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Spectral asymptotics for quantum graphs with equal edge lengths. (English) Zbl 1149.34021
The equation
\[ -y''+qy=\lambda y, \]
is considered on a finite graph with \(N\) edges, each of unit length, and Kirchhoff boundary conditions at each non-terminal node. At each terminal node either Dirichlet, \(y(\nu)=0\), or Neumann, \(y'(\nu)=0\), boundary conditions are imposed. In addition, for simplicity, it is assumed that there are no edges with only one vertex.
This boundary value problem can be rewritten as a Sturmian system
\[ -Y''+QY=\lambda Y \]
with boundary conditions
\[ [B_1, B_2, B_3, B_4][Y(0)^T, Y'(0)^T, Y(1)^T, Y'(1)^T]^T=0 \]
for suitable \(2N\times N\) matrices \(B_1,B_2,B_3,B_4\). Let \(\zeta_j, j=1,\dots,2N,\) be the eigenvalues of \([B_3+B_4, B_1-B_2]^{-1}[B_1+B_2, B_3-B_4]\) repeated according to multiplicity. Here, \(\zeta_j=e^{i\omega_j}\) where \(0< \omega_j\leq 2\pi\). Setting \(\omega_{j,n}=2n\pi+\omega_j, n\geq 0,\) it is shown that eigenvalues \(\lambda_{j,n}\) of the boundary value problem on the graph are given by \(\sqrt{\lambda_{j,n}}=\omega_{j,n}+O(1/n),\) and more precisely that
\[ \lim_{n\to\infty}\sum_{\omega_{i,n}= \omega_{j,n}}[\lambda_{i,n}-\omega_{i,n}^2]=\Lambda_j(q), \]
where \(\Lambda_j(q)\) is a linear combination of \(\int_0^1 q_j\,dx, j=1,\dots,N.\) The methods used in the asymptotic approximations are given for arbitrary order approximation.

MSC:
34B45 Boundary value problems on graphs and networks for ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
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