Extinct Ghosts in Potential Theory

To resolve the difficulties that arise if the $P$ and/or $P^{\prime }$ Regge trajectories pass through $J=0\mathrm{}$ at a negative value of the center-of-mass energy squared, Chew has conjectured that the determinant of the physical $D$ matrix does indeed vanish, but that the $N$ matrix is such as to lead to vanishing residues at the pole. We investigate whether this phenomenon of a simultaneous zero of the $N$ and $D$ functions can occur in potential theory. Standard arguments exclude this possibility for sufficiently well-behaved potentials. However, it is easy to explicitly construct amplitudes which do involve a coincident zero. Using the Gel'fand-Levitan-Marchenko equations, we derive a representation for the potential in terms of the Fredholm determinant of the integral operator that appears in the $\frac{N}{D}$ equations. We show that if the $s$-wave amplitude has coincident zeros, the corresponding potential behaves like $\frac{1}{r^2}$ near the origin; conversely, such potentials give rise, in general, to coincident zeros. However, these zeros are unrelated to any Regge trajectory, so that (except perhaps for potentials which diverge more strongly than $\frac{1}{r^2}$ at the origin) the phenomenon hypothesized by Chew cannot occur in potential theory.