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

We study anisotropic fractional discrete Laplacians ΔZᵈr⃗ with exponents r⃗ᵈ\0\ on ²(Zᵈ). 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=ΔZᵈr⃗+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 $S(λ)$, prove the optical theorem, and, under a standard trace--class assumption on W, establish the Birman--Krein formula S(λ)=exp(-2πi\,ξ(λ)).