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" In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operator beginequation*Au(x)=- Delta Delta u(x)+V(x)u(x), endequation* for all x in R^n, in the Hilbert space H=L_2(R^n,H_1) with the operator potential V(x) in C^1(R^n,L(H_1)), where L(H_1) is the space of all bounded linear operators on the Hilbert space H_1, while Delta Delta u is the biharmonic differential operator and beginequation* Delta u=- sum_i,j=1^n frac1 sqrt det g frac partial partial x_i left[ sqrt det gg^-1(x) frac partial u partial x_j right] endequation* is the Laplace-Beltrami differential operator in R^n. Here g(x)=(g_ij(x)) is the Riemannian matrix, while g^-1(x) is the inverse of the matrix g(x). Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation Au=- Delta Delta u+V(x)u(x)=f(x) in the Hilbert space H where f(x) in H as an application of the separation approach. ", number=1, journal=Journal of Partial Differential Equations, year=2016, month=Mar., pages=59–70