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

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.