Sharp time decay estimates for the discrete Klein-Gordon equation

We establish sharp time decay estimates for the Klein–Gordon equation on the cubic lattice in dimensions d = 2, 3, 4. The ℓ¹ → ℓ^∞ dispersive decay rate is |t|^−3/4 for d = 2, |t|^−7/6 for d = 3 and |t|^−3/2 log|t| for d = 4. These decay rates are faster than conjectured by Kevrekidis and Stefanov (2005). The proof relies on oscillatory integral estimates and proceeds by a detailed analysis of the singularities of the associated phase function. We also prove new Strichartz estimates and discuss applications to nonlinear PDEs and spectral theory.