  The local quasiparticle spectra around the vortex core of a d-wave
superconductor in the mixed state is studied by solving the lattice
Bogoliubov-de Gennes equations self-consistently. It is shown that in addition
to the zero-energy states, there also exist core states of high angular
momentum. A nonmagnetic unitary impurity sitting at the core center is
nondestructive to these core states and the zero-energy resonant state induced
by itself is still visible in the local quasiparticle spectrum. The calculated
imaging shows a fourfold ``star'' shape of the local density of states whose
orientation is energy dependent. It is also found that, although the
zero-energy states are extended, the core states of finite energy level could
be localized.
