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Essential Self-Adjointness and the \(L^2\)-Liouville Property

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Abstract

We discuss connections between the essential self-adjointness of a symmetric operator and the constancy of functions which are in the kernel of the adjoint of the operator. We then illustrate this relationship in the case of Laplacians on both manifolds and graphs. Furthermore, we discuss the Green’s function and when it gives a non-constant harmonic function which is square integrable.

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Acknowledgements

The authors are grateful to Marcel Schmidt for a careful reading of the manuscript and for helpful comments. R.K.W. would also like to thank Józef Dodziuk for his support and for many fruitful discussions. Furthermore, the authors would like to thank the referees for useful comments and Isaac Pesenson for the invitation to submit this article.

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Bobo Hua

  2. Shanghai Center for Mathematical Sciences, Jiangwan Campus, Fudan University, No. 2005 Songhu Road, Shanghai, 200438, China

    Bobo Hua

  3. Department of Mathematics, Faculty of Science, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan

    Jun Masamune

  4. Graduate Center of the City University of New York, 365 Fifth Avenue, New York, NY, 10016, USA

    Radosław K. Wojciechowski

  5. York College of the City University of New York, 94-20 Guy R. Brewer Blvd., Jamaica, NY, 11451, USA

    Radosław K. Wojciechowski

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  1. Bobo Hua
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  2. Jun Masamune
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  3. Radosław K. Wojciechowski
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Correspondence to Radosław K. Wojciechowski.

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B.H. is supported by NSFC, Grants Numbers 11831004 and 11926313. J.M. is supported in part by JSPS KAKENHI Grant Numbers 18K03290 and 17H01092. R.W. is supported by PSC-CUNY Awards, jointly funded by the Professional Staff Congress and the City University of New York, and the Collaboration Grant for Mathematicians, funded by the Simons Foundation.

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Hua, B., Masamune, J. & Wojciechowski, R.K. Essential Self-Adjointness and the \(L^2\)-Liouville Property. J Fourier Anal Appl 27, 26 (2021). https://doi.org/10.1007/s00041-021-09833-2

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