|XXV Workshop on Geometric Methods in Physics||2-8.07.2006|
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Regular And Chaotic Quantum Scattering on Lagrange Surfaces of Few Body Systems
The quantum multichannel scattering theory for the few-body systems is formulated on the Lagrange surfaces. Such a method automatically separates internal and external motions and for Hamiltonian of body-system in the intrinsic space irrespective of number of particles allows finding a universal and simple form. Using this arrangement with combining of natural collision coordinates and writing the full wave function in coupled-channel form it is proved that the 3D multi-channel quantum scattering problem can be treated in the same way as the inelastic single-arrangement problem which is described by the system of ordinary first order differential equations. This system of equations is solved with the help of the R-matrix propagation method which simultaneously gives the full wave function and all S-matrix elements without further calculation.
When the classical analog of the few-body system is chaotic the quantization should be conducted on the classical and complex-classical trajectory tubes which are generated by the system of geodesic equations on the Lagrange surfaces. It was shown that such a type of quantization presents a generalization of Maslov semiclassical quantization method. The simplification of quantum chaotic scattering approach and its reducing to the inelastic single-arrangement problem is carried out by the similar way. In the present work the criteria of quantum chaos initiation and its connection to the geometric characters of Lagrange surfaces are investigated in detail.