Espectro e dimensão Hausdorff de operadores bloco-Jacobi com perturbações esparsas distribuídas aleatoriamente

Abstract

In this work we attempt to caracterize the spectrum of a class of limited block–Jacobi operators defined in l(Λ,C) (Λ : Z+×{0, 1, . . . , L−1} represents a strip of width L ≥ 2 on the semi–plane Z+) subject to a sparse perturbation (which means that the distance between the “barries” grow geometrically with their distance to the origin) randomly distributed. Such operators are defined as Kronecker sums of unidimensional Jacobi matrices J , each one acting in different directions of the space. We prove, by means of a block–diagonalization of the operator, that its most relevant spectral properties depend on the caracterization of the “mixture measure” 1 L ∑L−1 j=0 μj, μj the spectral measure of the Jacobi matrix J j = J + 2 cos(2πj/L)I. For this, we must characterize at first each one of the measures μj, exploiting and improving some well known techniques developed in the study of unidimensional sparse operators. We prove, for instance, that the sequence of Prufer angles (variables which parametrize the solutions of the eigenvalue equation) are uniform distributed on the interval [0, π), a result which gives us condition to determine the average asymptotic behavior of the solutions of the eigenvalue equation. Such result, in association with the techniques developed by Marchetti et. al. in [MWGA] and with an adaptation of Last–Simon [LS1] criteria for sparse operator, permit us to prove the existence of a sharp transition between singular continuous and pure point spectra. Following on, we use the results from Jitomirskaya–Last of [JL] and obtain the exact Hausdorff dimension of the measure μj, given by αj = 1 + 4(1−p2)2 p2(4−(λ−2 cos(2πj/L))2) (λ ∈ [−2, 2]), recovering an analogous result due to Zlatos in [Z]. At last, we adapt these results to the mixture measure of the block–Jacobi matrix, obtaining α = minj∈I(λ) αj, I(λ) : {m ∈ {0, 1, . . . , L − 1} : λ ∈ [−2 + 2 cos(2πj/L), 2 + 2 cos(2πj/L)]}, as its exact Hausdorff dimension. We study as well identical models with sub and super geometric sparsities conditions, obtaining a pure point spectrum (with null Hausdorff dimension) in the first case, and a purely singular continuous spectrum (such that its Hausdorff dimension is 1) in the second. Finally, we prove the existence of a transition between pure point and singular continuous spectra in a model with sub–geometric sparsity whose Hausdorff dimension related to the spectral measure is null.