Dr. Elke Rosenberger

wissenschaftliche Mitarbeiterin

Kontakt
Raum:
2.09.2.07
Telefon:
+49 331 977-1258



Sprechzeiten:  nach Vereinbarung

...
  • Spectral Theory
  • Difference Operators
  • Pseudodifferential Calculus
  • Jump Processes
  • Tunneling
  • Finsler Geometry

Publikationen

2018 | Tunneling for a class of Difference Operators: Complete Asymptotics | Markus Klein, Elke Rosenberger Zeitschrift: Annales Henri Poincare Verlag: Springer Verlag Seiten: 3511-3559 Band: 19(11) Link zur Publikation, Link zum Preprint, https://rdcu.be/9Gyk

Tunneling for a class of Difference Operators: Complete Asymptotics

Autoren: Markus Klein, Elke Rosenberger (2018)

We analyze a general class of difference operators H_\varepsilon = T_\varepsilon +V_\varepsilo on \ell^2((\varepsilon\mathbb{Z})^d), where
V_\varepsilon is a multi-well potential and \varepsilon 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 h_0(x, ξ) of H_\varepsilon) connecting the two minima and the case where
the minimal geodesics form an k+1 dimensional manifold, k ≥ 1. These results on the tunneling
problem are as sharp as the classical results for the Schrödinger operator in [Helffer, Sjöstrand,
1984]. Technically, our approach is pseudodifferential and we adapt techniques from [Helffer,
Sjöstrand, 1988] and [Helffer, Parisse, 1994] to our discrete setting.

Zeitschrift:
Annales Henri Poincare
Verlag:
Springer Verlag
Seiten:
3511-3559
Band:
19(11)

2018 | The Tunneling Effect for a Class of Difference Operators | Markus Klein, Elke Rosenberger Zeitschrift: Reviews in Mathematical Physics (RMP) Verlag: World Scientific Publishing Seiten: 1830002 (1-42) Band: Vol.30, No. 4 Link zur Publikation

The Tunneling Effect for a Class of Difference Operators

Autoren: Markus Klein, Elke Rosenberger (2018)

We analyze a general class of self-adjoint difference operators H_\varepsilon = T_\varepsilon

+ V_\varepsilon on \ell^2(\varepsilon\mathbb{Z}^d), whereMimetex cannot convert this formula<tex>V_ε is a multi-well

potential and ε is a small parameter.

We review some preparatory results on tunneling of the authors, needed for
our presentation of new sharp results on tunneling on the level of complete asymptotic expansions.

The wells are decoupled by introducing certain Dirichlet operators on regions containing only one
potential well. Then the eigenvalue problem for the Hamiltonian H_\varepsilon is treated as a small perturbation of these comparison
problems.
After constructing a Finslerian distance d induced by H_\varepsilon 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_\varepsilon converge to
the first n eigenvalues of the direct sum of harmonic oscillators on \mathbb{R}^d located
at the 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_\varepsilon. These are obtained
from eigenfunctions or quasimodes for the
operator H_\varepsilon, acting on L^2(\mathbb{R}^d), via restriction to the lattice \varepsilon\mathbb{Z}^d.

Tunneling is then described by a certain interaction matrix, similar to the analysis for the Schrödinger operator, the remainder is exponentially small and roughly quadratic compared with the
interaction matrix.
We give weighted \ell^2-estimates for the difference of eigenfunctions of Dirichlet-operators in
neighbourhoods 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 Schrödinger operator given by Helffer and Sjöstrand.

Zeitschrift:
Reviews in Mathematical Physics (RMP)
Verlag:
World Scientific Publishing
Seiten:
1830002 (1-42)
Band:
Vol.30, No. 4

2015 | Agmon estimates for the difference of exact and approximate Dirichlet eigenfunctions for difference operators | Markus Klein, Elke Rosenberger Zeitschrift: Asymptotic Analysis Verlag: IOS Press Seiten: 61-89 Band: 97 Link zur Publikation

Agmon estimates for the difference of exact and approximate Dirichlet eigenfunctions for difference operators

Autoren: Markus Klein, Elke Rosenberger (2015)

We analyze a general class of difference operators H_\varepsilon = T_\varepsilon + V_\varepsilon on \ell^2(\varepsilon \mathbb{Z}^d), where V_\varepsilon is a multi-well
potential and \varepsilon is a small parameter. We  construct approximate eigenfunctions in
neighbourhoods of the different wells and give weighted \ell^2-estimates for the difference of these
and the exact eigenfunctions of the associated Dirichlet-operators.

Zeitschrift:
Asymptotic Analysis
Verlag:
IOS Press
Seiten:
61-89
Band:
97

2014 | Agmon-type estimates for a class of jump processes | Markus Klein, Christian Leonard, Elke Rosenberger Zeitschrift: Mathematische Nachrichten Seiten: 2021 – 2039 Band: 287, Nr. 1 Link zur Publikation

Agmon-type estimates for a class of jump processes

Autoren: Markus Klein, Christian Leonard, Elke Rosenberger (2014)

In the limit \varepsilon \to 0 we analyze the generators H_\varepsilon of families of reversible jump processes in R^d 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 C^2 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 \varepsilon Z^d.

Although our final interest is in the (sub)stochastic jump process, technically this is a pure analysis paper, inspired by PDE techniques.

Zeitschrift:
Mathematische Nachrichten
Seiten:
2021 – 2039
Band:
287, Nr. 1

2013 | Asymptotic eigenfunctions for Schrödinger operators on a vector bundle | Matthias Ludewig, Elke Rosenberger Link zum Preprint

Asymptotic eigenfunctions for Schrödinger operators on a vector bundle

Autoren: Matthias Ludewig, Elke Rosenberger (2013)

In the limit ℏ→0, we analyze a class of Schr\"odinger operators H = ℏ2 L + ℏ W + V idEh acting on sections of a vector bundle Eh 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 ∈ M. We construct quasimodes of WKB-type near p for eigenfunctions associated with the low lying eigenvalues of H. These are obtained from eigenfunctions of the associated harmonic oscillator Hp,ℏ at p, acting on C(TpM, Ehp).

2012 | Tunneling for a class of difference operators | Markus Klein, Elke Rosenberger Zeitschrift: Annales Henri Poincare Seiten: 1231 – 1269 Band: 13, Nr. 5 Link zur Publikation

Tunneling for a class of difference operators

Autoren: Markus Klein, Elke Rosenberger (2012)

We analyze a general class of difference operators H_\varepsilon = T_\varepsilon + V_\varepsilon acting on the square summable function on \varepsilon Z^d, where V_\varepsilon is a multi-well potential and \varepsilon 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 H_\varepsilon as a small perturbation of these comparison problems. We describe tunneling by a certain interaction matrix, similar to the analysis for the Schrödinger operator (see Helffer-Sjöstrand (1982)), and estimate the remainder, which is exponentially small and roughly quadratic compared with the interaction matrix.

Zeitschrift:
Annales Henri Poincare
Seiten:
1231 – 1269
Band:
13, Nr. 5

2011 | Asymptotic Eigenfunctions for a class of difference operators | Markus Klein, Elke Rosenberger Zeitschrift: Asymptotic Analysis Seiten: 1 – 36 Band: 73, Nr. 1-2 Link zur Publikation

Asymptotic Eigenfunctions for a class of difference operators

Autoren: Markus Klein, Elke Rosenberger (2011)

We analyze a general class of difference operators H_\varepsilon = T_\varepsilon+ V_\varepsilon on \ell^2(\varepsilon\mathbb{Z}^d), where V_\varepsilon is a one-well potential and \varepsilon is a small parameter.

We construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low lying eigenvalues of H_\varepsilon. These are obtained from eigenfunctions or quasimodes for the operator H_\varepsilon, acting on L^2(R^d), via restriction to the lattice \varepsilon Z^d.

Zeitschrift:
Asymptotic Analysis
Seiten:
1 – 36
Band:
73, Nr. 1-2

2009 | Harmonic Approximations of difference operators | Markus Klein, Elke Rosenberger Zeitschrift: Journal of Functional Analysis Seiten: 3409 – 3453 Band: 257, Nr. 1 Link zur Publikation

Harmonic Approximations of difference operators

Autoren: Markus Klein, Elke Rosenberger (2009)

For a general class of difference operators H_\varepsilon = T_\varepsilon + V_\varepsilon on \ell^2(\varepsilon\mathbf{Z}^d).
where V_\varepsilon is a multi-well potential and \varepsilon is a small parameter, we analyze the asymptotic behavior as \varepsilon\to 0 of the (low-lying) eigenvalues and eigenfunctions.

We show that the first n eigenvalues of H_\varepsilon converge to the first n eigenvalues of the direct sum of harmonic oscillators on \mathbf{R}^d located at the several wells. Our proof is microlocal.

Zeitschrift:
Journal of Functional Analysis
Seiten:
3409 – 3453
Band:
257, Nr. 1

2008 | Agmon-Type Estimates for a class of Difference Operators | Markus Klein, Elke Rosenberger Zeitschrift: Ann. Henri Poincare Verlag: Birkhäuser Verlag Basel Seiten: 1177 - 1215 Band: 9 Link zur Publikation

Agmon-Type Estimates for a class of Difference Operators

Autoren: Markus Klein, Elke Rosenberger (2008)

We analyze a general class of self-adjoint difference operators H_\varepsilon = T_\varepsilon + V_\varepsilon on \ell^2(\varepsilon{\mathbf Z}^d), where V_\varepsilon is a one-well potential and \varepsilon is a small parameter.

We construct a Finslerian distance d induced by H_\varepsilon and show that short integral curves are geodesics. Then we show that Dirichlet eigenfunctions decay exponentially with a rate controlled by the Finsler distance to the well. This is analog to semiclassical Agmon estimates for Schrödinger operators.

Zeitschrift:
Ann. Henri Poincare
Verlag:
Birkhäuser Verlag Basel
Seiten:
1177 - 1215
Band:
9

2006 | Asymptotic spectral analysis and tunnelling for a class of difference operators | Elke Rosenberger Buchtitel: Dissertation Link zur Publikation

Asymptotic spectral analysis and tunnelling for a class of difference operators

Autoren: Elke Rosenberger (2006)

We analyze the asymptotic behavior in the limit \varepsilon \to 0 for a wide class of difference operators H_\varepsilon = T_\varepsilon + V_\varepsilon with underlying multi-well potential. They act on the square summable functions on the lattice \varepsilon \mathbb{Z}^d. We start showing the validity of an harmonic approximation and construct WKB-solutions at the wells. Then we construct a Finslerian distance d induced by H_\varepsilon and show that short integral curves are geodesics and d gives the rate for the exponential decay of Dirichlet eigenfunctions. In terms of this distance, we give sharp estimates for the interaction between the wells and construct the interaction matrix.

Buchtitel:
Dissertation
  • Dr. Elke Rosenberger, Raum 2.09.2.07, Tel. 0331 977 1258, email: erosen*at*uni-potsdam.de
  • Dr. Barbara Schneider, Raum 2.28.1.104, Tel. 0331 977 2961, email: barbara.schneider*at*uni-potsdam.de
  • Dr. Alice Boit, Tel. 0331977 1907, email: aboit*at*uni-potsdam.de

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