Spectral expansion series with parenthesis for the nonself-adjoint periodic differential operators

Abstract

In this paper we construct the spectral expansion for the differential operator generated in \begin{document}$L_{2}(-∞, ∞)$\end{document} by ordinary differential expression of arbitrary order with periodic complex-valued coefficients by introducing new concepts as essential spectral singularities and singular quasimomenta and using the series with parenthesis. Moreover, we find a criteria for which the spectral expansion coincides with the Gelfand expansion for the self-adjoint case.