On the spectrum of the Laplacian of a left-invariant metric on a Lie group
type: Item
instance of: scholarly article
title: On the spectrum of the Laplacian of a left-invariant metric on a Lie group
zbMATH Open document ID: 1182.22005
author: Noël Lohoué
published in: Monatshefte für Mathematik
publication date: 2009-11-06T00:00:00Z
review text: On considère un groupe de Lie \(G\) connexe, non compact, de décomposition de Levi \(G=HS\); on suppose que le radical résoluble \(H\) de \(G\) est connexe. On considère un système de Hörmander \(\widetilde {X_1},\dots,\widetilde {X_n}\) de champs de vecteurs sur \(G\) invariant à gauche et l'on pose \(L=\sum_{i=1}^n{\widetilde X_i}^2\) le sous-laplacien associé. On montre que le spectre de \(L\) est un intervalle \([\sigma,\infty]\) avec, éventuellement, des valeurs propres de multiplicité infinies. Dans certains cas particuliers, on montre aussi qu'il est absolument continu; en général l'intervalle \([\sigma,\infty]\) est le spectre essentiel. Cette étude a débuté pour la première fois avec \textit{K. Furutani, K. Sagami} et \textit{N. Otsuki} [Commun. Partial Differ. Equations 18, No. 3--4, 533--555 (1993; Zbl 0791.58096)].
reviewed by: Béchir Dali
Mathematics Subject Classification ID: 22E15
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 32F45
Mathematics Subject Classification ID: 43A05
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 5627053
zbMATH Keywords: Lie group
zbMATH Keywords: invariant metric
zbMATH Keywords: Laplacian
zbMATH Keywords: spectrum
MaRDI profile type: Publication
full work available at URL: s00605-009-0137-8
OpenAlex ID: W2051697275
cites work: The \(L^2\)-index theorem for homogeneous spaces of Lie groups
cites work: The spectrum of the laplacian on a certain nilpotent lie group
cites work: Q5526485
cites work: The plancherel formula for group extensions
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DOI: 10.1007/S00605-009-0137-8
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description: scientific article; zbMATH DE number 5627053