黎曼流形上的协变薛定谔算子和 L2- 消失特性

Pub Date : 2024-09-13 DOI:10.1016/j.difgeo.2024.102191
Ognjen Milatovic
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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。
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Covariant Schrödinger operator and L2-vanishing property on Riemannian manifolds

Let M be a complete Riemannian manifold satisfying a weighted Poincaré inequality, and let E be a Hermitian vector bundle over M equipped with a metric covariant derivative ∇. We consider the operator HX,V=+X+V, where is the formal adjoint of ∇ with respect to the inner product in the space of square-integrable sections of E, X is a smooth (real) vector field on M, and V is a fiberwise self-adjoint, smooth section of the endomorphism bundle EndE. We give a sufficient condition for the triviality of the L2-kernel of HX,V. As a corollary, putting X0 and working in the setting of a Clifford module equipped with a Clifford connection ∇, we obtain the triviality of the L2-kernel of D2, where D is the Dirac operator corresponding to ∇. In particular, when E=ΛCkTM and D2 is the Hodge–deRham Laplacian on (complex-valued) k-forms, we recover some recent vanishing results for L2-harmonic (complex-valued) k-forms.

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