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General quantum surface-of-section method. (English) Zbl 0860.58019

Summary: 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 \(f\)-dimensional configuration space (CS) along the \((f-1)\)-dimensional 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 \((E- \widehat H)^{-1}\) 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 \((\hbar \to 0)\) the quantum Poincaré mapping converges to the Bogomolny’s propagator and explain how the higher-order semiclassical corrections can be obtained systematically.

MSC:

53D50 Geometric quantization
81S10 Geometry and quantization, symplectic methods
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