Bounds on Scattering Phase Shifts: Static Central Potentials

It has recently been shown that rigorous upper bounds on scattering lengths can be obtained by adding to the Kohn variational expression certain integrals involving approximate wave functions for each of the negative-energy states. For potentials which vanish identically beyond a certain point, it is possible to extend the method to positive-energy scattering; one obtains upper bounds on ${(-kcot\eta )}^{-1\mathrm{}}$, where $\eta$ is the phase shift. In addition to the negative-energy states one must now take into account a finite number of states with positive energies lying below the scattering energy. The states in this associated energy eigenvalue problem are defined by the imposition of certain boundary conditions on the wave functions. A second approach, involving an associated potential-strength eigenvalue problem, is also used. The second method includes the first as a special case and, more significantly, can be extended to scattering by compound systems. If some states are not accounted for, a bound on $cot\eta$ is not obtained; nevertheless it is still possible to obtain a rigorous lower bound on $\eta$. Upper bounds on $\eta$ may also be obtained, but in a way which is probably not too useful for many-body scattering problems.