Simultaneous global exact controllability in projection of infinite 1D bilinear Schr\"odinger equations

TL;DR

This paper establishes the conditions under which infinite bilinear Schrödinger equations on a segment can be controlled exactly in projection, both locally and globally, with implications for finite-dimensional approximations and density matrices.

Contribution

It proves simultaneous local and global exact controllability in projection for infinite bilinear Schrödinger equations, providing explicit examples and linking finite and infinite controllability.

Findings

Proves local and global controllability in projection for infinite BSE.

Shows equivalence between controllability of infinite BSE in projection and finite BSE.

Rephrases controllability results in terms of density matrices.

Abstract

The aim of this work is to study the controllability of infinite bilinear Schr\"odinger equations on a segment. We consider the equations (BSE) $i\partial_t\psi^{j}=-\Delta\psi^j+u(t)B\psi^j$ in the Hilbert space $L^2((0,1),\mathbb{C})$ for every $j\in\mathbb{N}^*$. The Laplacian $-\Delta$ is equipped with Dirichlet homogeneous boundary conditions, $B$ is a bounded symmetric operator and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. We prove the simultaneous local and global exact controllability of infinite (BSE) in projection. The local controllability is guaranteed for any positive time and we provide explicit examples of $B$ for which our theory is valid. In addition, we show that the controllability of infinite (BSE) in projection onto suitable finite dimensional spaces is equivalent to the controllability of a finite number of (BSE) (without projecting). In conclusion, we rephrase our controllability results in terms of density matrices.