Large eigenvalues and traces of Sturm-Liouville equations on star-shaped graphs
type: Item
instance of: scholarly article
title: Large eigenvalues and traces of Sturm-Liouville equations on star-shaped graphs
zbMATH Open document ID: 1168.34023
author: Chuan-Fu Yang
author: Jianxin Yang
published in: Methods and Applications of Analysis
publication date: 2008-08-28T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.maa/1215442822
review text: The authors consider a Sturm-Liouville problem on a star-shaped graph with \(d\) edges each of length \(\pi\), Kirchhoff boundary conditions at the common node and either Dirichlet or Neumann boundary conditions at the terminal nodes. The potential \(q\) is assumed to be \(C^1\) on each edge. If the potential \(q\) restricted to the edge \(j\) is denoted \(q_j\), then in the Dirichlet case it is shown that the eigenvalues are given by \[ \begin{aligned}\lambda_{n,d}= \left(n-\frac{1}{2}\right)^2+\frac{1}{d}\sum_{j=1}^{d}\bar{q}_j+O\left(\frac{1}{n^2}\right), &\quad n\in\mathbb N,\\ \lambda_{n,j}= n^2+2c_j+O\left(\frac{1}{n^2}\right),&\quad j=1,\dots,d-1,\;n\in\mathbb N, \end{aligned} \] where \(\bar{q}_j\) is the mean value of \(q_j\) and \(c_j, j=1,\dots,d-1\), are roots of a given \(d-1\) order polynomial. If \(\mu_{n,j}\), \(j=1,\dots,d\), \(n\in\mathbb N\), denote the eigenvalues of the same problem but with \(q\) replaced by \(0\), then the following regularised trace formula holds \[ 4d\sum_{n=1}^\infty\bigg[\sum_{j=1}^d(\lambda_{n,j}-\mu_{n,j})-\sum_{j=1}^d \bar{q}_j\bigg]= \sum_{j=1}^d \big[(2-d)q_j(\pi)-dq_j(0) +2(d-1)\bar{q}_j\big]. \] Analogous formulae are given for the Neumann case.
Mathematics Subject Classification ID: 34B45
Mathematics Subject Classification ID: 34L20
Mathematics Subject Classification ID: 47E05
zbMATH DE Number: 5317404
zbMATH Keywords: Sturm-Liouville
zbMATH Keywords: eigenvalue asymptotics
zbMATH Keywords: regularised trace
zbMATH Keywords: graphs
zbMATH Keywords: trees
reviewed by: Bruce A. Watson
MaRDI profile type: Publication
DOI: 10.4310/MAA.2007.V14.N2.A4
description: scientific article; zbMATH DE number 5317404