嵌入度量图上Kirchhoff拉普拉斯算子的上特征值界

IF 1 3区数学 Q1 MATHEMATICS

Journal of Spectral Theory Pub Date : 2020-04-07 DOI:10.4171/jst/388

Marvin Plumer

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引用次数: 7

摘要

根据紧度量图的亏格g，我们导出了紧度量图上基尔霍夫-拉普拉斯算子的特征值的上界。如果$g=0$，即度量图是平面的，则可以进一步改进这些上界。我们的结果是基于基尔霍夫拉普拉斯算子和特定的组合加权拉普拉斯算子之间的谱对应关系。为了利用这种对应关系，我们还证明了加权组合拉普拉斯算子的特征值的新估计，这些估计以前只在加权情况下已知。

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Upper eigenvalue bounds for the Kirchhoff Laplacian on embedded metric graphs

We derive upper bounds for the eigenvalues of the Kirchhoff Laplacian on a compact metric graph depending on the graph's genus g. These bounds can be further improved if $g = 0$, i.e. if the metric graph is planar. Our results are based on a spectral correspondence between the Kirchhoff Laplacian and a particular a certain combinatorial weighted Laplacian. In order to take advantage of this correspondence, we also prove new estimates for the eigenvalues of the weighted combinatorial Laplacians that were previously known only in the weighted case.

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来源期刊

Journal of Spectral Theory MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

2.00

自引率

0.00%

发文量

期刊介绍： The Journal of Spectral Theory is devoted to the publication of research articles that focus on spectral theory and its many areas of application. Articles of all lengths including surveys of parts of the subject are very welcome. The following list includes several aspects of spectral theory and also fields which feature substantial applications of (or to) spectral theory. Schrödinger operators, scattering theory and resonances; eigenvalues: perturbation theory, asymptotics and inequalities; quantum graphs, graph Laplacians; pseudo-differential operators and semi-classical analysis; random matrix theory; the Anderson model and other random media; non-self-adjoint matrices and operators, including Toeplitz operators; spectral geometry, including manifolds and automorphic forms; linear and nonlinear differential operators, especially those arising in geometry and physics; orthogonal polynomials; inverse problems.