Trace-class approach in scattering problems for perturbations of media

We consider the operators $H_0=M_0^{-1}(x)P(D)$ and $H=M^{-1}(x)P(D)$ where $M_0(x)$ and $M(x)$ are positively definite bounded matrix-valued functions and $P(D)$ is an elliptic differential operator. Our main result is that the wave operators for the pair $H_0$, $H$ exist and are complete if the difference $M(x)-M_0(x)=O(\mathrm{\mid }x{\mathrm{\mid }}^{-rho})$, $rho>d$, as $\mathrm{\mid }x\mathrm{\mid }toinfty$. Our point is that no special assumptions on $M_0(x)$ are required. Similar results are obtained in scattering theory for the wave equation.