[Paper Review] $L^p$ continuity of wave operators in $\Bbb Z$

Scipio Cuccagna|ArXiv.org|Sep 16, 2008

Advanced Mathematical Physics Problems6 references3 citations

TL;DR

This paper establishes $L^p$ boundedness and isomorphism properties of wave operators $W$ and $Z$ for discrete Schrödinger operators on $\mathbb{Z}$, showing that $W: \ell^p \to \ell^p_c(H)$ and $Z: \ell^p_c(H) \to \ell^p$ extend as isomorphisms for $1 < p < \infty$ when $q \in \ell^{1,1}$ without resonances at 0 or 4, and for $q \in \ell^{1,2}$ when resonances are present. For $p=1,\infty$, the operators extend boundedly if and only if the transmission coefficients satisfy $T(0) = T(\pi) = 1$, under stronger decay $q \in \ell^{1,2+\sigma}$ with $\sigma > 0$. The results extend dispersive estimates and generalize continuous-space theory to the discrete setting using Jost function techniques and Fourier analysis on the torus.

ABSTRACT

We recover for discrete Schrödinger operators on the lattice Z, stronger analogues of the results by Weder, Galtabiar and Yajima and by D'Ancona and Fanelli on R.

Motivation & Objective

• To establish the $L^p$ boundedness and isomorphism properties of wave operators $W$ and $Z$ for discrete Schrödinger operators on $\mathbb{Z}$.
• To determine the minimal decay conditions on the potential $q$ for the wave operators to extend boundedly to $\ell^p$ spaces, particularly for $p=1$ and $p=\infty$.
• To characterize the role of resonances at the spectral edges $\lambda=0$ and $\lambda=4$ in the boundedness of wave operators.
• To extend the theory of wave operators from the continuous case ($\mathbb{R}$) to the discrete case ($\mathbb{Z}$), using analogous Jost function and Fourier analysis techniques.
• To provide dispersive estimates for solutions of the Klein-Gordon equation $u_{tt} + Hu + m^2u = 0$ via the $\ell^1 \to \ell^\infty$ boundedness of $W$ and $Z$.

Proposed method

• The wave operators $W$ and $Z$ are defined as strong limits of $e^{itH}e^{it\Delta}$ and $e^{-it\Delta}e^{-itH}$ in $\ell^2$, and their $L^p$ extensions are analyzed via spectral and Fourier-analytic techniques.
• The paper uses the Fourier transform $F_0[u](\theta) = \frac{1}{\sqrt{2\pi}} \sum_{n \in \mathbb{Z}} e^{-in\theta} u(n)$ to analyze the spectral properties of $H$ and relate them to the transmission coefficient $T(\theta)$.
• Jost functions $f_\pm(n,\theta)$ are used to construct solutions to the eigenvalue equation $Hu = zu$, with asymptotic behavior $f_\pm(n,\theta) \sim e^{\mp in\theta}$ as $n \to \pm\infty$, and their generating functions $m_\pm(n,\theta)$ are analyzed via Neumann series.
• The wave operator $W$ is decomposed into four components $V_1, V_2, V_3, V_4$, each involving integrals over $\theta \in \mathbb{T}$, with coefficients depending on $m_\pm(\mu,\theta)$, $T(\theta)$, and boundary terms at $\theta = 0, \pi$.
• Boundedness of $W$ on $\ell^1$ and $\ell^\infty$ is shown to be equivalent to $T(0) = T(\pi) = 1$, derived from the vanishing of the $V_j$ terms and the structure of the resolvent at spectral edges.
• The proof relies on estimates from [CT], [GS], and [DT], adapted to the discrete setting, and uses the fact that $q \in \ell^{1,\sigma}$ implies decay of the resolvent kernel and spectral projections.

Experimental results

Research questions

• RQ1 Under what conditions on the potential $q$ do the wave operators $W$ and $Z$ extend as bounded operators on $\ell^p$ for $1 < p < \infty$?
• RQ2 What is the minimal decay rate of $q$ required for $W$ and $Z$ to be bounded on $\ell^1$ and $\ell^\infty$?
• RQ3 How do resonances at $\lambda = 0$ and $\lambda = 4$ affect the $L^p$ boundedness of the wave operators?
• RQ4 What role does the transmission coefficient $T(\theta)$ play in the $\ell^1 \to \ell^\infty$ boundedness of $W$?
• RQ5 Can the discrete wave operator theory on $\mathbb{Z}$ be made equivalent to the continuous theory on $\mathbb{R}$ in terms of $L^p$ boundedness and dispersive estimates?

Key findings

• For $q \in \ell^{1,1}$ and no resonances at $\lambda = 0,4$, the wave operators $W$ and $Z$ extend as isomorphisms from $\ell^p$ to $\ell^p_c(H)$ for all $1 < p < \infty$.
• When $H$ has resonances at $\lambda = 0$ or $\lambda = 4$, the wave operators extend as isomorphisms for $q \in \ell^{1,2}$ and $1 < p < \infty$.
• For $p = 1$ and $p = \infty$, the wave operators extend boundedly if and only if $T(0) = T(\pi) = 1$, under the stronger assumption $q \in \ell^{1,2+\sigma}$ with $\sigma > 0$.
• The $\ell^1 \to \ell^\infty$ boundedness of $W$ is optimal and implies dispersive estimates for the Klein-Gordon equation with $m^2 > 0$.
• The vanishing of the components $V_1, V_2, V_3, V_4$ in the operator decomposition is equivalent to $T(0) = T(\pi) = 1$, which is necessary and sufficient for $\ell^1$ and $\ell^\infty$ boundedness.
• The number of eigenvalues of $H$ is finite and bounded by $4 + \|\nu q(\nu)\|_{\ell^1}$ when $q \in \ell^{1,1}$, with the proof relying on a discrete Sturm oscillation theorem and sign-change counting.