@article{BeckusPinchover2020,
  author    = {Beckus, Siegfried and Pinchover, Yehuda},
  title     = {Shnol-type theorem for the Agmon ground state},
  series = {Journal of spectral theory},
  volume    = {10},
  journal   = {Journal of spectral theory},
  number    = {2},
  publisher = {EMS Publishing House},
  address   = {Z{\"u}rich},
  issn      = {1664-039X},
  doi       = {10.4171/JST/296},
  pages     = {355 -- 377},
  year      = {2020},
  abstract  = {LetH be a Schrodinger operator defined on a noncompact Riemannianmanifold Omega, and let W is an element of L-infinity (Omega; R). Suppose that the operator H + W is critical in Omega, and let phi be the corresponding Agmon ground state. We prove that if u is a generalized eigenfunction ofH satisfying vertical bar u vertical bar <= C-phi in Omega for some constant C > 0, then the corresponding eigenvalue is in the spectrum of H. The conclusion also holds true if for some K is an element of Omega the operator H admits a positive solution in (Omega) over bar = Omega \ K, and vertical bar u vertical bar <= C psi in (Omega) over bar for some constant C > 0, where psi is a positive solution of minimal growth in a neighborhood of infinity in Omega. Under natural assumptions, this result holds also in the context of infinite graphs, and Dirichlet forms.},
  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{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}
}
@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{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}
}