Tuesday, January 9, 2018 - 10:15 in D5-153
Approximation of of the Sierpinski Gasket by discrete and metric graphs: norm convergence of resolvents and spectra
A talk in the Geometric Analysis Seminar series by
Jan Simmer from Universität Trier
| Abstract: |
In the present talk we will briefly discuss the abstract setting of quasi-unitary equivalence, a generalized norm resolvent convergence for self-adjoint operators (bounded from below) acting in different Hilbert spaces. More precisely this convergence will be expressed for the corresponding symmetric quadratic forms.
Then, we apply this theory to the Sierpinski Gasket (SG) with its standard energy form and show that the approximating sequence of discrete graphs for SG with discrete energy forms and also a related sequence of SG-like metric graphs with its canonic energy forms are close to each other in the above mentioned sense.
From the abstract theory we now conclude that functions of the Laplacian, like the heat operator or the spectral projection, converge as well in operator norm. It is well known that the latter implies the convergence of the spectra and eigenfunctions. (This is a joint work with Olaf Post, Trier.)
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