Permuting quantum eigenmodes by a quasi-adiabatic motion of a potential wall

We study the Schrödinger equation itψ = -Δψ + Vψ on L2((0,1),C) where V is a very high and localized potential wall. We consider the process where the position and the height of the wall change as follows: First, the potential increases from zero to a very large value, and so a narrow potential wall is formed and almost splits the interval into two parts; then, the wall moves to a different position, after which the height of the wall decreases to zero again. We show that even though the rate of 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 instantaneous energy eigenstates. Furthermore, we consider potentials with several narrow walls and 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 Schrödinger equation by means of a soft, quasi-adiabatic variation of the potential.