Some Results on Ordinary Differential Operators with Periodic Coefficients

TL;DR

This paper investigates the spectral properties of ordinary differential operators with periodic coefficients, proving irreducibility of the Floquet matrix's characteristic polynomial and analyzing eigenspaces of multipoint problems.

Contribution

It establishes the irreducibility of the characteristic polynomial over meromorphic functions and characterizes eigenspaces in multipoint eigenvalue problems for periodic differential operators.

Findings

Characteristic polynomial of Floquet matrix is irreducible over meromorphic functions

Eigenspaces of multipoint problems are spanned by Floquet solutions

Discussion of conjectures and open questions in the field

Abstract

For a general ordinary differential operator $\mathcal{L}$ with periodic coefficients we prove that the characteristic polynomial of the Floquet matrix is irreducible over the field of meromorphic functions. We also consider a multipoint eigenvalue problem and show that its eigenspaces are spanned by pure or generalized Floquet solutions. Finally, at the end of the paper we mention some relevant conjectures and open questions.