---
_id: '59344'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title>\r\n          <jats:p>For a finite graph,
    we establish natural isomorphisms between eigenspaces of a Laplace operator acting
    on functions on the edges and eigenspaces of a transfer operator acting on functions
    on one-sided infinite non-backtracking paths. Interpreting the transfer operator
    as a classical dynamical system and the Laplace operator as its quantization,
    this result can be viewed as a <jats:italic>quantum-classical correspondence</jats:italic>.
    In contrast to previously established quantum-classical correspondences for the
    vertex Laplacian which exclude certain exceptional spectral parameters, our correspondence
    is valid for all parameters. This allows us to relate certain spectral quantities
    to topological properties of the graph such as the cyclomatic number and the 2-colorability.
    The quantum-classical correspondence for the edge Laplacian is induced by an edge
    Poisson transform on the universal covering of the graph which is a tree of bounded
    degree. In the special case of regular trees, we relate both the vertex and the
    edge Poisson transform to the representation theory of the automorphism group
    of the tree and study associated operator valued Hecke algebras.</jats:p>"
author:
- first_name: Christian
  full_name: Arends, Christian
  last_name: Arends
- first_name: Jan
  full_name: Frahm, Jan
  last_name: Frahm
- first_name: Joachim
  full_name: Hilgert, Joachim
  id: '220'
  last_name: Hilgert
citation:
  ama: Arends C, Frahm J, Hilgert J. Edge Laplacians and Edge Poisson Transforms for
    Graphs. <i>Potential Analysis</i>. Published online 2025. doi:<a href="https://doi.org/10.1007/s11118-024-10184-y">10.1007/s11118-024-10184-y</a>
  apa: Arends, C., Frahm, J., &#38; Hilgert, J. (2025). Edge Laplacians and Edge Poisson
    Transforms for Graphs. <i>Potential Analysis</i>. <a href="https://doi.org/10.1007/s11118-024-10184-y">https://doi.org/10.1007/s11118-024-10184-y</a>
  bibtex: '@article{Arends_Frahm_Hilgert_2025, title={Edge Laplacians and Edge Poisson
    Transforms for Graphs}, DOI={<a href="https://doi.org/10.1007/s11118-024-10184-y">10.1007/s11118-024-10184-y</a>},
    journal={Potential Analysis}, publisher={Springer Science and Business Media LLC},
    author={Arends, Christian and Frahm, Jan and Hilgert, Joachim}, year={2025} }'
  chicago: Arends, Christian, Jan Frahm, and Joachim Hilgert. “Edge Laplacians and
    Edge Poisson Transforms for Graphs.” <i>Potential Analysis</i>, 2025. <a href="https://doi.org/10.1007/s11118-024-10184-y">https://doi.org/10.1007/s11118-024-10184-y</a>.
  ieee: 'C. Arends, J. Frahm, and J. Hilgert, “Edge Laplacians and Edge Poisson Transforms
    for Graphs,” <i>Potential Analysis</i>, 2025, doi: <a href="https://doi.org/10.1007/s11118-024-10184-y">10.1007/s11118-024-10184-y</a>.'
  mla: Arends, Christian, et al. “Edge Laplacians and Edge Poisson Transforms for
    Graphs.” <i>Potential Analysis</i>, Springer Science and Business Media LLC, 2025,
    doi:<a href="https://doi.org/10.1007/s11118-024-10184-y">10.1007/s11118-024-10184-y</a>.
  short: C. Arends, J. Frahm, J. Hilgert, Potential Analysis (2025).
date_created: 2025-04-04T08:02:14Z
date_updated: 2025-04-04T08:02:34Z
doi: 10.1007/s11118-024-10184-y
language:
- iso: eng
publication: Potential Analysis
publication_identifier:
  issn:
  - 0926-2601
  - 1572-929X
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Edge Laplacians and Edge Poisson Transforms for Graphs
type: journal_article
user_id: '220'
year: '2025'
...
---
_id: '53540'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title><jats:p>This note is concerned with two
    families of operators related to the fractional Laplacian, the first arising from
    the Caffarelli-Silvestre extension problem and the second from the fractional
    heat equation. They both include the Poisson semigroup. We show that on a complete,
    connected, and non-compact Riemannian manifold of non-negative Ricci curvature,
    in both cases, the solution with <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^1$$</jats:tex-math><mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                <mml:msup>\r\n
    \                 <mml:mi>L</mml:mi>\r\n                  <mml:mn>1</mml:mn>\r\n
    \               </mml:msup>\r\n              </mml:math></jats:alternatives></jats:inline-formula>
    initial data behaves asymptotically as the mass times the fundamental solution.
    Similar long-time convergence results remain valid on more general manifolds satisfying
    the Li-Yau two-sided estimate of the heat kernel. The situation changes drastically
    on hyperbolic space, and more generally on rank one non-compact symmetric spaces:
    we show that for the Poisson semigroup, the convergence to the Poisson kernel
    fails -but remains true under the additional assumption of radial initial data.</jats:p>"
author:
- first_name: Efthymia
  full_name: Papageorgiou, Efthymia
  id: '100325'
  last_name: Papageorgiou
citation:
  ama: Papageorgiou E. Large-Time Behavior of Two Families of Operators Related to
    the Fractional Laplacian on Certain Riemannian Manifolds. <i>Potential Analysis</i>.
    Published online 2023. doi:<a href="https://doi.org/10.1007/s11118-023-10109-1">10.1007/s11118-023-10109-1</a>
  apa: Papageorgiou, E. (2023). Large-Time Behavior of Two Families of Operators Related
    to the Fractional Laplacian on Certain Riemannian Manifolds. <i>Potential Analysis</i>.
    <a href="https://doi.org/10.1007/s11118-023-10109-1">https://doi.org/10.1007/s11118-023-10109-1</a>
  bibtex: '@article{Papageorgiou_2023, title={Large-Time Behavior of Two Families
    of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds},
    DOI={<a href="https://doi.org/10.1007/s11118-023-10109-1">10.1007/s11118-023-10109-1</a>},
    journal={Potential Analysis}, publisher={Springer Science and Business Media LLC},
    author={Papageorgiou, Efthymia}, year={2023} }'
  chicago: Papageorgiou, Efthymia. “Large-Time Behavior of Two Families of Operators
    Related to the Fractional Laplacian on Certain Riemannian Manifolds.” <i>Potential
    Analysis</i>, 2023. <a href="https://doi.org/10.1007/s11118-023-10109-1">https://doi.org/10.1007/s11118-023-10109-1</a>.
  ieee: 'E. Papageorgiou, “Large-Time Behavior of Two Families of Operators Related
    to the Fractional Laplacian on Certain Riemannian Manifolds,” <i>Potential Analysis</i>,
    2023, doi: <a href="https://doi.org/10.1007/s11118-023-10109-1">10.1007/s11118-023-10109-1</a>.'
  mla: Papageorgiou, Efthymia. “Large-Time Behavior of Two Families of Operators Related
    to the Fractional Laplacian on Certain Riemannian Manifolds.” <i>Potential Analysis</i>,
    Springer Science and Business Media LLC, 2023, doi:<a href="https://doi.org/10.1007/s11118-023-10109-1">10.1007/s11118-023-10109-1</a>.
  short: E. Papageorgiou, Potential Analysis (2023).
date_created: 2024-04-17T13:17:37Z
date_updated: 2024-04-17T13:19:59Z
department:
- _id: '555'
doi: 10.1007/s11118-023-10109-1
keyword:
- Analysis
language:
- iso: eng
publication: Potential Analysis
publication_identifier:
  issn:
  - 0926-2601
  - 1572-929X
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Large-Time Behavior of Two Families of Operators Related to the Fractional
  Laplacian on Certain Riemannian Manifolds
type: journal_article
user_id: '100325'
year: '2023'
...
---
_id: '37662'
author:
- first_name: Margit
  full_name: Rösler, Margit
  id: '37390'
  last_name: Rösler
- first_name: Piotr
  full_name: Graczyk, Piotr
  last_name: Graczyk
- first_name: Tomasz
  full_name: Luks, Tomasz
  last_name: Luks
citation:
  ama: Rösler M, Graczyk P, Luks T. On the Green Function and Poisson Integrals of
    the Dunkl Laplacian. <i>Potential Analysis</i>. 2018;48(3):337-360. doi:<a href="https://doi.org/10.1007/s11118-017-9638-6">10.1007/s11118-017-9638-6</a>
  apa: Rösler, M., Graczyk, P., &#38; Luks, T. (2018). On the Green Function and Poisson
    Integrals of the Dunkl Laplacian. <i>Potential Analysis</i>, <i>48</i>(3), 337–360.
    <a href="https://doi.org/10.1007/s11118-017-9638-6">https://doi.org/10.1007/s11118-017-9638-6</a>
  bibtex: '@article{Rösler_Graczyk_Luks_2018, title={On the Green Function and Poisson
    Integrals of the Dunkl Laplacian}, volume={48}, DOI={<a href="https://doi.org/10.1007/s11118-017-9638-6">10.1007/s11118-017-9638-6</a>},
    number={3}, journal={Potential Analysis}, publisher={Springer Science and Business
    Media LLC}, author={Rösler, Margit and Graczyk, Piotr and Luks, Tomasz}, year={2018},
    pages={337–360} }'
  chicago: 'Rösler, Margit, Piotr Graczyk, and Tomasz Luks. “On the Green Function
    and Poisson Integrals of the Dunkl Laplacian.” <i>Potential Analysis</i> 48, no.
    3 (2018): 337–60. <a href="https://doi.org/10.1007/s11118-017-9638-6">https://doi.org/10.1007/s11118-017-9638-6</a>.'
  ieee: 'M. Rösler, P. Graczyk, and T. Luks, “On the Green Function and Poisson Integrals
    of the Dunkl Laplacian,” <i>Potential Analysis</i>, vol. 48, no. 3, pp. 337–360,
    2018, doi: <a href="https://doi.org/10.1007/s11118-017-9638-6">10.1007/s11118-017-9638-6</a>.'
  mla: Rösler, Margit, et al. “On the Green Function and Poisson Integrals of the
    Dunkl Laplacian.” <i>Potential Analysis</i>, vol. 48, no. 3, Springer Science
    and Business Media LLC, 2018, pp. 337–60, doi:<a href="https://doi.org/10.1007/s11118-017-9638-6">10.1007/s11118-017-9638-6</a>.
  short: M. Rösler, P. Graczyk, T. Luks, Potential Analysis 48 (2018) 337–360.
date_created: 2023-01-20T09:25:41Z
date_updated: 2023-01-24T22:16:02Z
department:
- _id: '555'
doi: 10.1007/s11118-017-9638-6
intvolume: '        48'
issue: '3'
keyword:
- Analysis
language:
- iso: eng
page: 337-360
publication: Potential Analysis
publication_identifier:
  issn:
  - 0926-2601
  - 1572-929X
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: On the Green Function and Poisson Integrals of the Dunkl Laplacian
type: journal_article
user_id: '37390'
volume: 48
year: '2018'
...
---
_id: '40072'
author:
- first_name: Tomasz
  full_name: Luks, Tomasz
  id: '58312'
  last_name: Luks
citation:
  ama: Luks T. Boundary Behavior of α-Harmonic Functions on the Complement of the
    Sphere and Hyperplane. <i>Potential Analysis</i>. 2013;39(1):29-67. doi:<a href="https://doi.org/10.1007/s11118-012-9321-x">10.1007/s11118-012-9321-x</a>
  apa: Luks, T. (2013). Boundary Behavior of α-Harmonic Functions on the Complement
    of the Sphere and Hyperplane. <i>Potential Analysis</i>, <i>39</i>(1), 29–67.
    <a href="https://doi.org/10.1007/s11118-012-9321-x">https://doi.org/10.1007/s11118-012-9321-x</a>
  bibtex: '@article{Luks_2013, title={Boundary Behavior of α-Harmonic Functions on
    the Complement of the Sphere and Hyperplane}, volume={39}, DOI={<a href="https://doi.org/10.1007/s11118-012-9321-x">10.1007/s11118-012-9321-x</a>},
    number={1}, journal={Potential Analysis}, publisher={Springer Science and Business
    Media LLC}, author={Luks, Tomasz}, year={2013}, pages={29–67} }'
  chicago: 'Luks, Tomasz. “Boundary Behavior of α-Harmonic Functions on the Complement
    of the Sphere and Hyperplane.” <i>Potential Analysis</i> 39, no. 1 (2013): 29–67.
    <a href="https://doi.org/10.1007/s11118-012-9321-x">https://doi.org/10.1007/s11118-012-9321-x</a>.'
  ieee: 'T. Luks, “Boundary Behavior of α-Harmonic Functions on the Complement of
    the Sphere and Hyperplane,” <i>Potential Analysis</i>, vol. 39, no. 1, pp. 29–67,
    2013, doi: <a href="https://doi.org/10.1007/s11118-012-9321-x">10.1007/s11118-012-9321-x</a>.'
  mla: Luks, Tomasz. “Boundary Behavior of α-Harmonic Functions on the Complement
    of the Sphere and Hyperplane.” <i>Potential Analysis</i>, vol. 39, no. 1, Springer
    Science and Business Media LLC, 2013, pp. 29–67, doi:<a href="https://doi.org/10.1007/s11118-012-9321-x">10.1007/s11118-012-9321-x</a>.
  short: T. Luks, Potential Analysis 39 (2013) 29–67.
date_created: 2023-01-25T15:50:45Z
date_updated: 2023-01-26T17:29:16Z
department:
- _id: '555'
doi: 10.1007/s11118-012-9321-x
extern: '1'
intvolume: '        39'
issue: '1'
language:
- iso: eng
page: 29-67
publication: Potential Analysis
publication_identifier:
  issn:
  - 0926-2601
  - 1572-929X
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Boundary Behavior of α-Harmonic Functions on the Complement of the Sphere and
  Hyperplane
type: journal_article
user_id: '58312'
volume: 39
year: '2013'
...