@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}
}
@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}
}