2025/04/04 by Huang, Sisi, Yao, Xiaohua
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2504.03290
It is known that the discrete Laplace operator Δ on the lattice ℤ satisfies the following sharp time decay estimate: ‖eitΔ‖ℓ1→ℓ∞\lesssim|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-Schrödinger operators of the form H=Δ2+V on the lattice space ℓ2(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 ℓ1-ℓ∞ dispersive estimates: ‖e-itHPac(H)‖ℓ1→ℓ∞\lesssim|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: ‖\rm cos(t√ H)Pac(H)‖ℓ1→ℓ∞+‖\frac\rm sin(t√ H)t√ HPac(H)‖ℓ1→ℓ∞\lesssim|t|-(1)/(3), t≠0.