On the smoothness of eigenfunctions of differential-difference operators with boundary conditions of the first type.

It is known that, unlike ordinary differential equations, the smoothness of generalized solutions of differential-difference equations can be violated at the inner points of the interval, even for the infinitely differentiable right-hand side. Boundary value problems for functional differential equations arise in many applications, in particular, in the problem of calming the control system with aftereffect. However, the question remained open for a long time: "Will generalized eigenfunctions of differential-difference operators maintain their smoothness over the entire interval or not?" In a recently published paper [1], it was shown that the smoothness of generalized eigenfunctions can be violated at the inner points of the interval.
In this report, the issue of the regularity of generalized eigenfunctions of differential-difference operators with boundary conditions of the first kind on a finite interval will be considered. We present some new (in comparison with [1]) necessary and sufficient conditions for preserving the smoothness of generalized eigenfunctions over the entire interval. Examples of both violation and preservation of the smoothness of generalized eigenfunctions will be considered.
References: [1] R. Y. Vorotnikov, A. L. Skubachevsky, “Smoothness of generalized proper functions of differential-difference operators on a finite- tervale”, Mat. Notes, 114:5 (2023), 679-701.