2.3 Zero crossings

The consequences of these results for the signs of R_l1,l2(E₁,E₂)can be most easily understood by looking at the first quadrant of the schematic illustration in Figure 7.1. From the result for the signs in the high energy limiting case described above we see that if R_l1,l2(E^-,E^-) < 0 for some choice of E^-, since R_l1,l2is continuous, there must be at least one zero (or more generally, an odd number of zero crossings) along any path, confined to the first quadrant without crossing the soft photon line, connecting (E^-,E^-)to $(E_1\rightarrow\infty,E^-)$. Similarly, if R_l1,l2(E⁺,E⁺) > 0then we predict an even number of zero crossings (possibly none) along such a path.

While the exact trajectory (or trajectories) in the (E₁,E₂) plane of the curve(s) of zero crossings probably requires detailed calculation, based on our previous discussion we can see that it is required that one such trajectory intersect any soft photon zero crossing. Thus a zero in the soft photon matrix element must persist in bremsstrahlung away from the soft photon limit. (How far is yet unknown; results from bound-free transitions suggest that this depends on the choice of potential and change of angular momentum in the transition [23].) Closed loops of zero crossings (or paths for which both ends extend out of the first quadrant) are allowed but paths which end in the first quadrant away from the the soft photon limit are not. We illustrate this in Figure 7.2.

We remember, as noted above, that Levinson's theorem for short range potentials which do not support l=0 virtual bound states at threshold requires a sign change in the soft photon matrix element R_l1,l2(E,E) for E>0 if the condition $\left\vert n_{l_1}-n_{l_2} \right\vert > 1$ on the difference of the number of bound states is satisfied. Similarly, the generalization to quantum defects for ions requires a soft photon zero crossing for E>0 if $\left\vert\mu_{l_<}(0)-\mu_{l_>}(0) + 1/2\right\vert \ge 1$. Thus in these cases an energy E^- where R_l1,l2(E^-,E^-) < 0 must exist and, since $R(E^-,\infty)>0$, there must be zero crossings in the bremsstrahlung matrix element R(E^-,E₂) for some E₂.

In the theory of bound-free transitions for a neutral atom (all states in a potential with an asymptotic ionic Latter tail) it is well known that there can be a sequence of zeros in the matrix elements for transitions between Rydberg states and the continuum. In Figure 7.1 we have included such a sequence (in the second quadrant) for illustrative purposes. One can anticipate a continuation of such a curve of zero matrix elements into the free-free regime for such an ionic potential.