Riemann surface approach to bound and resonant states: Exotic resonant states for a central rectangular potential - art. no. 032716

An approach to bound and resonant states in scattering by a central potenti al gV(r), g is an element of C, based on a global analysis of S-matrix pole s, 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 su rface R-g((l)), into sheets and the construction of the Riemann sheets imag es 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 e ach state (l,n) of the quantum system one associates a sheet Sigma(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 proper ties 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 wa ves and due to the local degeneracy with respect to l, is discussed.