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Behavior of Andreev gap states in a quantum dot with Coulomb repulsionsymmetrically attached to superconducting leads is studied via the perturbationexpansion in the interaction strength. We find the exact asymptotic form of thespin-symmetric solution for the Andreev states continuously approaching theFermi level. We thereby derive a critical interaction at which the Andreevstates at zero temperature merge at the Fermi energy, being the upper bound forthe $0-\pi$ transition. We show that the spin-symmetric solution becomesdegenerate beyond this interaction, in the $\pi$ phase, and the Andreev statesdo not split unless the degeneracy is lifted. We further demonstrate that thedegeneracy of the spin-symmetric state extends also into the $0$ phase in whichthe solutions with zero and non-zero frequencies of the Andreev states maycoexist.

Spin-symmetric solution of an interacting quantum dot attached to superconducting leads: Andreev states and the $0-\pi$ transition