Journal of Statistical Physics

, Volume 150, Issue 5, pp 889–907 | Cite as

Extremal Theory for Spectrum of Random Discrete Schrödinger Operator. III. Localization Properties



In this paper, we study the asymptotic localization properties with high probability of the Kth eigenfunction (associated with the Kth largest eigenvalue, K⩾1 fixed) of the multidimensional Anderson Hamiltonian in torus V increasing to the whole of lattice. Denote by z K,V V the site at which the Kth largest value of potential is attained. It is well-known that if the tails of potential distribution are heavier than the double exponential function and satisfies additional regularity and continuity conditions at infinity, then the Kth eigenfunction is asymptotically delta-function at the site z τ(K),V (localization centre) for some random τ(K)=τ V (K)⩾1. We study the asymptotic behavior of the index τ V (K) by distinguishing between three cases of the tails of potential distribution: (i) for the “heavy tails” (including Gaussian), τ V (K) is asymptotically bounded; (ii) for the light tails, but heavier than the double exponential, the index τ V (K) unboundedly increases like |V|o(1); (iii) finally, for the double exponential tails with high disorder, the index τ V (K) behaves like a power of |V|. For Weibull’s and fractional-double exponential types distributions associated with the case (ii), we obtain the first order expansion formulas for logτ V (K).


Anderson Hamiltonian Random potential Localization Largest eigenvalues and eigenfunctions Localization centres Convergence in probability 



I thank the referees for their comments on improving the presentation of this paper. I also thank the referees for bringing Refs. [15, 16, 20, 24, 25] to my attention.


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© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.VU Institute of Mathematics and InformaticsVilniusLithuania

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