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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 \vert V_2 \right \vert$ 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

DOI: https://doi.org/10.1090/S0002-9939-2014-11859-4
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