  We present two complementary methods to calculate the Andreev bound state
energies of a single-level quantum dot connected to superconducting leads
described by the superconducting impurity Anderson model. The first method,
which is based on a mapping to a low-energy model, can be utilized to extract
the Andreev bound state energies from finite-temperature, imaginary-time
quantum Monte Carlo data without the necessity of any analytic continuation
technique. The second method maps the full model on an exactly solvable
superconducting atomic limit with renormalized parameters. As such, it
represents a fast and reliable method for a quick scan of the parameter space.
We demonstrate that after adding a simple band correction this method can
provide predictions for measurable quantities, including the Josephson current,
that are in a solid quantitative agreement with precise results obtained by the
numerical renormalization group and quantum Monte Carlo.
