On the completeness of a pair of biorthogonally conjugated systems of functions

In this paper we studied the spectral problem for an ordinary second order differential equation on a finite interval with a discontinuous coefficient of the highest derivative. At the ends of the segment the boundary conditions of the first kind are given. We found eigenvalues with their asymptotic behavior as the roots of the transcendental equation. The system of eigenfunctions is the trigonometric sine on one half of the segment, and the hyperbolic sine on the other. The system of eigenfunctions is not orthogonal in the space of square integrable functions. The corresponding biorthogonal system of functions was built as a solution to the dual problem. In the proof of the completeness of the biorthogonal system we used well known Keldysh theorem about the completeness of the eigenfunctions system of a nonselfadjoint operator.