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Alfonso Gracia-saz
The symbol of a function of a pseudo-differential operator
(Le symbole d'une fonction d'un opérateur pseudo-différentiel)
Annales de l'institut Fourier, 55 no. 7 (2005), p. 2257-2284, doi: 10.5802/aif.2161
Article PDF | Reviews MR 2207384 | Zbl 1091.53062 | 1 citation in Cedram
Class. Math.: 53D55, 81S10
Keywords: Deformation quantization, Moyal product, Weyl quantization, Bohr-Sommerfeld, symbol, diagrammatic technique

Résumé - Abstract

We give an explicit formula for the symbol of a function of an operator. Given a pseudo-differential operator $\widehat{A}$ on $L^2({\Bbb{R}}^{N})$ with symbol $A \in {{\cal{C}}^{\infty}(T^* {\Bbb{R}}^{N})}$ 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 $\hbar$.

Bibliography

[1] M. Andersson & J. Sjöstrand, “Functional calculus for non-commuting operators with real spectra via an iterated Cauchy formula”, arXiv:math.SP/0303024, 2003 arXiv |  Zbl 1070.47009
[2] P.N. Argyres, “The Bohr-Sommerfeld quantization rule and the Weyl correspondence”, Physics 2 (1965), p. 131-139
[3] F. Bayen, M. Flato, C. Fronsdal & D. Sternheimer A. Lichnerowicz, “Deformation theory and quantization I-II”, Ann. Phys. 111 (1978)  MR 496158 |  Zbl 0377.53025
[4] M. Cargo, A. Gracia- Saz, R.G. Littlejohn, M.W. Reinsch & P. de M. Rios, “Quantum normal forms, Moyal star product and Bohr-Sommerfeld approximation”, J. Phys. A, Math. and Gen. 38 (2005), p. 1977-2004, arXiv:math-ph/0409039 Article |  MR 2124376 |  Zbl 02151996
[5] L. Charles, “Berezin-Toeplitz operators, a semi-classical approach”, Comm. Math. Phys. 239 (2003), p. 1-28 Article |  MR 1997113 |  Zbl 1059.47030
[6] Y. Colin de Verdière, “Bohr-Sommerfeld rules to all order”, to appear in Henri Poincaré Acta, 2004  Zbl 1080.81029
[7] E.B. Davies, Spectral theory and differential operators, Cambridge Studies in Advanced Mathematics 42, Cambridge University Press, 1995  MR 1349825 |  Zbl 0893.47004
[8] A. Grigis & J. Sjöstrand, Microlocal analysis for differential operators, 196, London Mathematical Society, 1994  MR 1269107 |  Zbl 0804.35001
[9] H.J. Groenewold, “On the principles of elementary quantum mechanics”, Physica (Amsterdam) 12 (1946), p. 405-460 Article |  MR 18562 |  Zbl 0060.45002
[10] B. Helffer & J. Sjöstrand, “Équation de Schrödinger avec champ magnétique et équation de Harper”, Springer Lecture Notes in Physics 345 (1989), p. 118-197 Article |  MR 1037319 |  Zbl 0699.35189
[11] A.C. Hirshfeld & P. Henselder, “Deformation quantization in the teaching of quantum mechanics”, Amer. J. Physics 70 (2002), p. 537-547, arXiv:quant-ph/ 0208163 Article |  MR 1897018
[12] V. Kathotia, “Kontsevich's universal formula for deformation quantization and the Campbell-Baker-Haussdorf formula, I”, Internat. J. Math. 11 (2000), p. 523-551, arXiv:math.QA/9811174 arXiv |  MR 1768172 |  Zbl 01629355
[13] M. Kontsevich, “Deformation quantization of Poisson manifolds I”, Lett. Math. Phys. 66 (2003), p. 157-216, arXiv:q-alg/9709040 Article |  MR 2062626 |  Zbl 1058.53065
[14] J. Loikkanen & C. Paufler, “Yang-Mills action from minimally coupled bosons on $\mathbb{R}^4$ and on the 4D Moyal plane,” 2004, arXiv:math-ph/0407039 arXiv |  Zbl 1076.58023
[15] J.E. Moya, “Quantum mechanics as a statistical theory”, Proc. Cambridge Phil. Soc. 45 (1949), p. 99-124 Article |  Zbl 0031.33601
[16] H. Omori, Y. Maeda, N. Miyazaki & A. Yoshioka, “Strange phenomena related to ordering problems in quantizations”, J. Lie Theory 13 (2003), p. 479-508  MR 2003156 |  Zbl 1046.53057
[17] M. Polyak, “Quantization of linear Poisson structures and degrees of maps” 2003, arXiv:math.GT/0210107 arXiv |  Zbl 1056.53060
[18] N.J.A. Sloane (editor), “The On-Line Encyclopedia of Integer Sequences”, http://www.research.att.com/%7enjas/sequences/ arXiv |  Zbl 1044.11108
[19] A. Voros, “Asymptotic $\hbar$-expansions of stationary quantum states”, Ann. Inst. H. Poincaré Sect. A (N.S.) 26 (1977), p. 343-403 Numdam |  MR 456138
[20] H. Weyl, “Gruppentheorie und Quantenmechanik”, Z. Phys. 46 (1928), p. 1-46  JFM 53.0848.02
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