Erdos L.
Lifschitz tail in a magnetic field: the nonclassical regime (improved
version)
(879K, .ps file)
ABSTRACT. We obtain the Lifschitz tail, i.e. the exact low energy asymptotics
of the integrated density of states (IDS) of the two dimensional
magnetic Schr\"odinger operator with a uniform magnetic field and
random Poissonian impurities. The single site potential is repulsive
and it has a finite but nonzero range. We show that the IDS is a
continuous function of the energy at the bottom of the spectrum.
This result complements the earlier (nonrigorous) calculations by
Br\'ezin, Gross and Itzykson which predict that the IDS is discontinuous
at the bottom of the spectrum for zero range (Dirac delta) impurities
at low density. We also elucidate the reason behind this apparent controversy.
Our methods involve magnetic localization techniques (both in space and
energy) in addition to a modified version of the "enlargement of obstacles"
method developed by A.-S. Sznitman. This work is an improved version
of the earlier paper under the same title (number 97-385).