@article{KleinRosenberger2018,
  author    = {Klein, Markus and Rosenberger, Elke},
  title     = {Tunneling for a class of difference operators},
  series = {Annales Henri Poincar{\´e} : a journal of theoretical and mathematical physics},
  volume    = {19},
  journal   = {Annales Henri Poincar{\´e} : a journal of theoretical and mathematical physics},
  number    = {11},
  publisher = {Springer International Publishing},
  address   = {Cham},
  issn      = {1424-0637},
  doi       = {10.1007/s00023-018-0732-0},
  pages     = {3511 -- 3559},
  year      = {2018},
  abstract  = {We analyze a general class of difference operators Hε=Tε+Vε on ℓ2((εZ)d), where Vε is a multi-well potential and ε is a small parameter. We derive full asymptotic expansions of the prefactor of the exponentially small eigenvalue splitting due to interactions between two "wells" (minima) of the potential energy, i.e., for the discrete tunneling effect. We treat both the case where there is a single minimal geodesic (with respect to the natural Finsler metric induced by the leading symbol h0(x,ξ) of Hε) connecting the two minima and the case where the minimal geodesics form an ℓ+1 dimensional manifold, ℓ≥1. These results on the tunneling problem are as sharp as the classical results for the Schr{\"o}dinger operator in Helffer and Sj{\"o}strand (Commun PDE 9:337-408, 1984). Technically, our approach is pseudo-differential and we adapt techniques from Helffer and Sj{\"o}strand [Analyse semi-classique pour l'{\´e}quation de Harper (avec application {\`a} l'{\´e}quation de Schr{\"o}dinger avec champ magn{\´e}tique), M{\´e}moires de la S.M.F., 2 series, tome 34, pp 1-113, 1988)] and Helffer and Parisse (Ann Inst Henri Poincar{\´e} 60(2):147-187, 1994) to our discrete setting.},
  language  = {en}
}
@article{KleinRosenberger2012,
  author    = {Klein, Markus and Rosenberger, Elke},
  title     = {Tunneling for a class of difference operators},
  series = {ANNALES HENRI POINCARE},
  volume    = {13},
  journal   = {ANNALES HENRI POINCARE},
  number    = {5},
  publisher = {Springer},
  address   = {Basel},
  issn      = {1424-0637},
  doi       = {10.1007/s00023-011-0152-x},
  pages     = {1231 -- 1269},
  year      = {2012},
  abstract  = {We analyze a general class of difference operators on where is a multi-well potential and is a small parameter. We decouple the wells by introducing certain Dirichlet operators on regions containing only one potential well, and we shall treat the eigenvalue problem for as a small perturbation of these comparison problems. We describe tunneling by a certain interaction matrix, similar to the analysis for the Schrodinger operator [see Helffer and Sjostrand in Commun Partial Differ Equ 9:337-408, 1984], and estimate the remainder, which is exponentially small and roughly quadratic compared with the interaction matrix.},
  language  = {en}
}
@unpublished{KleinRosenberger2012,
  author    = {Klein, Markus and Rosenberger, Elke},
  title     = {Tunneling for a class of difference operators},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-56989},
  year      = {2012},
  abstract  = {We analyze a general class of difference operators containing a multi-well potential and a small parameter. We decouple the wells by introducing certain Dirichlet operators on regions containing only one potential well, and we treat the eigenvalue problem as a small perturbation of these comparison problems. We describe tunneling by a certain interaction matrix similar to the analysis for the Schr{\"o}dinger operator, and estimate the remainder, which is exponentially small and roughly quadratic compared with the interaction matrix.},
  language  = {en}
}
@article{NeidhardtZagrebnov1998,
  author    = {Neidhardt, Hagen and Zagrebnov, Valentin A.},
  title     = {Trotter-Kato product formula and symmetrically-normed ideals},
  year      = {1998},
  language  = {en}
}
@article{NeidhardtZagrebnov1999,
  author    = {Neidhardt, Hagen and Zagrebnov, Valentin A.},
  title     = {Trotter-Kato product formula and operator-norm convergence},
  year      = {1999},
  language  = {en}
}
@article{Zagrebnov2020,
  author    = {Zagrebnov, Valentin},
  title     = {Trotter product formula on Hilbert and Banach spaces for operator-norm convergence},
  series = {Lectures in pure and applied mathematics},
  journal   = {Lectures in pure and applied mathematics},
  publisher = {Universit{\"a}tsverlag Potsdam},
  address   = {Potsdam},
  isbn      = {978-3-86956-485-2},
  issn      = {2199-4951},
  doi       = {10.25932/publishup-47197},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-471971},
  pages     = {23 -- 34},
  year      = {2020},
  language  = {en}
}
@article{BrungsGraeter2000,
  author    = {Brungs, Hans and Gr{\"a}ter, Joachim},
  title     = {Trees and Valuation Rings},
  year      = {2000},
  language  = {en}
}
@article{DeneckePabhapote2001,
  author    = {Denecke, Klaus-Dieter and Pabhapote, Nittiya},
  title     = {Tree-recognizers and tree-hyperrecognizers},
  year      = {2001},
  language  = {en}
}
@article{ArwornDenecke2001,
  author    = {Arworn, Srichan and Denecke, Klaus-Dieter},
  title     = {Tree Transformations defined by Hypersubstitutions},
  issn      = {1509 - 9415},
  year      = {2001},
  language  = {en}
}
@article{DeneckeLeeratanavalee2003,
  author    = {Denecke, Klaus-Dieter and Leeratanavalee, Sorasak},
  title     = {Tree transformations defined by generalized hypersubstitutions},
  year      = {2003},
  language  = {en}
}