Bilinear control of discrete spectrum Schr\"odinger operators

TL;DR

This paper investigates the controllability of Schr"odinger equations with bilinear control, demonstrating approximate controllability for systems with discrete spectra under certain conditions.

Contribution

It establishes approximate controllability for bilinear Schr"odinger systems with discrete spectra using topological irreducibility methods.

Findings

Proves approximate controllability for systems with simple discrete spectrum

Uses topological irreducibility of operator sets

Identifies conditions on control operator B

Abstract

The bilinear control problem of the Schr\"odinger equation $i\frac{\partial}{\partial t}\psi(t)$ $=(A+u(t) B)\psi(t)$, where $u(t)$ is the control function, is investigated through topological irreducibility of the set $\mathfrak{M}=\{e^{-it (A+u B)}, u\in \mathbb{R}, t>0\}$ of bounded operators. This allows to prove the approximate controllability of such systems when the uncontrolled Hamiltonian $A$ has a simple discrete spectrum and under an appropriate assumption on $B$.