A weak spectral condition for the controllability of the bilinear Schr\"odinger equation with application to the control of a rotating planar molecule

TL;DR

This paper establishes an approximate controllability result for the bilinear Schrödinger equation with weaker spectral conditions, extending control to density matrices and applying it to rotating planar molecules.

Contribution

It introduces a less restrictive spectral condition for controllability, removes boundedness constraints on the control operator, and extends results to density matrices.

Findings

Proves approximate controllability under weaker spectral conditions.

Extends controllability to density matrices.

Provides estimates for the control's L^1 norm.

Abstract

In this paper we prove an approximate controllability result for the bilinear Schr\"odinger equation. This result requires less restrictive non-resonance hypotheses on the spectrum of the uncontrolled Schr\"odinger operator than those present in the literature. The control operator is not required to be bounded and we are able to extend the controllability result to the density matrices. The proof is based on fine controllability properties of the finite dimensional Galerkin approximations and allows to get estimates for the $L^{1}$ norm of the control. The general controllability result is applied to the problem of controlling the rotation of a bipolar rigid molecule confined on a plane by means of two orthogonal external fields.