Spectral and Scattering-Theory for Wave-Propagation in Perturbed Stratified Media

AbstractThe wave equation (∂2t + Ĥ)u = 0 with Ĥ ≔ −f2(z)∇zg(z)∇z, z = (x, y) ∈ Rm+n describes the propagation of acoustical waves as well as that of electromagnetic waves in perturbed stratified media. The L∞(Rm+n)-functions f and g are short-range (Enss-type) perturbations of other L∞(Rn)-functions f0 and g0, respectively. A generalized version of the conjugate operator method is used in order to show that the spectrum of Ĥis absolutely continuous and equal to [0, ∞), except possibly for a set of eigenvalues which may accumulate at zero, infinity, or a discrete set of thresholds. Away from this set an optimal limiting absorption principle for Ĥ is established. Moreover, the existence and the completeness as well as an invariance principle for the wave operators between Ĥ and Ĥ0 are proven, where Ĥ0 ≔−f20(y)∇zg0(y)∇z