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

Consider the one-dimensional discrete Schrödinger operator Hθ:(Hθq)n=−(qn+1+qn−1)+V(θ+nω)qn,n∈Z, with ω∈Rd Diophantine, and V a real-analytic function on Td=(R/2πZ)d. For V sufficiently small, we prove the dispersive estimate: for every ϕ∈ℓ1(Z),(1)‖e−itHθϕ‖ℓ∞≤K0|ln⁡ε0|a(ln⁡ln⁡(2+〈t〉))2d〈t〉13‖ϕ‖ℓ1,〈t〉:=1+t2, with a and K0 two absolute constants and ε0 an analytic norm of V. The estimate holds for every θ∈Td.