On the spectral theory and dispersive estimates for a discrete Schr\"{o}dinger equation in one dimension

TL;DR

This paper develops the spectral theory for a one-dimensional discrete Schrödinger operator with decaying potentials, establishing spectral properties and dispersive estimates that are sharp and extend previous results.

Contribution

It advances the spectral analysis and dispersive estimates for discrete Schrödinger operators, including new sharp decay rates and handling general potentials with decay conditions.

Findings

Spectrum consists of finitely many eigenvalues and absolutely continuous spectrum.

Dispersive decay estimate of order t^{-3/2} for the evolution operator.

New sharp dispersive estimate of order t^{-1/3} in the L^1 to L^∞ norm.

Abstract

Based on the recent work \cite{KKK} for compact potentials, we develop the spectral theory for the one-dimensional discrete Schr\"odinger operator $$ H \phi = (-\De + V)\phi=-(\phi_{n+1} + \phi_{n-1} - 2 \phi_n) + V_n \phi_n. $$ We show that under appropriate decay conditions on the general potential (and a non-resonance condition at the spectral edges), the spectrum of $H$ consists of finitely many eigenvalues of finite multiplicities and the essential (absolutely continuous) spectrum, while the resolvent satisfies the limiting absorption principle and the Puiseux expansions near the edges. These properties imply the dispersive estimates $$ \|e^{i t H} P_{\rm a.c.}(H)\|_{l^2_{\sigma} \to l^2_{-\sigma}} \lesssim t^{-3/2} $$ for any fixed $\sigma > {5/2}$ and any $t > 0$, where $P_{\rm a.c.}(H)$ denotes the spectral projection to the absolutely continuous spectrum of $H$. In addition, based on the scattering theory for the discrete Jost solutions and the previous results in \cite{SK}, we find new dispersive estimates $$ \|e^{i t H} P_{\rm a.c.}(H) \|_{l^1\to l^\infty}\lesssim t^{-1/3}. $$ These estimates are sharp for the discrete Schr\"{o}dinger operators even for $V = 0$.