TY - JOUR A1 - Keller, Matthias A1 - Schwarz, Michael T1 - The Kazdan-Warner equation on canonically compactifiable graphs JF - Calculus of variations and partial differential equations N2 - 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. Y1 - 2018 U6 - https://doi.org/10.1007/s00526-018-1329-7 SN - 0944-2669 SN - 1432-0835 VL - 57 IS - 2 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Güneysu, Batu A1 - Keller, Matthias T1 - Scattering the Geometry of Weighted Graphs JF - Mathematical physics, analysis and geometry : an international journal devoted to the theory and applications of analysis and geometry to physics N2 - Given two weighted graphs (X, b(k), m(k)), k = 1, 2 with b(1) similar to b(2) and m(1) similar to m(2), we prove a weighted L-1-criterion for the existence and completeness of the wave operators W-+/- (H-2, H-1, I-1,I-2), where H-k denotes the natural Laplacian in l(2)(X, m(k)) w.r.t. (X, b(k), m(k)) and I-1,I-2 the trivial identification of l(2)(X, m(1)) with l(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. KW - Graphs KW - Laplacian KW - Scattering theory Y1 - 2018 U6 - https://doi.org/10.1007/s11040-018-9285-1 SN - 1385-0172 SN - 1572-9656 VL - 21 IS - 3 PB - Springer CY - Dordrecht ER - TY - JOUR A1 - Fischer, Florian A1 - Keller, Matthias T1 - Riesz decompositions for Schrödinger operators on graphs JF - Journal of mathematical analysis and applications N2 - We study superharmonic functions for Schrodinger operators on general weighted graphs. Specifically, we prove two decompositions which both go under the name Riesz decomposition in the literature. The first one decomposes a superharmonic function into a harmonic and a potential part. The second one decomposes a superharmonic function into a sum of superharmonic functions with certain upper bounds given by prescribed superharmonic functions. As application we show a Brelot type theorem. KW - Potential theory KW - Green's function KW - Schrödinger operator KW - Weighted KW - graph KW - Subcritical KW - Greatest harmonic minorant Y1 - 2021 U6 - https://doi.org/10.1016/j.jmaa.2020.124674 SN - 0022-247X SN - 1096-0813 VL - 495 IS - 1 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Keller, Matthias A1 - Pinchover, Yehuda A1 - Pogorzelski, Felix T1 - Optimal Hardy inequalities for Schrodinger operators on graphs JF - Communications in mathematical physics N2 - For a given subcritical discrete Schrodinger operator H on a weighted infinite graph X, we construct a Hardy-weight w which is optimal 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 Schrodinger operators on general weighted graphs. Y1 - 2018 U6 - https://doi.org/10.1007/s00220-018-3107-y SN - 0010-3616 SN - 1432-0916 VL - 358 IS - 2 SP - 767 EP - 790 PB - Springer CY - New York ER - TY - JOUR A1 - Keller, Matthias A1 - Lenz, Daniel A1 - Münch, Florentin A1 - Schmidt, Marcel A1 - Telcs, Andras T1 - Note on short-time behavior of semigroups associated to self-adjoint operators JF - Bulletin of the London Mathematical Society N2 - 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. Y1 - 2016 U6 - https://doi.org/10.1112/blms/bdw054 SN - 0024-6093 SN - 1469-2120 VL - 48 SP - 935 EP - 944 PB - Oxford Univ. Press CY - Oxford ER - TY - JOUR A1 - Keller, Matthias A1 - Münch, Florentin A1 - Pogorzelski, Felix T1 - Geometry and spectrum of rapidly branching graphs JF - Mathematische Nachrichten N2 - We study graphs whose vertex degree tends to infinity 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. (C) 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim KW - Graph Laplacians KW - discrete spectrum KW - eigenvalue asymptotics KW - isoperimetric estimates KW - stochastic completeness Y1 - 2016 U6 - https://doi.org/10.1002/mana.201400349 SN - 0025-584X SN - 1522-2616 VL - 289 SP - 1636 EP - 1647 PB - Wiley-VCH CY - Weinheim ER - TY - JOUR A1 - Keller, Matthias A1 - Mugnolo, Delio T1 - General Cheeger inequalities for p-Laplacians on graphs JF - Theoretical ecology N2 - 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 uses 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. (C) 2016 Elsevier Ltd. All rights reserved. KW - Cheeger inequalities KW - Spectral theory of graphs KW - Intrinsic metrics for Dirichlet forms Y1 - 2016 U6 - https://doi.org/10.1016/j.na.2016.07.011 SN - 0362-546X SN - 1873-5215 VL - 147 SP - 80 EP - 95 PB - Elsevier CY - Oxford ER - TY - JOUR A1 - Keller, Matthias A1 - Pinchover, Yehuda A1 - Pogorzelski, Felix T1 - From hardy to rellich inequalities on graphs JF - Proceedings of the London Mathematical Society N2 - We show how to deduce Rellich inequalities from Hardy inequalities on infinite graphs. Specifically, the obtained Rellich inequality gives an upper bound on a function by the Laplacian of the function in terms of weighted norms. These weights involve the Hardy weight and a function which satisfies an eikonal inequality. The results are proven first for Laplacians and are extended to Schrodinger operators afterwards. KW - 35R02 KW - 39A12 (primary) KW - 26D15 KW - 31C20 KW - 35B09 KW - 58E35 (secondary) Y1 - 2020 U6 - https://doi.org/10.1112/plms.12376 SN - 0024-6115 SN - 1460-244X VL - 122 IS - 3 SP - 458 EP - 477 PB - Wiley CY - Hoboken ER - TY - JOUR A1 - Güneysu, Batu A1 - Keller, Matthias T1 - Feynman path integrals for magnetic Schrödinger operators on infinite weighted graphs JF - Journal d'analyse mathématique N2 - We prove a Feynman path integral formula for the unitary group exp(-itL(nu,theta)), t >= 0, associated with a discrete magnetic Schrodinger operator L-nu,L-theta on a large class of weighted infinite graphs. As a consequence, we get a new Kato-Simon estimate vertical bar exp(- itL(nu,theta))(x,y)vertical bar <= exp( -tL(-deg,0))(x,y), which controls the unitary group uniformly in the potentials in terms of a Schrodinger semigroup, where the potential deg is the weighted degree function of the graph. Y1 - 2020 U6 - https://doi.org/10.1007/s11854-020-0110-y SN - 0021-7670 SN - 1565-8538 VL - 141 IS - 2 SP - 751 EP - 770 PB - The Magnes Press, the Hebrew Univ. CY - Jerusalem ER - TY - JOUR A1 - Keller, Matthias A1 - Pinchover, Yehuda A1 - Pogorzelski, Felix T1 - Criticality theory for Schrödinger operators on graphs JF - Journal of spectral theory N2 - We study Schrodinger operators given by positive quadratic forms on infinite graphs. From there, we develop a criticality theory for Schrodinger operators on general weighted graphs. KW - green function KW - ground state KW - positive solutions KW - discrete Schrodinger KW - operators KW - weighted graphs Y1 - 2019 U6 - https://doi.org/10.4171/JST/286 SN - 1664-039X SN - 1664-0403 VL - 10 IS - 1 SP - 73 EP - 114 PB - European Mathematical Society CY - Zürich ER - TY - JOUR A1 - Keller, Matthias A1 - Schwarz, Michael T1 - Courant’s nodal domain theorem for positivity preserving forms JF - Journal of spectral theory N2 - 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. KW - Nodal domain KW - eigenfunction KW - Dirichlet form KW - compact resolvent Y1 - 2020 U6 - https://doi.org/10.4171/JST/292 SN - 1664-039X SN - 1664-0403 VL - 10 IS - 1 SP - 271 EP - 309 PB - EMS Publishing House CY - Zürich ER - TY - JOUR A1 - Keller, Matthias A1 - Lenz, Daniel A1 - Schmidt, Marcel A1 - Schwarz, Michael T1 - Boundary representation of Dirichlet forms on discrete spaces JF - Journal de Mathématiques Pures et Appliquées N2 - 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. (C) 2018 Elsevier Masson SAS. KW - Dirichlet form KW - Royden boundary KW - Infinite graph KW - Harmonic measure KW - Trace Dirichlet form Y1 - 2019 U6 - https://doi.org/10.1016/j.matpur.2018.10.005 SN - 0021-7824 SN - 1776-3371 VL - 126 SP - 109 EP - 143 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Keller, Matthias A1 - Pinchover, Yehuda A1 - Pogorzelski, Felix T1 - An improved discrete hardy inequality JF - The American mathematical monthly : an official publication of the Mathematical Association of America N2 - In this note, we prove an improvement of the classical discrete Hardy inequality. Our improved Hardy-type inequality holds with a weight w which is strictly greater than the classical Hardy weight w(H)(n) : 1/(2n)(2), where N. KW - Primary 26D15 Y1 - 2018 U6 - https://doi.org/10.1080/00029890.2018.1420995 SN - 0002-9890 SN - 1930-0972 VL - 125 IS - 4 SP - 347 EP - 350 PB - Taylor & Francis Group CY - Philadelphia ER - TY - JOUR A1 - Keller, Matthias A1 - Liu, Shiping A1 - Peyerimhoff, Norbert T1 - A note on eigenvalue bounds for non-compact manifolds JF - Mathematische Nachrichten N2 - In this article we prove upper bounds for the Laplace eigenvalues lambda(k) below the essential spectrum for strictly negatively curved Cartan-Hadamard manifolds. Our bound is given in terms of k(2) and specific geometric data of the manifold. This applies also to the particular case of non-compact manifolds whose sectional curvature tends to -infinity, where no essential spectrum is present due to a theorem of Donnelly/Li. The result stands in clear contrast to Laplacians on graphs where such a bound fails to be true in general. KW - Cheeger inequality KW - eigenvalues KW - Laplacian KW - negative curvature KW - Riemannian manifold Y1 - 2021 U6 - https://doi.org/10.1002/mana.201900209 SN - 0025-584X SN - 1522-2616 VL - 294 IS - 6 SP - 1134 EP - 1139 PB - Wiley-VCH CY - Weinheim ER - TY - JOUR A1 - Keller, Matthias A1 - Münch, Florentin T1 - A new discrete Hopf-Rinow theorem JF - Discrete Mathematics N2 - 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. Y1 - 2019 U6 - https://doi.org/10.1016/j.disc.2019.03.014 SN - 0012-365X SN - 1872-681X VL - 342 IS - 9 SP - 2751 EP - 2757 PB - Elsevier CY - Amsterdam ER -