arXiv 0804.19632008-04-11math-phmath.MP

一维离散薛定谔方程的谱理论和色散估计

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

D. E. Pelinovsky, A. Stefanov

DOI：: 10.1063/1.3005597

摘要：

在最近关于紧致势的\cite{KKK}工作的基础上，我们发展了一维离散薛定谔算子$$ H ϕ= (-\De + V)ϕ=-(ϕ_{n+1} + ϕ_{n-1} - 2 ϕ_n) + V_n ϕ_n. $$的谱理论。（和在光谱边缘的非共振条件），$H$的光谱由200多个有限重数的本征值和基本的（绝对连续）谱，而预解式满足极限吸收原理和边缘附近的Pui-xix展开式。这些性质意味着对任何固定的$σ> {5/2}$和任何$t > 0$的色散估计$$ \|e^{i t H} P_{\rm a.c.}(H)\|_{l^2_σ \to l^2_{-σ}} \lesssim t^{-3/2} $$，其中$P_{\rm a.c.}(H)$表示$H$的绝对连续谱的谱投影。此外，基于离散Jost解的散射理论和\cite{SK}中的结果，我们得到了新的色散估计$$ \|e^{i t H} P_{\rm a.c.}(H) \|_{l^1\to l^\infty}\lesssim t^{-1/3}. $$这些估计对于离散Schr dinger算子甚至对于$V = 0$都是尖锐的.

英文摘要：

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 $$ \|e^{i t H} P_{\rm a.c.}(H)\|_{l^2_σ \to l^2_{-σ}} \lesssim t^{-3/2} $$ for any fixed $σ> {5/2}$ 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 Schrödinger operators even for $V = 0$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文