Spectral asymptotics for periodic fourth-order operators

Abstract

We consider the operator d4/dt4+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 nonreal resonances for specific potentials. We determine the asymptotics of the periodic and antiperiodic spectra and of the resonances at high energy. We show that there exist two types of gaps: (1) stable gaps, where the endpoints are periodic and antiperiodic 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 antiperiodic spectra together determine the spectrum of our operator. Finally, we show that for small potentials V ≠ 0 the spectrum in the lowest band has multiplicity 4 and the bottom of the spectrum is a resonance, and not a periodic (or antiperiodic) eigenvalue.