Optimal integrability threshold for Gibbs measures associated with focusing NLS on the torus

We study an optimal mass threshold for normalizability of the Gibbs measures associated with the focusing mass-critical nonlinear Schrodinger equation on the one-dimensional torus. In an influential paper, Lebowitz et al. (J Stat Phys 50(3-4):657-687, 1988) proposed a critical mass threshold given by the mass of the ground state on the real line. We provide a proof for the optimality of this critical mass threshold. The proof also applies to the two-dimensional radial problem posed on the unit disc. In this case, we answer a question posed by Bourgain and Bulut (Ann Inst H Poincare Anal Non Lineaire 31(6):1267-1288, 2014) on the optimal mass threshold. Furthermore, in the one-dimensional case, we show that the Gibbs measure is indeed normalizable at the optimal mass threshold, thus answering an open question posed by Lebowitz et al. (1988). This normalizability at the optimal mass threshold is rather striking in view of the minimal mass blowup solution for the focusing quintic nonlinear Schrodinger equation on the one-dimensional torus.

Automated summary

The study focuses on the optimal mass threshold for normalizability of Gibbs measures associated with the mass-critical nonlinear Schrodinger equation on a one-dimensional torus. The proof of optimality for the critical mass threshold also applies to a two-dimensional radial problem. Moreover, it is shown that the Gibbs measure is normalizable at the optimal mass threshold in the one-dimensional case, addressing an open question posed by Lebowitz et al.