Directional Ballistic Transport in Quantum Waveguides

TL;DR

This paper investigates directional ballistic transport in quantum waveguides with mixed periodic and localized potentials, revealing surface states that exhibit ballistic motion in certain directions.

Contribution

It introduces a Floquet theory framework and reformulates the eigenvalue problem as a Fredholm problem to analyze surface state transport properties.

Findings

Surface states show ballistic transport in periodic directions.

Transport is weakly confined and absent in non-periodic directions.

Develops a novel Floquet theory for surface states.

Abstract

We study the transport properties of Schr\"odinger operators on $\mathbb{R}^d$ with potentials that are periodic in some directions and compactly supported in the others. Such systems are known to produce surface states that are weakly confined near the support of the potential. We show that a natural set of surface states exhibits directional ballistic transport, characterized by ballistic transport in the periodic directions and its absence in the others. To prove this, we develop a Floquet theory that captures the analytic variation of surface states. The main idea consists of reformulating the eigenvalue problem for surface states as a Fredholm problem via the Dirichlet-to-Neumann map.