Construction of a characteristic determinant for one type of eigenvalue problems under integral perturbation of two boundary conditions

Abstract

It is well known that the system of eigenfunctions of an operator given by a formally self-adjoint
differential expression, with arbitrary self-adjoint boundary conditions providing a discrete spectrum,
forms an orthonormal basis. In many papers, the question on saving basis properties under
some (weak in a certain sense) perturbation of the initial operator has been investigated. For the
case of an arbitrary ordinary differential operator, when unperturbed boundary conditions are
strongly regular, the question of the stability of the basis property of root vectors under their
integral perturbation is positively solved in papers of A.A. Shkalikov. In a series of our previous
papers, we have considered the question of constructing a characteristic determinant and of the
stability of the basis property of the root vectors under the integral perturbation of one of the
boundary conditions. Almost all possible types of the boundary conditions that are regular but not
strongly regular have been considered. In the present paper, a spectral problem for the multiple
differentiation operator under the integral perturbation of one type boundary conditions being
regular but not strongly regular is considered. In contrast to the previous papers we consider a
case when the integral perturbation is present in both boundary conditions. The first main result
of the paper is to construct a characteristic determinant of the spectral problem. Based on the
obtained formula, we come to the conclusion about the asymptotic behavior of eigenvalues and
eigenfunctions of the problem. The second main result of the paper is to justify the Riesz basis
property of the system of root functions of the problem under consideration under the integral
perturbation of two boundary conditions.