Lifschitz tail in a magnetic field: The nonclassical regime

L. Erdös, Probability Theory and Related Fields 112 (1998) 321–371.

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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 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 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.
Publishing Year
Date Published
1998-11-01
Journal Title
Probability Theory and Related Fields
Volume
112
Issue
3
Page
321 - 371
IST-REx-ID

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Erdös L. Lifschitz tail in a magnetic field: The nonclassical regime. Probability Theory and Related Fields. 1998;112(3):321-371. doi:10.1007/s004400050193
Erdös, L. (1998). Lifschitz tail in a magnetic field: The nonclassical regime. Probability Theory and Related Fields, 112(3), 321–371. https://doi.org/10.1007/s004400050193
Erdös, László. “Lifschitz Tail in a Magnetic Field: The Nonclassical Regime.” Probability Theory and Related Fields 112, no. 3 (1998): 321–71. https://doi.org/10.1007/s004400050193.
L. Erdös, “Lifschitz tail in a magnetic field: The nonclassical regime,” Probability Theory and Related Fields, vol. 112, no. 3, pp. 321–371, 1998.
Erdös L. 1998. Lifschitz tail in a magnetic field: The nonclassical regime. Probability Theory and Related Fields. 112(3), 321–371.
Erdös, László. “Lifschitz Tail in a Magnetic Field: The Nonclassical Regime.” Probability Theory and Related Fields, vol. 112, no. 3, Springer, 1998, pp. 321–71, doi:10.1007/s004400050193.

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