Variational bounds from matrix Padé approximants in potential scattering

We fully develop the content of the Schwinger variational principle for the phase shifts in potential scattering. We introduce matrix Padé approximations built up from the perturbation expansion of the Green's function. They appear to lead to a new type of (Padé) approximation when optimized through the variational principle. These new approximations, which are no longer rational fractions in the expansion parameter, appears to have the full analytical richness of the exact solution. For the case of a nonchanging-sign potential these new types of approximations provide the best bounds to the phase shifts and bound states. The extension to arbitrarily singular potentials is also discussed. A numerical example confirms the extreme efficiency of the method Typically, for values of the coupling giving rise to one or two bound states the phase shifts are obtained within ${10}^{-3\mathrm{}}$ of their exact values, and this on the full range of energy, by including only the first and second Born terms of the perturbation series.