  We apply the unbiased weak-coupling continuous time quantum Monte Carlo
(CTQMC) method to review the physics of a single magnetic impurity coupled to
s-wave superconducting leads described by the BCS reduced Hamiltonian. As a
function of the superconducting gap $\Delta$, we study the signature of the
first order transition between the singlet and doublet (local moment) states on
various quantities. In particular we concentrate on the Josephson current with
0 to $\pi$ phase shift, the crossing of the Andreev bound states in the single
particle spectral function, as well as the local dynamical spin structure
factor. Within DMFT, this impurity problem provides a link to the periodic
Anderson model with superconducting conduction electrons (BCS-PAM). The first
order transition observed in the impurity model is reproduced in the BCS-PAM
and is signalized by the crossing of the low energy excitations in the local
density of states. The momentum resolved single particle spectral function in
the singlet state reveals the coherent, Bloch-like, superposition of Andreev
bound states. In the doublet or local moment phase the single particle spectral
function is characterized by incoherent quasiparticle excitations.
