Stieltjes Strings Research Articles

The deformations problem for the Stieltjes strings system with a nonlinear condition

The deformations problem for the Stieltjes strings system with a nonlinear condition

Model of deformations of a Stieltjes string system with a nonlinear condition

In the present paper we study a model of deformations for a system of n Stieltjes strings located along a geometric graph-star with a nonlinear condition at the node

Inverse Spectral Problem for a Star Graph of Stieltjes Strings with Given Numbers of Masses on the Edges

Inverse Spectral Problem for a Star Graph of Stieltjes Strings with Given Numbers of Masses on the Edges

On inverse problem for tree of Stieltjes strings

For a given metric tree and two strictly interlacing sequences of numbers there exits a distribution of point masses on the edges (which are Stieltjes strings) such that one of the sequences is the spectrum of the spectral problem with the Neumann condition at the root of the tree while the second sequence is the spectrum of the spectral problem with the Dirichlet condition at the root.

Direct and inverse spectral problems for a star graph of Stieltjes strings damped at a pendant vertex

A spectral problem occurring in description of small transverse vibrations of a star graph of Stieltjes strings is considered. At all but one pendant vertices Dirichlet conditions are imposed which mean that these vertices are clamped. One vertex (the root) can move with damping in the direction orthogonal to the equilibrium position of the strings. We describe the spectrum of such spectral problem. The corresponding inverse problem lies in recovering the values of point masses and the lengths of the intervals between the masses using the spectrum and some other parameters. We propose conditions on a sequence of complex numbers and a collection of real numbers to be the spectrum of a problem we consider and the lengths of the edges, correspondingly.

On the Minimum Number of Distinct Eigenvalues in the Problem for a Tree Formed by Stieltjes Strings

We consider spectral problems related to small vibrations of a tree formed by Stieltjes strings. It is shown that the minimum number of distinct eigenvalues of this problem is equal to the maximum length (measured as the number of point masses) of paths in this tree.

Three spectra problem for Stieltjes string equation and Neumann conditions

Spectral problems are considered which appear in description of small transversal vibrations of Stieltjes strings. It is shown that the eigenvalues of the Neumann-Neumann problem, i.e. the problem with the Neumann conditions at both ends of the string interlace with the union of the spectra of the Neumann-Dirichlet problems, i.e. problems with the Neumann condition at one end and Dirichlet condition at the other end on two parts of the string. It is shown that the spectrum of Neumann-Neumann problem on the whole string, the spectrum of Neumann-Dirichlet problem on the left part of the string, all but one eigenvalues of the Neumann-Dirichlet problem on the right part of the string and total masses of the parts uniquely determine the masses and the intervals between them.

Compressed Resolvents and Reduction of Spectral Problems on Star Graphs

In this paper a two-step reduction method for spectral problems on a star graph with n+1 edges e_{0}, e_{1}, ldots , e_{n} and a self-adjoint matching condition at the central vertex v is established. The first step is a reduction to the problem on the single edge e_0 but with an energy depending boundary condition at v. In the second step, by means of an abstract inverse result for Q-functions, a reduction to a problem on a path graph with two edges e_0, widetilde{e}_1 joined by continuity and Kirchhoff conditions is given. All results are proved for symmetric linear relations in an orthogonal sum of Hilbert spaces. This ensures wide applicability to various different realizations, in particular, to canonical systems and Krein strings which include, as special cases, Dirac systems and Stieltjes strings. Employing two other key inverse results by de Branges and Krein, we answer e.g. the following question: If all differential operators are of one type, when can the reduced system be chosen to consist of two differential operators of the same type?

On the Relationship Between the Multiplicities of Eigenvalues in Finite- and Infinite-Dimensional Problems on Graphs

It is shown that some results concerning the multiplicities of eigenvalues of the spectral problem that describes small transverse vibrations of a star graph of Stieltjes strings and the multiplicities of the eigenvalues of tree-patterned matrices can be used for the description of possible multiplicities of the normal eigenvalues (bound states) of the Sturm–Liouville operator on a star graph.

Location and multiplicities of eigenvalues for a star graph of Stieltjes strings

The equations of motion of a star graph of Stieltjes strings with prescribed number of masses on each edge, with or without a mass at the central vertex, lead to a system of second order difference equations. At the central vertex Dirichlet or Neumann conditions are imposed while all pendant vertices are subject to Dirichlet conditions. We establish necessary and sufficient conditions on the location and multiplicities of two (finite) sequences of numbers and to be the corresponding Dirichlet and Neumann eigenvalues. Moreover, we derive necessary and sufficient conditions for one (finite) sequence to be the Neumann eigenvalues of such a star graph. Here the possible multiplicities play a key role; the conditions on them are formulated by means of the notion of vector majorization. Our results include, as a special case, some earlier results for star-patterned matrix inverse problems where only multiplicities, not the location of eigenvalues, are prescribed.

Dirichlet–Neumann inverse spectral problem for a star graph of Stieltjes strings

We solve two inverse spectral problems for star graphs of Stieltjes strings with Dirichlet and Neumann boundary conditions, respectively, at a selected vertex called root. The root is either the central vertex or, in the more challenging problem, a pendant vertex of the star graph. At all other pendant vertices Dirichlet conditions are imposed; at the central vertex, at which a mass may be placed, continuity and Kirchhoff conditions are assumed. We derive conditions on two sets of real numbers to be the spectra of the above Dirichlet and Neumann problems. Our solution for the inverse problems is constructive: we establish algorithms to recover the mass distribution on the star graph (i.e. the point masses and lengths of subintervals between them) from these two spectra and from the lengths of the separate strings. If the root is a pendant vertex, the two spectra uniquely determine the parameters on the main string (i.e. the string incident to the root) if the length of the main string is known. The mass distribution on the other edges need not be unique; the reason for this is the non-uniqueness caused by the non-strict interlacing of the given data in the case when the root is the central vertex. Finally, we relate of our results to tree-patterned matrix inverse problems.

Two Inverse Spectral Problems for a Class of Singular Krein Strings

We solve the inverse problem from the spectral measure and the inverse three-spectra problem for the class of singular Krein strings on a finite interval with trace class resolvents. In particular, this includes a complete description of all possible spectral measures and three (Dirichlet) spectra associated with this class of Krein strings. The solutions of these inverse problems are obtained by approximation with Stieltjes strings.

A polynomial identity and its application to inverse spectral problems in Stieltjes strings

A polynomial identity and its application to inverse spectral problems in Stieltjes strings

Existence of a tree of Stieltjes strings corresponding to two given spectra

Transversal vibrations of a plane rooted tree of Stieltjes strings are considered with Dirichlet boundary conditions at each pendant vertex. Continuity and Kirchhoff conditions are imposed at each interior vertex except the root. The Dirichlet problem is that with the Dirichlet condition at the root, and the Neumann problem is that with continuity and Kirchhoff conditions at the root. It is shown that strict interlacing is a sufficient condition for two sequences of real numbers to be the spectra of Neumann and Dirichlet problems generated by Stieltjes string recurrent relations on any prescribed tree.