The shifted wave equation on Damek--Ricci spaces and on homogeneous trees

TL;DR

This paper explicitly solves the shifted wave equation on Damek--Ricci spaces and homogeneous trees, exploring Huygens' principle and employing harmonic analysis techniques like Asgeirsson's theorem and the inverse dual Abel transform.

Contribution

It provides explicit solutions to the shifted wave equation on Damek--Ricci spaces and homogeneous trees, extending harmonic analysis methods to these settings.

Findings

Explicit solutions to the shifted wave equation on Damek--Ricci spaces

Analysis of Huygens' principle in these contexts

Extension of harmonic analysis techniques to discrete structures

Abstract

We solve explicitly the shifted wave equation on Damek--Ricci spaces, using Asgeirsson's theorem and the inverse dual Abel transform. As an application, we investigate Huygens' principle. A similar analysis is carried out in the discrete setting of homogeneous trees.