Explicit approximate controllability of the Schr\"odinger equation with a polarizability term

TL;DR

This paper demonstrates the approximate controllability of a Schrödinger equation with a polarizability term by extending finite-dimensional stabilization techniques to an infinite-dimensional setting, using oscillating controls and Lyapunov functions.

Contribution

It introduces a novel approach to control Schrödinger equations with polarizability, extending finite-dimensional stabilization methods to infinite dimensions with oscillating controls.

Findings

Proves semi-global weak H^2 stabilization of the averaged system.

Shows solutions of the Schrödinger and averaged equations remain close over finite horizons.

Establishes approximate controllability to the ground state for the polarizability system.

Abstract

We consider a controlled Schr\"odinger equation with a dipolar and a polarizability term, used when the dipolar approximation is not valid. The control is the amplitude of the external electric field, it acts non linearly on the state. We extend in this infinite dimensional framework previous techniques used by Coron, Grigoriu, Lefter and Turinici for stabilization in finite dimension. We consider a highly oscillating control and prove the semi-global weak $H^2$ stabilization of the averaged system using a Lyapunov function introduced by Nersesyan. Then it is proved that the solutions of the Schr\"odinger equation and of the averaged equation stay close on every finite time horizon provided that the control is oscillating enough. Combining these two results, we get approximate controllability to the ground state for the polarizability system.