Self-Adjointness, \(\varvec{m}\)-Accretivity, and Separability for Perturbations of Laplacian and Bi-Laplacian on Riemannian Manifolds

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Abstract

Let \(\Delta _{A}\) and \(\Delta ^2_{A}\) be the magnetic Laplacian and magnetic bi-Laplacian (with a smooth magnetic field A) on a geodesically complete Riemannian manifold M and let V be a real-valued function on M. We give a sufficient condition for the essential self-adjointness of \(\Delta _{A}+V\) on the space of smooth compactly supported functions on M. Additionally, we provide sufficient conditions for the m-accretivity of the operator sum \(T^{(p)}_{\Delta _{A}}+T^{(p)}_{V}\) and the self-adjointness of \(T^{(2)}_{\Delta ^2_{A}}+T^{(2)}_{V}\), where \(T^{(p)}_{\Delta _{A}}\), \(T^{(p)}_{\Delta ^2_{A}}\), \(T^{(p)}_{V}\) are the “maximal” operators in \(L^p(M)\), \(1<p<\infty \), corresponding to \(\Delta _{A}\), \(\Delta ^2_{A}\), and V. As a consequence of these results, we obtain the separation property for \(\Delta _{A}+V\) in \(L^{p}(M)\) and the same property for \(\Delta ^2_{A}+V\) in \(L^2(M)\). In some results pertaining to \(\Delta ^2_{A}+V\), we assume that the Ricci curvature of M is bounded from below by a (possibly unbounded) non-positive function depending on the distance from a reference point.

Keywords

Bi-Laplacian Laplacian m-accretivity Perturbation Riemannian manifold Self-adjointness Separation 

Mathematics Subject Classification

35P05 47B25 58J50 

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

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

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