 02120 Christian Ferrari, Nicolas Macris
 Intermixture of extended edge and localized bulk energy levels
in macroscopic Hall systems
(333K, Postscript)
Mar 13, 02

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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.
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