Sturm-Liouville Theory Past and Present

SturmLiouville Theory Past and Present Werner O Amrein Andreas M Hinz David B Pearson (Editors) Birkhäuser Verlag Basel Boston Berlin • • Editors: Werner O Amrein Section de Physique Université de Genève 24, quai Ernest-Ansermet 1211 Genève Switzerland Werner.Amrein@physics.unige.ch Andreas M Hinz Mathematisches Institut Universität München Theresienstrasse 39 D-80333 München Germany Andreas.Hinz@mathematik.uni-muenchen.de David P Pearson Department of Mathematics University of Hull Cottingham Road Hull HU6 7RX United Kingdom D.B.Pearson@hull.ac.uk 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 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 987654321 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 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 September 1803 To commemorate the 200th anniversary of his birth, an international colloquium in recognition of Sturm’s major contributions to science took place at the University of Geneva, Switzerland, following a proposal by Andreas Hinz The colloquium was held from 15 to 19 September 2003 and attended by more than 60 participants from 16 countries It was organized by Werner Amrein of the Department of Theoretical Physics and Jean-Claude Pont, leader of the History of Science group of the University of Geneva The meeting was divided into two parts In the first part, historians of science discussed the many contributions of Charles Sturm to mathematics and physics, including his pedagogical work The second part of the colloquium was then devoted to Sturm-Liouville theory The impact and development of this theory, from the death of Sturm to the present day, was the subject of a series of general presentations by leading experts in the field, and the colloquium concluded with a workshop covering recent research in this highly active area This drawing together of historical presentations with seminars on current mathematical research left participants in no doubt of the degree to which Sturm’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 organizers had 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 Pont will edit a volume1 containing the collected works of Sturm accompanied by a biographical review as well as abundant historical and technical comments provided by the contributors to the first part of the meeting The present volume is a collection of twelve refereed articles relating to the second part of the colloquium It contains, in somewhat extended form, the survey lectures on Sturm-Liouville theory given by the invited speakers; these are the first The Collected Works of Charles Fran¸ cois Sturm, J.-C Pont, editor (in preparation) 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, besides being a tribute to Charles Fran¸cois Sturm, will be a useful resource 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 scientific work in this volume As regards Sturm-Liouville theory, some aspects of Sturm’s original approach are presented in the contributions to the present book, and a more detailed discussion will be given in the article by Jesper Lă utzen and Angelo Mingarelli in the companion volume Of course, the more recent literature concerned with this theory and its applications is strikingly vast (on the day of writing, MathSciNet yields 1835 entries having the term “Sturm-Liouville” in their title); it is therefore unavoidable that there may be certain aspects of the theory which are not sufficiently covered here The articles in this volume can be read essentially independently The authors have included cross-references to other contributions In order to respect the style and habits of the authors, the editors did not ask them to use a uniform standard for notations and conventions of terminology For example, the reader should take note that, according to author, inner products may be anti-linear in the first or 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 The colloquium would not have been possible without support from numerous individuals and organizations Financial contributions were received from various divisions of the University of Geneva (Commission administrative du Rectorat, Facult´e des Lettres, Facult´e des Sciences, Histoire et Philosophie des Sciences, Section de Physique), from the History of Science Museum and the City of Geneva, the Soci´et´e Acad´emique de Gen`eve, the Soci´et´e de Physique et d’Histoire Naturelle de Gen`eve, the Swiss Academy of Sciences and the Swiss National Science Foundation 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 undertook most of the secretarial work, Jan Lacki and Andreas Malaspinas for technical support, Dani`ele Chevalier, Laurent Freland, Serge Richard and Rafael Tiedra de Aldecoa for attending to the needs of the speakers and other participants Special thanks are due to Jean-Claude Pont for his enthusiastic collaboration over a period 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 giving of their time We thank Norrie Everitt, Hubert Kalf, Karl Michael Schmidt, Charles Stuart and Peter Wittwer who freely gave their scientific advice, Serge Richard who undertook the immense task of preparing manuscripts for the publishers, and Christian Clason for further technical help We are much indebted to Thomas Hemping from Birkhăauser Verlag for continuing support in a fruitful and rewarding partnership The cover of this book displays, in Liouville’s handwriting, the original formulation 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 paper, which is discussed here by W.N Everitt on pages 47–50, was presented to the Paris Acad´emie des sciences on May 1837 and published in Comptes rendus de l’Acad´emie des sciences, Vol IV (1837), 675–677, as well as in Journal de Math´ematiques Pures et Appliqu´ees, Vol (1837), 220–223 The original manuscript, with the title “Analyse d’un M´emoire sur le d´eveloppement des fonctions en s´eries, dont les diff´erents termes sont assujettis a` satisfaire `a une mˆeme ´equation diff´erentielle lin´eaire contenant un param`etre variable”, is preserved in the archives 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 lAlg`ebre J Lă utzen 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 320 W.N Everitt Semi-periodic boundary conditions on (−1/2, +1/2), i.e., y(−1/2) = −y(+1/2) y (−1/2) = −y (+1/2) √ √ −1 With β = cos ((1 + 33)/16) and γ = cos−1 ((1 − 33)/16) these are all simple and given by, for n = 0, 1, 2, λ4n = (2nπ + β)2 ; λ4n+1 = (2nπ + γ)2 ; λ4n+2 = (2(n + 1)π − γ)2 ; λ4n+3 = (2(n + 1)π − β)2 For the general theory of periodic differential boundary value problems see [26]; for a special case with discontinuous coefficients see [45] 51 Morse equation This differential equation has exponentially small and large coefficients; the differential equation is −y (x) + (9 exp(−2x) − 18 exp(−x))y(x) = λy(x) for all x ∈ (−∞, +∞) Endpoint classification in the space L2 (−∞, +∞): Endpoint Classification −∞ LP +∞ LP This differential equation on the interval (−∞, ∞) is studied in [5, Example 6]; the spectrum has exactly three negative, simple eigenvalues, and a continuous spectrum on [0, ∞); the eigenvalues are given explicitly by λn = −(n − 2.5)2 for n = 0, 1, 52 Morse rotation equation This differential equation is considered in [5] and is given as −y (x) + (2x−2 − 2000(2e(x) − e(x)2 ))y(x) = λy(x) for all x ∈ (0, +∞), where e(x) = exp(−1.7(x − 1.3)) for all x ∈ (0, +∞) Endpoint classification in the space L2 (0, +∞) Endpoint Classification LP +∞ LP This classical problem on the interval (0, ∞) has a continuous spectrum on [0, ∞) and exactly 26 negative eigenvalues; it provides an invaluable numerical test for computer programs A Catalogue of Sturm-Liouville Equations 321 53 Brusencev/Rofe-Beketov equations 53.1 Example The Sturm-Liouville differential equation −(x4 y (x)) − 2x2 y(x) = λy(x) for all x ∈ (0, ∞) is considered in the paper [19]; this example provides a LC case with special properties Endpoint classification in L2 (0, +∞): Endpoint Classification LP +∞ LCNO For the endpoint +∞ in the LCNO classification the boundary condition functions u, v are determined by Endpoint u +∞ x−1 v x−2 53.2 Example The Sturm-Liouville differential equation −y (x) − x10 + x4 sign(sin(x)) y(x) = λy(x) for all x ∈ [0, ∞) is considered in the paper [73]; this example provides a LC case with special properties Endpoint classification in L2 (0, +∞): Endpoint Classification R +∞ LCO For the endpoint +∞ in the LCO classification the boundary condition functions u, v may be determined as the real and imaginary parts of the expression x−5/2 exp(ix6 /6)Y (x) for all x ∈ [1, ∞), where the function Y (·) is the solution of the integral equation, for x ∈ [1, ∞), Y (x) = + i ∞ x t−1 sign(sin(t)) + 35 −7 t exp i (t − x6 ) − Y (t) dt The solution Y (·) of this integral equation may be obtained by the iteration method of successive approximations; in this process it has to be noted that the integrals concerned are only conditionally convergent 322 W.N Everitt 54 Slavyanov equations In the important text [77] the authors give a systematic presentation of a unified theory of special functions based on singularities of linear ordinary differential equations in the complex plane C In particular, in [77, Chapter 3], there is to be found an authoritative account of the definition and properties of the Heun differential equation In [77, Chapter 4] there is a chapter devoted to physical applications, including the use of the Heun differential equation, resulting from the application of separation techniques to boundary value problems for linear partial differential equations From this chapter we have selected three examples of Sturm-Liouville differential equations; each equation contains a number of symbols denoting physical constants and parameters which are given here without explanation To allow the quoted examples to be given in Sturm-Liouville form the notation for one of these parameters has been changed to play the role of the spectral parameter λ ∈ C The resulting Sturm-Liouville examples given below have not yet been considered for their endpoint classification, nor for their boundary condition functions if required for LC endpoints 54.1 Example The hydrogen-molecule ion problem, see [77, Chapter 4, Section 4.1.3], gives the two differential equations: − (1 − η )Y (η) + n2 (1 − η )−1 − µ Y (η) = λη Y (η) for all η ∈ (−1, 1) and − (1 − ξ )X (ξ) + κξ + n2 (1 − ξ ) − µ X(ξ) = λξ X(ξ) for all ξ ∈ (1, ∞) 54.2 Example The Teukolsky equations in astrophysics gives the equation, see [77, Chapter 4, Section 4.2.1]: −((1 − u2 )X (u)) + + (m − 2u)2 (1 − u2 )−1 − 4aωu − a2 ω u2 X(u) = λX(u) for all u ∈ (−1, 1) 54.3 Example The theory of tunnelling in double-well potentials, see [77, Chapter 4, Section 4.4], gives the differential equation −y (x) + V (x)y(x) = λy(x) for all x ∈ (−∞, ∞) with the potential V determined by V (x) = −A(sech2 (x + x0 ) + sech2 (x − x0 )) for all x ∈ (−∞, ∞); here A is a number and x0 is a parameter A Catalogue of Sturm-Liouville Equations 323 55 Fuel cell equation This Sturm-Liouville differential equation −(xy (x)) − x3 y(x) = λxy(x) for all x ∈ (0, b] plays an important role in a fuel cell problem as discussed in the paper [6] Endpoint classification in the space L2 ((0, b); x): Endpoint Classification LCNO b R For the LCNO endpoint at the u, v boundary condition functions can be taken as, see [6, Section 8]: u(x) = and v(x) = ln(x) for all x ∈ (0, b] Various boundary value problems are considered in [6, Section 8]; the technical requirements of the fuel cell problem require a study of the analytic properties of these boundary value problems, as the endpoint b tends to zero 56 Shaw equation This Sturm-Liouville differential equation is considered in the paper [76] and has the form y (x) − Q(x)y(x) = λy(x) for all x ∈ (0, ∞) where Q(x) = A − B exp(−Cx) + Dx−2 for all (0, ∞) for positive real numbers A, B, C, D with D ≥ 3/4 Endpoint classification in L2 (0, +∞): Endpoint Classification LP +∞ LP In the paper [76] the following specific values for A, B, C, D are used in connection with the chemical photodissociation of methyl iodide: A = 19362.8662 B = 19362.8662 × 46.4857 C = 1.3 D = 2.0 324 W.N Everitt 57 Plum equation This Sturm-Liouville equation is one of the first to be considered for numerical computation using interval arithmetic: the equation is −y (x) + 100 cos2 (x)y(x) = λy(x) for all x ∈ (−∞, +∞) Endpoint classification in L2 (−∞, +∞): Endpoint Classification −∞ LP +∞ LP In [71] the first seven eigenvalues for periodic boundary conditions on the interval [0, π], i.e., y(0) = y(π) y (0) = y (π), are computed using a numerical homotopy method together with interval arithmetic; rigorous bounds for these seven eigenvalues are obtained 58 Sears-Titchmarsh equation This differential equation is considered in detail in [79, Chapter IV, Section 4.14] and [75]; the equation is −y (x) − exp(2x)y(x) = λy(x) for all x ∈ (−∞, ∞) and has solutions of the form, using the Bessel function Jν and writing σ + it, y(x, λ) = Jis (exp(x)) for all x ∈ (−∞, ∞); √ λ=s= in the space L2 (−∞, ∞) this equation is LP at −∞ and is LCO at +∞ This differential equation is then another example of equations derived from the original Bessel differential equation This Sears-Titchmarsh differential equation is the Liouville form, see Section above, of the Sturm-Liouville equation −(xy (x)) − xy(x) = λx−1 y(x) for all x ∈ (0, +∞) In the space L2 ((0, ∞); x−1 ) this differential equation is LP at and is LCO at +∞ Endpoint classification in L2 ((0, ∞); x−1 ): Endpoint Classification LP +∞ LCO For the LCO endpoint +∞ the boundary condition functions can be chosen as, for all x ∈ (0, +∞), u(x) = x−1/2 (cos(x) + sin(x)) v(x) = x−1/2 (cos(x) − sin(x)) A Catalogue of Sturm-Liouville Equations 325 For details of boundary value problems for this Sturm-Liouville equation, on [1, ∞) see [8, Example 4] For problems on [1, ∞) the spectrum is simple and discrete but unbounded both above and below, since the endpoint +∞ is LCO Numerical results are given in [11, Data base file xamples.tex; example 6] 59 Zettl equation This differential equation is closely linked to the classical Fourier equation 8; −(x1/2 y (x)) = λx−1/2 y(x) for all x ∈ (0, +∞) Endpoint classification in L2 ((0, +∞); x−1/2 ) Endpoint Classification R +∞ LP This is a devised example to illustrate the computational difficulties of regular problems which have mild (integrable) singularities, in this example at the endpoint of (0, ∞) The differential equation gives p(0) = and w(0) = ∞ but nevertheless is a regular endpoint in the Lebesgue integral sense; however this endpoint does give difficulties in the computational sense The Liouville normal form of this equation is the Fourier equation, see Section above; thus numerical results for this problem can be checked against numerical results from (i) an R problem, (ii) the roots of trigonometrical equations, and (iii) as an LCNO problem (see below) There are explicit solutions of this equation given by √ √ √ cos(2x1/2 λ) ; sin(2x1/2 λ)/ λ If is treated as an LCNO endpoint then u, v boundary condition functions are u(x) = 2x1/2 v(x) = The regular Dirichlet condition y(0) = is equivalent to the singular condition [y, u](0) = Similarly the regular Neumann condition (py )(0) = is equivalent to the singular condition [y, v](0) = The following indicated boundary value problems have the given explicit formulae for the eigenvalues: y(0) = or [y, u](0) = 0, and y(1) = gives λn = ((n + 1)π)2 /4 (n = 0, 1, ) (py )(0) = or [y, v](0) = 0, and (py )(1) = gives λn = (n + 12 )π /4 (n = 0, 1, ) 326 W.N Everitt 60 Remarks The author has made use of an earlier collection of examples of SturmLiouville differential equations drawn up by Bailey, Everitt and Zettl, in connection with the development and testing of the computer program SLEIGN2; see [8] and [10] The author has made use of major collections of Sturm-Liouville differential equations from Pryce [69] and [70], and from Fulton, Pruess and Xie [38] and [68] This catalogue will continue to be developed; the author welcomes corrections to the present form, and information about additional examples to extend the scope, of the catalogue 61 Acknowledgments The author is grateful to Werner Amrein (University of Geneva, Switzerland), David Pearson (University of Hull, England, UK) and other colleagues who organized the Sturm meeting, held at the University of Geneva in September 2003, which brought together so many scientists working in and making application of Sturm-Liouville theory The author is grateful to many colleagues who commented on the earlier drafts and sent information concerning possible examples to include in this catalogue: Werner Amrein, Paul Bailey, Richard Cooper, Desmond Evans, Charles Fulton, Fritz Gesztesy, Don Hinton, Hubert Kalf, Lance Littlejohn, Clemens Markett, Marco Marletta, Lawrence Markus, David Pearson, Michael Plum, John Pryce, Fiodor Rofe-Beketov, Ken Shaw, Barry Simon, Sergei Slavyanov, Rudi Weikard, Tony Zettl The author is especially indebted to Barry Simon, Fritz Gesztesy, Clemens Markett, John Pryce, Fiodor Rofe-Beketov and Rudi Weikard who supplied detailed information for the examples in Sections 11, 24, 25, 26, 31, 40, 41, 49 and 53 The author thanks the editors of the proceedings volume, for the coming publication of the Sturm-Liouville manuscripts, for checking the final draft of this catalogue; their careful scrutiny led to many improvements in the text, and to the correction of not a small number of errors 62 The future As mentioned above it is hoped to continue this catalogue as a database for SturmLiouville differential equations The main contributor to the assessment and extension of successive drafts of this catalogue is Fritz Gesztesy, who has agreed to join with the author in continuing to update the content of this database A Catalogue of Sturm-Liouville Equations 327 Together we ask that all proposals for enhancing and extending the catalogue be sent to both of us, if possible by e-mail and LaTeX file The author’s affiliation data is to be found at the end of this paper; the corresponding data for Fritz Gesztesy is: Fritz Gesztesy Department of Mathematics University of Missouri Columbia, MO 65211, USA e-mail: fritz@math.missouri.edu fax: ++ 573 882 1869 References [1] M Abramowitz and I.A Stegun, Handbook of mathematical functions, Dover Publications, Inc., New York, 1972 [2] N.I Akhiezer and I.M Glazmann, Theory of linear operators in Hilbert space: I and II, Pitman and Scottish Academic Press, London and Edinburgh, 1981 [3] R.A 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and G.N Watson, Modern analysis, Cambridge University Press, 1950 [82] A Zettl, Computing continuous spectrum, in Trends and Developments in Ordinary Differential Equations, 393–406, Y Alavi and P Hsieh editors, World Scientific, 1994 [83] A Zettl, Sturm-Liouville problems, in Spectral Theory and Computational Methods of Sturm-Liouville problems, 1–104, Lecture Notes in Pure and Applied Mathematics 191, Marcel Dekker, Inc., New York, 1997 W Norrie Everitt School of Mathematics and Statistics University of Birmingham Edgbaston Birmingham B15 2TT, UK e-mail: w.n.everitt@bham.ac.uk Index absolutely continuous spectrum, 92–97, 100, 101, 220 absolutely continuous subspace, 100, 104 Airy equation, 289 algebro-geometric equations, 307–312 almost Mathieu operator, 102, 105–113 α-continuous measure, 104, 105 α-continuous spectrum, 105 α-singular measure, 104, 105 α-singular spectrum, 105 Anderson localization, 111 approximation of singular problems, 84–95 Aronszajn theorem, 221 Ashbaugh-Brown-Hinton theorem, 17 Aubry duality, 107, 112 decomposition method, 85 deficiency index, 56, 57, 79–82 degenerate spectrum, 128, 129 Denisov-Rakhmanov theorem, 41 difference equations, 30–34, 105–112 Dirac operators, 129, 261, 262, 267 disconjugacy, 8, 15–17 discrete spectrum, 90–92 disfocal, Dixon, A.C., 54 Dunford-Schwartz equation, 303–305 eigendifferentials, 53 eigenfunction expansion, 49, 53, 61, 65–69, 150–153 embedded singular spectrum, 221–229 essential spectrum, 79, 87, 219 essential support, 125 Euler equation, backward continuation, 177–181 Bailey equation, 301 Bargmann potentials, 312–314 Bargmann’s bound, 40 Behnke-Goerisch equation, 302 Bessel equation, 284–289, 305, 308, 315, 324 bifurcation, 205–213 Borg theorem, 229, 237, 247 Borg type results, 230, 231, 251–262 bounded solution, 123–124 Boyd equation, 302, 303 Brusencev/Rofe-Beketov equations, 321 buckling of a column, 203 Byers-Harris-Kwong theorem, 17 focal point, formation of multiple zeros, 179 forward continuation, 178 Fourier equation, 280 fuel cell equation, 323 Gel’fand-Levitan theorem, 101 generalised transform, 165 generalized Fourier expansion, 153 generalized Fourier transform, 160–162, 164, 166, 168 global bifurcation theorem, 203, 206, 207 Goldsheid-Molchanov-Pastur model, 133 Green’s formula, 140, 276 Green’s function, 83, 87, 167, 168 Cantor spectrum, 109–111 coexistence, 222–229 collapse of multiple zeros, 180, 181 comparison theorem, 4, 8–10, 19, 34 complex domains, 13, 14, 322 conjugate point, 8, 15, 16 continuous spectrum, 53, 54, 56 curve shortening, 190 Halvorsen equation, 314 Harper operator, 105 Hartman theorem, 11 Hartman-Wintner theorem, 12 Hausdorff dimension, 104, 105, 191 333 334 Hausdorff measure, 104, 105 Hawking-Penrose theorem, 12 heat equation, 49, 186 Herglotz function, 59, 63, 149 Hermite equation, 189, 192, 291, 292 Heun equation, 297–299 higher-order equations, 15–19, 183 Hill’s equation, 301 Hille theorem, 11 Hochstadt-Lieberman theorem, 229, 238 Hochstadt-Lieberman type results, 230–232, 263–267 Hofstadter model, 105 hydrogen atom, 305–307 hypergeometric equation, 281–283, 295, 309 indefinite weights, 20, 21 initial value problem, 141 inverse methods, 101, 229–232, 251–263 Jacobi equation, 292296, 309 Jacobi matrix, 3134, 101, 130 Jă orgens equation, 315 Kolmogorov-Petrovskii-Piskunov analysis, 186, 187 Korteweg-de Vries equation, 307, 312 Krall equation, 315 Kummer equation, 283 Lagrange bracket, 77 Lagrange identity, 77 Laguerre equation, 296, 297 Laguerre polynomials, 188 Lam´e equation, 7, 299, 300, 309, 310 Laplace tidal wave equation, 316 Latzko equation, 317 Lebesgue integration, 54–56 Legendre equation, 290, 291 limit-circle, 53, 56, 64, 79–82, 142–148, 277 non-oscillatory, 278 oscillatory, 278 limit-point, 53, 56, 64, 79–82, 141, 277 Liouville number, 109–111 Liouville transformation, 280 Littlewood-McLeod equation, 317 Lohner equation, 318 London theorem, 14 Lyapunov theorem, 14 m-coefficient, 62–66, 71 m-function, 146–148, 169, 219, 220, 231, 261 Index Makarov conjecture, 222 Mathieu equation, 301, 302 maximal operator, 57, 77–80, 141, 276 Meissner equation, 319 metal-insulator transition, 111 minimal operator, 57, 77–80, 141, 276 minimal support, 125–128, 220 monodromy matrix, 238, 244250 Morse equation, 320 Morse rotation equation, 320 Mă uller-Pfeier theorem, 17 multiple zeros, 176–181 N -soliton potential, 310–312 Nehari theorems, 11, 14, 202 Nevanlinna function, 59, 63, 149 non-oscillatory, 8, 123 nonlinear equations, 186–191, 201–214 orthogonal polynomials, 30–34 orthogonality, 48 oscillation, 10–21, 32–40 oscillatory, 8, 15, 16, 123 parabolic PDEs, 186–192 parabolic point, 123 parameter dependent boundary conditions, 19, 20 periodic coefficients, 14, 111, 301, 319 Pick function, 59, 63, 149, 227 Picone identity, 8, 18 Plum equation, 324 point spectrum, 53, 54, 87–91, 100 porous medium equation, 189, 191 principal solution, 123 Pră ufer transformation, Pryce-Marletta equation, 318 pure-point subspace, 100 quantum Hall effect, 106 quasi-derivative, 55, 57, 76 Rabinowitz bifurcation theorem, 205 Rakhmanov theorem, 41 random Schră odinger operators, 130 recurrent subspace, 103 reectionless potentials, 311 regular endpoint, 76, 277 Reid Roundabout theorem, 18 Rellich equation, 286, 307, 315, 316 resolvent, 58, 60, 64, 83, 84, 87, 153–168 Rofe-Beketov theorem, 41 rotation numbers, 36, 37 Index Schră odinger operators, 81, 82, 313 Sears-Titchmarsh equation, 302, 324 self-adjoint, 83 self-adjoint extensions, 58, 80 self-adjointness, 82 separation theorem, 4, 19 Shaw equation, 323 sign changes, 175, 176, 181 singular continuous spectrum, 100, 101, 105, 111–115, 122 continuous subspace, 100 endpoint, 76, 277 singularity formation, 178, 192 Slavyanov equations, 322 SLEIGN2, 2, 316, 318, 319, 326 sparse potential, 113–115 spectral density, 219 function, 123, 125, 127, 219 measure, 100, 219 multiplicity, 128 representation, 92–97 spectral representation, 165 spectrum, 70, 71, 100, 123, 124 Stone, M.H., 55–58 strong resolvent convergence, 85–87 Sturm -Hurwitz theorem, 184, 185, 188, 190 -Liouville systems, 16–19, 182, 239–264 comparison theorem, 4, 8, 34 oscillation theorem, 32–37, 39, 40 separation theorem, 4, 19 subordinacy, 94, 125–133 subordinate solution, 126 ten Martini problem, 110, 111 Titchmarsh, E.C., 58–68 Titchmarsh-Weyl contributions, 68–71 transfer matrix, 132 transient subspace, 103, 104 travelling waves, 186 typical Diophantine properties, 109–112 uniform nonsubordinacy, 95 uniqueness theorems, 213, 229–232, 245–267 variational approach, 16–20, 183, 204, 205, 211 von Neumann-Wigner potential, 132 weight function, 275 335 Weyl H., 53 Weyl’s alternative, 79 Weyl, H., 51–54 Whittaker equation, 299 Wintner theorem, 11 Zakharov-Shabat system, zeros, 4, 5, 34–37, 39, 40, 175–183, 189 Zettl equation, 325 ... overview of the field We have refrained from presenting details of Sturm s life and his other scientific work in this volume As regards Sturm- Liouville theory, some aspects of Sturm s original approach... 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. .. 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