  In this paper we study the electronic properties of carbon nanocones with one
and two nappes, with pentagonal and heptagonal defects in their lattices. We
use the continuum model, which is based on a Dirac-like Hamiltonian with the
topological effects described by localized non-Abelian gauge field fluxes. We
develop a geometrical approach that can describe the two nappes of the double
cone surface simultaneously, by extending the radial coordinate to the complete
set of real numbers. We show that, for some combinations of different
nanocones, forming the double conical surface, the local density of states near
the apex of the cone does not vanish at the Fermi energy and presents a strong
dependence on the angular momentum. We also obtain the energy spectrum for
finite-sized nanocones and verify that it depends on the choice of topological
defect on the surface, which suggests that a double nanocone can be used to
control the electronic transport in carbon-based electronic devices.
