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Quantification et analyse pseudo-différentielle. (Quantization and pseudodifferential analysis). (English) Zbl 0646.58025
The authors start from a general scheme of a mathematical theory of quantization which relates between a transitive Lie group \(\Gamma\) of diffeomorphisms of a phase space \(\Pi\), a projective unitary representation of \(\Gamma\) on a Hilbert space H, and a space L of selfadjoint operators on H which have as symbols \(C^{\infty}\)-functions on \(\Pi\). In order to handle the quantization problems, they develop various analytical machineries: Besse calculus, Fuchs calculus, etc. By studying the case where the phase space \(\Pi\) is the Poincaré half- plane, it is shown that, generally, asymptotic formulas do not appear. An important result of the paper is the formula for the composition of symbols established in Section 5. Consequences of this formula are discussed.
Reviewer: I.Vaisman

MSC:
58J40 Pseudodifferential and Fourier integral operators on manifolds
81Q99 General mathematical topics and methods in quantum theory
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