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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
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
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
Linking options:
https://www.mathnet.ru/eng/sm3892https://doi.org/10.1070/SM2008v199n07ABEH003954 https://www.mathnet.ru/eng/sm/v199/i7/p123
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Abstract page: | 635 | Russian version PDF: | 399 | English version PDF: | 14 | References: | 71 | First page: | 14 |
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