  We provide an asymptotic completeness criterion and a representation formula
for the scattering matrix of the scattering couple $(A_B,A)$, where both $A$
and $A_B$ are self-adjoint operator and $A_B$ formally corresponds to adding to
$A$ two terms, one regular and the other singular. In particular, our abstract
results apply to the couple $(\Delta_B,\Delta)$, where $\Delta$ is the free
self-adjoint Laplacian in $L^2(\mathbb{R}^3)$ and $\Delta_B$ is a self-adjoint
operator in a class of Laplacians with both a regular perturbation, given by a
short-range potential, and a singular one describing boundary conditions (like
Dirichlet, Neumann and semi-transparent $\delta$ and $\delta'$ ones) at the
boundary of a open, bounded Lipschitz domain. The results hinge upon a limiting
absorption principle for $A_B$ and a Krein-like formula for the resolvent
difference $(-A_B+z)^{-1}-(-A+z)^{-1}$ which puts on an equal footing the
regular (here, in the case of the Laplacian, a Kato-Rellich potential suffices)
and the singular perturbations.
