Gaps in the spectrum of 1d periodic selfadjoint operator of order 4

Vadim Tkachenko Ben-Gurion University of the Negev, Israel

Spectrum of a self-adjoint differential operator with periodic coefficients is formed by a system of spectral bands separated by gaps. In 1975 V.Marchenko and I.Ostrovskii demonstrated how to open collapsed gaps and to close sufficiently small gaps in the spectrum of Hill’s operator

H = −\frac{d^{2}}{dx^{2}} + q(x), q(x + π) = q(x), x ∈ ℝ,

with small perturbations of its potential q.

We describe a method of treating the same problem for a self-adjoint operator

H = \frac{d^{4}}{dx^{4}} + \frac{d}{dx}p(x)\frac{d}{dx} + q(x), p(x + π) = p(x), q(x + π) = q(x), x ∈ ℝ.