TY - JOUR

T1 - Wave propagation on Riemannian symmetric spaces

AU - 'Olafsson, G.

AU - Schlichtkrull, H.

PY - 1992/8/1

Y1 - 1992/8/1

N2 - Let X = G K be a Riemannian symmetric space of the noncompact type and let LX be the Laplace-Beltrami operator on X. We consider on X × R a differential equation of the type LXu = P(∂t)u, where P(∂t) is a second order differential operator in t ε{lunate} R. Using the Radon transform on X we relate Huygens' principle for the solutions u to this equation, to the corresponding question for the solutions v to the equation LAv = (P(∂t) + R)v on a maximal flat subspace A, where R is a certain explicit constant. In particular we conclude that Huygens' principle holds for solutions to the (modified) wave equation on X obtained by letting P(∂t) = ∂t2 - R, when X is odd-dimensional and G has one conjugacy class of Cartan subgroups.

AB - Let X = G K be a Riemannian symmetric space of the noncompact type and let LX be the Laplace-Beltrami operator on X. We consider on X × R a differential equation of the type LXu = P(∂t)u, where P(∂t) is a second order differential operator in t ε{lunate} R. Using the Radon transform on X we relate Huygens' principle for the solutions u to this equation, to the corresponding question for the solutions v to the equation LAv = (P(∂t) + R)v on a maximal flat subspace A, where R is a certain explicit constant. In particular we conclude that Huygens' principle holds for solutions to the (modified) wave equation on X obtained by letting P(∂t) = ∂t2 - R, when X is odd-dimensional and G has one conjugacy class of Cartan subgroups.

UR - http://www.scopus.com/inward/record.url?scp=38249009303&partnerID=8YFLogxK

U2 - 10.1016/0022-1236(92)90107-T

DO - 10.1016/0022-1236(92)90107-T

M3 - Journal article

AN - SCOPUS:38249009303

VL - 107

SP - 270

EP - 278

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 2

ER -