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Boundary Value Problems, Weyl Functions, and Differential Operators.

Bibliographic Details
Main Author: Behrndt, Jussi.
Other Authors: Hassi, Seppo., de Snoo, Henk.
Format: eBook
Language:English
Published: Cham : Springer International Publishing AG, 2020.
Edition:1st ed.
Series:Monographs in Mathematics Series
Subjects:
Electronic books.
Online Access:Click to View
Table of Contents:
  • Intro
  • 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.

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