---
_id: '51206'
abstract:
- lang: eng
text: "We present a numerical algorithm for the computation of invariant Ruelle\r\ndistributions
on convex co-compact hyperbolic surfaces. This is achieved by\r\nexploiting the
connection between invariant Ruelle distributions and residues\r\nof meromorphically
continued weighted zeta functions established by the authors\r\ntogether with
Barkhofen (2021). To make this applicable for numerics we express\r\nthe weighted
zeta as the logarithmic derivative of a suitable parameter\r\ndependent Fredholm
determinant similar to Borthwick (2014). As an additional\r\ndifficulty our transfer
operator has to include a contracting direction which\r\nwe account for with techniques
developed by Rugh (1992). We achieve a further\r\nimprovement in convergence speed
for our algorithm in the case of surfaces with\r\nadditional symmetries by proving
and applying a symmetry reduction of weighted\r\nzeta functions."
author:
- first_name: Philipp
full_name: Schütte, Philipp
id: '50168'
last_name: Schütte
- first_name: Tobias
full_name: Weich, Tobias
id: '49178'
last_name: Weich
orcid: 0000-0002-9648-6919
citation:
ama: Schütte P, Weich T. Invariant Ruelle Distributions on Convex-Cocompact Hyperbolic
Surfaces -- A Numerical Algorithm via Weighted Zeta Functions. arXiv:230813463.
Published online 2023.
apa: Schütte, P., & Weich, T. (2023). Invariant Ruelle Distributions on Convex-Cocompact
Hyperbolic Surfaces -- A Numerical Algorithm via Weighted Zeta Functions. In
arXiv:2308.13463.
bibtex: '@article{Schütte_Weich_2023, title={Invariant Ruelle Distributions on Convex-Cocompact
Hyperbolic Surfaces -- A Numerical Algorithm via Weighted Zeta Functions}, journal={arXiv:2308.13463},
author={Schütte, Philipp and Weich, Tobias}, year={2023} }'
chicago: Schütte, Philipp, and Tobias Weich. “Invariant Ruelle Distributions on
Convex-Cocompact Hyperbolic Surfaces -- A Numerical Algorithm via Weighted Zeta
Functions.” ArXiv:2308.13463, 2023.
ieee: P. Schütte and T. Weich, “Invariant Ruelle Distributions on Convex-Cocompact
Hyperbolic Surfaces -- A Numerical Algorithm via Weighted Zeta Functions,” arXiv:2308.13463.
2023.
mla: Schütte, Philipp, and Tobias Weich. “Invariant Ruelle Distributions on Convex-Cocompact
Hyperbolic Surfaces -- A Numerical Algorithm via Weighted Zeta Functions.” ArXiv:2308.13463,
2023.
short: P. Schütte, T. Weich, ArXiv:2308.13463 (2023).
date_created: 2024-02-06T20:58:35Z
date_updated: 2024-02-11T19:56:01Z
department:
- _id: '10'
- _id: '623'
- _id: '548'
external_id:
arxiv:
- '2308.13463'
language:
- iso: eng
publication: arXiv:2308.13463
status: public
title: Invariant Ruelle Distributions on Convex-Cocompact Hyperbolic Surfaces --
A Numerical Algorithm via Weighted Zeta Functions
type: preprint
user_id: '49178'
year: '2023'
...
---
_id: '53410'
abstract:
- lang: eng
text: AbstractWe consider a geodesic billiard system
consisting of a complete Riemannian manifold and an obstacle submanifold with
boundary at which the trajectories of the geodesic flow experience specular reflections.
We show that if the geodesic billiard system is hyperbolic on its trapped set
and the latter is compact and non-grazing, the techniques for open hyperbolic
systems developed by Dyatlov and Guillarmou (Ann Henri Poincaré 17(11):3089–3146,
2016) can be applied to a smooth model for the discontinuous flow defined by the
non-grazing billiard trajectories. This allows us to obtain a meromorphic resolvent
for the generator of the billiard flow. As an application we prove a meromorphic
continuation of weighted zeta functions together with explicit residue formulae.
In particular, our results apply to scattering by convex obstacles in the Euclidean
plane.
author:
- first_name: Benjamin
full_name: Delarue, Benjamin
id: '70575'
last_name: Delarue
- first_name: Philipp
full_name: Schütte, Philipp
id: '50168'
last_name: Schütte
- first_name: Tobias
full_name: Weich, Tobias
id: '49178'
last_name: Weich
orcid: 0000-0002-9648-6919
citation:
ama: Delarue B, Schütte P, Weich T. Resonances and Weighted Zeta Functions for Obstacle
Scattering via Smooth Models. Annales Henri Poincaré. 2023;25(2):1607-1656.
doi:10.1007/s00023-023-01379-x
apa: Delarue, B., Schütte, P., & Weich, T. (2023). Resonances and Weighted Zeta
Functions for Obstacle Scattering via Smooth Models. Annales Henri Poincaré,
25(2), 1607–1656. https://doi.org/10.1007/s00023-023-01379-x
bibtex: '@article{Delarue_Schütte_Weich_2023, title={Resonances and Weighted Zeta
Functions for Obstacle Scattering via Smooth Models}, volume={25}, DOI={10.1007/s00023-023-01379-x},
number={2}, journal={Annales Henri Poincaré}, publisher={Springer Science and
Business Media LLC}, author={Delarue, Benjamin and Schütte, Philipp and Weich,
Tobias}, year={2023}, pages={1607–1656} }'
chicago: 'Delarue, Benjamin, Philipp Schütte, and Tobias Weich. “Resonances and
Weighted Zeta Functions for Obstacle Scattering via Smooth Models.” Annales
Henri Poincaré 25, no. 2 (2023): 1607–56. https://doi.org/10.1007/s00023-023-01379-x.'
ieee: 'B. Delarue, P. Schütte, and T. Weich, “Resonances and Weighted Zeta Functions
for Obstacle Scattering via Smooth Models,” Annales Henri Poincaré, vol.
25, no. 2, pp. 1607–1656, 2023, doi: 10.1007/s00023-023-01379-x.'
mla: Delarue, Benjamin, et al. “Resonances and Weighted Zeta Functions for Obstacle
Scattering via Smooth Models.” Annales Henri Poincaré, vol. 25, no. 2,
Springer Science and Business Media LLC, 2023, pp. 1607–56, doi:10.1007/s00023-023-01379-x.
short: B. Delarue, P. Schütte, T. Weich, Annales Henri Poincaré 25 (2023) 1607–1656.
date_created: 2024-04-11T12:30:14Z
date_updated: 2024-04-11T12:37:34Z
department:
- _id: '548'
doi: 10.1007/s00023-023-01379-x
intvolume: ' 25'
issue: '2'
keyword:
- Mathematical Physics
- Nuclear and High Energy Physics
- Statistical and Nonlinear Physics
language:
- iso: eng
page: 1607-1656
publication: Annales Henri Poincaré
publication_identifier:
issn:
- 1424-0637
- 1424-0661
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Resonances and Weighted Zeta Functions for Obstacle Scattering via Smooth Models
type: journal_article
user_id: '70575'
volume: 25
year: '2023'
...
---
_id: '31059'
abstract:
- lang: eng
text: In this article we prove meromorphic continuation of weighted zeta functions
in the framework of open hyperbolic systems by using the meromorphically continued
restricted resolvent of Dyatlov and Guillarmou (2016). We obtain a residue formula
proving equality between residues of weighted zetas and invariant Ruelle distributions.
We combine this equality with results of Guillarmou, Hilgert and Weich (2021)
in order to relate the residues to Patterson-Sullivan distributions. Finally we
provide proof-of-principle results concerning the numerical calculation of invariant
Ruelle distributions for 3-disc scattering systems.
author:
- first_name: Philipp
full_name: Schütte, Philipp
id: '50168'
last_name: Schütte
- first_name: Tobias
full_name: Weich, Tobias
id: '49178'
last_name: Weich
orcid: 0000-0002-9648-6919
- first_name: Sonja
full_name: Barkhofen, Sonja
id: '48188'
last_name: Barkhofen
citation:
ama: Schütte P, Weich T, Barkhofen S. Meromorphic Continuation of Weighted Zeta
Functions on Open Hyperbolic Systems. Communications in Mathematical Physics.
2023;398:655-678. doi:https://doi.org/10.1007/s00220-022-04538-z
apa: Schütte, P., Weich, T., & Barkhofen, S. (2023). Meromorphic Continuation
of Weighted Zeta Functions on Open Hyperbolic Systems. Communications in Mathematical
Physics, 398, 655–678. https://doi.org/10.1007/s00220-022-04538-z
bibtex: '@article{Schütte_Weich_Barkhofen_2023, title={Meromorphic Continuation
of Weighted Zeta Functions on Open Hyperbolic Systems}, volume={398}, DOI={https://doi.org/10.1007/s00220-022-04538-z},
journal={Communications in Mathematical Physics}, author={Schütte, Philipp and
Weich, Tobias and Barkhofen, Sonja}, year={2023}, pages={655–678} }'
chicago: 'Schütte, Philipp, Tobias Weich, and Sonja Barkhofen. “Meromorphic Continuation
of Weighted Zeta Functions on Open Hyperbolic Systems.” Communications in Mathematical
Physics 398 (2023): 655–78. https://doi.org/10.1007/s00220-022-04538-z.'
ieee: 'P. Schütte, T. Weich, and S. Barkhofen, “Meromorphic Continuation of Weighted
Zeta Functions on Open Hyperbolic Systems,” Communications in Mathematical
Physics, vol. 398, pp. 655–678, 2023, doi: https://doi.org/10.1007/s00220-022-04538-z.'
mla: Schütte, Philipp, et al. “Meromorphic Continuation of Weighted Zeta Functions
on Open Hyperbolic Systems.” Communications in Mathematical Physics, vol.
398, 2023, pp. 655–78, doi:https://doi.org/10.1007/s00220-022-04538-z.
short: P. Schütte, T. Weich, S. Barkhofen, Communications in Mathematical Physics
398 (2023) 655–678.
date_created: 2022-05-04T12:27:46Z
date_updated: 2026-02-18T10:41:07Z
department:
- _id: '10'
- _id: '548'
- _id: '623'
- _id: '15'
doi: https://doi.org/10.1007/s00220-022-04538-z
external_id:
arxiv:
- '2112.05791'
intvolume: ' 398'
language:
- iso: eng
page: 655-678
publication: Communications in Mathematical Physics
status: public
title: Meromorphic Continuation of Weighted Zeta Functions on Open Hyperbolic Systems
type: journal_article
user_id: '49178'
volume: 398
year: '2023'
...
---
_id: '31057'
abstract:
- lang: eng
text: In this paper we give an overview over some aspects of the modern mathematical
theory of Ruelle resonances for chaotic, i.e. uniformly hyperbolic, dynamical
systems and their implications in physics. First we recall recent developments
in the mathematical theory of resonances, in particular how invariant Ruelle distributions
arise as residues of weighted zeta functions. Then we derive a correspondence
between weighted and semiclassical zeta functions in the setting of negatively
curved surfaces. Combining this with results of Hilgert, Guillarmou and Weich
yields a high frequency interpretation of invariant Ruelle distributions as quantum
mechanical matrix coefficients in constant negative curvature. We finish by presenting
numerical calculations of phase space distributions in the more physical setting
of 3-disk scattering systems.
article_number: '244007'
article_type: review
author:
- first_name: Sonja
full_name: Barkhofen, Sonja
id: '48188'
last_name: Barkhofen
- first_name: Philipp
full_name: Schütte, Philipp
id: '50168'
last_name: Schütte
- first_name: Tobias
full_name: Weich, Tobias
id: '49178'
last_name: Weich
orcid: 0000-0002-9648-6919
citation:
ama: 'Barkhofen S, Schütte P, Weich T. Semiclassical formulae For Wigner distributions.
Journal of Physics A: Mathematical and Theoretical. 2022;55(24). doi:10.1088/1751-8121/ac6d2b'
apa: 'Barkhofen, S., Schütte, P., & Weich, T. (2022). Semiclassical formulae
For Wigner distributions. Journal of Physics A: Mathematical and Theoretical,
55(24), Article 244007. https://doi.org/10.1088/1751-8121/ac6d2b'
bibtex: '@article{Barkhofen_Schütte_Weich_2022, title={Semiclassical formulae For
Wigner distributions}, volume={55}, DOI={10.1088/1751-8121/ac6d2b},
number={24244007}, journal={Journal of Physics A: Mathematical and Theoretical},
publisher={IOP Publishing Ltd}, author={Barkhofen, Sonja and Schütte, Philipp
and Weich, Tobias}, year={2022} }'
chicago: 'Barkhofen, Sonja, Philipp Schütte, and Tobias Weich. “Semiclassical Formulae
For Wigner Distributions.” Journal of Physics A: Mathematical and Theoretical
55, no. 24 (2022). https://doi.org/10.1088/1751-8121/ac6d2b.'
ieee: 'S. Barkhofen, P. Schütte, and T. Weich, “Semiclassical formulae For Wigner
distributions,” Journal of Physics A: Mathematical and Theoretical, vol.
55, no. 24, Art. no. 244007, 2022, doi: 10.1088/1751-8121/ac6d2b.'
mla: 'Barkhofen, Sonja, et al. “Semiclassical Formulae For Wigner Distributions.”
Journal of Physics A: Mathematical and Theoretical, vol. 55, no. 24, 244007,
IOP Publishing Ltd, 2022, doi:10.1088/1751-8121/ac6d2b.'
short: 'S. Barkhofen, P. Schütte, T. Weich, Journal of Physics A: Mathematical and
Theoretical 55 (2022).'
date_created: 2022-05-04T12:23:11Z
date_updated: 2024-02-06T20:40:45Z
department:
- _id: '623'
- _id: '548'
- _id: '10'
doi: 10.1088/1751-8121/ac6d2b
external_id:
arxiv:
- '2201.04892'
intvolume: ' 55'
issue: '24'
language:
- iso: eng
publication: 'Journal of Physics A: Mathematical and Theoretical'
publisher: IOP Publishing Ltd
status: public
title: Semiclassical formulae For Wigner distributions
type: journal_article
user_id: '49178'
volume: 55
year: '2022'
...
---
_id: '31058'
abstract:
- lang: eng
text: We consider a geodesic billiard system consisting of a complete Riemannian
manifold and an obstacle submanifold with boundary at which the trajectories of
the geodesic flow experience specular reflections. We show that if the geodesic
billiard system is hyperbolic on its trapped set and the latter is compact and
non-grazing the techniques for open hyperbolic systems developed by Dyatlov and
Guillarmou can be applied to a smooth model for the discontinuous flow defined
by the non-grazing billiard trajectories. This allows us to obtain a meromorphic
resolvent for the generator of the billiard flow. As an application we prove a
meromorphic continuation of weighted zeta functions together with explicit residue
formulae. In particular, our results apply to scattering by convex obstacles in
the Euclidean plane.
author:
- first_name: Philipp
full_name: Schütte, Philipp
id: '50168'
last_name: Schütte
- first_name: Tobias
full_name: Weich, Tobias
id: '49178'
last_name: Weich
orcid: 0000-0002-9648-6919
- first_name: Benjamin
full_name: Delarue, Benjamin
last_name: Delarue
citation:
ama: Schütte P, Weich T, Delarue B. Resonances and weighted zeta functions for obstacle
scattering via smooth models. Published online 2021.
apa: Schütte, P., Weich, T., & Delarue, B. (2021). Resonances and weighted
zeta functions for obstacle scattering via smooth models.
bibtex: '@article{Schütte_Weich_Delarue_2021, title={Resonances and weighted zeta
functions for obstacle scattering via smooth models}, author={Schütte, Philipp
and Weich, Tobias and Delarue, Benjamin}, year={2021} }'
chicago: Schütte, Philipp, Tobias Weich, and Benjamin Delarue. “Resonances and Weighted
Zeta Functions for Obstacle Scattering via Smooth Models,” 2021.
ieee: P. Schütte, T. Weich, and B. Delarue, “Resonances and weighted zeta functions
for obstacle scattering via smooth models.” 2021.
mla: Schütte, Philipp, et al. Resonances and Weighted Zeta Functions for Obstacle
Scattering via Smooth Models. 2021.
short: P. Schütte, T. Weich, B. Delarue, (2021).
date_created: 2022-05-04T12:25:58Z
date_updated: 2022-05-17T12:05:52Z
department:
- _id: '10'
- _id: '548'
external_id:
arxiv:
- '2109.05907'
language:
- iso: eng
status: public
title: Resonances and weighted zeta functions for obstacle scattering via smooth models
type: preprint
user_id: '50168'
year: '2021'
...
---
_id: '31302'
author:
- first_name: Philipp
full_name: Schütte, Philipp
id: '50168'
last_name: Schütte
citation:
ama: Schütte P. Numerically Investigating Residues of Weighted Zeta Functions
on Schottky Surfaces.; 2019.
apa: Schütte, P. (2019). Numerically Investigating Residues of Weighted Zeta
Functions on Schottky Surfaces.
bibtex: '@book{Schütte_2019, title={Numerically Investigating Residues of Weighted
Zeta Functions on Schottky Surfaces}, author={Schütte, Philipp}, year={2019} }'
chicago: Schütte, Philipp. Numerically Investigating Residues of Weighted Zeta
Functions on Schottky Surfaces, 2019.
ieee: P. Schütte, Numerically Investigating Residues of Weighted Zeta Functions
on Schottky Surfaces. 2019.
mla: Schütte, Philipp. Numerically Investigating Residues of Weighted Zeta Functions
on Schottky Surfaces. 2019.
short: P. Schütte, Numerically Investigating Residues of Weighted Zeta Functions
on Schottky Surfaces, 2019.
date_created: 2022-05-17T13:41:53Z
date_updated: 2024-02-19T06:21:23Z
department:
- _id: '548'
language:
- iso: eng
status: public
supervisor:
- first_name: Tobias
full_name: Weich, Tobias
id: '49178'
last_name: Weich
orcid: 0000-0002-9648-6919
- first_name: Joachim
full_name: Hilgert, Joachim
id: '220'
last_name: Hilgert
title: Numerically Investigating Residues of Weighted Zeta Functions on Schottky Surfaces
type: mastersthesis
user_id: '49063'
year: '2019'
...
---
_id: '31301'
author:
- first_name: Philipp
full_name: Schütte, Philipp
id: '50168'
last_name: Schütte
citation:
ama: Schütte P. Identifying and Realizing Symmetries in Quantum Walks - Symmetry
Classes and Quantum Walks.; 2017.
apa: Schütte, P. (2017). Identifying and Realizing Symmetries in Quantum Walks
- Symmetry Classes and Quantum Walks.
bibtex: '@book{Schütte_2017, title={Identifying and Realizing Symmetries in Quantum
Walks - Symmetry Classes and Quantum Walks}, author={Schütte, Philipp}, year={2017}
}'
chicago: Schütte, Philipp. Identifying and Realizing Symmetries in Quantum Walks
- Symmetry Classes and Quantum Walks, 2017.
ieee: P. Schütte, Identifying and Realizing Symmetries in Quantum Walks - Symmetry
Classes and Quantum Walks. 2017.
mla: Schütte, Philipp. Identifying and Realizing Symmetries in Quantum Walks
- Symmetry Classes and Quantum Walks. 2017.
short: P. Schütte, Identifying and Realizing Symmetries in Quantum Walks - Symmetry
Classes and Quantum Walks, 2017.
date_created: 2022-05-17T13:40:30Z
date_updated: 2022-05-17T13:42:20Z
department:
- _id: '548'
- _id: '288'
language:
- iso: eng
status: public
supervisor:
- first_name: Tobias
full_name: Weich, Tobias
id: '49178'
last_name: Weich
orcid: 0000-0002-9648-6919
- first_name: Christine
full_name: Silberhorn, Christine
id: '26263'
last_name: Silberhorn
title: Identifying and Realizing Symmetries in Quantum Walks - Symmetry Classes and
Quantum Walks
type: bachelorsthesis
user_id: '50168'
year: '2017'
...