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Spectre des réseaux topologiques finis. (The spectrum of finite topological networks). (French) Zbl 0644.35076
We establish different estimates of the eigenvalues of the laplacian on a finite topological network, adapting certain classical results [see L. E. Payne, SIAM Review 9, 453-488 (1967; Zbl 0154.126)].
Let \(\{-\lambda_ n\}_{n\in {\mathbb{N}}}\) denote the spectrum of the laplacian on a finite topological network; using the min-max principle and a generalized Poisson summation formula, we can study the asymptotic behaviour of \(\lambda_ n\) and we can show that Weyl’s formula still holds, in other words, if L denotes the length of the network i.e. the sum of the lengths of the branches of this network, then \(\lim_{n\to +\infty}\lambda_ n/(n^ 2\prod^ 2/L^ 2)=1.\)
In certain particular cases, this result was also obtained by F. Ali Mehmeti [Integral Equations Oper. Theory 9, 753-766 (1986; Zbl 0617.35022)].
We also establish the Cheeger inequality and the Faber-Krahn inequality, this last one means that of all topological networks of given length, it is the interval which has the lowest strictly positive eigenvalue.
Finally, using the ideas developed by G. N. Hile and M. H. Protter [Indiana Univ. Math. J. 29, 523-538 (1980; Zbl 0454.35064)], we show that on a tree, the ratio \(\lambda_{n+1}/\lambda_ n\) is uniformly bounded. In particular, the ratio \(\lambda_ 2/\lambda_ 1\) is bounded by \(2+\sqrt{5}\).
Reviewer: S.Nicaise

35P15 Estimates of eigenvalues in context of PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation