  Searching 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 $\alpha$ in the complex momentum plane for a complex extension of real
potentials by a phase factor $e^{i\alpha}$. 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 $2\pi$, $4\pi$, and $\infty$. The
appearance of non-recurrent behavior of the trajectory for the change of phase
$\Delta \alpha=2\pi$ is clearly related with the existence of resonance poles
for real repulsive potentials. Dynamical changes of trajectory structure are
examined.
