Generalized Solutions and Spectrum for Dirichlet Forms on Graphs

Abstract

We study the connection of the existence of solutions with certain properties and the spectrum of operators in the framework of regular Dirichlet forms on infinite graphs. 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.

Volume growth and bounds for the essential spectrum for Dirichlet forms

“…We shortly discuss a strategy for approaching this case: we need a positive generalized harmonic function u, that is, E(u, ϕ) = 0 for all ϕ ∈ D, where u is assumed to be locally in the domain of E (this space is introduced in [5] as D * loc ). Such a function exists in many settings (see, for example, [3,11,21]); the result guaranteeing the existence of such a function is often referred to as an Allegretto-Piepenbrink-type theorem. Then, by a ground state representation (see [5,Theorem 10.1]), one obtains a form E u with vanishing killing term such that E = E u on the intersection of their domains.…”

“…Then, by a ground state representation (see [5,Theorem 10.1]), one obtains a form E u with vanishing killing term such that E = E u on the intersection of their domains. Now we can apply the methods above for E u to derive the result for E. However, as shown in [11], there are examples of non-locally finite weighted graphs that do not have such a generalized harmonic function. Therefore, it would be interesting to find sufficient conditions under which the approach above could be carried out.…”

Criticality theory for Schrödinger operators on graphs

“…However, once one has shown the existence of a positive harmonic function the results can be carried over using the ground state transform. Unfortunately, this can be guaranteed a priori only for locally finite graphs, see [10].…”

“…For q ≥ 0 and λ 0 (H (m) ) such a result is found in [10], and for λ ess 0 (H (m) ) see [15]. For q = 0 and locally finite graphs see [3], and for Dirichlet forms see [5,22].…”

“…Clearly, on finite graphs there are no positive (H − λ)-harmonic functions for λ < λ 0 (H (m) ) but only (H − λ)-superharmonic functions. In [10] an example is given that shows that in the non locally finite setting there might be only (H −λ)-superharmonic functions for λ < λ 0 (H (m) ) either.…”