Riesz decompositions for Schrödinger operators on graphs

Fischer, Florian; Keller, Matthias

doi:10.1016/j.jmaa.2020.124674

Cited by 5 publications

(4 citation statements)

References 27 publications

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“…Remark 2.10. These Riesz decompositions are the fundamental tools to develop a Choquet-Martin boundary theory which generalises the existing theory in the probabilistic case, see [Woe00], to all graphs with corresponding subcritical Schrödinger operator, see [Fis18,Chapter 5]. For instance, Theorem 2.4 is the crucial step to get the so-called discrete Poisson-Martin integral representation.…”

Section: Resultsmentioning

confidence: 99%

Riesz Decompositions for Schrödinger Operators on Graphs

Fischer,

Keller

2019

Preprint

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We study superharmonic functions for Schrödinger operators on general weighted graphs. Specifically, we prove two decompositions which both go under the name Riesz decomposition in the literature. The first one decomposes a superharmonic function into a harmonic and a potential part. The second one decomposes a superharmonic function into a sum of superharmonic functions with certain upper bounds given by prescribed superharmonic functions. As application we show a Brelot type theorem.

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Section: Resultsmentioning

confidence: 99%

Riesz Decompositions for Schrödinger Operators on Graphs

Fischer,

Keller

2019

Preprint

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“…Trofimov [54] proved that every infinite locally finite vertex-transitive graph has a nonconstant harmonic function, which grows at most exponentially with respect to the distance to a base point. Tointon [8,10,17,20,43,51]. A Schrödinger operator with a nonnegative potential on a finite digraph is either a Laplacian matrix or a perturbed Laplacian matrix.…”

Section: Schrödinger Operators On Countable Digraphsmentioning

confidence: 99%

Three conjectures of Ostrander on digraph Laplacian eigenvectors

Zhao

2021

Art Discrete Appl. Math.

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Ostrander proposed three conjectures on the connections between topological properties of a weighted digraph and combinatorial properties of its Laplacian eigenvectors. We verify one of his conjectures, give counterexamples to the other two, and then seek for related valid connections and generalizations to Schrödinger operators on countable digraphs.We suggest the open question of deciding if the countability assumption can be dropped from our main results.

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“…In the present paper we exploit the fact that they have joint harmonic functions, see Section 3.1. This idea has been used very recently by Fischer and Keller (2021) in the study of the Riesz decomposition for superharmonic functions of graph Laplacians.…”

Section: Introductionmentioning

confidence: 99%

Decay of harmonic functions for discrete time Feynman–Kac operators with confining potentials

Cygan¹,

Kaleta²,

Mateusz³

2022

ALEA

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We propose and study a certain discrete time counterpart of the classical Feynman-Kac semigroup with a confining potential in a countably infinite space. For a class of long range Markov chains which satisfy the direct step property we prove sharp estimates for functions which are (sub-, super-)harmonic in infinite sets with respect to the discrete Feynman-Kac operators. These results are compared with respective estimates for the case of a nearest-neighbour random walk which evolves on a graph of finite geometry. We also discuss applications to the decay rates of solutions to equations involving graph Laplacians and to eigenfunctions of the discrete Feynman-Kac operators. We include such examples as non-local discrete Schrödinger operators based on fractional powers of the nearest-neighbour Laplacians and related quasi-relativistic operators. Finally, we analyse various classes of Markov chains which enjoy the direct step property and illustrate the obtained results by examples.

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Riesz decompositions for Schrödinger operators on graphs

Cited by 5 publications

References 27 publications

Riesz Decompositions for Schrödinger Operators on Graphs

Riesz Decompositions for Schrödinger Operators on Graphs

Three conjectures of Ostrander on digraph Laplacian eigenvectors

Decay of harmonic functions for discrete time Feynman–Kac operators with confining potentials

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