The discrete bi-Laplacian on Z has sharp |t|^{-1/4} decay, and with decaying potentials and no embedded positive eigenvalues, the continuous spectral part of the evolution still decays at the same rate.
Agmon, Spectral properties of Schr¨ odinger operators and scattering theory, Ann
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Decay estimates for discrete bi-Laplace operators with potentials on the lattice $\mathbb{Z}$
The discrete bi-Laplacian on Z has sharp |t|^{-1/4} decay, and with decaying potentials and no embedded positive eigenvalues, the continuous spectral part of the evolution still decays at the same rate.