On smoothness of generalized eigenfunctions for differential-difference operators

Unlike ordinary differential equations with constant coefficients, smoothness of generalized solutions to boundary value problems for neutral differential–difference equations on a finite interval $(0,d)$ can be violated and preserved only on some subintervals. However, for generalized eigenfunctions of differential-difference operators problem of smoothness remained open. In this lecture we obtain the necessary and sufficient conditions of smoothness for eigenfunctions of differential–difference operators.