A Martin boundary connected with the ∞-volume limit of the focussing cubic Schrödinger equation

Summary

The existence of a change of phase in the micro-canonical ensemble for the focussing cubic Schrödinger system was suggested by Lebowitz-Rose-Speer [1989]. Chorin [1994] disputes their numerical evidence; his own is based on a more sophisticated approximation to the micro-canonical distribution and leads him to the opposite conclusion. Perhaps the source of this contradictory testimony is the fact, proved here, that the ∞-volume limit does not exist at any temperature 0 < T < ∞ or density 0 < D < ∞. This does not preclude a more or less dramatic change in the ensemble, from high to low temperatures, but it does guarantee the existence of several distinct ∞-volume Gibbs states. These are related to a sort of “boundary” of the type introduced by Martin [1941] for the description of classical harmonic functions in general 3-dimensional regions, as will be explained below.

This work was performed at the Courant Institute of Mathematical Sciences with the partial support of the National Science Foundation, under NSF Grant No. DMS-9112664 which is gratefully acknowledged.