We give an explicit formula for the $L^2$ analytic torsion of the finite metric cone over an oriented compact connected Riemannian manifold. We provide an interpretation of the different factors appearing in this formula. We prove that the analytic t
orsion of the cone is the finite part of the limit obtained collapsing one of the boundaries, of the ratio of the analytic torsion of the frustum to a regularising factor. We show that the regularising factor comes from the set of the non square integrable eigenfunctions of the Laplace Beltrami operator on the cone.
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 t
he 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.
We compute the analytic torsion of a cone over a sphere of dimension 1, 2, and 3, and we conjecture a general formula for the cone over an odd dimensional sphere.
