Riemann surface approach to bound and resonant states: Exotic resonant states for a central rectangular potential

An approach to bound and resonant states in scattering by a central potential gV(r), g∈C, based on a global analysis of S-matrix poles, is presented. The global method involves the construction of the Riemann surface R_g^(l) over the g plane on which the pole function k=k^(l)(g) is single valued and analytic. This implies the division of the Riemann surface R_g^(l) into sheets and the construction of the Riemann sheets images in the k plane. By keeping the sheets of the Riemann surface apart, the single pole laying on each sheet image in the k plane is identified. With each state (l,n) of the quantum system one associates a sheet Σ_n^(l) of the Riemann surface R_g^(l). A new quantum number n with a topological meaning is introduced in order to label a pole and the corresponding state (l,n). All S-matrix poles for a central rectangular potential gV(r), with l=0, 1, 2, 3, and 4, are analyzed by using the global method. A new class of resonant state poles, having unusual properties, is identified. The properties of these resonant state poles (exotic poles) and of the corresponding resonant states are studied. A new type of resonance in the cross section, associated with the cooperative contribution from three adjacent partial waves and due to the local degeneracy with respect to l, is discussed.