Abstract
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 \(RU_tSU_t^*\) where \(R,S\) are two symmetries and \(U_t\) 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 \(RU_tSU_t^*\) possesses two atoms at \(\pm1\) and an \(L^\infty\)-density on the unit circle \(\mathbb{T}\), for every \(t>0\). Next, via a Szegő type transform of this law, we obtain a full description of the spectral distribution of \(PU_tQU_t^*\) beyond the \(\tau(P)=\tau(Q)=1/2\) case. Finally, we give some specializations for which these measures are explicitly computed.