  The poles of the quantum scattering matrix (S-matrix) in the complex momentum
plane have been studied extensively. Bound states give rise to S-matrix poles,
and other poles correspond to non-normalizable anti-bound, resonance and
anti-resonance states. They describe important physics, but their locations can
be difficult to find. In pioneering work, Nussenzveig performed the analysis
for a square well/wall, and plotted the flow of the poles as the potential
depth/height varied. More than fifty years later, however, little has been done
in the way of direct generalization of those results. We point out that today
we can find such poles easily and efficiently, using numerical techniques and
widely available software. We study the poles of the scattering matrix for the
simplest piece-wise flat potentials, with one and two adjacent (non-zero)
pieces. For the finite well/wall the flow of the poles as a function of the
depth/height recovers the results of Nussenzveig. We then analyze the flow for
a potential with two independent parts that can be attractive or repulsive, the
two-piece potential. These examples provide some insight into the complicated
behavior of the resonance, anti-resonance and anti-bound poles.
