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Controllability of a quantum particle in a moving potential well. (English) Zbl 1188.93017
Summary: We consider a nonrelativistic charged particle in a 1D moving potential well. This quantum system is subject to a control, which is the acceleration of the well. It is represented by a wave function solution of a Schrödinger equation, the position of the well together with its velocity. We prove the following controllability result for this bilinear control system: given $$\psi_{0}$$ close enough to an eigenstate and $$\psi_{f}$$ close enough to another eigenstate, the wave function can be moved exactly from $$\psi_{0}$$ to $$\psi_{f}$$ in finite time. Moreover, we can control the position and the velocity of the well. Our proof uses moment theory, a Nash–Moser implicit function theorem, the return method and expansion to the second order.

##### MSC:
 93B05 Controllability 93C20 Control/observation systems governed by partial differential equations 35Q55 NLS equations (nonlinear Schrödinger equations)
##### Keywords:
controllability; Schrödinger; Nash-Moser
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##### References:
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