Sturm- Liouville Theory Past and Present Werner O.Amrein Andreas M.Hinz David B.Pearson (Editors) Birkhäuser Verlag Basel Boston Berlin • • Editors: Werner O.Amrein Andreas M.Hinz Section de Physique Mathematisches Institut Université de Genève Universität München 24,quai Ernest-Ansermet Theresienstrasse 39 1211 Genève 4 D-80333 München Switzerland Germany [email protected] [email protected] David P.Pearson Department of Mathematics University of Hull Cottingham Road Hull HU6 7RX United Kingdom [email protected] 2000 Mathematical Subject Classification 34B24,34C10,34L05,34L10,01A55,01A10 A CIP catalogue record for this book is available from the Library of Congress,Washington D.C.,USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. ISBN 3-7643-7066-1 Birkhäuser Verlag,Basel – Boston – Berlin This work is subject to copyright.All rights are reserved,whether the whole or part of the material is concerned, specifically the rights of translation,reprinting,re-use of illustrations,recitation,broadcasting,reproduction on microfilms or in other ways,and storage in data banks.For any kind of use permission of the copyright owner must be obtained. © 2005 Birkhäuser Verlag,P.O.Box 133,CH-4010 Basel,Switzerland Part of Springer Science+Business Media Cover design:Micha Lotrovsky,CH-4106 Therwil,Switzerland Printed on acid-free paper produced of chlorine-free pulp.TCF°° Printed in Germany ISBN-10:3-7643-7066-1 ISBN-13:978-3-7643-7066-4 9 8 7 6 5 4 3 2 1 www.birkhauser.ch Contents Preface ................................................................... vii Scientific Lectures given at the Sturm Colloquium ......................... x Introduction (David Pearson) ............................................. xiii Don Hinton Sturm’s 1836 Oscillation Results. Evolution of the Theory ........... 1 Barry Simon Sturm Oscillation and Comparison Theorems ........................ 29 W. Norrie Everitt Charles Sturm and the Development of Sturm-Liouville Theory in the Years 1900 to 1950 ............................................ 45 Joachim Weidmann Spectral Theory of Sturm-Liouville Operators. Approximation by Regular Problems ................................. 75 Yoram Last Spectral Theory of Sturm-Liouville Operators on Infinite Intervals: A Review of Recent Developments ................................... 99 Daphne Gilbert Asymptotic Methods in the Spectral Analysis of Sturm-Liouville Operators ........................................ 121 Christer Bennewitz and W. Norrie Everitt The Titchmarsh-Weyl Eigenfunction Expansion Theorem for Sturm-Liouville Differential Equations ............................... 137 Victor A. Galaktionov and Petra J. Harwin Sturm’s Theorems on Zero Sets in Nonlinear Parabolic Equations .... 173 Chao-Nien Chen A Survey of Nonlinear Sturm-Liouville Equations .................... 201 Rafael del R´ıo Boundary Conditions and Spectra of Sturm-Liouville Operators ...... 217 vi Contents Mark M. Malamud Uniqueness of the Matrix Sturm-Liouville Equation given a Part of the Monodromy Matrix, and Borg Type Results ............ 237 W. Norrie Everitt A Catalogue of Sturm-Liouville Differential Equations ............... 271 Index ..................................................................... 333 Preface Charles Fran¸cois Sturm, through his papers published in the 1830’s,is considered to be the founder of Sturm-Liouville theory. He was born in Geneva in Septem- ber 1803. To commemorate the 200th anniversary of his birth, an international colloquium in recognitionof Sturm’s major contributions to science took place at the UniversityofGeneva,Switzerland,followingaproposalbyAndreasHinz.The colloquiumwasheldfrom15to19September 2003andattendedbymorethan60 participantsfrom16countries.ItwasorganizedbyWernerAmreinofthe Depart- mentofTheoreticalPhysicsandJean-ClaudePont,leaderoftheHistoryofScience groupofthe UniversityofGeneva.Themeeting wasdividedintotwoparts.Inthe firstpart,historiansofsciencediscussedthe manycontributionsofCharlesSturm tomathematicsandphysics,includinghispedagogicalwork.Thesecondpartofthe colloquium was then devoted to Sturm-Liouville theory. The impact and develop- mentofthistheory,fromthedeathofSturmtothepresentday,wasthesubjectof aseriesofgeneralpresentationsbyleadingexpertsinthefield,andthecolloquium concluded with a workshopcovering recent researchin this highly active area. This drawing together of historical presentations with seminars on current mathematicalresearchleftparticipantsinnodoubtofthedegreetowhichSturm’s original ideas are continuing to have an impact on the mathematics of our own times. The format of the conference provided many opportunities for exchange of ideas and collaboration and might serve as a model for other multidisciplinary meetings. The organizershad decided not to publish proceedings of the meeting in the usual form (a complete list of scientific talks is appended, however). Instead it was planned to prepare, in conjunction with the colloquium, a volume containing a complete collection of Sturm’s published articles and a volume presenting the various aspects of Sturm-Liouville theory at a rather general level, accessible to the non-specialist. Thus Jean-Claude Pontwill edit a volume1 containing the col- lected works of Sturm accompanied by a biographical review as well as abundant historicalandtechnicalcommentsprovidedbythecontributorstothe firstpartof the meeting. The present volume is a collection of twelve refereed articles relating to the secondpartofthecolloquium.Itcontains,insomewhatextendedform,thesurvey lecturesonSturm-Liouvilletheorygivenbytheinvitedspeakers;thesearethefirst 1The Collected Works of Charles Franc¸ois Sturm,J.-C.Pont,editor(inpreparation). viii Preface six papers of the book. To complement this range of topics, the editors invited a few participants in the colloquium to provide a review or other contribution in an area related to their presentation and which should cover some important aspects of current interest. The volume ends with a comprehensive catalogue of Sturm-Liouville differential equations. At the conclusion of the Introduction is a brief description of the articles in the book, placing them in the context of the developing theory of Sturm-Liouville differential equations. We hope that these articles,besidesbeingatributetoCharlesFranc¸oisSturm,willbeausefulresource for researchers,graduate students and others looking for an overview of the field. We have refrained from presenting details of Sturm’s life and his other sci- entific work in this volume. As regards Sturm-Liouville theory, some aspects of Sturm’s original approachare presented in the contributions to the present book, and a more detailed discussion will be given in the article by Jesper Lu¨tzen and Angelo Mingarelliin the companionvolume. Ofcourse,the more recent literature concerned with this theory and its applications is strikingly vast (on the day of writing,MathSciNetyields1835entrieshavingtheterm“Sturm-Liouville”intheir title); it is therefore unavoidable that there may be certain aspects of the theory which are not sufficiently covered here. Thearticlesinthisvolumecanbereadessentiallyindependently.Theauthors have included cross-referencesto other contributions.In order to respectthe style and habits of the authors,the editors did not askthem to use a uniform standard for notationsandconventionsofterminology.Forexample,the readershouldtake note that, accordingto author,inner products may be anti-linearinthe firstor in the second argument, and deficiency indices are either single natural numbers or pairs of numbers. Moreover, there are some differences in terminology as regards spectral theory. Thecolloquiumwouldnothavebeenpossiblewithoutsupportfromnumerous individuals and organizations. Financial contributions were received from various divisions of the University of Geneva (Commission administrative du Rectorat, Facult´edesLettres,Facult´edesSciences,HistoireetPhilosophiedesSciences,Sec- tion de Physique), from the History of Science Museum and the City of Geneva, theSoci´et´eAcad´emiquedeGen`eve,theSoci´et´edePhysiqueetd’HistoireNaturelle de Gen`eve, the Swiss Academy of Sciences and the Swiss National Science Foun- dation. To all these sponsors we express our sincere gratitude. We also thank the various persons who volunteered to take care of numerous organizational tasks in relation with the colloquium, in particular Francine Gennai-Nicole who under- tookmostofthesecretarialwork,JanLackiandAndreasMalaspinasfortechnical support, Dani`ele Chevalier, Laurent Freland, Serge Richard and Rafael Tiedra de Aldecoa for attending to the needs of the speakers andother participants.Special thanksareduetoJean-ClaudePontforhisenthusiasticcollaborationoveraperiod of more than three years in the entire project, as well as to all the speakers of the meeting for their stimulating contributions. As regards the present volume, we are grateful to our authors for all the efforts they have put into the project, as well as to our referees for generously Preface ix givingoftheirtime.WethankNorrieEveritt,HubertKalf,KarlMichaelSchmidt, Charles Stuart and Peter Wittwer who freely gave their scientific advice, Serge Richard who undertook the immense task of preparing manuscripts for the pub- lishers, and Christian Clason for further technical help. We are much indebted to ThomasHempflingfromBirkh¨auserVerlagforcontinuingsupportinafruitfuland rewarding partnership. The cover of this book displays, in Liouville’s handwriting, the original for- mulation by Sturm and Liouville, in the manuscript of their joint 1837 paper, of the regular second-order boundary value problem on a finite interval. The pa- per, which is discussed here by W.N. Everitt on pages 47–50, was presented to the Paris Acad´emie des sciences on 8 May 1837 and published in Comptes ren- dus de l’Acad´emie des sciences, Vol. IV (1837), 675–677,as well as in Journal de Math´ematiques Pures et Appliqu´ees, Vol. 2 (1837), 220–223. The original manu- script,withthetitle“Analysed’unM´emoiresurled´eveloppementdesfonctionsen s´eries, dont les diff´erents termes sont assujettis a` satisfaire `a une mˆeme ´equation diff´erentiellelin´eairecontenantunparam`etrevariable”,ispreservedinthearchives of the Acad´emie des sciences to whom we are much indebted for kind permission to reproduce an extract. Geneva, September 2004 Werner Amrein Andreas Hinz David Pearson Scientific Lectures given at the Sturm Colloquium J. Dhombres Charles Sturm et la G´eom´etrie H. Sinaceur Charles Sturm et l’Alg`ebre J. Lu¨tzen The history of Sturm-Liouville theory, in particular its early part A. Mingarelli Two papers by Sturm (1829 and 1833) are considered in the light of their impact on his famous 1836 Memoir P. Radelet Charles Sturm et la M´ecanique E.J. Atzema Charles Sturm et l’Optique J.-C. Pont Charles Sturm, Daniel Colladon et la compressibilit´e de l’eau D. Hinton Sturm’s 1836 Oscillation Results: Evolution of the Theory B. Simon Sturm Oscillation and Comparison Theorems and Some Applications W.N. Everitt The Development of Sturm-Liouville Theory in the Years 1900 to 1950 J. Weidmann Spectral Theory of Sturm-Liouville Operators; Approximation of Singular Problems by Regular Problems Y. Last Spectral Theory of Sturm-Liouville Operators on Infinite Intervals: A Review of Recent Developments D. Gilbert Asymptotic Methods in the Spectral Analysis of Sturm-Liouville Operators E. Sanchez Palencia Singular Perturbations with Limit Essential Spectrum and Complexification of the Solutions