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

Abstract

In this work, we study the Schrödinger 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 $\eta :[0,T]⟶R^{+}$ and $a_j:[0,T]⟶(0,1)$, $j=1,...,J$. We show how to permute the energy associated to different eigenmodes of the Schrödinger equation via suitable choice of the functions $\eta$ 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.