Covariant Schrödinger operator and L2-vanishing property on Riemannian manifolds

Abstract

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 X≡0 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=ΛCkT⁎M 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.