Resolvent Expansions at All Spectral Thresholds for Three-Dimensional Discrete Schrödinger Operators

This paper studies resolvent expansions at all four spectral thresholds for the discrete Schrödinger operator $H=-\mathrm{\Delta }+V$ on ${\mathbb{Z}}^3$, where $\mathrm{\Delta }$ denotes the discrete Laplacian on ${\mathbb{Z}}^3$ and $V(n)$ is a real-valued decaying potential of ${\mathbb{Z}}^3$. At the thresholds 0, 4, 8, and 12, we first give a complete classification of the threshold types and derive the corresponding resolvent expansions in weighted ${\mathrm{\ell }}^2$-spaces, including the non-generic cases of resonances and eigenvalues. As a result, we establish the Jensen-Kato type time-decay estimates of the evolution $e^{itH}{P}_{ac}(H)$ to discrete Schrödinger equations with thresholds singularities on ${\mathbb{Z}}^3$, with decay $O(t^{-3\mathrm{/}2})$ when all thresholds are regular and $O(t^{-1\mathrm{/}2})$ otherwise. These results provide a three-dimensional lattice analogue of the Jensen–Kato classical analysis at zero energy threshold on ${\mathbb{R}}^3$.