Blower, Gordon and Brett, Caroline and Doust, Ian
(2014)
*Logarithmic Sobolev inequalities and spectral concentration for the cubic Shrödinger equation.*
Stochastics, 86 (6).
pp. 870-881.
ISSN 1744-2508

## Abstract

The nonlinear Schr\"odinger equation $\NLSE(p, \beta)$, $-iu_t=-u_{xx}+\beta \vert u\vert^{p-2} u=0$, 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 $\Omega_N= \{ \phi\in \Lp^2(\mT) \,:\, \Vert \phi \Vert^2_{\Lp^2} \leq N\}$ 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.