Optimal Wegner estimates for random Schrödinger operators on metric graphs

Gruber, Michael J.; Helm, Mario; Veselić, Ivan

doi:10.1090/pspum/077/2459884

Cited by 4 publications

(6 citation statements)

References 12 publications

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“…In fact, the Wegner estimates needed for this purpose are much weaker than those necessary to establish regularity of the IDS. This has been discussed in the context of random quantum graphs in Section 3.2 of [GHV08].…”

Section: Introductionmentioning

confidence: 99%

“…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%

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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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Section: Introductionmentioning

confidence: 99%

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

Self Cite

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“…The paper [14] studied the situation where each edge carries a random potential and showed the Anderson localization near the bottom of the spectrum. Some generalizations were then obtained in [20,21]. The case of random coupling was considered by the present authors in [25]; recently we learned on an earlier paper [10] where some preliminary estimates for the same model were obtained.The present Letter is devoted to the study of quantum graphs spanned by the Z d -lattice where the edge lengths are random independent identically distributed variables.…”

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confidence: 99%

Localization on Quantum Graphs with Random Edge Lengths

Klopp

Pankrashkin

2009

Lett Math Phys

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The spectral properties of the Laplacian on a class of quantum graphs with random metric structure are studied. Namely, we consider quantum graphs spanned by the simple Z d -lattice with δ-type boundary conditions at the vertices, and we assume that the edge lengths are randomly independently identically distributed. Under the assumption that the coupling constant at the vertices does not vanish, we show that the operator exhibits the Anderson localization near the spectral edges situated outside a certain forbidden set. MSC 2000: 81Q10; 35R60; 47B80; 60H25Up to now, the most studied models of disordered media are either discrete (tight-binding) Hamiltonians or continuous Schrödinger operators. In the last decade, there is an increasing interest in the analysis of quantum Hamiltonians on so-called quantum graphs, i.e. differential operators on singular one-dimensional spaces, see the collection of papers [6,13,15,27]. A quantum graph is composed of one-dimensional differential operators on the edges and boundary conditions at the vertices describing coupling of edges. Such operators provide an effective model for the study of various phenomena in the condensed matter physics admitting an experimental verification, and there is natural question about the influence of random perturbations in such systems [39]. There are numerous possibilities to introduce randomness: combinatorial structure, coefficients of differential expression, coupling, metric, etc. Being locally of one-dimensional nature and admitting a complex global shape, quantum graphs take an intermediate position between the one-dimensional and higher dimensional Schrödinger operators. It seems that the paper [26] considering the random necklace model was the first one discussing random interactions and the Anderson localization in the quantum graph setting. Later, these results were generalized for radial tree configurations [22], where Anderson localization at all energies was proved. Both papers used a machinery specific for one-dimensional operators. The paper [1] addressed the spectral analysis on quantum tree graphs with random edge lengths; it appears that the Anderson localization does not hold near the bottom of the spectrum at least in the weak disorder limit and one always has some absolutely continuous spectrum. Another important class of quantum graphs is given by Z d -lattices. The paper [14] studied the situation where each edge carries a random potential and showed the Anderson localization near the bottom of the spectrum. Some generalizations were then obtained in [20,21]. The case of random coupling was considered by the present authors in [25]; recently we learned on an earlier paper [10] where some preliminary estimates for the same model were obtained.The present Letter is devoted to the study of quantum graphs spanned by the Z d -lattice where the edge lengths are random independent identically distributed variables. We consider the free Laplacian on each edge with δ-type boundary conditions and show, under certain technical assump...

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“…Note that W 1,2 (X) and W 2,2 (X) are sometimes referred to as decoupled or maximal Sobolev spaces, see e.g. [9,19]. Other Sobolev spaces can also be found in the literature.…”

Section: Vol 62 (2008) Eigenfunction Expansion 543mentioning

confidence: 99%