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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.

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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