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

Abstract

In this paper we consider n coupled Schrödinger equations −d2ydt2+u(ωt)y=Ey,y∈Rn, where E=diag(λ12,…,λn2) 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 supn→∞lnqn+1qn, 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.