Uniform convergence in von Neumann's ergodic theorem in absence of a spectral gap

Jonathan Ben-Artzi, Baptiste Morisse

Published 2019-02-11Version 1

Von Neumann's original proof of the ergodic theorem is revisited. A convergence rate is established under the assumption that one can control the density of the spectrum of the underlying self-adjoint operator when restricted to suitable subspaces. Explicit rates are obtained when the bound is polynomial or logarithmic, with applications to the linear Schr\"odinger and wave equations. In particular, decay estimates for time-averages of solutions are shown.