Complex extension of potentials and trajectories of poles of the S-matrix element in the complex momentum plane

TL;DR

This paper investigates how the poles of the S-matrix in the complex momentum plane evolve under a complex phase extension of potentials, revealing new periodic and non-recurrent behaviors linked to resonance phenomena.

Contribution

It introduces a complex extension of potentials with a phase factor to analyze pole trajectories, uncovering novel periodicities and behaviors in the pole spectrum.

Findings

Trajectories exhibit 2π, 4π, and infinite periodicities.

Non-recurrent trajectories are linked to resonance poles in repulsive potentials.

Dynamical changes in trajectory structures are characterized.

Abstract

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.