A linearized model of quantum transport with interface conditions in the adiabatic regime

TL;DR

This paper develops a modified quantum transport model with interface conditions that enable accurate adiabatic approximations of resonant states, providing a rigorous framework for analyzing adiabatic transport in quantum wells.

Contribution

It introduces a novel h-dependent interface condition approach that yields small perturbations and precise asymptotics in the adiabatic quantum transport problem.

Findings

Small perturbation of dynamics due to interface conditions

Polynomially small error in adiabatic variable asymptotics

Extension to non-autonomous adiabatic evolution

Abstract

We introduce a modified model where h-dependent artificial interface conditions, occurring at the boundary of an interaction region, allow to obtain adiabatic approximations for the relevant resonant states connected to the quantum transport problem. Under positivity assumptions on the potential, we show that this modification produces a small perturbation of the dynamics on any time scale. The result extends to the corresponding non-autonomous case, allowing us to consider the adiabatic evolution problem in the modified setting. In this framework, we give an expansion formula for the small-h asymptotics of the adiabatic variable, where the error introduced on the adiabatic dynamics by the interface conditions is polynomially small w.r.t. h, independently from the adiabatic time scale. This result provides with a rigorous mathematical framework for the interface conditions approach to the analysis of the adiabatic transport problem in the quantum wells regime.