Table Of ContentMultiparameter
Eigenvalue
Problems
Sturm-Liouville Theory
Multiparameter
Eigenvalue
Problems
Sturm-Liouville Theory
F. V. Atkinson
University of Toronto
,
Toronto, Ontario Canada
Angelo B. Mingarelli
Carleton University
Ottawa, Ontario, Canada
CRC Press
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Library of Congress Cataloging‑in‑Publication Data
Atkinson, F. V.
Multiparameter eigenvalue problems : Sturm-Liouville theory / F.V. Atkinson, Angelo
B. Mingarelli.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-4398-1622-6 (hardback)
1. Sturm-Liouville equation. 2. Eigenvalues. I. Mingarelli, Angelo B. (Angelo
Bernardo), 1952- II. Title.
QA372.A843 2010
512.9’436--dc22 2010038203
Visit the Taylor & Francis Web site at
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Preface
A literary executor messes with the literary legacy of the author
whoseworkshewasmeanttoprotectwithunintendedconsequences.1
Thetaskofresurrectingamanuscriptwrittenbythehandofoneoftheforemost
experts in differential equations in the school of Hardy-Titchmarsh, Frederick
(Derick)Valentine Atkinson,hasbeenadauntingone. Derickdiedonthe13th
day of November 2002, in Toronto, Canada, after a long illness that kept him
confined to a bed and unable to work for some 10 years.
He was born in Pinner, Middlesex, on the 25th of January 1916 of a father
whowasawell-readjournalistandamotherwhowasahomemaker. Wewould
certainly be led astray if we were to confine ourselves at this point on further
biographical items. The interested reader may of course consult the elaborate
obituary2 or the brief note to the Royal Society of Edinburgh3 for further
biographical and professional details.
Unlike Kafka and Hemingway, Atkinson left no written record forbidding the
publication of any unpublished works after his death. Thus, soon after the
funeral services were held, the task of publishing his “opera omnia” became
the order of the day, a seed that had been germinating since at least 1972.
The family held that a complete work existed by the end of July 1992 (one
month prior to his debilitating stroke) and that this manuscript was ready for
publication. Thefinaldocumentwasetchedonasmanyas8x3.5in. IBM-DOS
formatted diskettes. Unfortunately, neither the manuscript nor these diskettes
have ever been found.
Wereviewhere,fortherecord,theeventsthatledtothispublication. Onmany
an occasion during the winter of 2003, the second author was always greeted
warmly by the Atkinson family during his frequent visits to Toronto. He was
1www.complete-review.com/reviews/moddeut/krugerm.htm, July1,2008.
2A glimpse into the life and times of F.V. Atkinson, Math. Nachr.,278(12–13) (2005),
1–29
3http://www.rse.org.uk/fellowship/obits/obits alpha/Atkinson fdv.pdf.
v
vi
encouraged to search for the missing disks through the wealth of material left
behind by Derick. This body of work consisted mainly of notes, books, scraps
of paper, even napkins bearing equations, many three-holed binders, 5.25 in.
floppy disks, and two personal computers (an old Amstrad and an IBM PC-
XT) with antiquated dot-matrix printers. At the suggestion of the family, we
contacted Prof. Paul Binding (Calgary) who, sometime in the spring of 2003,
kindly returned some disks he had obtained in the past and a stack of some
300 manuscript pages of what appeared to be an old, but definitely not final
draft of what would lead to this work.
The problem of sewing this quilt of a book together would become the most
challenging task. At our disposal we had many 5.25 in. floppy disks with
files labeled *.T3V and dated anywhere between January and July 1992, as
many as four manuscript “Table of Contents,” and four essentially different
butfrequentlyoverlappingtypewritten/printeddraftsofthebook(withatotal
of about 1500 pages). After some forensic work, we managed to decipher that
*.T3V indicated files that were “T3 volumes” and so had been written in a
nowdefunctsoftwarecalledScientificWord(cid:13)c madebyTCISoftwareResearch,
a one-time precursor of the modern-day Scientific Workplace(cid:13)c produced by
MacKichan. Fortunately for this adventure, my own father had left behind an
old and still working IBM PC-XT running IBM-DOS 5.0 and equipped with
dualfloppydrives(onewideandoneshort). Itwasperfect. Afterhavingcopied
the old Atkinson files for the sake of posterity, the task was now to find a way
to “read” them.
SoitappearedthatweshouldsecureaworkingcopyofScientificWord. Itcame
asasurprisewhenwerealizedthatDerickactuallykepttherequiredsoftwarein
a dusty oldbox ononeofhis basementoffice bookshelves. Installationwasnot
successful at first since we needed an old monochrome display screen as well,
and, in addition, the printer drivers were absent and impossible to find for the
oldprinterswehadatourdisposal. Managingourwayaroundthis problemby
finding an old monochrome monitor the installation then succeeded, and the
ancient language was mastered anew. After reading those old DOS help files,
the files were opened so that we could view the final days of the work that
Atkinson left behind, unfinished, unpolished, and we suspect only some 65%
of the way to a complete text. It was clear that many sections were missing
and others were incomplete. Occasionally, paragraphs were cut short in the
middle of a sentence, and the excitement quickly turned to a mild, though
not complete, state of despair. After all, at least we had something, and this
coupled with the hardcopies that were floating around should allow one to
“complete” the book. Of course, this was all theoretical until we embarked
upon the actual task of assimilating all the material into a coherent whole. In
the end, we found and compiled at least four manuscripts of this book, which
we name individually herewith: the Tracing Paper version, the Black Binder
version, the Daisy Wheel version, and the Green Binder version.
vii
The TracingPaperversionconsistsofmanyloose(unnumbered)leavesoftrac-
ingpaperfilledwithmathematicalequationsandtheoremsscatteredaboutthe
Atkinson Nachlass. Taking into account the fading and almost complete disin-
tegrationofthe paper,the watermark,the hand-insertedsymbols,anddefinite
use of a typewriter as opposed to either an IBM selectric or PC, we would
identify this document as the oldest of all; estimating its creation at or about
the early 1970s,possibly even the late 1960s.
The BlackBinder versionis familiar to the secondauthoras this is the version
Atkinson carried with him and kept in his office in the mid-1970s when he
was Derick’s doctoral student. We observed that this black binder version
consisted of eight chapters whose headings agreed almost exactly (save for the
final chapter) with the Contents of Volume II on p. ix of Atkinson’s already
publishedVolumeI(AcademicPress,1972). Hencethisversionisessentiallyin
complete agreement with what Atkinson expected to submit to the publishers
back in 1972, when he had agreed to publish Volume II of his Multiparameter
Eigenvalue Problems (i.e., the book that led to the present text). However, it
is now clear that he was dissatisfied with the contents of this version and so
did not submit the manuscript for publication although he did release copies
of parts of this volume to colleagues via documented private communications.
The next version, dubbed the Daisy Wheel version, is so-called because it was
printed off a Daisy Wheel printer; it is of later date, perhaps from the early
to mid-1980s when such printers were fashionable. It differs from the Black
Binder version essentially in its contents; new ideas permeate the text, some
sections are omitted and new ones are inserted. Yet the corresponding work
in our possession is still incomplete relative to the table of contents for this
specific version.
Now comes the Green Binder version (GB); we thought at first that this may
have been the final version since it agreed almost verbatim with the contents
of the 5.25 in. diskettes dating from January to March–April 1992 with a few
files dating to July 1992 (containing mostly references). Printed off a high-
quality dot matrix printer we used this version as the working version for the
publication of this monograph. It consisted of 12 chapters with titles that
differed from what he had expected to submit in 1972. Indeed, this was a
substantially different book.
In retrospect, one thing became clear and very sad. Atkinson was definitely
strugglingfiercely with the completionof the book during the spring andsum-
mer of 1992, months before his collapse in August of the same year. We can
surmise this because the Green Binder version makes reference to equations
and sections that sometimes appear in the previous versions (Daisy Wheel,
BlackBinder,etc.) andthiswithdifferentnumbering! Inaddition,onegathers
from the floppies that some files were saved in the early hours of the morning
and this on many consecutive days. As a result, GB became filled with hun-
viii
dreds of misprints and typographical errors as one had to refer to the older
versions constantly for what he had intended to say. Like the writers of the
NewTestamentinChristiandoctrine,wesuspectthatAtkinsonhadsomeolder
manuscripts before him when he wrote up the Green Binder version and not
unlike those writers he lifted the material from his older texts and fused them
into the GB. Yet, like Schubert’s majorsymphony, GB remained“unfinished.”
Still, in agreement with the family’s views, we believe that there must have
been another (later) set of diskettes other than the ones in our possession. In
some places in GB, there are lines of text that do not appear anywhere in the
disksthatwefound. ThusthesepagesinGBmusthavebeenprintedoffalater
(but certainlynotfinal)setofdiskettes. Itisunfortunatethatwe,andhistory,
willneverseewhatAtkinsonhadinmindasafinalmanuscript. Thisworkcan
only be a good approximationat best.
Now for the layout of the book: Because this book never had an actual title
we decided to give it the natural name appearing on the cover as it was to be
considered a continuation of Atkinson’s Volume I. However, the “Volume II”
wasdroppedbythepublisher. Thetableofcontents(ToC)usedhereismainly
drawn from GB version’s ToC with some sections omitted as they were either
incomplete or missing entirely from either the diskettes and/or the hard copy.
No Preface/Introduction was ever found for any of the versions; all existing
drafts consisted solely of solid mathematics with preambles to the sections
being addressed. More details on the differences between this text and the
documents referred to can be found in the Notes at the end of every chapter.
On this subject of “Notes,” we observe that Atkinson was fond of the Notes
idea. Indeed,herehecouldusehisencyclopedicknowledgeofthefieldtobefree
tooutlinehisreviewsandimpressionsofcompletedworkandintroducelinesfor
futureresearch. Unfortunately,wecannotevenbegintoaddressthisissueinhis
styleandinthis manner. Thistragedyleavesonewithanotherapproximation;
the Notes in this text, written by the second author, were reconstructed from
Atkinson’s own original notes for the few chapters for which he actually had
germs of notes. We have also added a section on Research Problems and have
illustrated the examples using Maple(cid:13)c. Finally, the bibliography has been
updated to the end of 2008, although we cannot make any claim as to its
thoroughness. We have tried to be faithful to the first author’s wishes for the
inclusion of references prior to 1992. Thus references to pre-1992 documents
aredrawndirectly fromAtkinson’sGBbibliography,andonlya fewadditional
ones that appeared to be relevant to the material presented are included (see
the notes and the bibliography for more information related to this item).
This text would certainly have had more impact had it appeared in 1992 or
1993whenitwasoriginallyexpectedtoappear(Atkinsonhadalreadycontacted
anotherpublisher atthat time). The interveningyearshaveseenmorethan80
papers in the subject of multiparameter spectral theory, a body of work that
supersedes this text in breadth and generality. Still, one of the triumphs of
this book lies in its exposition. It is written in the solid style typical of his
ix
texts, filled with ideas and techniques that have inspired researchers from far
and wide. It can be considered a lasting tribute to his contributions to this
subject over a span of more than 40 years.
Thebookisintendedforsenior-levelgraduatestudentsandresearchersindiffer-
entialequationsalike. Itcan,however,bereadbystrongseniorundergraduates
withanhonorsbackground,yetitmaybeslow-goingbuttherewardsarehigh.
I trust the reader will learn from this book and then contribute to the area as
a result: This would have been Atkinson’s wish, and it is certainly mine. Any
remaining typographical and other errors are the responsibility of the second
author so, if you get the chance, please write so that we can update this by
means of a web-based erratum.
I am indebted to a number of people for assistance in this project: To Dusja
AtkinsonandherchildrenLeslie andVivienne fortheir recollectionofthe man
that we all knew and admired and for their insights into the final stages of
this book. In particular, my profound gratitude goes to Vivienne Atkinson
Chisholm without whose constant encouragement over the past six years this
work would likely have never been completed. For technical support, I am
grateful to Antonio Rodriguez, his assistant Francisco, and to the Department
of Mathematics of the University of Las Palmas in Gran Canaria for building
me andprovidingme accesstoapureMS-DOS5.0machinewithawidefloppy
diskdrivesoastogainpassagetocriticaloriginaldocumentationrelatedtothe
task at hand. As well, I acknowledge,with thanks, Professor J. M. Pacheco of
the UniversityofLas Palmasfor aninvitationas visiting professorthere inthe
years2006–2008,duringwhichtimethisreconstructionbecameareality. Ialso
thank George Pearson of MacKichan Support, MacKichan Software, Inc., for
providing critical updates of Scientific Word and T3 to TEX software bundled
in its package. I must acknowledge with thanks a grant in partial support of
thisprojectfromtheOfficeoftheVice-PresidentofResearchandInternational
at Carleton University for the years (2002–2005). I also express my deep ap-
preciation to my partner Karen for her unfailing support through these past
few years, and to my daughter Oliviana–both of whom were there caring for
my younger children so that I could pursue this venture uninterrupted by the
daily requirements of raising a family while overseas.
Finally,Iwishtothankthefollowingreviewersfortheirtime: ProfessorsAfgan
Aslanov and Larry Turyn for reading overthe first few chapters and providing
helpful comments, suggestions, and correcting numerous typographical errors.
My most profound gratitude goes to Professor Hans Volkmer who painstak-
ingly read the entire manuscript over a period of a mere few months thereby
correcting interminable errors both mathematical and grammatic.
Angelo B. Mingarelli
Sardina del Norte, Gran Canaria, Spain
[email protected]
Contents
1 Preliminaries and Early History 1
1.1 Main results of Sturm-Liouville theory . . . . . . . . . . . . . . 1
1.2 General hypotheses for Sturm-Liouville
theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Transformations of linear second-order equations . . . . . . . . 4
1.4 Regularization in an algebraic case . . . . . . . . . . . . . . . . 6
1.5 The generalized Lam´e equation . . . . . . . . . . . . . . . . . . 6
1.6 Klein’s problem of the ellipsoidal shell . . . . . . . . . . . . . . 8
1.7 The theorem of Heine and Stieltjes . . . . . . . . . . . . . . . . 9
1.8 The later work of Klein and others . . . . . . . . . . . . . . . . 10
1.9 The Carmichael program . . . . . . . . . . . . . . . . . . . . . 11
1.10 Research problems and open questions . . . . . . . . . . . . . . 13
2 Some Typical Multiparameter Problems 17
2.1 The Sturm-Liouville case . . . . . . . . . . . . . . . . . . . . . 17
2.2 The diagonal and triangular cases. . . . . . . . . . . . . . . . . 19
2.3 Transformations of the parameters . . . . . . . . . . . . . . . . 20
2.4 Finite difference equations . . . . . . . . . . . . . . . . . . . . . 21
xi
