On "Do the attractive bosons condense?" by N. K. Wilkin, J. M. F. Gunn, R. A. Smith

Original · EN

Using Perron-Frobenius theorem, we prove that the results by Wilkin, Gunn and Smith [1] for the ground states of N Bose atoms rotating at the angular momentum L in a harmonic atomic trap with frequency omega interacting via attractive delta²(r) forces, are valid for a broad class of predominantly attractive interactions V(r), not necessarily attractive for any r. The sufficient condition for the interaction is that all the two-body matrix elements <z₁ᵏ z₂ˡ |V| z₂ᵐ z₁ⁿ> allowed by the conservation of angular momentum k+l = m+n, are negative. This class includes, in particular, the Gaussian attraction of arbitrary radius, -1/r - Coulomb and log(r)-Coulomb forces, as well as all the short-range R << omega⁻¹/² interactions satisfying inequality int d²r V(r) < 0. There is no condensation at L>> 1, and the angular momentum is concentrated in the collective ``center-of-mass'' mode.