@phdthesis{Rosenberger2006,
  author    = {Rosenberger, Elke},
  title     = {Asymptotic spectral analysis and tunnelling for a class of difference operators},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-7393},
  school      = {Universit{\"a}t Potsdam},
  year      = {2006},
  abstract  = {We analyze the asymptotic behavior in the limit epsilon to zero for a wide class of difference operators H_epsilon = T_epsilon + V_epsilon with underlying multi-well potential. They act on the square summable functions on the lattice (epsilon Z)^d. We start showing the validity of an harmonic approximation and construct WKB-solutions at the wells. Then we construct a Finslerian distance d induced by H and show that short integral curves are geodesics and d gives the rate for the exponential decay of Dirichlet eigenfunctions. In terms of this distance, we give sharp estimates for the interaction between the wells and construct the interaction matrix.},
  subject      = {Mathematische Physik},
  language  = {en}
}
@article{KleinRosenberger2011,
  author    = {Klein, Markus and Rosenberger, Elke},
  title     = {Asymptotic eigenfunctions for a class of difference operators},
  series = {Asymptotic analysis},
  volume    = {73},
  journal   = {Asymptotic analysis},
  number    = {1-2},
  publisher = {IOS Press},
  address   = {Amsterdam},
  issn      = {0921-7134},
  doi       = {10.3233/ASY-2010-1025},
  pages     = {1 -- 36},
  year      = {2011},
  abstract  = {We analyze a general class of difference operators H(epsilon) = T(epsilon) + V(epsilon) on l(2)((epsilon Z)(d)), where V(epsilon) is a one-well potential and epsilon is a small parameter. We construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low lying eigenvalues of H(epsilon). These are obtained from eigenfunctions or quasimodes for the operator H(epsilon), acting on L(2)(R(d)), via restriction to the lattice (epsilon Z)(d).},
  language  = {en}
}