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On the Weyl representation of metaplectic operators. (English) Zbl 1115.81048

Summary: We study the Weyl representation of metaplectic operators associated to a symplectic matrix having no non-trivial fixed point, and justify a formula suggested in earlier work of B. Mehlig and M. Wilkinson [Ann. Phys. (8) 10, No. 6–7, 541–559 (2001; Zbl 1033.81050)]. We give precise calculations of the associated Maslov-type indices; these indices intervene in a crucial way in Gutzwiller’s formula of semiclassical mechanics, and are simply related to an index defined by C. Conley and E. Zehnder [Commun. Pure Appl. Math. 37, 207–253 (1984; Zbl 0559.58019)].

MSC:

81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
43A32 Other transforms and operators of Fourier type
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References:

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[17] Wong M.W. (1998). Weyl Transforms, Springer · Zbl 0908.44002
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