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Proceedings of the American Mathematical Society

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


Author: Batu Güneysu
Journal: Proc. Amer. Math. Soc. 142 (2014), 1289-1300
MSC (2010): Primary 47B25, 58J35; Secondary 60H30
DOI: https://doi.org/10.1090/S0002-9939-2014-11859-4
Published electronically: January 27, 2014
MathSciNet review: 3162250
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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.


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Additional Information

Batu Güneysu
Affiliation: Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany
Email: gueneysu@math.hu-berlin.de

Received by editor(s): September 19, 2011
Received by editor(s) in revised form: May 10, 2012
Published electronically: January 27, 2014
Communicated by: Varghese Mathai
Article copyright: © Copyright 2014 American Mathematical Society