[Paper Review] Weighted estimates for the discrete Laplacian on the cubic lattice

Evgeny Korotyaev, Jacob Schach Møller|arXiv (Cornell University)|Jan 13, 2017

Spectral Theory in Mathematical Physics14 references5 citations

TL;DR

This paper establishes weighted $\ell^q$ estimates for the discrete Schrödinger operator on the cubic lattice $\mathbb{Z}^d$, $d \geq 3$, by leveraging Bessel function representations of the propagator kernel and recent optimal bounds on Bessel functions. It derives improved dispersive and resolvent estimates beyond those implied by standard time-decay, enabling sharp spectral and scattering results for potentials in $\ell^p(\mathbb{Z}^d)$ with $p \geq 1$, particularly via Birman-Schwinger arguments.

ABSTRACT

We consider the discrete Laplacian $Δ$ on the cubic lattice $\mathbb Z^d$, and deduce estimates on the group $e^{itΔ}$ and the resolvent $(Δ-z)^{-1}$, weighted by $\ell^q(\mathbb Z^d)$-weights for suitable $q\geq 2$. We apply the obtained results to discrete Schrödinger operators in dimension $d\geq 3$ with potentials from $\ell^p(\mathbb Z^d)$ with suitable $p\geq 1$.

Motivation & Objective

• To derive improved dispersive estimates for the propagator $e^{it\Delta}$ on the cubic lattice $\mathbb{Z}^d$ with $d \geq 3$ using weighted $\ell^q$ norms.
• To establish sharper resolvent estimates for $(\Delta - z)^{-1}$ than those implied by dispersive estimates alone, exploiting pointwise decay of the resolvent kernel.
• To apply these estimates to discrete Schrödinger operators $H = \Delta + V$ with real-valued potentials $V \in \ell^p(\mathbb{Z}^d)$, $p \geq 1$, to analyze spectral and scattering properties.
• To extend classical Kato-type estimates from the continuous case to the discrete setting on $\mathbb{Z}^d$, particularly by analyzing the role of Bessel functions in the kernel representation.

Proposed method

• Represent the propagator kernel $e^{it\Delta}(n,m)$ as a product of Bessel functions using the momentum space representation of the discrete Laplacian.
• Apply recent optimal estimates on Bessel functions by Krasikov and Landau to control the pointwise decay of the kernel and derive weighted $\ell^q$ bounds.
• Use discrete Young's inequality and $\ell^p$-type convolution estimates to bound the action of weighted operators on the propagator and resolvent.
• Employ Birman-Schwinger type arguments to deduce spectral consequences, such as absence of embedded eigenvalues and bounds on the number of discrete eigenvalues.
• Analyze the resolvent kernel near threshold energies $\tau(H) = (2\mathbb{Z} + d) \cap [-d,d]$ using detailed asymptotic expansions in momentum space.
• Combine $L^p$-based estimates with interpolation and duality to derive uniform bounds in the time and spectral parameters.

Experimental results

Research questions

• RQ1 Can dispersive estimates for the discrete Laplacian on $\mathbb{Z}^d$ be improved by incorporating weighted $\ell^q$ norms on the potential?
• RQ2 Why do standard dispersive estimates fail to imply optimal resolvent bounds in the discrete case, unlike in the continuous case?
• RQ3 What role do Bessel functions play in the pointwise decay of the discrete propagator and resolvent kernels on $\mathbb{Z}^d$?
• RQ4 How can optimal estimates on Bessel functions be used to derive sharp $\ell^q$-weighted bounds for the discrete Schrödinger operator?
• RQ5 What spectral consequences follow from these refined estimates, particularly regarding embedded eigenvalues and the number of discrete eigenvalues?

Key findings

• The paper establishes a dispersive estimate of the form $\|u e^{it\Delta} u\| \leq C_{d,q} |t|^{-d/q} \|u\|_{\ell^q}^2$ for $q \geq 2$, with improved time-decay due to weighted $\ell^q$ norms.
• Resolvent estimates are shown to be strictly better than those implied by dispersive estimates, with bounds of the form $\|u(-\Delta - \lambda)^{-1}u\| \leq C_{d,\varepsilon}(\|u\|_{\ell^{d-\varepsilon}}^2 + \|u\|_{\ell^{d+\varepsilon}}^2)$ for $\lambda \in \mathbb{C} \setminus [0,\infty)$.
• The authors derive a sharp estimate for the convolution kernel involving $\ell^p$-weights, showing decay like $t^{-\beta}$ for $\beta < 1$, using discrete Young's inequality and Bessel function decay.
• For potentials $V \in \ell^p(\mathbb{Z}^d)$ with $p \geq 1$, the method yields bounds on the number of discrete eigenvalues and proves absence of embedded eigenvalues in $(-d,d)$, extending results from finite- and compactly supported potentials.
• The analysis reveals that the resolvent kernel's pointwise decay is critical for obtaining optimal estimates, and that this decay is governed by Bessel functions whose asymptotics are tightly controlled via recent inequalities.
• The paper confirms that the discrete case requires a more refined approach than the continuous case, as dispersive estimates alone do not yield optimal resolvent bounds, necessitating direct analysis of the kernel's decay structure.