Sparse block-Jacobi matrices with arbitrarily accurate Hausdorff dimension

Abstract

We show that the Hausdorff dimension of the spectral measure of a class of deterministic, i.e. nonrandom, block-Jacobi matrices may be determined with any degree of precision, improving a result of Zlatoš [Andrej Zlatoš, Sparse potentials with fractional Hausdorff dimension, J. Funct. Anal. 207 (2004) 216-252].

Spectral Packing Dimensions through Power-Law Subordinacy

“…Corollary 19 is actually a dual to Corollary 3.7 in Carvalho et al [3], a result about Hausdorff dimension first proposed in the context of Schrödinger sparse operators. It follows, from Theorem 14, that τ (A ∩ ·) is α-packing singular.…”

Spectral Properties of Block Jacobi Matrices

“…In the scalar case, the subordination theory (see, e.g., [6]) implies that in fact the spectrum of A is purely absolutely continuous on . Unfortunately, a subordination theory for the nonscalar case has not been formulated (but there is some progress, see [5]). We expect that in our case the spectrum of A is, similarly to the scalar case, purely absolutely continuous of the maximal multiplicity on .…”

“…For each nonzero α ∈ H ⊕ H, there is a unique generalized eigenvector u such that 5 (u 0 , u 1 ) t = α. If the recurrence relation (6) holds also for n = 0, with the convention that a −1 = u −1 = 0, then u is a formal eigenvector of the matrix A associated with z.…”

Sparse one-dimensional discrete Dirac operators II: Spectral properties

“…1,4,12,16,20,28, and 31). Pearson 21 was the first to recognize the utility of this kind of potential on the construction of Schrödinger operators with singular continuous spectrum (see Chapter 13 in Ref.…”

“…20, but with off-diagonal perturbations. Carvalho et al 1 extended this operator to the strip := Z + × {0, 1, . .…”