## ABSTRACT

Here our goal is to develop a method for calculating any nanotubes, regardless of their chirality and the number of atoms in the translation cells. As noted earlier, all the carbon single-walled nanotubes can be constructed by rolling up a single graphite sheet and the structures of tubules can be visualized as a conformal mapping of a two-dimensional graphitic lattice on the surface of a cylinder. One can make such a seamless tubule without any special distortion of their bonding angles other than the introduction of curvature to the carbon hexagons through the rolling process. Each tubule can be labeled by the pair of integers (n
_{1}, n
_{2}) where n
_{1} ≥ n
_{2} ≥ 0, which, together with C-C bond length determine the nanotubes geometry (Fig. 1.1).
Dispersion diagram for an empty 1D lattice with screw axis for ω = 2π/3. The indices label the values of <italic>P</italic> and k<sub>Φ</sub>, e.g., 2, 1 corresponds to dispersion curves with <italic>P</italic> = 2 and k<sub>Φ</sub> =1. Energy <italic>E</italic>(k) is in units of π/h.
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