@article{59344,
  abstract     = {{<jats:title>Abstract</jats:title>
          <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       = {{Arends, Christian and Frahm, Jan and Hilgert, Joachim}},
  issn         = {{0926-2601}},
  journal      = {{Potential Analysis}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{Edge Laplacians and Edge Poisson Transforms for Graphs}}},
  doi          = {{10.1007/s11118-024-10184-y}},
  year         = {{2025}},
}

@article{53540,
  abstract     = {{<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">
                <mml:msup>
                  <mml:mi>L</mml:mi>
                  <mml:mn>1</mml:mn>
                </mml:msup>
              </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       = {{Papageorgiou, Efthymia}},
  issn         = {{0926-2601}},
  journal      = {{Potential Analysis}},
  keywords     = {{Analysis}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds}}},
  doi          = {{10.1007/s11118-023-10109-1}},
  year         = {{2023}},
}

@article{37662,
  author       = {{Rösler, Margit and Graczyk, Piotr and Luks, Tomasz}},
  issn         = {{0926-2601}},
  journal      = {{Potential Analysis}},
  keywords     = {{Analysis}},
  number       = {{3}},
  pages        = {{337--360}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{On the Green Function and Poisson Integrals of the Dunkl Laplacian}}},
  doi          = {{10.1007/s11118-017-9638-6}},
  volume       = {{48}},
  year         = {{2018}},
}

@article{40072,
  author       = {{Luks, Tomasz}},
  issn         = {{0926-2601}},
  journal      = {{Potential Analysis}},
  number       = {{1}},
  pages        = {{29--67}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{Boundary Behavior of α-Harmonic Functions on the Complement of the Sphere and Hyperplane}}},
  doi          = {{10.1007/s11118-012-9321-x}},
  volume       = {{39}},
  year         = {{2013}},
}