文摘
The theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic differential operator on Rn 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 f0ba52d4" title="Click to view the MathML source">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 05079b194b264fe6a983" title="Click to view the MathML source">δ′-type, assigned either on a f0ec7d079c0bd738d2ba4da74a3c" title="Click to view the MathML source">(n−1) dimensional compact boundary 05f09921" title="Click to view the MathML source">Γ=∂Ω or on a relatively open part Σ⊂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.