doi: 10.3934/eect.2020075

A remark on the attainable set of the Schrödinger equation

CNRS & Laboratoire interdisciplinaire Carnot de Bourgogne, UMR 6303 Université de Bourgogne Franche-Comté, 9 Av. A. Savary, 21078 Dijon Cedex, France

Received  November 2019 Revised  April 2020 Published  June 2020

We discuss the set of wavefunctions $ \psi_V(t) $ that can be obtained from a given initial condition $ \psi_0 $ by applying the flow of the Schrödinger operator $ -\Delta + V(t,x) $ and varying the potential $ V(t,x) $. We show that this set has empty interior, both as a subset of the sphere in $ L^2( \mathbb{R}^d) $ and as a set of trajectories.

Citation: Jonas Lampart. A remark on the attainable set of the Schrödinger equation. Evolution Equations & Control Theory, doi: 10.3934/eect.2020075
References:
[1]

J. M. BallJ. E. Marsden and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control Optim., 20 (1982), 575-597.  doi: 10.1137/0320042.  Google Scholar

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K. Beauchard and J.-M. Coron, Controllability of a quantum particle in a moving potential well, J. Funct. Anal., 232 (2006), 328-389.  doi: 10.1016/j.jfa.2005.03.021.  Google Scholar

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K. Beauchard and C. Laurent, Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control, J. Math. Pures Appl., 94 (2010), 520-554.  doi: 10.1016/j.matpur.2010.04.001.  Google Scholar

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N. Boussaïd, M. Caponigro and T. Chambrion, Regular propagators of bilinear quantum systems, J. Funct. Anal., 278 (2020), 108412, 66 pp, arXiv: 1406.7847. doi: 10.1016/j.jfa.2019.108412.  Google Scholar

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N. Boussaid, M. Caponigro and T. Chambrion, On the Ball–Marsden–Slemrod obstruction in bilinear control systems, 2019 IEEE 58th Conference on Decision and Control (CDC), 2019, arXiv: 1903.05846. doi: 10.1109/CDC40024.2019.9029511.  Google Scholar

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H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.  Google Scholar

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T. Chambrion, P. Mason, M. Sigalotti and U. Boscain, Controllability of the discrete-spectrum Schrödinger equation driven by an external field, Ann. Inst. H. Poincaré C, 26 (2009), 329–349. doi: 10.1016/j.anihpc.2008.05.001.  Google Scholar

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T. Chambrion and L. Thomann, A topological obstruction to the controllability of nonlinear wave equations with bilinear control term, SIAM J. Control Optim., 57 (2019), 2315-2327.  doi: 10.1137/18M1215207.  Google Scholar

[10]

T. Chambrion and L. Thomann, On the bilinear control of the Gross-Pitaevskii equation, Ann. Inst. H. Poincaré C, 37 (2020), 605–626. doi: 10.1016/j.anihpc.2020.01.001.  Google Scholar

[11]

J. Diestel and J. Uhl, Vector Measures, vol. 15 of Mathematical surveys, American Mathematical Society, 1977.  Google Scholar

[12]

S. Fournais, J. Lampart, M. Lewin and T. Ø. Sørensen, Coulomb potentials and Taylor expansions in time-dependent density-functional theory, Phys. Rev. A, 93 (2016), 062510. doi: 10.1103/PhysRevA.93.062510.  Google Scholar

[13]

R. L. FrankM. LewinE. H. Lieb and R. Seiringer, Strichartz inequality for orthonormal functions, J. Eur. Math. Soc., 16 (2014), 1507-1526.  doi: 10.4171/JEMS/467.  Google Scholar

[14]

S. Lang, Real and Functional Analysis, vol. 142 of Graduate Texts in Mathematics, 3rd edition, Springer, 1993. doi: 10.1007/978-1-4612-0897-6.  Google Scholar

[15]

P. Mason and M. Sigalotti, Generic controllability properties for the bilinear Schrödinger equation, Commun. Partial Diff. Eq., 35 (2010), 685-706.  doi: 10.1080/03605300903540919.  Google Scholar

[16]

V. Nersesyan, Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications, Ann. Inst. H. Poincaré C, 27 (2010), 901–915. doi: 10.1016/j.anihpc.2010.01.004.  Google Scholar

[17]

I. Rodnianski and T. Tao, Effective limiting absorption principles, and applications, Commun. Math. Phys., 333 (2015), 1-95.  doi: 10.1007/s00220-014-2177-8.  Google Scholar

[18]

E. Runge and E. K. Gross, Density-functional theory for time-dependent systems, Phys. Rev. Lett., 52 (1984), 997. doi: 10.1103/PhysRevLett.52.997.  Google Scholar

[19]

G. Turinici, On the controllability of bilinear quantum systems, in Mathematical Models and Methods for ab Initio Quantum Chemistry (eds. M. Defrancesci and C. Le Bris), vol. 74 of Lecture Notes in Chemistry, Springer, 2000, 75–92. doi: 10.1007/978-3-642-57237-1_4.  Google Scholar

show all references

References:
[1]

J. M. BallJ. E. Marsden and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control Optim., 20 (1982), 575-597.  doi: 10.1137/0320042.  Google Scholar

[2]

K. Beauchard and J.-M. Coron, Controllability of a quantum particle in a moving potential well, J. Funct. Anal., 232 (2006), 328-389.  doi: 10.1016/j.jfa.2005.03.021.  Google Scholar

[3]

K. Beauchard and C. Laurent, Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control, J. Math. Pures Appl., 94 (2010), 520-554.  doi: 10.1016/j.matpur.2010.04.001.  Google Scholar

[4]

K. Beauchard and C. Laurent, Local exact controllability of the 2D-Schrödinger-Poisson system, J. École Polytechnique, 4 (2016), 287–336. doi: 10.5802/jep.44.  Google Scholar

[5]

N. Boussaïd, M. Caponigro and T. Chambrion, Regular propagators of bilinear quantum systems, J. Funct. Anal., 278 (2020), 108412, 66 pp, arXiv: 1406.7847. doi: 10.1016/j.jfa.2019.108412.  Google Scholar

[6]

N. Boussaid, M. Caponigro and T. Chambrion, On the Ball–Marsden–Slemrod obstruction in bilinear control systems, 2019 IEEE 58th Conference on Decision and Control (CDC), 2019, arXiv: 1903.05846. doi: 10.1109/CDC40024.2019.9029511.  Google Scholar

[7]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.  Google Scholar

[8]

T. Chambrion, P. Mason, M. Sigalotti and U. Boscain, Controllability of the discrete-spectrum Schrödinger equation driven by an external field, Ann. Inst. H. Poincaré C, 26 (2009), 329–349. doi: 10.1016/j.anihpc.2008.05.001.  Google Scholar

[9]

T. Chambrion and L. Thomann, A topological obstruction to the controllability of nonlinear wave equations with bilinear control term, SIAM J. Control Optim., 57 (2019), 2315-2327.  doi: 10.1137/18M1215207.  Google Scholar

[10]

T. Chambrion and L. Thomann, On the bilinear control of the Gross-Pitaevskii equation, Ann. Inst. H. Poincaré C, 37 (2020), 605–626. doi: 10.1016/j.anihpc.2020.01.001.  Google Scholar

[11]

J. Diestel and J. Uhl, Vector Measures, vol. 15 of Mathematical surveys, American Mathematical Society, 1977.  Google Scholar

[12]

S. Fournais, J. Lampart, M. Lewin and T. Ø. Sørensen, Coulomb potentials and Taylor expansions in time-dependent density-functional theory, Phys. Rev. A, 93 (2016), 062510. doi: 10.1103/PhysRevA.93.062510.  Google Scholar

[13]

R. L. FrankM. LewinE. H. Lieb and R. Seiringer, Strichartz inequality for orthonormal functions, J. Eur. Math. Soc., 16 (2014), 1507-1526.  doi: 10.4171/JEMS/467.  Google Scholar

[14]

S. Lang, Real and Functional Analysis, vol. 142 of Graduate Texts in Mathematics, 3rd edition, Springer, 1993. doi: 10.1007/978-1-4612-0897-6.  Google Scholar

[15]

P. Mason and M. Sigalotti, Generic controllability properties for the bilinear Schrödinger equation, Commun. Partial Diff. Eq., 35 (2010), 685-706.  doi: 10.1080/03605300903540919.  Google Scholar

[16]

V. Nersesyan, Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications, Ann. Inst. H. Poincaré C, 27 (2010), 901–915. doi: 10.1016/j.anihpc.2010.01.004.  Google Scholar

[17]

I. Rodnianski and T. Tao, Effective limiting absorption principles, and applications, Commun. Math. Phys., 333 (2015), 1-95.  doi: 10.1007/s00220-014-2177-8.  Google Scholar

[18]

E. Runge and E. K. Gross, Density-functional theory for time-dependent systems, Phys. Rev. Lett., 52 (1984), 997. doi: 10.1103/PhysRevLett.52.997.  Google Scholar

[19]

G. Turinici, On the controllability of bilinear quantum systems, in Mathematical Models and Methods for ab Initio Quantum Chemistry (eds. M. Defrancesci and C. Le Bris), vol. 74 of Lecture Notes in Chemistry, Springer, 2000, 75–92. doi: 10.1007/978-3-642-57237-1_4.  Google Scholar

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