Zeros of the scattering amplitude

Problem of the zeroes of elastic forward scattering amplitude is studied in the case when the amplitude possesses the unphysical region and arbitrary number of poles. With the aid of the argument principle, the formulas for calculating the number of zeroes are obtained. It is shown that the number of zeroes depends on the sign in the asymptotics of the expression $A (E)$ which is possible to find using either the data on the total cross-section in the low-energy region or any suggestions about the high-energy behaviour of the amplitude. In particular, the $\pi N$ and $KN$-amplitudes are considered.