Decay estimates for discrete bi-Schrödinger operators 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 $\mathrm{\mid }t{\mathrm{\mid }}^{-\frac{1}{2}}$ decay in the continuous case on $\mathbb{R}$. However in this paper, we have showed 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 these free decay estimates, this paper further investigates the discrete bi-Schrödinger operators of the form $H={\mathrm{\Delta }}^2+V$ on the lattice space ${\mathrm{\ell }}^2(\mathbb{Z})$, where $V(n)$ is a real valued potential of $\mathbb{Z}$. Under suitable decay conditions on $V$ and assuming that both 0 and 16 are regular spectral points of $H$, we establish the following sharp ${\mathrm{\ell }}^1-{\mathrm{\ell }}^{\mathrm{\infty }}$ dispersive estimates: $${\parallel e^{-itH}{P}_{ac}(H)\parallel }_{{\mathrm{\ell }}^1\to {\mathrm{\ell }}^{\mathrm{\infty }}}\lesssim \mathrm{\mid }t{\mathrm{\mid }}^{-\frac{1}{4}},t\mathrm{\ne }0,$$ where $P_{ac}(H)$ denotes the spectral projection onto the absolutely continuous spectrum space of $H$. Additionally, the following decay estimates for beam equation are also derived: $$\mathrm{\parallel }\mathrm{c}\mathrm{o}\mathrm{s}(t\sqrt{H})P_{ac}(H){\mathrm{\parallel }}_{{\mathrm{\ell }}^1\to {\mathrm{\ell }}^{\mathrm{\infty }}}+{\parallel \frac{\mathrm{s}\mathrm{i}\mathrm{n}(t\sqrt{H})}{t\sqrt{H}}{P}_{ac}(H)\parallel }_{{\mathrm{\ell }}^1\to {\mathrm{\ell }}^{\mathrm{\infty }}}\lesssim \mathrm{\mid }t{\mathrm{\mid }}^{-\frac{1}{3}},t\mathrm{\ne }0.$$