TY  - JOUR
A1  - Klein, Markus
A1  - Rosenberger, Elke
T1  - The tunneling effect for Schrödinger operators on a vector bundle
JF  - Analysis and mathematical physics
N2  - 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.
KW  - Laplace-type operator
KW  - Vector bundle
KW  - WKB-expansion
KW  - Quasimodes
KW  - Tunneling
KW  - Spectral gap
KW  - Complete asymptotics
Y1  - 2021
U6  - https://doi.org/10.1007/s13324-021-00485-5
SN  - 1664-2368
SN  - 1664-235X
VL  - 11
IS  - 2
PB  - Springer International Publishing AG
CY  - Cham (ZG)
ER  - 
TY  - JOUR
A1  - Klein, Markus
A1  - Rosenberger, Elke
T1  - Tunneling for a class of difference operators
BT  - Complete asymptotics
JF  - Annales Henri Poincaré : a journal of theoretical and mathematical physics
N2  - 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 Schrödinger operator in Helffer and Sjöstrand (Commun PDE 9:337–408, 1984). Technically, our approach is pseudo-differential and we adapt techniques from Helffer and Sjöstrand [Analyse semi-classique pour l’équation de Harper (avec application à l’équation de Schrödinger avec champ magnétique), Mémoires de la S.M.F., 2 series, tome 34, pp 1–113, 1988)] and Helffer and Parisse (Ann Inst Henri Poincaré 60(2):147–187, 1994) to our discrete setting.
Y1  - 2018
U6  - https://doi.org/10.1007/s00023-018-0732-0
SN  - 1424-0637
SN  - 1424-0661
VL  - 19
IS  - 11
SP  - 3511
EP  - 3559
PB  - Springer International Publishing
CY  - Cham
ER  - 
TY  - JOUR
A1  - Klein, Markus
A1  - Rosenberger, Elke
T1  - Agmon estimates for the difference of exact and approximate Dirichlet eigenfunctions for difference operators
JF  - Asymptotic analysis
N2  - 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.
Y1  - 2016
U6  - https://doi.org/10.3233/ASY-151343
SN  - 0921-7134
SN  - 1875-8576
VL  - 97
SP  - 61
EP  - 89
PB  - IOS Press
CY  - Amsterdam
ER  - 
TY  - JOUR
A1  - Klein, Markus
A1  - Rama, Juliane
T1  - Time asymptotics of e(-ith(kappa)) for analytic matrices and analytic perturbation theory
JF  - Asymptotic analysis
N2  - 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.
KW  - resonances
KW  - exponential decay
KW  - long-time corrections
KW  - Fermi golden rule
KW  - analytic perturbation theory
Y1  - 2014
U6  - https://doi.org/10.3233/ASY-141226
SN  - 0921-7134
SN  - 1875-8576
VL  - 89
IS  - 3-4
SP  - 189
EP  - 233
PB  - IOS Press
CY  - Amsterdam
ER  - 
TY  - JOUR
A1  - Klein, Markus
A1  - Leonard, Christian
A1  - Rosenberger, Elke
T1  - Agmon-type estimates for a class of jump processes
JF  - Mathematische Nachrichten
N2  - 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.
KW  - Decay of eigenfunctions
KW  - semiclassical Agmon estimate
KW  - Finsler distance
KW  - jump process
KW  - Dirichlet-form
Y1  - 2014
U6  - https://doi.org/10.1002/mana.201200324
SN  - 0025-584X
SN  - 1522-2616
VL  - 287
IS  - 17-18
SP  - 2021
EP  - 2039
PB  - Wiley-VCH
CY  - Weinheim
ER  - 
TY  - JOUR
A1  - Klein, Markus
A1  - Rosenberger, Elke
T1  - Asymptotic eigenfunctions for a class of difference operators
JF  - Asymptotic analysis
N2  - 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).
KW  - difference operator
KW  - tunneling
KW  - WKB-expansion
KW  - quasimodes
Y1  - 2011
U6  - https://doi.org/10.3233/ASY-2010-1025
SN  - 0921-7134
VL  - 73
IS  - 1-2
SP  - 1
EP  - 36
PB  - IOS Press
CY  - Amsterdam
ER  - 
TY  - JOUR
A1  - Klein, Markus
A1  - Rosenberger, Elke
T1  - Tunneling for a class of difference operators
JF  - ANNALES HENRI POINCARE
N2  - 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.
Y1  - 2012
U6  - https://doi.org/10.1007/s00023-011-0152-x
SN  - 1424-0637
VL  - 13
IS  - 5
SP  - 1231
EP  - 1269
PB  - Springer
CY  - Basel
ER  - 
TY  - JOUR
A1  - Klein, Markus
A1  - Rosenberger, Elke
T1  - Harmonic approximation of difference operators
N2  - 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.
Y1  - 2009
UR  - http://www.sciencedirect.com/science/journal/00221236
U6  - https://doi.org/10.1016/j.jfa.2009.09.004
SN  - 0022-1236
ER  - 
TY  - JOUR
A1  - Jaksic, V.
A1  - Jung, K.
A1  - Klein, Markus
A1  - Seiler, R.
T1  - Corrections to quantized charge transport in quantum hall systems
Y1  - 1994
ER  - 
TY  - JOUR
A1  - Klein, Markus
T1  - Hall conductance of Riemann surfaces
Y1  - 1994
ER  - 
TY  - JOUR
A1  - Bürklin, Wilhelm
A1  - Klein, Markus
A1  - Ruß, Achim
T1  - Dimensionen des Wertewandels : eine empirische Längsschnittanalyse zur Dimensionalität und der Wandlungsdynamik gesellschaftlicher Wertorientierungen
Y1  - 1994
ER  - 
TY  - JOUR
A1  - Bürklin, Wilhelm
A1  - Klein, Markus
A1  - Ruß, Achim
T1  - Postmaterieller oder anthropozentrischer Wertewandel? : eine Erwiderung auf Ronald Inglehart und Hans-Dieter Klingemann
Y1  - 1996
ER  - 
TY  - JOUR
A1  - Klein, Markus
T1  - On the Born-Oppenheimer approximation of diatomic wave operators : II. Singular potentials
Y1  - 1997
ER  - 
TY  - JOUR
A1  - Klein, Markus
T1  - The Born-Oppenheimer Expansion : eigenvalues, eigenfunctions and low-energy scattering
Y1  - 1999
SN  - 3-540-65106-3
ER  - 
TY  - JOUR
A1  - Klein, Markus
A1  - Bovier, Anton
A1  - Eckhoff, Michael
A1  - Gayrard, Véronique
T1  - Metastability and small eigenvalues in Markov chains
Y1  - 2000
ER  - 
TY  - JOUR
A1  - Klein, Markus
T1  - Parametrization of periodic weighted operators in terms of gap lengths
Y1  - 2000
ER  - 
TY  - JOUR
A1  - Gräter, Joachim
A1  - Klein, Markus
T1  - The Principal Axis Theorem for Holomorphic Functions
Y1  - 2000
ER  - 
TY  - JOUR
A1  - Bovier, Anton
A1  - Eckhoff, Michael
A1  - Gayrard, Veronique
A1  - Klein, Markus
T1  - Metastability in stochastic dynamics of disordered mean-field models
Y1  - 2001
SN  - 0178-8051
ER  - 
TY  - JOUR
A1  - Bovier, Anton
A1  - Eckhoff, Michael
A1  - Gayrard, Veronique
A1  - Klein, Markus
T1  - Metastability and low-Lying spectra in reversible Markov chains
Y1  - 2002
ER  - 
TY  - JOUR
A1  - Bovier, Anton
A1  - Eckhoff, Michael
A1  - Gayrard, Veronique
A1  - Klein, Markus
T1  - Metastability in reversible diffusion processes : I. Sharp asymptotics for capacities and exit times
N2  - 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
Y1  - 2004
SN  - 1435-9855
ER  - 
TY  - JOUR
A1  - Klein, Markus
A1  - Korotyaev, Evgeni
A1  - Pokrovski, A.
T1  - Spectral asymptotics of the harmonic oscillator perturbed by bounded potentials
N2  - 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.
Y1  - 2005
SN  - 1424-0637
ER  - 
TY  - JOUR
A1  - Bovier, Anton
A1  - Gayrard, Veronique
A1  - Klein, Markus
T1  - Metastability in reversible diffusion processes : II. Precise asymptotics for small eigenvalues
N2  - 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
Y1  - 2005
SN  - 1435-9855
ER  - 
TY  - JOUR
A1  - Baake, Ellen
A1  - Baake, Michael
A1  - Bovier, Anton
A1  - Klein, Markus
T1  - An asymptotic maximum principle for essentially linear evolution models
N2  - 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
Y1  - 2005
SN  - 0303-6812
ER  -