Friedrich (Fritz) Gesztesy
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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. -
Quasi Regular Dirichlet Forms and the Stochastic Quantization Problem
Quasi regular Dirichlet forms and the stochastic quantization problem Sergio Albeverio ∗† Zhi Ming Ma ‡ Michael R¨ockner †§ Dedicated to Masatoshi Fukushima for his 80th birthday 7.4.14 Abstract After recalling basic features of the theory of symmetric quasi regu- lar Dirichlet forms we show how by applying it to the stochastic quan- tization equation, with Gaussian space-time noise, one obtains weak solutions in a large invariant set. Subsequently, we discuss non sym- metric quasi regular Dirichlet forms and show in particular by two simple examples in infinite dimensions that infinitesimal invariance, does not imply global invariance. We also present a simple example of non-Markov uniqueness in infinite dimensions. 1 Introduction The theory of symmetric Dirichlet forms is the natural extension to higher dimensional state spaces of the classical Feller theory of stochastic processes on the real line. Through ground breaking work by Beurling and Deny (1958- 59), [BD58], [BD59], [Den70], [Sil74], [Sil76], and Fukushima (since 1971), [Fu71a], [Fu71b], [Fu80], [FOT11], [CF12], it has developed into a powerful method of combining analytic and functional analytic, as well as poten- tial theoretic and probabilistic methods to handle the basic construction of stochastic processes under natural conditions about the local characteris- arXiv:1404.2757v2 [math.PR] 30 Apr 2014 tics (e.g., drift and diffusion coefficients), avoiding unnecessary regularity conditions. Such processes arise in a number of applied problems where coefficients can be singular, locally or at infinity. For detailed expositions of the gen- eral theory, mainly concentrated on finite dimensional state spaces, see ∗Inst. Appl. Mathematics and HCM, University of Bonn; CERFIM (Locarno); [email protected] †BiBoS (Bielefeld, Bonn) ‡Inst. -
Sergio Albeverio
Sergio Albeverio Academic career 1966 Dr. rer. nat., ETH Zurich,¨ Switzerland (supervi- sor: R. Jost, M. Fierz) 1962 - 1971 Assistant Professor, Lecturer, Research Fel- low: ETH Zurich¨ (Switzerland), Imperial Colle- ge London (England, UK), Princeton University (NJ, USA) 1972 - 1977 Professor (Marseille, Oslo, and Naples) 1977 - 1979 Professor (H3), University of Bielefeld 1979 - 1997 Professor (H4), University of Bochum 1997 - 2008 Professor (H4/C4), University of Bonn 1997 - 2009 Professor and Director of Mathematics Sect., Accademia di Achitettura, USI, Mendrisio, Swit- zerland Since 2008 Professor Emeritus Longer research stays and invited professor- ships at many universities and research cen- ters in Europe, China, Japan, Mexico, Rus- sia/USSR, Saudi Arabia, USA Honours 1969 - 1971 SNF Research Awards 1993 Max Planck Research Award Since 2002 Listed in the ISI Highly Cited Scientists 2002 Doctor honoris causa, Oslo University, Norway (Bicentennial of N. H. Abel) 2002 Visiting Professorship (over several years) per chiara fama, University of Trento 2003 Bonn University Prize for an interdisciplinary project “Extreme events in natural and artificial systems” 2011 - 2015 Chair Professorship in Mathematics, King Fahd University of Petroleum and Mi- nerals (KFUPM) Offers 1988 Chair of Mathematics, per chiara fama, Roma University II, Italy Invited Lectures Plenary lecturer at over 230 international meetings, including: 1977 International Congress on Mathematical Physics, Rome, Italy 1983 International Congress on Mathematical Physics, Boulder, CO, USA 1986 International Congress on Mathematical Physics, Marseille, France 1988 International Congress on Mathematical Physics, Swansea, Wales, UK 1994 N. Wiener Memorial Symposium, East Lansing, MI, USA 1995 S. Lefshetz Memorial Lecture, UNAM, Mex. Soc. Math./AMS, Mexico City 2000 Conference in Honor of S. -
Arxiv:1903.00779V1 [Math.SP] 2 Mar 2019 O Clrsh¨Dne Prtr ( Schr¨Odinger Operators Scalar for Rn O P29177
THE INVERSE APPROACH TO DIRAC-TYPE SYSTEMS BASED ON THE A-FUNCTION CONCEPT FRITZ GESZTESY AND ALEXANDER SAKHNOVICH Abstract. The principal objective in this paper is a new inverse approach to general Dirac-type systems of the form ′ y (x,z)= i(zJ + JV (x))y(x,z) (x ≥ 0), ⊤ where y = (y1,...,ym) and (for m1, m2 ∈ N) Im1 0m1×m2 0m1 v J = , V = ∗ , m1 + m2 =: m, 0m2×m1 −Im2 v 0m2 m ×m for v ∈ C1([0, ∞)) 1 2 , modeled after B. Simon’s 1999 inverse approach to half-line Schr¨odinger operators. In particular, we derive the A-equation associated to this Dirac-type system in the (z-independent) form x ∂ ∂ ∗ A(x,ℓ)= A(x,ℓ)+ A(x − t, ℓ)A(0,ℓ) A(t, ℓ) dt (x ≥ 0,ℓ ≥ 0). ∂ℓ ∂x ˆ0 Given the fundamental positivity condition ST > 0 in (1.14) (cf. (1.13) for details), we prove that this integro-differential equation for A( · , · ) is uniquely solvable for initial conditions 1 m ×m A( · , 0) = A( · ) ∈ C ([0, ∞)) 2 1 , m ×m and the corresponding potential coefficient v ∈ C1([0, ∞)) 1 2 can be recovered from A( · , · ) via ∗ v(ℓ) = −iA(0,ℓ) (ℓ ≥ 0). Contents 1. Introduction 1 2. Preliminaries 6 3. The A-Function for Dirac-Type Systems: Part I 7 4. The A-Function for Dirac-Type Systems: Part II 9 5. The A-Equation for Dirac-Type Systems 13 6. The Inverse Approach 15 arXiv:1903.00779v1 [math.SP] 2 Mar 2019 Appendix A. Various Results in Support of Step 2 in the Proof of Theorem 5.1 19 References 29 1. -
Scientific Report for 2012
Scientific Report for 2012 Impressum: Eigent¨umer,Verleger, Herausgeber: The Erwin Schr¨odingerInternational Institute for Mathematical Physics - University of Vienna (DVR 0065528), Boltzmanngasse 9, A-1090 Vienna. Redaktion: Joachim Schwermer, Jakob Yngvason. Supported by the Austrian Federal Ministry of Science and Research (BMWF) via the University of Vienna. Contents Preface 3 The Institute and its Mission . 3 Scientific activities in 2012 . 4 The ESI in 2012 . 7 Scientific Reports 9 Main Research Programmes . 9 Automorphic Forms: Arithmetic and Geometry . 9 K-theory and Quantum Fields . 14 The Interaction of Geometry and Representation Theory. Exploring new frontiers. 18 Modern Methods of Time-Frequency Analysis II . 22 Workshops Organized Outside the Main Programmes . 32 Operator Related Function Theory . 32 Higher Spin Gravity . 34 Computational Inverse Problems . 35 Periodic Orbits in Dynamical Systems . 37 EMS-IAMP Summer School on Quantum Chaos . 39 Golod-Shafarevich Groups and Algebras, and the Rank Gradient . 41 Recent Developments in the Mathematical Analysis of Large Systems . 44 9th Vienna Central European Seminar on Particle Physics and Quantum Field Theory: Dark Matter, Dark Energy, Black Holes and Quantum Aspects of the Universe . 46 Dynamics of General Relativity: Black Holes and Asymptotics . 47 Research in Teams . 49 Bruno Nachtergaele et al: Disordered Oscillator Systems . 49 Alexander Fel'shtyn et al: Twisted Conjugacy Classes in Discrete Groups . 50 Erez Lapid et al: Whittaker Periods of Automorphic Forms . 53 Dale Cutkosky et al: Resolution of Surface Singularities in Positive Characteristic . 55 Senior Research Fellows Programme . 57 James Cogdell: L-functions and Functoriality . 57 Detlev Buchholz: Fundamentals and Highlights of Algebraic Quantum Field Theory . -
A New Approach to Inverse Spectral Theory, II. General Real Potentials
Annals of Mathematics, 152 (2000), 593–643 A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure∗ By Fritz Gesztesy and Barry Simon Abstract We continue the study of the A-amplitude associated to a half-line d2 2 Schr¨odinger operator, 2 + q in L ((0, b)), b . A is related to the Weyl- − dx ≤∞ Titchmarsh m-function via m( κ2)= κ a A(α)e 2ακ dα+O(e (2a ε)κ) for − − − 0 − − − all ε> 0. We discuss five issues here. First,R we extend the theory to general q in L1((0, a)) for all a, including q’s which are limit circle at infinity. Second, we prove the following relation between the A-amplitude and the spectral measure 1 ρ: A(α) = 2 ∞ λ− 2 sin(2α√λ) dρ(λ) (since the integral is divergent, this − −∞ formula has toR be properly interpreted). Third, we provide a Laplace trans- form representation for m without error term in the case b < . Fourth, we ∞ discuss m-functions associated to other boundary conditions than the Dirichlet boundary conditions associated to the principal Weyl-Titchmarsh m-function. Finally, we discuss some examples where one can compute A exactly. 1. Introduction In this paper we will consider Schr¨odinger operators d2 (1.1) + q −dx2 in L2((0, b)) for 0 <b< or b = and real-valued locally integrable q. arXiv:math/9809182v2 [math.SP] 1 Sep 2000 ∞ ∞ There are essentially four distinct cases. ∗This material is based upon work supported by the National Science Foundation under Grant No. -
Contemporary Mathematics 526
CONTEMPORARY MATHEMATICS 526 Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena The 2008–2009 Research Program on Nonlinear Partial Differential Equations Centre for Advanced Study at the Norwegian Academy of Science and Letters Oslo, Norway Helge Holden Kenneth H. Karlsen Editors American Mathematical Society http://dx.doi.org/10.1090/conm/526 Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena CONTEMPORARY MATHEMATICS 526 Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena The 2008–2009 Research Program on Nonlinear Partial Differential Equations Centre for Advanced Study at the Norwegian Academy of Science and Letters Oslo, Norway Helge Holden Kenneth H. Karlsen Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Abel Klein Martin J. Strauss 2000 Mathematics Subject Classification. Primary 35L65, 35Q30, 35G25, 35J70, 35K65, 35B65, 39A14, 35B25, 35L05, 65M08. Library of Congress Cataloging-in-Publication Data Det Norske Videnskaps-Akademi. Research Program on Nonlinear Partial Differential Equations (2008–2009 : Centre for Advanced Study at the Norwegian Academy of Science and Letters) Nonlinear partial differential equations and hyperbolic wave phenomena : 2008–2009 Research Program on Nonlinear Partial Differential Equations, Centre for Advanced Study at the Norwegian Academy of Science and Letters, Oslo, Norway / [edited by] Helge Holden, Kenneth H. Karlsen. p. cm. — (Contemporary mathematics ; v. 526) Includes -
Notices of the American Mathematical Society
OF THE 1994 AMS Election Special Section page 7 4 7 Fields Medals and Nevanlinna Prize Awarded at ICM-94 page 763 SEPTEMBER 1994, VOLUME 41, NUMBER 7 Providence, Rhode Island, USA ISSN 0002-9920 Calendar of AMS Meetings and Conferences This calendar lists all meetings and conferences approved prior to the date this issue insofar as is possible. Instructions for submission of abstracts can be found in the went to press. The summer and annual meetings are joint meetings with the Mathe· January 1994 issue of the Notices on page 43. Abstracts of papers to be presented at matical Association of America. the meeting must be received at the headquarters of the Society in Providence, Rhode Abstracts of papers presented at a meeting of the Society are published in the Island, on or before the deadline given below for the meeting. Note that the deadline for journal Abstracts of papers presented to the American Mathematical Society in the abstracts for consideration for presentation at special sessions is usually three weeks issue corresponding to that of the Notices which contains the program of the meeting, earlier than that specified below. Meetings Abstract Program Meeting# Date Place Deadline Issue 895 t October 28-29, 1994 Stillwater, Oklahoma Expired October 896 t November 11-13, 1994 Richmond, Virginia Expired October 897 * January 4-7, 1995 (101st Annual Meeting) San Francisco, California October 3 January 898 * March 4-5, 1995 Hartford, Connecticut December 1 March 899 * March 17-18, 1995 Orlando, Florida December 1 March 900 * March 24-25, -
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. -
Nontrivial Attractors in a Model Related to the Three-Body Quantum Problem
Acta Applicandae Mathematicae 48: 113±184, 1997. 113 c 1997 Kluwer Academic Publishers. Printed in the Netherlands. Nontrivial Attractors in a Model Related to the Three-Body Quantum Problem SERGIO ALBEVERIO1,2 and KONSTANTIN A. MAKAROV1,3 1FakultatÈ furÈ Mathematik, Ruhr UniversitatÈ Bochum, D-44780 Bochum, Germany 2BiBoS; SFB 237 Bochum-Essen-Dusseldorf;È CERFIM, Locarno, Italy 3Department of Computational and Mathematical Physics, St. Petersburg University, 198094 St. Petersburg, Russia (Received: 29 January 1996) Abstract. A model of a quantum mechanical system related to the three-body problem is studied. The model is de®ned in terms of a symmetric pseudo-differential operator (PDO) with unbounded symbol. The entire family of self-adjoint extensions of this operator is studied using harmonic analysis. A regularization procedure for this PDO is introduced and the spectral properties of the operators obtained in this way are investigated. The limit behavior of the regularized operators when the regularization parameter is removed is analyzed and a nontrivial attractor is exhibited. Mathematics Subject Classi®cations (1991). 47A20, 47B35, 81Q10. Key words: extension theory of symmetric operators, convolutions, pseudo-differential operators, harmonic analysis, E®mov effect, three-body quantum problem, regularization, attractors, Hamil- tonians which are not lower semibounded, spectral analysis, Lagrangian planes. 0. Introduction In 1933, Wigner [1] remarked that nuclear forces must be of very short range and very strong. Two years later, Bethe and Peierls [2] in order to model such forces introduced point potentials that vanish except at the origin and so correspond to zero range forces. The introduction of zero-range potentials gave a powerful impulse to study quantum mechanical systems with singular interaction, many of which being also solvable [3]. -
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. -
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-Universit¨atM¨unchen, 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 ◦ Poincar´ePrize, 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.