Zeros of the Partial-Wave Scattering Amplitude

Under the general assumptions of analyticity, unitarity, temperedness, and normal threshold behavior, the relation between the number of zeros and the high-energy behavior of the partial-wave scattering amplitude is studied. If the left-hand-cut discontinuity makes a finite number of sign changes, the maximum and minimum numbers of zeros can be determined by the high-energy upper and lower bound of the partial-wave respectively. In particular, for $C{\mathrm{}\left|s\right|\mathrm{}}^{-1+\epsilon }<\left|f_l\mathrm{}\right(s\left)\right|<C$, the number of zeros is determined. After having determined the number of zeros, the number of subtractions as well as the sign of the scattering length is obtained for given number and character of zeros of the left-hand discontinuity $\Delta f_l\mathrm{}\left(s\right)\mathrm{}$.