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

  • Henry P. McKean


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.


Schrodinger Equation Gibbs State Formal Density Martin Boundary Positive Harmonic Function 
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Copyright information

© Springer-Verlag Tokyo 1996

Authors and Affiliations

  • Henry P. McKean
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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