On spectral gaps of a Laplacian in a strip with a bounded periodic perturbation

In the work we consider the Laplacian subject to the Dirichlet condition in an infinite planar strip perturbed by a periodic operator. The perturbation is introduced as an arbitrary bounded periodic operator in $L_2$ on the periodicity cell; then this operator is extended periodically on the entire strip.
We study the band spectrum of such operator. The main obtained result is the absence of the spectral gaps in the lower part of the spectrum for a sufficiently small potential. The upper bound for the period ensuring such result is written explicitly as a certain number. It also involves a certain characteristics of the perturbing operator, which can be nonrigorously described as “the maximal oscillation of the perturbation”. We also explicitly write out the length of the part of the spectrum, in which the absence of the gaps is guaranteed. Such result can be regarded as a partial proof of the strong Bethe-Sommerfeld conjecture on absence of internal gaps in the band spectra of periodic operators for sufficiently small periods.