  By analytically solving the Bogoliubov-de Gennes equations we study the
fermion bound states at the center of the core of a vortex in a two-dimensional
superconductor. We consider three kinds of 2D superconducting models: (a) a
standard type II superconductor in the mixed state with low density of vortex
lines, (b) a superconductor with strong spin-orbit coupling locking the spin
parallel to the momentum and (c) a superconductor with strong spin-orbit
coupling locking the spin perpendicular to the momentum. The 2D superconducting
states are induced via proximity effect between an $s$-wave superconductor and
the surface states of a strong topological insulator. In case (a) the energy
gap for the excitations is of order $\Delta_{\infty}^2/(2E_F)$, while for cases
(b) and (c) a zero-energy Majorana state arises together with an equally spaced
($\Delta^2_{\infty}/E_F$) sequence of fermion excitations. The spin-momentum
locking is key to the formation of the Majorana state. We present analytical
expressions for the energy spectrum and the wave functions.
