The symbol of a function of a pseudo-differential operator

[Le symbole d'une fonction d'un opérateur pseudo-différentiel]

Gracia-saz, Alfonso

Annales de l'Institut Fourier, Tome 55 (2005), p. 2257-2284 / Harvested from Numdam

Résumé

On obtient une formule explicite pour le symbole d’une fonction d’un opérateur. À partir d’un opérateur pseudo-différentiel $\widehat{A}$ sur $L^2({\mathbb {R}}^{N})$ avec symbole $A \in {{\mathcal {C}}^{\infty }(T^* {\mathbb {R}}^{N})}$ et une fonction lisse $f$, nous obtenons le symbole de $f(\widehat{A})$ en termes de $A$. Comme application, les règles de quantification de Bohr-Sommerfeld sont calculées explicitement à l’ordre 4 en $\hbar $.

Abstract

We give an explicit formula for the symbol of a function of an operator. Given a pseudo-differential operator $\widehat{A}$ on $L^2({\mathbb {R}}^{N})$ with symbol $A \in {{\mathcal {C}}^{\infty }(T^* {\mathbb {R}}^{N})}$ and a smooth function $f$, we obtain the symbol of $f(\widehat{A})$ in terms of $A$. As an application, Bohr-Sommerfeld quantization rules are explicitly calculated at order 4 in $\hbar $.

Détail

Publié le : 2005-01-01

MR 2207384 | Zbl 1091.53062 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.2161
Classification:  53D55,  81S10
Mots clés: quantification par déformation, produit de Moyal, quantification de Weyl, Bohr-Sommerfeld, symbole, technique diagrammatique

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