Differential Equations and Mathematical Physics: Proceedings of an International Conference held in Birmingham, Alabama, USA March 3–8, 1986 PDF
Preview Differential Equations and Mathematical Physics: Proceedings of an International Conference held in Birmingham, Alabama, USA March 3–8, 1986
Lecture Notes ni Mathematics Edited by .A Dold and .B Eckmann 1285 I.W. Knowles .Y Sait5 )sdE( laitnereffiD snoitauqE dna Mathematical scisyhP Proceedings of na International Conference held ni Birmingham, Alabama, USA, March 3-8, 1986 Springer-Verlag Heidelberg Berlin NewYork London Paris oykoT Editors lan W. Knowles Yoshimi Salt6 Department of Mathematics, The University of Alabama at Birmingham Birmingham, AL 35294, USA Mathematics Subject Classification (1980): 34, 35, 42, 73, 76, 78, 81 ISBN 3-540-18479-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-18479-1 Springer-Verlag New York Berlin Heidelberg This work subject is to copyright. whether All reserved, the rights are wohro le part of the material concerned, is specifically the rights of translation, reprinting, re-use of recitation, illustrations, broadcasting, reproduction microfilms on or storage in ways, and other in data banks. Duplication of this publication or pads thereof permitted the the provisions under German of only is Copyright Law of September 9, version 1965, its in of 24, June tg85, a copyright and fee always must be fall the Violations paid. under prosecution act of the Copyright German Law. © Springer-Verlag Berlin Heidelberg 1987 Printed ni Germany and binding: Printing Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 This volume is respectfully dedicated to Professor Tosio Kato on the occasion of his seventieth birthday. PREFACE This volume forms a permanent record of lectures given at the International Conference on Differential Equations and Mathematical Physics held at the University of Alabama at Birmingham during March 3-8, 1986. The conference was supported by about 250 mathematicians from the following countries: Belgium, Canada, Czechoslovakia, Denmark, Egypt, Finland, France, Hungary, India, Ireland, Japan, Kuwait, Nigeria, Norway, P.R. of China, South Africa, Sweden, Switzerland, The Netherlands, the U.K., the U.S.A., and West Germany. Its main purpose was to provide a forum for the discussion of recent developments in the theory of ordinary and partial differential equations, both linear and non-linear, with particular reference to work relating to the equations of mathematical physics. Invited one-hour lectures were given by P. Deift, R. DiPerna, W.N. Everitt, C. Foias, T. Kato, S. Kotani, A. Majda, J. Mawhin, J. McLaughlin, J. HcLeod, C. Morawetz, R. Newton, R. Phillips, .M Reed, I. Sigal, and B. Simon. The remainder o£ the program consisted of invited one-half hour lectures. On behalf of the participants, the conference directors acknowledge, with gratitude, the generous financial support provided by the U.S. National Science Foundation, under grant number DMS-8516772, the Department of Mathematics and the Graduate School, University of Alabama at Birmingham, and the College o£ Arts and Sciences, the Graduate School and the Office of Academic Affairs, University of Alabama. eW acknowledge also the valuable support provided by the ether members of the conference committee: Robert Kauffman, Roger Lewis, and Fred Martens from UAB, and Richard Brown and James Ward from .AU As always, the committee is much indebted to the faculty, staff, and graduate students of the Department of Mathematics at BAU for their manifold contributions; here, we wish to make particular mention of Mrs. Eileen Schauer for undertaking the onerous task of typing much of the conference material, including many of the the papers appearing in this volume. Ian .W Knowles Yoshimi Saito Conference Directors O 0 ~ Analytical Solutions for Ordinary and Partial Differential Equations G. Adom~an ........................................................ On the Ratio of the First owT Eigenvalues of Schr~dinger Operators with Positive Potentials Mark S. hshbaugh and Rafael E:enguria .............................. 16 Inverse Scattering for Self-Adjoint n th Order Differential Operators on the Line R. Beals, P. DeCft, and C. Tomet .................................. 26 On the Dirichlet Index Conjecture R.C. Bro~at and Don Htnton ......................................... 39 Three-Dimensional Inverse Scattering ~argaret Cheney, James H. Rose, and Br~artDeFacio ................. 46 Spectral Properties o£ SchrUdinger Operators with Trapping Potentials in the Semi-Classical Limit J.~. Corabes, Ph. Brtet, and P. Ductos ............................. 55 Discrete Sets of Coherent States and their Use in Signal Analysis Ingr~dDaubechtes ................................................. 73 Information, Uncertainty and the Singular Value Decomposition of the Filtered Fourier Transformation B. DeFaato and O. Brander ......................................... 83 On Schrodinger Operators with yon Neumann-Wigner Type Potentials A~len Devtnatz, Richard ~oecke~, and Peter Rejto .................. 91 Nonlinear Conservative Systems RonaLd J. DZPerna ................................................. 99 The Initial Value Problem for the Nonlinear Equations for Zero Mach Number Combustion Pedro EmbZd ....................................................... llO Longtime Solutions for a Class of Convection Diffusion Systems W,E. Fttzgtbbon ................................................... 119 A Closed Form for the Symbol of the Resolvent Parametri× of an Elliptic Operator S. A. Futltng and G. Kennedy ...................................... 126 IIV Existence and Finite-Dimensionality of Attractors for the Landau-Lifschitz Equations Tepper L. Gt[[ and W.W. ZacharN ................................... 134 The Coulomb Potential in Higher Dimensions Jerome A. Coldstein and Gtsele Ruiz Rteder ........................ 143 A Regularity Theorem for Differential Equations Of Constant Strength Ghdrun Gudmundsdotttr ............................................. 150 Intermittent Bifurcation of Vortex Flows Karl Gustafson .................................................... 155 Remarks on the Scattering Problem for Nonlinear Schr~dinger Equations Nakao Hayasht and Yoshto TsutsumL ................................. 162 Asymptotics of Solutions and Spectra of Perturbed Periodic Hamiltonian syS snret D. B. Btnton ond J. K. Sham ....................................... 169 Behavior o£ Eigenfunctions and the Spectrum of Schr~dingerOperators Andreas .M Htnz ................................................... 175 Shape Resonances in Quantum Mechanics P.D. BtsIop and I. .M Siga[ ....................................... 180 Random PerturhationTheory and Quantum Chaos James S. HomLand .................................................. 197 Path Integral for a Weyl Quantized Relativistic Hamiltonian and the Nonrelativistie Limit Problem Takashi Ichtnose .................................................. 205 Scattering with Penetrable Wall Potentials Teruo llzebe and Shtn-tcht Shtmada ................................. 211 Conmlutator Methods and Asymptotic Completeness for a New Class of Stark Effect Hamiltonians Arne Jensen ........................................................ 215 Asymptotics of the Titchmarsh-Weyl m-Coefficient for Integrable Potentials, II Hans G. Kaper and Man KamKmong ................................... 222 On the Difference Between Eigenvalues of Sturm-Liouville Operators and the Semiclassical Limit Werner Kitsch ..................................................... 230 Finite Element Approximation to Singular Minimizers and Applications to Cavitation in Non-Linear Elasticity Greg KnomLes ...................................................... 236 On Relating Generalized Expansions to Fourier Integrals M.A. Kon, L.A. Raphael, and i.E. Young ............................ 248 Vlll Link Between Periodic Potentials and Random Potentials in One-Dimensional Schr~dinger Operators tn~h-b cht Ko rant ................................................... 256 Undressing of Odd Pseudodifferential Operators B.A. Kupersbmidt .................................................. 270 Some Mathematical Aspects of the Wavemaker Theory K. G. Lamb and G. Tent~ ........................................... 276 Integro-Differential Equations Associated with Piecewise Deterministic Processes Summnne .M Lenhart and Yu-Chung Liao .............................. 282 Ambrosetti-Prodi Type Results in Nonlinear Boundary Value Problems lean Mom~t n ....................................................... 290 Transmutation of Analytic and Harmonic Functions Peter A. McCoy .................................................... 314 Some Solved and Unsolved Canonical Problems of Diffraction Theory Erhard Metster .................................................... 320 Runge-Kutta Schemes and Numerical Instabilites: The Logistic Equation RonaLd E. Mtakens ................................................. 337 Structure of Positive Solutions to (-h+V)u = 0 in n R Minoru Murata ..................................................... 342 An Extension of Lavine's Formula for Time-Delay Shu Nahamura ...................................................... 348 Some Open Questions in Multi-Dimensional Inverse Problems Roger G. Newton ................................................... 352 Radially Symmetric Solutions of a Monge-Ampere Equation Arising in a Reflector biapping Problem .V OLtker dna P. Waltman .......................................... 361 Scattering Theory for the Wave Equation on a Hyperbolic Manifold Ralph Phillips, Betttna Wtskott, and ALex ooW ..................... 375 On Unsteady Flow in a Two-Dimensional Cascade with In-Passage Shocks Kernel K. Purl ..................................................... 391 On the Absorption of Singularities in Dissipative Nonlinear Equations Jeffrey Rauch and Michael C. Reed ................................. 403 Feedback Control for an Abstract Parabolic Equation Rouben Rostomtan, Tahao Nambu, and Thomas I. Seidman .............. 408 Approximate Solution of Random Differential Equation M. Sambandhom and Negash Medhin ................................... 417 Geometric Properties and Bounds for Positive Solutions of Semilinear Elliptic Equations KLaus Schmt t t ..................................................... 426 XI Asymptotic Completeness Multiparticle of Scattering I.~. Stgal and A. Softer .......................................... 435 On a Multi-Dimensional Inverse Problem Related to the Gel'£and-Levitan Theory Takasht Su~t .................................................... 473 Fundamental Solution of the Poisson-Boltzroann Equation Kyrtl Ttntarev .................................................... 480 Examples of Exponential Decay of Eigenfunctions of biagnetic Schr~dinger Operators Jun Uchtyama ...................................................... 486 Spatially Localized Free Vibrations of Certain Semilinear Wave Equations on ~2: Recent Results and Open Problems. Pterre-A. Yutllermot .............................................. 493 LECTURES NOT APPEARING IN THE PROCEEDINGS A Nonlinear Eigenvalue Problem in Astrophysical Magnetohydrodynamics John A. Adam (Old Dominion University, U.S.A.) Existence of Non-Trivial Periodic Solutions of a Certain Third-Order Non-Linear Differential Equation Anthony UNt Afuwape (University of Ife, NIGERIA) Stabilization of Solutions for a Class o£ Degenerate Equations in Divergence Form in One Space Dimension .N Althakos (University of Tennessee, U.S.A.) Spectral Properties o£ Indefinite Elliptic Problems W. Allegretto (University of Alberta, CANADA) Quasilinear Parabolic Systems .H Amann (Universitat Zurich, SWITZERLAND) Convergence Properties o£ Strongly-Damped Semilinear Wave Equations ]oe[ .D Aurtn (University of North Carolina-Charlotte, U.S.A.) On Smoothness o£ Solutions of Elliptic Equations in n-Dimensional Nonsmooth Domains .A Azzam (Kuwait University, KUWAIT) Singular Elliptic Operators with Discrete Spectra .] .Y Baxtey (Wake Forest University, U.S.A.) and .R .O Chapman (Oxford University, U.K.) Numerical Solution of Nonlinear Parabolic Variational and Quasi-Variational Inequalities S. .A Belbas (University o£ Alabama, U.S.A.) The Kolomogoroff-Arnold-Moses Theorem in Schrodinger's Equation lean Bellissard (California Institute of Technology, U.S.A.) The Limiting Absorption Principle for Differential Operators with Short-Range Perturbations ~atanta Ben-Artzt (University of California, Los Angeles, U.S.A.) Random Wave Operators Marc A. Berger (Georgia Institute of Technology, U.S.A.) Singular Linear Differential and Difference Operators in the Complex Plane and in B~-Algebras .C .E BL~tghetmer (McMaster University, CANADA) Resonance Regions Determined by a Projection operator Formulation i~3brE Brandas dna Eri~ lhadgnE (Uppsala University, )NEDEWS IX Martingale Drifts and Yang-Mills Quantum Fields John .L Chatltfour (Indiana University, U.S.A.) Computational Methods for the Thomas-Fermi Equation C. .Y Char (University of Southwestern Louisiana, U.S.A.) Action Principles and Periodic Solutions of Hamiltonian Systems Frox~ .H Clark (Universite de Montreal, CANADA) Connection Formulae for the Second Painleve Transcendent .P .A Clarkson (Clarkson University, U.S.A.) Symplectic Geometry and Fixed Point Theory K.R. Colvin (California Polytechnic State University, U.S.A.) A Problem in the Many Body Quantum Mechanics of Coulomb Systems Joseph Conlon (University of Missouri-Columbia, U.S.A.) Continental Shelf Wave Scattering by a Semi-Infinite Coastline A.M.J. Davis (University of Alabama, U.S.A.) Linearization o£ Boundary Problems with Spectral Parameter in the Boundary Condition I and II .A DtJ~smaand H.S.¥. de S~oo (University of Groningen, THE h~ETHERL~) On a Differential Equation with a Small Parameter James .A Donaldson (Howard University, U.S.A.) The Coulomb Potential Revisited Ertk Enqdahl and Er~i Brondas (Uppsala University, SWEDEN) Sobolev Embeddings for Generalised Ridged Domains .W .D Evans (University College, Cardiff, U.K.) Differential Operators in Locally Integrable Spaces .W N. Everttt (University of Birmingham, U.K.), R.R. Ashurou (State University-Tashkent, U.S.S.R.) On the Existence of Positive Solutions for "Indefinite" Elliptic Problems of Second Order J. F[eckinger (Universite Paul Sabatier, )ECNARF Application of Energy Functions to Predator-Prey Systems .H I. Freedman (University of Alberta, )ADANAC A Random Walk for the Solution Sought .H .A Freedman (University of Alaska, U.S.A.) Limit-Point Criteria for Not Necessarily Symmetric Quasi-Differential Expressions .H Frentzen (Universitat Gesamthochschule Essen, WEST GEP~) Boundary Value Problems for a Class of Kinetic Equations with Nonsymmetric Collision operators .A Gancb~v, .W Greenberg, and .C van der Mee (VPI &SU, U.S.A.)
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