Small eigenvalues of the low temperature linear relaxation Boltzmann equation with a confining potential

TL;DR

This paper analyzes the spectral properties of the linear relaxation Boltzmann equation with a confining potential, providing resolvent estimates and insights into the return to equilibrium using advanced mathematical techniques.

Contribution

It introduces new resolvent estimates and a novel approach to understanding the return to equilibrium for the kinetic model with a confining potential.

Findings

Resolved spectral estimates for the associated operator

Established return to equilibrium rates

Applied scaling and hypocoercivity methods

Abstract

We study the linear relaxation Boltzmann equation, a simple semiclassical kinetic model. We provide a resolvent estimate for an associated non-selfadjoint operator as well as an estimate on the return to equilibrium. This is done using a scaling argument and non-semiclassical hypocoercive estimate.