Taiwanese Journal of Mathematics

Reducibility, Lyapunov Exponent, Pure Point Spectra Property for Quasi-periodic Wave Operator

Jing Li

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In the present paper, it is shown that the linear wave equation subject to Dirichlet boundary condition \[ u_{tt} - u_{xx} + \varepsilon V(\omega t, x) u = 0, \quad u(t,-\pi) = u(t,\pi) = 0 \] can be changed by a symplectic transformation into \[ v_{tt} - v_{xx} + \varepsilon M_{\xi} v = 0, \quad v(t,-\pi) = v(t,\pi) = 0, \] where $V$ is finitely smooth and time-quasi-periodic potential with frequency $\omega \in \mathbb{R}^n$ in some Cantor set of positive Lebeague measure and where $M_{\xi}$ is a Fourier multiplier. Moreover, it is proved that the corresponding wave operator $\partial_t^2 - \partial_x^2 + \varepsilon V(\omega t, x)$ possesses the property of pure point spectra and zero Lyapunov exponent.

Article information

Taiwanese J. Math., Volume 24, Number 2 (2020), 377-411.

Received: 11 December 2018
Revised: 4 April 2019
Accepted: 12 May 2019
First available in Project Euclid: 21 May 2019

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Zentralblatt MATH identifier

Primary: 37K55: Perturbations, KAM for infinite-dimensional systems 35P05: General topics in linear spectral theory 81Q15: Perturbation theories for operators and differential equations

reducibility quasi-periodic wave operator KAM theory finitely smooth Lyapunov exponent pure-point spectrum


Li, Jing. Reducibility, Lyapunov Exponent, Pure Point Spectra Property for Quasi-periodic Wave Operator. Taiwanese J. Math. 24 (2020), no. 2, 377--411. doi:10.11650/tjm/190505. https://projecteuclid.org/euclid.twjm/1558404261

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  • P. Baldi, M. Berti and R. Montalto, KAM for autonomous quasi-linear perturbations of KdV, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 6, 1589–1638.
  • D. Bambusi, Reducibility of $1$-d Schrödinger equation with time quasiperiodic unbounded perturbations II, Comm. Math. Phys. 353 (2017), no. 1, 353–378.
  • D. Bambusi and S. Graffi, Time quasi-periodic unbounded perturbations of Schrödinger operators and KAM methods, Comm. Math. Phys. 219 (2001), no. 2, 465–480.
  • D. Bambusi, B. Grébert, A. Maspero and D. Robert, Reducibility of the quantum harmonic oscillator in $d$-dimensions with polynomial time-dependent perturbation, Anal. PDE 11 (2018), no. 3, 775–799.
  • M. Berti and P. Bolle, Sobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential, Nonlinearity 25 (2012), no. 9, 2579–2613.
  • N. N. Bogoljubov, Ju. A. Mitropoliskii and A. M. Samoĭlenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer-Verlag, New York, 1976.
  • J. Bourgain, Growth of Sobolev norms in linear Schrödinger equations with quasi-periodic potential, Comm. Math. Phys. 204 (1999), no. 1, 207–247.
  • J. Bourgain and M. Goldstein, On nonperturbative localization with quasi-periodic potential, Ann. of Math. (2) 152 (2000), no. 3, 835–879.
  • L. Chierchia and D. Qian, Moser's theorem for lower dimensional tori, J. Differential Equations 206 (2004), no. 1, 55–93.
  • M. Combescure, The quantum stability problem for time-periodic perturbations of the harmonic oscillator, Ann. Inst. H. Poincaré Phys. Théor. 47 (1987), no. 1, 63–83.
  • P. Duclos and P. Š\vtovíček, Floquet Hamiltonians with pure point spectrum, Comm. Math. Phys. 177 (1996), no. 2, 327–347.
  • P. Duclos, P. Š\vtovíček and M. Vittot, Perturbation of an eigen-value from a dense point spectrum: a general Floquet Hamiltonian, Ann. Inst. H. Poincaré Phys. Théor. 71 (1999), no. 3, 241–301.
  • H. L. Eliasson and S. B. Kuksin, On reducibility of Schrödinger equations with quasiperiodic in time potentials, Comm. Math. Phys. 286 (2009), no. 1, 125–135.
  • D. Fang, Z. Han and W.-M. Wang, Bounded Sobolev norms for Klein-Gordon equations under non-resonant perturbation, J. Math. Phys. 55 (2014), no. 12, 121503, 11 pp.
  • R. Feola and M. Procesi, Quasi-periodic solutions for fully nonlinear forced reversible Schrödinger equations, J. Differential Equations 259 (2015), no. 7, 3389–3447.
  • B. Grébert and E. Paturel, On reducibility of quantum harmonic oscillator on $\mathbb{R}^{d}$ with quasiperiodic in time potential, arXiv:1603.07455.
  • B. Grébert and L. Thomann, KAM for the quantum harmonic oscillator, Comm. Math. Phys. 307 (2011), no. 2, 383–427.
  • D. Jackson, The Theory of Approximation, Reprint of the 1930 original, American Mathematical Society Colloquium Publications 11, American Mathematical Society, Providence, RI, 1994.
  • S. Klein, Anderson localization for the discrete one-dimensional quasi-periodic Schrödinger operator with potential defined by a Gevrey-class function, J. Funct. Anal. 218 (2005), no. 2, 255–292.
  • S. B. Kuksin, Nearly integrable infinite-dimensional Hamiltonian systems, Lecture Notes in Mathematics 1556, Springer-Verlag, New York, 1993.
  • Z. Liang and X. Wang, On reducibility of 1d wave equation with quasiperiodic in time potentials, J. Dynam. Differential Equations 30 (2018), no. 3, 957–978.
  • J. Liu and X. Yuan, Spectrum for quantum Duffing oscillator and small-divisor equation with large-variable coefficient, Comm. Pure Appl. Math. 63 (2010), no. 9, 1145–1172.
  • A. Maspero and D. Robert, On time dependent Schrödinger equations: global well-posedness and growth of Sobolev norms, J. Funct. Anal. 273 (2017), no. 2, 721–781.
  • R. Montalto, A reducibility result for a class of linear wave equations on $\mathbb{T}^{d}$, Int. Math. Res. Not. IMRN 2019 (2019), no. 6, 1788–1862.
  • J. Pöschel, A KAM-theorem for some nonlinear partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), no. 1, 119–148.
  • D. A. Salamon, The Kolmogorov-Arnold-Moser theorem, Math. Phys. Electron. J. 10 (2004), no. 3, 37 pp.
  • D. Salamon and E. Zehnder, KAM theory in configuration space, Comment. Math. Helv. 64 (1989), no. 1, 84–132.
  • Y. Sun, J. Li and B. Xie, Reducibility for wave equations of finitely smooth potential with periodic boundary conditions, Journal of Differential Equations 266 (2019), 2762–2804.
  • W.-M. Wang, Pure point spectrum of the Floquet Hamiltonian for the quantum harmonic oscillator under time quasi-periodic perturbations, Comm. Math. Phys. 277 (2008), no. 2, 459–496.
  • Z. Wang and Z. Liang, Reducibility of 1D quantum harmonic oscillator perturbed by a quasiperiodic potential with logarithmic decay, Nonlinearity 30 (2017), no. 4, 1405–1448.
  • X. Yuan and K. Zhang, A reduction theorem for time dependent Schrödinger operator with finite differentiable unbounded perturbation, J. Math. Phys. 54 (2013), no. 5, 052701, 23 pp.
  • E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems I, II, Comm. Pure Appl. Math. 28 (1975), 91–140; 29 (1976), no. 1, 49–111.