The symbol of a function of a pseudo-differential operator

Gracia-saz, Alfonso

Annales de l'Institut Fourier, Tome 55 (2005) no. 7, pp. 2257-2284.

On obtient une formule explicite pour le symbole d’une fonction d’un opérateur. À partir d’un opérateur pseudo-différentiel $\widehat{A}$ sur $L^2\left({ℝ}^N\right)$ avec symbole $A\in {𝒞}^{\infty }\left(T^{*}{ℝ}^N\right)$ et une fonction lisse $f$, nous obtenons le symbole de $f\left(\widehat{A}\right)$ en termes de $A$. Comme application, les règles de quantification de Bohr-Sommerfeld sont calculées explicitement à l’ordre 4 en $\hslash$.

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

Classification : 53D55, 81S10
Keywords: Deformation quantization, Moyal product, Weyl quantization, Bohr-Sommerfeld, symbol, diagrammatic technique, Deformation quantization, Moyal product, Weyl quantization, Bohr-Sommerfeld, symbol, diagrammatic technique
Mot clés : quantification par déformation, produit de Moyal, quantification de Weyl, Bohr-Sommerfeld, symbole, technique diagrammatique

[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

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

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

[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., Volume 38 (2005), pp. 1977-2004 (arXiv:math-ph/0409039) | DOI | MR | Zbl

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

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

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

[8] Grigis, A.; Sjöstrand, J. Microlocal analysis for differential operators, 196, Lecture Note, London Mathematical Society, 1994 | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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