  We study the self adjoint extensions of a class of non maximal multiplication
operators with boundary conditions. We show that these extensions correspond to
singular rank one perturbations (in the sense of \cite{AK}) of the Laplace
operator, namely the formal Laplacian with a singular delta potential, on the
half space. This construction is the appropriate setting to describe the
Casimir effect related to a massless scalar field in the flat space time with
an infinite conducting plate and in the presence of a point like "impurity". We
use the relative zeta determinant (as defined in \cite{Mul} and \cite{SZ}) in
order to regularize the partition function of this model. We study the analytic
extension of the associated relative zeta function, and we present explicit
results for the partition function, and for the Casimir force.
