Wave Equations with Finite Velocity of Propagation

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 ${\text {SL}}(2,{\mathbf {R}})$.