Remarks on Variational Bounds in Scattering Theory

The lower variational bound on the eigenphase shifts obtained by Sugar and Blankenbecler (SB) is shown to be formally equivalent to the upper variational bound ${\left({\mathbf{K}}^{-1\mathrm{}}\right)}_{\mathrm{up}}$ on the inverse of the reactance matrix ${\mathbf{K}}^{-1\mathrm{}}$ that had been obtained earlier. By formally equivalent we mean that if the identical trial function is used in the two formulations, the eigenphase shifts contained in ${\left({\mathbf{K}}^{-1\mathrm{}}\right)}_{\mathrm{up}}$ are identical with those determined by the SB formulation. A still more recent variational bound by Rosenberg (R) is shown to be identical, for most purposes, to the SB result, and therefore also formally equivalent to the original variational bound. The SB and R approaches may nevertheless have certain advantages, since they suggest different actual numerical procedures. A lower variational bound on ${\mathbf{K}}^{-1\mathrm{}}$ is obtained by means of the SB technique for determining upper bounds on the eigenphase shifts.