Acoustic waves in perturbed stratified fluids: A spectral theory. (English) Zbl 0675.35065

The authors study the acoustic wave propagator in perturbed stratified fluids, i.e. \[ H=-c^ 2(x,y)\rho (x,y)(\nabla \cdot (1/\rho (x,y))\nabla) \] to be regarded as a perturbation of \[ H_ 0=-c^ 2_ 0(y)\{\Delta_ x+\rho_ 0(y)(\partial /\partial y)((1/\rho_ 0(y))\partial /\partial y)\} \] where \((x,y)\in {\mathbb{R}}^ n\times {\mathbb{R}}\). For \(n=2\) y denotes the depth, \(\rho_ 0\) the density and \(c_ 0\) the speed of sound.
Under certain assumptions on \(\rho_ 0\) and \(c_ 0\) the authors establish a “limiting absorption principle” for the resolvent of \(H_ 0\) and a certain class of its perturbation. This means roughly that the resolvent operator is uniformly continuous up to the spectrum in same suitable operator topology.
It should be noted that the assumptions on \(\rho_ 0\) and \(c_ 0\) are considerably relaxed with respect to other recent works on this subject.
Reviewer: F.Rosso


35P05 General topics in linear spectral theory for PDEs
35B20 Perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
76Q05 Hydro- and aero-acoustics