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Lifschitz tail in a magnetic field: The nonclassical regime. (English) Zbl 0921.60099
Summary: We obtain the Lifschitz tail, i.e. the exact low energy asymptotics of the integrated density of states (IDS) of the two-dimensional magnetic Schrödinger 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 E. Brézin, D. J. Gross and C. Itzykson [Nucl. Phys. B 235, 24-44 (1984)] 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.

MSC:
60K40 Other physical applications of random processes
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
82D30 Statistical mechanical studies of random media, disordered materials (including liquid crystals and spin glasses)
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