Nanoribbons in external electric fields

We consider the Schr\"odinger operator on nanoribbons (tight-binding models) in an external electric potentials $V$. The corresponding electric field is perpendicular to the axis of the nanoribbon. If V=0, then the spectrum of the Schr\"odinger operator consists of two spectral bands and the flat band (i.e., the eigenvalue with infinite multiplicity) between them. If we switch on an weak electric potential $V\to 0$, then we determine the asymptotics of the spectral bands for small fields. In particular, we describe all potentials when the unperturbed eigenvalue remains the flat band and when one becomes the small band of the continuous spectrum.