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 $H_{X,V}={\nabla }^{†}\nabla +{\nabla }_X+V$, where ${\nabla }^{†}$ 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 $\mathrm{End}E$. 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 ∇, we obtain the triviality of the $L^2$-kernel of $D^2$, where D is the Dirac operator corresponding to ∇. In particular, when $E={\Lambda }_C^k{T}^{⁎}M$ 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.