Open Access

# Spectral problem for a graph of symmetric Stieltjes strings

### Abstract

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.

Key words: Dirichlet boundary condition, Neumann boundary condition, Lagrange interpolating polynomial, Nevanlinna function, Stieltjes function, graph.

### Article Information

 Title Spectral problem for a graph of symmetric Stieltjes strings Source Methods Funct. Anal. Topology, Vol. 20 (2014), no. 2, 164-174 MathSciNet MR3242864 zbMATH 1313.39031 Milestones Received 29/09/2013 Copyright The Author(s) 2014 (CC BY-SA)

### Authors Information

V. Pivovarchik
South Ukrainian National Pedagogical University, 26 Staroportofrankovskaya, Odessa, 65020, Ukraine

### Citation Example

V. Pivovarchik and O. Taystruk, Spectral problem for a graph of symmetric Stieltjes strings, Methods Funct. Anal. Topology 20 (2014), no. 2, 164-174.

### BibTex

@article {MFAT710,
AUTHOR = {Pivovarchik, V. and Taystruk, O.},
TITLE = {Spectral problem for a graph of symmetric Stieltjes strings},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {20},
YEAR = {2014},
NUMBER = {2},
PAGES = {164-174},
ISSN = {1029-3531},
MRNUMBER = {MR3242864},
ZBLNUMBER = {1313.39031},
URL = {http://mfat.imath.kiev.ua/article/?id=710},
}