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

Contents
Preface
1 Introduction
2 Linear Relations in Hilbert Spaces
	1.1
Elementary facts about linear relations
	1.2
Spectra, resolvent sets, and points of regular type
	1.3
Adjoint relations
	1.4
Symmetric relations
	1.5
Self-adjoint relations
	1.6
Maximal dissipative and accumulative relations
	1.7
Intermediate extensions and von Neumann\'s formulas
	1.8
Adjoint relations and indefinite inner products
	1.9
Convergence of sequences of relations
	1.10
Parametric representations for relations
	1.11
Resolvent operators with respect to a bounded operator
	1.12
Nevanlinna families and their representations
3 Boundary Triplets and Weyl Functions
	2.1
Boundary triplets
	2.2
Boundary value problems
	2.3 Associated γ-fields and Weyl functions
	2.4
Existence and construction of boundary triplets
	2.5
Transformations of boundary triplets
	2.6
Kreın\'s formula for intermediate extensions
	2.7
Kreın\'s formula for exit space extensions
	2.8
Perturbation problems
4 Spectra, Simple Operators, and Weyl Functions
	3.1
Analytic descriptions of minimal supports of Borel measures
	3.2
Growth points of finite Borel measures
	3.3
Spectra of self-adjoint relations
	3.4
Simple symmetric operators
	3.5
Eigenvalues and eigenspaces
	3.6
Spectra and local minimality
	3.7
Limit properties of Weyl functions
	3.8
Spectra and local minimality for self-adjoint extensions
5 Operator Models for Nevanlinna Functions
	4.1
Reproducing kernel Hilbert spaces
	4.2
Realization of uniformly strict Nevanlinna functions
	4.3
Realization of scalar Nevanlinna functions via L2-space models
	4.4
Realization of Nevanlinna pairs and generalized resolvents
	4.5
Kreın\'s formula for exit space extensions
	4.6
Orthogonal coupling of boundary triplets
6 Boundary Triplets and Boundary Pairs for Semibounded Relations
	5.1
Closed semibounded forms and their representations
	5.2
Ordering and monotonicity
	5.3
Friedrichs extensions of semibounded relations
	5.4
Semibounded self-adjoint extensions and their lower bounds
	5.5
Boundary triplets for semibounded relations
	5.6
Boundary pairs and boundary triplets
7 Sturm–Liouville Operators
	6.1
Sturm–Liouville differential expressions
	6.2
Maximal and minimal Sturm–Liouville differential operators
	6.3
Regular and limit-circle endpoints
	6.4
The case of one limit-point endpoint
	6.5
The case of two limit-point endpoints and interface conditions
	6.6
Exit space extensions
	6.7
Weyl functions and subordinate solutions
	6.8
Semibounded Sturm–Liouville expressions in the regular case
	6.9
Closed semibounded forms for Sturm–Liouville equations
	6.10
Principal and nonprincipal solutions of Sturm–Liouville equations
	6.11
Semibounded Sturm–Liouville operators and the limit-circle case
	6.12
Semibounded Sturm–Liouville operators and the limit-point case
	6.13
Integrable potentials
8 Canonical Systems of Differential Equations
	7.1
Classes of integrable functions
	7.2
Canonical systems of differential equations
	7.3
Regular and quasiregular endpoints
	7.4
Square-integrability of solutions of real canonical systems
	7.5
Definite canonical systems
	7.6
Maximal and minimal relations for canonical systems
	7.7
Boundary triplets for the limit-circle case
	7.8
Boundary triplets for the limit-point case
	7.9
Weyl functions and subordinate solutions
	7.10
Special classes of canonical systems
9 Schrödinger Operators on Bounded Domains
	8.1
Rigged Hilbert spaces
	8.2
Sobolev spaces, C2-domains, and trace operators
	8.3
Trace maps for the maximal Schrödinger operator
	8.4
A boundary triplet for the maximal Schrödinger operator
	8.5
Semibounded Schrödinger operators
	8.6
Coupling of Schrödinger operators
	8.7
Bounded Lipschitz domains
Integral Representations of Nevanlinna Functions
	A.1
Borel transforms and their Stieltjes inversion
	A.2
Scalar Nevanlinna functions
	A.3
Operator-valued integrals
	A.4
Operator-valued Nevanlinna functions
	A.5
Kac functions
	A.6
Stieltjes and inverse Stieltjes functions
Self-adjoint Operators and Fourier Transforms
	B.1
The scalar case
	B.2
The vector case
Sums of Closed Subspaces in Hilbert Spaces
Factorization of Bounded Linear Operators
Notes
Bibliography
List of Symbols
Index