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

In this work, we study the Schrödinger equation i∂tψ = −∆ψ + η(t) ∑J j=1 δx=aj(t)ψ on L((0, 1),C) where η : [0, T ] −→ R and aj : [0, T ] −→ (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 aj . 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.