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

TL;DR

This paper demonstrates how to permute energy among eigenmodes of the Schr"odinger equation with moving Dirac potentials by controlling the potential's strength and position, supported by theoretical proofs and numerical simulations.

Contribution

It introduces a method to control and permute energy among eigenmodes in a Schr"odinger equation with moving Dirac potentials, extending previous control techniques.

Findings

Successful permutation of energy among eigenmodes.

Convergence of Galerkin approximation.

Numerical simulations confirming control effectiveness.

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.