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Sbornik: Mathematics, 2008, Volume 199, Issue 7, Pages 1071–1087
DOI: https://doi.org/10.1070/SM2008v199n07ABEH003954
(Mi sm3892)
 

This article is cited in 6 scientific papers (total in 6 papers)

The kernel of Laplace-Beltrami operators with zero-radius potential or on decorated graphs

A. A. Tolchennikov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
References:
Abstract: An isomorphism is described for the kernel of the Laplace operator $\Delta^{\!\Lambda}$ (determined by a Lagrangian plane $\Lambda\subset\mathbb C^k\oplus\mathbb C^k$) with potential $\sum_{j=1}^kc_j\delta_{q_j}(x)$ on a manifold. The isomorphism is given by $\Gamma\colon\ker\Delta^{\!\Lambda}\to\Lambda\cap\nobreak L$, where $L$ is an (explicitly calculated) Lagrangian plane. A similar isomorphism also holds for the Laplace operator on a decorated graph. The inequality $1\le\dim\ker\Delta^{\!\Lambda_0}\le n-v+2$ is established for the Laplace operator $\Delta^{\!\Lambda_0}$ on a decorated graph (obtained by decorating a connected finite graph with $n$ edges and $v$ vertices) with ‘continuity’ conditions. It is also shown that the quantity $n-v+1-\dim\ker\Delta^{\!\Lambda_0}$ does not decrease when new edges or manifolds are added.
Bibliography: 12 titles.
Received: 31.05.2007 and 01.04.2008
Russian version:
Matematicheskii Sbornik, 2008, Volume 199, Number 7, Pages 123–138
DOI: https://doi.org/10.4213/sm3892
Bibliographic databases:
UDC: 517.984.68+515.168.5+517.956.227
MSC: 47B25, 58J05
Language: English
Original paper language: Russian
Citation: A. A. Tolchennikov, “The kernel of Laplace-Beltrami operators with zero-radius potential or on decorated graphs”, Mat. Sb., 199:7 (2008), 123–138; Sb. Math., 199:7 (2008), 1071–1087
Citation in format AMSBIB
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\by A.~A.~Tolchennikov
\paper The kernel of Laplace-Beltrami operators
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\pages 123--138
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  • https://doi.org/10.1070/SM2008v199n07ABEH003954
  • https://www.mathnet.ru/eng/sm/v199/i7/p123
  • This publication is cited in the following 6 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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