2008/11/30 by Daniel Lenz, Norbert Peyerimhoff, Olaf Post +2 · 18 citations
Mathematics · Physics and Astronomy · #Classification of discontinuities #Density of states #Discrete mathematics #Eigenfunction #Eigenvalues and eigenvectors #Ergodicity #Graph #Lattice (music) #Mathematical analysis #Mathematics #Physics #Quantum chaos and dynamical systems #Quantum graph #Quantum mechanics #Random Matrices and Applications #Spectral Theory in Mathematical Physics #Statistical physics #Statistics #TRACE (psycholinguistics) #math-ph #math.MP #math.SP
paper · pdf · doi:10.1007/s11040-009-9059-x
published in Mathematical Physics Analysis and Geometry 12(3), 219-254 (Springer Science+Business Media) · 31 pages, 2 figures; introduction extended, references updated
arxiv created 2009/04/25 · openalex publication_date 2009/05/11 · arxiv updated 2015/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 length 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.