  We study the algebra of differential operators on non-compact simply
connected harmonic manifolds and provide sufficient conditions for them to have
a radial fundamental solution and be surjective on the space of smooth
function. Furthermore, we show that the algebra of differential operators that
commute with taking averages over geodesic spheres is generated by the
Laplacian. As an application of this, we show that the heat-semi group is dense
in the radial $L^1$ space of a non-compact simply connected harmonic manifold.
In the process, we provide a characterisation of the eigenfunctions of the
Laplacian and therefore of the eigenfunctions of all differential operators
commuting with taking spherical averages.
