The landscape function on ${\mathbb{R}}^d$

Consider the Schrödinger operator $-\mathrm{△}+\lambda V$ with non-negative iid random potential $V$ of strength $\lambda >0$. 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 $-\mathrm{△}+\lambda V$ using Agmon’s positivity method, rank-one perturbation in dimensions $d\ge 3$, and first-passage percolation in dimensions $d=1,2$.