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In this paper, we consider the high dimensional Schrödinger equation −d2ydt2+u(t)y=Ey,y∈ℝn,$ -\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=diag(λ12,…,λn2)$E=… Click to show full abstract
In this paper, we consider the high dimensional Schrödinger equation −d2ydt2+u(t)y=Ey,y∈ℝn,$ -\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=diag(λ12,…,λn2)$E= \text {diag}({\lambda _{1}^{2}}, \ldots , {\lambda _{n}^{2}})$ is a diagonal matrix with λj>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 λj,j=1,…,n, there exist n pairs of conjugate quasi-periodic solutions.
Journal Title: Journal of Dynamical and Control Systems
Year Published: 2017
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