Separation of bi-harmonic differential operators on Riemannian manifolds

Consider the bi-harmonic differential expression of the form A = Delta Delta + q on a complete Riemannian manifold (M, g) with metric g, where Delta is the Laplacian on M and q >= 0 is a locally square integrable function on M. In the terminology of Everitt and Giertz, the differential expression A is said to be separated in L-2(M) if for all u is an element of L-2(M) such that Au is an element of L-2(M), we have qu is an element of L-2(M). In this paper we give sufficient conditions for A to be separated in L-2(M).