Equipartition of Energy for Waves in Symmetric Space

Let X = G=K be an odd-dimensional semisimple Riemannian symmetric space of the noncompact type, and suppose that all Cartan subgroups of G are conjugate. Let u be a real-valued classical solution of the modiied wave equation u tt = ((+ k)u on R X, the Cauchy data of which are supported in a closed metric ball of radius a at time t = 0. Here t is the coordinate on R, is the (nonpositive deenite) Laplace-Beltrami operator on X, and k is a positive constant depending on the root structure of the Lie algebra of G. We show that the (t-independent) energy functional of u is eventually (for jtj a) partitioned into equal potential and kinetic parts; speciically, half the integrals over X of u 2 t and jduj 2 ? ku 2 respectively, where d is the exterior derivative in X. The proof uses Helgason's Paley-Wiener Theorem for X, the classical Paley-Wiener Theorem, and properties of Harish-Chandra's c function.