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 Uspekhi Mat. Nauk, 2004, Volume 59, Issue 3(357), Pages 115–150 (Mi umn738)

Some problems of the qualitative Sturm–Liouville theory on a spatial network

Yu. V. Pokornyi, V. L. Pryadiev

Voronezh State University

Abstract: An analogue of the Sturm oscillation theory of the distribution of the zeros of eigenfunctions is constructed for the problem
Lu\overset{def}{=}-\frac d{d\Gamma}(pu')+qu=\lambda mu, \qquad u|_{\partial\Gamma}=0 \tag{1}
on a spatial network $\Gamma$ (in other terms, $\Gamma$ is a metric graph, a CW complex, a stratified locally one-dimensional manifold, a branching space, a quantum graph, and so on), where $\partial\Gamma$ is the family of boundary vertices of $\Gamma$. At interior points of the edges of $\Gamma$ the quasi-derivative $\displaystyle\frac d{d\Gamma}(pu')$ has the classical form $(pu')'$, and at interior nodes it is assumed that
$$\frac d{d\Gamma}(pu')=-\sum_\gamma\alpha_\gamma(a)u'_\gamma(a),$$
where the summation is taken over the edges $\gamma$ incident to the node $a$ and, for an edge $\gamma$, $u'_\gamma (a)$ stands for the ‘endpoint’ derivative of the restriction $u_\gamma (x)$ of the function $u\colon\Gamma\to\mathbb R$ to $\gamma$. Despite the branching argument, which is a kind of intermediate type between the one-dimensional and multidimensional cases, the outward form of the results turns out to be quite classical. The classical nature of the operator $L$ is clarified, and exact analogues of the maximum principle and of the Sturm theorem on alternation of zeros are established, together with the sign-regular oscillation properties of the spectrum of the problem (1) (including the simplicity and positivity of the points of the spectrum and also the number of zeros and their alternation for the eigenfunctions).

DOI: https://doi.org/10.4213/rm738

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English version:
Russian Mathematical Surveys, 2004, 59:3, 515–552

Bibliographic databases:

UDC: 517.927
MSC: Primary 34B24, 34B45; Secondary 34B10, 05C99, 35Q99

Citation: Yu. V. Pokornyi, V. L. Pryadiev, “Some problems of the qualitative Sturm–Liouville theory on a spatial network”, Uspekhi Mat. Nauk, 59:3(357) (2004), 115–150; Russian Math. Surveys, 59:3 (2004), 515–552

Citation in format AMSBIB
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• https://doi.org/10.4213/rm738
• http://mi.mathnet.ru/eng/umn/v59/i3/p115

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