  Behavior of Andreev gap states in a quantum dot with Coulomb repulsion
symmetrically attached to superconducting leads is studied via the perturbation
expansion in the interaction strength. We find the exact asymptotic form of the
spin-symmetric solution for the Andreev states continuously approaching the
Fermi level. We thereby derive a critical interaction at which the Andreev
states at zero temperature merge at the Fermi energy, being the upper bound for
the $0-\pi$ transition. We show that the spin-symmetric solution becomes
degenerate beyond this interaction, in the $\pi$ phase, and the Andreev states
do not split unless the degeneracy is lifted. We further demonstrate that the
degeneracy of the spin-symmetric state extends also into the $0$ phase in which
the solutions with zero and non-zero frequencies of the Andreev states may
coexist.
