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The landscape function on $\mathbb R^d$

Year: 2023
Type: preprint
Abstract: Consider the Schrödinger operator $-\triangle+λV$ 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 $-\triangle+λV$ using Agmon's positivity method, rank-one perturbation in dimensions $d\ge 3$, and first-passage percolation in dimensions $d=1,2$.
DOI: 10.48550/arxiv.2307.11182
Institutions: Université Paris-Saclay, Laboratoire de Mathématiques d'Orsay, Laboratoire de Mathématiques Blaise Pascal, European Union, Université Libre de Bruxelles, Institut Universitaire de France, Sorbonne Université, University of Minnesota
Language: English
Open Access status: green
Awards: G280662651, Geometric and Harmonic Analysis with Interdisciplinary Applications, G434337789, G4937468798, RAISE-TAQS: The Hidden Structure of the Disorder in Quantum Systems, G7641127039, G7839567693, Corrector equations and random operators
OpenAlex ID: W4385208660
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