Small-time bilinear control of Schr\"odinger equations with application to rotating linear molecules

TL;DR

This paper investigates small-time controllability of linear Schrödinger equations on Riemannian manifolds, specifically the 2D sphere, demonstrating approximate controllability among eigenfunctions in arbitrarily small times.

Contribution

It extends controllability results from nonlinear to linear Schrödinger equations on manifolds, with a focus on the sphere and applications to rotating molecules.

Findings

Achieves approximate controllability among eigenfunctions on S^2

Demonstrates small-time controllability for linear Schrödinger equations

Applies results to control of rotating linear molecules

Abstract

In [14] Duca and Nersesyan proved a small-time controllability property of nonlinear Schr\"odinger equations on a d-dimensional torus $\mathbb{T}^d$. In this paper we study a similar property, in the linear setting, starting from a closed Riemannian manifold. We then focus on the 2-dimensional sphere $S^2$, which models the bilinear control of a rotating linear top: as a corollary, we obtain the approximate controllability in arbitrarily small times among particular eigenfunctions of the Laplacian of $S^2$.