We prove that Gibbs measures based on 1D defocusing nonlinear Schr{\"o}dinger
functionals with sub-harmonic trapping can be obtained as the mean-field/large
temperature limit of the corresponding grand-canonical ensemble for many
bosons. The limit measure is supported on Sobolev spaces of negative regularity
and the corresponding density matrices are not trace-class. The general proof
strategy is that of a previous paper of ours, but we have to complement it with
Hilbert-Schmidt estimates on reduced density matrices.Comment: Minor changes and precision
