Normalization in Banach scale Lie algebras via mould calculus and applications

TL;DR

This paper develops a perturbative normalization scheme for resonant problems in Banach scale Lie algebras, unifying classical and quantum cases, and provides estimates on the difference between quantum and classical normal forms.

Contribution

It introduces a novel normalization method using mould calculus within Banach scale Lie algebras, unifying classical and quantum normal forms and estimating their differences.

Findings

Established a precise estimate for quantum-classical normal form difference, proportional to the square of Planck's constant.

Developed a perturbative normalization scheme applicable to resonant problems in Banach scale Lie algebras.

Unified classical and quantum normalization frameworks for direct comparison.

Abstract

We study a perturbative scheme for normalization problems involving resonances of the unperturbed situation, and therefore the necessity of a non-trivial normal form, in the general framework of Banach scale Lie algebras (this notion is defined in the article). This situation covers the case of classical and quantum normal forms in a unified way which allows a direct comparison. In particular we prove a precise estimate for the difference between quantum and classical normal forms, proven to be of order of the square of the Planck constant. Our method uses mould calculus (recalled in the article) and properties of the solution of a universal mould equation studied in a preceding paper.