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

Abstract

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.

Chambers’s Formula for the Graphene and the Hou Model with Kagome Periodicity and Applications

“…Helffer and P. Kerdelhué are partially supported by the ANR programme NOSEVOL. B. Helffer thanks Gianluca Panati for useful discussions, Gian Michele Graf for the transmission of [1], and Norbert Peyerimoff for the transmission of [22]. We also thank the referee for informing us of [17].…”

“…Note that this model appears in [22,Proposition 3.2] who refers to [21] in the context of quantum graphs. Other models also appear when γ = 0 in [17] as mentioned to us by the referee.…”

Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space

“…Moreover, N is strictly increasing on the absolutely continuous spectrum of ∆ X, , which is explicitly given in [40,Cor. 3.4].…”

Some Abstract Wegner Estimates with Applications

“…However the greater generality does not require additional effort, and we believe it could be useful for models not considered here. For example, the case ζ = −2 appears in the model of [35].…”

“…Previous estimates appeared in [35] and [30], both assumed that P = ⊗ e∈E µ e , with µ e = h e (λ)dλ, but their bounds were linear in |Λ|. Our proof heavily relies on the analysis of [30].…”