Relation between time-domain diffusion and resolvent Green’s function heuristics

Ascertain whether the Gaussian random-walk approximation for the long-time propagator |e^{itH}_{0x}|^2 and the spatial Green’s function scaling |R(E+iη)_{0x}|^2 ≈ λ^2 |x|^{2−d} (as η → 0) rigorously imply one another for the discrete Anderson Hamiltonian H = Δ + λV on Zd; in particular, determine whether either heuristic can be derived from the other under controlled assumptions.

Background

In discussing diffusive scaling limits, the notes present two related heuristics: a Gaussian random-walk picture for the propagator |e^{itH}_{0x}|^2 leading to r(t) ≈ λ^{-1} t^{1/2}, and a resolvent-based estimate |R(E+iη)_{0x}|^2 ≈ λ^2 |x|^{2−d}. These are often treated as complementary perspectives on diffusion/delocalization.

However, the authors point out that a rigorous implication in either direction is not currently established within the presented framework, highlighting a conceptual gap in connecting time-domain and resolvent-domain descriptions.