Decomposition of Resonant Scatterers by Surfaces of Section

Abstract

Scattering on the energy shell is viewed here as the relation between the bound states of the Hamiltonian, restricted to sections on leads that are asymptotically independent, far away from the interaction region. The decomposition is achieved by sectioning this region and adding new leads, thus generating two new scatterers. So a resonant scatterer, whose $\bS$-matrix has sharp energy peaks, can be resolved into a pair of scatterers with smooth energy dependence. The resonant behaviour is concentrated in a spectral determinant obtained from a dissipative section map. The semiclassical limit of this theory coincides with the orbit resummation previously proposed by Georgeot and Prange. A numerical example for a semiseparable scatterer is investigated, revealing the accurate portrayal of the Wigner time delay by the spectral determinant.