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...

Full description

Saved in:
Bibliographic Details
Published inJournal of functional analysis Vol. 232; no. 2; pp. 328 - 389
Main Authors Beauchard, Karine, Coron, Jean-Michel
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.03.2006
Elsevier
Subjects
Analysis of PDEs
Controllability
Mathematics
Nash–Moser
Schrödinger
35Q55
Controllability
Schrödinger
93C20
Nash–Moser
93B05
Nash-Moser
Online AccessGet full text
ISSN0022-1236
1096-0783
DOI10.1016/j.jfa.2005.03.021

Cover

More Information
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 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.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2005.03.021