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

Abstract

We study the one-dimensional Schrödinger equation in a bounded interval with several Dirac potentials supported in some moving points $ x = a_j(t) $, $ 1\leq j\leq J $, in the time interval $ t\in[0,T] $. We give a convergent numerical method to approximate the solutions of this equation. This is applied to construct explicit control functions $ \eta(t) $, the intensity of the Diracs, and their trajectories $ a_j(t) $ to achieve any prescribed permutation of the energy associated to a finite number of eigenmodes.