Finiteness of the Number of Positive-α Landau Surfaces in Bounded Portions of the Physical Region

It is shown that if the spectrum of physical particle rest masses contains neither accumulation points nor the zero point, then the number of different positive‐α Landau surfaces that enter any bounded portion of the physical region of any multiple‐particle scattering process is finite. This implies that if the physical‐region singularities of scattering functions are confined to the closure of the set of points lying on positive‐α Landau surfaces, then the scattering functions are analytic at almost all points of the physical region. The proof is made by proving an equivalent property of systems of classical point particles scattering via point interactions.