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Zigzag nanoribbons in external electric fields. (English) Zbl 1194.35478

Summary: We consider the Schrödinger operator on nanoribbons (tight-binding models) in an external electric potential \(V\) on the plane. The corresponding electric field is perpendicular to the axis of the nanoribbon. If \(V=0\), then the spectrum of the Schrödinger operator consists of two non-flat bands and one flat band (an eigenvalue with infinite multiplicity) between them. If we switch on a weak electric potential \(V\to 0\), then there are two cases: (1) this eigenvalue splits into the small spectral band. We determine the asymptotics of the spectral bands for small fields. (2) the unperturbed eigenvalue remains the flat band. We describe all potentials when the unperturbed eigenvalue remains the flat band and when one splits into the small band of the continuous spectrum.

MSC:

35R02 PDEs on graphs and networks (ramified or polygonal spaces)
35J10 Schrödinger operator, Schrödinger equation
35P05 General topics in linear spectral theory for PDEs
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
78A30 Electro- and magnetostatics
82D80 Statistical mechanics of nanostructures and nanoparticles
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