Covariant Schr\"odinger Operator and L²-Vanishing Property on Riemannian Manifolds

Let M be a complete Riemannian manifold satisfying a weighted Poincar\'e 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 End E. We give a sufficient condition for the triviality of the L²-kernel of HX,V. As a corollary, putting X≡ 0 and working in the setting of a Clifford bundle 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=ΛᵏT^*M and D² is the Hodge--deRham Laplacian on k-forms, we recover some recent vanishing results for L²-harmonic k-forms.