Ballistic Transport and Absolute Continuity of One-Frequency Schrödinger Operators

For the solution u ( t ) to the discrete Schrödinger equation i d d t u n ( t ) = - ( u n + 1 ( t ) + u n - 1 ( t ) ) + V ( θ + n α ) u n ( t ) , n ∈ Z , with α ∈ R \ Q and V ∈ C ω ( T , R ) , we consider the growth rate with t of its diffusion norm ⟨ u ( t ) ⟩ p : = ∑ n ∈ Z ( n p + 1 ) | u n ( t ) | 2 1 2 , and the (non-averaged) transport exponents β u + ( p ) : = lim sup t → ∞ 2 log ⟨ u ( t ) ⟩ p p log t , β u - ( p ) : = lim inf t → ∞ 2 log ⟨ u ( t ) ⟩ p p log t . We will show that, if the corresponding Schrödinger operator has purely absolutely continuous spectrum, then β u ± ( p ) = 1 , provided that u (0) is well localized.