Boundary representation of Dirichlet forms on discrete spaces

Keller, Matthias; Lenz, Daniel; Schmidt, Marcel; Schwarz, Michael

doi:10.1016/j.matpur.2018.10.005

Cited by 7 publications

(18 citation statements)

References 35 publications

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“…Clearly, the forms t G and t (N ) G coincide if and only if H is the unique positive self-adjoint extension of H 0 (this in particular holds if H 0 is essentially self-adjoint). We are not aware of a description of the self-adjoint operator H N associated with the form t (N ) G if the pre-minimal operator has nontrivial deficiency indices (however, see the recent work [13,38]). Moreover, to the best of our knowledge, the description of deficiency indices of H 0 and its self-adjoint extensions is a widely open problem.…”

Section: )mentioning

confidence: 99%

Spectral estimates for infinite quantum graphs

Kostenko

Nicolussi

2018

Calc. Var.

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We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Our definition of the isoperimetric constant is purely combinatorial and thus it establishes connections with the combinatorial isoperimetric constant, one of the central objects in spectral graph theory and in the theory of simple random walks on graphs. The latter enables us to prove a number of criteria for quantum graphs to be uniformly positive or to have purely discrete spectrum. We demonstrate our findings by considering trees, antitrees and Cayley graphs of finitely generated groups.2010 Mathematics Subject Classification. Primary 34B45; Secondary 35P15; 81Q35.

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Section: )mentioning

confidence: 99%

Spectral estimates for infinite quantum graphs

Kostenko

Nicolussi

2018

Calc. Var.

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“…Moreover, in this case the end compactification G of G coincides with several other spaces, among them the metric completion of G and the Royden compactification of a related discrete graph (see [27,Corollary 4.22] and also [26, p. 1526]). Let us also stress that the latter was employed recently in a description of Markovian extensions of discrete Laplacians [41]. The metric completion G was considered in connection with quantum graphs in [10,11]; however, G can have a rather complicated structure if vol(G) = ∞ and a further analysis usually requires additional assumptions.…”

Section: Introductionmentioning

confidence: 99%

“…Let us finish this introduction with two more comments concerning Theorem 6.11. First of all, results related to Theorem 6.11 were proven recently in [41], which provides a description of Markovian extensions for discrete Laplacians on graphs in terms of Royden's boundary. On the one hand, taking into account certain close relationships between quantum graphs and discrete Laplacians (see [20, §4]), one can easily obtain the results analogous to Theorem 4.1 and Theorem 6.11 for a particular class of discrete Laplacians on G d defined by the following expression…”

Section: Introductionmentioning

confidence: 99%

Self-adjoint and Markovian extensions of infinite quantum graphs

Kostenko,

Mugnolo,

Nicolussi

2019

Preprint

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We investigate self-adjoint extensions of the minimal Kirchhoff Laplacian on an infinite metric graph. More specifically, the main focus is on the relationship between graph ends and the space of self-adjoint extensions of the corresponding minimal Kirchhoff Laplacian H 0 . First, we introduce the notion of finite and infinite volume for (topological) ends of a metric graph and then establish a lower bound on the deficiency indices of H 0 in terms of the number of finite volume graph ends. This estimate is sharp and we also find a necessary and sufficient condition for the equality between the number of finite volume graph ends and the deficiency indices of H 0 to hold. Moreover, it turns out that finite volume graph ends play a crucial role in the study of Markovian extensions of H 0 . In particular, we show that the minimal Kirchhoff Laplacian admits a unique Markovian extension exactly when every topological end of the underlying metric graph has infinite volume. In the case of finitely many finite volume ends (for instance, the latter includes Cayley graphs of a large class of finitely generated infinite groups) we are even able to provide a complete description of all Markovian extensions of H 0 .

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“…Since this type of construction is essentially known, see e.g. [3, Proof of Theorem 6.6.5] or [11] for Dirichlet forms on graphs, we only give a brief sketch and leave details to the reader.…”

Section: Reflected Regular Dirichlet Formsmentioning

confidence: 99%

A Note on Reflected Dirichlet Forms

Schmidt

2018

Potential Anal

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In this paper we give an algebraic construction of the (active) reflected Dirichlet form. We prove that it is the maximal Silverstein extension whenever the given form does not possess a killing part and we prove that Dirichlet forms need not have a maximal Silverstein extension if a killing is present. For regular Dirichlet forms we provide an alternative construction of the reflected process on a compactification (minus one point) of the underlying space.

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Boundary representation of Dirichlet forms on discrete spaces

Abstract: We describe the set of all Dirichlet forms associated to a given infinite graph in terms of Dirichlet forms on its Royden boundary. Our approach is purely analytical and uses form methods.

Cited by 7 publications

References 35 publications

Spectral estimates for infinite quantum graphs

Spectral estimates for infinite quantum graphs

Self-adjoint and Markovian extensions of infinite quantum graphs

A Note on Reflected Dirichlet Forms

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