Laplacians with singular perturbations supported on hypersurfaces

We review the main results of our recent work on singular perturbations supported on bounded hypersurfaces. Our approach consists in using the theory of self-adjoint extensions of restrictions to build self-adjoint realizations of the $n$-dimensional Laplacian with linear boundary conditions on (a relatively open part of) a compact hypersurface. This allows one to obtain Krein-like resolvent formulae where the reference operator coincides with the free self-adjoint Laplacian in $\mathbb{R}^n$, providing in this way with an useful tool for the scattering problem from a hypersurface. As examples of this construction, we consider the cases of Dirichlet and Neumann boundary conditions assigned on an unclosed hypersurface.