Logarithmic Sobolev inequalities and spectral concentration for the cubic Schr\"odinger equation

TL;DR

This paper proves logarithmic Sobolev inequalities for Gibbs measures associated with the nonlinear Schrödinger equation, leading to spectral data concentration results for random potentials in certain regimes.

Contribution

It establishes logarithmic Sobolev inequalities for Gibbs measures of the NLSE in the focusing case, enabling spectral data concentration for random potentials.

Findings

LSI holds for eta<0, 2 extless p extless 4 on \Omega_N

Restricted LSI for 4 extless p extless 6 on compact subsets

Spectral data concentrate near mean values for random potentials

Abstract

The nonlinear Schr\"odinger equation NLSE(p, \beta), -iu_t=-u_{xx}+\beta | u|^{p-2} u=0, arises from a Hamiltonian on infinite-dimensional phase space \Lp^2(\mT). For p\leq 6, Bourgain (Comm. Math. Phys. 166 (1994), 1--26) has shown that there exists a Gibbs measure \mu^{\beta}_N on balls \Omega_N= {\phi \in \Lp^2(\mT) : | \phi |^2_{\Lp^2} \leq N} in phase space such that the Cauchy problem for NLSE(p,\beta) is well posed on the support of \mu^{\beta}_N, and that \mu^{\beta}_N is invariant under the flow. This paper shows that \mu^{\beta}_N satisfies a logarithmic Sobolev inequality for the focussing case \beta <0 and 2\leq p\leq 4 on \Omega_N for all N>0; also \mu^{\beta} satisfies a restricted LSI for 4\leq p\leq 6 on compact subsets of \Omega_N determined by H\"older norms. Hence for p=4, the spectral data of the periodic Dirac operator in \Lp^2(\mT; \mC^2) with random potential \phi subject to \mu^{\beta}_N are concentrated near to their mean values. The paper concludes with a similar result for the spectral data of Hill's equation when the potential is random and subject to the Gibbs measure of KdV.