Riemann surface approach to bound and resonant states: application to a two-channel model with square potentials

Abstract

The Riemann surface approach to bound and resonant states is generalized to the case of a two-channel model. The approach consists of the construction of the Riemann surface Rg of the S-matrix pole function k = k(g) over the g-plane, where g is the strength of the complex potential in each of the two channels. On the Riemann surface Rg the pole function k = k(g) is single valued and analytic. The branch points of the pole function k = k(g) and their k-plane images are asymptotically and numerically determined and analysed as a function of the strength of the coupling potential. It is shown that besides the branch points which originate from the branch points for the uncoupled channels, there are new branch points which are intrinsic to the coupling. According to the Riemann surface approach to each bound or resonant state a sheet of the Riemann surface Rg is associated. All the natural modes (bound and resonant states) of the system are identified and treated in a unified way. The Riemann surface approach allows introducing two new quantum numbers (m, n) with topological meaning which label each bound or resonant state. The approach provides a new insight into the nature of the resonant states for two coupled channels. The distinct origin and the relationship of the Feshbach and Newton–Fonda resonant states are clarified.