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Huygens' principle for wave equations on symmetric spaces

Authors
Journal
Journal of Functional Analysis
0022-1236
Publisher
Elsevier
Publication Date
Volume
107
Issue
2
Identifiers
DOI: 10.1016/0022-1236(92)90108-u
Disciplines
  • Mathematics

Abstract

Abstract Let X = G K be a symmetric space, L X the Laplace-Betrami operator on X, and 2ϱ the sum of the positive restricted roots of X (with multiplicity). Using the Fourier transform on X we write down the solution of the modified wave equation ∂ 2u ∂t 2 = (L X + ¦ϱ¦ 2)u on X . It is proved, using the Paley-Wiener theorem for this Fourier transform ( S. Helgason, Ann. of Math. 98 (1973), 451–480) that the equation satisfies Huygens' principle if dim X is odd and all Cartan subgroups of G are conjugate.

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