Beyond Adiabatic Elimination: Systematic Expansions

TL;DR

This paper reformulates adiabatic elimination as a perturbation expansion, providing systematic improvements and convergence proofs, and addressing hermiticity issues in effective Hamiltonians within quantum systems.

Contribution

It introduces a systematic expansion framework for adiabatic elimination using invariant manifold formalism, connecting it with quantum perturbation methods and proving convergence.

Findings

Established a formal connection between adiabatic elimination and singular perturbation theory.

Proved convergence of the systematic expansions under certain energy scale conditions.

Addressed hermiticity concerns in improved effective Hamiltonians.

Abstract

We restate the adiabatic elimination approximation as the first term in a singular perturbation expansion. We use the invariant manifold formalism for singular perturbations in dynamical systems to identify systematic improvements on adiabatic elimination, connecting with well established quantum mechanical perturbation methods. We prove convergence of the expansions when energy scales are well separated. We state and solve the problem of hermiticity of improved effective hamiltonians.