Generalized Eigenfunctions and Spectral Theory for Strongly Local Dirichlet Forms

Lenz, Daniel; Stollmann, Peter; Veselić, Ivan

doi:10.1007/978-3-7643-9994-8_6

Cited by 10 publications

(12 citation statements)

References 57 publications

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“…Furthermore, using (8) we get f ∆(χ n Ψ)dµ = f Ψ∆χ n dµ + 2 (f dχ n , dΨ)dµ + Let us add some comments on Theorem 2.11: The property "C -positivity preservation" and thus also the BMS-conjecture both admit an equivalent formulation which makes sense on an arbitrary local Dirichlet space [23,17]. On the other hand, the concept of sequences of cut-off functions does not immediately extend to this setting, so that it would be very interesting to find a proof of the fact that geodesic completeness and a nonnegative Ricci curvature imply L q -positivity preservation for any q ∈ [1, ∞] which does not use sequences of cut-off functions.…”

Section: 4mentioning

confidence: 99%

Sequences of Laplacian Cut-Off Functions

Güneysu

2014

J Geom Anal

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Abstract. We derive several new applications of the concept of sequences of Laplacian cut-off functions on Riemannian manifolds (which we prove to exist on geodesically complete Riemannian manifolds with nonnegative Ricci curvature): In particular, we prove that this existence implies L q -estimates of the gradient, a new density result of smooth compactly supported functions in Sobolev spaces on the whole L q -scale, and a slightly weaker and slightly stronger variant of the conjecture of Braverman, Milatovic and Shubin on the nonnegativity of L 2 -solutions f of (−∆ + 1)f ≥ 0. The latter fact is proved within a new notion of positivity preservation for Riemannian manifolds which is related to stochastic completeness.

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Section: 4mentioning

confidence: 99%

Sequences of Laplacian Cut-Off Functions

Güneysu

2014

J Geom Anal

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“…Recently in [LSV] a result of this type was proven in the very general context of strongly local Dirichlet forms. For further discussion and references we refer to [LSV2]. In contrast to the situations discussed so far Dirichlet forms on discrete space are not strongly local.…”

Section: Introductionmentioning

confidence: 99%

“…There, the condition of subexponential growth is measured in terms of the so-called intrinsic metric. See also [LSV2] for further background and references. For quantum graphs a Shnol' theorem is proven in [Ku].…”

Section: Introductionmentioning

confidence: 99%

Generalized Solutions and Spectrum for Dirichlet Forms on Graphs

Haeseler

Keller

2011

Random Walks, Boundaries and Spectra

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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.

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“…(a) Under suitable assumptions it is possible to show a converse to the previous theorem i.e. every λ admitting an subexponentially bounded generalized eigenfunction must then belong to the spectrum of L. This type of result is known as Shnol theorem (see [8,14,34,35] for recent results of this type for operators arising from Dirichlet forms and further references).…”

Section: Metric Measure Spaces and Finer Growth Propertiesmentioning

confidence: 90%

Expansion in generalized eigenfunctions for Laplacians on graphs and metric measure spaces

Lenz

Teplyaev

2015

Trans. Amer. Math. Soc.

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Abstract. We consider an arbitrary selfadjoint operator in a separable Hilbert space. To this operator we construct an expansion in generalized eigenfunctions, in which the original Hilbert space is decomposed as a direct integral of Hilbert spaces consisting of general eigenfunctions. This automatically gives a Plancherel type formula. For suitable operators on metric measure spaces we discuss some growth restrictions on the generalized eigenfunctions. For Laplacians on locally finite graphs the generalized eigenfunctions are exactly the solutions of the corresponding difference equation. Contents

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Generalized Eigenfunctions and Spectral Theory for Strongly Local Dirichlet Forms

Abstract: Abstract. We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples.

Cited by 10 publications

References 57 publications

Sequences of Laplacian Cut-Off Functions

Sequences of Laplacian Cut-Off Functions

Generalized Solutions and Spectrum for Dirichlet Forms on Graphs

Expansion in generalized eigenfunctions for Laplacians on graphs and metric measure spaces

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