Graph‐like spaces approximated by discrete graphs and applications. Issue 11 (16th December 2021)

• Record Type: Journal Article
• Authors: Post, Olaf; Simmer, Jan
• Is Part Of: Mathematische Nachrichten. Volume 294:Issue 11(2021)
• Page Start: 2237
• Page End: 2278
• Publication Date: 2021-12-16
• Subjects: approximation by discrete graphs -- discretisation of energy forms -- norm convergence of Laplace resolvents -- Primary: 47A58 -- Secondary: 28A80 -- 47A07 -- 58J50; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201900108
• Languages: English
• ISSNs: 0025-584X

Abstract

Abstract: We define a distance between energy forms on a graph‐like metric measure space and on a suitable discrete weighted graph using the concept of quasi‐unitary equivalence. We apply this result to metric graphs, graph‐like manifolds (e.g. a small neighbourhood of an embedded metric graph) or pcf self‐similar fractals as metric measure spaces with energy forms associated with canonical Laplacians, e.g., the Kirchhoff Laplacian on a metric graph resp. the (Neumann) Laplacian on a manifold (with boundary), and express the distance of the associated energy forms in terms of geometric quantities. In particular, we show that there is a sequence of domains converging to a pcf self‐similar fractal such that the corresponding (Neumann) energy forms converge to the fractal energy form. As a consequence, the spectra and suitable functions of the associated Laplacians converge, the latter in operator norm.