Analytical methods in problems of completeness of eigenfunctions of differential operators

The first part of the talk will present an analytical approach to solving problems on the completeness of eigenvectors of general non-self-adjoint operators in Hilbert space. The method takes origin from the works of J.Birkhoff, T.Carleman, and M.V.Keldysh. Its modernization was carried out in the recent work of the author of the talk.
The second part will discuss the problems of completeness of eigenfunctions of ordinary differential operators both on the segment and on the half-axis. I shall discuss analytical challenges arrising to solving these problems and methods to overcome them. Some open problems will be set up.