Abstract
This paper studies the completeness of root function systems of third-order differential operators on a finite interval under general two-point boundary conditions. The operator is generated by a third-order linear differential expression in $$L_{2}(0,1)$$ and in general, is neither self-adjoint nor invertible. The main goal is to identify boundary conditions that extend the classical Birkhoff regular class while preserving completeness of root functions. To this end, the Wronskian matrix and its second-order minors are analyzed, and a general representation of the characteristic determinant is obtained in terms of boundary matrices and solutions of the adjoint equation. On this basis, the concept of non-degenerate boundary conditions is introduced: boundary conditions are called non-degenerate if the characteristic determinant is not constant with respect to the spectral parameter. The paper also develops a normalization procedure reducing general boundary conditions to a canonical form and derives a resolvent representation suitable for asymptotic estimates. As a principal result, a sufficient condition for completeness of the root functions is established via a lower estimates for the characteristic determinant along a complex ray.