  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 $L^{2}(M,{\mathscr B},m)$
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.
