@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}
}
@unpublished{KleinRosenberger2012,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}
}
@misc{KleinRosenberger2018,
  author    = {Klein, Markus and Rosenberger, Elke},
  title     = {The tunneling effect for a class of difference operators},
  series = {Reviews in Mathematical Physics},
  volume    = {30},
  journal   = {Reviews in Mathematical Physics},
  number    = {4},
  publisher = {World Scientific},
  address   = {Singapore},
  issn      = {0129-055X},
  doi       = {10.1142/S0129055X18300029},
  pages     = {42},
  year      = {2018},
  abstract  = {We analyze a general class of self-adjoint difference operators H-epsilon = T-epsilon + V-epsilon on l(2)((epsilon Z)(d)), where V-epsilon is a multi-well potential and v(epsilon) is a small parameter. We give a coherent review of our results on tunneling up to new sharp results on the level of complete asymptotic expansions (see [30-35]). Our emphasis is on general ideas and strategy, possibly of interest for a broader range of readers, and less on detailed mathematical proofs. The wells are decoupled by introducing certain Dirichlet operators on regions containing only one potential well. Then the eigenvalue problem for the Hamiltonian H-epsilon is treated as a small perturbation of these comparison problems. After constructing a Finslerian distance d induced by H-epsilon, we show that Dirichlet eigenfunctions decay exponentially with a rate controlled by this distance to the well. It follows with microlocal techniques that the first n eigenvalues of H-epsilon converge to the first n eigenvalues of the direct sum of harmonic oscillators on R-d located at several wells. In a neighborhood of one well, 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). Tunneling is then described by a certain interaction matrix, similar to the analysis for the Schrodinger operator (see [22]), the remainder is exponentially small and roughly quadratic compared with the interaction matrix. We give weighted l(2)-estimates for the difference of eigenfunctions of Dirichlet-operators in neighborhoods of the different wells and the associated WKB-expansions at the wells. In the last step, we derive full asymptotic expansions for interactions between two "wells" (minima) of the potential energy, in particular for the discrete tunneling effect. Here we essentially use analysis on phase space, complexified in the momentum variable. These results are as sharp as the classical results for the Schrodinger operator in [22].},
  language  = {en}
}