  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.
