Controllability of a quantum particle in a moving potential well
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...
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| Published in | Journal of functional analysis Vol. 232; no. 2; pp. 328 - 389 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Inc
15.03.2006
Elsevier |
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| Online Access | Get full text |
| ISSN | 0022-1236 1096-0783 |
| DOI | 10.1016/j.jfa.2005.03.021 |
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| Abstract | 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
ψ
0
close enough to an eigenstate and
ψ
f
close enough to another eigenstate, the wave function can be moved exactly from
ψ
0
to
ψ
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. |
|---|---|
| AbstractList | 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
ψ
0
close enough to an eigenstate and
ψ
f
close enough to another eigenstate, the wave function can be moved exactly from
ψ
0
to
ψ
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. 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 an initial condition close enough to an eigenstate and a target close enough to another eigenstate, the wave function can be moved exactly from the first one to the second one 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. |
| Author | Coron, Jean-Michel Beauchard, Karine |
| Author_xml | – sequence: 1 givenname: Karine surname: Beauchard fullname: Beauchard, Karine email: Karine.Beauchard@math.u-psud.fr organization: Département de Mathématique, Bât. 425, Université Paris-Sud, 91405 Orsay, France – sequence: 2 givenname: Jean-Michel surname: Coron fullname: Coron, Jean-Michel email: jean-Michel.Coron@math.u-psud.fr organization: Département de Mathématique, Bât. 425, Université Paris-Sud, 91405 Orsay, France |
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| Snippet | 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... |
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| Title | Controllability of a quantum particle in a moving potential well |
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