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 zK,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 zτ(K),V (localization centre) for some random τ(K)=τV(K)⩾1. We study the asymptotic behavior of the index τV(K) by distinguishing between three cases of the tails of potential distribution: (i) for the “heavy tails” (including Gaussian), τV(K) is asymptotically bounded; (ii) for the light tails, but heavier than the double exponential, the index τV(K) unboundedly increases like |V|o(1); (iii) finally, for the double exponential tails with high disorder, the index τV(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 logτV(K).