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

The theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic differential operator on RnRn with linear boundary conditions on (a relatively open part of) a compact hypersurface. Our approach allows to obtain Kreĭn-like resolvent formulae where the reference operator coincides with the “free” operator with domain H2(Rn)H2(Rn); 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, δ   and δ′δ′-type, assigned either on a (n−1)(n−1) dimensional compact boundary Γ=∂ΩΓ=∂Ω or on a relatively open part Σ⊂ΓΣ⊂Γ. 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.