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

Abstract

Consider the one-dimensional discrete Schrödinger operator $H_{\theta }$: $$(H_{\theta }q{)}_n=-(q_{n+1}+q_{n-1})+V(\theta +n\omega )q_n,n\in Z,$$ with $\omega \in R^d$ Diophantine, and $V$ a real-analytic function on $T^d=(R/2\pi Z{)}^d$. For $V$ sufficiently small, we prove the dispersive estimate: {for every $\varphi \in {\ell }^1(Z)$,} $$\Vert e^{-\mathrm{i}t{H}_{\theta }}\varphi {\Vert }_{{\ell }^{\infty }}\le K_0\frac{|\ln {\epsilon }_0{|}^{a(\ln \ln (2+⟨t⟩){)}^{2}d}}{⟨t{⟩}^{\frac{1}{3}}}\Vert \varphi {\Vert }_{{\ell }^1},⟨t⟩:=\sqrt{1+t^2},$$ {with $a$ and $K_0$ two absolute constants} and ${\epsilon }_0$ an analytic norm of $V$. The estimate holds for every $\theta \in T^d$.

References in corpus (3)

• The absolutely continuous spectrum of the almost Mathieu operator
• On the spectral theory and dispersive estimates for a discrete Schrödinger equation in one dimension
• On asymptotic stability of standing waves of discrete Schrödinger equation in $\mathbb{Z}$