Achieving energy permutation of modes in the Schr\"odinger equation with moving Dirac potentials

Abstract

In this work, we study the Schr\"odinger equation $i\partial_t\psi=-\Delta\psi+\eta(t)\sum_{j=1}^J\delta_{x=a_j(t)}\psi$ 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 Schr\"odinger 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.