C ONTEMPORARY M ATHEMATICS 476 ´ APORTACIONES MATEMATICAS SOCIEDAD MATEMÁTICA MEXICANA Fourth Summer School in Analysis and Mathematical Physics Topics in Spectral Theory and Quantum Mechanics May 2005 Universidad Nacional Autóónnnooommmaaa dddeee MMMéééxxxiiicccooo Cuernavaca, México Carlos Villegas-Blas Editor American Mathematical Society This page intentionally left blank Fourth Summer School in Analysis and Mathematical Physics Topics in Spectral Theory and Quantum Mechanics This page intentionally left blank C ONTEMPORARY M ATHEMATICS 476 APORTACIONES MATEMA´TICAS SOCIEDAD MATEMÁTICA MEXICANA Fourth Summer School in Analysis and Mathematical Physics Topics in Spectral Theory and Quantum Mechanics May 2005 Universidad Nacional Autónoma de México Cuernavaca, México Carlos Villegas-Blas Editor American Mathematical Society Providence, Rhode Island Editorial Board of Contemporary Mathematics Dennis DeTurck, managing editor George Andrews Abel Klein Martin J. Strauss Editorial Board of Aportaciones Matem´aticas Guillermo Pastor and Luz de Tesesa, Managing Editors Marcelo Aguilar Luis Hern´andez-Lamoneda Jos´e Seade Jos´e Mar´ıa Gonza´lez-Barrios Max Neumann Martha Takane Luis Gorostiza Sergio Rajsbaum Jorge Velasco 2000 Mathematics Subject Classification. Primary81–02, 82D30, 35P05, 81Q99. Library of Congress Cataloging-in-Publication Data SummerSchoolinanalysisandMathematicalPhysics(4th: 2005: Cuernavaca,Mexico) FourthSummerSchoolinAnalysisandMathematicalPhysics: topicsinspectraltheoryand quantum mechanics : Fourth Summer School in Analysis and MathematicalPhysics, May 2005, Cuernavaca,Mexico/CarlosVillegas-Blas,editor. p.cm. —(Contemporarymathematics,ISSN0271-4132;476)(Aportacionesmatem´aticas/ SociedadMatem´aticaMexicana) Includesbibliographicalreferences. ISBN978-0-8218-4064-1(alk.paper) 1. Spectral theory (Mathematics)—Congresses. 2. Schr¨odinger operator—Congresses. 3.Quantumtheory—Congresses. I.Villegas-Blas,Carlos,1964– II.SociedadMatem´aticaMex- icana. III.Title. IV.Title: Topicsinspectraltheoryandquantummechanics. QC20.7.S64S76 2005 515(cid:1).7222—dc22 2008028976 Copying and reprinting. Materialinthis bookmaybe reproducedby anymeansforedu- cationaland scientific purposes without fee or permissionwith the exception of reproduction by servicesthatcollectfeesfordeliveryofdocumentsandprovidedthatthecustomaryacknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercialuseofmaterialshouldbeaddressedtotheAcquisitionsDepartment,AmericanMath- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can [email protected]. Excludedfromtheseprovisionsismaterialinarticlesforwhichtheauthorholdscopyright. In suchcases,requestsforpermissiontouseorreprintshouldbeaddresseddirectlytotheauthor(s). (Copyrightownershipisindicatedinthenoticeinthelowerright-handcornerofthefirstpageof eacharticle.) (cid:1)c 2008bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. Copyrightofindividualarticlesmayreverttothepublicdomain28years afterpublication. ContacttheAMSforcopyrightstatusofindividualarticles. PrintedintheUnitedStatesofAmerica. (cid:1)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 131211100908 Contents Introduction vii Isoperimetric Inequalities for Eigenvalues of the Laplace Operator Rafael D. Benguria and Helmut Linde 1 Lectures on Random Schro¨dinger Operators Peter D. Hislop 41 Quantum Mechanics, The Stability of Matter and Quantum Electrodynamics Elliott H. Lieb 131 v This page intentionally left blank Introduction Quantum mechanics started at the beginning of the last century as a revolu- tionary theory to explain certain experimental facts which were in contradiction with the classical physics theory. One of the very basic aspects of this new theory was to recognize the wave-like behavior of particles. As a consequence, the study of characteristic frequencies of a given system plays a key role in the theory. In mathematical terms, the spectral theory of Schr¨odinger operators turned out to be part of the base of Quantum Mechanics. This volume is about three topics strongly related to the development of spec- tral theory. It consists of three expository articles written by outstanding re- searchers in mathematical physics and spectral theory: Rafael D. Benguria, Peter D.Hislop, ElliottH.LiebandHelmutLinde. Thearticlesarebasedonthelectures delivered by Benguria, Hislop and Lieb in the IV Summer School in analysis and mathematicalphysics thattookplaceattheInstituteofMathematics, Universidad Nacional Auto´nomade M´exico, Cuernavaca, May 2005. The main goalof boththe lectures and the articles is to link the basic knowledge of a graduate student in mathematics with three current research topics in mathematical physics: Isoperi- metric inequalities for eigenvalues of the Laplace Operator, Random Schro¨dinger Operators and Stability of Matter. The above-mentioned authors have made de- cisive contributions to their corresponding research fields. Thus graduate students will find very well written articles guiding and introducing them into current re- search topics and specialist researchers will find information on recent progress in some areas of mathematical physics written by some of the people who have made it. Finally let me mention that the article written by Elliott H. Lieb has already appeared in a previous publication: The Stability of Matter and Quantum Elec- trodynamics, the Jahresbericht of the German Math. Soc. JB 106, 93-110 (2004) (Teubner). Professor Lieb based his lectures on this article and has been so kind to let us reproduce it in this volume in order to make it more complete as the proceedings of the above-mentioned summer school. Carlos Villegas Blas, editor. Universidad Nacional Auto´noma de M´exico, Instituto de Matem´aticas, Unidad Cuernavaca. email: [email protected] vii This page intentionally left blank