  There is a remarkable relation between two kinds of phase space distributions
associated to eigenfunctions of the Laplacian of a compact hyperbolic manifold:
It was observed in \cite{AZ} that for compact hyperbolic surfaces
$X_{\Gamma}=\Gamma\backslash\mathbb{H}$ Wigner distributions $\int_{S^*
X_{\Gamma}} a dW_{ir_j} = < \mathrm{Op}(a)\phi_{ir_j},\phi_{ir_j}
>_{L^2(X_{\Gamma})}$ and Patterson--Sullivan distributions $PS_{ir_j}$ are
asymptotically equivalent as $r_j\to\infty$. We generalize the definitions of
these distributions to all rank one symmetric spaces of noncompact type and
introduce off-diagonal elements $PS_{\lambda_j,\lambda_k}$. Further, we give
explicit relations between off-diagonal Patterson--Sullivan distributions and
off-diagonal Wigner distributions and describe the asymptotic relation between
these distributions.
