Even order periodic operators on the real line

TL;DR

This paper analyzes the spectral properties of even order periodic differential operators on the real line, revealing the structure of their spectrum, branch points, and asymptotics at high energy levels.

Contribution

It introduces a detailed spectral analysis of even order periodic operators, including the Lyapunov function on a Riemann surface and high-energy asymptotics.

Findings

Spectrum has multiplicity two at high energy

All gap endpoints are eigenvalues or real branch points

Operator spectrum has infinitely many gaps with finitely many non-real branch points

Abstract

We consider $2p\ge 4$ order differential operator on the real line with a periodic coefficients. The spectrum of this operator is absolutely continuous and is a union of spectral bands separated by gaps. We define the Lyapunov function, which is analytic on a p-sheeted Riemann surface. The Lyapunov function has real or complex branch points. We prove the following results: (1) The spectrum at high energy has multiplicity two. (2) Endpoints of all gaps are periodic (or anti-periodic) eigenvalues or real branch points. (3) The spectrum of operator has an infinite number of open gaps and there exists only a finite number of non-real branch points for some specific coefficients (the generic case). (4) The asymptotics of the periodic, anti-periodic spectrum and branch points are determined at high energy.