{"title":"Point interactions for 3D sub-Laplacians","authors":"Riccardo Adami , Ugo Boscain , Valentina Franceschi , Dario Prandi","doi":"10.1016/j.anihpc.2020.10.007","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper we show that, for a sub-Laplacian Δ on a 3-dimensional manifold <em>M</em>, no point interaction centered at a point <span><math><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mi>M</mi></math></span> exists. When <em>M</em> is complete w.r.t. the associated sub-Riemannian structure, this means that Δ acting on <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><mo>(</mo><mi>M</mi><mo>∖</mo><mo>{</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>}</mo><mo>)</mo></math></span> is essentially self-adjoint in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>M</mi><mo>)</mo></math></span><span>. A particular example is the standard sub-Laplacian on the Heisenberg group<span>. This is in stark contrast with what happens in a Riemannian manifold </span></span><em>N</em>, whose associated Laplace-Beltrami operator acting on <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><mo>(</mo><mi>N</mi><mo>∖</mo><mo>{</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>}</mo><mo>)</mo></math></span> is never essentially self-adjoint in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>N</mi><mo>)</mo></math></span>, if <span><math><mi>dim</mi><mo></mo><mi>N</mi><mo>≤</mo><mn>3</mn></math></span><span><span>. We then apply this result to the Schrödinger evolution of a thin molecule, i.e., with a vanishing </span>moment of inertia, rotating around its center of mass.</span></p></div>","PeriodicalId":55514,"journal":{"name":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","volume":"38 4","pages":"Pages 1095-1113"},"PeriodicalIF":1.8000,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.anihpc.2020.10.007","citationCount":"7","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales De L Institut Henri Poincare-Analyse Non Lineaire","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0294144920301128","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 7
Abstract
In this paper we show that, for a sub-Laplacian Δ on a 3-dimensional manifold M, no point interaction centered at a point exists. When M is complete w.r.t. the associated sub-Riemannian structure, this means that Δ acting on is essentially self-adjoint in . A particular example is the standard sub-Laplacian on the Heisenberg group. This is in stark contrast with what happens in a Riemannian manifold N, whose associated Laplace-Beltrami operator acting on is never essentially self-adjoint in , if . We then apply this result to the Schrödinger evolution of a thin molecule, i.e., with a vanishing moment of inertia, rotating around its center of mass.
期刊介绍:
The Nonlinear Analysis section of the Annales de l''Institut Henri Poincaré is an international journal created in 1983 which publishes original and high quality research articles. It concentrates on all domains concerned with nonlinear analysis, specially applicable to PDE, mechanics, physics, economy, without overlooking the numerical aspects.