  Consider the Dirichlet-Laplacian in $\Omega:= (0,L)\times (0,H)$ and choose
another open set $\omega\subset \Omega$. The estimate $0<C_{\omega}\leq
R_{\omega}(u):=\frac{\Vert u\Vert^{2}_{L^{2}(\omega)}}{\Vert
u\Vert^{2}_{L^{2}(\Omega)}}\leq \frac{vol(\omega)}{vol(\omega)}$, for all the
eigenfunctions, is well known. This is no longer true for an inhomogeneous
elliptic selfadjoint operator $A$. In this work we create a partition among the
set of eigenfunctions: $\forall \omega$, the eigenfunctions satisfy
$R_{omega}>C_{\omega}>0,\exists \omega, \omega\not=\emptyset$, such that $\inf
R_{\omega}(u)=0$,and we wish to characterize these two sets. For two patterns
we give a sufficient condition, sometimes necessary. As our operator
corresponds to a layered media we can give another representation of its
spectrum: i.e. a subset of points of $R\times R$ that leads to the suggested
partition and others connected results: micro local interpretation, default
measures,... Section 4.1 of the previous version was not correct, now it is
corrected, many proofs are simplified and a new general result is added.
