Boundary relations

Description and Aim

Numerous physical and technical processes can be described by differential equations. The mathematical treatment of the corresponding initial-value and boundary-value problems uses methods from operator theory and functional analysis. Typically the properties of the solutions of initial-value and boundary-value problems depend on the spectral properties of the associated differential operators. Hence operator theoretic methods in spectral theory are intimately connected with the analysis of differential equations and their applications. These operator theoretic methods give rise to the extension theory in the title. On the other hand the theory of boundary triplets offers a convenient technique for dealing with boundary-value problems involving ordinary differential operators, partial differential operators, and equations with operator-valued coefficients. Boundary triplets were first introduced some thirty years ago and nowadays they are extensively used also in mathematical physics, for instance, for Schrödinger operators on quantum graphs and for scattering problems. Their popularity has been widened due to the fundamental contributions by Derkach and Malamud on the area of extension theory of abstract symmetric operators who associated with a boundary triplet an analytic object, the so-called Weyl function. The original notion of boundary triplet is sometimes too restrictive because the corresponding Weyl function belongs to a class of functions, which is in general too small when dealing with infinite-dimensional problems. This objection led to introduction of the new notions of boundary relations and associated Weyl families with a proper generality for dealing with various problems appearing in the extension theory of general symmetric operators. The theory of boundary triplets and boundary relations is presently being further developed with applications to, for instance, elliptic operators and to non-standard boundary conditions. Parallel to the development of boundary relations, analogous notions have been independently developed in the area of system theory.  The aim of the workshop is to bring together specific mathematicians from operator theory, functional analysis, and system theory involved with these recent developments. The aim is to pay attention to the general field of extension theory with a special emphasis on boundary relations and boundary triplets and their applications to boundary value problems for ordinary and partial differential equations, as well as to consider the intimate connections with system theory. So the workshop brings together people working in these areas, with an aim to create new views and ideas for some further progress in the outlined area of the workshop.