A new theory is presented for phase shifts resulting from scattering by atoms with attractive potentials, i.e., the threehyphen;turninghyphen;point problem. A simple formula, valid in and below the threehyphen;turninghyphen;point region is derivedeegr;equals;eegr;0minus;12Tanminus;1lcub;cosagr;sinbgr;sol;lsqb;lpar;minus;1rpar;nsinagr;minus;cosbgr;rsqb;rcub;plus;thgr;lsqb;lpar;minus;1rpar;nsinagr;minus;cosbgr;rsqb;12pgr;,where the angle agr; is related to the Stokes constant for the two outer turning points; bgr; is proportional to twice the action integral minus pgr; times the number of WBKJ wavefunction nodes in the inner region, (nminus;1); and tgr; is a unit step function which is zero for negative arguments. The theory predicts virtual energy states for incident energies in the threehyphen;turninghyphen;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.
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