution10 of the variational problem

supV# ° }v7+vz El (V) I leads to C'7,1 > 2

(y- 1/2ry-1/2 y + 1/2)

(Keller [79], later rediscovered by Lieb and Thirring [118]). In particular, C.y,1 > 1 for y < 2 and C1,1 > 2/V' > 1.154, which should be compared to the Eden-Foias upper bound on C1,1. C1/2,1 < 00 (Weidl [167]). The sharp result C3/2,1 = 1 follows from the lower bound Cy,d > 1 and a sum rule for one-dimensional Schrodinger operators from the theory of the KdV equation. It reads 3 16

foo V(x)2 dx =

jEj I3/2

+ "scattering data".

oo

As noted on page 115 of [57], the contribution from the scattering data is nonnegative, so one can drop it to get an inequality for the moment of the negative 2 eigenvalues.11 It remains to note that L ,2,1 = 1s Lieb and Thirring [118] did not settle the critical case ry = 2 in one dimension. The question whether C112,1 is finite or not was open for twenty years until Weidl [167] showed that C1/2,1 < 4.02. But, despite some considerable interest, and in contrast to other results on sharp inequalities (see, e.g., [24, 25, 99] on Sobolev 10See also the very nice discussion in [15].

"It might be amusing to note that dropping the contribution of the eigenvalues gives an upper bound for the contribution of the scattering data which was the key to proof of a conjecture of Kiselev-Last-Simon [82] on the ac spectrum of one-dimensional Schrodinger operators with L2 potentials by Deift and Killip [33]. Sum rules have also turned out to be instrumental in the study of other related spectral problems; see [80, 81, 85, 124, 136, 137].

BOUND STATES IN QUANTUM MECHANICS

475

inequalities), the only sharp bound for the Lieb-Thirring inequalities for more than twenty years was the original result by Lieb and Thirring. This was especially tantalizing since, depending on the dimension, there are obvious conjectures for the sharp Lieb-Thirring constants,

CONJECTURE 5. In dimension d > 3, C1,d = 1. In particular, the ThomasFermi-type bound for the kinetic energy of N fermions should hold with the Thomas-

Fermi constant. In one dimension, 'y-y

for2<7<2

Cry,l=2(y

\y+

2

(with C1/2,1 = lim,y_..,l/z C.y,1 = 2) and the extremizers in the one-bound-state variational problem studied by Keller [79] and by Lieb and Thirring [118] should also be

extremizers in the Lieb-Thirring inequality. That is, up to scaling and translations, the extremizing potentials V in (13) for d = 1 and 2 < y < a are of the form

X1/4 ))-2.

V(x) _ -yz 11/4 (cosh)

Yz

In particular, for -y = 2, CI/z,I = 2 and the extremizing potential should be a multiple and translate of a delta function. For the first moment one should have C1,1 = 2/v < 1.155. Shortly after Weidl's result, Hundertmark, Lieb, and Thomas proved the y = 2 part of the one-dimensional Lieb-Thirring conjecture in [70]. THEOREM 6 (Hundertmark-Lieb-Thomas [70]). Suppose v is a non-negative measure with v(ll) < oo and let El < Ez < E3 < . < 0 be the negative eigenvalues counting multiplicity of the Schrodinger operator -ate - v (if any) given by the corresponding quadratic form. Then 00

< 2 v(l ) %=1

with equality if and only if the measure v is a single Dirac measure. Since Li,2,1 = 4, this shows C112,1 = 2, confirming the left endpoint of the one-dimensional Lieb-Thirring Conjecture 5. It might be interesting to note that Schmincke, [138], proved a corresponding sharp lower bound for one-dimensional Schrodinger operators with a potential. That is, for -8,2,, + V, he showed 00

-4J V(x)dx<E. 1

i=1

Schmincke's proof uses Schrodinger's factorization method (see also [30, 32]) and has been extended in [126] to some higher moments.

1.7. The Laptev-Weidl Extension of the Lieb-Thirring Bounds. The last few years saw a dramatic increase in our understanding of the Lieb-Thirring inequalities: C.y,d = 1 for y > 2 all d (Laptev and Weidl [91]).

C1,d < 2 for all d (Hundertmark, Laptev, and Weidl [69]). Hence also Cy,d < 2 for 1 < y < a , by monotonicity in y.

D. HUNDERTMARK

476

Cod < 81 all d > 3 (Hundertmark [68]). The key observation of Laptev and Weidl [91], which was already noted in Laptev [88], was to do something seemingly crazy: Extend the Lieb-Thirring inequalities from scalar to operator-valued potentials. They considered Schrodinger operators

of the form -A ® lg + V on the Hilbert space L2 (Rd, C), where V is now an operator-valued potential with values V(x) in the bounded self-adjoint operators on the auxiliary Hilbert space 9, and asked whether a bound of the form

trL2(Rd,g)(-® lg + V)ry

cop (2x)d

ffddx

+V(x))ry

(20)

RdRd

holds. Or, again doing the t; integral explicitly with the help of the spectral theorem and scaling, trL2(ad,g)(-A ® lg + V)ry < LOP d J d dx trg(V(x)ry+d/2).

(21)

This seemingly purely technical extension of the Lieb-Thirring inequality turned out to be the key in proving at least a part of the Lieb-Thirring conjecture. In fact, using this type of inequality, Laptev and Weidl noticed the following12 monotonicity properties of the operator-valued Lieb-Thirring constants Cop in the dimension:

1
and

op op 1
(22)

Assume that one knows C3j21 = 1. Then using the first part of (22) repeatedly, we get

1GC3/22

- C3/21C3/21 -1 1=1

that is, C3/22,2 = 1 and hence

1 G(3/23-c3/21c3/22=1 also. Thus an obvious induction in d shows 1 < C3/2 d - C3/2 103/2 d-1

and with the monotonicity in y, one concludes

Cry1 for y>3/2and all dEN. The beauty (and simplicity!) of this observation is that C3/211 = 1 is well-known (already in [118]) and Laptev and Weidl were able to prove that C3j21 = 1 by extending the Buslaev-Faddeev-Zakharov sum rules for the KdV equation [23, 173] to matrix-valued potentials. THEOREM 7 (Laptev-Weidl [91]). One has op C3/2 1 - 1.

121n fact, they observed that Lry d = Lc11Lry+2,d-1 by explicitly multiplying F-functions. So if (21) holds with the classical constant for some -y, then using monotonicity in -y, one can start an induction in the dimension argument. We prefer to avoid multiplying P-functions and to think of this in terms of the Cry d's, using a Fubini-type argument on the operator side instead.

BOUND STATES IN QUANTUM MECHANICS

477

For a nice alternative proof, which avoids the use of sum rules, see [14]. However, as beautiful as this result is, it sheds no light on the physically most important constants C1,d. Recently, Benguria and Loss [15] developed a new viewpoint on the Lieb-Thirring conjecture in one dimension. They connected a simplified "2 bound state version" with an isoperimetric problem for ovals in the plane. However, no progress has been made so far using this viewpoint but it has led to to very interesting mathematical problems; see, for example, [22]. The second submultiplicativity property of Cry d, together with the sharp bound due to Laptev and Weidl, shows that Clop < C1 P C3j2 1 = C1 P . Again by induction, this implies C1d
In particular, if the one-dimensional version of the Lieb-Thirring conjecture (see Conjecture 5) were true for operator-valued potentials, the uniform bound C1 d < 1.16 would follow.

Using ideas from [70], Hundertmark, Laptev, and Weidl extended the sharp bound in the critical case in one dimension to operator-valued potentials [69]. Together with the Aizenman-Lieb monotonicity in 'y, this gives the uniform bound Cop < C°pl = 2, for 1 < -y < 2. Together with the sharp result from Laptev and Weidl, this gives ,

-

THEOREM 8 (Hundertmark-Laptev-Weidl [69]). The bounds C.y,d < 2

for l < ry and all d > 1,

C.y,d :54

for! < y < 1 and all d > 2

on the Lieb-Thirring constants hold.

Moreover, the same estimates for the Lieb-Thirring constants for magnetic Schrodinger operators hold. This follows from the proof of these estimates: One strips off one dimension, but in one dimension there are no magnetic fields, since any "vector" potential in one dimension is gauge equivalent to the zero potential. Thus, by induction, in the dimension, the magnetic vector potential simply drops out, see [69] for details. It soon became obvious that one is not restricted to stripping off only one dimension at a time. More precisely, one has the following two general submultiplicativity properties of C" THEOREM 9 (Hundertmark [68]). For 1 < n < d,

1
-

pp

7,n

and

op

C +n/2 d-n'

In particular, stripping off n = 3 dimensions, and using that Co+3/2,d-3 = 1 for d > 3 by the Laptev-Weidl result, one immediately sees that Cop < Cry 3

for ally > O and d > 3.

(23)

Laptev asked the question [89] (see also [90]) whether, in particular, the CwikelLieb-Rozenblum estimate holds for Schrodinger operators with operator-valued potentials. The proof of this fact was also given in [68]:

D. HUNDERTMARK

478

THEOREM 10 (Hundertmark [68]). Let be some auxiliary Hilbert space and V a potential in Ld/2(Rd Sd/2(g)) with Sd/2 the Schatten-von Neumann operator ideal on G. Then the operator -0 ® 1g + V has a finite number N of negative eigenvalues. Furthermore, one has the bound

N < Lo d L trg(V(x)dl2) dx with

Lo,d < (21rKd)dLp d,

where the constant Kq is given by 8 1-2/q (27r)_d/(T)

Kq =

2

\1+ g-

l/

1/q

This shows Co,d < (27r Kd)d. The constant Kd is exactly the one given by Cwikel [31]. The proof of the above theorem is by extending Cwikel's method to an operator-valued setting. Thereby one recovers Cwikel's bound Co 3 < 81, which is 17 times larger than Lieb's estimate (however, for the scalar case). Nevertheless, using the submultiplicativity, it gives the uniform bound Cod < Co s < 81 which, by monotonicity in ry, also extends to moments 0 < ry < 2. We will not discuss the by now rather big literature on Lieb-Thirring inequalities for the Pauli operator (see, e.g., the review article by Erdos [40]), nor the quite extensive literature on quantum graphs (see [86]), nor the results on quantum wave guides (see, e.g., [39, 42, 43]), but we would like to mention one more recent and, at least for us, rather surprising result on Lieb-Thirring inequalities.

1.8. The Ekholm-Frank Result. It is well known that an attractive potential does not necessarily produce a bound state in three and more dimensions. This follows, for example, from Hardy's inequality, which says that in dimension three and more, the sharp operator inequality (d - 2)2 < 41x12

(24)

holds. Using this, one can refine the usual Lieb-Thirring inequalities in the following way: Using Hardy's inequality, for any e E (0, 1), one has

_

-0+V>-e0+(1-e)(4I

x12)2

+V.

Thus with the known Lieb-Thirring inequality, one gets

tr(-0 +V)ry < =

(d 41x12)2 C )a,d/2 JfRdXRd (1512 + (1 - e)

+V(x)J7

(2

L.y,de_

d/2 d

fm

C(1

- e)

(d - 2)2 41x12

+ V(X) )

'y+d/2

dx.

Of course, as a -> 0, the constant in front of the integral diverges. Very recently, Ekholm and Frank established that one can nevertheless take the limit e -> 0. They proved the rather surprising result

BOUND STATES IN QUANTUM MECHANICS

479

THEOREM 11 (Ekholm-Frank [38]). For moments -y > 0, the inequality

tr(-0 + V)ry <

2) liEdxRd

(41x12)2 +V(x))ry d dx

(2

= LEd f d (

(d - 2)2 4IxI2

\'y+d/2

+V(x) I

dx

/

holds in dimension three and more.

Thus, as far as moments are concerned, only the part of the potential below the critical Hardy potential is responsible for bound states. This amounts to an infinite phase-space renormalization on the level of the Lieb-Thirring inequality. Note that the Ekholm-Frank bound cannot hold for ry = 0. Also, effective bounds on CEd (resp. LEd) are not known.

2. Multi-Particle Coulomb Schrodinger Operators 2.1. The Coulomb Hamiltonian. The Hamiltonian for N electrons in the field of M nuclei is given by

H=HN,z,R=T+VC=T+VC,

(26)

where T = EN1 -Aj is the kinetic energy of N electrons, and (27)

Vc = Vee. + Vee

with

NM j=1 a=1

Z

Ix -Ra

the electron-nucleus interaction, 1

Vee = i

Ixi

<j

the electron-electron repulsion. Sometimes, one also considers Vnn =

za_ E Ra - RI' 0<0 I

the repulsion of the M nuclei at positions R = (R1i ... , RM) E R3M. We keep the position of the nuclei fixed, for simplicity. The electrons have spin q. In real life q = 2. Thus the N-electron operator HN,Z,R is defined on /feermions. ^ N (L2 (R2), C4), the antisymmetric subspace of L2([83N,CgN), since electrons are Note that in the units we use, the ground state energy of Hydrogen is 4. Zhislin's theorem [174] (see also [147] for a simple proof) guarantees the existence of a ground state if N < Ea Za + 1. One of the problems is to compute the ground state and the ground state energy, that is, N

EQ (N, Z, R) = inf

((VI, HN,z,R'O) 10 E / \(L2(R3, C4)),

11011

= 1)

(28)

and the minimizing groundstate wave function. The catch is that, although the Schrodinger equation is a linear equation, even the problem of two electrons in the field of one nucleus is not exactly solvable. Moreover, due to the exponential scaling

D. HUNDERTMARK

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of the degrees of freedom, there are no efficient methods to solve the Schrodinger equation approximately, even for moderate numbers of electrons. Nevertheless, one can give a rather complete answer in the limit of large atoms or molecules. As shown by Lieb and Simon, Thomas-Fermi theory becomes exact in this case.

2.2. Thomas-Fermi Theory. Thomas-Fermi theory is a simplification of the usual Schrodinger equation. For an excellent review and proofs of results, see [97]. A fundamental object in this theory (or approximation) is not the wave function any more, but the so-called single-particle density. Given a normalized wave function O c AN L2 (R', Co), recall that its single particle density is given by

p,p (x) = N

J

10(x, a1, x2, a2, ... , XN, QN)I2 dx2 ... dxN.

(29)

The electron nucleus interaction is easily expressed in terms of the density, M

(V, Ven) _ -

Z

j=1 JR-

Ix

- Rj I pp (x) dx.

(30)

The kinetic energy T = (0, Hoo) cannot be expressed by density p, but it can be approximately expressed by the density. For a free electron gas in a container of sidelength L in d dimensions, with periodic boundary conditions, say, the ground state energy is given by

(2)2 7r

T= (n,o)EZdx{1,..., q}

I<no

In the limit N - oo, one has

N=

do = gwdn0

g

1

111n I<no

(, ,a)EZdx{1,...,q} InI<"o

with wd the volume of the sphere of radius one. Also, again in the limit N z T = (Lv)

g

n2 do = Inl<no

d

(27r)2

2

d d +2

W dd

L

oo,

nd+2 0

N(d+2)/d

d + 2 (gwd)2/d

L2

Thus the kinetic energy density should be given by

T Ld

_

d

(2ir)2

d + 2 (gwd)2/d

N (d+2)/d Ld

q-2/dKa Fp(d+2)/d

with the density p = N/Ld and the Thomas-Fermi constant Ka F = dd+z (id d For slowly varying densities, the above suggest that the kinetic energy of N electrons (in the large N limit) is well-approximated by f TF = d with Kd - d + 2 /d2 (31) T = q 2/d K TF J d p(x) (d+2)/d dx d

BOUND STATES IN QUANTUM MECHANICS

481

In fact, the duality of the Lieb-Thirring bound and the kinetic energy bound for fermions shows that, as an inequality, this holds with the Thomas-Fermi constant, if the Lieb-Thirring inequality is true with the classical constant, that is, if C1,d = 1. Now let us concentrate on the physical relevant case of three space dimensions. Similarly as for the kinetic energy, the Coulomb repulsion of the electrons cannot be expressed by the density alone, but, keeping fingers crossed, it should be wellapproximated by the Coulomb integral. In fact, up to a small error, the Coulomb integral is a lower bound; see Lieb-Oxford [111] and Lieb [96]:

2<j x

xa

>2

fj

y) 3xR3

dxdy - cLOJ 3 PG(x)413 dx, (32)

where 1.234 < cLO < 1.68. For an alternative derivation, with a slight loss in the constants, see, for example, [121].

Ignoring the error terms and putting the above together, one is led to the Thomas-Fermi energy functional

S(P) = S(P, Z, R) = 4 2/3x3

F

f

/ P(x)5/3 dx - J 3 V(x)p(x) dx 3

P, (x)P,G (y) dxdy

+a

(33)

x-y with V(x) = EM1 ZRn for Coulomb matter. N(p) = fE3 p(x) dx = N is the J R3XR3

number of electrons (although in this formulation it does not have to be an integer). The functional is well-defined on the spaces S = {pI p > 0, p E L5/3(R3) n LI (R3)} Sa = {p E SI f p(x) dx < A}, and Saa = {p E SI f p(x) dx = Al. The Thomas-Fermi energy is given by ET F (N, Z, R) = inf {E(p, Z, R) I p E SaN }.

(34)

P

The catch with this definition is that Saa is not a convex set. But for N < Zn, it is shown in [116] that the minimum over the convex set SN exists and gives the same energy as ET F (N, R, Z). This follows from the strict convexity of the Thomas-Fermi functional. One can show (see [116, 97]) that for Coulomb matter, the function A --3 ETF (A, Z, R) is strictly convex on [0, E Z,,,] and constant if A is bigger than the total charge of the nuclei. Moreover, for A less or equal to the total charge, the minimizing density of the TF functional exists and is unique and is a solution of the so-called Thomas-Fermi equations. The Thomas-Fermi energy functional and the Thomas-Fermi energy have a natural scaling. For any 6 > 0, let p6 (X) = 62P(61/3x). Then S(P8, 6Z, 6-1/3R) = 6713E(p, Z, R).

(35)

So for a Coulomb system with nuclear charges Z = (Z1,. .. , ZM) at positions R = (R1,. .. , RM), this implies the scaling law ETF(6N, 6Z, 6-113R) = 67/3ETF(N, Z, R).

(36)

The main quantum mechanical limit theorem, proved by Lieb and Simon in [116] is that asymptotically, the Thomas-Fermi energy gives the correct answer for the ground state energy of atoms or molecules with a large nuclear charge.

D. HUNDERTMARK

482

THEOREM 12 (Lieb-Simon [116]). lim b

EQ (SZ)

1/3 R)

6713

00

= ETF(.A, Z, R).

Moreover, the quantum m-point densities, suitably rescaled, converge to a product p(x1) ... p(xm,) of Thomas-Fermi densities. Originally, the above result was proved using a decomposition of the space into boxes together with DirichletNeumann decoupling techniques. Motivated by a result of Thirring [165], Lieb used coherent states techniques in [97]. In addition, much more is known by now about lower order terms in the asymptotic for large atoms and in part for molecules. The energy has the asymptotic EQ(Z, R) = -aTF 1: Z,7/3 + 2 n

ZN + CiZ513 + O(Z5/3-E) n

(here aTF is the Thomas-Fermi prediction). The second term in this asymptotic is the Scott correction (see [140]); for molecules it was formulated in [97]. The second term asymptotic was established in a series of papers for atoms by Hughes [67] and Siedentop and Weikard [143, 142, 144]. For molecules it was established by Ivrii and Sigal [77]; see also [160]. There are also some results on the convergence of suitably scaled ground states, the so-called strong Scott conjecture; see [73, 74, 75]. The next higher correction due to Dirac and Schwinger was established for atoms in a monumental series of papers by Fefferman and Seco [54]. The Fefferman-Seco proof is a two-step proof. First, one reduces the problem, with an error less than O(Z5/3-E), to the study of an effective one-particle Hamiltonian which is given by the mean-field approximation. The second step then

consists of a very detailed analysis of the bound states of this mean-field Hamiltonian. Based on his work on error bounds for the Hartree-Fock approximation in [5], Bach gave in [6] a much simpler proof of the first step, the reduction to the mean-field Hamiltonian. For a variation on Bach's proof, see [63]. Bach's proof also works for molecules and not only atoms, but the spectral properties of the meanfield Hamiltonian have been established to the needed accuracy only for atoms. In this case, the mean-field Hamiltonian is rotationally symmetric and the study of its bound states, within accuracy of can be done by a quite delicate WKB O(Z5/3-E)

analysis. The abovementioned landmark paper of Simon and Lieb established many more

interesting results for the Thomas-Fermi theory of matter, for example, a rigorous treatment of the Thomas-Fermi theory of solids, also Teller's no-binding result for Thomas-Fermi theory. It says that, taking the nucleus-nucleus repulsion into account, the Thomas-Fermi energy of a molecule is always bigger than the sum of the energies of the individual atoms. Thus binding, which is due to the outermost electrons in atoms, is not correctly described by TF theory. For any nuclear charge vector Z = (Z1, ... , ZM) and positions of nuclei R = (R1, ... , RM), define

eTF(A,Z,R) =ETF(A,Z,R)+ E n-1

the energy with nucleus-nucleus repulsion. Then

ZnZm

A, - Rm l

BOUND STATES IN QUANTUM MECHANICS

483

THEOREM 13 (Teller's no-binding theorem). For any strictly positive Z = (Z1,.. ZM), that is, Zj > for all j, and R and A > 0, one has . ,

eTF(A, Z, R) > min [eTF(A', ZA, RA) + eTF(A - A', ZB, RB) 0<1\1<1\

for any decomposition ZA = (Z1, ... , Zk), ZB = (Zk+1, ... , Zn), and similarly for RA and RB .

This theorem is at the heart of the proof of stability of matter by Lieb and Thirring. The no-binding lower bound and the behavior of the Thomas-Fermi energy in terms of A together with scaling implies, in particular, that K

K

ETF(Zn, Zn) = ETF(1, 1) j ZZ/3.

eTF(A, Z, R) >n=1

(37)

n=1

By scaling, one has

E (1,1)=- K with K = q-2/3K3 F. Numerically, it is known that ETF = 2.21. So with the classical value of KTF and q = 2 for real electrons, one has ETF(1,1) > -0.385. On the other hand, it is known by now that molecules do bind in modifications of the Thomas-Fermi model, for example, the Thomas-Fermi-von Weizsacker model, as was established by Catto and Lions [26].

2.3. Stability of Matter. Following Lieb and Thirring, Teller's no-binding theorem for Thomas-Fermi theory and the kinetic energy lower bound in Theorem 1

has a beautiful application to the stability-of-matter problem. A basic fact of astrophysics claims that bulk matter undergoes gravitational collapse in the absence of nuclear forces. Onsager asked in the 1930's why bulk matter does not undergo an electrostatic collapse, too. This is due to the Pauli principle. The first proof of this fact is due to Dyson and Lennard in the 1960's [34, 35]. However their proof was rather complicated. On the other hand, this defect of Thomas-Fermi theory is in turn a chance to give a physically motivated and very appealing proof of stability of quantum mechanical matter, as soon as one can give a lower bound of the quantum mechanical energy of an antisymmetric N-fermion wave function in terms of the Thomas-Fermi energy of its one-particle density pp. The lower bound provided in Theorem 1 was the main tool of Lieb and Thirring [117], leading to a much simpler proof of stability of matter than the original one by Dyson and Lenard. At the same time, by relating the stability-of-matter question to the Thomas-Fermi model, it provided considerable new physical insight in the mechanism which prevents this collapse. (See [95, 102, 103] for a nice presentation of the physical and mathematical aspects of this problem.) Also, it turned out that not only the existence of these bounds but good estimates on the constants in these inequalities are of considerable importance for a rigorous understanding of the properties of matter [28, 46, 47, 62]. We will only sketch this application here. Note that the kinetic energy lower bound and the Lieb-Oxford bound give the following lower bound for the quantum

D. HUNDERTMARK

484

mechanical energy (V),

(HC + Vnn)

£KF

(PV))

-c LO Jf

P'P (x)4/3 dx.

Here £KF is the Thomas-Fermi functional with KTF in the kinetic energy term Recall that q = 2 for real electrons and that the replaced by equality C1,d = 1 is a long-standing open conjecture. The best known bounds so far say that C1,d < 2. (gC1,d)-2/3KTF.

Using Holder's inequality, one has f PO (x)4/3 dx <

(f(x) dx l

1/2

1/2

)

\f Pp (x)5/3 dx

< 4 N + ry

I

fp(x)/3 dx

Thus, replacing y by cLOy

(HC + Vnn)V)) > £KF(p) -

)2

(C

4

yN - fp(x5/3dx

(cL0)z

yN.

4

Taking the no-binding result of Thomas-Fermi theory into account, the right-hand side is bound below by e

K

LO 2

TF

1/y E Zn/3 n

- (c 4) yN =

TFN CK

1 1/y

7/3 C L. N n - c }Nl

l

with c = (cLC)2/(4ETF) < 0.3193. Maximizing w.r.t. y leads to the bound (0, (HC + Vnn)W)

-(gC1,3)2/3 KTF N (

\z E 7/3/NJt + n

Together with the easy bound 2ab < a2 + b2, one gets

/

ETF

(0, (HC + Vnn) `V) >- -(gC1,3)2/3 KTF (1 + C) I N + \

Znl3 l 1: n

which, in the case of real fermions q = 2 and the estimates C1,3 < 2 and c < 0.3193, gives (

, (HC + Vnn)O) ? -0.804 I N +

\

Zn7/3 f . n

/

(38)

As shown by Thirring [165], it is also known that, asymptotically, ThomasFermi theory is, indeed, the correct lower bound for the stability-of-matter result, K

Zn7/3(1 + O(Z 2/33),

(0, (Hc + Vnn,)4b) > -0.385 n=1

2/33) depend on the number of electrons. where the constants in the error term O(Zz Somewhat better error estimates were established in [123]. In particular, the robust estimate in (38) is not far from the truth. Note also that, following the route leading to (38), the best possible lower estimate for the Coulomb energy is -0.5078(N + Zn7/3) (using the conjectured value C1,3 = 1 and q = 2). En

BOUND STATES IN QUANTUM MECHANICS

485

REMARK 14. On can also use Thomas-Fermi theory to get a lower bound for the electron-electron repulsion: Reshuffling the terms in the stability result for Thomas-Fermi theory,13 1

J

-

p5/3 dx

1

x - yl

7=1

p(y) dy + D(p, p) + i

1

1
>

ET FyN

gives

?

1

1
Noting (,0,

ENI

b, E

1 x' -1 Y1 p(y) dy - D(p, p) - ETFyN - I/

j=1

f -s'

y1

1

i<j Ixi -x.71

fP5/3dx.

p(y) dyb) = 2D(pp, p) and choosing p= pp, one sees

b > D(pV,, pp) - TFyN - 1

y

/

f

P5/3

dx.

This way one gets the lower bound

(Hc + Vnn))

>-

JF K

)2

N 1+

Zn/3 /N n

Again, using Cauchy-Schwartz, a similar lower bound as before follows from this, with the factor 1 + c replaced by 2. The "best possible" lower bound for the energy of real matter this way is

/

('0, (HC + Vnn)0) >- -0.771 N + E Za/3

\

n

(see, e.g., [20, 122]), which should be compared with (38). Of course, this is not best possible, since, as demonstrated above, using the Lieb-Oxford bound improves the estimate quite a bit. We will not discuss other approaches to this circle of problems [60, 61, 44], nor the extension of the stability results to other models of real matter, for example, the Pauli operator [56, 106, 110, 48, 49], or relativistic models [52, 119, 120, 109, 113, 112], and, more recently, matter interacting with quantized electromagnetic fields [50, 51, 107, 108], but point the interested reader to the excellent reviews by Lieb [104, 105]. There is also a growing literature on stability/instability of the relativistic electron-positron field in Hartree-Fock approximation; see, for example, [7, 8, 9, 21, 64, 65, 711.

3. More on Bound States for Atoms It is known that an N-electron Coulomb system has a bound state as long as the number of electrons is lower than the total charge of the nuclei (plus one). This is known as Zhislin's theorem [174]. In nature, one can observe once negatively charged free atoms, but not twice or more negatively charged atoms. Formulated conservatively, one expects the maximal negative ionization to be independent of 13This was the route originally taken by Lieb and Thirring in [117].

D. HUNDERTMARK

486

the nucleus charge Z. This is known as the ionization conjecture (see, e.g., the review article by Simon [155]). For a more precise form of this conjecture, let

E(-)+ :: Z

HN,

j=1

1
1 (39)

Nlxi-xjI

be the Hamiltonian for N electrons in the field of an atom of charge Z. For fermions, the Hilbert space is AN L2(ll 3 (C2) Define

E(N, Z) = inf o,(HN,Z)

(40)

I = I(N, Z) = inf Oess(HN,Z) - info (HN,Z).

(41)

and the ionization energy By the HVZ theorem, Uess(HN, Z) = [E (N - 1, Z), oo), so

I(N, Z) = E(N - 1, Z) - E(N, Z). Note that HN,Z has a bound state, a discrete eigenvalue below its essential spectrum, if and only if I(N, Z) > 0. Due to electrostatic reasons, one expects that there should be an Ncr(Z) such

that

E(N, Z) = E(N - 1, Z) for all N > Ncr(Z), that is, the atom can bind only a finite number of electrons. The picture is not that simple, however. Heuristically, ignoring all many-body effects, the potential felt by the Nth electron is given by the effective potential Ueff(x)

-ICI +

f

x

1 ylpG(y)dy,

(42)

where p,, is the single-particle density of the other N - 1 electrons. The physics of the system should be somehow described by the effective Hamiltonian (43) Heff = -0 + Ueff. When N increases, pp increases and hence Ueff should increase at least in some average sense. Moreover, when N > Z, Newton's theorem implies Ueff(x) > 0 for large x. But for x near zero, Ueff is very negative since, due to the uncertainty principle, the electrons cannot concentrate too much close to zero. Hence the attraction of the nucleus is never fully screened. Nevertheless, eventually, Ueff will cease to have a bound state. Note that there is a big difference for fermions and bosons from this viewpoint: Whereas in order to bind N bosonic "electrons" the effective one-particle Hamiltonian should have at least one bound state, for fermionic systems, due to the Pauli principle, if one wants to bind N electrons the effective one particle Hamiltonian needs to have at least N bound states. For real atoms, this Nth bound state seems to disappear precisely when N - Z + 1. Bosons, however, should be much more easily bound to an atom than fermions. Thus, one expects the atomic Hamiltonian (39) not to have a fermionic bound

state if N > Ncr = Z + Q, where Z is the total charge of the nucleus and Q > 0 some fixed positive number, hopefully of the order of one. This innocent-looking conjecture, known as the ionization conjecture (see, e.g., the review article by Simon [155]), has withstood all attempts to prove it, even for very large but fixed excess charge Q. Only partial results are known.

BOUND STATES IN QUANTUM MECHANICS

487

3.1. Ruskai-Sigal-type Results. We are far from understanding the ionization conjecture rigorously, at least for fermions. One of the first results in this direction is the result by Ruskai and Sigal.

THEOREM 15 (Ruskai [134, 135]; Sigal [145, 146]). For any Z, there exist Ncr(Z) such that

I(N, Z) = E(N - 1, Z) - E(N, Z) = 0 for all N > Ncr(Z). In addition, for fermions, one has the bound lim sup

2

Z--oc

This theorem was proven independently by Ruskai [134, 135] and Sigal [145, 146]. A huge improvement of this theorem is due to Lieb. He showed THEOREM 16 (Lieb [101]). Independently of the statistics of the particles,

Ncr(Z) < 2Z + 1. In fact, Lieb also treats molecules and other refinements like magnetic fields and relativistic kinetic energies; see also Ichinose [76]. Another improvement of the Ruskai-Sigal bound is given by Lieb, Sigal, Simon, and Thirring [114]. They showed that large atoms are asymptotically neutral, lim Ncr(Z) = 1. Z

(44)

Z-00

Unfortunately, their proof used a compactness argument in the construction of suitable localizing functions and did not give any quantitative information or error estimates for finite nuclear charges Z. The first quantitative result seems to have been given by Fefferman and Seco in 1990. They proved THEOREM 17 (Fefferman-Seco [53]).

Ncr(Z) < Z + O(Z')

with a =

47 56

Shortly afterwards, this approach was simplified by by Sigal, Seco, and Solovej [141] who gave the estimate THEOREM 18. One has the bound

I(N, Z) < C1Z4o,13 - C2(N - Z)Zc,/3 for the ionization energy and as a consequence also

Ncr(Z) < Z + CZ-Here a = 47/56 as in the Fefferman-Seco theorem. Nevertheless, these bounds are still far from the (expected) truth, since Ncr(Z)Z and I(Z, Z) should be bounded in Z.

D. HUNDERTMARK

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3.2. Existence of Highly Negative Ions for Bosonic Atoms. As discussed above, for an atom to bind N electrons, the effective one particle Hamiltonian Heff should have at least N bound states. For bosons, however, it is enough

that it has at least one bound state. Thus a "bosonic" atom should be able to bind many more particles than one might naively expect from purely electrostatic reasons. This is indeed the case, THEOREM 19 (Benguria-Lieb [12]). For bosonic atoms, lim inf N,ZZ) z--oo

> 1+7.

Here 0 < -y < 1 is defined via the ground state of the Hartree functional: Let /' be the unique positive solution of the non-linear Hartree equation

-off - (iT - I,0I2*0 = 0. Then 1,0(x) I2

(1 +'Y)Z = f

dx.

R3

Benguria (see [97] for a reference) showed that 0 < ry < 1. Moreover, numerically it is known that ry = 0.21; see [11]. Later, Solovej proved a corresponding upper bound using similar ideas as in [141],

THEOREM 20 (Solovej [158]). For bosonic atoms,

N,, (Z) < (1 + 7)Z + CZ719. In particular, NrZ

lim

Z-oo

Z)

= 1+7.

3.3. Solution of the Ionization Conjecture in Hartree-Fock Theory. In Hartree-Fock theory (or better in the Hartree-Fock approximation), one does not consider the full N-body Hilbert space AN L2 ([83, C2), but restricts the attention to pure Slater determinants

0=vln...AVN

(45)

where the single-particle orbitals v? E L2(ll3,(C2). In other words, the density matrix

Y_ J ...

P(X,Ce,X2, 0'2,

.

, xN, 0'N) b (y, )3, x2) Q2 ... , xn,, ON) dx2... dxN (46)

is assumed to be a projection operator. Note that, for arbitrary

CAL 2 (]E83, C2),

the associated density matrix -y p is a bounded operator with 0 < rye < 1 on L2(R3

C2)

The Hartree-Fock energy of an atom of charge Z with N electrons is then defined to be

E HF = EHE (N, Z) =

inf (0, HN,z')

(47)

BOUND STATES IN QUANTUM MECHANICS

A little bit of calculation shows that for Slater determinants

489

,

(V), HN,Zb) = EHF('y )

with EHF(y) = trL2(a3,c2)

((-A - xI )y) + D(y) - Ex(7)

(48)

I

where trc2 (-Y (x, x)y(y, y)) dxdy

D(y) = z ff Ix - yI is the direct part of the Coulomb energy and

Ex(-y) = 2 ff

trC2I Iry(, )I2

dxdy

the exchange term. In particular, EHF(N, Z) = inf {,,HF(,y)I'y*'y ='y, try = N}.

The existence of a minimizing projection operator y for N < Z + 1 was proven by Lieb and Simon [115]. Moreover, given the Hartree-Fock functional, one can relax the assumption that y is a projection operator, as was shown by Lieb [98] EHF(N, Z) = inf {,,1F(y)10 < y < 1, tr(y) = N}. For a simple proof of this see [5]. For the Hartree-Fock approximation, Solovej gave a proof showing the ionization energy and maximum surcharge an atom can bind to be bounded uniformly in Z.

THEOREM 21 (Solovej [159]). For a neutral atom, the ionization energy in Hartree-Fock approximation is bounded, that is, IHF(Z, Z) = EHF(Z - 1, Z) - EHF(Z, Z) = O(1) as Z - oo.

Moreover, there exists a finite Q > 0 such that for all N > Z + Q, there are no minimizers for the Hartree-Fock functional among N-dimensional projections.

A related result for the Thomas-Fermi-von Weizsacker model was shown in [13]. Unfortunately, all results of this precise form are, so far, for models which shed no light on the Schrodinger case.

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Titles in This Series 76

Fritz Gesztesy (Managing editor), Percy Deift, Cherie Galvez, Peter Perry, and Wilhelm Schlag, Editors, Spectral theory and mathematical physics: A Festschrift in honor of Barry Simon's 60th birthday, Parts 1 and 2 (California Institute of Technology, Pasadena, CA, March 27-31, 2006)

75 Solomon Friedberg (Managing editor), Daniel Bump, Dorian Goldfeld, and Jeffrey Hoffstein, Editors, Multiple Dirichlet series, automorphic forms, and analytic number theory (Bretton Woods, New Hampshire, July 11-14, 2005)

74 Benson Farb, Editor, Problems on mapping class groups and related topics, 2006

73 Mikhail Lyubich and Leon Takhtajan, Editors, Graphs and patterns in mathematics and theoretical physics (Stony Brook University, Stony Brook, NY, June 14-21, 2001)

72 Michel L. Lapidus and Machiel van Frankenhuijsen, Editors, Fractal geometry and applications: A jubilee of Benoit Mandelbrot, Parts 1 and 2 (San Diego, California, 2002 and Ecole Normale Superieure de Lyon, 2001)

71 Gordana Matic and Clint McCrory, Editors, Topology and Geometry of Manifolds (University of Georgia, Athens, Georgia, 2001)

70 Michael D. Fried and Yasutaka Ihara, Editors, Arithmetic fundamental groups and noncommutative algebra (Mathematical Sciences Research Institute, Berkeley, California, 1999)

69 Anatole Katok, Rafael de la Llave, Yakov Pesin, and Howard Weiss, Editors, Smooth ergodic theory and its applications (University of Washington, Seattle, 1999)

68 Robert S. Doran and V. S. Varadarajan, Editors, The mathematical legacy of Harish-Chandra: A celebration of representation theory and harmonic analysis (Baltimore, Maryland, 1998)

67 Wayne Raskind and Charles Weibel, Editors, Algebraic K-theory (University of Washington, Seattle, 1997)

66 Robert S. Doran, Ze-Li Dou, and George T. Gilbert, Editors, Automorphic forms, automorphic representations, and arithmetic (Texas Christian University, Fort Worth, 1996)

65 M. Giaquinta, J. Shatah, and S. R. S. Varadhan, Editors, Differential equations: La Pietra 1996 (Villa La Pietra, Florence, Italy, 1996)

64 G. Ferreyra, R. Gardner, H. Hermes, and H. Sussmann, Editors, Differential geometry and control (University of Colorado, Boulder, 1997)

63 Alejandro Adem, Jon Carlson, Stewart Priddy, and Peter Webb, Editors, Group representations: Cohomology, group actions and topology (University of Washington, Seattle, 1996)

62 JAnos Kollar, Robert Lazarsfeld, and David R. Morrison, Editors, Algebraic geometry-Santa Cruz 1995 (University of California, Santa Cruz, July 1995)

61 T. N. Bailey and A. W. Knapp, Editors, Representation theory and automorphic forms (International Centre for Mathematical Sciences, Edinburgh, Scotland, March 1996)

60 David Jerison, I. M. Singer, and Daniel W. Stroock, Editors, The legacy of Norbert Wiener: A centennial symposium (Massachusetts Institute of Technology, Cambridge, October 1994)

59 William Arveson, Thomas Branson, and Irving Segal, Editors, Quantization, nonlinear partial differential equations, and operator algebra (Massachusetts Institute of Technology, Cambridge, June 1994)

58 Bill Jacob and Alex Rosenberg, Editors, K-theory and algebraic geometry: Connections with quadratic forms and division algebras (University of California, Santa Barbara, July 1992)

57 Michael C. Cranston and Mark A. Pinsky, Editors, Stochastic analysis (Cornell University, Ithaca, July 1993)

56 William J. Haboush and Brian J. Parshall, Editors, Algebraic groups and their generalizations (Pennsylvania State University, University Park, July 1991)

TITLES IN THIS SERIES

55 Uwe Jannsen, Steven L. Kleiman, and Jean-Pierre Serre, Editors, Motives (University of Washington, Seattle, July/August 1991)

54 Robert Greene and S. T. Yau, Editors, Differential geometry (University of California, Los Angeles, July 1990)

53 James A. Carlson, C. Herbert Clemens, and David R. Morrison, Editors, Complex geometry and Lie theory (Sundance, Utah, May 1989)

52 Eric Bedford, John P. D'Angelo, Robert E. Greene, and Steven G. Krantz, Editors, Several complex variables and complex geometry (University of California, Santa Cruz, July 1989)

51 William B. Arveson and Ronald G. Douglas, Editors, Operator theory/operator algebras and applications (University of New Hampshire, July 1988)

50 James Glimm, John Impagliazzo, and Isadore Singer, Editors, The legacy of John von Neumann (Hofstra University, Hempstead, New York, May/June 1988)

49 Robert C. Gunning and Leon Ehrenpreis, Editors, Theta functions - Bowdoin 1987 (Bowdoin College, Brunswick, Maine, July 1987)

48 R. O. Wells, Jr., Editor, The mathematical heritage of Hermann Weyl (Duke University, Durham, May 1987)

47 Paul Fong, Editor, The Arcata conference on representations of finite groups (Humboldt State University, Arcata, California, July 1986)

46 Spencer J. Bloch, Editor, Algebraic geometry - Bowdoin 1985 (Bowdoin College, Brunswick, Maine, July 1985)

45 Felix E. Browder, Editor, Nonlinear functional analysis and its applications (University of California, Berkeley, July 1983)

44 William K. Allard and Frederick J. Almgren, Jr., Editors, Geometric measure theory and the calculus of variations (Humboldt State University, Arcata, California, July/August 1984)

43 Francois Treves, Editor, Pseudodifferential operators and applications (University of Notre Dame, Notre Dame, Indiana, April 1984)

42 Anil Nerode and Richard A. Shore, Editors, Recursion theory (Cornell University, Ithaca, New York, June/July 1982) 41 Yum-Tong Siu, Editor, Complex analysis of several variables (Madison, Wisconsin, April 1982)

40 Peter Orlik, Editor, Singularities (Humboldt State University, Arcata, California, July/August 1981)

39 Felix E. Browder, Editor, The mathematical heritage of Henri Poincare (Indiana University, Bloomington, April 1980)

38 Richard V. Kadison, Editor, Operator algebras and applications (Queens University, Kingston, Ontario, July/August 1980)

37 Bruce Cooperstein and Geoffrey Mason, Editors, The Santa Cruz conference on finite groups (University of California, Santa Cruz, June/July 1979)

36 Robert Osserman and Alan Weinstein, Editors, Geometry of the Laplace operator (University of Hawaii, Honolulu, March 1979)

35 Guido Weiss and Stephen Wainger, Editors, Harmonic analysis in Euclidean spaces (Williams College, Williamstown, Massachusetts, July 1978)

34 D. K. Ray-Chaudhuri, Editor, Relations between combinatorics and other parts of mathematics (Ohio State University, Columbus, March 1978)

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