  We report on several new basic properties of a parabolic dot in the presence
of a magnetic field. The ratio between the potential strength and the Landau
level (LL) energy spacing serves as the coupling constant of this problem. In
the weak coupling limit the energy spectrum in each Hilbert subspace of an
angular momentum consists of discrete LLs of graphene. In the intermediate
coupling regime non-resonant states form a closely spaced energy spectrum. We
find, counter-intuitively, that resonant quasi-boundstates of both positive and
negative energies exist in the spectrum. The presence of resonant
quasi-boundstates of negative energies are a unique property of massless Dirac
fermions. As the strong coupling limit is approached resonant and non-resonant
states transform into anomalous states, whose probability densities develop a
narrow peak inside the well and another broad peak under the potential barrier.
These properties may investigated experimentally by measuring optical
transition energies that can be described by a scaling function of the coupling
constant.
