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Izv. Akad. Nauk SSSR Ser. Mat., 1983, Volume 47, Issue 5, Pages 999–1029 (Mi izv1434)  

This article is cited in 10 scientific papers (total in 10 papers)

Pseudodifferential operators and a canonical operator in general symplectic manifolds

M. V. Karasev, V. P. Maslov

Abstract: A calculus of $h$-pseudodifferential operators with symbols on $\mathfrak X$ is defined modulo $O(h^2)$ on a closed symplectic manifold $(\mathfrak X,\omega)$ under the condition that $[\omega]/(2\pi h)-\varkappa/4 \in H^2(\mathfrak X,\mathbf Z)$. The class $\varkappa\in H^2(\mathfrak X,\mathbf Z)$ is described. On Lagrangian submanifolds $\Lambda\subset\mathfrak X$ a class in $H^1(\Lambda,\mathbf U(1))$ obstructing the definition of a canonical operator on $\Lambda$ is found. It is shown that an analogus calculus of pseudodifferential operators can be constructed with respect to homogeneity from an action of the group $\mathbf R_+$ on $\mathfrak X$.
Bibliography: 22 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1984, 23:2, 277–305

Bibliographic databases:

UDC: 517.9
MSC: Primary 35S05, 58F05, 58G15; Secondary 47G05, 53C15, 55N30, 55S35, 58F06, 70D10, 70G35
Received: 14.06.1982

Citation: M. V. Karasev, V. P. Maslov, “Pseudodifferential operators and a canonical operator in general symplectic manifolds”, Izv. Akad. Nauk SSSR Ser. Mat., 47:5 (1983), 999–1029; Math. USSR-Izv., 23:2 (1984), 277–305

Citation in format AMSBIB
\by M.~V.~Karasev, V.~P.~Maslov
\paper Pseudodifferential operators and a~canonical operator in general symplectic manifolds
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1983
\vol 47
\issue 5
\pages 999--1029
\jour Math. USSR-Izv.
\yr 1984
\vol 23
\issue 2
\pages 277--305

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    This publication is cited in the following articles:
    1. V. P. Maslov, “Non-standard characteristics in asymptotic problems”, Russian Math. Surveys, 38:6 (1983), 1–42  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. M. V. Karasev, “Asymptotic behavior of the spectrum of mixed states for self-consistent field equations”, Theoret. and Math. Phys., 61:1 (1984), 1034–1040  mathnet  crossref  mathscinet  zmath  isi
    3. M. V. Karasev, V. P. Maslov, “Asymptotic and geometric quantization”, Russian Math. Surveys, 39:6 (1984), 133–205  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    4. M. V. Karasev, “Asymptotics of the eigenvalues of operators with a poisson algebra of symmetries in the highest symbol”, Funct. Anal. Appl., 18:2 (1984), 140–142  mathnet  crossref  mathscinet  zmath  isi
    5. M. V. Karasev, “Poisson symmetry algebras and the asymptotics of spectral series”, Funct. Anal. Appl., 20:1 (1986), 17–26  mathnet  crossref  mathscinet  zmath  isi
    6. Hideki Omori, Yoshiaki Maeda, Akira Yoshioka, “Weyl manifolds and deformation quantization”, Advances in Mathematics, 85:2 (1991), 224  crossref
    7. M. V. Karasev, M. B. Kozlov, “Representations of Compact Semisimple Lie Algebras over Lagrangian Submanifolds”, Funct. Anal. Appl., 28:4 (1994), 238–246  mathnet  crossref  mathscinet  zmath  isi
    8. V. G. Bagrov, V. V. Belov, M. F. Kondrat'eva, “The semiclassical approximation in quantum mechanics. A new approach”, Theoret. and Math. Phys., 98:1 (1994), 34–38  mathnet  crossref  mathscinet  zmath  isi
    9. Frank Geshwind, Nets Hawk Katz, “Pseudodifferential operators onSU(2)”, The Journal of Fourier Analysis and Applications, 3:2 (1997), 193  crossref  mathscinet
    10. J. Tosiek, P. Brzykcy, “States in the Hilbert space formulation and in the phase space formulation of quantum mechanics”, Annals of Physics, 2013  crossref
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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