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The $N/D$ method is used to study the $S_{11}$ channel low energy $\pi N$scattering amplitude. The input of left cuts are obtained from variousphenomenological models. With the aid of the production representation, thetotal phase shifts can be decomposed into different contributions, and itfurther reveals that the existence of subthreshold resonance $N^*(890)$ doesn'tdepend on the details of the dynamical input. Additionally, it is found thatthere exist virtual states in partial waves, which are induced by the $u$channel nucleon exchanges. These virtual states accumulate at the end point ofthe $u$ channel segment cut. The end point is hence the essential singularityof the full amplitude on the second sheet of complex $s$ plane.

The $N/D$ study on the singularity structure of $\pi N$ scattering amplitudes