Spectral Packing Dimensions through Power-Law Subordinacy

Abstract

We offer a method of classification of spectral measures of discrete one-dimensional Schrödinger operators with respect to packing measures, which can be seen as dual to results for Hausdorff measures in subordinacy theory. We apply this method to classes of sparse operators, and give an example whose spectral measure has different Hausdorff and packing dimensions, and others for which such dimensions coincide. Some dynamical motivations are also mentioned.

Packing Subordinacy With Application to Spectral Continuity

“…We are interested in packing dimensional properties of spectral measures for discrete Schrödinger operators H, in l 2 (Z), of the form (Hψ)(n) = ψ(n + 1) + ψ(n − 1) + V(n)ψ(n), (1.1) with (real) potentials V = {V(n)}. First, we extend some results from (the partial) packing subordinacy theory for one-dimensional operators on the half-line [3] to the whole-line case. This was initially proposed to provide information about packing dimensional properties of spectral measures and it was an adaptation of the (Hausdorff) power-law subordinacy introduced by Jitomirskaya and Last in [14,15].…”

“…There is a mistake in the discussion in [3] (in Theorem 14 there) and currently one guarantees that only the implication in (2.1) holds true (there is no proof or counterexample to the converse statement). However, we emphasize that (2.1) is exactly what we need in this work.…”

“…According to results in [1], we can also apply Theorem 4.1 to operators with sparse potentials. We reconsider here the class of operators H α ϕ [3,14,27] defined by the action (1.1) in l 2 (Z + ), along with a phase boundary condition…”

“…where B = (x j ) j = (2 j j ) j . It is known that the restriction of its spectral measure to the interval (−2, 2) is 1-packing dimensional [3,4] for every boundary phase ϕ. with |P(n)| ≤ C(1 + n) −p for all n and some C > 0, p > min{3/(2α), (2 − α)/α}. Then, the restriction of the spectral measure of the operator H P,α ϕ to (−2, 2) is also 1-packing dimensional, for all boundary phase ϕ ∈ (−π/2, π/2].…”

“…In this appendix we recall some definitions and concepts regarding Hausdorff and packing measures, and fix notation. Most of the material presented here is based on [3,9,14,20,23,25].…”