Sharp Decay estimates for the discrete Fourth-Order Schrödinger Equation

We establish sharp time-decay estimates for the discrete fourth-order Schrödinger equation on lattices of arbitrary dimension, extending the author’s previous work \cite{C24}. In higher dimensions, scalar Fourier factorization reduces the analysis to product estimates for one-dimensional oscillatory integrals. Since this method does not yield the sharp decay exponents in low dimensions, we also use Newton polyhedra to obtain the required uniform estimates. In particular, the sharp decay rate for the biharmonic Schrödinger propagator is $\mathrm{\mid }t{\mathrm{\mid }}^{-d\mathrm{/}4}$.