Integrability and Huygens' principle on symmetric spaces
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 other spaces \(X\), namely, to spaces \(X\) of type \(BC_n\) with even multiplicities of roots \(\pm e_i \pm e_j\). These shift operators (found by Opdam and Heckman) allow to reduce the problem to that for spaces which are much simpler.
scientific article; zbMATH DE number 934320
scientific article