Third order operator with small periodic coefficients

TL;DR

This paper analyzes a third order differential operator with small periodic coefficients, revealing conditions under which its spectrum has varying multiplicity and determining the asymptotics of spectral intervals.

Contribution

It establishes the spectral multiplicity structure of the operator and provides asymptotic descriptions of spectral intervals under minimal coefficient conditions.

Findings

Spectrum is absolutely continuous and covers the entire real line.

Spectrum has either multiplicity one or a small interval with multiplicity three.

Asymptotics of the spectral interval with higher multiplicity are determined.

Abstract

We consider the third order operator with small 1-periodic coefficients on the real line. The spectrum of the operator is absolutely continuous and covers all real line. Under the minimal conditions on the coefficients we show that there are two possibilities: 1) The spectrum has multiplicity one except for a small spectral nonempty interval with multiplicity three. Moreover, the asymptotics of the small interval is determined. 2) All spectrum has multiplicity one only.