A note on self-adjoint extensions of the Laplacian on weighted graphs

Xueping Huang; Matthias Keller; Jun Masamune; Radosław K. Wojciechowski

doi:10.1016/j.jfa.2013.06.004

A note on self-adjoint extensions of the Laplacian on weighted graphs

Xueping Huang, Matthias Keller, Jun Masamune, Radosław K. Wojciechowski

SCI - Mathematics

Research output: Contribution to journal › Article › peer-review

59 Citations (Scopus)

Abstract

We study the uniqueness of self-adjoint and Markovian extensions of the Laplacian on weighted graphs. We first show that, for locally finite graphs and a certain family of metrics, completeness of the graph implies uniqueness of these extensions. Moreover, in the case when the graph is not metrically complete and the Cauchy boundary has finite capacity, we characterize the uniqueness of the Markovian extensions.

Original language	English
Pages (from-to)	1556-1578
Number of pages	23
Journal	Journal of Functional Analysis
Volume	265
Issue number	8
DOIs	https://doi.org/10.1016/j.jfa.2013.06.004
Publication status	Published - 2013 Oct 15

Keywords

Essential self-adjointness
Intrinsic metrics
Laplacians
Weighted graphs

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10.1016/j.jfa.2013.06.004

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title = "A note on self-adjoint extensions of the Laplacian on weighted graphs",

abstract = "We study the uniqueness of self-adjoint and Markovian extensions of the Laplacian on weighted graphs. We first show that, for locally finite graphs and a certain family of metrics, completeness of the graph implies uniqueness of these extensions. Moreover, in the case when the graph is not metrically complete and the Cauchy boundary has finite capacity, we characterize the uniqueness of the Markovian extensions.",

keywords = "Essential self-adjointness, Intrinsic metrics, Laplacians, Weighted graphs",

author = "Xueping Huang and Matthias Keller and Jun Masamune and Wojciechowski, {Rados{\l}aw K.}",

note = "Funding Information: M.K. enjoyed various inspiring discussion with Daniel Lenz and gratefully acknowledges the financial support from the German Research Foundation (DFG) . R.K.W. thanks J{\'o}zef Dodziuk for numerous insights and acknowledges the financial support of the FCT under project PTDC/MAT/101007/2008 and of the PSC-CUNY Awards , jointly funded by the Professional Staff Congress and the City University of New York . The authors are grateful to Ognjen Milatovic for a careful reading of the manuscript and suggestions. ",

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doi = "10.1016/j.jfa.2013.06.004",

language = "English",

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T1 - A note on self-adjoint extensions of the Laplacian on weighted graphs

AU - Huang, Xueping

AU - Keller, Matthias

AU - Masamune, Jun

AU - Wojciechowski, Radosław K.

N1 - Funding Information: M.K. enjoyed various inspiring discussion with Daniel Lenz and gratefully acknowledges the financial support from the German Research Foundation (DFG) . R.K.W. thanks Józef Dodziuk for numerous insights and acknowledges the financial support of the FCT under project PTDC/MAT/101007/2008 and of the PSC-CUNY Awards , jointly funded by the Professional Staff Congress and the City University of New York . The authors are grateful to Ognjen Milatovic for a careful reading of the manuscript and suggestions.

PY - 2013/10/15

Y1 - 2013/10/15

N2 - We study the uniqueness of self-adjoint and Markovian extensions of the Laplacian on weighted graphs. We first show that, for locally finite graphs and a certain family of metrics, completeness of the graph implies uniqueness of these extensions. Moreover, in the case when the graph is not metrically complete and the Cauchy boundary has finite capacity, we characterize the uniqueness of the Markovian extensions.

AB - We study the uniqueness of self-adjoint and Markovian extensions of the Laplacian on weighted graphs. We first show that, for locally finite graphs and a certain family of metrics, completeness of the graph implies uniqueness of these extensions. Moreover, in the case when the graph is not metrically complete and the Cauchy boundary has finite capacity, we characterize the uniqueness of the Markovian extensions.

KW - Essential self-adjointness

KW - Intrinsic metrics

KW - Laplacians

KW - Weighted graphs

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SP - 1556

EP - 1578

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 8

ER -

A note on self-adjoint extensions of the Laplacian on weighted graphs

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Keywords

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