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  - 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  - INPR
A1  - Klein, Markus
A1  - Rosenberger, Elke
T1  - Tunneling for a class of difference operators
N2  - 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 Schrödinger operator, and estimate the remainder, which is exponentially small and roughly quadratic compared with the interaction matrix.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 1 (2012) 5 
KW  - semi-classical difference operator
KW  - tunneling
KW  - interaction matrix
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-56989
ER  - 
TY  - JOUR
A1  - Neidhardt, Hagen
A1  - Zagrebnov, Valentin A.
T1  - Trotter-Kato product formula and symmetrically-normed ideals
Y1  - 1998
ER  - 
TY  - JOUR
A1  - Neidhardt, Hagen
A1  - Zagrebnov, Valentin A.
T1  - Trotter-Kato product formula and operator-norm convergence
Y1  - 1999
ER  - 
TY  - JOUR
A1  - Zagrebnov, Valentin
T1  - Trotter product formula on Hilbert and Banach spaces for operator-norm convergence
JF  - Lectures in pure and applied mathematics
KW  - random point processes
KW  - statistical mechanics
KW  - stochastic analysis
Y1  - 2020
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-471971
SN  - 978-3-86956-485-2
SN  - 2199-4951
SN  - 2199-496X
SP  - 23
EP  - 34
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - JOUR
A1  - Brungs, Hans
A1  - Gräter, Joachim
T1  - Trees and Valuation Rings
Y1  - 2000
ER  - 
TY  - JOUR
A1  - Denecke, Klaus-Dieter
A1  - Pabhapote, Nittiya
T1  - Tree-recognizers and tree-hyperrecognizers
Y1  - 2001
ER  - 
TY  - JOUR
A1  - Arworn, Srichan
A1  - Denecke, Klaus-Dieter
T1  - Tree Transformations defined by Hypersubstitutions
Y1  - 2001
SN  - 1509 - 9415
ER  - 
TY  - JOUR
A1  - Denecke, Klaus-Dieter
A1  - Leeratanavalee, Sorasak
T1  - Tree transformations defined by generalized hypersubstitutions
Y1  - 2003
ER  -