# Normalization in Banach scale Lie algebras via mould calculus and applications

@article{Paul2016NormalizationIB, title={Normalization in Banach scale Lie algebras via mould calculus and applications}, author={T. Paul and D. Sauzin}, journal={arXiv: Analysis of PDEs}, year={2016} }

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… Expand

#### 4 Citations

Normalization in Lie algebras via mould calculus and applications

- Mathematics
- 2016

We establish Écalle’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… Expand

Rayleigh–Schrödinger series and Birkhoff decomposition

- Physics, Mathematics
- 2018

We derive new expressions for the Rayleigh–Schrödinger series describing the perturbation of eigenvalues of quantum Hamiltonians. The method, somehow close to the so-called dimensional… Expand

Hopf algebra techniques to handle dynamical systems and numerical integrators

- Mathematics
- 2016

In a series of papers the present authors and their coworkers have developed a family of algebraic techniques to solve a number of problems in the theory of discrete or continuous dynamical systems… Expand

Time Dependent Quantum Perturbations Uniform in the Semiclassical Regime

- Mathematics, Physics
- 2021

We present a time dependent quantum perturbation result, uniform in the Planck constant for potential whose gradient is bounded a.e..We show also that the classical limit of the perturbed quantum… Expand

#### References

SHOWING 1-10 OF 33 REFERENCES

Normalization in Lie algebras via mould calculus and applications

- Mathematics
- 2016

We establish Écalle’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… Expand

Spectral asymptotics via the semiclassical Birkhoff normal form

- Mathematics, Physics
- 2006

This article gives a simple treatment of the quantum Birkhoff normal form for semiclassical pseudo-differential operators with smooth coefficients. The normal form is applied to describe the discrete… Expand

Quantum singular complete integrability

- Mathematics, Physics
- 2014

We consider some perturbations of a family of pairwise commuting linear quantum Hamiltonians on the torus with possibly dense pure point spectra. We prove that the Rayleigh-Schr{\"o}dinger… Expand

Normal Forms and Quantization Formulae

- Mathematics
- 1999

Abstract:We consider the Schrödinger operator , where as , is Gevrey of order and has a unique non-degenerate minimum. A quantization formula up to an error of order is obtained for all… Expand

Some Remarks about Semiclassical Trace Invariants and Quantum Normal Forms

- Physics, Mathematics
- 2009

In this paper we explore the connection between semi-classical and quantum Birkhoff canonical forms (BCF) for Schrödinger operators. In particular we give a “non-symbolic” operator theoretic… Expand

Rigidity around Poisson submanifolds

- Mathematics
- 2014

We prove a rigidity theorem in Poisson geometry around compact Poisson submanifolds, using the Nash–Moser fast convergence method. In the case of one-point submanifolds (fixed points), this implies a… Expand

Rigidity of Hamiltonian actions on Poisson manifolds

- Mathematics
- 2012

Abstract This paper is about the rigidity of compact group actions in the Poisson context. The main result is that Hamiltonian actions of compact semisimple type are rigid. We prove it via a… Expand

Levi decomposition for smooth Poisson structures

- Mathematics
- 2002

We prove the existence of a local smooth Levi decomposition for smooth Poisson structures and Lie algebroids near a singular point. This Levi decomposition is a kind of normal form or partial… Expand

Quantization of the classical Lie algorithm in the Bargmann representation

- Physics
- 1991

Abstract For any polynomial perturbation of a d -dimensional system of non-resonant harmonic oscillators it is proved, writing the classical Hamiltonian in complex coordinates and the corresponding… Expand

Convergence of a quantum normal form and an exact quantization formula

- Mathematics, Physics
- 2011

Abstract The operator − i ℏ ω ⋅ ∇ on L 2 ( T l ) , quantizing the linear flow of diophantine frequencies ω = ( ω 1 , … , ω l ) over T l , l > 1 , is perturbed by the operator quantizing a function V… Expand