Logarithmic Sobolev inequalities and spectral concentration for the cubic Schrödinger equation

Pub Date : 2013-08-16 DOI:10.1080/17442508.2014.882924
G. Blower, C. Brett, I. Doust
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引用次数: 4

Abstract

The nonlinear Schrödinger equation , , arises from a Hamiltonian on infinite-dimensional phase space . For , Bourgain (Comm. Math. Phys. 166 (1994), 1–26) has shown that there exists a Gibbs measure on balls in phase space such that the Cauchy problem for is well posed on the support of , and that is invariant under the flow. This paper shows that satisfies a logarithmic Sobolev inequality (LSI) for the focusing case and on for all N>0; also satisfies a restricted LSI for on compact subsets of determined by Hölder norms. Hence for p = 4, the spectral data of the periodic Dirac operator in with random potential subject to 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 Korteweg–de Vries.
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对数Sobolev不等式和三次Schrödinger方程的谱浓度
非线性Schrödinger方程由无限维相空间上的哈密顿量产生。For, Bourgain (Comm. Math)。Phys. 166(1994), 1-26)证明了在相空间中球上存在一个吉布斯测度,使得柯西问题在的支持下被很好地提出,并且在流动下是不变的。本文证明了在聚焦情况下满足对数Sobolev不等式(LSI),在所有N>0的情况下满足对数Sobolev不等式;也满足由Hölder规范确定的紧子集的限制LSI。因此,当p = 4时,受随机势约束的周期狄拉克算子的谱数据集中在其均值附近。本文对希尔方程的谱数据作了类似的结论,当势是随机的,并服从Korteweg-de Vries的Gibbs测度。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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