Asymptotic Properties of the Wave Function for a Bound Nonrelativistic Three-Body System

Abstract

The asymptotic properties of a Schrödinger wave function which represents the bound ground state of a system of three interacting particles are examined. It is assumed that the interaction can be described by a static potential which is the sum of three two-body potentials and one three-body potential, where the potentials have the property that if any one of the particles is separated from the other two by a distance which tends to infinity, then the part of the potential energy which depends on the position of that particle tends to zero. The problem is treated nonrelativistically. Decreasing exponential bounds on the ground-state three-body wave function are established in configuration space. It is shown that these bounds depend only on the masses of the three particles, on the ground-state energy of the three-body system, and on the lower bounds on the spectra of the Hamiltonians for the three two-body systems arising when one of the three particles is removed and the remaining two interact through the corresponding two-body potential. It is furthermore shown that the three-body wave function in momentum space admits of analytic continuation into a tube-region in complex momentum space, which tube-region is the product of real momentum space and a convex bounded region in imaginary momentum space. The tube-region is explicitly determined. The implications of the results for the theory of the vertex function in quantum field theory are discussed. The relevancy of the results obtained in this paper to the variational computation of energy levels and wave functions of three-body systems is also briefly discussed. In the course of the derivation some generalizations are pointed out; in particular, the final results remain valid for a class of more general interactions, which cannot be described by static potentials. Likewise, the results remain valid for a class of spin-dependent interactions.