A regular differential operator with perturbed boundary condition

Abstract

The operator ℒ₀ generated by a linear ordinary differential expression of nth order and regular boundary conditions of general form is considered on a closed interval. The characteristic determinant of the spectral problem for the operator ℒ₁, where ℒ₁ is an operator with the integral perturbation of one of its boundary conditions, is constructed, assuming that the unperturbed operator ℒ₀ possesses a system of eigenfunctions and associated functions generating an unconditional basis in L₂(0, 1). Using the obtained formula, we derive conclusions about the stability or instability of the unconditional basis properties of the system of eigenfunctions and associated functions of the problem under an integral perturbation of the boundary condition. The Samarskii–Ionkin problem with integral perturbation of its boundary condition is used as an example of the application of the formula.