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

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.