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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Wave equations with finite velocity of propagation


Author: Stephen J. Berman
Journal: Trans. Amer. Math. Soc. 188 (1974), 127-148
MSC: Primary 35L05; Secondary 43A70
MathSciNet review: 0330781
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Abstract: If B is a selfadjoint translation-invariant operator on the space $ {L^2}$ of complex-valued functions on n-dimensional Euclidean space which are square-summable with respect to Lebesgue measure, then the wave equation $ {d^2}F/d{t^2} + {B^2}F = 0$ has the solution $ F(t) = (\cos tB)f + ((\sin tB)/B)g$, for f and g in $ {L^2}$. In the classical case in which $ - {B^2}$ is the Laplacian, this solution has finite velocity of propagation in the sense that (letting supp denote support of a function) $ {\text{supp}}\;F(t) \subset ({\text{supp}}\;f \cup {\text{supp}}\;g) + {K_t}$ for all f and g and some compact set $ {K_t}$ independent of f and g. We show that a converse holds, namely, if $ \cos \;tB$ has finite velocity of propagation (that is, if $ {\text{supp}}\,(\cos tB)f \subset {\text{supp}}\;f + {K_t}$ for all f and some compact $ {K_t}$) for three values of t whose reciprocals are independent over the rationals, then $ {B^2}$ must be a second order differential operator.

If Euclidean space is replaced by a locally compact abelian group which does not contain the real line as a subgroup, then $ \cos \;tB$ has finite velocity of propagation for all t if and only if it is convolution with a distribution $ {T_t}$ such that all $ {T_t}$ are supported on a compact open subgroup.

Problems of a similar nature are discussed for compact connected abelian groups and for the nonabelian group $ {\text{SL}}(2,{\mathbf{R}})$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0330781-6
PII: S 0002-9947(1974)0330781-6
Keywords: Wave equation, locally compact abelian group, translation-invariant operators, Laplacian, velocity of propagation, distributions on a LCA group
Article copyright: © Copyright 1974 American Mathematical Society