Semiclassical Theory of the Three-Turning-Point Problem

A new theory is presented for phase shifts resulting from scattering by atoms with attractive potentials, i.e., the three-turning-point problem. A simple formula, valid in and below the three-turning-point region is derived η=η0−12Tan−1{cosα sinβ/[(−1)nsinα−cosβ]}+θ[(−1)nsinα−cosβ]12π,where the angle α is related to the Stokes constant for the two outer turning points; β is proportional to twice the action integral minus π times the number of WBKJ wavefunction nodes in the inner region, (n−1); and τ is a unit step function which is zero for negative arguments. The theory predicts virtual energy states for incident energies in the three-turning-point range whose location compares well with numerical quantum calculations. The resonance portion of the phase shifts shows only semiquantitative comparison with the quantum results.