Riemann surface approach to bound and resonant states for a nonlocal potential

Abstract

The Riemann surface approach to bound and resonant states is extended to the case of a separable nonlocal potential that is constant on a certain domain of the inner region and vanishes in the rest of the domain. The approach consists in the construction of the Riemann surface \documentclass[12pt]{minimal}\begin{document}$R_g$\end{document}Rg of the S-matrix pole function k = k(g) over the g-plane, where g is the strength of the complex nonlocal potential. On the Riemann surface \documentclass[12pt]{minimal}\begin{document}$R_g$\end{document}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 determined and analyzed as a function of the position of the region of nonlocality. The Riemann surface of the S-matrix pole function is constructed. According to the Riemann surface approach to each bound or resonant state a sheet of the Riemann surface \documentclass[12pt]{minimal}\begin{document}$R_g$\end{document}Rg is associated. All the natural modes (bound and resonant states) of the system are identified and treated in a unified way. The nonlocal potential generates narrow resonant states that cannot be produced by a local potential.