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A discrete Hardy’s uncertainty principle and discrete evolutions

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Abstract

In this paper we give a discrete version of Hardy’s uncertainty principle, by using complex variable arguments, as in the classical proof of Hardy’s principle. Moreover, we give an interpretation of this principle in terms of decaying solutions to the discrete Schrödinger and heat equations.

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References

  1. N. B. Andersen, A connection between the uncertainty principles on the real line and on the circle, arXiv:1307.4904v1 [math.FA].

  2. J. A. Barceló, L. Fanelli, S. Gutiérrez, A. Ruiz and M. C. Vilela, Hardy uncertainty principle and unique continuation properties of covariant Schrödinger flows, J. Funct. Anal. 264 (2013), 2386–2425.

    Article  MathSciNet  MATH  Google Scholar 

  3. P. Carruthers and M. M. Nieto, Phase and Angle variables in quantum mechanics, Rev. Modern Phys. 40 (1968), 411–440.

    Article  Google Scholar 

  4. B. Cassano and L. Fanelli, Sharp Hardy uncertainty principle and Gaussian profiles of covariant Schrödinger evolutions, Trans. Amer.Math. Soc. 367 (2015), 2213–2233.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Cowling, L. Escauriaza, C. E. Kenig, G. Ponce and L. Vega, The Hardy uncertainty principle revisited, Indiana Univ. Math. J. 59 (2010), 2007–2025.

    Article  MathSciNet  MATH  Google Scholar 

  6. H. Dym and H. P. McKean Jr., Fourier Series and Integrals, Probability and Mathematical Statistics, Vol. 14, Academic Press, New York–London, 1972.

  7. L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega, On uniqueness properties of solutions of Schrödinger equations, Comm. Partial Differential Equations 31 (2006), 1811–1823.

    Article  MathSciNet  MATH  Google Scholar 

  8. L. Escauriaza, C.E. Kenig, G. Ponce, and L. Vega, Hardy’s uncertainty principle, convexity and Schrödinger equations, J. Eur. Math. Soc. 10 (2008), 883–907.

    Article  MATH  Google Scholar 

  9. L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega, The sharp Hardy uncertainty principle for Schrödinger evolutions, Duke Math. J. 155 (2010), 163–187.

    Article  MathSciNet  MATH  Google Scholar 

  10. L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega, Uniqueness properties of solutions to Schrödinger equations, Bull. Amer. Math. Soc. 49 (2012), 415–422.

    Article  MathSciNet  MATH  Google Scholar 

  11. L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega, Hardy uncertainty principle, convexity and parabolic evolutions, Comm. Math. Phys., Springer, Berlin–Heidelberg, 346 (2016), 667–678.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Fernández-Bertolin, Discrete uncertainty principles and virial identities, Appl. Comput. Harmon. Anal. 40 (2016), 229–259.

    Article  MathSciNet  MATH  Google Scholar 

  13. T. N. T. Goodman, and S. S. Goh, Uncertainty principles and optimality on circles and spheres, in Advances in Constructive Approximation: Vanderbilt 2003, Mod. Methods Math., Nashboro Press, Brentwood, TN, 2004, pp. 207–218.

    Google Scholar 

  14. L. Ignat and E. Zuazua, Convergence rates for dispersive approximation schemes to nonlinear Schrödinger equatons, J. Math. Pures Appl. (9) 98 (2012), 479–517.

    Article  MathSciNet  MATH  Google Scholar 

  15. Ph. Jaming, Yu. Lyubarskii, E. Malinnikova, and K.-M. Perfekt, Uniqueness for discrete Schrödinger evolutions, Rev. Mat. Iberoamericana 34 (2018), 949–966.

    Article  MathSciNet  MATH  Google Scholar 

  16. B. Ya. Levin, Lectures on Entire Functions, Translations ofMathematical Monographs, American Mathematical Society, Providence, RI, 1996.

    Book  Google Scholar 

  17. NIST Digital Library ofMathematical Functions. http://dlmf.nist.gov/, Release 1.0.10 of 2015-08-07.

  18. E. M. Stein and R. Shakarchi, Princeton Lecture in Analysis II. Complex Analysis, Princeton University Press, Princeton, NJ, 2003.

    Google Scholar 

  19. J. Strikwerda, Finite Difference Schemes and Partial Differential Equations, SIAM, Philadelphia, PA, 2004.

    MATH  Google Scholar 

  20. G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge; Macmillan, New York, 1944.

    MATH  Google Scholar 

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Correspondence to Aingeru Fernández-Bertolin.

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Fernández-Bertolin, A. A discrete Hardy’s uncertainty principle and discrete evolutions. JAMA 137, 507–528 (2019). https://doi.org/10.1007/s11854-019-0002-1

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  • DOI: https://doi.org/10.1007/s11854-019-0002-1

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