Magnetic Bi-harmonic differential operators on Riemannian manifolds and the separation problem

In this paper we obtain sufficient conditions for the bi-harmonic differential operator A = Δ_E² + q to be separated in the space L² (M) on a complete Riemannian manifold (M,g) with metric g, where Δ_E is the magnetic Laplacian onM and q ≥ 0 is a locally square integrable function on M. Recall that, in the terminology of Everitt and Giertz, the differential operator A is said to be separated in L² (M) if for all u ∈ L² (M) such that Au ∈ L² (M) we have Δ_E²u ∈ L² (M) and qu ∈ L² (M).