Semiclassics for a dissipative quantum map

We present a semiclassical analysis for a dissipative quantum map with an a rea-nonpreserving classical limit, We show that in the limit (h) over bar - -> 0 the tract of an arbitrary natural power of the propagator is dominated by contributions from periodic orbits of the corresponding classical dissi pative motion. We derive trace formulae of the Gutzwiller type for such qua ntum maps. In comparison to Tabor's formula for area-preserving maps, both classical action and stability prefactor are modified by the dissipation. W e evaluate the traces explicitly in the case of a dissipative kicked top wi th integrable classical motion and find good agreement with numerical resul ts. (C) 1999 Elsevier Science B.V. All rights reserved.