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:
| [9] | Gutzwiller M.C. (1990). Chaos in Classical and Quantum Mechanics. Interdisciplinary Applied Mathematics, Springer-Verlag · Zbl 0727.70029 |
| [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 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
