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Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

A locally compact separable metric space together with a regular Dirichlet form is called a Dirichlet space.

There is a strong interplay between geometric properties of the Dirichlet space, spectral features of the generator of the Dirichlet form and stochastic features of the associated Markov process. The project studies this interplay focusing on global properties viz. on properties of the geometry "far out" and corresponding spectral and stochastic features.

One approach is centered around the compactification via the Royden boundary, boundary terms and Greens formulae. The other approach is centered around harmonic functions and (generalized) eigenfunctions. Both approaches capture geometry "far out" via specific tools and concepts. The approaches are strongly related and exhibiting the relationship will lead to additional insights.

The project will focus on the non-smooth non-local situation of graphs.

## Publications

Given two weighted graphs $(X,b_k,m_k)$, $k=1,2$ with $b_1\sim b_2$ and $m_1\sim m_2$, we prove a weighted $L^1$-criterion for the existence and completeness of the wave operators $W_{\pm}(H_{2},H_1, I_{1,2})$, where $H_k$ denotes the natural Laplacian in $\ell^2(X,m_k)$ w.r.t. $(X,b_k,m_k)$ and $I_{1,2}$ the trivial identification of $\ell^2(X,m_1)$ with $\ell^2(X,m_2)$. In particular, this entails a general criterion for the absolutely continuous spectra of $H_1$ and $H_2$ to be equal.

 Journal Math. Phys. Anal. Geom. Pages 21-28 Link to preprint version

We study the Kazdan-Warner equation on canonically compactifiable graphs. These graphs are distinguished as analytic properties of Laplacians on these graphs carry a strong resemblance to Laplacians on open pre-compact manifolds.

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## Team Members

Prof. Dr. Matthias Keller
Universität Potsdam
mkeller(at)math.uni-potsdam.de

Prof. Dr. Daniel Lenz