Kato's inequality and form boundedness of Kato potentials on arbitrary Riemannian manifolds

Abstract

Let $(M,g)$ be a Riemannian manifold with Laplace-Beltrami operator $-\Delta$ and let $E\to M$ be a Hermitian vector bundle with a Hermitian covariant derivative $\nabla$. Furthermore, let H(0) denote the Friedrichs realization of $\nabla^*\nabla$ and let $V$ be a potential. We prove that $V^-$ is H(0)-form bounded with bound $<1$, if the function $\max\sigma(V^-)$ is in the Kato class of $(M,g)$. In particular, this gives a sufficient condition under which one can define the form sum $H(V):=H(0)\dotplus V$ on arbitrary Riemannian manifolds.