Boundary Value Problems, Weyl Functions, and Differential Operators (Monographs in Mathematics (108))

775 Pages·2020·3.698 MB·English

by  Jussi Behrndt, Seppo Hassi, Henk de Snoo

Monographs in Mathematics
108
Jussi Behrndt
Seppo Hassi
Henk de Snoo
Boundary
Value Problems,
Weyl Functions,
and Differential
Operators

Monographs in Mathematics
Volume 108
Series Editors
Herbert Amann, Universität Zürich, Zürich, Switzerland
Jean-Pierre Bourguignon, IHES, Bures-sur-Yvette, France
William Y. C. Chen, Nankai University, Tianjin, China
Associate Editor
Huzihiro Araki, Kyoto University, Kyoto, Japan
John Ball, Heriot-Watt University, Edinburgh, UK
Franco Brezzi, Università degli Studi di Pavia, Pavia, Italy
Kung Ching Chang, Peking University, Beijing, China
Nigel Hitchin, University of Oxford, Oxford, UK
Helmut Hofer, Courant Institute of Mathematical Sciences, New York, USA
Horst Knörrer, ETH Zürich, Zürich, Switzerland
Don Zagier, Max-Planck-Institut, Bonn, Germany
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Jussi Behrndt Seppo Hassi Henk de Snoo
(cid:129) (cid:129)
Boundary Value Problems,
Weyl Functions,
and Differential Operators
JussiBehrndt SeppoHassi
Institut für Angewandte Mathematik Mathematics andStatistics
Technische UniversitätGraz University of Vaasa
Graz,Austria Vaasa,Finland
Henk deSnoo
Bernoulli Institute for Mathematics
ComputerScienceandArtificialIntelligence
University of Groningen
Groningen,The Netherlands
ISSN 1017-0480 ISSN 2296-4886 (electronic)
Monographsin Mathematics
ISBN978-3-030-36713-8 ISBN978-3-030-36714-5 (eBook)
https://doi.org/10.1007/978-3-030-36714-5
MathematicsSubjectClassification(2010): 47A,47B,47E,47F,34B,34L,35P,81C,93B
©TheEditor(s)(ifapplicable)andTheAuthor(s)2020.Thisbookisanopenaccesspublication.
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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Linear Relations in Hilbert Spaces
1.1 Elementary facts about linear relations . . . . . . . . . . . . . . . 11
1.2 Spectra, resolvent sets, and points of regular type . . . . . . . . . 23
1.3 Adjoint relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.4 Symmetric relations . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.5 Self-adjoint relations . . . . . . . . . . . . . . . . . . . . . . . . . . 48
1.6 Maximal dissipative and accumulative relations . . . . . . . . . . . 58
1.7 Intermediate extensions and von Neumann’s formulas . . . . . . . 65
1.8 Adjoint relations and indefinite inner products . . . . . . . . . . . 74
1.9 Convergence of sequences of relations . . . . . . . . . . . . . . . . 79
1.10 Parametric representations for relations . . . . . . . . . . . . . . . 87
1.11 Resolvent operators with respect to a bounded operator . . . . . . 96
1.12 Nevanlinna families and their representations . . . . . . . . . . . . 100
2 Boundary Triplets and Weyl Functions
2.1 Boundary triplets . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
2.2 Boundary value problems . . . . . . . . . . . . . . . . . . . . . . . 115
2.3 Associated γ-fields and Weyl functions . . . . . . . . . . . . . . . . 118
2.4 Existence and construction of boundary triplets . . . . . . . . . . 126
2.5 Transformations of boundary triplets . . . . . . . . . . . . . . . . 134
2.6 Kre˘ın’s formula for intermediate extensions . . . . . . . . . . . . . 148
2.7 Kre˘ın’s formula for exit space extensions . . . . . . . . . . . . . . 155
2.8 Perturbation problems . . . . . . . . . . . . . . . . . . . . . . . . . 163
3 Spectra, Simple Operators, and Weyl Functions
3.1 Analytic descriptions of minimal supports of Borel measures . . . 169
3.2 Growth points of finite Borel measures. . . . . . . . . . . . . . . . 178
3.3 Spectra of self-adjoint relations . . . . . . . . . . . . . . . . . . . . 183
vi Contents
3.4 Simple symmetric operators . . . . . . . . . . . . . . . . . . . . . . 188
3.5 Eigenvalues and eigenspaces. . . . . . . . . . . . . . . . . . . . . . 196
3.6 Spectra and local minimality . . . . . . . . . . . . . . . . . . . . . 203
3.7 Limit properties of Weyl functions . . . . . . . . . . . . . . . . . . 212
3.8 Spectra and local minimality for self-adjoint extensions . . . . . . 218
4 Operator Models for Nevanlinna Functions
4.1 Reproducing kernel Hilbert spaces . . . . . . . . . . . . . . . . . . 223
4.2 Realization of uniformly strict Nevanlinna functions . . . . . . . . 235
4.3 Realization of scalar Nevanlinna functions via L2-space models . . 252
4.4 Realization of Nevanlinna pairs and generalized resolvents . . . . . 261
4.5 Kre˘ın’s formula for exit space extensions . . . . . . . . . . . . . . 270
4.6 Orthogonal coupling of boundary triplets . . . . . . . . . . . . . . 274
5 Boundary Triplets and Boundary Pairs for Semibounded Relations
5.1 Closed semibounded forms and their representations . . . . . . . . 282
5.2 Ordering and monotonicity . . . . . . . . . . . . . . . . . . . . . . 300
5.3 Friedrichs extensions of semibounded relations . . . . . . . . . . . 311
5.4 Semibounded self-adjoint extensions and their lower bounds . . . . 319
5.5 Boundary triplets for semibounded relations . . . . . . . . . . . . 332
5.6 Boundary pairs and boundary triplets . . . . . . . . . . . . . . . . 343
6 Sturm–Liouville Operators
6.1 Sturm–Liouville differential expressions . . . . . . . . . . . . . . . 366
6.2 Maximal and minimal Sturm–Liouville differential operators . . . 380
6.3 Regular and limit-circle endpoints . . . . . . . . . . . . . . . . . . 388
6.4 The case of one limit-point endpoint . . . . . . . . . . . . . . . . . 397
6.5 The case of two limit-point endpoints and interface conditions . . 412
6.6 Exit space extensions . . . . . . . . . . . . . . . . . . . . . . . . . 421
6.7 Weyl functions and subordinate solutions . . . . . . . . . . . . . . 425
6.8 Semibounded Sturm–Liouville expressions in the regular case . . . 434
6.9 Closed semibounded forms for Sturm–Liouville equations . . . . . 442
6.10 Principal and nonprincipal solutions
of Sturm–Liouville equations . . . . . . . . . . . . . . . . . . . . . 454
6.11 Semibounded Sturm–Liouville operators and
the limit-circle case . . . . . . . . . . . . . . . . . . . . . . . . . . 469
6.12 Semibounded Sturm–Liouville operators and
the limit-point case . . . . . . . . . . . . . . . . . . . . . . . . . . 477
6.13 Integrable potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 483
7 Canonical Systems of Differential Equations
7.1 Classes of integrable functions . . . . . . . . . . . . . . . . . . . . 500
7.2 Canonical systems of differential equations . . . . . . . . . . . . . 504
Contents vii
7.3 Regular and quasiregular endpoints . . . . . . . . . . . . . . . . . 510
7.4 Square-integrability of solutions of real canonical systems . . . . . 513
7.5 Definite canonical systems. . . . . . . . . . . . . . . . . . . . . . . 520
7.6 Maximal and minimal relations for canonical systems . . . . . . . 525
7.7 Boundary triplets for the limit-circle case . . . . . . . . . . . . . . 534
7.8 Boundary triplets for the limit-point case . . . . . . . . . . . . . . 543
7.9 Weyl functions and subordinate solutions . . . . . . . . . . . . . . 559
7.10 Special classes of canonical systems . . . . . . . . . . . . . . . . . 566
8 Schr¨odinger Operators on Bounded Domains
8.1 Rigged Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . 577
8.2 Sobolev spaces, C2-domains, and trace operators . . . . . . . . . . 581
8.3 Trace maps for the maximal Schr¨odinger operator . . . . . . . . . 588
8.4 A boundary triplet for the maximal Schr¨odinger operator . . . . . 600
8.5 Semibounded Schr¨odinger operators . . . . . . . . . . . . . . . . . 611
8.6 Coupling of Schr¨odinger operators . . . . . . . . . . . . . . . . . . 616
8.7 Bounded Lipschitz domains . . . . . . . . . . . . . . . . . . . . . . 624
A Integral Representations of Nevanlinna Functions
A.1 Borel transforms and their Stieltjes inversion . . . . . . . . . . . . 631
A.2 Scalar Nevanlinna functions . . . . . . . . . . . . . . . . . . . . . . 636
A.3 Operator-valued integrals . . . . . . . . . . . . . . . . . . . . . . . 645
A.4 Operator-valued Nevanlinna functions . . . . . . . . . . . . . . . . 655
A.5 Kac functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663
A.6 Stieltjes and inverse Stieltjes functions . . . . . . . . . . . . . . . . 668
B Self-adjoint Operators and Fourier Transforms
B.1 The scalar case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678
B.2 The vector case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
C Sums of Closed Subspaces in Hilbert Spaces . . . . . . . . . . . . . . . . 691
D Factorization of Bounded Linear Operators . . . . . . . . . . . . . . . . 699
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769
Preface
This monograph is about boundary value problems, Weyl functions, and differ-
ential operators. It grew out of a number of courses and seminars on functional
analysis,operatortheory,anddifferentialequations,whichtheauthorshavegiven
over a long period of time at various institutions. The project goes back to 2005
withacourseonextensiontheoryofsymmetricoperators,boundarytriplets,and
Weyl functions given at TU Berlin, while an extended form of the course was
presented in 2006/2007 at the University of Groningen. Many more such courses
and seminars, often on special topics, would follow at TU Berlin, Jagiellonian
University in Krak´ow, and, since 2011, at TU Graz.
Theauthorswishtothankallthestudents,PhDstudents,andpostdocswho
haveattendedtheselectures;theircriticalquestionsandcommentshaveledtonu-
merous improvements. They have shown that lectures at the blackboard provide
the ultimate test for the quality of the material. In particular, we mention Bern-
hard Gsell, Markus Holzmann, Christian Ku¨hn, Vladimir Lotoreichik, Jonathan
Rohleder, Peter Schlosser, Philipp Schmitz, Simon Stadler, Alef Sterk, and Rudi
Wietsma. It is our experience that the individual chapters of this monograph can
be used (with small additions from some of the other chapters) for independent
courses on the respective topics.
The book has benefited from our collaboration with many different col-
leagues. We would like to single out our friends and faithful coauthors Yuri Ar-
linski˘ı,VladimirDerkach,PeterJonas,MatthiasLanger,AnnemarieLuger,Mark
Malamud, Hagen Neidhardt, Franek Szafraniec, Carsten Trunk, Henrik Winkler,
and Harald Woracek. Special thanks go to Fritz Gesztesy, Gerd Grubb, Heinz
Langer, and James Rovnyak, who have responded to our queries concerning his-
torical developments and references.
Wegratefullyacknowledgethesupportofthefollowinginstitutions:Deutsche
Forschungsgemeinschaft, Jagiellonian University, TU Berlin, and TU Graz. We
would like to thank the Mathematisches Forschungsinstitut Oberwolfach and the
Mittag-Leffler Institute in Djursholm for their hospitality during the final stages
of the preparation of this book. Finally, we are indebted to the Austrian Science
Fund(GrantPUB683-Z)andtheUniversityofVaasaforfundingtheopenaccess
publication of this monograph.
Jussi Behrndt, Seppo Hassi, and Henk de Snoo
Introduction
In this monograph the theory of boundary triplets and their Weyl functions is
developed and applied to the analysis of boundary value problems for differential
equationsandgeneraloperatorsinHilbertspaces.Concreteillustrationsbymeans
ofweightedSturm–Liouvilledifferentialoperators,canonicalsystemsofdifferential
equations,andmultidimensionalSchr¨odingeroperatorsareprovided.Theabstract
notions of boundary triplets and Weyl functions have their roots in the theory of
ordinary differential operators; they appear in a slightly different context also in
the treatment of partial differential operators.
Beforedescribingthecontentsofthemonographitmaybehelpfultoexplain
theideasinthistextbymeansofthefollowingsimpleSturm–Liouvilledifferential
expression
d2
L=− +V, (1)
dx2
whereitisassumedthatthepotentialV isarealmeasurablefunction.Thecontext
inwhichthisdifferentialexpressionwillbeplacedservesasanexampleaswellas
a motivation. The first step is to associate with L some differential operators in
a suitable Hilbert space. Assume, e.g., that (1) is given on the positive half-line
R+ =(0,∞)andassumeforsimplicitythattherealfunctionV isbounded.Define
the linear space D by
max
D =(cid:2)f ∈L2(R+):f,f(cid:2) absolutely continuous, Lf ∈L2(R+)(cid:3)
max
and define the minimal operator S associated with L by
Sf =−f(cid:2)(cid:2)+Vf, domS =(cid:2)f ∈D :f(0)=f(cid:2)(0)=0(cid:3).
max
Then S is a closed densely defined symmetric operator L2(R+); in fact, it is the
closureof(thegraphof)therestrictionofS toC∞(R+).Itcanbeshownthatthe
0
adjoint operator S∗ is given by
S∗f =−f(cid:2)(cid:2)+Vf, domS∗ =D ,
max
whichisusuallycalledthemaximaloperatorassociatedwithL.Roughlyspeaking,
S is a two-dimensional restriction of S∗ by means of the boundary conditions
© The Author(s) 2020 1
J. Behrndt et al., Boundary Value Problems, Weyl Functions,
and Differential Operators, Monographs in Mathematics 108,
https://doi.org/10.1007/978-3-030-36714-5_1