Andreev bound states form inside a short Josephson junction when quasiparticles reflect off both interfaces.
E±(φ) = ±Δ√(1 − τ sin²(φ/2)). The supercurrent is I(φ) = (eΔ/2ℏ) · τ sin(φ) / √(1−τsin²(φ/2)).
At τ=1 (ideal junction): ABS touches zero at φ=π (fermion parity crossing). White dot marks current operating point.