Figotin A., Klein A.
Localization of Classical Waves I: Acoustic Waves
(138K, LaTeX)
ABSTRACT. We consider classical acoustic waves in a medium
described by a position dependent mass density $\varrho (x)$. We assume
that $\varrho (x)$ is a random perturbation of a periodic function
$\varrho_{0}(x)$ and that the periodic acoustic operator $A_{0} =-\nabla
\cdot \frac 1{\varrho_{0} (x)}\nabla$ 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 $A = -\nabla\cdot\frac 1
{\varrho (x)} \nabla$ on $L^2(\Bbb{R}^d)$. We prove that, in the random
medium described by $\varrho(x)$, the random operator $A$ exhibits
Anderson localization inside the gap in the spectrum of $A_{0}$ . This
is shown even in situations when the gap is totally filled by the
spectrum of the random operator; we can prescribe random environments
that ensure localization in almost the whole gap.