Generalized Solutions and Spectrum for Dirichlet Forms on Graphs

By Sebastian Haeseler, Matthias Keller. 2010

76 citations recorded by the source corpus.

In the framework of regular Dirichlet forms we consider operators on infinite graphs. We study the connection of the existence of solutions with certain properties and the spectrum of the operators. In particular we prove a version of the Allegretto-Piepenbrink theorem which says that positive (super)-solutions to a generalized eigenvalue equation exist exactly for energies not exceeding the infimum of the spectrum. Moreover we show a version of Shnol’s theorem, which says that existence of solutions satisfying a growth condition with respect to a given boundary measure implies that the corresponding energy is in the spectrum.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6909e159d1218d62e2991c861d2d32682e07e9b1

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A note on self-adjoint extensions of the Laplacian on weighted graphs (2012, background): “lutions to (∆+λ)u = 0 for λ > 0. From this fact we can infer Theorems 1 and 2. 3.1. Leibniz rule and Green’s formula. The following auxiliary lemmas are well known in various other situations, see [11, 23, 24, 28, 29, 31, 32]. However, they do not hold on graphs without further assumptions. Here,”
Graphs of finite measure (2013, methodology): “e form on ℓ2(X,m) whose domain D = D(Q) satisﬁes Cc(X) ⊆ D ⊆ De∩ℓ2(X,m) and which satisﬁes Q(u,v) = Qe(u,v), for u,v ∈ D. For such a form we deﬁne Q(u) := Q(u,u) for u ∈ D and Q(u) := ∞ for u ∈ D. In [19, 20] an ‘integration by parts’ was shown that allows one to pair functions in Feand C c(X) via Q”
Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory (2010, background): “This basically unifies the corresponding results of [12, 17] for graphs and strongly local forms respectively.”
Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation (2011, background): “For a related study of basic spectral properties in terms of generalized solutions we refer the reader to [17].”