Decay estimates for discrete bi-Schr\"{o}dinger operators on the lattice Z

It is known that the discrete Laplace operator Δ on the lattice Z satisfies the following sharp time decay estimate: \|eitΔ\|_¹→∞|t|-1/3, t≠0, which is slower than the usual |t|-1/2 decay in the continuous case on R. However in this paper, we have showed that the discrete bi-Laplacian Δ² on Z actually exhibits the same sharp decay estimate |t|-1/4 as its continuous counterpart. In view of these free decay estimates, this paper further investigates the discrete bi-Schr\"{o}dinger operators of the form H=Δ²+V on the lattice space ²(Z), where $V(n)$ is a real valued potential of 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 ¹-∞ dispersive estimates: \|e-itHPac(H)\|_¹→∞|t|-1/4, t≠0, where Pac(H) denotes the spectral projection onto the absolutely continuous spectrum space of H. Additionally, the following decay estimates for beam equation are also derived: \| cos(t√ H)Pac(H)\|_¹→∞+\| sin(t√ H)t√ HPac(H)\|_¹→∞|t|-1/3, t≠0.