Nodal structure and phase shifts of zero-incident-energy wave functions: Multiparticle single-channel scattering

For potential scattering, with δL(k) the phase shift modulo {π} for an incident wave number k, Levinson's theorem gives δL(0)-δL({∞}) in terms of NL, the number of bound states of angular momentum L, for δL(k) assumed to be a continuous function of k. NL also determines the number of nodes of the zero-energy wave function uL(r). A knowledge of the nodal structure and of the absolute value of δL(0) is very useful in theoretical studies of low-energy potential scattering. Two preliminary attempts, one formal and one ``physical,'' are made to extend the above results to single-channel scattering by a compound system initially in its ground state. The nodal structure will be of greater interest to us here than an extension of Levinson's theorem.The formal approach is applied to e⁺-H and e⁺-He scattering. Both H and He have zero orbital angular momentum and a nodeless ground-state wave function ψT. An effective one-body wave function uL for the positron incident with zero kinetic energy can be constructed by factoring out the spin and Euler-angle dependence of the full scattering wave function {Ψ} and projecting the remaining ``radial'' function RL onto ψT.The nodal surfaces of RL are shown to divide configuration space into at most NL+1 subdomains, where NL is the number of composite bound states of the given L. Since NL=0 for e⁺-H and e⁺-He, it follows that uL is nodeless and that δL(0)=0, for all L. Partial but useful information on the nodal structure of {Ψ} for e^--H scattering is also deduced. Interestingly, nodal surfaces exist which are not consistent with a naive generalization of (bound-state) one-dimensional Sturm-Liouville theory.The physical arguments, based on the qualitative concept of an effective central potential seen by each target particle and by the incident particle P, strengthen a previous surmise on the value of δLJ(0) for e^±-atom scattering and for the scattering of neutrons or protons by a heavy nucleus; L and J are the quantum numbers of the incident P, and ψT is assumed to have zero spin and zero orbital angular momentum, but ψT need not be nodeless, and P need not be distinguishable. Roughly, the surmise is that δLJ(0)=KLJ{π}, where, for the given L and J, KLJ is the number of composite bound states plus the number of one-particle states excluded by the Pauli principle.