  We study the Schr\"odinger equation $i\partial_t\psi=-\Delta\psi+V\psi$ on
$L^2((0,1),\mathbb{C})$ where $V$ is a very high and localized potential wall.
We aim to perform permutations of the eigenmodes and to control the solution of
the equation. We consider the process where the position and the height of the
potential wall change as follows. First, the potential increases from zero to a
very large value, so a narrow potential wall is formed that almost splits the
interval into two parts; then the wall moves to a different position, after
which the height of the wall decays to zero again. We show that even though the
rate of the variation of the potential's parameters can be arbitrarily slow,
this process alternates adiabatic and non-adiabatic dynamics, leading to a
non-trivial permutation of the eigenstates. Furthermore, we consider potentials
with several narrow walls and we show how an arbitrarily slow motion of the
walls can lead the system from any given state to an arbitrarily small
neighborhood of any other state, thus proving the approximate controllability
of the above Schr\"odinger equation by means of a soft, quasi-adiabatic
variation of the potential.
