  We investigate the spectrum of finite-length carbon nanotubes in the presence
of onsite and nearest-neighbor superconducting pairing terms. A one-dimensional
ladder-type lattice model is developed to explore the low-energy spectrum and
the nature of the electronic states. We find that zero energy edge states can
emerge in zigzag class carbon nanotubes as a combined effect of
curvature-induced Dirac point shift and strong superconducting coupling between
nearest-neighbor sites. The chiral symmetry of the system is exploited to
define a winding number topological invariant. The associated topological phase
diagram shows regions with nontrivial winding number in the plane of chemical
potential and superconducting nearest-neighbor pair potential (relative to the
onsite pair potential). A one-dimensional continuum model reveals the
topological origin of the zero energy edge states: A bulk-edge correspondence
is proven, which shows that the condition for nontrivial winding number and
that for the emergence of edge states are identical. For armchair class
nanotubes, the presence of edge states in the superconducting gap depends on
the nanotube's boundary shape. For the minimal boundary condition, the
emergence of the subgap states can also be deduced from the winding number.
