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

In this work, we study the Schrödinger equation $i\partial_tψ = −\Delta\psi + \eta(t)\sum^J_{j=1}\delta_{x=a_j (t)}\psi$ on $L^2 ((0, 1), C)$ where $η : [0, T ]\rightarrow R^+$ and $a_j : [0, T ] \rightarrow (0, 1)$, $j = 1, ..., J$. We show how to permute the energy associated to different eigenmodes of the Schrödinger equation via suitable choice of the functions $η$ 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.