On the weak coupling limit of the periodic quantum Lorentz gas

TL;DR

This paper investigates the weak coupling limit of the periodic quantum Lorentz gas, revealing that some observables follow a trivial transport equation while others depend on unresolved regularity properties at resonant momenta.

Contribution

It introduces the use of the sewing lemma to derive the kinetic scaling limit for almost every momentum, highlighting partial progress and open questions in the field.

Findings

Some observables follow a trivial transport equation in the limit.

Limit behavior depends on unresolved regularity at resonant momenta.

The sewing lemma is used to derive the kinetic scaling limit.

Abstract

We report partial progress on the weak coupling limit behavior of observables for the periodic quantum Lorentz gas. Our results indicate that for certain observables, the limit behavior is trivial and can be described via a transport equation, while for other observables, the existence of the limit hinges on the regularity properties at resonant momenta of a certain Bloch-Wigner transform. We are currently unable to prove or disprove this regularity property, and so the weak coupling limit for these observables remains an open question. A novelty of this work is the use of the sewing lemma in the derivation of the kinetic scaling limit for almost every mometum.