Sparse one-dimensional discrete Dirac operators II: Spectral properties

Abstract

We study spectral properties of some discrete Dirac operators with nonzero potential only at some sparse and suitably randomly distributed positions. As observed in the corresponding Schrödinger operators, we determine the Hausdorff dimension of its spectral measure and identify a sharp spectral transition from point to singular continuous. C

Spectral and dynamical properties of sparse one-dimensional continuous Schrödinger and Dirac operators

“…This is exactly the same problem present in [1,3,22]. By using the ergodic theorem (see [17, Theorem 1.1]), we may substitute, in the asymptotic limit N → ∞, the average (2.16) by the integral…”

“…Using again the arguments presented in [1,22] (see, in particular, [22, § 4]), one may prove the following.…”

“…Let Σ ess denote the essential spectrum of both operators, β the sparsity parameter and r(E) the asymptotic behaviour of the norm T D (x, 0; E) , which is given by (3.14) for H c D (v, φ) and by [1] , φ), the proof follows the same steps, excepting some minor details.…”

“…We may compare the spectral properties of H D (v, v, φ) with its discrete counterpart, studied in [1]. Let H c D (v, φ) represent the continuous Dirac operator in Definition 1.2 (with v 1 = v 2 ) and define H d D (v, φ) as the discrete Dirac operator (see [8,9])…”

Spectral Packing Dimensions through Power-Law Subordinacy

“…There are other examples of sparse operators with spectral measures of exact local dimension that can be obtained as above; see [3] for an extension of [17,27] to a strip of Z 2 + , [2] for a Dirac operator analog to [27], and [1] for a continuous counterpart of [2,27].…”