Controllability of localised quantum states on infinite graphs through bilinear control fields

Abstract

In this work, we consider the bilinear Schrödinger equation (BSE) $i{\mathrm{\partial }}_t\psi =-\mathrm{\Delta }\psi +u\left(t\right)B\psi$ in the Hilbert space $L^2(\mathcal{G},\mathbb{C})$ with $\mathcal{G}$ an infinite graph. The Laplacian $-\mathrm{\Delta }$ is equipped with self-adjoint boundary conditions, B is a bounded symmetric operator and $u\in L^2\left(\right(0,T),\mathbb{R})$ with T>0. We study the well-posedness of the (BSE) in suitable subspaces of $D\left(|\mathrm{\Delta }{|}^{3/2}\right)$ preserved by the dynamics despite the dispersive behaviour of the equation. In such spaces, we study the global exact controllability and the ‘energetic controllability’. We provide examples involving for instance infinite tadpole graphs.

Keywords:

• Bilinear control
• infinite graph
• Schrödinger equation

2010 Mathematics Subject Classifications:

• 35Q40
• 93B05
• 93C05

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Kaïs Ammari http://orcid.org/0000-0003-2920-4106

Alessandro Duca http://orcid.org/0000-0001-7060-1723

Additional information

Funding

The second author has been financially supported by the ISDEEC project by Agence Nationale de la Recherche ANR-16-CE40-0013.