Eigenfunctions of Periodic Differential Operators Analytic in a Strip

TL;DR

This paper investigates the eigenfunctions of periodic differential operators that are analytic in a strip, establishing conditions under which these eigenfunctions form a complete set in a Hardy-Hilbert space.

Contribution

It proves the completeness of eigenfunctions for regular second order operators with matrix coefficients under a positive real part condition in a strip.

Findings

Eigenfunctions may be complete in narrow strips but not in wider ones.

Completeness is guaranteed for certain second order operators with matrix coefficients.

Positive real part condition on the leading coefficient ensures completeness.

Abstract

Ordinary differential operators with periodic coefficients analytic in a strip act on a Hardy-Hilbert space of analytic functions with inner product defined by integration over a period on the boundary of the strip. Simple examples show that eigenfunctions may form a complete set for a narrow strip, but completeness may be lost for a wide strip. Completeness of the eigenfunctions in the Hardy-Hilbert space is established for regular second order operators with matrix-valued coefficients when the leading coefficient satisfies a positive real part condition throughout the strip.