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We demonstrate how S-matrix poles manifest themselves as the physicalspectrum near the upper threshold in the context of the two-channel uniformizedMittag-Leffler expansion, an expression written as a sum of pole terms under anappropriate variable where the S-matrix is made single-valued (uniformization).We show that the transition of the spectrum is continuous as a S-matrix polemoves across the boundaries of the complex energy Riemann sheets and that thephysical spectrum peaks at or near the upper threshold when the S-matrix poleis positioned sufficiently close to it on the uniformized plane. There is noessential difference on which sheet the pole is positioned. What is importantis the existence of a pole near the upper threshold and the distance betweenthe pole and the physical region, not on which complex energy sheet the pole ispositioned. We also point out that when the pole is close to the upperthreshold, the complex pole does not have the usual meaning of the resonance.Neither the real part represents the peak energy, nor the imaginary partrepresents the half width. Subsequently, we try to understand the currentstatus of $Z(3900)$ from the viewpoint of the uniformized Mittag-Lefflerexpansion reflecting in particular, Phys.Rev.Lett.117, 242001 (2016) in whichthey concluded that $Z(3900)$ is not a conventional resonance but a thresholdcusp. We point out that their results turn out to indicate the existence ofS-matrix poles near the $\bar D D^*$ threshold, which is most likely the originof the peak found in their calculation of the near-threshold spectrum. In orderto support our argument, we set up a separable potential model which sharescommon behavior of poles near the $\bar D D^*$ threshold to the above-mentionedreference and show in our model that the structures near the $\bar D D^*$threshold are indeed caused by these near-threshold poles.

Near-threshold Spectrum from Uniformized Mittag-Leffler Expansion -Pole Structure of $Z(3900)$-