Decay estimates for discrete bi-Laplace operators with potentials on the lattice $\mathbb{Z}$

It is known that the discrete Laplace operator $\mathrm{\Delta }$ on the lattice $\mathbb{Z}$ satisfies the following sharp time decay estimate: $$\parallel e^{it\mathrm{\Delta }}{\parallel }_{{\mathrm{\ell }}^1\to {\mathrm{\ell }}^{\mathrm{\infty }}}\lesssim \mathrm{\mid }t{\mathrm{\mid }}^{-\frac{1}{3}},t\mathrm{\ne }0,$$ which is slower than the usual $O(\mathrm{\mid }t{\mathrm{\mid }}^{-\frac{1}{2}})$ decay in the continuous case on $\mathbb{R}$. However, this paper shows that the discrete bi-Laplacian ${\mathrm{\Delta }}^2$ on $\mathbb{Z}$ actually exhibits the same sharp decay estimate $\mathrm{\mid }t{\mathrm{\mid }}^{-\frac{1}{4}}$ as its continuous counterpart. In view of the free decay estimate, we further investigate the discrete bi-Schrödinger operators of the form $H={\mathrm{\Delta }}^2+V$ on the lattice space ${\mathrm{\ell }}^2(\mathbb{Z})$, 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 ${\mathrm{\ell }}^2$-weighted spaces. Based on these results above, we establish the following sharp ${\mathrm{\ell }}^1-{\mathrm{\ell }}^{\mathrm{\infty }}$ decay estimates for all different resonances types of $H$ under suitable decay conditions on $V$: $$\parallel e^{-itH}{P}_{ac}(H){\parallel }_{{\mathrm{\ell }}^1\to {\mathrm{\ell }}^{\mathrm{\infty }}}\lesssim \mathrm{\mid }t{\mathrm{\mid }}^{-\frac{1}{4}},t\mathrm{\ne }0,$$ 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: $$\mathrm{\parallel }\cos (t\sqrt{H})P_{ac}(H){\mathrm{\parallel }}_{{\mathrm{\ell }}^1\to {\mathrm{\ell }}^{\mathrm{\infty }}}+\parallel \frac{\sin (t\sqrt{H})}{t\sqrt{H}}{P}_{ac}(H){\parallel }_{{\mathrm{\ell }}^1\to {\mathrm{\ell }}^{\mathrm{\infty }}}\lesssim \mathrm{\mid }t{\mathrm{\mid }}^{-\frac{1}{3}},t\mathrm{\ne }0.$$