{"title":"黎曼流形上的协变薛定谔算子和 L2- 消失特性","authors":"Ognjen Milatovic","doi":"10.1016/j.difgeo.2024.102191","DOIUrl":null,"url":null,"abstract":"<div><p>Let <em>M</em> be a complete Riemannian manifold satisfying a weighted Poincaré inequality, and let <span><math><mi>E</mi></math></span> be a Hermitian vector bundle over <em>M</em> equipped with a metric covariant derivative ∇. We consider the operator <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>X</mi><mo>,</mo><mi>V</mi></mrow></msub><mo>=</mo><msup><mrow><mi>∇</mi></mrow><mrow><mi>†</mi></mrow></msup><mi>∇</mi><mo>+</mo><msub><mrow><mi>∇</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>+</mo><mi>V</mi></math></span>, where <span><math><msup><mrow><mi>∇</mi></mrow><mrow><mi>†</mi></mrow></msup></math></span> is the formal adjoint of ∇ with respect to the inner product in the space of square-integrable sections of <span><math><mi>E</mi></math></span>, <em>X</em> is a smooth (real) vector field on <em>M</em>, and <em>V</em> is a fiberwise self-adjoint, smooth section of the endomorphism bundle <span><math><mi>End</mi><mspace></mspace><mi>E</mi></math></span>. We give a sufficient condition for the triviality of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-kernel of <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>X</mi><mo>,</mo><mi>V</mi></mrow></msub></math></span>. As a corollary, putting <span><math><mi>X</mi><mo>≡</mo><mn>0</mn></math></span> and working in the setting of a Clifford module equipped with a Clifford connection ∇, we obtain the triviality of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-kernel of <span><math><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, where <em>D</em> is the Dirac operator corresponding to ∇. In particular, when <span><math><mi>E</mi><mo>=</mo><msubsup><mrow><mi>Λ</mi></mrow><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msubsup><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>M</mi></math></span> and <span><math><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> is the Hodge–deRham Laplacian on (complex-valued) <em>k</em>-forms, we recover some recent vanishing results for <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-harmonic (complex-valued) <em>k</em>-forms.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Covariant Schrödinger operator and L2-vanishing property on Riemannian manifolds\",\"authors\":\"Ognjen Milatovic\",\"doi\":\"10.1016/j.difgeo.2024.102191\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <em>M</em> be a complete Riemannian manifold satisfying a weighted Poincaré inequality, and let <span><math><mi>E</mi></math></span> be a Hermitian vector bundle over <em>M</em> equipped with a metric covariant derivative ∇. We consider the operator <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>X</mi><mo>,</mo><mi>V</mi></mrow></msub><mo>=</mo><msup><mrow><mi>∇</mi></mrow><mrow><mi>†</mi></mrow></msup><mi>∇</mi><mo>+</mo><msub><mrow><mi>∇</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>+</mo><mi>V</mi></math></span>, where <span><math><msup><mrow><mi>∇</mi></mrow><mrow><mi>†</mi></mrow></msup></math></span> is the formal adjoint of ∇ with respect to the inner product in the space of square-integrable sections of <span><math><mi>E</mi></math></span>, <em>X</em> is a smooth (real) vector field on <em>M</em>, and <em>V</em> is a fiberwise self-adjoint, smooth section of the endomorphism bundle <span><math><mi>End</mi><mspace></mspace><mi>E</mi></math></span>. We give a sufficient condition for the triviality of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-kernel of <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>X</mi><mo>,</mo><mi>V</mi></mrow></msub></math></span>. As a corollary, putting <span><math><mi>X</mi><mo>≡</mo><mn>0</mn></math></span> and working in the setting of a Clifford module equipped with a Clifford connection ∇, we obtain the triviality of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-kernel of <span><math><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, where <em>D</em> is the Dirac operator corresponding to ∇. In particular, when <span><math><mi>E</mi><mo>=</mo><msubsup><mrow><mi>Λ</mi></mrow><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msubsup><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>M</mi></math></span> and <span><math><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> is the Hodge–deRham Laplacian on (complex-valued) <em>k</em>-forms, we recover some recent vanishing results for <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-harmonic (complex-valued) <em>k</em>-forms.</p></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-09-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0926224524000846\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0926224524000846","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
假设 M 是满足加权波恩卡列不等式的完整黎曼流形,假设 E 是 M 上的赫尔墨斯向量束,并配有度量协变导数∇。我们考虑算子 HX,V=∇†∇+∇X+V,其中∇† 是∇关于 E 的平方可积分截面空间内积的形式邻接,X 是 M 上的光滑(实)向量场,V 是内形束 EndE 的纤维自交光滑截面。我们给出了 HX,V 的 L2 内核三性的充分条件。作为推论,假设 X≡0 并在配备了克利福德连接∇的克利福德模块的环境中工作,我们会得到 D2 的 L2 内核的三性,其中 D 是对应于∇的狄拉克算子。特别是,当 E=ΛCkT⁎M 和 D2 是(复值)k 形式上的霍奇-德拉姆拉普拉卡时,我们恢复了 L2 谐波(复值)k 形式的一些最新消失结果。
Covariant Schrödinger operator and L2-vanishing property on Riemannian manifolds
Let M be a complete Riemannian manifold satisfying a weighted Poincaré inequality, and let be a Hermitian vector bundle over M equipped with a metric covariant derivative ∇. We consider the operator , where is the formal adjoint of ∇ with respect to the inner product in the space of square-integrable sections of , X is a smooth (real) vector field on M, and V is a fiberwise self-adjoint, smooth section of the endomorphism bundle . We give a sufficient condition for the triviality of the -kernel of . As a corollary, putting and working in the setting of a Clifford module equipped with a Clifford connection ∇, we obtain the triviality of the -kernel of , where D is the Dirac operator corresponding to ∇. In particular, when and is the Hodge–deRham Laplacian on (complex-valued) k-forms, we recover some recent vanishing results for -harmonic (complex-valued) k-forms.