Separation for the biharmonic differential operator in the Hilbert space associated with the existence and uniqueness theorem

In this paper, we have studied the separation for the following biharmonic differential operator:Au=ΔΔu+V(x)u(x),x∈Rn, in the Hilbert space H=L2(Rn,H1) with the operator potential V(x)∈C1(Rn,L(H1)), where L(H1) is the space of all bounded linear operators on the Hilbert space H1 and ΔΔu is the biharmonic differential operator, while Δu=∑i=1n∂2u∂xi2 is the Laplace operator in Rn. Moreover, we have studied the existence and uniqueness of the solution of the biharmonic differential equationAu=ΔΔu+V(x)u(x)=f(x) in the Hilbert space H, where f(x)∈H.