Consider the bi-harmonic differential expression of the form A D 44 C q on a complete Riemannian manifold .M; g/ with metric g; where 4 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 2 L 2 .M / such that Au 2 L 2 .M /, we have qu 2 L 2 .M /. In this paper we give sufficient conditions for A to be separated in L 2 .M /.

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