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 |
References:
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[10] | Howe. R. (1988). The Oscillator Semigroup. Proc. of Symposia in Pure Mathematics 48, Amer. Math. Soc. pp. 61–132 · Zbl 0687.47034 |
[13] | Nostre-Marques R.C., Piccione P., Tausk D.V. (2001). On the Morse and the Maslov index for periodic geodesics of arbitrary causal character, Differential Geometry and its Applications (Opava 2001), Math. Publ. 3, Silesian Univ. Opava, pp. 343–358 · Zbl 1034.53043 |
[17] | Wong M.W. (1998). Weyl Transforms, Springer · Zbl 0908.44002 |
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