An algebra of pseudo-differential operators and quantum mechanics in phase space
Grossmann, A. ; Loupias, Guy ; Stein, Elias M.
Annales de l'Institut Fourier, Tome 18 (1968), p. 343-368 / Harvested from Numdam
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Nous étudions une algèbre 𝒫 de fonctions infiniment différentiables définies sur l’espace de phase et satisfaisant des conditions de croissance à l’infini. Le produit dans 𝒫 est la transformée de Fourier symplectique de la convolution gauche. On montre que 𝒫 est une généralisation naturelle de l’algèbre des opérateurs pseudodifférentiels.

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     author = {Grossmann, A. and Loupias, Guy and Stein, Elias M.},
     title = {An algebra of pseudo-differential operators and quantum mechanics in phase space},
     journal = {Annales de l'Institut Fourier},
     volume = {18},
     year = {1968},
     pages = {343-368},
     doi = {10.5802/aif.305},
     mrnumber = {42 \#2327},
     zbl = {0176.45102},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1968__18_2_343_0}
}

Grossmann, A.; Loupias, Guy; Stein, Elias M. An algebra of pseudo-differential operators and quantum mechanics in phase space. Annales de l'Institut Fourier, Tome 18 (1968) pp. 343-368. doi : 10.5802/aif.305. http://gdmltest.u-ga.fr/item/AIF_1968__18_2_343_0/

[1] H. Weyl, " The theory of groups and quantum mechanics ", London, Methnen, (1931).

E. C. G. Sudarshan, " Structure of Dynamical Theories ", Lectures in Theoretical Physics, Brandeis, 1961, (Benjamin).

[2] G. Loupias and S. Miracle-Sole, " C*-algèbres des systèmes canoniques : I ", Commun. Math. Phys., 2, 31-48 (1966). | MR 34 #3357 | Zbl 0144.23401

[3] J. J. Kohn and L. Nirenberg, Commun. Pure and Appl. Math., 18, 269 (1965). | Zbl 0171.35101

[4] D. Kastler, Commun. Math. Phys., 1, 14 (1965). | Zbl 0137.45601

[5] J. C. T. Pool, J. Math. Phys., 7, 66 (1966). | Zbl 0139.45903

[6] J. E. Moyal, Proc. Cambridge Phil. Soc., 45, 99 (1949). | Zbl 0031.33601

[7] J. Von Neumann, Mathematische Annalen, 104, 570 (1931).

[8] V. Bargmann, Commun. Pure and Appl. Math., 14, 187 (1961). | Zbl 0107.09102

I. E. Segal, Illinois J. Math., 6, 500 (1962). | Zbl 0106.42804

[9] R. Schatten, Norm Ideal of completely continuous operators, (Van Nostrand), (1960). | MR 22 #9878 | Zbl 0090.09402

[10] I. E. Segal, Math. Scand., 13, 31 (1963). | Zbl 0208.39002

[11] G. Loupias and S. Miracle-Sole, " C*-algèbres des systèmes canoniques: II ", Ann. I.H.P., VI, 1, 39 (1967). | Numdam | MR 37 #3823 | Zbl 0168.23505

[12] T. F. Jordan and E. C. G. Sudarshan, Review Mod. Phys., 33, 515 (1961). | Zbl 0118.44003

[13] G. Loupias, " Sur le formalisme de la convolution gauche ", Thesis, Marseille (1966).

[14] E. Wigner, Phys. Rev., 40, 749 (1932). | Zbl 0004.38201

[15] L. Cohen, Journal of Math. Phys., 5, 781 (1966).

[16] L. Hörmander, Commun. Pure and Appl. Math., 18, 501 (1965). | Zbl 0125.33401

[17] A. Erdelyi, Asymptotic expansions, Dover Publications (1956). | MR 17,1202c | Zbl 0070.29002

[18] D. Iagolnitzer, S-Matrix and classical description of interactions. Preprint C.E.N. Saclay (France).

[19] G. Mackey, Mathematical Foundations of Quantum Mechanics, Benjamin 1963. | Zbl 0114.44002