Graph-like spaces approximated by discrete graphs and applications

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.