Interior Spectral Windows and Transport for Discrete Fractional Laplacians on $d$-Dimensional Hypercubic Lattices

We study anisotropic fractional discrete Laplacians ${\mathrm{\Delta }}_{{\mathbb{Z}}^d}^{\vec{\mathbf{r}}}$ with exponents $\vec{\mathbf{r}}\in {\mathbb{R}}^d\setminus \{0\}$ on ${\mathrm{\ell }}^2({\mathbb{Z}}^d)$. We establish a Mourre estimate on compact energy intervals away from thresholds. As consequences we derive a Limiting Absorption Principle in weighted spaces, propagation estimates (minimal velocity and local decay), and the existence and completeness of local wave operators for perturbations $H={\mathrm{\Delta }}_{{\mathbb{Z}}^d}^{\vec{\mathbf{r}}}+W(Q)$, where $W$ is an anisotropically decaying potential of long–range type. In the stationary scattering framework we construct the on–shell scattering matrix $S(\lambda )$, prove the optical theorem, and, under a standard trace–class assumption on $W$, establish the Birman–Krein formula $\det S(\lambda )=\exp (-2\pi i\xi (\lambda ))$.