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

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

2011-05-03

arxiv.org/abs/1105.0532v3

Physics

Mathematical Physics

extended version

Scientific paper

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.

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