September  2021, 10(3): 461-469. 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  September 2021 Early access  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 and Control Theory, 2021, 10 (3) : 461-469. 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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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