- “Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation” by Matthias Keller, Daniel Lenz (arXiv:1101.2979v1 [math.FA]; 2011.01.15) – http://arxiv.org/abs/1101.2979
- Abstract: We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus on phenomena related to unboundedness of the Laplacians. This includes (failure of) essential selfadjointness, absence of essential spectrum and stochastic incompleteness.
- “Note on basic features of large time behaviour of heat kernels” by Matthias Keller, Daniel Lenz, Hendrik Vogt, Radosław Wojciechowski (arXiv:1101.0373v1 [math.FA]; 2011.01.11) – http://arxiv.org/abs/1101.0373
- Abstract: Large time behaviour of heat semigroups (and more generally, of positive selfadjoint semigroups) is studied. Convergence of the semigroup to the ground state and of averaged logarithms of kernels to the ground state energy is shown in the general framework of positivity improving selfadjoint semigroups. This framework includes Laplacians on manifolds, metric graphs and discrete graphs.