{"title":"三维三次NLS在能量空间中的定量推导和散射","authors":"Xuwen Chen, Justin Holmer","doi":"10.1007/s40818-022-00126-5","DOIUrl":null,"url":null,"abstract":"<div><p>We consider the derivation of the defocusing cubic nonlinear Schrödinger equation (NLS) on <span>\\({\\mathbb {R}}^{3}\\)</span> from quantum <i>N</i>-body dynamics. We reformat the hierarchy approach with Klainerman-Machedon theory and prove a bi-scattering theorem for the NLS to obtain convergence rate estimates under <span>\\(H^{1}\\)</span> regularity. The <span>\\(H^{1}\\)</span> convergence rate estimate we obtain is almost optimal for <span>\\(H^{1}\\)</span> datum, and immediately improves if we have any extra regularity on the limiting initial one-particle state.</p></div>","PeriodicalId":36382,"journal":{"name":"Annals of Pde","volume":"8 2","pages":""},"PeriodicalIF":2.4000,"publicationDate":"2022-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40818-022-00126-5.pdf","citationCount":"7","resultStr":"{\"title\":\"Quantitative Derivation and Scattering of the 3D Cubic NLS in the Energy Space\",\"authors\":\"Xuwen Chen, Justin Holmer\",\"doi\":\"10.1007/s40818-022-00126-5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We consider the derivation of the defocusing cubic nonlinear Schrödinger equation (NLS) on <span>\\\\({\\\\mathbb {R}}^{3}\\\\)</span> from quantum <i>N</i>-body dynamics. We reformat the hierarchy approach with Klainerman-Machedon theory and prove a bi-scattering theorem for the NLS to obtain convergence rate estimates under <span>\\\\(H^{1}\\\\)</span> regularity. The <span>\\\\(H^{1}\\\\)</span> convergence rate estimate we obtain is almost optimal for <span>\\\\(H^{1}\\\\)</span> datum, and immediately improves if we have any extra regularity on the limiting initial one-particle state.</p></div>\",\"PeriodicalId\":36382,\"journal\":{\"name\":\"Annals of Pde\",\"volume\":\"8 2\",\"pages\":\"\"},\"PeriodicalIF\":2.4000,\"publicationDate\":\"2022-05-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s40818-022-00126-5.pdf\",\"citationCount\":\"7\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annals of Pde\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s40818-022-00126-5\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Pde","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s40818-022-00126-5","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Quantitative Derivation and Scattering of the 3D Cubic NLS in the Energy Space
We consider the derivation of the defocusing cubic nonlinear Schrödinger equation (NLS) on \({\mathbb {R}}^{3}\) from quantum N-body dynamics. We reformat the hierarchy approach with Klainerman-Machedon theory and prove a bi-scattering theorem for the NLS to obtain convergence rate estimates under \(H^{1}\) regularity. The \(H^{1}\) convergence rate estimate we obtain is almost optimal for \(H^{1}\) datum, and immediately improves if we have any extra regularity on the limiting initial one-particle state.
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