We present a scanning tunneling spectroscopy study on quasiparticle states in vortex cores in Bi$_2$Sr$_2$CaCu$_2$O$_{8+delta}$. The energy of the observed vortex core states shows an approximately linear scaling with the superconducting gap in the region just outside the core. This clearly distinguishes them from conventional localized core states, and is a signature of the mechanism responsible for their discrete appearance in high-temperature superconductors. The energy scaling of the vortex core states also suggests a common nature of vortex cores in Bi$_2$Sr$_2$CaCu$_2$O$_{8+delta}$ and YBa$_2$Cu$_3$O$_{7-delta}$. Finally, the observed vortex core states do not show any dependence on the applied magnetic field in the range from 1 to 6 T.
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