Wave equations with finite velocity of propagation

Abstract

If B is a selfadjoint translation-invariant operator on the space L 2 {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 {d^2}F/d{t^2} + {B^2}F = 0 has the solution F ( t ) = ( cos ⁡ t B ) f + ( ( sin ⁡ t B ) / B ) g F(t) = (\cos tB)f + ((\sin tB)/B)g , for f and g in L 2 {L^2} . In the classical case in which − B 2 - {B^2} is the Laplacian, this solution has finite velocity of propagation in the sense that (letting supp denote support of a function) supp F ( t ) ⊂ ( supp f ∪ supp g ) + K t {\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 {K_t} independent of f and g. We show that a converse holds, namely, if cos t B \cos \;tB has finite velocity of propagation (that is, if supp ( cos ⁡ t B ) f ⊂ supp f + K t {\text {supp}}\,(\cos tB)f \subset {\text {supp}}\;f + {K_t} for all f and some compact K t {K_t} ) for three values of t whose reciprocals are independent over the rationals, then B 2 {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 t B \cos \;tB has finite velocity of propagation for all t if and only if it is convolution with a distribution T t {T_t} such that all T t {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 SL ( 2 , R ) {\text {SL}}(2,{\mathbf {R}}) .