A characterization of singular packing subspaces with an application to limit-periodic operators

TL;DR

This paper introduces a new way to characterize singular packing subspaces of bounded self-adjoint operators and shows that, generically, limit-periodic Schrödinger operators have spectral measures with maximal upper packing dimension, implying quasiballistic behavior.

Contribution

It provides a novel characterization of singular packing subspaces and demonstrates that, generically, limit-periodic operators exhibit spectral measures with upper packing dimension one.

Findings

Set of operators with spectral measures of upper packing dimension one is a $G_\delta$ set.

Spectral measures of generic limit-periodic Schrödinger operators have upper packing dimension one.

Such operators are quasiballistic in a generic sense.

Abstract

A new characterization of the singular packing subspaces of general bounded self-adjoint operators is presented, which is used to show that the set of operators whose spectral measures have upper packing dimension equal to one is a $G_\delta$ (in suitable metric spaces). As an application, it is proven that, generically (in space of continuous sampling functions), spectral measures of the limit-periodic Schr\"odinger operators have upper packing dimensions equal to one. Consequently, in a generic set, these operators are quasiballistic.