V. N. Pivovarchik

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.

The (main) spectral problem for a star graph with three edges composed of Stieltjes strings is considered with the Dirichlet conditions at the pendant vertices. In addition we consider the Dirichlet-Neumann problem on the first edge (Problem 2) and the Dirichlet-Dirichlet problem on the union of the second and the third strings (Problem 3). It is shown that the spectrum of the main problem interlace (in a non-strict sense) with the union of spectra of Problems 2 and 3. The inverse problem lies in recovering the masses of the beads (point masses) and the lengths of the intervals between them using the spectra of the main problem and of Problems 2 and 3. Conditions on three sequences of numbers are proposed sufficient to be the spectra of the main problem and of Problems 2 and 3, respectively.

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 find conditions on two sequences of positive numbers that are sufficient for the sequences to be the Neumann and the Dirichlet spectra of a Krein string such that Barcilon’s formula holds true.

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.

A spectral problem generated by the Stieltjes string recurrence relations with a finite number of point masses on a connected graph is considered with Neumann conditions at pendant vertices and continuity and Kirchhoff conditions at interior vertices. The strings on the edges are supposed to be the same and symmetric with respect to the midpoint of the string. The characteristic function of such a problem is expressed via characteristic functions of two spectral problems on an edge: one with Dirichlet conditions at the both ends and the other one with the Neumann condition at one end and the Dirichlet condition at the other end. This permits to find values of the point masses and the lengths of the subintervals into which the masses divide the string from knowing the spectrum of the problem on the graph and the length of an edge. If the number of vertices is less than five then the spectrum uniquely determines the form of the graph.

Known connection between discrete and continuous Laplacians in case of same symmetric potential on the edges of a quantum graph is used to construct characteristic functions of quantum graphs and to find some parameters of graphs using spectra of boundary value problems.

We describe the spectrum of the problem generated by the Stieltjes string recurrence relations on a figure-of-eight graph. The continuity and the force balance conditions are imposed at the vertex of the graph. It is shown that the eigenvalues of such (main) problem are interlaced with the elements of the union of sets of eigenvalues of the Dirichlet problems generated by the parts of the string which correspond to the loops of the figure-of-eight graph. Also the eigenvalues of the main problem are interlaced with the elements of the union of sets of eigenvalues of the periodic problems generated by the same parts of the string.

We solve the inverse spectral problem for a star graph of Stieltjes strings (these are threads bearing a finite number of point masses) with the pendant ends fixed, i.e., we recover the masses and lengths of the intervals between them from the spectra of small transverse vibrations of the graph together with the spectra of the Dirichlet problems on the edges and the total lengths of the edges.

Small transversal vibrations of the Stieltjes string, i.e., an elastic thread bearing point masses is considered for the case of one end being fixed and the other end moving with viscous friction in the direction orthogonal to the equilibrium position of the string. The inverse problem of recovering the masses, the lengths of subintervals and the coefficient of damping by the spectrum of vibrations of such a string and its total length is solved.

We investigate the subclass of symmetric indefinite Hermite-Biehler functions which is obtained from positive definite Hermite-Biehler functions by means of the square-transform. It is known that functions of this class can be characterized in terms of location of their zeros. We give another, more elementary and geometric, proof of this result. The present proof employs a `shifting-of-zeros' perturbation method. We apply our results to obtain information on the eigenvalues of a concrete boundary value problems.