Christian Ferrari, Nicolas Macris
Intermixture of extended edge and localized bulk energy levels 
in macroscopic Hall systems
(333K, Postscript)

ABSTRACT.  We study the spectrum of a random Schr\" odinger operator for an 
electron submitted to a magnetic field in a finite but macroscopic two 
dimensional system of linear dimensions equal to $L$. The $y$ direction 
is periodic and in the $x$ direction the electron is confined by two 
smooth increasing boundary potentials. The eigenvalues of the Hamiltonian 
are classified according to their associated quantum mechanical current 
in the $y$ direction. Here we look at an interval of energies inside the 
first Landau band of the random operator for the infinite plane. In this 
energy interval, with large probability, there exist O(L) eigenvalues 
with positive or negative currents of O(1). Between each of these there 
exist O(L^2) eigenvalues with infinitesimal current O(e^{-cB(log L)^2}). 
We explain what is the relevance of this analysis to the integer quantum 
Hall effect.