A note on magnetic-field induced level-density condensation in a two-dimensional electron gas with point scatterers

The density-of-states (DOS) for a magnetized (B) two-dimensional electron gas (2DEG) containing point scatterers of arbitrary strengths, concentration ($n_s$) and distribution is analyzed. It is shown from the first principles that for $n_s \leq B/\Phi_o \equiv n_B$, the areal density of flux quanta $\Phi_o \equiv hc/e$, the DOS retains the extensive degeneracy characteristic of the Landau levels, but reduced by a factor $(1 - {n_s}/{n_B})$. This elementary but exact result gives a level condensation for magnetic field $B > n_s\Phi_o$, as first noted by Br\'{e}zin {\em et al.}. Its implications for the Integral Quantum Hall Effect and for Random Matrix Theory are pointed out.