Studying the properties of carbon nanotubes with the functional integration method

We obtain an exact, closed, self-consistent system of equations for describing nanotubes that takes electron and oscillation subsystems in the collective variables into account. Collective excitations in nanotubes are described by the quantum numbers $n$, $m$, and $k$, where $n$ – $1$ is the number of radial modes, $m$ is the number of azimuthal modes, and $k$ – $1$ is the number of longitudinal modes of the wave function. The results obtained approximate the experimental data better than those obtained by the method of linear combinations of atomic orbitals.