Amplitude Bounds in the Ladder Graph Approximation

We consider forward scattering in the ladder approximation for a trilinear scalar interaction. The corresponding Bethe-Salpeter integral equation for the absorptive part of the amplitude is of the Volterra type; and the kernel and inhomogeneous term are both positive. We exploit these special features in order to set upper bounds on the absorptive amplitude for arbitrary values of the coupling constant $g$. Two different techniques are described. For large scattering energies the bounds obtained imply corresponding bounds on the value of the leading Regge pole $\alpha \mathrm{}\left(0\right)\mathrm{}$. In the limit of weak coupling our upper bound on $\alpha \mathrm{}\left(0\right)\mathrm{}$ is linear in $g^2$ and in fact coincides exactly with the known weak-coupling result. In the limit of strong coupling our upper bound varies as the square root of $g^2$. The correctness of this feature is discussed on the analogy with the Schrödinger problem of binding in a potential field.