Characterization of eigenfunctions of the Laplace-Beltrami operator through heat propagation in small time
type: Item
instance of: scholarly article
title: Characterization of eigenfunctions of the Laplace-Beltrami operator through heat propagation in small time
zbMATH Open document ID: 1451.58012
author: Muna Naik
author: Rudra P. Sarkar
published in: Monatshefte für Mathematik
publication date: 2020-07-14T00:00:00Z
review text: Let \(X\) be a Riemann manifold and \(\Delta\) the positive Laplace-Beltrami operator on \(X\). \(e^{-t\Delta}\) denotes the heat propagator. \(\Delta f=cf\) always implies \(e^{-t\Delta}f=e^{-tc}f \forall t>0 \). In this note the authors endeavour is to explore the converse of this. They show: For \(n\geq 2\) and \( p > \frac{2n}{n-1}\), if \(f \in L^p (\mathbb R^n)\) satisfies \( e^{-t\Delta_{\mathbb R^n}} f = e^{-tc} f \) for some \(c > 0, t > 0\), then \(\Delta_\mathbb R^n f = cf \). If \(f \in L^\infty (\mathbb R^n)\) satisfies \( e^{-t\Delta_{\mathbb R^n}} f = e^{-tc} f \) for some \(c \geq 0, t > 0\), then \(\Delta_\mathbb R^n f = cf \). For a rank one Riemannian symmetric space of noncompact type \(X\) the situation is different. The authors show that the answer to such a question is affirmative only if \(t\) is within a sharp range \(0 < t < T \), where the critical time \(T \) depends on the proposed eigenvalue \(c\) and the integrability of the eigenfunction (or temperedness of the eigendistribution). In fact, \(f \in L^p(X)\) in the above discussion can be replaced by appropriately tempered distributions. It appears that the tempered distributions are ideal objects in this setup as they are less restrictive.
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 7220396
zbMATH Keywords: eigenfunction of the Laplace-Beltrami operator
zbMATH Keywords: Riemannian symmetric spaces
zbMATH Keywords: heat kernel
reviewed by: Jorge Vargas
MaRDI profile type: Publication
full work available at URL: s00605-020-01437-0
OpenAlex ID: W3038065904
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DOI: 10.1007/S00605-020-01437-0
description: scientific article; zbMATH DE number 7220396