Boundary Triplets, Weyl Functions, and the Kreĭn Formula

Abstract

This chapter contains a short review of the theory of boundary triplets, and the corresponding Weyl functions, of symmetric operators in Hilbert and Kreȋn spaces. The theory of generalized resolvents of such operators is exposed from the point of view of boundary triplets approach. Applications to different continuation problems related to the extension theory of Kreȋn space symmetric operators are discussed.

Functional model for extensions of symmetric operators and applications to scattering theory

“…Clearly, a maximal dissipative operator is closed. We next describe the boundary triple approach to the extension theory of symmetric operators with equal deficiency indices (see in [14] a review of the subject). This approach is particularly useful in the study of self-adjoint extensions of differential operators of second order.…”

Compressed Resolvents and Reduction of Spectral Problems on Star Graphs

“…For this we need some notation and basic properties of symmetric linear relations, boundary triplets and corresponding Titchmarsh-Weyl functions, see e.g. [34,Chapter 14], [9,10].…”

Effective Behaviour of Critical-Contrast PDEs: Micro-resonances, Frequency Conversion, and Time Dispersive Properties. I

“…where A is an operator in a Hilbert space H (e.g., a Laplacian would lead to the classical Green's identity with Γ 0 and Γ 1 defined as traces of the function and of its normal derivative on the boundary, respectively) and (H, Γ 0 , Γ 1 ) is the boundary triple, consisting of an auxiliary Hilbert space (e.g., the L 2 space over the boundary) and a pair of operators onto H. Unfortunately, it is well known that such setup (although possible) encounters problems in the PDE setting (we refer the reader to a review [32] of the state of the art in the theory, see also Section 2 below), thus necessitating an alternative, for which we have selected [58] as a natural fit for the analysis we have in mind. In order to put the results of our work into the correct context, we mention that already in the extensively studied setup of double-porosity models (see Model I, Section 2) we are not only able to develop the operatortheoretical approach, but also to extend the existing results on spectral convergence in at least two ways: firstly, by ascertaining the rate convergence as O(ε 2/3 ), and secondly, by disposing of the assumption on the eigenvectors of the Dirichlet Laplacian on the soft component, which was previously considered essential.…”