Spectral and scattering theory in perturbed stratified fluids
The author considers the spectral and scattering theory of the acoustic propagator of a perturbed stratified fluid. This involves the study of the operator \(A=-c^ 2(x,y)\Delta\); under suitable assumptions on the measurable function c, he proves that the spectrum of A coincides with the essential spectrum, and consists of [0,\(\infty)\). There is no singular continuous spectrum, the eigenvalues of A on \({\mathbb{R}}^+\setminus \{\cup_{k=1,\infty}\mu_ k\}\) have finite multiplicity and can only accumulate at zero and \(\{\) \(\cup_{k=1,\infty}\mu_ k\}\). The numbers \(\mu_ k\) are the thresholds at which a new guided wave of the unperturbed operator appears. Although this result is not completely new, the proof given by the author in the present paper is really simpler and definitely shorter than the known one. The generalized Fourier maps \(F_ A^{\pm}\) and the scattering operator S are then investigated. In particular it is proved that S is unitary and that the wave operators can be obtained as suitable strong limits of a function of both the perturbed and unperturbed acoustic operator.
scientific article; zbMATH DE number 3963450
scientific article