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Poissonian obstacles with Gaussian walls discriminate between classical and quantum Lifshits tailing in magnetic fields. (English) Zbl 1006.82012
Summary: We investigate the leading low-energy falloff of the integrated density of states of a charged quantum particle in the Euclidean plane subject to a perpendicular constant magnetic field and repulsive impurities randomly distributed according to Poisson’s law. This so-called magnetic Lifshits tail was determined by K. Broderix, D. Hundertmark, W. Kirsch and H. Leschke [The fate of Lifshits tails in magnetic fields, J. Stat. Phys. 80, 1–22 (1995; Zbl 1081.82572)] for algebraically decaying and by L. Erdős [Lifschitz tail in a magnetic field: The nonclassical regime, Probab. Theory Relat. Fields 112, 321–371 (1998; Zbl 0921.60099)] for compactly supported single-impurity potentials. While the result in the first case coincides with the corresponding classical one, the Lifshits tail in Erdős’ case exhibits a genuine quantum behavior. Building on both works, we determine magnetic Lifshits tails for a wide class of positive impurity potentials with a leading long-distance decay in between these limiting cases. Gaussian decay may be shown to discriminate between classical and quantum behavior. The Lifshits tail caused by Gaussian decay reveals power-law falloff with an exponent not yet completely determined.

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
47N55 Applications of operator theory in statistical physics (MSC2000)
60K40 Other physical applications of random processes
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47B80 Random linear operators
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