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

Published online by Cambridge University Press:  13 April 2020

JONATHAN BEN-ARTZI
Affiliation:
School of Mathematics, Cardiff University, Cardiff, CF24 4AG, UK email Ben-ArtziJ@cardiff.ac.uk, MorisseB@cardiff.ac.uk
BAPTISTE MORISSE
Affiliation:
School of Mathematics, Cardiff University, Cardiff, CF24 4AG, UK email Ben-ArtziJ@cardiff.ac.uk, MorisseB@cardiff.ac.uk
Corresponding

Abstract

Von Neumann’s original proof of the ergodic theorem is revisited. A uniform 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, with applications to the linear Schrödinger and wave equations. In particular, decay estimates for time averages of solutions are shown.

Type
Original Article
Copyright
© The Author(s) 2020. Published by Cambridge University Press

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