A. Mantile (Uni. Reims)
The abstract theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of Schrödinger-type operators with linear boundary/interface conditions on (a relatively open part of) a compact hypersurface in nD. Our approach allows to obtain Krein-like resolvent formulae where the reference operator coincides with the free operator corresponding to our model, i.e.: the Schrödinger operator defined on regular functions without interface conditions. This provides an useful tool for the scattering problem from a hypersurface. Schatten-von Neumann estimates for the difference of the powers of resolvents (of the free and the perturbed models) yield the existence and completeness of the wave operators of the associated scattering systems, while a limiting absorption principle for singular perturbations allows us to obtain the generalized eigenfunctions and the scattering matrix; both these objects are written in terms of operator-valued Weyl functions.
This is a joint work with A. Posilicano and M. Sini.