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arXiv · 2307.11182

The landscape function on $\mathbb R^d$

Guy David·Antoine Gloria·Svitlana Mayboroda
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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 $d=1,2$.

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Guy David, Antoine Gloria, Svitlana Mayboroda. 2023-07-20. The landscape function on $\mathbb R^d$. https://arxiv.org/abs/2307.11182

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