  The origin of the vortex-core states in s-wave and d_{x^2-y^2}-wave
superconductors is investigated by means of some selected numerical
experiments. By relaxing the self-consistency condition in the Bogoliubov-de
Gennes equations and tuning the order parameter in the core region, it is shown
that the suppression of the superfluid density in the core is not a necessary
condition for the core states to form. This excludes ``potential well'' types
of interpretations for the core states. The topological defect in the phase of
the order parameter, however, plays a crucial role. This observation is
explained by considering the effect of the vortex supercurrent on the
Bogoliubov quasiparticles, and illustrated by comparing conventional vortices
with multiply-quantized vortices and vortex-antivortex pairs. The core states
are also found to be extremely robust against random phase disorder.
