  Engineering effective p-wave superconductors hosting Majorana quasiparticles
(MQPs) is nowadays of particular interest, also in view of the possible
utilization of MQPs in fault-tolerant topological quantum computation. In quasi
one-dimensional systems, the parameter space for topological superconductivity
is significantly reduced by the coupling between transverse modes. Together
with the requirement of achieving the topological phase under experimentally
feasible conditions, this strongly restricts in practice the choice of systems
which can host MQPs. Here we demonstrate that semiconducting carbon nanotubes
(CNTs) in proximity with ultrathin s-wave superconductors, e.g. exfoliated
NbSe$_2$, satisfy these needs. By precise numerical tight-binding calculations
in the real space we show the emergence of localized zero-energy states at the
CNT ends above a critical value of the applied magnetic field. Knowing the
microscopic wave functions, we unequivocally demonstrate the Majorana nature of
the localized states. An accurate analytical model Hamiltonian is used to
calculate the topological phase diagram.
