Logarithmic Sobolev inequalities and spectral concentration for the cubic Schrödinger equation

Abstract

The nonlinear Schrödinger equation NLSE(p, β), -iu_t=-u_{xx}+β| 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 μ^β_N on balls Ω_N= {ϕ\in \Lp^2(\mT) : | ϕ|^2_{\Lp^2} \leq N} in phase space such that the Cauchy problem for NLSE(p,β) is well posed on the support of μ^β_N, and that μ^β_N is invariant under the flow. This paper shows that μ^β_N satisfies a logarithmic Sobolev inequality for the focussing case β<0 and 2\leq p\leq 4 on Ω_N for all N>0; also μ^β satisfies a restricted LSI for 4\leq p\leq 6 on compact subsets of Ω_N determined by Hölder norms. Hence for p=4, the spectral data of the periodic Dirac operator in \Lp^2(\mT; \mC^2) with random potential ϕsubject to μ^β_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.

This is the authors' manuscript version (11 pages). An improved version appears in Stochastics An International Journal of Probability and Stochastic Processes