  This work is concerned with operators of the type A = --c$\Delta$ acting in
domains $\Omega$ := $\Omega$ ' x (0, H) $\subseteq$ R^d x R ^+. The diffusion
coefficient c > 0 depends on one coordinate y $\in$ (0, H) and is bounded but
may be discontinuous. This corresponds to the physical model of ''layered
media'', appearing in acoustics, elasticity, optical fibers... Dirichlet
boundary conditions are assumed. In general, for each $\epsilon$ > 0, the set
of eigenfunctions is divided into a disjoint union of three subsets : Fng
(non-guided), Fg (guided) and Fres (residual). The residual set shrinks as
$\epsilon$ $\rightarrow$ 0. The customary physical terminology of
guided/non-guided is often replaced in the mathematical literature by
concentrating/non-concentrating solutions, respectively. For guided waves, the
assumption of ''layered media'' enables us to obtain rigorous estimates of
their exponential decay away from concentration zones. The case of non-guided
waves has attracted less attention in the literature. While it is not so
closely connected to physical models, it leads to some very interesting
questions concerning oscillatory solutions and their asymptotic properties.
Classical asymptotic methods are available for c(y) $\in$ C 2 but a lesser
degree of regularity excludes such methods. The associated eigenfunctions (in
Fng) are oscillatory. However, this fact by itself does not exclude the
possibility of ''flattening out'' of the solution between two consecutive
zeros, leading to concentration in the complementary segment. Here we show it
cannot happen when c(y) is of bounded variation, by proving a ''minimal
amplitude hypothesis''. However the validity of such results when c(y) is not
of bounded variation (even if it is continuous) remains an open problem.
