{"title":"Upper eigenvalue bounds for the Kirchhoff Laplacian on embedded metric graphs","authors":"Marvin Plumer","doi":"10.4171/jst/388","DOIUrl":null,"url":null,"abstract":"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.","PeriodicalId":48789,"journal":{"name":"Journal of Spectral Theory","volume":" ","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2020-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Spectral Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/jst/388","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 7
Abstract
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.
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.
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