Spectral resolution of differential operators associated with symmetric hyperbolic systems

Abstract

Spectral properties of the operator A=-i∑_j=1 ⁿA_j∂/∂jare derived and the results are applied to study the perturbed opearator A=E(x)^-1∑_j=1 ⁿA_j∂/∂j. Here the A _j's are constand, real symmmetric.k×k matrices. Aside from this no further assumption is made. An expression for the spectral family {E(λ)}of A is given in terms of the Fourier transform and the spectral resolution of the matrixA(p)=∑^j=1=1P _{A j}. The spectrum of A is proved to be absolutely continuous, unless the equation det(τl-A(p))=0 possesses roots τ=τ(p) which vanish for all P. in this case λ=0 is an eigenvalue of a, but still the spectral measure E(s) is absolutely continuous when restricted to the Borel sets S which do not contain the point λ=0. The existence of the wave operator is established, thus proving that Ap is unitarily equivalent to a part of A.