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), ‖e−itHjavax.xml.bind.JAXBElement@451f543ϕ‖ℓjavax.xml.bind.JAXBElement@31c1caec≤K0[Formula presented]‖ϕ‖ℓjavax.xml.bind.JAXBElement@6e0bf990,〈t〉:=1+t2, with a and K0 two absolute constants and ε0 an analytic norm of V. The estimate holds for every θ∈Td.