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

ESAIM: Control, Optimisation and Calculus of Variations (2008)

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

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topBeauchard, Karine. "Controllability of a quantum particle in a 1D variable domain." ESAIM: Control, Optimisation and Calculus of Variations 14.1 (2008): 105-147. <http://eudml.org/doc/244959>.

@article{Beauchard2008,

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 $\phi $ 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},

language = {eng},

number = {1},

pages = {105-147},

publisher = {EDP-Sciences},

title = {Controllability of a quantum particle in a 1D variable domain},

url = {http://eudml.org/doc/244959},

volume = {14},

year = {2008},

}

TY - JOUR

AU - Beauchard, Karine

TI - Controllability of a quantum particle in a 1D variable domain

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2008

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 $\phi $ 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

UR - http://eudml.org/doc/244959

ER -

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