Jump phenomenon induced by potential strength variation and the influence of exotic resonant states

The jump phenomenon between two S-matrix poles induced by a small potential strength variation is studied in the framework of the nonrelativistic scattering by a central potential gV(r), g∈C. Let R_g^(l) denote the Riemann surface over the complex g plane, on which the pole function k=k^(l)(g) is single valued and analytic. By associating a sheet Σ_n^(l) of R_g^(l) and its k-plane image Σ_n^′(l) to a state with the quantum numbers (l,n), the jump between the states (l,n) and (l,m), induced by the potential strength variation, is understood as the jump between the sheet images Σ_n^′(l) and Σ_m^′(l). The rules for the jumps (l,n)(l,m) as a consequence of a small potential strength variation are deduced. The influence of the exotic resonant states on the selection rules for the jumps is discussed. The occurrence of the local degeneracy with respect to l of the resonant levels (l-1,n), (l,m), and (l+1,p) for the rectangular central potential and the level stability with respect to the potential strength variation are also discussed.