| Abstract: |
| In this paper we consider the high dimensional Schr\{o}dinger equation $ -\frac{d^{2}y}{dt^{2}} + S(t)y= Ey , y\in\mathbb{R}^{n}, $
where $E= \mbox{diag}(\lambda_{1}^{2}, \ldots, \lambda_{n}^{2})$
is a diagonal matrix, $S(t)$ is a real analytic quasi-periodic symmetric matrix with basic frequencies $\omega= (1, \alpha),$ where $\alpha$ is irrational,
it is proved that for most of sufficiently large $\lambda_{j}, j=1, \ldots, n, $
the Schr\{o}dinger equation has n pairs of conjugate quasi-periodic solutions, if the basic frequencies satisfy that $ 0\leq\beta(\alpha) < r, $ where $\beta(\alpha)$ measures how Liouvillean $\alpha$ is, $r$ is the initial analytic radius. |
|