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We consider the one dimensional cubic nonlinear Schr{\"o}dinger equation withtrapping potential behaving like |x| s (s > 1) at infinity. We construct Gibbsmeasures associated to the equation and prove that the Cauchy problem isglobally well-posed almost surely on their support. Consequently, the Gibbsmeasure is indeed invariant under the flow of the equation. We also address theconstruction and invariance of canonical Gibbs measures, conditioned on the L 2mass.

Invariant Gibbs measures for 1D NLS in a trap