Dynamics of semigroups generated by analytic functions of the Laplacian on Homogeneous Trees
Let $\Psi $ be a non-constant complex-valued analytic function defined on a connected, open set containing the $L^p$-spectrum of the Laplacian $\mathcal{L}$ on a homogeneous tree. In this paper we give a necessary and sufficient condition for the semigroup $T(t)=e^{t\Psi (\mathcal{L})}$ to be chaoti...
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Académie des sciences
2023-01-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.382/ |
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| author | Kumar, Pratyoosh Rano, Sumit Kumar |
| author_facet | Kumar, Pratyoosh Rano, Sumit Kumar |
| author_sort | Kumar, Pratyoosh |
| collection | DOAJ |
| description | Let $\Psi $ be a non-constant complex-valued analytic function defined on a connected, open set containing the $L^p$-spectrum of the Laplacian $\mathcal{L}$ on a homogeneous tree. In this paper we give a necessary and sufficient condition for the semigroup $T(t)=e^{t\Psi (\mathcal{L})}$ to be chaotic on $L^{p}$-spaces. We also study the chaotic dynamics of the semigroup $T(t)=e^{t(a\mathcal{L}+b)}$ separately and obtain a sharp range of $b$ for which $T(t)$ is chaotic on $L^{p}$-spaces. It includes some of the important semigroups such as the heat semigroup and the Schrödinger semigroup. |
| format | Article |
| id | doaj-art-5b530b83cb454a3f96107c7695004da3 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-01-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-5b530b83cb454a3f96107c7695004da32025-02-07T11:06:07ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-01-01361G111310.5802/crmath.38210.5802/crmath.382Dynamics of semigroups generated by analytic functions of the Laplacian on Homogeneous TreesKumar, Pratyoosh0Rano, Sumit Kumar1Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, 781039, IndiaDepartment of Mathematics, Indian Institute of Technology Guwahati, Guwahati, 781039, IndiaLet $\Psi $ be a non-constant complex-valued analytic function defined on a connected, open set containing the $L^p$-spectrum of the Laplacian $\mathcal{L}$ on a homogeneous tree. In this paper we give a necessary and sufficient condition for the semigroup $T(t)=e^{t\Psi (\mathcal{L})}$ to be chaotic on $L^{p}$-spaces. We also study the chaotic dynamics of the semigroup $T(t)=e^{t(a\mathcal{L}+b)}$ separately and obtain a sharp range of $b$ for which $T(t)$ is chaotic on $L^{p}$-spaces. It includes some of the important semigroups such as the heat semigroup and the Schrödinger semigroup.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.382/ |
| spellingShingle | Kumar, Pratyoosh Rano, Sumit Kumar Dynamics of semigroups generated by analytic functions of the Laplacian on Homogeneous Trees Comptes Rendus. Mathématique |
| title | Dynamics of semigroups generated by analytic functions of the Laplacian on Homogeneous Trees |
| title_full | Dynamics of semigroups generated by analytic functions of the Laplacian on Homogeneous Trees |
| title_fullStr | Dynamics of semigroups generated by analytic functions of the Laplacian on Homogeneous Trees |
| title_full_unstemmed | Dynamics of semigroups generated by analytic functions of the Laplacian on Homogeneous Trees |
| title_short | Dynamics of semigroups generated by analytic functions of the Laplacian on Homogeneous Trees |
| title_sort | dynamics of semigroups generated by analytic functions of the laplacian on homogeneous trees |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.382/ |
| work_keys_str_mv | AT kumarpratyoosh dynamicsofsemigroupsgeneratedbyanalyticfunctionsofthelaplacianonhomogeneoustrees AT ranosumitkumar dynamicsofsemigroupsgeneratedbyanalyticfunctionsofthelaplacianonhomogeneoustrees |