Wave operators in a multistratified strip

One investigates the scattering theory for the positive self-adjoint operat or H = -del . rho del acting in H = L-2(Omega) with Omega = Omega' x R and Omega' a bounded open set in Rn-1, n greater than or equal to 2. The real-v alued function rho belongs to L-infinity(Omega), is bounded from below by c > 0 and there exist real-valued functions rho(1) and rho(2) in L-infinity( Omega) such that rho - rho(j), j = 1, 2, is a short range perturbation of r ho(j) when (-1)(j)x(n) --> +infinity. One assumes rho(j) = rho((j))x1(R), j = 1, 2, with rho((j)) is an element of L-infinity(Omega') bounded from bel ow by c > 0. One proves the existence and completeness of the generalized w ave operators Omega(j)(+/-) = s - lim e(itH) chi(j)(e-itHj), j = 1, 2, with H-j = -del . rho(j)del and chi(j) : Omega --> R equal to 1 if (-1)(j)x(n) > 0 and to 0 if (-1)(j)x(n) < 0. The ranges of W-j(+/-) := (Omega(j)(+/-))* are characterized so that H+/- := Ran W-1(-/+) := Ran W-2(+/-) and H = H- + H+. The scattering operator can then de defined.