Integrability and Huygens' principle on symmetric spaces
type: Item
instance of: scholarly article
title: Integrability and Huygens' principle on symmetric spaces
zbMATH Open document ID: 0859.43004
DOI: 10.1007/BF02099451
author: Oleg Chalykh
author: A. P. Veselov
published in: Communications in Mathematical Physics
publication date: 1997-03-31T00:00:00Z
review text: Let \(X=G/K\) be a Riemannian symmetric space, \({\mathcal L}_X\) the Laplace-Beltrami operator on \(X\). Consider the following equations: \[ \text{(a)}\quad \varphi_{tt}-{\mathcal L}_X\varphi + c\varphi=0,\quad \text{(b)}\quad {\mathcal L}_X\varphi=0,\quad \text{(c)}\quad ({\mathcal L}_X-c) \varphi=0,\quad \text{(d)}\quad u_t={\mathcal L}_X u \] (the modified wave equation, the Laplace equation, the modified Laplace equation, the heat equation, respectively). Here \(c\) is a constant which for irreducible \(X\) is equal to \(\pm|\rho|^2\), where \(\rho\) is half of the sum of positive restricted roots with multiplicities and ``+'' or ``\(-\)'' is taken for the compact or noncompact type, respectively. Assume that all multiplicities of restricted roots are even. In this paper explicit formulae for the fundamental solutions of the equations (a), (b), (c), (d) are given. These formulae for (a) in the odd-dimensional case imply that in this case Huygens' principle (in Hadamard's sense) for (a) holds. For \(X\) of noncompact type the last result was obtained by \textit{L. E. Solomatina} [Sov. Math. 30, 108-111 (1986); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1986, 72-74 (1986; Zbl 0639.35070)] and also by \textit{G. Ólafsson} and \textit{H. Schlichtkrull} [J. Funct. Anal. 107, 270-278 (1992; Zbl 0757.58037)] and by \textit{S. Helgason} [J. Funct. Anal. 107, 279-288 (1992; Zbl 0757.58036)]. For compact semisimple Lie groups and for their duals Huygens' principle was established by \textit{S. Helgason} [Integral geometry, Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Brunswick/Maine 1984, Contemp. Math. 63, 151-177 (1987; Zbl 0614.53049)]. In various special cases the formulae for (b), (c), (d) where known earlier. The main instrument to write the formulae mentioned above is the shift operators for the radial part of \({\mathcal L}_X\). Shift operators are also used to extend the results (the formulae for the fundamental solutions for (a), (b), (c), (d)) to some o…
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 58J45
Mathematics Subject Classification ID: 58J47
zbMATH DE Number: 934320
zbMATH Keywords: shift operators
zbMATH Keywords: Riemannian symmetric space
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: wave equation
zbMATH Keywords: Laplace equation
zbMATH Keywords: heat equation
zbMATH Keywords: Huygens' principle
reviewed by: V. F. Molchanov
MaRDI profile type: Publication
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