Asymptotic Completeness and S-Matrix for Singular Perturbations

We give a criterion of asymptotic completeness and provide a representation of the scattering matrix for the scattering couple $(A_0,A)$, where $A_0$ and $A$ are semi-bounded self-adjoint operators in $L^2(M,\mathcal{B},m)$ such that the set $\{u\in D(A_0)\cap D(A):A_{0}u=Au\}$ is dense. No sort of trace-class condition on resolvent differences is required. Applications to the case in which $A_0$ corresponds to the free Laplacian in $L^{2}({\mathbb R}^{n})$and$A$ describes the Laplacian with self-adjoint boundary conditions on rough compact hypersurfaces are given.