Large eigenvalues and traces of Sturm-Liouville equations on star-shaped graphs
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.
scientific article; zbMATH DE number 5317404