The landscape function on $\mathbb R^d$

TL;DR

This paper proves the existence, uniqueness, and exponential decay of the landscape function for a Schrödinger operator with random potential on  space, using probabilistic and analytical techniques.

Contribution

It establishes the existence and decay properties of the landscape function for random Schrödinger operators on  space, a novel result in mathematical physics.

Findings

Existence and uniqueness of the landscape function on  space.

Exponential decay of correlations and Green function.

Application of Agmon's method, rank-one perturbation, and first-passage percolation.

Abstract

Consider the Schr\"odinger operator $-\triangle+\lambda V$ 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 $-\triangle+\lambda V$ using Agmon's positivity method, rank-one perturbation in dimensions $d\ge 3$, and first-passage percolation in dimensions $d=1,2$.