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Time-fractional Schrödinger equation

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Abstract

We propose a time-fractional extension of the Schrödinger equation that keeps the main mechanical and quantum properties of the classical Schrödinger equation. This extension is shown to be equivalent to another well identified time first-order PDE with fractional Hamiltonian.

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References

  1. Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, and Frank Neubrander. Vector-valued Laplace transforms and Cauchy problems, volume 96 of Monographs in Mathematics. Birkhäuser/Springer, Basel, second edition, 2011.

  2. Ana Bernardis, Francisco J. Martín-Reyes, Pablo Raúl Stinga, and José L. Torrea. Maximum principles, extension problem and inversion for nonlocal one-sided equations. J. Differential Equations, 260(7):6333–6362, 2016.

    Article  MathSciNet  Google Scholar 

  3. Kai Diethelm. The analysis of fractional differential equations, volume 2004 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2010.

  4. Hassan Emamirad and Arnaud Rougirel. Solution operators of three time variables for fractional linear problems. Mathematical Methods in the Applied Sciences, 40(5):1553–1572, 2017.

    Article  MathSciNet  Google Scholar 

  5. Hassan Emamirad and Arnaud Rougirel. Time fractional linear problems on \(L^2(\mathbb{R}^d)\). Bull. Sci. Math., 144:1–38, 2018.

    Article  MathSciNet  Google Scholar 

  6. Brian C. Hall. Quantum theory for mathematicians, volume 267 of Graduate Texts in Mathematics. Springer, New York, 2013.

  7. Marko Kowalski. Spectral Theory in Hilbert Spaces. http://www.math.ethz.ch/~kowalski/spectral-theory.pdf, 2009.

  8. Nick Laskin. Fractional Schrödinger equation. Phys. Rev. E (3), 66(5):056108, 7, 2002.

    Article  MathSciNet  Google Scholar 

  9. E. R. Love. Fractional Integration, and Almost Periodic Functions. Proc. London Math. Soc. (2), 44(5):363–397, 1938.

    Article  MathSciNet  Google Scholar 

  10. Yuri Luchko. Fractional Schrödinger equation for a particle moving in a potential well. J. Math. Phys., 54(1):012111, 10, 2013.

    Article  MathSciNet  Google Scholar 

  11. Mark Naber. Time fractional Schrödinger equation. J. Math. Phys., 45(8):3339–3352, 2004.

    Article  MathSciNet  Google Scholar 

  12. Michael Reed and Barry Simon. Methods of modern mathematical physics. I. Academic Press, Inc., New York, second edition, 1980. Functional analysis.

  13. Konrad Schmüdgen. Unbounded self-adjoint operators on Hilbert space, volume 265 of Graduate Texts in Mathematics. Springer, 2012.

  14. Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev. Fractional integrals and derivatives. Gordon and Breach Science Publishers, Yverdon, 1993.

    MATH  Google Scholar 

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Correspondence to Arnaud Rougirel.

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This research was in part supported by a Grant from IPM # 91470221.

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Emamirad, H., Rougirel, A. Time-fractional Schrödinger equation. J. Evol. Equ. 20, 279–293 (2020). https://doi.org/10.1007/s00028-019-00525-5

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  • DOI: https://doi.org/10.1007/s00028-019-00525-5

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