General quantum surface-of-section method

Abstract

A new method for exact quantization of general bound Hamiltonian systems is presented. It is the quantum analogue of the classical Poincare surface-of-section (SOS) reduction of classical dynamics. The quantum Poincare 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-dimensional configuration space (CS) the along the (f-1)-dimensional configurational SOS and attaching the flat quasi-one-dimensional waveguides instead. The quantum Poincare mapping has fixed points at the eigenenergies of the original bound Hamiltonian. The energy-dependent quantum propagator (E-H)-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 (h(cross) to 0) the quantum Poincare mapping converges to the Bogomolny propagator and explain how the higher-order semiclassical corrections can be obtained systematically.