Low-Energy Scattering by a Compound System: Positrons on Atomic Hydrogen

The formalism, developed by Kato, which gives upper and lower bounds on the phase shift for the scattering of a particle by a center of force has been generalized to apply to scattering by a compound system. Particularly simple and useful results are obtained for the case of zero energy scattering where no composite bound state exists. As a first example, because of its relative simplicity, the problem of low-energy positron scattering from atomic hydrogen, with zero total orbital angular momentum, is studied. It is shown that at zero energy the ordinary Kohn variational calculation, which ignores second order contributions, provides an upper bound on the scattering length, from which a bound on the cross section is deduced. For nonzero energies a bound on the phase shift may similarly be obtained, but for a fictitious problem with cutoff potentials. If the energy is sufficiently small (less than 3 ev, say) the error thus incurred is expected to be negligibly small. Numerical calculations performed at $k=0\mathrm{}$ and $k{a}_0=0.2\mathrm{}$ lead to the result that at these energies the effects of polarization are large enough to cause the positron to be on the whole attracted to the hydrogen atom.