On System of Root Vectors of Perturbed Regular Second-Order Differential Operator Not Possessing Basis Property

This article delves into the spectral problem associated with a multiple differentiation operator that features an integral perturbation of boundary conditions of one specific type, namely, regular but not strengthened regular. The integral perturbation is characterized by the function p x , which belongs to the space L 2 0 , 1 . The concept of problems involving integral perturbations of boundary conditions has been the subject of previous studies, and the spectral properties of such problems have been examined in various early papers. What sets the problem under consideration apart is that the system of eigenfunctions for the unperturbed problem (when p x ≡ 0 ) lacks the property of forming a basis. To address this, a characteristic determinant for the spectral problem has been constructed. It has been established that the set of functions p x , for which the system of eigenfunctions of the perturbed problem does not constitute an unconditional basis in L 2 0 , 1 , is dense within the space L 2 0 , 1 . Furthermore, it has been demonstrated that the adjoint operator shares a similar structure.