Spectral distribution of the free Jacobi process, revisited

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 Szeg\H{o} 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.