Ballistic Motion in One-Dimensional Quasi-Periodic Discrete Schrödinger Equation

Abstract

For the solution q(t) to the one-dimensional continuous Schrödinger equationwith ω ∈ R d satisfying a Diophantine condition, and V a real-analytic function on we consider the growth rate of the diffusion norm q(t)for any non-zero initial condition q(0) ∈ H 1 (R) with q(0) D < ∞. We prove that q(t) D grows linearly with t if V is sufficiently small.

Dispersive estimate for quasi-periodic Schrödinger operators on 1-d lattices

“…the second of which is obtained from an estimate of an oscillatory integral (Lemma 4.1 of [Zha16]) which is very close to the estimate given in Lemma 3.3 of the present paper.…”

“…with some β n,n ∆ which can be shown to fulfill the estimates claimed in the statement (for the details see [Zha16]). Thus one gets K n = n ∆ β n,n ∆ sin n ∆ ρ, J n = n ∆ β n,n ∆ cos n ∆ ρ.…”

“…It turns out that this is possible, and that, if one uses such "normalized" eigenfunctions to define a spectral transform, then such a transformation is not unitary, but it is bounded with a bounded inverse and thus suffices to get the result. This was done and proved in [Zha16]. Here we just recall the needed results.…”

“…The price to pay is that S is no more unitary, but turns out to be just a bounded transformation with bounded inverse. For completeness we add now the details of the construction of K and J , while we refer to [Zha16] for the details of the proofs of the estimates. First we remark that one can construct Bloch-waves of Schrödinger operator H θ on Σ using the reducibility of Schrödinger cocycle.…”

“…First we remark that one can construct Bloch-waves of Schrödinger operator H θ on Σ using the reducibility of Schrödinger cocycle. Indeed, with an additional transform (see (3.17) of [Zha16]), one can findZ : Σ × 2T d → SL(2, C) and B : Σ → SL(2, C), with two eigenvalues e ±iρ , such that…”