Covariant Schrodinger operator and L2-vanishing property on Riemannian manifolds

Let M be a complete Riemannian manifold satisfying a weighted Poincar & eacute; inequality, and let E be a Hermitian vector bundle over M equipped with a metric covariant derivative del. We consider the operator H-X,H-V = del(dagger)del + del(X) + V, where del(dagger) is the formal adjoint of del 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 epsilon. We give a sufficient condition for the triviality of the L-2-kernel of HX,V. As a corollary, putting X equivalent to 0 and working in the setting of a Clifford module equipped with a Clifford connection del, we obtain the triviality of the L-2-kernel of D-2, where D is the Dirac operator corresponding to del. In particular, when epsilon = Lambda(k)(C) 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. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.