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Rights statement: The final, definitive version of this article has been published in the Journal, Stochastics, 86 (6), 2014, © Informa Plc

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## Logarithmic Sobolev inequalities and spectral concentration for the cubic Shrödinger equation

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Journal publication date 2014 Stochastics 6 86 12 870-881 Published 19/03/14 English

### 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.

### Bibliographic note

Submitted for publication, and posted on ArXiv 1308:3649 The final, definitive version of this article has been published in the Journal, Stochastics, 86 (6), 2014, © Informa Plc