Lifschitz tail in a magnetic field: the nonclassical regime |
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Authors: | László Erd?s |
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Institution: | (1) Courant Institute, New York University, 251 Mercer Street, New York, NY 10012, USA e-mail: erdos@cims.nyu.edu, US |
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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?dinger 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 Brézin, 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.
Received: 20 July 1997 / Revised version: 20 April 1998 |
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Keywords: | Mathematics Subject Classification (1991): 60K40 82B44 82D30 |
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