2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/172775Searching for infrastructure of the quantum mechanical system, we study trajectories of the s-wave poles of the S-matrix element with respect to a real phase $α$ in the complex momentum plane for a complex extension of real potentials by a phase factor $e^{iα}$. This complex extension relates the pole spectrum of the physical system with a potential to the spectrum of another system with the potential of the same shape but of opposite sign. There appear trajectories with the periodicity of $2π$, $4π$, and $\infty$. The appearance of non-recurrent behavior of the trajectory for the change of phase $Δα=2π$ is clearly related with the existence of resonance poles for real repulsive potentials. Dynamical changes of trajectory structure are examined.10 pages, 7 figuresQuantum Physicstext