Some scattering problems for the multidimensional biharmonic operator are studied. The operator is perturbed by first and zero order perturbations, which maybe complex-valued and singular. We show that the solutions to direct scattering problem satisfy a Lippmann-Schwinger equation, and that this integral equation has a unique solution in the weighted Sobolev space H 2 −δ. The main result of this paper is the proof of Saito's formula, which can be used to prove a uniqueness theorem for the inverse scattering problem. The proof of Saito's formula is based on norm estimates for the resolvent of the direct operator in H 1 −δ .
I n this paper we investigate two boundary d u e problems on the finite interval with potentials that may have non-integable singularity at the origin. The first problem allows unique determination of potential by only one spectrum for certain values of the parameter in one of the boundary conditions. The second problem deals with the Dirichlet problem for potentials which are the sum of Bessel's main part with constant bounded from below and non-integrable potential. Uniqueness results are proved for both problems.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.