The symbol of a function of a pseudo-differential operator

Gracia-saz, Alfonso

Annales de l'Institut Fourier, Tome 55 (2005) no. 7, pp. 2257-2284.

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

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

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