Advertisement

Russian Journal of Mathematical Physics

, Volume 26, Issue 1, pp 109–121 | Cite as

On a Heavy Quantum Particle

  • D. V. TreschevEmail author
  • O. E. ZubelevichEmail author
Article
  • 11 Downloads

Abstract

We consider the Schrödinger equation for a particle on a flat n-torus with a bounded potential depending on time. The mass of the particle equals 1/μ2, where μ is a small parameter. We show that the Sobolev Hν-norms of the wave function grow approximately as tν on the time interval t ∈ [−t*, t*], where t* is slightly less than O(1/μ).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D. Bambusi, “Reducibility of 1D Schrödinger Equation with Time Quasiperiodic Unbounded Perturbations. I.,” Trans. Amer. Math. Soc. 370 (3), 1823–1865 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    D. Bambusi, B. Grebert, A. Maspero, and D. Robert, “Growth of Sobolev Norms for Abstract Linear Schrödinger Equations,” arXiv:1706.09708 [math.AP].Google Scholar
  3. 3.
    D. Bambusi, B. Grebert, A. Maspero, and D. Robert, “Reducibility of the Quantum Harmonic Oscillator in d–Dimensions with Polynomial Time Dependent Perturbation,” Anal. PDE, 11 (3), 775–799 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    J. Bourgain, “Growth of Sobolev Norms in Linear Schrödinger Equations with Quasi–Periodic Potential,” Comm. Math. Phys. 204 (1), 207–247 (1999).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    J. Bourgain, “On Growth of Sobolev Norms in Linear Schrödinger Equations with Smooth Time–Dependent Potentials,” J. Anal. Math. 77 (1), 315–348 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    J. Bourgain, “Estimates on Green’s Functions, Localization and the Quantum Kicked Rotor Model,” Ann. of Math. 156 (249), (2002).Google Scholar
  7. 7.
    M. Combescure, “The Quantum Stability Problem for Time–Periodic Perturbations of the Harmonic Oscillator,” Ann. Inst. H. Poincaré Phys. Théor. 47 (1), 63–83 (1987).MathSciNetzbMATHGoogle Scholar
  8. 8.
    L. Chierchia and J. You, “KAM–Tiri for 1D Nonlinear Wave Equations with Periodic Boundary Conditions,” Comm. Math. Phys. 211 (2), 497–525 (2000).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    D. Fang, Z. Han, and W.–M. Wang, “Bounded Sobolev Norms for Klein–Gordon Equations under Non–Resonant Perturbation,” J. Math. Phys. 55 (12), (2014).Google Scholar
  10. 10.
    B. Grébert and E. Paturel, “KAM for the Klein–Gordon Equation on Sd,” ArXiv e–prints, arXiv: 1601.00610 (2016).Google Scholar
  11. 11.
    B. Grébert and E. Paturel, “On Reducibility of Quantum Harmonic Oscillator on ℝd with Quasiperiodic in Time Potential,” ArXiv e–prints, arXiv: 1603.07455 (2016).Google Scholar
  12. 12.
    B. Grébert and L. Thomann, “KAM for the Quantum Harmonic Oscillator,” Comm. Math. Phys. 307 (2), 383–427 (2011).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    F. M. Izrailev and D. L. Shepelyansky, “Quantum Resonans for a Rotator in Nonlinear Periodic Field,” Dokl. Phys. 43 (3), (1980).Google Scholar
  14. 14.
    S. Kuksin, “Growth and Oscillations of Solutions of Nonlinear Schrödinger Equation,” Comm. Math. Phys. 178 (2), 265–280 (1996).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    S. B. Kuksin, “Spectral Properties of Solutions for Nonlinear PDEs in the Turbulent Regime,” GAFA Geom. Funct. Anal. 9, 141–184 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    S. Kuksin and A. Neishtadt, “On Quantum Averaging, Quantum KAM, and Quantum Diffusion,” Russ. Math. Surveys 68 (2), 335–348 (2013).ADSCrossRefzbMATHGoogle Scholar
  17. 17.
    A. Maspero, “Lower Bounds on the Growth of Sobolev Norms in Some Linear Time Dependent Schrödinger Equations,” arXiv:1801.06813v2 [math.AP].Google Scholar
  18. 18.
    R. Montalto, “A Reducibility Result for a Class of Linear Wave Equations on Td,” IMRN, rnx167, DOI: 10.1093/imrn/rnx167/ (2017).Google Scholar
  19. 19.
    M. V. Safonov, “The Abstract Cauchy–Kovalevskaya Theorem in a Weighted Banach Space,” Comm. Pure Appl. Math. 48, 629–643 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    M. E. Taylor, Partial Differential Equations (III Springer, 2012).Google Scholar
  21. 21.
    W.–M. Wang, “Pure Point Spectrum of the Floquet Hamiltonian for the Quantum Harmonic Oscillator under Time Quasi–Periodic Perturbations,” Comm. Math. Phys. 277 (2), 459–496 (2008).ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia
  2. 2.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

Personalised recommendations