How to express basis conditions and conditions for the equiconvergence with trigonometric series of expansions related to non-self-adjoint differential operators

We established that neither conditions for the basis property of the system of root functions of a non-self-adjoint differential operator, nor conditions for the equiconvergence of the expansion in root functions with the trigonometric series can be formulated in terms of any type of boundary conditions (or even for a specific form of them). We propose to express these conditions in terms of the first components of asymptotic expansions for the eigenvalues λn and for the root functions by powers of λ−1/2n.