{"title":"区间上半经典薛定谔本征函数的一致观测","authors":"Camille Laurent, Matthieu Léautaud","doi":"10.2140/tunis.2023.5.125","DOIUrl":null,"url":null,"abstract":"We consider eigenfunctions of a semiclassical Schr{\\\"o}dinger operator on an interval, with a single-well type potential and Dirichlet boundary conditions. We give upper/lower bounds on the L^2 density of the eigenfunctions that are uniform in both semiclassical and high energy limits. These bounds are optimal and are used in an essential way in a companion paper in application to a controllability problem. The proofs rely on Agmon estimates and a Gronwall type argument in the classically forbidden region, and on the description of semiclassical measures for boundary value problems in the classically allowed region. Limited regularity for the potential is assumed.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2022-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":"{\"title\":\"Uniform observation of semiclassical\\nSchrödinger eigenfunctions on an interval\",\"authors\":\"Camille Laurent, Matthieu Léautaud\",\"doi\":\"10.2140/tunis.2023.5.125\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider eigenfunctions of a semiclassical Schr{\\\\\\\"o}dinger operator on an interval, with a single-well type potential and Dirichlet boundary conditions. We give upper/lower bounds on the L^2 density of the eigenfunctions that are uniform in both semiclassical and high energy limits. These bounds are optimal and are used in an essential way in a companion paper in application to a controllability problem. The proofs rely on Agmon estimates and a Gronwall type argument in the classically forbidden region, and on the description of semiclassical measures for boundary value problems in the classically allowed region. Limited regularity for the potential is assumed.\",\"PeriodicalId\":36030,\"journal\":{\"name\":\"Tunisian Journal of Mathematics\",\"volume\":\"1 1\",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2022-03-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"6\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Tunisian Journal of Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2140/tunis.2023.5.125\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Tunisian Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2140/tunis.2023.5.125","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Uniform observation of semiclassical
Schrödinger eigenfunctions on an interval
We consider eigenfunctions of a semiclassical Schr{\"o}dinger operator on an interval, with a single-well type potential and Dirichlet boundary conditions. We give upper/lower bounds on the L^2 density of the eigenfunctions that are uniform in both semiclassical and high energy limits. These bounds are optimal and are used in an essential way in a companion paper in application to a controllability problem. The proofs rely on Agmon estimates and a Gronwall type argument in the classically forbidden region, and on the description of semiclassical measures for boundary value problems in the classically allowed region. Limited regularity for the potential is assumed.