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

Abstract

It is known that the discrete Laplace operator $\Delta$ on the lattice $\mathbb{Z}$ satisfies the following sharp time decay estimate: $${\Vert e^{it\Delta }\Vert }_{{\ell }^1\to {\ell }^{\infty }}\lesssim |t{|}^{-\frac{1}{3}},t\ne 0,$$ which is slower than the usual $|t{|}^{-\frac{1}{2}}$ decay in the continuous case on $\mathbb{R}$. However in this paper, we have showed that the discrete bi-Laplacian ${\Delta }^2$ on $\mathbb{Z}$ actually exhibits the same sharp decay estimate $|t{|}^{-\frac{1}{4}}$ as its continuous counterpart. In view of these free decay estimates, this paper further investigates the discrete bi-Schrödinger operators of the form $H={\Delta }^2+V$ on the lattice space ${\ell }^2(\mathbb{Z})$, 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 ${\ell }^1-{\ell }^{\infty }$ dispersive estimates: $${\Vert e^{-itH}{P}_{ac}(H)\Vert }_{{\ell }^1\to {\ell }^{\infty }}\lesssim |t{|}^{-\frac{1}{4}},t\ne 0,$$ 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: $$\Vert \cos (t\sqrt{H})P_{ac}(H){\Vert }_{{\ell }^1\to {\ell }^{\infty }}+{\Vert \frac{\sin (t\sqrt{H})}{t\sqrt{H}}{P}_{ac}(H)\Vert }_{{\ell }^1\to {\ell }^{\infty }}\lesssim |t{|}^{-\frac{1}{3}},t\ne 0.$$

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