@article{KellerLiuPeyerimhoff2021,
  author    = {Keller, Matthias and Liu, Shiping and Peyerimhoff, Norbert},
  title     = {A note on eigenvalue bounds for non-compact manifolds},
  series = {Mathematische Nachrichten},
  volume    = {294},
  journal   = {Mathematische Nachrichten},
  number    = {6},
  publisher = {Wiley-VCH},
  address   = {Weinheim},
  issn      = {0025-584X},
  doi       = {10.1002/mana.201900209},
  pages     = {1134 -- 1139},
  year      = {2021},
  abstract  = {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.},
  language  = {en}
}
@misc{KellerPinchoverPogorzelski2020,
  author    = {Keller, Matthias and Pinchover, Yehuda and Pogorzelski, Felix},
  title     = {From hardy to rellich inequalities on graphs},
  series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  number    = {3},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-54214},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-542140},
  pages     = {22},
  year      = {2020},
  abstract  = {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.},
  language  = {en}
}
@article{KellerPinchoverPogorzelski2019,
  author    = {Keller, Matthias and Pinchover, Yehuda and Pogorzelski, Felix},
  title     = {Criticality theory for Schr{\"o}dinger operators on graphs},
  series = {Journal of spectral theory},
  volume    = {10},
  journal   = {Journal of spectral theory},
  number    = {1},
  publisher = {European Mathematical Society},
  address   = {Z{\"u}rich},
  issn      = {1664-039X},
  doi       = {10.4171/JST/286},
  pages     = {73 -- 114},
  year      = {2019},
  abstract  = {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.},
  language  = {en}
}
@article{KellerPinchoverPogorzelski2020,
  author    = {Keller, Matthias and Pinchover, Yehuda and Pogorzelski, Felix},
  title     = {From hardy to rellich inequalities on graphs},
  series = {Proceedings of the London Mathematical Society},
  volume    = {122},
  journal   = {Proceedings of the London Mathematical Society},
  number    = {3},
  publisher = {Wiley},
  address   = {Hoboken},
  issn      = {0024-6115},
  doi       = {10.1112/plms.12376},
  pages     = {458 -- 477},
  year      = {2020},
  abstract  = {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.},
  language  = {en}
}
@article{GueneysuKeller2020,
  author    = {G{\"u}neysu, Batu and Keller, Matthias},
  title     = {Feynman path integrals for magnetic Schr{\"o}dinger operators on infinite weighted graphs},
  series = {Journal d'analyse math{\´e}matique},
  volume    = {141},
  journal   = {Journal d'analyse math{\´e}matique},
  number    = {2},
  publisher = {The Magnes Press, the Hebrew Univ.},
  address   = {Jerusalem},
  issn      = {0021-7670},
  doi       = {10.1007/s11854-020-0110-y},
  pages     = {751 -- 770},
  year      = {2020},
  abstract  = {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.},
  language  = {en}
}
@article{FischerKeller2021,
  author    = {Fischer, Florian and Keller, Matthias},
  title     = {Riesz decompositions for Schr{\"o}dinger operators on graphs},
  series = {Journal of mathematical analysis and applications},
  volume    = {495},
  journal   = {Journal of mathematical analysis and applications},
  number    = {1},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0022-247X},
  doi       = {10.1016/j.jmaa.2020.124674},
  pages     = {22},
  year      = {2021},
  abstract  = {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.},
  language  = {en}
}
@article{KellerSchwarz2020,
  author    = {Keller, Matthias and Schwarz, Michael},
  title     = {Courant's nodal domain theorem for positivity preserving forms},
  series = {Journal of spectral theory},
  volume    = {10},
  journal   = {Journal of spectral theory},
  number    = {1},
  publisher = {EMS Publishing House},
  address   = {Z{\"u}rich},
  issn      = {1664-039X},
  doi       = {10.4171/JST/292},
  pages     = {271 -- 309},
  year      = {2020},
  abstract  = {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.},
  language  = {en}
}
@article{KellerPinchoverPogorzelski2018,
  author    = {Keller, Matthias and Pinchover, Yehuda and Pogorzelski, Felix},
  title     = {An improved discrete hardy inequality},
  series = {The American mathematical monthly : an official publication of the Mathematical Association of America},
  volume    = {125},
  journal   = {The American mathematical monthly : an official publication of the Mathematical Association of America},
  number    = {4},
  publisher = {Taylor \& Francis Group},
  address   = {Philadelphia},
  issn      = {0002-9890},
  doi       = {10.1080/00029890.2018.1420995},
  pages     = {347 -- 350},
  year      = {2018},
  abstract  = {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.},
  language  = {en}
}
@article{KellerPinchoverPogorzelski2018,
  author    = {Keller, Matthias and Pinchover, Yehuda and Pogorzelski, Felix},
  title     = {Optimal Hardy inequalities for Schrodinger operators on graphs},
  series = {Communications in mathematical physics},
  volume    = {358},
  journal   = {Communications in mathematical physics},
  number    = {2},
  publisher = {Springer},
  address   = {New York},
  issn      = {0010-3616},
  doi       = {10.1007/s00220-018-3107-y},
  pages     = {767 -- 790},
  year      = {2018},
  abstract  = {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.},
  language  = {en}
}
@article{KellerSchwarz2018,
  author    = {Keller, Matthias and Schwarz, Michael},
  title     = {The Kazdan-Warner equation on canonically compactifiable graphs},
  series = {Calculus of variations and partial differential equations},
  volume    = {57},
  journal   = {Calculus of variations and partial differential equations},
  number    = {2},
  publisher = {Springer},
  address   = {Heidelberg},
  issn      = {0944-2669},
  doi       = {10.1007/s00526-018-1329-7},
  pages     = {18},
  year      = {2018},
  abstract  = {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.},
  language  = {en}
}