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<p>We give two characterizations for the essential self-adjointness of the weighted Laplacian on birth–death chains. The first involves the edge weights and vertex measure and is classically known; however, we give another proof using stability results, limit point-limit circle theory and the connection between essential self-adjointness and harmonic functions. The second characterization involves a new notion of capacity. Furthermore, we also analyze the essential self-adjointness of Schrödinger operators, use the characterizations for birth–death chains and stability results to characterize essential self-adjointness for star-like graphs, and give some connections to the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11040_2025_9498_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\ell ^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ℓ</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-Liouville property.</p>

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Essential Self-Adjointness of the Laplacian on Weighted Graphs: Harmonic Functions, Stability, Characterizations and Capacity

  • 期刊论文

  • 发表日期：

    2025年5月26日

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  • Atsushi Inoue,
  • Sean Ku,
  • Jun Masamune,
  • Radosław K. Wojciechowski

摘要

We give two characterizations for the essential self-adjointness of the weighted Laplacian on birth–death chains. The first involves the edge weights and vertex measure and is classically known; however, we give another proof using stability results, limit point-limit circle theory and the connection between essential self-adjointness and harmonic functions. The second characterization involves a new notion of capacity. Furthermore, we also analyze the essential self-adjointness of Schrödinger operators, use the characterizations for birth–death chains and stability results to characterize essential self-adjointness for star-like graphs, and give some connections to the \(\ell ^2\) ℓ 2 -Liouville property.