@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}
}
@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{KleinRama2014,
  author    = {Klein, Markus and Rama, Juliane},
  title     = {Time asymptotics of e(-ith(kappa)) for analytic matrices and analytic perturbation theory},
  series = {Asymptotic analysis},
  volume    = {89},
  journal   = {Asymptotic analysis},
  number    = {3-4},
  publisher = {IOS Press},
  address   = {Amsterdam},
  issn      = {0921-7134},
  doi       = {10.3233/ASY-141226},
  pages     = {189 -- 233},
  year      = {2014},
  abstract  = {In quantum mechanics the temporal decay of certain resonance states is associated with an effective time evolution e(-ith(kappa)), where h(.) is an analytic family of non-self-adjoint matrices. In general the corresponding resonance states do not decay exponentially in time. Using analytic perturbation theory, we derive asymptotic expansions for e(-ith(kappa)), simultaneously in the limits kappa -> 0 and t -> infinity, where the corrections with respect to pure exponential decay have uniform bounds in one complex variable kappa(2)t. In the Appendix we briefly review analytic perturbation theory, replacing the classical reference to the 1920 book of Knopp [Funktionentheorie II, Anwendungen und Weiterfuhrung der allgemeinen Theorie, Sammlung Goschen, Vereinigung wissenschaftlicher Verleger Walter de Gruyter, 1920] and its terminology by standard modern references. This might be of independent interest.},
  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}
}
@article{JaksicJungKleinetal.1994,
  author    = {Jaksic, V. and Jung, K. and Klein, Markus and Seiler, R.},
  title     = {Corrections to quantized charge transport in quantum hall systems},
  year      = {1994},
  language  = {en}
}
@article{Klein1994,
  author    = {Klein, Markus},
  title     = {Hall conductance of Riemann surfaces},
  year      = {1994},
  language  = {en}
}
@article{BuerklinKleinRuss1994,
  author    = {B{\"u}rklin, Wilhelm and Klein, Markus and Ruß, Achim},
  title     = {Dimensionen des Wertewandels : eine empirische L{\"a}ngsschnittanalyse zur Dimensionalit{\"a}t und der Wandlungsdynamik gesellschaftlicher Wertorientierungen},
  year      = {1994},
  language  = {de}
}
@article{BuerklinKleinRuss1996,
  author    = {B{\"u}rklin, Wilhelm and Klein, Markus and Ruß, Achim},
  title     = {Postmaterieller oder anthropozentrischer Wertewandel? : eine Erwiderung auf Ronald Inglehart und Hans-Dieter Klingemann},
  year      = {1996},
  language  = {de}
}
@article{Klein1997,
  author    = {Klein, Markus},
  title     = {On the Born-Oppenheimer approximation of diatomic wave operators : II. Singular potentials},
  year      = {1997},
  language  = {en}
}
@article{Klein1999,
  author    = {Klein, Markus},
  title     = {The Born-Oppenheimer Expansion : eigenvalues, eigenfunctions and low-energy scattering},
  isbn      = {3-540-65106-3},
  year      = {1999},
  language  = {en}
}
@article{KleinBovierEckhoffetal.2000,
  author    = {Klein, Markus and Bovier, Anton and Eckhoff, Michael and Gayrard, V{\´e}ronique},
  title     = {Metastability and small eigenvalues in Markov chains},
  year      = {2000},
  language  = {en}
}
@article{Klein2000,
  author    = {Klein, Markus},
  title     = {Parametrization of periodic weighted operators in terms of gap lengths},
  year      = {2000},
  language  = {en}
}
@article{GraeterKlein2000,
  author    = {Gr{\"a}ter, Joachim and Klein, Markus},
  title     = {The Principal Axis Theorem for Holomorphic Functions},
  year      = {2000},
  language  = {en}
}
@article{BovierEckhoffGayrardetal.2001,
  author    = {Bovier, Anton and Eckhoff, Michael and Gayrard, Veronique and Klein, Markus},
  title     = {Metastability in stochastic dynamics of disordered mean-field models},
  issn      = {0178-8051},
  year      = {2001},
  language  = {en}
}
@article{BovierEckhoffGayrardetal.2002,
  author    = {Bovier, Anton and Eckhoff, Michael and Gayrard, Veronique and Klein, Markus},
  title     = {Metastability and low-Lying spectra in reversible Markov chains},
  year      = {2002},
  language  = {en}
}
@article{BovierEckhoffGayrardetal.2004,
  author    = {Bovier, Anton and Eckhoff, Michael and Gayrard, Veronique and Klein, Markus},
  title     = {Metastability in reversible diffusion processes : I. Sharp asymptotics for capacities and exit times},
  issn      = {1435-9855},
  year      = {2004},
  abstract  = {We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form -epsilonDelta+ delF(.) del on R-d or subsets of R-d, where F is a smooth function with finitely many local minima. In analogy to previous work on discrete Markov chains, we show that metastable exit times from the attractive domains of the minima of F can be related, up to multiplicative errors that tend to one as epsilon down arrow 0, to the capacities of suitably constructed sets. We show that these capacities can be computed, again up to multiplicative errors that tend to one, in terms of local characteristics of F at the starting minimum and the relevant saddle points. As a result, we are able to give the first rigorous proof of the classical Eyring - Kramers formula in dimension larger than 1. The estimates on capacities make use of their variational representation and monotonicity properties of Dirichlet forms. The methods developed here are extensions of our earlier work on discrete Markov chains to continuous diffusion processes},
  language  = {en}
}
@article{KleinKorotyaevPokrovski2005,
  author    = {Klein, Markus and Korotyaev, Evgeni and Pokrovski, A.},
  title     = {Spectral asymptotics of the harmonic oscillator perturbed by bounded potentials},
  issn      = {1424-0637},
  year      = {2005},
  abstract  = {Consider the operator T = -d(2)/dx(2) + x(2) + q(x) in L-2 (R), where q is a real function with q' and integral(0)(x) q(s) ds bounded. The spectrum of T is purely discrete and consists of simple eigenvalues. We determine their asymptotics mu(n) = (2n + 1) + (2 pi)(-1) integral(-pi)(pi) q(root 2n+1 sin theta)d theta + O(n(-1/3)) and we extend these results for complex q.},
  language  = {en}
}
@article{BovierGayrardKlein2005,
  author    = {Bovier, Anton and Gayrard, Veronique and Klein, Markus},
  title     = {Metastability in reversible diffusion processes : II. Precise asymptotics for small eigenvalues},
  issn      = {1435-9855},
  year      = {2005},
  abstract  = {We continue the analysis of the problem of metastability for reversible diffusion processes, initiated in [BEGK3], with a precise analysis of the low-lying spectrum of the generator. Recall that we are considering processes with generators of the form -epsilonDelta + delF(.) del on R-d or subsets of Rd, where F is a smooth function with finitely many local minima. Here we consider only the generic situation where the depths of all local minima are different. We show that in general the exponentially small part of the spectrum is given, up to multiplicative errors tending to one, by the eigenvalues of the classical capacity matrix of the array of capacitors made of balls of radius epsilon centered at the positions of the local minima of F. We also get very precise uniform control on the corresponding eigenfunctions. Moreover, these eigenvalues can be identified with the same precision with the inverse mean metastable exit times from each minimum. In [BEGK3] it was proven that these mean times are given, again up to multiplicative errors that tend to one, by the classical Eyring- Kramers formula},
  language  = {en}
}
@article{BaakeBaakeBovieretal.2005,
  author    = {Baake, Ellen and Baake, Michael and Bovier, Anton and Klein, Markus},
  title     = {An asymptotic maximum principle for essentially linear evolution models},
  issn      = {0303-6812},
  year      = {2005},
  abstract  = {Recent work on mutation-selection models has revealed that, under specific assumptions on the fitness function and the mutation rates, asymptotic estimates for the leading eigenvalue of the mutation-reproduction matrix may be obtained through a low-dimensional maximum principle in the limit N --> infinity (where N, or N-d with d greater than or equal to 1, is proportional to the number of types). In order to extend this variational principle to a larger class of models, we consider here a family of reversible matrices of asymptotic dimension N-d and identify conditions under which the high-dimensional Rayleigh-Ritz variational problem may be reduced to a low-dimensional one that yields the leading eigenvalue up to an error term of order 1/N. For a large class of mutation-selection models, this implies estimates for the mean fitness, as well as a concentration result for the ancestral distribution of types},
  language  = {en}
}