Bounds for a Class of Bethe-Salpeter Amplitudes

For a certain wide class of kernels involving trilinear coupling of scalar particles, the absorptive part of the Bethe-Salpeter amplitude for forward scattering is bounded from above and below. The bounds are expressed in the form $B_1{s}^{{\alpha }_1}<~A\left(s\right)\mathrm{}<~B_2{S}^{{\alpha }_2}$, where $s$ is the squared c.m. energy and $B_1$ and $B_2$ are positive constants. Expressions for the exponents ${\alpha }_1$ and ${\alpha }_2$ are given as functions of the coupling constant $g$. For the straight ladder model, ${\alpha }_1$ and ${\alpha }_2$ coincide for all values of $g$, the common expression agreeing with an exact result of Nakanishi. For the more complicated models, ${\alpha }_1$ and ${\alpha }_2$ do not in general coincide. However, in the strong-coupling limit $g\to \infty$, we find that $\frac{{\alpha }_2}{{\alpha }_1}\to 1$; moreover, the common asymptotic behavior ${\alpha }_{1,2}{\to }_{g\to \infty }\frac{g}{4\pi m}$ is the same for all the models, including the straight-ladder model.