Table Of ContentLecture Notes ni
Mathematics
detidE yb .A Dold dna .B nnamkcE
448
Spectral Theory dna
laitnereffiD snoitauqE
Proceedings of eht muisopmyS
held ta ,eednuD Scotland, 1-19 July, 4791
Edited yb .W .N Everitt
11111111]]
galreV-regnirpS
Berlin.Heidelberg-NewYork 1975
Prof. William .N Everitt
Department of Mathematics
University of Dundee
Dundee DD1 4 HN/Scotland
Library of Congress Cataloging in Publication Data
Symposium on Spectral Theory and Differential Equations,
University of Dundee, 1974.
Spectral theory and differential equations.
(Lecture notes in mathematics 448) ;
Bibliography: p.
Includes index.
.i Differential equations--Congresses. .2 Differen-
tial opez'atons--Congresses. -3 Spectral theory (Mathe-
matics)--Congresses. .I Everitt, William Norrie.
.II Title. III. Series: Lecture notes in mathematics
(Berlin) ; 448.
QA3.L28 no. 448 [QA371] 510'.8s [515'.35] 75-6675
Subject Classifications (1970):
AMS 26A84, 26A86, 34B15, 34B20, 34B25,
34C15, 34G05, 35B10, 35B46, 35C25, 35D05, 35D10, 35G05, 35J05, 35J10, 35J15,
S5J25, 35J45, 35J55, 35P05, 35P10, 35P15, 35P25, 35Q10, 47A05, 47A10, 47A25,
47A55, 47B25, 47B40, 47B44, 47E05, 47F05, 49A05, 49A10, 49G20, 65L10, 65L15,
A05, 76A10, B K05, 76 76 L99, 65 65 ,01 76D05, D 76 ,01 76D ,51 78A20, 78A25
ISBN 3-540-07150-4 Springer-Verlag Berlin • Heidelberg- New York
ISBN 0-387-07150-4 Springer-Verlag New York' Heidelberg • Berlin
This work is subject to copyright. All rights are reserved, whether the whole
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© by Springer-Verlag Berlin • Heidelberg 1975. Printed ni Germany.
Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
This volume is dedicated to the life,
work and memory of
KONRAD J~RGENS
1926-1974
PREFACE
These Proceedings form a record of the lectures given to the
Symposium on Spectral Theory and Differential Equations held at the
University of Dundee, Scotland during the period I to 91 July 1974.
The Symposium was organised on behalf of the Science Research
Council of the United Kingdom. The Council provided financial support
which made it possible to bring together many of the foremost workers
in various aspects of the application of spectral analysis to the theory
of both ordinary and partial differential equations. Without this
support from the Council it would have been impossible to bring the
Symposium into being.
The Symposium was attended by 60 mathematicians from the following
countries: BRD (Germany), Canada, Japan, Sweden, The Netherlands, the
United Kingdom and the United States of America.
Three mathematicians, Professors .E A. Coddington, Tosio Kato
and Joachim Weidmann, were each invited to give a series of four lectures
and ot prepare manuscripts which appear in this volume. Twelve other
mathematicians were invited to give single lectures and papers resulting
from these lectures are also collected in this volume. Additionally
Professor F S Rofe-Beketov, of Kharkov, the Ukraine, USSR, sent a
manuscript for inclusion in this collection.
The Symposium was organised, on behalf of the Science Research
Council, by the following Committee: W. N. Everitt (Chairman): .I M. Michael
and .B .D Sleeman (Organising Secretaries).
On behalf of the Con~ittee I express our keen appreciation to
all mathematicians who took part in the work of the Symposium. In
particular I thank all those who delivered lectures and supplied
manuscripts for this volume. Special thanks are due to Professor Weidmann
who took over responsibility for the lectures on the spectral theory of
partial differential operators when it became clear that the late
Professor JSrgens would not be able to attend the Sympsoium.
The Co~nittee thanks: the University of Dundee for generously
providing facilities which made it possible to hold the Symposium in Dundee .
The Wardens and Staff of Chalmers Hall and Peterson House for accommodating
many of those attending the Symposium; and many colleagues and research
students in the Department of Mathematics, for help received. Mr E R Dawson
of the Department of Mathematics gave particular advice in the preparation
of manuscripts for this volume, and helped in several other ways.
We are grateful to the Staff of the Mathematics Committee of
the Science Research Council for much help and advice; in particular
Miss Jean Melville and Dr K D Crosbie. Also to Professor D E Edmands
(University of Sussex) in his capacity as Adviser to the Council for the
Symposium.
The Committee extends very special gratitude to Mrs Norah Thompson,
Secretary in the Department of Mathematics, for her significant contribution
to the work of the Symposium. It would have been difficult, at times
impossible, to cope with the volume of secretarial work without her help;
many of the papers appearing in ahis volume bear witness to her ability
as a typist of mathematical manuscripts.
Finally it is a pleasure for me to thank once again my colleagues
Brian Sleeman and lan Michael for their sustained efforts over several
months; and for their patience and cheerfulness in the face of responsibility
for so much of the organisation of the Symposium. I am grateful to both
of them.
VI
One mathematician was sorely missed from the Symposium. By July
of 1974 the tall, commanding figure and personality of Konrad JSrgens
was no more; he died in April of that year as a result of a terminal
illness which had had a quite unexpected onset in the summer of 1973.
Professor JBrgens had earlier accepted our invitation to deliver a
series of lectures to the Symposium but as 1974 advanced it became clear
that he would be unable to attend. The loss sustained by mathematics,
and in particular the study of differential equations, as a result of
his untimely death will be felt not only in his own country but in many
parts of the world. It is for others in different circumstances to
speak and write in greater detail of his outstanding contributions to our
discipline. Here my colleagues and I dedicate, so appropriately, this
volume to the memory of a mathematician who contributed with such
significance to spectral theory and differential equations. Konrad JSrgens
was indeed a colleague and a friend to many of us and we are the poorer
for losing him at the height of his powers and influence.
October 1974 W N Everitt
CONTENTS
Survey papers
E. A. Coddin~ton:
Spectral theory of ordinary differential operators ................
T. Kato:
Quasi-linear equations of evolution, with applications to partial
differential equations ........................................... 25
J. Weidmann:
Spectral theory of partial differential operators ................ 17
Papers
L. Collatz:
Different applications of convex and nonconvex optimization,
especially to differential equations ............................. 112
M. .S P. Eastham:
Results and problems in the spectral theory of periodic
differential equations ........................................... 126
~ Evans:
Sobolev embeddings ............................................... 136
.N Everitt:
Integral inequalities and spectral theory ......................... 148
W. N. Everitt and M. Giertz:
On the deficiency indices of powers of formally symmetric
differential expressions ......................................... 167
H. Kalf~ U.-W. Schmincke~ J. Walter and R. WUst:
On the spectral theory of Schr~dinger and Dirac operators with
strongly singular potentials ..................................... 182
IIIV
S. T. Kuroda:
Scattering theory for differential operators, III; exterior
problems .......................................................... 227
J. B. McLeod:
Swirling flow ..................................................... 242
A. Plei~el:
A survey of spectral theory for pairs of ordinary differential
operators .. ........................................ .... ........... 256
F. .S Rofe-Beketov:
Deficiency indices and properties of spectrum of some classes
of differential operators ......................................... 273
K. Schmitt:
Eigenvalue problems for nonlinear second order differential
equations ......................................................... 294
B. D. Sleeman:
Left-definite multiparameter eigenvalue problems .................. 307
Lectures delivered to the Symposium
Series of four lectures
.E A, Coddinston:
Spectral theory of ordinary differential operators
T. Kato:
Quasi-linear equations of evolution, with applications to partial
differential equations
.J Weidmann:
Spectral theory of partial differential operators
Single Lectures
,J ,S Bradley:
Integral inequalities and spectral theory
.L Collatz:
Optimisation and differential equations
.M .S .P Eastham
Results and problems in spectral theory of periodic differential
equations
W D Evans
Sobelev embeddlngs
M. Giertz:
On the deficiency indices of powers of symmetric differential
expressions
.H Kalf:
On the spectral theory of Dirac and SchrSdinger operators
.S .T Kuroda:
Scattering theory for differential operators
.J .B McLeod:
Swirling flow
X
.A PleiSel:
A survey of spectral theory for pairs of differential operators
K Schmitt:
Eigenvalue problems for non-linear ordinary differential equations
.B .D Sleeman:
Left-definite multi-parameter eigenvalue problems
.W Velte:
On variational problems with unilateral constraints .~
