@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{BeckusEliaz2021,
  author    = {Beckus, Siegfried and Eliaz, Latif},
  title     = {Eigenfunctions growth of R-limits on graphs},
  series = {Journal of spectral theory / European Mathematical Society},
  volume    = {11},
  journal   = {Journal of spectral theory / European Mathematical Society},
  number    = {4},
  publisher = {EMS Press, an imprint of the European Mathematical Society - EMS - Publishing House GmbH, Institut f{\"u}r Mathematik, Technische Universit{\"a}t},
  address   = {Berlin},
  issn      = {1664-039X},
  doi       = {10.4171/JST/389},
  pages     = {1895 -- 1933},
  year      = {2021},
  abstract  = {A characterization of the essential spectrum of Schrodinger operators on infinite graphs is derived involving the concept of R-limits. This concept, which was introduced previously for operators on N and Z(d) as "right-limits," captures the behaviour of the operator at infinity. For graphs with sub-exponential growth rate, we show that each point in sigma(ss)(H) corresponds to a bounded generalized eigenfunction of a corresponding R-limit of H. If, additionally, the graph is of uniform sub-exponential growth, also the converse inclusion holds.},
  language  = {en}
}