The symbol of a function of a pseudo-differential operator
Annales de l'Institut Fourier, Volume 55 (2005) no. 7, p. 2257-2284
We give an explicit formula for the symbol of a function of an operator. Given a pseudo-differential operator A ^ on L 2 ( N ) with symbol A𝒞 (T * 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 .
On obtient une formule explicite pour le symbole d’une fonction d’un opérateur. À partir d’un opérateur pseudo-différentiel A ^ sur L 2 ( N ) avec symbole A𝒞 (T * N ) et une fonction lisse f, nous obtenons le symbole de f(A ^) en termes de A. Comme application, les règles de quantification de Bohr-Sommerfeld sont calculées explicitement à l’ordre 4 en .
DOI : https://doi.org/10.5802/aif.2161
Classification:  53D55,  81S10
Keywords: Deformation quantization, Moyal product, Weyl quantization, Bohr-Sommerfeld, symbol, diagrammatic technique
Keywords: Deformation quantization, Moyal product, Weyl quantization, Bohr-Sommerfeld, symbol, diagrammatic technique
@article{AIF_2005__55_7_2257_0,
     author = {Gracia-saz, Alfonso},
     title = {The symbol of a function of a pseudo-differential operator},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {7},
     year = {2005},
     pages = {2257-2284},
     doi = {10.5802/aif.2161},
     mrnumber = {2207384},
     zbl = {1091.53062},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2005__55_7_2257_0}
}
The symbol of a function of a pseudo-differential operator. Annales de l'Institut Fourier, Volume 55 (2005) no. 7, pp. 2257-2284. doi : 10.5802/aif.2161. https://aif.centre-mersenne.org/item/AIF_2005__55_7_2257_0/

[1] Andersson, M.; Sjöstrand, J. Functional calculus for non-commuting operators with real spectra via an iterated Cauchy formula (arXiv:math.SP/0303024, 2003) | Zbl 1070.47009

[2] Argyres, P.N. The Bohr-Sommerfeld quantization rule and the Weyl correspondence, Physics, Tome 2 (1965), pp. 131-139

[3] Bayen, F.; Flato, M.; Fronsdal, C.; A. Lichnerowicz, D. Sternheimer Deformation theory and quantization I-II, Ann. Phys., Tome 111 (1978), pp. 61-110, 111-151 | MR 496158 | Zbl 0377.53025

[4] Cargo, M.; Saz, A. Gracia-; Littlejohn, R.G.; Reinsch, M.W.; Rios, P. De M. Quantum normal forms, Moyal star product and Bohr-Sommerfeld approximation, J. Phys. A, Math. and Gen., Tome 38 (2005), pp. 1977-2004 (arXiv:math-ph/0409039) | Article | MR 2124376 | Zbl 02151996

[5] Charles, L. Berezin-Toeplitz operators, a semi-classical approach, Comm. Math. Phys., Tome 239 (2003), pp. 1-28 | Article | MR 1997113 | Zbl 1059.47030

[6] Verdière, Y. Colin De Bohr-Sommerfeld rules to all order (2004) (to appear in Henri Poincaré Acta) | Zbl 1080.81029

[7] Davies, E.B. Spectral theory and differential operators, Cambridge University Press, Cambridge Studies in Advanced Mathematics, Tome 42 (1995) | MR 1349825 | Zbl 0893.47004

[8] Grigis, A.; Sjöstrand, J. Microlocal analysis for differential operators, London Mathematical Society, 196 (1994) | MR 1269107 | Zbl 0804.35001

[9] Groenewold, H.J. On the principles of elementary quantum mechanics, Physica (Amsterdam), Tome 12 (1946), pp. 405-460 | Article | MR 18562 | Zbl 0060.45002

[10] Helffer, B.; Sjöstrand, J. Équation de Schrödinger avec champ magnétique et équation de Harper, Springer Lecture Notes in Physics, Tome 345 (1989), pp. 118-197 | Article | MR 1037319 | Zbl 0699.35189

[11] Hirshfeld, A.C.; Henselder, P. Deformation quantization in the teaching of quantum mechanics, Amer. J. Physics, Tome 70 (2002), pp. 537-547 (arXiv:quant-ph/ 0208163) | Article | MR 1897018

[12] Kathotia, V. Kontsevich's universal formula for deformation quantization and the Campbell-Baker-Haussdorf formula, I, Internat. J. Math., Tome 11 (2000), pp. 523-551 (arXiv:math.QA/9811174) | MR 1768172 | Zbl 01629355

[13] Kontsevich, M. Deformation quantization of Poisson manifolds I, Lett. Math. Phys., Tome 66 (2003), pp. 157-216 (arXiv:q-alg/9709040) | Article | MR 2062626 | Zbl 1058.53065

[14] Loikkanen, J.; Paufler, C. Yang-Mills action from minimally coupled bosons on 4 and on the 4D Moyal plane, (2004) (arXiv:math-ph/0407039) | Zbl 1076.58023

[15] Moya, J.E. Quantum mechanics as a statistical theory, Proc. Cambridge Phil. Soc., Tome 45 (1949), pp. 99-124 | Article | Zbl 0031.33601

[16] Omori, H.; Maeda, Y.; Miyazaki, N.; Yoshioka, A. Strange phenomena related to ordering problems in quantizations, J. Lie Theory, Tome 13 (2003), pp. 479-508 | MR 2003156 | Zbl 1046.53057

[17] Polyak, M. Quantization of linear Poisson structures and degrees of maps (2003) (arXiv:math.GT/0210107) | Zbl 1056.53060

[18] (Editor), N.J.A. Sloane The On-Line Encyclopedia of Integer Sequences (http://www.research.att.com/%7enjas/sequences/) | Zbl 1044.11108

[19] Voros, A. Asymptotic -expansions of stationary quantum states, Ann. Inst. H. Poincaré Sect. A (N.S.), Tome 26 (1977), pp. 343-403 | Numdam | MR 456138

[20] Weyl, H. Gruppentheorie und Quantenmechanik, Z. Phys., Tome 46 (1928), pp. 1-46 | JFM 53.0848.02