On the spectral Hausdorff dimension of 1D discrete Schr\"odinger operators under power decaying perturbations

@article{Bazo2016OnTS,
  title={On the spectral Hausdorff dimension of 1D discrete Schr\"odinger operators under power decaying perturbations},
  author={Vanderl{\'e}a Rodrigues Baz{\~a}o and Silas L. Carvalho and C'esar R. de Oliveira},
  journal={arXiv: Mathematical Physics},
  year={2016}
}
We show that spectral Hausdorff dimensional properties of discrete Schr\"oodinger operators with (1) Sturmian potentials of bounded density and (2) a class of sparse potentials are preserved under suitable polynomial decaying perturbations, when the spectrum of these perturbed operators have some singular continuous component. 
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References

SHOWING 1-10 OF 31 REFERENCES
Stability of singular spectral types under decaying perturbations
Half-line eigenfunction estimates and singular continuous spectrum of zero Lebesgue measure
We consider discrete one-dimensional Schr\"odinger operators with strictly ergodic, aperiodic potentials taking finitely many values. The well-known tendency of these operators to have purely…
Power-law subordinacy and singular spectra I. Half-line operators
We study Hausdorff-dimensional spectral properties of certain “whole-line” quasiperiodic discrete Schrodinger operators by using the extension of the Gilbert–Pearson subordinacy theory that we…
Singular continuous spectrum under rank one perturbations and localization for random hamiltonians
We consider a selfadjoint operator, A, and a selfadjoint rank-one projection, P, onto a vector, φ, which is cyclic for A. In terms of the spectral measure dμAφ, we give necessary and sufficient…
Pure point spectrum under 1-parameter perturbations and instability of Anderson localization
We consider a selfadjoint operator,A, and a selfadjoint rank-one projection,P, onto a vector, φ, which is cyclic forA. We study the set of all eigenvalues of the operatorAt=A+tP (t∈∝) that belong to…
On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators
Operators with singular continuous spectrum: II. Rank one operators
For an operator,A, with cyclic vector ϕ, we studyA+λP, whereP is the rank one projection onto multiples of ϕ. If [α,β] ⊂ spec (A) andA has no a.c. spectrum, we prove thatA+λP has purely singular…
α-Continuity Properties of One-Dimensional Quasicrystals
Abstract: We apply the Jitomirskaya-Last extension of the Gilbert-Pearson theory to discrete one-dimensional Schrödinger operators with potentials arising from generalized Fibonacci sequences. We…
On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints
Synopsis The theory of subordinacy is extended to all one-dimensional Schrödinger operatorsfor which the corresponding differential expression L = – d2/(dr2) + V(r) is in the limit point case at both…
Uniform Spectral Properties of One-Dimensional Quasicrystals, III. α-Continuity
Abstract: We study the spectral properties of one-dimensional whole-line Schrödinger operators, especially those with Sturmian potentials. Building upon the Jitomirskaya–Last extension of the…
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