Ballistic Transport for the Schr\"odinger Operator with Limit-Periodic or Quasi-periodic Potential in Dimension Two

TL;DR

This paper proves ballistic transport in two-dimensional Schr"odinger operators with limit-periodic or quasi-periodic potentials, extending spectral theory and eigenfunction analysis to establish transport properties under certain regularity conditions.

Contribution

It demonstrates ballistic transport for these operators in two dimensions, utilizing spectral analysis and advanced mathematical techniques to connect spectral properties with transport behavior.

Findings

Existence of ballistic transport in 2D Schr"odinger operators with specific potentials

Detailed spectral structure of eigenvalues and eigenfunctions

Application of stationary phase and integration by parts in proofs

Abstract

We prove the existence of ballistic transport for the Schr\"odinger operator with limit-periodic or quasi-periodic potential in dimension two. This is done under certain regularity assumptions on the potential which have been used in prior work to establish the existence of an absolutely continuous component and other spectral properties. The latter include detailed information on the structure of generalized eigenvalues and eigenfunctions. These allow to establish the crucial ballistic lower bound through integration by parts on an appropriate extension of a Cantor set in momentum space, as well as through stationary phase arguments.