Localization of classical waves I: Acoustic waves

Abstract

We consider classical acoustic waves in a medium described by a position dependent mass density ϱ(x). We assume that ϱ(x) is a reandom perturbation of a periodic function ϱ₀(x) and that the periodic acoustic operator$A_0 = - \nabla \cdot \tfrac{1}{{\varrho _0 (x)}}\nabla$ has a gap in the spectrum. We prove the existence of localized waves, i.e., finite energy solutions of the acoustic equations with the property that almost all of the wave's energy remains in a fixed bounded region of space at all times, with probability one. Localization of acoustic waves is a consequence of Anderson localization for the self-adjoint operators$A = - \nabla \cdot \tfrac{1}{{\varrho _0 (x)}}\nabla$ onL ²(ℝ^d). We prove that, in the random medium described by ϱ(x), the random operatorA exhibits Anderson localization inside the gap in the spectrum ofA ₀. This is shown even in situations when the gap is totally filled by the spectrum of the random opertor; we can prescribe random environments that ensure localization in almost the whole gap.