Spectral Expansion Series with Parenthesis for the Nonself-adjoint Periodic Differential Operators

TL;DR

This paper develops a spectral expansion method with parenthesis for nonself-adjoint periodic differential operators, introducing new concepts like essential spectral singularities and singular quasimomenta, and compares it to Gelfand expansion.

Contribution

It introduces a novel spectral expansion series with parenthesis for arbitrary order nonself-adjoint periodic differential operators, including criteria for self-adjoint cases.

Findings

Constructed spectral expansion series with parenthesis.

Defined essential spectral singularities and singular quasimomenta.

Established criteria for spectral expansion equivalence with Gelfand expansion.

Abstract

In this paper we construct the spectral expansion for the differential operator generated in all real line 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.