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American Mathematical Society Distribution Center 35 Monticello Place, Pawtucket, RI 02861 USA AMERICAN MATHEMATICAL SOCIETY A Comprehensive Course in Analysis Part 1: Real Analysis Barry Simon, California Institute of Technology, Pasadena, CA Part 2A: Basic Complex Analysis Part 2B: Advanced Complex Analysis A Comprehensive Course in Analysis by Poincaré Prize winner Barry Simon is a five-volume set that can serve as a graduate-level anal- Part 3: Harmonic Analysis ysis textbook with a lot of additional bonus information, including hundreds of problems and numerous notes that extend the text and Part 4: Operator Theory provide important historical background. Depth and breadth of Set: 2015; 3259 pages; Hardcover; ISBN: 978-1-4704-1098-8; List US$350; exposition make this set a valuable reference source for almost all AMS members US$280; Order code SIMON-SET areas of classical analysis. For more information go to www.ams.org/simon-set Hear from the author himself at www.facebook.com/simon.analysis. Also of interest: Analysis Functional Analysis An Epsilon of Room, I: Second Edition An Introduction Real Analysis Elliott H. Lieb, Princeton University, Yuli Eidelman and Vitali Milman, pages from year three of a mathemat- NJ, and Michael Loss, Georgia Institute Tel Aviv University, Israel, and Antonis ical blog of Technology, Atlanta, GA Tsolomitis, University of the Aegean, Terence Tao, University of California, Samos, Greece Los Angeles, CA Graduate Studies in Mathematics, Volume 14; 2001; 348 pages; Hardcover; ISBN: Graduate Studies in Mathematics, Graduate Studies in Mathematics, 978-0-8218-2783-3; List US$48; AMS members Volume 66; 2004; 322 pages; Hardcover; ISBN: Volume 117; 2010; 349 pages; Hardcover; ISBN: US$38.40; Order code GSM/14.R 978-0-8218-3646-0; List US$65; AMS members 978-0-8218-5278-1; List US$66; AMS members US$52; Order code GSM/66 US$52.80; Order code GSM/117 Or Order by Phone: facebook.com/amermathsoc (800)321-4267 (U.S. & Canada), @amermathsoc (401)455-4000 (Worldwide) plus.google.com/+AmsOrg.

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View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Caltech Authors PROOF OF THE STRONG SCOTT CONJECTURE FOR CHANDRASEKHAR ATOMS RUPERT L. FRANK, KONSTANTIN MERZ, HEINZ SIEDENTOP, AND BARRY SIMON Dedicated to Yakov Sinai on the occasion of his 85th birthday. Abstract. We consider a large neutral atom of atomic number Z, taking rel- ativistic effects into account by assuming the dispersion relation pc2p2 + c4. We study the behavior of the one-particle ground state density on the length scale Z−1 in the limit Z; c ! 1 keeping Z=c fixed and find that the spher- ically averaged density as well as all individual angular momentum densities separately converge to the relativistic hydrogenic ones. This proves the gen- eralization of the strong Scott conjecture for relativistic atoms and shows, in particular, that relativistic effects occur close to the nucleus. Along the way we prove upper bounds on the relativistic hydrogenic density. 1. Introduction 1.1. Some results on large Z-atoms. The asymptotic behavior of the ground state energy and the ground state density of atoms with large atomic number Z have been studied in detail in non-relativistic quantum mechanics. Soon after the advent of quantum mechanics it became clear that the non- relativistic quantum multi-particle problem is not analytically solvable and of in- creasing challenge with large particle number. This problem was addressed by Thomas [51] and Fermi [11, 12] by developing what was called the statistical model of the atom. The model is described by the so-called Thomas{Fermi functional (Lenz [28]) Z ZZ TF 3 5=3 Z 1 ρ(x)ρ(y) (1) EZ (ρ) := 10 γTFρ(x) − ρ(x) dx + dx dy ; 3 jxj 2 3 3 jx − yj R R ×R | {z } =:D[ρ] 2 2=3 where γTF = (6π =q) is a positive constant depending on the number q of spin states per electron, i.e., physically 2.
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On a Paley-Wiener Theorem for the Zs-Akns Scattering Transform
University of Kentucky UKnowledge Theses and Dissertations--Mathematics Mathematics 2013 ON A PALEY-WIENER THEOREM FOR THE ZS-AKNS SCATTERING TRANSFORM Ryan D. Walker University of Kentucky, [email protected] Right click to open a feedback form in a new tab to let us know how this document benefits ou.y Recommended Citation Walker, Ryan D., "ON A PALEY-WIENER THEOREM FOR THE ZS-AKNS SCATTERING TRANSFORM" (2013). Theses and Dissertations--Mathematics. 9. https://uknowledge.uky.edu/math_etds/9 This Doctoral Dissertation is brought to you for free and open access by the Mathematics at UKnowledge. It has been accepted for inclusion in Theses and Dissertations--Mathematics by an authorized administrator of UKnowledge. For more information, please contact [email protected]. STUDENT AGREEMENT: I represent that my thesis or dissertation and abstract are my original work. Proper attribution has been given to all outside sources. I understand that I am solely responsible for obtaining any needed copyright permissions. I have obtained and attached hereto needed written permission statements(s) from the owner(s) of each third-party copyrighted matter to be included in my work, allowing electronic distribution (if such use is not permitted by the fair use doctrine). I hereby grant to The University of Kentucky and its agents the non-exclusive license to archive and make accessible my work in whole or in part in all forms of media, now or hereafter known. I agree that the document mentioned above may be made available immediately for worldwide access unless a preapproved embargo applies. I retain all other ownership rights to the copyright of my work.
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On the Evolution of Scattering Data Under Perturbations of the Toda Lattice
ON THE EVOLUTION OF SCATTERING DATA UNDER PERTURBATIONS OF THE TODA LATTICE DENIZ BILMAN AND IRINA NENCIU Abstract. We present the results of an analytical and numerical study of the long-time behavior for certain Fermi-Pasta-Ulam (FPU) lattices viewed as perturbations of the completely integrable Toda lattice. Our main tools are the direct and inverse scattering transforms for doubly-infinite Jacobi matrices, which are well-known to linearize the Toda flow. We focus in particular on the evolution of the associated scattering data under the perturbed vs. the unperturbed equations. We find that the eigenvalues present initially in the scattering data converge to new, slightly perturbed eigenvalues under the perturbed dynamics of the lattice equation. To these eigenvalues correspond solitary waves that emerge from the solitons in the initial data. We also find that new eigenvalues emerge from the continuous spectrum as the lattice system is let to evolve under the perturbed dynamics. 1. Introduction The purpose of this work is to numerically investigate the long time behavior of solutions to certain perturbations of the completely integrable Toda lattice. In its wider context, this prob- lem lies within a set of fundamental questions concerning the qualitative features of solutions to nonlinear Hamiltonian partial differential or difference equations (PDEs or P∆Es, respectively). It is well known that in these types of equations several different phenomena can appear over long times, among them blow-up, scattering to the free evolution, or the emergence of stable nonlinear structures, such as solitary waves and breather solutions. A detailed description of any such phe- nomenon often depends on the precise structure of the equation in question.
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November 10, 2019 CURRICULUM VITAE Barry Simon IBM Professor
November 10, 2019 CURRICULUM VITAE Barry Simon IBM Professor of Mathematics and Theoretical Physics California Institute of Technology Pasadena, CA 91125 DATE OF BIRTH: April 16, 1946 EDUCATION A.B., summa cum laude, Harvard College, 1966 Ph.D. (Physics), Princeton University, 1970 POSITIONS AND TEACHING EXPERIENCE ◦ 1969/70 { Instructor, Department of Mathematics, Princeton ◦ 1970/72 { Assistant Professor, Departments of Mathematics and Physics, Princeton ◦ 1972/76 { Associate Professor, Departments of Mathematics and Physics, Princeton ◦ 1972/73 { On leave as Visiting Professor at IHES, Bures-sur-Yvette, France and the ETH, Zurich, Switzerland ◦ 1976/81 { Professor, Departments of Mathematics and Physics, Prince- ton ◦ 1976/77 { On leave as Visiting Professor at Department of Physics, Yeshiva University, NYC ◦ 1980/81 { On leave as a Sherman Fairchild Distinguished Visiting Scholar at California Institute of Technology ◦ 1981/ { Professor of Mathematics and Theoretical Physics, California Institute of Technology ◦ 1984/ { IBM Professor of Mathematics and Theoretical Physics, Cali- fornia Institute of Technology ◦ 2016/ { IBM Professor of Mathematics and Theoretical Physics, Emer- itus, California Institute of Technology HONORARY DEGREES ◦ D.Sc.h.c., Technion-Israel Institute of Technology, 1999 ◦ Honorary Fellow, University of Wales Swansea, 2006 1 ◦ D.h.c., Ludwig-Maximilians-UniversitätMünchen, 2014 HONORS ◦ 1965 Putnam Competition Winner (top five group) ◦ Sloan Foundation Fellow, 1972{76 ◦ Medal of International Academy of Atomic and Molecular Science, 1981 ◦ Fellow, American Physical Society, 1981 ◦ Stampacchia Prize, 1982 ◦ Guggenheim Fellowship, 1988{1989 ◦ Corresponding Member, Austrian Academy of Sciences, 1990 ◦ Fellow, American Academy of Arts and Sciences, 2005 ◦ PoincaréPrize, 2012 ◦ Fellow, American Math. Society, 2013 ◦ Simons Foundation Fellowship in Mathematics, 2013 ◦ Jew in the City Orthodox Allstars, 2014 (previous honoraries include Joe Lieberman and Jack Lew) ◦ Bolyai Prize of the Hungarian Academy of Sciences, 2015 ◦ American Math.
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Real Analysis a Comprehensive Course in Analysis, Part 1
Real Analysis A Comprehensive Course in Analysis, Part 1 Barry Simon Real Analysis A Comprehensive Course in Analysis, Part 1 http://dx.doi.org/10.1090/simon/001 Real Analysis A Comprehensive Course in Analysis, Part 1 Barry Simon Providence, Rhode Island 2010 Mathematics Subject Classiﬁcation. Primary 26-01, 28-01, 42-01, 46-01; Secondary 33-01, 35-01, 41-01, 52-01, 54-01, 60-01. For additional information and updates on this book, visit www.ams.org/bookpages/simon Library of Congress Cataloging-in-Publication Data Simon, Barry, 1946– Real analysis / Barry Simon. pages cm. — (A comprehensive course in analysis ; part 1) Includes bibliographical references and indexes. ISBN 978-1-4704-1099-5 (alk. paper) 1. Mathematical analysis—Textbooks. I. Title. QA300.S53 2015 515.8—dc23 2014047381 Copying and reprinting. Individual readers of this publication, and nonproﬁt libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright.
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Meetings of the MAA Ken Ross and Jim Tattersall
Meetings of the MAA Ken Ross and Jim Tattersall MEETINGS 1915-1928 “A Call for a Meeting to Organize a New National Mathematical Association” was DisseminateD to subscribers of the American Mathematical Monthly and other interesteD parties. A subsequent petition to the BoarD of EDitors of the Monthly containeD the names of 446 proponents of forming the association. The first meeting of the Association consisteD of organizational Discussions helD on December 30 and December 31, 1915, on the Ohio State University campus. 104 future members attendeD. A three-hour meeting of the “committee of the whole” on December 30 consiDereD tentative Drafts of the MAA constitution which was aDopteD the morning of December 31, with Details left to a committee. The constitution was publisheD in the January 1916 issue of The American Mathematical Monthly, official journal of The Mathematical Association of America. Following the business meeting, L. C. Karpinski gave an hour aDDress on “The Story of Algebra.” The Charter membership included 52 institutions and 1045 inDiviDuals, incluDing six members from China, two from EnglanD, anD one each from InDia, Italy, South Africa, anD Turkey. Except for the very first summer meeting in September 1916, at the Massachusetts Institute of Technology (M.I.T.) in CambriDge, Massachusetts, all national summer anD winter meetings discussed in this article were helD jointly with the AMS anD many were joint with the AAAS (American Association for the Advancement of Science) as well. That year the school haD been relocateD from the Back Bay area of Boston to a mile-long strip along the CambriDge siDe of the Charles River.
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Statistical Theory of the Atom in Momentum Space
STATISTICAL THEORY OF THE ATOM IN MOMENTUM SPACE dissertation an der fakultat¨ fur¨ mathematik, informatik und statistik der ludwig-maximilians-universitat¨ munchen¨ eingereicht von Verena von Conta am 19.11.2014 1. Gutachter: Prof. Dr. Heinz Siedentop 2. Gutachter: Prof. Dr. Simone Warzel Tag der Disputation: 13.04.2015 Zusammenfassung Im Jahre 1992 fuhrte¨ Englert [3] ein Impuls-Energie-Funktional fur¨ Atome ein und erörterte seinen Zusammenhang mit dem Thomas- Fermi-Funktional (Lenz [8]). Wir integrieren dieses Modell in eine ma- thematische Umgebung. Unter unseren Resultaten findet sich ein Be- weis fur¨ die Existenz und Eindeutigkeit einer minimierenden Impuls- dichte fur¨ dieses Impuls-Energie-Funktional; des Weiteren untersuchen wir einige Eigenschaften des Minimierers, darunter auch den Zusam- menhang mit der Euler-Gleichung. Wir verknupfen¨ die Minimierer des Thomas-Fermi-Funktionals mit dem Impuls-Energie-Funktional von Englert durch explizite Transfor- mationen. Wie sich herausstellt, können auf diese Weise bekannte Er- gebnisse aus dem Thomas-Fermi-Modell direkt auf das von uns be- trachtete Modell ubertragen¨ werden. Wir erhalten sogar die Aquiva-¨ lenz der beiden Funktionale bezuglich¨ ihrer Minima. Abschließend betrachten wir impulsabhängige Störungen. Insbeson- dere zeigen wir, dass die atomare Impulsdichte fur¨ große Kernladung in einem bestimmten Sinne gegen den Minimierer des Impuls-Energie- Funktionals konvergiert. Die vorliegende Arbeit basiert auf Zusammenarbeit mit Prof. Dr. Heinz Siedentop. Wesentliche Inhalte werden ebenfalls in einer ge- meinsamen Publikation [27] erscheinen. Abstract In 1992, Englert [3] found a momentum energy functional for atoms and discussed the relation to the Thomas-Fermi functional (Lenz [8]). We place this model in a mathematical setting.
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History of RSB Interview: Francesco Guerra January 26, 2021, 8:30Am-10:30Pm (EST)
History of RSB Interview: Francesco Guerra January 26, 2021, 8:30am-10:30pm (EST). Final revision: May 10, 2021 Interviewers: Patrick Charbonneau, Duke University, [email protected] Francesco Zamponi, ENS-Paris Location: Over Zoom, from Prof. Guerra’s second home in Genova, Italy. How to cite: P. Charbonneau, History of RSB Interview: Francesco Guerra, transcript of an oral history conducted 2021 by Patrick Charbonneau and Francesco Zamponi, History of RSB Project, CAPHÉS, École normale supérieure, Paris, 2021, 27 p. https://doi.org/10.34847/nkl.05bd6npc PC: Hello, Professor Guerra. Thank you very much for joining us. As we've dis- cussed, today we will be mostly discussing the history of replica symmetry breaking in a broad sense. Before we get there, however, we wanted to ask you a few questions about your background and interests. In particular, could you tell us what led you to be interested in physics and to pursue a Laurea degree in theoretical physics? FG: Like many people in Italy, in the first two years of university I was enrolled in engineering, like Ettore Majorana1 for example. Then I found that I was more interested in basic scientific questions, so I decided to shift to phys- ics. It was the start in Naples at the time—it was ‘62—of new, big enter- prises in physics. In Naples, there was the old institute of physics and they were involved mostly in classical physics, spectroscopy, solid state physics, and so on. Then, in the middle of the ‘50s, modern physics underwent an impressive development. My big boss at the time was Eduardo Caianiello2.
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Barry Simon and the János Bolyai International Mathematical Prize
Barry Simon and the JánosBolyai International Mathematical Prize Vilmos Totik∗ February 26, 2016 1 Introduction The recipient of the 2015 JánosBolyai International Mathematical Prize was Barry Simon (IBM Professor California Institute of Technology). The prize is given every 5 years by the Hungarian Academy of Sciences on the recommen- dation of a 10 member international committee (this is the only international prize of the Academy). It was established in 1902 for the 100th birth anniver- sary of the great Hungarian mathematician JánosBolyai, one of the founders of non-Euclidean geometry, and the first two awardees were Henry Poincaré(1905) and David Hilbert (1910). Then came World War I and the prize was not given until 2000, when the Academy renewed it. It is commonly accepted that, since there is no Nobel prize in mathematics, part of the original intention was to have a prestigious substitute that honors high quality mathematical work. In the renewed form the prize is given for monographs of high impact written in the preceding 10-15 years. In 2000 Saharon Shelah was the recipient for his book \Cardinal Arithmetic", in 2005 Mikhail Gromov got it for the monograph \Metric structures for Riemannian and non-Riemannian spaces", and the 2010 awardee was Yurii Manin for this work \Frobenius manifolds, quantum coho- mology, and moduli spaces". Barry Simon received the Bolyai Prize for his monumental two-volume trea- tise \Orthogonal Polynomials on the Unit Circle" published by the American Mathematical Society in the Colloquium Publications series in 2005. Simon does not need much introduction: he is one of the most cited mathematicians; the author of 21 monographs that has had profound influence on various fields of physics, mathematical physics and mathematics; among others he is the recipi- ent of the PoincaréPrize (2012), the Leroy P.
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Publications My Most Important Works Are [7], [11], [20], [27], [29], [30]
Publications My most important works are [7], [11], [20], [27], [29], [30]. My personal favorite is [29], for which a preliminary version is available from my home page. Close runners up are [10], [18], and [31]. Published: 32 Analyticity of extremizers to the Airy-Strichartz inequality, with Shuanglin Shao. Bull. Lond. Math. Soc. 44 (2012), no. 2, 336–352. 31 On non-local variational problems with lack of compactness related to non-linear optics, with Young-Ran Lee. J. Nonlinear Sci. 22 (2012), no. 1, 1–38. 30 Super-exponential decay of diffraction managed solitons, with Young-Ran Lee. Comm. Math. Phys. 309 (2012), no. 1, 1–21. 29 Exponential decay of dispersion management solitons, with Burak Erdogan and Young-Ran Lee. Math. Res. Lett. 18 (2011), no. 1, 11–24. 28 Heat-flow monotonicity of Strichartz norms, with Jonathan Bennett, Neal Bez, and Anthony Carbery. Anal. PDE 2 (2009), no. 2, 147–158. 27 Decay estimates and smoothness for solutions of the dispersion managed non-linear Schrödinger equation, with Young-Ran Lee. Comm. Math. Phys. 286 (2009), no. 3, 851–873. DOI 10.1007/s00220-008-0612-4. 26 A unified theoretical framework for fluctuating-charge models in atom-space and in bond- space, with Jiahao Chen and Todd Martinez, 44pp. J. Chem. Phys. 129, 214113 (2008); DOI:10.1063/1.3021400 25 Improved Hardy-Sobolev inequalities, with Alex Balinsky, Des Evans, and Roger Lewis. Ba- nach J. Math. Anal. 2 (2008), no. 2, 94–106. 24 A short introduction to Anderson localization, to appear in the proceedings of the LMS Meet- ing on Analysis and Stochastics of Growth Processes, Bath, September 2006, 25 pp.
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Relativistic Strong Scott Conjecture: a Short Proof
RELATIVISTIC STRONG SCOTT CONJECTURE: A SHORT PROOF RUPERT L. FRANK, KONSTANTIN MERZ, AND HEINZ SIEDENTOP Abstract. We consider heavy neutral atoms of atomic number Z modeled with kinetic energy (c2p2 + c4)1/2 − c2 used already by Chandrasekhar. We study the behavior of the one-particle ground state density on the length scale Z−1 in the limit Z,c →∞ keeping Z/c fixed. We give a short proof of a recent result by the authors and Barry Simon showing the convergence of the density to the relativistic hydrogenic density on this scale. 1. Introduction A simple description exhibiting some qualitative features of atoms of large atomic number Z with N electrons and with q spin states each is offered by the Chan- drasekhar operator N Z 1 N (1) c2∆ + c4 c2 + in L2(R3 : Cq) − ν − − x x x ν=1 ν 1 ν µ ν=1 X p | | ≤ν<µX≤N | − | ^ where c denotes the velocity of light. It is defined as the Friedrichs extension of N ∞ R3 Cq the corresponding quadratic form with form domain ν=1 C0 ( : ). By Kato’s inequality [7, Chapter 5, Equation (5.33)], it follows that the form is bounded from below if and only if Z/c 2/π (see also Herbst [5, TheoremV 2.5] and Weder [14]). N ≤ 1/2 R3N Cq N 2 R3 Cq For Z/c < 2/π its form domain is H ( : ) ν=1 L ( : ). Since there is no interaction involving the electron’s spin present,∩ we set q = 1 for notational simplicity. Moreover, we are restricting ourselves toV neutral atoms N = Z and fix γ = Z/c (0, 2/π).
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Equilibrium Measures and Capacities in Spectral Theory
Inverse Problems and Imaging Web site: http://www.aimSciences.org Volume 1, No. 4, 2007, 713–772 EQUILIBRIUM MEASURES AND CAPACITIES IN SPECTRAL THEORY Barry Simon∗ California Institute of Technology Pasadena, CA 91125 (Communicated by Lassi Päivärinta) Abstract. This is a comprehensive review of the uses of potential theory in studying the spectral theory of orthogonal polynomials. Much of the article focuses on the Stahl–Totik theory of regular measures, especially the case of OPRL and OPUC. Links are made to the study of ergodic Schrödinger op- erators where one of our new results implies that, in complete generality, the spectral measure is supported on a set of zero Hausdorff dimension (indeed, of capacity zero) in the region of strictly positive Lyapunov exponent. There are many examples and some new conjectures and indications of new research directions. Included are appendices on potential theory and on Fekete–Szeg˝o theory. 1. Introduction. This paper deals with applications of potential theory to spectral and inverse spectral theory, mainly to orthogonal polynomials especially on the real line (OPRL) and unit circle (OPUC). This is an area that has traditionally impacted both the orthogonal polynomial community and the spectral theory community with insufficient interrelation. The OP approach emphasizes the procedure of going from measure to recursion parameters, that is, the inverse spectral problem, while spectral theorists tend to start with recursion parameters and so work with direct spectral theory. Potential theory ideas in the orthogonal polynomial community go back at least to a deep 1919 paper of Faber [35] and a seminal 1924 paper of Szeg˝o[107] with crit- ical later developments of Kalmár [63] and Erdös–Turán [34].
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