BoundaryValueProblems, WeylFunctions, DiffentialOperators

The present monograph belongs to the area of mathematical analysis, functional analysis and operator theory, as well as the theory of differential equations. Furthermore, there are close connections to mathematical physics and mathematical system theory. A general class of boundary value problems for linear operators in Hilbert spaces is systematically investigated with the help of modern functional analytic techniques. The abstract treatment is directly inspired by the theory and applications of differential equations, and, in turn, the general results are applied to ordinary and partial differential operators. The methods are based on the notion of boundary triplets and corresponding Weyl functions. With the help of these techniques the spectral properties of the associated operators, and hence the solvability of the underlying boundary value problem, can be investigated and completely characterized. The monograph consists of an abstract part (Chapters 1-5) and an applied part (Chapters 6-8), as well as an Appendix and a comprehensive list of references with some further notes. After a self- contained survey on linear relations in Hilbert spaces in Chapter 1 boundary triplets and Weyl functions for symmetric operators and relations are studied in detail in Chapter 2. The abstract theory is further developed in Chapter 3, where a complete spectral characterization of the self- adjoint extensions is provided. In Chapter 4 operator models in reproducing kernel Hilbert spaces for the representation of Weyl functions are discussed. The important special case of semibounded operators and relations, as well as the interplay with the corresponding quadratic forms is investigated in Chapter 5. Eventually, the abstract concepts are applied to Sturm-Liouville operators in Chapter 6, to a class of canonical systems of differential equations in Chapter 7, and to multidimensional Schrödinger operators on bounded domains in Chapter 8. In these applications the boundary triplet typically consists of Dirichlet and Neumann boundary mappings defined on the domain of the maximal differential operator, and the corresponding abstract Weyl function coincides with the Titchmarsh-Weyl m-function or the Dirichlet-to-Neumann map.