# Controllablity of a quantum particle in a 1D variable domain

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

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

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topBeauchard, 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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