Asymptotic Behavior of the Inelasticity Parameter

We examine the asymptotic behavior of the inelasticity parameter $\eta$ under the assumption that an infinite number of 2-particle channels open up as energy becomes infinite. We investigate in detail three specific models, which can be handled analytically, and find that in general $\eta \left(\infty \right)=1\mathrm{}$ in these models. We speculate that this may in fact be the general asymptotic behavior of $\eta$ in the physical situations. We also observe that a purely imaginary elastic-scattering amplitude can be realized even as $\eta \to 1$.