An algebra of pseudo-differential operators and quantum mechanics in phase space
Annales de l'Institut Fourier, Tome 18 (1968) no. 2, pp. 343-368.

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.

@article{AIF_1968__18_2_343_0,
     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},
     pages = {343--368},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {18},
     number = {2},
     year = {1968},
     doi = {10.5802/aif.305},
     mrnumber = {42 #2327},
     zbl = {0176.45102},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.305/}
}
TY  - JOUR
AU  - Grossmann, A.
AU  - Loupias, Guy
AU  - Stein, Elias M.
TI  - An algebra of pseudo-differential operators and quantum mechanics in phase space
JO  - Annales de l'Institut Fourier
PY  - 1968
SP  - 343
EP  - 368
VL  - 18
IS  - 2
PB  - Institut Fourier
PP  - Grenoble
UR  - https://www.numdam.org/articles/10.5802/aif.305/
DO  - 10.5802/aif.305
LA  - en
ID  - AIF_1968__18_2_343_0
ER  - 
%0 Journal Article
%A Grossmann, A.
%A Loupias, Guy
%A Stein, Elias M.
%T An algebra of pseudo-differential operators and quantum mechanics in phase space
%J Annales de l'Institut Fourier
%D 1968
%P 343-368
%V 18
%N 2
%I Institut Fourier
%C Grenoble
%U https://www.numdam.org/articles/10.5802/aif.305/
%R 10.5802/aif.305
%G en
%F 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) no. 2, pp. 343-368. doi : 10.5802/aif.305. https://www.numdam.org/articles/10.5802/aif.305/

[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 | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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

  • Kisil, Vladimir V. Cross-Toeplitz operators on the Fock–Segal–Bargmann spaces and two-sided convolutions on the Heisenberg group, Annals of Functional Analysis, Volume 14 (2023) no. 2 | DOI:10.1007/s43034-022-00249-7
  • Kumar, Amit; Prasad, Akhilesh; Jain, Pankaj The Weyl correspondence in the linear canonical transform domain, Filomat, Volume 37 (2023) no. 22, p. 7431 | DOI:10.2298/fil2322431k
  • Aniello, Paolo Group-Covariant Stochastic Products and Phase-Space Convolution Algebras, International Journal of Theoretical Physics, Volume 62 (2023) no. 4 | DOI:10.1007/s10773-023-05338-4
  • Petersson, Albin Fourier Characterizations and Non-triviality of Gelfand–Shilov Spaces, with Applications to Toeplitz Operators, Journal of Fourier Analysis and Applications, Volume 29 (2023) no. 3 | DOI:10.1007/s00041-023-10009-3
  • Gorokhovsky, Alexander; van Erp, Erik The Heisenberg calculus, index theory and cyclic cohomology, Advances in Mathematics, Volume 399 (2022), p. 108229 | DOI:10.1016/j.aim.2022.108229
  • Kumar, Amit; Prasad, Akhilesh Wigner-Ville distribution function in the framework of linear canonical transform, Journal of Pseudo-Differential Operators and Applications, Volume 13 (2022) no. 3 | DOI:10.1007/s11868-022-00471-w
  • Beckner, William Symmetry in Fourier Analysis: Heisenberg Group to Stein–Weiss Integrals, The Journal of Geometric Analysis, Volume 31 (2021) no. 7, p. 7008 | DOI:10.1007/s12220-020-00589-7
  • Cornean, Horia; Helffer, Bernard; Purice, Radu Spectral analysis near a Dirac type crossing in a weak non-constant magnetic field, Transactions of the American Mathematical Society (2021) | DOI:10.1090/tran/8402
  • Beberok, Tomasz; Budzyński, Piotr; Kang, Dong-O Compact vectorial Toeplitz operators on the Segal-Bargmann space, Journal of Mathematical Analysis and Applications, Volume 481 (2020) no. 2, p. 123460 | DOI:10.1016/j.jmaa.2019.123460
  • Lord, Steven; Sukochev, Fedor A.; Zanin, Dmitriy Advances in Dixmier traces and applications, Advances in Noncommutative Geometry (2019), p. 491 | DOI:10.1007/978-3-030-29597-4_9
  • Aniello, Paolo Square Integrable Representations, An Invaluable Tool, Coherent States and Their Applications, Volume 205 (2018), p. 17 | DOI:10.1007/978-3-319-76732-1_2
  • Fischer, Jens V. Four Particular Cases of the Fourier Transform, Mathematics, Volume 6 (2018) no. 12, p. 335 | DOI:10.3390/math6120335
  • Huang, Jizheng; Li, Weiwei; Wang, Yaqiong Hardy-Sobolev Spaces Associated with Twisted Convolution, Journal of Function Spaces, Volume 2017 (2017), p. 1 | DOI:10.1155/2017/5692746
  • Beckner, William Functionals for multilinear fractional embedding, Acta Mathematica Sinica, English Series, Volume 31 (2015) no. 1, p. 1 | DOI:10.1007/s10114-015-4321-6
  • Kouneiher, Joseph; Sidharth, Burra G. Mass Generation Without the Higgs Mechanism, International Journal of Theoretical Physics, Volume 54 (2015) no. 9, p. 3044 | DOI:10.1007/s10773-015-2542-1
  • Huang, Jizheng; Xing, Zhou New real-variable characterizations of Hardy spaces associated with twisted convolution, Journal of Inequalities and Applications, Volume 2015 (2015) no. 1 | DOI:10.1186/s13660-015-0687-3
  • Bommier-Hato, Hélène; Engliš, Miroslav; Youssfi, El-Hassan Dixmier trace and the Fock space, Bulletin des Sciences Mathématiques, Volume 138 (2014) no. 2, p. 199 | DOI:10.1016/j.bulsci.2013.04.009
  • Aniello, Paolo Operators versus functions: from quantum dynamical semigroups to tomographic semigroups, Journal of Physics: Conference Series, Volume 474 (2013), p. 012005 | DOI:10.1088/1742-6596/474/1/012005
  • Estrada, Ricardo; Fulling, Stephen A; Mera, Fernando D Surface vacuum energy in cutoff models: pressure anomaly and distributional gravitational limit, Journal of Physics A: Mathematical and Theoretical, Volume 45 (2012) no. 45, p. 455402 | DOI:10.1088/1751-8113/45/45/455402
  • Taylor, Michael E. Pseudodifferential Operators, Partial Differential Equations II, Volume 116 (2011), p. 1 | DOI:10.1007/978-1-4419-7052-7_1
  • Catană, Viorel The Heat Kernel and Green Function of the Generalized Hermite Operator, and the Abstract Cauchy Problem for the Abstract Hermite Operator, Pseudo-Differential Operators: Analysis, Applications and Computations (2011), p. 155 | DOI:10.1007/978-3-0348-0049-5_9
  • Molahajloo, Shahla The heat kernel of the τ-twisted Laplacian, Journal of Pseudo-Differential Operators and Applications, Volume 1 (2010) no. 3, p. 293 | DOI:10.1007/s11868-010-0012-3
  • HUANG, JIZHENG SOME CHARACTERIZATIONS OF HARDY SPACES ASSOCIATED WITH TWISTED CONVOLUTION, Bulletin of the Australian Mathematical Society, Volume 79 (2009) no. 3, p. 405 | DOI:10.1017/s0004972709000021
  • Aniello, Paolo Star products: a group-theoretical point of view, Journal of Physics A: Mathematical and Theoretical, Volume 42 (2009) no. 47, p. 475210 | DOI:10.1088/1751-8113/42/47/475210
  • Catană, Viorel Products of Two-Wavelet Multipliers and Their Traces, Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations (2009), p. 195 | DOI:10.1007/978-3-0346-0198-6_11
  • Athanassoulis, Agissilaos G. Exact equations for smoothed Wigner transforms and homogenization of wave propagation, Applied and Computational Harmonic Analysis, Volume 24 (2008) no. 3, p. 378 | DOI:10.1016/j.acha.2007.06.006
  • Aniello, P; Man'ko, V I; Marmo, G Frame transforms, star products and quantum mechanics on phase space, Journal of Physics A: Mathematical and Theoretical, Volume 41 (2008) no. 28, p. 285304 | DOI:10.1088/1751-8113/41/28/285304
  • Wallet, J-C Noncommutative induced gauge theories on Moyal spaces, Journal of Physics: Conference Series, Volume 103 (2008), p. 012007 | DOI:10.1088/1742-6596/103/1/012007
  • de Gosson, Maurice Phase-Space Weyl Calculus and Global Hypoellipticity of a Class of Degenerate Elliptic Partial Differential Operators, New Developments in Pseudo-Differential Operators (2008), p. 1 | DOI:10.1007/978-3-7643-8969-7_1
  • de Goursac, A.; Tanasa, A.; Wallet, J.-C. Vacuum configurations for renormalizable non-commutative scalar models, The European Physical Journal C, Volume 53 (2008) no. 3, p. 459 | DOI:10.1140/epjc/s10052-007-0465-6
  • Athanassoulis, Agissilaos G., 2007 International Conference - Days on Diffraction (2007), p. 13 | DOI:10.1109/dd.2007.4531981
  • DE GOSSON, MAURICE METAPLECTIC REPRESENTATION, CONLEY–ZEHNDER INDEX, AND WEYL CALCULUS ON PHASE SPACE, Reviews in Mathematical Physics, Volume 19 (2007) no. 10, p. 1149 | DOI:10.1142/s0129055x07003152
  • de Goursac, A.; Wallet, J.-C.; Wulkenhaar, R. Noncommutative induced gauge theory, The European Physical Journal C, Volume 51 (2007) no. 4, p. 977 | DOI:10.1140/epjc/s10052-007-0335-2
  • Wong, M. W. Weyl Transforms, the Heat Kernel and Green Function of a Degenerate Elliptic Operator, Annals of Global Analysis and Geometry, Volume 28 (2005) no. 3, p. 271 | DOI:10.1007/s10455-005-1148-x
  • Boggiatto, Paolo; Toft, Joachim Embeddings and compactness for generalized Sobolev–Shubin spaces and modulation spaces, Applicable Analysis, Volume 84 (2005) no. 3, p. 269 | DOI:10.1080/00036810412331297253
  • Loikkanen, Juha; Paufler, Cornelius Yang–Mills action from minimally coupled bosons on R4 and on the four-dimensional Moyal plane, Journal of Mathematical Physics, Volume 46 (2005) no. 3 | DOI:10.1063/1.1839277
  • Gayral, Victor; Iochum, Bruno The spectral action for Moyal planes, Journal of Mathematical Physics, Volume 46 (2005) no. 4 | DOI:10.1063/1.1855401
  • Wildberger, N. J. Weyl quantization and a symbol calculus for abelian groups, Journal of the Australian Mathematical Society, Volume 78 (2005) no. 3, p. 323 | DOI:10.1017/s1446788700008569
  • Wong, M.W. Trace-Class Weyl Transforms, Recent Advances in Operator Theory and its Applications (2005), p. 469 | DOI:10.1007/3-7643-7398-9_24
  • Toft, Joachim Positivity properties in noncommutative convolution algebras with applications in pseudo-differential calculus, Bulletin des Sciences Mathématiques, Volume 127 (2003) no. 2, p. 101 | DOI:10.1016/s0007-4497(02)00003-9
  • Toft, Joachim An Embedding Result for Some General Symbol Classes in the Weyl Calculus, Jean Leray ’99 Conference Proceedings (2003), p. 219 | DOI:10.1007/978-94-017-2008-3_16
  • Toft, Joachim Continuity properties in non-commutative convolution algebras, with applications in pseudo-differential calculus, Bulletin des Sciences Mathématiques, Volume 126 (2002) no. 2, p. 115 | DOI:10.1016/s0007-4497(01)01089-2
  • Dubois-Violette, Michel; Kriegl, Andreas; Maeda, Yoshiaki; Michor, Peter W. Smooth *-Algebras, Progress of Theoretical Physics Supplement, Volume 144 (2001), p. 54 | DOI:10.1143/ptps.144.54
  • Du, Jingde; Wong, M. W. A trace formula for Weyl transforms, Approximation Theory and its Applications, Volume 16 (2000) no. 1, p. 41 | DOI:10.1007/bf02845227
  • Toft, Joachim Regularizations, decompositions and lower bound problems in the weyl calculus, Communications in Partial Differential Equations, Volume 25 (1999) no. 7-8, p. 1201 | DOI:10.1080/03605300008821548
  • Gr�chenig, Karlheinz; Heil, Christopher Modulation spaces and pseudodifferential operators, Integral Equations and Operator Theory, Volume 34 (1999) no. 4, p. 439 | DOI:10.1007/bf01272884
  • Durán, Ana; Estrada, Ricardo An extension of the integral test, Proceedings of the American Mathematical Society, Volume 127 (1999) no. 6, p. 1745 | DOI:10.1090/s0002-9939-99-04911-4
  • Durán, A.L; Estrada, R; Kanwal, R.P Extensions of the Poisson Summation Formula, Journal of Mathematical Analysis and Applications, Volume 218 (1998) no. 2, p. 581 | DOI:10.1006/jmaa.1997.5767
  • Matz, Gerald; Hlawatsch, Franz Time-frequency transfer function calculus (symbolic calculus) of linear time-varying systems (linear operators) based on a generalized underspread theory, Journal of Mathematical Physics, Volume 39 (1998) no. 8, p. 4041 | DOI:10.1063/1.532495
  • McQuarrie, B. R.; Osborn, T. A.; Tabisz, G. C. Semiclassical Moyal quantum mechanics for atomic systems, Physical Review A, Volume 58 (1998) no. 4, p. 2944 | DOI:10.1103/physreva.58.2944
  • Estrada, Ricardo The Cesàro behaviour of distributions, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, Volume 454 (1998) no. 1977, p. 2425 | DOI:10.1098/rspa.1998.0265
  • Beckner, William Sharp inequalities and geometric manifolds, The Journal of Fourier Analysis and Applications, Volume 3 (1997) no. S1, p. 825 | DOI:10.1007/bf02656488
  • Taylor, Michael E. Pseudodifferential Operators, Partial Differential Equations II, Volume 116 (1996), p. 1 | DOI:10.1007/978-1-4757-4187-2_1
  • Sternheimer, Daniel Star Products: Their Ubiquity and Unicity, Modern Group Theoretical Methods in Physics (1995), p. 255 | DOI:10.1007/978-94-015-8543-9_23
  • Shenoy, R.G.; Parks, T.W. The Weyl correspondence and time-frequency analysis, IEEE Transactions on Signal Processing, Volume 42 (1994) no. 2, p. 318 | DOI:10.1109/78.275605
  • Zweifel, P. F. The Wigner transform and the Wigner-poisson system, Transport Theory and Statistical Physics, Volume 22 (1993) no. 4, p. 459 | DOI:10.1080/00411459308203824
  • Simon, Barry The Weyl transform and 𝐿^𝑝 functions on phase space, Proceedings of the American Mathematical Society, Volume 116 (1992) no. 4, p. 1045 | DOI:10.1090/s0002-9939-1992-1100663-7
  • Sheu, Albert Jeu-Liang Quantization of the PoissonSU(2) and its Poisson homogeneous space — The 2-sphere, Communications in Mathematical Physics, Volume 135 (1991) no. 2, p. 217 | DOI:10.1007/bf02098041
  • Prástaro, Agostino Quantum geometry of PDE's, Reports on Mathematical Physics, Volume 30 (1991) no. 3, p. 273 | DOI:10.1016/0034-4877(91)90063-s
  • Cotlar, Mischa; Sadosky, Cora Two-parameter lifting theorems and double Hilbert transforms in commutative and non-commutative settings, Journal of Mathematical Analysis and Applications, Volume 150 (1990) no. 2, p. 439 | DOI:10.1016/0022-247x(90)90115-v
  • Dubois-Violette, Michel On the theory of quantum groups, Letters in Mathematical Physics, Volume 19 (1990) no. 2, p. 121 | DOI:10.1007/bf01045882
  • Bohnké, G. Besov-Sobolev Algebras of Symbols, Wavelets (1990), p. 216 | DOI:10.1007/978-3-642-75988-8_19
  • Narcowich, Francis J. Distributions of ℏ-positive type and applications, Journal of Mathematical Physics, Volume 30 (1989) no. 11, p. 2565 | DOI:10.1063/1.528537
  • Estrada, Ricardo; Gracia-Bondía, José M.; Várilly, Joseph C. On asymptotic expansions of twisted products, Journal of Mathematical Physics, Volume 30 (1989) no. 12, p. 2789 | DOI:10.1063/1.528514
  • Meladze, G. A.; Shubin, M. A. Proper uniform pseudodifferential operators on unimodular Lie groups, Journal of Soviet Mathematics, Volume 45 (1989) no. 5, p. 1421 | DOI:10.1007/bf01097159
  • Meladze, G. A.; Shubin, M. A. A functional calculus of pseudodifferential operators on unimodular Lie groups, Journal of Soviet Mathematics, Volume 47 (1989) no. 4, p. 2607 | DOI:10.1007/bf01105914
  • Antoine, J.-P. Poincaré Coherent States and Relativistic Phase Space Analysis, Wavelets (1989), p. 221 | DOI:10.1007/978-3-642-97177-8_20
  • Bohnké, Georges Algebres de type Besov-Sobolev, Harmonic Analysis, Volume 1359 (1988), p. 113 | DOI:10.1007/bfb0086592
  • Gracia-Bondía, José M.; Várilly, Joseph C. Algebras of distributions suitable for phase-space quantum mechanics. I, Journal of Mathematical Physics, Volume 29 (1988) no. 4, p. 869 | DOI:10.1063/1.528200
  • Ichinose, Takashi Path integral for a Weyl quantized relativistic Hamiltonian and the nonrelativistic limit problem, Differential Equations and Mathematical Physics, Volume 1285 (1987), p. 205 | DOI:10.1007/bfb0080598
  • Diner, Simon Introduction from Quantum Physics to Quantum Technology, Information Complexity and Control in Quantum Physics (1987), p. 1 | DOI:10.1007/978-3-7091-2971-5_1
  • Ichinose, Takashi The nonrelativistic limit problem for a relativistic spinless particle in an electromagnetic field, Journal of Functional Analysis, Volume 73 (1987) no. 2, p. 233 | DOI:10.1016/0022-1236(87)90067-x
  • Narcowich, Francis J. The problem of moments in the phase-space formulation of quantum mechanics, Journal of Mathematical Physics, Volume 28 (1987) no. 12, p. 2873 | DOI:10.1063/1.527687
  • Berger, C. A.; Coburn, L. A. Toeplitz operators on the Segal-Bargmann space, Transactions of the American Mathematical Society, Volume 301 (1987) no. 2, p. 813 | DOI:10.1090/s0002-9947-1987-0882716-4
  • Phong, D. H.; Stein, E. M. Hilbert integrals, singular integrals, and Radon transforms I, Acta Mathematica, Volume 157 (1986) no. 0, p. 99 | DOI:10.1007/bf02392592
  • Ichinose, Takashi; Tamura, Hiroshi Imaginary-time path integral for a relativistic spinless particle in an electromagnetic field, Communications In Mathematical Physics, Volume 105 (1986) no. 2, p. 239 | DOI:10.1007/bf01211101
  • Osborn, T. A.; Molzahn, F. H. Structural connections between two semiclassical approximations: The WKB and Wigner-Kirkwood approximations, Physical Review A, Volume 34 (1986) no. 3, p. 1696 | DOI:10.1103/physreva.34.1696
  • Ichinose, Takashi; Tamura, Hiroshi Path integral for the Weyl quantized relativistic Hamiltonian, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Volume 62 (1986) no. 3 | DOI:10.3792/pjaa.62.91
  • Ratcliff, G Symbols and orbits for 3-step nilpotent Lie groups, Journal of Functional Analysis, Volume 62 (1985) no. 1, p. 38 | DOI:10.1016/0022-1236(85)90018-7
  • Christ, Michael; Geller, Daryl Singular integral characterizations of Hardy spaces on homogeneous groups, Duke Mathematical Journal, Volume 51 (1984) no. 3 | DOI:10.1215/s0012-7094-84-05127-5
  • Howe, Roger; Ratcliff, Gail; Wildberger, Norman Symbol mappings for certain nilpotent groups, Lie Group Representations III, Volume 1077 (1984), p. 288 | DOI:10.1007/bfb0072342
  • Bibliography, Mathematical and Conceptual Foundations of 20Th-Century Physics, Volume 100 (1984), p. 515 | DOI:10.1016/s0304-0208(08)70666-2
  • Geller, D.; Stein, E. M. Estimates for singular convolution operators on the Heisenberg group, Mathematische Annalen, Volume 267 (1984) no. 1, p. 1 | DOI:10.1007/bf01458467
  • Springborg, Michael On semiclassical approximations to Wigner's phase space function, Physica A: Statistical Mechanics and its Applications, Volume 126 (1984) no. 1-2, p. 259 | DOI:10.1016/0378-4371(84)90153-5
  • Hansen, F. Quantum mechanics in phase space, Reports on Mathematical Physics, Volume 19 (1984) no. 3, p. 361 | DOI:10.1016/0034-4877(84)90008-9
  • Janssen, A. J. E. M. A Note on Hudson’s Theorem about Functions with Nonnegative Wigner Distributions, SIAM Journal on Mathematical Analysis, Volume 15 (1984) no. 1, p. 170 | DOI:10.1137/0515014
  • Emch, Gérard G. Geometric dequantization and the correspondence problem, International Journal of Theoretical Physics, Volume 22 (1983) no. 5, p. 397 | DOI:10.1007/bf02083286
  • Prosser, Reese T. On the correspondence between classical and quantum mechanics. I, Journal of Mathematical Physics, Volume 24 (1983) no. 3, p. 548 | DOI:10.1063/1.525726
  • Janssen, A. J. E. M.; Zelditch, Steven Szegő limit theorems for the harmonic oscillator, Transactions of the American Mathematical Society, Volume 280 (1983) no. 2, p. 563 | DOI:10.1090/s0002-9947-1983-0716838-1
  • Hermann, Robert Quantum mechanics and geometric analysis on manifolds, International Journal of Theoretical Physics, Volume 21 (1982) no. 10-11, p. 803 | DOI:10.1007/bf01856874
  • Mauceri, Giancarlo; Picardello, Massimo A; Ricci, Fulvio A hardy space associated with twisted convolution, Advances in Mathematics, Volume 39 (1981) no. 3, p. 270 | DOI:10.1016/0001-8708(81)90004-9
  • Howe, Roger On the role of the Heisenberg group in harmonic analysis, Bulletin of the American Mathematical Society, Volume 3 (1980) no. 2, p. 821 | DOI:10.1090/s0273-0979-1980-14825-9
  • Howe, Roger Quantum mechanics and partial differential equations, Journal of Functional Analysis, Volume 38 (1980) no. 2, p. 188 | DOI:10.1016/0022-1236(80)90064-6
  • Mauceri, Giancarlo The Weyl transform and bounded operators on Lp(Rn), Journal of Functional Analysis, Volume 39 (1980) no. 3, p. 408 | DOI:10.1016/0022-1236(80)90035-x
  • Hörmander, L. The weyl calculus of pseudo‐differential operators, Communications on Pure and Applied Mathematics, Volume 32 (1979) no. 3, p. 359 | DOI:10.1002/cpa.3160320304
  • Grossmann, A. Geometry of real and complex canonical transformations in quantum mechanics, Group Theoretical Methods in Physics, Volume 79 (1978), p. 162 | DOI:10.1007/3-540-08848-2_9
  • Burdet, G.; Perrin, M. Weyl quantization and metaplectic representation, Group Theoretical Methods in Physics, Volume 79 (1978), p. 515 | DOI:10.1007/3-540-08848-2_63
  • Voros, A. An algebra of pseudodifferential operators and the asymptotics of quantum mechanics, Journal of Functional Analysis, Volume 29 (1978) no. 1, p. 104 | DOI:10.1016/0022-1236(78)90049-6
  • Antonets, M. A. The classical limit for Weyl quantization, Letters in Mathematical Physics, Volume 2 (1978) no. 3, p. 241 | DOI:10.1007/bf00406411
  • Combe, Ph.; Rodriguez, R.; Sirugue-Collin, M.; Sirugue, M. CLASSICAL FUNCTIONS ASSOCIATED WITH SOME GROUPS OF AUTOMORPHISMS OF THE WEYL GROUP, Group Theoretical Methods in Physics (1977), p. 349 | DOI:10.1016/b978-0-12-637650-0.50035-8
  • Kaiser, Gerald Phase-space approach to relativistic quantum mechanics. I. Coherent-state representation for massive scalar particles, Journal of Mathematical Physics, Volume 18 (1977) no. 5, p. 952 | DOI:10.1063/1.523376
  • Burdet, G.; Perrin, M. Weyl quantization and metaplectic representation, Letters in Mathematical Physics, Volume 2 (1977) no. 2, p. 93 | DOI:10.1007/bf00398573
  • Grossmann, A. Parity operator and quantization of δ-functions, Communications in Mathematical Physics, Volume 48 (1976) no. 3, p. 191 | DOI:10.1007/bf01617867
  • Cressman, R An evolution equation in phase space and the Weyl correspondence, Journal of Functional Analysis, Volume 22 (1976) no. 4, p. 405 | DOI:10.1016/0022-1236(76)90006-9
  • Andersson, Stig I. Unitary implementation of second quantized dynamics of hyperbolic type, Reports on Mathematical Physics, Volume 9 (1976) no. 3, p. 393 | DOI:10.1016/0034-4877(76)90069-0
  • Liu, Kuang Chi An expansion theorem for the twisted product with applications, Journal of Mathematical Physics, Volume 16 (1975) no. 3, p. 724 | DOI:10.1063/1.522588
  • Poulsen, Ebbe Thue Quantization in Hamiltonian particle mechanics, Functional Analysis and its Applications, Volume 399 (1974), p. 396 | DOI:10.1007/bfb0063593
  • Tilgner, Hans Algebraical comparison of classical and quantum polynomial observables, International Journal of Theoretical Physics, Volume 7 (1973) no. 1, p. 67 | DOI:10.1007/bf02412661

Cité par 108 documents. Sources : Crossref