{"title":"Logarithmic Sobolev inequalities and spectral concentration for the cubic Schrödinger equation","authors":"G. Blower, C. Brett, I. Doust","doi":"10.1080/17442508.2014.882924","DOIUrl":null,"url":null,"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.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2013-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1080/17442508.2014.882924","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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.