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By F Calogero
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19) 41 BOUNDS O N THE SCATTERING PHASE SHIFT These bounds may be compared with that provided by Wigner’s theorem [29], 6, 2 -R (20) which is valid, however, only for potentials that vanish identically beyond the range R and only at high energy (if bound states are present; otherwise it holds at all energies [30]). Moreover, Wigner’s theorem provides only a lower bound on the phase shift; this bound depends on the range of the potential, not on its strength. T h e bounds given here also depend on the strength of the potential, and are obviously more stringent than Wigner’s bound if the energy is sufficiently large.
Phase function for an attractive potential. I S r 24 CHAPTER 5 possible, in general, to infer from the values of the scattering phase shift whether a bound state does or does not exist. However, by examining the behavior of the phase shift as a function of k and of 1 and by comparing it with graphs such as those discussed here and below, one may try to build up a rough image of the potential. I n comparing the results for different values of the angular momentum 1 it should be noted that different scales have been used in Figs.
7. D-Wave phase function for a repulsive potential. 8. Phase function for a repulsive potential. 0 THE PHASE FUNCTION. EXAMPLES 27 similar in shape to the attractive one discussed above, except that its range is shorter and its strength greater; but of course the behavior of the phase function is very different, especially at small k, because the interaction has changed from attractive to repulsive. However, there are certain common features, notably that at short distances the phase function increases with k and that the accumulation of the phase begins at larger values of r the larger the value of I and the smaller the value of k.