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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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<mark>Journal publication date</mark>2014
<mark>Journal</mark>Stochastics
Issue number6
Volume86
Number of pages12
Pages (from-to)870-881
Publication StatusPublished
Early online date19/03/14
<mark>Original language</mark>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