Decay estimates for discrete bi-Laplace operators with potentials on the lattice $\mathbb{Z}$
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arXiv
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2025
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| _version_ | 1866918075027161088 |
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| author | Huang, Sisi Yao, Xiaohua |
| author_facet | Huang, Sisi Yao, Xiaohua |
| contents | It is known that the discrete Laplace operator $Δ$ on the lattice $\mathbb{Z}$ satisfies the following sharp time decay estimate: $$\big\|e^{itΔ}\big\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{3}},\quad t\neq0,$$ which is slower than the usual $ O(|t|^{-\frac{1}{2}})$ decay in the continuous case on $\mathbb{R}$. However, this paper shows that the discrete bi-Laplacian $Δ^2$ on $\mathbb{Z}$ actually exhibits the same sharp decay estimate $|t|^{-\frac{1}{4}}$ as its continuous counterpart.
In view of the free decay estimate, we further investigate the discrete bi-Schrödinger operators of the form $H=Δ^2+V$ on the lattice space $\ell^2(\mathbb{Z})$, where $V$ is a class of real-valued decaying potentials on $\mathbb{Z}$. First, we establish the limiting absorption principle for $H$, and then derive the full asymptotic expansions of the resolvent of $H$ near the thresholds $0$ and $16$, including resonance cases. In particular, we provide a complete characterizations of the different resonance types in $\ell^2$-weighted spaces.
Based on these results above, we establish the following sharp $\ell^1-\ell^{\infty}$ decay estimates for all different resonances types of $H$ under suitable decay conditions on $V$: $$\big\|e^{-itH}P_{ac}(H)\big\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{4}},\quad t\neq0,$$ where $P_{ac}(H)$ denotes the spectral projection onto the absolutely continuous spectrum space of $H$. Additionally, the decay estimates for the evolution flow of discrete beam equation are also derived: $$\|{\cos}(t\sqrt H)P_{ac}(H)\|_{\ell^1\rightarrow\ell^{\infty}}+\Big\|\frac{{\sin}(t\sqrt H)}{t\sqrt H}P_{ac}(H)\Big\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{3}},\quad t\neq0.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23119 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decay estimates for discrete bi-Laplace operators with potentials on the lattice $\mathbb{Z}$ Huang, Sisi Yao, Xiaohua Analysis of PDEs It is known that the discrete Laplace operator $Δ$ on the lattice $\mathbb{Z}$ satisfies the following sharp time decay estimate: $$\big\|e^{itΔ}\big\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{3}},\quad t\neq0,$$ which is slower than the usual $ O(|t|^{-\frac{1}{2}})$ decay in the continuous case on $\mathbb{R}$. However, this paper shows that the discrete bi-Laplacian $Δ^2$ on $\mathbb{Z}$ actually exhibits the same sharp decay estimate $|t|^{-\frac{1}{4}}$ as its continuous counterpart. In view of the free decay estimate, we further investigate the discrete bi-Schrödinger operators of the form $H=Δ^2+V$ on the lattice space $\ell^2(\mathbb{Z})$, where $V$ is a class of real-valued decaying potentials on $\mathbb{Z}$. First, we establish the limiting absorption principle for $H$, and then derive the full asymptotic expansions of the resolvent of $H$ near the thresholds $0$ and $16$, including resonance cases. In particular, we provide a complete characterizations of the different resonance types in $\ell^2$-weighted spaces. Based on these results above, we establish the following sharp $\ell^1-\ell^{\infty}$ decay estimates for all different resonances types of $H$ under suitable decay conditions on $V$: $$\big\|e^{-itH}P_{ac}(H)\big\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{4}},\quad t\neq0,$$ where $P_{ac}(H)$ denotes the spectral projection onto the absolutely continuous spectrum space of $H$. Additionally, the decay estimates for the evolution flow of discrete beam equation are also derived: $$\|{\cos}(t\sqrt H)P_{ac}(H)\|_{\ell^1\rightarrow\ell^{\infty}}+\Big\|\frac{{\sin}(t\sqrt H)}{t\sqrt H}P_{ac}(H)\Big\|_{\ell^1\rightarrow\ell^{\infty}}\lesssim|t|^{-\frac{1}{3}},\quad t\neq0.$$ |
| title | Decay estimates for discrete bi-Laplace operators with potentials on the lattice $\mathbb{Z}$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.23119 |