3 years ago

# Controllability of localized quantum states on infinite graphs through bilinear control fields.

Kaïs Ammari, Alessandro Duca

In this work, we consider the bilinear Schr\"odinger equation (\ref{mainx1}) $i\dd_t\psi=-\Delta\psi+u(t)B\psi$ in the Hilbert space $L^2(\Gi,\C)$ with $\Gi$ an infinite graph. The Laplacian $-\Delta$ is equipped with self-adjoint boundary conditions, $B$ is a bounded symmetric operator and $u\in L^2((0,T),\R)$ with $T>0$. We study the well-posedness of the (\ref{mainx1}) in suitable subspaces of $D(|\Delta|^{3/2})$ preserved by the dynamics despite the dispersive behaviour of the equation. In such spaces, we study the global exact controllability and the {\virgolette{energetic controllability}}. We provide examples involving for instance infinite tadpole graphs.

Publisher URL: http://arxiv.org/abs/1811.04273

DOI: arXiv:1811.04273v1

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