Riemann Surfaces of Some Static Ispersion Models and Projective Spaces

The S-matrix in the static limit of a dispersion relation has a finite order N and is a matrix of meromorfic functions of energy ω in the plane with cuts ( −∞ , − 1] , [+1 , + ∞ ). In the elastic case it reduces to N functions S i ( ω ) connected by the crossing symmetry matrix A. The problem of analytical continuation of S i ( ω ) from the physical sheet to unphysical ones can be treated as a nonlinear system of difference equations. It is shown that a global analisis of this system can be carried out effectively in projective spaces P N and P N +1 . The connection between spases P N and P N +1 is discussed.