Renormalization effects in interacting quantum dots coupled to   superconducting leads
  We study subgap transport through an interacting quantum dot tunnel coupled to one normal and two superconducting leads. To check the reliability of an approximation of an infinitely-large gap $\Delta$ in the superconducting leads and weak tunnel coupling to the normal lead, we perform a $1/\Delta$ expansion, and we analyze next-to-leading order corrections in the tunnel coupling to the normal lead. Furthermore, we propose a resummation approach to calculate the Andreev bound states for finite $\Delta$. The results are substantially more accurate than those obtained by mean-field treatments and favorably compare with numerical exact results. 
David Futterer, Jacek Swiebodzinski, Michele Governale, and J\"urgen   K\"onig
cond-mat.mes-hall