Equipartition of energy in geometric scattering theory

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 $\mathbb{R}^{3}$.