Spectral asymptotics for periodic fourth order operators

TL;DR

This paper analyzes the spectral properties of a periodic fourth order differential operator, introducing a Lyapunov function with branch points called resonances, and characterizing the spectrum's asymptotics and gap structure.

Contribution

It develops a detailed spectral analysis of the operator, including the existence of resonances, asymptotics at high energy, and the structure of spectral gaps, extending classical results to fourth order operators.

Findings

Existence of real and non-real resonances for specific potentials.

Asymptotic behavior of spectrum and resonances at high energy.

Identification of stable and unstable spectral gaps.

Abstract

We consider the operator ${d^4dt^4}+V$ on the real line with a real periodic potential $V$. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define a Lyapunov function which is analytic on a two sheeted Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove the existence of real as well as non-real resonances for specific potentials. We determine the asymptotics of the periodic and anti-periodic spectrum and of the resonances at high energy. We show that there exist two type of gaps: 1) stable gaps, where the endpoints are periodic and anti-periodic eigenvalues, 2) unstable (resonance) gaps, where the endpoints are resonances (i.e., real branch points of the Lyapunov function above the bottom of the spectrum). We also show that the periodic and anti-periodic spectrum together determine the spectrum of our operator. Finally, we show that for small potentials $V\ne 0$ the spectrum in the lowest band has multiplicity 4 and the bottom of the spectrum is a resonance, and not a periodic (or anti-periodic) eigenvalue.