Controllablity of a quantum particle in a 1D variable domain

Karine Beauchard

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 14, Issue: 1, page 105-147
  • ISSN: 1292-8119

Abstract

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We consider a quantum particle in a 1D infinite square potential well with variable length. It is a nonlinear control system in which the state is the wave function ϕ of the particle and the control is the length l(t) of the potential well. We prove the following controllability result : given φ 0 close enough to an eigenstate corresponding to the length l = 1 and φ f close enough to another eigenstate corresponding to the length l=1, there exists a continuous function l : [ 0 , T ] → ℝ + * with T > 0, such that l(0) = 1 and l(T) = 1, and which moves the wave function from φ 0 to φ f in time T. In particular, we can move the wave function from one eigenstate to another one by acting on the length of the potential well in a suitable way. Our proof relies on local controllability results proved with moment theory, a Nash-Moser implicit function theorem and expansions to the second order.

How to cite

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Beauchard, Karine. "Controllablity of a quantum particle in a 1D variable domain." ESAIM: Control, Optimisation and Calculus of Variations 14.1 (2010): 105-147. <http://eudml.org/doc/90860>.

@article{Beauchard2010,
abstract = { We consider a quantum particle in a 1D infinite square potential well with variable length. It is a nonlinear control system in which the state is the wave function ϕ of the particle and the control is the length l(t) of the potential well. We prove the following controllability result : given $\phi_\{0\}$ close enough to an eigenstate corresponding to the length l = 1 and $\phi_\{f\}$ close enough to another eigenstate corresponding to the length l=1, there exists a continuous function $l:[0,T] \rightarrow \mathbb\{R\}^\{*\}_\{+\}$ with T > 0, such that l(0) = 1 and l(T) = 1, and which moves the wave function from $\phi_\{0\}$ to $\phi_\{f\}$ in time T. In particular, we can move the wave function from one eigenstate to another one by acting on the length of the potential well in a suitable way. Our proof relies on local controllability results proved with moment theory, a Nash-Moser implicit function theorem and expansions to the second order. },
author = {Beauchard, Karine},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Controllability; Schrödinger equation; Nash-Moser theorem; moment theory; controllability},
language = {eng},
month = {3},
number = {1},
pages = {105-147},
publisher = {EDP Sciences},
title = {Controllablity of a quantum particle in a 1D variable domain},
url = {http://eudml.org/doc/90860},
volume = {14},
year = {2010},
}

TY - JOUR
AU - Beauchard, Karine
TI - Controllablity of a quantum particle in a 1D variable domain
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 14
IS - 1
SP - 105
EP - 147
AB - We consider a quantum particle in a 1D infinite square potential well with variable length. It is a nonlinear control system in which the state is the wave function ϕ of the particle and the control is the length l(t) of the potential well. We prove the following controllability result : given $\phi_{0}$ close enough to an eigenstate corresponding to the length l = 1 and $\phi_{f}$ close enough to another eigenstate corresponding to the length l=1, there exists a continuous function $l:[0,T] \rightarrow \mathbb{R}^{*}_{+}$ with T > 0, such that l(0) = 1 and l(T) = 1, and which moves the wave function from $\phi_{0}$ to $\phi_{f}$ in time T. In particular, we can move the wave function from one eigenstate to another one by acting on the length of the potential well in a suitable way. Our proof relies on local controllability results proved with moment theory, a Nash-Moser implicit function theorem and expansions to the second order.
LA - eng
KW - Controllability; Schrödinger equation; Nash-Moser theorem; moment theory; controllability
UR - http://eudml.org/doc/90860
ER -

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