The shifted Wave equation on non flat harmonic manifolds

TL;DR

This paper provides explicit solutions and analysis of the shifted wave equation on non-flat harmonic manifolds, extending classical results with new integral transform techniques and asymptotic properties.

Contribution

It introduces an explicit solution representation for the shifted wave equation on harmonic manifolds using the inverse dual Abel transform and explores Fourier analysis and asymptotic behavior.

Findings

Explicit solution via inverse dual Abel transform

Paley-Wiener type theorem established

Asymptotic Huygens principle demonstrated

Abstract

We solve the shifted wave equation \begin{align*} \frac{\partial^2}{\partial t^2}\varphi(x,t)=(\Delta_x+\rho^2)\varphi(x,t) \end{align*} on a non compact simply connected harmonic manifold with mean curvature of the horospheres $2\rho>0$. 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.