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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. (English) Zbl 1267.35177

In this nice paper under review the authors 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 ones presented so far in the previous literature. The control operator is not required to be bounded and the authors were able to extent 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 \(L^1\) 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.

MSC:

35Q41 Time-dependent Schrödinger equations and Dirac equations
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q93 PDEs in connection with control and optimization
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