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 in the complex plane with the cuts (-,-1] and [+1,+), 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.