Agmon estimates for Schrödinger operators on graphs

Keller, Matthias; Pogorzelski, Felix

doi:10.48550/arxiv.2104.04737

Cited by 2 publications

(2 citation statements)

References 25 publications

(36 reference statements)

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“…There is also a recent work of Keller & Pogorzelski [16] who study Agmon estimates in the more general setting of weighted, infinite graphs where the Agmon distance is given in terms of Hardy weights. We also note the work of Akduman & Pankov [3,4] on metric graphs, the work of Harrell & Maltsev [11] on quantum graphs, Damanik, Fillman & Sukhtaiev [7] on tree graphs and Klein & Rosenberger [17,18], Mandich [20] and Wang & Zhang [24] on Z d .…”

Section: Related Resultsmentioning

confidence: 99%

An Agmon estimate for Schrödinger operators on Graphs

Steinerberger¹

2022

Preprint

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The Agmon estimate shows that eigenfunctions of Schrödinger operators, −∆φ + V φ = Eφ, decay exponentially in the 'classically forbidden' region where the potential exceeds the energy level {x : V (x) > E}. Moreover, the size of |φ(x)| is bounded in terms of a weighted (Agmon) distance between x and the allowed region. We derive such a statement on graphs when −∆ is replaced by the Graph Laplacian L = D − A: we identify an explicit Agmon metric and prove a pointwise decay estimate in terms of the Agmon distance.

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Section: Related Resultsmentioning

confidence: 99%

An Agmon estimate for Schrödinger operators on Graphs

Steinerberger¹

2022

Preprint

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“…In a sense, Agmon's technique may be regarded as an elliptic version of Davies' perturbation method, commonly used to obtain Gaussian heat kernel bounds, see [7,8] and references therein. Recently, Agmon's method has been transferred into the graph setting to study eigenfunctions of discrete Schrödinger operators in [43].…”

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confidence: 99%

First passage percolation with long-range correlations and applications to random Schrödinger operators

Andrés¹,

Prévost²

2021

Preprint

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We consider first passage percolation (FPP) with passage times generated by a general class of models with long-range correlations on Z d , d ≥ 2, including discrete Gaussian free fields, Ginzburg-Landau ∇φ interface models or random interlacements as prominent examples. We show that the associated time constant is positive, the FPP distance is comparable to the Euclidean distance, and we obtain a shape theorem. We also present two applications for random conductance models (RCM) with possibly unbounded and strongly correlated conductances. Namely, we obtain a Gaussian heat kernel upper bound for RCMs with a general class of speed measures, and an exponential decay estimate for the Green function of RCMs with random killing measures.

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Agmon estimates for Schrödinger operators on graphs

Abstract: We prove decay estimates for generalized eigenfunctions of discrete Schrödinger operators on weighted infinite graphs in the spirit of Agmon.

Cited by 2 publications

References 25 publications

An Agmon estimate for Schrödinger operators on Graphs

An Agmon estimate for Schrödinger operators on Graphs

First passage percolation with long-range correlations and applications to random Schrödinger operators

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