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

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

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## 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 ${\phi }_{0}$ 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 $l:\left[0,T\right]\to {ℝ}_{+}^{*}$ 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.

## 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 -

## References

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1. F. Albertini and D. D'Alessandro, Notions of controllability for bilinear multilevel quantum systems. IEEE Trans. Automat. Control48 (2003) 1399–1403.
2. S. Alinhac and P. Gérard, Opérateurs pseudo-différentiels et théorème de Nash-Moser. Intereditions (Paris), collection Savoirs actuels (1991).
3. C. Altafini, Controllability of quantum mechanical systems by root space decomposition of su(n). J. Math. Phys.43 (2002) 2051–2062.
4. J.M. Ball, J.E. Marsden and M. Slemrod, Controllability for distributed bilinear systems. SIAM J. Control Optim.20 (1982).
5. L. Baudouin, A bilinear optimal control problem applied to a time dependent Hartree-Fock equation coupled with classical nuclear dynamics. Portugaliae Matematica (N.S.)63 (2006) 293–325.
6. L. Baudouin and J. Salomon, Constructive solution of a bilinear control problem. C.R. Math. Acad. Sci. Paris342 (2006) 119–124.
7. L. Baudouin, O. Kavian and J.-P. Puel, Regularity for a Schrödinger equation with singular potential and application to bilinear optimal control. J. Differential Equations216 (2005) 188–222.
8. K. Beauchard, Local controllability of a 1-D beam equation. SIAM J. Control Optim. (to appear).
9. K. Beauchard, Local Controllability of a 1-D Schrödinger equation. J. Math. Pures Appl.84 (2005) 851–956.
10. K. Beauchard and J.-M. Coron, Controllability of a quantum particle in a moving potential well. J. Functional Analysis232 (2006) 328–389.
11. R. Brockett, Lie theory and control systems defined on spheres. SIAM J. Appl. Math.25 (1973) 213–225.
12. E. Cancès, C. Le Bris and M. Pilot, Contrôle optimal bilinéaire d'une équation de Schrödinger. C.R. Acad. Sci. Paris, Série I330 (2000) 567–571.
13. J.-M. Coron, Global asymptotic stabilization for controllable systems without drift. Math. Control Signals Systems5 (1992) 295–312.
14. J.-M. Coron, Contrôlabilité exacte frontière de l'équation d'Euler des fluides parfaits incompressibles bidimensionnels. C. R. Acad. Sci. Paris317 (1993) 271–276.
15. J.-M. Coron, On the controllability of 2-D incompressible perfect fluids. J. Math. Pures Appl.75 (1996) 155–188.
16. J.-M. Coron, Local Controllability of a 1-D Tank Containing a Fluid Modeled by the shallow water equations. ESAIM: COCV8 (2002) 513–554.
17. J.-M. Coron, On the small-time local controllability of a quantum particule in a moving one-dimensional infinite square potential well. C.R. Acad. Sci., Série I342 (2006) 103–108.
18. J.-M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with critical lengths. J. Eur. Math. Soc.6 (2004) 367–398.
19. J.-M. Coron and A. Fursikov, Global exact controllability of the 2D Navier-Stokes equation on a manifold without boundary. Russ. J. Math. Phys.4 (1996) 429–448.
20. A.V. Fursikov and O.Yu. Imanuvilov, Exact controllability of the Navier-Stokes and Boussinesq equations. Russian Math. Surveys54 (1999) 565–618.
21. O. Glass, On the controllability of the 1D isentropic Euler equation. J. European Mathematical Society9 (2007) 427–486.
22. O. Glass, Exact boundary controllability of 3-D Euler equation. ESAIM: COCV5 (2000) 1–44.
23. O. Glass, On the controllability of the Vlasov-Poisson system. J. Differential Equations195 (2003) 332–379.
24. G. Gromov, Partial Differential Relations. Springer-Verlag, Berlin-New York-London (1986).
25. A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire. J. Math. Pures Appl.68 (1989) 457–465.
26. L. Hörmander, On the Nash-Moser Implicit Function Theorem. Annales Academiae Scientiarum Fennicae (1985) 255–259.
27. T. Horsin, On the controllability of the Burgers equation. ESAIM: COCV3 (1998) 83–95.
28. R. Ilner, H. Lange and H. Teismann, Limitations on the control of Schrödinger equations. ESAIM: COCV 12 (2006) 615–635.
29. T. Kato, Perturbation Theory for Linear operators. Springer-Verlag, Berlin, New-York (1966).
30. W. Krabs, On moment theory and controllability of one-dimensional vibrating systems and heating processes. Springer – Verlag (1992).
31. I. Lasiecka and R. Triggiani, Optimal regularity, exact controllability and uniform stabilization of Schrödinger equations with Dirichlet controls. Differential Integral Equations5 (1992) 571–535.
32. I. Lasiecka, R. Triggiani and X. Zhang, Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carlemann estimates. J. Inverse Ill Posed-Probl.12 (2004) 183–231.
33. G. Lebeau, Contrôle de l'équation de Schrödinger. J. Math. Pures Appl.71 (1992) 267–291.
34. Machtyngier, Exact controllability for the Schrödinger equation. SIAM J. Contr. Opt.32 (1994) 24–34.
35. M. Mirrahimi and P. Rouchon, Controllability of quantum harmonic oscillators. IEEE Trans. Automat. Control49 (2004) 745–747.
36. E. Sontag, Control of systems without drift via generic loops. IEEE Trans. Automat. Control40 (1995) 1210–1219.
37. G. Turinici, On the controllability of bilinear quantum systems, in Mathematical Models and Methods for Ab Initio Quantum Chemistry, C. Le Bris and M. Defranceschi Eds., Lect. Notes Chemistry 74, Springer (2000).
38. E. Zuazua, Remarks on the controllability of the Schrödinger equation. CRM Proc. Lect. Notes33 (2003) 193–211.

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