O. M. Martynyuk

The largest possible multiplicity of an eigenvalue of a spectral problem generated by the Stieltjes string equations on a metric tree is $p_{pen}-1$, where $p_{pen}$ is the number of pendant vertices. We propose how to find the second largest possible multiplicity of an eigenvalue of such a problem. This multiplicity depends on the numbers of point masses on the edges of the trees.

Максимально можлива кратність власного значення спектральної задачі, породженої рівняннями струни Стілтьєса на метричному дереві, дорівнює $p_{pen}-1$, де $p_{pen}$ — кількість висячих вершин. Ми пропонуємо, як знайти другу за величиною кратність власного значення такої задачі. Ця кратність залежить від кількості точкових мас на ребрах дерев.

Recurrence relations of the second order on the edges of a metric connected graph together with boundary and matching conditions at the vertices generate a spectral problem for a self-adjoint finite-dimensional operator. This spectral problem describes small transverse vibrations of a graph of Stieltjes strings. It is shown that if the graph is cyclically connected and the number of masses on each edge is not less than 3 then the maximal multiplicity of an eigenvalue is $\mu+1$ where $\mu$ is the cyclomatic number of the graph. If the graph is not cyclically connected and each edge of it bears at least one point mass then the maximal multiplicity of an eigenvalue is expressed via $\mu$, the number of edges and the number of interior vertices in the tree obtained by contracting all the cycles of the graph into vertices.

We consider a generalization of the three spectral inverse problem, that is, for given spectrum of the Dirichlet-Dirichlet problem (the Sturm-Liouville problem with Dirichlet conditions at both ends) on the whole interval $[0,a]$, parts of spectra of the Dirichlet-Neumann and Dirichlet-Dirichlet problems on $[0,a/2]$ and parts of spectra of the Dirichlet-Newman and Dirichlet-Dirichlet problems on $[a/2,a]$, we find the potential of the Sturm-Liouville equation.