Quasi-periodic solutions of n coupled Schrödinger equations with Liouvillean basic frequencies.

In this paper we consider n coupled Schrödinger equations − d 2 y d t 2 + u (ω t) y = E y , y ∈ R n , where E = diag ( λ 1 2 , ... , λ n 2 ) is a diagonal matrix, u(ωt) is a real analytic quasi-periodic symmetric matrix. If the basic frequencies ω = (1, α), where α is irrational, it is proved that for most of sufficiently large λj, j = 1, ..., n, all the solutions of n coupled Schrödinger equations are bounded. Furthermore, if the basic frequencies satisfy that 0 ≤ β(α) < r, where β (α) = lim sup n → ∞ ln q n + 1 q n , qn is the sequence of denominations of the best rational approximations for α ∈ R \ Q , r is the initial analytic radius, we obtain the existence of n pairs of conjugate quasi-periodic solutions for most of sufficiently large λj, j = 1, ..., n. [ABSTRACT FROM AUTHOR]