Trends and Tricks in Spectral Theory

99 Pages·2007·0.89 MB·english

by  Lyonell Boulton

XX ESCUELA VENEZOLANA DE MATEMÁTICAS
n
viti
e
L
el
a
h
c
Mi
d
n
a
n
o
ult
o
B
ell
n
o
y
L TrENDS AND TrICkS
IN SpECTrAL ThEOry
Lyonell Boulton
Michael Levitin
y
r
O
E
h
T

L
A
r
T
C
E
p
S

N
I

S
k
C
I
r
T

D
N
A
Ediciones IVIC

S
D
N
E
r
T
MÉrIDA, VENEZUELA, 2 AL 7 DE SEpTIEMBrE DE 2007
XX ESCUELA VENEZOLANA DE MATEMA´TICAS
Trends and Tricks
in Spectral Theory
1
Lyonell Boulton
Michael Levitin1,2
1Heriot-Watt University, 2Cardiff University
[email protected], [email protected]
ME´RIDA, VENEZUELA, 2 AL 7 DE SEPTIEMBRE DE 2007
XX ESCUELA VENEZOLANA DE MATEMA´TICAS
La Escuela Venezolana de Matem´aticas es una actividad de los postgrados
en matem´aticas de las siguientes instituciones: Centro de Estudios Avan-
zados del Instituto Venezolano de Investigaciones Cient´ıficas, Facultad de
Ciencias de la Universidad Central de Venezuela, Facultad de Ciencias de la
Universidad de Los Andes, Universidad Sim´on Bol´ıvar, Universidad Centro
Occidental Lisandro Alvarado y Universidad de Oriente, y se realiza bajo el
auspicio de la Asociaci´on Matem´atica Venezolana.
La XX ESCUELA VENEZOLANA DE MATEMA´TICAS recibio´ financiamiento de
la Academia de Ciencias F´ısicas, Matem´aticas y Naturales, la Corporacio´n
Andina de Fomento (CAF), el Fondo Nacional de Ciencia, Tecnolog´ıa e In-
novaci´on(FONACIT),laFundacio´nTALVEN,elInstitutoVenezolanodeIn-
vestigaciones Cient´ıficas (Departamento de Matem´aticas y Ediciones IVIC),
la Universidad de los Andes (CEP, CDCHT, Facultad de Ciencias y Depar-
tamento de Matem´aticas) y el Rectorado de la Universidad Centroccidental
Lisandro Alvarado.
2000 Mathematics Subject Classification: 35P05,(35P15, 65N25).
cEdiciones IVIC
(cid:13)
Instituto Venezolano de Investigaciones Cient´ıficas
Trends and Tricks in Spectral Theory
Lyonell Boulton y Michael Levitin
Dise˜no y edicio´n: Escuela Venezolana de Matem´aticas
Preprensa e impresio´n: Editorial Texto
Dep´osito legal lf66020076002490
ISBN 978-980-261-086-0
Caracas, Venezuela
2007
iii
Preface
Thespectraofoperators,underdifferentnames,manifestthemselvesalmost
everywhere in our everyday life. The colour of light we see is related to
the spectra of atoms and molecules. The tones and overtones of musical
instruments we hear are determined by the spectra of strings and drums.
The resonances produced by cars running over a bridge —such as in the
recently collapsed viaduct number 1 of the Caracas-La Guaira highway, are
predicted by the spectral analysis of beams and suspension cables.
The spectral analysis of differential operators is an area of research that
has been active for over one hundred years. In these lecture notes we have
deliberately picked only some particular “trends” in this theory. Many good
surveys are available, and it makes no sense even to attempt to compete
against them. As modus operandi we only provide a rough overview of the
theoretical aspects, and focus mostly on describing techniques and “tricks”
that havebeen successfully used to solve some long standingproblems, sev-
eral of which are among the most important ones in mathematical analysis
in the last century.
The book is organised in two main parts. The first part comprises Chap-
ters 1 and 2. They serve as an overview of the basics of spectral theory and
the theory of differential operators. We strongly recommend that even the
readers familiar with this material attempt all the problems appearing along
thetext. Althoughtheyarenotusuallydifficult,theyareoftennon-standard
and form an integral part of these notes.
Chapters3-6maybereadindependentlyfromeachother,withoccasional
cross-references. However, Chapters 3-5 discuss in detail various aspects
involving the spectrum of Laplace operators, so they are closely connected.
Chapter 6 may be read immediately after Chapter 2.
The list of references is not supposed to be complete; the readers are
advised to consult these books and papers for further literature sources.
iv
Weacknowledge withthanksthefinancialsupportoftheFundacio´nTAL-
VEN and the several organisations which provided funding for the Escuela.
Part of the research presented in this book was carried out under the grant
of the Leverhulme Trust, we acknowledge their support. Michael Levitin is
grateful for the hospitality of Isaac Newton Institute, Cambridge, where a
large part of these notes was written.
Finally,we would like tothankStellaBrassesco, Carlos Di Prisco, and the
other organisers of the XX Escuela Venezolana de Matem´aticas, for their
involvement in the preparation of this event.
Contents
Preface iii
1 Elements of spectral theory 1
1.1 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Linear operators . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Self-adjointness . . . . . . . . . . . . . . . . . . . . 5
1.1.4 Symmetric quadratic forms . . . . . . . . . . . . . . 5
1.2 Spectral problems . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 The spectrum . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Discrete and essential spectrum. . . . . . . . . . . . 7
1.3 Differential operators . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 Ellipticity . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Sobolev spaces . . . . . . . . . . . . . . . . . . . . 10
1.3.3 Dirichlet and Neumann boundary conditions . . . . . 11
1.4 Further spectral results . . . . . . . . . . . . . . . . . . . . 12
1.4.1 The spectral theorem . . . . . . . . . . . . . . . . . 12
1.4.2 Separation of variables . . . . . . . . . . . . . . . . 14
2 Variational techniques 17
2.1 The Rayleigh-Ritz principle . . . . . . . . . . . . . . . . . . 17
2.1.1 The Rayleigh quotient . . . . . . . . . . . . . . . . . 17
2.1.2 The min-max principle . . . . . . . . . . . . . . . . 18
2.1.3 The Rayleigh-Ritz principle . . . . . . . . . . . . . . 20
2.2 The projection method . . . . . . . . . . . . . . . . . . . . 21
2.2.1 Weak eigenvalue problems . . . . . . . . . . . . . . 21
2.2.2 Estimating the Mathieu characteristic values . . . . . 22
v
vi CONTENTS
2.3 The finite element method . . . . . . . . . . . . . . . . . . 23
2.3.1 Finite element spaces . . . . . . . . . . . . . . . . . 24
2.3.2 Piecewise linear elements in 1D . . . . . . . . . . . . 24
2.3.3 Finite elements in 2D . . . . . . . . . . . . . . . . . 26
2.3.4 Higher order elements elements . . . . . . . . . . . . 29
3 Basic estimates of eigenvalues. Counting function 31
3.1 Elementary estimates and fundamental tools . . . . . . . . . 31
3.1.1 Domain monotonicity for the Dirichlet Laplacian . . 32
3.1.2 Dirichlet-Neumann bracketing . . . . . . . . . . . . 33
3.2 Counting function . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.1 Weyl’s one-term asymptotics . . . . . . . . . . . . . 36
3.2.2 Method of proof — counting the squares . . . . . . 37
3.2.3 Two-term asymptotics . . . . . . . . . . . . . . . . 39
3.2.4 Fractal boundary . . . . . . . . . . . . . . . . . . . 40
3.3 An inequality between Dirichlet and Neumann eigenvalues . 44
3.3.1 Statement and proof . . . . . . . . . . . . . . . . . 44
3.3.2 Dirichlet-to-Neumann map . . . . . . . . . . . . . . 45
4 Isoperimetric and universal estimates of eigenvalues 47
4.1 Isoperimetric estimates of eigenvalues . . . . . . . . . . . . 47
4.1.1 Symmetrisation. Faber-Krahn inequality . . . . . . . 47
4.1.2 Other isoperimetric inequalities. Ratios of eigenval-
ues, numerics . . . . . . . . . . . . . . . . . . . . . 49
4.2 Universal estimates of eigenvalues . . . . . . . . . . . . . . 52
4.2.1 Payne-Po´lya-Weinberger and Yang’s inequalities for
the Dirichlet Laplacian . . . . . . . . . . . . . . . . 52
4.2.2 General commutator method . . . . . . . . . . . . . 53
4.2.3 Polya’s conjecture . . . . . . . . . . . . . . . . . . . 55
5 Isospectrality and symmetry tricks 57
5.1 Can one hear the shape of a drum? . . . . . . . . . . . . . . 57
5.1.1 Heattraceasymptotics,heatinvariants,Kac’squestion 57
5.1.2 NegativeanswertoKac’squestion—exampleoftwo
isospectral domains by Gordon, Webb and Wolpert . 58
5.2 Can one hear boundary conditions? . . . . . . . . . . . . . 59
5.2.1 Zarembaproblem,mixedDirichlet-Neumannisospec-
trality and transplantation tricks . . . . . . . . . . . 59
CONTENTS vii
5.2.2 Symmetry tricks . . . . . . . . . . . . . . . . . . . . 64
5.2.3 Symmetries and isospectrality . . . . . . . . . . . . . 67
6 Eigenvalue enclosures and spectral pollution 71
6.1 Spectral pollution . . . . . . . . . . . . . . . . . . . . . . . 71
6.1.1 Spurious eigenvalues . . . . . . . . . . . . . . . . . 71
6.1.2 Why does the projection method pollute? . . . . . . 73
6.1.3 The distance function from the spectrum and the
Davies-Plum approach . . . . . . . . . . . . . . . . 75
6.2 The quadratic projection method . . . . . . . . . . . . . . . 77
6.2.1 Weak formulation of the method . . . . . . . . . . . 78
6.2.2 The Shargorodsky theorem . . . . . . . . . . . . . . 78
6.2.3 Quadratic matrix eigenvalue problems . . . . . . . . 80
6.2.4 Spectral exactness . . . . . . . . . . . . . . . . . . . 82
Bibliography 85
viii
Chapter 1
Elements of spectral theory
In this first chapter, we:
Describe briefly the elements of spectral theory.
•
Fix the notation which will be used in this course.
•
Introduce various canonical operators which will be the centre
•
of interest of later chapters.
1.1 Operators
Operators and their spectral properties are the main objects to be studied
in this course. The audience is assumed to be familiar with the elements
of linear algebra, functional analysis and operator theory. For the benefit
of those who need further reading on the topics discussed here, we include
some details on standard references.
1.1.1 Hilbert spaces
Throughout this course, the calligraphic letters, such as and , refer to
H L
generic separable Hilbert spaces over the field of complex numbers, C. The
inner product on will be denoted by , : C and the norm
H h· ·i H×H −→
by u := u,u 1/2, for any u .
k k h i ∈ H
Recall that , is a sesquilinear form:
h· ·i
αu+βv,w = α u,w +β v,w and u,v = v,u ,
h i h i h i h i h i
1