Dispersive estimates for periodic discrete one-dimensional Schrödinger operators

Mi, Yue; Zhao, Zhiyan

doi:10.1090/proc/15699

Dispersive estimates for periodic discrete one-dimensional Schrödinger operators
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by Yue Mi and Zhiyan Zhao PDF

Proc. Amer. Math. Soc. 150 (2022), 267-277 Request permission

Abstract:

In this paper, we consider the periodic discrete one-dimensional Schrödinger operator \begin{equation*} (H_V\psi )_n=-(\psi _{n+1}+\psi _{n-1})+V_n\psi _n,\quad n\in \mathbb {Z}, \end{equation*} with $V_{n+N}=V_n$, $\forall \ n\in \mathbb {Z}$, for some $N\in \mathbb {N}^*$ and $V=\{V_j\}_{j=0}^{N-1}\in \mathbb {R}$. We show the dispersive estimate \begin{equation*} \|e^{-\mathrm {i}tH_V}\psi \|_{\ell ^\infty }\lesssim \|\psi \|_{\ell ^1} (1+|t|)^{-\min \left \{\frac 13,\frac 1{N+1}\right \}},\quad \forall \ \psi \in \ell ^1(\mathbb {Z}),\quad \forall \ t\in \mathbb {R}. \end{equation*}

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