  Motivated by a problem of one mode approximation for a non-linear evolution
with charge accumulation in potential wells, we consider a general linear
adiabatic evolution problem for a semi-classical Schr\"odinger operator with a
time dependent potential with a well in an island. In particular, we show that
we can choose the adiabatic parameter $\varepsilon $ with $\ln\varepsilon
\asymp -1/h$, where $h$ denotes the semi-classical parameter, and get adiabatic
approximations of exact solutions over a time interval of length $\varepsilon
^{-N}$ with an error ${\cal O}(\varepsilon ^N)$. Here $N>0$ is arbitrary.
