  We develop a general perturbative framework based on a superconducting atomic
limit for the description of Andreev bound states (ABS) in interacting quantum
dots connected to superconducting leads. A local effective Hamiltonian for
dressed ABS, including both the atomic (or molecular) levels and the induced
proximity effect on the dot is argued to be a natural starting point. A
self-consistent expansion in single-particle tunneling events is shown to
provide accurate results even in regimes where the superconducting gap is
smaller than the atomic energies, as demonstrated by a comparison to recent
Numerical Renormalization Group calculations. This simple formulation may have
bearings for interpreting Andreev spectroscopic experiments in superconducting
devices, such as STM measurements on carbon nanotubes, or radiative emission in
optical quantum dots.
