Essential self-adjointness for covariant tri-harmonic operators on manifolds and the separation problem

Abstract

Consider the tri-harmonic differential expression $L_{V}^{\nabla}u=\left(\nabla^{+}\nabla\right)^{3}u+Vu$, on sections of a hermitian vector bundle over a complete Riemannian manifold $\left(M,g\right)$ with metric $g$, where $\nabla$ is a metric covariant derivative on bundle E over complete Riemannian manifold, $\nabla^{+}$ is the formal adjoint of $\nabla$ and $V$ is a self adjoint bundle on $E$. We will give conditions for $L_{V}^{\nabla}$ to be essential self-adjoint in $L^{2}\left(E\right).$ Additionally, we provide sufficient conditions for $L_{V}^{\nabla}$ to be separated in $L^{2}\left( E\right)$. According to Everitt and Giertz, the differential operator $L_{V}^{\nabla}$ is said to be separated in $L^{2}\left( E\right) $ if for all $u$ $\in L^{2}\left( E\right)$ such that $L_{V}^{\nabla}u\in L^{2}\left( E\right) $, we have $Vu\in L^{2}\left( E\right)$.