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  - 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  -