  Self-consistent perturbation expansion up to the second order in the
interaction strength is used to study a single-level quantum dot with local
Coulomb repulsion attached asymmetrically to two generally different
superconducting leads. At zero temperature and wide range of other parameters
the spin-symmetric version of the expansion yields excellent results for the
position of the $0-\pi$ impurity quantum phase transition boundary and
Josephson current together with the energy of Andreev bound states in the
$0$-phase as confirmed by numerical calculations using the Numerical
Renormalisation Group method. We analytically prove that the method is
charge-conserving as well as thermodynamically consistent. Explicit formulas
for the position of the $0-\pi$ phase boundary are presented for the
Hartree-Fock approximation as well as for its variant called Generalized Atomic
Limit. It is shown that the Generalized Atomic Limit can be used as a quick
estimate for the position of the phase boundary at half-filling in a broad
range of parameters. We apply our second order perturbation method to the
interpretation of the existing experimental data on the phase boundary with
very satisfactory outcome suggesting that the so far employed heavy numerical
tools such as Numerical Renormalization Group and/or Quantum Monte Carlo are
not necessary in a class of generic situations and can be safely replaced by a
perturbative approach.
