  We demonstrate how S-matrix poles manifest themselves as the physical
spectrum near the upper threshold in the context of the two-channel uniformized
Mittag-Leffler expansion, an expression written as a sum of pole terms under an
appropriate variable where the S-matrix is made single-valued (uniformization).
We show that the transition of the spectrum is continuous as a S-matrix pole
moves across the boundaries of the complex energy Riemann sheets and that the
physical spectrum peaks at or near the upper threshold when the S-matrix pole
is positioned sufficiently close to it on the uniformized plane. There is no
essential difference on which sheet the pole is positioned. What is important
is the existence of a pole near the upper threshold and the distance between
the pole and the physical region, not on which complex energy sheet the pole is
positioned. We also point out that when the pole is close to the upper
threshold, the complex pole does not have the usual meaning of the resonance.
Neither the real part represents the peak energy, nor the imaginary part
represents the half width. Subsequently, we try to understand the current
status of $Z(3900)$ from the viewpoint of the uniformized Mittag-Leffler
expansion reflecting in particular, Phys.Rev.Lett.117, 242001 (2016) in which
they concluded that $Z(3900)$ is not a conventional resonance but a threshold
cusp. We point out that their results turn out to indicate the existence of
S-matrix poles near the $\bar D D^*$ threshold, which is most likely the origin
of the peak found in their calculation of the near-threshold spectrum. In order
to support our argument, we set up a separable potential model which shares
common behavior of poles near the $\bar D D^*$ threshold to the above-mentioned
reference and show in our model that the structures near the $\bar D D^*$
threshold are indeed caused by these near-threshold poles.
