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

In the present paper, it is shown that the linear wave equation subject to Dirichlet boundary condition u(tt) - u(xx) + epsilon V (wt, x)u = 0, u(t, -pi) = u(t, pi) = 0 can be changed by a symplectic transformation into v(tt) - v(xx) + epsilon M(xi)v = 0, v(t, -pi) = v(t, pi) = 0, where V is finitely smooth and time-quasi-periodic potential with frequency omega is an element of 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 at partial derivative(2)(t) - partial derivative(2)(x) + epsilon V (omega t, x) possesses the property of pure point spectra and zero Lyapunov exponent.