On " Do the Attractive Bosons Condense? " by Typeset Using Revt E X

Using Perron-Frobenius theorem, we prove that the results by Wilkin, Gunn and Smith [1] for the ground states of rotating N Bose atoms in a harmonic atomic trap with frequency ω interacting via attractive δ2(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 〈z̄1z̄2V zm 2 zn 1 〉 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≪ω−1/2 interactions satisfying inequality ∫ d2~rV (r) < 0. There is no condensation, and the angular momentum is concentrated in the collective “center-of-mass” mode. PACS numbers: 03.75.Fi, 32.80.Pj, 67.40.Db, 03.65.Fd Typeset using REVTEX 1 Recently, Wilkin, Gunn and Smith have shown in Ref. [1] that the minimum energy states at given angular momentum L of a system of N bosons in a 2d spherically symmetric harmonic trap with frequency ω= 1, interacting via attractive δ(r)-forces have, in the weak coupling limit, the form ΨL = G ( k=N