The shifted Wave equation on non flat harmonic manifolds

Abstract

We solve the shifted wave equation \begin{align*} \frac{\partial^2}{\partial t^2}φ(x,t)=(Δ_x+ρ^2)φ(x,t) \end{align*} on a non compact simply connected harmonic manifold with mean curvature of the horospheres $2ρ>0$. We give an explicit representation of the solution as the inverse dual Abel transform of the spherical means of there initial conditions using the local injectivity of the Abel transform and symmetry properties of the spherical mean value operator. Furthermore we investigate the wave equation using the Fourier transform on harmonic manifolds of rank one. Additionally we show an analogous of the classical Paley-Wiener theorem and use it to show an asymptotic Huygens principle as well as asymptotic equidistribution of the energy of a solution of the shifted wave equation under assumptions on the $\mathbf{c}$-function.