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 ${\nabla }^{\ast }\nabla$ 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.

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