The peak model for the triplet extensions and their transformations to the reference Hilbert space in the case of finite defect numbers

Abstract

We develop the so-called peak model for the triplet extensions of supersingular perturbations in the case of a not necessarily semibounded symmetric operator with finite defect numbers. The triplet extensions in scales of Hilbert spaces are described by means of abstract boundary conditions. The resolvent formulas of Krein-Naimark type are presented in terms of the $\gamma$-field and the abstract Weyl function. By applying certain scaling transformations to the triplet extensions in an intermediate Hilbert space we investigate the obtained operators acting in the reference Hilbert space and we show the connection with the classical extensions