×

zbMATH — the first resource for mathematics

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.

MSC:
53D55 Deformation quantization, star products
81S10 Geometry and quantization, symplectic methods
47G30 Pseudodifferential operators
58J40 Pseudodifferential and Fourier integral operators on manifolds
Software:
OEIS
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML arXiv
References:
[1] Andersson, M.; Sjöstrand, J., Functional calculus for non-commuting operators with real spectra via an iterated Cauchy formula · Zbl 1070.47009
[2] Argyres, P. N., The Bohr-Sommerfeld quantization rule and the Weyl correspondence, Physics, 2, 131-139, (1965)
[3] Bayen, F.; Flato, M.; Fronsdal, C.; A. Lichnerowicz, D. Sternheimer, Deformation theory and quantization I-II, Ann. Phys., 111, 61-110, 111-151, (1978) · 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., 38, 1977-2004, (2005) · Zbl 1073.81056
[5] Charles, L., Berezin-Toeplitz operators, a semi-classical approach, Comm. Math. Phys., 239, 1-28, (2003) · Zbl 1059.47030
[6] Verdière, Y. Colin de, Bohr-Sommerfeld rules to all order, (2004) · Zbl 1080.81029
[7] Davies, E. B., Spectral theory and differential operators, 42, (1995), Cambridge University Press · Zbl 0893.47004
[8] Grigis, A.; Sjöstrand, J., Microlocal analysis for differential operators, (1994), London Mathematical Society · Zbl 0804.35001
[9] Groenewold, H. J., On the principles of elementary quantum mechanics, Physica (Amsterdam), 12, 405-460, (1946) · Zbl 0060.45002
[10] Helffer, B.; Sjöstrand, J., Équation de Schrödinger avec champ magnétique et équation de harper, Springer Lecture Notes in Physics, 345, 118-197, (1989) · Zbl 0699.35189
[11] Hirshfeld, A. C.; Henselder, P., Deformation quantization in the teaching of quantum mechanics, Amer. J. Physics, 70, 537-547, (2002) · Zbl 1219.81170
[12] Kathotia, V., Kontsevich’s universal formula for deformation quantization and the Campbell-Baker-haussdorf formula, I, Internat. J. Math., 11, 523-551, (2000) · Zbl 1110.53308
[13] Kontsevich, M., Deformation quantization of Poisson manifolds I, Lett. Math. Phys., 66, 157-216, (2003) · Zbl 1058.53065
[14] Loikkanen, J.; Paufler, C., Yang-Mills action from minimally coupled bosons on \(\mathbb{R}^4\) and on the 4D Moyal plane,, (2004) · Zbl 1076.58023
[15] Moya, J. E., Quantum mechanics as a statistical theory, Proc. Cambridge Phil. Soc., 45, 99-124, (1949) · Zbl 0031.33601
[16] Omori, H.; Maeda, Y.; Miyazaki, N.; Yoshioka, A., Strange phenomena related to ordering problems in quantizations, J. Lie Theory, 13, 479-508, (2003) · Zbl 1046.53057
[17] Polyak, M., Quantization of linear Poisson structures and degrees of maps, (2003) · Zbl 1056.53060
[18] (editor), N. J.A. Sloane, The on-line encyclopedia of integer sequences · Zbl 1044.11108
[19] Voros, A., Asymptotic \(ℏ\)-expansions of stationary quantum states, Ann. Inst. H. Poincaré Sect. A (N.S.), 26, 343-403, (1977)
[20] Weyl, H., Gruppentheorie und quantenmechanik, Z. Phys., 46, 1-46, (1928) · JFM 53.0848.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.