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On the Control Problem for Schrödinger Operators on Tori

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Geometric Aspects of Functional Analysis

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2116))

Abstract

We consider the linear Schrödinger equation on the three dimensional torus with a bounded spatially dependent potential and prove controllability. This extends the earlier work due to Burq, Zworski and the author in the two dimensional case.

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References

  1. A. Anantharaman, F. Macia, Semi-classical measures for the Schrödinger equation on the torus. JEMS 16(6), 1253–1288 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. C. Bardos, G. Lebeau, J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30, 1024–1065 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Bourgain, Restriction phenomena for lattice subsets and applications to nonlinear evolution equations I, Schrödinger equations. GAFA 3(2), 107–156 (1993)

    MathSciNet  MATH  Google Scholar 

  4. J. Bourgain, Green’s Function Estimates for Lattice Schrödinger Equations and Applications. Annals of Mathematics Studies, vol. 158 (Princeton University Press, Princeton, 2005)

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  5. J. Bourgain, Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces. Israel J. Math. 193, 441–450 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Bourgain, C. Demeter, New bounds for the discrete Fourier restriction to the sphere in four and five dimensions. IMRN (to appear)

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  7. J. Bourgain, N. Burq, M. Zworski, Control for Schrödinger operators on 2-tori: rough potentials. J. Eur. Math. Soc. 15(5), 1597–1628 (2013)

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  8. N. Burq, M. Zworski, Control for Schrödinger equations on tori. Math. Res. Lett. 19, 309–324 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Research supported in part by NSF Grants DMS 1301619.

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Authors and Affiliations

  1. School of Mathematics, Institute for Advanced Study, Princeton, NJ, 08540, USA

    Jean Bourgain

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  1. Jean Bourgain
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Correspondence to Jean Bourgain .

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Editors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    Bo'az Klartag

  2. Technion - Israel Institute of Technology, Haifa, Israel

    Emanuel Milman

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Bourgain, J. (2014). On the Control Problem for Schrödinger Operators on Tori. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2116. Springer, Cham. https://doi.org/10.1007/978-3-319-09477-9_8

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