We give an explicit formula for the symbol of a function of an operator. Given a pseudo-differential operator on with symbol and a smooth function , we obtain the symbol of in terms of . As an application, Bohr-Sommerfeld quantization rules are explicitly calculated at order 4 in .
On obtient une formule explicite pour le symbole d’une fonction d’un opérateur. À partir d’un opérateur pseudo-différentiel sur avec symbole et une fonction lisse , nous obtenons le symbole de en termes de . Comme application, les règles de quantification de Bohr-Sommerfeld sont calculées explicitement à l’ordre 4 en .
Keywords: Deformation quantization, Moyal product, Weyl quantization, Bohr-Sommerfeld, symbol, diagrammatic technique, Deformation quantization, Moyal product, Weyl quantization, Bohr-Sommerfeld, symbol, diagrammatic technique
Mot clés : quantification par déformation, produit de Moyal, quantification de Weyl, Bohr-Sommerfeld, symbole, technique diagrammatique
@article{AIF_2005__55_7_2257_0, author = {Gracia-saz, Alfonso}, title = {The symbol of a function of a pseudo-differential operator}, journal = {Annales de l'Institut Fourier}, pages = {2257--2284}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {55}, number = {7}, year = {2005}, doi = {10.5802/aif.2161}, mrnumber = {2207384}, zbl = {1091.53062}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2161/} }
TY - JOUR AU - Gracia-saz, Alfonso TI - The symbol of a function of a pseudo-differential operator JO - Annales de l'Institut Fourier PY - 2005 SP - 2257 EP - 2284 VL - 55 IS - 7 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2161/ DO - 10.5802/aif.2161 LA - en ID - AIF_2005__55_7_2257_0 ER -
%0 Journal Article %A Gracia-saz, Alfonso %T The symbol of a function of a pseudo-differential operator %J Annales de l'Institut Fourier %D 2005 %P 2257-2284 %V 55 %N 7 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2161/ %R 10.5802/aif.2161 %G en %F AIF_2005__55_7_2257_0
Gracia-saz, Alfonso. The symbol of a function of a pseudo-differential operator. Annales de l'Institut Fourier, Volume 55 (2005) no. 7, pp. 2257-2284. doi : 10.5802/aif.2161. http://www.numdam.org/articles/10.5802/aif.2161/
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