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A Weak Spectral Condition for the Controllability of the Bilinear Schrödinger Equation with Application to the Control of a Rotating Planar Molecule

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Abstract

In this paper we prove an approximate controllability result for the bilinear Schrödinger equation. This result requires less restrictive non-resonance hypotheses on the spectrum of the uncontrolled Schrödinger 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.

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Authors and Affiliations

  1. Centre National de la Recherche Scientifique (CNRS) http://www.cnrs.fr

    U. Boscain

  2. CMAP, École Polytechnique, Route de Saclay, 91128, Palaiseau Cedex, France

    U. Boscain & M. Sigalotti

  3. INRIA, Centre de Recherche Saclay, Team GECO http://www.inria.fr/en/teams/geco

    U. Boscain & M. Sigalotti

  4. INRIA, Centre de Recherche Nancy-Grand Est, Team CORIDA http://www.inria.fr/en/teams/corida

    M. Caponigro & T. Chambrion

  5. Institut Élie Cartan, UMR 7502 Nancy-Université/CNRS, BP 239, Vandœuvre-lès-Nancy, 54506, France

    M. Caponigro & T. Chambrion

Authors
  1. U. Boscain
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  2. M. Caponigro
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  3. T. Chambrion
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  4. M. Sigalotti
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Corresponding author

Correspondence to U. Boscain.

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Communicated by I. M. Sigal

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Boscain, U., Caponigro, M., Chambrion, T. et al. A Weak Spectral Condition for the Controllability of the Bilinear Schrödinger Equation with Application to the Control of a Rotating Planar Molecule. Commun. Math. Phys. 311, 423–455 (2012). https://doi.org/10.1007/s00220-012-1441-z

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  • DOI: https://doi.org/10.1007/s00220-012-1441-z

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