Reducibility for wave equations of finitely smooth potential with periodic boundary conditions

TL;DR

This paper proves the reducibility of wave equations with finitely smooth, quasi-periodic potentials under periodic boundary conditions, transforming them into systems with pure point spectra and zero Lyapunov exponents.

Contribution

It extends reducibility results to wave equations with finitely smooth potentials, showing spectral properties and stability under periodic boundary conditions.

Findings

Wave equations are reducible to constant coefficient Hamiltonian systems.

The wave operator has pure point spectrum and zero Lyapunov exponent.

The results apply to finitely smooth, quasi-periodic potentials with Diophantine frequencies.

Abstract

In the present paper, the reducibility is derived for the wave equations with finitely smooth and time-quasi-periodic potential subjects to periodic boundary conditions. More exactly, the linear wave equation $u_{tt}-u_{xx}+Mu+\varepsilon (V_0(\omega t)u_{xx}+V(\omega t, x)u)=0,\;x\in \mathbb{R}/2\pi \mathbb{Z}$ can be reduced to a linear Hamiltonian system of a constant coefficient operator which is of pure imaginary point spectrum set, where $V$ is finitely smooth in $(t, x)$, quasi-periodic in time $t$ with Diophantine frequency $\omega\in \mathbb{R}^{n},$ and $V_0$ is finitely smooth and quasi-periodic in time $t$ with Diophantine frequency $\omega\in \mathbb{R}^{n},$ Moreover, it is proved that the corresponding wave operator possesses the property of pure point spectra and zero Lyapunov exponent.