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 $R{U}_{t}S{U}_t^{\ast }$ 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 $R{U}_{t}S{U}_t^{\ast }$ possesses two atoms at $\pm 1$ and an $L^{\mathrm{\infty }}$-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 $P{U}_{t}Q{U}_t^{\ast }$ beyond the $\tau (P)=\tau (Q)=1\mathrm{/}2$ case. Finally, we give some specializations for which these measures are explicitly computed.