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.