A note on the extension of the Dinaburg–Sinai theorem to higher dimension

Abstract

The system $-d^2y/dt^2+u(t)y=Ey\ (y\in R^n)$ is considered, where $E=\text{diag} (\lambda_1^2,\, \lambda_2^2,\dots,\lambda_n^2)$ is a diagonal matrix with $\lambda_j>0\ (j=1,2,\dots, n)$ being regarded as parameters, and u a real analytic quasi-periodic symmetric matrix. Suppose that the basic frequencies of u satisfy a Diophantine condition. Then, for most of sufficiently large $\lambda_j\ (j=1,2,\dots,n)$, the system has n pairs of conjugate quasi-periodic solutions.