Normalization in Banach scale Lie algebras via mould calculus and applications

Thierry Paul; David Sauzin

doi:10.3934/dcds.2017191

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2017, Volume 37, Issue 8: 4461-4487. Doi: 10.3934/dcds.2017191

This issue Previous Article On the uniqueness of solution to generalized Chaplygin gas Next Article Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system

Normalization in Banach scale Lie algebras via mould calculus and applications

Thierry Paul and
David Sauzin

1.
CMLS, Ecole polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France
2.
CNRS UMR 8028 – IMCCE, Observatoire de Paris, 75014 Paris, France

Received: July 2016

Revised: March 2017

Published: May 2017

This work has been partially carried out thanks to the support of the A*MIDEX project (n^o ANR-11-IDEX-0001-02) funded by the "Investissements d'Avenir" French Government program, managed by the French National Research Agency (ANR). D.S.'s work has received funding from the French National Research Agency under the reference ANR-12-BS01-0017.

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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.

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Mathematics Subject Classification: Primary: 37J40, 81Q15; Secondary: 81Q20.

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