# Dr. Elke Rosenberger

### wissenschaftliche Mitarbeiterin

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

Sprechzeiten:  nach Vereinbarung

#### Forschungsgebiete

• 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 RosenbergerZeitschrift: Annales Henri PoincareVerlag: Springer VerlagSeiten: 3511-3559Band: 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 <tex>H_\varepsilon = T_\varepsilon +V_\varepsilo</tex> on <tex>\ell^2((\varepsilon\mathbb{Z})^d)</tex>, where
<tex>V_\varepsilon</tex> is a multi-well potential and <tex>\varepsilon</tex> 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<tex> h_0(x, ξ)</tex> of <tex>H_\varepsilon</tex>) 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 RosenbergerZeitschrift: Reviews in Mathematical Physics (RMP)Verlag: World Scientific PublishingSeiten: 1830002 (1-42)Band: Vol.30, No. 4Link 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 <tex>H_\varepsilon = T_\varepsilon
</tex>

<tex>+ V_\varepsilon</tex> on <tex>\ell^2(\varepsilon\mathbb{Z}^d)</tex>, where<tex> </tex><tex><tex>V_ε</tex></tex> 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 <tex>H_\varepsilon</tex> is treated as a small perturbation of these comparison
problems.
After constructing a Finslerian distance d induced by <tex>H_\varepsilon</tex> 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 <tex>H_\varepsilon</tex> converge to
the first n eigenvalues of the direct sum of harmonic oscillators on <tex>\mathbb{R}^d</tex> 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<tex> H_\varepsilon</tex>. These are obtained
from eigenfunctions or quasimodes for the
operator <tex>H_\varepsilon</tex>, acting on <tex>L^2(\mathbb{R}^d)</tex>, via restriction to the lattice <tex>\varepsilon\mathbb{Z}^d</tex>.

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<tex> \ell^2</tex>-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 RosenbergerZeitschrift: Asymptotic AnalysisVerlag: IOS PressSeiten: 61-89Band: 97Link 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 <tex>H_\varepsilon = T_\varepsilon + V_\varepsilon</tex> on <tex>\ell^2(\varepsilon \mathbb{Z}^d</tex>), where <tex>V_\varepsilon</tex> is a multi-well
potential and <tex>\varepsilon</tex> is a small parameter. We  construct approximate eigenfunctions in
neighbourhoods of the different wells and give weighted <tex>\ell^2</tex>-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 RosenbergerZeitschrift: Mathematische NachrichtenSeiten: 2021 – 2039Band: 287, Nr. 1Link zur Publikation

### Agmon-type estimates for a class of jump processes

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

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

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 RosenbergerLink 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 RosenbergerZeitschrift: Annales Henri PoincareSeiten: 1231 – 1269Band: 13, Nr. 5Link zur Publikation

### Tunneling for a class of difference operators

#### Autoren: Markus Klein, Elke Rosenberger (2012)

We analyze a general class of difference operators <tex>H_\varepsilon = T_\varepsilon + V_\varepsilon</tex> acting on the square summable function on <tex>\varepsilon</tex><tex> Z^d</tex>, where <tex>V_\varepsilon</tex> is a multi-well potential and <tex>\varepsilon</tex> 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 <tex>H_\varepsilon</tex> 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 RosenbergerZeitschrift: Asymptotic AnalysisSeiten: 1 – 36Band: 73, Nr. 1-2Link zur Publikation

### Asymptotic Eigenfunctions for a class of difference operators

#### Autoren: Markus Klein, Elke Rosenberger (2011)

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

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

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

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

### Harmonic Approximations of difference operators

#### Autoren: Markus Klein, Elke Rosenberger (2009)

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

We show that the first n eigenvalues of <tex>H_\varepsilon</tex> converge to the first n eigenvalues of the direct sum of harmonic oscillators on <tex>\mathbf{R}^d</tex> 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 RosenbergerZeitschrift: Ann. Henri PoincareVerlag: Birkhäuser Verlag BaselSeiten: 1177 - 1215Band: 9Link 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 <tex>H_\varepsilon = T_\varepsilon + V_\varepsilon</tex> on <tex>\ell^2(\varepsilon{\mathbf Z}^d)</tex>, where <tex>V_\varepsilon</tex> is a one-well potential and <tex>\varepsilon</tex> is a small parameter.

We construct a Finslerian distance d induced by<tex> H_\varepsilon</tex> 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 RosenbergerBuchtitel: DissertationLink 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 <tex>\varepsilon \to 0</tex> for a wide class of difference operators <tex>H_\varepsilon = T_\varepsilon + V_\varepsilon</tex> with underlying multi-well potential. They act on the square summable functions on the lattice <tex>\varepsilon \mathbb{Z}^d</tex>. 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 <tex>H_\varepsilon</tex> 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

#### Team der Gleichstellungsbeautragten der Mathematisch-Naturwissenschaftlichen Fakultät

• 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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