## Prof. Dr. Matthias Keller

### Project leader

Universität Potsdam

E-mail: mkeller(at)math.uni-potsdam.de

Telephone: +49 331 977-2259

Homepage: https://www.math.uni-potsdam.de/professu…

## Project

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

## Publications within SPP2026

We describe the set of all Dirichlet forms associated to a given infinite graph in terms of Dirichlet forms on its Royden boundary. Our approach is purely analytical and uses form methods.

Journal | Journal de Mathématiques Pures et Appliquées. (9) |

Volume | 126 |

Pages | 109--143 |

Link to preprint version | |

Link to published version |

**Related project(s):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

In this note we study the eigenvalue growth of infinite graphs with discrete spectrum. We assume that the corresponding Dirichlet forms satisfy certain Sobolev-type inequalities and that the total measure is finite. In this sense, the associated operators on these graphs display similarities to elliptic operators on bounded domains in the continuum. Specifically, we prove lower bounds on the eigenvalue growth and show by examples that corresponding upper bounds can not be established.

Journal | to appear in Proceedings of the American Mathematical Society |

Link to preprint version |

**Related project(s):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

In this note we prove an optimal volume growth condition for stochastic completeness of graphs under very mild assumptions. This is realized by proving a uniqueness class criterion for the heat equation which is an analogue to a corresponding result of Grigor'yan on manifolds. This uniqueness class criterion is shown to hold for graphs that we call globally local, i.e., graphs where we control the jump size far outside. The transfer from general graphs to globally local graphs is then carried out via so called refinements.

**Related project(s):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

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 |

**Related project(s):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

We introduce a notion of nodal domains for positivity preserving forms. This notion generalizes the classical ones for Laplacians on domains and on graphs. We prove the Courant nodal domain theorem in this generalized setting using purely analytical methods.

Journal | to appear in Journal of Spectral Theory |

Link to preprint version | |

Link to published version |

**Related project(s):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

We study magnetic Schrödinger operators on graphs. We extend the notion of sparseness of graphs by including a magnetic quantity called the frustration index. This notion of magnetic sparse turn out to be equivalent to the fact that the form domain is an \(\ell^2\) space. As a consequence, we get criteria of discreteness for the spectrum and eigenvalue asymptotics.

**Related project(s):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces

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.

**Related project(s):****19**Boundaries, Greens formulae and harmonic functions for graphs and Dirichlet spaces