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

Yu. V. Pokornyi ; V. L. Pryadiev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 59 (2004) no. 3, pp. 515-552

Résumé

An analogue of the Sturm oscillation theory of the distribution of the zeros of eigenfunctions is constructed for the problem \begin{equation} Lu\overset{\text{def}}{=}-\frac d{d\Gamma}(pu')+qu=\lambda mu, \qquad u\big|_{\partial\Gamma}=0 \tag{1} \end{equation} 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).