Continuity of spectral averaging

TL;DR

This paper investigates how the continuity properties of a measure used in spectral averaging of rank one perturbations influence the resulting spectral measure, extending previous results beyond Lebesgue measure.

Contribution

It generalizes Kotani's trick by analyzing spectral measure averages with respect to measures with various degrees of Hausdorff measure continuity.

Findings

Continuity of the measure $ u$ affects the continuity of the spectral average $ppa$.

Results extend Kotani's trick to measures beyond Lebesgue measure.

Provides conditions under which spectral averaging preserves certain continuity properties.

Abstract

We consider averages $\kappa$ of spectral measures of rank one perturbations with respect to a $\sigma$-finite measure $\nu$. It is examined how various degrees of continuity of $\nu$ with respect to $\alpha$-dimensional Hausdorff measures ($0 \leq \alpha \leq 1$) are inherited by $\kappa$. This extends Kotani's trick where $\nu$ is simply the Lebesgue measure.