Dynamical Properties of Quasi-periodic Schrödinger Equations

摘要: This thesis deals with the investigation of dynamical properties quasiperiodic Schrodinger equations. It contains following two papers: Paper I. Positive Lyapunov exponent for a class 1-D equations — discrete case. For nonconstant C1 potential function V : T → R and large λ, we prove that an almost full measure set irrational frequencies ω ∈ energies E R, all lying in spectrum operator (Hθu)n = −(un+1 + un−1) λV (θ nω)un, (maximal) associated equation Hθu Eu, is positive. Moreover, these energies, projective flow corresponding to fundamental solution system ( un un+1 ) 0 1 −1 nω) −E )( un−1 , shown be minimal. II. continuum We bottom (Hθu)(t) − d2 dt2 u(t) (t, θ ωt)u(t), positive \ Q, provided λ T2 satisfies some regularity conditions. also u u′ )′ ωt) obtained from minimal cases.