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

Abstract:

Let $M$ be a Riemannian manifold and let $E\to M$ be a Hermitian vector bundle with a Hermitian covariant derivative $\nabla$. Furthermore, let $H(0)$ denote the Friedrichs extension of $\nabla ^*\nabla /2$ and let $V:M\to \mathrm {End}(E)$ be a potential. We prove that if $V$ has a decomposition of the form $V=V_1-V_2$ with $V_j\geq 0$, $V_1$ locally integrable and $\left | V_2 \right |$ in the Kato class of $M$, then one can define the form sum $H(V):=H(0)\dotplus V$ in $\Gamma _{\mathsf {L}^2}(M,E)$ without any further assumptions on $M$. Applications to quantum physics are discussed.