Dispersion laws for vibrational excitations in an arbitrary polygon or a curved two-dimensional honeycomb lattice are calculated. It is shown that curvature leads to a relative increase of longitudinal vibration frequencies for long-wave vectors <$E q> and their decrease for <$E q> belonging to the boundary of the first Brillouin zone. As for transverse vibrations, their frequencies can only decrease. The initial touching of two vibrational zones of a planar honeycomb lattice is eliminated on its bending in a nanotube, so that the energy gap, that is proportional to the squared ratio of the lattice constant to the cylinder radius, emerges in the spectrum.

• Розенбаум Віктор Михайлович (фізико-математичні науки)

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