Boundary triplets and Weyl functions. Recent developments

Abstract

Selfadjoint extensions of a closed symmetric operator in a Hilbert space with equal deficiency indices are described by means of ordinary boundary triplets. In certain problems the more general notion of a boundary triplet of bounded type is needed. It will be shown that such triplets correspond in a certain way with the, in general infinite dimensional, graph perturbations of selfadjoint operators or relations. However, when considering selfadjoint exit spaces extensions for a symmetric operator by means of the so-called Kreĭn's formula one meets the notion of boundary relation (even when the deficiency indices are finite). Whereas ordinary boundary triplets and boundary triplets of bounded type correspond to bounded unitary or unitary operators in a Kreĭn space, respectively, boundary relations correspond to general unitary relations in a Kreĭn space, which are not necessarily single-valued. It is shown that the study of isometric relations in a Kreĭn space has useful applications. This present overview of recent developments includes illustrations, for instance, by means of elliptic differential operators and Schrödinger operators with local point interactions.