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April, 2020 Reducibility, Lyapunov Exponent, Pure Point Spectra Property for Quasi-periodic Wave Operator
Jing Li
Taiwanese J. Math. 24(2): 377-411 (April, 2020). DOI: 10.11650/tjm/190505

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.

Citation

Download Citation

Jing Li. "Reducibility, Lyapunov Exponent, Pure Point Spectra Property for Quasi-periodic Wave Operator." Taiwanese J. Math. 24 (2) 377 - 411, April, 2020. https://doi.org/10.11650/tjm/190505

Information

Received: 11 December 2018; Revised: 4 April 2019; Accepted: 12 May 2019; Published: April, 2020
First available in Project Euclid: 21 May 2019

zbMATH: 07192940
MathSciNet: MR4078203
Digital Object Identifier: 10.11650/tjm/190505

Subjects:
Primary: 35P05, 37K55, 81Q15

Rights: Copyright © 2020 The Mathematical Society of the Republic of China

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Vol.24 • No. 2 • April, 2020
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