  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.
