Mutual transformation among bound, virtual and resonance states in one-dimensional rectangular potentials

A detailed analysis has been made by R.Zavin and N.Moiseyev(2004 J. Phys. A: Math, Gen, \textbf{37} 4619) for the change of bound states into resonance states via coalescence of virtual states in a one-dimensional symmetric rectangular attractive potential as it becomes shallow, with convergent wave functions of virtual and resonance states by the complex scaling method. As a complement to such an analysis, we discuss some global features of the pole spectrum of the S-matrix by using a complex extension of the real potential $V^{(\mathrm{real})}$ to $e^{i\alpha}V^{(\mathrm{real})}$ with a real phase $\alpha$. We show the structures of trajectories of poles developed for the change of $\alpha$ in the complex momentum plane, which is useful to understand the mutual transformation among the bound, virtual and resonance states.