On the spectral theory and dispersive estimates for a discrete Schrödinger equation in one dimension

Abstract

Based on the recent work \cite{KKK} for compact potentials, we develop the spectral theory for the one-dimensional discrete Schrödinger operator $$ H ϕ= (-\De + V)ϕ=-(ϕ_{n+1} + ϕ_{n-1} - 2 ϕ_n) + V_n ϕ_n. $$ 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 $$\Vert e^{itH}{P}_{a.c.}(H){\Vert }_{l_{\sigma }^2\to l_{-\sigma }^2}\lesssim t^{-3/2}$$ for any fixed $\sigma >5/2$ and any $t>0$, where $P_{a.c.}(H)$ 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 $$\Vert e^{itH}{P}_{a.c.}(H){\Vert }_{l^1\to l^{\infty }}\lesssim t^{-1/3}.$$ These estimates are sharp for the discrete Schrödinger operators even for $V=0$.

References in corpus (4)

• Dispersive estimates for Schrodinger operators in dimensions one and three
• Excitation Thresholds for Nonlinear Localized Modes on Lattices
• Asymptotic behavior of small solutions for the discrete nonlinear Schrödinger and Klein-Gordon equations
• Asymptotic stability of small solitons to 1D NLS with potential

Cited by in corpus (17)

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• Asymptotic stability of small bound state of nonlinear quantum walks
• Properties of the Scattering Matrix and Dispersion Estimates for Jacobi Operators
• Existence and asymptotic stability of quasi-periodic solution of discrete NLS with potential in $\mathbb{Z}$
• $L^p$ continuity of wave operators in $\mathbb{Z}$
• Solitons and Gibbs measures for nonlinear Schroedinger equations
• Dispersive Properties for Discrete Schrodinger Equations
• Dispersive estimate for quasi-periodic Schrödinger operators on 1-$d$ lattices
• Stabilization of small solutions of discrete NLS with potential having two eigenvalues
• Large-time asymptotic behavior of the infinite system of harmonic oscillators
• A complete classification of threshold properties for one-dimensional discrete Schrödinger operators
• Dispersion estimates for 1D discrete equations
• Orbitally but not asymptotically stable ground states for the discrete NLS