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Asymptotic behavior of solutions to nonlinear Schrödinger equations with time-dependent harmonic potentials

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Abstract

In this study, we examine the asymptotic behavior of solutions to nonlinear Schrödinger equations with time-dependent harmonic oscillators and prove the time-decay property of solutions in the case of a long range power type nonlinearity.

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References

  1. Adachi, T., Kawamoto, M.: Quantum scattering in a periodically pulsed magnetic field, Ann. Henri Poincaré, 17, 2409–2438 (2016).

    Article  MathSciNet  Google Scholar 

  2. Barab, J.E.: Nonexistence of asymptotic free solutions for a nonlinear Schrödinger equation, J. Math. Phys., 25, 3270–3273 (1984).

    Article  MathSciNet  Google Scholar 

  3. Carles, R.: Nonlinear Schrödinger equations with repulsive harmonic potential and applications, SIAM J. on Math. Anal., 35, 823–843 (2003).

    Article  MathSciNet  Google Scholar 

  4. Carles, R.: Nonlinear Schrödinger equation with time dependent potential, Commun. Math. Sci., 9, 937–964 (2011).

    Article  MathSciNet  Google Scholar 

  5. Carles, R., Silva, J. D.: Large time behavior in nonlinear Schrödinger equation with time dependent potential, Comm. Math. Sci., 13, 443–460 (2015).

    Article  Google Scholar 

  6. Cazenave, T., Wessler, F.B.: The Cauchy problem for the critical nonlinear Schrödinger equations \(H^s\), Nonlinear Anal., 14, 807–836 (1990).

    Article  MathSciNet  Google Scholar 

  7. Dodson, B.: Global well-posedness and scattering for the defocusing, \(L^2\)-critical nonlinear Schrödinger equation when \(d \ge 3\), J. of the A. M. S., 25, 429–463 (2012).

    MATH  Google Scholar 

  8. Geluk, J. L., Marić, V., Tomić, M.: On regularly varying solutions of second order linear differential equations, Differential and Integral Equ., 6, 329–336, (1993).

    MathSciNet  MATH  Google Scholar 

  9. Ginibre, J., Ozawa, T., Velo, G.: On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. de l’L.H.P. Phys. théorique, 60 211–239 (1994).

    MATH  Google Scholar 

  10. Ginibre, J., Velo, G.: Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pure. Appl., 64, 363–401 (1985).

    MATH  Google Scholar 

  11. Hani, Z., Thomann, L.: Asymptotic behavior of the nonlinear Schrödinger equation with harmonic trapping, Comm. in P. D. E., 69, 1727–1776 (2016).

    MATH  Google Scholar 

  12. Hayashi, N., Naumkin, P.: Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations, American J. of Math., 120, 369–389 (1998).

    Article  MathSciNet  Google Scholar 

  13. Hayashi, N., Ozawa, T.: Scattering theory in the weighted \(L^2({\mathbb{R}}^n)\) spaces for some Schrödinger equations, Ann. Inst. H. Poincaré, Phys. Théor., 48, 17–37 (1988)

    MathSciNet  MATH  Google Scholar 

  14. Kawamoto, M.: Quantum scattering with time-decaying harmonic oscillators, preprint (arXiv:1704.03714).

  15. Kawamoto, M.: Mourre theory for time-periodic magnetic fields, J. Funct. Anal., 277, 1–30 (2019).

    Article  MathSciNet  Google Scholar 

  16. Kawamoto, M.: Strichartz estimates for Schrödinger operators with square potential with time-dependent coefficients, preprint (arXiv:1805.07991).

  17. Kawamoto, M., Yoneyama, T.: Strichartz estimates for harmonic potential with time-decaying coefficient, J. Evol. Eqn. 18, 127–142 (2017) .

    Article  MathSciNet  Google Scholar 

  18. Korotyaev, E. L.: On scattering in an external, homogeneous, time-periodic magnetic field, Math. USSR-Sb., 66, 499–522 (1990).

    Article  MathSciNet  Google Scholar 

  19. Masaki, S., Miyazaki, H., Uriya, K.: Long-range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity in three space dimensions, to appear in Transaction of the A.M.S.

  20. Naito, M.: Asymptotic behavior of solutions of second order differential equations with integrable coefficients, Trans. A.M.S., 282, 577–588, (1984).

    Article  MathSciNet  Google Scholar 

  21. Ozawa, T.: Long range scattering for nonlinear Schrödinger equations in one space dimension, Com. Math. Phys., 139, 479–493 (1991).

    Article  Google Scholar 

  22. Sagawa, Y., Sunagawa, H.: The lifespan of small solutions to cubic derivative nonlinear Schrödinger equations in one dimension, Discrete and Cont. Dyn. Sys., 36, 5743–5761 (2016).

    Article  Google Scholar 

  23. Shimomura, A.: Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities, Comm. in P. D. E., 31, 1407–1423 (2004).

    Article  Google Scholar 

  24. Strauss, W. A.: Nonlinear scattering theory, scattering theory in Math. Physics, Reidel, Dordrecht, 1974, pp. 53–78.

  25. Tsutsumi, Y., Yajima, K.: The asymptotic behavior of nonlinear Schrödinger equations, Bull. Amer. Math. Soc., 11, 186–188 (1984).

    Article  MathSciNet  Google Scholar 

  26. Willett, D.: On the oscillatory behavior of the solutions of second order linear differential equations, Ann. Polon. Math., 21, 175–194, (1969).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Engineering for Production, Graduate School of Science and Engineering, Ehime University, 3 Bunkyo-cho, Matsuyama, Ehime, 790-8577, Japan

    Masaki Kawamoto

  2. Department of Mathematics, Graduate School of Science, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan

    Ryo Muramatsu

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  1. Masaki Kawamoto
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  2. Ryo Muramatsu
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Correspondence to Masaki Kawamoto.

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Kawamoto, M., Muramatsu, R. Asymptotic behavior of solutions to nonlinear Schrödinger equations with time-dependent harmonic potentials. J. Evol. Equ. 21, 699–723 (2021). https://doi.org/10.1007/s00028-020-00597-8

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  • DOI: https://doi.org/10.1007/s00028-020-00597-8

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