Controllability of a quantum particle in a 1D variable domain

Karine Beauchard

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

  • 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. "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 $T &gt; 0$, 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 $T &gt; 0$, 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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