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Über dieses Buch

This open access book presents a comprehensive survey of modern operator techniques for boundary value problems and spectral theory, employing abstract boundary mappings and Weyl functions. It includes self-contained treatments of the extension theory of symmetric operators and relations, spectral characterizations of selfadjoint operators in terms of the analytic properties of Weyl functions, form methods for semibounded operators, and functional analytic models for reproducing kernel Hilbert spaces. Further, it illustrates these abstract methods for various applications, including Sturm-Liouville operators, canonical systems of differential equations, and multidimensional Schrödinger operators, where the abstract Weyl function appears as either the classical Titchmarsh-Weyl coefficient or the Dirichlet-to-Neumann map.

The book is a valuable reference text for researchers in the areas of differential equations, functional analysis, mathematical physics, and system theory. Moreover, thanks to its detailed exposition of the theory, it is also accessible and useful for advanced students and researchers in other branches of natural sciences and engineering.

Inhaltsverzeichnis

Frontmatter

Open Access

Chapter 1. Introduction

Abstract
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 equations and general operators in Hilbert spaces.
Jussi Behrndt, Seppo Hassi, Henk De Snoo

Open Access

Chapter 2. Linear Relations in Hilbert Spaces

Abstract
A linear relation from one Hilbert space to another Hilbert space is a linear subspace of the product of these spaces. In this chapter some material about such linear relations is presented and it is shown how linear operators, whether densely defined or not, fit in this context.
Jussi Behrndt, Seppo Hassi, Henk De Snoo

Open Access

Chapter 3. Boundary Triplets and Weyl Functions

Abstract
The basic properties of boundary triplets for closed symmetric operators or relations in Hilbert spaces are presented. These triplets give rise to a parametrization of the intermediate extensions of symmetric relations, in particular of the self-adjoint extensions.
Jussi Behrndt, Seppo Hassi, Henk De Snoo

Open Access

Chapter 4. Spectra, Simple Operators, and Weyl Functions

Abstract
In this chapter the spectrum of a self-adjoint operator or relation will be completely characterized in terms of the analytic behavior and the limit properties of the Weyl function.
Jussi Behrndt, Seppo Hassi, Henk De Snoo

Open Access

Chapter 5. Operator Models for Nevanlinna Functions

Abstract
The classes of Weyl functions and more generally of Nevanlinna functions will be studied from the point of view of reproducing kernel Hilbert spaces.
Jussi Behrndt, Seppo Hassi, Henk De Snoo

Open Access

Chapter 6. Boundary Triplets and Boundary Pairs for Semibounded Relations

Abstract
Semibounded relations in a Hilbert space automatically have equal defect numbers, so that there are always self-adjoint extensions. In this chapter the semibounded self-adjoint extensions of a semibounded relation will be investigated.
Jussi Behrndt, Seppo Hassi, Henk De Snoo

Open Access

Chapter 7. Sturm-Liouville Operators

Abstract
Second-order Sturm-Liouville differential expressions generate self-adjoint differential operators in weighted L2-spaces on an interval (a, b).
Jussi Behrndt, Seppo Hassi, Henk De Snoo

Open Access

Chapter 8. Canonical Systems of Differential Equations

Abstract
Boundary value problems for regular and singular canonical systems of differential equations are investigated.
Jussi Behrndt, Seppo Hassi, Henk De Snoo

Open Access

Chapter 9. Schrödinger Operators on Bounded Domains

Abstract
For the multi-dimensional Schrödinger operator -∆+V with a bounded real potential V on a bounded domain \( \varOmega \subset {\mathbb{R}}^{n} \) with a C2-smooth boundary a boundary triplet and a Weyl function will be constructed.
Jussi Behrndt, Seppo Hassi, Henk De Snoo

Backmatter

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