On spectral properties of a periodic problem with an integral perturbation of the boundary condition

In this paper we consider the spectral problem for the Schrödinger equation with an integral perturbation in the periodic boundary conditions. The unperturbed problem is assumed to have the system of eigenfunctions and associated functions forming a Riesz basis in $L_2(0,1)$. We construct the characteristic determinant of the spectral problem. We show that the basis property of the system of root functions of the problem may fail to be satisfied under an arbitrarily small change in the kernel of the integral perturbation.