• Previous Article
    New results on controllability of fractional evolution systems with order $ \alpha\in (1,2) $
  • EECT Home
  • This Issue
  • Next Article
    A blow – up result for the semilinear Moore – Gibson – Thompson equation with nonlinearity of derivative type in the conservative case
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

[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

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

[1]

Karine Beauchard, Morgan Morancey. Local controllability of 1D Schrödinger equations with bilinear control and minimal time. Mathematical Control & Related Fields, 2014, 4 (2) : 125-160. doi: 10.3934/mcrf.2014.4.125

[2]

Felipe Hernandez. A decomposition for the Schrödinger equation with applications to bilinear and multilinear estimates. Communications on Pure & Applied Analysis, 2018, 17 (2) : 627-646. doi: 10.3934/cpaa.2018034

[3]

Dan-Andrei Geba, Evan Witz. Revisited bilinear Schrödinger estimates with applications to generalized Boussinesq equations. Electronic Research Archive, 2020, 28 (2) : 627-649. doi: 10.3934/era.2020033

[4]

Camille Laurent. Internal control of the Schrödinger equation. Mathematical Control & Related Fields, 2014, 4 (2) : 161-186. doi: 10.3934/mcrf.2014.4.161

[5]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[6]

Kai Wang, Dun Zhao, Binhua Feng. Optimal nonlinearity control of Schrödinger equation. Evolution Equations & Control Theory, 2018, 7 (2) : 317-334. doi: 10.3934/eect.2018016

[7]

Thomas Duyckaerts, Carlos E. Kenig, Frank Merle. Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1275-1326. doi: 10.3934/cpaa.2015.14.1275

[8]

Manuel González-Burgos, Sergio Guerrero, Jean Pierre Puel. Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation. Communications on Pure & Applied Analysis, 2009, 8 (1) : 311-333. doi: 10.3934/cpaa.2009.8.311

[9]

Klaus-Jochen Engel, Marjeta Kramar FijavŽ. Exact and positive controllability of boundary control systems. Networks & Heterogeneous Media, 2017, 12 (2) : 319-337. doi: 10.3934/nhm.2017014

[10]

Jamel Ben Amara, Hedi Bouzidi. Exact boundary controllability for the Boussinesq equation with variable coefficients. Evolution Equations & Control Theory, 2018, 7 (3) : 403-415. doi: 10.3934/eect.2018020

[11]

José R. Quintero, Alex M. Montes. On the exact controllability and the stabilization for the Benney-Luke equation. Mathematical Control & Related Fields, 2020, 10 (2) : 275-304. doi: 10.3934/mcrf.2019039

[12]

Mo Chen, Lionel Rosier. Exact controllability of the linear Zakharov-Kuznetsov equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 3889-3916. doi: 10.3934/dcdsb.2020080

[13]

Gökçe Dİlek Küçük, Gabil Yagub, Ercan Çelİk. On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 503-512. doi: 10.3934/dcdss.2019033

[14]

Juan Belmonte-Beitia, Víctor M. Pérez-García, Vadym Vekslerchik, Pedro J. Torres. Lie symmetries, qualitative analysis and exact solutions of nonlinear Schrödinger equations with inhomogeneous nonlinearities. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 221-233. doi: 10.3934/dcdsb.2008.9.221

[15]

M. Burak Erdoǧan, William R. Green. Dispersive estimates for matrix Schrödinger operators in dimension two. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4473-4495. doi: 10.3934/dcds.2013.33.4473

[16]

Jibin Li, Yan Zhou. Bifurcations and exact traveling wave solutions for the nonlinear Schrödinger equation with fourth-order dispersion and dual power law nonlinearity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3083-3097. doi: 10.3934/dcdss.2020113

[17]

Patrick Martinez, Judith Vancostenoble. Exact controllability in "arbitrarily short time" of the semilinear wave equation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 901-924. doi: 10.3934/dcds.2003.9.901

[18]

Arnaud Heibig, Mohand Moussaoui. Exact controllability of the wave equation for domains with slits and for mixed boundary conditions. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 367-386. doi: 10.3934/dcds.1996.2.367

[19]

Abdelmouhcene Sengouga. Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints. Evolution Equations & Control Theory, 2020, 9 (1) : 1-25. doi: 10.3934/eect.2020014

[20]

Peng Gao. Global exact controllability to the trajectories of the Kuramoto-Sivashinsky equation. Evolution Equations & Control Theory, 2020, 9 (1) : 181-191. doi: 10.3934/eect.2020002

2019 Impact Factor: 0.953

Metrics

  • PDF downloads (17)
  • HTML views (115)
  • Cited by (0)

Other articles
by authors

[Back to Top]