Covariant Schrödinger Operator and $L^2$-Vanishing Property on Riemannian Manifolds

Abstract

Let $M$ be a complete Riemannian manifold satisfying a weighted Poincaré inequality, and let $\mathcal{E}$ be a Hermitian vector bundle over $M$ equipped with a metric covariant derivative $\nabla$. We consider the operator $H_{X,V}={\nabla }^{†}\nabla +{\nabla }_X+V$, where ${\nabla }^{†}$ is the formal adjoint of $\nabla$ with respect to the inner product in the space of square-integrable sections of $\mathcal{E}$, $X$ is a smooth (real) vector field on $M$, and $V$ is a fiberwise self-adjoint, smooth section of the endomorphism bundle $\text{End }\mathcal{E}$. We give a sufficient condition for the triviality of the $L^2$-kernel of $H_{X,V}$. As a corollary, putting $X\equiv 0$ and working in the setting of a Clifford module equipped with a Clifford connection $\nabla$, we obtain the triviality of the $L^2$-kernel of $D^2$, where $D$ is the Dirac operator corresponding to $\nabla$. In particular, when $\mathcal{E}={\Lambda }_{\mathbb{C}}^k{T}^{\ast }M$ and $D^2$ is the Hodge--deRham Laplacian on (complex-valued) $k$-forms, we recover some recent vanishing results for $L^2$-harmonic (complex-valued) $k$-forms.

We streamlined the exposition in subsection 3.2. We added the remark 3.2. We streamlined the exposition in remark 3.5. We corrected a few typos