Achieving energy permutation of modes in the Schrodinger equation with moving Dirac potentials

In this work, we study the Schrodinger equation $ipartial_tpsi=-Deltapsi+eta(t)sum_{j=1}^Jdelta_{x=a_j(t)}psi$ on $L^2((0,1),C)$ where $eta:[0,T]longrightarrow R^+$ and $a_j:[0,T]longrightarrow (0,1)$, $j=1,...,J$. We show how to permute the energy associated to different eigenmodes of the Schrodinger equation via suitable choice of the functions $eta$ and $a_j$. To the purpose, we mime the control processes introduced in [17] for a very similar equation where the Dirac potential is replaced by a smooth approximation supported in a neighborhood of $x=a(t)$. We also propose a Galerkin approximation that we prove to be convergent and illustrate the control process with some numerical simulations.