Characterization of eigenfunctions of the Laplace–Beltrami operator through heat propagation in small time

On rank one Riemannian symmetric spaces of noncompact type, which accommodates all hyperbolic spaces, we show that characterization of the eigenfunctions/eigendistributions of the Laplace–Beltrami operator, through the action of the heat operator is possible only when we confine in a small time. The results are different from their counter parts in the Euclidean spaces. All results and their proofs extend to Damek–Ricci spaces.