The symbol of a function of a pseudo-differential operator

Alfonso Gracia-saz[1]

  • [1] University of California at Berkeley, Department of Mathematics, Berkeley CA 94720-3840 (USA)

Annales de l'institut Fourier (2005)

  • Volume: 55, Issue: 7, page 2257-2284
  • ISSN: 0373-0956

Abstract

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

How to cite

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Gracia-saz, Alfonso. "The symbol of a function of a pseudo-differential operator." Annales de l'institut Fourier 55.7 (2005): 2257-2284. <http://eudml.org/doc/116254>.

@article{Gracia2005,
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 $.},
affiliation = {University of California at Berkeley, Department of Mathematics, Berkeley CA 94720-3840 (USA)},
author = {Gracia-saz, Alfonso},
journal = {Annales de l'institut Fourier},
keywords = {Deformation quantization; Moyal product; Weyl quantization; Bohr-Sommerfeld; symbol; diagrammatic technique; deformation quantization; Moyal products; Bohr-Sommerfeld symbol; diagrammatic technique.},
language = {eng},
number = {7},
pages = {2257-2284},
publisher = {Association des Annales de l'Institut Fourier},
title = {The symbol of a function of a pseudo-differential operator},
url = {http://eudml.org/doc/116254},
volume = {55},
year = {2005},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 7
SP - 2257
EP - 2284
AB - 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 $.
LA - eng
KW - Deformation quantization; Moyal product; Weyl quantization; Bohr-Sommerfeld; symbol; diagrammatic technique; deformation quantization; Moyal products; Bohr-Sommerfeld symbol; diagrammatic technique.
UR - http://eudml.org/doc/116254
ER -

References

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