Riemann Surfaces of Some Static Dispersion Models and Projective Spaces

We show that the analytic continuation of the $S$-matrix elements, which are meromorphic functions of the energy $\omega$ in the complex plane with the cuts $(-\infty ,-1]$, $[+1,+\infty )$, from the physical sheet to nonphysical ones results in a system of nonlinear difference equations. A global analysis of this system is performed in the projective spaces $P_{N}$ and $P_{N+1}$. We discuss the connection between the spaces $P_{N}$ and $P_{N+1}$ and obtain some particular solutions of the initial system.