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

TL;DR

This paper establishes conditions under which Kato potentials on Riemannian manifolds are form bounded, enabling the definition of Schrödinger-type operators with potentials in geometric settings.

Contribution

It proves that negative parts of potentials in the Kato class are form bounded on arbitrary Riemannian manifolds, extending previous results to more general geometric contexts.

Findings

Negative potentials in the Kato class are H(0)-form bounded with bound <1.

Allows defining form sums of Laplace-Beltrami operators and potentials on any Riemannian manifold.

Provides a geometric criterion for Schrödinger operator well-posedness.

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.