Phase Representation and High-Energy Behavior of the Forward Scattering Amplitude

By making use of the phase representation, the relation between the high-energy behavior of the symmetric forward scattering amplitude $F\left(x\right)\mathrm{}$ and the asymptotic properties of the ratio $cot\delta =\frac{\mathrm{Re}F}{\mathrm{Im} F}$ is discussed. Starting assumptions are dispersion relations and the Greenberg-Low bound. Lower bounds as well as upper bounds are derived. Under the assumption of the Froissart bound, it is shown that $cot\delta$ cannot stay indefinitely greater than an arbitrarily small positive number. Also if the total cross section decreases steadily to a finite limit, but slower than $\frac{\mathrm{const}}{E}$ the real part must tend to $-\infty$. The results are discussed in connection with those of Khuri and Kinoshita. The unsymmetric case is also treated by the same methods.