Adiabatic excitation of a confined one-dimensional particle with a variable $\delta$-wall

It is shown that adiabatic cycles excite a quantum particle, which is confined in a one-dimensional region and is initially in an eigenstate. During the cycle, a $delta$-wall is applied and varied its strength and position. After the completion of the cycle, the state of the particle arrives another eigenstate. It is also shown that an arbitrary pair of eigenstates can be connected through a combination of adiabatic cycles. A detailed argument is provided for the case that the particle is confined by an infinite square well. This is an example of exotic quantum holonomy in Hamiltonian systems.