Abstract
We consider the spectral problem for the random Schrödinger operator on the multidimensional lattice torus increasing to the whole of lattice, with an i.i.d. potential (Anderson Hamiltonian). We prove complete Poisson-type limit theorems for the (normalized) eigenvalues and their locations, provided that the upper tails of the distribution of potential decay at infinity slower than the double exponential tails. For the fractional-exponential tails, the strong influence of the parameters of the model on a specification of the normalizing constants is described.
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Astrauskas, A.: On high-level exceedances of i.i.d. random fields. Stat. Probab. Lett. 52, 271–277 (2001)
Astrauskas, A.: Extremal theory for spectrum of random discrete Schrödinger operator. Preprint No. 2002-26, Institute of Mathematics and Informatics, Vilnius (2002)
Astrauskas, A., Molchanov, S.A.: Limit theorems for the ground states of the Anderson model. Funct. Anal. Appl. 26, 305–307 (1992)
Grenkova, L.N., Molchanov, S.A., Sudarev, J.N.: On the basic states of one-dimensional disordered structures. Commun. Math. Phys. 90, 101–124 (1983)
Grenkova, L.N., Molchanov, S.A., Sudarev, J.N.: The structure of the edge of the multidimensional Anderson model spectrum. Theoret. Math. Phys. 85(1), 1033–1039 (1990)
Kallenberg, O.: Random Measures, 4th edn. Academic, New York (1986)
Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Processes. Springer, New York (1983)
Resnick, S.: Extreme Values, Regular Variation and Point Processes. Springer, Berlin (1987)
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Astrauskas, A. Poisson-Type Limit Theorems for Eigenvalues of Finite-Volume Anderson Hamiltonians. Acta Appl Math 96, 3–15 (2007). https://doi.org/10.1007/s10440-007-9096-z
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DOI: https://doi.org/10.1007/s10440-007-9096-z