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

## TL;DR

This paper revisits von Neumann's ergodic theorem, establishing uniform convergence rates without requiring a spectral gap, with applications to Schrödinger and wave equations.

## Contribution

It introduces a method to obtain uniform convergence rates based on spectral density control, extending ergodic results beyond spectral gap assumptions.

## Key findings

- Explicit polynomial decay rates for time-averages of solutions
- Applications to linear Schrödinger and wave equations demonstrating decay estimates
- Uniform convergence results without spectral gap assumptions

## 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 Schr\"odinger and wave equations. In particular, decay estimates for time-averages of solutions are shown.