Remove the logarithmic loss in the two-cluster asymptotic error estimate

Determine whether the logarithmic factor in the quantitative convergence error for the asymptotic factorization of an N-particle atomic ground state in the expanding two-cluster regions can be removed.

Background

The main theorem establishes an error of order O(r_N^{-2+\alpha}f_\alpha(r_N)) for the approximation of the N-particle ground state by the product of the (N−1)-particle ground state and the one-particle factor u(x_N), where f_\alpha(r_N)=\ln r_N for 0<\alpha<1 and f_0(r_N)=(\ln r_N)^2. The paper explains that one logarithmic factor arises from the technical choice t=R\ln|x_N| in an intermediate estimate. It therefore leaves unresolved whether this logarithmic loss is an artifact of the proof and can be eliminated.