General Quantum Surface of Section Method

A new method for exact quantization of general bound Hamiltonian systems is presented. It is the quantum analogue of the classical Poincaré Surface Of Section (SOS) reduction of classical dynamics. The quantum Poincaré mapping is shown to be the product of the two generalized (non-unitary but compact) on-shell scattering operators of the two scattering Hamiltonians which are obtained from the original bound one by cutting the f -dim configuration space (CS) along (f−1)−dim configurational SOS and attaching the flat quasi-one-dimensional waveguides instead. The quantum Poincaré mapping has fixed points at the eigenenergies of the original bound Hamiltonian. The energy dependent quantum propagator (E−Ĥ)−1 can be decomposed in terms of the four energy dependent propagators which propagate from and/or to CS to and/or from configurational SOS (which may generally be composed of many disconnected parts). I show that in the semiclassical limit (h̄ → 0) the quantum Poincaré mapping converges to the Bogomolny’s propagator and explain how the higher order semiclassical corrections can be obtained systematically. PACS numbers: 03.65.Db, 03.65.Ge, 02.30.+g, 05.45.+b Submitted to Journal of Physics A: Mathematical and General e-mail: [email protected] 1