Study on the Basis Properties of Eigenfunctions for Differential Problems with Conjugation

This paper investigates the spectral properties of a second-order differential operator with a discontinuity point and conjugation conditions containing a spectral parameter. The analysis is carried out within the framework of functional analysis and basis theory in Banach and Hilbert spaces. Sufficient conditions are established for the completeness, minimality, and basis properties of the system of eigenfunctions and associated functions. It is proved that the root functions form a basis in the corresponding Lebesgue spaces and a Riesz basis in the Hilbert space setting. The obtained results provide a rigorous characterization of spectral expansions and contribute to the theory of discontinuous differential operators with nonclassical conjugation conditions.