Uniform resolvent estimates for the discrete Schrödinger operator in dimension three
The author considers the discrete Laplacian \[ H_0 u (x) = - \sum_{|x-y| = 1} \left( u(y) - u(x) \right) \] acting on functions \(u \in \ell^2(\mathbb{Z}^3)\). Its main result are the following two statements: (i) Let \(p \in [1,5/4]\), \(r \in [1,10/3]\), and define for \(\epsilon > 0\) the set \(D_{\epsilon} = \cap_{k=1,2,3} \{ z \in \mathbb{C} \ | \ | z-4k | \geq \epsilon \}\). Then the resolvent estimates \begin{align*} \sup_{z \in D_{\epsilon} \backslash \mathbb{R} } \left\Vert \frac{1}{H_0-z} \right\Vert_{B(\ell^p(\mathbb{Z}^3)),\ell^{p'}(\mathbb{Z}^3))} < +\infty \quad \text{ and } \quad \sup_{z \in D_{\epsilon} \backslash \mathbb{R}, \ \Vert W_j \Vert_{\ell^r(\mathbb{Z}^3)} = 1 } \left\Vert W_1 \frac{1}{H_0-z} W_2 \right\Vert_{B(\ell^2(\mathbb{Z}^3))} < +\infty \end{align*} hold away from the threshold (\(1/p+1/p' = 1\)). (ii) Let \(p \in [1,6/5]\), \(r \in [1,3]\). Then the resolvent estimates \begin{align*} \sup_{z \in \mathbb{C} \backslash \mathbb{R} } \left\Vert \frac{1}{H_0-z} \right\Vert_{B(\ell^p(\mathbb{Z}^3)),\ell^{p'}(\mathbb{Z}^3))} < +\infty \quad \text{ and } \quad \sup_{z \in \mathbb{C} \backslash \mathbb{R}, \ \Vert W_j \Vert_{\ell^r(\mathbb{Z}^3)} = 1 } \left\Vert W_1 \frac{1}{H_0-z} W_2 \right\Vert_{B(\ell^2(\mathbb{Z}^3))} < +\infty \end{align*} hold near the threshold. For a previous result by the author we refer to [\textit{Y. Tadano} and \textit{K. Taira}, Trans. Am. Math. Soc. 372, No. 7, 5243--5262 (2019; Zbl 1422.81103)]. While the result in [loc. cit.] depends on an endpoint Strichartz estimate in [\textit{A. Stefanov} and \textit{P. G. Kevrekidis}, Nonlinearity 18, No. 4, 1841--1857 (2005; Zbl 1181.35266)], the proof of the above statement uses the strategy in [\textit{J.-C. Cuenin}, Ann. Henri Poincaré 20, No. 5, 1501--1516 (2019; Zbl 1412.35205)], that is, the Fourier decay of the surface measure for the Fermi surface is calculated. To that end, the author improves a result in [\textit{L. Erdős} and \textit{M. Salmhofer}, Math. Z. 257, No. 2, 261--294 (2007; Zbl 1122.42003)].
scientific article; zbMATH DE number 7484002
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