Taiwanese Journal of Mathematics

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

Jing Li

Advance publication

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Abstract

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

Source
Taiwanese J. Math., Advance publication (2019), 35 pages.

Dates
First available in Project Euclid: 21 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1558404261

Digital Object Identifier
doi:10.11650/tjm/190505

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

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

Citation

Li, Jing. Reducibility, Lyapunov Exponent, Pure Point Spectra Property for Quasi-periodic Wave Operator. Taiwanese J. Math., advance publication, 21 May 2019. doi:10.11650/tjm/190505. https://projecteuclid.org/euclid.twjm/1558404261


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