Normalization in Lie algebras via mould calculus and applications

TL;DR

This paper develops an abstract Lie-theoretic framework using mould calculus to solve normalization problems in dynamical systems, providing explicit solutions and applications to normal forms in classical and quantum contexts.

Contribution

It introduces a novel Lie-theoretic approach with mould calculus to explicitly solve normalization problems in dynamical systems.

Findings

Explicit solutions to normalization problems via mould equations

Construction of formal normal forms for vector fields and Hamiltonian systems

Demonstration of convergence of quantum Birkhoff forms to classical forms

Abstract

We establish Ecalle's mould calculus in an abstract Lie-theoretic setting and use it to solve a normalization problem, which covers several formal normal form problems in the theory of dynamical systems. The mould formalism allows us to reduce the Lie-theoretic problem to a mould equation, the solutions of which are remarkably explicit and can be fully described by means of a gauge transformation group. The dynamical applications include the construction of Poincar{\'e}-Dulac formal normal forms for a vector field around an equilibrium point, a formal infinite-order multiphase averaging procedure for vector fields with fast angular variables (Hamiltonian or not), or the construction of Birkhoff normal forms both in classical and quantum situations. As a by-product we obtain, in the case of harmonic oscillators, the convergence of the quantum Birkhoff form to the classical one, without any Diophantine hypothesis on the frequencies of the unperturbed Hamiltonians.