Spectral distribution of the free Jacobi process, revisited

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We obtain a description for the spectral distribution of the free Jacobi process for any initial pair of projections. This result relies on a study of the unitary operator RUₜSUₜ* where R,S are two symmetries and Uₜ a free unitary Brownian motion, freely independent from {R,S}. In particular, for non-null traces of R and S, we prove that the spectral measure of RUₜSUₜ* possesses two atoms at ±1 and an L∞-density on the unit circle T, for every t>0. Next, via a Szegő type transform of this law, we obtain a full description of the spectral distribution of PUₜQUₜ* beyond the τ(P)=τ(Q)=1/2 case. Finally, we give some specializations for which these measures are explicitly computed.