On High Dimensional Schrödinger Equation with Quasi-Periodic Potentials.

In this paper, we consider the high dimensional Schrödinger equation $ -\frac {d^{2}y}{dt^{2}} + u(t)y= Ey, y\in \mathbb {R}^{n}, $ where u( t) is a real analytic quasi-periodic symmetric matrix, $E= \text {diag}({\lambda _{1}^{2}}, \ldots , {\lambda _{n}^{2}})$ is a diagonal matrix with λ >0, j=1,..., n, being regarded as parameters, and prove that if the basic frequencies of u satisfy a Bruno-Rüssmann's non-resonant condition, then for most of sufficiently large λ , j=1,..., n, there exist n pairs of conjugate quasi-periodic solutions. [ABSTRACT FROM AUTHOR]