@article{LudewigRosenberger2020,
  author    = {Ludewig, Matthias and Rosenberger, Elke},
  title     = {Asymptotic eigenfunctions for Schr{\"o}dinger operators on a vector bundle},
  series = {Reviews in mathematical physics},
  volume    = {32},
  journal   = {Reviews in mathematical physics},
  number    = {7},
  publisher = {World Scientific},
  address   = {Singapore},
  issn      = {0129-055X},
  doi       = {10.1142/S0129055X20500208},
  pages     = {28},
  year      = {2020},
  abstract  = {In the limit (h) over bar -> 0, we analyze a class of Schr{\"o}dinger operators H-(h) over bar = (h) over bar L-2 + (h) over barW + V .id(epsilon) acting on sections of a vector bundle epsilon over a Riemannian manifold M where L is a Laplace type operator, W is an endomorphism field and the potential energy V has a non-degenerate minimum at some point p is an element of M. We construct quasimodes of WKB-type near p for eigenfunctions associated with the low-lying eigenvalues of H-(h) over bar. These are obtained from eigenfunctions of the associated harmonic oscillator H-p,H-(h) over bar at p, acting on smooth functions on the tangent space.},
  language  = {en}
}
@article{KleinRosenberger2021,
  author    = {Klein, Markus and Rosenberger, Elke},
  title     = {The tunneling effect for Schr{\"o}dinger operators on a vector bundle},
  series = {Analysis and mathematical physics},
  volume    = {11},
  journal   = {Analysis and mathematical physics},
  number    = {2},
  publisher = {Springer International Publishing AG},
  address   = {Cham (ZG)},
  issn      = {1664-2368},
  doi       = {10.1007/s13324-021-00485-5},
  pages     = {35},
  year      = {2021},
  abstract  = {In the semiclassical limit (h) over bar -> 0, we analyze a class of self-adjoint Schrodinger operators H-(h) over bar = (h) over bar L-2 + (h) over barW + V center dot id(E) acting on sections of a vector bundle E over an oriented Riemannian manifold M where L is a Laplace type operator, W is an endomorphism field and the potential energy V has non-degenerate minima at a finite number of points m(1),... m(r) is an element of M, called potential wells. Using quasimodes of WKB-type near m(j) for eigenfunctions associated with the low lying eigenvalues of H-(h) over bar, we analyze the tunneling effect, i.e. the splitting between low lying eigenvalues, which e.g. arises in certain symmetric configurations. Technically, we treat the coupling between different potential wells by an interaction matrix and we consider the case of a single minimal geodesic (with respect to the associated Agmon metric) connecting two potential wells and the case of a submanifold of minimal geodesics of dimension l + 1. This dimension l determines the polynomial prefactor for exponentially small eigenvalue splitting.},
  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}
}
@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{KleinRosenberger2016,
  author    = {Klein, Markus and Rosenberger, Elke},
  title     = {Agmon estimates for the difference of exact and approximate Dirichlet eigenfunctions for difference operators},
  series = {Asymptotic analysis},
  volume    = {97},
  journal   = {Asymptotic analysis},
  publisher = {IOS Press},
  address   = {Amsterdam},
  issn      = {0921-7134},
  doi       = {10.3233/ASY-151343},
  pages     = {61 -- 89},
  year      = {2016},
  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 multi-well potential and epsilon is a small parameter. We construct approximate eigenfunctions in neighbourhoods of the different wells and give weighted l(2)-estimates for the difference of these and the exact eigenfunctions of the associated Dirichlet-operators.},
  language  = {en}
}
@article{KleinLeonardRosenberger2014,
  author    = {Klein, Markus and Leonard, Christian and Rosenberger, Elke},
  title     = {Agmon-type estimates for a class of jump processes},
  series = {Mathematische Nachrichten},
  volume    = {287},
  journal   = {Mathematische Nachrichten},
  number    = {17-18},
  publisher = {Wiley-VCH},
  address   = {Weinheim},
  issn      = {0025-584X},
  doi       = {10.1002/mana.201200324},
  pages     = {2021 -- 2039},
  year      = {2014},
  abstract  = {In the limit 0 we analyse the generators H of families of reversible jump processes in Rd associated with a class of symmetric non-local Dirichlet-forms and show exponential decay of the eigenfunctions. The exponential rate function is a Finsler distance, given as solution of a certain eikonal equation. Fine results are sensitive to the rate function being C2 or just Lipschitz. Our estimates are analogous to the semiclassical Agmon estimates for differential operators of second order. They generalize and strengthen previous results on the lattice Zd. Although our final interest is in the (sub)stochastic jump process, technically this is a pure analysis paper, inspired by PDE techniques.},
  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}
}
@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}
}
@article{KleinRosenberger2009,
  author    = {Klein, Markus and Rosenberger, Elke},
  title     = {Harmonic approximation of difference operators},
  issn      = {0022-1236},
  doi       = {10.1016/j.jfa.2009.09.004},
  year      = {2009},
  abstract  = {For a general class of difference operators H-epsilon = T-epsilon + V-epsilon on l(2) ((epsilon Z)(d)), where V- epsilon is a multi-well potential and a is a small parameter. we analyze the asymptotic behavior as epsilon -> 0 of the (low-lying) eigenvalues and eigenfunctions. We show that the first it eigenvalues of H converge to the first it eigenvalues of the direct suns of harmonic oscillators oil R-d located at the several wells. Our proof is microlocal.},
  language  = {en}
}
@unpublished{KleinLeonardRosenberger2012,
  author    = {Klein, Markus and L{\´e}onard, Christian and Rosenberger, Elke},
  title     = {Agmon-type estimates for a class of jump processes},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-56995},
  year      = {2012},
  abstract  = {In the limit we analyze the generators of families of reversible jump processes in the n-dimensional space associated with a class of symmetric non-local Dirichlet forms and show exponential decay of the eigenfunctions. The exponential rate function is a Finsler distance, given as solution of certain eikonal equation. Fine results are sensitive to the rate functions being twice differentiable or just Lipschitz. Our estimates are similar to the semiclassical Agmon estimates for differential operators of second order. They generalize and strengthen previous results on the lattice.},
  language  = {en}
}