Uniqueness of the restoration of boundary conditions differential operator on a set of spectra

Abstract

In this paper, we pose the inverse problem of reconstructing the boundary conditions of a fourthorder
differential operator on a finite interval from the set of spectra of four related operators.
The uniqueness theorem for the restoration of boundary functions from the set of spectra of four
related operators is proved. A fourth-order fixed linear differential expression is considered with
arbitrary intensely regular two-point boundary conditions. It is believed that complete information
about eigenvalues and eigenfunctions is known about this operator. Then we alternately perturb
the boundary conditions. First, an integral perturbation is added only to the first boundary condition.
Then we perturb the first and second boundary conditions with integral terms. Thus, four
related boundary value problems are constructed. The inverse problem is to reconstruct the added
integral perturbations of the boundary conditions from the four spectra of related boundary value
problems. The uniqueness of the restoration of integral perturbations is proved. Note that integral
perturbations may contain derivatives of solutions. However, the order of the derivative is subject
to natural restrictions. In the case of multipoint boundary value problems, the results are greatly
simplified.