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The m-accretivity of covariant Schrödinger operators with unbounded drift

  • Ognjen MilatovicEmail author
Article
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Abstract

We work in the context of a geodesically complete Riemannian n-manifold M with a Hermitian vector bundle \(\mathcal {V}\) over M, equipped with a metric covariant derivative \(\nabla \). We study the operator \(H:=\nabla ^{\dagger }\nabla +\nabla _{X}+V\), where \(\nabla ^{\dagger }\) is the formal adjoint of \(\nabla \), the symbol \(\nabla _{X}\) stands for the action of \(\nabla \) along a smooth vector field X on M, and V is a locally bounded section of the endomorphism bundle \({\text {End}}\mathcal {V}\). We show that under certain conditions on X and V, the closure \(\overline{H|_{C_{c}^{\infty }(\mathcal {V})}}\) of \(H|_{C_{c}^{\infty }(\mathcal {V})}\) in \(L^p(\mathcal {V})\), where \(1<p<\infty \), is a maximal accretive operator. We also show that \(\overline{H|_{C_{c}^{\infty }(\mathcal {V})}}\) coincides with the “maximal” realization of H in \(L^p(\mathcal {V})\).

Keywords

Bochner Laplacian Covariant Schrödinger operator Drift m-Accretivity Riemannian manifold 

Mathematics Subject Classification

35P05 47B44 58J05 

Notes

Acknowledgements

We thank Hemanth Saratchandran for writing the proof of part (iv) of Lemma 2. We also use this opportunity to express our gratitude to the referees for valuable comments and suggestions and for drawing our attention to the paper [21].

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of North FloridaJacksonvilleUSA

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