Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces

The abstract theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic operator on $\mathbb{R}^{n}$ with linear boundary conditions on (a relatively open part of) a compact hypersurface. Our approach allows to obtain Krein-like resolvent formulas where the reference operator coincides with the "free" operator with domain $H^{2}(\mathbb{R}^{n})$; this provides an useful tool for the scattering problem from a hypersurface. Concrete examples of this construction are developed in connection with the standard boundary conditions, Dirichlet, Neumann, Robin, $\delta$ and $\delta^{\prime}$-type, assigned either on a $n-1$ dimensional compact boundary $\Gamma=\partial\Omega$ or on a relatively open part $\Sigma\subset\Gamma$. Schatten-von Neumann estimates for the difference of the powers of resolvents of the free and the perturbed operators are also proven; these give existence and completeness of the wave operators of the associated scattering systems.