2025/06/29 by Huang, Sisi, Xiaohua Yao, Yao, Xiaohua
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2506.23119
openalex publication_date 2025/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 O(|t|-(1)/(2)) decay in the continuous case on ℝ. However, this paper shows 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 the free decay estimate, we further investigate the discrete bi-Schrödinger operators of the form H=Δ2+V on the lattice space ℓ2(ℤ), where V is a class of real-valued decaying potentials on 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 ℓ2-weighted spaces. Based on these results above, we establish the following sharp ℓ1-ℓ∞ decay estimates for all different resonances types of H under suitable decay conditions on V: ‖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 decay estimates for the evolution flow of discrete beam equation are also derived: ‖cos(t√ H)Pac(H)‖ℓ1→ℓ∞+‖\fracsin(t√ H)t√ HPac(H)‖ℓ1→ℓ∞\lesssim|t|-(1)/(3), t≠0.