Scattering by periodically moving obstacles

Suppose $x\in\mathbf R^n$, $L_0(\partial_t,\partial_x)$ is a homogeneous hyperbolic matrix, $U_0(t)$ is the operator taking the Cauchy data for the system $L_0u=0$ for $t=0$ into the corresponding data at time $t$, and $U(t)$ is the analogous operator constructed from the exterior mixed problem for the hyperbolic system $Lu=0$. It is assumed that the boundary of the domain and the coefficients of the operator $L$ are periodic in $t$ with period $T$, $L=L_0$ for $|x|\gg1$, the noncapturing condition is satisfied, the matrix $L_0(0,\partial_x)$ is elliptic, and the energy of solutions of the exterior problem is uniformly bounded for $t\geqslant 0$.
Under these conditions it is proved that the space $H$ generated by the eigenfunctions of the monodromy operator $V=U(T)$ with eigenvalues on the unit circle is finite dimensional; for initial data $f$ with compact support the asymptotics of the solution $U(t)f$ of the exterior problem as $t\to\infty$ is obtained; in particular, it is shown that $U(t)f\sim U(t)Pf$, $t\to\infty$, where $P$ is the operator of projection onto $H$; and existence of the wave operators constructed on the basis of $U_0(t)$ and $U(t)$ and of the scattering operator is proved.