Abstract

Some inverse scattering problems for the three-dimensional biharmonic operator are considered. The operator is perturbed by first and zero order perturbations, which may be complex-valued and singular. We show the existence of the scattering solutions in the Sobolev space \(W^1_{\infty }(R^3)\). One of the main result of this paper is the proof of analogue of Saito’s formula (in different form as known before), which can be used to prove a uniqueness theorem for the inverse scattering problem. Another main result is to obtain the estimates for the kernel of the resolvent of the direct operator in \(W^1_{\infty}\) and to prove the reconstruction formula for the unknown coefficients of this perturbation.