Gibbs measures based on 1D (an)harmonic oscillators as mean-field limits

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.