Invertibility of Helmholtz operators for nonhomogeneous medias

Abstract

The paper is devoted to the investigation of the Helmholtz operators describing the propagation of acoustic waves in non‐homogeneous space. We consider the operator A with a wave number k such that where k0 is a positive function, k± are complex constants with ℑ︁(k)>0. The Helmholtz operator A with such wave number describes the propagation of acoustic waves in the waveguides being no homogeneous layer between two absorbing half‐space. We prove that the operator A has an inverse operator A‐1 bounded in the Hilbert space L2(ℝn). Our proof is based on the limit operators method.We also consider the construction of the inverse operator for the Helmholtz operator Aε with the density ρ = ρ(xn) depending on xn only and wave number k0 = k0(εx′, xn) depending on a small parameter ε>0 which characterizes the slowness of variation of the wave number in the horizontal direction. Copyright © 2009 John Wiley & Sons, Ltd.