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Lifshitz tails for
generalized alloy-type random Schrödinger operators
Frédéric Klopp and Shu Nakamura |
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Vol. 3 (2010), No. 4, 409–426
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Abstract |
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We study Lifshitz tails for random Schrödinger operators where the random potential is alloy-type in the sense that the single site potentials are independent, identically distributed, but they may have various function forms. We suppose the single site potentials are distributed in a finite set of functions, and we show that under suitable symmetry conditions, they have a Lifshitz tail at the bottom of the spectrum except for special cases. When the single site potential is symmetric with respect to all the axes, we give a necessary and sufficient condition for the existence of Lifshitz tails. As an application, we show that certain random displacement models have a Lifshitz singularity at the bottom of the spectrum, and also complete our previous study (2009) of continuous Anderson type models. |
Keywords
random Schrödinger operators, sign-indefinite potentials,
Lifshitz tail
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Mathematical Subject Classification 2000
Primary: 35P20, 47B80, 47N55, 81Q10, 82B44
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Milestones
Received: 30 March 2009
Revised: 18 February 2010
Accepted: 4 April 2010
Published: 8 September 2010
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Authors |
| Frédéric Klopp | |
| Institut Universitaire de France | LAGA, U.M.R. 7539 C.N.R.S Institut Galilée, Université de Paris-Nord 99 Avenue Jean-Baptiste Clément 93430 Villetaneuse France |
| Shu Nakamura | |
| Graduate School of Mathematical
Sciences University of Tokyo 3-8-1 Komaba, Meguro-ku Tokyo 153-8914 Japan |
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