{"title":"Heisenberg群上Schrödinger算子的局部色散和Strichartz估计","authors":"H. Bahouri, I. Gallagher","doi":"10.4208/cmr.2021-0101","DOIUrl":null,"url":null,"abstract":"It was proved by H. Bahouri, P. G{\\'e}rard and C.-J. Xu in [9] that the Schr{\\\"o}dinger equation on the Heisenberg group $\\mathbb{H}^d$, involving the sublaplacian, is an example of a totally non-dispersive evolution equation: for this reason global dispersive estimates cannot hold. This paper aims at establishing local dispersive estimates on $\\mathbb{H}^d$ for the linear Schr{\\\"o}dinger equation, by a refined study of the Schr{\\\"o}dinger kernel $S_t$ on $\\mathbb{H}^d$. The sharpness of these estimates is discussed through several examples. Our approach, based on the explicit formula of the heat kernel on $\\mathbb{H}^d$ derived by B. Gaveau in [20], is achieved by combining complex analysis and Fourier-Heisenberg tools. As a by-product of our results, we establish local Strichartz estimates and prove that the kernel $S_t$ concentrates on quantized horizontal hyperplanes of $\\mathbb{H}^d$.","PeriodicalId":66427,"journal":{"name":"数学研究通讯","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2020-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Local Dispersive and Strichartz Estimates for the Schrödinger Operator on the Heisenberg Group\",\"authors\":\"H. Bahouri, I. Gallagher\",\"doi\":\"10.4208/cmr.2021-0101\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"It was proved by H. Bahouri, P. G{\\\\'e}rard and C.-J. Xu in [9] that the Schr{\\\\\\\"o}dinger equation on the Heisenberg group $\\\\mathbb{H}^d$, involving the sublaplacian, is an example of a totally non-dispersive evolution equation: for this reason global dispersive estimates cannot hold. This paper aims at establishing local dispersive estimates on $\\\\mathbb{H}^d$ for the linear Schr{\\\\\\\"o}dinger equation, by a refined study of the Schr{\\\\\\\"o}dinger kernel $S_t$ on $\\\\mathbb{H}^d$. The sharpness of these estimates is discussed through several examples. Our approach, based on the explicit formula of the heat kernel on $\\\\mathbb{H}^d$ derived by B. Gaveau in [20], is achieved by combining complex analysis and Fourier-Heisenberg tools. As a by-product of our results, we establish local Strichartz estimates and prove that the kernel $S_t$ concentrates on quantized horizontal hyperplanes of $\\\\mathbb{H}^d$.\",\"PeriodicalId\":66427,\"journal\":{\"name\":\"数学研究通讯\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-12-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"数学研究通讯\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4208/cmr.2021-0101\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"数学研究通讯","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4208/cmr.2021-0101","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Local Dispersive and Strichartz Estimates for the Schrödinger Operator on the Heisenberg Group
It was proved by H. Bahouri, P. G{\'e}rard and C.-J. Xu in [9] that the Schr{\"o}dinger equation on the Heisenberg group $\mathbb{H}^d$, involving the sublaplacian, is an example of a totally non-dispersive evolution equation: for this reason global dispersive estimates cannot hold. This paper aims at establishing local dispersive estimates on $\mathbb{H}^d$ for the linear Schr{\"o}dinger equation, by a refined study of the Schr{\"o}dinger kernel $S_t$ on $\mathbb{H}^d$. The sharpness of these estimates is discussed through several examples. Our approach, based on the explicit formula of the heat kernel on $\mathbb{H}^d$ derived by B. Gaveau in [20], is achieved by combining complex analysis and Fourier-Heisenberg tools. As a by-product of our results, we establish local Strichartz estimates and prove that the kernel $S_t$ concentrates on quantized horizontal hyperplanes of $\mathbb{H}^d$.