On the inverse problem of a fourth-order self-adjoint binomial operator

Let L be the binomial operator L = (d/dx)(4) + q(x) acting an (0,pi) with Dirichlet boundary conditions. We study the associated inverse spectral problem under the assumptions that q is symmetric, i.e., q(pi-x) = q(x). Our analysis is inspired by the well-known work of Borg for the Sturm-Liouville case, We first derive the eigenfunction asymptotics by an approach that is different from the ones used in the second-order case (WKB, etc.). These asymptotics are then used to obtain the local uniqueness of the inverse problem.