 Bright and dark state reduction of a star-coupled Hamiltonian

Suppose one state $|0⟩$ couples to degenerate states $|1⟩,\dots ,|m⟩$ with amplitudes $v_1,\dots ,v_m$, while the degenerate states do not couple to each other. The normalized bright state proportional to ${\sum }_j{v}_j|j⟩$ is the only combination coupled to $|0⟩$. Every orthogonal combination is a dark state and remains an exact eigenstate at the degenerate energy. The remaining spectrum comes from a two-dimensional Hamiltonian whose off-diagonal entry is $({\sum }_j|v_j{|}^2{)}^{1/2}$.