Validity of the Phase-Shift Representations

Our previous discussion on the validity of the phase-shift representation is further developed. An example treated in a previous work based on a separable-potential model is given in an improved method to show that such a representation fails in more general cases than those considered recently by Bander, Coulter, and Shaw, based on a simplified $\frac{N}{D}$ model. It is also shown that ${\delta }_I\mathrm{}\left(s\right)\mathrm{}$, the imaginary part of the elastic phase shift, is a boundary value of a function which connects the elastic $S$-matrix element in the physical sheet to that in other Riemann sheets. Consequently, ${\delta }_I\mathrm{}\left(s\right)\mathrm{}$ should be so chosen as to satisfy certain mathematical conditions as well as to fit experiments. The simplest example of such a function is given.