Slowly changing potential problems in Quantum Mechanics: Adiabatic Theorems, Ergodic Theorems, and Scattering

TL;DR

This paper investigates the long-term behavior of slowly varying Hamiltonians in quantum mechanics, providing new proofs and theorems related to adiabatic processes, ergodic properties, and scattering theory.

Contribution

It introduces a novel multi-time scale averaging method and proves a new uniform ergodic theorem for slowly changing unitary operators.

Findings

New proof of the Adiabatic Theorem in the gapless case

Development of a uniform ergodic theorem for slowly changing unitaries

Application to scattering theory and classical propagation estimates

Abstract

We employ the recently developed multi-time scale averaging method to study the large time behavior of slowly changing (in time) Hamiltonians. We treat some known cases in a new way, such as the Zener problem, and we give another proof of the Adiabatic Theorem in the gapless case. We prove a new Uniform Ergodic Theorem for slowly changing unitary operators. This theorem is then used to derive the adiabatic theorem, do the scattering theory for such Hamiltonians, and prove some classical propagation estimates and Asymptotic Completeness.