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The symbol of a function of a pseudodifferential operator. (English) Zbl 1091.53062
Let $$\widehat A$$ be a pseudodifferential operator in $$L^2({\mathbb R}^N)$$ which admits a self-adjoint extension. Let $$A\in C^{\infty}({\mathbb R}^{2N})$$ be the Weyl symbol of $$A$$. Let $$f :{\mathbb R}\rightarrow {\mathbb R}$$ be a smooth function and let $${\widehat B} =f({\widehat A})$$ be an operator with symbol $$B$$. The aim of this paper is to give a realistically computable formula for $$B$$ in terms of $$A$$. The author obtains the following formula $B=\sum _{\Gamma}\left({i\hbar\over 2}\right)^{E} {c_{\Gamma}\over S_{\Gamma}}{\lambda }_{\Gamma}(A){1\over V !}f^{(V)}(A)\tag{F}$ where the sum is taken over a set of finite graphs. Here $$V$$, $$E$$, $$S_{\Gamma}$$ and $$c_{\Gamma}$$ are numbers depending on a graph $$\Gamma$$ and $${\lambda }_{\Gamma}(A)$$ is a polynomial in the derivatives of $$A$$. The main step in the derivation of (F) is an expression for the iterated Moyal star product $$C_1\ast C_2\ast\dots\ast C_n$$ where $$C_1,C_2,\dots,C_n$$ are symbols.
The author also considers the problem of computing the symbol $$B$$ of an operator $${\widehat B}=F({\widehat A_1},\dots,{\widehat A_n})$$ where $${\widehat A_1},\dots,{\widehat A_n}$$ are commuting operators with symbols $$A_1,\dots,A_n$$ and gives some applications of (F) to Bohr-Sommerfeld quantization rules and to star exponentials of quadratic forms.

##### MSC:
 53D55 Deformation quantization, star products 81S10 Geometry and quantization, symplectic methods 47G30 Pseudodifferential operators 58J40 Pseudodifferential and Fourier integral operators on manifolds
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