Pontryagin space methods in singular perturbation theory

The study of perturbations of self-adjoint operators in Hilbert spaces is a wide and important area in operator theory. Many such problems arise from mathematical physics and are usually investigated by Hilbert space methods. In the treatment of certain classes of perturbations there often appears a scalar (or matrix) function from the so- called Nevanlinna class, which plays an important role, in particular, in the study of the spectral properties of the perturbation. However, if the perturbation is singular, then the corresponding function may only belong to a generalized Nevanlinna class. This class is closely connected with the theory of operators in spaces with an indefinite inner product, more precisely, in Pontryagin spaces. Within the proposed project it is planned to bring together these two fields and investigate more closely the interplay between singular perturbation theory and the theory of generalized Nevanlinna functions or - in other words - to use Pontryagin space methods in perturbation theory of operators in Hilbert spaces.