Variational bounds on transition amplitudes

Upper and lower variational bounds on the transition amplitudes of the impact-parameter model for ion-atom scattering were recently reported. A comparison of these bounds with those given by Spruch leads to an improved first-order variational bound on the second-order error term in the variational principle recently given. In addition, the second-order variational bound obtained by Spruch from an apparently different second-order error term also follows from this second-order error term. Finally, as Spruch has suggested, the second-order bound is simply given as one-half the first-order bound squared for those amplitudes corresponding to a rearrangement from state $n$ of one arrangement channel to state $n$ of another arrangement channel.