Abstract
We discuss the Krein-von Neumann extensions of three Laplacian-type operators—on discrete graphs, quantum graphs, and domains. In passing we present a class of one-dimensional elliptic operators such that for any \(n\in \mathbb {N}\) infinitely many elements of the class have \(n\)-dimensional null space.
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Notes
- 1.
We stress that \(\fancyscript{I}\fancyscript{R}^{-1}\fancyscript{I}^T\) is the discrete Laplacian of on \(\mathsf {G}\) with respect to the weight \(\rho ^{-1}\), whereas in Sect. 3 we have considered the discrete Laplacian with respect to the weight \(\rho \). We can think of weights \(\rho ,\rho ^{-1}\) as resistances (proportional to a wire’s length) and conductances (inversely proportional to a wire’s length), respectively. We need not care about realizations of \(\fancyscript{I}\fancyscript{R}^{-1}\fancyscript{I}^T\), since \(\mathsf {V}\) is finite and hence \(\fancyscript{I}\fancyscript{R}^{-1}\fancyscript{I}^T\) is bounded by assumption.
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Acknowledgments
The author is supported by the Land Baden–Württemberg in the framework of the Juniorprofessorenprogramm—research project on “Symmetry methods in quantum graphs.” The author is grateful to Matthias Keller (Jena) for interesting discussions.
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Mugnolo, D. (2015). Some Remarks on the Krein-von Neumann Extension of Different Laplacians. In: Banasiak, J., Bobrowski, A., Lachowicz, M. (eds) Semigroups of Operators -Theory and Applications. Springer Proceedings in Mathematics & Statistics, vol 113. Springer, Cham. https://doi.org/10.1007/978-3-319-12145-1_5
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