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.
Categories: math.SP math-ph math.MP