The first roadblock in developing a statistical mechanics for focussing cubic Schrodinger (considered first on the circle of perimeter L) is the unfortunate fact that the Hamiltonian is unbounded from below, causing the Gibbs' prescribed canonical ensemble to be unnormalizable. This divergence prompted Lebowitz- Rose-Speer to introduce the micro-canonical ensemble by fixing the empirical mean-square = N. Note this makes good mechanical sense as the mean-square is preserved by the flow. Our interest lies in identifying the thermodynamic limit(s): take N = DL for fixed density D, and ask what comes out when L tends to infinity.

Numerical experiments carried out by Lebowitz et al were interpreted to indicate a change of phase in the limit; the ensemble favoring radiation/solitons at low/high values of $D$ or inverse temperature. McKean later put forward a proof that the full thermodynamic limit did not exist. We believe both are mistaken and plan to provide what evidence we have to the contrary. The advertised two guesses are that the micro- canonical ensemble collapses in the limit onto the unit mass on the zero path with fluctuations resembling a White Noise.