Equipartition of energy in geometric scattering theory

TL;DR

This paper demonstrates that radiation fields' properties imply energy equipartition in various geometric wave equations, including on scattering and hyperbolic manifolds, as well as Schwarzschild spacetime and certain semilinear equations.

Contribution

It provides a simple argument linking radiation fields to energy distribution, establishing equipartition results across multiple geometric settings and wave equations.

Findings

Energy equipartition on scattering manifolds

Equipartition on asymptotically hyperbolic manifolds

Results for energy-critical semilinear wave equation

Abstract

In this note, we use an elementary argument to show that the existence and unitarity of radiation fields implies asymptotic partition of energy for the corresponding wave equation. This argument establishes the equipartition of energy for the wave equation on scattering manifolds, asymptotically hyperbolic manifolds, asymptotically complex hyperbolic manifolds, and the Schwarzschild spacetime. It also establishes equipartition of energy for the energy-critical semilinear wave equation on $\mathbb{R}^{3}$.