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

Sadybekov M. Imanbaev N.
October 2023 Multidisciplinary Digital Publishing Institute (MDPI)

Mathematics
2023 #11 Issue 20

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 (Formula presented.), which belongs to the space (Formula presented.). 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 (Formula presented.)) 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 (Formula presented.), for which the system of eigenfunctions of the perturbed problem does not constitute an unconditional basis in (Formula presented.), is dense within the space (Formula presented.). Furthermore, it has been demonstrated that the adjoint operator shares a similar structure.

basis property , characteristic determinant , eigenvalue , integral perturbation of boundary conditions , second-order differential operator , system of root vectors

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Institute of Mathematics and Mathematical Modeling, Almaty, 050010, Kazakhstan
Depatment of Mechanics and Mathematics, Al-Farabi Kazakh National University, Almaty, 050040, Kazakhstan
Faculty of Physics and Mathematics, South-Kazakhstan State Pedagogical University, Shymkent, 160012, Kazakhstan

Institute of Mathematics and Mathematical Modeling
Depatment of Mechanics and Mathematics
Faculty of Physics and Mathematics

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