Simultaneous local exact controllability of 1D bilinear Schrödinger equations

Abstract

We consider N independent quantum particles, in an infinite square potential well coupled to an external laser field. These particles are modelled by a system of linear Schrödinger equations on a bounded interval. This is a bilinear control system in which the state is the N-tuple of wave functions. The control is the real amplitude of the laser field. For N=1, Beauchard and Laurent proved local exact controllability around the ground state in arbitrary time. We prove, under an extra generic assumption, that their result does not hold in small time if N is greater or equal than 2. Still, for N=2, we prove using Coron's return method that local controllability holds either in arbitrary time up to a global phase or exactly up to a global delay. We also prove that for N greater or equal than 3, local controllability does not hold in small time even up to a global phase. Finally, for N=3, we prove that local controllability holds up to a global phase and a global delay.

References in corpus (3)

• Controllability of the discrete-spectrum Schrodinger equation driven by an external field
• Lyapunov control of a quantum particle in a decaying potential
• Growth of Sobolev norms and controllability of Schrödinger equation

Cited by in corpus (5)

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• Permuting quantum eigenmodes by a quasi-adiabatic motion of a potential wall