The modulus of continuity of Wegner estimates for random Schrödinger operators on metric graphs

Gruber, Michael J.; Veselić, Ivan

doi:10.1515/rose.2008.001

Cited by 4 publications

(5 citation statements)

References 15 publications

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“…(3) Our Wegner estimate reproduces the (arbitrary) modulus of continuity of the single site distribution. In particular our results unify and extend the results of [HV07] and [GV07].…”

Section: Introductionsupporting

confidence: 88%

“…The third section describes two important applications of such estimates, namely consequences for the modulus of continuity of the IDS and localisation proofs via multiscale analysis, with an application to log-Hölder continuous single site distributions. The last section contains proofs of our new key lemma and the main theorem, as well as a few lemmata taken from [KV02], [HV07], [GV07], and [GLV07], adapted to the new context: arbitrary modulus of continuity, partial covering conditions. M.G.…”

Section: Introductionmentioning

confidence: 99%

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Optimal Wegner estimates for random Schrödinger operators on metric graphs

Gruber¹,

Helm²,

Veselić³

2008

Analysis on Graphs and Its Applications

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Abstract. We consider Schrödinger operators with a random potential of alloy type on infinite metric graphs which obey certain uniformity conditions. For single site potentials of fixed sign we prove that the random Schrödinger operator restricted to a finite volume subgraph obeys a Wegner estimate which is linear in the volume and reproduces the modulus of continuity of the single site distribution. This improves and unifies earlier results for alloy type models on metric graphs.We discuss applications of Wegner estimates to bounds on the modulus of continuity for the integrated density of states of ergodic Schrödinger operators, as well as to the proof of Anderson localisation via the multiscale analysis.

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“…(3) Our Wegner estimate reproduces the (arbitrary) modulus of continuity of the single site distribution. In particular our results unify and extend the results of [HV07] and [GV07].…”

Section: Introductionsupporting

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Optimal Wegner estimates for random Schrödinger operators on metric graphs

Gruber¹,

Helm²,

Veselić³

2008

Analysis on Graphs and Its Applications

Self Cite

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“…These estimates are closely related to the finite rank estimates mentioned earlier in the context of the approximability of the IDS. For Schrödinger operators on metric graphs where the randomness enters via the potential, Wegner estimates have been proved in [HV07,GV08,GHV08]. In the recent preprint [KP09] a Wegner estimate for a model with Z d -structure and random edge lengths has been established.…”

Section: Introductionmentioning

confidence: 99%

Continuity of the Integrated Density of States on Random Length Metric Graphs

Lenz

Peyerimhoff

Post

et al. 2009

Math Phys Anal Geom

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We establish several properties of the integrated density of states for random quantum graphs: Under appropriate ergodicity and amenability assumptions, the integrated density of states can be defined using an exhaustion procedure by compact subgraphs. A trace per unit volume formula holds, similarly as in the Euclidean case. Our setting includes periodic graphs. For a model where the edge lengths are random and vary independently in a smooth way we prove a Wegner estimate and related regularity results for the integrated density of states.These results are illustrated for an example based on the Kagome lattice. In the periodic case we characterise all compactly supported eigenfunctions and calculate the position and size of discontinuities of the integrated density of states.

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“…In fact, in [12] for a class of alloy-type random potentials the Lipschitz-continuity of the IDS was established. In [10] we show for a different class of random potentials how one can estimate the modulus of continuity of the IDS.…”

Section: Theoremmentioning

confidence: 99%

Uniform existence of the integrated density of states for random Schrödinger operators on metric graphs over Zd

Gruber

Lenz

Veselić

2007

Journal of Functional Analysis

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We consider ergodic random Schrödinger operators on the metric graph Z d with random potentials and random boundary conditions taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable. This limit, the integrated density of states, can be expressed by a closed Shubin-Pastur type trace formula. It supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions. Among other examples we discuss random magnetic fields and percolation models.

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The modulus of continuity of Wegner estimates for random Schrödinger operators on metric graphs

Cited by 4 publications

References 15 publications

Optimal Wegner estimates for random Schrödinger operators on metric graphs

Optimal Wegner estimates for random Schrödinger operators on metric graphs

Continuity of the Integrated Density of States on Random Length Metric Graphs

Uniform existence of the integrated density of states for random Schrödinger operators on metric graphs over Zd

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