A Martin boundary connected with the ∞-volume limit of the focussing cubic Schrödinger equation
The existence of a change of phase in the micro-canonical ensemble for the focussing cubic Schrödinger system was suggested by Lebowitz-Rose-Speer . Chorin  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  for the description of classical harmonic functions in general 3-dimensional regions, as will be explained below.
KeywordsSchrodinger Equation Gibbs State Formal Density Martin Boundary Positive Harmonic Function
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- 1.Bourgain, J.: The Fourier transform restriction phenomena for certain lattice subsets and applications to non-linear evolution equations. IAS preprint (1994), to appear 1995.Google Scholar
- 2.Chorin, A.: Univ. CA, Berkeley preprint (1994).Google Scholar
- 3.Krylov, N. V., Non-Linear Elliptic and Parabolic Equations of Second Order, Riedel, Boston, Tokyo, 1987.Google Scholar
- 4.Lebowitz, J H. Rose, and E Speer: Statistical mechanics of the non-linear Schrödinger equation (2). J. Stat. Phys 54 (1989), 17–56.Google Scholar
- 7.McKean, H. P. and K. Vaninsky: Statistical mechanics of non-linear wave equations: the petit and micro-canonical ensembles. Trends and Perspective Appl. Math., ed. L. Sirovich, Springer-Verlag, New York, 1994.Google Scholar
- 8.Statistical mechanics of non-linear wave equations (4): cubic Schrödinger. Comm. Math. Phys. 168 (1995), 479–491.Google Scholar
- 9.Williams, D.: Diffusions, Markov Processes, and Martingales. J. Wiley & Sons, New York, 1979.Google Scholar