Asymptotics of resonances and geometry of resonance states in the problem of scattering of acoustic waves by a spherically symmetric inhomogeneity of the density

Abstract

We study resonances and resonance states in the problem of the scattering of acoustic waves by a spherically symmetric inhomogeneity of the medium using the Lax-Phillips approach. The density of the medium is assumed to be constant outside the sphere $r=a$ and to have either (i) a finite jump on this sphere, or (ii) a singularity corresponding to infinite rarefaction, or (iii) a singularity corresponding to infinite condensation. We show that, for each value of the angular momentum $\ell$, there exists an infinite sequence of resonances accumulating to infinity. Explicit asymptotic formulae for resonances and resonance states are given. In case (i) the resonances are asymptotically close to a horizontal line in the upper half-plane, in case (ii) (or (iii)) they converge to the real axis and are asymptotically close to the eigenvalues of the Dirichlet (or Neumann) problem in the interior of the scatterer. We give an explicit description of all main elements of the Lax-Phillips scheme: incoming and outgoing subspaces, spectral representation and $S$-matrix. Using this description, the Foias-Sz.-Nagy theory of contraction operators and the above-mentioned asymptotic formulae for resonances, we show that in all three cases (i)--(iii) the resonance states form a Riesz basis in the orthogonal complement to the incoming and outgoing subspaces.