Schroedinger equations with potentials generated by shifts and skew-shifts

In this talk we give an exposition of the ideas, methods and techn ological tools developed in last ten years for the analysis of one-dimensional d ifference Schroedinger equations with potentials generated by shift and sk ew-shift H(x, ω)ϕ (n) ≡ −ϕ(n − 1) − ϕ ;(n + 1) + V (Tω^n(x))ϕ(n) = Eϕ(n) , n ∈ Z. Here Tω is a shift Tω (x) = x + ω, x, ω ∈ Tν , or a skew-shift, Tω (x, y) = (x + ω, y + x), (x, y) ∈ T^2 , ω ∈ T, and T^nω , n ∈ Z, stands for the n-th iteration of T . We state also the central open problems in this field.

 

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