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

Consider the tri-harmonic differential expression L-Vu(& nabla;) = (& nabla; +& nabla;)(3)u + Vu, on sections of a hermitian vector bundle over a complete Riemannian manifold (M, g) 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 (E) . Additionally, we provide sufficient conditions for L-V(& nabla;) to be separated in L-2 (E). According to Everitt and Giertz, the differential operator L-V(& nabla;) is said to be separated in L-2 (E) if for all u is an element of L-2 (E) such that L(V)(& nabla; )u is an element of L-2 (E), we have V u is an element of L-2 (E).