Consider the tri-harmonic differential expression L ∇ V u = ∇ + ∇ 3 u + V u, on sections of a hermitian vector bundle over a complete Riemannian manifold (M, g) with metric g, where ∇ is a metric covariant derivative on bundle E over complete Riemannian manifold, ∇ + is the formal adjoint of ∇ and V is a self adjoint bundle on E. We will give conditions for L ∇ V to be essential self-adjoint in L 2 (E) . Additionally, we provide sufficient conditions for L ∇ V to be separated in L 2 (E). According to Everitt and Giertz, the differential operator