The study establishes existence of landscape function and exponential decay of correlations in random potentials, highlighting key methods like Agmon’s positivity.
Consider the Schrödinger operator -▵+λV with non-negative iid random potential V of strength λ>0. We prove existence and uniqueness of the associated landscape function on the whole space, and show that its correlations decay exponentially. As a main ingredient we establish the (annealed and quenched) exponential decay of the Green function of -▵+λV using Agmon’s positivity method, rank-one perturbation in dimensions d≥3, and first-passage percolation in dimensions d=1,2.
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David et al. (2025) studied this question.
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