# Prof. Dr. Matthias Keller

### Professor

Kontakt
Raum:
2.09.2.18
Telefon:
+49 331 977-2259
Fax:
+49 331 977-2899

#### Sprechzeiten:

nach Vereinbarung

## Research interests

• Dirichlet forms and unbounded operators on graphs
• Spectral and geometric properties of non-positively curved graphs
• Spectral theory of random Schrödinger operators

## Preprints

Criticality theory for Schrödinger operators on graphs (mit Yehuda Pinchover, Felix Pogorzelski)

A note on eigenvalue bounds for non-compact manifolds (mit Shiping Liu, Norbert Peyerimhoff)

Magnetic sparseness and Schrödinger operators on graphs (mit Michel Bonnefont, Sylvain Golénia,  Shiping Liu, Florentin Münch)

Sobolev-Type Inequalities and Eigenvalue Growth on Graphs mit Finite Measure (mit Bobo Hua,  Michael Schwarz, Melchior Wirth)

Boundary representation of Dirichlet forms on discrete spaces (mit Daniel Lenz, Marcel Schmidt, Michael Schwarz

Courant's Nodal Domain Theorem for Positivity Preserving Forms (mit Michael Schwarz)

On the uniqueness class, stochastic completeness and volume growth for graphs (mit Xueping Huang, Marcel Schmidt).

## PhD thesis

"On the spectral theory of operators on trees"
or at arxiv.org: here

## Extended abstracts and lecture notes

On Cheeger's inequality for graphs, Oberwolfach Report No. 7/2015, February 2015.

An overview of curvature bounds and spectral theory of planar tessellations, Proceedings of the CIRM Meeting, 3 nr 1, Discrete Curvature; theory and applications, 2013.

Absolutely continuous spectrum on trees-random potentials, random hopping and Galton-Watson trees  Oberwolfach Report No. 50/2011, October 2011.

"Curvature and spectrum on graphs" Oberwolfach Report No. 02/2012, January 2012.

"A minicourse on the L^p spectrum of graphs", Bizerte, Tunesia, March 2014.

## Publications

2019 | A new discrete Hopf-Rinow theorem | Matthias Keller, Florentin Münch Zeitschrift: Discrete Mathematics Seiten: 2751-2757 Band: 342 Link zur Publikation, Link zum Preprint

### A new discrete Hopf-Rinow theorem

#### Autoren: Matthias Keller, Florentin Münch (2019)

We prove a version of the Hopf–Rinow theorem with respect to path metrics on discrete spaces. The novel aspect is that we do not a priori assume local finiteness but isolate a local finiteness type condition, called essentially locally finite, that is indeed necessary. As a side product we identify the maximal weight, called the geodesic weight, generating the path metric in the situation when the space is complete with respect to any of the equivalent notions of completeness proven in the Hopf–Rinow theorem. As an application we characterize the graphs for which the resistance metric is a path metric induced by the graph structure.

Zeitschrift:
Discrete Mathematics
Seiten:
2751-2757
Band:
342

2018 | Scattering the Geometry of Weighted Graphs | Güneysu, Batu; Keller, Matthias Zeitschrift: Math. Phys. Anal. Geom. Seiten: 21-28 Band: 21 Link zur Publikation, Link zum Preprint

### Scattering the Geometry of Weighted Graphs

#### Autoren: Güneysu, Batu; Keller, Matthias (2018)

Given two weighted graphs (X, bk, mk), k = 1,2 with b1b2 and m1m2, we prove a weighted L1-criterion for the existence and completeness of the wave operators W±(H2, H1, I1,2), where Hk denotes the natural Laplacian in 2(X, mk) w.r.t. (X, bk, mk) and I1,2 the trivial identification of 2(X, m1) with 2(X, m2). In particular, this entails a general criterion for the absolutely continuous spectra of H1 and H2 to be equal.

Zeitschrift:
Math. Phys. Anal. Geom.
Seiten:
21-28
Band:
21

2018 | Optimal Hardy inequalities for Schrödinger operators on graphs | Keller, Matthias; Pinchover, Yehuda; Pogorzelski, Felix Zeitschrift: Comm. Math. Phys. Seiten: 767–790 Band: 358 Link zur Publikation, Link zum Preprint

### Optimal Hardy inequalities for Schrödinger operators on graphs

#### Autoren: Keller, Matthias; Pinchover, Yehuda; Pogorzelski, Felix (2018)

For a given subcritical discrete Schr\"odinger operator $H$ on a  weighted infinite graph $X$, we construct a Hardy-weight $w$ which is in the following sense. The operator $H - \lambda w$ is subcritical in $X$ for all $\lambda < 1$, null-critical in $X$ for $\lambda = 1$, and supercritical near any neighborhood of infinity in $X$ for any $\lambda > 1$.  Our results rely on a criticality theory for Schr\"odinger operators on general weighted graphs.

Zeitschrift:
Comm. Math. Phys.
Seiten:
767–790
Band:
358

2018 | The Kazdan-Warner equation on canonically compactifiable graphs | Keller, Matthias; Schwarz, Michael Zeitschrift: Calc. Var. Partial Differential Equations Seiten: 18 pp. Band: 57 Link zur Publikation, Link zum Preprint,

### The Kazdan-Warner equation on canonically compactifiable graphs

#### Autoren: Keller, Matthias; Schwarz, Michael (2018)

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.

Zeitschrift:
Calc. Var. Partial Differential Equations
Seiten:
18 pp.
Band:
57

2018 | An improved discrete Hardy inequality | Matthias Keller, Yehuda Pinchover, Felix Pogorzelski Zeitschrift: American Mathematics Monthly Seiten: 347–350 Band: 125 Link zur Publikation, Link zum Preprint

### An improved discrete Hardy inequality

#### Autoren: Matthias Keller, Yehuda Pinchover, Felix Pogorzelski (2018)

We improve the classical discrete Hardy inequality

<nobr>n=1a2n(12)2n=1(a1+a2++ann)2,</nobr>

where <nobr>n=1</nobr> is any sequence of non-negative real numbers.

Zeitschrift:
American Mathematics Monthly
Seiten:
347–350
Band:
125

2017 | Sectional curvature of polygonal complexes with planar substructures | Matthias Keller, Norbert Peyerimhoff, Felix Pogorzelski Zeitschrift: Advances in Mathematics Seiten: 1070--1107. Band: 307 Link zur Publikation, Link zum Preprint

### Sectional curvature of polygonal complexes with planar substructures

#### Autoren: Matthias Keller, Norbert Peyerimhoff, Felix Pogorzelski (2017)

In this paper we introduce a class of polygonal complexes for which we can define a notion of sectional combinatorial curvature. These complexes can be viewed as generalizations of 2-dimensional Euclidean and hyperbolic buildings. We focus on the case of non-positive and negative combinatorial curvature. As geometric results we obtain a Hadamard-Cartan type theorem, thinness of bigons, Gromov hyperbolicity and estimates for the Cheeger constant. We employ the latter to get spectral estimates, show discreteness of the spectrum in the sense of a Donnelly-Li type theorem and present corresponding eigenvalue asymptotics. Moreover, we prove a unique continuation theorem for eigenfunctions and the solvability of the Dirichlet problem at infinity.

Zeitschrift:
Seiten:
1070--1107.
Band:
307

2017 | Global properties of Dirichlet forms in terms of Green's formula | aeseler, Sebastian; Keller, Matthias; Lenz, Daniel; Masamune, Jun; Schmidt, Marcel; Zeitschrift: Calc. Var. Partial Differential Equations Seiten: 56-124. Band: 56 Link zum Preprint

### Global properties of Dirichlet forms in terms of Green's formula

#### Autoren: aeseler, Sebastian; Keller, Matthias; Lenz, Daniel; Masamune, Jun; Schmidt, Marcel; (2017)

Zeitschrift:
Calc. Var. Partial Differential Equations
Seiten:
56-124.
Band:
56

2017 | Geometric and spectral consequences of curvature bounds on tessellations | Matthias Keller Reihe: Lecture Notes in Mathematics Verlag: Springer Buchtitel: Modern Approaches to Discrete Curvature Band: 2184 Link zur Publikation, Link zum Preprint

### Geometric and spectral consequences of curvature bounds on tessellations

#### Autoren: Matthias Keller (2017)

Reihe:
Lecture Notes in Mathematics
Verlag:
Springer
Buchtitel:
Modern Approaches to Discrete Curvature
Band:
2184

2016 | Note on uniformly transient graphs | Matthias Keller, Daniel Lenz, Marcel Schmidt, Radosław K. Wojciechowski Zeitschrift: Revista Iberoamericana Seiten: 831–860 Band: 33 Link zur Publikation, Link zum Preprint

### Note on uniformly transient graphs

#### Autoren: Matthias Keller, Daniel Lenz, Marcel Schmidt, Radosław K. Wojciechowski (2016)

We study a special class of graphs with a strong transience feature called uniform transience. We characterize uniform transience via a Feller-type property and via validity of an isoperimetric inequality. We then give a further characterization via equality of the Royden boundary and the harmonic boundary and show that the Dirichlet problem has a unique solution for such graphs. The Markov semigroups and resolvents (with Dirichlet boundary conditions) on these graphs are shown to be ultracontractive. Moreover, if the underlying measure is finite, the semigroups and resolvents are trace class and their generators have $\ell^p$ independent pure point spectra (for $1 \leq p \leq \infty$). Examples of uniformly transient graphs include Cayley graphs of hyperbolic groups as well as trees and Euclidean lattices of dimension at least three. As a surprising consequence, the Royden compactification of such lattices turns out to be the one-point compacitifcation and the Laplacians of such lattices have pure point spectrum if the underlying measure is chosen to be finite.

Zeitschrift:
Revista Iberoamericana
Seiten:
831–860
Band:
33

2016 | Note on short time behavior of semigroups associated to selfadjoint operators | Matthias Keller, Daniel Lenz, Florentin Münch, Marcel Schmidt, Andras Telcs Zeitschrift: Bullettin of the London Mathematical Society, to appear Link zum Preprint

### Note on short time behavior of semigroups associated to selfadjoint operators

#### Autoren: Matthias Keller, Daniel Lenz, Florentin Münch, Marcel Schmidt, Andras Telcs (2016)

We present a simple observation showing that the heat kernel on a locally finite graph behaves for short times $t$ roughly like $t^d$, where $d$ is the combinatorial distance. This is very different from the classical Varadhan type behavior on manifolds. Moreover, this also gives that short time behavior and global behavior of the heat kernel are governed by two different metrics whenever the degree of the graph is not uniformly bounded.

Zeitschrift:
Bullettin of the London Mathematical Society, to appear

2016 | General Cheeger inequalities for p-Laplacians on graphs | Matthias Keller, Delio Mugnolo Zeitschrift: Nonlinear Analysis: Theory, Methods & Applications Seiten: 80–95 Band: 147 Link zur Publikation, Link zum Preprint

### General Cheeger inequalities for p-Laplacians on graphs

#### Autoren: Matthias Keller, Delio Mugnolo (2016)

We prove Cheeger inequalities for p-Laplacians on finite and infinite weighted graphs. Unlike in previous works, we do not impose boundedness of the vertex degree, nor do we restrict ourselves to the normalized Laplacian and, more generally, we do not impose any boundedness assumption on the geometry. This is achieved by a novel definition of the measure of the boundary which is using the idea of intrinsic metrics. For the non-normalized case, our bounds on the spectral gap of p-Laplacians are already significantly better for finite graphs and for infinite graphs they yield non-trivial bounds even in the case of unbounded vertex degree. We, furthermore, give upper bounds by the Cheeger constant and by the exponential volume growth of distance balls.

Zeitschrift:
Nonlinear Analysis: Theory, Methods & Applications
Seiten:
80–95
Band:
147

2016 | A Feynman–Kac–Itô formula for magnetic Schrödinger operators on graphs | Batu Güneysu, Matthias Keller, Marcel Schmidt Zeitschrift: Probability Theory and Related Fields Seiten: 365-399 Band: 165 Link zur Publikation, Link zum Preprint

### A Feynman–Kac–Itô formula for magnetic Schrödinger operators on graphs

#### Autoren: Batu Güneysu, Matthias Keller, Marcel Schmidt (2016)

In this paper we prove a Feynman–Kac–Itô formula for magnetic Schrödinger operators on arbitrary weighted graphs. To do so, we have to provide a natural and general framework both on the operator theoretic and the probabilistic side of the equation. On the operator side we identify a very general class of potentials that allows the definition of magnetic Schrödinger operators. On the probabilistic side, we introduce an appropriate notion of stochastic line integrals with respect to magnetic potentials. Apart from linking the world of discrete magnetic operators with the probabilistic world through the Feynman–Kac–Itô formula, the insights from this paper gained on both sides should be of an independent interest. As applications of the Feynman–Kac–Itô formula, we prove a Kato inequality, a Golden–Thompson inequality and an explicit representation of the quadratic form domains corresponding to a large class of potentials.

Zeitschrift:
Probability Theory and Related Fields
Seiten:
365-399
Band:
165

2015 | Geometry and spectrum of rapidly branching graphs | Matthias Keller, Felix Pogorzelski, Florentin Münch Zeitschrift: Mathematische Nachrichten Seiten: 1636–1647 Band: 289 Link zur Publikation, Link zum Preprint

### Geometry and spectrum of rapidly branching graphs

#### Autoren: Matthias Keller, Felix Pogorzelski, Florentin Münch (2015)

We study graphs whose vertex degree tends and which are, therefore, called rapidly branching. We prove spectral estimates, discreteness of spectrum, first order eigenvalue and Weyl asymptotics solely in terms of the vertex degree growth. The underlying techniques are estimates on the isoperimetric constant. Furthermore, we give lower volume growth bounds and we provide a new criterion for stochastic incompleteness.

Zeitschrift:
Mathematische Nachrichten
Seiten:
1636–1647
Band:
289

2015 | Cheeger inequalities for unbounded graph Laplacians | Frank Bauer, Matthias Keller, Radoslaw Wojciechowski Zeitschrift: Journal of the European Mathematical Society Seiten: 259–271 Band: 17 Link zur Publikation, Link zum Preprint

### Cheeger inequalities for unbounded graph Laplacians

#### Autoren: Frank Bauer, Matthias Keller, Radoslaw Wojciechowski (2015)

We use the concept of intrinsic metrics to give a new definition for an isoperimetric constant of a graph. We use this novel isoperimetric constant to prove a Cheeger-type estimate for the bottom of the spectrum which is nontrivial even if the vertex degrees are unbounded.

Zeitschrift:
Journal of the European Mathematical Society
Seiten:
259–271
Band:
17

2015 | Eigenvalue asymptotics for Schrödinger operators on sparse graphs | Michel Bonnefont, Sylvain Golénia, Matthias Keller Zeitschrift: Annales de l'Institut Fourier Seiten: 1969-1998 Band: 65 Link zur Publikation, Link zum Preprint

### Eigenvalue asymptotics for Schrödinger operators on sparse graphs

#### Autoren: Michel Bonnefont, Sylvain Golénia, Matthias Keller (2015)

We consider Schrödinger operators on sparse graphs. The geometric definition of sparseness turn out to be equivalent to a functional inequality for the Laplacian. In consequence, sparseness has in turn strong spectral and functional analytic consequences. Specifically, one consequence is that it allows to completely describe the form domain. Moreover, as another consequence it leads to a characterization for discreteness of the spectrum. In this case we determine the first order of the corresponding eigenvalue asymptotics.

Zeitschrift:
Annales de l'Institut Fourier
Seiten:
1969-1998
Band:
65

2015 | Diffussion determines the recurrent graph | Matthias Keller, Daniel Lenz, Marcel Schmidt, Melchior Wirth Zeitschrift: Advances in Mathematics Seiten: 364–398 Band: 269 Link zur Publikation, Link zum Preprint

### Diffussion determines the recurrent graph

#### Autoren: Matthias Keller, Daniel Lenz, Marcel Schmidt, Melchior Wirth (2015)

We consider diffusion on discrete measure spaces as encoded by Markovian semigroups arising from weighted graphs. We study whether the graph is uniquely determined if the diffusion is given up to order isomorphism. If the graph is recurrent then the complete graph structure and the measure space are determined (up to an overall scaling). As shown by counterexamples this result is optimal. Without the recurrence assumption, the graph still turns out to be determined in the case of normalized diffusion on graphs with standard weights and in the case of arbitrary graphs over spaces in which each point has the same mass. These investigations provide discrete counterparts to studies of diffusion on Euclidean domains and manifolds initiated by Arendt and continued by Arendt/Biegert/ter Elst and Arendt/ter Elst. A crucial step in our considerations shows that order isomorphisms are actually unitary maps (up to a scaling) in our context.

Zeitschrift:
Seiten:
364–398
Band:
269

2015 | Graphs of finite measure | Agelos Georgakopoulos, Sebastian Haeseler, Matthias Keller, Daniel Lenz, Radosław Wojciechowski Zeitschrift: Journal de Mathématiques Pures et Appliquées Seiten: 1093–1131 Band: 103 Link zur Publikation, Link zum Preprint

### Graphs of finite measure

#### Autoren: Agelos Georgakopoulos, Sebastian Haeseler, Matthias Keller, Daniel Lenz, Radosław Wojciechowski (2015)

We consider weighted graphs with an infinite set of vertices. We show that boundedness of all functions of finite energy can be seen as a notion of ‘relative compactness’ for such graphs and study sufficient and necessary conditions for this property in terms of various metrics. We then equip graphs satisfying this property with a finite measure and investigate the associated Laplacian and its semigroup. In this context, our results include the trace class property for the semigroup, uniqueness and existence of solutions to the Dirichlet Problem with boundary arising from the natural compactification, an explicit description of the domain of the Dirichlet Laplacian, convergence of the heat semigroup for large times as well as stochastic incompleteness and transience of the corresponding random walk in continuous time.

Zeitschrift:
Journal de Mathématiques Pures et Appliquées
Seiten:
1093–1131
Band:
103

2015 | Intrinsic Metrics on Graphs: A Survey | Matthias Keller Reihe: Springer Proceedings in Mathematics & Statistics Verlag: Springer Buchtitel: Mathematical Technology of Networks Seiten: 81-119 Band: 128

### Intrinsic Metrics on Graphs: A Survey

#### Autoren: Matthias Keller (2015)

A few years ago various disparities for Laplacians on graphs and manifolds were discovered. The corresponding results are mostly related to volume growth in the context of unbounded geometry. Indeed, these disparities can now be resolved by using so called intrinsic metrics instead of the combinatorial graph distance. In this article we give an introduction to this topic and survey recent results in this direction. Specifically, we cover topics such as Liouville type theorems for harmonic functions, essential selfadjointness, stochastic completeness and upper escape rates. Furthermore, we determine the spectrum as a set via solutions, discuss upper and lower spectral bounds by isoperimetric constants and volume growth and study p-independence of spectra under a volume growth assumption.

Reihe:
Springer Proceedings in Mathematics & Statistics
Verlag:
Springer
Buchtitel:
Mathematical Technology of Networks
Seiten:
81-119
Band:
128

2015 | An invitation to trees of finite cone type: random and deterministic operators | Matthias Keller, Daniel Lenz, Simone Warzel Zeitschrift: Markov Processes and Related Filelds Seiten: 557-574 Band: 21 Link zur Publikation, Link zum Preprint

### An invitation to trees of finite cone type: random and deterministic operators

#### Autoren: Matthias Keller, Daniel Lenz, Simone Warzel (2015)

Trees of finite cone type have appeared in various contexts. In particular, they come up as simplified models of regular tessellations of the hyperbolic plane. The spectral theory of the associated Laplacians can thus be seen as induced by geometry. Here we give an introduction focusing on background and then turn to recent results for (random) perturbations of trees of finite cone type and their spectral theory.

Zeitschrift:
Markov Processes and Related Filelds
Seiten:
557-574
Band:
21

2013 | On the l^p spectrum of Laplacians on graphs | Frank Bauer, Bobo Hua, Matthias Keller Zeitschrift: Advances in Mathematics Seiten: 717–735 Band: 248 Link zur Publikation, Link zum Preprint

### On the l^p spectrum of Laplacians on graphs

#### Autoren: Frank Bauer, Bobo Hua, Matthias Keller (2013)

We study the p-independence of spectra of Laplace operators on graphs arising from regular Dirichlet forms on discrete spaces. Here, a sufficient criterion is given solely by a uniform subexponential growth condition. Moreover, under a mild assumption on the measure we show a one-sided spectral inclusion without any further assumptions. We study applications to normalized Laplacians including symmetries of the spectrum and a characterization for positivity of the Cheeger constant. Furthermore, we consider Laplacians on planar tessellations for which we relate the spectral p-independence to assumptions on the curvature.

Zeitschrift:
Seiten:
717–735
Band:
248

2013 | Harmonic functions of general graph Laplacians | Bobo Hua, Matthias Keller Zeitschrift: Calculus of Variations and Partial Differential Equations Seiten: 343-362 Band: 51 Link zur Publikation, Link zum Preprint

### Harmonic functions of general graph Laplacians

#### Autoren: Bobo Hua, Matthias Keller (2013)

We study harmonic functions on general weighted graphs which allow for a compatible intrinsic metric. We prove an $$L^$$ Liouville type theorem which is a quantitative integral $$L^$$ estimate of harmonic functions analogous to Karp’s theorem for Riemannian manifolds. As corollaries we obtain Yau’s $$L^$$-Liouville type theorem on graphs, identify the domain of the generator of the semigroup on $$L^$$ and get a criterion for recurrence. As a side product, we show an analogue of Yau’s $$L^$$ Caccioppoli inequality. Furthermore, we derive various Liouville type results for harmonic functions on graphs and harmonic maps from graphs into Hadamard spaces.

Zeitschrift:
Calculus of Variations and Partial Differential Equations
Seiten:
343-362
Band:
51

2013 | Volume growth and bounds for the essential spectrum for Dirichlet forms | Sebastian Haeseler, Matthias Keller, Radoslaw Wojciechowski Zeitschrift: Journal of the London Mathematical Society Seiten: 883-898 Band: 88 Link zur Publikation, Link zum Preprint

### Volume growth and bounds for the essential spectrum for Dirichlet forms

#### Autoren: Sebastian Haeseler, Matthias Keller, Radoslaw Wojciechowski (2013)

We consider operators arising from regular Dirichlet forms with vanishing killing term. We give bounds for the bottom of the (essential) spectrum in terms of exponential volume growth with respect to an intrinsic metric. As special cases, we discuss operators on graphs. When the volume growth is measured in the natural graph distance (which is not an intrinsic metric), we discuss the threshold for positivity of the bottom of the spectrum and finiteness of the bottom of the essential spectrum of the (unbounded) graph Laplacian. This threshold is shown to lie at cubic polynomial growth.

Zeitschrift:
Journal of the London Mathematical Society
Seiten:
883-898
Band:
88

2013 | Spectral Analysis of Certain Spherically Homogeneous Graphs | Jonathan Breuer, Matthias Keller Zeitschrift: Operators and Matrices Seiten: 825-847 Band: 7 Link zur Publikation, Link zum Preprint

### Spectral Analysis of Certain Spherically Homogeneous Graphs

#### Autoren: Jonathan Breuer, Matthias Keller (2013)

We study operators on rooted graphs with a certain spherical homogeneity. These graphs are called path commuting and allow for a decomposition of the adjacency matrix and the Laplacian into a direct sum of Jacobi matrices which reflect the structure of the graph. Thus, the spectral properties of the adjacency matrix and the Laplacian can be analyzed by means of the elaborated theory of Jacobi matrices. For some examples which include antitrees, we derive the decomposition explicitly and present a zoo of spectral behavior induced by the geometry of the graph. In particular, these examples show that spectral types are not at all stable under rough isometries.

Zeitschrift:
Operators and Matrices
Seiten:
825-847
Band:
7

2013 | A note on self-adjoint extensions of the Laplacian on weighted graphs | Xueping Huang, Matthias Keller, Jun Masamune, Radoslaw Wojciechowski Zeitschrift: Journal of Functional Analysis Seiten: 1556-1578 Band: 265 Link zur Publikation, Link zum Preprint

### A note on self-adjoint extensions of the Laplacian on weighted graphs

#### Autoren: Xueping Huang, Matthias Keller, Jun Masamune, Radoslaw Wojciechowski (2013)

We study the uniqueness of self-adjoint and Markovian exten

sions of the Laplacian

on weighted graphs. We first show that, for locally finite grap

hs and a certain family of metrics,

completeness of the graph implies uniqueness of these exten

sions. Moreover, in the case when the

graph is not metrically complete and the Cauchy boundary has

finite capacity, we characterize

the uniqueness of the Markovian extensions

Zeitschrift:
Journal of Functional Analysis
Seiten:
1556-1578
Band:
265

2013 | On the spectral theory of trees with finite cone type | Matthias Keller, Daniel Lenz, Simone Warzel Zeitschrift: Israel Journal of Mathematics Seiten: 107-135 Band: 194 Link zur Publikation, Link zum Preprint

### On the spectral theory of trees with finite cone type

#### Autoren: Matthias Keller, Daniel Lenz, Simone Warzel (2013)

We study basic spectral features of graph Laplacians associated with a class of rooted trees which contains all regular trees. Trees in this class can be generated by substitution processes. Their spectra are shown to be purely absolutely continuous and to consist of finitely many bands. The main result gives stability of the absolutely continuous spectrum under sufficiently small radially label symmetric perturbations for non-regular trees in this class. In sharp contrast, the absolutely continuous spectrum can be completely destroyed by arbitrary small radially label symmetric perturbations for regular trees in this class.

Zeitschrift:
Israel Journal of Mathematics
Seiten:
107-135
Band:
194

2012 | Absolutely continuous spectrum for multi-type Galton Watson trees | Matthias Keller Zeitschrift: Annales Henri Poincaré Seiten: 1745-1766 Band: 13 Link zur Publikation, Link zum Preprint

### Absolutely continuous spectrum for multi-type Galton Watson trees

#### Autoren: Matthias Keller (2012)

We consider multi-type Galton Watson trees that are close to a tree of finite cone type in distribution. Moreover, we impose that each vertex has at least one forward neighbor. Then, we show that the spectrum of the Laplace operator exhibits almost surely a purely absolutely continuous component which is included in the absolutely continuous spectrum of the tree of finite cone type.

Zeitschrift:
Annales Henri Poincaré
Seiten:
1745-1766
Band:
13

2012 | Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions | Sebastian Haeseler, Matthias Keller, Daniel Lenz, Radoslaw Wojciechowski Zeitschrift: Journal of Spectral theory Seiten: 397-432 Band: 2 Link zur Publikation, Link zum Preprint

### Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions

#### Autoren: Sebastian Haeseler, Matthias Keller, Daniel Lenz, Radoslaw Wojciechowski (2012)

We study Laplacians associated to a graph and single out a class of such operators with special regularity properties. In the case of locally finite graphs, this class consists of all selfadjoint, non-negative restrictions of the standard formal Laplacian and we can characterize the Dirichlet and Neumann Laplacians as the largest and smallest Markovian restrictions of the standard formal Laplacian. In the case of general graphs, this class contains the Dirichlet and Neumann Laplacians and we describe howthesemay differ fromeach other, characterize when they agree, and study connections to essential selfadjointness and stochastic completeness.

Finally, we study basic common features of all Laplacians associated to a graph. In particular, we characterize when the associated semigroup is positivity improving and present some basic estimates on its long term behavior. We also discuss some situations in which the Laplacian associated to a graph is unique and, in this context, characterize its boundedness.

Zeitschrift:
Journal of Spectral theory
Seiten:
397-432
Band:
2

2012 | Absolutely continuous spectrum for random operators on trees of finite cone type | Matthias Keller, Daniel Lenz, Simone Warzel Zeitschrift: Journal d'Analyse Mathematique Seiten: 363-396. Band: 118 Link zur Publikation, Link zum Preprint

### Absolutely continuous spectrum for random operators on trees of finite cone type

#### Autoren: Matthias Keller, Daniel Lenz, Simone Warzel (2012)

We study the spectrum of random operators on a large class of trees. These trees have finitely many cone types and they can be constructed by a substitution rule. The random operators are perturbations of Laplace type operators either by random potentials or by random hopping terms, i.e., perturbations of the off-diagonal elements. We prove stability of arbitrary large parts of the absolutely continuous spectrum for sufficiently small but extensive disorder.

Zeitschrift:
Journal d'Analyse Mathematique
Seiten:
363-396.
Band:
118

2012 | Dirichlet forms and stochastic completeness of graphs and subgraphs | Matthias Keller, Daniel Lenz Zeitschrift: Journal für die reine und angewandte Mathematik (Crelle's Journal) Seiten: 189-223 Band: 2012 Link zur Publikation, Link zum Preprint

### Dirichlet forms and stochastic completeness of graphs and subgraphs

#### Autoren: Matthias Keller, Daniel Lenz (2012)

We study Laplacians on graphs and networks via regular Dirichlet forms. We give a sufficient geometric condition for essential selfadjointness and explicitly determine the generators of the associated semigroups on all ℓp, 1 ≦ p < ∞, in this case. We characterize stochastic completeness thereby generalizing all earlier corresponding results for graph Laplacians. Finally, we study how stochastic completeness of a subgraph is related to stochastic completeness of the whole graph.

Zeitschrift:
Journal für die reine und angewandte Mathematik (Crelle's Journal)
Seiten:
189-223
Band:
2012

2011 | Generalized solutions and spectrum for Dirichlet forms on graphs | Sebastian Haeseler, Matthias Keller Reihe: Progress in Probability Verlag: Birkhäuser Buchtitel: Random Walks, Boundaries and Spectra Seiten: 181-201 Link zur Publikation, Link zum Preprint

### Generalized solutions and spectrum for Dirichlet forms on graphs

#### Autoren: Sebastian Haeseler, Matthias Keller (2011)

In the framework of regular Dirichlet forms we consider operators on infinite graphs. We study the connection of the existence of solutions with certain properties and the spectrum of the operators. In particular we prove a version of the Allegretto-Piepenbrink theorem which says that positive (super)-solutions to a generalized eigenvalue equation exist exactly for energies not exceeding the infimum of the spectrum. Moreover we show a version of Shnol’s theorem, which says that existence of solutions satisfying a growth condition with respect to a given boundary measure implies that the corresponding energy is in the spectrum.

Reihe:
Progress in Probability
Verlag:
Birkhäuser
Buchtitel:
Random Walks, Boundaries and Spectra
Seiten:
181-201

2011 | Curvature, geometry and spectral properties of planar graphs | Matthias Keller Zeitschrift: Discrete & Computational Geometry Seiten: 500-525 Band: 46 Link zur Publikation, Link zum Preprint

### Curvature, geometry and spectral properties of planar graphs

#### Autoren: Matthias Keller (2011)

We introduce a curvature function for planar graphs to study the connection between the curvature and the geometric and spectral properties of the graph. We show that non-positive curvature implies that the graph is infinite and locally similar to a tessellation. We use this to extend several results known for tessellations to general planar graphs. For non-positive curvature, we show that the graph admits no cut locus and we give a description of the boundary structure of distance balls. For negative curvature, we prove that the interiors of minimal bigons are empty and derive explicit bounds for the growth of distance balls and Cheeger’s constant. The latter are used to obtain lower bounds for the bottom of the spectrum of the discrete Laplace operator. Moreover, we give a characterization for triviality of the essential spectrum by uniform decrease of the curvature. Finally, we show that non-positive curvature implies the absence of finitely supported eigenfunctions for nearest neighbor operators.

Zeitschrift:
Discrete & Computational Geometry
Seiten:
500-525
Band:
46

2011 | Cheeger constants, growth and spectrum of locally tessellating planar graphs (with Norbert Peyerimhoff), | Matthias Keller, Norbert Peyerimhoff Zeitschrift: Mathematische Zeitschrift Seiten: 871-886 Band: 268 Link zur Publikation, Link zum Preprint

### Cheeger constants, growth and spectrum of locally tessellating planar graphs (with Norbert Peyerimhoff),

#### Autoren: Matthias Keller, Norbert Peyerimhoff (2011)

In this article, we study relations between the local geometry of planar graphs (combinatorial curvature) and global geometric invariants, namely the Cheeger constants and the exponential growth. We also discuss spectral applications.

Zeitschrift:
Mathematische Zeitschrift
Seiten:
871-886
Band:
268

2010 | Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation | Matthias Keller, Daniel Lenz Zeitschrift: Mathematical Modeling of Natural Phenomena Buchtitel: "Spectral Problems" Seiten: 198-204 Band: 5 Link zur Publikation, Link zum Preprint

### Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation

#### Autoren: Matthias Keller, Daniel Lenz (2010)

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.

Zeitschrift:
Mathematical Modeling of Natural Phenomena
Buchtitel:
"Spectral Problems"
Seiten:
198-204
Band:
5

2010 | The essential spectrum of the Laplacian on rapidly branching tessellations | Matthias Keller Zeitschrift: Mathematische Annalen Seiten: 51-66 Band: 346 Link zur Publikation, Link zum Preprint

### The essential spectrum of the Laplacian on rapidly branching tessellations

#### Autoren: Matthias Keller (2010)

In this paper, we characterize absence of the essential spectrum of the Laplacian under a hyperbolicity assumption for general graphs. Moreover, we present a characterization for absence of the essential spectrum for planar tessellations in terms of curvature.

Zeitschrift:
Mathematische Annalen
Seiten:
51-66
Band:
346

## Places

University of Potsdam since 10/2015 Professor

Technion Haifa 03-07/2019 Visiting Associate Professor, 02-03/2015 Visiting Assistant Professor

Hebrew University Jerusalem  10/2012-09/2013 and 02-06/2011 Post-Doc

Friedrich Schiller University Jena 10/2008-09/2015 Post-Doc and PhD Student

Princeton University 10/2007-05/2008 Visiting Student/Research Collaborator

TU Chemnitz 07/2006-09/2008 PhD Student

## Education

Habilitation 15/2015, Friedrich Schiller University Jena

PhD 12/2010, Friedrich Schiller University Jena

Diploma 06/2006, TU Chemnitz

## Grants and Projects

DFG Project "Hardy inequalities on graphs and Dirichlet spaces" (joint with Y. Pinchover and F. Pogorzelski) since 05/2019

DFG Project "Boundaries, Greens formulae and  harmonic functions for graphs and Dirichlet spaces - follow up" (joint with D. Lenz and Marcel Schmidt) since 02/2018

DFG Project within the priority programme "Geometry at infinity",  "Boundaries, Greens formulae and  harmonic functions for graphs and Dirichlet spaces" (joint with D. Lenz) since 06/2017

Programme to support junior researchers in obtaining third-party funding, Line A  (Advanced), 09/2014-09/2015

Golda Meir Fellowship 10/2012 - 09/2013

Short visit grant ESF 07/2012

DFG Project "Geometry of discrete spaces and spectral theory of non-local operators" (joint with D. Lenz) since 07/2012

PhD Fellowship Klaus Murmann Foundation (sdw)  07/2007 - 06/2010

## Selected Talks

June 2019, Geometric aspects of harmonic analysis and spectral theory, Discrete spectrum for graphs, Technion Haifa.

January 2019, Spectral Methods in Mathematical Physics, On optimal Hardy inequalities on graphs, Mittag-Leffler-Institute.

September 2018, Upper curvature bounds and spectral theory, 240 minute course, Summer school Generalized Curvatures GenCurv2018, EPFL Lausanne.

March 2017, TSIMF Sanya China, Curvatures of Graphs, Simplicial Complexes and Metric Spaces Workshop, Sectional curvature of polygonal complexes with planar substructures.

August 2016, Euler Institute St. Petersburg, OTAMP 2016, Optimal Hardy inequalities on graphs.

June 2015, BIRS Banff, Groups, Graphs and Stochastic Processes, 'On the compactification of graphs ... the Royden compactification revisited'

March 2014, Université de Carthage, Bizerte, Tunesia, Cours pour Doctorants, 'L^p Spectrum of Graphs' (lecture notes) and Journée-WorkShop "Géométrie et Analyse sur les Graphes" 'Curvature and Spectrum on Tessellating Graphs'

November 2013, C.I.R.M. Luminy, A colloquium on discrete curvature, 'On the spectral theory of negatively curved planar graphs'

July 2013, LMS - EPSRC Durham Symposium, Graph Theory and Interactions 'On negative curvature and spectrum of graph Laplacians' (Video)

August 2012, Conference Spectral Theory and  Differential Operators at TU Graz, "Volume growth and spectra of Dirichlet forms"

Januar 2012, Oberwolfach Mini-Workshop: 'Boundary Value Problems and Spectral Geometry', "Curvature and spectrum on graphs"

October 2011, Oberwolfach Workshop 'Correlations and Interactions for Random Quantum Systems', "Absolutely continuous spectrum on trees"

July 2010, Isaac Newton Institute for Mathematical Sciences Cambridge, Workshop on Analysis on Graphs and its Applications, "Absolutely continuous spectrum for trees of finite forward cone type" (Video)

July 2009, St. Kathrein, "Alp-Workshop": "Random trees and absolutely continuous spectrum"

## (Co-)Supervision

Christian Scholz,  Topic on heat equation on graph, PhD

Florentin Münch, Discrete Ricci curvature, diameter bounds and rigidity, PhD, submitted.

Michael Schwarz, Nodal Domains and Boundary Representation for Dirichlet Forms, PhD, submitted.

Florian Fischer, Existence of certain positive solutions in criticality theory, Master

Florian Fischer, Riesz Decompositions and Martin Compactification Theory for Schrödinger Operators on Graphs, Master 2018

Christian Scholz, Boundary conditions and resolvent limits of graphs, Master 2017.

Sarah Burchert, The Puisseux expansion, Bachelor 2017.

Katja Ohde, Geometry of rapidly growing graphs, Bachelor 2016.

Florentin Münch, Li-Yau inequalities on finite graphs, Master Jena 2014

Melchior Wirth, Does diusion determine the graph structure?, Bachelor Jena 2013.

Oliver Siebert, Spectra of lamplighter random walks associated with percolation, Bachelor Jena 2013

Florentin Münch, Ultrametrische Cantormengen und Ränder von Bäumen (Ultra metric Cantor sets and boundaries of trees), Bachelor Jena 2012

Ricardo Kalke, Ricci curvature on graphs, Staatsexamen Jena 2012

## (Co-)Organized Scientific meetings

Oberwolfach Mini-Workshop, Recent Progress in Path Integration on Graphs and Manifolds, with Batu Güneysu, Kazumasu Kuwada, Anton Thalmeier, April 2019.

Two Day Workshop, Dirichlet forms on graphs, Friedrich Schiller University Jena, June 2018.

Conference Analysis and Geometry on Graphs and Manifolds, Potsdam University 2017.

Workshop on Discrete Analysis, Fudan University Shanghai, August 2016.

Workshop on Spectral Geometry, University of Potsdam, January 2016.

One Day Workshop 2015 "New Directions in Mathematical Physics and beyond", Jena January 14th 2015 (organized by Gerhard Bräunlich, Matthias Keller, Markus Lange, Marcel Schmidt)

International Conference Fractal Geometry and Stochastics V, in Tabarz, local organizing committee

Geometric aspects in probability and analysis September 14th 2013 in Jena (organized by Matthias Keller, Daniel Lenz and Marcel Schmidt)

One Day Workshop "Schrödingeroperators - December 8th 2011, Friedrich Schiller Universität Jena

Graduate student symposium within the Summer school on "Graphs and spectra" held at the TU Chemnitz in the week 18--23 July 2011, for more information see here

Graduate student symposium September 15-16th 2010, Friedrich Schiller University Jena

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